**Laminate microstructures in finite-strain crystal plasticity**

The analysis and simulation of microstructures in solids has gained crucial importance, virtue of the influence of all microstructural characteristics on a material's macroscopic, mechanical behavior. In particular, the arrangement of dislocations and other lattice defects to particular structures and patterns on the microscale as well as the resultant inhomogeneous distribution of localized strain can result in a highly altered mechanical behavior. Energetic models predicting the mechanical properties are commonly based on thermodynamic variational principles. Modeling the material response in finite-strain crystal plasticity very often results in a nonconvex variational problem, so that the minimizing deformation fields are no longer continuous but exhibit small-scale fluctuations related to probability distributions of deformation gradients to be calculated via energy relaxation. This results in fine structures which can be interpreted as the observed microstructures.

Experimental findings indicate that very often it is unfavorable for a material to accommodate an imposed macroscopic deformation gradient by a homogeneous deformation field but rather by forming microstructural patterns that mix different homogeneous states of minimal energy. This has been observed, e.g., in the context of phase transformations and for deformation twin structures, or for dislocation walls in single-crystals. An interesting feature of all these microstructures is that they tend to form similar spatial patterns, hinting at a universal underlying mechanism. Microstructures in physical reality exhibit an enormous variety in appearance, the simplest example being a spatial lamination of different domains; i.e. regions with different deformation states are separated by parallel, planar interfaces. The microstructural internal variables and the deformation differ from domain to domain with the only constraint that the deformation must ensure compatibility at interfaces.

From an energetic viewpoint, the free energy in models of finite-strain crystal plasticity often lacks (quasi)convexity, which implies the nonexistence of minimizers. Instead of undergoing a homogeneous deformation, the material breaks up into multiple domains to further reduce the energy, thus following the path of the relaxed energy (the quasiconvex hull). We approximate the relaxed energy by a rank-one-convex envelope, which corresponds to the energy of a first-order laminate microstructure. The resulting laminate structure is studied during time-continuous evolution for monotonic and cylic load experiments. The relaxation-based continuum model predicts the onset of lamination, the development of the laminate microstructure (i.e., of all laminate characteristics) and its impact on the stres-strain response. Results indicate an interesting behavior during cyclic loading, when - within a certain number of load cycles - a permanent laminate forms that concentrates plastic deformation to thin bands yo exhibit many characteristics of the experimentally observed persistent slip bands.

Current research aims at extending the established model to represent more complex, higher-order
laminates and more general microstructures, to account for interfacial energies and their
influence of the isotropic and kinematic hardening, and to couple the relaxation-based
model with **continuum dislocation approaches**.

#### Nonconvexity, energy relaxation, laminates

A more detailed description of laminate microstructures in metals, the causing principles,
the mathematical theory and the modeling approach applied to simulate the origin and evolution
of laminate microstructures in finite-strain crystal plasticity can be found
**here**.

#### References:

- D. M. Kochmann, K. Hackl:

**Time-continuous evolution of laminate microstructures in finite crystal plasticity: a variational approach**,

*Continuum Mech. Thermodyn.***23**(2011), 63-85. - K. Hackl, D. M. Kochmann,
**An incremental strategy for modeling laminate microstructures in finite plasticity - energy reduction, laminate orientation and cyclic behavior**, in: R. de Borst, E. (Eds.),*Lecture Notes in Applied and Computational Mechanics*, Springer (2010), pp. 117-134. - D. M. Kochmann, K. Hackl,
**Time-continuous evolution of microstructures in finite plasticity**, in: K. Hackl (Ed.),*Variational Concepts with Applications to the Mechanics of Materials*, IUTAM Bookseries, Springer (2010), pp. 117-130. - D. M. Kochmann, K. Hackl,
**Influence of hardening on the cyclic behavior of laminate microstructures in finite crystal plasticity**,*Techn. Mech.***30**(2010), 387-400. - K. Hackl, D. M. Kochmann,
**Relaxed potentials and evolution equations for inelastic microstructures**, in: B. Daya-Reddy (Ed.),*Theoretical, Computational and Modeling Aspects of Inelastic Media*, IUTAM Bookseries, Springer (2009), pp. 27-39.