Directional distortional hardening at large plastic deformations

This paper extents the directional distortional hardening model of Feigenbaum and Dafalias (2007) into the range of large plastic deformations. This model allows the yield surface to deform such that a region of high curvature develops approximately in the direction of loading and a region of flattening develops on the opposite side. To extend this model into large deformations and in order to ensure positive dissipation and objectivity, hardening rules are derived from thermodynamic conditions in terms of corotational rates. Since this model includes a fourth order tensor-valued hardening internal variable, the corotational rates for fourth order tensors are examined in this work employing the concept of plastic spin. Several choices for plastic spins are presented and used for the simulation of the response under simple shear loading up to 1000% strain.     

Isotropic hardening in micropolar plasticity

Experimental evidence for length scale effects in plasticity has been provided, e.g., by Fleck et al. (Acta Metall. Mater. 42:475–487, 1994). Results from torsional loadings on copper wires, when appropriately displayed, indicated that, for the same shear at the outer radius, the normalized torque increased with decreasing specimen radius. Modeling of the constitutive behavior in the framework of micropolar plasticity is a possibility to account for length scale effects. The present paper is concerned with this possibility and deals with the theory developed by Grammenoudis and Tsakmakis (Contin. Mech. Thermodyn. 13:325–363, 2001; Int. J. Numer. Methods Eng. 62:1691–1720, 2005; Proc. R. Soc. 461:189–205, 2005). Both isotropic and kinematic hardening are present in that theory, with isotropic hardening being captured in a unified manner. Here, we discuss isotropic hardening composed of two parts, responsible for strain and gradient effects, respectively.     

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.     

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.     

Distortional hardening plasticity model for paperboard

A distortional hardening elasto-plastic model at finite strains suitable for modeling of orthotropic materials is presented. As a prototype material, paperboard is considered. An in-plane model is established. The model developed is motivated from non-proportional loading tests on paperboard where the paperboard is pre-strained in one direction and then loaded in the perpendicular direction. A softening effect is revealed in the pre-strained samples. The observed experimental findings cannot be accurately predicted by current models for paperboard. To be able to model the softening effects, a yield surface based on multiple hardening variables is introduced. It is shown that the model parameters can be obtained from simple uniaxial experiments. The model is implemented in a finite element framework which is used to illustrate the behavior of the model at some specific loading situations and is compared with strain fields obtained from Digital Image Correlation experiments.     

Strain hardening plasticity with bauschinger effect

Summary The plastic behaviour of metals is studied, taking into account the strain bardening and the Bauschinger effect.Basic assumptions go back to the slip theory of plasticity and many features of this theory are developed and thoroughly examined. The Bauschinger effect is introduced on every couple of planes and the stress-strain relations are given in incremental form. The plastic strain properties corresponding to the theoretical model are studied with special reference to the evolution of the yield surface.The paper concludes by applying the proposed theory to the tensile test with stress reversal and to the characterization of the subsequent yield surfaces in an ideal tension-torsion test. The results agree, at least qualitatively, with the experimental results.

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Sommario In questa Nota viene studiato il comportamento plastico dei metalli prendendo in conto l'influenza dell'incrudimento e l'effetto Bauschinger.Partendo dai punti di base della teoria degli slip si sviluppano ed approfondiscono molti aspetti di questa, tra l'altro introducendo l'effetto Bauschinger su ogni coppia di piani, e si forniscono in forma incrementale le relazioni tensioni-deformazioni.Successivamente vengono studiate le proprietà della deformazione plastica che consegue a tale formulazione con particolare riferimento al problema dell'evoluzione della superficie di plasticizzazione.Il lavoro si conclude con l'applicazione della teoria al caso della prova a trazione con inversione di sforzo e alla caratterizzazione delle superfici susseguenti di plasticizzazione nella prova ideale combinata di trazione e torsione ritrovando accordo, almeno dal punto di vista qualitativo, con ricerche sperimentali in merito.

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Study supported by the C.N.R.     

An analysis of exhaustion hardening in micron-scale plasticity

The rate-dependent behavior of micron-scale model planar crystals is investigated using the framework of mechanism-based discrete dislocation plasticity. Long-range interactions between dislocations are accounted for through elasticity. Mechanism-based constitutive rules are used to represent the short-range interactions between dislocations, including dislocation multiplication and dislocation escape at free surfaces. Emphasis is laid on circumstances where the deformed samples are not statistically homogeneous. The calculations show that dimensional constraints selectively set the operating dislocation mechanisms, thus giving rise to the phenomenon of exhaustion hardening whereby the applied strain rate is predominantly accommodated by elastic deformation. When conditions are met for this type of hardening to take place, the calculations reproduce some interesting qualitative features of plastic deformation in microcrystals, such as flow intermittency over coarse time-scales and large values of the flow stress with no significant accumulation of dislocation density. In addition, the applied strain rate is varied down to 0.1 s⁻¹ and is found to affect the rate of exhaustion hardening.     

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 separated law of hardening in strain gradient plasticity

This paper presents a separated law of hardening in plasticity with strain gradient effects. The value of the length parameter ℓ contained in this model was estimated from the experimental data for copper. The project supported by the National Natural Science Foundation of China     

Nonlocal gradient-dependent modeling of plasticity with anisotropic hardening

This work addresses the formulation of the thermodynamics of nonlocal plasticity using the gradient theory. The formulation is based on the nonlocality energy residual introduced by Eringen and Edelen (1972). Gradients are introduced for those variables associated with isotropic and kinematic hardening. The formulation applies to small strain gradient plasticity and makes use of the evanescent memory model for kinematic hardening. This is accomplished using the kinematic flux evolution as developed by Zbib and Aifantis (1988). Therefore, the present theory is a four nonlocal parameter-based theory that accounts for the influence of large variations in the plastic strain, accumulated plastic strain, accumulated plastic strain gradients, and the micromechanical evolution of the kinematic flux. Using the principle of virtual power and the laws of thermodynamics, thermodynamically-consistent equations are derived for the nonlocal plasticity yield criterion and associated flow rule. The presence of higher-order gradients in the plastic strain is shown to enhance a corresponding history variable which arises from the accumulation of the plastic strain gradients. Furthermore, anisotropy is introduced by plastic strain gradients in the form of kinematic hardening. Plastic strain gradients can be attributed to the net Burgers vector, while gradients in the accumulation of plastic strain are responsible for the introduction of isotropic hardening. The equilibrium between internal Cauchy stress and the microstresses conjugate to the higher-order gradients frames the yield criterion, which is obtained from the principle of virtual power. Microscopic boundary conditions, associated with plastic flow, are introduced to supplement the macroscopic boundary conditions of classical plasticity. The nonlocal formulation developed here preserves the classical assumption of local plasticity, wherein plastic flow direction is governed by the deviatoric Cauchy stress. The theory is applied to the problems of thin films on both soft and hard substrates. Numerical solutions are presented for bi-axial tension and simple shear loading of thin films on substrates.     

Anisotropic hardening in single crystals and the plasticity of polycrystals

Based on the dislocation structures developed during plastic deformation, an anisotropic hardening law is developed to describe the latent hardening behavior of slip systems under multislip. This theory incorporates the concept of isotropic hardening, kinematic hardening, and the two-parameter representation; it automatically includes the strength differential between the forward and reversed slips and between the acute and obtuse cross slips. The self-hardening modulus of a slip system is found to be “associated” with the latent hardening law involved, and, based on some experimental evidence, two specific sets of self-hardening modulus are suggested. An important feature of this associated modulus is that the slip system with a soft latent hardening (e.g., the reversed system with a Bauschinge effect) will have an enhanced self-hardening modulus. This newly developed hardening law, together with its associated latent hardening moduli, is then applied to examine the strain-hardening behavior of a polycrystal. Although crystals with a stronger latent hardening will, in general, also lead to a stranger strain-hardening for the polycrystal, the stress-strain behavior of the polycrystal using the kinematic hardening law of single crystals is found to be not necessarily softer than that using the isotropic hardening law. Within the range of experimentally measured latent hardening ratio of slip systems, the anisotropic theory is also used to calculate the motion of yield surface of a polycrystal. The general results, employing four selected types of anisotropic hardening, all show the essential features of experimental observations by Phillips and his co-workers. The application is highlighted with a reasonably successful quantitative modeling of initial and subsequent yield surfaces of an aluminum.     

Crack growth resistance in non-perfect plasticity: Isotropic versus kinematic hardening

The Strain Energy Density Theory is applied for analyzing energy dissipation and crack growth in the three-point bending specimen when the material behavior follows a multilinear strain-hardening stress-strain relationship. The problem is solved through the application of incremental theory of plasticity and finite element method.The rate of change of the strain energy density factor S with crack length a is verified to be governed by the relation . Results are obtained for isotropic and kinematic hardening. Moreover, the effects of loading step and specimen size are pointed out.     

Original ArticleKinematic hardening rules in finite plasticity Part II: Some examples

We consider finite plasticity laws which are represented by means of dual variables and satisfy the intrinsic dissipation inequality. Within two families of dual variables, two different formulations for a kinematic hardening rule, referred to as Model 1 and Model 2, are proposed. In order to discuss basic properties of the two models, the predicted response for some simple deformations is calculated. It turns out that e.g. with respect to the simple shear and simple torsion the two models differ mainly in the effects of second order. Received June 14, 1995     

Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation

A strain gradient dependent crystal plasticity approach is used to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. Material points are considered as aggregates of grains, subdivided into several fictitious grain fractions: a single crystal volume element stands for the grain interior whereas grain boundaries are represented by bi-crystal volume elements, each having the crystallographic lattice orientations of its adjacent crystals. A relaxed Taylor-like interaction law is used for the transition from the local to the global scale. It is relaxed with respect to the bi-crystals, providing compatibility and stress equilibrium at their internal interface. During loading, the bi-crystal boundaries deform dissimilar to the associated grain interior. Arising from this heterogeneity, a geometrically necessary dislocation (GND) density can be computed, which is required to restore compatibility of the crystallographic lattice. This effect provides a physically based method to account for the additional hardening as introduced by the GNDs, the magnitude of which is related to the grain size. Hence, a scale-dependent response is obtained, for which the numerical simulations predict a mechanical behaviour corresponding to the Hall-Petch effect. Compared to a full-scale finite element model reported in the literature, the present polycrystalline crystal plasticity model is of equal quality yet much more efficient from a computational point of view for simulating uniaxial tension experiments with various grain sizes.     

Propagation of compaction waves in metal foams exhibiting strain hardening

One-dimensional models for compaction of cellular materials exhibiting strain hardening are proposed for two different impact scenarios. The models reveal the characteristic features of deformation under the condition of decreasing velocity during the impact event. It was established that an unloading plastic wave of strong discontinuity propagates in the foam and it has a significant dynamic effect on the foam compaction and energy absorption. The proposed models are based on the actual experimentally derived stress strain curves. The compaction mechanism in three aluminium based foam materials, two of them with relatively low density – Alporas and Cymat with 9% and 9.3% relative density, respectively and a higher density Cymat foam with 21% relative density, is analysed. Numerical simulations were carried out to verify the proposed models.The predictions of the proposed models are compared with published analytical models of compaction of cellular materials which assume a predefined densification strain. It is shown that the approximation of a cellular material with significant strain hardening by the Rigid Perfectly-Plastic-Locking (RPPL) model can lead to an overestimation of the energy absorption capacity for the observed stroke due to the non-uniform strains along the compacted zone of the actual material in contrast to the predefined constant densification strain in the RPPL model. The assumption of a constant densification strain leads also to an overestimation of the maximum stress, which occurs under impact.     

Integration of self-consistent polycrystal plasticity with dislocation density based hardening laws within an implicit finite element framework: Application to low-symmetry metals

We present an implementation of the viscoplastic self-consistent (VPSC) polycrystalline model in an implicit finite element (FE) framework, which accounts for a dislocation-based hardening law for multiple slip and twinning modes at the micro-scale grain level. The model is applied to simulate the macro-scale mechanical response of a highly anisotropic low-symmetry (orthorhombic) crystal structure. In this approach, a finite element integration point represents a polycrystalline material point and the meso-scale mechanical response is obtained by the mean-field VPSC homogenization scheme. We demonstrate the accuracy of the FE-VPSC model by analyzing the mechanical response and microstructure evolution of α-uranium samples under simple compression/tension and four-point bending tests. Predictions of the FE-VPSC simulations compare favorably with experimental measurements of geometrical changes and microstructure evolution. Specifically, the model captures accurately the tension–compression asymmetry of the material associated with twinning, as well as the rigidity of the material response along the hard-to-deform crystallographic orientations.     

Original ArticleKinematic hardening rules in finite plasticity Part I: A constitutive approach

Plasticity laws exhibiting non-linear kinematichardening are considered within the framework of infinitesimal deformations. The evolution equations governing the response of kinematic hardening are derived as sufficient conditions in order for the intrinsic dissipation inequality to be satisfied in every process. With a view to the extension to finite deformations, two basic possibilities are proposed. In every case, an isotropic elasticity law with respect to the so-called plastic intermediate configuration is assumed to hold. The theory applicable to finite deformations is based on the concept of so-called dual variables and associated time derivatives. Thus, the main difference between the present work and other contributions in this area is the choice of the variables used to formulate the theory. In fact, using dual variables, hardening rules are derived as sufficient conditions for the intrinsic dissipation inequality to be satisfied in every process. This is quite analogous to the case of infinitesimal deformation, but now the hardening rules take a very specific form which is explained in the paper. Received June 14, 1995