Necessity of law of balance/equilibrium of moment of moments in non-classical continuum theories for fluent continua

Abstract

In the non-classical theories for fluent continua, the presence of internal rotation rates and their gradients arising due to the velocity gradient tensor necessitate existence of moment tensor. The Cauchy moment tensor acting on the faces of the deformed tetrahedron (derived using Cauchy principle) and the gradients of the rates of total rotations are a rate of work conjugate pair in addition to the rate of work conjugate Cauchy stress tensor and rate of strain tensor. It is well established that in such non-classical continuum theories the Cauchy stress tensor is non-symmetric and the antisymmetric components of the Cauchy stress tensor are balanced by the gradients of the Cauchy moment tensor, balance of angular momenta balance law. In the non-classical continuum theories incorporating internal rotation rates and conjugate Cauchy moment tensor that are absent in classical continuum theories, the fundamental question is “are the conservation and balance laws used in classical continuum mechanics sufficient to ensure dynamic equilibrium of the deforming volume of matter?" If one only considers conservation and balance laws used in classical continuum theories, then the Cauchy moment tensor is non-symmetric. Thus, requiring constitutive theories for the symmetric as well as non-symmetric Cauchy moment tensors. The work presented in this paper shows that when the thermodynamically consistent constitutive theories are used for the symmetric as well as antisymmetric Cauchy moment tensors, non-physical and spurious solutions result even in simple flows. This suggests that perhaps the additional conjugate tensors resulting due to the presence of internal rotation rates, namely the Cauchy moment tensor and the antisymmetric part of Cauchy stress tensor, must obey some additional law or restriction so that the spurious behavior is precluded. The paper demonstrates that in the non-classical theory with internal rotation rates considered here the balance of moment of moments balance law and the equilibrium of moment of moments are in fact identical. When this balance law is considered, the Cauchy moment tensor becomes symmetric, hence eliminating the constitutive theory for the antisymmetric Cauchy moment tensor and thereby eliminating spurious and non-physical solutions. The necessity of this balance/equilibrium law is established theoretically, its derivation is presented using rate considerations, and its necessity is also demonstrated by a model problem using thermoviscous incompressible fluid as an example. The findings reported in this paper hold for all fluent continua, compressible as well as incompressible.