弹性力学的混合方程和Hamilton正则方程

本文指出,在弹性力学基本方程中,按变量分类的位移方程,应力方程以外的第三种混合方程,以及按运算子分类的微分方程、变分原理以外的第三种Hamilton方程,它们正好是对应的。本文讨论了静力的和动力的情况以及它们可能的应用。     

On the general expression of Fredholm Integral Equations Method in elasticity

In this paper, the concept of covering domain is introduced to develop a general expression for the Fredholm Integral Equations Method, by which elasticity problems of arbitrarily shaped bodies loaded by external forces can be solved. Some special expressions are given for a body with non-zero remote stresses, or subjected to some concentrated forces on its boundary. The relationship between the loading forces and solutions are also discussed. Some analytical solutions can be obtained for simple cases. When numerical computations are needed for the solution, the method proves to have high precision and fast convergency.     

弹性理论几类导数边界积分方程之间的变换关系

导数场边界积分方程通常难以应用，因为存在着超奇异主值积分的计算障碍。弹性理论中有几类不同的位移导数边界积分方程，本文采用算子δij和∈ij(排列张量)作用于这些导数边界积分方程，做一系列变换，原有的超奇异积分被正则化为强奇异积分获解。从而建立了这些位移导数边界积分方程之间的转换关系，它们均可以归结为自然边界积分方程。自然边界积分方程仅存在容易计算的Cauchy主值积分。自然边界积分方程分析可直接获得边界应力和位移导数。     

Fredholm integral equation for the multiple circular arc crack problem in plane elasticity

Summary An elementary solution for the multiple circular arc problem is obtained in this paper. The elementary solution is defined as a particular case of the single circular arc crack problem, in which remote stresses are equal to zero, and two pairs of concentrated forces are applied at a prescribed point of crack face. By using the principle of superposition, Fredholm integral equation for the multiple circular arc problem in plane elasticity is obtained. The suggested approach is illustrated by several numerical examples. If a smaller arc crack is surrounded by a larger arc crack, the stress intensity factors for the former become rather small. The phenomenon of shielding is illustrated by examples. Accepted for publication 17 September 1996     

Differentiation of Energy Functionals in Two-Dimensional Elasticity Theory for Solids with Curvilinear Cracks

This paper considers the equations of two-dimensional elasticity theory in nonsmooth domains. The domains contain curvilinear cracks of variable length. On the crack faces, conditions are specified in the form of inequalities describing mutual nonpenetration of the crack faces. It is proved that the solutions of equilibrium problems with a perturbed crack converge to the solution of the equilibrium problem with an unperturbed crack in the corresponding space. The derivative of the energy functional with respect to the length of a curvilinear crack is obtained.     

New Transforms for the Resolving Equations in Elastic Theory and New Integral Transforms, with Applications to Boundary-Value Problems of Mechanics

This paper reviews results from an analysis of exact boundary-value solutions of static and dynamic elasticity obtained by the method of integral transforms. Consideration is given to the solutions of problems for a half-space and a blunted hollow cone, the heat conduction problem for a cone, and the diffraction problem for a cone with a hole along the generatrix. Solutions of mechanics problems are analyzed and the method of integral transforms is generalized     

On the Interior Transmission Problem in Nondissipative, Inhomogeneous, Anisotropic Elasticity

In this paper, the interior transmission problem for the non absorbing, anisotropic and inhomogeneous elasticity is investigated. The direct scattering problem for the penetrable inhomogeneous, anisotropic and nondissipative scatterer is first studied and the existence and uniqueness of its solution are established. In the sequel, the interior transmission problem in its classical and weak form is presented and suitable variational formulations of it are settled. Finally, it is proved that the interior transmission eigenvalues constitute a discrete set. This revised version was published online in June 2006 with corrections to the Cover Date.     

横观各向同性圆柱体端部问题的三维弹性理论解

从位移的通解出发，用分离变量法得到横观各向同性圆柱体的位移和应力的特征函数展开式，并把位移势函数的解用付里叶积分的形式表示。利用留数运算，该积分解可以转换成类似于特征函数的展开式。通过混合端部边界问题，得到与特征函数解成双正交关系的另一组函数。利用这种双正交关系，可以处理不同的端部边界问题。     

The Fredholm Alternative for Functional Differential Equations of Mixed Type

We prove a Fredholm alternative theorem for a class of asymptotically hyperbolic linear differential difference equations of mixed type. We also establish the cocycle property and the spectral flow property for such equations, providing an effective means of calculating the Fredholm index. Such systems can arise from equations which describe traveling waves in a spatial lattice.     

弹性体表面分层压缩失稳问题的数学弹性力学解

采用数学弹性力学的稳定平衡方程并结合富氏积分变换的方法研究了含表面平行裂纹的弹性体在压缩载荷下的表面分层失稳问题。导出了一级显式的精确齐次奇异积分方程组,然后.通过Gauss-Chebyshev积分公式,得到一组齐次代数方程组,从而求出临界压缩载荷。并将结果与经典的材料力学梁板稳定的研究方法所得结果进行了比较,指出经典方法误差太大而不适于求解此问题。最后,利用数学弹性力学解求出的等效弹性支承常数给出一个简单精确的临界压缩载荷计算公式。     

Invariance of the Equilibrium Equations of Finite Elasticity for Compressible Materials

For homogeneous, isotropic, compressible nonlinearly elastic materials, a wide class of strain-energy density functions are obtained that leave the equations of equilibrium invariant under simple scaling transformations of the material and spatial coordinates. These strain-energy densities are homogeneous functions of the principal stretches. Several illustrative examples of particular strain-energies are provided. For axisymmetric problems, the invariance discussed here ensures that the equations of equilibrium can be solved by quadratures and thus often leads to analytic solutions in parametric or closed-form. Mathematics Subject Classifications (2000) 74B20, 74G55.     

Uniqueness of Equilibrium Solutions in Second-Order Gradient Nonlinear Elasticity

We consider a three-dimensional elastic body whose material response function depends not only on the gradient of the deformation, but also on its second gradient. Using the elastic energy-momentum tensor as derived by Eshelby [2] we generalize a well-known uniqueness result of Knops and Stuart [8] for a Dirichlet boundary value problem associated with this response function.     

Piezoelectric Analogy of Generalized Torsion in Anisotropic Elasticity

We consider a piezoelectric body bounded by a cylindrical surface in which all cross sections are of the same geometry. Suppose on the lateral surface the body is loaded in such a way that the stress and electric displacement do not vary along the axial direction. In addition the end of the cylinder is also subjected to forces reducing to bending moments, twisting moment, axial force and electric charge. We follow Lekhnitskii's formalism to characterize the deformation of the considered problem. It is found that, when the solid possesses a material symmetry plane normal to the axial direction, the simplified field equations together with suitably chosen boundary conditions are entirely analogous to those of a generalized torsion problem in anisotropic elasticity. In particular, we show that by setting a linkage between two sets of 21 material constants, any problem in one field can be resolved as another problem in the other area. This revised version was published online in August 2006 with corrections to the Cover Date.     

Generalization of the Stroh Formalism to 3-Dimensional Anisotropic Elasticity

A method of analysis for three-dimensional anisotropic elasticity is developed, based on Stroh's two-dimensional formalism. By use of the present method a three-dimensional solution can be constructed by the corresponding two-dimensional solution. The method is applied to solve several problems concerning a point force or a dislocation loop. Existing results for these problems are recovered in a unified manner and additional new results are obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Method for Solving Boundary-Value Problems of Linear Viscoelasticity for Anisotropic Composites

A new method is proposed to solve some problems of linear viscoelasticity for anisotropic bodies. The method uses a branching continued fraction to approximate an irrational multivariable function. Such an approach allows obtaining a linear operator as an approximation of a multivariable operator function. The deformation of a cracked composite body with a plastic matrix is analyzed as an example. Both composite components are assumed to exhibit viscoelastic properties     

基于三维弹性理论的约束阻尼结构振动阻尼特性分析

本文根据三维弹性理论建立了多层弹性－粘弹性阻尼复合板自由振动的运动方程．该方程所根据的力学模型包括了影响阻尼复合板振动的几乎全部变形因素，包括弹性层和粘弹性层的平面剪切、横向剪切、纵向拉伸以及粘弹性层在厚度方向的胀缩变形。用该方程对工程声学问题所关注的中高频域结构损耗因子进行了数值计算。此外，对“附加型”和“成品型”两种结构型式约束阻尼板损耗因子的差异，尤其是阻尼层的厚度方向胀缩变形耗能对结构损耗因子的影响，进行了计算讨论。     

基于非局部弹性理论的纳米板横向振动

本文利用非局部弹性理论研究了单层石墨烯的纳米板的横向自由振动响应．通过迭代法推导了非局部应力表达,进一步通过哈密顿原理推导了纳米板的控制方程,应用纳维解法得到四边简支纳米板振动固有频率的数值解,并将本文研究结果与已有文献结果进行对比,进一步讨论了小尺寸效应,以及纳米板的三维尺寸和半波数对振动频率的影响．结果表明：非局部效应的存在使得纳米板的等效刚度和固有频率降低;半波数的增加则使得纳米板的固有频率提高．相关分析结果对基于二维纳米材料的新设备的设计和优化具有重要意义．     

Corner Singularities and Regularity of Weak Solutions for the Two-Dimensional Lamé Equations on Domains with Angular Corners

This paper is concerned with corner singularities of weak solutions of boundary value problems in the theory of plane linearized elasticity. The presence of angular corner points or points at which the type of boundary conditions changes yields generally local singularities in the solution. This singular behavior in the vicinity of such points can be described with the help of asymptotic singular representations for the solution, which essentially depend on the zeros of certain transcendental functions. These transcendental functions will be derived and analyzed for all ten possible combinations of boundary conditions, generated by the four basic ones, prescribing in the tangential and normal direction of the boundary, respectively, either the displacement or the tractions. The regularity of the corresponding weak solutions will be investigated. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the regularization method of the first kind of Fredholm integral equation with a complex kernel and its application

The regularized integrodifferential equation for the first kind of Fredholm integral equation with a complex kernel is derived by generalizing the Tikhonov regularization method and the convergence of approximate regularized solutions is discussed. As an application of the method, an inverse problem in the two-demensional wave-making problem of a flat plate is solved numerically, and a practical approach of choosing optimal regularization parameter is given. Project supported by the National Natural Science Foundation, of China     

A Variational Approach to the Fredholm Alternative for the p-Laplacian near the First Eigenvalue

We are concerned with the existence of a weak solution to the degenerate quasi-linear Dirichlet boundary value problem