On shakedown analysis in hardening plasticity

The extension of classical shakedown theorems for hardening plasticity is interesting from both theoretical and practical aspects of the theory of plasticity. This problem has been much discussed in the literature. In particular, the model of generalized standard materials gives a convenient framework to derive appropriate results for common models of plasticity with strain-hardening. This paper gives a comprehensive presentation of the subject, in particular, on general results which can be obtained in this framework. The extension of the static shakedown theorem to hardening plasticity is presented at first. It leads by min-max duality to the definition of dual static and kinematic safety coefficients in hardening plasticity. Dual static and kinematic approaches are discussed for common models of isotropic hardening of limited or unlimited kinematic hardening. The kinematic approach also suggests for these models the introduction of a relaxed kinematic coefficient following a method due to Koiter. Some models for soils such as the Cam-clay model are discussed in the same spirit for applications in geomechanics. In particular, new appropriate results concerning the variational expressions of the dual kinematic coefficients are obtained.     

Modeling aspects of low plastic strain amplitude multiaxial cyclic plasticity in conventional and ultrafine grain nickel

Aspects of the cyclically saturated responses of initially annealed, conventional grain size (average grain diameter of approximately 50 μm) and electrodeposited, ultrafine grain (grains from 20 to 500 nm) nickel to reversed proportional and 90° out-of-phase axial-torsional, strain-controlled cycling at a nominally constant equivalent inelastic strain amplitude of approximately 100 μ strain are reported. An anisotropic, axial-torsional subspace version of Abdel-Karim and Ohno’s kinematic hardening model is presented. Within the framework of conventional small strain, rate-independent plasticity, this approach is used to model the responses. An anhysteretic, phenomenolically based, magnetomechanical model is coupled to the rate-independent plasticity model to include the cyclic magnetostriction response. The kinematic hardening parameter determination scheme, using the proportional path responses, is described. The model correlations achieved are presented and the ability of the resulting models to capture the 90° out-of-phase responses is examined. The model parameter sets, as determined from the proportional responses, require small changes to result in more accurate correlation of the 90° out-of-phase responses and the implications of this are discussed. The relative values of the model parameters between the two materials reflect the initial microstructures. Persistent mean stresses associated with mean total strains imposed are successfully modeled for the proportional strain path responses but not for the 90° out-of-phase responses.     

Multiaxial ratcheting with advanced kinematic and directional distortional hardening rules

Ratcheting is defined as the accumulation of plastic strains during cyclic plastic loading. Modeling this behavior is extremely difficult because any small error in plastic strain during a single cycle will add to become a large error after many cycles. As is typical with metals, most constitutive models use the associative flow rule which states that the plastic strain increment is in the direction normal to the yield surface. When the associative flow rule is used, it is important to have the shape of the yield surface modeled accurately because small deviations in shape may result in large deviations in the normal to the yield surface and thus the plastic strain increment in multi-axial loading. During cyclic plastic loading these deviations will accumulate and may result in large errors to predicted strains.This paper compares the bi-axial ratcheting simulations of two classes of plasticity models. The first class of models consists of the classical von Mises model with various kinematic hardening (KH) rules. The second class of models introduce directional distortional hardening (DDH) in addition to these various kinematic hardening rules. Directional distortion describes the formation of a region of high curvature on the yield surface approximately in the direction of loading and a region of flattened curvature approximately in the opposite direction. Results indicate that the addition of directional distortional hardening improves ratcheting predictions, particularly under biaxial stress controlled loading, over kinematic hardening alone.     

Shakedown theory for elastic plastic kinematic hardening bodies

Shakedown static and kinematic theorems for elastic–plastic (generally nonlinear) kinematic hardening solids are derived in classical (path-independence) spirit with new constructions. The generally plastic-deformation-history-dependent hardening curve is assumed to be limited by the initial yield stress and ultimate yield strength, and to obey a positive hysteresis postulate for closed plastic cycles, but else can be arbitrary and unspecified. The theorems reveal that the shakedown of structures is not affected by the particular form of the hardening curve, but just by the initial and ultimate yield stresses. While the ultimate yield strength is clearly defined macroscopically and attached to the incremental collapse mode with unbounded plastic deformations, the initial yield stress, which is responsible for the bounded cyclic plasticity collapse mode, should not be taken as the convenient one at a fixed amount of plastic deformation (0.2%), but is suggested to be taken as low as the fatigue limit to preserve the classical load-history-independence spirit of the shakedown theorems. Otherwise, for our pragmatic application purpose, it may be given empirical values between the low fatigue limit and high ultimate yield stresses according to particular loading processes considered, which may range anywhere between the high-cycle and low-cycle ones. The theorems appear as simple as those of Melan and Koiter for perfect plasticity but applied to the much larger class of more realistic kinematic hardening materials.     

Dual-phase steel sheets under cyclic tension–compression to large strains: Experiments and crystal plasticity modeling

In this work, we develop a physically-based crystal plasticity model for the prediction of cyclic tension–compression deformation of multi-phase materials, specifically dual-phase (DP) steels. The model is elasto–plastic in nature and integrates a hardening law based on statistically stored dislocation density, localized hardening due to geometrically necessary dislocations (GNDs), slip-system-level kinematic backstresses, and annihilation of dislocations. The model further features a two level homogenization scheme where the first level is the overall response of a two-phase polycrystalline aggregate and the second level is the homogenized response of the martensite polycrystalline regions. The model is applied to simulate a cyclic tension–compression–tension deformation behavior of DP590 steel sheets. From experiments, we observe that the material exhibits a typical decreasing hardening rate during forward loading, followed by a linear and then a non-linear unloading upon the load reversal, the Bauschinger effect, and changes in hardening rate during strain reversals. To predict these effects, we identify the model parameters using a portion of the measured data and validate and verify them using the remaining data. The developed model is capable of predicting all the particular features of the cyclic deformation of DP590 steel, with great accuracy. From the predictions, we infer and discuss the effects of GNDs, the backstresses, dislocation annihilation, and the two-level homogenization scheme on capturing the cyclic deformation behavior of the material.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

Constitutive modeling of cyclic plasticity deformation of a pure polycrystalline copper

Cyclic plasticity experiments were conducted on a pure polycrystalline copper and the material was found to display significant cyclic hardening and nonproportional hardening. An effort was made to describe the cyclic plasticity behavior of the material. The model is based on the framework using a yield surface together with the Armstrong–Frederick type kinematic hardening rule. No isotropic hardening is considered and the yield stress is assumed to be a constant. The backstress is decomposed into additive parts with each part following the Armstrong–Frederick type hardening rule. A memory surface in the plastic strain space is used to account for the strain range effect. The Tanaka fourth order tensor is used to characterize nonproportional loading. A set of material parameters in the hardening rules are related to the strain memory surface size and they are used to capture the strain range effect and the dependence of cyclic hardening and nonproportional hardening on the loading magnitude. The constitutive model can describe well the transient behavior during cyclic hardening and nonproportional hardening of the polycrystalline copper. Modeling of long-term ratcheting deformation is a difficult task and further investigations are required.     

A cyclic plasticity model for single crystals

The Armstrong–Frederick type kinematic hardening rule was invoked to capture the Bauschinger effect of the cyclic plastic deformation of a single crystal. The yield criterion and flow rule were built on individual slip systems. Material memory was introduced to describe strain range dependent cyclic hardening. The experimental results of copper single crystals were used to evaluate the cyclic plasticity model. It was found that the model was able to accurately describe the cyclic plastic deformation and properly reflect the dislocation substructure evolution. The well-known three distinctive regimes in the cyclic stress–strain curve of the copper single crystals oriented for single slip can be reproduced by using the model. The model can predict the enhanced hardening for crystals oriented for multislip, showing the model's ability to describe anisotropic cyclic plasticity. For a given loading history, the model was able to capture not only the saturated stress–strain response but also the detailed transient stress–strain evolution. The model was used to predict the cyclic plasticity under a high–low loading sequence. Both the stress–strain responses and the microstructural evolution can be appropriately described through the slip system activation.     

Modeling multi-axial deformation of planar anisotropic elasto-plastic materials,part I: Theory

In sheet metal forming processes local material points can experience multi-axial and multi-path loadings. Under such loading conditions, conventional phenomenological material formulations are not capable to predict the deformation behavior within satisfying accuracy. While micro-mechanical models have significantly improved the understanding of the deformation processes under such conditions, these models require large sets of material data to describe the micromechanical evolution and quite enormous computation expenses for industrial applications. To reduce the drawbacks of phenomenological material models under the multi-path loadings a new anisotropic elasto-plastic material formulation is suggested. The model enables the anisotropic yield surface to grow (isotropic hardening), translate (kinematic hardening) and rotate (rotation of the anisotropy axes) with respect to the deformation, while the shape of the yield surface remains essentially unchanged.Essentially, the model is formulated on the basis of an Armstrong–Frederick type kinematic hardening, the plastic spin theory for the reorientation of the symmetry axes of the anisotropic yield function, and additional terms coupling these expressions. The capability of the model is illustrated with multi-path loading simulations in ‘tension-shear’ and ‘reverse-shear’ to assess its performance with ‘cross’ hardening and ‘Bauschinger’ effects.     

A constitutive model of cyclic plasticity

This paper addresses a constitutive model of cyclic plasticity with special emphasis on the yield-point phenomena. In order to point out the deformation characteristics of a mild steel, four types of experiments were conducted, i.e. uniaxial tension at several crosshead speeds, cyclic straining, and stress- and strain-controlled ratchetting. A viscoplastic constitutive model of cyclic plasticity is proposed on the premise that the phenomena of sharp yield point and the subsequent abrupt yield drop result from rapid dislocation multiplication and the stress-dependence of dislocation velocity. Besides, cyclic plasticity behavior, such as the Bauschinger effect, cyclic hardening/softening characteristics and ratchet-strain accumulation, is described by some kinematic and isotropic hardening rules. The cyclic stress–strain responses predicted by this model agree well with the corresponding experimental results.     

A three-invariant cap model with isotropic–kinematic hardening rule and associated plasticity for granular materials

In this paper, a three-invariant cap model is developed for the isotropic–kinematic hardening and associated plasticity of granular materials. The model is based on the concepts of elasticity and plasticity theories together with an associated flow rule and a work hardening law for plastic deformations of granulars. The hardening rule is defined by its decomposition into the isotropic and kinematic material functions. The constitutive elasto-plastic matrix and its components are derived by using the definition of yield surface, material functions and non-linear elastic behavior, as function of hardening parameters. The model assessment and procedure for determination of material parameters are described. Finally, the applicability of proposed plasticity model is demonstrated in numerical simulation of several triaxial and confining pressure tests on different granular materials, including: wheat, rape, synthetic granulate and sand.     

Strain gradient plasticity modeling of the cyclic behavior of laminate microstructures

Two recently proposed Helmholtz free energy potentials including the full dislocation density tensor as an argument within the framework of strain gradient plasticity are used to predict the cyclic elastoplastic response of periodic laminate microstructures. First, a rank-one defect energy is considered, allowing for a size-effect on the overall yield strength of micro-heterogeneous materials. As a second candidate, a logarithmic defect energy is investigated, which is motivated by the work of Groma et al. (2003). The properties of the back-stress arising from both energies are investigated in the case of a laminate microstructure for which analytical as well as numerical solutions are derived. In this context, a new regularization technique for the numerical treatment of the rank-one potential is presented based on an incremental potential involving Lagrange multipliers. The results illustrate the effect of the two energies on the macroscopic size-dependent stress–strain response in monotonic and cyclic shear loading, as well as the arising pile-up distributions. Under cyclic loading, stress–strain hysteresis loops with inflections are predicted by both models. The logarithmic potential is shown to provide a continuum formulation of Asaro's type III kinematic hardening model. Experimental evidence in the literature of such loops with inflections in two-phased FFC alloys is provided, showing that the proposed strain gradient models reflect the occurrence of reversible plasticity phenomena under reverse loading.     

准脆性材料的弹塑性各向异性损伤耦合本构模型研究

针对准脆性材料的非线性特征：强度软化和刚度退化、单边效应、侧限强化和拉压软化、不可恢复变形、剪胀及非弹性体胀,在热动力学框架内,建立了准脆性材料的弹塑性与各向异性损伤耦合的本构关系。对准脆性材料的变形机理和损伤诱发的各向异性进行了诠释,并给出了损伤构形和有效构形中各物理量之间的关系。在有效应力空间内,建立了塑性屈服准则、拉压不同的塑性随动强化法则和各向同性强化法则。在损伤构形中,采用应变能释放率,建立了拉压损伤准则、拉压不同的损伤随动强化法则和各向同性强化法则。基于塑性屈服准则和损伤准则,构建了塑性势泛函和损伤势泛函,并由正交性法则,给出了塑性和损伤强化效应内变量的演化规律,同时,联立塑性屈服面和损伤加载面,给出了塑性流动和损伤演化内变量的演化法则。将损伤力学和塑性力学结合起来,建立了应变驱动的应力-应变增量本构关系,给出了本构数值积分的要点。以单轴加载-卸载往复试验识别和校准了本构材料常数,并对单轴单调试验、单轴加载-卸载往复试验、二轴受压、二轴拉压试验和三轴受压试验进行了预测,并与试验结果作了比较,结果表明,所建本构模型对准脆性材料的非线性材料性能有良好的预测能力。     

Benchmark experiments and characteristic cyclic plasticity deformation

Key issues in cyclic plasticity modeling are discussed based upon representative experimental observations on several commonly used engineering materials. Cyclic plasticity is characterized by the Bauschinger effect, cyclic hardening/softening, strain range effect, nonproporitonal hardening, and strain ratcheting. Additional hardening is identified to associate with ratcheting rate decay. Proper modeling requires a clear distinction among different types of cyclic plasticity behavior. Cyclic hardening/softening sustains dependent on the loading amplitude and loading history. Strain range effect is common for most engineering metallic materials. Often, nonproportional hardening is accompanied by cyclic hardening, as being observed on stainless steels and pure copper. A clarification of the two types of material behavior can be made through benchmark experiments and modeling technique. Ratcheting rate decay is a common observation on a number of materials and it often follows a power law relationship with the number of loading cycles under the constant amplitude stress controlled condition. Benchmark experiments can be used to explore the different cyclic plasticity properties of the materials. Discussions about proper modeling are based on the typical cyclic plasticity phenomena obtained from testing several engineering materials under various uniaxial and multiaxial cyclic loading conditions. Sufficient experimental evidence points to the unambiguous conclusion that none of the hardening phenomena (cyclic hardening/softening, strain range effect, nonproportional hardening, and strain hardening associated with ratcheting rate decay) is isotropic in nature. None of the hardening behavior can be properly modeled with a change in the yield stress.     

Ratcheting of cyclically hardening and softening materials: II. Multiaxial behavior

Part II of this study is concerned with ratcheting phenomena of cyclically hardening and softening materials under biaxial, cyclic loading. Two sets of biaxial experiments were performed on carbon steel 1018 and stainless steel 304 thin-walled tubes. In the first tyoe of experiment, a constant internal pressure was prescribed while the tubes were cycled axially in a strain-symmetric fashion. This causes ratcheting in the circumferential direction. In the second type of experiment, the axial cycling was carried out under stress control. This loading history results in simultaneous ratcheting in the axial and circumferential directions. In the case of stainless steel 304, the nonproportionality of these loading histories was found to induce significant hardening in addition to that recorded in unaxial loading. Cyclic hardening was found to reduce the rate of ratcheting. In the case of carbon steel 1018, the nonproportionality of the loading paths was found not to influence the induced softening. Cyclic softening in the axial and circumferential directions were found to be uncoupled.The time-independent cyclic plasticity models developed in Part I, suitably extended to multiaxial loading, were used to simulate the biaxial ratcheting experiments. Two methods for modeling the additional hardening/softening of the material due to nonproportional loading, developed by previous investigators, were incorporated in the models. The prediction of circumferential ratcheting is shown again to be sensitive to the kinematic hardening rule of the yield surface incorporated in the models. The performance of the models in predicting the biaxial ratcheting results was found to be rather poor. Several reasons for this poor performance are identified and suggestions for future improvements are made.     

A strain space nonlinear kinematic hardening/softening plasticity model

A strain space plasticity theory based on the nonlinear kinematic hardening and softening rule is developed in order to accommodate work-hardening, work-softening, and elastic-perfectly plastic materials with one set of constitutive equations, and to facilitate strain controlled calculations. A generalized hardening/softening parameter is proposed, and the potential of linking the parameter to micro-mechanical material changes is discussed. The theory is used to investigate work-softening materials numerically and highlights a need for additional experimental results in this area.     

Anatomy of coupled constitutive models for ratcheting simulation

This paper critically evaluates the performance of five constitutive models in predicting ratcheting responses of carbon steel for a broad set of uniaxial and biaxial loading histories. The models proposed by Prager, Armstrong and Frederick, Chaboche, Ohno-Wang and Guionnet are examined. Reasons for success and failure in simulating ratcheting by these models are elaborated. The bilinear Prager and the nonlinear Armstrong-Frederick models are found to be inadequate in simulating ratcheting responses. The Chaboche and Ohno-Wang models perform quite well in predicting uniaxial ratcheting responses; however, they consistently overpredict the biaxial ratcheting responses. The Guionnet model simulates one set of biaxial ratcheting responses very well, but fails to simulate uniaxial and other biaxial ratcheting responses. Similar to many earlier studies, this study also indicates a strong influence of the kinematic hardening rule or backstress direction on multiaxial ratcheting simulation. Incorporation of parameters dependent on multiaxial ratcheting responses, while dormant for uniaxial responses, into Chaboche-type kinematic hardening rules may be conducive to improve their multiaxial ratcheting simulations. The uncoupling of the kinematic hardening rule from the plastic modulus calculation is another potentially viable alternative. The best option to achieve a robust model for ratcheting simulations seems to be the incorporation of yield surface shape change (formative hardening) in the cyclic plasticity model.     

Evaluation of advanced anisotropic models with mixed hardening for general associated and non-associated flow metal plasticity

The main objective of this paper is to develop a generalized finite element formulation of stress integration method for non-quadratic yield functions and potentials with mixed nonlinear hardening under non-associated flow rule. Different approaches to analyze the anisotropic behavior of sheet materials were compared in this paper. The first model was based on a non-associated formulation with both quadratic yield and potential functions in the form of Hill’s (1948). The anisotropy coefficients in the yield and potential functions were determined from the yield stresses and r-values in different orientations, respectively. The second model was an associated non-quadratic model (Yld2000-2d) proposed by Barlat et al. (2003). The anisotropy in this model was introduced by using two linear transformations on the stress tensor. The third model was a non-quadratic non-associated model in which the yield function was defined based on Yld91 proposed by Barlat et al. (1991) and the potential function was defined based on Yld89 proposed by Barlat and Lian (1989). Anisotropy coefficients of Yld91 and Yld89 functions were determined by yield stresses and r-values, respectively. The formulations for the three models were derived for the mixed isotropic-nonlinear kinematic hardening framework that is more suitable for cyclic loadings (though it can easily be derived for pure isotropic hardening). After developing a general non-associated mixed hardening numerical stress integration algorithm based on backward-Euler method, all models were implemented in the commercial finite element code ABAQUS as user-defined material subroutines. Different sheet metal forming simulations were performed with these anisotropic models: cup drawing processes and springback of channel draw processes with different drawbead penetrations. The earing profiles and the springback results obtained from simulations with the three different models were compared with experimental results, while the computational costs were compared. Also, in-plane cyclic tension–compression tests for the extraction of the mixed hardening parameters used in the springback simulations were performed for two sheet materials.     

A cyclic anisotropic-plasticity model for metal matrix composites

Based on a six parameter general anisotropic yield surface proposed earlier by Voyiadjis and Thiagarajan (An Anisotropic Yield Surface Model for Directionally Reinforced Metal Matrix Composites, Int. J. Plasticity [1995]), a cyclic plasticity model to model the behavior of directionally reinforced metal matrix composite, has been proposed here. Apart from being able to model different initial yielding behavior along different stress directions, a number of features have been incorporated into the plasticity model. They include the usage of a proposed non-associative flow rule, kinematic hardening rule of Phillips type, a modified form of the bounding surface model for modelling the cyclic behavior, and the usage of a proposed form for evaluating the plastic modulus for anisotropic materials. Previous experimental data have been used for the evaluation of the yield surface parameters as well as those for the determination of the plastic modulus. The stress-strain results generated from the model have then been compared with those from the experiments. The behavior of the model under certain simulated cyclic loading situations has also been presented.