Analogues of the Kolosov–Muskhelishvili General Representation Formulas and Cauchy–Riemann Conditions in the Theory of Elastic Mixtures

Abstract

Analogues of the well-known Kolosov–Muskhelishvili formulas of general representations are obtained for nonhomogeneous equations of statics in the case of the theory of elastic mixtures. It is shown that in this theory the displacement and stress vector components, as well as the stress tensor components, are represented through four arbitrary analytic functions. The usual Cauchy–Riemann conditions are generalized for homogeneous equations of statics in the theory of elastic mixtures.

Time Impact on Residue in a Non-homogeneous Equation of Statics in the Theory of Elastic Mixture

“…Replacing the two right hand terms of equation ( 10) with the right hand terms of equation ( 11) and (12); to have,…”

“…To make equation ( 1) solvable, we shall as in [ 12]. Let…”

Explicit Solutions of the Basic Boundary Value Problems of Statics of the Elastic Mixture Theory for an Annulus

“…The basic homogeneous equations of statics of the elastic mixture theory are written in terms of displacement components as follows [1]:…”

“…Then we obtain a solution of the first boundary value problem in the form of a series. For these series together with their first derivatives to be absolutely and uniformly convergent it is sufficient that the functions f (ϕ) and F (ϕ) satisfy the Hölder condition with an exponent α > 1 2 . Solutions obtained under such conditions are regular in an annulus.…”

“…Substituting the defined coefficients into formula (3.1), we obtain the stress vector in the form of series for the annulus R 1 < ρ < R 2 . Series (3.2) is absolutely and uniformly convergent if the given vectors f and F satisfy the Hölder condition with an exponent α > 1 2 . Thus we have proved the following Theorem 2.…”