A total Lagrangian Galerkin free element method for finite deformation in hyperelastic materials

• A total Lagrangian Galerkin free element method is proposed for finite deformation. • Hyperelastic materials are considered calculated by the Galerkin free element method. • 2D and 3D nonlinear problems are solved to verify the accuracy. • Favorable results can be obtained using the proposed method for the nearly incompressible materials. In this research, a total Lagrangian Galerkin free element method (GFrEM) is proposed for the analysis of finite deformation in hyperelastic materials. This method derives the total Lagrangian formulation using the initial configuration as the reference. The mechanical behavior of hyperelastic materials is modeled by the non-Hookean strain energy function. Since Lagrangian isoparametric elements are freely formed in GFrEM by collocation nodes with their surrounding nodes, intrinsic boundary conditions can be imposed simply as in the finite elements method. In addition, the Galerkin method was used to ensure the stability of the results when constructing the equations for each collocation node. The validity and convergence of the proposed method are verified by several two- and three-dimensional numerical examples that include bending, compression, and torsion of hyperelastic materials. The example of nearly incompressible material shows that GFrEM remains highly accurate even with large deformations where the FEM cannot converge.