TITLE:
Rate Constitutive Theories of Orders n and 1n for Internal Polar Non-Classical Thermofluids without Memory
AUTHORS:
Karan S. Surana, Stephen W. Long, J. N. Reddy
KEYWORDS:
Rate Constitutive Theories, Non-Classical Thermofluids, Without Memory, Convected Time Derivatives, Internal Rotation Gradient Tensor, Generators and Invariants, Cauchy Moment Tensor
JOURNAL NAME:
Applied Mathematics,
Vol.7 No.16,
October
31,
2016
ABSTRACT: In recent papers, Surana et al. presented
internal polar non-classical Continuum theory in which velocity gradient tensor
in its entirety was incorporated in the conservation and balance laws. Thus,
this theory incorporated symmetric part of the velocity gradient tensor (as
done in classical theories) as well as skew symmetric part representing varying
internal rotation rates between material points which when resisted by
deforming continua result in dissipation (and/or storage) of mechanical work.
This physics referred as internal polar physics is neglected in classical
continuum theories but can be quite significant for some materials. In another
recent paper Surana et al. presented ordered rate constitutive theories for
internal polar non-classical fluent continua without memory derived using
deviatoric Cauchy stress tensor and conjugate strain rate tensors of up to
orders n and Cauchy moment tensor and its conjugate symmetric part of the first
convected derivative of the rotation gradient tensor. In this constitutive
theory higher order convected derivatives of the symmetric part of the rotation
gradient tensor are assumed not to contribute to dissipation. Secondly, the
skew symmetric part of the velocity gradient tensor is used as rotation rates
to determine rate of rotation gradient tensor. This is an approximation to true
convected time derivatives of the rotation gradient tensor. The resulting
constitutive theory: (1) is incomplete as it neglects the second and higher
order convected time derivatives of the symmetric part of the rotation gradient
tensor; (2) first convected derivative of the symmetric part of the rotation
gradient tensor as used by Surana et al. is only approximate; (3) has
inconsistent treatment of dissipation due to Cauchy moment tensor when compared
with the dissipation mechanism due to deviatoric part of symmetric Cauchy
stress tensor in which convected time derivatives of up to order n are
considered in the theory. The purpose of this paper is to present ordered rate
constitutive theories for deviatoric Cauchy strain tensor, moment tensor and
heat vector for thermofluids without memory in which convected time derivatives
of strain tensors up to order n are conjugate with the Cauchy stress tensor and
the convected time derivatives of the symmetric part of the rotation gradient
tensor up to orders 1n are conjugate with the moment tensor. Conservation and
balance laws are used to determine the choice of dependent variables in the
constitutive theories: Helmholtz free energy density Φ, entropy density η,
Cauchy stress tensor, moment tensor and heat vector. Stress tensor is
decomposed into symmetric and skew symmetric parts and the symmetric part of
the stress tensor and the moment tensor are further decomposed into equilibrium
and deviatoric tensors. It is established through conjugate pairs in entropy
inequality that the constitutive theories only need to be derived for symmetric
stress tensor, moment tensor and heat vector. Density in the current
configuration, convected time derivatives of the strain tensor up to order n,
convected time derivatives of the symmetric part of the rotation gradient
tensor up to orders 1n, temperature gradient tensor and temperature are
considered as argument tensors of all dependent variables in the constitutive
theories based on entropy inequality and principle of equipresence. The
constitutive theories are derived in contravariant and covariant bases as well
as using Jaumann rates. The nth and 1nth order rate constitutive theories for
internal polar non-classical thermofluids without memory are specialized for n
= 1 and 1n = 1 to demonstrate fundamental differences in the constitutive
theories presented here and those used presently for classical thermofluids
without memory and those published by Surana et al. for internal polar
non-classical incompressible thermofluids.