Cyclic multiaxial and shear finite deformation responses of OFHC Cu. Part II: An extension to the KHL model and simulations

In this part, the Khan–Huang–Liang (KHL) constitutive model was extended to account for kinematic hardening characteristic behavior of materials. The extended model is then generalized and used to simulate experimental response of oxygen free high conductivity (OFHC) copper under cyclic shear straining and biaxial tension–torsion (multiaxial ratchetting) experiments presented in Part I (Khan et al., 2007). In addition, a new modification for the non-linear kinematic hardening rule of Karim–Ohno (Abdel-Karim and Ohno, 2000) is proposed to simulate multiaxial ratchetting behaviors. Although, the kinematic hardening contributes the most to the response, it is shown that, the loading rate effect, and a coupled isotropic and kinematic hardening effect should also be considered while simulating the multiaxial ratchetting behavior of OFHC copper. Furthermore, the newly modified kinematic hardening rules is able to fairly well simulate the multiaxial ratchetting experiments under different loading conditions, irrespective of the value of applied axial tensile stress, shear strain amplitude, pre-cyclic hardening and/or loading sequence.     

An evaluation for several kinematic hardening rules on prediction of multiaxial stress-controlled ratchetting

This study evaluates the performance of several non-linear kinematic hardening rules in predicting the various biaxial ratchetting experiments of stainless steel (SS) 304L under various stress-controlled histories performed by Hassan et al. (2008). The non-linear kinematic hardening rules proposed by 9, 32, 33 and 160, 19, 12 and 13 and the different rules of Abdel-Karim (2009) are examined and carefully scrutinized. The considered kinematic hardening rules range from the simple classical ones to more detailed rules, which incorporate additional terms and/or parameters to simulate different factors that affect ratchetting. It is shown that none of the examined kinematic hardening rules is general enough to simulate all of the ratchetting responses for the experiments under consideration.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part II: A very high work hardening aluminum alloy (annealed 1100 Al)

Results are presented on the evolution of subsequent yield surfaces with finite deformation in a very high work hardening annealed 1100 aluminum alloy. In Part I [Khan, A.S., Kazmi, R., Stoughton, T., Pandey, A., 2009a. Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part 1: a very low work hardening aluminum alloy (Al-6061–T6511) 25, 1611–1625.] of this paper, similar results are presented for a very low work hardening aluminum alloy. Those results were very different from the present ones, and all the results were for proportional loading paths. The subsequent yield surfaces are determined in tension, free end torsion and combined tension–torsion proportional and non-proportional loading paths, using 10 με deviation from linearity definition of yield. Yield surfaces are also determined after linear, bi-linear, and non-linear unloading paths after finite deformation under tension, free end torsion, and combined tension–torsion loading. The initial yield surface is closer to the von-Mises surface and the subsequent yield surfaces show distortion, expansion, positive cross-effect, and “nose” in the loading direction. Additionally, the subsequent yield surfaces after non-proportional loading paths show shrinkage and compounded distortion. The yield surfaces after unloading depict strong anisotropy, positive cross-effect and exhibits different proportion of distortion in each loading conditions. The Young’s and shear modulus decrease with plastic deformation and this decrease is much less than those reported in the published literature.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part-I: A very low work hardening aluminum alloy (Al6061-T6511)

In the present study, the initial and subsequent yield surfaces in Al 6061-T6511, based on 10 με deviation from linearity definition of yield, are presented. The subsequent yield surfaces are determined during tension, free end torsion, and combined tension–torsion proportional loading paths after reaching different levels of strains. The yield surfaces are also obtained after linear, bi-linear and non-linear unloading paths after finite plastic deformation. The initial yield surface is very close to the von-Mises yield surface and the subsequent yield surfaces undergo translation and distortion. In the case of this low work hardening material, the size of the yield surfaces is smaller and negative cross-effect is observed with finite plastic deformation. The subsequent yield have a usual “nose” in the loading direction and flattened shape in the reverse loading direction; the observed nose is more dominant in the case of tension and combined tension–torsion loading than in torsional loading. The size of the yield surfaces after unloading is smaller than the initial yield surface but larger than subsequent yield surfaces obtained during prior loading, show much smaller cross-effect, and the shape of these yield surfaces depends strongly on the loading and unloading paths. Elastic constants (Young’s and shear moduli) are also measured within each subsequent yield surfaces. Evolution of these constants with finite deformation is also presented. The decrease of the two moduli is found to be much smaller than reported earlier in tension by Cleveland and Ghosh [Cleveland, R.M., Ghosh, A.K., 2002. Inelastic effects on springback in metals. Int. J. Plast. 18, 769–785]. Part-II and III [(Khan et al., 2009a) and (Khan et al., 2009b)] of the papers will include experimental results on annealed 1100 Al (a very high work hardening material) and on both Al alloys (Al6061-T6511 and annealed 1100 Al) in tension- tension stress space, respectively. The results for both cases are quite different than the ones that are presented in this paper.     

Constitutive modelling of cyclic plasticity of metal particulate-reinforced brittle matrix composites under compression-compression loading

The Mori-Tanaka approach is used to modelling metal particulate-reinforced brittle matrix composites under cyclic compressive loading. The J₂-flow theory is considered as the relevant physical law of plastic flow in inclusions. Ratchetting of the composite is prevented by the strong constraint exerted by the matrix on the inclusions, even under the assumption of evanescent kinematic hardening. However, the weakening constraint power of the matrix caused by microfracture damage around inclusions is closely coupled with the plasticity of inclusion and leads to ratchetting even when the plastic deformation of inclusions is described by an isotropic hardening rule. A detailed parametric study has revealed that ratchetting is followed by either plastic or elastic shakedown, depending on the load amplitude, composite parameters and the mean length of microcracks.     

Meso-mechanical constitutive model for ratchetting of particle-reinforced metal matrix composites

Based on the Eshelby equivalent inclusion theory and Mori-Tanaka averaging method, a meso-mechanical cyclic elasto-plastic constitutive model is proposed to predict the ratchetting of particle-reinforced metal matrix composites. In the proposed model, a Hill-typed incremental formulation is used to simulate the elasto-plastic responses of the composites during cyclic loading with assumptions of elastic particle, elasto-plastic metal matrix and perfect interfacial bond between metal matrix and particles. A new nonlinear kinematic hardening rule extended from the Ohno-Abdel-Karim model [M. Abdel-Karim, N. Ohno, Kinematic hardening model suitable for ratchetting with steady-state, Int. J. Plasticity 16 (2000) 225-240] is employed to describe the ratchetting of metal matrix which dominates the ratchetting of the composites. With further assumption of spherical particles, the proposed meso-mechanical cyclic constitutive model is verified by comparing the predicted uniaxial ratchetting of SiC_P/6061Al composites with corresponding experiments obtained at room temperature [G.Z. Kang, Uniaxial time-dependent ratchetting of SiC_P/6061Al alloy composites at room and high temperature, Comp. Sci. Tech. 66 (2006) 1418-1430]. In the meantime, the effects of different tangent operators employed in the numerical implementation of the proposed model, i.e., continuum (or elasto-plastic) tangent operator C^ep and algorithmic (or consistent) one C^alg, on the predicted ratchetting are also discussed. It is concluded that the proposed model predicts the uniaxial ratchetting of SiC_P/6061Al composites at room temperature reasonably.     

A two-surface model describing ratchetting behaviors and transient hardening under nonproportional loading

A new superposed rule of Mroz's kinematic hardening rule and Ziegler's kinematic hardening rule based on two-surface model is proposed in the paper. Some experimental results on ratchetting of 2014-T6 aluminum alloy are predicted very well under multiaxial loading. In addition the conformability of the model is discussed for transient cyclic hardening under two kinds of nonproportional cyclic loading paths, i.e. square and rhombic path. The project supported by the National Natural Science Foundation of China     

An advancement in cyclic plasticity modeling for multiaxial ratcheting simulation

In a search for a constitutive model for ratcheting simulations, the models by Chaboche, Ohno–Wang, McDowell, Jiang–Sehitoglu, Voyiadjis–Basuroychowdhury and AbdelKarim–Ohno are evaluated against a set of uniaxial and biaxial ratcheting responses. With the assumption of invariant shape of the yield surface during plastic loading, the ratcheting simulations for uniaxial loading are primarily a function of the plastic modulus calculation, whereas the simulations for multiaxial loading are sensitive to the kinematic hardening rule of a model. This characteristic of the above mentioned models is elaborated in this paper. It is demonstrated that if all parameters of the kinematic hardening rule are determined from uniaxial responses only, these parameters primarily enable a better plastic modulus calculation. However, in this case the role of the kinematic hardening rule in representing the ratcheting responses for multiaxial loading is under-appreciated. This realization motivated many researchers to incorporate multiaxial load dependent terms or parameters into the kinematic hardening rule. This paper evaluates some of these modified rules and finds that none is general enough to simulate the ratcheting responses consistently for the experiments considered. A modified kinematic hardening rule is proposed using the idea of Delobelle and his co-workers in the framework of the Chaboche model. This new rule introduces only one multiaxial load dependent parameter to the Chaboche model, but performs the best in simulating all the ratcheting responses considered.     

An elastic–plastic constitutive law for the description of uniaxial and multiaxial ratchetting

An elastic–plastic constitutive model is proposed to describe 1-D and 2-D ratchetting. The model is also able to give correct results for 2-D ratchetting when only uniaxial identification is used, while no special threshold or parameter is used for the case of non-proportional loading. The original feature of this model consist in the introduction of a ratchetting stress (material characteristic) along with the maximal stress supported in the history of loading and the plastic strain at the last unloading. In this paper uniaxial and 3-D formulations have been described based on a numerical implementation in the software Code_Aster. Uniaxial and also multiaxial identifications have been used. Simulations have been realized for proportional and non-proportional homogeneous cases, as well as for structures under anisothermal thermomechanical loading. The results of a benchmark on a structure, comparing experiment, simulations by this model and some other phenomenological models, and a polycrystalline model are presented. An analysis of error margin due to the choice of Mises criterion is exposed.     

A critical state sand model with elastic–plastic coupling

Soil elastic moduli are highly pressure-dependent. Experimental findings have indicated that the elastic shear modulus of sands depends on p^χ, where p is mean principal effective stress and χ is a non-dimensional parameter. χ practically remains unchanged for shear strains less than 10⁻⁵ where the mechanical behavior is purely elastic. However, experiments have revealed that the emergence of plasticity for shear strains larger than 10⁻⁵ provokes a gradual increase in χ. Technically, this observation is an elastic–plastic coupling effect in which plasticity causes to change the elastic characteristics. Here, this issue is considered in hyper-elasticity framework in conjunction with a critical state compatible bounding surface plasticity platform for granular soils. To this aim, constitutive equations linking χ to a proper kinematic hardening parameter are presented. Then, using the proposed approach, a hyper-elastic theory is modified to consider the mentioned elastic–plastic coupling effect in the whole domain of the elastoplastic behavior. Adopting the improved hyper-elasticity necessitates the modification of a number of basic plasticity platform elements. In this regard, dilatancy and plastic hardening modulus of the bounding surface platform are modified. Successful performance of the modified constitutive model is presented against experimental data of loading/unloading triaxial tests.     

Kinematic hardening model suitable for ratchetting with steady-state

A new kinematic hardening model useful for simulating the steady-state in ratchetting is developed within the framework of the strain hardening and dynamic recovery format. The model is formulated to have two kinds of dynamic recovery terms, which operate at all times and only in a critical state, respectively. The model is examined on the basis of nonproportional experiments of Modified 9Cr–1Mo steel at 550°C and IN738LC at 850°C. The experiments include multiaxial, as well as uniaxial, ratchetting, multiaxial cyclic stress relaxation, and nonproportional cyclic straining along a butterfly-type strain path. It is shown that the model is successful in simulating the experiments, and that the model is featured by the capability of representing appropriately the steady-state in ratchetting under multiaxial and uniaxial cyclic loading.     

Uniaxial time-dependent ratchetting: Visco-plastic model and finite element application

Based on a visco-plastic model, a time-dependent formulation was introduced. The model can describe as time-dependent ratchetting in term of revising the Abdel-Karim–Ohno nonlinear kinematic hardening rule [M. Abdel-Karim, N. Ohno, Kinematic hardening model suitable for ratchetting with steady-state, Int. J. Plast. 16 (2000) 225–240] by a static recovery term. It is shown that the simulated results are in good agreement with the corresponding experiment results of SS304 stainless steel [G.Z. Kang, Q.H. Kan, J. Zhang, Time-dependent ratchetting experiments of SS304 stainless steel, Int. J. Plast. 22 (2006) 858–894]. Then, the proposed model with static recovery term was implemented into the finite element package. Based on the radial return method and backward Euler’s integration, a new implicit stress integration algorithm was proposed, and a new expression of consistent tangent modulus was derived. Finally, the reasonability of such implementation was verified by some numerical samples.     

Cyclic ratchetting of 1070 steel under multiaxial stress states

Cyclic ratchetting behavior of 1070 steel is studied under proportional and nonproportional loading with specific emphasis on the ratchetting rate decay mechanisms for large numbers of loading cycles. Under proportional loading, where the principal stress directions are unchanged, the ratchetting evolves in the mean stress direction. Under nonproportional loading, however, the ratchetting direction is determined by the loading path and can be different from the mean stress direction. The ratchetting rate decreases with increasing loading cycles, displaying a power law relationship with the number of loading cycles. The experimental ratchetting results indicate that under cyclic loading the material exhibits a tendency toward complying with a linear hardening rule with concomitant hysteresis loop closure. Based on the fundamental framework of plasticity theory and detailed evaluation of the stress-strain behaviors, the ratchetting can be classified into two basic types; Type I, which is identifiable with proportional loading where the ratchetting is due to the different values of the plastic modulus function at the symmetric loading points with respect to the mean stress state, and Type II, which represents nonproportional loading where the ratchetting is driven by the noncoincidence of the plastic strain rate vector and the translation direction of the yield surface (backstress rate vector). The Armstrong-Frederick-based plasticity models modified by Chaboche et al. and Bower are ill-suited for describing the experimental results of both types of ratchetting. The Ohno-Wang model, which introduces a threshold concept, can account for the ratchetting rate decay of Type II ratchetting, providing results that agree with experimental observations. Modification may be needed for the Ohno-Wang model so that the model can better describe Type I ratchetting.     

Analysis of sheet metal formability through isotropic and kinematic hardening models

The present paper aims at analysing the sheet metal formability through several isotropic and kinematic hardening models. Specifically, a special attention is paid to the physically-based hardening model of Teodosiu and Hu (1995), which accounts for the anisotropic work-hardening induced by the microstructural evolution at large strains, as well as to some more conventional hardening models, including the isotropic Swift strain-hardening power law, and the Voce saturation strain-hardening law, combined with a non-linear kinematic hardening described by the Armstrong–Frederick law. The onset of localized necking is simulated by an advanced sheet metal forming limit model which connects, through the Marciniak–Kuczinsky analysis, the hardening models with the anisotropic yield criterion Yld2000-2d (Barlat et al., 2003). Both linear and complex strain paths are taken into account. The selected material is a DC06 steel sheet. The validity of each model is assessed by comparing the predicted forming limits with experimental results carefully obtained on this steel. The origin of discrepancy in the predicted results using different hardening models is thoroughly analyzed.     

Calculations of fracture process zones on meso-scale in notched concrete beams subjected to three-point bending

The paper describes investigations on fracture process zones (FPZ) at meso-scale in notched concrete beams subjected to quasi-static three-point bending. The simulations were carried out with the FEM using isotropic damage constitutive model enhanced by a characteristic length of micro-structure by means of a non-local theory. Concrete was modelled as a random heterogeneous three-phase material. The effect of the beam size, aggregate distribution, aggregate density, aggregate shape, aggregate size and characteristic length on the width and shape of FPZ and load-displacement curve was numerically investigated. The numerical results were compared with own test results using Digital Image Correlation method (Skar?yński et al., 2009a), the tests by Le Bellěgo et al., (2003) and the size effect law by Ba?ant (2004).     

Kinematic hardening rules with critical state of dynamic recovery,part II: Application to experiments of ratchetting behavior

The kinematic hardening rules formulated in Part I of this work (i.e, Models I and II) are applied to ratchetting experiments of Modified 9Cr-1Mo steel done by Tanaka et al. as well as to a nonproportional experiment of OFHC copper by Lamba and Sidebottom. It is shown the Models I and II have the capability of simulating ratchetting behavior well because they can predict much less accumulation of ratchetting strain under uniaxial and multiaxial loadings than the Armstrong and Frederick model. It is also shown that if ratchetting strain is negligible, Models I and II may give nearly the same predictions as the Armstrong and Frederick model.