复变量移动最小二乘法及其应用

提出了复变量移动最小二乘法，并详细讨论了基于正交基函数的复变量移动最小二乘 法. 然后，将复变量移动最小二乘法和弹性力学的边界无单元法结合，提出了弹性力学的复 变量边界无单元法，推导了相应的公式，并给出了数值算例. 基于正交基函数的复变量移动 最小二乘法的优点是不形成病态方程组、精度高，所形成的无网格方法计算量小. 复变量边 界无单元法是边界积分方程的无网格方法的直接列式法，容易引入边界条件，且具有更高的 精度.     

Structure of the constitutive relations for hereditarily elastic materials reinforced by hard fibers

The problem of simplifying the nonlinear hereditary elasticity relations is considered for strongly anisotropic materials such as fiber-reinforced composites. This is done using their property that the material stiffness is high along the reinforcement and is low in the cross-sectional direction. The material is considered transversally isotropic. The simplification is performed by analyzing asymptotic representations of creep relations. Relations of various degrees of accuracy for various types of composites and stress states are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 120–127, May–June, 2005.     

Stress Concentration at Interacting Spherical Inclusions in a Transversely Isotropic Body

An elastic-equilibrium problem is rigorously solved for a transversely isotropic body containing a finite number of arbitrarily arranged and oriented transversely isotropic spherical inclusions or an infinite periodic array of such inclusions. The solution is found based on the superposition principle for general solutions of single-particle problems and the addition theorems for partial vector solutions of the elastic equilibrium equations. The numerical results presented allow assessing the influence of matrix anisotropy on the stress concentration between inclusions     

On axisymmetric motion of an incompressible elastic medium under impact loading

The method of matched asymptotic expansions was employed to obtain approximate solutions to the one-dimensional boundary-value problems of nonlinear dynamic elasticity theory of impact loading on the surface of a cylindrical cavity of an incompressible medium that causes antiplane motion or torsion of the medium. The expansion of the solution in the near-front region is based on solutions of evolution equations different from the equations for quasi-simple waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 144–151, November–December, 2006.     

Spatial Estimates for the Constrained Anisotropic Elastic Cylinder

This paper establishes spatial estimates in a prismatic (semi-infinite) cylinder occupied by an anisotropic homogeneous linear elastic material, whose elasticity tensor is strongly elliptic. The cylinder is maintained in equilibrium under zero body force, zero displacement on the lateral boundary and pointwise specified displacement over the base. The other plane end is subject to zero displacement (when the cylinder is finite, say). The limiting case of a semi-infinite cylinder is also considered and zero displacement on the remote end (at large distance) is not assumed in this case. A first approach is developed by considering two mean-square cross-sectional measures of the displacement vector whose spatial evolution with respect to the axial variable is studied by means of a technique based on a second-order differential inequality. Conditions on the elastic constants are derived that show the cross-sectional measures exhibit alternative behaviour and in particular for the semi-infinite cylinder that there is either at least exponential growth or at most exponential decay. A second approach considers cross-sectional integrals involving the displacement and its gradient and furnishes information upon the spatial evolution, without restricting the range of strongly elliptic elastic constants. Such models are principally based upon a first-order differential inequality as well as on one of second order. The general results are explicitly presented for transversely isotropic materials and graphically illustrated for a cortical bone.     

The relation between the Valanis-Landel and classical strain-energy functions

Necessary and sufficient conditions are derived for the strain-energy function of an isotropic elastic solid, regarded as a function of the strain invariants, to be expressible in the Valanis-Landel form, both when the material is compressible and when it is incompressible. In the case when the Valanis-Landel strain-energy function is a polynomial in squares of the principal extension ratios, explicit representations as polynomials in the basic isotropic strain invariants are obtained.     

A nonlocal theory for brittle fracture

In this paper, a nonlocal theory of fracture for brittle materials has been systematically developed, which is composed of the nonlocal elastic stress fields of Griffith cracks of mode-I, II and III, the asymptotic forms of the stress fields at the neighborhood of the crack tips, and the maximum tensile stress criterion for brittle fracture. As an application of the theory, the fracture criteria of cracks of mode-I, II, III and mixed mode I–II, I–III are given in detail and compared with some experimental data and the theoretical results of minimum strain energy density factor.     

Exact solutions of some general nonlinear wave equations in elasticity

A similarity analysis of a nonlinear wave equation in elasticity is studied; in particular, one with anharmonic corrections. The symmetry transformation give rise to exact solutions via the method of invariants. In some cases, graphical figure of the solutions are presented. Furthermore, we consider some cases wherein the velocities of the longitudinal and transversal plane waves are variable. Finally, a brief discussion on how a symmetry analysis on a perturbation of the elasticity equation can be pursued.     

Averaging of a Layered Elastic Medium with Low Dynamic Dissipation at the Interlayer Boundary

Averaged relations for a layered medium with dynamic dissipation at the interlayer boundary are constructed for dynamic problems of longitudinal shear and twodimensional theory of elasticity. For low dissipation, the averaged boundaryvalue problem is shown to be singularly perturbed. The degeneration of the boundaryvalue problem is studied qualitatively.