# Schedule of the workshop: Analysis and computation of microstructure in finite plasticity

## Monday, May 4

9:00 - 9:10 |
Opening |

9:10 - 10:10 |
Michael Ortiz: Optimal scaling in plasticity and fracture |

10:10 - 10:50 |
Dietmar Gallistl: Numerical algorithms for the simulation of finite plasticity with microstructures |

10:50 - 11:20 |
Coffee break |

11:20 - 11:40 |
Georg Dolzmann: On the theory of relaxation with constraints on the determinant |

11:40 - 12:00 |
Carolin Kreisbeck: Homogenization of layered materials with rigid components in single-slip finite plasticity |

12:00 - 12:40 |
Klaus Hackl: Rate-independent versus viscous evolution of laminate microstructures in finite crystal plasticity |

12:40 - 14:20 |
Lunch |

14:20 - 15:20 |
Mark Peletier: Variational convergence for the analysis of boundary layers in dislocation pileups |

15:20 - 16:00 |
Christian Miehe: Variational gradient plasticity: Local-global updates, regularization and laminate microstructures in singlecrystals |

16:00 - 16:30 |
Coffee break |

16:30 - 17:30 |
Adriana Garroni: Derivation of the line tension energy for dislocations in 3D |

17:30 - 18:10 |
Alexander Mielke: Variational approaches and methods for dissipative material models with multiple scales |

18:10 - |
Poster session with snacks |

## Tuesday, May 5

9:00 - 9:40 |
Stefan Müller: Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations |

9:40 - 10:20 |
Patrick Dondl: Laminate microstructures in plasticity: relaxation, energy estimates, and a comparison with experiment |

10:20 - 10:50 |
Coffee break |

10:50 - 11:30 |
Lisa Scheunemann: Construction of statistically similar RVEs |

11:30 - 12:30 |
Henryk Petryk: The energy approach to microstructure formation in rate-independent plasticity |

12:30 - 12:40 |
Conclusion |

12:40 - |
Lunch |

# Abstracts

## Adriana Garroni: Derivation of the line tension energy for dislocations in 3D

In the understanding of plastic behaviour of metals a fundamental role isplayed by dislocations. These are line defects in the crystalline structure that favor the slip along slip planes, known to be the main mechanismfor plastic deformation. These defects interact, move and organize in complex structures producing other important effects. Owing to this fundamental role, dislocations have been extensively studied by theoretical, experimental and computational means. The bulk of this extensive body of literature regards dislocations as line defects in otherwise linear-elastic crystals. This approach classically requires a regularization of the coreof dislocations in order to capture the behaviour of the elastic interactions at a discrete level.

I will present a result in collaboration with S. Conti and M. Ortiz in which we prove that the classical line-tension approximation for dislocations in crystals, i.e., the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length, follows as the Γ-limit of regularized linear-elasticity as the lattice parameter becomes increasingly small. We consider two regularizations of the theory of linear-elastic dislocations: a core-cutoff and a mollification of the dislocation measure. We show that both regularizations give the same energy in the limit, namely, an energy defined on matrix-valued divergence-free measures concentrated on lines. The corresponding self-energy per unit length, which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the self-energy per unit length obtained from the classical theory of the prelogarithmic factor of linear-elastic straight dislocations. Microstruture can occur at small scales resulting in a further relaxation.

## Michael Ortiz: Optimal scaling in plasticity and fracture

Materials deforming plastically often exhibit deviations from volume scaling and their behavior depends on the size of the sample, be it the grain size, wire diameter, film thickness, or some other limiting feature size. This dependence is sometimes referred to as size effect. A classical manifestation of this size effect is the Hall-Petch relation. Likewise, the fracture energy of ductile materials scales with area, as opposed to volume, yet another indication of the existence of an intrinsic material length scale. Material models which are sensitive to the size of the sample are necessarily nonlocal and contain intrinsic length-scale parameters. Some classical models, including strain-gradient elasticity and plasticity, introduce the requisite intrinsic length scale by allowing the energy density to depend on deformation gradients. The central question brought forth therefrom concerns the connection between strain-gradient models, scaling and fracture and the ability of the models to account for the experimental record. Just about the only mathematical tool available at present for elucidating that connection, and an imperfect tool at that, is optimal scaling. For ductile single crystals, accounting for dislocation line energies renders their plastic behavior nonlocal. Explicit constructions then give rise to several optimal scaling laws that are observed experimentally in different regimes. For polycrystalline metals, when hardening exponents are given values consistent with observation, the deformation-theoretical energy is found to exhibit sublinear growth and, therefore, to relax to zero. A regularization of the strain-gradient plasticity type has the effect of introducing an intrinsic length scale into the energy. Constructions based on the introduction of void sheets are then found to result in fracture scaling. In a similar vein, the damage of polymers can be modeled by means of chain elasticity regularized through strain-gradient elasticity. Based on this type of model, constructions mimicking the crazing mechanism are again found to result in fracture scaling.

## Mark Peletier: Variational convergence for the analysis of boundary layers in dislocation pileups

Plasticity, the permanent deformation that one observes in metals, is the net effect of the movement of a large number of microscopic defects in the atomic lattice. These defects, called dislocations, are curve-like topological mismatches, and migrate through the metal under the influence of internal and external forces. Macroscopic, permanent, deformation arises through the concerted movement of a large number of these dislocations.

It is a major challenge to connect a microscopic description of dislocation movement on one hand with models of macroscopic plastic behaviour on the other hand. At this stage we are not able to do this; there is a major gap between the models at these different spatial and temporal scales. Part of the difficulty lies in the complex interactions between dislocations: they attract and repel each other, and form complex higher-level structures that appear to play an important role in determining the macroscopic behaviour.

In this talk I will report on a much more modest result. Dislocations 'pile up' at boundaries of the grains, through which they can not easily pass. These pileups give rise to a net force on the grain boundary, which can have macroscopic consequences. In this talk we study these pileups, and give a description at two levels, corresponding to a 'bulk' behaviour and a boundary-layer behaviour.

The methods we use are those of variational calculus, specifically Gamma-convergence at two different scales. These yield descriptions of the behaviour, both in the bulk and in the boundary layer, in terms of limiting energies.

This work is together with Adriana Garroni, Patrick van Meurs, and Lucia Scardia.

## Henry Petryk: The energy approach to microstructure formation in rate-independent plasticity

Spontaneous emergence of non-uniform deformation patterns and the formation and evolution of microstructures can be related to an intrinsic instability of macroscopically uniform deformation of the material, or shortly to material instability. The essence of the energy approach to material instability problems lies in the incremental minimization of the rate-independent work that is not quasiconvex. However, for dissipative solids this is not automatically a valid approach, as sometimes claimed, since the path dependence of energy dissipation can destroy a potential structure of the incremental problem. It is shown that if multiple mechanisms of inelastic deformation operate simultaneously and interact then the normality structure of governing equations does not guarantee correctness of incremental energy minimization. In particular, in finite-strain plasticity of metal crystals deformed by multislip, a kind of symmetrization of the interaction matrix for active mechanisms is also required.

The minimization of incremental work evaluated to second-order terms has been implemented in the computational algorithms for finite deformation plasticity. A number of examples have been calculated which demonstrate computational effectiveness of the approach in predicting the formation of microstructures. The examples include emergence of shear bands in polycrystalline metals and deformation banding in ductile single crystals. While previous works in the literature on modelling of microstructures were frequently focused on finite single-slip in subdomains, a new constitutive algorithm is presented that deals with arbitrary multiple-slip in 3D fcc crystals. The ability of the algorithm to eliminate the long-standing ambiguity in selection of active slip-systems inherent in rate-independent modeling is demonstrated. A comparison of predicted microstructures to experimental observations is provided.