Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells (original) (raw)

2007, International Journal of Solids and Structures

We formulate the exact, resultant equilibrium conditions for the non-linear theory of branching and self-intersecting shells. The conditions are derived by performing direct through-the-thickness integration in the global equilibrium conditions of continuum mechanics. At each regular internal and boundary point of the base surface our exact, local equilibrium equations and dynamic boundary conditions are equivalent, as expected, to the ones known in the literature. As the new equilibrium relations we derive the exact, resultant dynamic continuity conditions along the singular surface curve modelling the branching and self-intersection as well as the dynamic conditions at singular points of the surface boundary. All the results do not depend on the size of shell thicknesses, internal through-the-thickness shell structure, material properties, and are valid for an arbitrary deformation of the shell material elements. boundary value problem. Such a general, dynamically and kinematically exact, six-scalar-field theory of regular shells, formulated with regard to a non-material weighted surface of mass taken as the shell base surface, was developed by Simmonds (1983, 1998) and , and with regard to a material surface arbitrary located within the shell-like body by Stumpf (1990), Chró ścielewski et al. (1992), and . For this general shell model efficient finite element algorithms were developed and many numerical examples of equilibrium, stability, and dynamics of regular and complex shell structures were presented by Chró ścielewski et al. works it was assumed that the region of shell irregularity (e.g., branching, self-intersection, stiffening, technological junction, etc.) is small as compared with other shell dimensions and its size can be ignored in deriving the resultant 2D equilibrium conditions. However, such an assumption brings an undefinable error into the resultant dynamic continuity conditions formulated along the singular surface curves modelling the irregularity regions. Therefore, such conditions cannot be regarded as exact implications of 3D equilibrium conditions of continuum mechanics.