Comparison of Solutions of the Problems of Elasticity Theory for Different Near-Boundary Elements

We consider three types of near-boundary elements: curvilinear quadrangles and families of arcs and points, and their application to a two-dimensional problem of the static theory of elasticity with boundary conditions in displacements. The comparison of the theoretical and numerical aspects of the obtained solutions is performed. It is shown that the accuracy of evaluation of the components of the vector of displacements increases in the following cases: in the case of complete quantization of the near-boundary region (but not in the case of its partial quantization or quantization with overlapping), in the case where the numbers of arcs and points increase, and in the case of simultaneous application of different types of near-boundary elements (specifically, quadrangles and families of arcs).     

四阶奇异半正边值问题的正解

本文在一个特殊锥上利用不动点指数定理,得到了一类四阶奇异半正边值问题正解的存在性,并给出了一个例子作为对所获结果的应用,推广和改进了一些已知结果。     

三阶奇异边值问题的多解性

利用锥上的不动点定理和Green函数的性质,本文在较弱条件下研究了三阶奇异非线性边值问题的多个正解的存在性。     

四阶奇异边值问题的正解

利用了上下解方法和不动点定理，得到了一类四阶奇异边值问题正解的存在性。     

三阶奇异边值问题对称正解的最优存在性

本文利用Leggett-Williams不动点定理,通过构造特殊的锥,得到了三阶奇异边值问题三个对称正解最优存在性的新结果,最后,通过具体的例子说明了我们所得结果的有效性。     

奇异二阶边值问题的正解

研究了奇异二阶边值问题 u″+λp(t) f (t,u) =0 ,αu(0 ) -βu′(0 ) =0 ,γu(1 ) +δu′(1 ) =0 ,给出了正解的存在性 ,改进和推广了一些已知结果     

On the Nonuniqueness of Solutions to the Nonlinear Equations of Elasticity Theory

One-dimensional selfsimilar problems for waves in an elastic half-space generated by a sudden change of the boundary stress (the “piston” problem) and problems of disintegration of an arbitrary discontinuity are considered. For the case when small-amplitude waves are generated in a medium with small anisotropy a qualitative analysis shows that these problems have nonunique solutions when it is assumed that the solutions involve Riemann waves and evolutionary discontinuities. The above-mentioned problems are considered as limits of properly formulated problems for visco-elastic media when the viscosity tends to zero or (what is the same) that time tends to infinity. It is numerically found that all above-mentioned inviscid solutions can represent the asymptotics of visco-elastic solutions. The type of asymptotics depends on those details of the visco-elastic problem formulation which are absent when formulating inviscid selfsimilar problems. Similar considerations are made for elastic media with dispersion along with dissipation which are manifested in small-scale processes. In such media the number of available asymptotics (as t→∞) for the above-mentioned solutions depends on a relation between dispersion and dissipation and can be large. Thus, two possible causes for the nonuniqueness of solutions to the equations of elasticity theory are investigated.     

奇异二阶微分方程边值问题解的性质及其应用

我们在较一般的条件下,对奇异二阶非线性微分方程边值问题解的性质进行刻划,得到了其G^1[0,1]正解与G[0,1]正解分别满足的与相应格林函数的关系,最后给了推论及其应用的例子。     

奇异二阶Neumann边值条件下脉冲微分方程的正解

本文利用锥中不动点指数理论研究了Banach空间中一类二阶非线性奇异脉冲微分方程的Neumann边值问题,得到了正解存在的一个充分条件,将已有文献的一些结果推广到了Neumann边值条件和脉冲微分方程.     

二阶奇异脉冲微分方程周期边值问题的正解

二阶微分方程边值问题可描述两端支撑弹性梁的形变等一些问题,其广泛的应用引起很多学者的关注.本文研究二阶周期边值问题,其中非线性项在端点处具有奇异性(非线性项可有不同的超次线性).通过构造一个特殊的锥,利用不动点指数定理得到了该问题的正解.最后给出一个例子说明主要结果.     

奇异二阶Neumann边值问题的正解

利用不动点指数得到了奇异二阶Neumann边值问题的正解,推广和改进了现有的一些结果。     

弹性接触问题的最小二乘混合变分形式

采用最小二乘法对弹性接触问题提出了一种新的混合变分形式，对该变分问题解的存在唯一性进行了论证。     

Singular Integral Equations in Two-Dimensional Problems of the Theory of Cracks

We present a brief survey of the results of investigations devoted to the application of the method of singular integral equations to the solution of two-dimensional problems of fracture mechanics and carried out at the Karpenko Physicomechanical Institute of the Ukrainian National Academy of Sciences. Special attention is given to integral equations on piecewise smooth closed or nonclosed contours appearing in the boundary-value problems for domains with corners. We propose a new method for the solution of equations of this sort taking into account the singularities of stresses at angular points. The modified singular integral equations obtained as a result have continuous kernels and right-hand sides, i.e., belong to the same type of equations as in the case of smooth contours.     

抽象空间中二阶非线性奇异边值问题的正解

本文在一个特殊的Banach空间中研究了一类二阶奇异边值问题正解的存在性,其中的非线性项含有一阶导数,并且在自变量的端点处和该空间的零元处具有奇异性.通过构造一个特殊的非空凸闭集,利用Kuratowskii非紧性测度理论和Monch不动点定理,获得了该类问题正解的存在性.与已有文献相比,本文没有利用近似逼近方法.因此,即使退化到纯量空间,本文所用方法也完全不同于已有文献,并且得到了更为有效的结果.最后给出了两个例子说明其应用.     

New Method for the Solution of Dynamic Problems of the Theory of Elasticity and Fracture Mechanics

We propose a new approach to the solution of dynamic problems of the theory of elasticity and fracture mechanics based on the application of the finite-difference method only with respect to time. In this case, the equations of motion are split into homogeneous and inhomogeneous systems of differential equations for the determination of displacements at time nodes. For trivial initial conditions, only the homogeneous system of differential equations is preserved (of the same type as in the dynamic problem in Laplace transforms). Its efficient solution can be obtained by the methods of boundary integral equations or boundary elements.     

Mixed Projection-Mesh Scheme of the Finite-Element Method for the Solution of Problems of the Elasticity Theory

A mixed projection-mesh scheme for solving the boundary-value problems of the elasticity theory has been formulated. Correctness and convergence of the mixed approximations for strains and displacements are studied. Application of the numerical integration has been analyzed, and the obtained results are given. The theory of generalized functions and the functional analysis methods have been used for the convergence and accuracy estimates. Iteration algorithms for solving the discrete problems have been proposed.     

Mixed Approximation Scheme of the Finite-Element Method for the Solution of Two-Dimensional Problems of the Elasticity Theory

For the solution of two-dimensional boundary-value problems of the elasticity theory, a triangular finite element, ensuring stability and convergence of mixed approximation, is proposed. The system of resolving equations of the mixed method is derived with account of strict satisfaction of static boundary conditions at the surface. To solve matrix equations of the mixed method, various algorithms of the conjugate-gradient method with the pre-conditional matrix have been considered. Numerical data on convergence and accuracy of the solution for a number of test problems of the elasticity theory and fracture mechanics are given. The results obtained by the conventional and mixed finite-element method approaches are compared.     

Contact Analysis for Solids Based on Linearly Conforming Radial Point Interpolation Method

To simulate the contact nonlinearity in 2D solid problems, a contact analysis approach is formulated using incremental form of the subdomain parametric variational principle (SPVP). The formulation is based on a linearly conforming radial point interpolation method (LC-RPIM) using nodal integration technique. Contact interface equations are also presented using a modified Coulomb frictional contact model and discretized by contact point-pairs. In the present approach, the global discretized system equations are transformed into a standard linear complementarity problem (LCP) that can be solved readily using the Lemke method. The present approach can simulate various contact behaviors including bonding/debonding, contacting/departing, and sticking/slipping. An intensive numerical study is performed to validate the proposed method via comparison with the ABAQUS^® and to investigate the effects of the various parameters used in computations. These parameters include normal and tangential adhesions, frictional coefficient, nodal density, the dimension of local nodal support domain, nodal irregularity, shape parameters used in the radial basis function and the external load. The numerical results have demonstrated that the present approach is accurate and stable for contact analysis of 2D solids.