Relevant bibliographies by topics / Non-Euclidean elasticity

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  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters

Academic literature on the topic 'Non-Euclidean elasticity'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Non-Euclidean elasticity"

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Kupferman, Raz, and Cy Maor. "A Riemannian approach to the membrane limit in non-Euclidean elasticity." Communications in Contemporary Mathematics 16, no. 05 (August 29, 2014): 1350052. http://dx.doi.org/10.1142/s0219199713500521.

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Non-Euclidean, or incompatible elasticity, is an elastic theory for pre-stressed materials, which is based on a modeling of the elastic body as a Riemannian manifold. In this paper we derive a dimensionally reduced model of the so-called membrane limit of a thin incompatible body. By generalizing classical dimension reduction techniques to the Riemannian setting, we are able to prove a general theorem that applies to an elastic body of arbitrary dimension, arbitrary slender dimension, and arbitrary metric. The limiting model implies the minimization of an integral functional defined over immersions of a limiting submanifold in Euclidean space. The limiting energy only depends on the first derivative of the immersion, and for frame-indifferent models, only on the resulting pullback metric induced on the submanifold, i.e. there are no bending contributions.
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Kupferman, Raz, and Yossi Shamai. "Incompatible elasticity and the immersion of non-flat Riemannian manifolds in Euclidean space." Israel Journal of Mathematics 190, no. 1 (October 24, 2011): 135–56. http://dx.doi.org/10.1007/s11856-011-0187-1.

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Minozzi, Manuela, Paola Nardinocchi, Luciano Teresi, and Valerio Varano. "Growth-induced compatible strains." Mathematics and Mechanics of Solids 22, no. 1 (August 6, 2016): 62–71. http://dx.doi.org/10.1177/1081286515570510.

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We studied the time evolution problem driven by growth for a non-Euclidean ball which at its initial state is equipped with a non-compatible distortion field. The problem is set within the framework of non-linear elasticity with large growing distortions. No bulk accretive forces are considered, and growth is only driven by the stress state. We showed that, when stress-driven growth is considered, distortions can evolve along different trajectories which share a common attracting manifold; moreover, they eventually attain a steady and compatible form, to which there corresponds a stress-free state of the ball.
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Sharma, P., and S. Ganti. "Gauge-field-theory solution of the elastic state of a screw dislocation in a dispersive (non-local) crystalline solid." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2056 (April 8, 2005): 1081–95. http://dx.doi.org/10.1098/rspa.2004.1403.

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The relaxed state of a type of topological defect (screw dislocation) located in a dispersive (non-local) elastic solid is discussed from a viewpoint of gauge field theory. The starting point of this work is the non-local elastic Lagrangian, that is, like its classic elastic counterpart, globally gauge invariant under the Euclidean group of transformations O (3)▹ (3). When compared with gauge solutions of the same problem predicated on the classical elastic Lagrangian, the present solution sheds some interesting insights into the nature of non-locality-gauge field interactions. Both the (3) gauge theory of dislocations (predicated on breaking of the translational symmetry) and the phenomenological non-local elasticity introduce their own respective characteristic length-scale parameters in the elastic equilibrium of dislocations while removing unphysical singularities. In the present work we show that, surprisingly, attempts to elucidate gauge interactions in a dispersive or non-local medium lead to functionally the same solution as in the gauge theory based on local elasticity, albeit, the gauge length-scale must be replaced by an effective length-scale measure. In particular, the non-local and the gauge length-scale combine in a nonlinear fashion to yield the aforementioned effective length-scale. Our results allow one to immediately write the solution of most screw dislocation problems in the gauge non-local theory of defects, provided the counterpart gauge solution based on classical elasticity is known.
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Lewicka, Marta, L. Mahadevan, and Mohammad Reza Pakzad. "The Föppl-von Kármán equations for plates with incompatible strains." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2126 (July 21, 2010): 402–26. http://dx.doi.org/10.1098/rspa.2010.0138.

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We provide a derivation of the Föppl-von Kármán equations for the shape of and stresses in an elastic plate with incompatible or residual strains. These might arise from a range of causes: inhomogeneous growth, plastic deformation, swelling or shrinkage driven by solvent absorption. Our analysis gives rigorous bounds on the convergence of the three-dimensional equations of elasticity to the low-dimensional description embodied in the plate-like description of laminae and thus justifies a recent formulation of the problem to the shape of growing leaves. It also formalizes a procedure that can be used to derive other low-dimensional descriptions of active materials with complex non-Euclidean geometries.
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Kosmas, Odysseas, Pieter Boom, and Andrey P. Jivkov. "On the Geometric Description of Nonlinear Elasticity via an Energy Approach Using Barycentric Coordinates." Mathematics 9, no. 14 (July 19, 2021): 1689. http://dx.doi.org/10.3390/math9141689.

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The deformation of a solid due to changing boundary conditions is described by a deformation gradient in Euclidean space. If the deformation process is reversible (conservative), the work done by the changing boundary conditions is stored as potential (elastic) energy, a function of the deformation gradient invariants. Based on this, in the present work we built a “discrete energy model” that uses maps between nodal positions of a discrete mesh linked with the invariants of the deformation gradient via standard barycentric coordinates. A special derivation is provided for domains tessellated by tetrahedrons, where the energy functionals are constrained by prescribed boundary conditions via Lagrange multipliers. The analysis of these domains is performed via energy minimisation, where the constraints are eliminated via pre-multiplication of the discrete equations by a discrete null-space matrix of the constraint gradients. Numerical examples are provided to verify the accuracy of the proposed technique. The standard barycentric coordinate system in this work is restricted to three-dimensional (3-D) convex polytopes. We show that for an explicit energy expression, applicable also to non-convex polytopes, the general barycentric coordinates constitute fundamental tools. We define, in addition, the discrete energy via a gradient for general polytopes, which is a natural extension of the definition for discrete domains tessellated by tetrahedra. We, finally, prove that the resulting expressions can consistently describe the deformation of solids.
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Yajima, Takahiro, and Hiroyuki Nagahama. "Finsler geometry of seismic ray path in anisotropic media." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2106 (March 11, 2009): 1763–77. http://dx.doi.org/10.1098/rspa.2008.0453.

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The seismic ray theory in anisotropic inhomogeneous media is studied based on non-Euclidean geometry called Finsler geometry. For a two-dimensional ray path, the seismic wavefront in anisotropic media can be geometrically expressed by Finslerian parameters. By using elasticity constants of a real rock, the Finslerian parameters are estimated from a wavefront propagating in the rock. As a result, the anisotropic parameters indicate that the shape of wavefront is expressed not by a circle but by a convex curve called a superellipse. This deviation from the circle as an isotropic wavefront can be characterized by a roughness of wavefront. The roughness parameter of the real rock shows that the shape of the wavefront is expressed by a fractal curve. From an orthogonality of the wavefront and the ray, the seismic wavefront in anisotropic media relates to a fractal structure of the ray path.
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Yavari, Arash, and Alain Goriely. "Weyl geometry and the nonlinear mechanics of distributed point defects." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2148 (September 5, 2012): 3902–22. http://dx.doi.org/10.1098/rspa.2012.0342.

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The residual stress field of a nonlinear elastic solid with a spherically symmetric distribution of point defects is obtained explicitly using methods from differential geometry. The material manifold of a solid with distributed point defects—where the body is stress-free—is a flat Weyl manifold, i.e. a manifold with an affine connection that has non-metricity with vanishing traceless part, but both its torsion and curvature tensors vanish. Given a spherically symmetric point defect distribution, we construct its Weyl material manifold using the method of Cartan's moving frames. Having the material manifold, the anelasticity problem is transformed to a nonlinear elasticity problem and reduces the problem of computing the residual stresses to finding an embedding into the Euclidean ambient space. In the case of incompressible neo-Hookean solids, we calculate explicitly this residual stress field. We consider the example of a finite ball and a point defect distribution uniform in a smaller ball and vanishing elsewhere. We show that the residual stress field inside the smaller ball is uniform and hydrostatic. We also prove a nonlinear analogue of Eshelby's celebrated inclusion problem for a spherical inclusion in an isotropic incompressible nonlinear solid.
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Chang, SooAh, Scott Chapman, and William Smith. "Elasticity in certain block monoids via the Euclidean table." Mathematica Slovaca 57, no. 5 (January 1, 2007). http://dx.doi.org/10.2478/s12175-007-0037-0.

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AbstractThis paper continues the study begun in [GEROLDINGER, A.: On non-unique factorizations into irreducible elements II, Colloq. Math. Soc. János Bolyai 51 (1987), 723–757] concerning factorization properties of block monoids of the form ℬ(ℤn, S) where S = $$\{ \bar 1,\bar a\} $$ (hereafter denoted ℬa(n)). We introduce in Section 2 the notion of a Euclidean table and show in Theorem 2.8 how it can be used to identify the irreducible elements of ℬa(n). In Section 3 we use the Euclidean table to compute the elasticity of ℬa(n) (Theorem 3.4). Section 4 considers the problem, for a fixed value of n, of computing the complete set of elasticities of the ℬa(n) monoids. When n = p is a prime integer, Proposition 4.12 computes the three smallest possible elasticities of the ℬa(p).
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Dissertations / Theses on the topic "Non-Euclidean elasticity"

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Gemmer, John Alan. "Shape Selection in the Non-Euclidean Model of Elasticity." Diss., The University of Arizona, 2012. http://hdl.handle.net/10150/223311.

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In this dissertation we investigate the behavior of radially symmetric non-Euclidean plates of thickness t with constant negative Gaussian curvature. We present a complete study of these plates using the Föppl-von Kármán and Kirchhoff reduced theories of elasticity. Motivated by experimental results, we focus on deformations with a periodic profile. For the Föppl-von Kármán model, we prove rigorously that minimizers of the elastic energy converge to saddle shaped isometric immersions. In studying this convergence, we prove rigorous upper and lower bounds for the energy that scale like the thickness t squared. Furthermore, for deformation with n-waves we prove that the lower bound scales like nt² while the upper bound scales like n²t². We also investigate the scaling with thickness of boundary layers where the stretching energy is concentrated with decreasing thickness. For the Kichhoff model, we investigate isometric immersions of disks with constant negative curvature into R³, and the minimizers for the bending energy, i.e. the L² norm of the principal curvatures over the class of W^2,2 isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H² into R³. In elucidating the connection between these immersions and the nonexistence/ singularity results of Hilbert and Amsler, we obtain a lower bound for the L^∞ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W^2,2, and numerically have lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grow approximately exponentially with the radius of the disc.
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Book chapters on the topic "Non-Euclidean elasticity"

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Lewicka, Marta. "Metric-induced Morphogenesis and Non-Euclidean Elasticity: Scaling Laws and Thin Film Models." In Parabolic Problems, 433–45. Basel: Springer Basel, 2011. http://dx.doi.org/10.1007/978-3-0348-0075-4_22.

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