Theory of rods (plates) constructed for non-classical models of a porous medium in deformable body mechanics


Аuthors

Egorova M. S.*, Kalyagin M. Y.**, Rabinsky L. N.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: egorovams@mai.ru
**e-mail: mukalyagin@yandex.ru

Abstract

The article presents a mathematical formulation for a generalized model of a gradient porous medium. Variational formulations of the traditional "incorrect" and variational "correct" theory of cylindrical bending of porous plates are considered as applied models. The "incorrect" theory differs from the "correct" one in that in it the classical cylindrical stiffness is corrected due to gradient stiffness, while in the correct formulation of the generalized theory of a porous medium there is no such effect. An original applied model belonging to the class of Timoshenko models is also proposed.
The development of accurate deformation models for thin-walled structures is critical in aerospace engineering, microelectronics, and biomechanics, where scale effects caused by material microstructure significantly affect mechanical properties. 
A generalized gradient model of a porous medium is proposed, combining Mindlin’s and Toupin’s approaches. It accounts for gradients of compatible/incompatible deformations and introduces algebraic porosity. 
Equilibrium equations and boundary conditions for the generalized model are derived, covering classical elasticity, algebraic porosity, and gradient theories as special cases. 
Correct variational formulations for cylindrical bending of plates are developed, eliminating singularities at small thicknesses. 
Traditional "incorrect" models distort bending stiffness via gradient terms, while the proposed formulation preserves physical adequacy. 
The model overcomes limitations of analogs, ensuring precise description of scale effects. Results are applicable to microstructured components in aerospace systems.

Keywords:

gradient models, porous medium, variational formulations, scale effects, Mindlin and Toupin models, cylindrical bending, correct and incorrect theories, pore defects, algebraic porosity

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