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Ductile damage evolution under cyclic non-proportional loading paths
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IGF Workshop “Fracture and Structural Integrity” IGF Workshop “Fracture and Structural Integrity”
Ductile damage evolution under cyclic non-proportional loading Ductile damage evolution under cyclic non-proportional loading paths XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal paths *,a a b Riccardo Fincato , Seiichiro Tsutsumi , Hideto Momii Thermo-mechanical modeling of a high pressure turbine blade of an *,a a Riccardo Fincato , Seiichiro Tsutsumi , Hideto Momiib JWRI, Osaka University, 11-1 Mihogaoka, Ibaraki 5670047, Osaka, Japan airplane gasKamitonno, turbine engine Osaka Corporation,4727-1 University, 11-1 Mihogaoka, Ibaraki 5670047, Osaka, Japan Japan Nikken JWRI, Engineering Nogata 822-0003, Fukuoka, a a
b
Nikken Engineering Corporation,4727-1 Kamitonno, Nogata 822-0003, Fukuoka, Japan
b
P. Brandãoa, V. Infanteb, A.M. Deusc*
AbstractaDepartment of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal Abstract b IDMEC, Department of Mechanical Superior Técnico, de Lisboa, Av. Rovisco Pais, 1, Lisboa, The characterization of ductile damageEngineering, evolution, Instituto and its description, haveUniversidade been the object of extensive research in 1049-001 the continuum Portugal Theccharacterization of ductile evolution, andworks its description, have beenGurson the object of extensive research in many the continuum damage mechanic field. Startingdamage from the pioneering of Lemaitre (1985), (1977), Rousselier (1987), different CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, damage mechanic Starting thethe pioneering works ofand Lemaitre (1985), (1977), (1987),asmany different models have been field. developed. In from detail, stress triaxiality the Lode angleGurson parameters haveRousselier been identified the two main Portugal models have developed. In detail, the stress theaLode parameters have been identified as the two main variables thatbeen affect the material ductility. The triaxiality literature and offers greatangle number of investigations under monotonic loading variables that affect athe material ductility. The literature a great number investigations under monotonic conditions, however, proper characterization of the damageoffers evolution under cyclicofloading or non-proportional loadingloading is still conditions, Abstracthowever, a proper characterization of the damage evolution under cyclic loading or non-proportional loading is still missing. missing. In this paper, an unconventional coupled elastoplastic and damage constitutive model with a Mohr-Coulomb failure criterion (Bai In this paper, an 2010) unconventional coupled elastoplastic and damagelaw constitutive model withto a Mohr-Coulomb failure criterion (Bai their operation, modern aircraft engine components are to increasingly demanding operating conditions, andDuring Wierzbicki, is presented. The ductile damage evolution issubjected modified in order consider the damage evolution under the 2010) high pressure turbine (HPT) blades. Such conditions cause these parts different types of time-dependent andespecially Wierzbicki, is presented. The ductile damage evolution is modified in order to consider the damage evolution non-proportional loading conditions. The idea is to compare the law damage evolution oftoa undergo steel bridge column subjected to under three degradation, one of which is creep. A in model the finite element method (FEM) was developed, in ordersubjected to be abletotothree predict non-proportional loading conditions. The idea istotoidentify compare damage evolution a steel bridge column different non-proportional loading paths orderusing thethe loading condition thatof compromise the structural integrity. the creep behaviour ofloading HPT blades. data records a specific provided by a commercial different non-proportional paths inFlight order to identify the(FDR) loadingfor condition that aircraft, compromise the structural integrity. aviation company, were used to obtain © 2018 The Authors. Published bythermal Elsevierand B.V.mechanical data for three different flight cycles. In order to create the 3D model © 2018 2018 The Authors. Published by Elsevier B.V.blade needed the responsibility FEM analysis, HPT scrap was scanned, its chemical composition and material properties were © Thefor Authors. Published by B.V.Italiano Peer-review under ofaElsevier the Gruppo Frattura (IGF) and ExCo. Peer-review under responsibility ofgathered the Gruppo Italiano Frattura (IGF)model ExCo. and different simulations were run, first with a simplified 3D obtained. The data that was was fed into the FEM Peer-review under responsibility of the Gruppo Italiano Frattura (IGF) ExCo. rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The Keywords: Ductile damage; elastoplasticity; non-proportional loading; overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a Keywords: Ductile damage; elastoplasticity; non-proportional loading; model can be useful in the goal of predicting turbine blade life, given a set of FDR data.
1. © Introduction 2016 The Authors. Published by Elsevier B.V. 1. Peer-review Introduction under responsibility of the Scientific Committee of PCF 2016. In the recent years, many authors tried to characterize the ductile behavior of metals by means of different types of In the recent many authors triedand to characterize the ductile behavior of metals by means of different types of Keywords: High years, Pressure Turbine Blade; Creep; Finite Element models, Method; 3D Model; Simulation. numerical models (coupled elastoplastic damage uncoupled models, phenomenological, etc.). The large numerical models (coupled elastoplastic and damage models, uncoupled models, phenomenological, etc.). The large Corresponding author. Tel.: 81-6-6879-8667. Corresponding Tel.: 81-6-6879-8667. E-mail address:author. [email protected] E-mail address: [email protected] 2452-3216 © 2018 The Authors. Published by Elsevier B.V. 2452-3216 © 2018 Authors. Published Elsevier B.V. Frattura (IGF) ExCo. Peer-review underThe responsibility of theby Gruppo Italiano Peer-review underauthor. responsibility the Gruppo Italiano Frattura (IGF) ExCo. * Corresponding Tel.: +351of218419991. E-mail address: [email protected] * *
2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016. 2452-3216 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Gruppo Italiano Frattura (IGF) ExCo. 10.1016/j.prostr.2018.06.022
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campaign of experimental investigations carried out in the last decades (Algarni et al., 2015; Bai et al., 2009; Bao and Treitler, 2004; Bao and Wierzbicki, 2004; Brünig et al., 2013; Gao et al., 2011; Papasidero et al., 2015) pointed out that the stress triaxiality and the Lode angle are the main two factors that affect the ductility behavior in metals. In fact, monotonic loading tests, for different geometries and under different loading conditions (uniaxial extension, pure shear, plane strain, uniaxial compression), proved a different failure behavior for the same material. On the other hand, the literature still lacks a proper characterization of the damage evolution under non-proportional loading paths. In the recent years, Faleskog and Barsoum (2013), Papasidero et al. (2014), Cortese et al. (2016) and Algarni et al. (2017) conducted a series of experiments aimed to identify the effect of the loading path on ductility. In detail, the work of Faleskog and Barsoum (2013), Papasidero et al. (2014) and Corstese et al. (2016) consisted in the application of several non-proportional loading paths, as the result of the combination of tension-torsion or compression-torsion, on steel and aluminum tubular samples. On the other hand, Algarni et al. tried to describe the crack formation on notched Iconel 718 bars in low cycle fatigue investigations. A common aspect that emerged from those previous works is that the deformation at fracture is higher when the load is proportional, suggesting that the damage accelerates whenever non-proportional loading conditions are triggered. The present paper aims to investigate the influence of the loading path on the ductile damage evolution. In detail, the numerical analyses focus the attention of the structural response of a steel bridge column subjected to various loading condition. A modified Mohr-Coulomb criterion (Bai and Wierzbicki, 2010) is adopted for the description of the damage behavior of SS400 steel pier, with a modification of the damage evolution law, in order to take into account the effect of the non-proportionality of the load. Next, section 2, the theoretical Damage Subloading surface model (i.e. DSS) is presented, with particular emphasis on the ductile damage criterion. The subsequent section 3 is divided into two parts; the first one deals with the calibration of the material parameters, whereas the second offers an overview of the effect of three non-proportional loadings on the bridge pier. Nomenclature
σ, σ α, α s σ α E F F0 H HD
R λ D σm
Mises η
f
a b
Cauchy stress, corotational stress rate back stress, corotational back stress rate similarity centre conjugate Cauchy stress conjugate back stress tensor of the elastic constants isotropic-hardening function initial size of the normal-yield surface isotropic hardening variable cumulative plastic variable (i.e. equivalent plastic strain) similarity ratio plastic multiplier ductile damage scalar variable mean stress von Mises stress stress triaxiality Lode angle Lode angle parameter (-1 < < 1) equivalent strain at fracture Heaviside step function: a b 0 if a b 0; a b 1 if a b 0
2. Theoretical approach The present section deals with the theoretical framework for the description of the elastoplastic and damage model named Damage Subloading Surface model (i.e. DSS). The DSS model was formulated from the unconventional plasticity model Extended Subloading Surface, presented by Hashiguchi in Hashiguchi (2009, 1989), and upgraded to
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include an additional internal variable for the description of the damaging behavior. In the next section 2.1 the main features of the DSS model are briefly reported. A detailed description of the constitutive equations goes beyond the purposes of this paper, the reader is referred to Fincato and Tsutsumi (2017a) for an exhaustive discussion. 2.1. The Damage Subloading Surface model The DSS model is a coupled elastoplastic and damage model that derives it formulation from a previous unconventional plasticity theory, named Extended subloading surface model. The main feature consists in the abolition of the neat distinction between the elastic and plastic domains, stating that irreversible deformations can be generated whenever the loading criterion is satisfied. In order to achieve this goal a second surface, named subloading surface, is generated by means of a similarity transformation from the conventional yield surface, here renamed normal yield-surface. The center of similarity is not fixed in the stress space but it can move following the plastic strain rate. This allows to the creation of close hysteresis loops during the unloading and subsequent reloading, as shown by the authors in Fincato and Tsutsumi (2017b), for a more realistic description of the material ratcheting in cyclic mobility and fatigue problems. The numerical model is developed within the framework of finite elastoplasticity, assuming a hypoelastic based plasticity. The total strain rate D can be additively decomposed into and elastic part De and a plastic part D p . The coupling of the elasticity with damage is obtained by means of the concept of the effective stress Kachanov (1958) as follows:
σ 1 D E : De 1 D E : (D D p )
(1)
The coupling of the plastic internal variables with the scalar isotropic damage variable is obtained following the approach suggested by Lemaitre (1985) and modifying the expression of the consistency conditions for the normalyield and subloading surface as follows:
f (σˆ ) (1 D)F (H );
f (σ) (1 D)RF (H ); σˆ σ α
(2)
The scalar function f of the stress tensor in Eq. (2) is assumed as the von Mises criterion. The plastic strain rate can be obtained from Eq. (2) as shown in Fincato and Tsutsumi (2017a), giving the following expression:
D p / (1 D) f σ / σ ; H ; H D / (1 D);
(3)
In order to complete the set of equations describing the material behavior, the following isotropic and kinematic hardening laws are reported in Eqs. (4)-(5). The isotropic hardening law is a modified version of the law presented in Hashiguchi and Yoshimaru (1995), with an additional linear contribution regulated by the constant K. Eq. (5) was initially proposed by Chaboche (1986), and considered the linear combination of N non-linear contributions. The present paper considers N = 2 contributions. The calibration of the material constants K, h1, h2, Ci, Bi will be discussed in section 3.2.
F ( H ) F0 K H H d F0 h1 1 eh2 H Hd n
αi 1,...N Ci D p Bi αi ; α αi , n
(4) (5)
i 1
2.2. The ductile damage criterion Following a well-consolidated approach adopted by the continuum damage mechanics an arbitrary stress state can be described by using two dimensionless parameters: the stress triaxiality and the Lode angle parameter (see Figure 1b), defined as in Eq. (6):
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(6)
In particular, the Lode angle parameter field of existence covers all the possible loading conditions, from the uniaxial extension to the uniaxial compression passing by the pure shear (or plane strain) condition where it assumes null value. Therefore, every pair of variables , , representing a unique stress state, can be used in the formulation
of fracture envelope in the MC criterion. This criterion has been widely and successfully used in the prediction of the failure behavior of granular materials such as rock, soil and concrete. However, recent works (Algarni et al., 2015; Bai and Wierzbicki, 2010; Rousselier and Luo, 2014), applied the MC criterion to metallic materials due to its characteristic of being able to catch an exponential decay of the ductility with the stress triaxiality and its Lode angle dependency. The full expression of the MC requires the calibration of eight material parameters, however, if a von Mises potential is accounted for, the failure envelope can be defined by the calibration of only four material constants A, N , c1 , c2 (Bai and Wierzbicki, 2010). The use of a von Mises potential should be limited to cases where the effect of the Lode angle on plasticity and the pressure effect on the yield-surface can be neglected. Nonetheless, it can give a good first approximation of the damaging behavior of the material. Therefore, in the present paper we adopted the following form of the failure envelope:
A 1 c 2 1 1 f cos c1 sin c 3 3 6 6 2
1 N
(7)
Based on the previous Eq. (7) the ductile damage increment can be written as:
2 3 D p T3 Dt D f ,
H d1 ;
p p Hi H D HT 2 3 D T3 D
(8)
Where d1 is an additional material parameter and H represent the Heaviside step function, introduced by the authors to allow the damage to evolve whenever the cumulative plastic strain exceeds the parameter d1. The term at the numerator in Eq. (8) has been modified to take into account an additional inelastic term generated by the stress rate component tangential to the plastic potential as in Figure 1a. HT represents the cumulative inelastic strain that actively contributes to the damage. Non-proportional loading, in fact, triggers a non-negligible component of the stress rate that is directed tangentially to the yield surface. The idea is to consider the contribution of the strain rate associated with the tangential component of the stress rate in the damage accumulation. Previous work of the authors (Fincato and Tsutsumi, 2017c; Hashiguchi and Tsutsumi, 2001; Momii et al., 2015; Tsutsumi and Kaneko, 2008) considered a similar approach to overcome the excessive stiffness predicted by the extended subloading surface model with associate flow rule during non-proportional loading. In this case, the effect of the tangential deviatoric strain rate Dt affects only the damage evolution and its contribution can be regulated by the material constant T3 (0 T3 1) . The computation of the deviatoric tangential stress rate is done similarly as in Fincato and Tsutsumi (2017c) and in Momii et al. (2015). Briefly, it is assumed that the deviatoric tangential strain rate is linearly related to the component of the stress rate tangential to the yield surface σt . The advantage of this hypothesis lies in its very simple form, suitable for the application to boundary-value problems and general loading conditions.
Dt
1 T1RT2 A σ ; A σ Iˆ t ; σt 2G 2G 1 T1RT2
(9)
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a)
5
b)
Figure 1 a) sketch of tangential and normal components of stress rate for a generic stress state on the plastic surface; b) schematic representation of a generic stress state in the principal stress space.
where Iˆ is the deviatoric tangential (or projection) tensor (Hashiguchi, 2009) and the variable A is computed accordingly to Momii et al. (2015). In detail, A is function of the similarity ratio R and two material parameters T1 and T2 that were set to 0.9 and 1.0 in this study, respectively. It is worth mentioning that the constant T1 can assume values between 0 T1 1 . In conclusion, if the tangential inelastic stretch is considered then the cumulative inelastic strain to fracture Hi becomes a function of D p and Dt as in the second of Eq. (8). 3. Numerical analyses The present section deal with the numerical analyses carried out implementing the previous constitutive equations via user subroutine for the commercial code Abaqus (ver. 6.14-5). The DSS model parameters were calibrated reproducing a uniaxial tensile load for the SS400 steel reported in Van Do et al. (2014) and subsequently adjusted to reproduce the behavior of a thin steel bridge pier analyzed by Nishikawa et al. (1998). The analyses reported in section 3.4 offer an additional investigation that aims to point out the effect on the damage evolution of three non-proportional bidirectional loading conditions. 3.1. Description of the FE model Nishikawa et al. (1998) conducted a series of experiments on thin steel piers in order to analyse the horizontal loadcarrying capacity of the structures subjected to an increasing unidirectional horizontal load. The schematic of the pier is reported in Figure 2a together with its FE modeling, the loading sequence is instead reported in Figure 2b. In order to save computational time just half of the column was considered, applying symmetric boundary conditions. Moreover, the experimental results showed that the plastic deformations, and the buckling, appear in a region close to the base of the pier. Therefore, the structure was modeled according to its real geometry until a height equal to two times the internal diameter from the bottom and the remaining part was modeled with a beam element. A similar strategy was adopted by Gao et al. (1998). The geometric specifications are reported in Table 1. The lower part of the pier was meshed with eight-node hexahedral elements with reduced integration (i.e. Abaqus C3D8R elements) and the nodes of the top cross section were connected to the beam with rigid links. A mesh refinement was considered close to the base of the pier, where the maximum accumulation of plastic deformation is expected. The minimum element size is 9.8 mm (circumference) x 2.25 mm (thickness) x 15 mm (axial), comparable with the minimum element size used by Van Do et al. (2014), who conducted a mesh size sensitivity analysis on the same study case. The total number of elements amount to 51841.
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The structure is subjected to two types of loads: a compressive load P, constant, and kept for the whole duration of the experiment and a horizontal load H, directed along the x direction and with an increasing amplitude. The idea of Nishikawa et al. was to reproduce the conditions of a seismic solicitation hitting a bridge pier, where P represented the dead load of the infrastructure over the steel column and H represented a simplify shock wave. The magnitude of P was set to be 0.124 the squash load Py. The parameters Hy and δy in Table 1 represent the horizontal load and horizontal displacement when the specimen yields close to the base.
a)
b)
Figure 2 a) Model and geometry of the bridge pier, b) loading sequence for the model calibration. Table 1. Structural parameters of the specimen.
h t D P/Py Hy δy
3,403 [mm] 8.70 [mm] 891.3 [mm] 0.124 414.9 [kN] 10.5
3.2. The material parameters calibration The DSS requires the calibration of 13 parameters for the elastoplastic behavior and 4 material constants for the definition of the MC failure envelope. Few material parameters, such as the Young’s modulus, Poisson’s ratio and the yield stress were assumed directly from Nishikawa’s experimental work. The remaining constants were calibrated minimizing the difference between the numerical and experimental curves obtained in uniaxial tensile tests in (Goto et al., 2010; Van Do et al., 2014). The results of the calibration are reported in the Figure 3 whereas Table 2 and 3 report the values of the parameters. The four constants Re, u, c and χ in Table 2 are proper of the subloading surface model. They regulate the amount of plastic deformation generated in the sub-yield state, the details are available in Hashiguchi (2009) and in Tsutsumi et al. (2006). The characterization of the MC criterion cannot be done by using the experimental results from one single test (i.e. uniaxial extension) since a unique definition of the failure envelope requires at least few points in the , , f space. Therefore, the ductile damage parameters were preliminary set for the uniaxial tensile test and then adjusted to fit the pier behavior. The blue curve in Figure 3 represent the final stage of calibration obtained after the adjustment of the damage parameters in the pier analyses. As it can be seen the FE simulation overestimate the material performances reported by Van Do et al. (2014), however the blue curve seems to be in good agreement with the results reported in Goto et al. (2010) for the SS400 steel. Moreover, the authors run some numerical analyses fitting the uniaxial curve in Van Do et al. (2014), however, the subsequent analyses on the pier sample gave an unsatisfactory
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approximation of the peak values in the first four cycles, where the damage contribution can be considered as negligible. Therefore, the set of constants in Table 2 was preferred for the material description. The value of the cumulative plastic strain defining the stress plateau Hd was taken directly from Goto et al. (2010)’s work, and it was supposed by the authors that the damage starts to affect the material response right after the plateau (i.e. d1 = Hd). At this point, it is worth mentioning that the uniaxial tensile curve in Figure 3 is not affected by the constant T3 in Eq. (8), since the loading is perfectly proportional. The calibration of T3 was obtained in the following analyses of the thin steel bridge.
Figure 3 Uniaxial behaviour of the SS400 steel. Table 2. Elastoplastic parameters for the DSS.
Young’s modulus Poisson’s ratio u F0 Re Hd K, h1, h2 C1, B1 C2, B2 c χ
Table 3. Damage parameters for the modified Mohr-Coulomb’s law.
A N c1 c2 d1
206000 [MPa] 0.3 750 294 [MPa] 0.4 0.0183 140 [MPa], 0.30, 17.0 2755 [MPa], 23.01 590 [MPa], 22.84 400 0.9 700 [MPa] 0.25 0.15 480 [MPa] 0.0183
3.3. Unidirectional loading The present section reports the results obtained in the simulation of the steel pier under a unidirectional cyclic loading condition. Figure 4a and b report the normalized horizontal load vs the normalized horizontal displacement for the DSS model consider the damage law of Eq. (8) without the contribution of the tangential inelastic stretch (i.e. green solid line, P-D law), with the contribution of the tangential inelastic stretch (i.e. solid blue line NP-D law) and without the damage (i.e. red curve). As it can be seen both the blue and the red lines can catch the maximum normalized
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horizontal load on the positive side but they tend to overestimate the load on the negative side. Moreover, after the fifth cycle, there is a general overestimation of the material performances. This drawback can be corrected accounting for the additional contribution triggered by the tangential components of the stress rate that allows the numerical response overlapping the experimental results in the last cycles. The material parameter T3 was set to be equal to 0.6. The same behavior can be observed in Figure 5a, where the peaks of the horizontal load are plotted against the corresponding maximum lateral displacements for the positive side of the loading.
b)
a)
c) Figure 4 a) Hysteresis loops for the pier in Figure 2a (proportional and non-proportional damage laws), b) hysteresis loops for the pier in Figure 2a (without damage), c) example of the absorbed energy during the first cycle.
All the curves overestimate the initial elastic response of the pier, probably due to some uncertainties on the elastic constants and the assumption adopted in the FE for modeling the real geometry of the pier. However, the degree of approximation of the peak of the horizontal load can be considered as satisfactory. The post-peak behavior is overestimated by the red and green curves but is very well described by the solution that considers the non-proportional damage evolution law. Nishikawa et al. (1998) evaluated the horizontal load-carrying capacity with an energy-based criterion, investigating the work done per loop by the horizontal force, as schematically displayed in Figure 4c. The normalization factor E0 is represented by the elastic work done by the horizontal force up to the point when the pier starts to yield:
E0 H y y / 2
(10)
Figure 5b shows the results obtained with the DSS model that point out, once again, that the simulation, carried out without considering the damage, cannot be assumed as reliable for the description of the structural behavior. The
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coupling of the elastoplastic and ductile damage internal variables offers a better agreement with the experimental data, especially if the tangential inelastic stretch contribution is considered. However, the graph in Figure 5b can give just a qualitative indication on the structural response since the area of the hysteresis loops is subjected to some imprecisions such as the impossibility to control experimentally the exact maximum (positive or negative) prescribed displacements. As a general tendency, the blue solid line can catch quite realistically the absorbed peak and the postpeak behavior of the pier.
a)
b)
c)
d)
Figure 5 a) Envelope of the positive side, b) energy absorption per cycle for the loading sequence of Figure 2b, c9 radial displacement around the base of the pier column, d) damage evolution (centroid) for three points at the base of the steel pier.
As an additional check, Figure 5c compares the numerical and experimental lateral displacement at the unloading point near the maximum loading per loop (i.e. y , 2 y , , 9 y ). The numerical results are in good agreement with the real radial displacements reported with red marks for the 3 y and 6 y loops reported in Van Do et al. (2014) and Gao et al. (1998). The maximum radial displacements take place at around 120 mm from the base of the pier, which seems to be slightly lower than the experimental evidence. However, similar numerical results were obtained by Van Do et al. (2014) and Gao et al. (1998) who investigated the same study case. One of the reasons for the discrepancy between the experimental and FE results might be due to the influence of the welded area at the base of the column, which was simulated as a pure ‘encastre’ in the numerical modeling.
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The local damage behavior of the pier was analyzed at three points around the base of the column as schematically indicated in Figure 5d. It has to be said that the authors could not find any experimental data to compare with the FE simulation and the local ductile damage evolution is reported as a consequence of the calibration on the global response of the structure. However, it is interesting to notice how the contribution of the inelastic tangential stretch accelerates the damage accumulation which can explain why the ductility seems to be higher when the load is proportional. For example, the point B is completely undamaged in the solid red line but it shows some damage evolution whenever the tangential inelastic stretch is considered. Moreover, due to the asymmetry of the MC failure envelope, the different damage accumulation in compression and tension can be appreciated. All the three points A, B and C refer to the centroid values of the elements located in the internal wall of the pier, where the numerical simulation showed the maximum damage contribution. 3.4. Bidirectional loading This last set of analyses aims to investigate which kind of non-proportional loading represents the most critical scenario for the steel column. Three different bidirectional loading conditions were applied with increasing amplitude, similarly to the previous unidirectional case. The first load is a squared type loading SQ, the second one is a circular type loading CR and the last one is butterfly type loading BA. Figure 6 schematically displays these new boundary conditions that have been also investigated in a previous work of the authors Momii et al. (2015). All the numerical analyses in this section were conducted considering the whole cylindrical section, and not half as in the previous case, due to the bidirectional nature of the load. However, the body was modeled with linear hexahedral elements with reduced integration (Abaqus C3D8R elements) up to a height equal to 2D from the base of the column, the remaining part was modeled with a beam element as in Figure 2a.
a)
b)
c)
Figure 6 a) Square type loading SQ, b) circular type loading CR, c) butterfly type loading BA.
The following Figure 7a reports the damage evolutions at the most damaged element (element centroid) for the different loading paths. The solid lines refer to the solutions obtained neglecting the contribution of the tangential inelastic stretch, instead the dotted lines refer to the solution obtained using the damage evolution law in Eq. (8) and setting the material parameter T3 = 0.6. As it can be seen the solid lines display almost the same level of damage at the end of the analyses, which is, more or less the same damage level of the unidirectional loading analysis in Figure 5d. A different trend is shown for the NP-D laws where the CR loading path generate the highest damage, followed by the butterfly type and the squared type. The SQ loading path manifest a marked effect of the non-proportionality of the load at the beginning of the damage evolution (i.e. third and fourth cycles). In the BA type of loading the contribution of the inelastic stretch seems less relevant at the beginning, however the damage grows with a higher rate than in the SQ case, and the two dotted lines cross each other at the beginning of the eight cycle. A different tendency was observed in the circular path, where the red line shows a non-linear increase until rapture (i.e. D ≈ 1).
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Figure 7 a) Damage evolution (centroid) for the most damaged element with P-D and NP-D laws, b) damage difference, for the most damaged element, between the NP-D law and the P-D law.
a)
b)
c)
d)
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f)
Figure 8 a) and b) Normalized force vs normalized displacement along x and y for the SQ path; c) and d) normalized force vs normalized displacement along x and y for the CR path; e) and f) normalized force vs normalized displacement along x and y for the BA path.
a)
b)
c)
d)
Figure 9 Cumulative inelastic strain Hi of Eq. (8); a) SQ path (Hi max ≈ 1.2), a) CR path (Hi max ≈ 1.37), a) BA path (Hi max ≈ 1.0), a) Uniaxial path (Hi max ≈ 1.14).
The reason for this behaviour might be duplex. First, the tangential contribution is a function of the similarity ratio R through the Eq. (9), which is constantly maximized in the CR path compared to the SQ and BA paths. Secondly, the CR path induces a constant rotation of the principal stress and strain axis, constantly triggering a tangential component of the stress rate. The comparison of the maximum damage level between the unidirectional and bidirectional loadings reveals that the paths SQ and BA have a slightly higher damage at the ninth cycle compared to the uniaxial loading, D = 0.615 and D = 0.66 respectively. The crack is potentially formed at the beginning of the eight cycle in the circular path. Figure 8 reports the normalized lateral force vs the normalized displacement along the x and y axes for the three bidirectional loading paths. Compared with the unidirectional loading the force peaks for the SQ path are slightly higher and the CR and BA paths showed lower values of the normalized force. Figure 9 shows the cumulative inelastic strain Hi of Eq. (8) for the three bidirectional paths and the unidirectional path of section 3.3. As it can be seen the inelastic strains of the unidirectional path are localized at the base of the pier along the loading direction. The three bidirectional paths show a distribution of the inelastic strain along the circumference of the column. In particular, the path CR displays higher values of Hi with a peak of 1.37, and this may explain the damage evolution of Figure 7.
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4. Conclusions The present paper introduced a coupled elastoplastic and damage model for the evaluation of a thin wall steel bridge subjected to unidirectional and bidirectional non-proportional loading. The ductile damage was modeled with a modified version of the Mohr-Coulomb failure criteria that proved to give reliable results not only for granular materials (i.e. soil, rock, concrete) but also for metallic materials (Algarni et al., 2017, 2015; Bai and Wierzbicki, 2010; Papasidero et al., 2015). Moreover, the ductile damage evolution law was modified by the authors to take into account a different accumulation during non-proportional loading paths. Experimental works (Algarni et al., 2017; Cortese et al., 2016; De Freitas et al., 2006; Papasidero et al., 2015) pointed out that the total deformation at fracture is higher during proportional loading, suggesting that, in case of non-proportional loading, a different mechanism for the damage accumulation has to be considered. In the present paper, the authors took into account a previous idea developed in Hashiguchi and Tsutsumi (2001, 1993), in Momii et al. (2015) and in Tsutsumi and Kaneko (2008) for explaining the acceleration of the damage during non-proportional loading. The damage rate is a function not only of the plastic strain rate, generated along the normal to the plastic potential, but also of a tangential inelastic stretch generated by a deviatoric component of the stress rate, tangential to the yield surface. It is worth mentioning that this additional contribution arises only during the non-coaxiality of the principal stress and strain direction and it is negligible whenever the coaxiality is re-established. This allows considering the different mechanisms of the ductile damage accumulation during proportional and non-proportional loading conditions. The main result of the paper can be summarized in the following points:
The calibration of the model parameters was done reproducing a uniaxial tensile test data for the SS400 steel. The elastoplastic constants, chosen in this work, approximate well the uniaxial behavior of the SS400 steel reported in Gao et al. (1998). A calibration based on the uniaxial behavior suggested in Van Do et al. (2014) was not able to model the horizontal load peaks of the first few cycles of the pier sample.
A single experimental test is not enough for the calibration of the MC constants, since the definition of the failure
envelope is a non-linear function in the , , f . Therefore, an adjustment of the MC constants was done during the calibration of the steel column subjected to unidirectional loading.
The investigations on the pier under unidirectional loading revealed that a better approximation of the real structure behavior can be achieved considering the contribution to the damage of the inelastic tangential stretch.
The analyses on the local behavior of the structure pointed out that the damage accumulation is accelerated considerably by the NP-D law (i.e. increments of around 50% at the end of the analysis). However, a future campaign of investigation on the crack formation at the base of the structure is needed to characterize the ductile damage behavior of the pier.
The application of a bidirectional load to the structure revealed that in the CR path the effect of the tangential inelastic stretch is more relevant that in the SQ and BA, which showed similar damage level to the one obtained in the unidirectional loading.
Future works will be focused to design experimental and numerical analyses for a better understanding of the ductile damage evolution in non-proportional loading since it represents a challenge still open in the CDM community. References Algarni, M., Bai, Y., Choi, Y., 2015. A study of Inconel 718 dependency on stress triaxiality and Lode angle in plastic deformation and ductile fracture. Eng. Fract. Mech. 147, 140–157. https://doi.org/10.1016/j.engfracmech.2015.08.007 Algarni, M., Choi, Y., Bai, Y., 2017. A unified material model for multiaxial ductile fracture and extremely low cycle fatigue of Inconel 718. Int. J. Fatigue 96, 162–177. https://doi.org/10.1016/j.ijfatigue.2016.11.033 Bai, Y., Teng, X., Wierzbicki, T., 2009. On the application of stress triaxiality formula for plane strain fracture testing. J. Eng. Mater. Technol.
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Riccardo et al.Integrity / Procedia Structural Integrity 9 (2018) 136–150 Author nameFincato / Structural Procedia 00 (2018) 000–000
149
131, 21002. https://doi.org/10.1115/1.3078390 Bai, Y., Wierzbicki, T., 2010. Application of extended Mohr–Coulomb criterion to ductile fracture. Int. J. Fract. 161, 1–20. https://doi.org/10.1007/s10704-009-9422-8 Bao, Y., Treitler, R., 2004. Ductile crack formation on notched Al2024-T351 bars under compression–tension loading. Mater. Sci. Eng. A 384, 385–394. https://doi.org/10.1016/j.msea.2004.06.056 Bao, Y., Wierzbicki, T., 2004. On fracture locus in the equivalent strain and stress triaxiality space. Int. J. Mech. Sci. 46, 81–98. https://doi.org/10.1016/j.ijmecsci.2004.02.006 Brünig, 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. Plast. 50, 49–65. https://doi.org/10.1016/j.ijplas.2013.03.012 Chaboche, J.L., 1986. Time-independent constitutive theories for cyclic plasticity. Int. J. Plast. 2, 149–188. https://doi.org/10.1016/07496419(86)90010-0 Cortese, L., Nalli, F., Rossi, M., 2016. A nonlinear model for ductile damage accumulation under multiaxial non-proportional loading conditions. Int. J. Plast. 85, 77–92. https://doi.org/10.1016/j.ijplas.2016.07.003 De Freitas, M., Reis, L., Li, B., 2006. Comparative study on biaxial low-cycle fatigue behaviour of three structural steels. Fatigue Fract. Eng. Mater. Struct. 29, 992–999. https://doi.org/10.1111/j.1460-2695.2006.01061.x Faleskog, J., Barsoum, I., 2013. Tension–torsion fracture experiments—Part I: Experiments and a procedure to evaluate the equivalent plastic strain. Int. J. Solids Struct. 50, 4241–4257. https://doi.org/10.1016/j.ijsolstr.2013.08.029 Fincato, R., Tsutsumi, S., 2017a. A return mapping algorithm for elastoplastic and ductile damage constitutive equations using the subloading surface method. Int. J. Numer. Methods Eng. https://doi.org/10.1002/nme.5718 Fincato, R., Tsutsumi, S., 2017b. Numerical study of a welded plate instability using the subloading surface model. Mar. Struct. 55. https://doi.org/10.1016/j.marstruc.2017.05.001 Fincato, R., Tsutsumi, S., 2017c. Ductile Damage Evolution under Non-Proportional Loading. J. Japan Soc. Civ. Eng. Ser. A2 (Applied Mech. 73, I_355-I_361. https://doi.org/10.2208/jscejam.73.I_355 Gao, S., Usami, T., Ge, H., 1998. Ductility Evaluation of Steel Bridge Piers with Pipe Sections. J. Eng. Mech. 124, 260–267. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:3(260) Gao, X., Zhang, T., Zhou, J., Graham, S.M., Hayden, M., Roe, C., 2011. On stress-state dependent plasticity modeling: Significance of the hydrostatic stress, the third invariant of stress deviator and the non-associated flow rule. Int. J. Plast. 27, 217–231. https://doi.org/10.1016/j.ijplas.2010.05.004 Goto, Y., Kumar, G.P., Kawanishi, N., 2010. Nonlinear Finite-Element Analysis for Hysteretic Behavior of Thin-Walled Circular Steel Columns with In-Filled Concrete. J. Struct. Eng. 136, 1413–1422. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000240 Gurson, A.L., 1977. Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I—Yield Criteria and Flow Rules for Porous Ductile Media. J. Eng. Mater. Technol. 99, 2. https://doi.org/10.1115/1.3443401 Hashiguchi, K., 2009. Elastoplasticity theory, in: Lecture Notes in Applied and Computational Mechanics. pp. 1–406. https://doi.org/10.1007/978-3-642-00273-1_1 Hashiguchi, K., 1989. Subloading surface model in unconventional plasticity. Int. J. Solids Struct. 25, 917–945. https://doi.org/10.1016/00207683(89)90038-3 Hashiguchi, K., Tsutsumi, S., 2001. Elastoplastic constitutive equation with tangential stress rate effect. Int. J. Plast. 17, 117–145. https://doi.org/10.1016/S0749-6419(00)00021-8 Hashiguchi, K., Tsutsumi, S., 1993. Fundamental requirements and formulation of elastoplastic constitutive equations with tangential plasticity. Int. J. Plast. 9, 525–549. https://doi.org/10.1016/0749-6419(93)90018-L Hashiguchi, K., Yoshimaru, T., 1995. A generalized formulation of the concept of nonhardening region. Int. J. Plast. 11, 347–365. https://doi.org/10.1016/S0749-6419(95)00003-8 Kachanov, L.M., 1958. Time of the rupture process under creep conditions. Izv Akad Nauk S S R Otd Tech Nauk 8, 26–31. https://doi.org/citeulike-article-id:5466815 Lemaitre, J., 1985. Coupled elasto-plasticity and damage constitutive equations. Comput. Methods Appl. Mech. Eng. 51, 31–49. https://doi.org/10.1016/0045-7825(85)90026-X Momii, H., Tsutsumi, S., Fincato, R., 2015. Cyclic and Tangential Plasticity Effects for the Buckling Behavior of a Thin Wall Pier under Multiaxial and Non-proportional Loading Conditions. Trans. JWRI 44. Nishikawa K., Yamamoto S., Natori T., Terao K., Yasunami H., Terada M., 1998. Retrofitting for seismic upgrading of steel bridge columns. Eng. Struct. 20, 540–551. https://doi.org/10.1016/S0141-0296(97)00025-4 Papasidero, J., Doquet, V., Mohr, D., 2015. Ductile fracture of aluminum 2024-T351 under proportional and non-proportional multi-axial
150
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15
loading: Bao–Wierzbicki results revisited. Int. J. Solids Struct. 69–70, 459–474. https://doi.org/10.1016/j.ijsolstr.2015.05.006 Papasidero, J., Doquet, V., Mohr, D., 2014. Determination of the Effect of Stress State on the Onset of Ductile Fracture Through Tension-Torsion Experiments. Exp. Mech. 54, 137–151. https://doi.org/10.1007/s11340-013-9788-4 Rousselier, G., 1987. Ductile fracture models and their potential in local approach of fracture. Nucl. Eng. Des. 105, 97–111. https://doi.org/10.1016/0029-5493(87)90234-2 Rousselier, G., Luo, M., 2014. A fully coupled void damage and Mohr–Coulomb based ductile fracture model in the framework of a Reduced Texture Methodology. Int. J. Plast. 55, 1–24. https://doi.org/10.1016/j.ijplas.2013.09.002 Tsutsumi, S., Kaneko, K., 2008. Constitutive response of idealized granular media under the principal stress axes rotation. Int. J. Plast. https://doi.org/10.1016/j.ijplas.2008.05.001 Tsutsumi, S., Toyosada, M., Hashiguchi, K., 2006. Extended Subloading Surface Model Incorporating Elastic Boundary Concept. J. Appl. Mech. 9, 455–462. https://doi.org/10.2208/journalam.9.455 Van Do, V.N., Lee, C.H., Chang, K.H., 2014. A nonlinear CDM model for ductile failure analysis of steel bridge columns under cyclic loading. Comput. Mech. https://doi.org/10.1007/s00466-013-0964-2
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IGF Workshop “Fracture and Structural Integrity” IGF Workshop “Fracture and Structural Integrity”
Ductile damage evolution under cyclic non-proportional loading Ductile damage evolution under cyclic non-proportional loading paths XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal paths *,a a b Riccardo Fincato , Seiichiro Tsutsumi , Hideto Momii Thermo-mechanical modeling of a high pressure turbine blade of an *,a a Riccardo Fincato , Seiichiro Tsutsumi , Hideto Momiib JWRI, Osaka University, 11-1 Mihogaoka, Ibaraki 5670047, Osaka, Japan airplane gasKamitonno, turbine engine Osaka Corporation,4727-1 University, 11-1 Mihogaoka, Ibaraki 5670047, Osaka, Japan Japan Nikken JWRI, Engineering Nogata 822-0003, Fukuoka, a a
b
Nikken Engineering Corporation,4727-1 Kamitonno, Nogata 822-0003, Fukuoka, Japan
b
P. Brandãoa, V. Infanteb, A.M. Deusc*
AbstractaDepartment of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal Abstract b IDMEC, Department of Mechanical Superior Técnico, de Lisboa, Av. Rovisco Pais, 1, Lisboa, The characterization of ductile damageEngineering, evolution, Instituto and its description, haveUniversidade been the object of extensive research in 1049-001 the continuum Portugal Theccharacterization of ductile evolution, andworks its description, have beenGurson the object of extensive research in many the continuum damage mechanic field. Startingdamage from the pioneering of Lemaitre (1985), (1977), Rousselier (1987), different CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, damage mechanic Starting thethe pioneering works ofand Lemaitre (1985), (1977), (1987),asmany different models have been field. developed. In from detail, stress triaxiality the Lode angleGurson parameters haveRousselier been identified the two main Portugal models have developed. In detail, the stress theaLode parameters have been identified as the two main variables thatbeen affect the material ductility. The triaxiality literature and offers greatangle number of investigations under monotonic loading variables that affect athe material ductility. The literature a great number investigations under monotonic conditions, however, proper characterization of the damageoffers evolution under cyclicofloading or non-proportional loadingloading is still conditions, Abstracthowever, a proper characterization of the damage evolution under cyclic loading or non-proportional loading is still missing. missing. In this paper, an unconventional coupled elastoplastic and damage constitutive model with a Mohr-Coulomb failure criterion (Bai In this paper, an 2010) unconventional coupled elastoplastic and damagelaw constitutive model withto a Mohr-Coulomb failure criterion (Bai their operation, modern aircraft engine components are to increasingly demanding operating conditions, andDuring Wierzbicki, is presented. The ductile damage evolution issubjected modified in order consider the damage evolution under the 2010) high pressure turbine (HPT) blades. Such conditions cause these parts different types of time-dependent andespecially Wierzbicki, is presented. The ductile damage evolution is modified in order to consider the damage evolution non-proportional loading conditions. The idea is to compare the law damage evolution oftoa undergo steel bridge column subjected to under three degradation, one of which is creep. A in model the finite element method (FEM) was developed, in ordersubjected to be abletotothree predict non-proportional loading conditions. The idea istotoidentify compare damage evolution a steel bridge column different non-proportional loading paths orderusing thethe loading condition thatof compromise the structural integrity. the creep behaviour ofloading HPT blades. data records a specific provided by a commercial different non-proportional paths inFlight order to identify the(FDR) loadingfor condition that aircraft, compromise the structural integrity. aviation company, were used to obtain © 2018 The Authors. Published bythermal Elsevierand B.V.mechanical data for three different flight cycles. In order to create the 3D model © 2018 2018 The Authors. Published by Elsevier B.V.blade needed the responsibility FEM analysis, HPT scrap was scanned, its chemical composition and material properties were © Thefor Authors. Published by B.V.Italiano Peer-review under ofaElsevier the Gruppo Frattura (IGF) and ExCo. Peer-review under responsibility ofgathered the Gruppo Italiano Frattura (IGF)model ExCo. and different simulations were run, first with a simplified 3D obtained. The data that was was fed into the FEM Peer-review under responsibility of the Gruppo Italiano Frattura (IGF) ExCo. rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The Keywords: Ductile damage; elastoplasticity; non-proportional loading; overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a Keywords: Ductile damage; elastoplasticity; non-proportional loading; model can be useful in the goal of predicting turbine blade life, given a set of FDR data.
1. © Introduction 2016 The Authors. Published by Elsevier B.V. 1. Peer-review Introduction under responsibility of the Scientific Committee of PCF 2016. In the recent years, many authors tried to characterize the ductile behavior of metals by means of different types of In the recent many authors triedand to characterize the ductile behavior of metals by means of different types of Keywords: High years, Pressure Turbine Blade; Creep; Finite Element models, Method; 3D Model; Simulation. numerical models (coupled elastoplastic damage uncoupled models, phenomenological, etc.). The large numerical models (coupled elastoplastic and damage models, uncoupled models, phenomenological, etc.). The large Corresponding author. Tel.: 81-6-6879-8667. Corresponding Tel.: 81-6-6879-8667. E-mail address:author. [email protected] E-mail address: [email protected] 2452-3216 © 2018 The Authors. Published by Elsevier B.V. 2452-3216 © 2018 Authors. Published Elsevier B.V. Frattura (IGF) ExCo. Peer-review underThe responsibility of theby Gruppo Italiano Peer-review underauthor. responsibility the Gruppo Italiano Frattura (IGF) ExCo. * Corresponding Tel.: +351of218419991. E-mail address: [email protected] * *
2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016. 2452-3216 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Gruppo Italiano Frattura (IGF) ExCo. 10.1016/j.prostr.2018.06.022
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campaign of experimental investigations carried out in the last decades (Algarni et al., 2015; Bai et al., 2009; Bao and Treitler, 2004; Bao and Wierzbicki, 2004; Brünig et al., 2013; Gao et al., 2011; Papasidero et al., 2015) pointed out that the stress triaxiality and the Lode angle are the main two factors that affect the ductility behavior in metals. In fact, monotonic loading tests, for different geometries and under different loading conditions (uniaxial extension, pure shear, plane strain, uniaxial compression), proved a different failure behavior for the same material. On the other hand, the literature still lacks a proper characterization of the damage evolution under non-proportional loading paths. In the recent years, Faleskog and Barsoum (2013), Papasidero et al. (2014), Cortese et al. (2016) and Algarni et al. (2017) conducted a series of experiments aimed to identify the effect of the loading path on ductility. In detail, the work of Faleskog and Barsoum (2013), Papasidero et al. (2014) and Corstese et al. (2016) consisted in the application of several non-proportional loading paths, as the result of the combination of tension-torsion or compression-torsion, on steel and aluminum tubular samples. On the other hand, Algarni et al. tried to describe the crack formation on notched Iconel 718 bars in low cycle fatigue investigations. A common aspect that emerged from those previous works is that the deformation at fracture is higher when the load is proportional, suggesting that the damage accelerates whenever non-proportional loading conditions are triggered. The present paper aims to investigate the influence of the loading path on the ductile damage evolution. In detail, the numerical analyses focus the attention of the structural response of a steel bridge column subjected to various loading condition. A modified Mohr-Coulomb criterion (Bai and Wierzbicki, 2010) is adopted for the description of the damage behavior of SS400 steel pier, with a modification of the damage evolution law, in order to take into account the effect of the non-proportionality of the load. Next, section 2, the theoretical Damage Subloading surface model (i.e. DSS) is presented, with particular emphasis on the ductile damage criterion. The subsequent section 3 is divided into two parts; the first one deals with the calibration of the material parameters, whereas the second offers an overview of the effect of three non-proportional loadings on the bridge pier. Nomenclature
σ, σ α, α s σ α E F F0 H HD
R λ D σm
Mises η
f
a b
Cauchy stress, corotational stress rate back stress, corotational back stress rate similarity centre conjugate Cauchy stress conjugate back stress tensor of the elastic constants isotropic-hardening function initial size of the normal-yield surface isotropic hardening variable cumulative plastic variable (i.e. equivalent plastic strain) similarity ratio plastic multiplier ductile damage scalar variable mean stress von Mises stress stress triaxiality Lode angle Lode angle parameter (-1 < < 1) equivalent strain at fracture Heaviside step function: a b 0 if a b 0; a b 1 if a b 0
2. Theoretical approach The present section deals with the theoretical framework for the description of the elastoplastic and damage model named Damage Subloading Surface model (i.e. DSS). The DSS model was formulated from the unconventional plasticity model Extended Subloading Surface, presented by Hashiguchi in Hashiguchi (2009, 1989), and upgraded to
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3
include an additional internal variable for the description of the damaging behavior. In the next section 2.1 the main features of the DSS model are briefly reported. A detailed description of the constitutive equations goes beyond the purposes of this paper, the reader is referred to Fincato and Tsutsumi (2017a) for an exhaustive discussion. 2.1. The Damage Subloading Surface model The DSS model is a coupled elastoplastic and damage model that derives it formulation from a previous unconventional plasticity theory, named Extended subloading surface model. The main feature consists in the abolition of the neat distinction between the elastic and plastic domains, stating that irreversible deformations can be generated whenever the loading criterion is satisfied. In order to achieve this goal a second surface, named subloading surface, is generated by means of a similarity transformation from the conventional yield surface, here renamed normal yield-surface. The center of similarity is not fixed in the stress space but it can move following the plastic strain rate. This allows to the creation of close hysteresis loops during the unloading and subsequent reloading, as shown by the authors in Fincato and Tsutsumi (2017b), for a more realistic description of the material ratcheting in cyclic mobility and fatigue problems. The numerical model is developed within the framework of finite elastoplasticity, assuming a hypoelastic based plasticity. The total strain rate D can be additively decomposed into and elastic part De and a plastic part D p . The coupling of the elasticity with damage is obtained by means of the concept of the effective stress Kachanov (1958) as follows:
σ 1 D E : De 1 D E : (D D p )
(1)
The coupling of the plastic internal variables with the scalar isotropic damage variable is obtained following the approach suggested by Lemaitre (1985) and modifying the expression of the consistency conditions for the normalyield and subloading surface as follows:
f (σˆ ) (1 D)F (H );
f (σ) (1 D)RF (H ); σˆ σ α
(2)
The scalar function f of the stress tensor in Eq. (2) is assumed as the von Mises criterion. The plastic strain rate can be obtained from Eq. (2) as shown in Fincato and Tsutsumi (2017a), giving the following expression:
D p / (1 D) f σ / σ ; H ; H D / (1 D);
(3)
In order to complete the set of equations describing the material behavior, the following isotropic and kinematic hardening laws are reported in Eqs. (4)-(5). The isotropic hardening law is a modified version of the law presented in Hashiguchi and Yoshimaru (1995), with an additional linear contribution regulated by the constant K. Eq. (5) was initially proposed by Chaboche (1986), and considered the linear combination of N non-linear contributions. The present paper considers N = 2 contributions. The calibration of the material constants K, h1, h2, Ci, Bi will be discussed in section 3.2.
F ( H ) F0 K H H d F0 h1 1 eh2 H Hd n
αi 1,...N Ci D p Bi αi ; α αi , n
(4) (5)
i 1
2.2. The ductile damage criterion Following a well-consolidated approach adopted by the continuum damage mechanics an arbitrary stress state can be described by using two dimensionless parameters: the stress triaxiality and the Lode angle parameter (see Figure 1b), defined as in Eq. (6):
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m Mises 1.0 6
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(6)
In particular, the Lode angle parameter field of existence covers all the possible loading conditions, from the uniaxial extension to the uniaxial compression passing by the pure shear (or plane strain) condition where it assumes null value. Therefore, every pair of variables , , representing a unique stress state, can be used in the formulation
of fracture envelope in the MC criterion. This criterion has been widely and successfully used in the prediction of the failure behavior of granular materials such as rock, soil and concrete. However, recent works (Algarni et al., 2015; Bai and Wierzbicki, 2010; Rousselier and Luo, 2014), applied the MC criterion to metallic materials due to its characteristic of being able to catch an exponential decay of the ductility with the stress triaxiality and its Lode angle dependency. The full expression of the MC requires the calibration of eight material parameters, however, if a von Mises potential is accounted for, the failure envelope can be defined by the calibration of only four material constants A, N , c1 , c2 (Bai and Wierzbicki, 2010). The use of a von Mises potential should be limited to cases where the effect of the Lode angle on plasticity and the pressure effect on the yield-surface can be neglected. Nonetheless, it can give a good first approximation of the damaging behavior of the material. Therefore, in the present paper we adopted the following form of the failure envelope:
A 1 c 2 1 1 f cos c1 sin c 3 3 6 6 2
1 N
(7)
Based on the previous Eq. (7) the ductile damage increment can be written as:
2 3 D p T3 Dt D f ,
H d1 ;
p p Hi H D HT 2 3 D T3 D
(8)
Where d1 is an additional material parameter and H represent the Heaviside step function, introduced by the authors to allow the damage to evolve whenever the cumulative plastic strain exceeds the parameter d1. The term at the numerator in Eq. (8) has been modified to take into account an additional inelastic term generated by the stress rate component tangential to the plastic potential as in Figure 1a. HT represents the cumulative inelastic strain that actively contributes to the damage. Non-proportional loading, in fact, triggers a non-negligible component of the stress rate that is directed tangentially to the yield surface. The idea is to consider the contribution of the strain rate associated with the tangential component of the stress rate in the damage accumulation. Previous work of the authors (Fincato and Tsutsumi, 2017c; Hashiguchi and Tsutsumi, 2001; Momii et al., 2015; Tsutsumi and Kaneko, 2008) considered a similar approach to overcome the excessive stiffness predicted by the extended subloading surface model with associate flow rule during non-proportional loading. In this case, the effect of the tangential deviatoric strain rate Dt affects only the damage evolution and its contribution can be regulated by the material constant T3 (0 T3 1) . The computation of the deviatoric tangential stress rate is done similarly as in Fincato and Tsutsumi (2017c) and in Momii et al. (2015). Briefly, it is assumed that the deviatoric tangential strain rate is linearly related to the component of the stress rate tangential to the yield surface σt . The advantage of this hypothesis lies in its very simple form, suitable for the application to boundary-value problems and general loading conditions.
Dt
1 T1RT2 A σ ; A σ Iˆ t ; σt 2G 2G 1 T1RT2
(9)
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a)
5
b)
Figure 1 a) sketch of tangential and normal components of stress rate for a generic stress state on the plastic surface; b) schematic representation of a generic stress state in the principal stress space.
where Iˆ is the deviatoric tangential (or projection) tensor (Hashiguchi, 2009) and the variable A is computed accordingly to Momii et al. (2015). In detail, A is function of the similarity ratio R and two material parameters T1 and T2 that were set to 0.9 and 1.0 in this study, respectively. It is worth mentioning that the constant T1 can assume values between 0 T1 1 . In conclusion, if the tangential inelastic stretch is considered then the cumulative inelastic strain to fracture Hi becomes a function of D p and Dt as in the second of Eq. (8). 3. Numerical analyses The present section deal with the numerical analyses carried out implementing the previous constitutive equations via user subroutine for the commercial code Abaqus (ver. 6.14-5). The DSS model parameters were calibrated reproducing a uniaxial tensile load for the SS400 steel reported in Van Do et al. (2014) and subsequently adjusted to reproduce the behavior of a thin steel bridge pier analyzed by Nishikawa et al. (1998). The analyses reported in section 3.4 offer an additional investigation that aims to point out the effect on the damage evolution of three non-proportional bidirectional loading conditions. 3.1. Description of the FE model Nishikawa et al. (1998) conducted a series of experiments on thin steel piers in order to analyse the horizontal loadcarrying capacity of the structures subjected to an increasing unidirectional horizontal load. The schematic of the pier is reported in Figure 2a together with its FE modeling, the loading sequence is instead reported in Figure 2b. In order to save computational time just half of the column was considered, applying symmetric boundary conditions. Moreover, the experimental results showed that the plastic deformations, and the buckling, appear in a region close to the base of the pier. Therefore, the structure was modeled according to its real geometry until a height equal to two times the internal diameter from the bottom and the remaining part was modeled with a beam element. A similar strategy was adopted by Gao et al. (1998). The geometric specifications are reported in Table 1. The lower part of the pier was meshed with eight-node hexahedral elements with reduced integration (i.e. Abaqus C3D8R elements) and the nodes of the top cross section were connected to the beam with rigid links. A mesh refinement was considered close to the base of the pier, where the maximum accumulation of plastic deformation is expected. The minimum element size is 9.8 mm (circumference) x 2.25 mm (thickness) x 15 mm (axial), comparable with the minimum element size used by Van Do et al. (2014), who conducted a mesh size sensitivity analysis on the same study case. The total number of elements amount to 51841.
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The structure is subjected to two types of loads: a compressive load P, constant, and kept for the whole duration of the experiment and a horizontal load H, directed along the x direction and with an increasing amplitude. The idea of Nishikawa et al. was to reproduce the conditions of a seismic solicitation hitting a bridge pier, where P represented the dead load of the infrastructure over the steel column and H represented a simplify shock wave. The magnitude of P was set to be 0.124 the squash load Py. The parameters Hy and δy in Table 1 represent the horizontal load and horizontal displacement when the specimen yields close to the base.
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Figure 2 a) Model and geometry of the bridge pier, b) loading sequence for the model calibration. Table 1. Structural parameters of the specimen.
h t D P/Py Hy δy
3,403 [mm] 8.70 [mm] 891.3 [mm] 0.124 414.9 [kN] 10.5
3.2. The material parameters calibration The DSS requires the calibration of 13 parameters for the elastoplastic behavior and 4 material constants for the definition of the MC failure envelope. Few material parameters, such as the Young’s modulus, Poisson’s ratio and the yield stress were assumed directly from Nishikawa’s experimental work. The remaining constants were calibrated minimizing the difference between the numerical and experimental curves obtained in uniaxial tensile tests in (Goto et al., 2010; Van Do et al., 2014). The results of the calibration are reported in the Figure 3 whereas Table 2 and 3 report the values of the parameters. The four constants Re, u, c and χ in Table 2 are proper of the subloading surface model. They regulate the amount of plastic deformation generated in the sub-yield state, the details are available in Hashiguchi (2009) and in Tsutsumi et al. (2006). The characterization of the MC criterion cannot be done by using the experimental results from one single test (i.e. uniaxial extension) since a unique definition of the failure envelope requires at least few points in the , , f space. Therefore, the ductile damage parameters were preliminary set for the uniaxial tensile test and then adjusted to fit the pier behavior. The blue curve in Figure 3 represent the final stage of calibration obtained after the adjustment of the damage parameters in the pier analyses. As it can be seen the FE simulation overestimate the material performances reported by Van Do et al. (2014), however the blue curve seems to be in good agreement with the results reported in Goto et al. (2010) for the SS400 steel. Moreover, the authors run some numerical analyses fitting the uniaxial curve in Van Do et al. (2014), however, the subsequent analyses on the pier sample gave an unsatisfactory
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approximation of the peak values in the first four cycles, where the damage contribution can be considered as negligible. Therefore, the set of constants in Table 2 was preferred for the material description. The value of the cumulative plastic strain defining the stress plateau Hd was taken directly from Goto et al. (2010)’s work, and it was supposed by the authors that the damage starts to affect the material response right after the plateau (i.e. d1 = Hd). At this point, it is worth mentioning that the uniaxial tensile curve in Figure 3 is not affected by the constant T3 in Eq. (8), since the loading is perfectly proportional. The calibration of T3 was obtained in the following analyses of the thin steel bridge.
Figure 3 Uniaxial behaviour of the SS400 steel. Table 2. Elastoplastic parameters for the DSS.
Young’s modulus Poisson’s ratio u F0 Re Hd K, h1, h2 C1, B1 C2, B2 c χ
Table 3. Damage parameters for the modified Mohr-Coulomb’s law.
A N c1 c2 d1
206000 [MPa] 0.3 750 294 [MPa] 0.4 0.0183 140 [MPa], 0.30, 17.0 2755 [MPa], 23.01 590 [MPa], 22.84 400 0.9 700 [MPa] 0.25 0.15 480 [MPa] 0.0183
3.3. Unidirectional loading The present section reports the results obtained in the simulation of the steel pier under a unidirectional cyclic loading condition. Figure 4a and b report the normalized horizontal load vs the normalized horizontal displacement for the DSS model consider the damage law of Eq. (8) without the contribution of the tangential inelastic stretch (i.e. green solid line, P-D law), with the contribution of the tangential inelastic stretch (i.e. solid blue line NP-D law) and without the damage (i.e. red curve). As it can be seen both the blue and the red lines can catch the maximum normalized
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horizontal load on the positive side but they tend to overestimate the load on the negative side. Moreover, after the fifth cycle, there is a general overestimation of the material performances. This drawback can be corrected accounting for the additional contribution triggered by the tangential components of the stress rate that allows the numerical response overlapping the experimental results in the last cycles. The material parameter T3 was set to be equal to 0.6. The same behavior can be observed in Figure 5a, where the peaks of the horizontal load are plotted against the corresponding maximum lateral displacements for the positive side of the loading.
b)
a)
c) Figure 4 a) Hysteresis loops for the pier in Figure 2a (proportional and non-proportional damage laws), b) hysteresis loops for the pier in Figure 2a (without damage), c) example of the absorbed energy during the first cycle.
All the curves overestimate the initial elastic response of the pier, probably due to some uncertainties on the elastic constants and the assumption adopted in the FE for modeling the real geometry of the pier. However, the degree of approximation of the peak of the horizontal load can be considered as satisfactory. The post-peak behavior is overestimated by the red and green curves but is very well described by the solution that considers the non-proportional damage evolution law. Nishikawa et al. (1998) evaluated the horizontal load-carrying capacity with an energy-based criterion, investigating the work done per loop by the horizontal force, as schematically displayed in Figure 4c. The normalization factor E0 is represented by the elastic work done by the horizontal force up to the point when the pier starts to yield:
E0 H y y / 2
(10)
Figure 5b shows the results obtained with the DSS model that point out, once again, that the simulation, carried out without considering the damage, cannot be assumed as reliable for the description of the structural behavior. The
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coupling of the elastoplastic and ductile damage internal variables offers a better agreement with the experimental data, especially if the tangential inelastic stretch contribution is considered. However, the graph in Figure 5b can give just a qualitative indication on the structural response since the area of the hysteresis loops is subjected to some imprecisions such as the impossibility to control experimentally the exact maximum (positive or negative) prescribed displacements. As a general tendency, the blue solid line can catch quite realistically the absorbed peak and the postpeak behavior of the pier.
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Figure 5 a) Envelope of the positive side, b) energy absorption per cycle for the loading sequence of Figure 2b, c9 radial displacement around the base of the pier column, d) damage evolution (centroid) for three points at the base of the steel pier.
As an additional check, Figure 5c compares the numerical and experimental lateral displacement at the unloading point near the maximum loading per loop (i.e. y , 2 y , , 9 y ). The numerical results are in good agreement with the real radial displacements reported with red marks for the 3 y and 6 y loops reported in Van Do et al. (2014) and Gao et al. (1998). The maximum radial displacements take place at around 120 mm from the base of the pier, which seems to be slightly lower than the experimental evidence. However, similar numerical results were obtained by Van Do et al. (2014) and Gao et al. (1998) who investigated the same study case. One of the reasons for the discrepancy between the experimental and FE results might be due to the influence of the welded area at the base of the column, which was simulated as a pure ‘encastre’ in the numerical modeling.
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The local damage behavior of the pier was analyzed at three points around the base of the column as schematically indicated in Figure 5d. It has to be said that the authors could not find any experimental data to compare with the FE simulation and the local ductile damage evolution is reported as a consequence of the calibration on the global response of the structure. However, it is interesting to notice how the contribution of the inelastic tangential stretch accelerates the damage accumulation which can explain why the ductility seems to be higher when the load is proportional. For example, the point B is completely undamaged in the solid red line but it shows some damage evolution whenever the tangential inelastic stretch is considered. Moreover, due to the asymmetry of the MC failure envelope, the different damage accumulation in compression and tension can be appreciated. All the three points A, B and C refer to the centroid values of the elements located in the internal wall of the pier, where the numerical simulation showed the maximum damage contribution. 3.4. Bidirectional loading This last set of analyses aims to investigate which kind of non-proportional loading represents the most critical scenario for the steel column. Three different bidirectional loading conditions were applied with increasing amplitude, similarly to the previous unidirectional case. The first load is a squared type loading SQ, the second one is a circular type loading CR and the last one is butterfly type loading BA. Figure 6 schematically displays these new boundary conditions that have been also investigated in a previous work of the authors Momii et al. (2015). All the numerical analyses in this section were conducted considering the whole cylindrical section, and not half as in the previous case, due to the bidirectional nature of the load. However, the body was modeled with linear hexahedral elements with reduced integration (Abaqus C3D8R elements) up to a height equal to 2D from the base of the column, the remaining part was modeled with a beam element as in Figure 2a.
a)
b)
c)
Figure 6 a) Square type loading SQ, b) circular type loading CR, c) butterfly type loading BA.
The following Figure 7a reports the damage evolutions at the most damaged element (element centroid) for the different loading paths. The solid lines refer to the solutions obtained neglecting the contribution of the tangential inelastic stretch, instead the dotted lines refer to the solution obtained using the damage evolution law in Eq. (8) and setting the material parameter T3 = 0.6. As it can be seen the solid lines display almost the same level of damage at the end of the analyses, which is, more or less the same damage level of the unidirectional loading analysis in Figure 5d. A different trend is shown for the NP-D laws where the CR loading path generate the highest damage, followed by the butterfly type and the squared type. The SQ loading path manifest a marked effect of the non-proportionality of the load at the beginning of the damage evolution (i.e. third and fourth cycles). In the BA type of loading the contribution of the inelastic stretch seems less relevant at the beginning, however the damage grows with a higher rate than in the SQ case, and the two dotted lines cross each other at the beginning of the eight cycle. A different tendency was observed in the circular path, where the red line shows a non-linear increase until rapture (i.e. D ≈ 1).
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Figure 7 a) Damage evolution (centroid) for the most damaged element with P-D and NP-D laws, b) damage difference, for the most damaged element, between the NP-D law and the P-D law.
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Figure 8 a) and b) Normalized force vs normalized displacement along x and y for the SQ path; c) and d) normalized force vs normalized displacement along x and y for the CR path; e) and f) normalized force vs normalized displacement along x and y for the BA path.
a)
b)
c)
d)
Figure 9 Cumulative inelastic strain Hi of Eq. (8); a) SQ path (Hi max ≈ 1.2), a) CR path (Hi max ≈ 1.37), a) BA path (Hi max ≈ 1.0), a) Uniaxial path (Hi max ≈ 1.14).
The reason for this behaviour might be duplex. First, the tangential contribution is a function of the similarity ratio R through the Eq. (9), which is constantly maximized in the CR path compared to the SQ and BA paths. Secondly, the CR path induces a constant rotation of the principal stress and strain axis, constantly triggering a tangential component of the stress rate. The comparison of the maximum damage level between the unidirectional and bidirectional loadings reveals that the paths SQ and BA have a slightly higher damage at the ninth cycle compared to the uniaxial loading, D = 0.615 and D = 0.66 respectively. The crack is potentially formed at the beginning of the eight cycle in the circular path. Figure 8 reports the normalized lateral force vs the normalized displacement along the x and y axes for the three bidirectional loading paths. Compared with the unidirectional loading the force peaks for the SQ path are slightly higher and the CR and BA paths showed lower values of the normalized force. Figure 9 shows the cumulative inelastic strain Hi of Eq. (8) for the three bidirectional paths and the unidirectional path of section 3.3. As it can be seen the inelastic strains of the unidirectional path are localized at the base of the pier along the loading direction. The three bidirectional paths show a distribution of the inelastic strain along the circumference of the column. In particular, the path CR displays higher values of Hi with a peak of 1.37, and this may explain the damage evolution of Figure 7.
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4. Conclusions The present paper introduced a coupled elastoplastic and damage model for the evaluation of a thin wall steel bridge subjected to unidirectional and bidirectional non-proportional loading. The ductile damage was modeled with a modified version of the Mohr-Coulomb failure criteria that proved to give reliable results not only for granular materials (i.e. soil, rock, concrete) but also for metallic materials (Algarni et al., 2017, 2015; Bai and Wierzbicki, 2010; Papasidero et al., 2015). Moreover, the ductile damage evolution law was modified by the authors to take into account a different accumulation during non-proportional loading paths. Experimental works (Algarni et al., 2017; Cortese et al., 2016; De Freitas et al., 2006; Papasidero et al., 2015) pointed out that the total deformation at fracture is higher during proportional loading, suggesting that, in case of non-proportional loading, a different mechanism for the damage accumulation has to be considered. In the present paper, the authors took into account a previous idea developed in Hashiguchi and Tsutsumi (2001, 1993), in Momii et al. (2015) and in Tsutsumi and Kaneko (2008) for explaining the acceleration of the damage during non-proportional loading. The damage rate is a function not only of the plastic strain rate, generated along the normal to the plastic potential, but also of a tangential inelastic stretch generated by a deviatoric component of the stress rate, tangential to the yield surface. It is worth mentioning that this additional contribution arises only during the non-coaxiality of the principal stress and strain direction and it is negligible whenever the coaxiality is re-established. This allows considering the different mechanisms of the ductile damage accumulation during proportional and non-proportional loading conditions. The main result of the paper can be summarized in the following points:
The calibration of the model parameters was done reproducing a uniaxial tensile test data for the SS400 steel. The elastoplastic constants, chosen in this work, approximate well the uniaxial behavior of the SS400 steel reported in Gao et al. (1998). A calibration based on the uniaxial behavior suggested in Van Do et al. (2014) was not able to model the horizontal load peaks of the first few cycles of the pier sample.
A single experimental test is not enough for the calibration of the MC constants, since the definition of the failure
envelope is a non-linear function in the , , f . Therefore, an adjustment of the MC constants was done during the calibration of the steel column subjected to unidirectional loading.
The investigations on the pier under unidirectional loading revealed that a better approximation of the real structure behavior can be achieved considering the contribution to the damage of the inelastic tangential stretch.
The analyses on the local behavior of the structure pointed out that the damage accumulation is accelerated considerably by the NP-D law (i.e. increments of around 50% at the end of the analysis). However, a future campaign of investigation on the crack formation at the base of the structure is needed to characterize the ductile damage behavior of the pier.
The application of a bidirectional load to the structure revealed that in the CR path the effect of the tangential inelastic stretch is more relevant that in the SQ and BA, which showed similar damage level to the one obtained in the unidirectional loading.
Future works will be focused to design experimental and numerical analyses for a better understanding of the ductile damage evolution in non-proportional loading since it represents a challenge still open in the CDM community. References Algarni, M., Bai, Y., Choi, Y., 2015. A study of Inconel 718 dependency on stress triaxiality and Lode angle in plastic deformation and ductile fracture. Eng. Fract. Mech. 147, 140–157. https://doi.org/10.1016/j.engfracmech.2015.08.007 Algarni, M., Choi, Y., Bai, Y., 2017. A unified material model for multiaxial ductile fracture and extremely low cycle fatigue of Inconel 718. Int. J. Fatigue 96, 162–177. https://doi.org/10.1016/j.ijfatigue.2016.11.033 Bai, Y., Teng, X., Wierzbicki, T., 2009. On the application of stress triaxiality formula for plane strain fracture testing. J. Eng. Mater. Technol.
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Riccardo et al.Integrity / Procedia Structural Integrity 9 (2018) 136–150 Author nameFincato / Structural Procedia 00 (2018) 000–000
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131, 21002. https://doi.org/10.1115/1.3078390 Bai, Y., Wierzbicki, T., 2010. Application of extended Mohr–Coulomb criterion to ductile fracture. Int. J. Fract. 161, 1–20. https://doi.org/10.1007/s10704-009-9422-8 Bao, Y., Treitler, R., 2004. Ductile crack formation on notched Al2024-T351 bars under compression–tension loading. Mater. Sci. Eng. A 384, 385–394. https://doi.org/10.1016/j.msea.2004.06.056 Bao, Y., Wierzbicki, T., 2004. On fracture locus in the equivalent strain and stress triaxiality space. Int. J. Mech. Sci. 46, 81–98. https://doi.org/10.1016/j.ijmecsci.2004.02.006 Brünig, 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. Plast. 50, 49–65. https://doi.org/10.1016/j.ijplas.2013.03.012 Chaboche, J.L., 1986. Time-independent constitutive theories for cyclic plasticity. Int. J. Plast. 2, 149–188. https://doi.org/10.1016/07496419(86)90010-0 Cortese, L., Nalli, F., Rossi, M., 2016. A nonlinear model for ductile damage accumulation under multiaxial non-proportional loading conditions. Int. J. Plast. 85, 77–92. https://doi.org/10.1016/j.ijplas.2016.07.003 De Freitas, M., Reis, L., Li, B., 2006. Comparative study on biaxial low-cycle fatigue behaviour of three structural steels. Fatigue Fract. Eng. Mater. Struct. 29, 992–999. https://doi.org/10.1111/j.1460-2695.2006.01061.x Faleskog, J., Barsoum, I., 2013. Tension–torsion fracture experiments—Part I: Experiments and a procedure to evaluate the equivalent plastic strain. Int. J. Solids Struct. 50, 4241–4257. https://doi.org/10.1016/j.ijsolstr.2013.08.029 Fincato, R., Tsutsumi, S., 2017a. A return mapping algorithm for elastoplastic and ductile damage constitutive equations using the subloading surface method. Int. J. Numer. Methods Eng. https://doi.org/10.1002/nme.5718 Fincato, R., Tsutsumi, S., 2017b. Numerical study of a welded plate instability using the subloading surface model. Mar. Struct. 55. https://doi.org/10.1016/j.marstruc.2017.05.001 Fincato, R., Tsutsumi, S., 2017c. Ductile Damage Evolution under Non-Proportional Loading. J. Japan Soc. Civ. Eng. Ser. A2 (Applied Mech. 73, I_355-I_361. https://doi.org/10.2208/jscejam.73.I_355 Gao, S., Usami, T., Ge, H., 1998. Ductility Evaluation of Steel Bridge Piers with Pipe Sections. J. Eng. Mech. 124, 260–267. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:3(260) Gao, X., Zhang, T., Zhou, J., Graham, S.M., Hayden, M., Roe, C., 2011. On stress-state dependent plasticity modeling: Significance of the hydrostatic stress, the third invariant of stress deviator and the non-associated flow rule. Int. J. Plast. 27, 217–231. https://doi.org/10.1016/j.ijplas.2010.05.004 Goto, Y., Kumar, G.P., Kawanishi, N., 2010. Nonlinear Finite-Element Analysis for Hysteretic Behavior of Thin-Walled Circular Steel Columns with In-Filled Concrete. J. Struct. Eng. 136, 1413–1422. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000240 Gurson, A.L., 1977. Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I—Yield Criteria and Flow Rules for Porous Ductile Media. J. Eng. Mater. Technol. 99, 2. https://doi.org/10.1115/1.3443401 Hashiguchi, K., 2009. Elastoplasticity theory, in: Lecture Notes in Applied and Computational Mechanics. pp. 1–406. https://doi.org/10.1007/978-3-642-00273-1_1 Hashiguchi, K., 1989. Subloading surface model in unconventional plasticity. Int. J. Solids Struct. 25, 917–945. https://doi.org/10.1016/00207683(89)90038-3 Hashiguchi, K., Tsutsumi, S., 2001. Elastoplastic constitutive equation with tangential stress rate effect. Int. J. Plast. 17, 117–145. https://doi.org/10.1016/S0749-6419(00)00021-8 Hashiguchi, K., Tsutsumi, S., 1993. Fundamental requirements and formulation of elastoplastic constitutive equations with tangential plasticity. Int. J. Plast. 9, 525–549. https://doi.org/10.1016/0749-6419(93)90018-L Hashiguchi, K., Yoshimaru, T., 1995. A generalized formulation of the concept of nonhardening region. Int. J. Plast. 11, 347–365. https://doi.org/10.1016/S0749-6419(95)00003-8 Kachanov, L.M., 1958. Time of the rupture process under creep conditions. Izv Akad Nauk S S R Otd Tech Nauk 8, 26–31. https://doi.org/citeulike-article-id:5466815 Lemaitre, J., 1985. Coupled elasto-plasticity and damage constitutive equations. Comput. Methods Appl. Mech. Eng. 51, 31–49. https://doi.org/10.1016/0045-7825(85)90026-X Momii, H., Tsutsumi, S., Fincato, R., 2015. Cyclic and Tangential Plasticity Effects for the Buckling Behavior of a Thin Wall Pier under Multiaxial and Non-proportional Loading Conditions. Trans. JWRI 44. Nishikawa K., Yamamoto S., Natori T., Terao K., Yasunami H., Terada M., 1998. Retrofitting for seismic upgrading of steel bridge columns. Eng. Struct. 20, 540–551. https://doi.org/10.1016/S0141-0296(97)00025-4 Papasidero, J., Doquet, V., Mohr, D., 2015. Ductile fracture of aluminum 2024-T351 under proportional and non-proportional multi-axial
150
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15
loading: Bao–Wierzbicki results revisited. Int. J. Solids Struct. 69–70, 459–474. https://doi.org/10.1016/j.ijsolstr.2015.05.006 Papasidero, J., Doquet, V., Mohr, D., 2014. Determination of the Effect of Stress State on the Onset of Ductile Fracture Through Tension-Torsion Experiments. Exp. Mech. 54, 137–151. https://doi.org/10.1007/s11340-013-9788-4 Rousselier, G., 1987. Ductile fracture models and their potential in local approach of fracture. Nucl. Eng. Des. 105, 97–111. https://doi.org/10.1016/0029-5493(87)90234-2 Rousselier, G., Luo, M., 2014. A fully coupled void damage and Mohr–Coulomb based ductile fracture model in the framework of a Reduced Texture Methodology. Int. J. Plast. 55, 1–24. https://doi.org/10.1016/j.ijplas.2013.09.002 Tsutsumi, S., Kaneko, K., 2008. Constitutive response of idealized granular media under the principal stress axes rotation. Int. J. Plast. https://doi.org/10.1016/j.ijplas.2008.05.001 Tsutsumi, S., Toyosada, M., Hashiguchi, K., 2006. Extended Subloading Surface Model Incorporating Elastic Boundary Concept. J. Appl. Mech. 9, 455–462. https://doi.org/10.2208/journalam.9.455 Van Do, V.N., Lee, C.H., Chang, K.H., 2014. A nonlinear CDM model for ductile failure analysis of steel bridge columns under cyclic loading. Comput. Mech. https://doi.org/10.1007/s00466-013-0964-2