TITLE:
A Nonlinear Micropolar Continuum Theory for Thermoviscoelastic Solid Medium Based on Classical Rotations
AUTHORS:
Karan S. Surana, Sri Sai Charan Mathi
KEYWORDS:
Nonclassical, Micropolar, Dissipation, Ordered Rate, Conservation and Balance Laws, Representation Theorem, Microviscous Dissipation, Microdissipation, Ordered Rate, Finite Deformation Theories, Finite Strain
JOURNAL NAME:
Applied Mathematics,
Vol.16 No.3,
March
27,
2025
ABSTRACT: This paper presents a nonlinear micropolar nonclassical mathematical continuum theory for finite deformation/finite strain deformation physics of compressible thermoviscoelastic solids based on classical rotations
c
Θ
and its rates. Stress and moment measures for finite deformation/finite strain physics are utilized in conjunction with the finite deformation/finite strain measures presented in ref. [1] to derive conservation and the balance law as well as the constitutive theories using conjugate pairs in the entropy inequality and the representation theorem. The nonlinear micropolar nonclassical continuum theory presented in this paper for thermoviscoelastic solid: (1) incorporates nonlinear ordered rate dissipation mechanism for the viscous medium based on rates of Green’s strain tensor up to order
n
. This is usual viscous dissipation (macrodissipation) in the solid medium due to the viscosity of the medium. (2) Also incorporates additional ordered rate dissipation mechanism due to microconstituents and the viscosity of medium, which depends upon rates of the symmetric part of the rotation gradient tensor up to order
n
˜
. We refer to this dissipation mechanism as microdissipation or microviscous dissipation. This dissipation mechanism is consistent with the deformation measure derived in ref. [1] for nonlinear micropolar nonclassical continuum theory. (3) With the assumption of small deformation, small strain, the nonlinear micropolar nonclassical continuum theory presented here reduces to a consistent linear micropolar nonclassical continuum theory with both mechanisms of dissipation. (4) In the absence of micropolar physics, the theory reduces to finite deformation/finite strain classical continuum theory for compressible thermoviscoelastic solid medium. The complete mathematical model consisting of the conservation and balance laws and the constitutive theories has closure without the conservation of micro inertia law needed in the micropolar theories of Eringen for closure. It has been shown that the balance of moment of moments is an essential balance law in all micropolar theories for achieving thermodynamic and mathematical consistency of the resulting linear micropolar theory. The balance of moment of moments balance law is necessary and has been successfully used in ref. [2] to derive a nonlinear micropolar theory for thermoelastic solid and is essential in this nonlinear micropolar nonclassical continuum theory for thermoviscoelastic solid based on classical rotations
c
Θ
presented in this paper. The nonlinear micropolar nonclassical continuum theory based on rotations
Θ
c
,
Θ
α
and
α
Θ
(neglecting
c
Θ
) is not considered in the present work due to the fact that the linear micropolar nonclassical continuum theory based on these rotations is thermodynamically and mathematically inconsistent [1].