INVARIANT PROPERTIES OF THE STRESS IN PLANE ELASTICITY AND EQUIVALENCE CLASSES OF COMPOSITES

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 J(x) is shifted uniformly by J(I)(lambda, -lambda), where lambda is an arbitrary constant and J(I)(kappa, mu) is the compliance tensor of a isotropic material with two-dimensional bulk and shear moduli kappa and mu. This invariance is explained from two simple observations: first, that in two dimensions the tensor J(I)(1/2, -1/2) acts to locally rotate the stress by 90-degrees 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 J* also gets shifted by J(I)(lambda, -lambda) when the constituent moduli are 'each shifted by J(I)(lambda, -lambda). This imposes constraints on the functional dependence of J* 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/lambda and the inverse shear modulus is shifted by -1/lambda, 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.