Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAnalysis of a model of elastic dislocations in geophysics
94
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We analyze a mathematical model of elastic dislocations with applications to geophysics, where by an elastic dislocation we mean an open, oriented Lipschitz surface in the interior of an elastic solid, across which there is a discontinuity of the displacement. We model the Earth as an infinite, isotropic, inhomogeneous, elastic medium occupying a half space, and assume only Lipschitz continuity of the Lame parameters. We study the well posedness of very weak solutions to the forward problem of determining the displacement by imposing traction-free boundary conditions at the surface, continuity of the traction and a given jump on the displacement across the fault. We employ suitable weighted Sobolev spaces for the analysis. We utilize the well posedness of the forward problem and unique-continuation arguments to establish uniqueness in the inverse problem of determining the dislocation surface and the displacement jump from measuring the displacement at the surface of the Earth. Uniqueness holds for tangential or normal jumps and under some geometric conditions on the surface.
rate research
Read More
In this paper we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum non-local energy $E_gamma$ modelling the interactions$-$at a typical length-scale of $1/gamma$$-$of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy $E_gamma$ in powers of $1/gamma$ by $Gamma$-convergence, in the limit $gammatoinfty$. While the zero-order term in the expansion, the $Gamma$-limit of $E_gamma$, captures the `bulk profile of the density of dislocation walls in the pile-up domain, the first-order term in the expansion is a `boundary-layer energy that captures the profile of the density in the proximity of the lock. This study is a first step towards a rigorous understanding of the behaviour of dislocations at obstacles, defects, and grain boundaries.
In this paper, we present a semiclassical description of surface waves or modes in an elastic medium near a boundary, in spatial dimension three. The medium is assumed to be essentially stratified near the boundary at some scale comparable to the wave length. Such a medium can also be thought of as a surficial layer (which can be thick) overlying a half space. The analysis is based on the work of Colin de Verdi`ere on acoustic surface waves. The description is geometric in the boundary and locally spectral beneath it. Effective Hamiltonians of surface waves correspond with eigenvalues of ordinary differential operators, which, to leading order, define their phase velocities. Using these Hamiltonians, we obtain pseudodifferential surface wave equations. We then construct a parametrix. Finally, we discuss Weyls formulas for counting surface modes, and the decoupling into two classes of surface waves, that is, Rayleigh and Love waves, under appropriate symmetry conditions.
A system of $n$ screw dislocations in an isotropic crystal undergoing antiplane shear is studied in the framework of linear elasticity. Imposing a suitable boundary condition for the strain, namely requesting the non-vanishing of its boundary integral, results in a confinement effect. More precisely, in the presence of an external strain with circulation equal to n times the lattice spacing, it is energetically convenient to have n distinct dislocations lying inside the crystal. The result is obtained by formulating the problem via the core radius approach and by studying the asymptotics as the core size vanishes. An iterative scheme is devised to prove the main result. This work sets the basis for studying the upscaling problem, i.e., the limit as $ntoinfty$, which is treated in [17].
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.
The work analyzes a one-dimensional viscoelastic model of blood vessel growth under nonlinear friction with surroundings, and provides numerical simulations for various growing cases. For the nonlinear differential equations, two sufficient conditions are proven to guarantee the global existence of biologically meaningful solutions. Examples with breakdown solutions are captured by numerical approximations. Numerical simulations demonstrate this model can reproduce angiogenesis experiments under various biological conditions including blood vessel extension without proliferation and blood vessel regression.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
