Bounds of M-eigenvalues and strong ellipticity conditions for elasticity tensors

2021 ◽  
pp. 1-14
Author(s):  
Shigui Li ◽  
Zhen Chen ◽  
Qilong Liu ◽  
Linzhang Lu
Keyword(s):  
Strong Ellipticity ◽  
Elasticity Tensors

scholarly journals New S-Type Bounds of M-Eigenvalues for Elasticity Tensors with Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Zhuanzhou Zhang ◽  
Jun He ◽  
Yanmin Liu ◽  
Zerong Ren
Keyword(s):  
Sufficient Conditions ◽  
Upper Bounds ◽  
Elasticity Tensor ◽  
Strong Ellipticity ◽  
Extreme Eigenvalues ◽  
Elasticity Tensors ◽  
The Given

In this paper, based on the extreme eigenvalues of the matrices arisen from the given elasticity tensor, S-type upper bounds for the M-eigenvalues of elasticity tensors are established. Finally, S-type sufficient conditions are introduced for the strong ellipticity of elasticity tensors based on the S-type M-eigenvalue inclusion sets.

scholarly journals Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

2021 ◽  
Author(s):  
V. Calisti ◽  
A. Lebée ◽  
A. A. Novotny ◽  
J. Sokolowski
Keyword(s):  
Circular Inclusion ◽  
Elasticity Tensor ◽  
Second Order ◽  
Topological Derivative ◽  
Microstructural Changes ◽  
Topological Derivatives ◽  
Spatial Dimensions ◽  
Elasticity Tensors ◽  
Derivatives Of ◽  
Optimization Of Microstructures

AbstractThe multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

A Note on Strong Ellipticity in Two-Dimensional Isotropic Elasticity

2012 ◽  
Vol 109 (1) ◽  
pp. 67-74 ◽  
Author(s):  
Domenico De Tommasi ◽  
Giuseppe Puglisi ◽  
Giuseppe Zurlo
Keyword(s):  
Two Dimensional ◽  
Strong Ellipticity ◽  
Isotropic Elasticity

scholarly journals Seismic wave propagation in anisotropic ice – Part 1: Elasticity tensor and derived quantities from ice-core properties

2015 ◽  
Vol 9 (1) ◽  
pp. 367-384 ◽  
Author(s):  
A. Diez ◽  
O. Eisen
Keyword(s):  
Seismic Waves ◽  
Dynamic Behaviour ◽  
Ice Core ◽  
Ice Cores ◽  
Seismic Wave Propagation ◽  
Elasticity Tensor ◽  
Reflection Coefficients ◽  
Seismic Velocities ◽  
Crystal Anisotropy ◽  
Elasticity Tensors

Abstract. A preferred orientation of the anisotropic ice crystals influences the viscosity of the ice bulk and the dynamic behaviour of glaciers and ice sheets. Knowledge about the distribution of crystal anisotropy is mainly provided by crystal orientation fabric (COF) data from ice cores. However, the developed anisotropic fabric influences not only the flow behaviour of ice but also the propagation of seismic waves. Two effects are important: (i) sudden changes in COF lead to englacial reflections, and (ii) the anisotropic fabric induces an angle dependency on the seismic velocities and, thus, recorded travel times. A framework is presented here to connect COF data from ice cores with the elasticity tensor to determine seismic velocities and reflection coefficients for cone and girdle fabrics. We connect the microscopic anisotropy of the crystals with the macroscopic anisotropy of the ice mass, observable with seismic methods. Elasticity tensors for different fabrics are calculated and used to investigate the influence of the anisotropic ice fabric on seismic velocities and reflection coefficients, englacially as well as for the ice–bed contact. Hence, it is possible to remotely determine the bulk ice anisotropy.

scholarly journals On the Strong Ellipticity of the Anisotropic Linearly Elastic Materials

2007 ◽  
Vol 87 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Stan Chiriţă ◽  
Alexandre Danescu ◽  
Michele Ciarletta
Keyword(s):  
Strong Ellipticity ◽  
Elastic Materials ◽  
Linearly Elastic Materials

Elasticity M-tensors and the strong ellipticity condition

2020 ◽  
Vol 373 ◽  
pp. 124982 ◽  
Author(s):  
Weiyang Ding ◽  
Jinjie Liu ◽  
Liqun Qi ◽  
Hong Yan
Keyword(s):  
Strong Ellipticity ◽  
Ellipticity Condition ◽  
Strong Ellipticity Condition

Strict convexity, strong ellipticity, and regularity in the calculus of variations

1980 ◽  
Vol 87 (3) ◽  
pp. 501-513 ◽  
Author(s):  
J. M. Ball
Keyword(s):  
Nonlinear Systems ◽  
Calculus Of Variations ◽  
Weak Solutions ◽  
Convex Functions ◽  
Strict Convexity ◽  
Mathematical Tool ◽  
Strong Ellipticity ◽  
Regularity Of Weak Solutions ◽  
Nonlinear Elastostatics

In this paper we investigate the connection between strong ellipticity and the regularity of weak solutions to the equations of nonlinear elastostatics and other nonlinear systems arising from the calculus of variations. The main mathematical tool is a new characterization of continuously differentiable strictly convex functions. We first describe this characterization, and then explain how it can be applied to the calculus of variations and to elastostatics.

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