Phenomenological model for the macroscopical material behavior of ferroelectric ceramics

A thermodynamically consistent phenomenological model for the simulation of the macroscopic behavior of ferroelectric polycrystalline ceramics is presented. It is based on the choice of microscopically motivated internal state variables, which describe the texture and the polarization state of the polycrystal. Saturation states are defined for the internal state variables. The linear material behavior is modelled by a transversely isotropic piezoelectric constitutive law, where the anisotropy is history dependent. For non-linear irreversible processes, a switching function and associated evolution rules are applied, satisfying the principle of maximum ferroelectric dissipation. Saturation is modelled by the use of energy-barrier functions in the electric enthalpy density function. Numerical examples demonstrate the capability of the proposed model, to predict the typical experimental results.     

Quasistatic Evolution in Perfect Plasticity for General Heterogeneous Materials

Inspired by some recent developments in the theory of small-strain heterogeneous elastoplasticity, we both revisit and generalize the formulation of the quasistatic evolutionary problem in perfect plasticity given by Francfort and Giacomini (Commun Pure Appl Math, 65:1185–1241, 2012). We show that their definition of the plastic dissipation measure is equivalent to an abstract one, where it is defined as the supremum of the dualities between the deviatoric parts of admissible stress fields and the plastic strains. By means of this abstract definition, a viscoplastic approximation and variational techniques from the theory of rate-independent processes give the existence of an evolution satisfying an energy-dissipation balance and consequently Hill’s maximum plastic work principle for an abstract and very large class of yield conditions.     

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.     

The effect of rotation on generalized micropolar thermoelasticity for a half-space under five theories

A model of the equations of a two-dimensional problem in a micropolar thermoelastic medium for a half-space whose surface is free and subjected to an instantaneous thermal point source is studied. The entire elastic medium is rotating with a uniform angular velocity. The formulation is applied under five theories of the generalized thermoelasticity: Lord–Shulman with one relaxation time, Green–Lindsay with two relaxation times, Green–Naghdi theory (of type II) without energy dissipation and Chandrasekharaiah–Tzou theory with dual-phase-lag, as well as the coupled theory. The normal mode analysis is used to obtain the exact expressions for the considered variables. The distributions of the considered variables are illustrated graphically. Comparisons are made with the results predicted by the five theories in the presence and absence of rotation.     

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     

A Variational Formulation of¶Rate-Independent Phase Transformations¶Using an Extremum Principle

We propose a rate-independent, mesoscopic model for the hysteretic evolution of phase transformations in shape-memory alloys. The model uses the deformation and phase-indicator function as basic unknowns and the potentials for the elastic energy and for the dissipation as constitutive laws. Using the associated functionals, admissible processes are defined to be the ones which are stable at all times and which satisfy the energy inequality.This concept leads to a natural time-incremental method which consists in a minimization problem. The mesoscopic model is obtained by a relaxation procedure. It leads to new functionals involving the cross-quasiconvexification of the elastic stored-energy density. For a special case involving two phases of linearized elastic materials we show that the incremental problem provides existence of admissible processes for the time-continuous problem, if we let the time-step go to 0. Dedicated to Erwin Stein on the occasion of his seventiethbirthday     

岩体结构非平衡演化的有效应力原理及长期稳定性分析

开挖卸荷后的天然岩体往往处于非平衡演化状态, 将直接影响岩体工程结构的正常运行、长期稳定和安全. 时效变形和损伤演化是岩体结构非平衡演化的核心. 在赖斯(Rice) 内变量热力学理论框架下, 提出了岩体结构非平衡演化的有效应力原理, 指出有效应力是总应力中能有效驱动结构演化的部分. 将内变量率形式的非弹性应变率方程和能量耗散率函数表示为有效应力形式, 并提出非弹性余能概念. 给定具体的余能密度函数和内变量演化方程, 得到了考虑损伤的内变量黏塑性应变率方程. 通过相似材料加卸载蠕变试验结果进行参数辨识, 并分别计算了内变量率形式和有效应力形式的黏塑性应变率、能量耗散率和非弹性余能, 并对其进行比较分析. 结果表明:在过渡蠕变和稳态蠕变阶段两种形式的方程计算的黏塑性应变率几乎相等, 但在加速蠕变阶段两者相差较大;非弹性余能和能量耗散率全域积分分别从驱动结构非平衡演化的内在潜力和实际效果的角度表征了结构的非平衡演化状态和演化趋势, 能量耗散率积分更合适用于评价岩体工程结构的长期稳定性. 最后以深埋地下洞室作为工程算例, 并对其长期稳定性进行分析.      

A material model for finite elasto-plastic deformations considering a substructure

Developing further the substructure models proposed by Mandel and Dafalias a thermodynamically consistent system of differential and algebraic equations is derived to describe anisotropic elasto-plastic material behavior at finite deformations. Based on the multiplicative split of the deformation gradient an appropriate material law is formulated applying the principle of the maximum of plastic dissipation. Generalized basic relations of this material model containing a relation of hyperelasticity, evolutional equations for the internal variables describing different kinds of hardening, and the yield condition are presented. The capacity of the proposed material model is demonstrated on the example of a sheet with a hole. Presenting the evolution of yield surfaces the capability of the model to describe anisotropic hardening behavior is shown.     

The principle of the minimum of the dissipation potential for non-isothermal processes

In this paper, we contribute to the methodology of material modeling by presenting a potential-based approach for non-isothermal inelastic processes. It is based on the principle of the minimum of the dissipation potential which was used previously only in the isothermal context. In contrast to the principle of maximum dissipation, the presented procedure results in mathematically simplified equations. Due to its variational character, the inclusion of constraints is very simple. After derivation of our method, we use the examples of non-isothermal perfect plasticity and shape memory alloys for demonstration of the validity and performance of the concept.     

耗散系统的虚功变分不等式及其应用

对于保守系统,能量变分原理为推导力学系统控制方程提供了简洁的途径.对于耗散系统,控制方程的建立往往需要引入经验的或理性的假定,增大了建模的难度.针对耗散系统,引入系统局部稳定的概念,并在此基础上,提出一类虚功变分不等式.这一不等式事实上揭示了耗散系统的一类虚功不等原理.该原理的物理含义为:使系统状态稳定的必要条件是,在该状态附近所有可能的虚拟路径上系统释放的势能不大于系统耗散的能量.研究表明:仅需结合虚功不等原理和能量守恒原理,即可导出准静态系统力学状态量的全部控制方程.作为应用,文章重新讨论了塑性力学,结合虚功不等原理与能量守恒原理,导出经典塑性力学的全部控制方程,并证明了经典的最大塑性耗散原理可以作为虚功不等原理的推论导出;同时,以Mohr-Coulomb强度准则为例,讨论了虚功不等原理在强度理论中的应用,说明基于应力的强度准则可以是基于能量的稳定性准则的推论.上述例子说明了虚功不等原理的广泛适用性和在建立耗散系统控制方程中的有效性.     

准脆性材料抗压强度能量平衡尺寸效应模型

针对现有尺寸效应模型难以体现准脆性材料完整的抗压强度尺寸效应变化规律及其内在机理, 本文通过分析准脆性材料单轴压缩破坏过程中能量输入、储存、整体和局部能量耗散, 建立体现整体和局部损伤的力学模型及描述上述能量演化过程的双线性名义和真实应力应变曲线, 在此基础上确定了名义应力最大时输入能量、储存弹性能、整体和局部能量耗散的表达式, 最后基于能量平衡原理建立抗压强度尺寸效应模型. 抗压强度能量平衡尺寸效应模型能完整体现名义抗压强度尺寸效应, 即随试样尺寸增大, 名义抗压强度在试样尺寸小于等于局部损伤区尺寸时为真实强度, 然后逐渐减小, 最终当试样尺寸趋于无穷大时趋于弹性极限强度; 抗压强度能量平衡尺寸效应模型也能同时体现高径比和试样直径对名义强度的影响, 其包含的参数具有明确的物理意义, 可以反映真实强度、弹性极限强度、名义损伤模量非线性、局部损伤区大小和方向对准脆性材料名义抗压强度尺寸效应的影响; 通过把抗压强度能量平衡尺寸效应模型和现有尺寸效应模型应用于预测各种材料尺寸效应试验和数值模拟数据, 结果表明: 抗压强度能量平衡尺寸效应模型能很好描述试验和数值模拟尺寸效应的非线性变化规律及内在机理, 和现有尺寸效应模型相比, 其总体平均误差最小, 且小于5%.      

Numerical simulation of inertial flow of heated and cooled viscoelastic fluids inside a planar sudden expansion channel: investigation of stresses effects on the total dissipation

In this paper, the inertial and non-isothermal flow of viscoelastic fluids in a planar channel with 1:3 sudden expansion has been simulated for Brinkman numbers in the range \( - \,20 \le Br \le 20 \). The mass, momentum and energy conservation equations with the non-linear form of Phan-Thien–Tanner constitutive equation are used to describe the behavior of heated and cooled viscoelastic fluids flow. The properties of fluid are assumed temperature-dependent and the viscous dissipation terms are considered in the energy equation. The object of the current paper is to investigate the stresses and their effects on heat generation via the viscous dissipation terms in the energy equation for inertial flow of heated and cooled viscoelastic fluids. Therefore, plots of streamlines, isothermal lines, normal stress (\( \tau_{xx} \)), normal-transverse stress (\( \tau_{yy} \)) and shear stress (\( \tau_{xy} \)), total dissipation, temperature and local Nusselt numbers have been drawn and examined in the channel expansion. The results show that for the asymmetric flow of heated and cooled viscoelastic fluids, the maximum values of total dissipation are located adjacent to the lower wall and at the centerline of the channel expansion. Also, by incrementing the Brinkman number in the hydrodynamically and thermally developing and fully developed zones, the values of total dissipation are increased.     

On impulsive processes

Consider impulsive processes that consists of between two impulses non-autonomous, non-linear continuous processes subjected to time-varying impulsive constraints. By extending compact processes defined by Dafermos for non-autonomous systems without impulses, a class of regular impulsive processes is provided and its applications are stressed by selected examples. A weak invariance principle and an invariance principle are established for the regular impulsive processes. The paper thus lays ground work for geometric theory of general impulsive processes.     

Overshoot of the non-equilibrium temperature in the shock wave structure of a rarefied polyatomic gas subject to the dynamic pressure

The shock wave structure in a rarefied polyatomic gas is analyzed on the basis of non-linear extended thermodynamics with 6 independent fields (ET6); the mass density, the velocity, the temperature and the dynamic pressure, which permits us to study the shock profile also for large Mach numbers. The first result of this paper is that the shock wave structure is substantially the same as that obtained previously from the linear theory for small or moderately large Mach numbers. Only for very large Mach numbers there exist some differences in the relaxation part of the profile between the model with a non-linear production term and the one with a linear production term. The mathematical reason of this behavior is due to the fact that the non-linear differential system has the same principal part of the linear one.The classical Meixner theory of relaxation processes with one internal variable is fully compatible with the ET6 theory and this fact gives us the explicit expressions of the internal variable and the non-equilibrium temperature in the Meixner theory in terms of the 6 fields, especially, of the dynamic pressure. By using the correspondence relation, the shock wave structure described by the ET6 theory is converted into the variables described by the Meixner theory. It is shown that the non-equilibrium Meixner temperature overshoots in a shock wave in contrast to the kinetic temperature. This implies that the temperature overshoot is a matter of definition of the non-equilibrium temperature.     

Cyclic plasticity modeling with the distribution of non-linear relaxations approach

Motivated by the distribution of non-linear relaxation (DNLR) approach, a phenomenological model is proposed in order to describe the cyclic plasticity behavior of metals under proportional and non-proportional loading paths with strain-controlled conditions. Such a model is based on the generalization of the Gibbs's relationship outside the equilibrium of uniform system and the use of the fluctuation theory to analyze the material dissipation due to its internal reorganization. The non-linear cyclic stress–strain behavior of metals notably under complex loading is of particular interest in this study. Since the hardening effects are described appropriately and implicitly by the model, thus, a host of inelastic behavior of metals under uniaxial and multiaxial cyclic loading paths are successfully predicted such as, Bauschinger, strain memory effects as well as additional hardening. After calibrating the model parameters for two metallic materials, the model has demonstrated obviously its ability to describe the cyclic elastic-inelastic behavior of the nickel base alloy Waspaloy and the stainless steel 316L. The model is then implemented in a commercial finite element code simulating the cyclic stress–strain response of a thin-walled tube specimen. The numerical responses are in good agreement with experimental results.     

Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible–irreversible coupling

Thermodynamic models for viscoplastic solids are often formulated in the context of continuum thermodynamics and the dissipation principle. The purpose of the current work is to show that models for such material behavior can also be formulated in the form of a General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC), see, e.g., Grmela and Öttinger (Phys Rev E, 56:6620–6632, 1997), Öttinger and Grmela (Phys Rev E, 56:6633–6655, 1997), Grmela (J Non-Newtonian Fluid Mech, 165:980–986, 2010). A GENERIC combines Hamiltonian-dynamics-based modeling of time-reversible processes with Onsager–Casimir-based modeling of time-irreversible processes. The result is a model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to model complex fluids, it has recently been applied to anisotropic inelastic solids in Eulerian (Hütter and Tervoort, in J Non-Newtonian Fluid Mech, 152:45–52, 2008; Hütter and Tervoort, in J Non-Newtonian Fluid Mech, 152:53–65, 2008; Hütter and Tervoort, in Adv Appl Mech, 42:254–317, 2008) and Lagrangian (Hütter and Svendsen, in J Elast 104:357–368, 2011) settings, as well as to damage mechanics. For simplicity, attention is focused in the current work on the case of thermoelastic viscoplasticity. Central to this formulation is a GENERIC-based form for the viscoplastic flow rule. A detailed comparison with the formulation based on continuum thermodynamics and the dissipation principle is given.     

On the dissipative zone in anisotropic damage models for concrete

This paper presents a three-dimensional model to simulate the behavior of plain concrete structures that are predominantly tensile loaded. This model, based on continuum damage mechanics, uses a symmetric second-order tensor as the damage variable, which permits the simulation of orthotropic degradation. The validity of the first and the second law of thermodynamics, as well as the validity of the principle of maximum dissipation rate, are required. That is attained by defining the loading functions in quantities that are thermodynamically conjugated to the damage variables. Furthermore, the evolution rule is derived by maximizing the energy dissipation rate. This formulation is regularized by means of the fracture energy approach by introducing a characteristic length. The basic and new idea in this paper is that the characteristic length should always coincide with the width of the dissipative zone appearing in the simulation. The integration points with increasing damage in one loading increment are the dissipative zone in this loading increment. The main objective of this paper is the convenient formulation of approaches for the characteristic length in order to attain the coincidence of the characteristic length with the width of the dissipative zone appearing in the simulation. It is shown that simulations are objective and yield good results if the requirement is fulfilled that the characteristic length in the constitutive law coincides with the width of the dissipative zone in the simulation.