Second order tensorial framework for 2D medium with open and closed cracks

•Any open and closed cracks system is represented by two second order tensors.•Initially isotropic medium with closed microcracks is found to be square symmetric.•Any closed cracks system is represented by one second order tensor.•Any closed cracks system is represented by two non orthogonal cracks.•Crack closure macroscopic model is proposed.

The tensorial nature of crack density of an initially isotropic 2D medium with open and closed cracks is studied by means of polar decomposition rewriting of standard micro-mechanics results. The question of both indicial and constitutive symmetries of different crack density tensors is addressed: for instance the standard fourth order crack density tensor DcDc is rari-constant (totally symmetric) and the fourth order closed cracks density tensor by which closed cracks are acting is found to have the square symmetry. The effect of cracks closure and sliding is accordingly shown to be represented by a second order tensor (δc) so that only two second order crack density tensors, do and δc, are needed for 2D medium with open and closed sliding cracks. Similarly to the open cracks case, any arbitrary closed crack system is shown to be represented by only two non orthogonal families of cracks. The question of macroscopic cracks closure conditions is finally studied. Present study leads to an approximate framework in which the only internal variable representative of physical cracks, open and closed, is second order cracks density tensor. Proposed second order tensorial framework is shown to be exact in the case of two orthogonal arrays of cracks, open and/or closed, it is approximate in the general case of many arrays of cracks, open and/or closed.