## Research Abstracts Online

January
2008 - March 2009

### University of Minnesota Twin Cities

Institute of Technology

Department of Aerospace Engineering and Mechanics

## PI: Ryan S. Elliott

### A Quasicontinuum for Multilattic Crystals Exhibiting Martensitic Phase Transformation: Cascading Cauchy-Born Kinematics

The quasicontinuum (QC) method is a multiscale method coupling atomistic regions with a surrounding continuum modeled within a nonlinear finite element formulation. The constitutive response in the continuum is obtained by application of Cauchy-Born (CB) kinematics to the underlying lattice and calculation of the energy and necessary gradients using the same interatomic potentials applied in the atomistic region. Application of QC to multilattice crystals is straightforward in principle, but a question arises as to which description of the crystal structure should be used with CB kinematics. Traditionally, CB kinematics has been interpreted as being applied to the simplest periodic structure that reproduces the crystal. However, with this definition the CB rule can "fail” since some deformations require an increase in the periodic size of the unit cell. The problem is that the minimum required cell size cannot be known a priori. Detection of such period extensions can be critical in many cases where martensitic phase transformations occur in the material. To address this issue within QC, a phonon stability analysis is performed locally within each finite element in the continuum region at the end of each load step. This analysis detects the onset of period extension and identifies a new minimal lattice description; a methodology referred to as "Cascading Cauchy-Born kinematics.”

This project studies applications to one- and 2.5-dimensional test problems. The methodology will be extended to three dimensions and used to study the interaction of phase transformations with defects and larger-scale material inhomogeneities. Materials of interest include shape-memory alloys, which have many applications as sensors and actuators, and silicon-dioxide and related materials used for optics applications.

### Group Members

Dipta Ghosh, Graduate Student

Venkata Guthikonda, Graduate Student

Vincent Jusuf, Graduate Student

Dan S. Karls, Supercomputing Institute Undergraduate Intern

Viacheslav Sorkin, Graduate Student