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Ratcheting behaviour of elasto-plastic thin-walled pipes under internal pressure and subjected to cyclic axial loading
Thin-Walled Structures 93 (2015) 102–111
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Ratcheting behaviour of elasto-plastic thin-walled pipes under internal pressure and subjected to cyclic axial loading Heraldo S. da Costa Mattos a,n, Jadyr M.A. Peres a, Marco Antonio C. Melo a,b a Laboratory of Theoretical and Applied Mechanics, Graduate Program in Mechanical Engineering, Universidade Federal Fluminense, Rua Passo da Pátria 156, 24210-240 Niterói, RJ, Brazil b Eletrobras—Eletronuclear, Rua da Candelária, no 65-61 andar, 20091-020 Rio de Janeiro, RJ, Brazil
art ic l e i nf o
a b s t r a c t
Article history: Received 5 September 2014 Received in revised form 25 February 2015 Accepted 11 March 2015
The present paper is concerned with the analysis of the ratcheting behaviour of elasto-plastic thinwalled pipes under internal pressure and subjected to cyclic axial loading. Understanding the behaviour of this kind of structure at different load levels is of critical importance in a range of engineering applications such as in the design of structural components of power and chemical reactors. Depending on the kinematic hardening, the pipe may exhibit a ratcheting behaviour in the circumferential direction, which leads to a progressive accumulation of deformation. Many different constitutive theories have been proposed to model the kinematic hardening under such kind of loading history. The present paper presents a simple local criterion to indicate whether or not the pipe may exhibit a progressive accumulation of deformation. Such criterion is independent of the choice of the evolution law adopted for the backstress tensor. As an example, a semi-analytic approach using a mixed nonlinear kinematic/ isotropic hardening model is proposed to be used in a preliminary analysis of this kind of structure. & 2015 Published by Elsevier Ltd.
Keywords: Thin-walled pipes Cyclic elasto-plasticity Multial Ratcheting Modelling
1. Introduction When a metallic component is subjected to cycles of mechanical loading beyond the elastic limit many important phenomena can occur, what may lead to structural failure. Different structural situations exist in which this combination of sustained or primary loading and secondary cyclic loading can lead to incremental collapse or what is known as ratcheting. For instance, pressurised metallic tubes under reversed bending [1] or pressurised metallic tubes subjected to cyclic push pull [2]. These structures under such load combinations are known to exhibit continued strain growth in the hoop direction. Understanding the behaviour of this kind of structure at different load levels is of critical importance in a range of engineering applications such as in the design of structural components of power and chemical reactors (primary heat transport system of nuclear power plants, for instance). The reliability of structural integrity prediction depends strongly on the physical adequacy of the elasto-plastic constitutive equations considered in the analysis. Many papers concerned with ratcheting failure mechanisms or with constitutive models for ratcheting have been performed in the last years. Since the classical works of Chaboche (see [3], for instance), most works were concerned with an adequate modelling of the kinematic hardening to improve the description of ratcheting effects and to include a better modelling
n
Corresponding author. Tel./fax: þ 55 21 2629 5585. E-mail address: [email protected] (H.S. da Costa Mattos).
http://dx.doi.org/10.1016/j.tws.2015.03.011 0263-8231/& 2015 Published by Elsevier Ltd.
of multiaxial behavior [4–7]. In [4], a complete model was developed, including isotropic hardening, to describe the ratcheting behaviour of 316L stainless steel at room temperature. In this study, one particular kinematic hardening rule was selected aiming at describing both the shape of the normal cyclic stress–strain relations and the ratcheting results. The main concern, as discussed in [5], was to propose rules that induce much less accumulation of uniaxial and multiaxial ratchetting strains than the Armstrong and Frederick rule. In [5], kinematic hardening rules formulated in a hardening/dynamic recovery format were examined for simulating racheting behaviour. These rules, characterized by decomposition of the kinematic hardening variable into components, are based on the assumption that each component has a critical state for its dynamic recovery to be fully activated. In the paper by Abdel Karim and Ohno [6], an alternative kinematic hardening model was proposed for simulating the steady-state in ratcheting within the framework of the strain hardening and dynamic recovery format. The model was formulated to have two kinds of dynamic recovery terms, which operate at all times and only in a critical state, respectively. The model is able of representing appropriately the steady-state in ratcheting under multiaxial and uniaxial cyclic loading. In [7], seven cyclic plasticity models for structural ratcheting response simulations were analysed: bilinear (Prager), multilinear (Besseling), Chaboche, Ohno–Wang, Abdel Karim–Ohno, modified Chaboche (Bari and Hassan) and modified Ohno–Wang (Chen and Jiao). Apparently, none of the models evaluated perform satisfactorily in simulating the straight pipe diameter change and circumferential strain ratcheting responses.
H.S. da Costa Mattos et al. / Thin-Walled Structures 93 (2015) 102–111
In particular, many experimental and numerical studies on ratcheting induced by reversed bending in straight pipes and elbows have been performed in the last years [1,8–10], and a detailed review can be found in [1,10]. In [1], ratcheting studies were carried out on Type 304LN stainless steel straight pipes and elbows subjected to steady internal pressure and cyclic bending load. Ratcheting behaviour of straight pipes and elbows were compared and it was generally inferred that ratcheting was more pronounced in straight pipes than in elbows. Shariati et al. [8] investigated the softening and ratcheting behaviours of SS316L cantilevered cylindrical shells under cyclic bending load. Accumulation of the plastic strain or ratcheting phenomenon occurred under force-control loading with nonzero mean force. It was verified that an increase of the mean force induces an increase in the accumulation of the plastic deformation and its rate. Plastic mechanism analyses of circular tubular members under cyclic loading were performed in [9]. This paper provides new methods of analyses for circular hollow sections subjected to a constant amplitude cyclic pure bending and a large axial compression–tension cycle. The local buckling analysis was performed using a rigid plastic mechanism analysis. Although ratcheting based criteria for integrity assessment of pressurized piping under severe loading can be found in codes [11], so far most constitutive models are unable to describe the complex nonlinear cyclic behaviour observed in this kind of problem. Besides, the analysis of the stress and strain fields usually requires the use of complex finite element codes, what can be a shortcoming for the effective use of these theories by designers. The present paper is concerned with the analysis of the coupled effect of kinematic and isotropic hardening on the multiaxial ratcheting behaviour of elasto-plastic thin-walled straight pipes under internal pressure and subjected to cyclic axial loading. An easily employable framework to be used in the analysis of structures of this type with complex material behaviour, as elasto-plasticity with both isotropic and kinematic hardening, is presented. A simple and efficient algorithm for approximating the solution is described. Simulations of AISI 316L steel and AU4G aluminium alloy pressure vessels at room temperature subjected to multiaxial loadings are presented and analysed. It is shown that cyclic behaviour is strongly dependent on the kinematic hardening, but also on the isotropic hardening. The main result of the paper is the proposition of a simple condition (involving the ratio of the circumferential component of the backstress tensor and the auxiliary variable related to the isotropic hardening) to indicate when the pipe may exhibit a ratcheting behaviour, which leads to a progressive accumulation of deformation. Such kind of criterion is independent of the evolution law adopted for the backstresss tensor. Therefore, it is important to emphasize that the paper is not focused on the evaluation of the physical adequacy of different constitutive models for predicting ratcheting behaviour. Although the Marquis–Chaboche elasto-plastic constitutive Eqs. (11) and (12) have been considered in the present study, such a condition for ratcheting to arise can be applied to any cyclic plasticity model with both isotropic and kinematic hardening.
2. Summary of the elasto-plastic constitutive equations The following set of elasto-plastic constitutive equations proposed by Marquis [12] is adequate to model the cyclic inelastic behaviour of metallic material at room temperature. A further discussion about these equations can be found in [13]. In the framework of small deformations and isothermal processes, besides the stress tensor σ and the strain tensor ε , the following auxiliary variables are also considered: the plastic strain tensor ε p , the accumulated plastic strain p and two other auxiliary variables ðX ; YÞ, respectively related to the kinematic hardening and to the isotropic hardening. A complete set of elasto-plastic constitutive equations is
103
given by
σ
νE E trðε ε p Þ 1 þ ðε ε p Þ ¼ ð1 þ νÞð1 2νÞ ð1 þ νÞ
ε_ p ¼
3 S X p_ 2J
ð1Þ
ð2Þ
2 X_ ¼ aε_ p bX p_ 3
ð3Þ
Y_ ¼ v2 ðv1 þ σ y YÞp_
ð4Þ
p_ Z 0;
F ¼ ðJ YÞ Z 0;
_ ¼0 pF
ð5Þ
with ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 3 X 3 u u3 X 3 t S X U S X ¼ ðS X ij Þ2 J¼ 2 2 i ¼ 1 j ¼ 1 ij
ð6Þ
where E is the young modulus, ν the Poisson’s ratio and σ y , v1 , v2 , a, b are positive constants that characterize the plastic behaviour of the material and they can be obtained from a simple tension-compression test [14]. 1 is the identity tensor, and trðA Þ is the trace of an arbitrary tensor A . S is the deviatoric stress given by 1 trðσ Þ1 S¼ σ 3
ð7Þ
J is the equivalent von Mises stress. X is an auxiliary variable related to the kinematic hardening (eventually called the backstress tensor) and it is introduced to account for the anisotropy introduced by the plastic deformation. Y is an auxiliary variable related to the isotropic hardening and models how the yield stress varies with plastic deformation. p is usually called the accumulated plastic strain and p_ can be interpreted as a Lagrange multiplier associated to the constraint F r 0. Function F characterizes the elasticity domain and the plastic yielding surface. From Eq. (2) it is possible to affirm that rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p p _p ¼ ε_ U ε_ ð8Þ 3 and, therefore, Z pðtÞ ¼ pðt ¼ 0Þ þ
t t¼0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 p ε_ ðζ Þ U ε_ p ðζ Þ dζ 3
ð9Þ
If F ¼ J Y o0, it comes that J oY. Hence, from the condition _ ¼ 0 in (5), it is possible to conclude that p_ ¼ 0. If p_ a 0, from the pF _ ¼ 0 it also comes that, necessarily, F ¼ 0. Besides, from condition pF Eqs. (2)–(4) it comes that, in this case, ε_ p a , ̇ a and Y_ a0. Therefore, the elasto-plastic material is characterized by an elastic domain in the stress space where yielding doesn’t occur (ε_ p ¼ ̇ ¼ ,p_ ¼ Y_ ¼ 0 if F o 0). Noting the eigenvalues of S and X , respectively by fS1 ; S2 ; S3 g and fX 1 ; X 2 ; X 3 g, the elastic domain can be represented in the space of the principal directions of the deviatoric stress pffiffiffiffiffiffiffiffias a sphere centred at the point fX 1 ; X 2 ; X 3 g with radius R ¼ 2=3Y. Generally the following initial conditions are used for a “virgin” material pðt ¼ 0Þ ¼ 0;
ε p ðt ¼ 0Þ ¼ X ðt ¼ 0Þ ¼ ;
Yðt ¼ 0Þ ¼ σ y
ð10Þ
From now on, the initial conditions (10) are assumed to hold in the analysis. It is also important to remark that the constitutive equations with conditions (10) and definition (7) imply that the principal directions the stress tensor, of the deviatoric stress tensor, of the plastic strain tensor and of the backstress tensor are the same.
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Eqs. (1)–(3) have the following form if expressed using the basis of the principal directions 3 X νE E ðεi εpi Þ ðεi εpi Þ þ ð1 þ νÞð1 2νÞ i ¼ 1 ð1 þ νÞ
ð11Þ
ε_ pi ¼ ðSi X i Þp_
3 2J
ð12Þ
2 X_ i ¼ aε_ pi bX i p_ 3
ð13Þ
σi ¼
with Si ði ¼ 1; 2or3Þ, εpi ði ¼ 1; 2or3Þ and X i ði ¼ 1; 2or3Þ being the principal components (eigenvalues) respectively of S , ε p and X . In this case, from Eq. (7), it is possible to obtain the following alternative expression qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J ¼ 3½ðSi X i Þ2 þ ðSi X i ÞðSj X j Þ þ ðSj X j Þ2 8 ði; jÞ; with a i j ð14Þ Using the evolution laws (2) and (3) and considering the initial conditions (10), it is also possible to verify that the following relation always holds (provided Sj a 0, εpj a 0 and X j a 0) S i εi X i ¼ ¼ 8 Sj εpj X j p
ði; j ¼ 1; 2; or 3Þ
ð15Þ
3. Thin-walled elasto-plastic cylinder under constant internal pressure and cyclic axial loading The goal of this section is to summarize the procedure to analyse a particular two-dimensional state of stress obtained from the general model. In this two-dimensional abstract context, the tensor σ ðx; tÞ at a given point x and a given time instant t is supposed to be given by the following expressions (in cylindrical coordinates) 2 3 σr 0 0 7 σ ¼6 4 0 σ θ 0 5 with σ r ¼ 0; σ θ ¼ αðtÞσ y ; σ z ¼ βðtÞσ y : 0 0 σz ð16Þ
σ r is the radial stress component, σ θ the circumferential stress component and σ z the axial stress component. All other components are considered to be equal to zero. α and β are arbitrary scalar functions of time such that αðt ¼ 0Þ ¼ 0 and βðt ¼ 0Þ ¼ 0. A theoretical analysis allows explaining the different possible cyclic plastic behaviours in if such kind of (local) stress history is supposed to occur. This analysis includes the case of thin-walled elasto-plastic cylinders under constant internal pressure and subjected to cyclic axial loading. In this case, the expressions relating the external loading and geometry with the axial and circumferential stress component σ z and may vary depending on the nature of the cyclic axial loading. Nevertheless, no matter the nature of the axial loading, the stress tensor at a given point will always have the abstract form presented in Eq. (16), since αðtÞ and βðtÞ are arbitrary scalar functions. For instance, in the case of an elasto-plastic cylinder with internal radius r i and thickness e submitted to an internal pressure P, the circumferential stress component (or hoop stress) σ θ and the circumferential strain component εθ are classically approximated in the framework of membrane theory of shells of revolution by the following expression (provided the internal radius r i and the thickness e are such that r i 4 10e and ovalisation is negligible) Pr σθ ¼ i; e
εθ ¼
Δr ri
ð17Þ
where Δr is the variation of the internal radius. To simplify the
analysis, the cylinder is assumed to be open-ended (studies analysing the effect of the closed ends in the cylinder integrity can be found in [15,16]). In a cyclic push and pull with axial force F t , the axial stress is homogeneous along a cross-section and approximated by the following expression: σ z F t =ð2π r i eÞ. In real straight pipes in operation, the cyclic axial stress component can be caused by a non-homogeneous temperature variation or by a bending moment [10]. The present paper is not concerned with the analysis of the equilibrium equations in these cases and the expression of both the axial stress component σ z and may vary depending on the material, the geometry and the loading history. In the case of cyclic bending, the axial stress is not homogeneous and complex expressions should be adopted. In the case of non isothermal problems, the fully coupled constitutive equations must be considered (see, [17], for instance). Nevertheless, if ovalisation is negligible, the hoop stress and the axial stress in a thin walled pipe can be considered uncoupled since the expressions presented in Eq. (17) are still reasonable. Besides, as it will be verified in the sequence, the particular expression relating the axial stress with material, geometry and external loading does not affect the basic goal of the paper, which is to identify a simple criterion for ratcheting to arise when the circumferential component of the backstress tensor exceed a given isotropic hardening parameter. No matter the pipe material and geometry, α must be a constant value in Eq. (16) for a fixed internal pressure of the pipe. Besides, since the goal is to study the ratcheting induced by a periodic axial loading, the axial stress is assumed to be a periodic function of time at a given material point and, consequently, β must be a periodic function of time. Since plasticity is rate-independent, the frequency of the axial loading does not affect the stress–strain behaviour and only maximum and minimum stress levels (and, therefore, the maximum and minimum values of the function β) are important in the analysis. Taking into account assumption (16) and using definition (7), it is possible to conclude that the radial, circumferential and axial components Sr , Sθ , Sz of the deviatoric stress tensor S are its only non-zero components 1 1 Sr ¼ ðσ θ þ σ z Þ ¼ ðαðtÞ þ βðtÞÞσ y 3 3
ð18:1Þ
1 1 Sθ ¼ ð2σ θ σ z Þ ¼ ð2αðtÞ β ðtÞÞσ y 3 3
ð18:2Þ
1 1 Sz ¼ ð2σ z σ θ Þ ¼ ð2β ðtÞ αðtÞÞσ y 3 3
ð18:3Þ
It is important to note that the principal components of the deviatoric stress and of the plastic deformation are not independent, which allows introducing additional simplifications in the equations. From Eqs. (15) and (18.1)–(18.3), it comes that (provided ð2α β Þ a 0) Sr εpr X r ðα þ β Þ ¼ ¼ ¼ ; Sθ εpθ X θ ðβ 2αÞ
Sz εpz X z ð2β αÞ ¼ ¼ ¼ Sθ εpθ X θ ð2α β Þ
ð19Þ
Using definition (14) and Eq. (19) it is possible to obtain the following equivalent expressions for the von Mises equivalent stress J¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3½ðSθ X θ Þ2 þ ðSθ X θ ÞðSz X z Þ þ ðSz X z Þ2
J ¼ Sθ X θ ηðtÞ
with
ð20:1Þ
ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u " 2 # u ð2β αÞ ð2β αÞ t þ ηðtÞ ¼ 3 1 þ ð2α βÞ ð2α βÞ ð20:2Þ
H.S. da Costa Mattos et al. / Thin-Walled Structures 93 (2015) 102–111
Thus, the behaviour in the circumferential direction is governed by the following system of equations 1 Sθ ¼ ð2αðtÞ βðtÞÞσ y ; 3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u " 2 # u ð2βðtÞ αðtÞÞ ð2βðtÞ αðtÞÞ ηðtÞ ¼ t3 1 þ þ ð2αðtÞ βðtÞÞ ð2αðtÞ βðtÞÞ
ð21:1Þ 3 3η ε_ pθ ¼ ðSθ X θ Þp_ ¼ Sgp_ 2J 2
( with
Sg ¼
ifðSθ X θ Þ Z 0
1;
otherwise
3ðηÞn þ 1 ðSgÞn Δp 2
ð25:1Þ
2 3
ð25:2Þ
ΔX θ ¼ aΔεpθ bðX θ Þn Δp ¼ ½aðηÞn þ 1 ðSgÞn bðX θ Þn Δp
ð21:3Þ
Y_ ¼ v2 ðv1 þ σ y YÞp_
ð21:4Þ _ ¼0 pF
ð21:5Þ
From Eqs. (18.2) and (21.2) it is possible to verify that the axial stress component affects the plastic strain in the circumferential direction. In the case of a tube under constant internal pressure, a significant variation of the axial stress component may cause a plastic flow in the circumferential direction.
4. Solution algorithm Eqs. (21.1)–(21.4) define a nonlinear system of ordinary differential equations coupled with the constraints (21.5). The basic mathematical problem considered in this paper is formed by Eqs. (21.1)–(21.4) together with the following initial conditions ð22Þ
The solution algorithm used to obtain a numerical approximation for this initial value problem is based on the operator split method [18–20]. The objective of this technique is to take advantage of some additive decomposition properties of the mathematical operator to solve a sequence of simpler problems instead of a unique complex problem. The approximation techniques based on the operator split method and associated product formula algorithms have been applied in various areas of Continuum Mechanics and, in particular, in elastoplasticity and elasto-viscoplasticity. 4.1. Time discretization Considering a sequence of time instants (since the elasto-plastic equations are rate independent, this definition is only used to establish a sequence of events) 0 o t 1 ot 2 o ⋯ o t n
ð25:3Þ
9 F n þ 1 ¼ ðSθ Þn þ 1 ðX θ Þn þ 1 ηn þ 1 ðYÞn þ 1 ) > > = F n þ 1 ðSθ Þn þ 1 ðX θ Þn ΔX θ ðηÞn þ 1 ðYÞn ΔY n )
> > ðSθ Þn þ 1 ðX θ Þn ðaηn þ 1 ðSgÞn bðX θ Þn ÞΔp ηn þ 1 ðYÞn v2 ðv1 ðYÞn þ σ y Þ Δp ;
ð26Þ and an analytical approximation of the increment Δp can be obtained using the condition ðFÞn þ 1 ¼ 0 when Δp a0: ðFÞn þ 1 ½ðSθ Þn þ 1 ðX θ Þn ðað
ηÞn þ 1 ðSgÞn bðX θ Þn ÞΔpðηÞn þ 1 ðSgÞn ðYÞn v2 ðv1 ðYÞn þ σ y Þ Δp
ð23Þ
and using the following notation for any function yðtÞ: yðt n Þ ¼ yn The main idea of the solution algorithm is to explore the fact that all time rates of the variables εpθ , X θ and Y, defined in Eqs. _ hence, the following (21.2)–(21.4), are linear functions of p, approximation can be considered
ð27Þ
If ðFÞn þ 1 ¼ 0, then ½ðSθ Þn þ 1 ðX θ Þn ηn þ 1 ðSgÞn Y n ¼ faη2n þ 1 bðX θ Þn ηn þ 1 ðSgÞn þ v2 ðv1 ðYÞn þ σ y Þg
Δp ) Δp
pðt ¼ 0Þ ¼ εpθ ðt ¼ 0Þ ¼ X θ ðt ¼ 0Þ ¼ 0; Yðt ¼ 0Þ ¼ σ y
The plastic function F can be expressed in terms of Δp:
2 X_ θ ¼ aε_ pθ bX θ p_ 3
F ¼ ð Sθ X θ ηðtÞÞ Y Z0;
Δεpθ ¼
ΔY ¼ v2 ðv1 ðYÞn þ σ y Þ Δp
ð21:2Þ
p_ Z 0;
with
1;
105
η
ðSθ Þn þ 1 ðX θ Þn ð Þn þ 1 ðSgÞn ðYÞn
Þ2n þ 1 bðX θ Þn ð Þn þ 1 ðSgÞn þ v2 v1 þ y ðYÞn
aðη
η
σ
ð28Þ
Once the increment Δp is known, all the variables can be obtained at the instant t n þ 1 using Eqs. (24.1)–(24.4), (25.1)–(25.3). The procedure can be improved using an iterative solution technique that uses this Euler scheme coupled with a projection scheme. The projection technique assures that F o δ at any time step, with δ 40 being the maximum admissible error in the computation of F in the plastic steps. Such a technique allows the computation of accurate results even if very large steps are used, what is crucial if the simulation of a great number of cycles is necessary.
4.2. Basic solution algorithm The basic solution algorithm is defined as follows (i) n ¼ 0 (ii) ðηÞn þ 1 , ðεpθ Þn , ðpÞn , ðX θ Þn , ðYÞn are known and 1 ðSθ Þn þ 1 ¼ ð2ðαÞn þ 1 ðβÞn þ 1 Þσ y ; ðηÞn þ 1 3vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u " 2 # u ð2ðβ Þn þ 1 ðαÞn þ 1 Þ ð2ðβÞn þ 1 ðαÞn þ 1 Þ t þ ¼ 3 1þ ð2ðαÞn þ 1 β ðtÞÞ ð2ðαÞn þ 1 ðβÞn þ 1 Þ (iii) Compute the auxiliary variables εpθ , p, X θ and Y: p p εθ ¼ ðεθ Þn , p ¼ ðpÞn , X θ ¼ ðX θ Þn , Y ¼ ðYÞn , ITER ¼ 1 (iv) ðSθ Þn þ 1 X ðηÞn þ 1 Y o δ? YES ) The step is elastic: ðεpθ Þn þ 1 ¼ εpθ , ðX θ Þn þ 1 ¼ X, Y n þ 1 ¼ Y, pn þ 1 ¼ p, n ¼ n þ 1 If n 4 NMAX, stop ) maximum number of time steps Go to (ii) NO ) The step is plastic:
ðεpθ Þn þ 1 ¼ ðεpθ Þn þ Δεpθ
ð24:1Þ
ðX θ Þn þ 1 ¼ ðX θ Þn þ ΔX θ
ð24:2Þ
ITER ¼ ITER þ 1
ðYÞn þ 1 ¼ ðYÞn þ ΔY
ð24:3Þ
ðpÞn þ 1 ¼ ðpÞn þ Δp
ð24:4Þ
if ITER 4 ICONT, stop ) Maximum number of iterations per time step go to (vi)
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(v) Computation of the increment Δp:
Y ¼ Y þ ΔY
ðSθ Þn þ 1 X θ ðηÞn þ 1 Sg Y n ðS Þ Xθ ) Δp ¼ Sg ¼ θ n þ 1
2 ðSθ Þ X að η Þ θ nþ1 n þ 1 bX θ ðηÞn þ 1 Sg þ v2 v1 þ σ y Y
p ¼ p þ Δp (viii) Go to (iv)
(vi) Computation of the increments Δεpθ , ΔX θ , ΔY: 3η Δεpθ ¼ n þ 1 Sg ΔpΔX θ ¼ ðaηn þ 1 Sg bX θ ÞΔp 2
ΔY ¼ v2 ðv1 Y þ σ y Þ Δp (vii) Update the variables εpθ , X θ , Y, p
Good results
are obtained taking δ ¼ σ y =100 , ICONT ¼ 5 and ΔSθ r σ y =100 5. Results and discussion
εθ ¼ εθ þ Δεθ
In order to better understand when ratcheting may occur in an arbitrary tube, it is considered an AISI 316L steel at room temperature. The material parameters in this case are
X θ ¼ X θ þ ΔX θ
σ y ¼ 300 MPa;
p
p
p
E ¼ 196 GPa;
ν ¼ 0:3;
Fig. 1. Loading histories considering Eq. (16).
Fig. 2. Curves Sθ εpθ for different amplitudes of the cyclic axial loading. AISI 316L steel at room temperature.
H.S. da Costa Mattos et al. / Thin-Walled Structures 93 (2015) 102–111
a ¼ 45 GPa;
b ¼ 60;
v1 ¼ 0;
v2 ¼ 0
ð29Þ
It can be observed that, for this kind of material, the hardening is purely kinematic (since v1 ¼ 0 and v2 ¼ 0). The goal is to demonstrate that the ratcheting phenomenon is essentially due to the kinematic hardening and thus it is chosen a material which does not present isotropic hardening (Y ¼ σ y ). In the analysis, the loading histories depicted in Fig. 1 (the functions αðtÞ and β ðtÞ, see (16)) are taken into account. Since the elasto-plastic constitutive are rate independent, the time is not important and it is only used to establish a sequence of events. Thus, t n ¼ ðt=ΔtÞ, where Δt is an arbitrary time interval. The parameters A and B are chosen in order to allow understanding the role of the stress components in the ratcheting behaviour. Parameter α is related to the internal pressure. Combining the expressions in (16) and (17), we have σ e Pr ð30Þ σ θ ¼ αðt n Þσ y ¼ i ) P ¼ α y ri e The maximum hoop stress is given by ðσ θ Þmax ¼ Aσ y ) P ¼ Aðσ y e=r i Þ (see Eq. (16)). Therefore, if A ¼ 1 then the maximum hoop stress is such that ðσ θ Þmax ¼ σ y and the internal pressure is given by P ¼ Aðσ y e=r i Þ. B 41 is the amplitude of the periodic function βðt n Þ ¼ πtn B sin 200 , with σ z ¼ βðt n Þσ y . Therefore, the maximum and minimum axial stress component are ðσ z Þmax ¼ Bσ y and ðσ z Þmin ¼ Bσ y . Significant ratcheting in the circumferential direction occurs for different values of B as it is shown in Figs. 2 and 3. Parameter B is related to the amplitude of the cyclic axial loading. Therefore, different amplitudes in the axial loading induce different ratcheting behaviour in circumferential direction. Since εpθ ¼ ðΔrÞp =r i ,
107
where ðΔrÞp is the plastic or irreversible variation of the internal radius, it may be concluded that the tube diameter will increase incrementally with the axial cyclic loading. Fig. 4 presents the evolution of the accumulated plastic strain for A ¼ 1 and different values of the parameter B. Under cyclic axial loading and unloading, the pipe exhibits hysteresis (a phase lag), which leads to a dissipation of mechanical energy. The progressive accumulation of plastic deformation can be explained by the strong kinematic hardening observed in this material. The analysis is very similar to the one performed in [21] for progressive accumulation of deformation observed in cyclic tension tests performed in epoxy polymers. The elastic region is the domain of stresses in which there is no plastic flow. As commented before, the elastic domain can be
Fig. 4. Evolution of p for different amplitudes of the cyclic axial loading. AISI 316L steel at room temperature.
Fig. 3. Evolution of εpθ for different amplitudes of the cyclic axial loading. AISI 316L steel at room temperature.
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Fig. 5. Influence of kinematic hardening in the accumulation of plastic deformation.
Fig. 6. Curves Sθ εpθ for different amplitudes of the cyclic axial loading. AU4G aluminium alloy at room temperature.
H.S. da Costa Mattos et al. / Thin-Walled Structures 93 (2015) 102–111
represented in the space of the principal directions of the deviatoric stress pffiffiffiffiffiffiffiffi as a sphere centred at the point fX 1 ; X 2 ; X 3 g with radius R ¼ 2=3Y. This domain varies with the plastic deformation process. In case studied in this paper, from the particular condition F ¼ Sθ X θ η Y o 0 it possible to define the elastic segment as the set of all possible Sθ such that Sθ X θ ðY=ηÞ o 0. The elastic segment has a length (2Y=η) and it is centred on X θ . The elastic segment also varies with the plastic deformation process. If ðY=ηÞ 4 X θ hysteresis with consequent accumulation of cyclic plastic deformation does not occur. For the pipe material under the prescribed loading conditions considered in this paper, the kinematic hardening variable can be greater that the isotopic hardening variable ðY=ηÞ and a negative plastic strain rate may occur even with a positive stress (ε_ pθ o 0 if S_ θ o0 even when Sθ 4 0). This fact explains why hysteresis and a progressive accumulation of deformation occur in a cyclic load–unload test as shown in Fig. 5. In the case of a material that presents both kinematic and isotropic hardening, the reasoning is similar and ratcheting occurs if the kinematic hardening parameter (the circumferential component of the backstress tensor) X θ exceeds the isotropic hardening parameter ðY=ηÞ. Therefore, a simple criterion involving the ratio of the circumferential component of the backstress tensor and the auxiliary variable related to the isotropic hardening indicates if the pipe may exhibit a ratcheting behaviour, which leads to a progressive accumulation of deformation. Ratcheting may occur if X θ 4 ðY=ηÞ ) ðX θ =YÞ o
1
η
109
different values of the parameter B. The material parameters in this case are
σ y ¼ 170 MPa; E ¼ 72 GPa; ν ¼ 0:32; a ¼ 22:5 GPa; b ¼ 125; v1 ¼ 100; v2 ¼ 1:6:
ð32Þ
As it can be verified, despite the isotropic hardening, the behaviour is similar than in the previous case. The consideration of a range of hoop stresses is perfectly possible within the model. This would permit a ratchet diagram to be developed—for any given set of hardening parameters. Fig. 9 shows the results of simulations considering the aluminium alloy with B ¼ 1:5 and different values of the parameter A. Significant ratcheting in the circumferential direction also occurs even when A o 1 ) σ θ o σ y . It is interesting to remark that, in all cases, after a few cycles, the increment of the accumulated plastic strain per cycle δp tends to a constant value (Fig. 10). Such a plastic increment per cycle can
ð31Þ
Figs. 6–8 show the results of simulations considering an AU4G aluminium alloy at room temperature. The loading history considering Eq. (16) is the same presented in Fig. 1, with A ¼ 1 and
Fig. 8. Evolution of p for different amplitudes of the cyclic axial loading. AU4G aluminium alloy at room temperature.
Fig. 7. Evolution of εpθ for different amplitudes of the cyclic axial loading. AU4G aluminium alloy at room temperature.
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Fig. 9. Curves Sθ εpθ for different amplitudes of the hoop stress. AU4G aluminium alloy at room temperature.
Fig. 10. Increment of the accumulated plastic strain per cycle δp.
be easily obtained using the proposed numerical procedure. A simple preliminary failure criterion would be to assume that the probability of failure is higher when the accumulated plastic strain is greater than a given limit, i.e. when p ðnδpÞ 4pmax .
6. Conclusion Metallic pipes under constant pressure and cyclic axial loading beyond the elastic limit may exhibit continued strain growth in the hoop direction, what is known as ratcheting, which leads to the accumulation of cyclic plastic deformation. The simple elastoplastic analysis presented in this paper allows explaining this kind of phenomenon. With an adequate cmanipulation of variables, the multiaxial problem can be reduced to the analysis in the
circumferential direction. It is shown that cyclic behaviour is strongly dependent on the kinematic hardening, but also on the isotropic hardening. A simple condition involving the ratio of the circumferential component of the backstress tensor and the auxiliary variable related to the isotropic hardening allows indicating when the pipe may exhibit a ratcheting behaviour. Although the Marquis–Chaboche elasto-plastic constitutive equations have been considered in the present study, the condition for ratcheting to arise is valid for any cyclic plasticity model with both isotropic and kinematic hardening, provided Eq. (15) holds. Besides, the solution algorithm can be easily adapted to other kinematic hardening rules provided the increment of the circumferential component of the backstress tensor ΔX θ can be expressed as a function of the plastic strain increment Δεpθ of the accumulated plastic strain increment Δp: ΔX θ ¼ f ðΔεpθ ; ΔpÞ. In this case, Eq. (25.2) would be replaced by the alternative expression. References [1] Vishnuvardhan S, Raghava G, Gandhi P, Saravanan M, Goyal S, Arora P, Gupta SK, Bhasin V. Ratcheting failure of pressurised straight pipes and elbows under reversed bending. Int J Press Vessels Pip 2013;105-106:79–89. [2] Kulkarni SC, Desai YM, Kant T, Reddy GR, Parulekar Y, Vaze KK. Uniaxial and biaxial ratchetting study of SA333 Gr. 6 steel at room temperature. Int J Press Vessels Pip 2003;80:179–85. [3] Chaboche JL. Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int J Plast 1989;5:247–302. [4] Chaboche JL. On some modifications of kinematic hardening to improve the description of ratcheting effects. Int J Plast 1991;7:661–78. [5] Ohno N, Wang JD. Kinematic hardening rules with critical state of dynamic recovery, Part I: Formulations and basic features for ratcheting behavior. Int J Plast 1993;9:375–90. [6] Abdel Karim M, Ohno N. Kinematic hardening model suitable for ratcheting with steady-state. Int J Plast 2000;16:225–40.
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[7] Rahman SM, Hassan T, Corona E. Evaluation of cyclic plasticity models in ratcheting simulation of straight pipes under cyclic bending and steady internal pressure. Int J Plast 2008;24:1756–91. [8] Shariati M, Kolasangiani K, Norouzi G, Shahnavaz A. Experimental study of SS316L cantilevered cylindrical shells under cyclic bending load. Thin Walled Struct 2014;82:124–31. [9] Elchalakani M. Plastic mechanism analyses of circular tubular members under cyclic loading. Thin Walled Struct 2007;45:1044–57. [10] Xiaohui Chen, Chen Xu, Dunji Yu, Bingjun Gao. Recent progresses in experimental investigation and finite element analysis of ratcheting in pressurized piping. Int J Press Vessels Pip 2013;101:113–42. [11] ASME. Boiler and pressure vessel code, section III, divison I, subsection NB, USA: American Society of Mechanical Engineers; 1998. [12] Marquis D. Modelisation et identification de l’ecrouissage anisotrope des metaux. These 3ème Cycle: Université Pierre et Marie Curie-Paris VI; 1979. [13] Lemaitre J, Chaboche JL. Mechanics of solid materials. London: Cambridge University Press; 1990.
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[14] da Costa Mattos HS, Soares Filho PF. Approximate local elasto-plastic solution for notched plates undergoing cyclic tensile loading. Mater Des 2010;31:4336–47. [15] da Costa Mattos HS, Paim LM, Reis JML. Analysis of burst tests and long-term hydrostatic tests in produced water pipelines. Eng Fail Anal 2012;22:128–40. [16] da Costa Mattos H, Silva ML. Failure pressure estimations for corroded pipelines. Mater Sci Forum 2013;758:65–76. [17] da Costa-Mattos H, Pacheco PMCL. Non-isothermal low-cycle fatigue analysis of elasto-viscoplastic materials. Mech Res Commun 2009;36:428–36. [18] Ames WF. Numerical methods for partial differential equations. Computer science and scientific computing. 3rd ed.Academic Press Inc; 1992. [19] Farlow SJ. Partial differential equations for scientists & engineers. John Wiley & Sons; 1982. [20] Ciarlet PG, Lions JL. Handbook of numerical analysis. North-Holland Elsevier Science 1990. [21] da Costa Mattos HS, Martins SA. Plastic behaviour of an epoxy polymer under cyclic tension. Polym Test 2013;32:1–8.
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Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Ratcheting behaviour of elasto-plastic thin-walled pipes under internal pressure and subjected to cyclic axial loading Heraldo S. da Costa Mattos a,n, Jadyr M.A. Peres a, Marco Antonio C. Melo a,b a Laboratory of Theoretical and Applied Mechanics, Graduate Program in Mechanical Engineering, Universidade Federal Fluminense, Rua Passo da Pátria 156, 24210-240 Niterói, RJ, Brazil b Eletrobras—Eletronuclear, Rua da Candelária, no 65-61 andar, 20091-020 Rio de Janeiro, RJ, Brazil
art ic l e i nf o
a b s t r a c t
Article history: Received 5 September 2014 Received in revised form 25 February 2015 Accepted 11 March 2015
The present paper is concerned with the analysis of the ratcheting behaviour of elasto-plastic thinwalled pipes under internal pressure and subjected to cyclic axial loading. Understanding the behaviour of this kind of structure at different load levels is of critical importance in a range of engineering applications such as in the design of structural components of power and chemical reactors. Depending on the kinematic hardening, the pipe may exhibit a ratcheting behaviour in the circumferential direction, which leads to a progressive accumulation of deformation. Many different constitutive theories have been proposed to model the kinematic hardening under such kind of loading history. The present paper presents a simple local criterion to indicate whether or not the pipe may exhibit a progressive accumulation of deformation. Such criterion is independent of the choice of the evolution law adopted for the backstress tensor. As an example, a semi-analytic approach using a mixed nonlinear kinematic/ isotropic hardening model is proposed to be used in a preliminary analysis of this kind of structure. & 2015 Published by Elsevier Ltd.
Keywords: Thin-walled pipes Cyclic elasto-plasticity Multial Ratcheting Modelling
1. Introduction When a metallic component is subjected to cycles of mechanical loading beyond the elastic limit many important phenomena can occur, what may lead to structural failure. Different structural situations exist in which this combination of sustained or primary loading and secondary cyclic loading can lead to incremental collapse or what is known as ratcheting. For instance, pressurised metallic tubes under reversed bending [1] or pressurised metallic tubes subjected to cyclic push pull [2]. These structures under such load combinations are known to exhibit continued strain growth in the hoop direction. Understanding the behaviour of this kind of structure at different load levels is of critical importance in a range of engineering applications such as in the design of structural components of power and chemical reactors (primary heat transport system of nuclear power plants, for instance). The reliability of structural integrity prediction depends strongly on the physical adequacy of the elasto-plastic constitutive equations considered in the analysis. Many papers concerned with ratcheting failure mechanisms or with constitutive models for ratcheting have been performed in the last years. Since the classical works of Chaboche (see [3], for instance), most works were concerned with an adequate modelling of the kinematic hardening to improve the description of ratcheting effects and to include a better modelling
n
Corresponding author. Tel./fax: þ 55 21 2629 5585. E-mail address: [email protected] (H.S. da Costa Mattos).
http://dx.doi.org/10.1016/j.tws.2015.03.011 0263-8231/& 2015 Published by Elsevier Ltd.
of multiaxial behavior [4–7]. In [4], a complete model was developed, including isotropic hardening, to describe the ratcheting behaviour of 316L stainless steel at room temperature. In this study, one particular kinematic hardening rule was selected aiming at describing both the shape of the normal cyclic stress–strain relations and the ratcheting results. The main concern, as discussed in [5], was to propose rules that induce much less accumulation of uniaxial and multiaxial ratchetting strains than the Armstrong and Frederick rule. In [5], kinematic hardening rules formulated in a hardening/dynamic recovery format were examined for simulating racheting behaviour. These rules, characterized by decomposition of the kinematic hardening variable into components, are based on the assumption that each component has a critical state for its dynamic recovery to be fully activated. In the paper by Abdel Karim and Ohno [6], an alternative kinematic hardening model was proposed for simulating the steady-state in ratcheting within the framework of the strain hardening and dynamic recovery format. The model was formulated to have two kinds of dynamic recovery terms, which operate at all times and only in a critical state, respectively. The model is able of representing appropriately the steady-state in ratcheting under multiaxial and uniaxial cyclic loading. In [7], seven cyclic plasticity models for structural ratcheting response simulations were analysed: bilinear (Prager), multilinear (Besseling), Chaboche, Ohno–Wang, Abdel Karim–Ohno, modified Chaboche (Bari and Hassan) and modified Ohno–Wang (Chen and Jiao). Apparently, none of the models evaluated perform satisfactorily in simulating the straight pipe diameter change and circumferential strain ratcheting responses.
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In particular, many experimental and numerical studies on ratcheting induced by reversed bending in straight pipes and elbows have been performed in the last years [1,8–10], and a detailed review can be found in [1,10]. In [1], ratcheting studies were carried out on Type 304LN stainless steel straight pipes and elbows subjected to steady internal pressure and cyclic bending load. Ratcheting behaviour of straight pipes and elbows were compared and it was generally inferred that ratcheting was more pronounced in straight pipes than in elbows. Shariati et al. [8] investigated the softening and ratcheting behaviours of SS316L cantilevered cylindrical shells under cyclic bending load. Accumulation of the plastic strain or ratcheting phenomenon occurred under force-control loading with nonzero mean force. It was verified that an increase of the mean force induces an increase in the accumulation of the plastic deformation and its rate. Plastic mechanism analyses of circular tubular members under cyclic loading were performed in [9]. This paper provides new methods of analyses for circular hollow sections subjected to a constant amplitude cyclic pure bending and a large axial compression–tension cycle. The local buckling analysis was performed using a rigid plastic mechanism analysis. Although ratcheting based criteria for integrity assessment of pressurized piping under severe loading can be found in codes [11], so far most constitutive models are unable to describe the complex nonlinear cyclic behaviour observed in this kind of problem. Besides, the analysis of the stress and strain fields usually requires the use of complex finite element codes, what can be a shortcoming for the effective use of these theories by designers. The present paper is concerned with the analysis of the coupled effect of kinematic and isotropic hardening on the multiaxial ratcheting behaviour of elasto-plastic thin-walled straight pipes under internal pressure and subjected to cyclic axial loading. An easily employable framework to be used in the analysis of structures of this type with complex material behaviour, as elasto-plasticity with both isotropic and kinematic hardening, is presented. A simple and efficient algorithm for approximating the solution is described. Simulations of AISI 316L steel and AU4G aluminium alloy pressure vessels at room temperature subjected to multiaxial loadings are presented and analysed. It is shown that cyclic behaviour is strongly dependent on the kinematic hardening, but also on the isotropic hardening. The main result of the paper is the proposition of a simple condition (involving the ratio of the circumferential component of the backstress tensor and the auxiliary variable related to the isotropic hardening) to indicate when the pipe may exhibit a ratcheting behaviour, which leads to a progressive accumulation of deformation. Such kind of criterion is independent of the evolution law adopted for the backstresss tensor. Therefore, it is important to emphasize that the paper is not focused on the evaluation of the physical adequacy of different constitutive models for predicting ratcheting behaviour. Although the Marquis–Chaboche elasto-plastic constitutive Eqs. (11) and (12) have been considered in the present study, such a condition for ratcheting to arise can be applied to any cyclic plasticity model with both isotropic and kinematic hardening.
2. Summary of the elasto-plastic constitutive equations The following set of elasto-plastic constitutive equations proposed by Marquis [12] is adequate to model the cyclic inelastic behaviour of metallic material at room temperature. A further discussion about these equations can be found in [13]. In the framework of small deformations and isothermal processes, besides the stress tensor σ and the strain tensor ε , the following auxiliary variables are also considered: the plastic strain tensor ε p , the accumulated plastic strain p and two other auxiliary variables ðX ; YÞ, respectively related to the kinematic hardening and to the isotropic hardening. A complete set of elasto-plastic constitutive equations is
103
given by
σ
νE E trðε ε p Þ 1 þ ðε ε p Þ ¼ ð1 þ νÞð1 2νÞ ð1 þ νÞ
ε_ p ¼
3 S X p_ 2J
ð1Þ
ð2Þ
2 X_ ¼ aε_ p bX p_ 3
ð3Þ
Y_ ¼ v2 ðv1 þ σ y YÞp_
ð4Þ
p_ Z 0;
F ¼ ðJ YÞ Z 0;
_ ¼0 pF
ð5Þ
with ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 3 X 3 u u3 X 3 t S X U S X ¼ ðS X ij Þ2 J¼ 2 2 i ¼ 1 j ¼ 1 ij
ð6Þ
where E is the young modulus, ν the Poisson’s ratio and σ y , v1 , v2 , a, b are positive constants that characterize the plastic behaviour of the material and they can be obtained from a simple tension-compression test [14]. 1 is the identity tensor, and trðA Þ is the trace of an arbitrary tensor A . S is the deviatoric stress given by 1 trðσ Þ1 S¼ σ 3
ð7Þ
J is the equivalent von Mises stress. X is an auxiliary variable related to the kinematic hardening (eventually called the backstress tensor) and it is introduced to account for the anisotropy introduced by the plastic deformation. Y is an auxiliary variable related to the isotropic hardening and models how the yield stress varies with plastic deformation. p is usually called the accumulated plastic strain and p_ can be interpreted as a Lagrange multiplier associated to the constraint F r 0. Function F characterizes the elasticity domain and the plastic yielding surface. From Eq. (2) it is possible to affirm that rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p p _p ¼ ε_ U ε_ ð8Þ 3 and, therefore, Z pðtÞ ¼ pðt ¼ 0Þ þ
t t¼0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 p ε_ ðζ Þ U ε_ p ðζ Þ dζ 3
ð9Þ
If F ¼ J Y o0, it comes that J oY. Hence, from the condition _ ¼ 0 in (5), it is possible to conclude that p_ ¼ 0. If p_ a 0, from the pF _ ¼ 0 it also comes that, necessarily, F ¼ 0. Besides, from condition pF Eqs. (2)–(4) it comes that, in this case, ε_ p a , ̇ a and Y_ a0. Therefore, the elasto-plastic material is characterized by an elastic domain in the stress space where yielding doesn’t occur (ε_ p ¼ ̇ ¼ ,p_ ¼ Y_ ¼ 0 if F o 0). Noting the eigenvalues of S and X , respectively by fS1 ; S2 ; S3 g and fX 1 ; X 2 ; X 3 g, the elastic domain can be represented in the space of the principal directions of the deviatoric stress pffiffiffiffiffiffiffiffias a sphere centred at the point fX 1 ; X 2 ; X 3 g with radius R ¼ 2=3Y. Generally the following initial conditions are used for a “virgin” material pðt ¼ 0Þ ¼ 0;
ε p ðt ¼ 0Þ ¼ X ðt ¼ 0Þ ¼ ;
Yðt ¼ 0Þ ¼ σ y
ð10Þ
From now on, the initial conditions (10) are assumed to hold in the analysis. It is also important to remark that the constitutive equations with conditions (10) and definition (7) imply that the principal directions the stress tensor, of the deviatoric stress tensor, of the plastic strain tensor and of the backstress tensor are the same.
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Eqs. (1)–(3) have the following form if expressed using the basis of the principal directions 3 X νE E ðεi εpi Þ ðεi εpi Þ þ ð1 þ νÞð1 2νÞ i ¼ 1 ð1 þ νÞ
ð11Þ
ε_ pi ¼ ðSi X i Þp_
3 2J
ð12Þ
2 X_ i ¼ aε_ pi bX i p_ 3
ð13Þ
σi ¼
with Si ði ¼ 1; 2or3Þ, εpi ði ¼ 1; 2or3Þ and X i ði ¼ 1; 2or3Þ being the principal components (eigenvalues) respectively of S , ε p and X . In this case, from Eq. (7), it is possible to obtain the following alternative expression qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J ¼ 3½ðSi X i Þ2 þ ðSi X i ÞðSj X j Þ þ ðSj X j Þ2 8 ði; jÞ; with a i j ð14Þ Using the evolution laws (2) and (3) and considering the initial conditions (10), it is also possible to verify that the following relation always holds (provided Sj a 0, εpj a 0 and X j a 0) S i εi X i ¼ ¼ 8 Sj εpj X j p
ði; j ¼ 1; 2; or 3Þ
ð15Þ
3. Thin-walled elasto-plastic cylinder under constant internal pressure and cyclic axial loading The goal of this section is to summarize the procedure to analyse a particular two-dimensional state of stress obtained from the general model. In this two-dimensional abstract context, the tensor σ ðx; tÞ at a given point x and a given time instant t is supposed to be given by the following expressions (in cylindrical coordinates) 2 3 σr 0 0 7 σ ¼6 4 0 σ θ 0 5 with σ r ¼ 0; σ θ ¼ αðtÞσ y ; σ z ¼ βðtÞσ y : 0 0 σz ð16Þ
σ r is the radial stress component, σ θ the circumferential stress component and σ z the axial stress component. All other components are considered to be equal to zero. α and β are arbitrary scalar functions of time such that αðt ¼ 0Þ ¼ 0 and βðt ¼ 0Þ ¼ 0. A theoretical analysis allows explaining the different possible cyclic plastic behaviours in if such kind of (local) stress history is supposed to occur. This analysis includes the case of thin-walled elasto-plastic cylinders under constant internal pressure and subjected to cyclic axial loading. In this case, the expressions relating the external loading and geometry with the axial and circumferential stress component σ z and may vary depending on the nature of the cyclic axial loading. Nevertheless, no matter the nature of the axial loading, the stress tensor at a given point will always have the abstract form presented in Eq. (16), since αðtÞ and βðtÞ are arbitrary scalar functions. For instance, in the case of an elasto-plastic cylinder with internal radius r i and thickness e submitted to an internal pressure P, the circumferential stress component (or hoop stress) σ θ and the circumferential strain component εθ are classically approximated in the framework of membrane theory of shells of revolution by the following expression (provided the internal radius r i and the thickness e are such that r i 4 10e and ovalisation is negligible) Pr σθ ¼ i; e
εθ ¼
Δr ri
ð17Þ
where Δr is the variation of the internal radius. To simplify the
analysis, the cylinder is assumed to be open-ended (studies analysing the effect of the closed ends in the cylinder integrity can be found in [15,16]). In a cyclic push and pull with axial force F t , the axial stress is homogeneous along a cross-section and approximated by the following expression: σ z F t =ð2π r i eÞ. In real straight pipes in operation, the cyclic axial stress component can be caused by a non-homogeneous temperature variation or by a bending moment [10]. The present paper is not concerned with the analysis of the equilibrium equations in these cases and the expression of both the axial stress component σ z and may vary depending on the material, the geometry and the loading history. In the case of cyclic bending, the axial stress is not homogeneous and complex expressions should be adopted. In the case of non isothermal problems, the fully coupled constitutive equations must be considered (see, [17], for instance). Nevertheless, if ovalisation is negligible, the hoop stress and the axial stress in a thin walled pipe can be considered uncoupled since the expressions presented in Eq. (17) are still reasonable. Besides, as it will be verified in the sequence, the particular expression relating the axial stress with material, geometry and external loading does not affect the basic goal of the paper, which is to identify a simple criterion for ratcheting to arise when the circumferential component of the backstress tensor exceed a given isotropic hardening parameter. No matter the pipe material and geometry, α must be a constant value in Eq. (16) for a fixed internal pressure of the pipe. Besides, since the goal is to study the ratcheting induced by a periodic axial loading, the axial stress is assumed to be a periodic function of time at a given material point and, consequently, β must be a periodic function of time. Since plasticity is rate-independent, the frequency of the axial loading does not affect the stress–strain behaviour and only maximum and minimum stress levels (and, therefore, the maximum and minimum values of the function β) are important in the analysis. Taking into account assumption (16) and using definition (7), it is possible to conclude that the radial, circumferential and axial components Sr , Sθ , Sz of the deviatoric stress tensor S are its only non-zero components 1 1 Sr ¼ ðσ θ þ σ z Þ ¼ ðαðtÞ þ βðtÞÞσ y 3 3
ð18:1Þ
1 1 Sθ ¼ ð2σ θ σ z Þ ¼ ð2αðtÞ β ðtÞÞσ y 3 3
ð18:2Þ
1 1 Sz ¼ ð2σ z σ θ Þ ¼ ð2β ðtÞ αðtÞÞσ y 3 3
ð18:3Þ
It is important to note that the principal components of the deviatoric stress and of the plastic deformation are not independent, which allows introducing additional simplifications in the equations. From Eqs. (15) and (18.1)–(18.3), it comes that (provided ð2α β Þ a 0) Sr εpr X r ðα þ β Þ ¼ ¼ ¼ ; Sθ εpθ X θ ðβ 2αÞ
Sz εpz X z ð2β αÞ ¼ ¼ ¼ Sθ εpθ X θ ð2α β Þ
ð19Þ
Using definition (14) and Eq. (19) it is possible to obtain the following equivalent expressions for the von Mises equivalent stress J¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3½ðSθ X θ Þ2 þ ðSθ X θ ÞðSz X z Þ þ ðSz X z Þ2
J ¼ Sθ X θ ηðtÞ
with
ð20:1Þ
ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u " 2 # u ð2β αÞ ð2β αÞ t þ ηðtÞ ¼ 3 1 þ ð2α βÞ ð2α βÞ ð20:2Þ
H.S. da Costa Mattos et al. / Thin-Walled Structures 93 (2015) 102–111
Thus, the behaviour in the circumferential direction is governed by the following system of equations 1 Sθ ¼ ð2αðtÞ βðtÞÞσ y ; 3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u " 2 # u ð2βðtÞ αðtÞÞ ð2βðtÞ αðtÞÞ ηðtÞ ¼ t3 1 þ þ ð2αðtÞ βðtÞÞ ð2αðtÞ βðtÞÞ
ð21:1Þ 3 3η ε_ pθ ¼ ðSθ X θ Þp_ ¼ Sgp_ 2J 2
( with
Sg ¼
ifðSθ X θ Þ Z 0
1;
otherwise
3ðηÞn þ 1 ðSgÞn Δp 2
ð25:1Þ
2 3
ð25:2Þ
ΔX θ ¼ aΔεpθ bðX θ Þn Δp ¼ ½aðηÞn þ 1 ðSgÞn bðX θ Þn Δp
ð21:3Þ
Y_ ¼ v2 ðv1 þ σ y YÞp_
ð21:4Þ _ ¼0 pF
ð21:5Þ
From Eqs. (18.2) and (21.2) it is possible to verify that the axial stress component affects the plastic strain in the circumferential direction. In the case of a tube under constant internal pressure, a significant variation of the axial stress component may cause a plastic flow in the circumferential direction.
4. Solution algorithm Eqs. (21.1)–(21.4) define a nonlinear system of ordinary differential equations coupled with the constraints (21.5). The basic mathematical problem considered in this paper is formed by Eqs. (21.1)–(21.4) together with the following initial conditions ð22Þ
The solution algorithm used to obtain a numerical approximation for this initial value problem is based on the operator split method [18–20]. The objective of this technique is to take advantage of some additive decomposition properties of the mathematical operator to solve a sequence of simpler problems instead of a unique complex problem. The approximation techniques based on the operator split method and associated product formula algorithms have been applied in various areas of Continuum Mechanics and, in particular, in elastoplasticity and elasto-viscoplasticity. 4.1. Time discretization Considering a sequence of time instants (since the elasto-plastic equations are rate independent, this definition is only used to establish a sequence of events) 0 o t 1 ot 2 o ⋯ o t n
ð25:3Þ
9 F n þ 1 ¼ ðSθ Þn þ 1 ðX θ Þn þ 1 ηn þ 1 ðYÞn þ 1 ) > > = F n þ 1 ðSθ Þn þ 1 ðX θ Þn ΔX θ ðηÞn þ 1 ðYÞn ΔY n )
> > ðSθ Þn þ 1 ðX θ Þn ðaηn þ 1 ðSgÞn bðX θ Þn ÞΔp ηn þ 1 ðYÞn v2 ðv1 ðYÞn þ σ y Þ Δp ;
ð26Þ and an analytical approximation of the increment Δp can be obtained using the condition ðFÞn þ 1 ¼ 0 when Δp a0: ðFÞn þ 1 ½ðSθ Þn þ 1 ðX θ Þn ðað
ηÞn þ 1 ðSgÞn bðX θ Þn ÞΔpðηÞn þ 1 ðSgÞn ðYÞn v2 ðv1 ðYÞn þ σ y Þ Δp
ð23Þ
and using the following notation for any function yðtÞ: yðt n Þ ¼ yn The main idea of the solution algorithm is to explore the fact that all time rates of the variables εpθ , X θ and Y, defined in Eqs. _ hence, the following (21.2)–(21.4), are linear functions of p, approximation can be considered
ð27Þ
If ðFÞn þ 1 ¼ 0, then ½ðSθ Þn þ 1 ðX θ Þn ηn þ 1 ðSgÞn Y n ¼ faη2n þ 1 bðX θ Þn ηn þ 1 ðSgÞn þ v2 ðv1 ðYÞn þ σ y Þg
Δp ) Δp
pðt ¼ 0Þ ¼ εpθ ðt ¼ 0Þ ¼ X θ ðt ¼ 0Þ ¼ 0; Yðt ¼ 0Þ ¼ σ y
The plastic function F can be expressed in terms of Δp:
2 X_ θ ¼ aε_ pθ bX θ p_ 3
F ¼ ð Sθ X θ ηðtÞÞ Y Z0;
Δεpθ ¼
ΔY ¼ v2 ðv1 ðYÞn þ σ y Þ Δp
ð21:2Þ
p_ Z 0;
with
1;
105
η
ðSθ Þn þ 1 ðX θ Þn ð Þn þ 1 ðSgÞn ðYÞn
Þ2n þ 1 bðX θ Þn ð Þn þ 1 ðSgÞn þ v2 v1 þ y ðYÞn
aðη
η
σ
ð28Þ
Once the increment Δp is known, all the variables can be obtained at the instant t n þ 1 using Eqs. (24.1)–(24.4), (25.1)–(25.3). The procedure can be improved using an iterative solution technique that uses this Euler scheme coupled with a projection scheme. The projection technique assures that F o δ at any time step, with δ 40 being the maximum admissible error in the computation of F in the plastic steps. Such a technique allows the computation of accurate results even if very large steps are used, what is crucial if the simulation of a great number of cycles is necessary.
4.2. Basic solution algorithm The basic solution algorithm is defined as follows (i) n ¼ 0 (ii) ðηÞn þ 1 , ðεpθ Þn , ðpÞn , ðX θ Þn , ðYÞn are known and 1 ðSθ Þn þ 1 ¼ ð2ðαÞn þ 1 ðβÞn þ 1 Þσ y ; ðηÞn þ 1 3vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u " 2 # u ð2ðβ Þn þ 1 ðαÞn þ 1 Þ ð2ðβÞn þ 1 ðαÞn þ 1 Þ t þ ¼ 3 1þ ð2ðαÞn þ 1 β ðtÞÞ ð2ðαÞn þ 1 ðβÞn þ 1 Þ (iii) Compute the auxiliary variables εpθ , p, X θ and Y: p p εθ ¼ ðεθ Þn , p ¼ ðpÞn , X θ ¼ ðX θ Þn , Y ¼ ðYÞn , ITER ¼ 1 (iv) ðSθ Þn þ 1 X ðηÞn þ 1 Y o δ? YES ) The step is elastic: ðεpθ Þn þ 1 ¼ εpθ , ðX θ Þn þ 1 ¼ X, Y n þ 1 ¼ Y, pn þ 1 ¼ p, n ¼ n þ 1 If n 4 NMAX, stop ) maximum number of time steps Go to (ii) NO ) The step is plastic:
ðεpθ Þn þ 1 ¼ ðεpθ Þn þ Δεpθ
ð24:1Þ
ðX θ Þn þ 1 ¼ ðX θ Þn þ ΔX θ
ð24:2Þ
ITER ¼ ITER þ 1
ðYÞn þ 1 ¼ ðYÞn þ ΔY
ð24:3Þ
ðpÞn þ 1 ¼ ðpÞn þ Δp
ð24:4Þ
if ITER 4 ICONT, stop ) Maximum number of iterations per time step go to (vi)
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(v) Computation of the increment Δp:
Y ¼ Y þ ΔY
ðSθ Þn þ 1 X θ ðηÞn þ 1 Sg Y n ðS Þ Xθ ) Δp ¼ Sg ¼ θ n þ 1
2 ðSθ Þ X að η Þ θ nþ1 n þ 1 bX θ ðηÞn þ 1 Sg þ v2 v1 þ σ y Y
p ¼ p þ Δp (viii) Go to (iv)
(vi) Computation of the increments Δεpθ , ΔX θ , ΔY: 3η Δεpθ ¼ n þ 1 Sg ΔpΔX θ ¼ ðaηn þ 1 Sg bX θ ÞΔp 2
ΔY ¼ v2 ðv1 Y þ σ y Þ Δp (vii) Update the variables εpθ , X θ , Y, p
Good results
are obtained taking δ ¼ σ y =100 , ICONT ¼ 5 and ΔSθ r σ y =100 5. Results and discussion
εθ ¼ εθ þ Δεθ
In order to better understand when ratcheting may occur in an arbitrary tube, it is considered an AISI 316L steel at room temperature. The material parameters in this case are
X θ ¼ X θ þ ΔX θ
σ y ¼ 300 MPa;
p
p
p
E ¼ 196 GPa;
ν ¼ 0:3;
Fig. 1. Loading histories considering Eq. (16).
Fig. 2. Curves Sθ εpθ for different amplitudes of the cyclic axial loading. AISI 316L steel at room temperature.
H.S. da Costa Mattos et al. / Thin-Walled Structures 93 (2015) 102–111
a ¼ 45 GPa;
b ¼ 60;
v1 ¼ 0;
v2 ¼ 0
ð29Þ
It can be observed that, for this kind of material, the hardening is purely kinematic (since v1 ¼ 0 and v2 ¼ 0). The goal is to demonstrate that the ratcheting phenomenon is essentially due to the kinematic hardening and thus it is chosen a material which does not present isotropic hardening (Y ¼ σ y ). In the analysis, the loading histories depicted in Fig. 1 (the functions αðtÞ and β ðtÞ, see (16)) are taken into account. Since the elasto-plastic constitutive are rate independent, the time is not important and it is only used to establish a sequence of events. Thus, t n ¼ ðt=ΔtÞ, where Δt is an arbitrary time interval. The parameters A and B are chosen in order to allow understanding the role of the stress components in the ratcheting behaviour. Parameter α is related to the internal pressure. Combining the expressions in (16) and (17), we have σ e Pr ð30Þ σ θ ¼ αðt n Þσ y ¼ i ) P ¼ α y ri e The maximum hoop stress is given by ðσ θ Þmax ¼ Aσ y ) P ¼ Aðσ y e=r i Þ (see Eq. (16)). Therefore, if A ¼ 1 then the maximum hoop stress is such that ðσ θ Þmax ¼ σ y and the internal pressure is given by P ¼ Aðσ y e=r i Þ. B 41 is the amplitude of the periodic function βðt n Þ ¼ πtn B sin 200 , with σ z ¼ βðt n Þσ y . Therefore, the maximum and minimum axial stress component are ðσ z Þmax ¼ Bσ y and ðσ z Þmin ¼ Bσ y . Significant ratcheting in the circumferential direction occurs for different values of B as it is shown in Figs. 2 and 3. Parameter B is related to the amplitude of the cyclic axial loading. Therefore, different amplitudes in the axial loading induce different ratcheting behaviour in circumferential direction. Since εpθ ¼ ðΔrÞp =r i ,
107
where ðΔrÞp is the plastic or irreversible variation of the internal radius, it may be concluded that the tube diameter will increase incrementally with the axial cyclic loading. Fig. 4 presents the evolution of the accumulated plastic strain for A ¼ 1 and different values of the parameter B. Under cyclic axial loading and unloading, the pipe exhibits hysteresis (a phase lag), which leads to a dissipation of mechanical energy. The progressive accumulation of plastic deformation can be explained by the strong kinematic hardening observed in this material. The analysis is very similar to the one performed in [21] for progressive accumulation of deformation observed in cyclic tension tests performed in epoxy polymers. The elastic region is the domain of stresses in which there is no plastic flow. As commented before, the elastic domain can be
Fig. 4. Evolution of p for different amplitudes of the cyclic axial loading. AISI 316L steel at room temperature.
Fig. 3. Evolution of εpθ for different amplitudes of the cyclic axial loading. AISI 316L steel at room temperature.
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Fig. 5. Influence of kinematic hardening in the accumulation of plastic deformation.
Fig. 6. Curves Sθ εpθ for different amplitudes of the cyclic axial loading. AU4G aluminium alloy at room temperature.
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represented in the space of the principal directions of the deviatoric stress pffiffiffiffiffiffiffiffi as a sphere centred at the point fX 1 ; X 2 ; X 3 g with radius R ¼ 2=3Y. This domain varies with the plastic deformation process. In case studied in this paper, from the particular condition F ¼ Sθ X θ η Y o 0 it possible to define the elastic segment as the set of all possible Sθ such that Sθ X θ ðY=ηÞ o 0. The elastic segment has a length (2Y=η) and it is centred on X θ . The elastic segment also varies with the plastic deformation process. If ðY=ηÞ 4 X θ hysteresis with consequent accumulation of cyclic plastic deformation does not occur. For the pipe material under the prescribed loading conditions considered in this paper, the kinematic hardening variable can be greater that the isotopic hardening variable ðY=ηÞ and a negative plastic strain rate may occur even with a positive stress (ε_ pθ o 0 if S_ θ o0 even when Sθ 4 0). This fact explains why hysteresis and a progressive accumulation of deformation occur in a cyclic load–unload test as shown in Fig. 5. In the case of a material that presents both kinematic and isotropic hardening, the reasoning is similar and ratcheting occurs if the kinematic hardening parameter (the circumferential component of the backstress tensor) X θ exceeds the isotropic hardening parameter ðY=ηÞ. Therefore, a simple criterion involving the ratio of the circumferential component of the backstress tensor and the auxiliary variable related to the isotropic hardening indicates if the pipe may exhibit a ratcheting behaviour, which leads to a progressive accumulation of deformation. Ratcheting may occur if X θ 4 ðY=ηÞ ) ðX θ =YÞ o
1
η
109
different values of the parameter B. The material parameters in this case are
σ y ¼ 170 MPa; E ¼ 72 GPa; ν ¼ 0:32; a ¼ 22:5 GPa; b ¼ 125; v1 ¼ 100; v2 ¼ 1:6:
ð32Þ
As it can be verified, despite the isotropic hardening, the behaviour is similar than in the previous case. The consideration of a range of hoop stresses is perfectly possible within the model. This would permit a ratchet diagram to be developed—for any given set of hardening parameters. Fig. 9 shows the results of simulations considering the aluminium alloy with B ¼ 1:5 and different values of the parameter A. Significant ratcheting in the circumferential direction also occurs even when A o 1 ) σ θ o σ y . It is interesting to remark that, in all cases, after a few cycles, the increment of the accumulated plastic strain per cycle δp tends to a constant value (Fig. 10). Such a plastic increment per cycle can
ð31Þ
Figs. 6–8 show the results of simulations considering an AU4G aluminium alloy at room temperature. The loading history considering Eq. (16) is the same presented in Fig. 1, with A ¼ 1 and
Fig. 8. Evolution of p for different amplitudes of the cyclic axial loading. AU4G aluminium alloy at room temperature.
Fig. 7. Evolution of εpθ for different amplitudes of the cyclic axial loading. AU4G aluminium alloy at room temperature.
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Fig. 9. Curves Sθ εpθ for different amplitudes of the hoop stress. AU4G aluminium alloy at room temperature.
Fig. 10. Increment of the accumulated plastic strain per cycle δp.
be easily obtained using the proposed numerical procedure. A simple preliminary failure criterion would be to assume that the probability of failure is higher when the accumulated plastic strain is greater than a given limit, i.e. when p ðnδpÞ 4pmax .
6. Conclusion Metallic pipes under constant pressure and cyclic axial loading beyond the elastic limit may exhibit continued strain growth in the hoop direction, what is known as ratcheting, which leads to the accumulation of cyclic plastic deformation. The simple elastoplastic analysis presented in this paper allows explaining this kind of phenomenon. With an adequate cmanipulation of variables, the multiaxial problem can be reduced to the analysis in the
circumferential direction. It is shown that cyclic behaviour is strongly dependent on the kinematic hardening, but also on the isotropic hardening. A simple condition involving the ratio of the circumferential component of the backstress tensor and the auxiliary variable related to the isotropic hardening allows indicating when the pipe may exhibit a ratcheting behaviour. Although the Marquis–Chaboche elasto-plastic constitutive equations have been considered in the present study, the condition for ratcheting to arise is valid for any cyclic plasticity model with both isotropic and kinematic hardening, provided Eq. (15) holds. Besides, the solution algorithm can be easily adapted to other kinematic hardening rules provided the increment of the circumferential component of the backstress tensor ΔX θ can be expressed as a function of the plastic strain increment Δεpθ of the accumulated plastic strain increment Δp: ΔX θ ¼ f ðΔεpθ ; ΔpÞ. In this case, Eq. (25.2) would be replaced by the alternative expression. References [1] Vishnuvardhan S, Raghava G, Gandhi P, Saravanan M, Goyal S, Arora P, Gupta SK, Bhasin V. Ratcheting failure of pressurised straight pipes and elbows under reversed bending. Int J Press Vessels Pip 2013;105-106:79–89. [2] Kulkarni SC, Desai YM, Kant T, Reddy GR, Parulekar Y, Vaze KK. Uniaxial and biaxial ratchetting study of SA333 Gr. 6 steel at room temperature. Int J Press Vessels Pip 2003;80:179–85. [3] Chaboche JL. Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int J Plast 1989;5:247–302. [4] Chaboche JL. On some modifications of kinematic hardening to improve the description of ratcheting effects. Int J Plast 1991;7:661–78. [5] Ohno N, Wang JD. Kinematic hardening rules with critical state of dynamic recovery, Part I: Formulations and basic features for ratcheting behavior. Int J Plast 1993;9:375–90. [6] Abdel Karim M, Ohno N. Kinematic hardening model suitable for ratcheting with steady-state. Int J Plast 2000;16:225–40.
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[7] Rahman SM, Hassan T, Corona E. Evaluation of cyclic plasticity models in ratcheting simulation of straight pipes under cyclic bending and steady internal pressure. Int J Plast 2008;24:1756–91. [8] Shariati M, Kolasangiani K, Norouzi G, Shahnavaz A. Experimental study of SS316L cantilevered cylindrical shells under cyclic bending load. Thin Walled Struct 2014;82:124–31. [9] Elchalakani M. Plastic mechanism analyses of circular tubular members under cyclic loading. Thin Walled Struct 2007;45:1044–57. [10] Xiaohui Chen, Chen Xu, Dunji Yu, Bingjun Gao. Recent progresses in experimental investigation and finite element analysis of ratcheting in pressurized piping. Int J Press Vessels Pip 2013;101:113–42. [11] ASME. Boiler and pressure vessel code, section III, divison I, subsection NB, USA: American Society of Mechanical Engineers; 1998. [12] Marquis D. Modelisation et identification de l’ecrouissage anisotrope des metaux. These 3ème Cycle: Université Pierre et Marie Curie-Paris VI; 1979. [13] Lemaitre J, Chaboche JL. Mechanics of solid materials. London: Cambridge University Press; 1990.
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