Professor Roger Fosdick

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 being used to investigate material stability of orthotropic elastic solids. The researchers have considered the equilibrium problem, with no body force, of a cylindrically orthotropic disk subject to a prescribed displacement along its boundary. In classical linear elasticity, the solution of this problem predicts material overlapping, which is not physically realistic. One way to prevent this anomalous behavior is to consider the minimization of the total potential energy of classical linear elasticity subjected to the local injectivity constraint. In the context of this constrained minimization theory, bifurcation occurs from a radially symmetric solution to a secondary solution. The researchers have analytical and computational results indicating that this secondary solution is rotationally symmetric. Although mathematically sound, this solution involves large rotations and, therefore, violates the basic hypotheses upon which classical linear elasticity is founded.

During 2024, the researchers are continuing this investigation by considering the minimization of the total potential energy of a fully nonlinear elastic material, also subjected to the local injectivity constraint. Here, there are no restrictions on the deformations undertaken by the solid. Thus, there should be a clearer picture about the existence of bifurcating solutions in elasticity. This research is of interest in the investigation of solids having radial microstructure, such as certain types of carbon fibers and wood. Another part of this research involves using bond-breakage damage criteria to model dynamic fracture of brittle materials using the peridynamic formulation. To study the capability of the damage models in predicting the formation and the propagation of cracks, the researchers have simulated numerically the experiment of a thin glass plate with an initial semi-crack under mode I loading. All the implemented models were able to grasp the main features of crack propagation, such as the crack propagation speed and the crack pattern. The group has also considered the experiment of a notched cement-mortar plate with a hole under quasi-static mode I loading. The solutions of the associated numerical problems yield convergent sequences of response forces in terms of displacements. In addition, crack patterns predicted by these solutions are in good agreement with crack patterns observed experimentally.