Related papers: Dimension reduction in the context of structured d…
Second-order structured deformations of continua provide an extension of the multiscale geometry of first-order structured deformations by taking into account the effects of submacroscopic bending and curving. We derive here an integral…
We extend the theory of structured deformations to the setting of linearized elasticity by providing an integral representation for the underlying energy that features bulk and surface contributions. Our derivation is obtained both via a…
We present some relaxation and integral representation results for energy functionals in the setting of structured deformations, with special emphasis given to the case of multi-level structured deformations. In particular, we present an…
The response of many materials to applied forces and boundary constraints depends upon internal geometric changes at multiple submacroscopic levels. Hierarchical structured deformations provide a mathematical setting for the description of…
A $\Gamma$-convergence analysis is used to perform a 3D-2D dimension reduction of variational problems with linear growth. The adopted scaling gives rise to a nonlinear membrane model which, because of the presence of higher order external…
We study the scaling properties of two-dimensional turbulence using dimensional analysis. In particular, we consider the energy spectrum both at large and small scales and in the "inertial ranges" for the cases of freely decaying and forced…
The classical low-dimensional models of thin structures are based on certain a priori assumptions on the three-dimensional deformation and/or stress fields, diverse in nature but all motivated by the smallness of certain dimensions with…
An energy for first-order structured deformations in the context of periodic homogenization is obtained. This energy, defined in principle by relaxation of an initial energy of integral type featuring contributions of bulk and interfacial…
This paper presents a method for reducing a three-dimensional gradient damage model to a one-dimensional model for slender rods (with a small radius-to-length ratio, $\delta = R/L \to 0$). The 3D model minimizes an energy functional that…
This paper investigates the homogenization, dimension reduction, and linearization of a composite plate subjected to external loading within the framework of non-linear elasticity problem. The total elastic energy of the problem is of order…
We consider a class of models describing an ensemble of identical interacting agents subject to multiplicative noise. In the thermodynamic limit, these systems exhibit continuous and discontinuous phase transitions in a, generally,…
The squeezing process of a three-dimensional quantum system by use of an external deformed one-body oscillator potential can also be described by the $d$-method, without external field and where the dimension can take non-integer values. In…
Measure structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure structured deformation is defined via relaxation departing either from energies associated with…
We investigate dynamics of large scale and slow deformations of layered structures. Starting from the respective model equations for a non-conserved system, a conserved system and a binary fluid, we derive the interface equations which are…
We present a collection of algorithms which utilize dimensional reduction to perform mesh refinement and study possibly singular solutions of time-dependent partial differential equations. The algorithms are inspired by constructions used…
Using a geometrically motivated 8-parameter ansatz through the thickness, we reduce a three-dimensional shell-like geometrically nonlinear Cosserat material to a fully two-dimensional shell model. Curvature effects are fully taken into…
We develop reduced, stochastic models for high dimensional, dissipative dynamical systems that relax very slowly to equilibrium and can encode long term memory. We present a variety of empirical and first principles approaches for model…
We investigate dimensional reduction, the property that the critical behavior of a system in the presence of quenched disorder in dimension d is the same as that of its pure counterpart in d-2, and its breakdown in the case of the…
Structured deformations provide a multiscale geometry that captures the contributions at the macrolevel of both smooth geometrical changes and non-smooth geometrical changes (disarrangements) at submacroscopic levels. For each (first-order)…
The rapidly evolving field of engineering design of functional surfaces necessitates sophisticated tools to manage the inherent complexity of high-dimensional design spaces. This survey paper offers a scoping review, i.e., a literature…