Numerical study of two-dimensional problems of nonsymmetric elasticity

We consider the algorithm of the finite element method for solving two-dimensional problems of nonsymmetric elasticity. We discuss the possibilities of the algorithm and its efficiency by comparing the numerical results with the well-known analytic solutions. We present the results obtained by solving the problem of tension of a plate weakened by a series of holes and the problem of tension for a plate with a central crack. The numerical results thus obtained are considered as an addition to the analytic solutions in the context of experimental justification of couple-stress effects arising under deformation of elastic materials and in the context of solving the identification problem for mechanical constants in nonsymmetric elasticity.     

Numerical solutions of two-dimensional anisotropic crack problems

A complete set of series form solutions of stress and displacement functions, including all higher order terms, around the crack tip for anisotropic crack problems have been newly derived by eigenfunction expansion approach. The analytical solutions of displacement functions were classified into four cases with respect to different types of complex parameters and different corresponding physical meanings. By employing these displacement functions as global interpolation functions, fractal two-level finite element method (F2LFEM) was applied to evaluate the stress intensity factors (SIFs) for various kinds of anisotropic crack problems. In the method of F2LFEM, the infinite number of nodal displacements was transformed to a small set of generalized coordinates by fractal transformation technique. New element matrices need not be generated and the singular numerical integration was avoided completely. Numerical examples of the four cases were studied and high accurate results of SIFs were obtained.     

Singular loadings in elasticity problems and singular solutions of the corresponding integral equations

The method of singular integral equations is an efficient method for the formulation and numerical solution of plane and antiplane, static and dynamic, isotropic and anisotropic elasticity problems. Here we consider three cases of singular loadings of the elastic medium: by a force, by a moment and by a loading distribution with a simple pole. These loadings cause corresponding singularities in the right-hand side function and in the unknown function of the integral equation. A method for the numerical solution of the singular integral equation under the above singular loadings is proposed and the validity of this equation at the singular points is investigated.     

Stress tensor symmetry and singular solutions in the theory of elasticity

In the plane problem of the theory of elasticity about a cantilever strip bending, we study the stress state near its fixed end. It is found that the solution singularity at the corner points does not have any physical nature and is generated by specific characteristics of the statement of the problem in which it is assumed that the stress tensor symmetry is violated at these points.     

Matrix inversions and singular solutions in the linear theories of elasticity

Elementary matrix multiplications and inversions lead to Galerkin type representations for the steady-state vibration equations of linear classical elasticity, thermoclasticity, Mindlin's couple stress and Eringen's micropolar theories of elasticity. The use of representations and the equations satisfied by potentials is further illustrated by obtaining singular solutions. Some of the results agree with known solutions.     

Function theoretic solutions to problems of orthotropic elasticity

The plane strain problem for a two dimensional orthotropic elastic body is investigated. In particular analytic representations for the solution of the displacement boundary value problem and the stress boundary value problems are found. To this end, the Navier equations are reduced by means of composite transformations to normal form. These are the so-called equations for bianalytic function of the type (k). The generalized Cauchy integral formula for this function theory is used to obtain representation formulae. A simplified method to solve these problems by bianalytic function theory is given for certain situations of plane strain for an orthotropic elastic body. AMS (MOS): 35A20, 35CO5, 35G15, 35J55.Applied Mathematics Institute Technical Report No. 140A, July 1983.The work of this author was supported in part by grant no. DE-AC01-81ER-10967 from the Department of Energy.     

Solution of statistical problems in elasticity theory in the singular approximation

A method of calculating elastic fields and effective moduli of microheterogeneous solids is developed in the random field theory. The solution is obtained in the form of an operator series, each term of which is constructed on the basis of the regular component of the second derivative tensor of the equilibrium Green function. The zeroth approximation of such a series consists of the local part of the interaction between inhomogeneity grains. The possibilities of the method are illustrated on the example of an isotropic mixture of two isotropic components.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 98–102, January–February, 1972.     

Numerical simulation of discontinuous solutions in nonlinear elasticity theory

A mathematical model of a nonlinear elastic medium is considered. Discontinuous solutions of the model are studied numerically for the one-dimensional case.     

Reducing three-dimensional elasticity problems to two-dimensional problems by approximating stresses and displacements by legendre polynomials

Shell equations are constructed in orthogonal curvilinear coordinates using approximations of stresses and displacements by Legendre polynomials. The order of the constructed system of differential equations is independent of whether stresses and displacements or their combination are specified on the shell surfaces, which provides the correct formulation of the surface conditions in terms of both displacements and stresses. This allows the system of differential equations of laminated shells to be constructed using matching conditions for displacements and stresses on the contact surfaces. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 179–190, May–June, 2007.     

The more general displacement solutions for the plane elasticity problems

In this paper,the author obtains the more general displacement solutions for theisotropic plane elasticity problems.The general solution obtained in ref.[1 ]is merelythe particular case of this paper,In comparison with ref.[1],the general solutions ofthis paper contain more arbitrary constants.Thus they may satisfy more boundaryconditions.     

Analytical solutions to general anti-plane shear problems in finite elasticity

This paper presents a pure complementary energy variational method for solving a general anti-plane shear problem in finite elasticity. Based on the canonical duality–triality theory developed by the author, the nonlinear/nonconvex partial differential equations for the large deformation problem are converted into an algebraic equation in dual space, which can, in principle, be solved to obtain a complete set of stress solutions. Therefore, a general analytical solution form of the deformation is obtained subjected to a compatibility condition. Applications are illustrated by examples with both convex and nonconvex stored strain energies governed by quadratic-exponential and power-law material models, respectively. Results show that the nonconvex variational problem could have multiple solutions at each material point, the complementary gap function and the triality theory can be used to identify both global and local extremal solutions, while the popular convexity conditions (including rank-one condition) provide mainly local minimal criteria and the Legendre–Hadamard condition (i.e., the so-called strong ellipticity condition) does not guarantee uniqueness of solutions. This paper demonstrates again that the pure complementary energy principle and the triality theory play important roles in finite deformation theory and nonconvex analysis.