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

It has been shown earlier that Noether symmetry does not admit a form of corresponding to an action in which is coupled to scalar-tensor theory of gravity or even for pure theory of gravity taking anisotropic model into account. Here, we prove that theory of gravity does not admit Noether symmetry even if it is coupled to tachyonic field and considering a gauge in addition. To handle such a theory, a general conserved current has been constructed under a condition which decouples higher-order curvature part from the field part. This condition, in principle, solves for the scale-factor independently. Thus, cosmological evolution remains independent of the form of the chosen field, whether it is a scalar or a tachyon.

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

Materials ◽

10.1115/imece2003-42828 ◽

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.

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

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2010.0149 ◽

2010 ◽

Vol 467 (2126) ◽

pp. 314-332 ◽

Cited By ~ 12

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.

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