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This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently...
This work is concerned with the reformulation of evolutionary problems in a
weak form enabling consideration of solutions that may exhibit
evolving microstructures. This reformulation is accomplished by
expressing the evolutionary problem in variational form, i.e., by
identifying a functional whose minimizers represent entire
trajectories of the system. The particular class of functionals under
consideration is derived by first defining a sequence of time-discretized
minimum problems and...
In the present paper, we motivate and describe a numerical approach in order to detect the creation of fractures in a facet of a crystal evolving by anisotropic mean curvature. The result appears to be in accordance with the known examples of facet-breaking. Graphical simulations are included.
We propose an unconditionally stable semi-implicit time discretization of the phase field crystal evolution. It is based on splitting the underlying energy into convex and concave parts and then performing H-1 gradient descent steps implicitly for the former and explicitly for the latter. The splitting is effected in such a way that the resulting equations are linear in each time step and allow an extremely simple implementation and efficient solution. We provide the associated stability and error...
In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller.
Let for . Let . Let . Let be a invertible bilipschitz function with , .
There exists positive constants and depending only on , , such that if and satisfies the...
In this paper we analyse the structure of approximate solutions to the compatible
two well problem with the constraint that the surface energy of the solution
is less than some fixed constant. We prove a quantitative estimate that can be seen as
a two well analogue of the Liouville theorem of Friesecke James Müller.
Let for .
Let . Let .
Let be a invertible bilipschitz
function with , .
There exists positive constants and depending only on σ, ,
such that if
and u satisfies...
Controlling growth at crystalline surfaces requires a detailed and quantitative understanding of the thermodynamic and kinetic parameters governing mass transport. Many of these parameters can be determined by analyzing the isothermal wandering of steps at a vicinal [“step-terrace”] type surface [for a recent review see [4]]. In the case of crystals one finds that these meanderings develop larger amplitudes as the equilibrium temperature is raised (as is consistent with the statistical mechanical...
Controlling growth at crystalline surfaces requires a detailed and quantitative understanding
of the thermodynamic and kinetic parameters governing mass transport. Many of these
parameters can be determined by analyzing the isothermal wandering of steps at a vicinal
[“step-terrace”] type surface [for a recent review see [4]]. In the case of orthodox
crystals one finds that these meanderings develop larger amplitudes as the equilibrium
temperature is raised (as is consistent with the statistical...
In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case....
In order to describe a solid which deforms smoothly in some region, but
non smoothly in some other region, many multiscale methods have recently
been proposed. They aim at coupling an atomistic model (discrete
mechanics) with a macroscopic model
(continuum mechanics).
We provide here a theoretical ground for such a coupling in a
one-dimensional setting. We briefly study the general case of a convex
energy, and next concentrate on
a specific example of a nonconvex energy, the Lennard-Jones case....
We give results for the approximation of a laminate with
varying volume fractions for multi-well energy minimization
problems modeling martensitic crystals that
can undergo either an orthorhombic
to monoclinic or a cubic to tetragonal transformation.
We construct energy minimizing sequences of deformations which satisfy
the corresponding boundary condition, and we
establish a series of error bounds in terms of the elastic energy
for the approximation of the limiting macroscopic
deformation and...
Asymptotic behavior of solutions of an area-preserving crystalline curvature flow equation is investigated. In this equation, the area enclosed by the solution polygon is preserved, while its total interfacial crystalline energy keeps on decreasing. In the case where the initial polygon is essentially admissible and convex, if the maximal existence time is finite, then vanishing edges are essentially admissible edges. This is a contrast to the case where the initial polygon is admissible and convex:...
The asymptotic behaviour for of the solutions to a one-dimensional model for thermo-visco-plastic behaviour is investigated in this paper. The model consists of a coupled system of nonlinear partial differential equations, representing the equation of motion, the balance of the internal energy, and a phase evolution equation, determining the evolution of a phase variable. The phase evolution equation can be used to deal with relaxation processes. Rate-independent hysteresis effects in the strain-stress...
We compare dewetting characteristics of a thin nonwetting solid film in the
absence of stress, for
two models of a wetting potential: the exponential and the algebraic.
The exponential model is a one-parameter (r) model, and the algebraic model is
a two-parameter (r, m)
model, where r is the ratio of the characteristic wetting length to the height
of the unperturbed film,
and m is the exponent of h (film height) in a smooth function that
interpolates the system's surface
energy above and below...
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