Linearized Elasticity as Γ-Limit of Finite Elasticity in the Case of Cracks

The small-deformation limit of finite elasticity is considered in presence of a fixed crack. The rescaled finite energies with the constraint of global injectivity are shown to Γ-converge to the linearized elastic energy with a local constraint of noninterpenetrability on the crack. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Methods of Micro–Macro Transitions in Nonlinear Elasticity

Continuum micromechanics deals with idealized materials where the macroscopic material response is modelled in an averaged or homogenized sense based on the information of the heterogeneous microstructure. In general, an efficient treatment of multiscale systems requires the application of equivalent structural problems where the constituents are governed by overall properties. The key contribution of this paper is the computational exploitation of variational methods for a numerical upper and lower bound estimation of the effective material response. We present aspects for the formulation of an appropriate minimizing principle yielding the displacement fluctuations on the microstructure and the local effective constitutive variables of the macrostructure depending on the choice whether we apply linear displacement, traction or periodic boundary conditions to the displacement fluctuations on the boundary of the microstructure. The proposed concept will be demonstrated in the scope of some representative model problems.     

ADirect Method for the Problem of the Solid Cylinder in Elasticity

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＳａｉｎｔＶｅｎａｎｔｐｒｏｐｏｓｅｄｔｈｅｆａｍｏｕｓｓｅｍｉ－ｉｎｖｅｒｓｅｓｏｌｕｔｉｏｎｍｅｔｈｏｄ［１］，ｔｈａｔｓｏｍｅａｐｐｒｏ－ｐｒｉａｔｅａｓｓｕｍｐｔｉｏｎｓｔｏｔｈｅｄｅｆｏｒｍａｔｉｏｎｓｈｏｕｌｄｂｅｍａｄｅｂｅｆｏｒｅｈａｎｄｔｏｆｉｎｄｔｈｅｓｏｌｕｔｉｏｎ，ａｆｔｅｒ－ｗａｒｄｓｃｈｅｃｋｉｎｇｔｈｅａｓｓｕｍｐｔｉｏｎｓｂｅｉｎｇｖａｌｉｄ．Ｔｈｅｒｅａｆｔｅｒ，ｔｈｅｓｅｍｉ－ｉｎｖｅｒｓｅｓｏｌｕｔｉｏｎｂｅｃｏｍｅｓｔ…     

Unique Continuation Property with Partial Information for Two-Dimensional Anisotropic Elasticity Systems

In this paper,we establish a novel unique continuation property for two-dimensional anisotropic elasticity systems with partial information.More precisely,given a homogeneous elasticity system in a connected open bounded domain,we investigate the unique continuation by assuming only the vanishing of one component of the solution in a subdomain.Using the corresponding Riemann function,we prove that the solution vanishes in the whole domain provided that the other component vanishes at one point up to its second derivatives.Further,we construct several examples showing the possibility of further reducing the additional information of the other component.This result possesses remarkable significance in both theoretical and practical aspects because the required data are almost halved for the unique determination of the whole solution.     

On the Pertodic Fundamental Problems of a Half Plane with Anisotropic Elasticity

IntroductionThe fundamental problems of a half plane with anisotropic elasticity andthe periodic fundamental problems of a half plane with isotropic elasticitywere respectively investigated by S.G.Leknitskii[1]and Lu Chien-ke[2].These problems are formulated on the cornplex z-plane in terms of     

On the Mathematical Theory of Plane Elasticity About Punctured Anisotropic Composite Media

ＯｎｔｈｅＭａｔｈｅｍａｔｉｃａｌＴｈｅｏｒｙｏｆＰｌａｎｅＥｌａｓｔｉｃｉｔｙＡｂｏｕｔＰｕｎｃｔｕｒｅｄＡｎｉｓｏｔｒｏｐｉｃＣｏｍｐｏｓｉｔｅＭｅｄｉａ￥ＬｉｕＳｈｉｑｉａｎｇ；ＬｉＸｉｎｇ（ＮｉｎｇｘｉａＵｎｉｖｅｒｓｉｔｙ，Ｙｉｎｃｈｕａｎ...     

Quadrilateral Finite Elements for Planar Linear Elasticity Problem with Large Lamé Constant

In this paper, we discuss the quadrilateral finite element approximation to the two-dimensional linear elasticity problem associated with a homogeneous isotropic elastic material. The optimal convergence of the finite element method is proved for both the $L^2$-norm and energy-norm, and in particular, the convergence is uniform with respect to the Laméconstant $\lambda$. Also the performance of the scheme does not deteriorate as the material becomes nearly incompressible. Numerical experiments are given which are consistent with our theory.     

Elasticity Solutions for Cylindrical Bending of Functionally Graded Piezoelectric Material Plates北大核心CSCD

功能梯度压电材料（FGPM）同时兼具功能梯度材料和压电材料特性,可为多功能或智能化轻质结构设计提供支撑,在诸多领域有着广泛的应用前景.将Mian和Spencer功能梯度板理论由功能梯度弹性材料推广到功能梯度压电材料,解析研究了FGPM板的柱面弯曲问题,其中,材料弹性常数、压电和介电参数沿板厚方向可以任意连续变化.最终,给出了FGPM板受横向均布荷载作用下柱面弯曲问题的弹性力学解.通过算例分析,重点讨论了压电效应对FGPM板静力响应的影响.     

The Contact Problems of the Mathematical Theory of Elasticity for Plates with an Elastic Inclusion

The contacts problem of the theory of elasticity and bending theory of plates for finite or infinite plates with an elastic inclusion of variable rigidity are considered. The problems are reduced to integral differential equation or to the system of integral differential equations with variable coefficient of singular operator. If such coefficient varies with power law we can manage to investigate the obtained equations, to get exact or approximate solutions and to establish behavior of unknown contact stresses at the ends of elastic inclusion.      

General Solution for the Two-Dimensional System of Static Lame’s Equations with an Asymmetric Elasticity Matrix

We study a two-dimensional system of equations of linear elasticity theory in the case when the symmetric stress and strain tensors are related by an asymmetric matrix of elasticity moduli or elastic compliances. The linear relation between stresses and strains is written in an invariant form which contains three positive eigenmodules in the two-dimensional case. Using a special eigenbasis in the strain space, it is possible to write the constitutive equations with a symmetric matrix, i.e., in the same way as in the case of hyperelasticity. We obtain a representation of the general solution of two-dimensional equations in displacements as a linear combination of the first derivatives of two functions which satisfy two independent harmonic equations. The obtained representation directly implies a generalization of the Kolosov–Muskhelishvili representation of displacements and stresses in terms of two analytic functions of complex variable. We consider all admissible values of elastic parameters, including the case when the system of differential equations may become singular. We provide an example of solving the problem for a plane with a circular hole loaded by constant stresses.     

An Integral Representation Result for the Γ-Limit of Functionals with Non-standard Growth Conditions in the Case of Elasticity

An integral representation result on regular functions is proved for the o -limit of a sequence of integral functionals defined in the vectorial case and modelled on elasticity theory functional Z z f (( x , e ( u )) dx where convex lagrangians satisfy a non-standard estimate $$ -c_{0} + c_{1} | \xi|^{\alpha }\leq f ( (x,\xi ) \leq c_{0} + c_{2} | \xi|^{\beta },\quad 1 \lt \alpha \leq \beta \lt \frac {n\alpha }{n-\alpha },\enskip c_{0}\geq 0,\enskip c_{1},c_{2} \gt 0. $$ When the limit integrand does not show Lavrent'ev phenomenon the representation result is also true on the whole space W 1, f ( z ; R n ).     

Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣Γ = 0 for Integrable Tensor-Coefficients G and Applications to Elasticity

Let be bounded Lipschitz and relatively open. We show that the solution to the linear first order system 1 : (1) vanishes if and , (e.g. ). We prove to be a norm if with , for some p, q > 1 with 1/p + 1/q = 1 and . We give a new proof for the so called ‘in-finitesimal rigid displacement lemma’ in curvilinear coordinates: Let , satisfy for some with . Then there are and a constant skew-symmetric matrix , such that . (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)