The elastoplastic response of poly(methyl methacrylate) to indentation

In our other recent work (Zheng & Hwang 1996), we established the explicit relations between the effective and the matrix elastic properties in two-dimensional (2D) elasticity, as the effective medium is comprised of an isotropic and homogeneous matrix, microcracks and holes of various shapes. The present paper extends the above result to the case for a composite material comprising either isotropic or anisotropic matrix and inclusions which have respectively continuously varying compliances, and for a variety of interface conditions. It is shown that the effective compliance depends upon a reduced list of the matrix and inclusion material constants. In particular, when the matrix and inclusions are all isotropic and respectively homogeneous, then E$_{\text{m}}$H (i.e. the defect compliance H which is the difference, S - S$_{\text{m}}$, between the effective compliance S and the matrix compliance S$_{\text{m}}$ multiplied by the matrix Young's modulus E$_{\text{m}}$) has a reduced dependence upon the E$_{\text{m}}$/E$_{\text{I}}$ and $\nu _{\text{m}}$ -$\nu _{\text{I}}$E$_{\text{m}}$/E$_{\text{I}}$ for the inclusion of Young's moduli E$_{\text{I}}$ and Poisson's ratios $\nu _{\text{I}}$ and the matrix Poisson's ratio $\nu _{\text{m}}$, rather than the full material constant list of E$_{\text{I}}$, $\nu _{\text{I}}$, E$_{\text{m}}$, $\nu _{\text{m}}$. Applications to various important types of matrix and inclusion materials are given. We also show that the generalized self-consistent method (Christenson & Lo 1979) provides solutions of the effective moduli for unidirectional fibre composites in compatible forms with the general dependence relations proved in the present paper.