On the Decomposition of Equations of Micropolar Elasticity and Thin Body Theory

Abstract

The motion equations of the micropolar theory of elasticity in displacements and rotations represented by the matrix differential tensor-operator for any inhomogeneous anisotropic materials. As a particular case the micropolar isotropic homogeneous materials with a center of symmetry is considered. In this case, the matrix differential tensor-operator of cofactors to the matrix differential tensor-operator of the motion equations is constructed. This constructed operator makes possible to decompose the equations. The equations are obtained separately with respect to the displacement and rotation vectors. Decomposed equations also obtained for a reduced medium. In this case, the equation with respect to the displacement vector is the same as the equation of the classical theory, and the equation with respect to the rotation vector has a similar form. In addition, in the absence of volume loads, the equations of the reduced medium do not depend on the properties of the material. This suggests that these equations can be used to identify the material constants of this medium. The cases under which the static boundary conditions are easily split are revealed. From the decomposed equations of the micropolar theories of elasticity, the corresponding decomposed equations of the static (quasistatic) problem of the theories of single-layer and multi-layer prismatic bodies of constant thickness in displacements and rotations are obtained. From the last systems of equations the equations in the moments of unknown vector functions with respect to any systems of orthogonal polynomials are derived. As a particular case, we obtain a system of equations of the eighth approximation in moments with respect to the system of Legendre polynomials, which decomposes into two systems. One of them is the system with respect to the even order moments of the unknown vector function, and the other system is the system with respect to odd order moments of the same functions. Based on the obtained operator of cofactors to the operator of any of these systems we get a high order (the order of the system depends on the order of approximation) elliptic type equation for each moment of the unknown vector function, which characteristic roots are easily found. Using Vekua’s method for solving such equations [66], we can obtain their analytical solution. Note also that the analytic method with the use of the orthogonal polynomial systems (Legendre and Chebyshev) in constructing the one-layer [2, 3, 7, 10, 15, 17, 18, 20–22, 63, 68, 69] and multilayer [4–6, 13, 60, 61] thin body theory was also applied by other authors. In this direction the authors had published the papers [24–31, 33–37, 41–45, 51–53], and others with the application of Legendre and Chebyshev polynomial systems. These expansions can be successfully used in constructing any thin body theory. Despite this, classical theories are far from perfect, and micropolar theories and theories of another rheology are very far from perfect.