Non-linear problems of connecting composite spatial bodies and thin shells, and variational methods for their solution

Abstract

The contact formulation of geometrically non-linear problems of connecting composite spatial bodies, as well as thin composite shells interconnected by a butt (rigidly or not, as a hinge, say) is considered. In this formulation, the composite body (shell) is separated into individual elements, appropriate interaction reactions are introduced into the consideration on the common interface, and an appropriate boundary value problem is formulated for each element. An artificial increase in the number of unknowns of the problem here results in a corresponding increase, in the number of equations because of the replacement of the static connection conditions for the elements by double the number of static boundary conditions on the common interface. While solving the problem, the interaction reactions are determined from the kinematic conditions for the element connections. It is shown that the advantage of such a formulation for problems of the connection of composite bodies is the considerable simplification of the application of methods of the direct calculus of variations for their solution. To this end, functionals referred to the class of generalized Lagrange functions for problems defined on discontinuous stress, deformation, and displacement fields /1–3/, to which the unknown interaction reactions are also referred together with the displacements to the number of functional arguments, are constructed for spatial composite bodies and composite shells. It is proved that the conditions for their stationarity yield variational equations from which static boundary conditions equivalent to the static connection conditions, and kinematic connection conditions follow in addition to the equilibrium equations and the static boundary conditions on the boundaries where the external static forces are given. According to these equations, the use of direct methods for solving the problems does not require the construction of coordinate functions for the displacements with preliminary satisfaction of the kinematic connection conditions.