# FOR 797: DFG Research Unit "Analysis and computation of microstructure in finite plasticity"

Subproject P5:

**Regularizations and relaxations of time-continuous problems in plasticity**

## Researchers:

Project Head: |
Prof. Dr. Alexander Mielke |
---|---|

Investigators: |
Dr. Dorothee Knees, Dr. Sebastian Heinz |

## Description:

The theory of finite-strain elastoplasticity has been developed quite rapidly
during the last decade. The major impulses for this were twofold:
on the one hand, the discovery that time-incremental problems can be formulated
as minimization problems and, on the other hand, the recent developments in the
field of microstructures generated by infimizing sequences of functionals.
In mathematical theory, the formation of microstructure is mostly treated via
global minimization for static problems. In contrast to that, our aim is to
derive models for the evolution of microstructure under slowly varying loads.

This project is devoted to the study of temporal evolution models for plasticity
and for systems with microstructure in general. Using spatial regularization via
higher gradients and temporal regularization via viscosity, we first want to
derive mathematical models that allow for an existence theory of solutions without
microstructure. The temporal regularization will lead to time-continuous solutions
and thus avoid the problems occuring through global minimization. Starting from
these solutions, we then generalize the recently developed energetic formulation
for rate-independend processes.

As a prelimenary step, this program will be studied via simplified model problems,
for which existence, uniqueness and convergence of numerical schemes can be
proven and tested. Finally, the more difficult case of geometrically exact
finite-strain elastoplasticity will be attacked. Funding began in 2007.

## Collaborations:

- Carsten Carstensen, Humboldt-Universität zu Berlin.
- Klaus Hackl, Ruhr-Universität Bochum.
- Michael Ortiz, California Institute of Technology, Firestone, USA.
- Tomás Roubíček, Mathematical Institute Charles University, Prague, Czech Republic.
- Martin Kružík, Institute of Information Theory and Automation, Academy of Sciences, Czech Republic.

## Activities:

**13th GAMM Seminar on Microstructures**

K. Hackl (Bochum), A. Mielke

Bochum, January 17 - 18, 2014**Oberwolfach Workshop on Variational Methods for Evolution**

A. Mielke, F. Otto (Leipzig), G. Savaré (Pavia), U. Stefanelli (Pavia)

Mathematisches Forschungsinstitut Oberwolfach, December 4 - 10, 2011**Autumn School on Mathematical Principles for and Advances in Continuum Mechanics**

P. M. Mariano (Firence), A. Mielke

Centro di Ricerca Matematica Ennio De Giorgi, Pisa, November 7 - 12, 2011**XVII International ISIMM Conference on Trends in Applications of Mathematics to Mechanics STAMM 2010**

W. H. Müller (TU Berlin), A. Mielke

Akademie Berlin-Schmöckwitz, August 30 - September 2, 2010**Oberwolfach Workshop on Microstructures in Solids**

M. Ortiz (Pasadena), A. Mielke

Mathematisches Forschungsinstitut Oberwolfach, March 14 - 20, 2010**Sixth GAMM Seminar on Microstructures**

A. Mielke, S. Conti (Bonn)

WIAS Berlin, January 12 - 13, 2007

## Publications:

- D. Knees, R. Kornhuber, Chr. Kraus, A. Mielke, J. Sprekels.
**C3 - Phase transformation and separation in solids.**

MATHEON - Mathematics for Key Technologies. M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann et. al., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 189-203 - A. Mielke, R. Rossi, G. Savaré.
**Balanced-Viscosity solutions for multi-rate systems.**

Preprint no. 2001, WIAS, Berlin, 2014. - A. Mielke, R. Rossi, G. Savaré.
**Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems.**

*J. Eur. Math. Soc. (JEMS)*, to appear. - A. Mielke, Ch. Ortner, Y. Şengül.
**An approach to nonlinear viscoelasticity via metric gradient flows.**

Preprint no. 1816, WIAS, Berlin, 2013. - S. Heinz.
**Quasiconvexity equals lamination convexity for isotropic sets of 2x2 matrices.**

*Adv. Calc. Var.*, to appear. - S. Heinz.
**On the structure of the quasiconvex hull in planar elasticity.**

*Calc. Var. Part. Diff. Eqns.*, to appear. - A. Mielke, S. Zelik.
**On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms.**

*Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)*, Vol. XIII, 67-135, 2014. - A. Mielke, R. Rossi, G. Savaré.
**Nonsmooth analysis of doubly nonlinear evolution equations.**

*Calc. Var. Partial Differ. Equ.*, 46, 253-310, 2013. - A. Mielke, U. Stefanelli.
**Linearized plasticity is the evolutionary Γ-limit of finite plasticity.**

*J. Eur. Math. Soc. (JEMS)*, 15(3), 923-948, 2013. - A. Mielke, R. Rossi, G. Savaré.
**Variational convergence of gradient flows and rate-independent evolutions in metric spaces.**

*Milan J. Math.*, 80, 381-410, 2012. - A. Mielke, L. Truskinovsky.
**From discrete visco-elasticity to continuum rate-independent plasticity: Rigorous results.**

*Arch. Ration. Mech. Anal.*, 203, 577-619, 2012. - K. Hackl, S. Heinz, A. Mielke.
**A model for the evolution of laminates in finite-strain elastoplasticity.**

*Z. Angew. Math. Mech.*, 92(11-12), 888-909, 2012. - A. Mielke, R. Rossi, G. Savaré.
**BV solutions and viscosity approximations of rate-independent systems.**

*ESAIM Control Optim. Calc. Var.*, 18, 36-80, 2012. - M. Liero, A. Mielke.
**An evolutionary elastoplastic plate model derived via Γ-convergence.**

*Math. Models Meth. Appl. Sci.*, 21(9), 1961-1986, 2011. - A. Mielke.
**On thermodynamically consistent models and gradient structures for thermoplasticity.**

*GAMM Mitteilungen*, 34(1), 51-58, 2011. - A. Mielke, U. Stefanelli.
**Weighted energy-dissipation functionals for gradient flows.**

*ESAIM Control Optim. Calc. Var.*, 17, 52-85, 2011. - A. Mielke.
**Complete-damage evolution based on energies and stresses.**

Special Issue "Thermomechanics and Phase Change" of*Discrete Cont. Dyn. Syst. Ser. S*, 4, 423-439, 2011. - A. Mielke.
**Existence theory for finite-strain crystal plasticity with gradient regularization.**

Proc. of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials. K. Hackl, ed., vol. 21 of IUTAM Bookseries, Springer, 171-183, 2010. - A. Mainik, A. Mielke.
**Global existence for rate-independend gradient plasticity at finite strain.**

*J. Nonlinear Sci.*, 19(3), 221-248, 2009. - R. Rossi, A. Mielke, G. Savaré.
**A metric approach to a class of doubly nonlinear evolution equations and applications.**

*Ann. Scuola Nor. Sup. Pisa Cl. Sci. (5)*, 7, 97-169, 2008. - S. Heinz.
**Quasiconvex functions can be approximated by quasiconvex polynomials.**

*ESAIM: Control, Optimi. Calc. Var.*, 14, 795-801, 2008. - A. Mielke, M. Ortiz.
**A class of minimum principles for characterizing the trajectories of dissipative systems.**

*ESAIM Control Optim. Calc. Var.*, 14, 494-516, 2008. - D. Knees.
**Global stress regularity of convex and some nonconvex variational problems.**

*Annali di Matematica*, 187, 157-184, 2007. - A. Mielke, T. Roubíček, U. Stefanelli.
**Gamma-limits and relaxations for rate-independent evolutionary problems.**

*Calc. Var.*, 31, 387-416, 2007. - A. Mielke.
**Deriving new evolution equations for microstructures via relaxation of variational incremental problems.**

*Comput. Mthods Appl. Mech. Engrg.*, 193, 5095-5127, 2004. - A. Mielke.
**Energetic formulation of multiplicative elasto-plasticity using dissipation distances.**

*Cont. Mech. Thermodynamics*, 15, 351-382, 2003. - C. Carstensen, K. Hackl, A. Mielke.
**Non-convex potentials and microstructures in finite-strain plasticity.**

*Proc. R. Soc. London*, A 458, 299-317, 2002. - S. Govindjee, A. Mielke, G. Hall.
**The free-energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis.**

*J. Mech. Physics Solids*, 50, 1897-1922, 2002.

Last modified: 2014-06-04 by Sebastian Heinz