Eventos Anais de eventos
COBEM 2023
27th International Congress of Mechanical Engineering
Numerical investigation of non-smooth solutions in finite elasticity
Submission Author:
Lucas Rocha , SP , Brazil
Co-Authors:
Adair Aguiar, Lucas Rocha
Presenter: Lucas Rocha
doi://10.26678/ABCM.COBEM2023.COB2023-1179
Abstract
We consider the problem of an elastic annular disk with uniform thickness in equilibrium in the absence of body force. The disk is fixed on its inner surface and compressed by a uniform pressure on its outer surface. The disk is made of a transversely isotropic material that has a radial axis of symmetry and a constitutive response that is stiffer in the radial direction than in the tangential direction. Material properties of this type are found in carbon fibers with radial microstructure and fiber-reinforced composites. In the context of the classical linear elasticity theory, the solution of this problem predicts large stresses and strains on the inner surface of the disk for a small enough inner radius. The large strains violate the hypothesis of infinitesimal strains upon which the theory is based. A natural constitutive extension of the linear to the nonlinear elasticity theory consists of considering that the disk is made of a transversely isotropic St Venant-Kirchhoff material. In this work, we formulate the disk problem as a minimization problem of the total potential energy functional. We first solve this minimization problem using a standard numerical procedure. We find that there is a critical value of pressure above which the numerical results are not convergent which coincides with the emergence of a jump discontinuity in the deformation gradient. In addition, this critical pressure yields an upper bound for the range of validity of the linear elasticity theory. We then propose a modified numerical procedure that yields convergent results.
Keywords
Nonlinear elasticity, Transverse isotropy, Minimization, Non-smooth deformation, Finite Element Method
