TITLE:
Nonlinear Deformation/Strains for 3M Continua and Consistency of Linear Micropolar Theories
AUTHORS:
Karan Surana, Sri Sai Charan Mathi
KEYWORDS:
Nonclassical, Micropolar, Dissipation, Ordered Rate, Conservation and Balance Laws, Representation Theorem, Microviscous Dissipation, Microdissipation, Finite Deformation Theories, Finite Strain
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.3,
March
26,
2025
ABSTRACT: In 3M continuum theories, micro deformations greatly influence macro deformation physics. Thus, in 3M theories we need a mechanism of micro deformation from which macro deformation can be derived. The one way to accomplish this is to assume that a material point contains a deformable director, the deformation of the director is representation of the microconstituent deformation in the material point, thus in essence a material point is deformable in 3M theories as opposed to classical continuum mechanics, in which material points are rigid. The thermodynamic principles of classical continuum mechanics are assumed to hold for micro deformation of the microconstituents leading to ‘integral-average’ definitions for macro deformation that can be used in the thermodynamics of the matter at the macro level. The principal element of this theory is to derive deformation/strain measures for a material point based in a single deformable director representing the macro deformation physics in the material point. This derivation is the first paper in which nonlinear deformation/strain measures are established for the micro as well as macro deformation physics. It is shown in this paper that, in currently published works, only deformation measures are possible and not strain measures, which is also true in our derivation presented here. Reasons for this are explained in the paper. Only the rate of work conjugate pairs in entropy inequality establish whether any of these measures are strain measures or can be made strain measures by simple modifications. Second part of the paper is devoted to the evaluation of various linear micropolar theories in the published literature based on the following considerations: 1) Are the linear form of the deformation measures derived in this paper utilized appropriately in the derivation of the theories? 2) Is the adequacy of conservation and balance laws of classical continuum mechanics and need for their modifications and perhaps the need for a new balance law, addressed satisfactorily? This is necessary due to presence of new micropolar physics over and beyond classical continuum mechanics 3) Are the derivations of constitutive theories supported by the representation theorem? 4) Are the conservation and balance laws and constitutive theories thermodynamically and mathematically consistent? 5) Lastly, do the complete mathematical models have closure?