Determination of a general solution of three-dimensional Lamé equations of elasticity theory

We integrate the Lamé equation and find new solutions in the case of three-dimensional elasticity theory, which are expressed in terms of harmonic functions. We prove that the solution obtained involves only three independent functions. In a curvilinear orthogonal coordinate system, a general solution of the Lamé equation is expressed in terms of three harmonic functions. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 109–116, January–March, 2006.     

A general solution to some plane problems of micropolar elasticity

We obtain a general solution to the field equations of plane micropolar elasticity for materials characterized by a hexagonal or equilateral triangular structure. These materials exhibit 3-fold symmetry in the plane and the elastic response is isotropic. Utilizing two displacement potential functions, the solution is obtained in terms of two analytic functions and a third function satisfying the modified homogeneous Helmholtz equation. Expressions for the two-dimensional components of displacement, stress, and couple stress, along with the resultant force on a contour, are presented. We observe that micropolar effects are most significant in material regions subjected to large deformation gradients. Specific results are presented for the classical crack problem, the half plane loaded uniformly on the surface, Flamant's problem, and the circular cylinder compressed by equal and opposite concentrated forces.     

Representing general solution of equations in theory of elasticity by harmonic functions

The general solution of the equations in the theory of elasticity is represented by seven harmonic functions, where there are only three harmonic functions independent of each other and every one of them has certain mechanics meaning. The examples applying the general solution to solve several simple inverse problems in elastostatics are presented.     

Canonical moduli and general solution of equations of a two-dimensional static problem of anisotropic elasticity

Equations of a two-dimensional static problem of anisotropic elasticity are brought to a simple form with the use of orthogonal and affine transformations of coordinates and corresponding transformations of mechanical quantities. It is proved that an arbitrary matrix of elasticity moduli containing six independent components can be always converted by a congruent transformation to a matrix with two independent components, which are called the canonical moduli. Depending on the relations between the canonical moduli, the determinant of the matrix of operators of equations in displacements is presented as a product of various quadratic terms. A general presentation of the solution of equations in displacements in the form of a linear combination of the first derivatives of two quasi-harmonic functions satisfying two independent equations is given. A symmetry operator (i.e., a formula of production of new solutions) is found to correspond to each presentation. In a three-dimensional case, the matrix of elasticity moduli with 21 independent components is congruent to a matrix with 12 independent canonical moduli.     

Thermoelastic problems in the half space—An application of the general solution in elasticity

In this paper, some thermoelastic problems in the half space are studied by using the general solutions of the elastic equations. The method presented here is extremely effective for the axisymmetric problems of the half space as well as the half plane problems.     

An elasticity solution of a nonhomogeneous half-plane problem

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Application of subregion function method in the method of weighted residuals

In this paper, a new concept of subregion function method is suggested. According to the boundary shape and the stiffness and loading conditions of the structure, the original zone of the structure is divided into some subregions, in each of which different trial functions may be adopted. Conditions of compatibilities between subregions are considered. Finally residual equations consisted of interior residuals, boundary residuals and coboundary residuals between subregions are given.A numerical example to illustrate the theory of this method is given.     

On a molecular statistical basis for Ogden's model of rubber elasticity

In this paper, a link is established between the statistical theory of long chain molecules and Ogden's phenomenological model of rubber elasticity. It has been shown by several authors in the past that many invariant-based phenomenological models for rubber-like materials are related to the classical statistical theories. The essential means to reach this reconciliation were methods to account for a non-affine deformation of polymer chains in the network, appropriate techniques to calculate their averaged response, and an approximation of the inverse Langevin function appearing in the non-Gaussian statistical theory. It is shown in this paper that the very same approach, if appropriately implemented, allows to express the strain-energy function of Ogden's material in terms of physical constants characterising the polymer chain and network, together with few additional parameters that account for the non-affine deformation of the polymer chains. Particularly, it is shown that Ogden's model can be represented as a non-affine non-Gaussian 3-chain model with topological constraints.     

A combined experimental and numerical method for the solution of generalized elasticity problems

This work presents the method for the investigation of three-dimensionally stressed bodies with arbitrary shape which are under the action of an outside system of arbitrary forces. The combined method is based on syntheses of photoelastic experimental methods (other experimental methods may also be used) and digital methods of discrete analysis. Experimental procedures are used for defining superfluous boundary conditions. The boundary-value problem with such boundary conditions is solved by numerical methods. This approach qualitatively changes the very essence of experimental methods and essentially widens their range. It reduces the amount of measurements required and, at the same time, allows one to obtain complete stress fields throughout a body in a short time. In comparison with numerical methods, the combined method increases the accuracy of problem solutions and, at the same time, reduces the time required for complete investigations.     

On the solution of plane boundary-value problems of elasticity in a rectangle

An algorithm for solving plane boundary-value problems of elasticity for a rectangular domain is expounded. The algorithm is based on a complex-valued representation of the general solution to the differential equations of the plane problem and on the use of Lagrange polynomials to satisfy the boundary conditions. The algorithm can quite easily be implemented in a computer program. This is probably the simplest way of solving boundary-value problems of this class __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 97–102, January 2006.     

A general method for the incremental solution of elastic-plastic problems

Summary A general iterative method for the incremental solution of elastic-plastic problems is described. The convergence of the procedure on the rigorous solution is demonstrated.

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Sommario Si espone un procedimento iterativo generale per la soluzione incrementale dei problemi elasto-plastici. Si dimostra la convergenza del procedimento verso la soluzione effettiva.

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The results presented in this paper were obtained in the course of research sponsored by the C.N.R., Gruppo Plasticità     

The built-in rectangular plate (general method of solution)

Summary The method of solution is based on the following considerations:In order to clamp a supported plate, it is necessary to act on its edge with bending couples that will produce, along the edge, rotations equal but contrary to those produced by the action of external loads which are taken as known.If now at one edge of a supported rectangular plate a distribution of bending couples is applied, along this edge a rotation is produced as a function of the point which we call a « direct rotation ». At the same time there appear on the other edges of the plate « indirect rotations » which are connected with the « direct » rotation by functional relations of integral nature.When the couples act along the whole perimeter we obtain for each edge a comprehensive rotation equal to the sum of the direct rotation produced by the couples on the edge in question and of the indirect ones produced by the couples applied to the other edges.From what has been said, this sum is a function of the four direct rotations, and if we wish to clamp the edge, it must take on a known value.The same considerations apply to the remaining edges, so that if the direct rotations are presumed to be principal unknowns -and here lies the main feature of the procedure — we obtain a system of four integral equations of the second kind with fixed limits.Assuming a means of separating the unknowns based on the consideration of the symmetrical and antisymmetrical components of the quantities in question, we come to tackle the real solution of the integral equations by means of the iteration method.From this the recurrent formulas are deduced and the convergence is analysed.When the direct rotations are known, we come to know the additional deformations which clamp the supported plate.At the end, two examples illustrate the procedure.First published in Italian in Giornale del Genio Civile, Nos. 5, 6, 1964.     

On uniqueness in finite elasticity with general loading

In many problems of interest the (Cauchy) surface traction is given as a function of position on the deformed surface. A class of loadings sufficiently general to include these problems is considered and within the context of finite elasticity a number of uniqueness results are established. A key ingredient is the result of Gurtin and Spector that uniqueness holds in any convex, stable set of deformations.

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Résumé Dans plusieurs problémes interessants la traction surfacique de Cauchy est donnée comme une fonction de la position dans la surface deformée. Une telle classe des charges, suffisamment générale, est considerée et un nombre des résultats d'unicité et établit dans le cadre d'élasticité non linéaire. Le résultat de Gurtin et Spector (unicité est valable sur un ensemble stable et convexe queleonque des déformations) est utilisé.