Análise computacional tridimensional de sólidos hookeanos utilizando a família de deformação seno hiperbólico generalizada

Abstract

The proposition of constitutive equations for the modeling of isotropic hyperelastic solids under moderate/large deformations is still a challenge in the scientific community, considering the nonlinear behavior they exhibit in this regime. This work, in a 3D context, computationally implements and investigates a constitutive equation derived from Hooke's law, which uses the generalized hyperbolic sine strain tensor and its corresponding energetically conjugated stress tensor. The positional formulation of the finite element method is employed, initially written considering a generic Lagrangian energetically conjugate pair. A simple and general strategy was developed for determining the derivatives of the hyperbolic sine strain tensor with respect to the deformation gradient. Additionally, the Newton-Raphson method is used to solve the nonlinear problem. To demonstrate the applicability of the formulation, some numerical examples are conducted. Using experimental results from axial extension, equibiaxial extension, and simple shear obtained from the literature, calibrations were performed to determine the constants of the proposed model, which has four parameters. They resulted in remarkable agreement between the analytical/numerical and experimental data. The versatility of the constructed model allowed for the modeling of rubbers, biological tissues, and foams. Finally, three numerical examples of non-homogeneous deformation, whose results qualitatively show an excellent performance of the formulation, are presented.