Decomposition of damage tensor in continuum damage mechanics

The principles of continuum damage mechanics are reviewed first for the case of uniaxial tension. The damage variable is then decomposed into two variables called crack damage variables and void damage variables. A consistent mathematical formulation is presented to decompose the damage tensor into two parts: one caused by voids and the other caused by cracks. In the first part of this work, isotropic damage in the uniaxial tension case is assumed. However, the generalization to three-dimensional states of damage is presented in the second part of this work, using tensorial damage variables. It is shown that the components of tensorial crack damage variables and void damage variables are not independent of each other, implying a coupling between the two damage mechanisms. This coupling may be obvious based on the physics of the problem, but a rigorous mathematical proof is given for it. Also, explicit relations governing the components of the crack and void damage variables are derived.