Analytical Solutions in the Theory of Thin Bodies

Abstract

Some questions about the parametrization of three-dimensional thin body with one small size under an arbitrary base surface and the changing of transverse coordinate from –1 to 1 are considered. The vector parametric equation of the thin body domain is given. In particular, we have defined the various families of bases and geometric characteristics generated by them. Expressions for the components of the second rank isotropic tensor are obtained. The representations of some differential operators, the equations of motion, and the constitutive relations of micropolar elasticity theory under the considered parametrization of the thin body domain are given. The inverse tensor operators to a tensor operator of the equations of motion in terms of displacements for an isotropic homogeneous material and to a stress operator are found. They allow decomposing equations and boundary conditions. The inverse matrix differential tensor operator to the matrix differential tensor operator of the equations of motion in displacements and rotations of the micropolar theory of elasticity is constructed for isotropic homogeneous materials with a symmetry center as well as for materials without a symmetry center. We obtain the equations with respect to displacement vector and rotation vector individually. As a special case, a reduced continuum is considered. Cases in which it is easy to invert the stress and the couple stress operator are found out. From the decomposed equations of classical (micropolar) theory of elasticity, the corresponding decomposed equations of quasistatic problems of theory of prismatic bodies with constant thickness in displacements (in displacements and rotations) are obtained. From these systems of equations, we derive the equations in moments of unknown vector functions with respect to any system of orthogonal polynomials. We obtain the systems of equations of various approximations (from zero to eighth order) in moments with respect to the systems of Legendre and second kind Chebyshev polynomials. The system splits and for each moment of unknown vector function we, obtain a high order elliptic type equation (the system order depends on the order of approximation), the characteristic roots of which can be easily found. Using the method of Vekua, their analytical solution is obtained. For micropolar theory of thin prismatic bodies with two small sizes and a the rectangular cross-section, the decomposed equations in moments of displacement and rotation vectors via an arbitrary system of polynomials (Legendre, Chebyshev) are obtained. Similar equations are also deduced for the reduced medium containing classical equation. The decomposed systems of equations of eight approximations for micropolar theory of multilayer prismatic bodies of constant thickness in moments of displacement and rotation vectors are obtained. Using Vekua method, we can find the analytical solutions for this system and for equations for the reduced medium.