A note on nonassociated plastic flow rules

The analytical properties of the constitutive equations in plasticity with a nonassociated flow rule are investigated. Under the assumption of small deformations the directional stiffness (and compliance) rule is considered and the relevant spectral properties of the tangent stiffness tensor are assessed. It is shown that the directional stiffness may be larger than the elastic. It may also be negative in the case of a formally perfectly plastic material and so the nonassociative flow rule represents (spurious) softening in terms of an associated flow rule. The issue of uniqueness at finite strains is briefly addressed, whereby use is made of the tangent stiffness tensor relating the velocity gradient to the first Piola-Kirchhoff stress rate. The relevant spectral properties, which generalise those from the small deformation case, are found explicit. A sufficient condition for uniqueness is given in terms of a critical (upper bound) value of the hardening modulus.     

On free energy-based formulations for kinematic hardening and the decomposition F = fpfe

Within the framework of linear plasticity, based on additive decomposition of the linear strain tensor, kinematical hardening can be described by means of extended potentials. The method is elegant and avoids the need for evolution equations. The extension of small strain formulations to the finite strain case, which is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, proved not straight forward. Specifically, the symmetry of the resulting back stress remained elusive. In this paper, a free energy-based formulation incorporating the effect of kinematic hardening is proposed. The formulation is able to reproduce symmetric expressions for the back stress while incorporating the multiplicative decomposition of the deformation gradient. Kinematic hardening is combined with isotropic hardening where an associative flow rule and von Mises yield criterion are applied. It is shown that the symmetry of the back stress is strongly related to its treatment as a truly spatial tensor, where contraction operations are to be conducted using the current metric. The latter depends naturally on the deformation gradient itself. Various numerical examples are presented.     

Application of corotational rates of the logarithmic strain in constitutive modeling of hardening materials at finite deformations

In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. The evolution equation for the kinematic hardening of this model relates the corotational rate of the back stress tensor to the corotational rate of the logarithmic strain tensor. Using Jaumann, Green–Naghdi, Eulerian and logarithmic corotational rates in the proposed constitutive model, stress–strain responses and subsequent yield surfaces are determined for rigid plastic kinematic and isotropic hardening materials in the simple shear problem at finite deformations.     

Stress-induced crystal preferred orientation in the poromechanics of in-pore crystallization

A non-hydrostatic stress field affects the orientation of crystals growing in the pore network of an elastic porous medium. The hypothesis of a hydrostatic state of stress within the crystal has been implicitly made in the recent extension of poromechanics to in-pore crystalization (Coussy, 2006). This underlying hypothesis is revisited on a small-scale conceptual model based on Eshelby's problem and shows that chemo-mechanical equilibrium requires that the crystal adapts its shape and orientation to the far-field stress, therefore resulting at equilibrium in a hydrostatic state of stress within the crystal. The optimum crystal shape as a function of the far-field stress is consistently investigated, highlighting limiting cases. The small scale model allows to understand the macroscopic effects associated with deviatoric stresses in the poromechanics of in-pore crystallization. Moreover, it provides the building block for an up-scaling of the macroscopic tangent poroelastic properties, which depend on both the current crystal saturation and the state of stress. A dilute micromechanical scheme illustrates the variation of the macroscopic Biot's coefficient tensor as a function of deviatoric stresses. A simple configuration akin to a potential laboratory experiment finally illustrates the strong induced anisotropy of the crystallization induced macroscopic strain when deviatoric stresses are applied to the material prior to crystallization.     

A note on divergence and flutter instabilities in elastic–plastic materials

Dynamic stability of uniform straining of a nonlinear elastic solid is known to require that all eigenvalues of the acoustic tensor associated with the tangent elastic moduli be real and nonnegative. The focus of this note is to what extent this conclusion applies to time-independent, elastoplastic materials. Nonlinearity of the elastic–plastic constitutive law imposes limits on validity of a solution to the linear problem for which the acoustic tensor is determined. The effect of those limits on the conclusions about instability is examined.     

基于高阶Cauchy-Born准则的单壁碳纳米管本构模型

提出了一种基于高阶Cauchy—Born准则建立单壁碳纳米管本构模型的方法。通过引入高阶变形梯度,合理地修正了传统Cauchy—Born准则在描述纳米管变形几何关系时所存在的缺陷。利用原子间相互作用势以及能量等效原理,得到了基于广义连续介质模型的单壁碳纳米管的本构关系。由此得到的本构参数不仅与变形梯度张量F,而且与其梯度F相关,因此是一种广义连续介质模型。利用这样的本构模型,本文还对单壁碳纳米管的杨氏模量进行了预测,并与采用其他方法得到的结果进行了对比,从而证实了所提出方法的有效性。     

Invariant Properties for Finding Distance in Space of?Elasticity Tensors

Using orthogonal projections, we investigate distance of a given elasticity tensor to classes of elasticity tensors exhibiting particular material symmetries. These projections depend on the orientation of the elasticity tensor; hence the distance is obtained as the minimization of corresponding expressions with respect to the action of the orthogonal group. These expressions are stated in terms of the eigenvalues of both the given tensor and the projected one. The process of minimization is facilitated by the fact that, as we prove, the traces of the corresponding Voigt and dilatation tensors are invariant under these orthogonal projections. For isotropy, cubic symmetry and transverse isotropy, we formulate algorithms to find both the orientation and the eigenvalues of the elasticity tensor endowed with a particular symmetry and closest to the given elasticity tensor.      

An illustration of the equivalence of the loss of ellipticity conditions in spatial and material settings of hyperelasticity

The loss of ellipticity indicated through the rank-one-convexity condition is elaborated for the spatial and material motion problem of continuum mechanics. While the spatial motion problem is characterized through the classical equilibrium equations parametrised in terms of the deformation gradient, the material motion problem is driven by the inverse deformation gradient. As such, it deals with material forces of configurational mechanics that are energetically conjugated to variations of material placements at fixed spatial points. The duality between the two problems is highlighted in terms of balance laws, linearizations including the consistent tangent operators, and the acoustic tensors. Issues of rank-one-convexity are discussed in both settings. In particular, it is demonstrated that if the rank-one-convexity condition is violated, the loss of well-posedness of the governing equations occurs simultaneously in the spatial and in the material motion context. Thus, the material motion problem, i.e. the configurational force balance, does not lead to additional requirements to ensure ellipticity. This duality of the spatial and the material motion approach is illustrated for the hyperelastic case in general and exemplified analytically and numerically for a hyperelastic material of Neo-Hookean type. Special emphasis is dedicated to the geometrical representation of the ellipticity condition in both settings.     

Effect of deviatoric nonassociativity on the failure prediction for elastic–plastic materials

This paper presents a theoretical study of the effect of nonassociativity of the plastic flow rule on the critical plastic modulus for discontinuous bifurcation in an elastic–plastic material. Nonassociativity in both the spherical and the deviatoric spaces are considered, with an emphasis on the effect of nonassociativity in the deviatoric space. A particular form of nonassociativity in the deviatoric space is introduced, where the projections of the plastic flow direction and the normal to the yield surface are assumed to have the same length but the projection of plastic flow direction is allowed to lag that of the normal by an angle. It is shown that even for the simple yield surface of von Mises, nonassociativity in the deviatoric space can lead to a bifurcation for a load parameter significantly lower than the value predicted with an associated flow rule.     

On loss of ellipticity in second-gradient hyper-elasticity of fibre-reinforced materials

This paper establishes a three-dimensional hyper-elasticity framework for studying the manner in which fibre bending stiffness affects current knowledge regarding the presence of weak discontinuity surfaces in unconstrained fibre-reinforced solid materials. This is achieved by considering and studying the loss of ellipticity of a new set of incremental partial differential equations, which emerges from the second-gradient, hyper-elasticity theory developed in [11] and yields its conventional theory counterpart as a particular case. It is accordingly seen that, besides the conventional acoustic tensor met in relevant symmetric hyper-elasticity studies, where fibres are assumed perfectly flexible, some new, higher-order acoustic tensor is involved and becomes dominant in the present situation. Nevertheless, the manner becomes also clear in which the present analysis reduces to and, hence, connects with loss of ellipticity concepts met in conventional hyper-elasticity. No particular form is assigned to the strain energy density of the material, which is kept general throughout this paper. Considerable elucidation of the outlined new issues and concepts is however achieved by focusing attention on plane deformations of hyper-elastic solids reinforced by a single family of straight fibres. This development concludes with a specific application which relates the present analysis with kink band formation in unidirectional fibrous composites containing fibres resistant in bending.     

Orientation dependence of stress distributions in polycrystals deforming elastoplastically under biaxial loadings

The influence of biaxiality of the loading on the crystallographic orientation dependence of crystal stress distributions is examined for polycrystalline solids deformed well into the elastoplastic regime. The examination is couched in terms of two decompositions of the stress. The first is a split of the tensor into its hydrostatic and deviatoric components; the second is a spectral decomposition of the deviatoric stress from which we express the relative values of the principal components as a function of the biaxiality of the stress. Using the framework provided by these decompositions, we investigate trends observed in the lattice strains in polycrystals subjected to biaxial loadings, comparing strains measured by neutron diffraction with finite element simulations. We conclude by showing how the orientation dependence of the stress distributions is influenced by the load biaxiality and by connecting features of the distributions to the elastic and plastic properties of the crystals. Implications of the results are discussed relative to the modeling of strain hardening and defect initiation.     

Construction of eigentensor columns in the linear micropolar theory of elasticity

The internal structure of the block tensor matrix of the elastic modulus tensors is studied for the case of micropolar theory. In particular, the problem of finding the eigenvalues and eigentensor columns of block tensor matrices is considered. The complete orthonormal system of eigentensor columns for a block tensor matrix is constructed. A number of definitions and theorems are formulated. Several newly introduced terms are used to propose various representations of the specific strain energy and the corresponding constitutive relations.     

A new exact integration method for the Drucker–Prager elastoplastic model with linear isotropic hardening

This paper presents the exact stress solution of the non-associative Drucker–Prager elastoplastic model governed by linear isotropic hardening rule. The stress integration is performed under constant strain-rate assumption and the derived formulas are valid in the setting of small strain elastoplasticity theory. Based on the time-continuous stress solution, a complete discretized stress updating algorithm is also presented providing the solutions for the special cases when the initial stress state is located in the apex and when the increment produces a stress path through the apex. Explicit expression for the algorithmically consistent tangent tensor is also derived. In addition, a fully analytical strain solution is also derived for the stress-driven case using constant stress-rate assumption. In order to get a deeper understanding of the features of these solutions, two numerical test examples are also presented.     

A parametric description of orthotropic plasticity in metal sheets

A polar-coordinate representation of the yield surface in principal stress space is utilized to formulate constitutive equations for plane-stress plasticity of orthotropic sheets. The yield function and the associated flow rule are analysed by taking account of the orientation of the principal stress axes, and conditions for internal consistency of the model are derived. An orthotropic yield criterion is proposed, which is devised as an extension of a previous isotropic yield function involving the second and third invariants of the deviatoric stress tensor. Comparisons with micro-macro computations and experimental measurements of yield surfaces are discussed.     

Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation

In a recent paper we examined the loss of ellipticity and its interpretation in terms of fiber kinking and other instability phenomena in respect of a fiber-reinforced incompressible elastic material. Here we provide a corresponding analysis for fiber-reinforced compressible elastic materials. The analysis concerns a material model which consists of an isotropic base material augmented by a reinforcement dependent on the fiber direction. The assessment of loss of ellipticity can be cast in terms of the eigenvalues of the acoustic tensors associated with the isotropic and anisotropic parts of the strain-energy function. For the anisotropic part, two different reinforcing models are examined and it is shown that, depending on the choice of model and whether the fiber is under compression or extension, loss of ellipticity may be associated with, in particular, a weak surface of discontinuity normal to or parallel to the deformed fiber direction or at an intermediate angle. Under compression the associated failure interpretations include fiber kinking and fiber splitting, while under extension fiber de-bonding and matrix failure are included.     

一种非统一黏塑性本构的有限元实现

在与率相关黏塑性本构模型的背应力分量中加入静态回复项通常用来更精确地描述材料应力松弛效应．本文基于包含静态回复项的非统一黏塑性本构模型,提出了对应的隐式应力积分算法．该模型可简化为两个关于偏背应力和等效应力张量的方程组,通过牛顿迭代法进行求解．在应力积分算法中采用了Return-Mapping求解策略．此外还推导出了材料的一致切线刚度矩阵来判断有限元分析收敛性．在此基础上开发出了适用于ABAQUS工程软件的UMAT用户材料子程序．文中选取了高温下镍合金材料在伴随应力松弛的循环加载路径下的试验结果来验证算法的有效性．从比较结果中可以发现：用户子程序即使在大时间增量下也具有良好的收敛性,静态回复项的引入使得模型对应力松弛效应的模拟更加准确．     

Computational aspects of the integration of the von Mises linear hardening constitutive laws

The problem of the integration of the von Mises linear kinematic and isotropic hardening constitutive equations is considered. A new numerical integration algorithm, a generalised trapezoidal rule, is proposed and discussed in detail. It is shown how the structure of the elastic-plastic constitutive equations of the, well known, backward difference and midpoint rules, leading to a symmetric consistent tangent modulus, can be adopted for this trapezoidal rule. On this base a unified treatment of the backward difference, midpoint, and trapezoidal rules is presented. An accuracy analysis is conducted by means of detailed isoerror maps so as to provide a comparison between different integration algorithms.     

A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: Modelling

A constitutive model for anisotropic elastoplasticity at finite strains is developed together with its numerical implementation. An anisotropic elastic constitutive law is described in an invariant setting by use of structural tensors and the elastic strain measure C_e. The elastic strain tensor as well as the structural tensors are assumed to be invariant in relation to superimposed rigid body rotations. An anisotropic Hill-type yield criterion, described by a non-symmetric Eshelby-like stress tensor and further structural tensors, is developed, where use is made of representation theorems for functions with non-symmetric arguments. The model also considers non-linear isotropic hardening. Explicit results for the specific case of orthotropic anisotropy are given. The associative flow rule is employed and the features of the inelastic flow rule are discussed in full. It is shown that the classical definition of the plastic material spin is meaningless in conjunction with the present formulation. Instead, the study motivates an alternative definition, which is based on the demand that such a quantity must be dissipation-free, as the plastic material spin is in the case of isotropy. Equivalent spatial formulations are presented too. The full numerical treatment is considered in Part II.     

Loading restrictions for the Blatz-Ko hyperelastic model—application to a finite element analysis

In this paper, we consider the Blatz-Ko constitutive model for compressible elastic solids. It is shown that the Cauchy stress tensor leads to a normal loading limitation. This limitation induces slow convergence of the Newton-Raphson algorithm near the maximum authorized normal loading value and divergence if this value is exceeded. In addition, convergence of the Newton-Raphson scheme also depends on ellipticity and strong ellipticity conditions. These various points are discussed in the case of a rectangular specimen subjected to a tensile load and modeled with finite elements.     

Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.