A 3D finite volume scheme for solving the updated Lagrangian form of hyperelasticity

Summary The finite volume discretization of nonlinear elasticity equations seems to be a promising alternative to the traditional finite element discretization as mentioned by Lee et al. [Computers and Structures (2013)]. In this work, we propose to solve the elastic response of a solid material by using a cell‐centered finite volume Lagrangian scheme in the current configuration. The hyperelastic approach is chosen for representing elastic isotropic materials. In this way, the constitutive law is based on the principle of frame indifference and thermodynamic consistency, which are imposed by mean of the Coleman–Noll procedure. It results in defining the Cauchy stress tensor as the derivative of the free energy with respect to the left Cauchy–Green tensor. Moreover, the materials being isotropic, the free‐energy is function of the left Cauchy–Green tensor invariants, which enable the use of the neo‐Hookean model. The hyperelasticity system is discretized using the cell‐centered Lagrangian scheme from the work of Maire et al. [J. Comput. Phys. (2009)]. The 3D scheme is first order in space and time and is assessed against three test cases with both infinitesimal displacements and large deformations to show the good accordance between the numerical solutions and the analytic ones. Copyright © 2016 John Wiley & Sons, Ltd. This paper presents the 3D extension of the EUCCLHYD scheme (Explicit Unstructured Cell‐Centered Lagrangian HYDrodynamics) [Maire SIAM 2007] for the numerical modeling of the hyperelasticity system at first order in space and time. The constitutive law is derived by means of a Coleman–Noll procedure in the case of isotropic neo‐Hookean materials. The scheme is validated on three test cases and is proved to have an inherent 3D nature when shearing is present such as in the oscillating beam problem (refer to Figure 1).