Mathematical foundations of fracture

A crack is a discontinuity, and classical continuum mechanics does not describe discontinuities gracefully: the strain is undefined exactly where the interesting physics is. There are several ways around that, and this project is about the mathematics behind one family of them, and about what it does and does not guarantee.

The strategy is to replace the derivative with an integral, so material points interact across a finite distance rather than only with their neighbors. Cracks then emerge from the model rather than being tracked by it. Peridynamics is the best-known formulation of that idea and the one most of this work uses, alongside finite element and finite difference discretizations and the variational tools needed to analyze them. Elsewhere in the group the same problems are attacked with phase-field and cohesive-zone formulations; none of the three is the answer to everything, which is part of the point.

What a crack does, and whether the model gets it right

The work with Robert Lipton examined the energy in the process zone and recovered the kinetic relation for crack-tip velocity in the mode-I problem. That is, the model reproduces classical linear elastic fracture mechanics in the limit where LEFM should hold, which is the least a fracture model should be asked to do. The theory is in Lipton and Jha 2021 and Jha and Lipton 2020.

The animation at the top of this page is that problem: mode-I propagation through a pre-cracked plate pulled outward at thin layers along its top and bottom edges, from the published study. Nothing in the model says where the crack should go or how fast it should travel — both come out of it.

Convergence and well-posedness

Much of the doctoral and postdoctoral work behind this project, under Robert Lipton at Louisiana State University, went into a priori error estimates for finite-difference and finite-element discretizations of nonlinear models of this kind, and into proving the underlying problems are well posed at all. Both matter for an unglamorous reason: without them, a simulation that produces a plausible crack gives you no way to tell whether you are looking at the material’s behavior or the mesh’s. The resulting papers are on the publications page.

The regularization has a length scale, and it is worth seeing what that scale does. The sharp discontinuity is replaced by a softening zone of high strain whose width is proportional to the horizon. Shrink the horizon and the zone shrinks with it; in the limit it collapses to a sharp crack. Figures 1–3 show that happening for three horizons.

Damage with memory

Not all failure happens in one loading. In this paper that work proposed a bond-based model with memory: if each cycle takes the material past a critical strain, it loses stiffness cycle after cycle, because the strength at the current time depends on the history rather than only the present state. That idea connects to the fatigue work in magnetic soft composites, where repeated loading changes interface strength in a different material system.