Keywords: Mindlin plate, finite elements, strain gradient notation, shear locking, spurious zero energy modes, parasitic shear.

An eight-node serendipity plate element based on Mindlin's plate theory is formulated using strain gradient notation. Strain gradient notation is a physically interpretable notation which allows for an a-priori evaluation of the finite element modeling capabilities [1]. It is possible to precisely identify spurious terms which are responsible for locking and afterwards remove those terms to render an error-free element.

In the present case, the shear strains polynomial expansions of the element are inspected for the identification of spurious terms associated with shear locking. Several spurious terms are identified and removed. In addition, compatible mode terms, which might be mistaken as spurious, are clearly identified by recognizing that they comprise compatibility equations. Those terms are retained in order to avoid the introduction of spurious zero energy modes.

Three example problems are solved to validate the procedure and to show the effectiveness of the corrected version of the finite element; namely, the pergola roof [2], a corner supported square plate [3], and a symmetric cross-ply laminated composite plate [4]. In each case, solutions provided by the model containing the spurious terms are compared to solutions provided by the corrected model. In the pergola roof problem, only transverse displacements are calculated. In the corner supported plate problem, transverse displacements and in-plane normal stresses are calculated. In the laminated composite plate problem, in-plane normal and transverse shear stresses are calculated. The solutions are displayed via curves which clearly demonstrate the differences in stiffness of the model with and without the spurious terms. Those numerical analyses demonstrate that the identified spurious terms are the cause of shear locking. Those analyses also demonstrate the effectiveness of the procedure employed to eliminate the spurious terms as, in general, solutions provided by the corrected model converge monotonically and faster to analytical solutions.

A limitation of the current work is that the element has not been designed to assume arbitrary shapes as its formulation does not embed a geometric mapping procedure such as that used in the isoparametric formulation. Nevertheless, it is shown that the element behaves quite well as a rectangular element after elimination of the spurious terms. The use of strain gradient notation may be viewed as advantageous as it allows spurious terms responsible for shear locking to be identified and eliminated a-priori. The authors claim to have demonstrated theoretically the sources of shear locking and spurious zero energy modes in the eight-node serendipity plate element, and to have built an efficient first-order shear deformation theory element to analyze rectangular plates.

1

J.O. Dow, "A Unified Approach to the Finite Element Method and Error Analysis Procedures", San Diego, CA: Academic Press, 1999.

2

T.J.R. Hughes, M. Cohen, M. Haroun, "Reduced and selective integration techniques in finite element analysis of plates", Nuclear Engineering and Design, 46, 203-222, 1978. doi:10.1016/0029-5493(78)90184-X

3

F. Gruttmann, W. Wagner, "A linear quadrilateral shell element with fast stiffness computation", Computer Methods in Applied Mechanics and Engineering, 194, 4279-4300, 2005. doi:10.1016/j.cma.2004.11.005

4

J.N. Reddy, "Mechanics of Laminated Composite Plates and Shells: Theory and Analysis", 2nd ed., Boca Raton, FL: CRC Press, 2004.

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