Phone:  (508) 8316124 

Dept. Fax:  (508) 8315824 
Office:  109B Stratton Hall

Email:  cjlarsen@wpi.edu 
US Mail:  Department of Mathematical Sciences
Worcester Polytechnic Institute 100 Institute Road Worcester, MA 01609 
Upcoming conference at CIRM on Shape Optimization and Isoperimetric and Functional Inequalities, November, 2016.
Recent workshop on Advances in the mathematical analysis of material defects in elastic solids, June, 2016.
Recent workshop at the Banff International Research Station on Variational Models of Fracture, May, 2016.
Recent workshop at the Lorentz Center on Microstructure evolution in materials: defects, cracks, and interfaces. April, 2016.
Recent undergraduate course at Park City Mathematics Institute ~ Institute for Advanced Study on "variational methods for materials science," JuneJuly, 2014. Abstract.
Video of IPAM lecture (December, 2012)
Lecture Notes for SISSA minicourse, "Mathematical Issues in Quasistatic and Dynamic Fracture Evolution" (March, 2013)
The main focus of my research has been on the interplay between bulk and surface energies in variational problems, motivated largely by questions in Materials Science (e.g., optimal design of composite materials, fracture mechanics, debonding of thin films). My current emphasis is on the evolution of damage and fracture, and in particular, on formulating and analyzing models for predicting damage sets and crack paths. The analysis requires (new) tools from the Calculus of Variations, Partial Differential Equations, and Geometric Measure Theory. Current support:
NSF 1616197. Additional support provided by ERC Advanced Grant 290888.
Below are recent papers with abstracts, by topic.
Dynamic Fracture Evolution
 G. Dal Maso, C. J. Larsen, and R. Toader. Existence for constrained dynamic Griffith
fracture with a weak maximal dissipation condition, JMPS, submitted.
There are very few existence results for fracture evolution, outside of globally minimizing quasistatic 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.
 C. J. Larsen and V. Slastikov. Dynamic cohesive fracture: models and analysis, Math. Models Methods Appl. Sci,24 (2014), pp. 18571875.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 onedimensional 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.
 G. Dal Maso and C. J. Larsen. Existence for wave equations on domains with arbitrary growing cracks, Rend. Lincei Mat. App., 22 (2011), pp. 387408. (special issue in honor of Giovanni Prodi)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.
 C. J. Larsen. Models for dynamic fracture based on Griffith's criterion,
in IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials (Klaus Hackl, ed.), Springer, 2010, pp. 131140.
There has been much recent progress in extending Griffith's criterion for crack growth into mathematical models for quasistatic crack evolution that are wellposed, 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 phasefield model of dynamic fracture, and introduce models for "sharp interface" dynamic fracture.

C. J. Larsen, C. Ortner, and E. Süli. Existence of
solutions to a regularized model of dynamic fracture,
Math. Models Methods Appl. Sci. 20 (2010), pp. 10211048.
Existence and convergence results are proved for a regularized model of dynamic brittle fracture based on the AmbrosioTortorelli approximation. We show that the timediscrete elastodynamics proposed by Bourdin, Larsen \& Richardson as a numerical model for dynamic fracture converges, as the timestep approaches zero, to a solution of the natural timecontinuous elastodynamics, and that this solution satisfies an energy balance. We emphasize that these models do not specify crackpaths a priori, but predict them, including such complicated behavior as kinking, crack branching, and so forth, in any spatial dimension.

B. Bourdin, C. J. Larsen, and C. L. Richardson. A timediscrete model for dynamic fracture based on crack regularization,
Int. J. Fracture 168 (2011), pp. 133143.
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 sharpinterface energy~\cite{AmbrosioTortorelli1990, AmbrosioTortorelli1992}; second, our condition for crack growth, based on Griffith's criterion, matches that in quasistatic settings~\cite{Bourdin}, where Griffith originally stated his criterion; third, solutions to our model converge, as the timestep tends to zero, to solutions of the correct continuous time model~\cite{LarsenOrtnerSuli}. 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 socalled "phasefield" ones. In particular, we explain why phasefield approaches are good for approximating free boundaries, but not the free discontinuity sets that model fracture.
Local Minimality and Material Defects (Fracture, Damage, and Plasticity)

I. Chenchiah and C. J. Larsen.
Quasistatic brittle damage evolution in elastic materials
with multiple damaged states, Arch. Ration. Mech. Anal. 215 (2015), pp. 831866..
We present energetic and strainthreshold models for the quasistatic evolution of brutal brittle damage for geometricallylinear 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., Thresholdbased quasistatic brittle damage evolution, Archive for Rational Mechanics and Analysis 194 (2), 585

C. J. Larsen.
A new variational principle for cohesive fracture and elastoplasticity, Mech. Re. Commun. (2013), http://dx.doi.org/10.1016/j.mechrescom.2013.10.025.
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 quasistatic solutions, even when global minima do not exist.

C. J. Larsen.
Local minimality and crack prediction in quasistatic Griffith fracture evolution, Discrete Contin. Dyn. Syst. Series S 6 (2013), pp. 121129.
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.

C. J. Larsen.
Epsilonstable quasistatic brittle fracture evolution,
Comm. Pure Appl. Math. 63 (2010), pp. 630654.
We introduce a new definition of stability, $\ep$stability, that implies local minimality and is robust enough for passing from discretetime to continuoustime quasistatic evolutions, even with very irregular energies. We use this to give the first existence result for quasistatic 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.
 A. Garroni and C. J. Larsen. Thresholdbased quasistatic brittle damage evolution, Arch. Ration. Mech. Anal. 194 (2009), pp. 585609.
We introduce models for static and quasistatic damage in elastic materials, based on a strain threshold, and then investigate the relationship between these threshold models and the energybased 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 energybased 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.
Globally Minimizing QuasiStatic Evolution

C. J. Larsen, M. Ortiz, and
C. Richardson. Fracture paths from front kinetics: relaxation and rateindependence, Arch. Ration. Mech. Anal. 193 (2009), pp. 539583.
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 \[\int_{F(t)} \psi(v(x,t)) d{\mathcal H}^{N2}(x)\] 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 quasistatic fracture, as well as in the new dissipation functionals of MielkeOrtiz. 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 rateindependent.
 C. J. Larsen, C. Richardson and M. Sarkis.
A level set method for the MumfordShah functional and fracture, Technical Report Serie A 581 (2008), Instituto de Matematica Pura e Aplicada, Brazil.
Existing level set methods for the MumfordShah functional have been incapable of obtaining certain features, such as cracktips and the presence of only triple junctions, which are known to occur in MumfordShah 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.
Gamma Convergence

A. Braides and C. J. Larsen.
Gammaconvergence for
stable states and local minimizers,
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. X (2011), 193206.
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.