Modified Young��s moduli of nano-materials taking into account the scale effects and vacancies

In this paper, the modified Young??s moduli in both directions are obtained for the two-dimensional single crystal body with a square lattice by using the continuum approach of continuum mechanics and taking into account only the interaction of neighboring atoms. Separately, an expression is obtained taking into account the linear defects such as vacancies. The values of effective Young??s moduli compared with the same values for an infinite crystal lattice. Analyses show that the influences of scale effects and vacancies on the Young??s moduli are considerable. In addition, it is shown that the effective Young??s moduli have three components: the macroscopic value; factors determining the scale effect; factors determining the vacancy. The last component is analogous to the parameter of the damage of the theory of fracture.     

Coalescence of Pore Columns by Domain Switching

The present paper studies the coalescence of pore columns in ferroelectric ceramics driven by back and forth domain switching under cyclic electric field. A finite element method that incorporates mass transfer capacity is formulated to simulate the evolution of point defects subjected to the kinetics of pore surface diffusion and domain wall migration. The merge of point defects provides a mechanism for the vacancy agglomeration that leads to the formation of large pores or microcracks.The project supported by the National Natural Science Foundation of China (10332020 and 10121202), and the National 973 Project (2004CB619304).     

Continuum modeling of charged vacancy migration in elastic dielectric solids, with application to perovskite thin films

A continuum theory describing the behavior of dielectric materials containing mobile, electrically charged vacancies is formulated. The theory is implemented to simulate diffusion, at the nanometer scale, of oxygen vacancies in acceptor-doped barium strontium titanate (BST) thin films in the paraelectric state. In the simulations, charged vacancies coalesce into boundary layers of large concentration at potential-free interfaces, with increases in the local electric field intensity emerging near such boundaries. Upon relating this increase to a reduction in the energy barrier for charge transmission from film to electrode at the interface, and accepting an inverse relationship between the concentrations of doping elements and mobile oxygen vacancies, the model shows agreement with observed trends of decreasing current losses with increased doping.     

THERMODYNAMIC VARIATIONAL APPROACH FOR CLIMB OF AN EDGE DISLOCATION

A general thermodynamic variational approach is applied to study the force on an edge dislocation, which drives the dislocation to climb. Our attention is focused on the physical mechanism responsible for dislocation climb. A dislocation in a material element climbs as a result of vacancies diffusing into or out from the dislocation core, with the dislocation acting as a source or a sink for vacancy diffusion in the material element. The basic governing equations for dislocation climb and the climb forces on the dislocation are obtained naturally as a result of the present thermodynamic variational approach.     

A phase field model for failure in interconnect lines due to coupled diffusion mechanisms

We describe a diffuse interface, or phase field model for simulating electromigration and stress-induced void evolution and growth in interconnect lines. Microstructural evolution is tracked by defining an order parameter, which takes on distinct uniform values within solid material and voids, and varying rapidly from one to the other over narrow interfacial layers associated with the void surfaces. The order parameter is governed by a form of the Cahn-Hilliard equation. An asymptotic analysis demonstrates that the zero contour of order parameter tracks the motion of a void evolving by coupled surface and lattice diffusion, driven by stress, electron wind and vacancy concentration gradients. Efficient finite element schemes are described to solve the modified Cahn-Hilliard equation, as well as the equations associated with the accompanying mechanical, electrical and bulk diffusion problems. The accuracy and convergence of the numerical scheme is investigated by comparing results to known analytical solutions. The method is applied to solve various problems involving void growth and evolution in representative interconnect geometries.     

Deformation, stress state and thermodynamic force for a growing void in an elastic–plastic material

The growth of a spherical void in an elastic–plastic body, subjected to external pressure or tension and a gas pressure as well as a surface stress at the void surface, is investigated. The deformation, strain and stress state in the full body is presented. In addition, the local and global energy terms are calculated. Finally the total thermodynamic force on the void surface as well as the total dissipation are evaluated and compared allowing the calculation of the mechanical contribution to void growth due to diffusion of vacancies generated by plastification or irradiation.     

Green’s function-based multiscale modeling of defects in a semi-infinite silicon substrate

We have developed a Green’s function (GF) based multiscale modeling of defects in a semi-infinite silicon substrate. The problem—including lattice defects and substrate surface, i.e., an extended defect, at different length scales—is first formulated within the theory of lattice statics. It is then reduced and solved by using a scale-bridging technique based on the Dyson’s equation that relates a defect GF to a reference GF and on the asymptotic relationship of the reference lattice-statics GF (LSGF) to the continuum GF (CGF) of the semi-infinite substrate. The reference LSGF is obtained approximately by solving the boundary-value problem of a super-cell of lattice subject to a unit point force and under a boundary condition given by the reference CGF. The Tersoff potential of silicon, germanium and their compounds is used to derive the lattice-level force system and force constants and further to derive the continuum-level elastic constants (of the bulk silicon, needed in the reference CGF). We have applied the method to solve for the lattice distortion of a single vacancy and a single germanium substitution. We have further calculated the relaxation energy in these cases and used it to examine the interaction of the point defects with the (traction-free) substrate surface and the interaction of a single vacancy with a relatively large germanium cluster in the presence of the substrate surface. In the first case, the point defects are found to be attracted to the substrate surface. In the second case, the single vacancy is attracted to the germanium cluster as well as to the substrate surface.     

应用通用权函数法计算热机载荷作用下三维界面裂纹的应力强度因子

热机载荷共同作用下双材料、复合材料中的裂纹扩展往往发生在界面处,并且工程中实际遇到的裂纹大多数是三维裂纹,传统的求解热冲击及机械载荷共同作用下界面裂纹应力强度因子的数值方法如有限元、边界元法计算量大,计算效率低。由于通用权函数仅仅与裂纹体的几何形状有关,与载荷、时间无关,求解应力强度因子时避免了反复的应力分析,计算效率大大提高, 通用权函数法非常适合计算复杂冲击载荷下应力强度因子分布的过渡过程。根据Betti互易原理,本文推导出了三维界面裂纹问题通用权函数法的普遍表达式,并给出了热机载荷共同作用下三维界面I型、Ⅱ型和Ⅲ型裂纹问题通用权函数法的有限元格式. 通过实例计算比较,表明此方法得到的结果可以达到满意的工程应用精度。     

Some issues associated with the intermediate space in single-crystal plasticity

In this study we show that some discussions of finite-deformation single-crystal plasticity are conceptually flawed in their focus on a set referred to as the intermediate configuration. Specifically, we prove that what is usually referred to as the intermediate configuration is not a configuration but instead a vector space that we term the intermediate space. We argue that when applied to single crystals this intermediate space represents the lattice.     

Numerical investigation of the slope discontinuities in large deformation finite element formulations

In many multibody system applications, the system components are made of structural elements that can have different orientations, leading to slope discontinuities. In this paper, a numerical investigation of a new procedure that can be used to model structures with slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is presented. This procedure can be applied to model slope discontinuities in the case of commutative rotations of gradient deficient elements that are used for modeling thin beam and plate structures. An important special case to which the proposed procedure can be applied is the case of all planar gradient deficient ANCF finite elements. The use of the proposed method leads to a constant orthogonal element transformation that describes an arbitrary initial configuration. As a consequence, one obtains, in the case of large commutative rotations and large deformations, a constant mass matrix for structures which have complex geometry. The procedure used in this investigation to model slope discontinuities requires the use of the concept of the intermediate finite element coordinate system. For each finite element, a new set of gradient coordinates that define, at the discontinuity node, the element deformation with respect to the intermediate element coordinate system is introduced. These new gradient coordinates are assumed to be equal for the two finite elements at the point of intersection. That is, the change of the gradients of two elements at the intersection point from their respective intermediate initial reference configuration is assumed to be the same. This procedure leads to a set of linear algebraic equations that define the orthogonal transformation matrix for the finite element. Numerical examples are presented in order to demonstrate the use of the proposed procedure for modeling slope discontinuities.     

Investigations of pipe-diffusion-based dislocation climb by discrete dislocation dynamics

A new numerical dislocation climb model based on incorporating the pipe diffusion theory (PDT) of vacancies with 3D discrete dislocation dynamics (DDD) is developed. In this model we hold that the climb rate of dislocations is determined by the gradient of the vacancy concentration on the segment, but not by the mechanical climb force as traditionally believed. The nodal forces on discrete dislocation segments in DDD simulation are transferred to PDT to calculate the vacancy concentration gradient. This transfer establishes a bridge connecting the DDD and PDT. The model is highly efficient and accurate. As verifications, two typical climb-involved examples are predicted, e.g. the activation of a Bardeen-Herring source as well as the shrinkage and annihilation of prismatic loops. Finally, the model is applied to study the breakup process of an infinite edge dislocation dipole into prismatic loops. This coupling methodology provides us a useful tool to intensively study the evolution of dislocation microstructures at high temperatures.     

Sublimation of a crystal slowed by sorption of foreign molecules from a gaseous medium

Numerical simulation of the sublimation of the surface of an ice crystal has shown that the presence of a non-condensable gas substantially reduces the sublimation rate. It has been found that the slowing of the sublimation due to the adsorption of gas molecules on the crystal surface that fill the vacancies during sublimation prevents the formation of vacancy islands on the surface of the crystal required for evaporation of the surface molecular layer. The available results of studies that explain the slowing of the sublimation by the presence of a diffusion layer formed in the vapor medium near the sublimating surface are supplemented with new data showing that molecular processes on the crystal surface play an equally important role as the diffusion layer. Cases have been found where crystal sublimation is accelerated by the sorption of gas molecules. The obtained results refining the theory of sublimation can be used to develop methods for controlling sublimation.     

Structure of defective crystals at finite temperatures: A quasi-harmonic lattice dynamics approach

In this paper we extend the classical method of lattice dynamics to defective crystals with partial symmetries. We start by a nominal defect configuration and first relax it statically. Having the static equilibrium configuration, we use a quasi-harmonic lattice dynamics approach to approximate the free energy. Finally, the defect structure at a finite temperature is obtained by minimizing the approximate Helmholtz free energy. For higher temperatures we take the relaxed configuration at a lower temperature as the reference configuration. This method can be used to semi-analytically study the structure of defects at low but non-zero temperatures, where molecular dynamics cannot be used. As an example, we obtain the finite temperature structure of two 180° domain walls in a 2D lattice of interacting dipoles. We dynamically relax both the position and polarization vectors. In particular, we show that increasing temperature the domain wall thicknesses increase.     

High-temperature discrete dislocation plasticity

A framework for solving problems of dislocation-mediated plasticity coupled with point-defect diffusion is presented. The dislocations are modeled as line singularities embedded in a linear elastic medium while the point defects are represented by a concentration field as in continuum diffusion theory. Plastic flow arises due to the collective motion of a large number of dislocations. Both conservative (glide) and nonconservative (diffusion-mediated climb) motions are accounted for. Time scale separation is contingent upon the existence of quasi-equilibrium dislocation configurations. A variational principle is used to derive the coupled governing equations for point-defect diffusion and dislocation climb. Superposition is used to obtain the mechanical fields in terms of the infinite-medium discrete dislocation fields and an image field that enforces the boundary conditions while the point-defect concentration is obtained by solving the stress-dependent diffusion equations on the same finite-element grid. Core-level boundary conditions for the concentration field are avoided by invoking an approximate, yet robust kinetic law. Aspects of the formulation are general but its implementation in a simple plane strain model enables the modeling of high-temperature phenomena such as creep, recovery and relaxation in crystalline materials. With emphasis laid on lattice vacancies, the creep response of planar single crystals in simple tension emerges as a natural outcome in the simulations. A large number of boundary-value problem solutions are obtained which depict transitions from diffusional to power-law creep, in keeping with long-standing phenomenological theories of creep. In addition, some unique experimental aspects of creep in small scale specimens are also reproduced in the simulations.     

两种流型下雾滴表面气溶胶粒子的扩散沉积

用浓度边界层理论分析了势流和Hadamard流情形下雾滴或云滴前向驻点处小粒子的扩散效应。结果表明：存在静电效应时，有限直径小粒子在雾滴表面的扩散通量与两个无量纲数参数有效；对于无外场作用时的点粒子情形，上述两种流型下的扩散沉积特征与粘性流结果亦不相同。     

基于中间构形的大变形弹塑性模型

在大变形弹塑性本构理论中,一个基本的问题是弹性变形和塑性变形的分解.通常采用两种分解方式,一是将变形率(或应变率)加法分解为弹性和塑性两部分,其中,弹性变形率与Kirchhoff应力的客观率通过弹性张量联系起来构成所谓的次弹性模型,而塑性变形率与Kirchhoff应力使用流动法则建立联系;另一种是基于中间构形将变形梯度进行乘法分解,它假定通过虚拟的卸载过程得到一个无应力的中间构形,建立所谓超弹性-塑性模型.研究了基于变形梯度乘法分解并且基于中间构形的大变形弹塑性模型所具有的若干性质,包括:在不同的构形上,塑性旋率的存在性、背应力的对称性、塑性变形率与屈服面的正交性以及它们之间的关系.首先,使用张量函数表示理论,建立了各向同性函数的若干特殊性质,并导出了张量的张量值函数在中间构形到当前构形之间进行前推后拉的简单关系式.然后,基于这些特殊性质和关系式,从热力学定律出发,建立模型在不同构形上的数学表达,包括客观率表示的率形式和连续切向刚度等,从而获得模型所具有的若干性质.最后,将模型与4种其他模型进行了比较分析.      

Sur le champ des vitesses en thermoplasticité finie

The purpose of this paper is the variational formulation of the rate problem in thermoplasticity or coupled thermoviscoplasticity, when the elastic-plastic transformation is finite. Following Hill's method we show that, for an elastic-plastic material with intermediate relaxed configuration, the nominal stress rate can be derived from a potential function expressed in terms of the gradient of the displacement rate; from this property we derive an extremum principle and a uniqueness criterion for the velocity field.     

A crack on the grain boundary — An analysis based on crystal plasticity theory

The influence of the mismatch of the lattice orientation on the deformation and stress fields of a crack located on the grain boundary is studied by means of the finite-element analysis taking account of finite deformatio and finite lattice rotation. The plane strain calculations for an fcc crystal subjected to mode I loading are performed on the basis of the crystalline plasticity described by a planar three-slip model. For the crack-tip shapes and the dominant deformation modes on slip systems, results of all the cases analysed here are in qualitative agreement with the earlier analytical and numerical solutions. Our results indicate that the lattice orientation difference may greatly influence the shear stress along the grain boundary which is related to grain-boundary sliding, while the normal stress along the grain boundary, which may induce cleavage fracture, is virtually insensitive to it. The influence of the lattice orientations on the crack-tip fields is also investigated under small-scale-yielding conditions and the comparison with the results of finite deformation is made.