On construction of approximate solutions of equations of nonlinear and nonshallow shells (Q2655989)

This work is concerned with the derivation of two-dimensional static shell equations from the equtions of three dimensional elasticity. To this end a particular curvilinear coordinate system generated by two Gaussian parameters of the midsurface of the shell and the third coordinate in the normal direction is adopted. This is of course a very well-known approach. Then the components of the stress and strain tensors are evaluated in this coordinate system, the exact nonlinear relations between the strain tensor and the displacement vector is assumed. Furthermore, in order to describe the constitutive relations for a homogeneous and isotropic elastic solid a cubic expression for the strain energy function is utilised. Hence stress-strain relations become quadratic. Then the relevant quantities are expanded into series of Legendre polynomials across the shell thickness so a hierarchy of field equations depending only two Gaussian parameters are obtained. By employing a small parameter that is the characteristic ratio of thickness to radius of curvature, an asymptotic expansion is used for the displacement vector to obtain approximate solutions for each member of the hierarcical field equations. By introducing a complex variable, the field equations are transformed to forms whose solutions can be found by analytical functions by imitating the method of Muskhelishvili.