A method of solving problems of the linear theory of elasticity

Abstract

A method is developed for solving linear boundary value problems, based on their interpretation in the spirit of functional analysis. In the special case of the theory of elasticity, the stress and strain fields areconsidered as elements of a real Hubert space of symmetric tensors of the second rank. On the basis of the second derivative of Green's tensor of the equilibrium equations, projection operators P ̄ and Q ̄ are constructed that satisfy the equation P ̄ + Q ̄ = I . The solution of the mixed boundary value problem is represented in the form of Neumann series, whose sufficient conditions for convergence are written in the form of operator inequalities which lend themselves to a simple interpretation in the language of energy functionals. By strengthening these conditions we can express them in terms of the closeness of the coefficients of the problem λ and λ c . A representation of the potential energy is given in the form of a certain functional which can always be expanded in series. The limits within which the exact value of the potential energy lies is obtained. The purpose of this paper is to develop a method for solving linear boundary value problem based on the formalism of Green's tensors on the one hand, and on the interpretation of these problems in the spirit of functional analysis, on the other. Consideration of the stress field α and strain field e as elements of a real Hilbert space H of symmetric tensors of the second rank and the introduction of the operators P and Q, constructed on the basis of Green's tensor G and acting in the space H, enable a transfer to be made from the equilibrium equation to a functional equation of the form (1.13). The iteration method often utilized to solve such equations results in a solution in the form of the Neumann series (1.15) whose convergence conditions are not always evident. It is assumed that the elastic properties of the medium under investigation are described by a symmetric fourth-rank tensor λ = λ (r). (Here and henceforth, the tensor subscripts are omitted, for simplicity, almost everywhere, and the vector quantities are denoted by heavy type. In the product A k B l of the tensor A k of rank k and the tensor B l of rank l the summation is over all subscripts of the tensor B l if l< k and over the n inner subscripts of the tensors A k and B l if k = l = 2 n). A medium for which the solution of the initial problem is known is used as the auxiliary medium (the comparison medium). Its elastic properties are described by the tensor λ c. Without limiting the generality, we consider λ and λ c symmetric operators (see Sect.2). This enables the method to be extended to viscoelastic media and a medium with a microstructure. Important relations are obtained in Sect.2 for the operators P and Q and their associated P ̄ , Q ̄ . It is shown that these belong to the class of projections. This circumstance exerts a substantial influence on the form of the convergence conditions for series (2.14). By rounding off the sufficient conditions for (3.4) and (3.5) to converge, we obtain conditions (3.7) and (3.8) (or (3.11)), whichwhen λ c μ c ≠ I turn out to be independent and can be satisfied simultaneously. A representation of the potential energy U in the form of functionals computed using the auxiliary fields σ c and ε c referred to the comparison medium, is given in Sect.4. It is shown in Sect.5 that the energy u' is representable in the form of the series (5.1) or (5.2) whose sign-definiteness depends on the properties of the functionals l k , or m k , respectively. In any case the limits within which the exact value of u' lies can be computed.