Mathematical model of micropolar elastic thin plates and their strength and stiffness characteristics

Abstract

A number of hypotheses were formulated using the properties of an asymptotic solution of boundary-value problems of the three-dimensional micropolar (moment asymmetric) theory of elasticity for areas with one geometrical parameter being substantially smaller than the other two (plates and shells). A general theory of bending deformation of micropolar elastic thin plates with independent fields of displacements and rotations is constructed. In the constructed model of a micropolar elastic plate, transverse shear strains are fully taken into account. A problem of determining the stress-strain state in bending deformation of micropolar elastic thin rectangular plates is considered. The numerical analysis reveals that plates made of a micropolar elastic material have high strength and stiffness characteristics.

Asymptotically Confirmed Hypotheses Method for the Construction of Micropolar and Classical Theories of Elastic Thin Shells

“…Question of reduction of three-dimensional static problem of asymmetric theory of elasticity for thin domain of the shell to two-dimensional problem is considered on the basis of asymptotic method with boundary layer [14], including the question of satisfaction of boundary conditions on shell edge Σ .…”

“…In papers [12]- [14] the mentioned idea is developed: on the basis of qualitative aspects of asymptotic solution adequate hypotheses are formulated and as a result static and dynamic applied theories of micropolar elastic thin shells and plates are constructed. The accepted hypotheses are the followings: 1) During the deformation initially straight and normal to the shell middle surface fibers rotate freely in space at an angle as a whole rigid body, without changing their length and without remaining perpendicular to the deformed middle surface.…”

“…It should be noted that in papers [17] [18] applied theories of micropolar elastic thin plates and bars, constructed in papers [14] [27], are justified on the basis of asymptotic method.…”

“…The main problem of the general theory of micropolar or classical elastic thin plates and shells is in approx-imate, but adequate reduction of three-dimensional boundary-value problem of the micropolar or classical theory of elasticity to two-dimensional problem. From our point of view, for achievement of this aim [12]- [14] during the construction of applied theories of thin plates and shells main results of the asymptotic solution of boundary-value or initial boundary-value problem of three-dimensional micropolar or classical theory of elasticity in corresponding thin domains can be used, which are formulated as hypotheses [15]- [18]. Micropolar and classical theories of elastic thin plates and shells, constructed on the basis of such approach, are asymptotically correct theories.…”

Asymptoticaly confirmed hypoteses metod for the construction of micropolar and classical theories of elastic thin shells:

“…Основная проблема общей теории микрополярных или классических упругих тонких пластин и оболочек заключается в приближённом, но адекватном сведении трёхмерной задачи микрополярной или классической теории упругости к двумерной краевой задаче. На наш взгляд, для этой цели уместен [12][13][14] при построении прикладной теории тонких пластин и оболочек использование в качестве гипотез основные особенности поведения асимптотического решения краевой или начальнокраевой задачи трёхмерной микрополярной или классической теории упругости в соответствующих тонких областях [15][16][17][18]. Построенные на таком подходе, как микрополярная, так и классическая теории упругих тонких пластин и оболочек будут асимптотически точными теориями.…”