TITLE:
Thermodynamically and Mathematically Consistent Linear Micromorphic Microcontinuum Theory for Solid Continua
AUTHORS:
Karan S. Surana, Sri Sai Charan Mathi
KEYWORDS:
Micromorphic, Micro, Macro, Deformation/Strain Measures, Conservation and Balance Laws, Balance of Moment of Moments, Integral-Average, Representation Theorem, Constitutive Theories
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.10,
October
31,
2025
ABSTRACT: This paper presents derivation of micro and macro conservation and balance laws and the constitutive theories for the linear elastic micromorphic theory, in which elasticity is considered for microconstituents, the solid medium, and for the interaction of microconstituents with the solid medium. The conservation and balance laws are initiated for micro deformation, followed by consistent “integral-average” definitions valid at the macro level. These permit the derivation of the conservation and balance laws at the macro level. Significant aspects of this theory are: 1) microconstituent rigid rotation physics is the same in all 3M theories. The rigid rotations of the microconstituents are in fact classical rotations; hence, they do not introduce three unknown degrees of freedom at the material point and also can not be part of the strain measures. Thus, in this theory, a microconstituent has only six unknown degrees of freedom, six independent components of the symmetric part of the micro deformation gradient tensor, as opposed to Eringen’s theory, in which all nine components of the micro deformation gradient tensor are unknown degrees of freedom. 2) The balance of moment of moments balance law is shown to be essential in all 3M theories and hence is considered here, due to which the Cauchy moment tensor is symmetric. This avoids a spurious constitutive theory for the moment tensor. 3) In the case of nonsymmetric macro Cauchy stress tensor, the constitutive theory is needed only for the symmetric part, as the skew-symmetric part is defined by the balance of angular momenta. 4) The smoothing weighting function
ϕ
(
α
)
for the microconstituent, as advocated by Eringen and used to multiply the balance of linear momenta of the micro deformation physics, has no thermodynamic, physical or mathematical basis; hence, it is not used in the present work. 5) In contrast with published works of Eringen and others, all constitutive tensors of rank two are always symmetric, hence always permitting the use of the representation theorem in deriving constitutive theories, ensuring the mathematical consistency of the resulting theories. 6) Conservation of micro inertia, necessary in Eringen’s theories to provide closure to the mathematical model, is neither needed nor used in the present work. The linear micromorphic theory derived here is compared with Eringen’s theory to identify differences, discuss and evaluate these for their validity based on thermodynamic and mathematical principles to ultimately determine the thermodynamic and mathematical consistency of the published micromorphic theories.