This paper presents an interesting approach to quasi nonlinear finite-strain viscoelasticity of rate type in terms of metric gradient flows. Utilizing the relations for the energy functional and the dissipation function, a dissipation matrix \(d\), a function of the dissipation density \(D\), is formed. The function \(D\) is connected to the dissipation potential \(R\). An incremental minimization formulation that minimizes \(u\) for small time steps is then considered. The conditions on the stored energy density \(W\) and the dissipation density \(D\) that are physically admissible and admit the existence for the function \(u\) are analyzed. A discussion of the modeling of viscoelasticity as a formal gradient system, energy functional and dissipation potential, dissipation distances and incremental minimization problems, along with some dissipation distances examples and a multidimensional existence theory, are presented in Section 2. Section 3 presents the one-dimensional Dirichlet case, state space and energy, generalized geodesics for the distance, one-dimensional distances derived using compositions, and the Hellinger (or the square-root) distance. The time-incremental minimization problem and discrete variational inequality topics are discussed is Section 4. Section 5 is devoted to the development of the time-continuous case, convergence in the square-root distance case, curves of maximum slope and the slope and the weak solutions of one-dimensional viscoelasticity. The paper is supplemented by a useful appendix that discusses the one-dimensional case with Neumann boundary conditions.
This is an interesting, highly mathematical paper. The results are established by stating and proving a number of theorems and lemmas. The reviewer enjoyed reading and reviewing the paper.