Pavel Belik, Graduate Student Researcher Tim Brule, Graduate Student Researcher Guido Kanschat, Supercomputing Institute Research Scholar Bo Li, Mathematics Department, University of California, Los Angeles, California Julia Liakhova, Graduate Student Researcher
The deformation of a active martensitic thin film. Such films are being developed for application in microvalves so that film open and closes a channel as the temperature cycles around the temperature of transformation.
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"Theory and Computation for the Microstructure Near the Interface Between Twinned Layers and a Pure Variant of Martensite," B. Li and M. Luskin, University of Minnesota Supercomputing Institute Research Report UMSI 99/29, March 1999. Publication in press. |
This research is developing computational methods for nonlinear partial differential equations that model the dynamics of the austenitic-martensitic transformation in active thin films. The computation of active thin films is essential to the development of micromachines for wide-ranging applications from medicine to aerospace. This computational project is developing the ability to simulate the behavior of shape memory materials, most of which undergo a martensitic transformation. These researchers are working to develop a computational model that can simulate the behavior of shape memory materials, most of which undergo a martensitic transformation. They are working to develop a computational model that can simulate the experiments currently being performed in Professor Richard James' laboratory on a Cu-Al-Ni shape memory alloy.
Energy-minimizing sequences of deformations for martensitic crystals modeled by the Ericksen-James elastic energy density often exhibit a microstructure-the simplest of which are layers in which the deformation gradient is nearly constant and across which the deformation gradient oscillates between energy wells-to allow the effective energy of a deformation to be that of a macroscopic or relaxed energy. A more detailed model including a surface energy can be used to obtain a length scale for the oscillations and to select among competing energy-minimizing sequences, some of which may exhibit branching. During the past several years, computational methods have been developed for the approximation of microstructure and a mathematical theory that has made the rigorous analysis of the numerical approximation of microstructures possible.
The linearization of the dynamical model about some strains leads to an ill-posed problem since the energy density is not convex. However, the nonlinear model need not be ill-posed since the energy density is convex in the neighborhood of the energy wells where the deformation gradient is usually confined. A major goal of this computational program is to show how microstructure develops as a consequence of this ill-posedness.
The model and numerical code are currently being extended to include thermal effects more accurately. Current work is studying the effect of the improved model on the propagation of the austenitic-martensitic interface. This research is also attempting to simulate recent laboratory experiments that exhibit hysteretic phenomena.
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URL: http://www.msi.umn.edu/about/publications/annualreport/ar2000/depts/IT/Math/luskin.html |
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