Time Derivatives Of Strain Tensor Research Articles

Rate Constitutive Theories of Orders n and 1n for Internal Polar Non-Classical Thermofluids without Memory

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.

The Rate Constitutive Equations and Their Validity for Progressively Increasing Deformation

The rate constitutive equations based on upper convected, lower convected, Jaumann, Truesdell and Green–Naghdi stress rates, etc. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] have been used for the solid matter and polymeric liquids when the mathematical models are derived employing conservation laws in the Eulerian description. The rate constitutive equations provide relationship between convected time derivatives of the stress tensor and the convected time derivatives of strain tensor through the constitution of the matter. It is well known that if one assumes constant properties of the matter, then the use of various rate constitutive equations yield different responses [9] when the strains and strain rates are not infinitesimal (finite deformation) even though the rate constitutive equations are objective or frame invariant. This has been rationalized by the argument that in order for the different rate constitutive equations to produce the same response, the material tensor must be different for each case [9]. Our point of view is rather pragmatic in the sense that if all rate constitutive models describe the behavior of the same matter in which material constants do not change during deformation, then surely there must be some inherent assumptions in their derivations that are responsible for this anomaly. The covariant and contravariant convected bases in the current configuration of the deforming matter provide two possible alternate means of defining the convected time derivatives of the contravariant and covariant Cauchy stress tensors as well as the strain tensors. The relationship between the convected time derivatives of the stress tensor, material tensor and convected time derivative of the strain tensor result in the rate constitutive equations. Thus there are at least two obvious approaches for deriving rate constitutive equations: one based on covariant description referred to as lower convected rate constitutive equations and the other based on a contravariant description referred to as upper convected rate constitutive equations. It can be shown that the other rate constitutive equations available (in the literature) can also be derived using these two basic descriptions by modifications that are either justified based on the physics of the deforming matter or mathematical manipulations. When the strains and the strain rates are small (closer to infinitesimal assumption), there is isomorphism (or equivalence) between the covariant and contravariant descriptions, hence in this case the two descriptions will yield identical results even though the explicit forms of the rate equation expressions in the two descriptions are different. It is shown that when the strains and the strain rates are finite (finite deformation), the isomorphism or equivalence between the two descriptions is lost. The paper demonstrates that with progressively increasing deformation leading to finite deformation only the contravariant description has physical basis, hence the rate constitutive equations derived using contravariant description remain valid whereas all others become spurious. Detailed mathematical development of the rate constitutive equations based on covariant, contravariant descriptions and others based on these two bases are presented to illustrate their validity and limitations. Numerical examples are also presented using Giesekus constitutive model for dense polymeric liquids (polymer melts) to demonstrate the validity of contravariant basis and the failures of covariant and others commonly used in the literature.