This paper is concerned with a result of homogenization of an integro-differential equation describing dislocation dynamics. Our model involves both an anisotropic Lévy operator of order 1 and a potential depending periodically on $u/\ep$. The limit equation is a non-local Hamilton-Jacobi equation, which is an effective plastic law for densities of dislocations moving in a single slip plane.
In this paper we consider the time dependent Peierls-Nabarro model in dimension one. This model is a semi-linear integro-differential equation associated to the half Laplacian. This model describes the evolution of phase transitions associated to dislocations. At large scale with well separated dislocations, we show that the dislocations move at a velocity proportional to the effective stress. This implies Orowan's law which claims that the plastic strain velocity is proportional to the product of the density of dislocations by the effective stress.
We study a generalization of the fully overdamped Frenkel-Kontorova model in dimension $n\geq 1.$ This model describes the evolution of the position of each atom in a crystal, and is mathematically given by an infinite system of coupled first order ODEs. We prove that for a suitable rescaling of this model, the solution converges to the solution of a Peierls-Nabarro model, which is a coupled system of two PDEs (typically an elliptic PDE in a domain with an evolution PDE on the boundary of the domain). This passage from the discrete model to a continuous model is done in the framework of viscos…
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 to the PN model with $γ$-surface for the dislocation in a bilayer system (e.g. bilayer graphene). We prove that the displacement field of and the total energy of the dislocation solution of the PN model are asymptotically close to those of the full atomistic model. Our work can be considered as a generalization of the analysis of the convergence from atomisti…
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 a crystal. The equation comprises the evolutive version of the classical Peierls-Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a well known equation called by Head \cite{H} "the equation of motion of the dislocation continuum". The limit equation is a model for the macroscopic c…
In this paper we study the relaxation process of Peierls-Nabarro dislocation model, which is a gradient flow with singular nonlocal energy and double well potential describing how the materials relax to its equilibrium with the presence of a dislocation. We prove the dynamic solution to Peierls-Nabarro model will converge exponentially to a shifted steady profile which is uniquely determined.
In this paper, we perform mathematical validation of the Peierls--Nabarro (PN) models, which are multiscale models of dislocations that incorporate the detailed dislocation core structure. We focus on the static and dynamic PN models of an edge dislocation. In a PN model, the total energy includes the elastic energy in the two half-space continua and a nonlinear potential energy across the slip plane, which is always infinite. We rigorously establish the relationship between the PN model in the full space and the reduced problem on the slip plane in terms of both governing equations and energy…
We study the well-posedness of the vector-field Peierls-Nabarro model for curved dislocations with a double well potential and a bi-states limit at far field. Using the Dirichlet to Neumann map, the 3D Peierls-Nabarro model is reduced to a nonlocal scalar Ginzburg-Landau equation. We derive an integral formulation of the nonlocal operator, whose kernel is anisotropic and positive when Poisson's ratio $ν\in(-\frac12, \frac13)$. We then prove that any bounded stable solutions to this nonlocal scalar Ginzburg-Landau equation has a 1D profile, which corresponds to the PDE version of flatness resul…
Plasticity of metals is the emergent phenomenon of many crystal defects (dislocations) which interact and move on microscopic time and length scales. Two of the commonly used models to describe such dislocation dynamics are the Peierls-Nabarro model and the so-called discrete dislocation dynamics model. However, the consistency between these two models is known only for a few number of dislocations or up to the first time at which two dislocations collide. In this paper we resolve these restrictions, and establish the consistency for any number of dislocations and without any restriction on th…
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 reduced non-local scalar Ginzburg-Landau equation with an anisotropic positive (if Poisson ratio belongs to $(-1/2,1/3)$) singular kernel is derived on the slip plane. We first prove that minimizers of the PN energy for this reduced scalar problem exist. Starting from $H^{1/2}$ regularity, we prove that these minimizers are smooth 1D profiles only depending on…
In this paper, we prove the convergence from the atomistic model to the Peierls--Nabarro (PN) model of two-dimensional bilayer system with complex lattice. We show that the displacement field of the dislocation solution of the PN model converges to the dislocation solution of the atomistic model with second-order accuracy. The consistency of PN model and the stability of atomistic model are essential in our proof. The main idea of our approach is to use several low-degree polynomials to approximate the energy due to atomistic interactions of different groups of atoms of the complex lattice.
We derive strain-gradient plasticity from a nonlocal phase-field model of dislocations in a plane. Both a continuous energy with linear growth depending on a measure which characterizes the macroscopic dislocation density and a nonlocal effective energy representing the far-field interaction between dislocations arise naturally as scaling limits of the nonlocal elastic interaction. Relaxation and formation of microstructures at intermediate scales are automatically incorporated in the limiting procedure based on $Γ$-convergence.
We construct multiple families of solitary standing waves of the discrete cubically nonlinear Schrödinger equation (DNLS) in dimensions $d=1,2$ and $3$. These states are obtained via a bifurcation analysis about the continuum (NLS) limit. One family consists {\it on-site symmetric} (vertex-centered) states; these are spatially localized solitary standing waves which are symmetric about any fixed lattice site. The other spatially localized states are {\it off-site symmetric}. Depending on the spatial dimension, these may be bond-centered, cell-centered, or face-centered. Finally, we show that t…
We study a class of discrete focusing nonlinear Schr{ö}dinger equations (DNLS) with general nonlocal interactions. We prove the existence of onsite and offsite discrete solitary waves, which bifurcate from the trivial solution at the endpoint frequency of the continuous spectrum of linear dispersive waves. We also prove exponential smallness, in the frequency-distance to the bifurcation point, of the Peierls-Nabarro energy barrier (PNB), as measured by the difference in Hamiltonian or mass functionals evaluated on the onsite and offsite states. These results extend those of the authors for the…
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 with dislocation densities. We show how each model can be deduced from the previous one at a smaller scale.
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 literature, we treat here the case in which such dislocation is not the superpositions of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well).
We revisit some recents results inspired by the Peierls-Nabarro model on edge dislocations for crystals which rely on the fractional Laplace representation of the corresponding equation. In particular, we discuss results related to heteroclinic, homoclinic and multibump patterns for the atom dislocation function, the large space and time scale of the solutions of the parabolic problem, the dynamics of the dislocation points and the large time asymptotics after possible dislocation collisions.
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 introduce a suitable rescaling of the evolution equation, using a small parameter $\varepsilon$. As $\varepsilon$ goes to zero, we show that the limit dynamics is characterized by a system of ODEs describing the motion of particles with two-body interactions. The interaction forces are in $1/x$ and correspond to the well-known interaction between dislocatio…
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 Laplace operator of order $2 s \in (0, 2)$ acting in one space dimension and the reaction is determined by a $1$-periodic multi-well potential. We construct solutions of these equations that represent the typical propagation of $N \ge 2$ equally oriented dislocations of size $1$. For large times, the dislocations occur around points that evolve according to a repu…
We consider the equation $$v_t=L_s v-W'(v)+σ_ε(t,x) \quad {\mbox{ in }} (0,+\infty)\times\R,$$ where $L_s$ is an integro-differential operator of order $2s$, with $s\in(0,1)$, $W$ is a periodic potential, and $σ_ε$ is a small external stress. The solution $v$ represents the atomic dislocation in the Peierls--Nabarro model for crystals, and we specifically consider the case $s\in(0,1/2)$, which takes into account a strongly nonlocal elastic term. We study the evolution of such dislocation function for macroscopic space and time scales, namely we introduce the function $$ v_ε(t,x):=v\left(\frac{…
