## 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 $\scr{G}$($\chi $) is shifted uniformly by $\ell^{\text{I}}$($\lambda $, -$\lambda $), where $\lambda $ is an arbitrary constant and $\scr{G}^{\text{I}}$($\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 $\scr{G}^{\text{I}}$($\frac{1}{2}$, -$\frac{1}{2}$) 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 $\ell^{\ast}$ also gets shifted by $\scr{G}^{\text{I}}$($\lambda $, -$\lambda $) when the constituent moduli are each shifted by $\ell^{\text{I}}$($\lambda $, -$\lambda $). This imposes constraints on the functional dependence of $\ell^{\ast}$ 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.