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Direct modeling of multi-axial fatigue failure for metals
Accepted Manuscript
Direct modeling of multi-axial fatigue failure for metals Heng Xiao, Zhao-Ling Wang PII: DOI: Reference:
S0020-7683(17)30320-7 10.1016/j.ijsolstr.2017.07.003 SAS 9649
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
1 February 2017 15 May 2017 3 July 2017
Please cite this article as: Heng Xiao, Zhao-Ling Wang, Direct modeling of multi-axial fatigue failure for metals, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.07.003
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Highlights • New elastoplasticity models are proposed toward direct modeling of metal fatigue
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failure • Conditions for yielding and loading-unloading need not be imposed but automatically incorporated
• Any damage-like variables and ad hoc failure criteria are not involved
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• Critical failure states are derived with a criterion in uni ed form
• Explicit, direct procedures for determining constitutive quantities based on uniaxial
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data
• Explicit, direct procedures for determining fatigue lives under stress cycles and ther-
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mal cycles
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Direct modeling of multi-axial fatigue failure for metals Heng Xiao1 , Zhao-Ling Wang1,2 Shanghai Institute of Applied Mathematics and Mechanics and State Key Laboratory for Advanced Special Steels, Shanghai University, Yanchang Road 149, 200072 Shanghai, China
2
School of Mathematics and Information Sciences, Weifang University, 261061 Weifang, Shandong Province, China
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1
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Abstract
Innovative elastoplastic J2 −flow models with nonlinear combined hardening are proposed for the purpose of simulating multi-axial thermo-coupled fatigue failure for metals and alloys, in a direct sense without involving any usual damage-like variables and any ad hoc failure criteria. As contrasted with usual models, the yield condition and the loading-
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unloading conditions need not be imposed as extrinsic coercive conditions but may be
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automatically incorporated into a free, smooth flow rule in a more realistic sense. Novel results in three respects are available directly from model predictions, namely, (i) complex features of thermo-coupled fatigue failure may be automatically represented by simple
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asymptotic properties of the hardening functions introduced, (ii) critical failure states may
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be derived with a criterion in unified form, and, in particular, (iii) direct procedures may be established for determining fatigue lives under either stress cycles or thermal cycles.
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Simulation results in the simplest cases of the proposed model are in good agreement with experimental data. Keywords: Metals, Finite deformation, Thermo-coupled effects, New elastoplastic equations, Fatigue failure, Direct simulation
Preprint submitted to International Journal of Solids and Structures
July 4, 2017
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Nomenclature D the stretching, given by the symmetric part of the velocity gradient
τ , τ˜ the Kirchhoff stress and its deviatoric part T the absolute temperature W complementary thermo-elastic potential
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G, B, E the shear, bulk and Young’s moduli
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d the magnitude of the stretching
D e , D p the elastic and the plastic part of the stretching α, ζ the back stress and its magnitude
J2 the magnitude of the effective stress (˜ τ − α)
ϑ the effective plastic work
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√
o log
τ
o log
,α
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L the scalar product of τ and (˜ τ − α)
the logarithmic rates of τ and α
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r∗ the stress limit
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ρ the plastic factor
κ0 the fatigue limit
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m the plastic index c, ξ Prager’s modulus and the hysteresis modulus ˘ the regularized plastic modulus h ˆ ∗ the plastic modulus h f˘, fˆ the strain- and stress-rate loading functions
K, K plastic slopes of the primary and reverse loading curves 3
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hm , τm uniaxial strain and stress at the start of softening ϑˆ threshold value of ϑ demarcating hardening and softening stages
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p(τ ), ps (τ ) shape functions for hardening and softening parts of tensile curve φ(τ ) bridging function linking primary and reverse loading curves rm the maximum stress limit β0 , ϑ0 parameters for softening behavior
κ0 , γ0 parameters for fatigue limit 1. Introduction
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c0 , ξ0 hardening parameters
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Whenever repeatedly undergoing cyclic deformations, metals and alloys may suffer gradual reduction in strength up to eventual failure, albeit the strength reduction at each
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cycle alone may be negligible. It is known (Pook 2007) that the just-mentioned metal fatigue behavior is mainly responsible for fracture and failure of metal components and
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parts in engineering structures. This fact becomes even more outstanding for hot sections of aircraft engines and gas turbines under extreme service conditions with elevated tem-
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perature and high-level stress (cf., e.g., Reed 2006). Toward effectively assessing safety and reliability of metal components and parts by predicting the service life under prescribed
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service conditions, rational and realistic modeling of the fatigue failure behavior for metals and alloys is accordingly essential. In the past decades, numerous investigations into materials fracture and failure have
been carried out from various standpoints. Usually, two approaches are used, including the fracture-mechanics-based approach and the damage-mechanics-based approach, in conjunction with various assumed variables and failure criteria. The former treats different types of embedded cracks from a discontinuity standpoint, while the latter considers the 4
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gradual strength degradation from a contiuum viewpoint. Surveys for certain earlier results may be found in, e.g., Hutchinson and Evans (2000) for the fracture respect and in Dvorak (2000), Krajcinovic (2000), Ritchie et al. (2000) and Ciavarella and Demelio
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(2001) for the damage and fatigue respect. Recent surveys are presented in, e.g., Susmel (2008, 2014), Rozumek and Macha (2009) and Macha and Nieslony (2012) for energy-based and critical-plane-based approaches to modeling fatigue failure, and Br¨ unig (2015) in the damage respect, as well as Zhang (2016) for thermomechanical fatigue of single crystal
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superalloys, and, in particular, in Chaboche (2008) for certain significant developments of elastoplasticity models under cyclic loading conditions.
Further results in both approaches have been obtained in most recent studies. Here, only certain representatives are mentioned below. On the one hand, results based on fracture discontinuities are presented in Jir´asek and Zimmermann (2001) for an embedded
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crack model, in Shyam et al. (2007) , Ural et al. (2009), Baletto et al. (2010) and Minh et
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al. (2012) for fatigue crack growth, in Stoughton and Yoon (2011) for failure criteria for sheet metals and Khan and Liu (2012a, b) for latest advances concerning strain rate and
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temperature effects, in Khoel et al. (2012), Chew (2014), Moura and Consalves (2014), Heitbreder et al. (2016) for cohesive zone models and Ottosen et al. (2016) for non-cohesive
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interface models, and in Cervera and Wu (2015) and Wu and Cervera (2015) for modeling of localized failure based on embedded and smeared discontinuities.
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On the other hand, recent results based on continuum damage mechanics since 2000 are presented in Armero and Oller( 2000) , Bruhns et al. (2001) and Alfredsson and Stigh (2004) for a general framework of continuum damage models, in Br¨ unig (2003a, b) , Bonora et al. (2005) and Badreddine et al. (2016) for ductile damage models, in Br¨ unig and Ricci (2005) and Br¨ unig et al. (2008) for a nonlocal model for anisotropically damaged metals and for a ductile damage criterion under multi-axial stress states, in Cazes et al. (2009) for a nonlocal cohesive model, in Driemeier et al. (2010) and Br¨ unig and Gerke (2011) for 5
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test data for combined effects of stress magnitude and triaxiality and for damage evolution simulation under dynamic loading, in Malcher et al. (2012) for a study of ductile fracture under various cases of triaxial stress, in Bonora et al. (2005), Steglich et al. (2005) and
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Br¨ unig et al. (2013) for micro-mechanical modeling, in Shojaei et al. (2013) for a study of brittle to ductile damage based on viscoplastic models, in Wu and Xu (2015) for a multicrack elastoplastic damage model for tensile cracking, and in Cortese et al. (2016) for ductile dame accumulation, etc.
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In the respect of fatigue behavior under cyclic loadings, certain most recent results may be found in Kang et al. (2008), Facheris et al. (2014), Ahmed et al. (2016), Barrett et al. (2016), Mohd et al. (2016), Xu et al. (2016) and many others. In particular, results in the thermomechanical respect may be found in, e.g., Egner (2012), Canadija and Mosler (2016), and Zhu et al. (2016). Moreover, results are reported for ratcheting effects in, e.g.,
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Hassan et al. (1992), Hassan and Kyriakides (1992, 1994a, b), Lee et al.. (2014) and many
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others.
According to usual approaches indicated above, crack development criteria should be
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assumed with sophisticated procedures for treating discontinuities and, moreover, damage variables should be introduced with assumed evolution equations. Namely, various forms
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of criteria for fracture and failure should be assumed on an ad hoc basis. This implies that issues concerning either additional ad hoc criteria or augmented constitutive structures for
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additional variables should be treated in experimental tests and model analyses. Since extreme cases of thermomechanical deformation behavior with strong discontinu-
ities are involved, undue complexities are expected in modeling fatigue failure. That may be particularly the case for thermo-coupled fatigue failure under combined changes of both stress and temperature. Indeed, in various cases close to and even just at fracture and failure, pronounced dissipation with appreciable thermal effects may be expected and, in particular, unlimitedly growing inelastic deformations may be induced in association with 6
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initiation of strong discontinuities. Generally, the four thermomechanical fields including deformation, temperature, heat flux and stress in a material body are inextricably coupled with one another and should be studied on the rigorous ground of the thermodynamic
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principles.
The above facts may suggest that large elastoplastic deformations with thermo-coupled effects should be the very features of fracture and failure behavior for metals and alloys. Since these features exhibit undue complexities as indicated above, results are usually
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presented for cases at small isothermal deformations. It appears that further studies for a broad case with finite deformation effects, in particular, with thermo-coupled effects, are rarely reported, except for some particular cases.
In this article, an alternative approach will be proposed toward simulating multi-axial thermo-coupled fatigue failure for metals and alloys, in a direct sense without involving
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any usual damage-like variables and any ad hoc failure criteria. This new, direct approach
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will be based on innovative elastoplastic J2 −flow models. Toward this objective, it is noted that usual elastoplasticity models with the yield condition and the loading-unloading
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conditions could not serve the purpose. For instance, various cases of fatigue failure under cyclic loading conditions are excluded from such models. In a most recent study (Xiao et
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al. 2014; Xiao 2014b, 2015), innovative thermo-coupled elastoplasticity models have been established from a fresh standpoint. As contrasted with usual elastoplasticity models,
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such new models of much simpler structure are totally free, in the sense that both the yield condition and the loading-unloading conditions need not be imposed as extrinsic coercive conditions but are automatically incorporated as inherent constitutive features into the models. It has been demonstrated that various cases of thermo-coupled fatigue failure under combined cyclic changes of stress and temperature may be derived as direct consequences from the new models. General constitutive frameworks are established in the foregoing references that are 7
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mainly devoted to bypassing certain issues associated with usual elastoplasticity models and, accordingly, only an initial study of the fatigue failure behavior is incorporated therein. This contribution may represent substantial developments both in generalizing the model
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and in deriving fatigue failure features, in a broad sense for multi-axial finite deformations with thermo-coupled effects and for both rate-independent and rate-dependent effects. Toward this goal, a substantial extension of the previous models will first be presented. It is then demonstrated that multi-axial thermo-coupled fatigue failure may be derived
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as inherent constitutive features of the new model. Novel results in three respects may be available directly from model predictions, namely, (i) complicated failure features may be automatically represented by certain simple asymptotic properties of the hardening functions introduced, (ii) critical failure states may be derived with a criterion in unified form, and, in particular, (iii) salient features of multi-axial fatigue failure under stress
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cycles and under thermal cycles may be disclosed with results for the fatigue life. Moreover,
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numerical examples for model predictions will be presented and compared with extensive test data for the purpose of model validation.
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The main content will be organized in seven sections. A substantial extension of the previous model will be presented in § 2 and the thermodynamic consistency will be demon-
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strated in §3. In §4, critical failure states will be disclosed and a unified criterion for such states will be derived from the model established. In §5, stress cycles with temperature
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held fixed will be taken into account and results for the fatigue life will be presented with direct, explicit procedures for determining the hardening moduli incorporated. In §6, thermal cycles will be taken into consideration and the fatigue life will be determined. In §7, numerical examples will be provided for the purpose of model validation. Finally,
discussions and remarks will be made in §8. To conclude this introduction, some notations are explained as follows. Let A, B and T be two symmetric 2nd-order tensors and a 4th-order tensor, respectively. The notations 8
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trA, |A|, A : B and T : A respresent the trace and the norm of A, the double-dot products between A and B and between T and B, namely (repeated indices below mean
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summation), trA = Aii , |A|2 = trA2 , A : B = Aij Bij , T : A = Tijkl Akl ei ⊗ ej .
In the above, (e1 , e2 , e3 ) are three orthonormal vectors and ei ⊗ ej is the tensor (dyadic)
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product of ei and ej .
2. New elastoplastic J2 −flow models with rate and thermal effects The objective of this section is to establish elastoplastic J2 −flow models into which the previous models (cf., Xiao et al. 2014 and Xiao 2014b, 2015) are included as particular
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cases. In what follows the procedures in deriving the main results are omitted and may be found in the references just indicated.
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Consider finite elastoplastic deformations with rate- and temperature-dependent effects. According to the Eulerian rate formulation of finite elastoplasticity (cf., Xiao et al. 2006),
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the total stretching D is a sum of an elastic and a plastic part, D e and D p , as shown
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below:
D = De + Dp .
(1)
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The objective rate equations of Eulerian type for D e and D p will be given below, separately. First, in a general case with thermal effects, the self-consistent Eulerian rate equation
for the elastic stretching D e may be given by (cf., Xiao et al. 2007, Xiao 2014b, 2015): De =
∂ 2 W o log ∂ 2W ˙ : τ + T, ∂τ 2 ∂τ ∂T
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(2)
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where T and τ are the absolute temperature and the Kirchhoff stress and
o log
is a complementary thermo-elastic potential and the τ rate of the Kirchhoff stress τ .
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W = W (τ , T ) .
is the corotational logarithmic
Details may be found in, e.g., Xiao et al. (1997). For
metals and alloys with small elastic strains, the potential W is of the quadratic form below: 1 1 tr˜ τ2 + (trτ )2 + θ(trτ )(T − T∗ ) . 4G 6B
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W =
(3)
In the above, τ˜ is the deviatoric part of τ , θ and T∗ are the thermal expansion coefficient and a reference temperature, and, besides, G and B are the shear modulus and the bulk
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modulus and dependent on the temperature T , namely,
B = B(T ) .
(4)
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G = G(T ),
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With Eq. (3), Eq. (2) for the elastic stretching D e reduces to 1 o log 1 G0 ˙ 1 B0 D = τ˜ + (trτ˙ )I − T τ˜ + θ − trτ T˙ I , 2G 3B 2G2 3 B2
(5)
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e
where G0 and K 0 are the derivatives of G and B with respect to the temperature T .
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Next, the objective rate equation for the plastic stretching D p , namely, the plastic flow
rule, is usually concerned with a few respects including formulations of the yield function, the hardening behavior and the loading-unloading conditions, as well as rate-dependent effects, etc. These will be explained below. According to usual theory of elastoplasticity, the central concept for plastic behavior is the yielding state. The plastic flow, i.e. D p , is induced only in the case when the yielding
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state is attained and maintained. As such, the loading-unloading conditions associated with a yield criterion should be introduced and imposed as extrinsic coercive conditions. Such conditions are characteristic of usual theory of elastoplasticity and formulated in terms
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of a yield function f∗ . The yield surface f∗ = 0 in the stress-temperature space changes itself both in size and in shape, known as hardening behavior. A scalar quantity, ϑ, and a traceless tensor quantity, known as the back stress and denoted α, both evolving with the development of plastic flow, are introduced to characterize the hardening behavior. Then,
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the yield function f∗ of von Mises type with both rate and thermal effects is formulated as follows:
1 1 τ − α|2 − r∗2 , f∗ = |˜ 2 3
(6)
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where the r∗ is referred to as the stress limit and of the form
(7)
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r∗ = r∗ (ϑ, d, ζ, T )
in a broad sense. In the above, the d and ζ are the stretching magnitude (cf., Xiao 2015)
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and
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and the back-stress magnitude, given by
d=
r
ζ=
r
2 |D| , 3
2 |α| . 3
(8)
(9)
Since the latter enters the stress limit r∗ in Eq. (7), the coupling effect is introduced (cf.,
Xiao 2014a, Xiao et al. 2016) between the size change and the translation of the yield surface.
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The equations of evolution for ϑ and α are as follows (Xiao 2014a, b, 2015): ϑ˙ = (˜ τ − α) : D p ,
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o α log = cD p − ξαϑ˙ .
(10)
(11)
The ϑ as specified by Eq. (10) is referred to as the effective plastic work and invariably accumulates with development of plastic flow, as will be shown in §4.1. In Eq. (11), the
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c and ξ, known as the Prager modulus and the hysteresis modulus, need not be constant but rely on ϑ and T and others, namely (cf., Xiao 2014a,b),
(12)
ξ = ξ(ϑ, T, τ , α) .
(13)
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c = c(ϑ, T, τ , α) ,
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As indicated in the previous study (Xiao 2014a,b, 2015), such moduli in a broad sense may be used to characterise complicated behavior displaying strong anisotropic hardening for
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metals and alloys, in particular, for shape memory alloys. For the rate-independent case, a normality flow rule for D p was first formulated in Hill
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(1968) based on Ilyushin postulate and a further development has been made in a study in Bruhns et al. (2005). In a series of studies (Xiao et al. 2014, Xiao 2014b), such a
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normality rule has been extended to give rise to smooth flow rules1 free of usual yield condition and loading-unloading conditions and most recently (Xiao 2015) this smooth flow rule has further been extended to general cases with both rate-dependent effects and thermal effects. Here, a normality flow rule of such nature is further proposed in a broad 1
Smooth elastic-to-plastic transitions may be realized by regularization procedures from a standpoint based on the asymptotic numerical method. This idea has been suggested by Zahrouni and coworkers (Zahrouni et al. 1998, Assidi et al. 2009).
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sense as follows: 1 D p = ρ J2−1 2
! f˘∗ f˘∗ τ − α) . + (˜ ˘∗ h ˘∗ h
(14)
property below:
2 ρ = ρ (x, ϑ) , x = 3J , 2r∗2 ρ(0, ϑ) = 0, ρ(1, ϑ) = 1 , ∂ρ > 0 . ∂x
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˘ are given by (cf., Xiao 2015) and, moreover, the f˘ and h
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In the above, the dimensionless factor ρ is referred to as the plastic factor and of the
(15)
(16)
˘ ∗ = 2G + h ˆ∗ , h
(17)
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∂f∗ ˙ ∂f∗ ˙ G0 T+ d + (J2 + L) T˙ , f˘∗ = 2G(˜ τ − α) : D + ∂T ∂d G
with
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ˆ ∗ = 2 r∗ ∂r∗ + c − ξL − 4 r∗ ∂r∗ 1.5ξζ − cζ −1 J −1 L , h 2 3 ∂ϑ 9 ∂ζ
(19)
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J2 = |˜ τ − α|2 , L = (˜ τ − α) : α .
(18)
ˆ and h ˘ are referred to as the plastic modulus and the regularized plastic modulus. The h
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For metals and alloys, the former is far smaller than the shear modulus 2G and, therefore,
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the latter is always positive, namely, ˘∗ > 0 . h
(20)
Physically, the above property arises from the following fact: the regularized plastic
˘ relates the plastic strain rate to the total strain rate instead of the stress rate. modulus h
ˆ changes its sign from hardening Irrespective of the fact that the usual plastic modulus h ˘ keeps positive. In the uniaxial stress-strain case, the positivity property of to softening, h ˘ may become evident from the foregoing fact. h 13
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According to usual models, the plastic factor ρ always vanishes for all stress levels below the stress limit r∗ , i.e., f∗ < 0, whereas it suddenly jumps to 1 whenever the stress level reaches the stress limit, i.e., f∗ = 0. As contrasted with this discontinuity, Eq.(15) means
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the following facts: (i) when f˘∗ > 0, the plastic flow may be induced at any non-vanishing stress level but becomes appreciable only when the stress limit is attained, (ii) the plastic flow grows with increasing stress level and agrees with the classical case as the stress level reaches the stress limit, and (iii) ratcheting effects under cyclic loading conditions are
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automatically incorporated in the new model.
In the above, a new elastoplastic J2 −flow model for finite elastoplastic deformations with both rate-dependent effects and thermal effects is established by the elastic rate equation (5) and the plastic rate equation (14) with Eqs. (15)-(20), as well as the evolution equations (10)-(11). As contrasted with usual elastoplastic models subjected to the coercive
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restrictions assumed for plastic flow and stipulated by the thermodynamic principles, the
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proposed model is free in the three respects below: (i) it is of much simpler structure than usual models and free in the sense that
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neither the yield condition nor the loading-unloading conditions for plastic flow need be imposed;
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(ii) it is also free in the sense of identically fulfilling the thermodynamic restriction stipulated by the second law (the Clausius-Duhm inequality) for any given
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forms of the constitutive quantities introduced, as will be demonstrated in the next section, and, furthermore, (iii) it is free of any failure criteria for the purpose of simulating the fatigue failure behavior, namely, it incorporates the fatigue failure behavior as an inherent constitutive feature, as will be shown in §4 (see also Xiao 2014b).
The above properties may be demonstrated by developing the relevant procedures in 14
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the previous study (Xiao et al. 2014, Xiao 2014b, 2015), as will be done in the new two sections. Here, consequences concering fatigue effects are further highlighted. According to a usual flow rule, no plastic strain is induced with the plastic factor ρ invariably vanishing
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prior to yielding and, hence, any fatigue effects under cyclic loading processes would be excluded and, in particular, that may be evident for cyclic loading processes with small stress amplitudes within the classical yield limit. As contrasted with the usual flow rule, the new flow rule Eq. (14) requires that a plastic factor ρ of the properties Eq. (15) should
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smoothly take values from 0 to 1 as the stress grows from 0 to the yield limit. Then, plastic strain is induced with small values of the plastic factor ρ prior to the yield limit and agrees with its classical counterpart with ρ = 1 just at yielding, as will be exemplified shortly by a simple form of ρ. As such, the new flow rule Eq. (14) with a plastic factor ρ of the properties Eq. (15) ensures that the effective plastic work ϑ is invariably growing in every
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cyclic process and, therefore, both high and low cycle fatigue effects may be in a natural,
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unified manner characterized with the strength quantities of asymptotic properties (cf., Eqs. (35)-(41) given later on). With this perhaps favourable property, the new flow rule
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Eq. (14) with no extrinsic coercive conditions related to the yield limit may be not only simpler but even more realistic than the usual flow rule.
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The model proposed here includes the previous models as particular cases. In fact, the free rate-independent and rate-dependent model in the previous work (cf., Xiao et al.
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2014, Xiao 2014b, 2015) may be respectively incorporated as the particular case when the stretching magnitude d is dropped out in Eq.(7) and the plastic factor ρ is given by
3J2 −m e ρ= 2r∗2
1−
3J2 2 2r∗
.
(21)
In the above, the m is a positive dimensionless parameter and referred to as the plastic index. With the plastic factor of the above simple form, the plastic index m plays an
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essential role in characterizing the fatigue behavior, in the sense that it is mainly responsible for accumulation of the effective plastic work ϑ in every thermomechanical process. For fairly large m > 0, the plastic factor ρ is very small whenever the stress point stays far
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away from the usual yield surface 12 J2 − 13 r∗2 = 0, whereas it becomes close to 1 only when the stress point stays close to this surface.
At elevated temperature, creep deformation with appreciable viscous effects may be expected. Such viscous flow behavior may be incorporated in the proposed model. In
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fact, consider the simplest case without the back stress effect, i.e., α ≡ O. At a constant ˜ D e =O elevated temperature with T˙ = 0, the plastic flow dominates, namely, D p = ˙ D, ˙ and, moreover, f∗ = 21 J2 − 13 r∗2 = 0. Then, from Eqs. (14)-(18) with c = 0, ξ = 0, ρ = 1, d˙ = 0, we obtain
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2 ˜ ˜ = r∗2 D, ϑ˙ σ 3
ϑ˙ = r∗ d.
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and, therefore,
˜ is the deviatoric Cauchy stress. In deriving the above, the volumetric deformation Here, σ
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has been neglected and the isotropic hardening modulus r∗ (∂r∗ /∂ϑ) has also been neglected
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as compared with the shear modulus G, since the latter is far greater than the former. It
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may be clear that the usual equation for viscous flow follows, namely, ˜ , ˜ = ηD σ
whenever the rate-dependent stress limit r∗ is reduced to r∗ = 1.5ηd at elevated temperature, where η is the viscosity parameter. The thermomechanical deformation behavior of metals and alloys may be represented by the constitutive quantities introduced in the proposed model, including two elastic
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constants, i.e., G and B, as well as four quantities for plastic behavior, i.e., the plastic factor ρ, the stress limit r∗ , the Prager modulus c and the hysteresis modulus ξ. The latter four jointly characterise the strength property of metals and referred to as the strength
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quantities. Of them, the plastic factor ρ ensures acumulation of the effective plastic work at any stress level and, accordingly, it plays an essential role in characterizing both the ratcheting effect and the fatigue failure under cyclic loading conditions2 .
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3. Thermodynamic consistency in explicit, identical sense
Let q, ψ, η be the heat flux per unit current area, the Holmholtz free energy, the specific entropy per unit reference volume, respectively. Then, the energy balance (i.e., the first law) and the Clausius-Duhem inequality (the second law) may be written in the forms:
(22)
D−
(23)
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τ : D − (ψ˙ + η T˙ ) = D , J q · ∇T ≥ 0 , T
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with D the intrinsic dissipation below:
D = T η˙ − (χ − J∇ · q) .
(24)
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In the above, χ is the heat supply per unit reference volume. An enhanced form of the Clausius-Duhem inequality Eq. (23) requires that the intrinsic
dissipation should always be non-negative, namely, the Planck inequality below should be 2
However, it may be noted that, according to a usual elastoplasticity model, the plastic factor ρ vanishes and plastic strain can not be induced at any stress level below the stress limit, thus excluding the occurrence of the ratcheting effect and the fatigue failure in most cases, in particular, the so-called high cycle fatigue under cyclic stresses below the initial yield stress.
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satisfied: D≥0
(25)
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for every possible thermodynamic process. We are going to demonstrate that the proposed model is thermodynamically consistent in an explicit, identical sense, namely, the free energy function ψ and the specific entropy function η may be presented in explicit forms, so that the second law with non-negative intrinsic dissipation is fulfilled for any given forms of the constitutive functions incorpo-
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rated. Toward this goal, a free energy function ψ and a specific entropy function η in explicit forms may be constructed by developing the procedure in Xiao et al. (2007), so that the Planck inequality Eq. (25) is identically fulfilled. These two explicit functions are presented as follows:
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∂W : τ − W + κ − ϕ(ϑ, T ) , ∂τ
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ψ = ψ0 (T ) +
η = −ψ00 (T ) +
∂ϕ ∂W + , ∂T ∂T
(26)
(27)
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where the ψ0 (T ) is the specific heat capacity of the material at issue, the κ is the usual plastic work given by
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κ˙ = τ : D p
(28)
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and the ϕ = ϕ(ϑ, T ) is a monotonically increasing function of the effective plastic work ϑ, namely,
ϕ(0, T ) = 0,
∂ϕ > 0. ∂ϑ
(29)
Then, with Eqs. (22) and (26)-(28), the intrinsic dissipation is given by 1 D= ρ 2
! f˘ f˘ ∂ϕ + ≥ 0. ˘ h ˘ ∂ϑ h 18
(30)
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Thus, the intrinsic dissipation is always non-negative and the Planck inequality is identically met. Generally, the temperature T , the heat flux q, the stress τ and the deformation gradient
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F are coupled with one another and governed by the elastoplastic constitutive equations presented in the last section as well as the reduced form of the first and second laws, namely, Eq. (22) with Eq. (30). On account of the fact that there inevitably emerges the positive intrinsic dissipation, thermal effects may always be coupled with elastoplastic deformations.
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In particular, that may be the case even with the environmental temperature held fixed. A well-known example is the heating effect due to plastic dissipation in metal bars under tension, as observed by Taylor and Quinney (1934, 1937). Such coupling effects may be treated in a thermodynamically consistent sense by working out the temperature and the heat flux coupled with the deformation based on the complete system of the governing
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indicated in the foregoing.
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equations from both the elastoplastic equations and the thermodynamic equations, as
PT
4. Critical failure states with a unified criterion In this section, we are going to treat fatigue failure effects in a broad sense. It will
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be shown that critical states for thermo-coupled fatigue failure under combined changes of stress and temperature may be derived directly from the proposed model and, then, a
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criterion for determining these states may be obtained as model prediction. In association with various cases of inhomogeneity at a small scale, it is observed in
fatigue testing that fatigue failure of different nature may be induced by various localized effects with microcracks initiating and propagating either at the surface or in the interior of a material sample. It is instrumental to have a general knowledge of the characteristic deformation behavior of the materials under isothermal loading and, generally, under thermomechanical loading up to fatigue failure. The essential nature in this respect 19
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is concerned with the involvement of multiple deformation micro-mechanisms at different temperatures and the dependence of these mechanisms on the histories of both the temperature and the applied load. Details in micromechanisms and macroscopic behavior may be
and Pook (2007). 4.1. Asymptotic properties of the strength quantities
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found in, e.g., Stouffer (1996), Levitin (1996), Suresh (1998), Frost (1999), Argon (2007),
As indicated from the outset, a basic fact concerning deformation behavior of materials
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is as follows: When repeatedly undergoing deformations, metals and alloys suffer continued loss of the strength up to eventual failure. Since the material strength is characterised jointly by the strength quantities indicated at the end of §2 and each such quantity changes itself with accumulation of the effective plastic work ϑ, with the foregoing basic fact we
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come to a perhaps essential understanding. Namely, the strength quantities characterising the strength property in the proposed model should go down and eventually become
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vanishing with accumulation of the effective plastic work. With the strength quantities of such asymptotic properties, the proposed model automatically incorporates fatigue failure
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as an inherent constitutive feature. It has been demonstrated in an initial study (Xiao 2014b) that fatigue failure may be in principle derived as direct consequences of the new
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elastoplasticity models proposed. In this study, however, the central issue below is left unanswered, namely, under what circumstances will fatigue failure be eventually attained
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and, in particular, how will the fatigue life be determined under cyclic changes of stress and/or temperature? Here, the central issue above will further be studied. For this purpose, we first need to
specify certain relevant properties of the strength quantities introduced. We first point out certain general properties. For normal stress levels below the stress
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limit introduced in the model, we deduce (cf., Eqs. (6) and (16))
⇐⇒
1.5J2 /r∗2 < 1
⇐⇒
1 − ρ(1.5J2 /r∗2 ) > 0 .
With Eq. (20), we have f˘∗ >0 ˘∗ h
⇐⇒
f˘∗ > 0 .
Moreover, from Eqs. (10), (14) and (19)1 we deduce ! f˘∗ f˘∗ + ≥ 0. ˘∗ h ˘∗ h
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1 ϑ˙ = ρ 2
(31)
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f∗ < 0
(32)
(33)
Then, it follows that the rate ϑ˙ is always non-negative and it is positive for f˘∗ > 0 and, accordingly, the effective plastic work ϑ will invariably grow and accumulate with devel-
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opment of any plastic flow.
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We next take two stages of deformation behavior into consideration. On the one side, a hardening stage emerges for the effective plastic work below a threshold value, denoted
ˆ ∗ > 0 for ϑ < ϑˆ . h
(34)
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ˆ For this stage, the plastic modulus h ˆ ∗ (cf., Eq. (18)) is positive, i.e., ϑ.
ˆ For this stage, the stress limit r∗ On the other side, a softening stage emerges for ϑ > ϑ.
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is of the following properties: it goes to vanish as ϑ → +∞, and the isotropic hardening modulus r∗ (∂r∗ /∂ϑ) becomes negative, as shown below: ∂r∗ < 0 for ϑ > ϑˆ , ∂ϑ r∗
∂r∗ < 0 for ϑ > ϑˆ , ∂ϑ
21
(35)
(36)
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lim r∗ = 0 .
(37)
ϑ→+∞
Besides, as ϑ grows to infinity, the Prager modulus c = cs > 0 goes also down to vanish and
∂c ≤ 0 for ϑ > ϑˆ , ∂ϑ lim cs = 0 ,
and
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ϑ→+∞
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the hysteresis modulus ξ = ξs > 0 goes up to exceed a positive constant ξ0 > 0, namely,
(38)
(39)
∂ξ ≥ 0 for ϑ > ϑˆ , ∂ϑ
(40)
lim ξs ≥ ξ0 > 0 .
(41)
ϑ→+∞
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The above asymptotic properties are implied just by the basic fact for realistic material behavior, as explained at the outset of this subsection, and have been introduced in the
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ˆ ∗ becomes previous work (Xiao 2014b). Finally, at the softening stage, the plastic modulus h
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negative, i.e.,
ˆ ∗ < 0 for ϑ > ϑˆ . h
(42)
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The strength quantities of the above properties with two stages may be combined in unified, explicit form. In fact, let the strength quantities be given by rh , ch , ξh at the
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hardening stage and by rs , cs , ξs at the softening stage. Then, the strength quantities for both the hardening and the softening stage may be given as follows:
ˆ ˆ r∗ = ϕ ϑ − ϑ rh + ϕ ϑ − ϑ rs , c = ϕ ϑˆ − ϑ ch + ϕ ϑ − ϑˆ cs ,
ˆ ˆ ξ = ϕ ϑ − ϑ ξh + ϕ ϑ − ϑ ξs , 22
(43)
(44) (45)
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where
1 x . ϕ(x) = 1 + tanh a 2 ϑˆ
(46)
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In the above, the positive dimensionless parameter a > 0 may be temperature-dependent. It is fairly large, so that, in a small neighborhood centered at x = 0, ϕ(x) changes rapidly from -1 to +1.
In the isothermal case, the strength quantities at the hardening and the softening stage are determinable from suitable uniaxial data, as will be given in the next section. Moreover,
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simple forms of the plastic factor ρ will also be given in the subsequent development. 4.2. Critical failure states
The asymptotic properties given in Eqs. (35)-(41) imply that asymptotic loss of the strength, viz., fatigue failure will eventually emerge as the effective plastic work ϑ becomes
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indefinitely large, as has been indicated in the previous study (Xiao 2014b). However, it
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does not appear to be realistic to treat fatigue failure based directly on such properties. Then arises the central issue indicated in the last subsection. It is the finding of critical
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failure states below that leads to a solution of this central issue. Consider a process of thermomechanical deformation in which both stress and temper-
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ature are changing. At the end of this process, there will emerge such a state, referred to as a critical failure state, that , at this state, the effective plastic work ϑ will grow at an
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infinite rate and, immediately following this state, there is only one possibility left for the deformation behavior, namely, indefinitely large deformation will be induced concomitantly as the stress magnitude goes down to vanish. It is clear that a critical failure state heralds the eventual fatigue failure. In the next subsection, it will be demonstrated that critical failure states exist and may be derived as model prediction.
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4.3. A criterion in unified form In what follows we are in a position to find out all critical failure states and derive a unified criterion for such states.
From Eqs. (1), (5) and (10) we deduce o
2Gϑ˙ = 2G(˜ τ − α) : D − (˜ τ − α) : τ log +
G0 (J2 + L) T˙ , G
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and from Eq. (16) we infer
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˘ ∗ (cf., Eqs. (16)-(17)). Toward this goal, we first derive an alternative expression for f˘∗ /h
∂f∗ ˙ ∂f∗ ¨ G0 2G(˜ τ − α) : D = f˘∗ − T− ϑ − (J2 + L) T˙ . ˙ ∂T G ∂ϑ
(47)
(48)
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On the other hand, for plastic flow with f˘∗ > 0, Eqs. (10), (14) and (19) together yield f˘∗ ϑ˙ = ρ . h˘∗
(49)
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where
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Hence, substituting Eqs. (48)-(49) into Eq. (47) we arrive at
2Gρ
f˘∗ = f˘∗ − fˆ∗ , ˘∗ h
(50)
∂f∗ o log ∂f∗ ˙ ∂f∗ ˙ T+ fˆ∗ = :τ + d. ∂τ ∂T ∂d
(51)
f˘∗ fˆ∗ = , ˘∗ ˆ∗ h 2G(1 − ρ) + h
(52)
We then obtain
ˆ ∗ is just the plastic modulus given by Eq. (18). Thus, Eq. (49) may be where the h
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reformulated in the form: ϑ˙ =
ρfˆ∗
ˆ∗ 2G(1 − ρ) + h
.
(53)
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Now it becomes clear that the effective plastic work ϑ may grow at infinite rate, whenever the denominator in Eq. (53) becomes vanishing, namely, ˆ∗ = 0 . 2G(1 − ρ) + h
(54)
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If the above equation is satisfied, then a critical failure state as explained in the last subsection is found. Below we are going to demonstrate that critical failure states are implied by the strength quantities of the asymptotic properties indicated in §4.1. From Eqs. (31) and (34) it follows that the left-hand side of Eq. (54) is always positive at the hardening stage and, therefore, critical failure states could not emerge at
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ˆ the first term, i.e., 2G(1 − ρ), is the hardening stage. At the softening stage with ϑ > ϑ,
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positive (cf., Eq. (31)), while the second term, the plastic modulus h∗ is negative. Since the factor (1 − ρ) may take any given value in the interval (0,1), it follows that solutions
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of Eq. (54) always exist and critical failure states are accordingly determined by Eq. (54). Furthermore, we demonstrate that the magnitude of the stress goes down to vanish with
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further plastic flow following a critical failure state. We first prove that will be the case for the magnitude of the effective stress, i.e., J2 . To this end, we evaluate the time rate of
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the function f∗ (cf., Eq. (6)). In a process of thermo-coup-led elastoplastic deformation, the changing rate of the function f is given by (cf., §4 in Xiao et al. 2007): ∂f o log ∂f ˙ ∂f ˙ ∂f o log T+ f˙ = :τ + ϑ+ :α . ∂τ ∂T ∂ϑ ∂α
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On the other hand, utilizing the equation (cf., Eqs. (1)-(2)) o
and the flow rule Eq. (14), we infer o
τ log
∂ 2W ˙ 1 ρ =S:D−S: T− ∂τ ∂T 2 J2
! f˘ f˘ + S : (˜ τ − α) . ˘ h ˘ h
(55)
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In the above, the S is the elastic rigidity tensor:
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τ log = S : (D − D p )
1 S = 2GI + (B − 2G)I ⊗ I , 3
where I and I are the 2nd- and 4th-order identity tensors, respectively. Substituting this
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eqs. (16) and (47)-(49), we deduce
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expression and Eqs. (10)-(11) into the former expression for the time rate f˙ and then using
(56)
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f˙∗ = (1 − ρ)f˘∗ .
From this and Eq. (31) we infer that f˙∗ > 0 for any plastic flow with f˘ > 0 and, therefore,
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f∗ starts with a negative value (cf., Eq. (31)) and goes to vanish as ϑ goes to infinity. It follows from this and Eq. (37) that J2 goes to vanish. On the other hand, after a critical
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failure state is attained, cs becomes vanishing (cf., Eq. (39)) and ξs is no less than ξ0 > 0 (cf., Eq. (41)). From these and Eq. (11) we derive the following equation: dζ 2 ≤ −ξ0 ζ 2 , dϑ where 1.5ζ 2 = |α|2 . Hence, we infer that the back stress goes to vanish as ϑ goes to infinity.
Finally, since both J2 (cf., Eq. (20)) and ζ goes to vanish, we conclude that the stress τ 26
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should go to vanish. The criterion given by Eq. (54) for critical failure states is applicable for a broad case of multi-axial thermo-coupled fatigue failure with rate-dependent effects. It is for all kinds
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of thermomechanical processes. Usually, two particular kinds of cyclic processes are taken into consideration, namely, cyclic processes of stress and cyclic processes of temperature. Further results for these two cases will be presented in the next two sections.
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5. Fatigue failure under multi-axial stress cycles
In this section, we are going to treat the isothermal case with T˙ = 0. It will be shown that the hardening quantities r, c and ξ may be determined based on suitable uniaxial data. Fatigue lives will be determined for cyclic processes of multi-axial stress.
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The main idea in deriving the results below are explained as follows. It will be demonstrated that, given uniaxial stress-strain data in primary monotone tension test and in
ED
reverse monotone compression test, the foregoing three quantities may actually be determined by explicit procedures, in the sense that the predictions of the proposed model for
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the two cases of uniaxial tension and compression in the foregoing can exactly reproduce these data. The main idea of achieving this is to express the two uniaxial curves in tension
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and compression (cf., Fig. 1) by two suitable functions as will be given by Eqs. (57)-(59) and then obtain an explicit expression of each hardening quantity at issue in terms of
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these two functions. Results are first derived for uniaxial tension and compression and, then, these uniaxial results are extended to multiaxial cases by means of direct procedures. These explicit procedures represent a direct extension of those introduced in most recent studies (Xiao 2014a; Xiao et al. 2016). Details will be omitted and may be found in these references.
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5.1. Determination of the strength quantities at the hardening stage As shown in Eqs. (7) and (12)-(13), generally the hardening quantities r∗ , c and ξ may be complicated functions of the three variables ϑ, τ and α for the isothermal case. Usually,
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it may be rather difficult to find out the forms of these functions and ad hoc forms have to be assumed for various purposes. In what follows, explicit, direct procedures will be proposed to obtain the hardening quantities (rh , ch , ξh ) and (rs , cs , ξs ) in Eqs. (43)-(45) based on suitable uniaxial data at the hardening and the softening stage, separately.
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Toward the above purpose, we consider suitable uniaxial stress-strain data at a very low strain rate, including those for the primary loading curve and the reverse loading curve, as depicted in Fig. 1. The former is generated by monotonically growing tensile strain up to failure, while the latter by unloading the axial stress from a certain point on the primary
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loading curve and then reversely loading the compressive stress. It should be noted that the primary loading curve starts at an initial yield point in classical sense, while the reverse
ED
loading curve at a reverse yield point. The two stages indicated in §4.1 now become clear: the part of the curve prior to the maximum stress τm corresponds to the hardening stage,
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while the other part after τm to the softening stage. Details are schematically shown in
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Fig. 1.
It should be indicated that here a very low strain rate means the rate-independent case
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with d = 0. In addition, the usual yield condition is attained along the primary loading curve and the reverse loading curve, namely, ρ = 1. Given suitable stress-strain data, the two curves in Fig. 1 may be determined by two
functions. The latter may be given in various forms.
For our purpose, it is essential to
prescribe these two curves by means of two functions described below. Following the idea suggested in a previous study (Xiao 2014a), we prescribe the hard-
28
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Fig. 1. Primary and reverse loading curves under tension and compression
ening part of the primary loading curve by a strain-stress function below:
(57)
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h = p(τ ), ϑ ≤ ϑˆ , p(r0 ) = r0 . E
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Here, ϑˆ is the effective plastic work at the maximum stress τm , E is Young’s modulus, τ and h are the axial stress and the axial strain at each point on the primary loading curve,
PT
and r0 is the initial yield stress in classical sense. Unlike the above procedure of prescribing the primary loading curve, we prescribe the
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reverse loading curve by means of an other procedure. For our purpose, a function, referred to as the bridging function for the primary and the reverse loading curve, is introduced as
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follows:
τ = φ(τ ) , φ(u0 ) = r0 .
(58)
where u0 is the stress at the intersecting point (cf., Fig. 1) between the reverse loading curve and the elastic line passing the origin. The above function establishes the correlation between the axial stresses τ and τ at each pair of points linked by an elastic unloading
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line, as shown in Fig. 1. By means of the bridging function we have 1 (τ − φ(τ )) E
(59)
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h = q(τ ) ≡ p(φ(τ )) +
for the axial strain h and the axial stress τ on the reverse loading curve. Thus, the reverse loading curve may be prescribed jointly by Eq. (57) and Eq. (58).
Whenever the two functions (cf., Eqs. (57)-(58)) for the primary and the reverse loading curve are available, the hardening quantities ch , ξh and rh at the hardening stage may be
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explicitly determined by these two functions, as will be done below.
For the monotone loading processes along the primary and the reverse loading curve, the following uniaxial relations may be derived (cf., Xiao 2014a, Xiao et al. 2016):
(60)
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K(τ ) , ∂rh ∂rh −1 −2 rh + 1.5(ch − ξh L) − rh ξh ζ − 1.5ch ζ rh L = ∂ϑ ∂ζ K(τ ) ,
where K(τ ) and K(τ ) are the plastic slopes of the primary loading curve and the reverse
dp 1 1 = − , dτ E K(τ )
(61)
1 dq 1 = − , K(τ ) dτ E
(62)
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loading curve, respectively, given by
with the Young’s modulus E = 3GB/(G + B). Eq. (60) merely applies to the uniaxial
case. A multi-axial extension may be given as follows (cf., Xiao et al. 2016):
rh
∂rh ∂rh + 1.5(ch − ξh L) − rh ξh ζ − 1.5ch ζ −1 rh−2 L = K(Λ, rh ) , ∂ϑ ∂ζ
30
(63)
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where
[Λ + rh ] [Λ + rh ] K(Λ, rh ) ≡ K (Λ + rh ) + 1 − K (Λ − rh ) , |Λ| + rh |Λ| + rh
(64)
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where Λ = 1.5L/rh and [x] = (x + |x|)/2. From Eq. (63) for general multi-axial cases, both ch and ξh are obtainable by following the idea of treating the singularity in Xiao (2014a) and Xiao et al. (2016). Results are as follows: ch =
r˜h0 K(−ζ r˜h0 , rh ) − rh rh0 K(Λ, rh ) − K(−ζ r˜h0 , rh ) − . ζ Λ + ζ r˜h0 1.5 − (˜ rh0 )2
In the above, rh0 =
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ξh rh =
K(−ζ r˜h0 , rh ) − rh rh0 , 1.5 − (˜ rh0 )2
∂rh , ∂ϑ
r˜h0 =
∂rh . ∂ζ
(65)
(66)
(67)
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Moreover, the rate-independent stress limit rh at the hardening stage is given by
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√ φ0 (0) − 1 −% rζ r0 + u0 −η rϑ r0 + u0 −η rϑ √ e 0 − 1.5 0 ζe 0 + g 1.5 ζ + e 0 , rh = 2 φ (0) + 1 2
(68)
PT
where % > 0 and η > 0 are two large parameters representing the localized effects at ζ = 0and at ϑ = 0, respectively, and the function r = g(α) is determined by the following
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parametric equations:
r = 1 (φ(τ ) − τ ) , 2 α = 1 (φ(τ ) + τ ) .
(69)
2
Details may be found in Xiao (2014a).
As shown in the previous work (Xiao 2014a, Xiao et al. 2016), a simple form of the shape
function (cf., Eq. (57)) for the primary loading curve may be given and then determined in fitting any given stress-strain data. Moreover, the shape function (cf., Eq. (59)) for the reverse loading curve is also available, whenever a suitable bridging function (cf., Eq. (58))
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is given. Usually, a linear function may serve the purpose, namely,
φ(τ ) = b1 (τ − u0 ) + r0 .
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In this case, we have
rh =
(70)
b (r + u ) r 0 − b1 u 0 √ b1 − 1 1 0 0 −η rϑ −% ζ + 1.5 ζ 1 − e r0 + e 0, b1 + 1 b1 + 1 b1 + 1
(71)
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The corresponding result in the previous work (Xiao 2014a, Xiao et al. 2016) is the particular case when u0 = 0.
5.2. Determination of the strength quantities at the softening stage After the maximum stress τm is attained, the uniaxial stress-strain curve will come to
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the softening part with negative slopes, as shown in Fig. 1. The softening part should be
ED
represented by another strain-stress function3 , as shown below: ϑ ≥ ϑˆ ,
(72)
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hs = ps (τ ),
with the following properties:
AC
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p0 (τ ) < 0 , s lim p (τ ) = +∞, τ →0
s
(73) limτ →0 p0s (τ )
= −∞ .
The moduli cs and ξs are still given by Eqs. (65)-(67), where ch , ξh and rh are replaced
3 It may be evident that the whole primary loading curve including both the hardening and the softening part can not be represented by a single strain-stress function.
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by their counterparts, i.e., cs , ξs and rs at the softening stage, namely,
ξs rs =
K(−ζ r˜s0 , rs ) − rs rs0 , 1.5 − (˜ rs0 )2
(74)
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cs =
r˜s0 K(−ζ r˜s0 , rs ) − rs rs0 K(Λ, rs ) − K(−ζ r˜s0 , rs ) − . ζ Λ + ζ r˜s0 1.5 − (˜ rs0 )2
(75)
where the K(Λ, r) is still given by Eq. (64) with the plastic slopes K(τ ) and K(τ ) below:
K(τ ) =
p0s (τ )
1 , − E −1
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K(τ ) =
b−1 s . p0s (bs τ ) − E −1
(76)
(77)
In the above, bs > 0 is a dimensionless parameter.
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Next, the stress limit rs is given by
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h i √ rs = 1 rm 1 − tanh β0 ϑ − 1 + 1.5 bbss −1 ζ, 2 ϑ0 +1 ϑ ≥ ϑˆ .
(78)
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In the above, rm > 0, β0 > 0 and ϑ0 > 0 are material parameters. The moduli cs and ξs and the stress limit given above characterise coupling effects for
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softening behavior, in which the shape funtion of the property indicated in Eq. (73) plays an essential role. A simple form of this function may be given as follows (cf., Xiao et al.
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2016):
h = ps (τ ) = hm
n 1 τ2 1 + − ln 2 , 0 < τ ≤ τm , λs τm
(79)
where hm and τm are the strain and the stress at the maximum stress point (cf., Fig. 1) and n > 0 and λs > 0 are dimensionless parameters. Details in deriving the above results may be found in Xiao et al. (2016). It may be demonstrated (cf., Xiao et al. 2016) that the quantities given meet the asymptotic 33
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properties prescribed by Eqs. (37), (39) and (41). 5.3. Determination of the fatigue life
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Consider a uniform deformation process consisting of stress cycles. Let τ 0 be any given multi-axial stress. At each cycle, the stress is changing from zero to τ 0 and then to uτ 0 and finally back to zero. Here, −1 ≤ u ≤ 0. A cycle with u = −1 is called a symmetric one. At each cycle, the plastic flow is induced in the two processes from 0 to τ 0 and from τ 0 to uτ 0 , separately.
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The fatigue life under the cyclic process at issue is the cycle number to failure, denoted N , for which a critical failure state is attained. It is determined by the following procedures: (i) Integrate the following system of the elastoplastic rate equations successively
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as from the first cycle:
ρJ2−1 fˆ ˆ∗ 2G(1−ρ)+h
(˜ τ − α) ,
ED
o 1 1 log ˜ D = (trτ˙ )I + τ + 3B 2G ρfˆ ϑ˙ = ˆ , 2G(1−ρ)+h ∗
(80)
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o log cρJ2−1 fˆ τ − α) − ξαϑ˙ , α = ˆ ∗ (˜ 2G(1−ρ)+h o fˆ = (˜ ∗ ˙ τ − α) : τ log − 23 r ∂r d ∂d
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for fˆ > 0 and, on the other hand, set fˆ = 0 in Eqs. (80)1−3 for fˆ ≤ 0; (ii) obtain the effective plastic work at the end of each cycle, denoted ϑ(n) for
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the n−th cycle;
(iii) the cycle number N for which a critical failure state is attained is determined by ϑ(N −1) > ϑ(N −2) , ϑ(N −1) > ϑ(N ) . 34
(81)
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The criterion Eq. (81) is implied by the following fact: According to Eq. (80)2 , the ϑ is increasing as the denominator is positive and decreasing as the latter becomes negative. As such, Eq. (81) means that, just at the N −th cycle, the denominator experiences a should be met with the occurrence of a critical failure state.
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change in sign from positive to negative values and, accordingly, the criterion Eq. (54)
As indicated before, the attainment of a critical failure state heralds the beginning of the asympotic strength loss, namely, asymptotic loss of the stress-bearing capacity. As
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such, immediately following the attainment of a critical failure state, the magnitude of the stress will invariably go down to vanish concomitantly with development of plastic flow. From this it follows that, whenever a critical failure state is attained under a cyclic stress process, the stress cycle could no longer be completed as in the case prior to this critical state. This may be explained as follows. A stress cycle means that the stress magnitude is
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allowed to grow to a given value known as the stress amplitude, as mentioned at the outset
ED
of this subsection. However, after a critical failure state the stress magnitude could no longer be allowed to grow to this amplitude but should go asymptotically down to vanish.
5.4. Remarks
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This phenomenon will be evidenced in Fig. 4 presented in §7.1.
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Results have been derived for cyclic loading cases. Generally, irregular or non-cyclic loading cases need be treated. Since the cycle number to failure becomes irrelevant in
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general, the fatigue life in usual sense should be replaced by the entire process from the start of loading up to the emergence of a critical failure state. Given uniaxial stress-strain data for the primary and the reverse loading curve as shown
in Fig. 1, the shape functions (cf., Eq. (57) and Eqs.(72)-(73)) for the hardening and the softening stage and the bridging function Eq. (58) may be prescribed for the purpose of fitting such data, as indicated in the previous work (Xiao 2014a, Xiao et al. 2016). Then, the results presented in §5.1 and §5.2 achieve double goals, namely, (i) the hardening 35
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quantities for a general multi-axial case are explicitly determined from these shape functions and (ii) the proposed model with these hardening quantities agree automatically with the uniaxial data at issue.
With the strength quantities identified from suitable uniaxial
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data, including those at the hardening and the softening stage demarcated by the effective plastic work ϑˆ at the uniaxial maximum stress, as given in §5.1-§5.2, the proposed model with a given form of the plastic factor ρ (cf., Eqs. (14)-(15)) is ready for obtaining results for multi-axial loading conditions and then comparing with multi-axial data. Here it is
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worthwhile to point out that it is essential to select a suitable form of the plastic factor ρ toward simulating both the ratcheting effect and the fatigue failure behavior. A simple form of ρ, given by Eq. (21), has been suggested in previous works (cf., e.g., Xiao et al. 2014; Xiao 2014b) and shown to be useful (Wang and Xiao 2014). Further developments
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of this form will be proposed in §7.
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6. Fatigue failure under thermal cycles
In this section, fatigue failure under thermal cycles will be taken into consideration. It
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may be of interest to know what features the fatigue behavior in this case will display. It is noted that results in this respect have been rarely reported, expected for experimental
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tests in some cases.
6.1. Governing equation for the back stress
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Consider a thermal process consisting of temperature cycles. At each cycle, the temper-
ature T grows from a given lower value, T1 , to a given higher value, T2 , and, then, returns to T1 . Let the back stress and the effective plastic work at the outset of the first cycle be α1 and ϑ1 , respectively. Moreover, a basic property for the stress limit r∗ is as follows: ∂r∗ < 0, ∂T 36
(82)
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namely, the stress limit r∗ goes down as the temperature goes up. We are going to derive a governing equation for the back stress α. With τ = O and Eqs. (1) and (5), we have
˜ . Dp = D
(83)
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D e = γ T˙ I ,
(84)
On account of the property Eq. (82) and the fact that ϑ˙ ≥ 0, we infer
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∂r∗ ˙ ∂r∗ ˙ f˘ = 3Gϑ˙ − r∗ T − r∗ d>0 ∂T ∂d
(85)
for T˙ > 0 and ϑ¨ ≤ 0. Hence, plastic flow will be induced, whenever the temperature grows from T1 to T2 at each thermal cycle. From this and Eqs. (14) and (32) we deduce
(86)
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f˘ D p = −ζ −2 ρ α . ˘ h
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Then, this and Eq. (49) produce
˙ . ˜ = −ζ −2 ϑα Dp = D
(87)
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Substituting this into Eq. (11), we derive o log
α
˙ . = (−cζ −2 − ξ)ϑα
(88)
Then, by using Eq. (88) and the equality o α : α log = 1.5ζ ζ˙ ,
37
(89)
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we arrive at: 1.5ζ ζ˙ = (−c − 1.5ξζ 2 )ϑ˙ .
(90)
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From Eq. (90) it follows that the magnitude of the back stress α will go down to vanish as the thermal cycle is progressing. In fact, for the heating process from T1 to T2 at the n−th cycle, Eq. (90) yields
2 − ζ1n
with Θ=
Z
t
t1n
4 2Θ ˙ ce ϑdt , 3
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ζ = e−Θ
s
Z
t
˙ , 2ξ ϑdt
(91)
(92)
t1n
2 where t1n and ζ1n are the initial values of the time t and the magnitude ζ at the start of
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the n−th cycle and t is any given instant in the heating process in the foregoing.
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6.2. Determination of the thermal fatigue life
We proceed to find out the cycle number for which a critical failure state will be
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attained. Toward this end, we first work out the effective plastic work at the end of each thermal cycle. At each cycle, with Eqs. (18)-(20), Eq. (53) and τ = O, we deduce that
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the ϑ is determined by the equation below:
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ϑ˙ =
∗ ˙ ∗ ˙ −ρr∗ ∂r T − ρr ∂r d ∂T ∂d . ∂r∗ ∗ 3G(1 − ρ) + (c + ζ 2 ξ) 1.5 − ζ −1 r∗ ∂r + r ∗ ∂ϑ ∂ζ
(93)
Given the time-dependent temperature T = T (t), how the ϑ will change with the time t
may be determined jointly from Eqs. (90) and (93). In particular, for the rate-independent case with the r∗ independent of the stretching magnitude d, Eq. (93) is substantially simplified and it is possible to obtain the ϑ as a function of the temperature T , as shown
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below: ∗ −ρr∗ ∂r dϑ ∂T = . ∂r∗ dT ∗ + r 3G(1 − ρ) + (c + ζ 2 ξ) 1.5 − ζ −1 r∗ ∂r ∗ ∂ϑ ∂ζ
(94)
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Again, the cycle number N , for which a critical failure state is attained, is determined by Eq. (81).
It should be pointed out that Eqs. (91)-(92) supplies a closed-form solution of the form ζ = ζ(ϑ) for the case when both the Prager modulus c and the hysteresis modulus ξ are dependent merely on the effective plastic work ϑ. Given the time-dependent temperature
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T = T (t), generally the solutions ζ = ζ(t) and ϑ = ϑ(t) should be obtained by integrating Eqs. (90) and (93). For the rate-independent case, the solutions of the form ζ = ζ(T ) and ϑ = ϑ(T ) may be determined from Eq. (91) and Eq. (94).
Eq. (93) displays strong nonlinearities coupled with rate-dependent effects. This feature
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is inherent in the thermal fatigue behavior, in the sense that it manifests itself for any form of temperature change and, in particular, that is the case even for temperature change at
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a constant rate.
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7. Numerical examples
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Numerical examples will be presented in this section. It will be shown that good agreement with test data may be achieved even in the simplest cases of the proposed model, namely, the hardening stage is neglected and the anisotropic hardening behavior is
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absent with α = 0.
Test data are usually given for uniaxial loading cases of cylindrical samples. Model
predictions for uniaxial stress cycles may be derived by applying the procedures in §4 and,
then, compared with test data in literature. For the uniaxial stress case, the J2 is given by 32 τ 2 with the axial Kirchhoff stress τ and, besides, the stretching magnitude d (cf., Eq. (8)) may be given by the axial strain rate h˙
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˙ as a good approximation4 . Namely, J2 = 23 τ 2 and d ≈ h. For the purpose of model validation, test data in four respects will be separately simulated, including monotone strain data with rate effects and those with the ratcheting
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effect and the fatigue limit effect, as well as data with the duplex effect from low to high cycle fatigue failure. Besides, model predictions will also be derived for fatigue lives under thermal cycles without and with a fixed uniaxial stress.
7.1. Monotone data with rate effects and model predictions under stress cycling
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We first take into account the low strain rate data for p92 steel in Giroux et al. (2010), which is concerned with a monotonically growing axial strain up to failure at the strain rate h˙ = 2.5 × 10−6 (Fig. 3 in the foregoing reference). In this case, the rate effect may be ρ=
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neglected. The plastic factor ρ (cf., Eq. (21)) and the stress limit r∗ are given by 1.5J2 −m(1−1.5J2 /r∗2 ) e r∗2
h
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r∗ = 1 rm 1 − tanh β0 2
,
ϑ ϑ0
i −1 ,
(95)
where the m is the plastic index (cf., Xiao et al. 2014; Xiao 2014b) and the rm , ϑ0 , β0
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represent the tensile strength, the critical value of ϑ, the slope at the softening part in the
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uniaxial stress-strain curve, respectively. The values of the parameters involved are found
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in fitting the foregoing data5 (cf., the lower curve in Fig. 2), as given below:
E = 120GPa, ν = 0.3, m = 1, rm = 338MPa, β0 = 2.9, ϑ0 = 58MPa .
Here and below, ν is the Poisson ratio. With these parameters, numerical results are
p ˙ with the lateral Hencky strain given In fact, the d in the uniaxial case is given by 2 (1 + 2s2 ) /3 h, by −sh. The ratio of the lateral to the axial Hencky strain, i.e., s, falls within the range (ν, 0.5). The ˙ value of the Poisson ratio ν is about 31 . As such, it follows that 0.9h˙ < d < h. 5 These data in Cauchy stress and engineering strain have been converted to those in Kirchhoff stress and Hencky strain. That will be the case for other data. 4
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obtained under the following type of uniaxial stress cycle: the axial stress τ grows from zero to a given amplitude, τ0 , and then is back to zero. The curve of the stress amplitude versus the cycle number to failure is depicted in Fig. 3 (the lower curve). Moreover, as
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an example, the stress-strain curves for all the stress cycles up to failure are shown in Fig. 4 for the stress amplitude τ0 = 200MPa. It may be clear from Fig. 4 that, immediately following a critical state, the stress can no longer be allowed to grow to the amplitude τ0 = 200MPa as in the case of each stress cycle prior to this state but goes invariably down
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to vanish concomitantly with indefinitely growing strain.
It is known (Xiao et al. 2014, Xiao 2014b) that the plastic index m plays an essential role in accumulation of the plastic strain and hence the effective plastic work. In Fig. 5, further results for the sensitivity analysis are presented for three values of the plastic index m, viz., m = 0.01, 0.3, 0.8. As is shown in Fig. 5, the fatigue life for a given stress
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amplitude goes up as the plastic index m goes up. This is in accord with the following fact
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(Xiao et al. 2014, 2014b): the greater the index m, the smaller the plastic strain at each cycle.
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Rate-dependent effects should be taken into consideration at high strain rate. In this case, each parameter in the stress limit r∗ in Eq. (95)2 relies on the equivalent strain-rate
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d. As illustration, consider the two data sets for monotone tension at the low strain-rate 2.5 × 10−6 s−1 and at the high strain rate 2.5 × 10−3 s−1 , as given in Giroux et al. (2010).
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The rate-dependent parameters are given below:
rm = 338 + 56800d (MPa), β0 = 2.9 + 680d, ϑ0 = 58 + 8800d (MPa).
As indicated before, the d may be given by the axial strain rate h˙ in the uniaxial case. The simulation results at the two strain rates h˙ = 2.5 × 10−6 s−1 and h˙ = 2.5 × 10−3 s−1 are shown for the monotone loading case in Fig. 2 and for the curves of fatigue life versus
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Fig. 2. Comparison with the monotone strain data (dots) from Giroux et al. (2010): tensile stress-strain curves at two strain rates 2.5 × 10−6 s−1 (lower curve) and 2.5 × 10−3 s−1 (upper curve)
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stress amplitude in Fig. 3. It follows from Fig. 2 and Fig. 3 that both the stress limit and the cycle number to failure increase with increasing strain rate.
Note here that the
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fatigue life in a direct sense, viz., the time to failure, decreases actually with increasing strain rate, since the latter is the cycle number to failure devided by the strain rate. That
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is the case for the model predictions shown in Fig. 2. The model prediction for the fatigue life, as shown in Fig. 3, agrees with the salient
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feature of the well-known S-N curve for the fatigue failure under uniaxial stress cycling. In general, the S-N curve is invariably inceasing with decreasing stress amplitude, as may be seen in Fig. 3. In particular, this monotone property should be exhibited even for vanishingly small stress amplitude, as shown in Fig. 3. Here, results for the fatigue failure and the racheting effect are merely presented as model predictions, since relevant data are absent in the foregoing reference. Comparisons with test data will be considered for various cases in the succeeding subsections. 42
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Fig. 3. Model prediction under stress cycling: stress amplitude versus cycle number to failure at two strain rates 2.5 × 10−6 s−1 (lower curve) and 2.5 × 10−3 s−1 (upper curve)
Fig. 4. Stress-strain curves up to failure under stress cycles at low strain rate 2.5 × 10−6 s−1
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Fig. 5. Parameter sensitivity analysis at low strain rate 2.5 × 10−6 s−1 : curves of fatigue life versus stress amplitude for three values of the plastic index (m =0.8, 0.3, 0.01 from top to bottom)
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It should be noted that uniform stress-strain data for monotone tensile strain test should be used for simulating the material softening up to failure. However, usually that is not the
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case for data with non-uniform strain and stress distributions after necking. This respect is studied in, e.g., Shaw and Kyriakides (1998). On account of this, data after necking
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are merely estimated values via certain averaging procedures and, as such, simulation is performed in an approximate sense. In fact, deviations of simulation results from such
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approximate data may become appreciable as failure is approached, as may be seen in Fig. 2.
7.2. Asymmetric stress cycling: racheting effect Simulation results are given for ratcheting effects and compared with the latest data in Fig. 9 in Lee et al. (2015) (see also Hassan and Kyriakides 1992) for S32750 super duplex stainless steel under the non-symmetric uniaxial stress cycling with a mean stress 44
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Fig. 6. Model prediction for the ratcheting effects under the non-symmetric uniaxial stress cycling with a mean stress of 60MPa and a stress amplitude of 636MPa
of 60MPa and a stress amplitude of 636MPa. The parameter values in Eqs. (95)1,2 are
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given below:
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E = 204GPa, ν = 0.3, m = 1.7, rm = 720MPa, β0 = 56, ϑ0 = 100MPa.
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(2014).
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Results are shown in Fig. 6 and compare well with the test results in Fig. 9 in Lee et al.
7.3. Asymmetric stress cycling: low-to-medium cycle failure with high tensile strength Low-to-medium cycle failure data by Koster et al. (2016) for the stainless steel X3CrNiMo
13-4 (AISI CA 6-NM) will be simulated, in which high tensile strength and non-symmetric stress cycles are involved (the ratio of the minimum to the maximum stress value is R = 0.1). The plastic factor ρ and the stress limit r∗ are still given by Eqs. (95)1,2 .
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Fig. 7. Comparison with the monotone strain data (dots) from Koster et al. (2016): tensile stress-strain curve
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The parameter values in fitting these data are as follows:
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E = 138GPa, ν = 0.3, m = 9.5, rm = 1250MPa,
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β0 = 3, ϑ0 = 195MPa, γ0 = 15.
Results are shown in Fig. 7 for monotone uniaxial extension and in Fig. 8 for the S − N
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curve of the cycle number to failure versus the stress amplitude. 7.4. Symmetric stress cycling: fatigue limit effect with low tensile strength As indicated at the end of §7.1, generally the S − N curve exhibits monotone property
with decreasing stress amplitude. In some cases, however, the cycle number to failure, N , may very rapidly grow as the stress amplitude S tends toward a certain value, known as fatigue limit or fatigue strength. In this subsection, data for NZ30K1 -T4 alloy under symmetric stress cycling, given in Li et al. (2015), are simulated, in which the foregoing 46
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Fig. 8. Comparison with the data (dots) under non-symmetric stress cycling (Koster et al. 2016): stress amplitude versus cycle number to failure
complexity with a fatigue limit is incorporated. As an example, use is made of the tensile
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stress-strain curve (cf., Fig. 8 (b) therein) and the S − N curve (cf., Fig. 16(a) therein) for symmetric stress cycles.
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The stress limit is still given by Eq. (95)2 , while the plastic factor ρ is now given by an
2 rm 1 1 + tanh γ0 x − 1 xem(x−1) ρ= 2 κ20
(96)
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extended form of Eq. (21) below:
with x = 1.5J2 /r∗2 . For a fairly large γ0 , the plastic factor ρ of the above form becomes very small as the stress amplitude does not exceed κ0 . The values of the parameters in the above form and Eq. (95)2 are found as follows:
E = 39GPa, ν = 0.3, m = 9, rm = 112MPa, β0 = 3.9 ,
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Fig. 9. Simulation of the monotone strain data (dots) from Li et al. (2015): tensile stress-strain curve
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ϑ0 = 90MPa, γ0 = 15, κ0 = 67MPa . Results in fitting the foregoing data are shown in Figs. 9 and 10. It may be seen in Fig. 10
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that the cycle number to failure becomes indefinitely large when the stress amplitude tends toward the fatigue limit k0 = 67MPa, which agrees with the remark at the outset of this
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subsection. At this point, however, it should be noted that, for the stress amplitude below a certain value, the slope of the S − N curve may become very small but never vanishing,
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namely, the S − N curve never becomes actually flat. That is the case for the proposed model with the asymptotic properties indicated in §4.1. Indeed, from Eq. (33) it follows that, for every cyclic deformation process, the effective plastic work ϑ will monotonically grow to become indefinitely large as the cyclic process is invariably progressing and, as a consequence, the cycle number to failure exists for any given value of the stress amplitude.
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Fig. 10. Simulation of the data (dots) under symmetric stress cycling (Li et al. 2015): stress amplitude versus cycle number to failure
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7.5. Symmetric stress cycling: duplex effect from low to high cycle failure In this subsection, model prediction will further be validated by simulating fatigue
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failure data that display another kind of complexity, referred to as the duplex effect. In the S − N curve, there emerges a plateau locating between two parts for low-to-medium
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cycle failure and for medium-to-high cycle failure. Details for this effect may be found in Mohd et al. (2016) and the relevant references therein.
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It appears that a unified simulation of the fatigue failure behavior with the above complex duplex effect poses a challenging issue. In fact, it is found (cf., e.g. Mohd et
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al. 2016) that the surface-induced crack nucleation mode and the subsurface-inclusioninduced crack nucleation mode are responsible for the foregoing two parts in the S − N curve, separately. This implies that the just mentioned two micro-mechanisms of distinct nature need be treated toward modeling the duplex effect in a unified sense. It is demonstrated here that fatigue failure with the duplex effect may be simulated based on the new model. Toward this objective, test data in Mohd et al. (2016) are treated, which supply data from low to high cycle failure for JIS SUS630 under symmetric 49
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Fig. 11. Simulation of the data (dots) from Mohd et al. (2016) with duplex effect from low to high cycle failure: stress amplitude versus cycle number to failure
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tension-compression stress cycling. Now the stress limit is still given by Eq. (95)2 and the
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plastic factor ρ by Eq. (95)1 with the plastic index m therein given below:
m = m1 + (m2 − m1 ) cosh ξ
−1 1.5J2 −1 , d20
(97)
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where the parameter d0 represents the stress level at the center of the plateau part. The
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plastic index m is very close to m1 outside the plateau part, whhereas it changes from m1 to m2 and then goes back to m1 along the plateau part.
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With the parameter values below:
E = 191GPa, ν = 0.3, rm = 1056MPa, ϑ0 = 300MPa, β0 = 5,
m1 = 12.7, m2 = 7, ξ = 50, d0 = 750MPa, the model prediction is calculated and compared with the data in the foregoing. Results are depicted in Fig. 11 and in good agreement with the test data. 50
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7.6. Fatigue lives under thermal cycles Suitable data for fatigue failure under thermal cycles are not available. Here, the model prediction for a cylindrical sample under thermal cycles is obtained by applying the results
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in §5.
For the sake of simplicity, the rate-independence case is treated and the plastic factor ρ is given by Eq. (21). The temperature-dependent stress limit r∗ and Prager modulus c are taken to be of the following forms:
(98)
1 ϑ T c = c0 1 − tanhβ −1 1 − tanh b −1 , 2 ϑ0 T0
(99)
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ϑ T 1 r∗ = rm 1 − tanhβ −1 1 − tanh b −1 , 4 ϑ0 T0
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and, besides, the hysteresis modulus ξ is taken to be constant, i.e., ξ = ξ0 . The thermal cycle considered is as follows: the temperature grows from the initial value
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T1 = 600K to a given higher value T2 and then goes back to T1 . The parameter values below are taken into account:
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G = 56154MPa, m = 5, rm = 100MPa, ϑ0 = 0.02MPa, β = 600,
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b = 2, T0 = 600K, c0 = 112.3MPa, ξ0 = 22/MPa,
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with the following initial values for the effective plastic work and the axial back stress: ϑ1 = 0, α1 = 3.37MPa.
Results are obtained by numerically integrating Eq. (90) and Eq. (94). The curve of the temperature amplitude T2 versus the cycle number to failure is depicted in Fig. 12 for pure thermal cycling and in Fig. 13 for thermal cycling with a fixed axial stress of 5MPa. As shown in Figs. 11 and 12, the cycle number to failure decreases as increasing temperature amplitude in the both cases considered and, moreover, the fatigue life in the presence of 51
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Fig. 12. Model prediction under thermal cycling: temperature amplitude versus cycle number to failure (the temperature at the start of each cycle is 600K)
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axial stress is considerably lower than that in the absence of axial stress. This initial result suggest that the fatigue life may considerably decrease under thermomechanical cyclic
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conditions in the presence of both stress and temperature. The model predictions here presented under thermal cycling can merely display certain
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qualitative features of thermal fatigue failure in a limited sense that both the coupling effect between stress and temperature and the viscous flow (creep) effect at elevated temperature
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are neglected. Comparisons with test data are not presented for lack of relevant data for the cases treated. Test data under thermomechanical cycling in literature show complex
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coupling effects and viscous effects. Toward simulating such data, the simple forms Eqs. (98)-(99) for the stress limit r∗ and the Prager’s modulus c should be extended and issues in extensive numerical integrations for large cycle numbers are involved. This aspect is among the topics that need be treated in further developments, as will be indicated in the next section.
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8. Concluding remarks
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Fig. 13. Model prediction under thermal cycling with a fixed axial stress of 5MPa: temperature amplitude versus cycle number to failure (the temperature at the start of each cycle is 600K)
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In the previous sections, new elastoplastic J2 −flow models in a free, smooth sense have been proposed for the purpose of achieving direct simulation of multi-axial thermo-coupled
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fatigue failure for metals and alloys. Both rate-dependent effects and thermo-coupled effects are covered in a broad sense. The new model is mainly based on two basic facts
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concerning deformation behaviors of metals and alloys. Namely, plastic strain may be smoothly induced at every stress level and becomes appreciable whenever a yielding state
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in a usual sense is approached, on the one side, and metals and alloys repeatedly undergoing deformations suffer continued loss of the strength up to eventual failure, on the other side. It is realized that two perhaps essential features of the macroscopic deformation behavior for metals and alloys are implied by the two basic facts just indicated. Namely, in every process of repeated deformation, the effective plastic work (cf., Eq. (10)) is invariably accumulating, and the constitutive quantities characterising the strength property are going down and eventually vanishing with acumulation of the effective plastic work. It has been 53
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demonstrated (Xiao 2014b) that, from the asymptotic properties just indicated, fatigue failure is derivable as a direct consequence of the proposed model and, accordingly, incorporated as an inherent constitutive feature into the proposed model. This would imply
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that, under any loading conditions, fatigue failure should automatically be predicted from the proposed model.
It is from the above fact that critical failure states with a unified criterion have further been derived as model prediction. In particular, results have been obtained for fatigue lives
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under isothermal stress cycles and under thermal cycles, separately. Explicit procedures have been introduced toward determining the constitutive quantities based on suitable uniaxial stress-strain data.
The main features of the direct approach proposed may be summarized as follows:
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(i) both ratcheting effects and fatigue effects are automatically incorporated as inherent constitutive features of the proposed model;
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(ii) the unified criterion for critical failure states is for all cases of multiaxial thermo-coupled fatigue failure, including
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(iii) all cases of fatigue failure under cyclic loadings, such as so-called low and
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high cycle fatigue and
(iv) all cases of fatigue failure under cyclic and non-cyclic loading conditions
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with constant and varying strain rates. Numerical examples for model validation have been presented and compared with test
data displaying different features of monotone and cyclic failure behaviors, including monotone failure behavior with rate effects, ratcheting effect under non-symmetric stress cycling, low and medium cycle failure with high tensile strength, fatigue limit effect with low tensile strength, as well as duplex effect from low to high cycle failure under symmetric stress cycling, etc. It has been demonstrated that good agreement with all such data, in particular, 54
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with those dispalying complex duplex effect, may be achieved even in the simplest case of the proposed model, namely, the back stress effect is totally ignored and only a simple form of the stress limit (cf., Eq. (95)2 ) with the asymptotic property Eq. (37) is presented
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for the softening effect.
In future study it may be of interest to obtain further results in simulating fatigue failure in six respects, including (i) suitable test data for both the primary and the reverse loading curve; (ii) suitable test data for fatigue failure at high strain rates and at elevated
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temperature; (iii) further simulations and model validation with appreciable hardening effects based on the data mentioned in the first respect; (iv) numerical simulations and model validation for fatigue failure at high strain rates, (v) further investigations into creep failure at elevated temperature; and (vi) extensive simulations and model validation under combined changes of stress and temperature and, in particular, under multiaxial loading
Acknowledgments
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conditions. Results in these further respects will be studied in future works.
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The authors would like to thank the two anonymous reviewers for their constructive comments, which have been instrumental in clarifying relevant issues.
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This study was carried out under the joint support of the funds from Natural Science Foundation of China (No.: 11372172; No.: 11542020) and from the science and technology
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development project launched by Weifang city (No.: 2015GX018). References
Ahmed, R., Barrett, P.R., Hassan, T., 2016. Unified viscoplasticity modeling for isothermal low-cycle fatigue and fatigue-creep stress-strain responses of Haynes 230. Int. J. Solids Struct. 88-89, 131-145. Alfredsson, K.S., Stigh, U., 2004. Continuum damage mechanics revised: A principle for mechanical and thermal equivalence. Int. J. Solids Struct. 41, 4025-4045. Argon, A.S., 2007. Strengthening Mechanisms in Crystal Plasticity. Oxford University Press, Oxford.
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Armero, F., Oller, S., 2000. A general framework for continuum damage models: I: infinitesimal plastic damage models in stress space. Int. J. Solids Struct. 37, 7409-7464. Assidi, M., Zahrouni, H., Damil, N., Michel Potier-Ferry, M., 2009. Regularization and perturbation technique to solve plasticity problems. Int. J. Mater. Form. 2, 1-14. Baletto, M.C., Pierres, E., Gravouli, A., 2010. A multi-model X-FEM strategy dedicated to frictional crack growth under cyclic fretting fatigue loadings. Int. J. Solids Struct. 47, 1405-1423. Badreddine, H., Yue, Z.M., Saansuni, K., 2016. Modeling of the induced plastic anisotropy fully coupled with ductile damage under finite strains. Int. J. Solids Struct. DOI: 10.1016/j/ijsolstr.2016.10.028. Barrett, P.R., Ahmed, R., Menon, M., Hassan, T., 2016. Isothermal low-cycle fatigue and fatigue-creep of Haynes 230. Int. J. Solids Struct. 88-89, 146-164. Bonora, N., Gentile, D., Pirondi, A., Newaz, G., 2005. Ductile damage evolution under triaxial state of stress: theory and experiments. Int. J. Plasticity 21, 981-1007. Bonora, N., Ruggiero, A., 2005. Micromechanical modelling of ductile cast iron incorporating damage. Part. Ferritic ductile cast iron. Int. J. Solids Struct. 42, 1401-1424. Bruhns, O.T., Xiao, H., Meyers, A., 2001. A self-consistent Eulerian rate type model for finite deformation elastoplasticity with isotropic damage. Int. J. Solids Struct. 38, 657-683. Bruhns, O.T., Xiao, H., Meyers, A., 2005. A weakened form of Ilyushin’s postulate and the structure of self-consistent Eulerian finite elastoplasticity. Int. J. Plasticity 21, 199-219. Br¨ unig, M., 2003a. An anisotropic ductile damage model based on irreversible thermodynamics. Int. J. Plasticity 19, 1679-1713. Br¨ unig, M., 2003b. Numerical analysis of anisotropic ductile continuum damage. Compt. Meth. Appl. Mech. Engrg. 192, 2749-2776. Br¨ unig, M., 2015. A continuum damage model based on experiments and numerical simulationsA review. In: Holm Altenbach, Tetsuya Matsuda, Dai Okumura (eds.), From Creep Damage Mechanics to Homogenization Methods - A Liber Amicorum to celebrate the brithday of Nobutada Ohno, Advanced Structural Materials Series, pp.19-35, Springer, Berlin. Br¨ unig, M., Ricci, S., 2005. Nonlocal continuum theory of anisotropically damaged metals. Int. J. Plasticity 21, 1346-1382. Br¨ unig, M., Chyra, O., Albrecht, D., Driemeier, L., Alves, M., 2008. A ductile damage criterion at various stress triaxialities. Int. J. Plasticity 24, 1731-1755. Br¨ unig, M., Gerke, S., 2011. Simulation of damage evolution in ductile metals undergoing dynamic loading conditions. Int. J. Plasticity 27, 1598-1617. Br¨ unig, M., Gerke, S., Hagenbrock, V., 2013. Micro-mechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage. Int. J. Plasticity 50, 49-65. Canadija, M., Mosler, J., 2016. A variational formulation for thermodynamically coupled low cycle fatigue at finite strains. Int. J. Solids Struct. 53, 388-398. Cazes, F., Coret, M., Combescure, A., Gravouil, A., 2009. A thermodynamic method for the construction of a cohesive law from a nonlocal damage model. Int. J. Solids Struct. 46, 1476-1490. Cervera, M., Wu, J.Y., 2015. On the conformity of strong, regularized, embedded and smeared discontinuity approaches for the modeling of localized failure in solids. Int. J. Solids Struct. 71, 19-38. Chew, H.B., 2014. Cohesive zone laws for fatigue crack growth: Numerical field projection of the micromechanical damage process in an elasto-plastic medium. Int. J. Solids Struct. 51, 1410-1420.
56
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Chaboche, J.L. 2008. A review of some plasticity and viscoplasticity constitutive theories. Int. J. Plasticity 24, 1642-1693. Ciavarella, M., Demelio, G., 2001. A review of analytical aspects of fretting fatigue, with extension to damage parameters and application to doveail joints. Int. J. Solids Struct. 38, 1791-1811. Cortese, L., Nalli, F., Rossi, M., 2016. A nonlinear model for ductile damage accumulation under multiaxial non-proportional loading conditions. Int. J. Plasticity 85, 77-92. Driemeier, L., Br¨ unig, M., Micheli G., Alves M., 2010. Experiments on stress-triaxiality dependence of material behavior of aluminum alloys. Mechanics of Materials 42, 207-217. Dvorak, G.J., 2000. Composite materials: Inelastic behavior, damage, fatigue and fracture. Int. J. Solids Struct. 37, 155-170. Egner, H., 2012. On the full coupling between thermo-plasticity and thermo-damage in thermodynamic modeling of dissipative materials. Int. J. Solids Struct. 49, 279-288. Facheris, G., Janssens K.G.F., Foletti, S., 2014. Multiaxial fatigue behavior of AISI 316L subjected to strain-controlled and ratcheting paths. Int. J. Fatigue 68, 195-208. Frost, N.E., Marsh, K.J., Pook, L.P., 1999. Metal Fatigue. Dover Publications, Inc., New York. Giroux, P., Dalle, F., Sauzay, M., Malaplate, J., Fournier, B., Gourgues-Lorenzon, A., 2010. Mechanical and microstructural stability of p92 steel under uniaxial tension at high temperature. Mater. Sci. Engng. A 527, 3984-3993. Hassan, T., Corona, E., Kyriakides, S., 1992. Ratcheting in cyclic plasticity, part II: multiaxial behavior. Int. J. Plasticity 8, 117146. Hassan, T., Kyriakides, S., 1992. Ratcheting in cyclic plasticity, part I: uniaxial behavior. Int. J. Plasticity 8, 91116. Hassan, T., Kyriakides, S., 1994a. Ratcheting of cyclically hardening and softening materials, part I: uniaxial behavior. Int. J. Plasticity 10, 149184. Hassan, T., Kyriakides, S., 1994b. Ratcheting of cyclically hardening and softening materials, part II: multiaxial behavior. Int. J. Plasticity 10, 185212. Heitbreder, T., Ottosen, N.S., Mosler, J., 2016. Consistent elastoplastic cohesive zone model at finite deformations−Variational formulation. International Journal of Solids and Structures, DOI: 10.1016/j.ijsolstr.2016.10.027. Hill, R., 1968. Constitutive inequalities for simple materials. J. Mech. Phys. Solids 16, 229-242. Hutchinson, J.W., Evans, A.G., 2000. Mechanics of materials: Top-down approaches to fracture. Acta Materialia 48, 125-135. Jir´ asek, M., Zimmermann, T., 2001. Embedded crack model. I: basic formulation. Int. J. Numer. Methods Eng. 50 (6), 1269-1290. Kang, G.Z., Liu, Y.J., Ding, J., 2008. Multiaxial ratcheting-fatigue interactions of annealed and tempered 42CrMo steels: Experimental observations. Int. J. Fatigue 30, 2104-2018. Khan, A.S., Liu, H.W., 2012a. A new approach for ductile fracture prediction on Al 2024-T351 alloy. Int. J. Plasticity 35, 1-12. Khan, A.S., Liu, H.W., 2012b. Strain rate and temperature dependent fracture criterion for isotropic and anisotropic metals. Int. J. Plasticity 37, 1-15. Krajcinovic, D., 2000. Damage mechanics: accomplishments, trends and needs. Int. J. Solids Struct. 37, 267-277. Khoel, A.R., Moslemi, H., Shavifi, M. 2012. Three-dimensional cohesive fracture modeling of non-planar crack growth using adaptive FE technique. Int. J. Solids Struct. 49, 2334-2348. Koster, M., Lis, A., Lee, W.J., Kenel, C., Leinenbach, C., 2016. Influence of elasticplastic base material properties on the fatigue and cyclic deformation behavior of brazed steel joints.
57
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Int. J. Fatigue 82, 49-59. Lee, C.-H., Do, V.N.V., Chang, K.-H., 2014. Analysis of uniaxial ratcheting behavior and cyclic mean stress relaxation of a duplex stainless steel. Int. J. Plasticity 62, 17-33. Levitin, V., 1996. High Temperature Strain of Metals and Alloys. WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. Li, Z.M., Wang, Q.G., Luo, A.A., Peng, L.M., Zhang, P., 2015. Fatigue behavior and life prediction of cast magnesium alloys. Mater. Sci. Engng. A 647, 113-126. Macha, E., Nieslony, A., 2012. Critical plane fatigue life models of materials and structures under multiaxial stationary random loading: The state-of-the-art in Opole Research Centre CESTI and directions of future activities. Int. J. Fatigue 39, 95-102. Malcher, L., Pires, F.M.A., C´esar de S` a, 2012. An assessment of isotropic constitutive models for ductile fracture under high and low stress triaxiality. Int. J. Plasticity 30-31, 81-115. Minh, B.L., Maitourmam, M.H., Doquet, V. 2012. A cyclic steady-state method for fatigue crack propagation: Evaluation of plasticity-induced crack closure in 3D. Int. J. Solids Struct. 49, 2301-2313. Mohd, S., Bhuiyan, Md. S., Nie, D.F., Otsuka, Y., Mutoh, Y., 2016. Fatigue strength scatter characteristics of JIS SUS630 stainless steel with duplex SN curve. Int. J. Fatigue 82, 371-378. Moura, M., Consalves, J.P.M. 2014. Cohesive zone model for high-cycle fatigue of adhesively bonded joints under model I loading. Int. J. Solids Struct. 51, 1123-1131. Ottosen, N.S., Rstinmaa, M., Mosler, J. 2016. Framework for non-cohesive interface models at finite displacement jumps and finite strains. J. Mech. Phys. Solids 90, 124-141. Pook, L., 2007. Metal Fatigue. Springer, Berlin. Reed, R.C., 2006. The Superalloys: Fundamentals and Applications. Cambridge Uni. Press, Cambridge. Ritchie, R.O., Gilbert, C.J., McNaney, J.N., 2000. Mechanics and mechanisms of fatigue damage and crack growth in advanced materials. Int. J. Solids Struct. 37, 311-329. Rozumek, D., Macha, E., 2009. A survey of failure criteria and parameters in mixed-mode fatigue crack growth. Mater. Sci. 45, 190-210. Shaw, J.A., Kyriakides, S., 1998. Initiation and propagation of localized deformation in elastoplastic strips under uniaxial tension. Int. J. Plasticity 13, 837-871. Shojaei, A., Voyadjis, G.Z., Tan, P.J., 2013. Viscoplastic constitutive theory for brittle to ductile damage in polycrystalline materials under dynamic loading. Int. J. Plasticity 48, 125-151. Shyam, A., Allison, J.E., Szczepanski, Pollock, T.M., Jones, J.W., 2007. Small fatigue crack growth in metallic materials: A model and its application to engineering alloys. Acta Materialia 55, 6606-6616. Steglich, D., Pirondi, A., Bonora, N., Brocks, W. 2005. Micromechanical modelling of cyclic plasticity incorporating damage. Int. J. Solids Struct. 42, 337-361. Stouffer, D.C., 1996. Inelastic Deformation of Metals. John Wiley & Sons, New York. Stoughton, T.B., Yoon, J.W., 2011. A new approach for failure criterion for sheet metals. Int. J. Plasticity 27, 440-459. Suresh, S., 1998. Fatigue of Materials. 2nd edition. Cambridge University Press, Cambridge. Susmel, L., 2008. The theory of critical distances: a review of its applications in fatigue. Eng. Fract. Mech.75, 17061724. Susmel, L., 2014. Four stress analysis strategies to use the Modified Whler Curve Method to perform the fatigue assessment of weldments subjected to constant and variable amplitude multiaxial fatigue loading. Int. J. Fatigue 67, 38-54.
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Taylor, G.I., Quinney, H., 1934. The latent energy remaining in a metal after cold working. Proc. R. Soc. London A143, 307-326. Taylor, G.I., Quinney, H., 1937. The latent heat remaining in a metal after cold working. Proc. R. Soc. London A163, 157-181. Ural, A., Krishnan, V.R., Papoulia, K.D. 2009. A cohesive zone model for fatigue crack growth allowing for crack retardation. Int. J. Solids Struct. 46, 2453-2462. Wang, Z.L., Xiao, H., 2015. A study of metal fatigue failure as inherent features of elastoplastic constitutive equations. In: Holm Altenbach, Tetsuya Matsuda, Dai Okumura (eds.), From Creep Damage Mechanics to Homogenization Methods - A Liber Amicorum to celebrate the brithday of Nobutada Ohno, Advanced Structural Materials Series, pp. 529-540, Springer, Berlin. Wu, J.Y., Cervera, M., 2015. On the equivalence between traction- and stress-based ap- proaches for the modeling of localized failure in solids. J. Mech. Phys. Solids 82, 137-163. Wu, J.Y., Xu, S.L., 2011. An augmented multicrack elastoplastic damage model for tensile cracking. Int. J. Solids Struct. 48, 2511-2528. Xiao, H., 2014a. An explicit, straightforward approach to modeling SMA pseudo-elastic hysteresis. Int. J. Plasticity 53, 228-240. Xiao, H., 2014b. Thermo-coupled elastoplasticity model with asymptotic loss of the material strength. Int. J. Plasticity 63, 211-228. Xiao, H., 2015. A direct, explicit simulation of finite strain multiaxial inelastic behavior of polymeric solids. Int. J. Plasticity 71, 146-169. Xiao, H., Bruhns, O.T., Meyers, A., 1997. Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica 124, 89-105. Xiao, H., Bruhns, O.T., Meyers, A., 2006. Elastoplasticity beyond small deformations. Acta Mechanica 182, 31-111. Xiao, H., Bruhns, O.T., Meyers, A., 2007. Thermodynamic laws and consistent Eulerian formulations of finite elastoplasticity with thermal effects. J. Mech. Phys. Solids 55, 338-365. Xiao, H., Bruhns, O.T., Meyers, A., 2014. Free rate-independent elastoplastic equations. ZAMM-J. Appl. Math. Mech. 94, 461-476. Xiao, H., Wang, X.M., Wang, Z.L., Yin, Z.N., 2016. Explicit, comprehensive modeling of multiaxial finite strain pseudo-elastic SMAs up to failure. Int. J. Solids Struct. 88-89, 215-226. Xu, L.Y., Nie, X., Fan, J.S., Tao, M.X., Ding, R., 2016. Cyclic hardening and softening behavior of the low yield point steel BLY160: Experimental response and constitutive modeling. Int. J. Plasticity 78, 44-63. Zahrouni, H., Potier-Ferry, M., Elasmar, H., Damil, N., 1998. Asymptotic numerical method ´ ements Finis 7, 841-869. for nonlinear constitutive laws. Revue Europ´eenne des El´ Zhang, W.J., 2016. Thermal-mechanical fatigue of single crystal superalloys: Achievements and challenges. Mater. Sci. Eng. A 650, 389-395. Zhu, Y.L., Kang, G.Z., Kan, Q.H., Bruhns, O.T., Liu, Y.J., 2016. Thermo-mechanically coupled cyclic elasto-viscoplastic constitutive model of metals: Theory and application. Int. J. Plasticity 70, 111-152.
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Direct modeling of multi-axial fatigue failure for metals Heng Xiao, Zhao-Ling Wang PII: DOI: Reference:
S0020-7683(17)30320-7 10.1016/j.ijsolstr.2017.07.003 SAS 9649
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
1 February 2017 15 May 2017 3 July 2017
Please cite this article as: Heng Xiao, Zhao-Ling Wang, Direct modeling of multi-axial fatigue failure for metals, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.07.003
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Highlights • New elastoplasticity models are proposed toward direct modeling of metal fatigue
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failure • Conditions for yielding and loading-unloading need not be imposed but automatically incorporated
• Any damage-like variables and ad hoc failure criteria are not involved
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• Critical failure states are derived with a criterion in uni ed form
• Explicit, direct procedures for determining constitutive quantities based on uniaxial
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data
• Explicit, direct procedures for determining fatigue lives under stress cycles and ther-
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mal cycles
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Direct modeling of multi-axial fatigue failure for metals Heng Xiao1 , Zhao-Ling Wang1,2 Shanghai Institute of Applied Mathematics and Mechanics and State Key Laboratory for Advanced Special Steels, Shanghai University, Yanchang Road 149, 200072 Shanghai, China
2
School of Mathematics and Information Sciences, Weifang University, 261061 Weifang, Shandong Province, China
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1
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Abstract
Innovative elastoplastic J2 −flow models with nonlinear combined hardening are proposed for the purpose of simulating multi-axial thermo-coupled fatigue failure for metals and alloys, in a direct sense without involving any usual damage-like variables and any ad hoc failure criteria. As contrasted with usual models, the yield condition and the loading-
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unloading conditions need not be imposed as extrinsic coercive conditions but may be
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automatically incorporated into a free, smooth flow rule in a more realistic sense. Novel results in three respects are available directly from model predictions, namely, (i) complex features of thermo-coupled fatigue failure may be automatically represented by simple
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asymptotic properties of the hardening functions introduced, (ii) critical failure states may
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be derived with a criterion in unified form, and, in particular, (iii) direct procedures may be established for determining fatigue lives under either stress cycles or thermal cycles.
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Simulation results in the simplest cases of the proposed model are in good agreement with experimental data. Keywords: Metals, Finite deformation, Thermo-coupled effects, New elastoplastic equations, Fatigue failure, Direct simulation
Preprint submitted to International Journal of Solids and Structures
July 4, 2017
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Nomenclature D the stretching, given by the symmetric part of the velocity gradient
τ , τ˜ the Kirchhoff stress and its deviatoric part T the absolute temperature W complementary thermo-elastic potential
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G, B, E the shear, bulk and Young’s moduli
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d the magnitude of the stretching
D e , D p the elastic and the plastic part of the stretching α, ζ the back stress and its magnitude
J2 the magnitude of the effective stress (˜ τ − α)
ϑ the effective plastic work
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√
o log
τ
o log
,α
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L the scalar product of τ and (˜ τ − α)
the logarithmic rates of τ and α
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r∗ the stress limit
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ρ the plastic factor
κ0 the fatigue limit
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m the plastic index c, ξ Prager’s modulus and the hysteresis modulus ˘ the regularized plastic modulus h ˆ ∗ the plastic modulus h f˘, fˆ the strain- and stress-rate loading functions
K, K plastic slopes of the primary and reverse loading curves 3
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hm , τm uniaxial strain and stress at the start of softening ϑˆ threshold value of ϑ demarcating hardening and softening stages
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p(τ ), ps (τ ) shape functions for hardening and softening parts of tensile curve φ(τ ) bridging function linking primary and reverse loading curves rm the maximum stress limit β0 , ϑ0 parameters for softening behavior
κ0 , γ0 parameters for fatigue limit 1. Introduction
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c0 , ξ0 hardening parameters
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Whenever repeatedly undergoing cyclic deformations, metals and alloys may suffer gradual reduction in strength up to eventual failure, albeit the strength reduction at each
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cycle alone may be negligible. It is known (Pook 2007) that the just-mentioned metal fatigue behavior is mainly responsible for fracture and failure of metal components and
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parts in engineering structures. This fact becomes even more outstanding for hot sections of aircraft engines and gas turbines under extreme service conditions with elevated tem-
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perature and high-level stress (cf., e.g., Reed 2006). Toward effectively assessing safety and reliability of metal components and parts by predicting the service life under prescribed
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service conditions, rational and realistic modeling of the fatigue failure behavior for metals and alloys is accordingly essential. In the past decades, numerous investigations into materials fracture and failure have
been carried out from various standpoints. Usually, two approaches are used, including the fracture-mechanics-based approach and the damage-mechanics-based approach, in conjunction with various assumed variables and failure criteria. The former treats different types of embedded cracks from a discontinuity standpoint, while the latter considers the 4
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gradual strength degradation from a contiuum viewpoint. Surveys for certain earlier results may be found in, e.g., Hutchinson and Evans (2000) for the fracture respect and in Dvorak (2000), Krajcinovic (2000), Ritchie et al. (2000) and Ciavarella and Demelio
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(2001) for the damage and fatigue respect. Recent surveys are presented in, e.g., Susmel (2008, 2014), Rozumek and Macha (2009) and Macha and Nieslony (2012) for energy-based and critical-plane-based approaches to modeling fatigue failure, and Br¨ unig (2015) in the damage respect, as well as Zhang (2016) for thermomechanical fatigue of single crystal
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superalloys, and, in particular, in Chaboche (2008) for certain significant developments of elastoplasticity models under cyclic loading conditions.
Further results in both approaches have been obtained in most recent studies. Here, only certain representatives are mentioned below. On the one hand, results based on fracture discontinuities are presented in Jir´asek and Zimmermann (2001) for an embedded
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crack model, in Shyam et al. (2007) , Ural et al. (2009), Baletto et al. (2010) and Minh et
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al. (2012) for fatigue crack growth, in Stoughton and Yoon (2011) for failure criteria for sheet metals and Khan and Liu (2012a, b) for latest advances concerning strain rate and
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temperature effects, in Khoel et al. (2012), Chew (2014), Moura and Consalves (2014), Heitbreder et al. (2016) for cohesive zone models and Ottosen et al. (2016) for non-cohesive
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interface models, and in Cervera and Wu (2015) and Wu and Cervera (2015) for modeling of localized failure based on embedded and smeared discontinuities.
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On the other hand, recent results based on continuum damage mechanics since 2000 are presented in Armero and Oller( 2000) , Bruhns et al. (2001) and Alfredsson and Stigh (2004) for a general framework of continuum damage models, in Br¨ unig (2003a, b) , Bonora et al. (2005) and Badreddine et al. (2016) for ductile damage models, in Br¨ unig and Ricci (2005) and Br¨ unig et al. (2008) for a nonlocal model for anisotropically damaged metals and for a ductile damage criterion under multi-axial stress states, in Cazes et al. (2009) for a nonlocal cohesive model, in Driemeier et al. (2010) and Br¨ unig and Gerke (2011) for 5
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test data for combined effects of stress magnitude and triaxiality and for damage evolution simulation under dynamic loading, in Malcher et al. (2012) for a study of ductile fracture under various cases of triaxial stress, in Bonora et al. (2005), Steglich et al. (2005) and
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Br¨ unig et al. (2013) for micro-mechanical modeling, in Shojaei et al. (2013) for a study of brittle to ductile damage based on viscoplastic models, in Wu and Xu (2015) for a multicrack elastoplastic damage model for tensile cracking, and in Cortese et al. (2016) for ductile dame accumulation, etc.
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In the respect of fatigue behavior under cyclic loadings, certain most recent results may be found in Kang et al. (2008), Facheris et al. (2014), Ahmed et al. (2016), Barrett et al. (2016), Mohd et al. (2016), Xu et al. (2016) and many others. In particular, results in the thermomechanical respect may be found in, e.g., Egner (2012), Canadija and Mosler (2016), and Zhu et al. (2016). Moreover, results are reported for ratcheting effects in, e.g.,
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Hassan et al. (1992), Hassan and Kyriakides (1992, 1994a, b), Lee et al.. (2014) and many
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others.
According to usual approaches indicated above, crack development criteria should be
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assumed with sophisticated procedures for treating discontinuities and, moreover, damage variables should be introduced with assumed evolution equations. Namely, various forms
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of criteria for fracture and failure should be assumed on an ad hoc basis. This implies that issues concerning either additional ad hoc criteria or augmented constitutive structures for
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additional variables should be treated in experimental tests and model analyses. Since extreme cases of thermomechanical deformation behavior with strong discontinu-
ities are involved, undue complexities are expected in modeling fatigue failure. That may be particularly the case for thermo-coupled fatigue failure under combined changes of both stress and temperature. Indeed, in various cases close to and even just at fracture and failure, pronounced dissipation with appreciable thermal effects may be expected and, in particular, unlimitedly growing inelastic deformations may be induced in association with 6
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initiation of strong discontinuities. Generally, the four thermomechanical fields including deformation, temperature, heat flux and stress in a material body are inextricably coupled with one another and should be studied on the rigorous ground of the thermodynamic
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principles.
The above facts may suggest that large elastoplastic deformations with thermo-coupled effects should be the very features of fracture and failure behavior for metals and alloys. Since these features exhibit undue complexities as indicated above, results are usually
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presented for cases at small isothermal deformations. It appears that further studies for a broad case with finite deformation effects, in particular, with thermo-coupled effects, are rarely reported, except for some particular cases.
In this article, an alternative approach will be proposed toward simulating multi-axial thermo-coupled fatigue failure for metals and alloys, in a direct sense without involving
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any usual damage-like variables and any ad hoc failure criteria. This new, direct approach
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will be based on innovative elastoplastic J2 −flow models. Toward this objective, it is noted that usual elastoplasticity models with the yield condition and the loading-unloading
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conditions could not serve the purpose. For instance, various cases of fatigue failure under cyclic loading conditions are excluded from such models. In a most recent study (Xiao et
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al. 2014; Xiao 2014b, 2015), innovative thermo-coupled elastoplasticity models have been established from a fresh standpoint. As contrasted with usual elastoplasticity models,
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such new models of much simpler structure are totally free, in the sense that both the yield condition and the loading-unloading conditions need not be imposed as extrinsic coercive conditions but are automatically incorporated as inherent constitutive features into the models. It has been demonstrated that various cases of thermo-coupled fatigue failure under combined cyclic changes of stress and temperature may be derived as direct consequences from the new models. General constitutive frameworks are established in the foregoing references that are 7
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mainly devoted to bypassing certain issues associated with usual elastoplasticity models and, accordingly, only an initial study of the fatigue failure behavior is incorporated therein. This contribution may represent substantial developments both in generalizing the model
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and in deriving fatigue failure features, in a broad sense for multi-axial finite deformations with thermo-coupled effects and for both rate-independent and rate-dependent effects. Toward this goal, a substantial extension of the previous models will first be presented. It is then demonstrated that multi-axial thermo-coupled fatigue failure may be derived
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as inherent constitutive features of the new model. Novel results in three respects may be available directly from model predictions, namely, (i) complicated failure features may be automatically represented by certain simple asymptotic properties of the hardening functions introduced, (ii) critical failure states may be derived with a criterion in unified form, and, in particular, (iii) salient features of multi-axial fatigue failure under stress
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cycles and under thermal cycles may be disclosed with results for the fatigue life. Moreover,
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numerical examples for model predictions will be presented and compared with extensive test data for the purpose of model validation.
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The main content will be organized in seven sections. A substantial extension of the previous model will be presented in § 2 and the thermodynamic consistency will be demon-
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strated in §3. In §4, critical failure states will be disclosed and a unified criterion for such states will be derived from the model established. In §5, stress cycles with temperature
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held fixed will be taken into account and results for the fatigue life will be presented with direct, explicit procedures for determining the hardening moduli incorporated. In §6, thermal cycles will be taken into consideration and the fatigue life will be determined. In §7, numerical examples will be provided for the purpose of model validation. Finally,
discussions and remarks will be made in §8. To conclude this introduction, some notations are explained as follows. Let A, B and T be two symmetric 2nd-order tensors and a 4th-order tensor, respectively. The notations 8
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trA, |A|, A : B and T : A respresent the trace and the norm of A, the double-dot products between A and B and between T and B, namely (repeated indices below mean
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summation), trA = Aii , |A|2 = trA2 , A : B = Aij Bij , T : A = Tijkl Akl ei ⊗ ej .
In the above, (e1 , e2 , e3 ) are three orthonormal vectors and ei ⊗ ej is the tensor (dyadic)
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product of ei and ej .
2. New elastoplastic J2 −flow models with rate and thermal effects The objective of this section is to establish elastoplastic J2 −flow models into which the previous models (cf., Xiao et al. 2014 and Xiao 2014b, 2015) are included as particular
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cases. In what follows the procedures in deriving the main results are omitted and may be found in the references just indicated.
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Consider finite elastoplastic deformations with rate- and temperature-dependent effects. According to the Eulerian rate formulation of finite elastoplasticity (cf., Xiao et al. 2006),
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the total stretching D is a sum of an elastic and a plastic part, D e and D p , as shown
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below:
D = De + Dp .
(1)
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The objective rate equations of Eulerian type for D e and D p will be given below, separately. First, in a general case with thermal effects, the self-consistent Eulerian rate equation
for the elastic stretching D e may be given by (cf., Xiao et al. 2007, Xiao 2014b, 2015): De =
∂ 2 W o log ∂ 2W ˙ : τ + T, ∂τ 2 ∂τ ∂T
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(2)
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where T and τ are the absolute temperature and the Kirchhoff stress and
o log
is a complementary thermo-elastic potential and the τ rate of the Kirchhoff stress τ .
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W = W (τ , T ) .
is the corotational logarithmic
Details may be found in, e.g., Xiao et al. (1997). For
metals and alloys with small elastic strains, the potential W is of the quadratic form below: 1 1 tr˜ τ2 + (trτ )2 + θ(trτ )(T − T∗ ) . 4G 6B
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W =
(3)
In the above, τ˜ is the deviatoric part of τ , θ and T∗ are the thermal expansion coefficient and a reference temperature, and, besides, G and B are the shear modulus and the bulk
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modulus and dependent on the temperature T , namely,
B = B(T ) .
(4)
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G = G(T ),
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With Eq. (3), Eq. (2) for the elastic stretching D e reduces to 1 o log 1 G0 ˙ 1 B0 D = τ˜ + (trτ˙ )I − T τ˜ + θ − trτ T˙ I , 2G 3B 2G2 3 B2
(5)
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e
where G0 and K 0 are the derivatives of G and B with respect to the temperature T .
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Next, the objective rate equation for the plastic stretching D p , namely, the plastic flow
rule, is usually concerned with a few respects including formulations of the yield function, the hardening behavior and the loading-unloading conditions, as well as rate-dependent effects, etc. These will be explained below. According to usual theory of elastoplasticity, the central concept for plastic behavior is the yielding state. The plastic flow, i.e. D p , is induced only in the case when the yielding
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state is attained and maintained. As such, the loading-unloading conditions associated with a yield criterion should be introduced and imposed as extrinsic coercive conditions. Such conditions are characteristic of usual theory of elastoplasticity and formulated in terms
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of a yield function f∗ . The yield surface f∗ = 0 in the stress-temperature space changes itself both in size and in shape, known as hardening behavior. A scalar quantity, ϑ, and a traceless tensor quantity, known as the back stress and denoted α, both evolving with the development of plastic flow, are introduced to characterize the hardening behavior. Then,
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the yield function f∗ of von Mises type with both rate and thermal effects is formulated as follows:
1 1 τ − α|2 − r∗2 , f∗ = |˜ 2 3
(6)
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where the r∗ is referred to as the stress limit and of the form
(7)
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r∗ = r∗ (ϑ, d, ζ, T )
in a broad sense. In the above, the d and ζ are the stretching magnitude (cf., Xiao 2015)
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and
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and the back-stress magnitude, given by
d=
r
ζ=
r
2 |D| , 3
2 |α| . 3
(8)
(9)
Since the latter enters the stress limit r∗ in Eq. (7), the coupling effect is introduced (cf.,
Xiao 2014a, Xiao et al. 2016) between the size change and the translation of the yield surface.
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The equations of evolution for ϑ and α are as follows (Xiao 2014a, b, 2015): ϑ˙ = (˜ τ − α) : D p ,
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o α log = cD p − ξαϑ˙ .
(10)
(11)
The ϑ as specified by Eq. (10) is referred to as the effective plastic work and invariably accumulates with development of plastic flow, as will be shown in §4.1. In Eq. (11), the
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c and ξ, known as the Prager modulus and the hysteresis modulus, need not be constant but rely on ϑ and T and others, namely (cf., Xiao 2014a,b),
(12)
ξ = ξ(ϑ, T, τ , α) .
(13)
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c = c(ϑ, T, τ , α) ,
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As indicated in the previous study (Xiao 2014a,b, 2015), such moduli in a broad sense may be used to characterise complicated behavior displaying strong anisotropic hardening for
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metals and alloys, in particular, for shape memory alloys. For the rate-independent case, a normality flow rule for D p was first formulated in Hill
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(1968) based on Ilyushin postulate and a further development has been made in a study in Bruhns et al. (2005). In a series of studies (Xiao et al. 2014, Xiao 2014b), such a
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normality rule has been extended to give rise to smooth flow rules1 free of usual yield condition and loading-unloading conditions and most recently (Xiao 2015) this smooth flow rule has further been extended to general cases with both rate-dependent effects and thermal effects. Here, a normality flow rule of such nature is further proposed in a broad 1
Smooth elastic-to-plastic transitions may be realized by regularization procedures from a standpoint based on the asymptotic numerical method. This idea has been suggested by Zahrouni and coworkers (Zahrouni et al. 1998, Assidi et al. 2009).
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sense as follows: 1 D p = ρ J2−1 2
! f˘∗ f˘∗ τ − α) . + (˜ ˘∗ h ˘∗ h
(14)
property below:
2 ρ = ρ (x, ϑ) , x = 3J , 2r∗2 ρ(0, ϑ) = 0, ρ(1, ϑ) = 1 , ∂ρ > 0 . ∂x
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˘ are given by (cf., Xiao 2015) and, moreover, the f˘ and h
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In the above, the dimensionless factor ρ is referred to as the plastic factor and of the
(15)
(16)
˘ ∗ = 2G + h ˆ∗ , h
(17)
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∂f∗ ˙ ∂f∗ ˙ G0 T+ d + (J2 + L) T˙ , f˘∗ = 2G(˜ τ − α) : D + ∂T ∂d G
with
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ˆ ∗ = 2 r∗ ∂r∗ + c − ξL − 4 r∗ ∂r∗ 1.5ξζ − cζ −1 J −1 L , h 2 3 ∂ϑ 9 ∂ζ
(19)
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J2 = |˜ τ − α|2 , L = (˜ τ − α) : α .
(18)
ˆ and h ˘ are referred to as the plastic modulus and the regularized plastic modulus. The h
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For metals and alloys, the former is far smaller than the shear modulus 2G and, therefore,
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the latter is always positive, namely, ˘∗ > 0 . h
(20)
Physically, the above property arises from the following fact: the regularized plastic
˘ relates the plastic strain rate to the total strain rate instead of the stress rate. modulus h
ˆ changes its sign from hardening Irrespective of the fact that the usual plastic modulus h ˘ keeps positive. In the uniaxial stress-strain case, the positivity property of to softening, h ˘ may become evident from the foregoing fact. h 13
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According to usual models, the plastic factor ρ always vanishes for all stress levels below the stress limit r∗ , i.e., f∗ < 0, whereas it suddenly jumps to 1 whenever the stress level reaches the stress limit, i.e., f∗ = 0. As contrasted with this discontinuity, Eq.(15) means
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the following facts: (i) when f˘∗ > 0, the plastic flow may be induced at any non-vanishing stress level but becomes appreciable only when the stress limit is attained, (ii) the plastic flow grows with increasing stress level and agrees with the classical case as the stress level reaches the stress limit, and (iii) ratcheting effects under cyclic loading conditions are
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automatically incorporated in the new model.
In the above, a new elastoplastic J2 −flow model for finite elastoplastic deformations with both rate-dependent effects and thermal effects is established by the elastic rate equation (5) and the plastic rate equation (14) with Eqs. (15)-(20), as well as the evolution equations (10)-(11). As contrasted with usual elastoplastic models subjected to the coercive
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restrictions assumed for plastic flow and stipulated by the thermodynamic principles, the
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proposed model is free in the three respects below: (i) it is of much simpler structure than usual models and free in the sense that
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neither the yield condition nor the loading-unloading conditions for plastic flow need be imposed;
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(ii) it is also free in the sense of identically fulfilling the thermodynamic restriction stipulated by the second law (the Clausius-Duhm inequality) for any given
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forms of the constitutive quantities introduced, as will be demonstrated in the next section, and, furthermore, (iii) it is free of any failure criteria for the purpose of simulating the fatigue failure behavior, namely, it incorporates the fatigue failure behavior as an inherent constitutive feature, as will be shown in §4 (see also Xiao 2014b).
The above properties may be demonstrated by developing the relevant procedures in 14
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the previous study (Xiao et al. 2014, Xiao 2014b, 2015), as will be done in the new two sections. Here, consequences concering fatigue effects are further highlighted. According to a usual flow rule, no plastic strain is induced with the plastic factor ρ invariably vanishing
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prior to yielding and, hence, any fatigue effects under cyclic loading processes would be excluded and, in particular, that may be evident for cyclic loading processes with small stress amplitudes within the classical yield limit. As contrasted with the usual flow rule, the new flow rule Eq. (14) requires that a plastic factor ρ of the properties Eq. (15) should
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smoothly take values from 0 to 1 as the stress grows from 0 to the yield limit. Then, plastic strain is induced with small values of the plastic factor ρ prior to the yield limit and agrees with its classical counterpart with ρ = 1 just at yielding, as will be exemplified shortly by a simple form of ρ. As such, the new flow rule Eq. (14) with a plastic factor ρ of the properties Eq. (15) ensures that the effective plastic work ϑ is invariably growing in every
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cyclic process and, therefore, both high and low cycle fatigue effects may be in a natural,
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unified manner characterized with the strength quantities of asymptotic properties (cf., Eqs. (35)-(41) given later on). With this perhaps favourable property, the new flow rule
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Eq. (14) with no extrinsic coercive conditions related to the yield limit may be not only simpler but even more realistic than the usual flow rule.
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The model proposed here includes the previous models as particular cases. In fact, the free rate-independent and rate-dependent model in the previous work (cf., Xiao et al.
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2014, Xiao 2014b, 2015) may be respectively incorporated as the particular case when the stretching magnitude d is dropped out in Eq.(7) and the plastic factor ρ is given by
3J2 −m e ρ= 2r∗2
1−
3J2 2 2r∗
.
(21)
In the above, the m is a positive dimensionless parameter and referred to as the plastic index. With the plastic factor of the above simple form, the plastic index m plays an
15
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essential role in characterizing the fatigue behavior, in the sense that it is mainly responsible for accumulation of the effective plastic work ϑ in every thermomechanical process. For fairly large m > 0, the plastic factor ρ is very small whenever the stress point stays far
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away from the usual yield surface 12 J2 − 13 r∗2 = 0, whereas it becomes close to 1 only when the stress point stays close to this surface.
At elevated temperature, creep deformation with appreciable viscous effects may be expected. Such viscous flow behavior may be incorporated in the proposed model. In
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fact, consider the simplest case without the back stress effect, i.e., α ≡ O. At a constant ˜ D e =O elevated temperature with T˙ = 0, the plastic flow dominates, namely, D p = ˙ D, ˙ and, moreover, f∗ = 21 J2 − 13 r∗2 = 0. Then, from Eqs. (14)-(18) with c = 0, ξ = 0, ρ = 1, d˙ = 0, we obtain
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2 ˜ ˜ = r∗2 D, ϑ˙ σ 3
ϑ˙ = r∗ d.
ED
and, therefore,
˜ is the deviatoric Cauchy stress. In deriving the above, the volumetric deformation Here, σ
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has been neglected and the isotropic hardening modulus r∗ (∂r∗ /∂ϑ) has also been neglected
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as compared with the shear modulus G, since the latter is far greater than the former. It
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may be clear that the usual equation for viscous flow follows, namely, ˜ , ˜ = ηD σ
whenever the rate-dependent stress limit r∗ is reduced to r∗ = 1.5ηd at elevated temperature, where η is the viscosity parameter. The thermomechanical deformation behavior of metals and alloys may be represented by the constitutive quantities introduced in the proposed model, including two elastic
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constants, i.e., G and B, as well as four quantities for plastic behavior, i.e., the plastic factor ρ, the stress limit r∗ , the Prager modulus c and the hysteresis modulus ξ. The latter four jointly characterise the strength property of metals and referred to as the strength
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quantities. Of them, the plastic factor ρ ensures acumulation of the effective plastic work at any stress level and, accordingly, it plays an essential role in characterizing both the ratcheting effect and the fatigue failure under cyclic loading conditions2 .
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3. Thermodynamic consistency in explicit, identical sense
Let q, ψ, η be the heat flux per unit current area, the Holmholtz free energy, the specific entropy per unit reference volume, respectively. Then, the energy balance (i.e., the first law) and the Clausius-Duhem inequality (the second law) may be written in the forms:
(22)
D−
(23)
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τ : D − (ψ˙ + η T˙ ) = D , J q · ∇T ≥ 0 , T
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PT
with D the intrinsic dissipation below:
D = T η˙ − (χ − J∇ · q) .
(24)
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In the above, χ is the heat supply per unit reference volume. An enhanced form of the Clausius-Duhem inequality Eq. (23) requires that the intrinsic
dissipation should always be non-negative, namely, the Planck inequality below should be 2
However, it may be noted that, according to a usual elastoplasticity model, the plastic factor ρ vanishes and plastic strain can not be induced at any stress level below the stress limit, thus excluding the occurrence of the ratcheting effect and the fatigue failure in most cases, in particular, the so-called high cycle fatigue under cyclic stresses below the initial yield stress.
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satisfied: D≥0
(25)
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for every possible thermodynamic process. We are going to demonstrate that the proposed model is thermodynamically consistent in an explicit, identical sense, namely, the free energy function ψ and the specific entropy function η may be presented in explicit forms, so that the second law with non-negative intrinsic dissipation is fulfilled for any given forms of the constitutive functions incorpo-
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rated. Toward this goal, a free energy function ψ and a specific entropy function η in explicit forms may be constructed by developing the procedure in Xiao et al. (2007), so that the Planck inequality Eq. (25) is identically fulfilled. These two explicit functions are presented as follows:
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∂W : τ − W + κ − ϕ(ϑ, T ) , ∂τ
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ψ = ψ0 (T ) +
η = −ψ00 (T ) +
∂ϕ ∂W + , ∂T ∂T
(26)
(27)
PT
where the ψ0 (T ) is the specific heat capacity of the material at issue, the κ is the usual plastic work given by
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κ˙ = τ : D p
(28)
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and the ϕ = ϕ(ϑ, T ) is a monotonically increasing function of the effective plastic work ϑ, namely,
ϕ(0, T ) = 0,
∂ϕ > 0. ∂ϑ
(29)
Then, with Eqs. (22) and (26)-(28), the intrinsic dissipation is given by 1 D= ρ 2
! f˘ f˘ ∂ϕ + ≥ 0. ˘ h ˘ ∂ϑ h 18
(30)
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Thus, the intrinsic dissipation is always non-negative and the Planck inequality is identically met. Generally, the temperature T , the heat flux q, the stress τ and the deformation gradient
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F are coupled with one another and governed by the elastoplastic constitutive equations presented in the last section as well as the reduced form of the first and second laws, namely, Eq. (22) with Eq. (30). On account of the fact that there inevitably emerges the positive intrinsic dissipation, thermal effects may always be coupled with elastoplastic deformations.
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In particular, that may be the case even with the environmental temperature held fixed. A well-known example is the heating effect due to plastic dissipation in metal bars under tension, as observed by Taylor and Quinney (1934, 1937). Such coupling effects may be treated in a thermodynamically consistent sense by working out the temperature and the heat flux coupled with the deformation based on the complete system of the governing
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indicated in the foregoing.
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equations from both the elastoplastic equations and the thermodynamic equations, as
PT
4. Critical failure states with a unified criterion In this section, we are going to treat fatigue failure effects in a broad sense. It will
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be shown that critical states for thermo-coupled fatigue failure under combined changes of stress and temperature may be derived directly from the proposed model and, then, a
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criterion for determining these states may be obtained as model prediction. In association with various cases of inhomogeneity at a small scale, it is observed in
fatigue testing that fatigue failure of different nature may be induced by various localized effects with microcracks initiating and propagating either at the surface or in the interior of a material sample. It is instrumental to have a general knowledge of the characteristic deformation behavior of the materials under isothermal loading and, generally, under thermomechanical loading up to fatigue failure. The essential nature in this respect 19
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is concerned with the involvement of multiple deformation micro-mechanisms at different temperatures and the dependence of these mechanisms on the histories of both the temperature and the applied load. Details in micromechanisms and macroscopic behavior may be
and Pook (2007). 4.1. Asymptotic properties of the strength quantities
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found in, e.g., Stouffer (1996), Levitin (1996), Suresh (1998), Frost (1999), Argon (2007),
As indicated from the outset, a basic fact concerning deformation behavior of materials
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is as follows: When repeatedly undergoing deformations, metals and alloys suffer continued loss of the strength up to eventual failure. Since the material strength is characterised jointly by the strength quantities indicated at the end of §2 and each such quantity changes itself with accumulation of the effective plastic work ϑ, with the foregoing basic fact we
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come to a perhaps essential understanding. Namely, the strength quantities characterising the strength property in the proposed model should go down and eventually become
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vanishing with accumulation of the effective plastic work. With the strength quantities of such asymptotic properties, the proposed model automatically incorporates fatigue failure
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as an inherent constitutive feature. It has been demonstrated in an initial study (Xiao 2014b) that fatigue failure may be in principle derived as direct consequences of the new
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elastoplasticity models proposed. In this study, however, the central issue below is left unanswered, namely, under what circumstances will fatigue failure be eventually attained
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and, in particular, how will the fatigue life be determined under cyclic changes of stress and/or temperature? Here, the central issue above will further be studied. For this purpose, we first need to
specify certain relevant properties of the strength quantities introduced. We first point out certain general properties. For normal stress levels below the stress
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limit introduced in the model, we deduce (cf., Eqs. (6) and (16))
⇐⇒
1.5J2 /r∗2 < 1
⇐⇒
1 − ρ(1.5J2 /r∗2 ) > 0 .
With Eq. (20), we have f˘∗ >0 ˘∗ h
⇐⇒
f˘∗ > 0 .
Moreover, from Eqs. (10), (14) and (19)1 we deduce ! f˘∗ f˘∗ + ≥ 0. ˘∗ h ˘∗ h
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1 ϑ˙ = ρ 2
(31)
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f∗ < 0
(32)
(33)
Then, it follows that the rate ϑ˙ is always non-negative and it is positive for f˘∗ > 0 and, accordingly, the effective plastic work ϑ will invariably grow and accumulate with devel-
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opment of any plastic flow.
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We next take two stages of deformation behavior into consideration. On the one side, a hardening stage emerges for the effective plastic work below a threshold value, denoted
ˆ ∗ > 0 for ϑ < ϑˆ . h
(34)
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ˆ For this stage, the plastic modulus h ˆ ∗ (cf., Eq. (18)) is positive, i.e., ϑ.
ˆ For this stage, the stress limit r∗ On the other side, a softening stage emerges for ϑ > ϑ.
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is of the following properties: it goes to vanish as ϑ → +∞, and the isotropic hardening modulus r∗ (∂r∗ /∂ϑ) becomes negative, as shown below: ∂r∗ < 0 for ϑ > ϑˆ , ∂ϑ r∗
∂r∗ < 0 for ϑ > ϑˆ , ∂ϑ
21
(35)
(36)
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lim r∗ = 0 .
(37)
ϑ→+∞
Besides, as ϑ grows to infinity, the Prager modulus c = cs > 0 goes also down to vanish and
∂c ≤ 0 for ϑ > ϑˆ , ∂ϑ lim cs = 0 ,
and
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ϑ→+∞
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the hysteresis modulus ξ = ξs > 0 goes up to exceed a positive constant ξ0 > 0, namely,
(38)
(39)
∂ξ ≥ 0 for ϑ > ϑˆ , ∂ϑ
(40)
lim ξs ≥ ξ0 > 0 .
(41)
ϑ→+∞
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The above asymptotic properties are implied just by the basic fact for realistic material behavior, as explained at the outset of this subsection, and have been introduced in the
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ˆ ∗ becomes previous work (Xiao 2014b). Finally, at the softening stage, the plastic modulus h
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negative, i.e.,
ˆ ∗ < 0 for ϑ > ϑˆ . h
(42)
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The strength quantities of the above properties with two stages may be combined in unified, explicit form. In fact, let the strength quantities be given by rh , ch , ξh at the
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hardening stage and by rs , cs , ξs at the softening stage. Then, the strength quantities for both the hardening and the softening stage may be given as follows:
ˆ ˆ r∗ = ϕ ϑ − ϑ rh + ϕ ϑ − ϑ rs , c = ϕ ϑˆ − ϑ ch + ϕ ϑ − ϑˆ cs ,
ˆ ˆ ξ = ϕ ϑ − ϑ ξh + ϕ ϑ − ϑ ξs , 22
(43)
(44) (45)
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where
1 x . ϕ(x) = 1 + tanh a 2 ϑˆ
(46)
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In the above, the positive dimensionless parameter a > 0 may be temperature-dependent. It is fairly large, so that, in a small neighborhood centered at x = 0, ϕ(x) changes rapidly from -1 to +1.
In the isothermal case, the strength quantities at the hardening and the softening stage are determinable from suitable uniaxial data, as will be given in the next section. Moreover,
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simple forms of the plastic factor ρ will also be given in the subsequent development. 4.2. Critical failure states
The asymptotic properties given in Eqs. (35)-(41) imply that asymptotic loss of the strength, viz., fatigue failure will eventually emerge as the effective plastic work ϑ becomes
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indefinitely large, as has been indicated in the previous study (Xiao 2014b). However, it
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does not appear to be realistic to treat fatigue failure based directly on such properties. Then arises the central issue indicated in the last subsection. It is the finding of critical
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failure states below that leads to a solution of this central issue. Consider a process of thermomechanical deformation in which both stress and temper-
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ature are changing. At the end of this process, there will emerge such a state, referred to as a critical failure state, that , at this state, the effective plastic work ϑ will grow at an
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infinite rate and, immediately following this state, there is only one possibility left for the deformation behavior, namely, indefinitely large deformation will be induced concomitantly as the stress magnitude goes down to vanish. It is clear that a critical failure state heralds the eventual fatigue failure. In the next subsection, it will be demonstrated that critical failure states exist and may be derived as model prediction.
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4.3. A criterion in unified form In what follows we are in a position to find out all critical failure states and derive a unified criterion for such states.
From Eqs. (1), (5) and (10) we deduce o
2Gϑ˙ = 2G(˜ τ − α) : D − (˜ τ − α) : τ log +
G0 (J2 + L) T˙ , G
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and from Eq. (16) we infer
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˘ ∗ (cf., Eqs. (16)-(17)). Toward this goal, we first derive an alternative expression for f˘∗ /h
∂f∗ ˙ ∂f∗ ¨ G0 2G(˜ τ − α) : D = f˘∗ − T− ϑ − (J2 + L) T˙ . ˙ ∂T G ∂ϑ
(47)
(48)
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On the other hand, for plastic flow with f˘∗ > 0, Eqs. (10), (14) and (19) together yield f˘∗ ϑ˙ = ρ . h˘∗
(49)
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where
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Hence, substituting Eqs. (48)-(49) into Eq. (47) we arrive at
2Gρ
f˘∗ = f˘∗ − fˆ∗ , ˘∗ h
(50)
∂f∗ o log ∂f∗ ˙ ∂f∗ ˙ T+ fˆ∗ = :τ + d. ∂τ ∂T ∂d
(51)
f˘∗ fˆ∗ = , ˘∗ ˆ∗ h 2G(1 − ρ) + h
(52)
We then obtain
ˆ ∗ is just the plastic modulus given by Eq. (18). Thus, Eq. (49) may be where the h
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reformulated in the form: ϑ˙ =
ρfˆ∗
ˆ∗ 2G(1 − ρ) + h
.
(53)
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Now it becomes clear that the effective plastic work ϑ may grow at infinite rate, whenever the denominator in Eq. (53) becomes vanishing, namely, ˆ∗ = 0 . 2G(1 − ρ) + h
(54)
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If the above equation is satisfied, then a critical failure state as explained in the last subsection is found. Below we are going to demonstrate that critical failure states are implied by the strength quantities of the asymptotic properties indicated in §4.1. From Eqs. (31) and (34) it follows that the left-hand side of Eq. (54) is always positive at the hardening stage and, therefore, critical failure states could not emerge at
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ˆ the first term, i.e., 2G(1 − ρ), is the hardening stage. At the softening stage with ϑ > ϑ,
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positive (cf., Eq. (31)), while the second term, the plastic modulus h∗ is negative. Since the factor (1 − ρ) may take any given value in the interval (0,1), it follows that solutions
PT
of Eq. (54) always exist and critical failure states are accordingly determined by Eq. (54). Furthermore, we demonstrate that the magnitude of the stress goes down to vanish with
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further plastic flow following a critical failure state. We first prove that will be the case for the magnitude of the effective stress, i.e., J2 . To this end, we evaluate the time rate of
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the function f∗ (cf., Eq. (6)). In a process of thermo-coup-led elastoplastic deformation, the changing rate of the function f is given by (cf., §4 in Xiao et al. 2007): ∂f o log ∂f ˙ ∂f ˙ ∂f o log T+ f˙ = :τ + ϑ+ :α . ∂τ ∂T ∂ϑ ∂α
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On the other hand, utilizing the equation (cf., Eqs. (1)-(2)) o
and the flow rule Eq. (14), we infer o
τ log
∂ 2W ˙ 1 ρ =S:D−S: T− ∂τ ∂T 2 J2
! f˘ f˘ + S : (˜ τ − α) . ˘ h ˘ h
(55)
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In the above, the S is the elastic rigidity tensor:
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τ log = S : (D − D p )
1 S = 2GI + (B − 2G)I ⊗ I , 3
where I and I are the 2nd- and 4th-order identity tensors, respectively. Substituting this
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eqs. (16) and (47)-(49), we deduce
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expression and Eqs. (10)-(11) into the former expression for the time rate f˙ and then using
(56)
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f˙∗ = (1 − ρ)f˘∗ .
From this and Eq. (31) we infer that f˙∗ > 0 for any plastic flow with f˘ > 0 and, therefore,
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f∗ starts with a negative value (cf., Eq. (31)) and goes to vanish as ϑ goes to infinity. It follows from this and Eq. (37) that J2 goes to vanish. On the other hand, after a critical
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failure state is attained, cs becomes vanishing (cf., Eq. (39)) and ξs is no less than ξ0 > 0 (cf., Eq. (41)). From these and Eq. (11) we derive the following equation: dζ 2 ≤ −ξ0 ζ 2 , dϑ where 1.5ζ 2 = |α|2 . Hence, we infer that the back stress goes to vanish as ϑ goes to infinity.
Finally, since both J2 (cf., Eq. (20)) and ζ goes to vanish, we conclude that the stress τ 26
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should go to vanish. The criterion given by Eq. (54) for critical failure states is applicable for a broad case of multi-axial thermo-coupled fatigue failure with rate-dependent effects. It is for all kinds
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of thermomechanical processes. Usually, two particular kinds of cyclic processes are taken into consideration, namely, cyclic processes of stress and cyclic processes of temperature. Further results for these two cases will be presented in the next two sections.
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5. Fatigue failure under multi-axial stress cycles
In this section, we are going to treat the isothermal case with T˙ = 0. It will be shown that the hardening quantities r, c and ξ may be determined based on suitable uniaxial data. Fatigue lives will be determined for cyclic processes of multi-axial stress.
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The main idea in deriving the results below are explained as follows. It will be demonstrated that, given uniaxial stress-strain data in primary monotone tension test and in
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reverse monotone compression test, the foregoing three quantities may actually be determined by explicit procedures, in the sense that the predictions of the proposed model for
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the two cases of uniaxial tension and compression in the foregoing can exactly reproduce these data. The main idea of achieving this is to express the two uniaxial curves in tension
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and compression (cf., Fig. 1) by two suitable functions as will be given by Eqs. (57)-(59) and then obtain an explicit expression of each hardening quantity at issue in terms of
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these two functions. Results are first derived for uniaxial tension and compression and, then, these uniaxial results are extended to multiaxial cases by means of direct procedures. These explicit procedures represent a direct extension of those introduced in most recent studies (Xiao 2014a; Xiao et al. 2016). Details will be omitted and may be found in these references.
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5.1. Determination of the strength quantities at the hardening stage As shown in Eqs. (7) and (12)-(13), generally the hardening quantities r∗ , c and ξ may be complicated functions of the three variables ϑ, τ and α for the isothermal case. Usually,
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it may be rather difficult to find out the forms of these functions and ad hoc forms have to be assumed for various purposes. In what follows, explicit, direct procedures will be proposed to obtain the hardening quantities (rh , ch , ξh ) and (rs , cs , ξs ) in Eqs. (43)-(45) based on suitable uniaxial data at the hardening and the softening stage, separately.
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Toward the above purpose, we consider suitable uniaxial stress-strain data at a very low strain rate, including those for the primary loading curve and the reverse loading curve, as depicted in Fig. 1. The former is generated by monotonically growing tensile strain up to failure, while the latter by unloading the axial stress from a certain point on the primary
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loading curve and then reversely loading the compressive stress. It should be noted that the primary loading curve starts at an initial yield point in classical sense, while the reverse
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loading curve at a reverse yield point. The two stages indicated in §4.1 now become clear: the part of the curve prior to the maximum stress τm corresponds to the hardening stage,
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while the other part after τm to the softening stage. Details are schematically shown in
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Fig. 1.
It should be indicated that here a very low strain rate means the rate-independent case
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with d = 0. In addition, the usual yield condition is attained along the primary loading curve and the reverse loading curve, namely, ρ = 1. Given suitable stress-strain data, the two curves in Fig. 1 may be determined by two
functions. The latter may be given in various forms.
For our purpose, it is essential to
prescribe these two curves by means of two functions described below. Following the idea suggested in a previous study (Xiao 2014a), we prescribe the hard-
28
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Fig. 1. Primary and reverse loading curves under tension and compression
ening part of the primary loading curve by a strain-stress function below:
(57)
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h = p(τ ), ϑ ≤ ϑˆ , p(r0 ) = r0 . E
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Here, ϑˆ is the effective plastic work at the maximum stress τm , E is Young’s modulus, τ and h are the axial stress and the axial strain at each point on the primary loading curve,
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and r0 is the initial yield stress in classical sense. Unlike the above procedure of prescribing the primary loading curve, we prescribe the
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reverse loading curve by means of an other procedure. For our purpose, a function, referred to as the bridging function for the primary and the reverse loading curve, is introduced as
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follows:
τ = φ(τ ) , φ(u0 ) = r0 .
(58)
where u0 is the stress at the intersecting point (cf., Fig. 1) between the reverse loading curve and the elastic line passing the origin. The above function establishes the correlation between the axial stresses τ and τ at each pair of points linked by an elastic unloading
29
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line, as shown in Fig. 1. By means of the bridging function we have 1 (τ − φ(τ )) E
(59)
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h = q(τ ) ≡ p(φ(τ )) +
for the axial strain h and the axial stress τ on the reverse loading curve. Thus, the reverse loading curve may be prescribed jointly by Eq. (57) and Eq. (58).
Whenever the two functions (cf., Eqs. (57)-(58)) for the primary and the reverse loading curve are available, the hardening quantities ch , ξh and rh at the hardening stage may be
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explicitly determined by these two functions, as will be done below.
For the monotone loading processes along the primary and the reverse loading curve, the following uniaxial relations may be derived (cf., Xiao 2014a, Xiao et al. 2016):
(60)
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K(τ ) , ∂rh ∂rh −1 −2 rh + 1.5(ch − ξh L) − rh ξh ζ − 1.5ch ζ rh L = ∂ϑ ∂ζ K(τ ) ,
where K(τ ) and K(τ ) are the plastic slopes of the primary loading curve and the reverse
dp 1 1 = − , dτ E K(τ )
(61)
1 dq 1 = − , K(τ ) dτ E
(62)
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loading curve, respectively, given by
with the Young’s modulus E = 3GB/(G + B). Eq. (60) merely applies to the uniaxial
case. A multi-axial extension may be given as follows (cf., Xiao et al. 2016):
rh
∂rh ∂rh + 1.5(ch − ξh L) − rh ξh ζ − 1.5ch ζ −1 rh−2 L = K(Λ, rh ) , ∂ϑ ∂ζ
30
(63)
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where
[Λ + rh ] [Λ + rh ] K(Λ, rh ) ≡ K (Λ + rh ) + 1 − K (Λ − rh ) , |Λ| + rh |Λ| + rh
(64)
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where Λ = 1.5L/rh and [x] = (x + |x|)/2. From Eq. (63) for general multi-axial cases, both ch and ξh are obtainable by following the idea of treating the singularity in Xiao (2014a) and Xiao et al. (2016). Results are as follows: ch =
r˜h0 K(−ζ r˜h0 , rh ) − rh rh0 K(Λ, rh ) − K(−ζ r˜h0 , rh ) − . ζ Λ + ζ r˜h0 1.5 − (˜ rh0 )2
In the above, rh0 =
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ξh rh =
K(−ζ r˜h0 , rh ) − rh rh0 , 1.5 − (˜ rh0 )2
∂rh , ∂ϑ
r˜h0 =
∂rh . ∂ζ
(65)
(66)
(67)
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Moreover, the rate-independent stress limit rh at the hardening stage is given by
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√ φ0 (0) − 1 −% rζ r0 + u0 −η rϑ r0 + u0 −η rϑ √ e 0 − 1.5 0 ζe 0 + g 1.5 ζ + e 0 , rh = 2 φ (0) + 1 2
(68)
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where % > 0 and η > 0 are two large parameters representing the localized effects at ζ = 0and at ϑ = 0, respectively, and the function r = g(α) is determined by the following
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parametric equations:
r = 1 (φ(τ ) − τ ) , 2 α = 1 (φ(τ ) + τ ) .
(69)
2
Details may be found in Xiao (2014a).
As shown in the previous work (Xiao 2014a, Xiao et al. 2016), a simple form of the shape
function (cf., Eq. (57)) for the primary loading curve may be given and then determined in fitting any given stress-strain data. Moreover, the shape function (cf., Eq. (59)) for the reverse loading curve is also available, whenever a suitable bridging function (cf., Eq. (58))
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is given. Usually, a linear function may serve the purpose, namely,
φ(τ ) = b1 (τ − u0 ) + r0 .
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In this case, we have
rh =
(70)
b (r + u ) r 0 − b1 u 0 √ b1 − 1 1 0 0 −η rϑ −% ζ + 1.5 ζ 1 − e r0 + e 0, b1 + 1 b1 + 1 b1 + 1
(71)
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The corresponding result in the previous work (Xiao 2014a, Xiao et al. 2016) is the particular case when u0 = 0.
5.2. Determination of the strength quantities at the softening stage After the maximum stress τm is attained, the uniaxial stress-strain curve will come to
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the softening part with negative slopes, as shown in Fig. 1. The softening part should be
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represented by another strain-stress function3 , as shown below: ϑ ≥ ϑˆ ,
(72)
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hs = ps (τ ),
with the following properties:
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p0 (τ ) < 0 , s lim p (τ ) = +∞, τ →0
s
(73) limτ →0 p0s (τ )
= −∞ .
The moduli cs and ξs are still given by Eqs. (65)-(67), where ch , ξh and rh are replaced
3 It may be evident that the whole primary loading curve including both the hardening and the softening part can not be represented by a single strain-stress function.
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by their counterparts, i.e., cs , ξs and rs at the softening stage, namely,
ξs rs =
K(−ζ r˜s0 , rs ) − rs rs0 , 1.5 − (˜ rs0 )2
(74)
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cs =
r˜s0 K(−ζ r˜s0 , rs ) − rs rs0 K(Λ, rs ) − K(−ζ r˜s0 , rs ) − . ζ Λ + ζ r˜s0 1.5 − (˜ rs0 )2
(75)
where the K(Λ, r) is still given by Eq. (64) with the plastic slopes K(τ ) and K(τ ) below:
K(τ ) =
p0s (τ )
1 , − E −1
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K(τ ) =
b−1 s . p0s (bs τ ) − E −1
(76)
(77)
In the above, bs > 0 is a dimensionless parameter.
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Next, the stress limit rs is given by
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h i √ rs = 1 rm 1 − tanh β0 ϑ − 1 + 1.5 bbss −1 ζ, 2 ϑ0 +1 ϑ ≥ ϑˆ .
(78)
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In the above, rm > 0, β0 > 0 and ϑ0 > 0 are material parameters. The moduli cs and ξs and the stress limit given above characterise coupling effects for
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softening behavior, in which the shape funtion of the property indicated in Eq. (73) plays an essential role. A simple form of this function may be given as follows (cf., Xiao et al.
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2016):
h = ps (τ ) = hm
n 1 τ2 1 + − ln 2 , 0 < τ ≤ τm , λs τm
(79)
where hm and τm are the strain and the stress at the maximum stress point (cf., Fig. 1) and n > 0 and λs > 0 are dimensionless parameters. Details in deriving the above results may be found in Xiao et al. (2016). It may be demonstrated (cf., Xiao et al. 2016) that the quantities given meet the asymptotic 33
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properties prescribed by Eqs. (37), (39) and (41). 5.3. Determination of the fatigue life
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Consider a uniform deformation process consisting of stress cycles. Let τ 0 be any given multi-axial stress. At each cycle, the stress is changing from zero to τ 0 and then to uτ 0 and finally back to zero. Here, −1 ≤ u ≤ 0. A cycle with u = −1 is called a symmetric one. At each cycle, the plastic flow is induced in the two processes from 0 to τ 0 and from τ 0 to uτ 0 , separately.
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The fatigue life under the cyclic process at issue is the cycle number to failure, denoted N , for which a critical failure state is attained. It is determined by the following procedures: (i) Integrate the following system of the elastoplastic rate equations successively
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as from the first cycle:
ρJ2−1 fˆ ˆ∗ 2G(1−ρ)+h
(˜ τ − α) ,
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o 1 1 log ˜ D = (trτ˙ )I + τ + 3B 2G ρfˆ ϑ˙ = ˆ , 2G(1−ρ)+h ∗
(80)
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o log cρJ2−1 fˆ τ − α) − ξαϑ˙ , α = ˆ ∗ (˜ 2G(1−ρ)+h o fˆ = (˜ ∗ ˙ τ − α) : τ log − 23 r ∂r d ∂d
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for fˆ > 0 and, on the other hand, set fˆ = 0 in Eqs. (80)1−3 for fˆ ≤ 0; (ii) obtain the effective plastic work at the end of each cycle, denoted ϑ(n) for
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the n−th cycle;
(iii) the cycle number N for which a critical failure state is attained is determined by ϑ(N −1) > ϑ(N −2) , ϑ(N −1) > ϑ(N ) . 34
(81)
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The criterion Eq. (81) is implied by the following fact: According to Eq. (80)2 , the ϑ is increasing as the denominator is positive and decreasing as the latter becomes negative. As such, Eq. (81) means that, just at the N −th cycle, the denominator experiences a should be met with the occurrence of a critical failure state.
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change in sign from positive to negative values and, accordingly, the criterion Eq. (54)
As indicated before, the attainment of a critical failure state heralds the beginning of the asympotic strength loss, namely, asymptotic loss of the stress-bearing capacity. As
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such, immediately following the attainment of a critical failure state, the magnitude of the stress will invariably go down to vanish concomitantly with development of plastic flow. From this it follows that, whenever a critical failure state is attained under a cyclic stress process, the stress cycle could no longer be completed as in the case prior to this critical state. This may be explained as follows. A stress cycle means that the stress magnitude is
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allowed to grow to a given value known as the stress amplitude, as mentioned at the outset
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of this subsection. However, after a critical failure state the stress magnitude could no longer be allowed to grow to this amplitude but should go asymptotically down to vanish.
5.4. Remarks
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This phenomenon will be evidenced in Fig. 4 presented in §7.1.
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Results have been derived for cyclic loading cases. Generally, irregular or non-cyclic loading cases need be treated. Since the cycle number to failure becomes irrelevant in
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general, the fatigue life in usual sense should be replaced by the entire process from the start of loading up to the emergence of a critical failure state. Given uniaxial stress-strain data for the primary and the reverse loading curve as shown
in Fig. 1, the shape functions (cf., Eq. (57) and Eqs.(72)-(73)) for the hardening and the softening stage and the bridging function Eq. (58) may be prescribed for the purpose of fitting such data, as indicated in the previous work (Xiao 2014a, Xiao et al. 2016). Then, the results presented in §5.1 and §5.2 achieve double goals, namely, (i) the hardening 35
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quantities for a general multi-axial case are explicitly determined from these shape functions and (ii) the proposed model with these hardening quantities agree automatically with the uniaxial data at issue.
With the strength quantities identified from suitable uniaxial
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data, including those at the hardening and the softening stage demarcated by the effective plastic work ϑˆ at the uniaxial maximum stress, as given in §5.1-§5.2, the proposed model with a given form of the plastic factor ρ (cf., Eqs. (14)-(15)) is ready for obtaining results for multi-axial loading conditions and then comparing with multi-axial data. Here it is
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worthwhile to point out that it is essential to select a suitable form of the plastic factor ρ toward simulating both the ratcheting effect and the fatigue failure behavior. A simple form of ρ, given by Eq. (21), has been suggested in previous works (cf., e.g., Xiao et al. 2014; Xiao 2014b) and shown to be useful (Wang and Xiao 2014). Further developments
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of this form will be proposed in §7.
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6. Fatigue failure under thermal cycles
In this section, fatigue failure under thermal cycles will be taken into consideration. It
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may be of interest to know what features the fatigue behavior in this case will display. It is noted that results in this respect have been rarely reported, expected for experimental
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tests in some cases.
6.1. Governing equation for the back stress
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Consider a thermal process consisting of temperature cycles. At each cycle, the temper-
ature T grows from a given lower value, T1 , to a given higher value, T2 , and, then, returns to T1 . Let the back stress and the effective plastic work at the outset of the first cycle be α1 and ϑ1 , respectively. Moreover, a basic property for the stress limit r∗ is as follows: ∂r∗ < 0, ∂T 36
(82)
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namely, the stress limit r∗ goes down as the temperature goes up. We are going to derive a governing equation for the back stress α. With τ = O and Eqs. (1) and (5), we have
˜ . Dp = D
(83)
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D e = γ T˙ I ,
(84)
On account of the property Eq. (82) and the fact that ϑ˙ ≥ 0, we infer
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∂r∗ ˙ ∂r∗ ˙ f˘ = 3Gϑ˙ − r∗ T − r∗ d>0 ∂T ∂d
(85)
for T˙ > 0 and ϑ¨ ≤ 0. Hence, plastic flow will be induced, whenever the temperature grows from T1 to T2 at each thermal cycle. From this and Eqs. (14) and (32) we deduce
(86)
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f˘ D p = −ζ −2 ρ α . ˘ h
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Then, this and Eq. (49) produce
˙ . ˜ = −ζ −2 ϑα Dp = D
(87)
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Substituting this into Eq. (11), we derive o log
α
˙ . = (−cζ −2 − ξ)ϑα
(88)
Then, by using Eq. (88) and the equality o α : α log = 1.5ζ ζ˙ ,
37
(89)
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we arrive at: 1.5ζ ζ˙ = (−c − 1.5ξζ 2 )ϑ˙ .
(90)
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From Eq. (90) it follows that the magnitude of the back stress α will go down to vanish as the thermal cycle is progressing. In fact, for the heating process from T1 to T2 at the n−th cycle, Eq. (90) yields
2 − ζ1n
with Θ=
Z
t
t1n
4 2Θ ˙ ce ϑdt , 3
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ζ = e−Θ
s
Z
t
˙ , 2ξ ϑdt
(91)
(92)
t1n
2 where t1n and ζ1n are the initial values of the time t and the magnitude ζ at the start of
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the n−th cycle and t is any given instant in the heating process in the foregoing.
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6.2. Determination of the thermal fatigue life
We proceed to find out the cycle number for which a critical failure state will be
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attained. Toward this end, we first work out the effective plastic work at the end of each thermal cycle. At each cycle, with Eqs. (18)-(20), Eq. (53) and τ = O, we deduce that
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the ϑ is determined by the equation below:
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ϑ˙ =
∗ ˙ ∗ ˙ −ρr∗ ∂r T − ρr ∂r d ∂T ∂d . ∂r∗ ∗ 3G(1 − ρ) + (c + ζ 2 ξ) 1.5 − ζ −1 r∗ ∂r + r ∗ ∂ϑ ∂ζ
(93)
Given the time-dependent temperature T = T (t), how the ϑ will change with the time t
may be determined jointly from Eqs. (90) and (93). In particular, for the rate-independent case with the r∗ independent of the stretching magnitude d, Eq. (93) is substantially simplified and it is possible to obtain the ϑ as a function of the temperature T , as shown
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below: ∗ −ρr∗ ∂r dϑ ∂T = . ∂r∗ dT ∗ + r 3G(1 − ρ) + (c + ζ 2 ξ) 1.5 − ζ −1 r∗ ∂r ∗ ∂ϑ ∂ζ
(94)
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Again, the cycle number N , for which a critical failure state is attained, is determined by Eq. (81).
It should be pointed out that Eqs. (91)-(92) supplies a closed-form solution of the form ζ = ζ(ϑ) for the case when both the Prager modulus c and the hysteresis modulus ξ are dependent merely on the effective plastic work ϑ. Given the time-dependent temperature
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T = T (t), generally the solutions ζ = ζ(t) and ϑ = ϑ(t) should be obtained by integrating Eqs. (90) and (93). For the rate-independent case, the solutions of the form ζ = ζ(T ) and ϑ = ϑ(T ) may be determined from Eq. (91) and Eq. (94).
Eq. (93) displays strong nonlinearities coupled with rate-dependent effects. This feature
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is inherent in the thermal fatigue behavior, in the sense that it manifests itself for any form of temperature change and, in particular, that is the case even for temperature change at
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a constant rate.
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7. Numerical examples
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Numerical examples will be presented in this section. It will be shown that good agreement with test data may be achieved even in the simplest cases of the proposed model, namely, the hardening stage is neglected and the anisotropic hardening behavior is
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absent with α = 0.
Test data are usually given for uniaxial loading cases of cylindrical samples. Model
predictions for uniaxial stress cycles may be derived by applying the procedures in §4 and,
then, compared with test data in literature. For the uniaxial stress case, the J2 is given by 32 τ 2 with the axial Kirchhoff stress τ and, besides, the stretching magnitude d (cf., Eq. (8)) may be given by the axial strain rate h˙
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˙ as a good approximation4 . Namely, J2 = 23 τ 2 and d ≈ h. For the purpose of model validation, test data in four respects will be separately simulated, including monotone strain data with rate effects and those with the ratcheting
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effect and the fatigue limit effect, as well as data with the duplex effect from low to high cycle fatigue failure. Besides, model predictions will also be derived for fatigue lives under thermal cycles without and with a fixed uniaxial stress.
7.1. Monotone data with rate effects and model predictions under stress cycling
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We first take into account the low strain rate data for p92 steel in Giroux et al. (2010), which is concerned with a monotonically growing axial strain up to failure at the strain rate h˙ = 2.5 × 10−6 (Fig. 3 in the foregoing reference). In this case, the rate effect may be ρ=
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neglected. The plastic factor ρ (cf., Eq. (21)) and the stress limit r∗ are given by 1.5J2 −m(1−1.5J2 /r∗2 ) e r∗2
h
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r∗ = 1 rm 1 − tanh β0 2
,
ϑ ϑ0
i −1 ,
(95)
where the m is the plastic index (cf., Xiao et al. 2014; Xiao 2014b) and the rm , ϑ0 , β0
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represent the tensile strength, the critical value of ϑ, the slope at the softening part in the
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uniaxial stress-strain curve, respectively. The values of the parameters involved are found
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in fitting the foregoing data5 (cf., the lower curve in Fig. 2), as given below:
E = 120GPa, ν = 0.3, m = 1, rm = 338MPa, β0 = 2.9, ϑ0 = 58MPa .
Here and below, ν is the Poisson ratio. With these parameters, numerical results are
p ˙ with the lateral Hencky strain given In fact, the d in the uniaxial case is given by 2 (1 + 2s2 ) /3 h, by −sh. The ratio of the lateral to the axial Hencky strain, i.e., s, falls within the range (ν, 0.5). The ˙ value of the Poisson ratio ν is about 31 . As such, it follows that 0.9h˙ < d < h. 5 These data in Cauchy stress and engineering strain have been converted to those in Kirchhoff stress and Hencky strain. That will be the case for other data. 4
40
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obtained under the following type of uniaxial stress cycle: the axial stress τ grows from zero to a given amplitude, τ0 , and then is back to zero. The curve of the stress amplitude versus the cycle number to failure is depicted in Fig. 3 (the lower curve). Moreover, as
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an example, the stress-strain curves for all the stress cycles up to failure are shown in Fig. 4 for the stress amplitude τ0 = 200MPa. It may be clear from Fig. 4 that, immediately following a critical state, the stress can no longer be allowed to grow to the amplitude τ0 = 200MPa as in the case of each stress cycle prior to this state but goes invariably down
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to vanish concomitantly with indefinitely growing strain.
It is known (Xiao et al. 2014, Xiao 2014b) that the plastic index m plays an essential role in accumulation of the plastic strain and hence the effective plastic work. In Fig. 5, further results for the sensitivity analysis are presented for three values of the plastic index m, viz., m = 0.01, 0.3, 0.8. As is shown in Fig. 5, the fatigue life for a given stress
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amplitude goes up as the plastic index m goes up. This is in accord with the following fact
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(Xiao et al. 2014, 2014b): the greater the index m, the smaller the plastic strain at each cycle.
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Rate-dependent effects should be taken into consideration at high strain rate. In this case, each parameter in the stress limit r∗ in Eq. (95)2 relies on the equivalent strain-rate
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d. As illustration, consider the two data sets for monotone tension at the low strain-rate 2.5 × 10−6 s−1 and at the high strain rate 2.5 × 10−3 s−1 , as given in Giroux et al. (2010).
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The rate-dependent parameters are given below:
rm = 338 + 56800d (MPa), β0 = 2.9 + 680d, ϑ0 = 58 + 8800d (MPa).
As indicated before, the d may be given by the axial strain rate h˙ in the uniaxial case. The simulation results at the two strain rates h˙ = 2.5 × 10−6 s−1 and h˙ = 2.5 × 10−3 s−1 are shown for the monotone loading case in Fig. 2 and for the curves of fatigue life versus
41
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Fig. 2. Comparison with the monotone strain data (dots) from Giroux et al. (2010): tensile stress-strain curves at two strain rates 2.5 × 10−6 s−1 (lower curve) and 2.5 × 10−3 s−1 (upper curve)
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stress amplitude in Fig. 3. It follows from Fig. 2 and Fig. 3 that both the stress limit and the cycle number to failure increase with increasing strain rate.
Note here that the
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fatigue life in a direct sense, viz., the time to failure, decreases actually with increasing strain rate, since the latter is the cycle number to failure devided by the strain rate. That
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is the case for the model predictions shown in Fig. 2. The model prediction for the fatigue life, as shown in Fig. 3, agrees with the salient
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feature of the well-known S-N curve for the fatigue failure under uniaxial stress cycling. In general, the S-N curve is invariably inceasing with decreasing stress amplitude, as may be seen in Fig. 3. In particular, this monotone property should be exhibited even for vanishingly small stress amplitude, as shown in Fig. 3. Here, results for the fatigue failure and the racheting effect are merely presented as model predictions, since relevant data are absent in the foregoing reference. Comparisons with test data will be considered for various cases in the succeeding subsections. 42
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Fig. 3. Model prediction under stress cycling: stress amplitude versus cycle number to failure at two strain rates 2.5 × 10−6 s−1 (lower curve) and 2.5 × 10−3 s−1 (upper curve)
Fig. 4. Stress-strain curves up to failure under stress cycles at low strain rate 2.5 × 10−6 s−1
43
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Fig. 5. Parameter sensitivity analysis at low strain rate 2.5 × 10−6 s−1 : curves of fatigue life versus stress amplitude for three values of the plastic index (m =0.8, 0.3, 0.01 from top to bottom)
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It should be noted that uniform stress-strain data for monotone tensile strain test should be used for simulating the material softening up to failure. However, usually that is not the
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case for data with non-uniform strain and stress distributions after necking. This respect is studied in, e.g., Shaw and Kyriakides (1998). On account of this, data after necking
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are merely estimated values via certain averaging procedures and, as such, simulation is performed in an approximate sense. In fact, deviations of simulation results from such
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approximate data may become appreciable as failure is approached, as may be seen in Fig. 2.
7.2. Asymmetric stress cycling: racheting effect Simulation results are given for ratcheting effects and compared with the latest data in Fig. 9 in Lee et al. (2015) (see also Hassan and Kyriakides 1992) for S32750 super duplex stainless steel under the non-symmetric uniaxial stress cycling with a mean stress 44
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Fig. 6. Model prediction for the ratcheting effects under the non-symmetric uniaxial stress cycling with a mean stress of 60MPa and a stress amplitude of 636MPa
of 60MPa and a stress amplitude of 636MPa. The parameter values in Eqs. (95)1,2 are
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given below:
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E = 204GPa, ν = 0.3, m = 1.7, rm = 720MPa, β0 = 56, ϑ0 = 100MPa.
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(2014).
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Results are shown in Fig. 6 and compare well with the test results in Fig. 9 in Lee et al.
7.3. Asymmetric stress cycling: low-to-medium cycle failure with high tensile strength Low-to-medium cycle failure data by Koster et al. (2016) for the stainless steel X3CrNiMo
13-4 (AISI CA 6-NM) will be simulated, in which high tensile strength and non-symmetric stress cycles are involved (the ratio of the minimum to the maximum stress value is R = 0.1). The plastic factor ρ and the stress limit r∗ are still given by Eqs. (95)1,2 .
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Fig. 7. Comparison with the monotone strain data (dots) from Koster et al. (2016): tensile stress-strain curve
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The parameter values in fitting these data are as follows:
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E = 138GPa, ν = 0.3, m = 9.5, rm = 1250MPa,
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β0 = 3, ϑ0 = 195MPa, γ0 = 15.
Results are shown in Fig. 7 for monotone uniaxial extension and in Fig. 8 for the S − N
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curve of the cycle number to failure versus the stress amplitude. 7.4. Symmetric stress cycling: fatigue limit effect with low tensile strength As indicated at the end of §7.1, generally the S − N curve exhibits monotone property
with decreasing stress amplitude. In some cases, however, the cycle number to failure, N , may very rapidly grow as the stress amplitude S tends toward a certain value, known as fatigue limit or fatigue strength. In this subsection, data for NZ30K1 -T4 alloy under symmetric stress cycling, given in Li et al. (2015), are simulated, in which the foregoing 46
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Fig. 8. Comparison with the data (dots) under non-symmetric stress cycling (Koster et al. 2016): stress amplitude versus cycle number to failure
complexity with a fatigue limit is incorporated. As an example, use is made of the tensile
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stress-strain curve (cf., Fig. 8 (b) therein) and the S − N curve (cf., Fig. 16(a) therein) for symmetric stress cycles.
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The stress limit is still given by Eq. (95)2 , while the plastic factor ρ is now given by an
2 rm 1 1 + tanh γ0 x − 1 xem(x−1) ρ= 2 κ20
(96)
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extended form of Eq. (21) below:
with x = 1.5J2 /r∗2 . For a fairly large γ0 , the plastic factor ρ of the above form becomes very small as the stress amplitude does not exceed κ0 . The values of the parameters in the above form and Eq. (95)2 are found as follows:
E = 39GPa, ν = 0.3, m = 9, rm = 112MPa, β0 = 3.9 ,
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Fig. 9. Simulation of the monotone strain data (dots) from Li et al. (2015): tensile stress-strain curve
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ϑ0 = 90MPa, γ0 = 15, κ0 = 67MPa . Results in fitting the foregoing data are shown in Figs. 9 and 10. It may be seen in Fig. 10
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that the cycle number to failure becomes indefinitely large when the stress amplitude tends toward the fatigue limit k0 = 67MPa, which agrees with the remark at the outset of this
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subsection. At this point, however, it should be noted that, for the stress amplitude below a certain value, the slope of the S − N curve may become very small but never vanishing,
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namely, the S − N curve never becomes actually flat. That is the case for the proposed model with the asymptotic properties indicated in §4.1. Indeed, from Eq. (33) it follows that, for every cyclic deformation process, the effective plastic work ϑ will monotonically grow to become indefinitely large as the cyclic process is invariably progressing and, as a consequence, the cycle number to failure exists for any given value of the stress amplitude.
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Fig. 10. Simulation of the data (dots) under symmetric stress cycling (Li et al. 2015): stress amplitude versus cycle number to failure
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7.5. Symmetric stress cycling: duplex effect from low to high cycle failure In this subsection, model prediction will further be validated by simulating fatigue
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failure data that display another kind of complexity, referred to as the duplex effect. In the S − N curve, there emerges a plateau locating between two parts for low-to-medium
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cycle failure and for medium-to-high cycle failure. Details for this effect may be found in Mohd et al. (2016) and the relevant references therein.
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It appears that a unified simulation of the fatigue failure behavior with the above complex duplex effect poses a challenging issue. In fact, it is found (cf., e.g. Mohd et
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al. 2016) that the surface-induced crack nucleation mode and the subsurface-inclusioninduced crack nucleation mode are responsible for the foregoing two parts in the S − N curve, separately. This implies that the just mentioned two micro-mechanisms of distinct nature need be treated toward modeling the duplex effect in a unified sense. It is demonstrated here that fatigue failure with the duplex effect may be simulated based on the new model. Toward this objective, test data in Mohd et al. (2016) are treated, which supply data from low to high cycle failure for JIS SUS630 under symmetric 49
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Fig. 11. Simulation of the data (dots) from Mohd et al. (2016) with duplex effect from low to high cycle failure: stress amplitude versus cycle number to failure
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tension-compression stress cycling. Now the stress limit is still given by Eq. (95)2 and the
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plastic factor ρ by Eq. (95)1 with the plastic index m therein given below:
m = m1 + (m2 − m1 ) cosh ξ
−1 1.5J2 −1 , d20
(97)
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where the parameter d0 represents the stress level at the center of the plateau part. The
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plastic index m is very close to m1 outside the plateau part, whhereas it changes from m1 to m2 and then goes back to m1 along the plateau part.
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With the parameter values below:
E = 191GPa, ν = 0.3, rm = 1056MPa, ϑ0 = 300MPa, β0 = 5,
m1 = 12.7, m2 = 7, ξ = 50, d0 = 750MPa, the model prediction is calculated and compared with the data in the foregoing. Results are depicted in Fig. 11 and in good agreement with the test data. 50
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7.6. Fatigue lives under thermal cycles Suitable data for fatigue failure under thermal cycles are not available. Here, the model prediction for a cylindrical sample under thermal cycles is obtained by applying the results
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in §5.
For the sake of simplicity, the rate-independence case is treated and the plastic factor ρ is given by Eq. (21). The temperature-dependent stress limit r∗ and Prager modulus c are taken to be of the following forms:
(98)
1 ϑ T c = c0 1 − tanhβ −1 1 − tanh b −1 , 2 ϑ0 T0
(99)
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ϑ T 1 r∗ = rm 1 − tanhβ −1 1 − tanh b −1 , 4 ϑ0 T0
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and, besides, the hysteresis modulus ξ is taken to be constant, i.e., ξ = ξ0 . The thermal cycle considered is as follows: the temperature grows from the initial value
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T1 = 600K to a given higher value T2 and then goes back to T1 . The parameter values below are taken into account:
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G = 56154MPa, m = 5, rm = 100MPa, ϑ0 = 0.02MPa, β = 600,
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b = 2, T0 = 600K, c0 = 112.3MPa, ξ0 = 22/MPa,
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with the following initial values for the effective plastic work and the axial back stress: ϑ1 = 0, α1 = 3.37MPa.
Results are obtained by numerically integrating Eq. (90) and Eq. (94). The curve of the temperature amplitude T2 versus the cycle number to failure is depicted in Fig. 12 for pure thermal cycling and in Fig. 13 for thermal cycling with a fixed axial stress of 5MPa. As shown in Figs. 11 and 12, the cycle number to failure decreases as increasing temperature amplitude in the both cases considered and, moreover, the fatigue life in the presence of 51
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Fig. 12. Model prediction under thermal cycling: temperature amplitude versus cycle number to failure (the temperature at the start of each cycle is 600K)
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axial stress is considerably lower than that in the absence of axial stress. This initial result suggest that the fatigue life may considerably decrease under thermomechanical cyclic
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conditions in the presence of both stress and temperature. The model predictions here presented under thermal cycling can merely display certain
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qualitative features of thermal fatigue failure in a limited sense that both the coupling effect between stress and temperature and the viscous flow (creep) effect at elevated temperature
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are neglected. Comparisons with test data are not presented for lack of relevant data for the cases treated. Test data under thermomechanical cycling in literature show complex
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coupling effects and viscous effects. Toward simulating such data, the simple forms Eqs. (98)-(99) for the stress limit r∗ and the Prager’s modulus c should be extended and issues in extensive numerical integrations for large cycle numbers are involved. This aspect is among the topics that need be treated in further developments, as will be indicated in the next section.
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8. Concluding remarks
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Fig. 13. Model prediction under thermal cycling with a fixed axial stress of 5MPa: temperature amplitude versus cycle number to failure (the temperature at the start of each cycle is 600K)
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In the previous sections, new elastoplastic J2 −flow models in a free, smooth sense have been proposed for the purpose of achieving direct simulation of multi-axial thermo-coupled
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fatigue failure for metals and alloys. Both rate-dependent effects and thermo-coupled effects are covered in a broad sense. The new model is mainly based on two basic facts
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concerning deformation behaviors of metals and alloys. Namely, plastic strain may be smoothly induced at every stress level and becomes appreciable whenever a yielding state
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in a usual sense is approached, on the one side, and metals and alloys repeatedly undergoing deformations suffer continued loss of the strength up to eventual failure, on the other side. It is realized that two perhaps essential features of the macroscopic deformation behavior for metals and alloys are implied by the two basic facts just indicated. Namely, in every process of repeated deformation, the effective plastic work (cf., Eq. (10)) is invariably accumulating, and the constitutive quantities characterising the strength property are going down and eventually vanishing with acumulation of the effective plastic work. It has been 53
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demonstrated (Xiao 2014b) that, from the asymptotic properties just indicated, fatigue failure is derivable as a direct consequence of the proposed model and, accordingly, incorporated as an inherent constitutive feature into the proposed model. This would imply
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that, under any loading conditions, fatigue failure should automatically be predicted from the proposed model.
It is from the above fact that critical failure states with a unified criterion have further been derived as model prediction. In particular, results have been obtained for fatigue lives
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under isothermal stress cycles and under thermal cycles, separately. Explicit procedures have been introduced toward determining the constitutive quantities based on suitable uniaxial stress-strain data.
The main features of the direct approach proposed may be summarized as follows:
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(i) both ratcheting effects and fatigue effects are automatically incorporated as inherent constitutive features of the proposed model;
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(ii) the unified criterion for critical failure states is for all cases of multiaxial thermo-coupled fatigue failure, including
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(iii) all cases of fatigue failure under cyclic loadings, such as so-called low and
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high cycle fatigue and
(iv) all cases of fatigue failure under cyclic and non-cyclic loading conditions
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with constant and varying strain rates. Numerical examples for model validation have been presented and compared with test
data displaying different features of monotone and cyclic failure behaviors, including monotone failure behavior with rate effects, ratcheting effect under non-symmetric stress cycling, low and medium cycle failure with high tensile strength, fatigue limit effect with low tensile strength, as well as duplex effect from low to high cycle failure under symmetric stress cycling, etc. It has been demonstrated that good agreement with all such data, in particular, 54
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with those dispalying complex duplex effect, may be achieved even in the simplest case of the proposed model, namely, the back stress effect is totally ignored and only a simple form of the stress limit (cf., Eq. (95)2 ) with the asymptotic property Eq. (37) is presented
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for the softening effect.
In future study it may be of interest to obtain further results in simulating fatigue failure in six respects, including (i) suitable test data for both the primary and the reverse loading curve; (ii) suitable test data for fatigue failure at high strain rates and at elevated
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temperature; (iii) further simulations and model validation with appreciable hardening effects based on the data mentioned in the first respect; (iv) numerical simulations and model validation for fatigue failure at high strain rates, (v) further investigations into creep failure at elevated temperature; and (vi) extensive simulations and model validation under combined changes of stress and temperature and, in particular, under multiaxial loading
Acknowledgments
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conditions. Results in these further respects will be studied in future works.
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The authors would like to thank the two anonymous reviewers for their constructive comments, which have been instrumental in clarifying relevant issues.
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This study was carried out under the joint support of the funds from Natural Science Foundation of China (No.: 11372172; No.: 11542020) and from the science and technology
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development project launched by Weifang city (No.: 2015GX018). References
Ahmed, R., Barrett, P.R., Hassan, T., 2016. Unified viscoplasticity modeling for isothermal low-cycle fatigue and fatigue-creep stress-strain responses of Haynes 230. Int. J. Solids Struct. 88-89, 131-145. Alfredsson, K.S., Stigh, U., 2004. Continuum damage mechanics revised: A principle for mechanical and thermal equivalence. Int. J. Solids Struct. 41, 4025-4045. Argon, A.S., 2007. Strengthening Mechanisms in Crystal Plasticity. Oxford University Press, Oxford.
55
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Armero, F., Oller, S., 2000. A general framework for continuum damage models: I: infinitesimal plastic damage models in stress space. Int. J. Solids Struct. 37, 7409-7464. Assidi, M., Zahrouni, H., Damil, N., Michel Potier-Ferry, M., 2009. Regularization and perturbation technique to solve plasticity problems. Int. J. Mater. Form. 2, 1-14. Baletto, M.C., Pierres, E., Gravouli, A., 2010. A multi-model X-FEM strategy dedicated to frictional crack growth under cyclic fretting fatigue loadings. Int. J. Solids Struct. 47, 1405-1423. Badreddine, H., Yue, Z.M., Saansuni, K., 2016. Modeling of the induced plastic anisotropy fully coupled with ductile damage under finite strains. Int. J. Solids Struct. DOI: 10.1016/j/ijsolstr.2016.10.028. Barrett, P.R., Ahmed, R., Menon, M., Hassan, T., 2016. Isothermal low-cycle fatigue and fatigue-creep of Haynes 230. Int. J. Solids Struct. 88-89, 146-164. Bonora, N., Gentile, D., Pirondi, A., Newaz, G., 2005. Ductile damage evolution under triaxial state of stress: theory and experiments. Int. J. Plasticity 21, 981-1007. Bonora, N., Ruggiero, A., 2005. Micromechanical modelling of ductile cast iron incorporating damage. Part. Ferritic ductile cast iron. Int. J. Solids Struct. 42, 1401-1424. Bruhns, O.T., Xiao, H., Meyers, A., 2001. A self-consistent Eulerian rate type model for finite deformation elastoplasticity with isotropic damage. Int. J. Solids Struct. 38, 657-683. Bruhns, O.T., Xiao, H., Meyers, A., 2005. A weakened form of Ilyushin’s postulate and the structure of self-consistent Eulerian finite elastoplasticity. Int. J. Plasticity 21, 199-219. Br¨ unig, M., 2003a. An anisotropic ductile damage model based on irreversible thermodynamics. Int. J. Plasticity 19, 1679-1713. Br¨ unig, M., 2003b. Numerical analysis of anisotropic ductile continuum damage. Compt. Meth. Appl. Mech. Engrg. 192, 2749-2776. Br¨ unig, M., 2015. A continuum damage model based on experiments and numerical simulationsA review. In: Holm Altenbach, Tetsuya Matsuda, Dai Okumura (eds.), From Creep Damage Mechanics to Homogenization Methods - A Liber Amicorum to celebrate the brithday of Nobutada Ohno, Advanced Structural Materials Series, pp.19-35, Springer, Berlin. Br¨ unig, M., Ricci, S., 2005. Nonlocal continuum theory of anisotropically damaged metals. Int. J. Plasticity 21, 1346-1382. Br¨ unig, M., Chyra, O., Albrecht, D., Driemeier, L., Alves, M., 2008. A ductile damage criterion at various stress triaxialities. Int. J. Plasticity 24, 1731-1755. Br¨ unig, M., Gerke, S., 2011. Simulation of damage evolution in ductile metals undergoing dynamic loading conditions. Int. J. Plasticity 27, 1598-1617. Br¨ unig, M., Gerke, S., Hagenbrock, V., 2013. Micro-mechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage. Int. J. Plasticity 50, 49-65. Canadija, M., Mosler, J., 2016. A variational formulation for thermodynamically coupled low cycle fatigue at finite strains. Int. J. Solids Struct. 53, 388-398. Cazes, F., Coret, M., Combescure, A., Gravouil, A., 2009. A thermodynamic method for the construction of a cohesive law from a nonlocal damage model. Int. J. Solids Struct. 46, 1476-1490. Cervera, M., Wu, J.Y., 2015. On the conformity of strong, regularized, embedded and smeared discontinuity approaches for the modeling of localized failure in solids. Int. J. Solids Struct. 71, 19-38. Chew, H.B., 2014. Cohesive zone laws for fatigue crack growth: Numerical field projection of the micromechanical damage process in an elasto-plastic medium. Int. J. Solids Struct. 51, 1410-1420.
56
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Chaboche, J.L. 2008. A review of some plasticity and viscoplasticity constitutive theories. Int. J. Plasticity 24, 1642-1693. Ciavarella, M., Demelio, G., 2001. A review of analytical aspects of fretting fatigue, with extension to damage parameters and application to doveail joints. Int. J. Solids Struct. 38, 1791-1811. Cortese, L., Nalli, F., Rossi, M., 2016. A nonlinear model for ductile damage accumulation under multiaxial non-proportional loading conditions. Int. J. Plasticity 85, 77-92. Driemeier, L., Br¨ unig, M., Micheli G., Alves M., 2010. Experiments on stress-triaxiality dependence of material behavior of aluminum alloys. Mechanics of Materials 42, 207-217. Dvorak, G.J., 2000. Composite materials: Inelastic behavior, damage, fatigue and fracture. Int. J. Solids Struct. 37, 155-170. Egner, H., 2012. On the full coupling between thermo-plasticity and thermo-damage in thermodynamic modeling of dissipative materials. Int. J. Solids Struct. 49, 279-288. Facheris, G., Janssens K.G.F., Foletti, S., 2014. Multiaxial fatigue behavior of AISI 316L subjected to strain-controlled and ratcheting paths. Int. J. Fatigue 68, 195-208. Frost, N.E., Marsh, K.J., Pook, L.P., 1999. Metal Fatigue. Dover Publications, Inc., New York. Giroux, P., Dalle, F., Sauzay, M., Malaplate, J., Fournier, B., Gourgues-Lorenzon, A., 2010. Mechanical and microstructural stability of p92 steel under uniaxial tension at high temperature. Mater. Sci. Engng. A 527, 3984-3993. Hassan, T., Corona, E., Kyriakides, S., 1992. Ratcheting in cyclic plasticity, part II: multiaxial behavior. Int. J. Plasticity 8, 117146. Hassan, T., Kyriakides, S., 1992. Ratcheting in cyclic plasticity, part I: uniaxial behavior. Int. J. Plasticity 8, 91116. Hassan, T., Kyriakides, S., 1994a. Ratcheting of cyclically hardening and softening materials, part I: uniaxial behavior. Int. J. Plasticity 10, 149184. Hassan, T., Kyriakides, S., 1994b. Ratcheting of cyclically hardening and softening materials, part II: multiaxial behavior. Int. J. Plasticity 10, 185212. Heitbreder, T., Ottosen, N.S., Mosler, J., 2016. Consistent elastoplastic cohesive zone model at finite deformations−Variational formulation. International Journal of Solids and Structures, DOI: 10.1016/j.ijsolstr.2016.10.027. Hill, R., 1968. Constitutive inequalities for simple materials. J. Mech. Phys. Solids 16, 229-242. Hutchinson, J.W., Evans, A.G., 2000. Mechanics of materials: Top-down approaches to fracture. Acta Materialia 48, 125-135. Jir´ asek, M., Zimmermann, T., 2001. Embedded crack model. I: basic formulation. Int. J. Numer. Methods Eng. 50 (6), 1269-1290. Kang, G.Z., Liu, Y.J., Ding, J., 2008. Multiaxial ratcheting-fatigue interactions of annealed and tempered 42CrMo steels: Experimental observations. Int. J. Fatigue 30, 2104-2018. Khan, A.S., Liu, H.W., 2012a. A new approach for ductile fracture prediction on Al 2024-T351 alloy. Int. J. Plasticity 35, 1-12. Khan, A.S., Liu, H.W., 2012b. Strain rate and temperature dependent fracture criterion for isotropic and anisotropic metals. Int. J. Plasticity 37, 1-15. Krajcinovic, D., 2000. Damage mechanics: accomplishments, trends and needs. Int. J. Solids Struct. 37, 267-277. Khoel, A.R., Moslemi, H., Shavifi, M. 2012. Three-dimensional cohesive fracture modeling of non-planar crack growth using adaptive FE technique. Int. J. Solids Struct. 49, 2334-2348. Koster, M., Lis, A., Lee, W.J., Kenel, C., Leinenbach, C., 2016. Influence of elasticplastic base material properties on the fatigue and cyclic deformation behavior of brazed steel joints.
57
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Int. J. Fatigue 82, 49-59. Lee, C.-H., Do, V.N.V., Chang, K.-H., 2014. Analysis of uniaxial ratcheting behavior and cyclic mean stress relaxation of a duplex stainless steel. Int. J. Plasticity 62, 17-33. Levitin, V., 1996. High Temperature Strain of Metals and Alloys. WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. Li, Z.M., Wang, Q.G., Luo, A.A., Peng, L.M., Zhang, P., 2015. Fatigue behavior and life prediction of cast magnesium alloys. Mater. Sci. Engng. A 647, 113-126. Macha, E., Nieslony, A., 2012. Critical plane fatigue life models of materials and structures under multiaxial stationary random loading: The state-of-the-art in Opole Research Centre CESTI and directions of future activities. Int. J. Fatigue 39, 95-102. Malcher, L., Pires, F.M.A., C´esar de S` a, 2012. An assessment of isotropic constitutive models for ductile fracture under high and low stress triaxiality. Int. J. Plasticity 30-31, 81-115. Minh, B.L., Maitourmam, M.H., Doquet, V. 2012. A cyclic steady-state method for fatigue crack propagation: Evaluation of plasticity-induced crack closure in 3D. Int. J. Solids Struct. 49, 2301-2313. Mohd, S., Bhuiyan, Md. S., Nie, D.F., Otsuka, Y., Mutoh, Y., 2016. Fatigue strength scatter characteristics of JIS SUS630 stainless steel with duplex SN curve. Int. J. Fatigue 82, 371-378. Moura, M., Consalves, J.P.M. 2014. Cohesive zone model for high-cycle fatigue of adhesively bonded joints under model I loading. Int. J. Solids Struct. 51, 1123-1131. Ottosen, N.S., Rstinmaa, M., Mosler, J. 2016. Framework for non-cohesive interface models at finite displacement jumps and finite strains. J. Mech. Phys. Solids 90, 124-141. Pook, L., 2007. Metal Fatigue. Springer, Berlin. Reed, R.C., 2006. The Superalloys: Fundamentals and Applications. Cambridge Uni. Press, Cambridge. Ritchie, R.O., Gilbert, C.J., McNaney, J.N., 2000. Mechanics and mechanisms of fatigue damage and crack growth in advanced materials. Int. J. Solids Struct. 37, 311-329. Rozumek, D., Macha, E., 2009. A survey of failure criteria and parameters in mixed-mode fatigue crack growth. Mater. Sci. 45, 190-210. Shaw, J.A., Kyriakides, S., 1998. Initiation and propagation of localized deformation in elastoplastic strips under uniaxial tension. Int. J. Plasticity 13, 837-871. Shojaei, A., Voyadjis, G.Z., Tan, P.J., 2013. Viscoplastic constitutive theory for brittle to ductile damage in polycrystalline materials under dynamic loading. Int. J. Plasticity 48, 125-151. Shyam, A., Allison, J.E., Szczepanski, Pollock, T.M., Jones, J.W., 2007. Small fatigue crack growth in metallic materials: A model and its application to engineering alloys. Acta Materialia 55, 6606-6616. Steglich, D., Pirondi, A., Bonora, N., Brocks, W. 2005. Micromechanical modelling of cyclic plasticity incorporating damage. Int. J. Solids Struct. 42, 337-361. Stouffer, D.C., 1996. Inelastic Deformation of Metals. John Wiley & Sons, New York. Stoughton, T.B., Yoon, J.W., 2011. A new approach for failure criterion for sheet metals. Int. J. Plasticity 27, 440-459. Suresh, S., 1998. Fatigue of Materials. 2nd edition. Cambridge University Press, Cambridge. Susmel, L., 2008. The theory of critical distances: a review of its applications in fatigue. Eng. Fract. Mech.75, 17061724. Susmel, L., 2014. Four stress analysis strategies to use the Modified Whler Curve Method to perform the fatigue assessment of weldments subjected to constant and variable amplitude multiaxial fatigue loading. Int. J. Fatigue 67, 38-54.
58
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Taylor, G.I., Quinney, H., 1934. The latent energy remaining in a metal after cold working. Proc. R. Soc. London A143, 307-326. Taylor, G.I., Quinney, H., 1937. The latent heat remaining in a metal after cold working. Proc. R. Soc. London A163, 157-181. Ural, A., Krishnan, V.R., Papoulia, K.D. 2009. A cohesive zone model for fatigue crack growth allowing for crack retardation. Int. J. Solids Struct. 46, 2453-2462. Wang, Z.L., Xiao, H., 2015. A study of metal fatigue failure as inherent features of elastoplastic constitutive equations. In: Holm Altenbach, Tetsuya Matsuda, Dai Okumura (eds.), From Creep Damage Mechanics to Homogenization Methods - A Liber Amicorum to celebrate the brithday of Nobutada Ohno, Advanced Structural Materials Series, pp. 529-540, Springer, Berlin. Wu, J.Y., Cervera, M., 2015. On the equivalence between traction- and stress-based ap- proaches for the modeling of localized failure in solids. J. Mech. Phys. Solids 82, 137-163. Wu, J.Y., Xu, S.L., 2011. An augmented multicrack elastoplastic damage model for tensile cracking. Int. J. Solids Struct. 48, 2511-2528. Xiao, H., 2014a. An explicit, straightforward approach to modeling SMA pseudo-elastic hysteresis. Int. J. Plasticity 53, 228-240. Xiao, H., 2014b. Thermo-coupled elastoplasticity model with asymptotic loss of the material strength. Int. J. Plasticity 63, 211-228. Xiao, H., 2015. A direct, explicit simulation of finite strain multiaxial inelastic behavior of polymeric solids. Int. J. Plasticity 71, 146-169. Xiao, H., Bruhns, O.T., Meyers, A., 1997. Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica 124, 89-105. Xiao, H., Bruhns, O.T., Meyers, A., 2006. Elastoplasticity beyond small deformations. Acta Mechanica 182, 31-111. Xiao, H., Bruhns, O.T., Meyers, A., 2007. Thermodynamic laws and consistent Eulerian formulations of finite elastoplasticity with thermal effects. J. Mech. Phys. Solids 55, 338-365. Xiao, H., Bruhns, O.T., Meyers, A., 2014. Free rate-independent elastoplastic equations. ZAMM-J. Appl. Math. Mech. 94, 461-476. Xiao, H., Wang, X.M., Wang, Z.L., Yin, Z.N., 2016. Explicit, comprehensive modeling of multiaxial finite strain pseudo-elastic SMAs up to failure. Int. J. Solids Struct. 88-89, 215-226. Xu, L.Y., Nie, X., Fan, J.S., Tao, M.X., Ding, R., 2016. Cyclic hardening and softening behavior of the low yield point steel BLY160: Experimental response and constitutive modeling. Int. J. Plasticity 78, 44-63. Zahrouni, H., Potier-Ferry, M., Elasmar, H., Damil, N., 1998. Asymptotic numerical method ´ ements Finis 7, 841-869. for nonlinear constitutive laws. Revue Europ´eenne des El´ Zhang, W.J., 2016. Thermal-mechanical fatigue of single crystal superalloys: Achievements and challenges. Mater. Sci. Eng. A 650, 389-395. Zhu, Y.L., Kang, G.Z., Kan, Q.H., Bruhns, O.T., Liu, Y.J., 2016. Thermo-mechanically coupled cyclic elasto-viscoplastic constitutive model of metals: Theory and application. Int. J. Plasticity 70, 111-152.
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