Peridynamics, Fracture, and Nonlocal Continuum Models

Most physical processes are the result of collective interactions across disparate length and time scales. The dynamic fracture of brittle solids is a particularly interesting collective interaction connecting large and small length scales. With the application of enough stress or strain to a sample of brittle material, atomistic-scale bonds will eventually snap, leading to fracture of the macroscopic specimen. The classic theory of dynamic fracture [7, 10] is based on the notion of a deformable continuum containing a crack. The crack is mathematically modeled as a branch cut that begins to move when an infinitesimal extension of the crack releases more energy than needed to create a fracture surface. Classic fracture theory, together with experiment, has been enormously successful in characterizing and measuring the resistance of materials to crack growth— and thereby enabling engineering design. However, the capability to quantitatively predict the dynamics of multiple propagating cracks that are free to nucleate, change course, bifurcate, and, indeed, stop if they choose lies completely outside the classic approach. Armed with supercomputers, contemporary science is engaged in the quest for a multiscale framework for quantitatively predicting the dynamics of multiple cracks that freely propagate and interact. Investigators realize the importance of quantifying the influence of macroscopic forces on the dynamics at the length scales at which atomic bonds are broken. Bottom-up approaches, recognizing the inherent discreteness of fracture through lattice models, have provided penetrating insight into the dynamics of the fracture processes [2,12,13,20]. Nevertheless, numerical simulations of fine-grained atomistic models, while offering important and necessary insight into the fracture process, do not scale up to finite-size samples with multiple freely propagating cracks. Complementary to the bottom-up approaches are top-down computational approaches that use cohesive zone elements [9, 22]. More recently, cohesive zones have been applied within the extended finite element method [1] to minimize the effects of mesh dependence on free crack paths. Current challenges facing these methods (indeed, all computational methods) include multiple growing cracks interacting in complex patterns. What remains elusive is an underlying continuum model that can seamlessly evolve both smooth and discontinuous deformation in a way that is useful for predicting free crack propagation. To be applicable, a model must be able to deliver quantifiable results and recover the classic results of fracture mechanics in situations in which it is known to hold. The peridynamic continuum model [17,18], a spatially nonlocal continuum theory, was introduced recently to fill this gap. Each material point interacts through short-range forces with other points inside a horizon of prescribed diameter δ. The short-range forces depend on the relative displacement between material points and are derived from a peridynamic potential specifying a kinematic constitutive relation. Within the recently developed nonlocal vector calculus framework [4], peridynamics can be viewed as nonlocal balance laws involving nonlocal fluxes defined between material domains that might not have a common boundary. This provides an alternative to standard approaches for circumventing the technicalities associated with the lack of sufficient regularity in local balance laws; by avoiding the explicit use of spatial derivatives, the approach allows for both smooth and discontinuous deformations. For short-range forces akin to elastic bonds that break when stretched beyond a critical point, the peridynamic formulation delivers remarkable simulations, capturing both crack branching (Figure 1) and multiple crack interactions (Figure 2). To test the theory of the peridynamic model, investigators have developed new mathematical results on its well-posedness and have assessed its connection to accepted continuum field theories. In a recent study, for linear elastic short-range forces and up-scaled linear peridynamics, which sent the peridynamic horizon δ to zero, the macroscopic limit of peridynamics was found to satisfy the classic equations of linear elasticity, with the macroscopic elastic moduli given by moments of the peridynamic nonlocal interaction kernel. Such relations can be established formally for smooth functions via simple Taylor expansions [6,19] and more rigorously in functional-analytic settings for solutions with minimal regularity [5,14]. Progress has also been made in developing a nonlocal calculus of variations for the analysis of variational and time-dependent problems subject to various nonlocal boundary conditions or, more precisely, Figure 1. Peridynamic simulation of dynamic fracture starting from a short edge crack. Areas where damage has occurred are shown in blue and green. The active process zones where damage is increasing are shown in red. Because the plate is stretched at a constant rate, the cracks see a higher and higher strain field ahead of them as they grow.