Tearing energy calculation in hyperelastic fracture mechanics using the local and global complex-variable finite element method. (15th March 2022) - Ytuarte, Ernest; Aguirre-Mesa, Andres M.; Ramirez-Tamayo, Daniel; Avila, David; Millwater, Harry; Restrepo, David; Montoya, Arturo
Abstract: The "local" and "global" complex-variable finite element methods were extended for computing the energy release rate (ERR) of materials undergoing nonlinear elastic deformation at both small and large strains. These methods compute the derivative of the strain energy with respect to crack length by applying a virtual crack extension through the imaginary domain of complex variables. The "global" complex finite element method (G-ZFEM) is a general sensitivity method that provides accurate ERR estimates regardless of the material and loading conditions in the cracked body. However, as G-ZFEM duplicates all the degrees of freedom of a finite element model to store information pertinent to the ERR, this analysis adds significant computational overhead. Local-based methods, analogous to the J -integral, can be employed to improve the computational efficiency of the fracture analysis. The "local" complex-variable finite element method (L-ZFEM) is a fracture-specific approach where complex-valued calculations are only conducted at a small group of elements surrounding the crack tip, adding minimum overhead to the analysis. The G-ZFEM and L-ZFEM methods were incorporated within the commercial finite element software ABAQUS through a two-dimensional user element that was linked to several material models, including Ramberg–Osgood, Neo-Hookean, and Mooney–Rivlin. The ERR, known as tearing energy (TE) in hyperelastic fracture problems, was obtained by the proposed methods in benchmark problems and found to be in excellent agreement with analytical formulations and the J -integral results. Unlike L-ZFEM and the J -integral, the G-ZFEM method does not exhibit variations in TE estimates with respect to the size or direction of the virtual crack extension in nonhomogeneous cracked bodies; thus, G-ZFEM estimates can be regarded as the reference tearing energy solutions. In addition, by using multidual or multicomplex algebra, the complex finite element methods allow the computation of arbitrary, higher-order derivatives of the strain energy with respect to the model's material, loading, or geometric properties, providing additional information to design structural components with maximum resistance to fracture. The complex-variable methods are versatile and can be easily adapted to any nonlinear elastic material formulation and extended to 3D in a straightforward manner.