基于分子动力学模拟的金属构件的弹-塑性分解方法

针对金属多晶材料构件的分子动力学（MD）模拟,本文提出了一种新的弹-塑性分解方法.文章将MD运动轨迹分解为结构变形和热振动,给出了计算结构变形的方法和近似公式;针对金属多晶材料构件的当前构型,给出了基于FCC|BCC晶胞和四原子占位的四面体单元相组合的连续变形函数及计算变形梯度的算法;利用原子-连续关联模型,发展了计算当前构型应力场和弹性张量的算法.分析了当构件承受过大载荷后在材料内部所产生的微观缺陷,并将其分类标定为位错、层错、挛晶界、晶界和空位等;对于层错和挛晶界讨论了在弹性卸载过程中应满足的刚体运动约束方程;利用极小势能原理构造了基于当前构型的弹性卸载算法,进而给出了完整的基于MD模拟的计算弹-塑性应变的算法.最后,针对单晶铜纳米线拉伸的MD模拟,计算了弹-塑性应变场,验证了本文方法的合理性.
本文提出的基于MD模拟的弹-塑性分解方法,为从微观到宏观的多尺度和多模型耦合计算提供了算法支撑.     

关于晶体旋转的误解及合理塑性应变速度

指出晶体是离散的而不是连续的材料,因此在极分解F=RU中的旋转R及F>F-1中的旋率W不是正确的。塑性变形速度若直接以滑移系中的滑移速度来表示,如γ>ν≠n,则差错将会产生。滑移后的晶格几何形状并不改变,基于此概念提出了连续介质力学范围内的一个简单途径来表达滑移产生的塑性应变速度。建议了大变形下的本构方程。     

《变形体非协调理论》

《应用数学和力学》讲座丛书第六部《变形体非协调理论》已由重庆出版出版.本书由我刊常务编委、北京大学郭仲衡教授和中山大学粱浩云副教授编著书中系统全面地介绍了线性的缺陷连续统理论、缺陷理论与非黎曼几何的对应变形体非协调理论的理性理论和位错与向错的规范场理论,以期读者对变形体协调理论的全貌有所了解,并为深入研究非协调问题打下初步的基础.全书共分十一章.前五章引进了三种不同程度的非协调性使线性的缺陷连续统理论从经典弹性理论脱胎出来,叙述采用绝对符号法.为能用缺陷的连续统理论去处理孤立位错和缺陷,又引进了线、面、体的δ函数.第六章介绍了位错和向错的运动.     

旋转流体中圆球沿旋转轴运动产生的扰动

本文用线化理论分析了整体旋转的理想流体中有一个圆球沿旋转轴作匀速运动时流体的扰动,基于旋轴对称流动的假设导出了决定运动稳定性的扰动压力方程和扰动流函数方程.用简正模法分析了扰动流函数方程,得出了非平凡中性扰动的波数与波速必须满足的约束条件,并求出了扰动的精确表达式.文中得出结论,中性扰动共有三种可能的形式.     

一种基于无屈服概念的粘塑性模型

结合位错运动的热激活理论,基于无屈服概念,提出了一组描述金属材料变形规律的弹/粘塑性本构方程.方程从总体上考虑了应变率、应变历史、应率变历史、硬化和温度等效应,具有较强的物理基础.恒温单轴条件下商业纯钛的力学性能的理论预测与实验结果相比较,存在着良好的一致性.     

纵波在率相关塑性软化材料中的传播

本文所推导出的率相关软化材料在塑性软化状态下所满足的运动方程是具有双曲型微分方程性质,从而证实了由于应变率效应,应力波在软化状态下仍将传播.这与率无关理论有着质的不同.从上述方程求解出半无限长杆端部受冲击载荷的瞬态解,结果表明,软化状态下的塑性波波速、软化区界面传播的距离与材料的率敏感性和软化特征有关.     

损伤材料本构关系的一种内蕴时间理论

在评述了现有的连续介质损伤力学(CDM)的严重缺陷之后,基于所提出的损伤对本构关系影响的物理机制和模型及在ε_(T_qD)空间中对损伤材料不可逆过程的热力学描述,本文提出了一组新型的弹塑性损伤本构方程。该方程符合本文第一作者所提出的耗散型材料本构方程形式不变性定律的条件,以及Lemaitre,Valanis的本构方程和经典塑性理论的主要结果可由它在简化条件下推出的属性。文中还简略提到了损伤演化方程、有限元算法、材料函数确定方法及其在弹塑性损伤场分析中的初步应用。文末强调了内蕴表征时间z*和φ在ε_TD与ε_Tq)子空间中分析复杂的损伤与非弹性变形耦合问题时的重要价值;并将本文所提出的模型推广得到了纤维增强复合材料的损伤本构方程。     

颗粒土中剪切带临界状态数学描述及其完全解

为正确模拟土体涉及剪切带演化的后失效力学响应,需采用包含细观特征长度的高阶连续介质力学模型.笔者利用前期所建立的微极亚塑性模型,对颗粒土中剪切带的发展过程进行了分析推导,得到了剪切带临界状态条件下关键变量所满足的非线性微分方程.该文展示了上述非线性微分方程的简要推导,重点讨论了该非线性微分方程的主要性质、主要参数变化范围和求解途径;通过对剪切带进一步的力学分析补充建立了一个能量方程,使问题具有确定解.在此基础上,应用数值积分求出了剪切带厚度因子和剪切内应力、变形率分布及剪切速度分布的完全解.其中剪切带厚度因子对于微极亚塑性模型细观参数的确定具有重要作用.     

推导刚体动力学方程的张量方法

本文提出了基于连续介质力学概念推导刚体动力学方程的张量方法,运用具有零共旋率的惯性张量的时间导数公式,证明了Lagrange方程、Nielsen方程、Gibbs-Appell方程、Kane方程和广义动量式Kane方程等五种方法的等价性,给出了角速度、角加速度之间的一些微分关系式.     

金属塑性变形中位错组态演化模型及其应用

将位错理论与耗散结构、协同学等非线性科学相结合,建立了金属塑性变形中位错组态演化模型。利用该模型,分析和模拟了金属塑性变形、静态回复和蠕变过程中,微观位错组态演化过程及与之相对应的宏观力学行为。结果表明:在一个统一的理论模型基础之上,分析和研究金属塑性变形中出现的各种现象和问题是可行的。     

A crystal plasticity framework based on the Continuum Dislocation Dynamics theory: relaxation of an idealized micropillar

The motion and interaction of dislocation lines are the physical basis of the plastic deformation of metals. Although ‘discrete dislocation dynamic’ (DDD) simulations are able to predict the kinematics of dislocation microstructure (i.e. the motion of dislocations in a given velocity field) and therefore the plastic behavior of crystals in small length scales, the computational cost makes DDD less feasible for systems larger than a few micro meters. To overcome this problem, the Continuum Dislocation Dynamics (CDD) theory was developed. CDD describes the kinematics of dislocation microstructure based on statistical averages of internal properties of dislocation systems. In this paper we present a crystal plasticity framework based on the CDD theory. It consists of two separate parts: a classical 3D elastic boundary value problem and the evolution of dislocation microstructure within slip planes according to the CDD constitutional equations. We demonstrate the evolution of dislocation density in a micropillar with a single slip plane. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some general results in the theory of crystallographic slip

Crystallographic slip of a Bravais lattice is analyzed utilizing the main results of a recently constructed theory of structured solids, where explicit account is taken of the influence of dislocation density identified in terms of Curl of plastic deformationG _p . In the present paper, the scope of the subject is enlarged to also include defects (other than dislocations) such as substitutional impurities and vacancies and it is shown that these point defects may also be characterized in terms of the plastic deformation fieldG _p . Several general results pertaining to the kinematics and kinetics of Crystallographic slip are proved within the scope of an appropriate constraint theory suitable for Crystallographic slip; the latter is motivated by the well-known basic mechanism of Crystallographic slip that constrains the admissible modes of plastic deformation. The constraint responses (or forces) that are necessary to maintain the active slip systems, as well as the conditions for the transitions between the slip systems, are determined. In spite of the nature of the assumption pertaining to the mechanism of Crystallographic slip on distinct slip systems, it is shown that the yield surface does not necessarily exhibit sharp corners. Instead, the shape of the yield surface is in the form of hyperplanes joined by round corners. In fact, the presence of sharp corners is mainly a result of the use of a special set of constitutive assumptions. The predictive capability of the theoretical results is further illustrated by using a two-dimensional crystal subjected to simple shear. The effect of the initial dislocation density on the response of the sheared-crystal is studied by carrying out detailed calculations for two substantially different initial dislocation densities. The calculations show that while the response of the crystal is sensitive to the initial dislocation density in the early stages of deformation, its influence diminishes with progressively larger deformations. Furthermore, the crystal exhibits a well-defined shear band which evolves naturally due to the presence of initial dislocation distribution and is easily visible at large deformations.     

Crysal Plasticity Finite Element Simulations based on Continuum Dislocation Dynamics

Plastic deformation of crystalline materials is the result of the motion and interaction of dislocations. Continuum dislocation dynamics (CDD) defines flux-type evolution equations of dislocation variables which can capture the kinematics of moving curved dislocations. Coupled with Orowan's law, which connects the plastic shear rate to the dislocation flux, CDD defines a dislocation density based material law for crystal plasticity. In the current work we provide simulations of a micro-bending experiment of a single crystal and compare the results qualitatively to those from discrete dislocation simulations from the literature. We show that CDD reproduces salient features from discrete dislocation simulations regarding the stress distribution, the dislocation density and the accumulated plastic shear, which would be hard to obtain from more traditional crystal plasticity constitutive laws. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

From Dislocation Motion to an Additive Velocity Gradient Decomposition, and Some Simple Models of Dislocation Dynamics

A mathematical theory of time-dependent dislocation mechanics of unrestricted geometric and material nonlinearity is reviewed. Within a ``small deformation" setting, a suite of simplified and interesting models consisting of a nonlocal Ginzburg Landau equation, a nonlocal level set equation, and a nonlocal generalized Burgers equation is derived. In the finite deformation setting, it is shown that an additive decomposition of the total velocity gradient into elastic and plastic parts emerges naturally from a micromechanical starting point that involves no notion of plastic deformation but only the elastic distortion, material velocity, dislocation density and the dislocation velocity. Moreover, a plastic spin tensor emerges naturally as well.     

Some details of the description of a disordered condensed system using the theory of defect states of orientational order

Using the theory of defect states of orientational order, we describe a disordered condensed system as an elastic medium with linear topological singularities. We show that elastic stress fields produced by linear disclinations are Abelian. In the quasistationary linear approximation, we obtain expressions for linear dislocation and disclination tensor potentials. We show that using the theory of defect states of orientational order, we can describe the α and β relaxations in a supercooled liquid as relaxation processes in the respective disclination and dislocation subsystems.     

A distributional approach to the geometry of 2D dislocations at the continuum scale

This paper develops a geometrical model of dislocations and disclinations in single crystals at the continuum scale, with use of the distribution theory to represent concentrated effects in the defect lines which in turn form the branching lines of the multiple-valued elastic displacement and rotation fields. Fundamental identities relating the incompatibility tensor to the dislocation and disclination densities are proved in the case of locally countably many parallel defect lines, under global 2D strain assumptions relying on the geometric measure theory. Our approach provides a new understanding of the continuum theory of defects and introduces the appropriate objective internal variables and the required mathematical framework for a rigorous homogenization of all relevant fields from mesoscopic to macroscopic scale.     

On the “relativistic” description of motion of soliton-like defects in elastic media

An analysis of the manner of establishing a relativistic micro-universe with respect to the motion of soliton-like defects in elastic media is performed. It is demonstrated that the change of variables in the elastic-dynamic equations holding the motion of a screw dislocation must be complemented by the contraction law for the displacement vector and that a theory based on Lorentz’s transformations is not the only possible framework for representing the motion of soliton-like defects.     

Distribution of dislocation density and residual stresses in plastically deformed specimens: numerical studies

A two-scale approach to the simulation of mechanical properties of metallic materials is considered. On the macroscopic level, the material behavior is described by a phenomenological model of finite strain viscoplasticity with nonlinear kinematic hardening. In particular, the process-induced plastic anisotropy is captured by backstresses. On the microstructural level, the so called “load path sensitive two-population dislocation cell model” is implemented. It describes an evolving dislocation cell structure with dislocation populations for dislocation cell walls and the cell interior. Owing to the coupling with the phenomenological plasticity model, it can describe the evolution of the dislocation densities depending on the load path. The applicability of the multiscale approach to the FEM simulation of severe plastic deformation processes such as Equal Channel Angular Pressing is demonstrated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A theory of dislocations and disclinations in elastic plates

The problem of the stress state of a thin elastic plate, containing dislocations and disclinations, is considered using Kirchhoff's theory. The problem of the equilibrium of a multiply connected plate with Volterra dislocations with specified characteristics is formulated. The problem of the flexure of an annular slab resulting from a screw dislocation and a twisting disclination is solved. The solutions of problems of concentrated (isolated) dislocations and disclinations in an unbounded plate as well as the dipoles of dislocations and disinclinations are found. It is shown that a screw dislocation in a thin plate is equivalent to the superposition of two orthogonal dipoles of torsional disclinations. By taking the limit from a discrete set of defects to their continuous distribution, a theory of thin plates with distributed dislocations and disclinations is constructed. Solutions of problems of the flexure of circular and elliptic plates with continuously distributed disclinations are obtained. An analogy is established between the problem of the flexure of a plate with defects and the plane problem of the theory of elasticity with mass forces, and also between a plane problem with dislocations and disclinations and the problem of the flexure of a plate with specified distributed loads.     

Linking Micro- and Macroscopic Aspects of a Dislocation Based Continuum Model by Investigating the Dislocation Double Pileup

In this contribution we discuss the problem of a dislocation double pileup at impenetrable boundaries which serves as a benchmark test to examine the continuum dislocation dynamics (CDD). We enhance a formulation for a numerical implementation and the mean-field theory to account for the ‘long-range’ stress field due to inhomogeneous plastic deformations by a suitable correction term. This allows us to handle the motion of dislocation densities at impenetrable boundaries and study the influence of dislocation interactions on a scale much smaller than the scale of the proposed mean-field stress. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)