, Volume 53, Issue 11–12, pp 2939–2972 | Cite as

Necessity of law of balance of moment of moments in non-classical continuum theories for solid continua

  • K. S. Surana
  • R. Shanbhag
  • J. N. Reddy


In the non-classical continuum theories for solid continua the presence of internal rotations and their gradients arising due to Jacobian of deformation and/or consideration of Cosserat rotations as additional unknown degrees of freedom at a material point necessitate existence of moment tensor. For small deformation, small strains theories, in Lagrangian description the Cauchy moment tensor and the rates of rotation gradients are rate of work conjugate pair in addition to the rate of work conjugate Cauchy stress tensor and the strain rate tensor. It is well established that in such non-classical theories the Cauchy stress tensor is non-symmetric and the antisymmetric components of the Cauchy stress tensor are balanced by gradients of the Cauchy moment tensor, the balance of angular momenta balance law. In the non-classical continuum theories incorporating internal rotations and conjugate moment tensor that are absent in the 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”. At this stage the Cauchy moment tensor remains non-symmetric if we only consider standard balance laws that are used in classical continuum theories. Thus, requiring constitutive theories for the symmetric as well as anti-symmetric Cauchy moment tensors. The work presented in this paper shows that when the thermodynamically consistent constitutive theories are used for symmetric as well as antisymmetric Cauchy moment tensor non physical and spurious solutions result even in simple model problems. This suggests that perhaps the additional conjugate tensors resulting due to presence of internal rotations, namely the Cauchy moment tensor and the antisymmetric part of the Cauchy stress stress tensor must obey some additional law or restriction so that the spurious behavior is precluded. This paper demonstrates that in the non-classical theory with internal rotations considered here the law of balance of moment of moments and the consideration of 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 law is established theoretically and is also demonstrated through model problems using thermoelastic solids with small strain small deformation as an example. The findings reported in this paper hold for thermoviscoelastic solids with and without memory as well as when deformation and strains are small. Extensions of the concepts presented here for finite deformation and finite strain will be presented in a follow up paper.


Non-classical continua Polar continua Lagrangian description Internal rotations Law of balance of moment of moments 



The support provided by the first and the third author’s university distinguished professorships is gratefully acknowledged. The financial support provided to the second author by the department of mechanical engineering and school of engineering is also acknowledged. The computational infrastructure of CML of the mechanical engineering department have been instrumental in performing the numerical studies presented in the paper.


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Authors and Affiliations

  1. 1.Mechanical EngineeringUniversity of KansasLawrenceUSA
  2. 2.Mechanical EngineeringTexas A&M UniversityCollege StationUSA

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