Calculation of the normal and shear compliances of a three-dimensional crack taking into account the contact between the crack surfaces
The calculation of the effective mechanical properties of materials with cracks is a long-standing problem in micromechanics. Usually each crack is modeled by a mathematical cut with traction free surfaces. In this case, the normal and shear compliances of the crack are approximately equal. Then the effective elastic properties of the material with randomly located and arbitrary oriented cracks are orthotropic. If the partial contact between the crack surfaces is taken into account, then the normal and shear compliances are generally different, and the effective properties are no longer orthotropic. This paper proposes a three-dimensional model for estimation of the effect of contacts on the normal and shear compliances of the crack. An infinite flat crack is considered. The surfaces of the crack are connected by a doubly periodic system of identical contacts. Each contact has the shape of a rectangular parallelepiped. The centers of the contacts form a square lattice. The influence of contact sizes and distance between contacts on the ratio of the normal and shear compliances of the crack is investigated. The compliances are calculated numerically using the finite element method. Calculations are carried out for a periodic cell containing a single contact. It is shown that in all considered cases, the normal compliance of the crack is less than shear compliance. The difference between the compliances increases with increasing contact height. A comparison with similar results obtained in a two-dimensional formulation is carried out. It is shown that in the two-dimensional case the ratio of compliances is larger than in the three-dimensional case.