Toward the thermodynamic modeling of reacting ionic mixtures

Based on the Müller–Liu entropy principle and the axioms of constitutive theory, a continuum model for reacting ionic mixtures is presented. The influence of microscopic structure on the mixture dynamics is taken into account through the thermodynamics of polar materials. Moreover, mechanical balance laws for classical mixtures under influence of electromagnetic fields and quasi-electrostatic Maxwell’s equations are briefly shown. With an appropriate constitutive model for a diluted and isotropic mixture of non-volatile solutes and by considering the same temperature field for all constituents, constraints on constitutive quantities are imposed, and the conditions for the thermodynamic equilibrium are established from the entropy principle. Furthermore, the nonlinear nature of chemical reactions as well as the reciprocal nature of some irreversible processes is highlighted. Unlike the classical approach for electrolyte solutions, the current constitutive model incorporates thermoelectric and electro-kinetic phenomena into the phenomenological equations, providing a more comprehensive approach of electrolyte solutions dynamics.     

Thermoelasto-viscous Materials

A thermoelasto-viscous material is defined by a set of constitutive laws in which the stress, entropy, heat flux and free energy are functions of the present configuration, temperature, temperature gradient and the rate of change of all three of these. Here these materials are presented within the framework of Walter Noll’s new theory of simple materials, so that the constitutive laws are specified without the use of a frame of reference. The Coleman-Noll procedure is carried out, and the symmetry group of the material is also discussed, both without using a frame of reference. It is then shown what form the constitutive laws of a fluid thermoelasto-viscous material take when a frame of reference is considered. Finally, the governing equations for these materials are explicitly obtained and discussed. The results in this paper may serve as a foundation for new and better mathematical models to deal with phenomena such as heat transfer, heat exchanges and thermophoresis.     

Simple Thermomechanical Materials with Memory

This paper presents a framework for formulating frame-free constitutive laws for simple materials accounting for thermal effects. It is shown how memory can be modeled within this framework and fading memory is given as an example. The restrictions that the second law of thermodynamics on the constitutive laws for materials with fading memory are found using the Coleman?CNoll procedure. It is shown that materials with fading memory, when undergoing a process that is long and slow enough can be approximated in some sense. Finally, equilibrium states are considered and under weak assumptions the familiar relations from thermostatics are obtained.     

Universal relations for non-linear magnetoelastic solids

In the light of recent and growing interest in the applications of magneto-sensitive elastomers and the corresponding theoretical analysis of their properties, this paper is devoted to the derivation of universal relations for these materials, that is connections between the components of a stress tensor and the components of the magnetic (induction) field vector that hold independently of the choice of constitutive law within a considered class of such laws. Here, attention is focussed on isotropic magnetoelastic materials. In particular, within this framework, it is shown that in general there is only one possible universal relation for these materials, but for particular classes of constitutive laws or for special deformations there can be more than one. The theory is exemplified by application to the problem of homogeneous triaxial deformation combined with a simple shear.     

The implicit standard material theory for modelling the nonassociative behaviour of metals

Summary  Modelling the elastoplastic or elastoviscoplastic behaviour of metallic materials exhibiting strain hardening and damage leads to complex nonassociative constitutive equations, sources of many theoretical and numerical troubles. The usual modelling of a nonassociative constitutive equation leads to the loss of the interesting and very useful properties of generalised standard materials deriving from the key concepts of convexity and normality. The argument that will be developed is that the bipotential concept is an appropriate answer. In the first part, after introducing the state variables generally used to describe the behaviour of metallic materials, the constitutive equations subjected to the principles of thermodynamics are derived from two potentials. The state potential gives the state laws, and the bipotential of dissipation delivers the evolution laws for state variables, through the implicit normality assumption. The second part is devoted to several particular applications to metal elastoplasticity and elastoviscoplasticity models. Received 29 March 2000; accepted for publication 26 September 2000     

橡胶类材料大变形本构关系及其有限元方法

讨论大变形拟不可压缩模胶类材料的本构关系及有限元分析方法，采用乘法分解，将变形梯度表示成等容和体积变形两部分，在此基础上，推导了克希荷夫应力和格林应力表示的Ｙｅｏｈ形式应变能模胶类材料的本构关系及数值处理方法，为处理不可压缩问题，采用三场变分原理，其中静水压力，体积膨胀，以及位移均作为独立变量进行处理，并指出该变分原理同胡－鹫津广义变分原理的联系，变形采用相容等参插值，压力及体积膨胀采用低阶插值，     

A Plane-Strain Formulation of Elastic-Plastic Constitutive Laws

ABSTRACT

Constitutive laws for elastic-plastic materials are derived by eliminating the transverse stress component on the basis of the plane-strain constraint. This leads to a fictitious hardening and temperature dependence of the loading function. For standard elastic-plastic materials the resulting laws are associated; however, the plastic strain state is represented by equivalent plastic-strain measures, which also account for transverse yielding. The new constitutive laws, together with the standard reduced form of the equilibrium and compatibility equations, permit the formulation of the plane-strain elastic-plastic analysis problem in terms of the in-plane stress components only. In the case of perfectly plastic materials, the subsequent plane-strain yield surfaces are contained within a domain bounded by a limit surface which represents the yield condition normally adopted in plane-strain limit analysis.     

Development of constitutive laws for thermomechanical behaviors of composites containing multi-type ellipsoidal reinforcements

Composite materials are widely used in industrial applications because of their excellent properties and behaviors. While a composite material is defined as a mixture of two or more different materials, many research works in the literature dealt with composites of only two constituents, which are matrix and one type of particles. On the other hand, the theoretical research works that dealt with more than two constituents are rare. Using some additives affects the sintering behavior, the tribological behavior and the fracture mechanics behavior of composites. For example, a suitable amount of additives as sintering aids, in the sintering process, could lower the sintering temperature, enhance phase wettability and bonding strength and improve the interlaminar fracture resistance of a composite. Therefore, it is worthwhile to develop the constitutive laws that describe the behavior of such composite materials. Accordingly, the aim of this paper is to modify the previous paper, Shabana (2003) [Shabana, Y.M., 2003. Incremental constitutive equation for discontinuously reinforced composites considering reinforcement damage and thermoelastoplasticity. Computational Materials Science 28, 31–40], in order to propose constitutive laws that predict the thermomechanical behavior of composites containing multi-type ellipsoidal reinforcements. This includes reinforcements with different materials and/or different shapes that are represented by aspect ratios. These constitutive laws not only predict the macroscopic and microscopic thermoelastoplastic behaviors of composites containing multi-type ellipsoidal reinforcements, but also characterize their different overall effective properties such as modulus of elasticity, Poison’s ratio and thermal expansion coefficient in different directions. Beside this, they are applicable for porous materials and composites with multiple reinforcements and porosities of different shapes and distributions. In the present numerical analyses, composites with two, three and four constituents considering different materials and aspect ratios as well as reinforcement damage are discussed.     

A micromechanics-based elastoplastic damage model for granular materials at low confining pressure

This paper is devoted to the formulation of a micromechanics-based constitutive model for granular materials under relatively low confining pressure. The constitutive formulation is performed within the general framework of homogenization for granular materials. However, new rigorous stress localization laws are proposed. Some local constitutive relations are established under the consideration of irreversible thermodynamics. Macroscopic plastic deformation is obtained by considering local plastic sliding in a limit number of families of contact planes. The plastic sliding at each contact plane is described by a non-associated plastic flow rule, taking into account pressure sensitivity and normal dilatancy. Nonlinear elastic deformation related to progressive compaction of contacts is also taken into account. Material softening is described by involving damage process related to degradation of microstructure fabric. The proposed model is applied to some typical granular materials (sands). The numerical predictions are compared with experimental data.     

On internal dissipation inequalities and finite strain inelastic constitutive laws: Theoretical and numerical comparisons

Internal dissipation always occurs in irreversible inelastic deformation processes of materials. The internal dissipation inequalities (specific mathematical forms of the second law of thermodynamics) determine the evolution direction of inelastic processes. Based on different internal dissipation inequalities several finite strain inelastic constitutive laws have been formulated for instance by Simo [Simo, J.C., 1992. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Computer Methods in Applied Mechanics and Engineering 99, 61–112]; Simo and Miehe [Simo, J.C., Miehe, C., 1992. Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Computer Methods in Applied Mechanics and Engineering 98, 41–104]; Lion [Lion, A., 1997. A physically based method to represent the thermo-mechanical behavior of elastomers. Acta Mechanica 123, 1–25]; Reese and Govindjee [Reese, S., Govindjee, S., 1998. A theory of finite viscoelasticity and numerical aspects. International Journal of Solids and Structures 35, 3455–3482]; Lin and Schomburg [Lin, R.C., Schomburg, U., 2003. A finite elastic–viscoelastic–elastoplastic material law with damage: theoretical and numerical aspects. Computer Methods in Applied Mechanics and Engineering 192, 1591–1627]; Lin and Brocks [Lin, R.C., Brocks, W., 2004. On a finite strain viscoplastic theory based on a new internal dissipation inequality. International Journal of Plasticity 20, 1281–1311]; and Lin and Brocks [Lin, R.C., Brocks, W., 2005. An extended Chaboche’s viscoplastic law at finite strains: theoretical and numerical aspects. Journal of Materials Science and Technology 21, 145–147]. These constitutive laws are consistent with the second law of thermodynamics. As the internal dissipation inequalities are described in different configurations or coordinate systems, the related constitutive laws are also formulated in the corresponding configurations or coordinate systems. Mathematically, these constitutive laws have very different formulations. Now, a question is whether the constitutive laws provide identical constitutive responses for the same inelastic constitutive problems. In the present work, four types of finite strain viscoelastic and endochronically plastic laws as well as three types of J₂-plasticity laws are formulated based on four types of dissipation inequalities. Then, they are numerically compared for several problems of homogeneous or complex finite deformations. It is demonstrated that for the same inelastic constitutive problem the stress responses are identical for deformation processes without rotations. In the simple shear deformation process with large rotation, the presented viscoelastic and endochronically plastic laws also show almost identical stress responses up to a shear strain of about 100%. The three laws of J₂-plasticity also produce the same shear stresses up to a shear strain of 100%, while different normal stresses are generated even at infinitesimal shear strains. The three J₂-plasticity laws are also compared at three complex finite deformation processes: billet upsetting, cylinder necking and channel forming. For the first two deformation processes similar constitutive responses are obtained, whereas for the third deformation process (with large global rotations) significant differences of constitutive responses can be observed.     

A principle of virtual work for combined electrostatic and mechanical loading of materials

The equations governing mechanics and electrostatics are formulated for a system in which the material deformations and electrostatic polarizations are arbitrary. A mechanical/electrostatic energy balance is formulated for this situation in terms of the electric enthalpy, in which the electric potential and the electric field are the independent variables, and charge and electric displacement, respectively, are the conjugate thermodynamic forces. This energy statement is presented in the form of a principle of virtual work (PVW), in which external virtual work is equated to internal virtual work. The resulting expression involves an internal material virtual work in which (1) material polarization is work-conjugate to increments of electric field, and (2) a combination of Cauchy stress, Maxwell stress and a product of polarization and electric field is work-conjugate to increments of strain. This PVW is valid for all material types, including those that are conservative and those that are dissipative. Such a virtual work expression is the basis for a rigorous formulation of a finite element method for problems involving the deformation and electrostatic charging of materials, including electroactive polymers and switchable ferroelectrics. The internal virtual work expression is used to develop the structure of conservative constitutive laws governing, for example, electroactive elastomers and piezoelectric materials, thereby determining the form of the Maxwell or electrostatic stress. It is shown that the Maxwell or electrostatic stress has a form fully constrained by the constitutive law and cannot be chosen independently of it. The structure of constitutive laws for dissipative materials, such as viscoelastic electroactive polymers and switchable ferroelectrics, is similarly determined, and it is shown that the Maxwell or electrostatic stress for these materials is identical to that for a material having the same conservative response when the dissipative processes in the material are shut off. The form of the internal virtual work is used further to develop the structure of dissipative constitutive laws controlled by rearrangement of material internal variables.     

Continuous symmetries and constitutive laws of dissipative materials within a thermodynamic framework of relaxation: Part I: Formal aspects

An analysis of the continuous symmetries of the constitutive laws of inelastic materials written within a thermodynamical framework of relaxation is performed. This framework relies on the generalization of Gibb’s relationship outside the equilibrium of a uniform system, and the use of the fluctuation theory to model the material dissipation due to its internal microstructure change [Cunat, C., 2001. The DNLR approach and relaxation phenomena. Part I – Historical account and DNLR formalism. Mech. Time-depend. Mater. 5, 39–65]. The approach leads to a viscoelastic like formulation for small deformations, and changes gradually for finite strains towards elastoviscoplasticity (with or without damage) via a dependence of characteristic times with the loading path, in a way similar to the endochronic approach developed by Valanis [Valanis, K.C., 1975. On the fundations of the endochronic theory of viscoplasticity. Arch. Mech. 27, 857–868]. The present thermodynamic framework has been previously applied to elastoviscoplastic materials under cyclic and non-proportional loadings [Dieng, L., Abdul-Latif, A., Haboussi, M., Cunat, C., 2005b. Cyclic plasticity modeling with the distribution of non-linear relaxations approach. Int. J. Plasticity 21, 353–379]. The constitutive laws split into the state laws relating intensive variables (thermodynamics forces) to extensive-like variables, and the complementary evolution laws of the internal variables associated to the dissipative mechanisms. An interpretation of a non-equilibrium thermodynamic approach of irreversible processes in terms of an extremum principle is proposed, associated to a Lagrangian functional. It is shown that one possible choice for the Lagrangian kernel is the material derivative of the internal energy density, augmented by a complementary term that accounts for the evolution laws of the internal variables. Interpreting the material behavior during the non-equilibrium evolution as the Euler–Lagrange equations of the resulting action integral, a differential condition expressing both the local and variational symmetries encapsulated into the Lagrangian formulation is formulated. It is further shown that both symmetry conditions are fully equivalent along the optimal path corresponding to the satisfaction of the constitutive laws. In terms of both practical and methodological aspects, the predictive nature of the symmetry analysis is highlighted, as a systematic tool for the exploitation of the constitutive response. Its performance and utility are exemplified by the construction of a time–temperature equivalence principle for a dry viscous polymer (PA66); the calculated shift factor is shown to well agree with the empirical shift factor given by Williams–Landel–Ferry (WLF) expression. A systematic interpretation of the calculated symmetry groups of the constitutive laws in terms of master curves for various plastic and viscoplastic materials shall be presented in a forthcoming contribution.     

广义参变本构模型分析

本文考察了力和(或)位移受给定条件控制的结构单元。这种控制条件可由线性化折线型本构关系表征,其本构关系是广义的,包括弹性、塑性、接触及其组合单元。借助于参变量变分原理和分级方法,具有广义本构关系单元的结构可以得解。所提出的方法还可直接用于解断裂问题。     

基于分子动力学模拟的多晶结构微观热-力耦合行为的计算

基于分子动力学模拟,建立了一套可用于表征微观下多晶结构热-力耦合行为的算法框架。该算法的要点是将连续模型和分子模拟耦合起来,并使守恒定律在微观连续模型和原子层次上都得到满足,与利用传统的连续介质力学建立晶界与晶粒的本构方程相比,本模型中的连续流是通过原子模型准确计算得到的,从而避免了使用经验的本构方程。     

A simple framework for full-network hyperelasticity and anisotropic damage

A formulation of a constitutive behaviour law is proposed for hyperelastic materials, such that damage induced anisotropy can be accounted for continuously. The full-network approach with directional damage is adopted as a starting point. The full-network law with elementary strain energy density based on the inverse Langevin is chosen as a reference law which is cast into the proposed framework. This continuum formalism is then rewritten using spherical harmonics to capture damage directionality. The proposed formalism allows for an efficient (and systematic) expansion of complex non-linear anisotropic constitutive laws. A low order truncated expression of the resulting behaviour is shown to reproduce accurately the stress-strain curves of the exact behaviour laws.     

Modelling of stress softening in elastomeric materials: foundations of simple theories

This work is concerned with formulation of constitutive relations for materials exhibiting the stress softening phenomenon (known as the Mullins effect) typical observed in elastomeric and other amorphous materials during loading–reloading cycles. It is assumed that microstructural changes in such materials during the deformation process can be represented by a single scalar-valued softening variable whose evolution is accompanied by microforces satisfying their own law of balance, besides the classical laws of mechanics underlying macroscopic deformation of a material. The constitutive equations are then derived in consistency with thermodynamics of irreversible processes with the restriction to purely mechanical theory. The general form of the derived constitutive equations is subsequently simplified through introduction of additional assumptions leading to various models of the stress softening phenomenon. As an illustration of the general theory, it is shown that the so-called pseudo-elastic model proposed in the literature may be derived without an ad hoc postulate of the variational principle.     

现代统一塑性理论

经典的本构理论不能精确描述材料表现出的率相关性和非线性本构关系,导致现代统一塑性理论的出现.各派现代统一塑性理论形式不相同,但最根本的观点接近:①材料变形由弹性变形与非弹性变形两部分组成,不再将非弹性变形区分为塑性变形与蠕变;②存在着两种物理参量:背应力X与粘滞系数K_0 σ—X构成非弹性形变的动力,K是非弹性应变的阻力,因而ε~(in)=f((σ-X)/K);③原始材料X= 0,K=K_0,K_0为较大的标量,所以加载初始表现出线弹性,续继加载X与K按演化规律变化导致非线性形变规律的出现.各派统一塑性理论的主要差异在于X与K的演化公式不同.各有其长亦有其短.在大量试验基础上取各派之长,舍各派之短,融合为一科学的理论是今后发展的方向.      

压电切口张开角和深度对其尖端力电损伤场的影响

基于三维各向异性压电损伤本构理论,导出了广义平面应力问题的损伤本构方程,并据此分析了压电薄板板边V形切口尖端附近的力电损伤,研究了切口张开角和深度对切口尖端力电损伤的影响规律．结果发现：和张开角对切口尖端损伤的影响相比,深度的影响更为明显;在张开角对切口尖端力损伤的影响规律方面,压电材料与一般弹塑性材料存在明显差异,原因在于压电切口尖端力电载荷比会随着深度的改变发生很大变化;不同深度下张开角与切口尖端力、电损伤关系曲线随着张开角的增大由发散逐渐会聚,不同张开角下深度与切口尖端力、电损伤关系曲线随着切口加深由会聚逐渐发散,并且电损伤曲线表现得更为明显．     

电磁热弹性介质的一些基本理论─电磁热动力理论

具有电磁热弹性耦合性质的介质是许多传感器及机敏结构或系统中的关键材料，本文则是力争对这种耦合材料的基本理论进行系统的研究。第一部分基于被推广的经典热动力学理论推导和归纳了电磁热弹性体的几个守恒方程、动力学方程、本构方程和热平衡方程。第二部分则是系统地讨论了在数值计算中具有极重要意义的电磁热弹性变分原理。     

A systematic procedure for constructing critical state models in three dimensions

A general procedure for developing constitutive models for frictional materials possessing a critical state is developed in a three-dimensional context. The procedure starts from the laws of thermo-dynamics, so that the first and second laws of thermo-dynamics are automatically satisfied. There is hence no need to invoke any extraneous stability postulates. The models involve a number of parameters, which can be interpreted in terms of micro-mechanical energy storage and dissipative mechanisms. In most cases non-associated flow rules are predicted and in some cases the yield surfaces are seen to have concave segments. The procedure is more general than that traditionally used for materials with non-associated flow rules, in that plastic potentials are not needed and not presumed to exist. In illustration, examples of families of models are given in which the critical state surface is either the Drucker–Prager or the Matsuoka–Nakai cone.