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Minimizers Of Nonlocal Polyconvex Energies In Nonlocal Hyperelasticity, José C. Bellido, Javier Cueto, Carlos Mora-Corral
Minimizers Of Nonlocal Polyconvex Energies In Nonlocal Hyperelasticity, José C. Bellido, Javier Cueto, Carlos Mora-Corral
Department of Mathematics: Faculty Publications
We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz’ fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola’s identity, the integration by parts of the determinant and the weak …
Nonlocal Helmholtz Decompositions And Connections To Classical Counterparts, Andrew Haar, Petronela Radu
Nonlocal Helmholtz Decompositions And Connections To Classical Counterparts, Andrew Haar, Petronela Radu
UCARE Research Products
In recent years nonlocal models have been successfully introduced in a variety of applications, such as dynamic fracture, nonlocal diffusion, flocking, and image processing. Thus, the development of a nonlocal calculus theory, together with the study of nonlocal operators has become the focus of many theoretical investigations. Our work focuses on a Helmholtz decomposition in the nonlocal (integral) framework. In the classical (differential) setting the Helmholtz decomposition states that we can decompose a three dimensional vector field as a sum of an irrotational function and a solenoidal function. We will define new nonlocal gradient and curl operators that allow us …
Local And Nonlocal Models In Thin-Plate And Bridge Dynamics, Jeremy Trageser
Local And Nonlocal Models In Thin-Plate And Bridge Dynamics, Jeremy Trageser
Department of Mathematics: Dissertations, Theses, and Student Research
This thesis explores several models in continuum mechanics from both local and nonlocal perspectives. The first portion settles a conjecture proposed by Filippo Gazzola and his collaborators on the finite-time blow-up for a class of fourth-order differential equations modeling suspension bridges. Under suitable assumptions on the nonlinearity and the initial data, a finite-time blowup is demonstrated as a result of rapid oscillations with geometrically growing amplitudes. The second section introduces a nonlocal peridynamic (integral) generalization of the biharmonic operator. Its action converges to that of the classical biharmonic as the radius of nonlocal interactions---the ``horizon"---tends to zero. For the corresponding …