Bifurcating solutions of anisotropic disk problem in a constrained minimization theory of elasticity - Rocha, Lucas Almeida (2021)
This work concerns the study of problems whose solutions in classical linear elasticity theory predict material overlapping, which is not physically admissible. To eliminate this anomalous behavior, we consider a theory that minimizes the total potential energy of the classical linear elasticity subject to the constraint that the deformation field is locally injective. We apply this theory, together with an interior penalty method and a standard finite element formulation, to obtain numerical solutions that do not exhibit material overlapping. We consider the problem of an anisotropic n-dimensional solid sphere, n = 2, 3, of radius Re compressed along its boundary in the context of this constrained minimization theory. First, we assume that the solutions for both cases, n = 2 and n = 3, are radially symmetric and reproduce results found in the literature with the aim of validating the computational procedure. We then assume the existence of a second solution of the disk problem (n = 2) that is rotationally symmetric and formulate this problem in a one-dimensional domain (0, Re), instead of the original two-dimensional domain. We compare results obtained from the numerical solution of this problem with computational results found in the literature, which were obtained by considering an asymmetric displacement field defined in a two-dimensional domain. Both solutions predict that the local injectivity constraint is active only in an annulus of inner radius Ra and outer radius Rb. Despite this good agreement, away from the center of the disk, the rotationally symmetric solution is similar to the classical solution and the asymmetric solution is similar to the radially symmetric solution of the constrained minimization theory. To verify the validity of our computational solution, we search numerically for asymmetric solutions defined in the original two-dimensional domain. In addition, we find an analytical expression for the rotationally symmetric solution in the interval (0, Ra) ∪ (Rb, Re) that depends on constants of integration whose values are determined from our computational results. Both approaches, computational and analytical, confirm the existence of the rotationally symmetric solution found computationally. Besides, our results clearly suggest that we must introduce a perturbation in the tangential displacement to obtain the rotationally symmetric solution. Otherwise, we obtain only the radially symmetric solution. Our results indicate that, for a fixed mesh, there are both a maximum shear modulus above which and a minimum load below which the rotationally symmetric solution cannot be obtained. It seems, however, that no threshold values exist in the limit case of an infinitely refined mesh. Moreover, the rotationally symmetric solution yields a lower value of the total potential energy functional when compared to the radially symmetric solution. Finally, we use a regular perturbation method to find approximate solutions of the disk problem in the context of the classical linear elasticity theory and verify that these solutions converge to the closed-form solution of the problem as a perturbation parameter tends to zero. This study aims to use this method to investigate more complex problems for which closed-form solutions are not known.