arXiv subjects

Mary Vaughan

Publications and source records attributed to Mary Vaughan.

7 recordsLinked to original sources

arXiv2026

Bond failure in peridynamics: Nonequivalence of critical stretch and critical energy density criteria

This paper rigorously analyzes bond failure in the peridynamic theory of solid mechanics, which is a fundamental component of fracture modeling. We compare analytically and numerically two common bond-failure criteria:~{\em critical stretch} and~{\em critical energy density}. In the former, bonds fail when they stretch to a critical value, whereas in the latter, bonds fail when the bond energy density exceeds a threshold. By focusing the analysis on bond-based models, we prove mathematically that the critical stretch criterion and the critical energy density criterion are not equivalent in general and result in different bond-breaking and fracture phenomena. Numerical examples showcase the striking differences between the effect of the two criteria on crack dynamics, including the crack tip evolution, crack propagation, and crack branching.

cond-mat.mtrl-sci↗

arXiv2025

Weighted nonlocal area functionals without the triangle inequality

We consider a weighted nonlocal area functional in which the coefficients do not satisfy the triangle inequality. In the context of three phase transitions, this means that one of the weights is larger than the sum of the other two, say $$\sigma_{-1,1} > \sigma_{-1,0} + \sigma_{0,1}.$$ We show that the energy can be reduced by covering interfaces between phases $-1$ and $1$ with a thin strip of phase $0$. Moreover, as the fractional parameter $s\nearrow1$, we prove that the nonlocal energies $\Gamma$-converge to a local area functional with different weights. The functional structure of this long-range interaction model is conceptually different from its classical counterpart, since the functional remains lower semicontinuous, even in the absence of the triangle inequality.

math.AP↗

arXiv2025

A new graph-directed construction of nonlocal energies on the unit interval

We present an analytic construction of nonlocal energies on the unit interval. The energies are defined using a new graph-directed construction of discrete energies on dyadic approximations of the interval. When the discrete jump kernels are comparable to the kernel of the fractional discrete Laplacian, we prove that the discrete energies Mosco converge and the limiting energy is equivalent to the fractional Gagliardo seminorm.

math.AP↗

arXiv2024

A threshold for higher-order asymptotic development of genuinely nonlocal phase transition energies

We study the higher-order asymptotic development of a nonlocal phase transition energy in bounded domains and with prescribed external boundary conditions. The energy under consideration has fractional order $2s \in (0,1)$ and a first-order asymptotic development in the $\Gamma$-sense as described by the fractional perimeter functional. We prove that there is no meaningful second-order asymptotic expansion and, in fact, no asymptotic expansion of fractional order $\mu > 2-2s$. In view of this range value for $\mu$, it would be interesting to develop a new asymptotic development for the $\Gamma$-convergence of our energy functional which takes into account fractional orders. The results obtained here are also valid in every space dimension and with mild assumptions on the exterior data.

math.AP↗

arXiv2024

Asymptotic expansion of a nonlocal phase transition energy

We study the asymptotic behavior of the fractional Allen--Cahn energy functional in bounded domains with prescribed Dirichlet boundary conditions. When the fractional power $s \in (0,\frac12)$, we establish establish the first-order asymptotic development up to the boundary in the sense of $\Gamma$-convergence. In particular, we prove that the first-order term is the nonlocal minimal surface functional. Also, we show that, in general, the second-order term is not properly defined and intermediate orders may have to be taken into account. For $s \in [\frac12,1)$, we focus on the one-dimensional case and we prove that the first order term is the classical perimeter functional plus a penalization on the boundary. Towards this end, we establish existence of minimizers to a corresponding fractional energy in a half-line, which provides itself a new feature with respect to the existing literature.

math.AP↗

arXiv2024

A convergence result for the derivation of front propagation in nonlocal phase field models

We prove that the mean curvature of a smooth surface in $\mathbb{R}^n$, $n\geq 2$, arises as the limit of a sequence of functions that are intrinsically related to the difference between an $n$- and $1$-dimensional fractional Laplacian of a phase transition. Depending on the order of the fractional Laplace operator, we recover the fractional mean curvature or the classical mean curvature of the surface. Moreover, we show that this is an essential ingredient for deriving the evolution of fronts in fractional reaction-diffusion equations such as those for atomic dislocations in crystals.

math.AP↗

arXiv2024

The discrete dislocation dynamics of multiple dislocation loops

We consider a nonlocal reaction-diffusion equation that physically arises from the classical Peierls-Nabarro model for dislocations in crystalline structures. Our initial configuration corresponds to multiple slip loop dislocations in $\mathbb{R}^n$, $n \geq 2$. After suitably rescaling the equation with a small phase parameter $\varepsilon>0$, the rescaled solution solves a fractional Allen-Cahn equation. We show that, as $\varepsilon \to 0$, the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature.

math.AP↗