A new, unified, theoretical framework for the formulation of general, nonlinear, single-scale shell theories

Abstract A new, general theoretical framework for the generation of single-scale shell theories is presented. The framework is developed for application to the analysis of the general laminated and monolithic shells. The proposed framework is intended to allow the accurate prediction of the effects of material nonlinearity and wave propagation effects through the thickness of a shell. The starting point for the framework is a general, single-scale description of the displacement field expressed in terms of arbitrary expansion functions through the thickness of the shell. The functional forms and orders for the expansion functions in the displacement field representation are arbitrary. The development of the governing equations for the theory is carried out using the general nonlinear equations of continuum mechanics referenced to the initial configuration within the context of general coordinate systems. The equations of motion and the lateral surface boundary conditions for the theory are derived using the method of moments over the domain of the expansion functions. The (arbitrary) top and bottom surface boundary conditions (BCs) are satisfied exactly. The interfacial constraints (continuity of tractions and (dis) continuity of displacements) are also satisfied exactly. Delamination effects are incorporated into the theory through the use of arbitrary functions relating the displacement jumps to appropriate state variables. These functions can be changed without the need for reformulation of the governing equations. The theory is formulated in a sufficiently general fashion that any type of history-dependent material model can be used to describe the history-dependent behavior of the material composing a layer without the need to reformulate the theory. The theoretical framework is unified in the sense that any type of desired single scale shell (smear/equivalent single layer (ESL), discrete layer, or zig-zag) theory can be obtained through suitable specialization of the framework. In the case of a smeared or ESL representation the domain of the displacement representation applies across the entire thickness of the shell. To generate a zig-zag theory within the context of the proposed framework is simply a matter of carrying out the interfacial analysis appropriate to the zig-zag assumptions and substituting the resulting displacement representation into the framework and then proceeding as with a smeared/ESL theory. In the case of a discrete layer analysis the displacement representations applies across each of the individual domains. The domains may correspond to several layers, a lamina, or a sublamina. Thus, the framework represents a comprehensive approach to modeling shells. The predictions of the theory are compared with the results obtained from an exact elastic solution for the static response of a sphere and the exact elastic solution for the dynamic response of monolithic sphere. Both exact solutions are based on the assumptions of spherically symmetric boundary conditions. It is shown that the theory is capable of providing accurate predictions for the pointwise (displacement, strain, and stress) fields distributions in laminated and monolithic shells. Furthermore, it is shown that the behavior of the theory is self-convergent and thus increasing the order of the analysis always converges the predictions to the correct answer.