On the continuous differentiability of state potentials for multiaxial stiffness recovery

The modeling of the micro-defect/micro-crack closure effect, also referred to as the quasi-unilateral effect or partial stiffness recovery, must be handled carefully in order to ensure the continuous differentiability of the (thermodynamics) state potential and the continuity of the elasticity law in terms of stress tensor versus elastic strain tensor. The key properties used to model such an effect properly for initially isotropic materials are recalled. We focus on an apparently innocuous term, the trace (squared) of the positive or negative part of the stress tensor. It is shown that such a term is not continuously differentiable, neither in 2D nor in 3D. When used in the expression of the thermodynamics potential, this term leads to a discontinuous elasticity law. The mechanical consequences of introducing positive/negative parts of stress or strain tensors into thermodynamics state potentials are substantiated. Incorrect and relevant modeling choices are finally inventoried.