Christopher J. Larsen

There are very few existence results for fracture evolution, outside of globally minimizing quasi-static evolutions. Dynamic evolutions are particularly problematic, due to the difficulty of showing energy balance, as well as of showing that solutions obey a maximal dissipation condition, or some similar condition that prevents stationary cracks from always being solutions. Here we introduce a new weak maximal dissipation condition and show that it is compatible with cracks constrained to grow smoothly on a smooth curve. In particular, we show existence of dynamic fracture evolutions satisfying this maximal dissipation condition, subject to the above smoothness constraints, and exhibit explicit examples to show that this maximal dissipation principle can indeed rule out stationary cracks as solutions.

Our goal in this paper is to initiate a mathematical study of dynamic cohesive fracture. Mathematical models of static cohesive fracture are quite well understood, and existence of solutions is known to rest on properties of the cohesive energy density $\psi$, which is a function of the jump in displacement. In particular, a relaxation is required (and a relaxation formula is known) if $\psi'(0^+)\not=\infty$. However, formulating a model for dynamic fracture when $\psi'(0^+)=\infty$ is not straightforward, compared to when $\psi'(0^+)$ is finite, and especially compared to when $\psi$ is smooth. We therefore formulate a model that is suitable when $\psi'(0^+)=\infty$ and also agrees with established models in the more regular case. We then analyze the one-dimensional case and show existence when a finite number of potential fracture points are specified a priori, independent of the regularity of $\psi$. We also show that if $\psi'(0^+)<\infty$, then relaxation is necessary without this constraint, at least for some initial data.

In this paper we formulate and study scalar wave equations on domains with arbitrary growing cracks. This includes a zero Neumann condition on the crack sets, and the only assumptions on these sets are that they have bounded surface measure and are growing in the sense of set inclusion. In particular, they may be dense, so the weak formulations must fall outside of the usual weak formulations using Sobolev spaces. We study both damped and undamped equations, showing existence and, for the damped equation, uniqueness and energy conservation.

There has been much recent progress in extending Griffith's criterion for crack growth into mathematical models for quasi-static crack evolution that are well-posed, in the sense that there exist solutions that can be numerically approximated. However, mathematical progress toward dynamic fracture (crack growth consistent with Griffith's criterion, together with elastodynamics) has been more meager. We describe some recent results on a phase-field model of dynamic fracture, and introduce models for "sharp interface" dynamic fracture.

Existence and convergence results are proved for a regularized model of dynamic brittle fracture based on the Ambrosio--Tortorelli approximation. We show that the time-discrete elastodynamics proposed by Bourdin, Larsen \& Richardson as a numerical model for dynamic fracture converges, as the time-step approaches zero, to a solution of the natural time-continuous elastodynamics, and that this solution satisfies an energy balance. We emphasize that these models do not specify crack-paths a priori, but predict them, including such complicated behavior as kinking, crack branching, and so forth, in any spatial dimension.

We propose a discrete time model for dynamic fracture based on crack regularization. The advantages of our approach are threefold: first, our regularization of the crack set has been rigorously shown to converge to the correct sharp-interface energy~\cite{Ambrosio-Tortorelli-1990, Ambrosio-Tortorelli-1992}; second, our condition for crack growth, based on Griffith's criterion, matches that in quasi-static settings~\cite{Bourdin}, where Griffith originally stated his criterion; third, solutions to our model converge, as the time-step tends to zero, to solutions of the correct continuous time model~\cite{Larsen-Ortner-Suli}. Furthermore, in implementing this model, we naturally recover several features, such as the elastic wave speed as an upper bound on crack speed, and crack branching for sufficiently rapid boundary displacements. We conclude by comparing our approach to so-called "phase-field" ones. In particular, we explain why phase-field approaches are good for approximating free boundaries, but not the free discontinuity sets that model fracture.

We present energetic and strain-threshold models for the quasi-static evolution of brutal brittle damage for geometrically-linear elastic materials. By allowing for anisotropic elastic moduli and multiple damaged states we present the issues for the first time in a truly elastic setting, and show that the threshold methods developed in (Garroni, A., Larsen, C. J., Threshold-based quasi-static brittle damage evolution, Archive for Rational Mechanics and Analysis 194 (2), 585

Variational methods for studying cohesive fracture and elastoplasticity have generally relied on mini- mizing an energy functional that is the sum of a stored elastic energy and a defect energy, corresponding to fracture or plasticity. The usual method for showing existence of minimizers is the Direct Method, whose success requires some properties of the defect energy that are not physically motivated, or in fact are contrary to physically desired properties. Here we introduce a new variational principle based on the idea of the defect, in the spirit of [9], reflecting the notion that these defects occur only if necessary in order for the elastic stress to be admissible, i.e., under the critical stress at which fracture or plasticity begins. The advantage is that the Direct Method only comes into play with a constraint on the defect set, which obviates some of the technical issues usually involved. The most significant advantage is that existence of global minimizers generally requires an infinite stress or strain threshold for plasticity or fracture, while our formulation is appropriate for finite thresholds. A further advantage is that the method produces local minimizers or locally stable states, rather than less physical global minimizers. General existence results will require new methods, but here we easily show existence in one dimension for both static and quasi-static solutions, even when global minima do not exist.

The mathematical analysis developed for energy minimizing fracture evolutions has been difficult to extend to locally minimizing evolutions. The reasons for this difficulty are not obvious, and our goal in this paper is to describe in some detail what precisely the issues are and why the previous analysis in fact cannot be extended to the most natural models based on local minimality. We also indicate how the previous methods can be modified for the analysis of models based on a recent definition of stability that is a bit stronger than local minimality.

We introduce a new definition of stability, $\ep$-stability, that implies local minimality and is robust enough for passing from discrete-time to continuous-time quasi-static evolutions, even with very irregular energies. We use this to give the first existence result for quasi-static crack evolutions that both predicts crack paths and produces states that are local minimizers at every time, but not necessarily global minimizers. The key ingredient in our model is the physically reasonable property, absent in global minimization models, that whenever there is a jump in time from one state to another, there must be a continuous path from the earlier state to the later along which the energy is almost decreasing. It follows that these evolutions are much closer to satisfying Griffith's criterion for crack growth than are solutions based on global minimization, and initiation is more physical than in global minimization models.

We introduce models for static and quasi-static damage in elastic materials, based on a strain threshold, and then investigate the relationship between these threshold models and the energy-based models introduced by Francfort and Marigo (1993) and Francfort and Garroni (2006). A somewhat surprising result is that, while classical solutions for the energy models are also threshold solutions, this is not the case for nonclassical solutions, i.e., solutions with microstructure. A new and arguably more physical definition of solutions with microstructure for the energy-based model is then given, in which the energy minimality property is satisfied by sequences of sets that generate the effective elastic tensors, rather than by the tensors themselves. We prove existence for this energy based problem, and show that these solutions are also threshold solutions. A byproduct of this analysis is that all local minimizers, in both the classical setting and for the new microstructure definition, are also global minimizers.

Crack fronts play a fundamental role in engineering models for fracture: they are the location of both crack growth and the energy dissipation due to growth. However, there has not been a rigorous mathematical definition of crack front, nor rigorous mathematical analysis predicting fracture paths using these fronts as the location of growth and dissipation. Here, we give a natural weak definition of crack front and front speed, and consider models of crack growth in which the energy dissipation is a function of the front speed, i.e., the dissipation rate at time $t$ is of the form \[\int_{F(t)} \psi(v(x,t)) d{\mathcal H}^{N-2}(x)\] where $F(t)$ is the front at time $t$ and $v$ is the front speed. We show how this dissipation can be used within existing models of quasi-static fracture, as well as in the new dissipation functionals of Mielke-Ortiz. An example of a constrained problem for which there is existence is shown, but in general, if there are no constraints or other energy penalties, this dissipation must be relaxed. We prove a general relaxation formula that gives the surprising result that the effective dissipation is always rate-independent.

Existing level set methods for the Mumford-Shah functional have been incapable of obtaining certain features, such as crack-tips and the presence of only triple junctions, which are known to occur in Mumford-Shah minimizers (and corresponding variational models for fracture). We introduce a new level set method for computing stationary points of certain free discontinuity problems that does obtain these critical features. Numerical experiments are presented to validate the new level set method.

We introduce new definitions of convergence, based on adding stability criteria to $\Gamma$-convergence, that are suitable in many cases for studying convergence of local minimizers.