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We are dealing with damage of brittle materials caused by growth of microcracks. In our model the cracks are penny-shaped. They can only enlarge but not heal. For a single crack a Rice–Griffith growth law is assumed: There is crack growth only if tension is applied normally to the crack surface, exceeding a critical value. Our aim is to investigate the effect of crack growth on macroscopic constitutive quantities. A possible approach taking into account such an internal structure within continuum mechanics is the mesoscopic theory. A distribution of crack lengths and crack orientations within the continuum element is introduced. Macroscopic quantities are calculated as averages with the distribution function. A macroscopic measure of the progressing damage, i.e., a damage parameter, is the average crack length. For this scalar damage parameter we derive an evolution equation. Due to the unilateral growth law for the single crack, it turns out that the form of this  differential equation  depends explicitly on the initial crack length distribution. In order to treat biaxial loading, it is necessary to introduce a tensorial damage parameter. We define a second-order tensor damage parameter in terms of the crack length and orientation distribution function.

Dynamics of the Size and Orientation Distribution of Microcracks and Evolution of Macroscopic Damage Parameters