Viscoelasticity, logarithmic stresses, and tensorial transport equations

We introduce models for viscoelastic materials, both solids and fluids, based on logarithmic stresses to capture the elastic contribution to the material response. The matrix logarithm allows to link the measures of strain, that naturally belong to a multiplicative group of linear transformations, to stresses, that are additive elements of a linear space of tensors. As regards the viscous stresses, we simply assume a Newtonian constitutive law, but the presence of elasticity and plastic relaxation makes the materials non-Newtonian. Our aim is to discuss the existence of weak solutions for the corresponding systems of partial differential equations in the nonlinear large-deformation regime. The main difficulties arise in the analysis of the transport equations necessary to describe the evolution of tensorial measures of strain. For the solid model, we only need to consider the equation for the left Cauchy-Green tensor, while for the fluid model, we add an evolution equation for the elastically-relaxed strain. Due to the tensorial nature of the fields, available techniques cannot be applied to the analysis of such transport equations. To cope with this, we introduce the notion of charted weak solution, based on non-standard a priori estimates, that lead to a global-in-time existence of solutions for the viscoelastic models in the natural functional setting associated with the energy inequality.