Constitutive potentials for dilutely voided nonlinear materials

Abstract

Constitutive relations are derived for an incompressible, isotropic power-law matrix material containing a dilute concentration of spherical voids. The derivation is made for a nonlinearly viscous material used to characterize steady creep. However, the theory applies equally well to small strain nonlinear elasticity (deformation theory), and an extension to a rate-independent flow theory is also discussed. The starting point and key element in the formulation is the potential function for an isolated spherical void in an infinite block of power-law material. Approximate, but accurate, representations for this potential function are given. The overall constitutive relation governing the behavior of the dilutely voided solid is obtained simply and directly using the void potential. An assessment of the range of validity of the dilute concentration results is obtained using numerical solutions to the problem of a spherical void centered in a sphere of finite radius made of the power-law material. The potential function is also given for a dilute concentration of aligned penny-shaped cracks in the same power-law material.