Abstract
This paper considers conservation and balance laws and the constitutive theories for non-classical viscous fluent continua without memory, in which internal rotation rates due to the velocity gradient tensor are incorporated in the thermodynamic framework. The constitutive theories for the deviatoric part of the symmetric Cauchy stress tensor and the Cauchy moment tensor are derived based on integrity. The constitutive theories for the Cauchy moment tensor are considered when the balance of moments of moments 1) is not a balance law and 2) is a balance law. The constitutive theory for heat vector based on integrity is also considered. Restrictions on the material coefficients in the constitutive theories for the stress tensor, moment tensor, and heat vector are established using the conditions resulting from the entropy inequality, keeping in mind that the constitutive theories derived here based on integrity are in fact nonlinear constitutive theories. It is shown that in the case of the simplest linear constitutive theory for stress tensor used predominantly for compressible viscous fluids, Stokes' hypothesis or Stokes' assumption has no thermodynamic basis, hence may be viewed incorrect. Thermodynamically consistent derivations of the restrictions on various material coefficients are presented for non-classical as well as classical theories that are applicable to nonlinear constitutive theories, which are inevitable if the constitutive theories are derived based on integrity.
Highlights
In fluent continua, velocities are observable quantities and the deformation physics is completely contained in the velocities ( v ) and the velocity gradient tensor ( L )
This paper considers conservation and balance laws and the constitutive theories for non-classical viscous fluent continua without memory, in which internal rotation rates due to the velocity gradient tensor are incorporated in the thermodynamic framework
1) We have shown that μ > 0 and λ > 0 are the only restrictions on the material coefficients μ and λ that are thermodynamically justified based on entropy inequality. 2) 2μ + 3λ = 0 (Stokes’ hypothesis) or 2μ + 3λ > 0 advocated by Rajagopal [56] have no thermodynamic basis, cannot be justified. 3) μ and λ are two independent material coefficients that must be determined from experiments for a fluid of interest
Summary
The notations used in this paper conform to Reference [32] but are different than conventional notations in continuum mechanics writings. Since the covariant base vectors are tangent to the deformed material lines, the convected time derivative of the covariant strain tensor is a physical measure of the strain rate tensor. We define σ (0) as contravariant Cauchy stress tensor, γ (1) as the first convected time derivative of the Green’s strain tensor These measures are physical as these are related to the faces and edges of the true deformed tetrahedron. Since g i and g i form reciprocal bases, we could use covariant directions for stress measure and contravarian directions for strain rate measures, i.e., σ (0) and γ (1) , covariant Cauchy stress tensor and contravariant strain rate tensor This is justified, in terms of physics, this description requires g i to be normal to the tetrahedron faces and g i to be the material line tangent vectors. When strain rates are small, the two measures are the same as the deformed and undeformed configurations are virtually the same
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