New Models for Viscoelastic Materials Using Logarithmic Stresses

Recent research by Gennaro Ciampa, Giulio G. Giusteri, and Alessio G. Soggiu introduces new models for viscoelastic materials, which include both solids and fluids, based on logarithmic stresses. This approach aims to better capture the elastic contributions to material responses. The study highlights the use of the matrix logarithm to connect strain measures, which are part of a multiplicative group of linear transformations, to stresses that are additive elements of a linear space of tensors.

The authors assume a Newtonian constitutive law for viscous stresses, but they note that the presence of elasticity and plastic relaxation renders these materials non-Newtonian. The primary focus of the paper is to discuss the existence of weak solutions for the corresponding systems of partial differential equations, particularly in the nonlinear large-deformation regime.

A significant challenge identified in the research is the analysis of transport equations necessary to describe the evolution of tensorial measures of strain. For the solid model, the study focuses on the left Cauchy-Green tensor, while for the fluid model, an additional evolution equation for elastically-relaxed strain is introduced. The authors propose a novel concept of charted weak solutions, utilizing non-standard a priori estimates, which leads to a global-in-time existence of solutions for these viscoelastic models within the context of energy inequality.

This work contributes to the mathematical understanding of viscoelastic materials and may have implications for various applications in material science and engineering, particularly in scenarios involving large deformations and complex material behaviors. The full paper is available for further reading at arXiv:2306.14049.