Mesoscopic Theory of Microcracks

The mesoscopic concept is a way to deal with complex materials with an internal structure within continuum mechanics. It consists of extending the domain of the balance equations by mesoscopic variables and of introducing a local distribution function of these variables as a statistical element. In our case microcracks are modelled as penny shaped and completely characterized by their diameter and the unit normal to the crack surface. Two examples of crack dynamics are given as well as a possible definition of a damage parameter. Orientational order parameters (fabric-alignment tensors) are defined and balance like dynamic equations for them are derived. 1. A model of microcracks Macroscopic failure of brittle materials is initiated by the propagation of micro-cracks. In a simple model the microcrack is described as a flat, rotation symmetric surface, a so called penny shaped crack. In addition we make here the following simplifying assumptions: (1) The diameter of the cracks is much smaller than the linear dimension of the continuum element. Under this assumption the cracks can be treated as an internal structure of the continuum element. The cracks are assumed small enough that there is a whole distribution of crack sizes and orientations in the volume element. (2) The cracks are fixed to the material. Therefore their motion is coupled to the motion of representative volume elements. (3) The cracks cannot rotate independently of the material, i. e. if they have a nonzero rotation velocity at all, this rotation velocity is determined by the antisymmetric part of the time derivative of the deformation gradient of the surrounding material and it does not depend on crack length and orientation. All cracks within a volume element move and rotate with the same velocity. (4) The number of cracks is fixed, there is no production of cracks, but very short cracks are preexisting in the virgin material. (5) The cracks cannot decrease area, but can only enlarge, meaning that cracks cannot heal. To summarize our model the microcrack is characterized by a unit vector n representing the orientation of the surface normal and by the radius l of the spherical crack surface. These parameters will be taken as the additional variables in the mesoscopic theory.