Mathematical Description of Anisotropic Damage State in Continuum Damage Mechanics.

A fictitious equivalent undamaged configuration B*t of an element of a damaged material is first introduced into a three-dimensional Riemannian manifold by excluding the irrelevant rigid rotation of the element. Then a new symmetric damage tensor for three-dimensional anisotropic damage is developed in terms of Finger's tensor of a fictitious deformation gradient from the current damaged configuration Bt to the equivalent undamaged configuration B*t. The damage tensor components are expressed in terms of the metric tensor components of the convective coordinate systems of Bt and B*t. Furthermore, the effective stress tensor of damage mechanics is defined by the equality of stress vectors acting on the damaged and equivalent undamaged plane elements, and the mechanical interpretation of the effective stress tensor and its symmetry condition are discussed. The symmetry condition is also formulated by means of a commutator tensor formed by the tensor pairs : damage tensor-Cauchy stress tensor and damage tensor-reversible gradient. Finally, the symmetrized effective stress tensor is introduced and interpreted.