Roger Fosdick, Associate Fellow

Dynamic Phase Transitions, Singularities, and Coexistent Phases in Elasticity

This group has continued its ongoing research on phase transitions, as well as investigating phases in elasticity. Purely mechanical theories of phase transitions require admissibility hypotheses—so-called “kinetic relations”—to be solvable. These kinetic relations, while ensuring uniqueness, have questionable physical justification. A thermomechanical model that includes capillarity, viscosity, and thermal conductivity is a more appropriate setting, and it is the preferred framework for this investigation.

In this setting, the idea is that the limiting solutions are physically relevant as the capillarity, viscosity, and thermal conductivity approach zero in a suitable manner. Because the free energy is nonconvex, the governing equations are of mixed type: hyperbolic in regions of pure single phase and elliptic in the intervening spinodal region. Continuing computations and analysis showed that this is a promising method of calculating dynamic phase transitions. Further investigations, including the determination of dependence on numerical methods, were pursued.

This group implemented several finite difference and finite element methods for solving these equations, and they developed their own approach. Part of the difficulty concerns optimizing the computations and addressing the questions of parallelization and alternate numerical techniques.

This program of research also considered singularities in elastostatics. The nonlinear theory of elasticity is more promising for the investigation of coexistent phase phenomena and singular behavior in the mechanics of materials than its linear counterpart. For solids, there are contemporary computational developments, iteration procedures, adaptive methods, and continuation techniques that are already being used successfully in the computation of regular boundary value problems that arise from such nonlinear theories. Some of these ideas were used in these investigations, but the emphasis of this program has been on the role of singularities in problems where solutions are not regular. The injectivity of the deformation map was of major concern here.

Finally, the researchers began to develop a program to consider coexistent phases in nonlocal elasticity. The goal of this program was to carry out a theoretical and computational investigation of stable, equilibrium coexistent phase structures in solids when subject to external load, environmental temperature, or electrical simulation. The long-range plan adopted by these researchers was to turn to dynamical issues in this non-local theory.

Research Group and Collaborators

Adair Roberto Aguiar, Wright, Logue, and Associates, Houston, Texas
John Ernie Dunn, Scientific Computing, Tempe, Arizona
Darren E. Mason, Department of Mechanics and Materials, Michigan State University, East Lansing, Michigan
Eric Petersen, Graduate Student Researcher
Gianni Royer, Civil Engineering Department, University of Parma, Parma, Italy
Ying Zhang, Department of Materials Science, Xiamen University, Fujian, People’s Republic of China