Path dependence and a general theory for the progressively fracturing solid

Abstract

Abstract A theory is developed for an ideal time-independent material in which nonlinear behaviour is due solely to degradation of stiffness during loading. Behaviour during unloading and reloading is linear elastic with the moduli being determined by the initial state of the material and the prior history of loading. The theory is developed partly by analogy with the theory of hardening plasticity. This involves an exchange of static and kinematic variables and the use of a fracture surface in deformation space to distinguish between deformation accompanied by linear or nonlinear behaviour. Central to the theory is the restriction to materials in which the change of stiffness due to an increment of deformation is independent of the path over which the increment is effected. This form of path independence in the small leads to a flow rule giving general relations between increments of stress and strain and also an expression for the changes in moduli accompanying a change in deformation. The resulting flow rule is associated in the sense that it follows from the form of the fracture surface. In general, though, the inelastic component of the stress increment does not exhibit the property of normality with the fracture surface. However, the normality property does hold for particular circumstances analogous to isotropic hardening in plasticity. For such a material, the theory takes an attractively simple form and could have applications for describing behaviour of fabrics made of brittle fibres and also degradation of stiffness in materials like rock and concrete.