Keywords: nine-parameter shell model, finite rotation, exact geometry solid-shell element.

The present paper deals with the development of the finite rotation exact geometry (EG) four-node solid-shell element with nine displacement degrees of freedom per node. The term EG reflects the fact that coefficients of the first and second fundamental forms are taken exactly at each element node. Therefore, no approximation of the reference surface is needed. The finite element formulation is based on the nine-parameter equivalent single-layer (ESL) theory. A conventional way for developing the higher-order shell theory is to utilize either quadratic or cubic series expansions in the thickness coordinate and to choose as unknowns the generalized displacements of the midsurface. Here, the nine-parameter shell model is developed using a new concept of interpolation surfaces (I-surfaces) inside the shell body. We introduce three I-surfaces, namely, bottom, middle and top surfaces and choose nine displacements with correspondence to these surfaces as fundamental shell unknowns. Such choice allows one to represent the higher-order shell formulation in a very compact form and to derive the non-linear strain-displacement relationships, which are invariant under arbitrarily large rigid-body shell motions in any convected curvilinear coordinate system. It is remarkable that the proposed ESL shell model permits the utilization of three-dimensional constitutive equations and eliminates the use of shear correction coefficients.

Allowing for that displacement vectors of I-surfaces are resolved in the reference surface frame, the proposed EG solid-shell element formulation has computational advantages compared to the conventional isoparametric solid-shell element formulations, since it reduces the computational cost of numerical integration in the evaluation of the stiffness matrix. This is due to the facts that, first, the tangent stiffness matrix derived requires only direct substitutions, i.e. no expensive numerical matrix inversion is needed. The latter is unusual for the isoparametric hybrid/mixed shell element formulations. Secondly, we use the efficient three-dimensional analytical integration that gives the possibility of employing coarse meshes. Besides, the proposed hybrid EG solid-shell element formulation permits additionally the utilization of very large load increments. Therefore, the EG four-node solid-shell element developed is robust and could be implemented for large scale computations of thin and thick laminated shell structures undergoing finite rotations.

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