**Phase field modeling of microstructure evolution**

A feature of interest in mechanics of materials is the formation and evolution of microstructural patterns, oftentimes locally minimizing sequences which derive from the non(quasi)convexity of a materialâ€™s thermodynamic free energy. For the prediction of microstructure evolution, we develop and employ computational homogenization techniques in order to determine the effective, macroscale properties resulting from microstructural mechanisms through physics-based multiscale modeling. From a numerical perspective, periodic homogenization allows us to observe the formation and development of microstructure, while seamlessly providing a route for obtaining effective constitutive response at the Representative Volume Element (RVE) level such as, e.g., the effective stress-strain behavior of polycrystalline magnesium.

*Figure: magnesium polycrystal exhibiting slip-twinning microstructure that results in the typical stress-strain behavior observed in experiments; shown is the comparison of full-field spectral model, the relaxed Taylor model of Chang and Kochmann (2014), and experimental data of Kelley and Hosford (1968).*

In contrast to our modeling efforts based on lamination (which use a sharp-interface description of domain patterning), diffuse-interface phase field models describe the microstructure through order parameters, thereby resolving interfaces and domain walls with full accuracy. Phase field models are well suited for periodic homogenization, and the consequent Euler-Lagrange equations can be recast into a Lippmann-Schwinger form with a mean term and a perturbation. The governing equations can then be solved naturally using a spectral method. However, the resulting lack of smoothness due to microstructure requires analytical reformulation in order for asymptotic consistency and uniform convergence, which are ongoing research thrusts.

The resulting iterative spectral method scales favourably in computational cost with the number of degrees of
freedom and, unlike in the Finite Element Method, avoids the assembly of large stiffness matrices.
Therefore, large systems such as polycrystals with large numbers of grains can be modeled efficiently
using these techniques. We employ such techniques to explore the phenomena of plasticity and twinning
in magnesium and magnesium alloys,
and the evolution of ferroelectric domain patterns
and associated electro-mechanical hysteresis in polycrystalline ferroelectric ceramics. Examples of the latter
are shown below (exemplary magnesium results are shown above).

*Figure: distribution of microstructural domain patterns in a 3D polycrystal of barium titanate (shown are the shear stress distribution, which highlights domain walls, and the distribution of the polarization).*

Of particular interest to us is the kinetics of domain pattern evolution, including the kinetics of domain wall motion in ferroelectrics and the rate-dependent microstructural mechanisms and texture evolution in polycrystalline magnesium. The below movies show results of domain wall motion under applied electric fields in single- and polycrystalline barium titanate under applied electric fields.

*Left: formation and motion of domain walls in a 2D barium titante single-crystal (first, walls and domains form from random initial polarizations; then, walls move under an applied electric field; shown is the shear stress). Right: formation and motion of domain walls in a barium titante polycrystal (again, domain walls form and subsequently move under an applied electric field; shown is the polarization distribution).*

#### References:

- Y. Chang, D. M. Kochmann.
A variational constitutive model for slip-twinning interactions in single- and polycrystalline magnesium,
*Int. J. Plasticity***73**(2015), 39-61. - V. Vidyasagar, D. M. Kochmann.
Using spectral methods to determine the effective response of polycrystalline ferroelectric ceramics during domain switching,
*in preparation*(2016).

#### Research Sponsor:

We gratefully acknowledge the support from the Army Research Labs (ARL) through the Materials in Extreme Dynamic Environments (MEDE) Collaborative Research Alliance.