A large time increment approach for cyclic viscoplasticity

The principles of a new approach for cyclic viscoplastic problems are described in this paper. A computational method of dealing with nonlinear mechanical behaviours, described by internal variables, is presented. This approach contrasts with the classical step-by-step method since it is an iterative procedure that accounts for the whole loading process in a single time increment. Special tools have been developed to simulate, within the large time increment method, several hundred cycles in a single increment. The feasibility of this strategy is shown through simple examples, with a large number of cycles in the case of viscoplastic behaviour.     

Proposition d'un paramètre énergétique de rupture pour les matériaux dissipatifs

We suggest an energetic fracture parameter for non elastic materials. This one is presented in a domain integral, and founded on the release rate of the total mechanical energy received by a notched solid, using a domain derivation method. This parameter is proposed for a large class of loading and materials described by internal variables. It is shown that this parameter is reduced to the Rice–Cherepanov integral when the material is either elastic or elastoplastic and submitted to proportional loading.     

Evolving internal length scales in plastic strain localization for granular materials

A numerical model in the Cosserat continuum for strain localization phenomena in granular materials is developed and proposed in this paper. The model assumes a constant internal length scale that is used to describe the shear band thickness. However, it is observed that the internal length scales need to change to accommodate the possible change in the contact surface between the particles, damage of the particles or/and any change in the local void ratio within the domain, which will change the shear band thickness. The mathematical formulations used in the present numerical model were equipped with evolution equations for the length scales through the Micropolar theory, those formulations are proposed and discussed in this paper. The evolution equations of the internal length scales describe any possible change in the contact surface between the particles, damage of the particles if exists and/or any change in the local void ratio within the domain. Hence, the strain localization described by the enhanced model with evolving internal length scales is more accurate and closer to the real solution. The solution for the shear bands thickness shows more accurate correlation with the experimental results and less dependency on the mesh size when such evolution equations are used. Moreover, the shear band thickness and inclination evolve during the deformation process.     

Simulation of damage evolution in ductile metals undergoing dynamic loading conditions

A set of constitutive equations for large rate-dependent elastic-plastic-damage materials at elevated temperatures is presented to be able to analyze adiabatic high strain rate deformation processes for a wide range of stress triaxialities. The model is based on the concepts of continuum damage mechanics. Since the material macroscopic thermo-mechanical response under large strain and high strain rate deformation loading is governed by different physical mechanisms, a multi-dissipative approach is proposed. It incorporates thermo-mechanical coupling effects as well as internal dissipative mechanisms through rate-dependent constitutive relations with a set of internal variables. In addition, the effect of stress triaxiality on the onset and evolution of plastic flow, damage and failure is discussed.Furthermore, the algorithm for numerical integration of the coupled constitutive rate equations is presented. It relies on operator split methodology resulting in an inelastic predictor-elastic corrector technique. The explicit finite element program LS-DYNA augmented by an user-defined material subroutine is used to approximate boundary-value problems under dynamic loading conditions. Numerical simulations of dynamic experiments with different specimens are performed and good correlation of numerical results and published experimental data is achieved. Based on numerical studies modified specimens geometries are proposed to be able to detect complex damage and failure mechanisms in Hopkinson-Bar experiments.     

A finite-strain model for anisotropic viscoplastic porous media: I – Theory

In this work, we propose an approximate homogenization-based constitutive model for estimating the effective response and associated microstructure evolution in viscoplastic (including ideally-plastic) porous media subjected to finite-strain loading conditions. The proposed model is based on the “second-order” nonlinear homogenization method, and is constructed in such a way as to reproduce exactly the behavior of a “composite-sphere assemblage” in the limit of hydrostatic loading and isotropic microstructure. However, the model is designed to hold for completely general three-dimensional loading conditions, leading to deformation-induced anisotropy, whose development in time is handled through evolution laws for the internal variables characterizing the instantaneous “ellipsoidal” state of the microstructure. In Part II of this study, results will be given for the instantaneous response and microstructure evolution in porous media for several representative loading conditions and microstructural configurations.     

Measurement and modeling of back stress at intermediate to high homologous temperatures

Unified creep-plasticity models often require a number of internal state variables to accurately capture the path dependence of rate- and temperature-dependent alloys especially at intermediate to high homologous temperatures. However, it is impossible to fully measure the internal state variables directly. Consequently, assumptions on the relative importance of the various state variables must be made when determining the material parameters and their evolution must be inferred. Any enhancements designed to better capture the evolution of the state variables are dependent on these assumptions and are generally limited. The strain transient dip test and the rapid load/unload test are two indirect experimental methods used to obtain an approximate measure of the evolutionary nature of the back stress in 60Sn–40Pb, a common rate- and temperature-dependent solder alloy. The rapid load/unload test gives a better measure of back stress because there is less time for the internal state to evolve during data acquisition. Modifications to the back stress evolution equations in the McDowell unified creep-plasticity model are proposed using these new measurements as guidance. The modified model can better capture the transients under cyclic loading and following strain-rate jumps.     

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。     

半空间饱和土中圆柱形衬砌内荷载引起的动力响应解答

地下衬砌结构经常会受到内部动荷载的作用,内荷载引起的衬砌结构的动应力集中备受关注。采用Laplace变换和波函数展开法,对半空间饱和土中均布突加荷载作用下圆柱形衬砌结构的动力响应进行了研究。对以大半径凸圆弧来近似半空间表面,将半空间饱和土中的波动方程展开成无穷级数的形式,应用Graff加法公式进行坐标变换,根据连续条件和边界条件,确定波函数展开式中的未知系数,求得了半空间饱和土中均布突加荷载作用下圆柱形衬砌的动力响应解答,讨论了衬砌埋深对圆柱形衬砌动力响应的影响。该问题的求解,为地下结构的动力分析提供了一种有效方法。     

A statistical analysis of the elastic distortion and dislocation density fields in deformed crystals

The statistical properties of the elastic distortion fields of dislocations in deforming crystals are investigated using the method of discrete dislocation dynamics to simulate dislocation structures and dislocation density evolution under tensile loading. Probability distribution functions (PDF) and pair correlation functions (PCF) of the simulated internal elastic strains and lattice rotations are generated for tensile strain levels up to 0.85%. The PDFs of simulated lattice rotation are compared with sub-micrometer resolution three-dimensional X-ray microscopy measurements of rotation magnitudes and deformation length scales in 1.0% and 2.3% compression strained Cu single crystals to explore the linkage between experiment and the theoretical analysis. The statistical properties of the deformation simulations are analyzed through determinations of the Nye and Kröner dislocation density tensors. The significance of the magnitudes and the length scales of the elastic strain and the rotation parts of dislocation density tensors are demonstrated, and their relevance to understanding the fundamental aspects of deformation is discussed.     

Complex loading of heterogeneous materials with redistribution of internal mass

The problem of inhomogeneous material with internal friction subjected to complex loading is solved in this paper. Different complex loadings are considered by continuously rotating the principal axes of the strain tensor. A hypoplastic model with internal variables is used to describe the internal structure of the material. The model can describe loading and unloading states. It accounts for essential non-linearity of constitutive equations. The finite element algorithm reduces the problem to a system of nonlinear algebraic equations of high order which are solved by the Newton's method. The results show a good qualitative and quantitative assessment of calculated parameters when compared with data obtained from experiments. Redistribution of internal mass takes place under loading such that a fractal structure is developed in time reflecting the material discontinuities.     

On the evolution of the linear material properties of PZT during loading history—an experimental study

Ferroelectrics exhibit material behavior which is strongly affected by its loading history. Among other phenomena, the coefficients describing the linear material behavior are known to change when the state of polarization is altered. There are several approaches to modeling ferroelectric/ferroelastic behavior. However, with all models, assumptions have to be made on how the linear coefficients depend on the state of polarization. Often the elastic and dielectric coefficients are defined to be constant for the sake of simplicity. Alternatively, their evolution and that of the piezoelectric constants are described rather intuitively, while systematic experimental data are sparse. The present study explores the impact of large signal mechanical and electrical loading on the low frequency linear response of a soft PZT ceramic. This is accomplished via cyclic tests with progressively increasing maximum electrical or mechanical load. Upon load reversal, the quasi-linear response is measured. Remanent polarization and remanent strain are used as internal variables to describe the material behavior as a function of loading history. While the dielectric permittivity κ₃₃ is shown to exhibit only minor variation, Young’s modulus and the piezoelectric coefficient d₃₃ change significantly in the course of loading.     

Modeling of Temperature and Strain-Rate Effects in Metals Using an Internal Variable Model

The thermo-viscoplastic behavior of three metals is characterized in a large range of loading conditions by using a new phenomenological constitutive model. The flow stress is decomposed into the sum of an effective stress with an internal stress depending upon an internal parameter which describes the strain hardening effect. The evolution of the internal stress is sensitive to the history of strain-rate and temperature. A systematic method is used for determining the model’s parameters. The model predictions show a good correlation with experimental data. Temperature history effects are especially analyzed.     

Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method

We propose a general formulation – which we believe to be new – for the mean-field homogenization of inclusion-reinforced elasto-viscoplastic composites assuming small strains. Our proposal is based on an interplay between constitutive equations and numerical algorithms, and the key ideas behind it are the following. The evolution equations for inelastic strain and internal variables at the beginning of each time interval are linearized around the ending time of the same interval. The linearized equations are then numerically integrated using a fully implicit backward Euler scheme. The obtained algebraic equations lead to an incrementally affine stress–strain relation which involves two important terms. The first one is the algorithmic tangent operator, obtained by consistent linearization of the time discretized constitutive equations. The second term is a new one and called an affine strain increment. The proposal leads to thermoelastic-like relations directly in the time domain, and not in the Laplace–Carson (L–C) one. There is no need for viscoelastic-type integral rewriting of the evolution equations, for L–C transforms, or for numerical inversion back from L–C to time domains. The proposed method can be readily applied to sophisticated elasto-viscoplastic models with an arbitrary set of scalar or tensor internal variables, and is valid for multi-axial, non-monotonic and non-proportional loading histories. The theory is applied in detail to a well-known constitutive model, and verified against finite element simulations of representative volume elements or unit cells, for a number of composite materials.     

An augmented multicrack elastoplastic damage model for tensile cracking

In this paper we present the general formulation and numerical aspects of an augmented multicrack elastoplastic damage model aiming to reflect the crack induced anisotropy in concrete like quasi-brittle materials. Consistent evolution laws for the involved internal variables are derived based on the augmented Lagrangian method. The (time) discrete formulation and the corresponding variational structure are investigated, with the Euler–Lagrangian equations defining the closest-point projection approximation of the proposed model. The numerical aspects, such as the stress updating algorithm and the algorithmic consistent tangent moduli, are also discussed in details. It is found that in the developed numerical algorithm the active loading surfaces are determined in such a posterior manner that potential numerical problems due to the iteratively updating procedure in classical algorithms can be avoided. The proposed model is applied to the modeling of tensile cracking in concrete. The behavior of a single crack is characterized by an elliptical cracking surface and a hyperbolic softening function, with the orientations of potential cracks determined by Mohr’s postulate. The model is verified by calculating the single point stress vs. strain relations of concrete under several typical proportional and non-proportional loading cases. Finally, two benchmark tests of concrete structures, i.e. four-point bending beam under cyclic loading (Hordijk, 1992) and double edge notched specimens under mixed tension/shear forces (Nooru-Mohamed, 1992), are numerically simulated. Both predicted load vs. displacement curves and crack patterns agree well with the experimental data.     

Simulation of the stochastic damage evolution process under low cycle fatigue by the finite element method

A numerical simulation of stochastic damage evolution process in the condition of low cycle fatigue loading is discussed. The relations between damage variables and micro-cracks are obtained by means of the micro-mechanics model of the representative volume element proposed by Lemaitre and Dufailly^[10]. The stochastic initial damage values are introduced in consideration of the inherent micro-defects in materials. The model combined with a finite element method is applied to simulate the damage evolution process under low cycle fatigue loading. The micro-cracks on the sur face of a specimen of 19Mn6 alloy steel are measured with a replica technique. The numerical results show that the nonhomogeneity of damage and the localization of the fatigue failure are well shown by the proposed simulations, and the fatigue lives are reasonably predicted.     

有限弹塑性变形的几何模型

以连续介质力学内变量理论为基础,建立了一个以材料内部微结构变量为底流形。材料外部变形状态为对应纤维的材料状态纤维丛模型,使材料的力学特性与模型的几何性质自然对应起来．在模型上讨论和分析了有限弹塑性变形中变形梯度的Ｌｅｅ和Ｃｌｉｆｔｏｎ的分解和联系,并证明了塑性变形为沿内变量演化在纤维丛的水平空间的运动由此获得了塑性变形随内变量演化的变化方程和塑性速率梯度与内变演化的协调关系．     

Dynamics of viscoelastoplastic soil under a moving wheel

A rate-dependent inelastic behavior or viscoelastoplastic behavior of soil subjected to dynamic loading of a moving rigid wheel is studied herein. A rational approach to this seemingly complex problem is treated in the context of continuum mechanics. Together with the concept of critical state soil mechanics for inelastic behavior we incorporate the so-called internal state variables for viscous effects. It is assumed that the free energy functional containing inelastic behavior may not be smooth for its entire domain of time histories. Rather, we assume a form of discretized free energy as a function of elastic strains, plastic strains, and internal or hidden variables of incremental quantity considered to be valid only for a small time interval or a fraction of loading increments. The finite element method is used to derive governing equations of motion. Exhaustive numerical examples are presented to demonstrate effectiveness of the present method.     

基于构型力内变量的界面损伤问题研究

本文基于材料构型力的基本理论和损伤力学中含有内变量的热力学框架,提出了新的损伤变量定义方式,为研究界面损伤问题提供了一种新思路。首先,基于双相弹性体的能量分析,给出界面材料构型力表达式,通过构型力的离散化方法,实现了其在有限元中的数值计算。其次,定义构型力为界面损伤内变量,进而提出一种新的损伤演化模型,并采用刚度劣化的方法,对该界面损伤模型进行数值实现。最后,通过对复合材料界面损伤问题（有裂纹或无裂纹）进行数值模拟,分析了其界面损伤发展趋势,探讨了此模型的合理性和优越性。基于构型力内变量的界面损伤模型,可为研究复合材料的界面损伤失效问题提供一种普适性的方法。     

无耦合条件下的多子系统静动态统一本构关系(上)--基本理论

引入了同步性的概念。按照是否同步对所有的内变量进行分组，得到了整个系统的子系统结构。证明了子系统间无耦合时各子系统内部一致性条件成立的判别准则，从而得到了可以统一描述子系统内部的时间相关、时间无关不可逆行为的内变量演化方程和加卸载准则。本构关系既适用于静态加载，也适用于动态加载。经典塑性本构关系、多屈服面本构关系及过应力型粘塑性本构关系只是本文的特例。     

无耦合条件下的多子系统静动态统一本构关系(下)--铝的双子系统静动态统一本构模型

构造了标量形式的无耦合条件下双子系统静动态统一本构模型.推导出第1和第2子系统中加载应变速率临界值ε·c1和ε·c2,当应变速率ε·分别低于和高于某个临界值时,相应子系统中的不可逆行为分别是与时间无关的和与时间相关的.由于当ε·跨越ε·c1和ε·c2时,内变量的求解公式发生变化,所以动态强度随ε·变化的规律发生变化.经与铝的实验结果比较确认,本构模型能够描述材料的多种静动态力学行为.