圆形界面刚性线夹杂的平面问题

研究了圆弧形界面刚性线夹杂的平面弹性问题．集中力作用于夹杂或基体中的任意点,并且无穷远处受均匀载荷作用．利用复变函数方法,得到了该问题的一般解答．当只含一条界面刚性线夹杂时,获得了分区复势函数和应力场的封闭形式解答,并给出刚性线端部奇异应力场的解析表达式．结果表明,在平面荷载下界面圆弧形刚性线夹杂尖端应力场和裂纹尖端相似具有奇异应力振荡性．对无穷远加载的情况,讨论了刚性线几何条件、加载条件和材料失配对端部场的影响．     

理想塑性晶体裂纹顶端渐近场

本文从晶体三维塑性流动理论出发,导出了双滑移理想塑性晶体平面应变问题曲基本方程。利用这些方程求得了静止裂纹顶端应力变形场。该场包含有弹性角形区并且整个应力变形场是连续的。进而导出了定常扩展裂纹顶端应力变形场。该场由五个角形区组成:裂纹前方有两个塑性区,它们的边界是速度场间断面。裂纹面附近有一个二次塑性区,中间是两个卸载弹性区,它们交界面也是个速度场间断面。该五个角形区不是唯一的。本文得到了一簇解答。最后本文分析了这些解答在面心立方和体心立方晶体中的应用。     

多变量样条元法分析弹性地基板的弯曲,振动与稳定问题

本文应用双三次乘积型二元B样条函数来构造弹性地基板的位移、弯矩和扭矩等多种场函数，由混合变分原理导出多变量样条无法方程．文中，对弹性地基板的弯曲、振动与稳定问题作了分析与计算．由于，本文方法设定了各自独立的场函数，因此，所算得的场未知量如位移、弯矩和扭矩值的精度均比较高。     

两种材料组成空间的弹性力学基本解

本文研究了由两种不同材料的半空间所组成弹性体的弹性力学基本解。应用三维弹性理论中的Papkovich-Neuber通解以及Kelvin特解,求解出了在空间内部作用有集中力时空间的弹性力学位移场。该位移场在两个半空间内部分别满足各自的位移平衡方程,在其交接面上满足位移及面力的连续条件。作为本文结果的几种特殊情况,半空间的Lorentz问题与Mindlin问题的解,以及Stokes流中类似问题的解均可从该解答中导出。     

受热弹性复合球体空化问题的分析

研究了由两种弹性固体材料组成的复合球体,在均匀变温场作用下的空化问题．采用了几何大变形的有限对数应变度量和Hooke弹性固体材料的本构关系,建立了问题的非线性数学模型．求出了复合球体大变形热弹性膨胀的参数形式的解析解．给出了空穴萌生时临界温度随几何参数和材料参数的变化曲线,以及空穴增长的分岔曲线．算例的数值结果指出:超过临界温度后空穴半径将迅速增大,并且空穴萌生时环向应力将成为无限大,这意味着如果内部球体是弹塑性材料,则会在空穴表面附近产生塑性变形而造成材料的局部损伤．另外,当内部球体材料的弹性接近于不可压时,复合球体可以在较低的变温下空化．     

半平面压电体的Green函数及其应用

本文研究半平面压电体在线力、电荷和位错作用下的弹性场和电场，即Green函数．基于各向异性弹性力学中的Stroh方法和解析延拓理论，推导了Green函数的封闭形式的解．作为解的应用，分析了含半无限裂纹的无限大压电介质的机电耦合场，给出了应力和电位移强度因子的解析表达式．     

复合材料层合板的三维非线性分析

本文提出了一种研究复合材料层合板壳三维问题的解析方法．该方法采用摄动方法和变分原理来满足三维弹性理论基本微分方程及限制条件，分析了受横向载荷作用的复合材料各向异性单层圆板及层合圆板的三维非线性问题．得到了高精确度的摄动级数解答．大量结果表明横向剪应力和横向正应力在层合板的三维非线性分析中是很重要的．     

压电复合材料圆形夹杂中螺型位错和界面裂纹的干涉分析

研究了无穷远纵向剪切和面内电场共同作用下,压电复合材料圆形夹杂中螺型位错与界面裂纹的电弹耦合干涉作用．运用Riemann-Schwarz 对称原理,并结合复变函数奇性主部分析方法,获得了该问题的一般解答．作为典型算例,求出了界面含一条裂纹时基体和夹杂区域复势函数和电弹性场的封闭形式解．应用广义Peach-Koehler公式,导出了位错力的解析表达式．分析了裂纹几何参数和材料的电弹性常数对位错力的影响规律．结果表明,界面裂纹对位错力和位错平衡位置有很强的扰动效应,当界面裂纹长度达到临界值时,可以改变位错力的方向．该结果可以作为格林函数研究圆形夹杂内裂纹和界面裂纹的干涉效应．其公式的退化结果与已有文献完全一致．     

Ⅰ阶梯度损伤理论

从热力学基本定律出发,将应变张量、标量损伤变量、损伤梯度作为Helmholtz自由能函数的状态变量,利用本构泛函展开法在自然状态附近作自由能函数的Taylor展开,未引入附加假设,推导出Ⅰ阶梯度损伤本构方程的一般形式．该形式在损伤为0时可退化为线弹性应力-应变本构方程,在损伤梯度为0时可退化为基于应变等效假设给出的线弹性局部损伤本构方程．一维解析解表明,随着应力增大,损伤场逐步由空间非周期解变为关于空间的类周期解,类周期解的峰值区域形成局部化带．局部化带内的损伤变量将不同于局部化带外的损伤变量,由此可以反映出介质的局部化特征．损伤局部化并不是与损伤同时发生,而是在损伤发生后逐渐显现出来,模型的局部化机制开始启动;损伤局部化的宽度同内部特征长度成正比．     

直线与圆锥曲线相交过定点问题的统一性质

如何利用坐标法简化解答，突破思维障碍，文[1]给出了解答问题的关键，获得了“完美”解答，读来颇受益．笔者从该问题的另一角度思考探究，得出直线与圆锥曲线过定点问题的一些性质，并从几何特征出发获得该问题的一般解法．     

Isotropic continuum damage/repair model for alveolar bone remodeling

Several authors have proposed mechanical models to predict long term tooth movement, considering both the tooth and its surrounding bone tissue as isotropic linear elastic materials coupled to either an adaptative elasticity behavior or an update of the elasticity constants with density evolution. However, tooth movements obtained through orthodontic appliances result from a complex biochemical process of bone structure and density adaptation to its mechanical environment, called bone remodeling. This process is far from linear reversible elasticity. It leads to permanent deformations due to biochemical actions. The proposed biomechanical constitutive law, inspired from Doblaré and García (2002) [30], is based on a elasto-viscoplastic material coupled with Continuum isotropic Damage Mechanics (Doblaré and García (2002) [30] considered only the case of a linear elastic material coupled with damage). The considered damage variable is not actual damage of the tissue but a measure of bone density. The damage evolution law therefore implies a density evolution. It is here formulated as to be used explicitly for alveolar bone, whose remodeling cells are considered to be triggered by the pressure state applied to the bone matrix. A 2D model of a tooth submitted to a tipping movement, is presented. Results show a reliable qualitative prediction of bone density variation around a tooth submitted to orthodontic forces.     

On Well Posedness in Continuum Interface Problems

By considering continuum interface problems like e.g. the modeling of composites, the possible loss of well posedness of the resulting Boundary Value Problem has to be considered, dependent on the choice of material laws in the bulk and in the interface. In this contribution, the problem is discussed for a bulk material connected to a rigid substrate by an interface layer. The isotropic bulk material is linear elastic while for the interface elasticity and elasticity with damage is investigated. A complex surface acoustic tensor is introduced by applying a decaying surface wave ansatz to the incremental boundary value problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of fatigue life in bending with rotation

The fracture and stress redistribution processes in a specimen stressed in cyclic bending with rotation are considered. The calculations are based on the assumption of linear damage accumulation and a linear dependence of the modulus of elasticity on the damage; the fatigue strength curve for cyclic tension-compression is employed. The calculated damage accumulation and crack penetration times are compared. Specific calculations are made for the case of PMMA fatigue under isothermal conditions.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 875–880, September–October, 1971.     

Multi-scale method for the quasi-periodic structures of composite materials

In this paper, we shall present a new second-order and two-scale approximate solution of boundary value problems for the linear elasticity systems with quasi-periodic structures by multi-scale analysis. The computation of strain and stress is presented here. This idea can apply to the damage computation of composite materials. Because the computation of the local stress by this method having more precision, it can provide some inspiration for construct optimal design. Finally numerical results show that the method presented in this paper is effective and reliable.     

Some remarks on the constant in the strengthened CBS inequality: Estimate for hierarchical finite element discretizations of elasticity problems

For a class of two‐dimensional boundary value problems including diffusion and elasticity problems, it is proved that the constants in the corresponding strengthened Cauchy‐Buniakowski‐Schwarz (CBS) inequality in the cases of two‐level hierarchical piecewise‐linear/piecewise‐linear and piecewise‐linear/piecewise‐quadratic finite element discretizations with triangular meshes differ by the factor 0.75. For plane linear elasticity problems and triangulations with right isosceles triangles, formulas are presented that show the dependence of the constant in the CBS inequality on the Poisson's ratio. Furthermore, numerically determined bounds of the constant in the CBS inequality are given for plane linear elasticity problems discretized by means of arbitrary triangles and for three‐dimensional elasticity problems discretized by means of tetrahedral elements. Finally, the robustness of iterative solvers for elasticity problems is discussed briefly. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 469–487, 1999     

A new type of full multigrid method for the elasticity eigenvalue problem

This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments.     

Covariance in linearized elasticity

In this paper we covariantly obtain all the governing equations of linearized elasticity. Our motivation is to see if one can make a connection between invariance (covariance) properties of the (global) balance of energy in nonlinear elasticity and those of its counterpart in linear elasticity. We start by proving a Green-Naghdi-Rivilin theorem for linearized elasticity. We do this by first linearizing energy balance about a given reference motion and then by postulating its invariance under isometries of the Euclidean ambient space. We also investigate the possibility of covariantly deriving a linearized elasticity theory, without any reference to the local governing equations, e.g. local balance of linear momentum. In particular, we study the consequences of linearizing covariant energy balance and covariance of linearized energy balance. We show that in both cases, covariance gives all the field equations of linearized elasticity.      

On parallel solution of linear elasticity problems: Part I: theory

The discretized linear elasticity problem is solved by the preconditioned conjugate gradient (pcg) method. Mainly we consider the linear isotropic case but we also comment on the more general linear orthotropic problem. The preconditioner is based on the separate displacement component (sdc) part of the equations of elasticity. The preconditioning system consists of two or three subsystems (in two or three dimensions) also called inner systems, each of which is solved by the incomplete factorization pcg-method, i.e., we perform inner iterations. A finite element discretization and node numbering giving a high degree of partial parallelism with equal processor load for the solution of these systems by the MIC(0) pcg method is presented. In general, the incomplete factorization requires an M-matrix. This property is studied for the elasticity problem. The rate of convergence of the pcg-method is analysed for different preconditionings based on the sdc-part of the elasticity equations. In the following two parts of this trilogy we will focus more on parallelism and implementation aspects. © 1998 John Wiley & Sons, Ltd.     

Wedge Dislocation in the Geometric Theory of Defects

We consider a wedge dislocation in the framework of elasticity theory and the geometric theory of defects. We show that the geometric theory quantitatively reproduces all the results of elasticity theory in the linear approximation. The coincidence is achieved by introducing a postulate that the vielbein satisfying the Einstein equations must also satisfy the gauge condition, which in the linear approximation leads to the elasticity equations for the displacement vector field. The gauge condition depends on the Poisson ratio, which can be experimentally measured. This indicates the existence of a privileged reference frame, which denies the relativity principle.     

考虑损伤的修正梯度弹性理论

梯度弹性理论在描述材料微结构起主导作用的力学行为时具有显著优势,将其与损伤理论相结合,可在材料破坏研究中考虑微结构的影响.基于修正梯度弹性理论,将应变张量、应变梯度张量和损伤变量作为Helmholtz自由能函数的状态变量,并在自然状态附近对自由能函数作Taylor展开,进而由热力学基本定律,推导出修正梯度弹性损伤理论本构方程的一般形式.编制有限元程序,模拟土样损伤局部化带的发展演化过程.结果表明,修正梯度弹性损伤理论消除了网格依赖性;损伤局部化带不是与损伤同时发生,而是在损伤发展到一定程度后再逐渐显现出来.