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Dynamics of Deformation Twinning Animation

When subjected to loads, a wide range of crystalline solids form microstructures in which parts of the crystal suffer large deformations involving different orientations (that is, either reflections or 180-degree rotations) of the original crystal lattice. Such microstructures are commonly reffered to as "twinned" and are of prominent importance in a variety of technologically significant applications, including those involving shape-memory devices, sensors, actuators, and damping mechanisms, fracture-resistant composites, and high-temperature superconductors.

Optical micrographs of twinned crystals reveal that the predominant twin morphology involves lamellar (or, lenticular) inclusions, with their major axes alligned parallel to the composition planes associated with the different orientations of the original lattice.

It is evident that any attempt to study lamellar microstructures must account for at least two spatial dimensions. This research is designed to gain understanding of nucleation and growth of twins in the simplest possible context that does so. That setting is antiplane shear, where the deformation is described by a scalar field, the "out-of-plane displacement". In particular, we restrict attention to twinning in monotomic BCC single-crystals and focus of the (1 1 2)[1 1 1] twinning mode, a mode common in such crystals. In this situation, the deformation lies parallel to the [1 1 1]-direction and the out-of-plane displacement depends on the spatial coordinates perpendicular to that direction; further, due to the three-fold symmetry of a cubic crystal about the [1 1 1]-direction, this setting allows for three distinct twin orientations and corresponding twin lamellae.

Our governing equations arise from the specialization of a theory for the dynamics of solid-solid phase transitions developed recently by Fried and Gurtin. In the context of antiplane shear that specialization results in a system of four (scalar) partial differential equations. The unknown fields appearing in this system consist of the out-of-plane displacement and three scalar order-parameters corresponding to the three types of twin lamellae. The equations governing these fields are strongly coupled and non-linear in their dependence on the order parameters.

We have developed efficient numerical methods for solving initial/boundary-value problems for this system of equations. Spatial and temporal discretization is based on finite differences. We use a fully implicit time stepping scheme and at each time step rely on nonsymmetric preconditioned conjugate gradient methods to solve the linear systems.

The output from this code was then used as the raw data for Data Explorer to produce a 4+ minute videotape animation.

This animation has been shown in conjunction with talks, seminars and presentations at the following:

• Transition Zone Analogues for Energy and Stress on Sharp Phase Interfaces, seminar, Instito di Meccanica Teorica ed Applicata, Universita di Udine, December, 1994.
• Transition Zone Analogues for Energy and Stress on Sharp Phase Interfaces, seminar, Dipartimento di Metodi e Modelli Matematica, Universita di Padova, December, 1994.
• Transition Zone Analogues for Energy and Stress on Sharp Phase Interfaces, talk, Trento Meeting on Models of Interphase Regions, December, 1994.
• Applications of a Regularized Theory for Solid-Solid Phase Transitions, talk, Caltech Meeting on Phase Transitions, January, 1995.
• Department of Mechanical and Aerospace Engineering, University of Arizona, February, 1995.
• Department of Mathematics, Carnegie-Mellon University, March, 1995.
• Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, April, 1995.
• Department of Materials Science and Mechanics, Michigan State University, May, 1995.
• Department of Mechanical Engineering, Massachusetts Institute of Technology, June, 1995.

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