Related papers: Long time behavior of dynamic solution to Peierls-…
We present a time-dependent Ginzburg-Landau model of nonlinear elasticity in solid materials. We assume that the elastic energy density is a periodic function of the shear and tetragonal strains owing to the underlying lattice structure.…
We consider a family of evolution equations that generalize the Peierls-Nabarro model for crystal dislocations. They can be seen as semilinear parabolic reaction-diffusion equations in which the diffusion is regulated by a fractional…
The Peierls-Nabarro (PN) model for dislocations is a hybrid model that incorporates the atomistic information of the dislocation core structure into the continuum theory. In this paper, we study the convergence from a full atomistic model…
We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian %This model describes the evolution of phase transitions associated to dislocations. whose solution represents the atom dislocation in…
We consider the equation $$v_t=L_s v-W'(v)+\sigma_\epsilon(t,x) \quad {\mbox{ in }} (0,+\infty)\times\R,$$ where $L_s$ is an integro-differential operator of order $2s$, with $s\in(0,1)$, $W$ is a periodic potential, and $\sigma_\epsilon$…
We focus on existence and rigidity problems of the vectorial Peierls-Nabarro (PN) model for dislocations. Under the assumption that the misfit potential on the slip plane only depends on the shear displacement along the Burgers vector, a…
We derive a continuum-level plasticity model for polycrystalline materials in the high energy density regime, based on a single dislocation density and single mobility mechanism, with an evolution model for the dislocation density. The…
Nonlinear relaxation dynamics of a vertically-vibrated granular pile is experimentally studied. In the experiment, the flux and slope on the relaxing pile are measured by using a high-speed laser profiler. The relation of these quantities…
This paper evaluates qualitatively as well as quantitatively the accuracy of a recently proposed Peierls--Nabarro Finite Element (PN-FE) model for dislocations by a direct comparison with an equivalent molecular statics simulation. To this…
In this paper we study the connection between four models describing dislocation dynamics: a generalized 2D Frenkel-Kontorova model at the atomic level, the Peierls-Nabarro model, the discrete dislocation dynamics and a macroscopic model…
This work rigorously implements a recent model of large-strain elasto-plastic evolution in single crystals where the plastic flow is driven by the movement of discrete dislocation lines. The model is geometrically and elastically nonlinear,…
Materials that undergo internal transformations are usually described in solid mechanics by multi-well energy functions that account for both elastic and transformational behavior. In order to separate the two effects, physicists use…
An extended Maxwell viscoelastic model with a relaxation parameter is studied from mathematical and numerical points of view. It is shown that the model has a gradient flow property with respect to a viscoelastic energy. Based on the…
We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous…
We study the relaxation dynamics of systems of straight, parallel crystal dislocations, starting from initially random and uncorrelated positions of the individual dislocations. A scaling model of the relaxation process is constructed by…
In the modeling of dislocations one is lead naturally to energies concentrated on lines, where the integrand depends on the orientation and on the Burgers vector of the dislocation, which belongs to a discrete lattice. The dislocations may…
We study the global existence of a unique strong solution and its large-time behavior of a two-phase fluid system consisting of the compressible isothermal Euler equations coupled with compressible isentropic Navier-Stokes equations through…
Dislocation is one of the most critical and fundamental crystal defects that dominate the mechanical behavior of crystalline solids, however, a quantitative determination of its character and property in experiments is quite challenging and…
We consider a reaction-diffusion equation with a half-Laplacian. In the case where the solution is independent on time, the model reduces to the Peierls-Nabarro model describing dislocations as transition layers in a phase field setting. We…
In this note we study the singular vanishing-viscosity limit of a gradient flow set in a finite-dimensional Hilbert space and driven by a smooth, but possibly non convex, time-dependent energy functional. We resort to ideas and techniques…