Constitutive modeling of temperature and strain rate dependent elastoplastic hardening materials using a corotational rate associated with the plastic deformation

In this paper, a constitutive model with a temperature and strain rate dependent flow stress (Bergstrom hardening rule) and modified Armstrong-Frederick kinematic evolution equation for elastoplastic hardening materials is introduced. Based on the multiplicative decomposition of the deformation gradient,new kinematic relations for the elastic and plastic left stretch tensors as well as the plastic deformation-dependent spin tensor are proposed. Also, a closed-form solution has been obtained for the elastic and plastic left stretch tensors for the simple shear problem.To evaluate model validity, results are compared with known experimental data for SUS 304 stainless steel, which shows a good agreement with the results of the proposed theoretical model.Finally, the stress-deformation curve, as predicted by the model, is plotted for the simple shear problem at room and elevated temperatures using the same material properties for AA5754-O aluminium alloy.     

A FE formulation for elasto-plastic materials with planar anisotropic yield functions and plastic spin

In this article a stress integration algorithm for shell problems with planar anisotropic yield functions is derived. The evolution of the anisotropy directions is determined on the basis of the plastic and material spin. It is assumed that the strains inducing the anisotropy of the pre-existing preferred orientation are much larger than subsequent strains due to further deformations. The change of the locally preferred orientations to each other during further deformations is considered to be neglectable. Sheet forming processes are typical applications for such material assumptions. Thus the shape of the yield function remains unchanged. The size of the yield locus and its orientation is described with isotropic hardening and plastic and material spin.The numerical treatment is derived from the multiplicative decomposition of the deformation gradient and thermodynamic considerations in the intermediate configuration. A common formulation of the plastic spin completes the governing equations in the intermediate configuration. These equations are then pushed forward into the current configuration and the elastic deformation is restricted to small strains to obtain a simple set of constitutive equations. Based on these equations the algorithmic treatment is derived for planar anisotropic shell formulations incorporating large rotations and finite strains. The numerical approach is completed by generalizing the Return Mapping algorithm to problems with plastic spin applying Hill’s anisotropic yield function. Results of numerical simulations are presented to assess the proposed approach and the significance of the plastic spin in the deformation process.     

A plasticity model for pressure-dependent anisotropic cellular solids

The initial and subsequent yield surfaces for an anisotropic and pressure-dependent 2D stochastic cellular material, which represents solid foams, are investigated under biaxial loading using finite element analysis. Scalar measures of stress and strain, namely characteristic stress and characteristic strain, are used to describe the constitutive response of cellular material along various stress paths. The coupling between loading path and strain hardening is then investigated in characteristic stress–strain domain. The nature of the flow rule that best describes the plastic flow of cellular solid is also investigated. An incremental plasticity framework is proposed to describe the pressure-dependent plastic flow of 2D stochastic cellular solids. The proposed plasticity framework adopts the anisotropic and pressure-dependent yield function recently introduced by Alkhader and Vural [Alkhader M., Vural M., 2009a. An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations. J. Mech. Phys. Solids 57(5), 871–890]. It has been shown that the proposed yield function can be simply calibrated using elastic constants and flow stresses under uniaixal loading. Comparison of stress fields predicted by continuum plasticity model to the ones obtained from FE analysis shows good agreement for the range of loading paths and strains investigated.     

A ductile damage criterion at various stress triaxialities

The paper discusses the effect of stress triaxiality on the onset and evolution of damage in ductile metals. A series of tests including shear tests and experiments on smooth and pre-notched tension specimens was carried out for a wide range of stress triaxialities. The underlying continuum damage model is based on kinematic definition of damage tensors. The modular structure of the approach is accomplished by the decomposition of strain rates into elastic, plastic and damage parts. Free energy functions with respect to fictitious undamaged configurations as well as damaged ones are introduced separately leading to elastic material laws which are affected by increasing damage. In addition, a macroscopic yield condition and a flow rule are used to adequately describe the plastic behavior. Numerical simulations of the experiments are performed and good correlation of tests and numerical results is achieved. Based on experimental and numerical data the damage criterion formulated in stress space is quantified. Different branches of this function are taken into account corresponding to different damage modes depending on stress triaxiality and Lode parameter. In addition, identification of material parameters is discussed in detail.     

Solution existence conditions for elastoplastic constitutive models of granular materials

In this paper, the conditions of solution existence for stress rates under given strain rates are investigated. The focus of the solution existence investigation is on the non-associated flow rule and elastic stress–strain relationship. Granular materials characterized with strong non-associated plastic flows are used as a particular example for analysis. Various flow rules for granular materials are analyzed, including Rowe’s, Roscoe’s flow rules and their modified versions. In the elastic stress–strain relationships of materials, the effects of Poisson’s ratio on solution existence are investigated. Both isotropic and anisotropic elasticity are considered. Given a granular material and its states, it is found that there exists a critical Poisson’s ratio for a particular non-associated flow rule. When the Poisson’s ratio of a material is above this critical Poisson’s ratio, its constitutive model is susceptible to solution non-existence. It is suggested that special attentions should be paid to the selection of material Poisson’s ratio and non-associated flow rule to ensure the existence of elastoplastic solutions.     

Numerical analysis and elastic–plastic deformation behavior of anisotropically damaged solids

The present paper is concerned with the numerical modelling of the large elastic–plastic deformation behavior and localization prediction of ductile metals which are sensitive to hydrostatic stress and anisotropically damaged. The model is based on a generalized macroscopic theory within the framework of nonlinear continuum damage mechanics. The formulation relies on a multiplicative decomposition of the metric transformation tensor into elastic and damaged-plastic parts. Furthermore, undamaged configurations are introduced which are related to the damaged configurations via associated metric transformations which allow for the interpretation as damage tensors. Strain rates are shown to be additively decomposed into elastic, plastic and damage strain rate tensors. Moreover, based on the standard dissipative material approach the constitutive framework is completed by different stress tensors, a yield criterion and a separate damage condition as well as corresponding potential functions. The evolution laws for plastic and damage strain rates are discussed in some detail. Estimates of the stress and strain histories are obtained via an explicit integration procedure which employs an inelastic (damage-plastic) predictor followed by an elastic corrector step. Numerical simulations of the elastic–plastic deformation behavior of damaged solids demonstrate the efficiency of the formulation. A variety of large strain elastic–plastic-damage problems including severe localization is presented, and the influence of different model parameters on the deformation and localization prediction of ductile metals is discussed.     

A plane stress anisotropic plastic flow theory for orthotropic sheet metals

A phenomenological theory is presented for describing the anisotropic plastic flow of orthotropic polycrystalline aluminum sheet metals under plane stress. The theory uses a stress exponent, a rate-dependent effective flow strength function, and five anisotropic material functions to specify a flow potential, an associated flow rule of plastic strain rates, a flow rule of plastic spin, and an evolution law of isotropic hardening of a sheet metal. Each of the five anisotropic material functions may be represented by a truncated Fourier series based on the orthotropic symmetry of the sheet metal and their Fourier coefficients can be determined using experimental data obtained from uniaxial tension and equal biaxial tension tests. Depending on the number of uniaxial tension tests conducted, three models with various degrees of planar anisotropy are constructed based on the proposed plasticity theory for power-law strain hardening sheet metals. These models are applied successfully to describe the anisotropic plastic flow behavior of 10 commercial aluminum alloy sheet metals reported in the literature.     

The appropriate corotational rate, exact formula for the plastic spin and constitutive model for finite elastoplasticity

The exact formulae for the plastic and the elastic spin referred to the deformed configuration are derived, where the plastic spin is a function of the plastic strain rate and the elastic spin a function of the elastic strain rate. With these exact formulae we determine the macroscopic substructure spin that allows us to define the appropriate corotational rate for finite elastoplasticity.Plastic, elastic and substructure spin are considered and simplified for various sub-classes of restricted elastic-plastic strains. It is shown that for the special cases of rigid-plasticity and hypoelasticity the proposed corotational rate is identical with the Green-Naghdi rate, while the ZarembaJaumann rate yields a good approximation for moderately large strains.To compare our exact plastic spin formula with the constitutive assumption for the plastic spin introduced by Dafalias and others, we simplify our result for small elastic-moderate plastic strains and introduce a simplest evolution law for kinematic hardening leading to the Dafalias formula and to an exact determination of its unknown coefficient. It is also shown that contrary to statements in the literature the plastic spin is not zero for vanishing kinematic hardening.For isotropic-elastic material with induced plastic flow undergoing isotropic and kinematic hardening constitutive and evolution laws are proposed. Elastic and plastic Lagrangean and Eulerian logarithmic strain measures are introduced and their material time derivatives and corotational rates, respectively, are considered. Finally, the elastic-plastic tangent operator is derived.The presented theory is implemented in a solution algorithm and numerically applied to the simple shear problem for finite elastic-finite plastic strains as well as for sub-classes of restricted strains. The results are compared with those of the literature and with those obtained by using other corotational rates.     

Orthotropic elastic–plastic material model for paper materials

Paper and paperboard generally exhibit anisotropic and non-linear mechanical material behaviour. In this work, the development of an orthotropic elastic–plastic constitutive model, suitable for modelling of the material behaviour of paper is presented. The anisotropic material behaviour is introduced into the model by orthotropic elasticity and an isotropic plasticity equivalent transformation tensor. A parabolic stress–strain relation is adopted to describe the hardening of the material. The experimental and numerical procedures for evaluation of the required material parameters for the model are described. Uniaxial tensile testing in three different inplane material directions provides the calibration of the material parameters under plane stress conditions. The numerical implementation of the material model is presented and the model is shown to perform well in agreement with experimentally observed mechanical behaviour of paper.     

A constitutive model for orthotropic elasto-plasticity at large strains

Summary The paper presents a thermodynamically consistent constitutive model for elasto-plastic analysis of orthotropic materials at large strain. The elastic and plastic anisotropies are assumed to be persistent in the material but the anisotropy axes can undergo a rigid rotation due to large plastic deformations. The orthotropic yield function is formulated in terms of the generally nonsymmetric Mandel stress tensor such that its skew-symmetric part is additionally taken into account. Special attention is focused on the convexity of the yield surface resulting in the nine-dimensional stress space. Of particular interest are new convexity conditions which do not appear in the classical theory of anisotropic plasticity. They impose additional constraints on the material constants governing the plastic spin. The role of the plastic spin is further studied in simple shear accompanied by large elastic and large plastic deformations. If the plastic spin is neglected, the shear stress response is characterized by oscillations with an amplitude strictly dependent on the degree of the plastic anisotropy.accepted for publication 2 March 2004     

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。     

An anisotropic elastoplastic constitutive formulation generalised for orthotropic materials

This paper presents a finite strain constitutive model to predict a complex elastoplastic deformation behaviour that involves very high pressures and shockwaves in orthotropic materials using an anisotropic Hill’s yield criterion by means of the evolving structural tensors. The yield surface of this hyperelastic–plastic constitutive model is aligned uniquely within the principal stress space due to the combination of Mandel stress tensor and a new generalised orthotropic pressure. The formulation is developed in the isoclinic configuration and allows for a unique treatment for elastic and plastic orthotropy. An isotropic hardening is adopted to define the evolution of plastic orthotropy. The important feature of the proposed hyperelastic–plastic constitutive model is the introduction of anisotropic effect in the Mie–Gruneisen equation of state (EOS). The formulation is further combined with Grady spall failure model to predict spall failure in the materials. The proposed constitutive model is implemented as a new material model in the Lawrence Livermore National Laboratory (LLNL)-DYNA3D code of UTHM’s version, named Material Type 92 (Mat92). The combination of the proposed stress tensor decomposition and the Mie–Gruneisen EOS requires some modifications in the code to reflect the formulation of the generalised orthotropic pressure. The validation approach is also presented in this paper for guidance purpose. The \({\varvec{\psi }}\) tensor used to define the alignment of the adopted yield surface is first validated. This is continued with an internal validation related to elastic isotropic, elastic orthotropic and elastic–plastic orthotropic of the proposed formulation before a comparison against range of plate impact test data at 234, 450 and \({\mathrm {895\,ms}}^{\mathrm {-1}}\) impact velocities is performed. A good agreement is obtained in each test.     

Development of plasticity and damage in vibrating structural elements performing guided rigid-body motions

Summary  A numerical algorithm for studying the development of plastic and damaged zones in a vibrating structural element with a large, guided rigid-body motion is presented. Beam-type elements vibrating in the small-strain regime are considered. A machine element performing rotatory motions, similar to an element of a slider-crank mechanism, is treated as a benchmark problem. Microstructural changes in the deforming material are described by the mesolevel variables of plastic strain and damage, which are consistently included into a macroscopic analysis of the overall beam motion. The method is based on an eigenstrain formulation, considering plastic strain and damage to contribute to an eigenstrain loading of a linear elastic background structure. Rigid-body coordinates are incorporated into this beam-type structural formulation, and an implicit numerical scheme is presented for iterative computation of the eigenstrains from the mesolevel constitutive behavior. Owing to the eigenstrain formulation, any of the existing constitutive models with internal variables could in principle be implemented. Linear elastic/perfectly plastic behavior is exemplarily treated in a numerical study, where plastic strain is connected to the Kachanov damage parameter by a simple damage law. Inelastic effects like plastic shakedown and damage-induced low-cycle rupture are shown to occur in the examplary problems. Received 1 September 1999; accepted for publication 9 March 2000     

On the modeling and simulation of induced anisotropy in polycrystalline metals with application to springback

The presence of initial, and the development of induced, anisotropic elastic and inelastic material behavior in polycrystalline metals, can be traced back to the influence of texture and dislocation substructural development on this behavior. As it turns out, via homogenization or other means, one can formulate effective models for such structure and its effect on the macroscopic material behavior with the help of the concept of evolving structure tensors. From the constitutive point of view, these quantities determine the material symmetry properties. Most importantly, all dependent constitutive fields (e.g., stress) are by definition isotropic functions of the independent constitutive variables, which include these evolving structure tensors. The evolution of these tensors during loading results in an evolution of the anisotropy of the material. From an algorithmic point of view, the current approach leads to constitutive models which are quite amenable to numerical implementation. To demonstrate the applicability of the resulting constitutive formulation, we apply it to the case of metal plasticity with combined hardening involving both deformation- and permanently induced anisotropy. Comparison of simulation results based on this model for the bending tension of aluminum-alloy sheet-metal strips with corresponding experimental ones show good agreement.     

Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions

In this paper we propose a formulation of polyconvex anisotropic hyperelasticity at finite strains. The main goal is the representation of the governing constitutive equations within the framework of the invariant theory which automatically fulfill the polyconvexity condition in the sense of Ball in order to guarantee the existence of minimizers. Based on the introduction of additional argument tensors, the so-called structural tensors, the free energies and the anisotropic stress response functions are represented by scalar-valued and tensor-valued isotropic tensor functions, respectively. In order to obtain various free energies to model specific problems which permit the matching of data stemming from experiments, we assume an additive structure. A variety of isotropic and anisotropic functions for transversely isotropic material behaviour are derived, where each individual term fulfills a priori the polyconvexity condition. The tensor generators for the stresses and moduli are evaluated in detail and some representative numerical examples are presented. Furthermore, we propose an extension to orthotropic symmetry.     

QUASI－FLOW CORNER THEORY ON LARGE PLASTIC DEFORMATION OF DUCTILE METALS AND ITS APPLICATIONS

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