Journal ArticleDOI
Asymptotic expansions by Γ-convergence
Andrea Braides,Lev Truskinovsky +1 more
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In this article, the authors propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them.Abstract:
Our starting point is a parameterized family of functionals (a ‘theory’) for which we are interested in approximating the global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this parameter is ‘small’ but finite. Since Γ-convergence may be non-uniform within the ‘theory’, we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated, and one has to deal with the so-called ‘size’ or ‘scale’ effects.read more
Citations
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Book
Local Minimization, Variational Evolution and Γ-Convergence
TL;DR: In this article, global minimization is replaced by local minimization as a selection criterion and convergence of local minimizers is achieved by a convergence of local optimizers, leading to small-scale stability.
Journal ArticleDOI
Metastability and Dynamics of Discrete Topological Singularities in Two Dimensions: A Γ-Convergence Approach
TL;DR: In this paper, a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements, is proposed, and it is shown that, if first the lattice spacing and then the time step of the minimizing movements tend to zero, the Vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg-Landau framework.
Journal ArticleDOI
The $\mathbf \Gamma$-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density
TL;DR: In this article, it was shown that the Ohta-kawasaki energy with Coulomb repulsion converges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel in the regime where one of the phases has very small volume fraction, thus creating small droplets of the minority phase in a "sea" of the majority phase.
Journal ArticleDOI
Variational Analysis of the Asymptotics of the XY Model
Roberto Alicandro,Marco Cicalese +1 more
TL;DR: In this paper, a variational analysis in the limit when the number of particles diverges is performed for the XY spin-type model and the appearance of vortex-like singularities can be described by properly scaling the energy of the system through a Γ-convergence procedure.
Journal ArticleDOI
The Γ-Limit of the Two-Dimensional Ohta-Kawasaki Energy. Droplet Arrangement via the Renormalized Energy
TL;DR: In this paper, the authors derived a Γ-expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems.
References
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Journal ArticleDOI
The Phenomena of Rupture and Flow in Solids
TL;DR: In this article, the authors investigated the effect of surface scratches on the mechanical strength of solids, and some general conclusions were reached which appear to have a direct bearing on the problem of rupture, from an engineering standpoint, and also on the larger question of the nature of intermolecular cohesion.
Book
Dynamical Theory of Crystal Lattices
Max Born,Kʿun Huang +1 more
TL;DR: Born and Huang's classic work on the dynamics of crystal lattices was published over thirty years ago, and it remains the definitive treatment of the subject as mentioned in this paper. But it is not the most complete work on crystal lattice dynamics.
Book ChapterDOI
The mathematical theory of equilibrium cracks in brittle fracture
TL;DR: In this paper, the authors present a unified view of the way basic problems in the theory of equilibrium cracks are formulated and discuss the results obtained thereby, and the object of the theory is the study of the equilibrium of solids in the presence of cracks.
Reference BookDOI
Asymptotics and Special Functions
TL;DR: A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool as discussed by the authors, and it can be found in many libraries.
MonographDOI
The Theory of Composites
TL;DR: Some of the greatest scientists including Poisson, Faraday, Maxwell, Rayleigh, and Einstein have contributed to the theory of composite materials Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients Although extensively studied for more than a hundred years, an explosion of ideas in the last five decades has dramatically increased our understanding of the relationship between the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective moduli which govern the macroscopic behavior as mentioned in this paper.