UC Berkeley

Peridynamics is a nonlocal reformulation of continuum mechanics aiming at modeling fracture and damage in solids. Unlike the finite element method which relies on a predefined mesh, peridynamics is established upon interactions among discrete material points within a given domain. There are two types of peridynamic formulations: bond-based and state-based. Bond-based peridynamics follows a ``bottom-up'' philosophy, where the bond-level mechanical interactions are determined first and their aggregated results emerge as the material behavior in macro-scale. In contrast, state-based peridynamics adopts a ``top-down'' approach where the macroscopic material response is first defined and subsequently distributed to bond-level interactions. This dissertation advances both formulations through new theoretical developments and practical engineering applications. The foundation of this work is the establishment of a unified bond-associated nonlocal deformation gradient. Based on this, a nonlocal-deformation-field-driven bond-based peridynamics framework is introduced. Compared to conventional bond-based formulation, this novel formulation offers (1) consistent kinematic relations in meso- and macro-scales, (2) a more versatile family of pairwise bond potentials that account for both bond stretch and shear, and (3) clear and consistent connections between meso-scale bond behaviors and macro-scale material responses. Furthermore, a unified framework of bond-associated state-based peridynamics is proposed. This framework employs the bond-associated nonlocal deformation gradient to construct nonlocal strain measure, which can be directly integrated into any constitutive models developed in continuum mechanics to obtain stress measures and hence force density state. Compared with conventional state-based formulation, the newly proposed approach eliminates the common issues of material instability or zero-energy mode systematically, while achieving the same accuracy in continuum mechanics problems as finite element method. Both newly-developed formulations are rigorously examined for linear and angular momentum balance, as well as material frame-indifference (stress objectivity). A comprehensive set of numerical examples, from elasticity and nonlinear cohesive fracture to dynamic failure, demonstrates the effectiveness of the methods. Overall, this work provides unified and robust frameworks for advanced bond-based and state-based peridynamic formulations, and it is intended to serve as a valuable reference for further development and application of peridynamics formulations in engineering problems.