## C05 - Effective models for materials and interfaces with multiple scales

**Head(s):** Prof. Dr. Alexander Mielke (WIAS)

**Project member(s):** Thomas Frenzel, Dr. Martin Heida, Artur Stephan

**Participating institution(s):** WIAS

### Project Summary

This project provides analytical techniques for discrete or continuous material models that depend on one or several small parameters. Special emphasis is given to systems that have a variational structure such as static minimisation problems or gradient-flow equations systems. Methods of static or evolutionary Gamma convergence are employed and further investigated, in particular EDP-convergence (i.e., in the sense of the energy dissipation principle). The small parameter may determine material properties via small layers or periodic, fractal, or stochastic material properties with a small correlation length. Applications involve diffusion in strongly heterogeneous media, elastic bulk materials with embedded interfaces along which Coulomb friction and other processes may occur.

More precisely we consider the topics:

- Gradient systems and evolutionary Gamma convergence, which provide tools for deriving effective models for systems with many scales. The connection between large-deviation principles for microscopic stochastic models and the gradient structures for the macroscopic deterministic models.

- Homogenisation of discrete elliptic operators (linear or non-linear) on regular or random graphs with random coefficients. Fractal homogenisation of elliptic problems with transmission on a fractal set of interfaces.

- Mathematical and thermodynamical modeling of evolutionary processes in bulk materials and in materials with interfaces. Periodic, fractal and stochastic homogenisation of evolutionary systems with variational structure.

- Rate-independent and rate-and-state friction between elastic bodies and its justification via dimension reduction.

- Connections between discrete chemical master equations and continuum descriptions like the reaction-rate equation. Hybrid models for reaction kinetics and reaction-diffusion systems.

### Project publications

Heida, M. and Kornhuber, R. and Podlesny, J.
(2020)
*Fractal homogenization of multiscale interface problems.*
Multiscale Modeling & Simulation, 18
(1).
pp. 294-314.
ISSN 1540-3459

Heida, M. and Neukamm, S. and Varga, M.
(2019)
*Stochastic homogenization of Λ-convex gradient flows.*
SFB 1114 Preprint in arXiv:1905.02562
.
pp. 1-28.
(Unpublished)

Franchi, B. and Heida, M. and Lorenzani, S.
(2019)
*A Mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation.*
SFB 1114 Preprint in arXiv:1904.11015
.
pp. 1-43.
(Unpublished)

Heida, M. and Nesenenko, S.
(2019)
*Stochastic homogenization of rate-dependent models of monotone type in plasticity.*
Asymptotic Analysis, 112
(3-4).
pp. 185-212.
ISSN 0921-7134

Donati, L. and Heida, M. and Weber, M. and Keller, B.
(2018)
*Estimation of the infinitesimal generator by square-root approximation.*
Journal of Physics: Condensed Matter, 30
(42).
p. 425201.
ISSN 0953-8984, ESSN: 1361-648X

Heida, M. and Patterson, R. I. A and Renger, M.
(2018)
*Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space.*
J. Evol. Equ.
.
pp. 1-42.
ISSN Online: 1424-3202 Print: 1424-3199

Heida, M.
(2018)
*Convergences of the squareroot approximation scheme to the Fokker–Planck operator.*
Mathematical Models and Methods in Applied Sciences, 28
(13).
pp. 2599-2635.
ISSN 0218-2025, ESSN: 1793-6314

Mielke, A. and Rossi, R. and Savaré, G.
(2018)
*Global existence results for viscoplasticity at finite strain.*
Archive for Rational Mechanics and Analysis, 227
(1).
pp. 423-475.
ISSN Print: 0003-9527; Online: 1432-0673

Heida, M. and Neukamm, S. and Varga, M.
(2017)
*Stochastic unfolding and homogenization.*
SFB 1114 Preprint at WIAS 12/2017
.
pp. 1-45.
(Unpublished)

Liero, M. and Mielke, A. and Savaré, G.
(2017)
*Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures.*
Invent. math.
.
pp. 1-149.
ISSN 1432-1297 (online)

Mielke, A. and Patterson, R. I. A and Peletier, M. A. and Renger, M.
(2017)
*Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics.*
SIAM Journal on Applied Mathematics, 77
(4).
pp. 1562-1585.
ISSN 1095-712X (online)

Heida, M.
(2017)
*Stochastic homogenization of rate-independent systems and applications.*
Continuum Mech. Thermodyn., 29
(3).
pp. 853-894.
ISSN 1432-0959 (online) 0935-1175 (print)

Gussmann, P. and Mielke, A.
(2017)
*Linearized elasticity as Mosco-limit of finite elasticity in the presence of cracks.*
Adv. Calc. Var.
.
(Submitted)

Flegel, F. and Heida, M. and Slowik, M.
(2017)
*Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps.*
SFB 1114 Preprint in arXiv:1702.02860
.
(Unpublished)

Heida, M. and Schweizer, B.
(2017)
*Stochastic homogenization of plasticity equations.*
ESAIM: Control, Optimisation and Calculus of Variations
.
pp. 1-30.
(Submitted)

Liero, M. and Mielke, A. and Peletier, M. A. and Renger, M.
(2017)
*On microscopic origins of generalized gradient structures.*
Discrete and Continuous Dynamical Systems - Series S, 10
(1).

Heida, M. and Mielke, A.
(2017)
*Averaging of time-periodic dissipation potentials in rate-independent processes.*
Discrete and Continuous Dynamical Systems - Series S, 10
(6).
pp. 1303-1327.

Mielke, A.
(2017)
*Three examples concerning the interaction of dry friction and oscillations.*
In:
Trends on Application of Mathematics to Mechanics.
Springer INdAM series.
(In Press)

Mielke, A. and Mittnenzweig, M.
(2017)
*Convergence to Equilibrium in Energy-Reaction–Diffusion Systems Using Vector-Valued Functional Inequalities.*
Journal of Nonlinear Science
.
pp. 1-42.
ISSN 1432-1467 (online)

Bonetti, E. and Rocca, E. and Rossi, R. and Thomas, M.
(2016)
*A rate-independent gradient system in damage coupled with plasticity via structured strains.*
ESAIM: Proceedings and Surveys, 54
.
pp. 54-69.

Liero, M. and Mielke, A. and Savaré, G.
(2016)
*Optimal Transport in Competition with Reaction: The Hellinger--Kantorovich Distance and Geodesic Curves.*
SIAM J. Math. Anal., 48
(2).
pp. 2869-2911.
ISSN 1095-7154 (online)

Mielke, A. and Peletier, M. A. and Renger, M.
(2016)
*A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility.*
Journal of Non-Equilibrium Thermodynamics, 41
(2).

Mielke, A. and Rossi, R. and Savaré, G.
(2016)
*Balanced-Viscosity solutions for multi-rate systems.*
Journal of Physics: Conference Series, 727
.
pp. 1-27.