Decompositions of a Higher-Order Tensor in Block Terms—Part I: Lemmas for Partitioned Matrices

2008 ◽  
Vol 30 (3) ◽  
pp. 1022-1032 ◽  
Author(s):  
Lieven De Lathauwer
Keyword(s):  
Higher Order ◽  
Order Tensor

Assessing the Use of Tensor Closure Methods With Orientation Distribution Reconstruction Functions

Materials ◽  
2003 ◽  
Author(s):  
David A. Jack ◽  
Douglas E. Smith
Keyword(s):  
Distribution Function ◽  
Fourth Order ◽  
Lower Order ◽  
Series Representation ◽  
Orientation Distribution ◽  
Higher Order ◽  
Interaction Coefficients ◽  
Order Tensor ◽  
Orientation Tensor ◽  
Orientation Tensors

Orientation tensors are widely used to describe fiber distri-butions in short fiber reinforced composite systems. Although these tensors capture the stochastic nature of concentrated fiber suspensions in a compact form, the evolution equation for each lower order tensor is a function of the next higher order tensor. Flow calculations typically employ a closure that approximates the fourth-order orientation tensor as a function of the second order orientation tensor. Recent work has been done with eigen-value based and invariant based closure approximations of the fourth-order tensor. The effect of using lower order tensors tensors in process simulations by reconstructing the distribution function from successively higher order orientation tensors in a Fourier series representation is considered. This analysis uses the property that orientation tensors are related to the series expansion coefficients of the distribution function. Errors for several closures are investigated and compared with errors developed when using a reconstruction from the exact 2nd, 4th, and 6th order orientation tensors over a range of interaction coefficients from 10−4 to 10−1 for several flow fields.

scholarly journals Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems

2010 ◽  
Vol 467 (2126) ◽  
pp. 314-332 ◽  
Author(s):  
Vincent Monchiet ◽  
Guy Bonnet
Keyword(s):  
Closed Form ◽  
Strain Field ◽  
Spherical Inclusion ◽  
Higher Order ◽  
Quadratic Polynomial ◽  
Elastic Matrix ◽  
Closed Form Expression ◽  
Form Expression ◽  
Order Tensor ◽  
Strain Fields

In this paper, the derivation of irreducible bases for a class of isotropic 2 n th-order tensors having particular ‘minor symmetries’ is presented. The methodology used for obtaining these bases consists of extending the concept of deviatoric and spherical parts, commonly used for second-order tensors, to the case of an n th-order tensor. It is shown that these bases are useful for effecting the classical tensorial operations and especially the inversion of a 2 n th-order tensor. Finally, the formalism introduced in this study is applied for obtaining the closed-form expression of the strain field within a spherical inclusion embedded in an infinite elastic matrix and subjected to linear or quadratic polynomial remote strain fields.

A Maximum Enhancing Higher-Order Tensor Glyph

2010 ◽  
Vol 29 (3) ◽  
pp. 1143-1152 ◽  
Author(s):  
T. Schultz ◽  
G. Kindlmann
Keyword(s):  
Higher Order ◽  
Order Tensor

Application of higher-order tensor theory for formulating enhanced continuum models

2000 ◽  
Vol 142 (1-4) ◽  
pp. 223-234 ◽  
Author(s):  
A. S. J. Suiker ◽  
C. S. Chang
Keyword(s):  
Higher Order ◽  
Continuum Models ◽  
Order Tensor ◽  
Tensor Theory
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