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Optimal paths play a fundamental role in numerous physical applications ranging from random polymers to brittle fracture, from the flow through porous media to information propagation. Here for the first time we explore the path that is…
The modeling of cracks is an important topic - both in engineering as well as in mathematics. Since crack propagation is characterized by a free boundary value problem (the geometry of the crack is not known beforehand, but part of the…
We investigate the existence of two nontrivial solutions for a poly-Laplacian system involving concave-convex nonlinearities and parameters with Dirichlet boundary condition on locally finite graphs. By using the mountain pass theorem and…
We are concerned with conservative systems $\ddot{q}=\nabla V(q), \; q\in\mathbb{R}^N$ for a general class of potentials $V\in C^1(\mathbb{R}^N)$. Assuming that a given sublevel set $\{V\leq c\}$ splits in the disjoint union of two closed…
Methods for studying droplets in models with quenched disorder are critically examined. Low energy excitations in two dimensional models are investigated by finding minimal energy interior excitations and by computing the effect of bulk…
We investigate the existence of ground states for a free energy functional on Cartan-Hadamard manifolds. The energy, which consists of an entropy and an interaction term, is associated to a macroscopic aggregation model that includes…
Stretching, drilling, and bending are the independent deformation modes of a thin shell, each of which has an individual energy content. When the energy content of a mode vanishes, that mode is neutral. We characterize all neutral modes of…
In a previous paper (Leblond et al., 2011), we proposed a theoretical interpretation of the experimentally well known instability of coplanar crack propagation in mode I+III. The interpretation relied on a stability analysis based on…
In this paper, we deal with an obstacle placement problem inside a disk, that can be formulated as an energy minimization problem with respect to the rotations of the obstacle about its center, with respect to the translations of the…
The dynamic energy balance is essential for earthquake studies. The energy balance approach is one of the most famous developments in fracture mechanics. To interpret seismological data, crack models and sliding on a frictional surface…
The dynamics of a crack propagating in an elastic inhomogeneous material is investigated. The variations of the average crack velocity with the external loading are measured for a brittle rock and are shown to display two distinct regimes:…
While cracks in isotropic homogeneous materials propagate straight, perpendicularly to the tensile axis, cracks in natural and synthetic composites deflect from a straight path, often increasing the toughness of the material. Here we…
Multifragmentation is the most extensively studied phenomena in this energy domain. In the past people have tried to develop various methods of clusterization[1]. Among these methods minimum spanning tree[1] is one of the fastest method. In…
We study the systematic numerical approximation of a class of Allen-Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an…
Predicting when rupture occurs or cracks progress is a major challenge in numerous elds of industrial, societal and geophysical importance. It remains largely unsolved: Stress enhancement at cracks and defects, indeed, makes the macroscale…
We investigate low energy structures of a lattice with dislocations in the context of nonlinear elasticity. We show that these low energy configurations exhibit in the limit a Cosserat-like behavior. Moreover, we give bounds from above and…
We investigate the shrinkage induced breakup of thin layers of heterogeneous materials attached to a substrate, a ubiquitous natural phenomenon with a wide range of potential applications. Focusing on the evolution of the fragment ensemble,…
The dynamics of tensile crack fronts restricted to advance in a plane are studied. In an ideal linear elastic medium, a propagating mode along the crack front with a velocity slightly less than the Rayleigh wave velocity, is found to exist.…
In this paper, we get the sharpest known to date lower bounds for the minimal Green energy of the compact harmonic manifolds of any dimension.
In viscoelastic materials, individually short-lived bonds collectively result in a mechanical resistance which is long-lived but finite, as ultimately cracks appear. Here we provide a microscopic mechanism by which cracks emerge from the…