A New Type of General Theory for the Development of Nonlinear, Multi-scale Plate Theories

Abstract

The analytical development of a new class of general, nonlinear, multiscale plate theories is presented. The multiscale nature of the theoretical framework is due to the use of a superposition of both general global and local displacement eects. The forms and the orders of both the global and local displacement fields are arbitrary. Using this global-local displacement field the governing equations of the theory are obtained by satisfying the governing equations of nonlinear continuum mechanics referenced to the initial configuration. In particular, the equations of motion and the lateral surface boundary conditions are derived by using the method of moments over the dierent scales subject to an orthogonality constraint. The theory satisfies the interfacical constraints (displacement (dis)continuity and traction continuity) and the top/bottom surface boundary conditions in the strong sense. Delamination eects are incorporated into the theory through the use of cohesive zone models (CZM). The theory is suciently general that any type of CZM can be used to model delamination initiation and growth at the dierent interfaces. Furthermore, the theory is developed in a suciently general fashion that any type of constitutive theory for inelastic material behavior can be incorporated. As a result of the formulation the global and local eects are fully coupled. It is shown that the theory is capable of providing accurate predictions for all of the fields in perfectly bonded and delaminated plates even for relatively low orders of displacement approximations. In particular, the theory is shown to provide accurate predictions for the transverse stresses where the predictions are obtained directly from the constitutive relations. This ability is particularly important when carrying out calculations using history-dependent material models and/or CZMs. Furthermore, these predictions satisfy in a strong sense the requirement of continuity of the stresses across all of the interfaces.