Fracture driving forces in finite strain corotational finite element methods

Abstract

The evaluation of 3D fracture driving forces under finite strain regimes is crucial for damage tolerance assessments of critical engineering structures. This technical note compares the performance and limitations of two complementary methodologies integrated within updated Lagrangian corotational solvers: the Eshelby-based domain integral method (formally equivalent to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G-\theta$$\end{document} formulation) and a direct boundary kinematic extraction approach. We demonstrate that while the continuous Eshelby-based domain integral theoretically preserves path-independence, its discrete finite element implementation struggles to maintain a generic workflow across all strain regimes. Under large-rotation, small-strain conditions, reconstructing the deformation gradient from nodal displacements is vulnerable to single-precision floating-point cancellation. Conversely, under large-strain, crack-tip blunting regimes, the domain integral is severely corrupted by volumetric locking on linear elements, in contrast to standard small-strain J-integral approximations using these elements. These limitations are discussed alongside a direct surface kinematic extraction method regularized via a second-order Total Variation (ADMM-TV2) scheme. Operating strictly on boundary displacement jumps, this kinematic approach offers an efficient, low-cost alternative that avoids volumetric reconstruction pathologies. Rather than advocating for a single superior methodology, we highlight the complementarity of these approaches, showing how boundary kinematics can reliably support volumetric integration on highly irregular meshes. Volumetric Eshelby integrals are prone to floating-point cancellation under finite rotations. Boundary kinematic TV2 extraction decouples fracture driving forces from element locking. Regularized surface kinematics accurately capture steep driving force gradients near free boundaries. Combined volumetric-kinematic toolchain validated on transient 3D contact gear mechanics. Volumetric Eshelby integrals are prone to floating-point cancellation under finite rotations. Boundary kinematic TV2 extraction decouples fracture driving forces from element locking. Regularized surface kinematics accurately capture steep driving force gradients near free boundaries. Combined volumetric-kinematic toolchain validated on transient 3D contact gear mechanics.