Spectral Decomposition of the Elasticity Tensor

1992 ◽  
Vol 59 (4) ◽  
pp. 762-773 ◽  
Author(s):  
S. Sutcliffe
Keyword(s):  
Spectral Decomposition ◽  
Dimensional Space ◽  
Anisotropic Elasticity ◽  
Elasticity Tensor ◽  
Symmetry Groups ◽  
Stored Energy ◽  
Equilibrium Equations ◽  
Classical Symmetry ◽  
Spectral Decompositions ◽  
Acoustic Tensor

The elasticity tensor in anisotropic elasticity can be regarded as a symmetric linear transformation on the nine-dimensional space of second-order tensors. This allows the elasticity tensor to be expressed in terms of its spectral decomposition. The structures of the spectral decompositions are determined by the sets of invariant subspaces that are consistent with material symmetry. Eigenvalues always depend on the values of the elastic constants, but the eigenvectors are, in part, independent of these values. The structures of the spectral decompositions are presented for the classical symmetry groups of crystallography, and numerical results are presented for representative materials in each group. Spectral forms for the equilibrium equations, the acoustic tensor, and the stored energy function are also derived.

SPECTRAL DECOMPOSITION OF R-MATRICES FOR EXCEPTIONAL LIE ALGEBRAS

1991 ◽  
Vol 06 (10) ◽  
pp. 923-927 ◽  
Author(s):  
S.M. SERGEEV
Keyword(s):  
Lie Algebras ◽  
Spectral Decomposition ◽  
Spectral Decompositions ◽  
Exceptional Lie Algebras ◽  
Exceptional Algebras ◽  
Vector Representations

In this paper spectral decompositions of R-matrices for vector representations of exceptional algebras are found.

scholarly journals Calculation of thin isotropic shells beyond the elastic limit by the method of elastic solutions

2018 ◽  
Vol 196 ◽  
pp. 01014 ◽  
Author(s):  
Avgustina Astakhova
Keyword(s):  
Elastic Limit ◽  
Physical Nonlinearity ◽  
Elasticity Tensor ◽  
Thin Shells ◽  
Equilibrium Equations ◽  
Spherical Shells ◽  
Plastic Deformations ◽  
Elastic Plastic ◽  
Internal Forces

The paper focuses on the model of calculation of thin isotropic shells beyond the elastic limit. The determination of the stress-strain state of thin shells is based on the small elastic-plastic deformations theory and the elastic solutions method. In the present work the building of the solution based on the equilibrium equations and geometric relations of linear theory of thin shells in curved coordinate system α and β, and the relations between deformations and forces based on the Hirchhoff-Lave hypothesis and the small elastic-plastic deformations theory are presented. Internal forces tensor is presented in the form of its expansion to the elasticity tensor and the additional terms tensor expressed the physical nonlinearity of the problem. The functions expressed the physical nonlinearity of the material are determined. The relations that allow to determine the range of elastic-plastic deformations on the surface of the present shell and their changing in shell thickness are presented. The examples of the calculation demonstrate the convergence of elastic-plastic deformations method and the range of elastic-plastic deformations in thickness in the spherical shell. Spherical shells with the angle of half-life regarding 90 degree vertical symmetry axis under the action of equally distributed ring loads are observed.

scholarly journals Spatial symmetry groups as sensorimotor guidelines

2008 ◽  
Vol 17 (5-6) ◽  
pp. 347-359
Author(s):  
Gin McCollum
Keyword(s):  
Symmetry Group ◽  
Motor Neurons ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Symmetry Groups ◽  
Coordinate Systems ◽  
Spatial Function ◽  
Logical Structures ◽  
Vestibular Complex ◽  
Causal Logic

While some aspects of neuroanatomical organization are related to packing and access rather than to function, other aspects of anatomical/physiological organization are directly related to function. The mathematics of symmetry groups can be used to determine logical structure in projections and to relate it to function. This paper reviews two studies of the symmetry groups of vestibular projections that are related to the spatial functions of the vestibular complex, including gaze, posture, and movement. These logical structures have been determined by finding symmetry groups of two vestibular projections directly from physiological and anatomical data. Logical structures in vestibular projections are distinct from mapping properties such as the ability to maintain two- and three-dimensional coordinate systems; rather, they provide anatomical/physiological foundations for these mapping properties. The symmetry group of the direct projection from the semicircular canal primary afferents to neck motor neurons is that of the cube (O, the octahedral group), which can serve as a discrete skeleton for coordinate systems in three-dimensional space. The symmetry group of the canal projection from the secondary vestibular afferents to the inferior olive and thence to the cerebellar uvula-nodulus is that of the square (D8), which can support coordinates for the horizontal plane. While the mathematical relationship between these symmetry groups and functions of the vestibular complex are clear, these studies open a larger question: what is the causal logic by which neural centers and their intrinsic organization affect each other and behavior? The relationship of vestibular projection symmetry groups to spatial function make them ideal projections for investigating this causal logic. The symmetry group results are discussed in relationship to possible ways they communicate spatial structure to other neural centers and format spatial functions such as body movements. These two projection symmetry groups suggest that all vestibular projections may have symmetry groups significantly related to function, perhaps all to spatial function.

Analysis and Visualization of the Contact Force Solution Space for Multi-Limbed Mobile Robots With Three Feet Contact

2003 ◽  
Author(s):  
Dennis W. Hong ◽  
Raymond J. Cipra
Keyword(s):  
Contact Force ◽  
Dimensional Space ◽  
Contact Point ◽  
Solution Process ◽  
Solution Space ◽  
Contact Forces ◽  
Static Equilibrium ◽  
Equilibrium Equations ◽  
Force Components ◽  
The Relationship

A new analytical method for determining, describing, and visualizing the solution space for the contact force distribution of multi-limbed robots with three feet in contact with the environment in three-dimensional space is presented. The foot contact forces are first resolved into strategically defined foot contact force components to decouple them for simplifying the solution process, and then the static equilibrium equations are applied to find certain contact force components and the relationship between the others. Using the friction cone equation at each foot contact point and the known contact force components, the problem is transformed into a geometrical one to find the ranges of contact forces and the relationship between them that satisfy the friction constraint. Using geometric properties of the friction cones and by simple manipulation of their conic sections, the whole solution space which satisfies the static equilibrium and friction constraints at each contact point can be found. Two representation schemes, the “force space graph” and the “solution volume representation,” are developed for describing and visualizing the solution space which gives an intuitive visual map of how well the solution space is formed for the given conditions of the system.

Spectral decomposition of anisotropic elasticity

2001 ◽  
Vol 150 (3-4) ◽  
pp. 237-261 ◽  
Author(s):  
P. S. Theocaris ◽  
D. P. Sokolis
Keyword(s):  
Spectral Decomposition ◽  
Anisotropic Elasticity

Application of symmetry groups of four-dimensional space in spectroscopy of crystals

2007 ◽  
Vol 102 (6) ◽  
pp. 857-866
Author(s):  
E. F. Kustov ◽  
V. G. Yarzhemsky ◽  
V. I. Nefedov
Keyword(s):  
Dimensional Space ◽  
Symmetry Groups

A Finite Element Implementation of Knowles Stored-Energy Function: Theory, Coding and Applications

2011 ◽  
Vol 58 (3) ◽  
pp. 319-346 ◽  
Author(s):  
Cyprian Suchocki
Keyword(s):  
Finite Element ◽  
Energy Function ◽  
Function Theory ◽  
Constitutive Models ◽  
Elasticity Tensor ◽  
Stored Energy ◽  
Energy Potential ◽  
Hyperelastic Materials ◽  
Finite Element Implementation ◽  
Living Tissues

A Finite Element Implementation of Knowles Stored-Energy Function: Theory, Coding and Applications This paper contains the full way of implementing a user-defined hyperelastic constitutive model into the finite element method (FEM) through defining an appropriate elasticity tensor. The Knowles stored-energy potential has been chosen to illustrate the implementation, as this particular potential function proved to be very effective in modeling nonlinear elasticity within moderate deformations. Thus, the Knowles stored-energy potential allows for appropriate modeling of thermoplastics, resins, polymeric composites and living tissues, such as bone for example. The decoupling of volumetric and isochoric behavior within a hyperelastic constitutive equation has been extensively discussed. An analytical elasticity tensor, corresponding to the Knowles stored-energy potential, has been derived. To the best of author's knowledge, this tensor has not been presented in the literature yet. The way of deriving analytical elasticity tensors for hyperelastic materials has been discussed in detail. The analytical elasticity tensor may be further used to develop visco-hyperelastic, nonlinear viscoelastic or viscoplastic constitutive models. A FORTRAN 77 code has been written in order to implement the Knowles hyperelastic model into a FEM system. The performance of the developed code is examined using an exemplary problem.

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