Boundary Value Problems of Plane Elasticity Involving Orientations of Displacements and Tractions

This paper presents unconventional formulations of boundary problems of plane elasticity formulated in terms of orientations of tractions and displacements on a closed contour separating internal and external domains as the boundary conditions. These are combined with the conditions of continuity of tractions or displacements across the boundary. Therefore the magnitudes of neither tractions nor displacements are assumed on the contour. Four boundary value problems for both external and internal domains are investigated by analyzing the solvability of the corresponding singular integral equations. It is shown that all considered problems can have non-unique solutions expressed as linear combinations of particular solutions and, hence, contain free arbitrary parameters, the number of which is finite and determined by the contour orientations of tractions and/or displacements     

Reducing the Order of the Boundary Condition in a Theory of Plane Strain Elasticity with Boundary Reinforcement

We consider an elastic solid whose boundary is partly reinforced in such a way that the coated part of the boundary can be represented by the union of a finite number of open curves. We show how the (nonstandard) boundary condition on the reinforcement can be integrated resulting in an equivalent (lower-order) boundary condition which, unlike its higher-order counterpart, permits analysis of the corresponding boundary value problems using the boundary integral equation method. Mathematics Subject Classifications (2000) 74B05, 45E05, 35J25     

On the Optimal Location of Singularities Arising in Variational Problems of Nonlinear Elasticity

Experiments on elastomers have shown that triaxial tension can induce a material to exhibit holes that were not previously evident. Analytic work in nonlinear elasticity has established that such cavity formation may indeed be an elastic phenomenon: sufficiently large prescribed boundary deformations yield a hole-creating deformation as the energy minimizer whenever the elastic energy is of slow growth. One of the many unanswered problems is where such holes will form. In this paper we suggest a new method, which is based upon asymptotics and linear elasticity, that can be used to determine the optimal location for hole creation. Using this method we show that, under reasonable hypotheses, the center is (locally) the best position for a solitary hole to form in an elastic ball. This revised version was published online in August 2006 with corrections to the Cover Date.     

弹性力学问题的虚边界元-配点法

本文提出虚边界方法,建立了离散化虚边界元-配点法,给出了离散化求系数的积分解析式。本文方法完全避免了边界奇异积分及其复杂耗时的运算,成功地提高了普通边界元法(以下简称边界元法)中边界附近区域内包括边界上解的精度,保留了边界元法的优点并扬弃了其弱点。边界元间接法是本文方法中的一个特例。数值算例表明,程序可靠,节省机时,计算精度较高。     

The Method of Integrodifferential Relations for Linear Elasticity Problems

Some possible modifications of the governing equations of the linear theory of elasticity are considered. The stress–strain relation is specified by an integral equality instead of the local Hooke’s law. The modified integrodifferential boundary value problem is reduced to the minimization of a nonnegative functional under differential constraints. A numerical algorithm based on polynomial approximations of unknown functions (stresses and displacements) is developed and applied to linear elasticity problems. The bilateral estimation criteria of solution errors are proposed in order to analyze the algorithm convergence rate. The numerical results obtained by applying the integrodifferential relation method and the conventional variational method are compared and discussed.     

A Study of Two-Dimensional Plane Elasticity Finite Elements for Optimal Design

ABSTRACT

Design of structures using a variety of two-dimensional finite elements is considered in this paper. An efficient technique of computing first-order derivatives of pointwise stress constraints for simple and higher-order two-dimensional (membrane) finite elements is presented. Computational aspects of design sensitivity vector calculation, using a semi-analytical method versus traditional methods are presented. Implementation of a fully-stressed design approach to find a suitable initial estimate leads to increased computational efficiency. These aspects of the design procedure are illustrated through analysis of numerical examples. Experience indicates that a suitable mix of low and higher-order elements yields the most efficient and accurate design model.     

An Average-Stretch Full-Network Model for Rubber Elasticity

Two constitutive models that are based on the classical non-Gaussian, Kuhn-Grün probability distribution function are reviewed. It is shown that all chains of a network cell structure comprised of a finite number of identical chains in an affine deformation referred to principal axes may have the same invariant stretch, if and only if the chains are oriented initially along any of eight directions forming the diagonals of a unit cube. The 4-chain tetrahedral and the 8-chain cubic cell structures are familiar admissible models having this property. An easy derivation of the constitutive equation for the Wu and van der Giessen full-network model of initially identical chains arbitrarily oriented in the undeformed state is presented. The constitutive equations for the neo-Hookean model, the 3 -chain model, and the equivalent 4- and 8-chain models are then derived from the Wu and van der Giessen equation. The squared chain stretch of an arbitrarily directed chain averaged over a unit sphere surrounding all chains radiating from a cross-link junction as its center is determined. An average-stretch, full-network constitutive equation is then derived by approximation of the Wu and van der Giessen equation. This result, though more general in that no special chain cell morphology is introduced, is the same as the constitutive equation for the 4- and 8-chain models. Some concluding remarks on extensions to amended models are presented.     

Unified Way for Dealing with Three-Dimensional Problems of Solid Elasticity

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Solving Problems of Elasticity of Conical Shells

The problem of a local action of normal pressure on a thin circular shell is solved by the method of asymptotic synthesis. A conical shell under the action of local radial loads is considered as an example.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 150–165, September–October, 2005.     

圆柱型各向异性弹性力学平面问题

本文对圆柱型各向异性弹性力学平面问题的基本方程进行了改写。在此基础上，导出了应力函数Ｇ和位移函数φ，它们满足相同的控制方程，比文〔１〕的应力函数Ｆ的控制方程要简单，便于求得特解，并有Ｆ＝ｒＧ的关系。还对若干经典问题进行了求解。     

Optimal Internal Stabilization of the Linear System of Elasticity

We address the non-linear optimal design problem which consists in finding the best position and shape of a feedback damping mechanism for the stabilization of the linear system of elasticity. Non-existence of classical designs are related to the over-damping phenomenon. Therefore, by means of Young measures, a relaxation of the original problem is proposed. Due to the vector character of the elasticity system, the relaxation is carried out through div-curl Young measures which let the analysis be direct and the dimension independent. Finally, the relaxed problem is solved numerically, and a penalization technique to recover quasi-optimal classical designs from the relaxed ones is implemented in several numerical experiments. A. Münch was partially supported by grants ANR-05-JC-0182-01 and ANR-07- JC-183284. P. Pedregal was supported by project MTM2004-07114 from Ministerio de Educación y Ciencia (Spain), and PAI05-029 from JCCM (Castilla-La Mancha). F. Periago was supported by projects MTM2004-07114 from Ministerio de Educación y Ciencia (Spain) and 00675/PI/04 from Fundación Séneca (Gobierno Regional de Murcia, Spain).     

Violation of the Complementing Condition and Local Bifurcation in Nonlinear Elasticity

The complementing condition (CC) is an algebraic compatibility requirement between the principal part of a linear elliptic differential operator and the principal part of the corresponding boundary operators. We study the implications of failure of the CC in the context of nonlinear elasticity. In particular we show that for axisymmetric deformations of cylinders and for any homogeneous isotropic material, failure of the CC is equivalent to the existence of sequences of possible bifurcation points accumulating at the point where the CC fails. For non axisymmetric deformations and for Hadamard–Green type materials, we show for axial compressions of the cylinder that the CC fails on a full interval of values of the loading parameter, and for the lateral compression problem it fails at least once.     

Development of the Boundary-Element Method for Three-Dimensional Problems of Static and Nonstationary Elasticity

The boundary-element method (BEM) applied to three-dimensional problems in the linear theory of elasticity is analyzed. The solutions of test problems for spherical and cubic cavities are used as examples to consider the basic aspects and difficulties of applying the traditional BEM to static and nonstationary three-dimensional problems. It is established that using Chebyshev polynomials in the Gaussian quadrature formula to evaluate the singular segments of surface integrals reduces the computation time by a factor of 2 to 3 without loss of accuracy compared with the traditional Gauss–Legendre formula. BEM-based approaches are proposed to solve three-dimensional problems in the linear theory of elasticity     

Inversion Symmetry of the Solutions of Boundary-Value Problems of Elasticity for a Half-Space

International Applied Mechanics - The inversion symmetry of the components of the displacement vector and stress tensor in the solution of the first boundary-value problem of elasticity for a...     

平面弹性力学充要的随机边界元

将平面弹性力学确定性的充分必要的边界积分方程推广到含材料常数随机的不确定问题中去，给出了位移的均值以及偏差的充分必要的边界积分方程。数值计算结果表明，和确定性的积分方程一样，习用的随机边界积分方程在退化尺度附近，无论是均值还是偏差都存在巨大的误差，而充要的随机边界积分方程则始终保持良好的精度     

Sensitivity Analysis of Problems of Elasticity with Unilateral Constraints*

ABSTRACT A unified approach is developed for sensitivity analyses of unilateral problems for discrete and continuous elastic structures. Sensitivity analyses are crucial in problems of redesign or optimization of structures. The topic covers, in this paper, the problem of determining the variation of structural response, subject to variations of ihe design, material, and loading. This type of problem is inherently nonlinear and nondifFerentiable for structures with unilateral constraints and, in general, only directional sensitivities can be obtained. Employing modern methods of functional analysis and the principle of minimum potential energy, a general result is presented for determination of sensitivities, and the implications for discrete or discretized structures are discussed in some detail. Four example problems are used to illustrate different aspects of the sensitivity analysis and to show how to handle variations of unilateral constraints.     

On the Boundary Value Problems of Linear Elasticity with Constraints

In this paper classic boundary value problems of linear elastostatics are studied. Displacement, mixed and traction type boundary conditions are considered for an internally constrained, non-homogeneous, anisotropic material. Existence of solutions and constraint stability results are presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

An Experimental Technique to Assess the Compactness of Slurry-Fuel Sediments

Experimental Techniques -     

Duality among Optimal Design Problems for Torsional Vibration

ABSTRACT

Four types of mass and frequency optimization problems are stated for free torsional vibration of thin-walled cylinders subject to constraints on wall thickness and frequencies of vibration. It is shown, using Pontryagin's method, that the mathematical structure of all four problems is similar and leads to identical classes of optimal thickness distributions. These duality relations are used in an example to construct an optimal frequency solution from the solutions for both maximum and minimum mass problems. General relations among the governing parameters for the four problems are stated. The results of Grinev and Filippov and of Thermann for the abnormal optimization problems are verfied as a specific limiting example of the general results.