N. B. Zhukova

On the applicability of the relations of the cubic timoshenko-type theory of shells to the study of the postcritical behavior of rods

The question of whether the nonlinear Timoshenko-type theory of shells can be applied to the study of the initial postcritical behavior of a rod under compression is considered. The Koiter asymptotic theory in the Budyanskii form is used. The exact solution of the problem is obtained and a formula for the coefficient of postcritical behavior allowing for the effect of lateral-shear strains is derived. It is shown that the expressions (specified to within cubic terms) for lateral-shear strains and curvature permit us to use the nonlinear theory of shells to analyze the initial supercritical behavior of rods

Application of the Timoshenko–Mindlin Theory to the Calculation of Nonlinear Deformation and Stability of Anisotropic Shells

The nonlinear deformation and stability of composite shells are estimated by using the Timoshenko–Mindlin theory of anisotropic shells. The resolving system of equations is presented in a mixed form in displacements, forces, and moments. For its derivation, a modified version of the generalized Hu–Washizu variational principle formulated in rates for a quasi-static problem is used. However, instead of differentiation with respect to time, displacements, stresses, and loads are assumed to depend on a parameter, for which it is advisable to take the length of the arc of equilibrium states, as demonstrated in some studies. On variation of this parameter, the shell-load system can occur either at regular or singular points. A boundary value problem is formulated in the form of a normal system of differential equations in the derivatives of displacements, forces, and moments. In the separation of variables, the Fourier series are used in a complex form. The boundary value problem is solved by the Godunov discrete orthogonalization method in the field of complex numbers. Then, the Cauchy problem is solved by using known methods. Using the methodology developed, an analysis of the influence of composite properties and parameters of the layered structures on the form of the equilibrium curves of cylindrical shells is carried out. The mechanical characteristics of the initial elementary layers of the reinforced material are determined by the micromechanics methods developed by Eshelby, Mori–Tanaka, and Vanin.