TITLE:
A Nonlinear Micromorphic Microcontinuum Theory with Nonlinear and Linear Microconstituent Kinematics for Thermoelastic Solids
AUTHORS:
Karan S. Surana, Chuong D. A. Nguyen
KEYWORDS:
Micromorphic, Micro, Macro, Finite Deformation, Finite Strain, Conservation and Balance Laws, Balance of Moment of Moments, Representation Theorem, Constitutive Theories, Integral-Average Definitions
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.3,
March
27,
2026
ABSTRACT: This paper presents conservation and balance laws for nonlinear micromorphic microcontinuum for thermoelastic solid medium in which nonlinear elasticity is considered for the microconstituents, for the solid medium as well as for the interaction of the microconstituents with the solid medium. The theory is based on completely deformable microconstituents with classical rotations as rigid body rotations of the microconstituents and the use of balance of moment of moments balance law essential for thermodynamic equilibrium. A check on closure of the mathematical model consisting of conservation and balance laws and the constitutive theory reveals that additional nine equations are needed for the closure of the mathematical model. It is shown in the paper that six of the nine equations can be extracted from the balance of angular momenta. We present two alternatives for obtaining the remaining three equations needed for closure. In this first case, one could use Eringen’s conservation of microinertia to obtain three additional equations; hence now we have closure. In the second alternative, if we only consider linear microconstituent kinematics, then we require only six additional equations; hence the mathematical model has closure with additional six equations extracted from the balance of angular momenta. Pros and cons of both approaches are discussed in the paper from the point of view of thermodynamic and mathematical consistency of the resulting theories. Since the microconstituents are deformable, we begin the derivation of the conservation and the balance laws for the microconstituents followed by integral-average definitions that facilitate the derivations of macro conservation and balance laws incorporating microconstituent kinematics. Constitutive theories are initiated using conjugate pairs in the entropy inequality in conjunction with axiom of causality and are derived using representation theorem, hence ensuring thermodynamic and mathematical consistency. Since classical rotation and the conjugate moments form a new kinematically conjugate pair in this theory, the balance of moment of moments balance law is necessitated by classical thermodynamics for thermodynamic equilibrium of the micromorphic solid medium and is used in the present work. In the derivation of conservation and balance laws for nonlinear micromorphic solid medium, we ensure that the modifications of the conservation and the balance laws of classical continuum mechanics are supported by classical thermodynamics. The thermodynamically and mathematically consistent nonlinear micromorphic theory presented here is compared with Eringen’s work.