A Generalized Peridynamic Model Based on Seth–Hill Bond-Strain Measures for Mixed-Mode Fracture

Abstract

In this study, a generalized peridynamic model incorporating Seth–Hill bond-strain measures is proposed to capture mixed-mode fracture behaviors. We begin with the reformulation of the model introduced by Tupek and Radovitzky within the ordinary state-based peridynamic (OSB-PD) framework, where we demonstrate that the three-dimensional shape tensor state satisfies an integral identity equivalent to the fourth-order symmetric identity tensor. Based on this identity, the shape tensor state tailored for two-dimensional problems is constructed, enabling the derivation of the corresponding scalar force state based on Seth–Hill bond-strain measures for linear elastic materials. This generalized model avoids unphysical material interpenetration and enables the decomposition of the scalar force state over the classic model. Moreover, a nonlocal work-conjugate stress tensor is developed for the first time by employing the reformulated scalar force state based on the principle of work conjugacy and the integral identity of the shape tensor state. Finally, the maximum principal stress and Drucker–Prager failure criteria are incorporated into the generalized OSB-PD framework to enable the simulation of mixed-mode brittle fracture. The accuracy and robustness of the proposed model are validated through several benchmark cases, demonstrating accurate stress evaluation and failure prediction. Notably, the model successfully captures complex crack coalescence patterns in rock subjected to uniaxial compression, underscoring its effectiveness in depicting mixed-mode fracture processes.