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)     

Elastoplastic torsion of a cylindrical rod for finite deformations

A solution of the problem of the torsion of a cylindrical rod was obtained in /1/ for a general, isotropic, incompressible elastic material. The present paper gives an analytical solution of the elastoplastic torsion problem for finite deformations, written in terms of quadratures of elliptic functions. The non-linear kinematics of elastoplastic deformation is introduced into the defining equations with the help of a multiplicative decomposition of the deformation gradient into elastic and plastic components /2, 3/. The elastic deformation and rate of plastic deformation are related to the state of stress of the body, in accordance with the defining Mooney-Rivlin equations /4/ and the law of flow for finite deformations associated with the Tresca yield condition /5/. A non-linear first-order partial differential equation and the initial data at the elastoplastic boundary are obtained in order to determine the angle of rotation within the plastic zone of the basis formed from the eigenvectors of the stress tensor, relative to the radial direction. The integration of the resulting equation is reduced to determining the general integral of the Ricatti equation with right-hand side determined from the angular velocity of flow of the material within the plastic zone. It is shown that neglecting the finiteness of the deformation leads to too high an estimate of the rigidity of the rod.     

立方准晶材料中的运动螺型位错

发展了立方准晶的位错弹性理论．通过引入位移势函数,使得立方准晶的反平面弹性动力学问题归结为求解两个波动方程,得到了运动螺型位错的位移场、应力场与能量的解析表达式及运动位错的速度极限．这些为研究此固体材料的塑性变形的物理机理提供了重要的信息．     

Elastoplastic deformation during projectile–wall collision

The Smoothed Particle Hydrodynamics method for elastic solid deformation is modified to include von Mises plasticity with linear isotropic hardening and is then used to investigate high speed collisions of elastic and elastoplastic bodies. The Lagrangian mesh-free nature of SPH makes is very well suited to these extreme deformation problems eliminating issues relating to poor element quality at high strains that limits finite element usage for these types of problems. It demonstrates excellent numerical stability at very high strains (of more than 200%). SPH can naturally track history dependent material properties such as the cumulative plastic strain and the degree of work hardening produced by its strain history. The high speed collisions modelled here demonstrate that the method can cope easily with collisions of multiple bodies and can also naturally resolve self-collisions of bodies undergoing high levels of plastic strain. The nature and the extent of the elastic and plastic deformation of a rectangular body impacting on an elastic wall and of an elastic projectile impacting on a thin elastic wall are investigated. The final plastically deformed shapes of the projectile and wall are compared for a range of material properties and the evolution of the maximum plastic strain throughout each collision and the coefficient of restitution are used to make quantitative comparisons. Both the elastoplastic projectile–elastic wall and the elastic projectile–elastoplastic wall type collisions have two distinct plastic flow regimes that create complex relationships between the yield stress and the responses of the solid bodies.     

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)     

Nonlocal damage-viscoplastic model for high temperature creep of single crystal superalloys

A new nonlocal damage-viscoplastic model for high temperature creep of single crystal superalloys is developed. It is based on the variational formulation consisting of free energy, plastic and damage dissipation potentials. Evolution equations for plastic strain and damage variables are derived from the minimum principle for dissipation potentials [1]. The model is capable of describing different stages of creep in a unified way. The evolution of dislocation densities of gamma and gamma prime phases in superalloys incorporates plastic deformation. It results in the time-dependence of the creep rate in primary and secondary creep. Tertiary creep is taken into account by introducing local and nonlocal damage variables. Herein the nonlocal one is considered as numerical treatment to remove mesh-dependence. Numerical results and comparisons with experimental data of the single crystal superalloy LEK94 are shown. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dislocation mechanism of the initial stage of plastic deformation of polyethylene

A unified treatment of the initial stages of the plastic deformation of the polyethylene single crystal — phase transformation from the orthorhombic to the monoclinic lattice, twinning, and crack formation — is proposed. This treatment, which makes use of dislocation theory, is based on an analysis of experiments on the deformation of the polyethylene single crystal.     

研究金属中激波构造与衰减的一个物理模型

本文给出了研究金属中激波构造与衰减的一个物理模型.为了建立高速形变下材料的本构方程和研究激波过渡带的构造,需要考虑二个独立的理论方面.首先,将比内能分解成弹性压缩能和弹性形变能,而将形变能作为弹性应变和熵的函数展开到三阶项,其中考虑了热与机械能的耦合效应.其次,从位错动力学角度建议了一个塑性松弛函数以便描述高温、高压下塑性流动的特性.另外,本文给出了一个常微分方程组用以计算定态激波过渡带中各状态变量的分布以及激波的厚度.倘若假定在激波上熵的跳跃可以忽略,并用Hugoniot压缩模量代替等熵压缩摸量,可以获得一个分析解.最后,本文还提出了求解平板对称碰撞中激波波头衰减的一个近似方法。     

微极原弹性物质体理论和非局部微极弹性介质的本构方程

本文提出了微极原弹性物质体的定义并利用虚功率原理导出了该类物质体的变分原理.利用上述同样思想和这里给出的微极原势的定义很自然地导出了非局部微极弹性介质的本构方程.     

Instability prevention at the modeling of large elasto-plastic strains under cyclic loading

The cyclic instability phenomenon is investigated at the modeling of large elasto-plastic strains. The possible causes of the cyclic instability and conditions ensuring cyclical stability of elasto-plastic models are analyzed for the case of large strains. Among the possible causes of the cyclic instability the following are considered: the method of strain decomposition on elastic and plastic parts; the constitutive law for the elastic deformation (hypo- and hyper-elasticity); the constitutive equation for the plastic deformation; the constitutive relation for the plastic spin; kinematic hardening law, in particular, the type of the objective rate in the generalized Prager's law. Predictions of 50 various models of the elasto-plastic material have been compared in order to find the causes of the cyclic instability. Two test problems are considered: cyclic simple shear, combined cyclic simple shear and tension-compression. Results of numerical experiments for the various material models are presented and discussed. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Three-dimensional Continuum Theory of Dislocation Plasticity - Modelling and Application to a Composite

The increasing demand for materials with well defined microstructure, which is accompanied by the advancing miniaturization of devices calls for physically motivated, dislocation-based continuum theories of plasticity. Only recently rigorous techniques have been developed for performing meaningful averages over systems of moving, curved dislocations, yielding evolution equations for a higher order dislocation density tensor. Our continuum dislocation theory allows for generalizing the planar system towards a three-dimensional system, where dislocations may have arbitrary orientation and curvature. With the inclusion of curvature, the theory naturally takes into account a deformation-induced increase in the overall dislocation density without having to invoke ad-hoc assumptions about dislocation sources. A numerical implementation and some benchmark tests of this continuum theory for dislocation dynamics has already been discussed in the literature. In this paper, we apply this continuum theory to composite materials, where we analyze a plastically deforming matrix with an elastic inclusion. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nature of the friction deformation of the surface layers of polymethyl methacrylate

The surface deformation of amorphous thermoplastics (polymethyl methacrylate) by a spherical steel indentor has been investigated at various sliding velocities. Small velocities correspond to elastic and forced-elastic deformation of the surface layers and asperities. At temperatures corresponding to the high-elastic state the deformed surface layer completely recovers its shape. As the sliding velocity increases, the forced-elastic deformation is localized in a thinner layer of plastic. Starting from a certain velocity, depending on the temperature and the activation energy for transition of the chain segments from one equilibrium position to another in the process of thermal motion, the deformation of the surface layers and asperities becomes purely elastic. In the event of elastic deformation at pressures above a certain value the surface layer of plastic suffers brittle fracture in the tensile zone behind the indentor.Mekhanika Polimerov, Vol. 4, No. 1, pp. 90–94, 1968.     

非局部非对称弹性固体理论

本文基于非局部连续统场论和非线性连续体力学理论,建立了非局部非对称弹性固体的非线性理论.它完善和发展了Eringen等人所建立的非局部弹性场论.将文献[1]中所建立的非局部非对称弹性力学的线性理论推广到有限变形.证明了在非局部弹性固体中存在着非局部体力矩.非局部体力矩引起应力的非对称性,而非局部体力矩则由原子晶格相互作用形成的共价键所产生的.并应用本文建立的理论合理地解释了平面横波和纵波色散系关的不相似性.     

Modeling of the response of elastic plastic materials treated as a mixture of hard and soft regions

Inelastic materials that form dislocation cells on being deformed are modeled as a constrained-mixture of plastically hard and soft regions by associating different natural states with these regions. The deformation gradient from the reference configuration to the natural configuration is identified as the plastic deformation tensor and the stress is measured from a changing set of natural configurations. Two sets of natural configurations are introduced: one for the hard phase and the other for the soft phase. The full elastic response of the body is determined by elastic responses from different natural configurations. The energy stored in the dislocation networks is explicitly accounted for in the Helmholtz potential. Within a specialized constitutive set up, the soft phase is assumed to be non-hardening while the hardening response of the hard phase is dependent upon the response of both the hard and soft phases. These special forms are used to model the response of the material that forms cellular structures when subjected to cyclic loading.     

带有摩擦的单边接触大变形问题的研究(Ⅰ)——增量变分方程

本文以非线性连续体几何场论为基本理论和方法,建立了拖带坐标下弹塑性大变形增量变分方程的更一般表示式.给出了二维、三维连续体接触边界变化率公式,得到了变边界接触大变形增量变分公式和速率型变分不等式,为有限元计算求解带有摩擦弹塑性大变形接触问题提供了理论基础.     

Mathematical modelling of plastic deformation instabilities with two delays

In this paper we present a new model describing the temporal evolution for multi-instabilities of plastic deformation of stressed monocrystal. This model generalizes that with a single delay presented in [12] and [13]. The plastic instability is due to the increase in dislocation density and the mutual interaction between dislocations, and can be explained by a phase shift, characterized by a time delay between the nucleation and the propagation of dislocations. Here, we consider the general case when several deformation-mechanisms are active, leading to several delays. Using a linear analysis, we deduce a differential equation with two delays. We present some results on existence and stability of the solution according to the characteristics of material and the two delays. Numerical examples for stability and instability of material close to a mean stress using the MATLAB software are also investigated.     

关于非局部场论的几个新观点及其在断裂力学中的应用(Ⅲ)——重论线性非局部弹性理论

证明了在线性非局部弹性力学中能量平衡方程是动量平衡方程的首次积分,论证了在非局部场论中局部化体力残余恒为零。详细推导了线性非局部弹性理论的本构方程,得到了反对称应力存在的新结果。     

Application of the Green Tensor to the Modeling of Solution-Precipitation Creep

Our aim is to present a mechanical model for solution-precipitation creep, a diffusive deformation process occurring in polycrystalline and granular materials. Within the scope of the model, a Lagrangian consisting of the elastic power and dissipation is proposed. The former is kept in a standard form typical for linear material behavior while the latter is assumed as a surface integral depending on the velocity of the material transport and the velocity of the motion of the boundary. The minimization with respect to the total deformations leads to the equilibrium equation which is solved analytically, by using Green's function. The evolution equations are obtained as the result of the minimization with respect to the internal variables and are solved via finite element method. The contribution closes with a numerical example showing the deformation of a polycrystal consisting of hexagonal crystals. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wave propagation in single-walled carbon nanotube under longitudinal magnetic field using nonlocal Euler–Bernoulli beam theory

In the present work, the effect of longitudinal magnetic field on wave dispersion characteristics of equivalent continuum structure (ECS) of single-walled carbon nanotubes (SWCNT) embedded in elastic medium is studied. The ECS is modelled as an Euler–Bernoulli beam. The chemical bonds between a SWCNT and the elastic medium are assumed to be formed. The elastic matrix is described by Pasternak foundation model, which accounts for both normal pressure and the transverse shear deformation. The governing equations of motion for the ECS of SWCNT under a longitudinal magnetic field are derived by considering the Lorentz magnetic force obtained from Maxwell’s relations within the frame work of nonlocal elasticity theory. The wave propagation analysis is performed using spectral analysis. The results obtained show that the velocity of flexural waves in SWCNTs increases with the increase of longitudinal magnetic field exerted on it in the frequency range; 0–20 THz. The present analysis also shows that the flexural wave dispersion in the ECS of SWCNT obtained by local and nonlocal elasticity theories differ. It is found that the nonlocality reduces the wave velocity irrespective of the presence of the magnetic field and does not influences it in the higher frequency region. Further it is found that the presence of elastic matrix introduces the frequency band gap in flexural wave mode. The band gap in the flexural wave is found to independent of strength of the longitudinal magnetic field.     

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.