An approach to nonlinear viscoelasticity via metric gradient flows
- Mielke, Alexander
- Ortner, Christoph
- Şengül, Yasemin
2010 Mathematics Subject Classification
- 74D10 35A15 35Q74 37L05 53C22
- Nonlinear viscoelasticity, gradient flow, dissipative distance, generalized geodesics
We formulate quasistatic nonlinear finite-strain viscoelasticity of rate-type as a gradient system. Our focus is on nonlinear dissipation functionals and distances that are related to metrics on weak diffeomorphisms and that ensure time-dependent frame-indifference of the viscoelastic stress. In the multidimensional case we discuss which dissipation distances allow for the solution of the time-incremental problem. Because of the missing compactness the limit of vanishing timesteps can only be obtained by proving some kind of strong convergence. We show that this is possible in the one-dimensional case by using a suitably generalized convexity in the sense of geodesic convexity of gradient flows. For a general class of distances we derive discrete evolutionary variational inequalities and are able to pass to the time-continuous in some case in a specific case.
- SIAM J. Math. Anal., 46 (2014) pp. 1317--1347.