Solution existence conditions for elastoplastic constitutive models of granular materials

In this paper, the conditions of solution existence for stress rates under given strain rates are investigated. The focus of the solution existence investigation is on the non-associated flow rule and elastic stress–strain relationship. Granular materials characterized with strong non-associated plastic flows are used as a particular example for analysis. Various flow rules for granular materials are analyzed, including Rowe’s, Roscoe’s flow rules and their modified versions. In the elastic stress–strain relationships of materials, the effects of Poisson’s ratio on solution existence are investigated. Both isotropic and anisotropic elasticity are considered. Given a granular material and its states, it is found that there exists a critical Poisson’s ratio for a particular non-associated flow rule. When the Poisson’s ratio of a material is above this critical Poisson’s ratio, its constitutive model is susceptible to solution non-existence. It is suggested that special attentions should be paid to the selection of material Poisson’s ratio and non-associated flow rule to ensure the existence of elastoplastic solutions.     

基于非关联流动规律的Gotoh屈服准则的参数确定方法

屈服准则对板料成形过程的理论解析、工艺优化和有限元模拟有着重要的影响. 通过提高屈服准则的各向异性表征能力, 可以确保成形过程的可靠性及实际预测的准确性. 本文基于非关联流动法则, 给出了Gotoh屈服准则一套全新的参数求解方法. 在结合常用屈服准则并考虑流动规律的基础上, 分别以5754O铝合金、DP980先进高强钢和SAPH440结构钢作为研究对象, 进行了不同加载路径下各向异性变形行为的预测. 根据Gotoh屈服准则推导的屈服函数、塑性势函数以及基于关联流动的理论函数计算出屈服应力和各向异性指数$r$值随加载角度的分布趋势, 进而针对平面应力状态的屈服轨迹展开分析, 验证了不同屈服准则和流动规律对各向异性屈服行为的预测精度. 理论与实验数据对比结果表明: 不同屈服准则针对同种板料在流动规律一致的情形下其表征各向异性的能力有显著差异; 相同屈服准则基于不同流动规律其表征能力也具有明显差别. 基于非关联流动的屈服准则能极大地提高精度, 各向异性表征能力显著加强. 相关结果能够为各向异性屈服准则在塑性成形领域的实际应用方案提供重要参考.      

History-sensitive accumulation rules for life-time prediction under variable creep loading

A general form of temporal strength conditions under variable creep loading is employed to formulate several new phenomenological accumulation rules based on the constant-loading durability diagram. Unlike the well-known Robinson rule of linear accumulation of partial life-times, the new rules allow to describe the life-time sensitivity to the load sequence observed in experiments. Comparison of the new rules with experimental data shows that they fit the data much more accurately than the Robinson rule.     

考虑道路信息和转向灯的可换道BML模型

车辆换道是司机为获得更好行驶条件而采用的常见措施,而转向灯对车辆换道行为有重要的指导作用.本文在BML (Biham-Middleton-Levine)模型的基础上加以改进,提出了综合考虑道路信息和前车转向灯影响的可换道BML模型.当车辆无法前行时,如满足换道条件,则将道路信息(车道密度及平均速度)和转向灯影响量化为车辆换道概率,确定车辆是否可以换道.通过数值模拟,研究了周期边界条件下车辆换道行为对有、无交通灯控制的两种BML模型发生相变的临界密度以及系统通行能力的影响.模拟结果表明对于无交通灯BML模型,引入换道规则可以明显提高系统发生相变的临界密度,在较小尺度下该临界密度接近有交通灯BML模型,换道效果明显,并发现了一种新的局部拥堵和自由流的共存相,讨论了该共存相的生成和演化机制.在较高密度下局部阻塞将演化为全局拥堵;对于有交通灯BML模型,引入换道规则对系统发生相变的临界密度没有明显的影响,但相变的过渡区域更窄.这表明有交通灯时,换道虽然可以改变局部交通特征,但难以显著影响交通系统的全局特征.     

A finite strain theory for elastic-plastic deformation

An elastic-plastic theory that is applicable when the elastic part of the strain is finite is proposed. A flow rule for an incompressible solid is obtained from Drucker's postulate [1]. Isothermal simple shear of a material which is neo-Hookean both before yielding and during elastic unloading after yielding is considered as an application of the theory. The problem is solved for two yield conditions and associated flow rules.     

Traction problem in electro-elasticity and bifurcation theory

This paper is the first of a series of two. It will deal with the problem of static traction problem with minor deformations for a material which is governed by the electrostriction phenomenon. Two approaches to this problem will be described. We can consider either the equilibrium equations which are naturally non-linear, or the equations after linearization. The linearization of equations must be done near a natural state. Locally, under some conditions, we can establish the existence and the uniqueness of the solutions. We use the local theorem of implicit functions. The problem can be approached more globally. If we consider the non-linear equations, we can use a natural principle of these equations: the independence of the choice of the observer, also known as objectivity property. This property makes it possible for us to take into account an action of the rotations group of the Euclidean space, and consequently to take into account all the trivial solutions. It is then possible to prove within the space of all configurations the existence of the non-linear equations solutions and to find their number.This work presents a thorough and detailed approach to a non-linear theory, the geometric arguments of which make it possible for us to prove the existence of all the solutions and to study their stability in the aggregate; this last aspect will be developed in the second paper. Not only can this theory anticipate the eventual existence of a stable solution, it can also anticipate that an unstable solution in terms of the elasticity can, thanks to the effect of an electric field, become stable in terms of the electro-elasticity.     

一类新的超弹性-循环塑性本构模型

目前,很多经典的超弹性-有限塑性本构模型已被提出,但由于超弹性理论中中间构型的引入使得随动硬化法则相对复杂,故多数文献均采用的是经典的Armstrong-Frederick(A-F)随动硬化法则.本文基于已有的本构理论,利用多机制过程的概念拓展了Lion塑性变形分解理论,明确提出了多重中间构型的概念,并在此基础上,对经典理论中客观性的定义进行了概念上的推广,使其更好地适用于超弹性本构理论分析,同时提出了一类新的超弹性-有限塑性本构模型.这类本构模型满足热动力学法则,且可融合多种小变形循环塑性理论中常用的随动硬化法则(如经典的A-F模型,Chaboche模型,Ohno-Wang(O-W)模型以及Karim-Ohno(K-O)模型等),使得小变形理论中背应力的加法分解性质及其演化的临界面阶跃特性在大变形领域中均有所体现,故本文提出的本构理论可看作是小变形循环塑性模型在大变形理论中的扩展.本文最后以K-O模型为例,对推荐模型进行了详细探讨,并与相应的次弹性模型进行了对比.      

电磁热效应止裂效果与裂纹走向关系的研究

采用理论、实验与数值分析的方法研究了在电磁热效应裂纹止裂中 ,由于裂纹走向不同导致裂纹尖端的温度场、温度梯度场分布状态的不同 ,并定量计算了它们的分布规律和差值。数值模拟计算应用了耦合场理论中热 电耦合 (焦耳热问题 )的分析方法。三方面的研究结果均表明 :在裂纹尖端处 ,由于电流产生的焦耳热源的作用 ,能够在很小的范围内熔化形成微小的焊口 ,遏制了裂纹的扩展 ;裂纹的走向是影响裂纹尖端温度场数值和温度梯度场变化即电磁热效应裂纹止裂效果的主要因素之一。     

弹塑性本构关系的Ilyushin应变空间理论研究进展

Ｉｌｙｕｓｈｉｎ提出五维偏应变矢量空间中的一般弹塑性本构理论，将应力表示为变形迹内蕴几何学参数的泛函，适合于描写复杂加载下金属材料的塑性响应特性。本文对其实验和理论两方面的研究进展作了综述，涉及关于塑性响应矢量特性的“局部确定性”假设，标量特性的“延心原理”假设及Ｉｌｙｕｓｈｉｎ关于矢量空间的“特殊各向同性”假设等的实验研究和验证，微分型和积分型本构模型的建立及所含本构泛函的形式和确定。     

Poincaré maps modeling and local orbital stability analysis of discontinuous piecewise affine periodically driven systems

This paper presents a methodology to study the local stability of periodic orbits (orbital stability) in switched discontinuous piecewise affine (DPWA) periodically driven multiple-input multiple-output (MIMO) systems. The switched system of interest has a bilinear state space representation where the controller inputs are binary signals taking values in the set {0,1}. These systems are characterized by a set of affine differential equations together with switching rules to commute between them. These switching rules are described by switching functions that are periodic in time and linear in state. The methodology is based on obtaining a discrete time model (Poincaré map), its steady state operation points, and its Jacobian matrix. This provides a powerful tool for studying their stability and to predict some kind of instability phenomena that the system can undergo like subharmonic oscillations. The proposed approach is applied to a power electronic circuit which toggles among six different system equations with five switching boundaries within a switching cycle. This work was supported by the Spanish Ministerio de Educación y Ciencia under Grant TEC-2004-05608-C02-02.     

316L不锈钢随动强化模型改进研究

本文对316L不锈钢进行了单轴与多轴非比例路径下的应力控制棘轮试验,考察了应力幅值、平均应力和加载历程对棘轮特性的影响。同时进行了应变控制循环试验以研究材料的应力松弛特性。试验结果表明轴向棘轮效应在对称剪切荷载下效果明显,同时棘轮应变随应力幅值和平均应力的增加而增加。研究了Chen-Jiao随动强化模型与Jiang-Sehitoglu随动强化模型采用的单轴与多轴参数对背应力分量增量方向的影响,将Chen-Jiao模型中的多轴系数替换为界面饱和率,并在此基础上引入新的参数对塑性模量系数进行修正,计算结果表明修正后的模型能提升应力控制下多轴棘轮的预测精度,并能很好的预测应力松弛现象,表明了新模型的正确性与有效性。     

Adaptive synchronization of an uncertain coupling complex network with time-delay

This paper proposes a new complex dynamical network model, in which the nodes are coupled with time-delay, and the inner coupling matrices are with uncertain forms. This model can describe the real world more realistically and can be widely used in practical engineering application. Synchronization in the proposed network model is analyzed by the Lyapunov stability theory and some adaptive controllers are designed to ensure that the proposed network achieve local and global synchronization, respectively. Theoretical analysis and numerical simulations fully verify the main results.     

Modifications to the Cauchy–Born rule: Applications in the deformation of single-walled carbon nanotubes

This paper presents a study of the Cauchy–Born (CB) rule as applied to the deformation analysis of single-walled carbon nanotubes (SWNTs) that are modeled as 2-dimensional manifolds. The C–C bond vectors in the SWNT are assumed to deform according to the local deformation gradient as per the CB rule or a modified version thereof. Aspects of the CB rule related to spatial inhomogeneity of the deformation gradient at the atomic scale are investigated in the context of a specific class of extension–twist deformation problems. Analytic expressions are derived for the deformed bond lengths using the standard CB rule as well as modified versions of the standard CB rule. Since the deformation map is conveniently prescribed in this work, it is possible to compare the performance of these deformation rules with the exact solution (i.e. the exact analytic expression for the deformed bond vectors) given directly by the deformation map. This approach provides insights into the CB rule and its possible modifications for use in more complicated deformations where an explicit deformation map is not available. Specifically, it is concluded that in the case of inhomogeneous deformations at the atomic scale for which the CB rule is only approximate (as demonstrated in Section 1 of this paper), the mean value theorem in calculus can be used as a guide to modify the CB rule and construct a more rigorous and accurate atomistic–continuum connection. The deformed bond lengths are used to formulate an enriched continuum hyperelastic strain energy density function based on interatomic potentials (the multi-body Tersoff–Brenner [Tersoff, J., 1988. New empirical approach for the structure and energy of covalent systems. Phys. Rev. B 37, 6991–7000; Brenner, D.W., 1990. Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys. Rev. B 42, 9458–9471] empirical interatomic potential for carbon-carbon bonds is used in this work). The deformation map (and hence the deformation gradient, the bond vectors and the continuum strain energy density) contains certain parameters, some of which are imposed and others determined as a result of energy minimization in the standard variational formulation. Numerical results for kinematic coupling and binding energy per atom are presented in the case of imposed extension and twist deformations on representative chiral, zig-zag and armchair nanotubes using the CB rule and its modifications. These results are compared with the exact solution based on the deformation map which serves as a basis for evaluating the efficacy of these deformation rules. The ideas presented in this paper can also be directly extended to other lattices.     

Kinematic hardening rules with critical state of dynamic recovery,part I: formulation and basic features for ratchetting behavior

Kinematic hardening rules formulated in a hardening/dynamic recovery format are examined for simulating rachetting behavior. These rules, characterized by decomposition of the kinematic hardening variable into components, are based on the assumption that each component has a critical state for its dynamic recovery to be activated fully. Discussing their basic features, the authors show that they can predict much less accumulation of uniaxial and multiaxial ratchetting strains than the Armstrong and Frederick rule. Comparisons with multilayer and multisurface models are made also, resulting in a finding that the simple one in the present rules is similar to the multilayer model with total strain rate replaced by inelastic (or plastic) strain rate. Part II of this work deals with applications to experiments.     

A plasticity theory without drucker's postulate,suitable for granular materials

A plasticity theory is introduced which starts with a dilatancy rule and a function of plastic strain rates which represents the energy dissipated during plastic deformation. Yield surfaces and flow rules are then derived using energy conservation and the theory of envelopes. This method allows valid plasticity theories to be derived for frictional materials, but gives results for non-frictional materials which are identical to those of the classical theories.A dissipation function which includes deformation by granule rearrangement and granule distortion is presented and used to obtain a range of yield surfaces and flow rules, which are similar to those used in the critical state theory of soil mechanics. The microstructural features which may control the governing parameters of the dissipation functions are discussed.     

Mechanics of soft active materials with phase evolution

This paper studies the mechanics of soft active materials where the actuation is generated due to the formation of phases that are stress-free at the moment of their creation and therefore experience no deformation in the associated configuration. Phase formation is a continuous time-dependent process, which results in individual phases forming at different times and in different configurations of the material body, and thus it is coupled with mechanical deformation. Subsequent deformation of the material body results in individual phases experiencing different states of deformation and the overall material response results from the combined responses of the individual phases weighted by their respective volume fractions. Therefore, a great challenge in modeling the mechanics of soft active materials with evolving phases is to track the deformation and evolution of individual phases formed at different times and in different configurations. In this paper, a generalized one-dimensional model framework is presented to address this challenge. However, this model proves to be computationally inefficient. In response, an effective phase model is developed that tracks the combined deformation histories of new phases through a single, effective deformation. Both the general and effective phase models are evaluated with two fundamentally distinct phase evolution rules for three common mechanical problems: extension, stress relaxation, and creep. The first evolution rule represents a discrete transition from one phase to another while the second rule corresponds to a general transition from several phases into one phase. The effective phase model demonstrates excellent agreement with the generalized theory for all three mechanical problems considered under both types of evolution rules.     

动力UH模型及其有限元应用

饱和砂土在循环载荷下具有复杂的应力应变关系, 通常表现出液化过程中的大变形以及往返活动性现象. 为简单有效地模拟上述特性, 在超固结UH模型的基础上, 将其扩展为可考虑砂土动力加载下的本构模型. 具体做法有3点: (1)改变屈服面椭圆长短轴之比, 将比值定义为反映应力诱导各向异性转轴斜率的函数; (2)引入旋转硬化规则, 用来反映应力诱导各向异性; (3)建立一个与旋转硬化规则以及临界状态特性相协调的统一硬化参数. 模型预测结果表明, 所提动力模型可简单、有效地用于砂土在动力载荷下应力应变关系的模拟. 最后将该动力UH模型嵌入到有限元软件中, 三维地基的动力加载模拟结果表明, 动力UH模型可方便地应用于岩土工程实践中.      

Comparisons of crystal hardening laws in multiple slip

This paper brings together and concisely reviews results from recent analytical investigations on single crystals (variously done alone or with students) in which predictions from different theoretical hardening laws are contrasted and compared with experimental studies. Finitely deforming f.c.c. crystals in both constrained and unconstrained multiple-slip configurations are considered. Four crystal hardening laws are given prominence. Two of these belong to a class of theories in which the physical hardening moduli relating rates-of-change of critical strengths (in the 24 crystallographically equivalent slip systems) to slip-rates are taken as symmetric. These are G.I. Taylor's classic isotropic hardening rule (proposed in 1923), which is almost universally adopted in the metallurgical literature for various approximate analyses of single and poly-crystal deformation, and a 2-parameter modification of Taylor's rule that has an empirical basis in the qualitative features of experimentally determined latent hardening in single slip. The other two hardening laws featured here belong to a class of theories that were introduced in 1977 by this author. This class requires the above moduli to be nonsymmetric and explicity dependent upon the current stress state in such a manner that the following consequences are assured. (1) The deformation-dependent hardening of latent slip systems necessarily develops anisotropically if there is relative rotation of gross material and underlying crystal lattice. (2) The theories admit self-adjoint boundary value problems for crystalline aggregates, hence a variational formulation. (The fact that symmetric physical hardening moduli do not permit variational formulations of polycrystalline problems was shown at the 1972 Warsaw Symposium.) The two members of this class considered here are the original (and simplest p possible) theory of rotation-dependent anisotropy, which was proposed by this author in 1977 and commonly has been referred to as the “simple theory,” and a modification of this theory introduced in 1982 by Peirce, Asaro and Needleman that lies between Taylor's rule and the simple theory in its predictions for finitely deforming f.c.c. crystals. (In a series of five papers during 1977–1979, the simple theory was shown to universally account for the experimental phenomenon of “overshooting” in single slip in both f.c.c. and b.c.c. crystals.) Theoretical results from the various hardening rules are contrasted and compared with finite strain experiments in the metallurgical literature. Both tensile-loaded crystals in 4, 6 and 8-fold symmetry orientations and compressively loaded crystals under conditions of channel die constraint are treated. A postulate of minimum plastic work introduced in 1981 plays a prominent role in the theoretical analyses, in many cases providing a unique solution to the slip system inequalities and deformation constraints (where applicable). The rather remarkable ability of the simple theory to reconcile diverse qualitative features of both constrained and unconstrained finited deformation of f.c.c. crystals is demonstrated. Finally, conditions for total loading (all systems active) in 6-fold symmetry are investigated, and certain concepts regarding the selection of active systems under prescribed straining are critically assessed.     

Corner Singularities and Regularity of Weak Solutions for the Two-Dimensional Lamé Equations on Domains with Angular Corners

This paper is concerned with corner singularities of weak solutions of boundary value problems in the theory of plane linearized elasticity. The presence of angular corner points or points at which the type of boundary conditions changes yields generally local singularities in the solution. This singular behavior in the vicinity of such points can be described with the help of asymptotic singular representations for the solution, which essentially depend on the zeros of certain transcendental functions. These transcendental functions will be derived and analyzed for all ten possible combinations of boundary conditions, generated by the four basic ones, prescribing in the tangential and normal direction of the boundary, respectively, either the displacement or the tractions. The regularity of the corresponding weak solutions will be investigated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Kinematic hardening rules in uncoupled modeling for multiaxial ratcheting simulation

An earlier paper by the authors evaluated the performance of several coupled models in simulating a series of uniaxial and biaxial ratcheting responses. This paper evaluates the performance of various kinematic hardening rules in an uncoupled model for the same set of ratcheting responses. A modified version of the Dafalias–Popov uncoupled model has been demonstrated to perform well for uniaxial ratcheting simulation. However, its performance in multiaxial ratcheting simulation is significantly influenced by the kinematic hardening rules employed in the model. Performances of eight different kinematic hardening rules, when engaged with the modified Dafalias–Popov model, are evaluated against a series of rate-independent multiaxial ratcheting responses of cyclically stabilized carbon steels. The kinematic hardening rules proposed by Armstrong–Frederick, Voyiadjis–Sivakumar, Phillips, Tseng–Lee, Kaneko, Xia–Ellyin, Chaboche and Ohno–Wang are examined. The Armstrong–Frederick rule performs reasonably for one type of the biaxial ratcheting response, but fails in others. The Voyiadjis–Sivakumar rule and its constituents, the Phillips and the Tseng–Lee rules, can not simulate the biaxial ratcheting responses. The Kaneko rule, composed of the Ziegler and the prestress directions, and the Xia–Ellyin rule, composed of the Ziegler and Mroz directions, also fail to simulate the biaxial ratcheting responses. The Chaboche rule, with three decomposed Armstrong–Frederick rules, performs the best for the whole set of ratcheting responses. The Ohno–Wang rule performs well for the data set, except for one biaxial response where it predicts shakedown with subsequent reversal of ratcheting.