Samira Boddin (Kassel): Balanced viscosity solutions for rate-independent systems with state-dependent dissipation and applications in non-associated plasticity

Zoom Link:
https://uni-kassel.zoom.us/j/96217091997?pwd=RVRaVVBRclFJYU9jczNZSWF3SXI2QT09

Meeting ID: 962 1709 1997
Passcode: cauchy

Abstract:

Various solution concepts for rate-independent systems have been developed, generally leading to different solutions. If the underlying energy functional is not convex or the dissipation potential depends not only on the rate but also on the state of the solution, a discontinuous evolution may be necessary. The concept of balanced viscosity (BV) solutions allows such temporal jumps and shows in addition a physically reasonable behavior.

In this talk, we assume the dissipation potential to depend Lipschitz-continuously on the state of the solution. We provide an approximation scheme with adaptive time-step size based on [1] and abstract conditions ensuring its convergence to BV solutions, cf. [2,3].

One possible application lies in plasticity models with non-associated flow rules, often used for granular media such as soils and rocks. Laborde proposed a generalization of the principle of maximum dissipation allowing to deduce a variational formulation of such models [4]. In particular, this leads to a state-dependent dissipation potential and therefore it fits into the setup of our theory.

Additionally, often a cap model is used. It modifies the yield surface to intersect the hydrostatic axis and thus bounds the elastic region. This way a compaction of the material under high hydrostatic pressure should be included. At last, we will discuss the cap model from a mathematical perspective.

[1] M. A. Efendiev and A. Mielke. On the rate-independent limit of systems with dry friction and small viscosity. Journal of Convex Analysis, vol. 13, pp. 151–167, 2006.

[2] M. Sievers. Convergence analysis of a local stationarity scheme for rate-independent systems. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 56(4), pp.1223-1253, 2022, https://doi.org/10.1051/m2an/2022034.

[3] A. Mielke, R. Rossi. Existence and uniqueness results for general rate-independent hysteresis problems. Mathematical Models and Methods in Applied Sciences, vol.17(1), pp. 81-123, 2007, https://doi.org/10.1142/S021820250700184X.

[4] J.-F. Babadjian, G. Francfort, M.G. Mora. Quasi-static Evolution in Nonassociative Plasticity: The cap Model. SIAM J. Math. Anal., vol. 44, pp. 245-292, 2012, https://doi.org/10.1137/110823511.

Mit freundlichen Grüßen,

Prof. Dr. Dorothee Knees