A new representation for the strain energy of anisotropic elastic materials with application to damage evolution in brittle materials

Abstract

In this paper, we develop a new representation of the strain energy of an elastic material using twenty-one scalar measures of strain. These measures are associated with material line elements which are directed along the six axes of symmetry of a regular icosahedron, and they include: six measures of axial strain and fifteen measures of the angular strain. When the strain energy is a quadratic function of strain, this representation yields twenty-one elastic moduli which naturally separate into only two physically different types. This is in contrast to the five physical types of moduli associated with the common rectangular Cartesian representation of the stiffness tensor. By formulating evolution equations for the elastic moduli, we describe general changes in the elastic state of a brittle material. Due to the physical nature of these moduli, we anticipate that the evolution equations will have simplifying features when expressed in terms of the new representation of the strain energy presented here. In general, these evolution equations are restricted by sufficient conditions which ensure that the change in the elastic state is a dissipative process in the sense that the second law of thermodynamics is satisfied. It is shown that not all changes of the elastic state can be interpreted as damage evolution, even if they are dissipative and cause reductions of the magnitudes of the twenty-one moduli. To ensure damage evolution, we propose an additional restriction on the strain energy. Furthermore, a simple modification of the strain energy is introduced to model the effect of crack closure. Specific equations for damage evolution which satisfy all necessary restrictions are presented. These equations characterize damage evolution in a material that exhibits a transition from an isotropic to an anisotropic elastic state. Examples are considered which show physically reasonable brittle material response.