Singularities, Constraints, and Stability in Elastostatics
The nonlinear theory of elasticity is more appropriate for the investigation of coexistent phase phenomena and singular behavior in the mechanics of materials than its linear counterpart. In the nonlinear theory, the governing system of equations can support the possibility of changes of type for certain applications that are not possible in the linear theory. Often, this is associated with the phenomena of instability and bifurcation, which leads to highly localized large deformations. 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. These researchers are using some of these ideas in their investigations, but the emphasis of this program is on the role of singularities in problems where solutions are not regular. The injectivity of the deformation map is of great concern here.
In ongoing work initiated in this program, analytical and numerical methods, such as the Finite Element Method, are used to investigate material stability of laminates with distinct homogeneous, isotropic, and nonlinearly elastic phases that alternate periodically. The materials of these phases, by themselves, do not present stability issues, and the laminates, which have bounded domains, are in equilibrium and subjected to deformations on their boundaries. Analytical and preliminary computational efforts show very good agreement and confirm results found elsewhere in the literature. These results indicate that the overall behavior of a laminate may be unstable even though the behavior of each underlying constituent is stable.
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