Invariant properties of the stress in plane elasticity and equivalence classes of composites

Abstract

Attention is drawn to the invariance of the stress field in a two-dimensional body loaded at the boundary by fixed forces when the compliance tensor S(x) is shifted uniformly by S 1 (λ, - λ), where λ is an arbitrary constant and S 1 ( k,u )is the compliance tensor of a isotropic material with two-dimensional bulk and shear moduli k and μ . This invariance is explained from two simple observations: first, That in two dimensions the tensor S (1/2, -1/2) acts to locally rotate the stress by 90° and the second that this rotated field is the symmetrized gradient of a vector field and therefore can be treated as a strain. For composite materials the invariance of the stress field implies that the effective compliance tensor S * also gets shifted by S 1 l( (λ, - λ) when the constituent moduli are each shifted by S (λ, - λ). This imposes constraints on the functional dependence of S * on the material moduli of the components. Applied to an isotropic composite of two isotropic components it implies that when the inverse bulk modulus is shifted by the constant 1/λ and the inverse shear modulus is shifted by — 1/λ, then the inverse effective bulk and shear moduli undergo precisely the same shifts. In particular it explains why the effective Young’s modulus of a two-dimensional media with holes does not depend on the Poisson’s ratio of the matrix material.