  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.
