基于微态方法的耦合韧性损伤的弹塑性本构模型

基于广义连续介质力学提出了一个热力学一致性的耦合微态韧性损伤的弹塑性本构模型。该模型遵循Forest的微态方法,在有限变形中提出引入额外的微态损伤因子及其一阶梯度以考虑材料的内部特征尺度。通过广义虚功原理得到了微态损伤的补充控制方程,对亥姆霍兹自由能进行扩展,得到了新的包含微态损伤变量的损伤能量释放率,在微态损伤的正则化作用下,采用隐式迭代更新局部损伤和应力等状态变量。基于Galerkin加权余量法,推导了以传统位移和微态损伤为基本未知量的有限元列式。利用该数值模型,对DP1000材料的单向拉伸实验和十字形零件的冲压实验进行了应变局部化与材料断裂的有限元分析。结果表明,该微态弹塑性损伤模型可以得到一致的有限元模拟响应曲线并收敛到实验曲线,从而避免发生网格依赖性问题。     

Micromorphic continuum. Part II: Finite deformation plasticity coupled with damage

It is demonstrated how a micromorphic plasticity theory may be formulated on the basis of multiplicative decompositions of the macro- and microdeformation gradient tensor, respectively. The theory exhibits non-linear isotropic and non-linear kinematic hardening. The yield function is expressed in terms of Mandel stress and double stress tensors, appropriately defined for micromorphic continua. Flow rules are derived from the postulate of Il’iushin and represent generalized normality conditions. Evolution equations for isotropic and kinematic hardening are introduced as sufficient conditions for the validity of the second law of thermodynamics in every admissible process. Finally, it is sketched how isotropic damage effects may be incorporated in the theory. This is done for the concept of effective stress combined with the hypothesis of strain equivalence.     

高温下混凝土的本构模拟及破坏分析

针对已建立的高温下混凝土中化学-热-水力-力学耦合过程分析的分级数学模型,发展了混凝土的化学-热-水力-力学(CTHM)耦合本构模型。在已有的Willam-Warnke弹塑性屈服准则基础上发展了考虑脱水和脱盐引起的材料损伤及化学塑性软化、塑性应变硬化/软化和吸力硬化的广义Willam-Warnke本构模型,模拟高温下混凝土的材料非线性行为。为保证全局守恒方程的Newton迭代过程的二阶收敛率,导出了非线性化学-热-水力-力学(CTHM)耦合本构模型的一致性切线模量矩阵。数值结果显示了本文所发展的化学-热-水力-力学(CTHM)耦合本构模型在模拟高温下混凝土中复杂破坏过程的能力和有效性。     

Micromorphic continuum Part I: Strain and stress tensors and their associated rates

Micropolar and micromorphic solids are continuum mechanics models, which take into account, in some sense, the microstructure of the considered real material. The characteristic property of such continua is that the state functions depend, besides the classical deformation of the macroscopic material body, also upon the deformation of the microcontinuum modeling the microstructure, and its gradient with respect to the space occupied by the material body. While micropolar plasticity theories, including non-linear isotropic and non-linear kinematic hardening, have been formulated, even for non-linear geometry, few works are known yet about the formulation of (finite deformation) micromorphic plasticity. It is the aim of the three papers (Parts I, II, and III) to demonstrate how micromorphic plasticity theories may be formulated in a thermodynamically consistent way.In the present article we start by outlining the framework of the theory. Especially, we confine attention to the theory of Mindlin on continua with microstructure, which is formulated for small deformations. After precising some conceptual aspects concerning the notion of microcontinuum, we work out a finite deformation version of theory, suitable for our aims. It is examined that resulting basic field equations are the same as in the non-linear theory of Eringen, which deals with a different definition of the microcontinuum. Furthermore, geometrical interpretations of strain and curvature tensors are elaborated. This allows to find out associated rates in a natural manner. Dual stress and double stress tensors, as well as associated rates, are then defined on the basis of the stress powers. This way, it is possible to relate strain tensors (respectively, micromorphic curvature tensors) and stress tensors (respectively, double stress tensors), as well as associated rates, independently of the particular constitutive properties.     

A new nonlocal bending model for Euler–Bernoulli nanobeams

This paper is concerned with the bending problem of nanobeams starting from a nonlocal thermodynamic approach. A new coupled nonlocal model, depending on two nonlocal parameters, is obtained by using a suitable definition of the free energy. Unlike previous approaches which directly substitute the expression of the nonlocal stress into the classical equilibrium equations, the proposed approach provides a methodology to recover nonlocal models starting from the free energy function. The coupled model can then be specialized to obtain a nanobeam formulation based on the Eringen nonlocal elasticity theory and on the gradient elastic model. The variational formulations are consistently provided and the differential equations with the related boundary conditions are thus derived. Nanocantilevers are solved in a closed-form and numerical results are presented to investigate the influence of the nonlocal parameters.     

Uniformly convergent numerical method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition

In this article, we consider a class of singularly perturbed differential equations of convection-diffusion type with nonlocal boundary conditions. A uniformly convergent numerical method is constructed via nonstandard finite difference and numerical integration methods to solve the problem. The nonlocal boundary condition is treated using numerical integration techniques. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be ϵ -uniformly convergent.     

Numerical implementation and assessment of the GLPD micromorphic model of ductile rupture

Just like all constitutive models involving softening, Gurson's classical model for porous ductile solids predicts unrealistic, unlimited localization of strain and damage. An improved variant of this model aimed at solving this problem has been proposed by Gologanu, Leblond, Perrin and Devaux (GLPD) on the basis of some refinement of Gurson's original homogenization procedure. The GLPD model is of “micromorphic” nature since it involves the second gradient of the macroscopic velocity and generalized macroscopic stresses of “moment” type, together with some characteristic “microstructural distance”. This work is devoted to its numerical implementation and the assessment of its practical relevance. This assessment is based on two criteria: absence of mesh size effects in finite element computations and agreement of numerical and experimental results for some typical experiments of ductile fracture. The GLPD model is found to pass both tests. It is therefore concluded that it represents a viable, although admittedly complex solution to the problem of unlimited localization in Gurson's model of ductile rupture.     

A thermodynamic consistent damage and healing model for self healing materials

Thermodynamics of the damage and the healing processes for viscoplastic materials is discussed in detail and constitutive equations for coupled inelastic-damage-healing processes are proposed in a thermodynamic consistent framework. Small deformation state is utilized and the kinematic and the isotropic hardening effects for the damage and healing processes are introduced into the governing equations. Two new yield surfaces for the damage and healing processes are proposed that take into account the isotropic hardening effect. The computational aspect for solving the coupled elasto-plastic-damage-healing problem is investigated, and the mechanical behavior of the proposed polymeric based self healing system is obtained. Uniaxial compression tests are implemented on a shape memory polymer based self healing system and the damage and the healing are captured by measurement of the changes in the modulus of elasticity. It is concluded that the proposed constitutive equations can model the damage and healing effectively and the mechanical behavior of a shape memory polymer based self healing system can be precisely modeled using this formulation.     

Nonlocal and gradient rate plasticity

This paper deals with a formulation of nonlocal and gradient plasticity with internal variables. The constitutive model complies with local internal variables which govern kinematic hardening and isotropic softening and with a nonlocal corrective internal variable defined either as the sum between a new internal variable and its spatial weighted average or as the gradient of a measure of plastic strain. The rate constitutive problem is cast in the framework provided by the convex analysis and the potential theory for monotone multivalued operators which provide the suitable tools to perform a theoretical analysis of such nonlocal and gradient problems. The validity of the maximum dissipation theorem is assessed and constitutive variational formulations of the rate model are provided. The structural rate problem for an assigned load rate is then formulated. The related variational formulation in the complete set of state variable is contributed and the methodology to derive variational formulations, with different combinations of the state variables, is explicitly provided. In particular the generalization to the present nonlocal and gradient model of the principles of Prager–Hodge, Greenberg and Capurso–Maier is presented. Finally nonlocal variational formulations provided in the literature are derived as special cases of the proposed model.     

Anti-plane problems of piezoelectric material with a micro-void or micro-inclusion based on micromorphic electroelastic theory

The linear theory of micromorphic electroelasticity, which incorporate the coupled electromechanical behavior into the framework of micromorphic continuum theory, is used to solve the anti-plane problems of piezoelectric media with a micro-void or micro-inclusion in this paper. The electromechanical field solutions for a transversely isotropic piezoelectric medium are derived in the context of micromorphic electroelasticity and a generalized characteristic length is introduced to describe the size effect. Anti-plane problems of an infinite piezoelectric medium containing a micro-void or micro-inclusion are analyzed. Numerical results reveal that the mechanical and electric fields predicted by the present model highly depend on the relative size of the micro-void or micro-inclusion with respect to the generalized characteristic length, which is obviously different from the classical prediction.     

A model for high temperature creep of single crystal superalloys based on nonlocal damage and viscoplastic material behavior

A model for high temperature creep of single crystal superalloys is developed, which includes constitutive laws for nonlocal damage and viscoplasticity. It is based on a variational formulation, employing potentials for free energy, and dissipation originating from plasticity and damage. Evolution equations for plastic strain and damage variables are derived from the well-established minimum principle for the dissipation potential. The model is capable of describing the different stages of creep in a unified way. Plastic deformation in superalloys incorporates the evolution of dislocation densities of the different phases present. It results in a time dependence of the creep rate in primary and secondary creep. Tertiary creep is taken into account by introducing local and nonlocal damage. Herein, the nonlocal one is included in order to model strain localization as well as to remove mesh dependence of finite element calculations. Numerical results and comparisons with experimental data of the single crystal superalloy LEK94 are shown.     

A theoretical and computational framework for anisotropic continuum damage mechanics at large strains

The main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of a fictitious, isotropic, undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient. Then, the corresponding Finger tensor – denoted as damage metric – constructs a second order, internal variable. Due to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration, the free energy function can be computed via push-forward operations within the nominal setting. Referring to the framework of standard dissipative materials, associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation. The numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account. Finally, some numerical examples demonstrate the applicability of the proposed framework.     

气泡在液体中运动过程的数值模拟

本文用数值方法预测气泡在液体中的百定常运动。运用位标函数进行界面的隐含跟踪并且与有限体积法相结合构成一种可行的计算方法。本文方法允许在界面处存在很大的物性差，而且较容易将表面张力引入控制方程。我们对气液两相流中单个气泡的运动进行了计算，得到了与实验结果符合很好的数值结果。     

A nonlocal model with strain-based damage

A thermodynamically consistent formulation of nonlocal damage in the framework of the internal variable theories of inelastic behaviours of associative type is presented. The damage behaviour is defined in the strain space and the effective stress turns out to be additively splitted in the actual stress and in the nonlocal counterpart of the relaxation stress related to damage phenomena. An important advantage of models with strain-based loading functions and explicit damage evolution laws is that the stress corresponding to a given strain can be evaluated directly without any need for solving a nonlinear system of equations. A mixed nonlocal variational formulation in the complete set of state variables is presented and is specialized to a mixed two-field variational formulation. Hence a finite element procedure for the analysis of the elastic model with nonlocal damage is established on the basis of the proposed two-field variational formulation. Two examples concerning a one-dimensional bar in simple tension and a two-dimensional notched plate are addressed. No mesh dependence or boundary effects are apparent.     

Elasto-plastic postbuckling of damaged orthotropic plates

Based on the elasto-plastic mechanics and continuum damage theory, a yield criterion related to spherical tensor of stress is proposed to describe the mixed hardening of damaged orthotropic materials. Its dimensionless form is isomorphic with the Mises criterion for isotropic materials. Furthermore, the incremental elasto-plastic damage constitutive equations and damage evolution equations are established. Based on the classical nonlinear plate theory, the incremental nonlinear equilibrium equations of orthotropic thin plates considering damage effect are obtained, and solved with the finite difference and iteration methods. In the numerical examples, the effects of damage evolution and initial deflection on the elasto-plastic postbuckling of orthotropic plates are discussed in detail.     

A stabilized finite element formulation for high‐speed inviscid compressible flows using finite calculus

This paper aims at the development of a new stabilization formulation based on the finite calculus (FIC) scheme for solving the Euler equations using the Galerkin FEM on unstructured triangular grids. The FIC method is based on expressing the balance of fluxes in a space–time domain of finite size. It is used to prevent the creation of instabilities typically present in numerical solutions due to the high convective terms and sharp gradients. Two stabilization terms, respectively called streamline term and transverse term, are added via the FIC formulation to the original conservative equations in the space–time domain. An explicit fourth‐order Runge–Kutta scheme is implemented to advance the solution in time. The presented numerical test examples for inviscid flows prove the ability of the proposed stabilization technique for providing appropriate solutions especially near shock waves. Although the derived methodology delivers precise results with a nearly coarse mesh, a mesh refinement technique is coupled to the solution process for obtaining a suitable mesh particularly in the high‐gradient zones. Copyright © 2014 John Wiley & Sons, Ltd.     

Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials

A Hashin-Shtrikman-Willis variational principle is employed to derive two exact micromechanics-based nonlocal constitutive equations relating ensemble averages of stress and strain for two-phase, and also many types of multi-phase, random linear elastic composite materials. By exact is meant that the constitutive equations employ the complete spatially-varying ensemble-average strain field, not gradient approximations to it as were employed in the previous, related work of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) and Drugan (J. Mech. Phys. Solids 48 (2000) 1359) (and in other, more phenomenological works). Thus, the nonlocal constitutive equations obtained here are valid for arbitrary ensemble-average strain fields, not restricted to slowly-varying ones as is the case for gradient-approximate nonlocal constitutive equations. One approach presented shows how to solve the integral equations arising from the variational principle directly and exactly, for a special, physically reasonable choice of the homogeneous comparison material. The resulting nonlocal constitutive equation is applicable to composites of arbitrary anisotropy, and arbitrary phase contrast and volume fraction. One exact nonlocal constitutive equation derived using this approach is valid for two-phase composites having any statistically uniform distribution of phases, accounting for up through two-point statistics and arbitrary phase shape. It is also shown that the same approach can be used to derive exact nonlocal constitutive equations for a large class of composites comprised of more than two phases, still permitting arbitrary elastic anisotropy. The second approach presented employs three-dimensional Fourier transforms, resulting in a nonlocal constitutive equation valid for arbitrary choices of the comparison modulus for isotropic composites. This approach is based on use of the general representation of an isotropic fourth-rank tensor function of a vector variable, and its inverse. The exact nonlocal constitutive equations derived from these two approaches are applied to some example cases, directly rationalizing some recently-obtained numerical simulation results and assessing the accuracy of previous results based on gradient-approximate nonlocal constitutive equations.     

Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour

An improved gradient-enhanced approach for softening elasto-plasticity is proposed, which in essence is fully nonlocal, i.e. an equivalent integral nonlocal format exists. The method utilises a nonlocal field variable in its constitutive framework, but in contrast to the integral models computes this nonlocal field with a gradient formulation. This formulation is considered ‘implicit’ in the sense that it strictly incorporates the higher-order gradients of the local field variable indirectly, unlike the common (explicit) gradient approaches. Furthermore, this implicit gradient formulation constitutes an additional partial differential equation (PDE) of the Helmholtz type, which is solved in a coupled fashion with the standard equilibrium condition. Such an approach is particularly advantageous since it combines the long-range interactions of an integral (nonlocal) model with the computational efficiency of a gradient formulation. Although these implicit gradient approaches have been successfully applied within damage mechanics, e.g. for quasi-brittle materials, the first attempts were deficient for plasticity. On the basis of a thorough comparison of the gradient-enhancements for plasticity and damage this paper rephrases the problem, which leads to a formulation that overcomes most reported problems. The two-dimensional finite element implementation for geometrically linear plain strain problems is presented. One- and two-dimensional numerical examples demonstrate the ability of this method to numerically model irreversible deformations, accompanied by the intense localisation of deformation and softening up to complete failure.