Volume no :16, Issue no: 1, August (2016)

A NON-CLASSICAL INTERNAL PLOAR CONTINUUM THEORY FOR FINITE DEFORMATION OF SOLIDS USING FIRST PIOLA-KIRCHHOFF STRESS TENSOR

Author's: K. S. Surana, A. D. Joy and J. N. Reddy
Pages: [1] - [41]
Received Date: June 21, 2016
Submitted by:
DOI: http://dx.doi.org/10.18642/jpamaa_7100121677

Abstract

This paper presents a non-classical internal polar continuum theory for finite deformation of isotropic, homogeneous compressible and incompressible solid continua. The classical continuum theories only incorporate partial physics of deformation in the thermodynamic framework. Since the Jacobian of deformation J is fundamental measure of deformation in solid continua, J in its entirety must be incorporated in the thermodynamic framework. Polar decomposition of J into right stretch tensor and pure rotation tensor R shows that entirety of J implies entirety of and R. The classical continuum theories for isotropic and homogeneous solid continua are derived purely using thus ignoring the influence of R altogether. The purpose of this research is to present a new and more complete thermodynamic framework for finite deformation of solids that incorporates complete deformation physics described by J. This can be accomplished by incorporating the additional physics due to R in the current theories as these theories already contain the physics due to We note that the rotation tensor R results due to deformation of solid continua, hence arises in all deforming solid continua. Thus, this theory can be referred to as internal polar non-classical theory for solid continua. The use of internal polar non-classical is appropriate as the theory considers internal rotations. If the varying internal rotations and the rotation rates are resisted by the solid continua, then there must exist internal moments that are conjugate to the rotations which together with rotations and rotation rates can result in additional energy storage, dissipation, and memory. Derivations of conservation and balance laws are presented for internal polar non-classical continuum theory for solid continua for finite deformation. Necessity of additional conservation and balance laws is discussed and their derivations are presented. The resulting mathematical model is compared with the mathematical models resulting from the current continuum theories for finite deformation to illustrate the differences in them due to incorporating the additional physics associated with R and thereby incorporating J in its entirety. The non-classical continuum theory for solid continua presented here is not to be confused with the micropolar theories, stress-couple theories, or strain gradient theories as demonstrated in this paper. The objective of the theory presented here is to present new thermodynamic framework for solid continua with finite deformation that is consistent with the deformation physics, which necessitates that J in its entirety must form the basis for derivation of conservation and balance laws. Since this internal polar non-classical continuum theory considers additional physics due to R, the resulting thermodynamic framework is more complete and consistent with the physics of deformation compared to the currently used thermodynamic framework.

Keywords

non-classical continuum theory, internal polar continuum theory, solid continua, Jacobian of deformation, polar decomposition, stretch tensor, rotation tensor, finite deformation.