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

The elastoplastic response of metals is a prime example of the close link between microstructure and macroscopic material properties. Two important microstructural featuers in metals are the grain structure and the dislocation network. Atoms in crystals are arranged in a regular lattice, i.e. atoms are arranged in a structured array of a repeated unit cell. In face-centered cubic metals (e.g., iron at elevated temperature), atoms are arranged to close-packed cubic unit cells as sketched below:

The elastic (i.e., reversible) response of the crystal is realized by a homogeneous deformation of the atomic lattice about the equilibrium positions of atoms. Thus, the elastic properties are determined by the lattice characteristics (e.g., lattice spacing, interatomic potentials, etc.), and the elastic deformation process can be idealized as in the following animation (neglecting the Poisson effect):

*(Click on figure, animation opens in new window.)*

The plastic (i.e., irreversible) deformation of metals stems from complex albeit important
microstructural rearrangements. The carriers of plastic deformation are called dislocations,
which are defects in the regular atomic lattice. Depending on the type of lattice distortion,
one differentiates between edge and screw dislocations. These dislocations can glide (and climb)
through the lattice and thereby accommodate plastic flow. The following picture illustrates
the idealized motion of edge dislocations (i.e., extra half-planes in the crystal lattice),
which glide through the lattice along specific planes into specific directions. These
close-packed planes are called the slip planes, on which edge dislocations slide into the
close-packed directions. The combination of slip plane (normal ** m**) and direction

**is called slip system (**

*s***,**

*s***). Possible slip systems in fcc crystals are included in the above figure of the atomic lattices.**

*m*

*(Click on figure, animation opens in new window.)*

Deformation of a metal at sufficiently large strains commonly results in both elastic and
plastic deformation. In finite-strain plasticity models, the deformation gradient ** F** (i.e., the linear
mapping between reference and deformed configuration) is decomposed multiplicatively into its elastic and its
plastic part,

**e and**

*F***p, respectively:**

*F*

The plastic part is accommodated by dislocation motion. We model the microstructural evolution
process from an energetic point of view, where the actual state of deformation is obtained
from thermodynamic minimum principles. On the one hand, we know that the actual deformation
state (i.e., all elastic variables) is obtained from the principle of minimum potential energy
(with essential boundary conditions **Φ**_{0} on the boundary **Γ**_{u}):

whereas the internal variables ** K** (which describe the microstructure at each point of the deformed
crystal) evolve in time according to the principle of maximum dissipation or the principle
of minimum dissipation potential (here,

**Δ**denotes the dissipation potential,

**Ψ**the free energy density,

**the total energy, and**

*I***Φ**the deformation mapping,

*t*is time):

Let us assume a solid with Neo-Hookean elastic energy and a single active slip system. Then, we can obtain the actual state of the deformation and of the internal variables for a given deformation path from the above minimum principles. Plotting the resulting condensed energy, e.g. for a simple shear experiment, we obtain the following paths of the total (condensed) energy (for various slip system orientations and hardening coefficients):

It becomes apparent that the energy landscape in these examples is non-convex (see, e.g., the bold blue curve). To overcome the non-(quasi)convexity and to reduce the energy, the crystal might form microstructures, i.e., fine-scale, minimizing sequences that are described in terms of the quasiconvex envelope: instead of remaining in a homogeneous deformation state, the material breaks up into multiple, compatible domains at different internal states, which can reduce the energy. Therefore, we can determine the actual microstructure that forms along a given deformation path, by replacing the energy and dissipation potentials by the quasiconvex counterparts, following the theory of relaxation. Since the quasiconvex hull is hard to compute in general, it is often replaced by the rank-one-convex envelope, which allows for a convenient geometric interpretation of the forming microstructure in terms of laminates. Laminate microstructures have been observed in numerous experiments and are employed in material models to approximate the quasiconvex hull:

Applying the thermondynamic minimum principles to the relaxed minimization problem (i.e., after replacing the energy and dissipation potentials by their relaxed counterparts), we can compute the evolution of microstructure for a given deformation path. For example, during a simple shear test with only a single active slip system, the following graphic describes the evolution of the microstructure at a material point. Out of the homogeneous, elastic single crystal forms a second laminate phase with high concentration of plastic strain, which increases in volume with increasing deformation and then decreases again, until the crystal is eventually homogeneous again, exhibiting high plastic deformation (lambda represents the volume fraction of the second laminate phase, and gamma denotes the plastic slips in phases 1 and 2):

A relaxation-based crystal plasticity model has been applied to various test cases, where the microstructural evolution in time could be simulated by the relaxation-based material model. Experimental investigations are used to validate the model and to provide further insight into the actual microstructural mechanisms. One interesting observation from the present modeling approach is the microstructures that arise from cyclic loading. Here, we observe the formation of a steady-state, saturated laminate after a certain number of load cycles, which exhibits many features typical of so-called persistent slip bands that have been reported from many experiments.

Much of this work was carried out within the DFG-funded Research Alliance *Microplast -
Analysis and computation of microstructure in finite plasticity* in collaboration with Prof. Klaus Hackl (Ruhr-University Bochum).

#### References:

- B. Klusemann, D. M. Kochmann.
Microstructural pattern formation in crystal plasticity during cyclic loading: relaxation vs. gradient plasticity,
*Comput. Methods Appl. Mech. Engrg.***278**(2014), 765-793. - 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.