) as well as rotations about y- and x-directions (��x and ��y, re

) as well as rotations about y- and x-directions (��x and ��y, resp.).The displacements of lamina subelement are expressible v=[Ni]vi;w=[No]wi,(3)where Ni and No are respectively the??asu=[Ni]ui; in-plane Lagrange shape function and out-of-plane polynomial shape function of a nonconforming rectangular element with 12 terms. In detail, ui Zotarolimus(ABT-578)? = u1u2u3u4T, vi = v1v2v3v4T, and wi = w1��1x��1yw2��2x��2yw3��3x��3yw4��4x��4yT. It follows that Bi = ?Ni and Bo = ?No, where ? is the derivative operator. The stiffness matrices of the lamina subelements are assembled in the local stiffness matrix of the laminate element as follows:KLAM=[KlowerKnullKnullKupper],(4)where KLAM[40��40] is the stiffness matrix of laminate element, Klower[20��20] is the stiffness matrix of 0�� lamina subelement, Knull[20��20] is the null matrix, and Kupper[20��20] is the stiffness matrix of 90�� lamina subelement.

Figure 1(c) shows the DOF of the laminate plate element developed in this study. The laminate plate element consists of 8 nodes, the numberings of which are ordered in anticlockwise fashion from bottom to top, and each node possesses 5 DOF, which is similar to that of lamina subelement.2.2. Stiffness Matrix of InterfaceWe adopt here for the interface layer a virtually zero-thickness interface element. There are eight nodes in the zero-thickness interface element, the node sequence of which is arranged in anticlockwise manner from bottom to top (refer Figure 1(b)).

Each node in the zero-thickness interface element contains 3 DOF, which are represented as w8}T,(6)dbot?v8?u8?w7?v7?u7?w6?v6?u6?w5?v5?w4}T,dtop��={u5?v4?u4?w3?v3?u3?w2?v2?u2?w1?v1?dtop=[N]dtop��,(5)wheredbot=ubotvbotwbotT,dtop=utopvtopwtopT,dbot��={u1??below:dbot=[N]dbot��; and dtop represent the displacements of nodes located at the bottom and the top surfaces of the interface element, respectively, and the subscripts in dbot�� and dtop�� are the nodal numbers of the interface element. Here, the Lagrange shape function for a 4-node quadrilateral element is employed for [N]. It should Cilengitide be noted that the shape function of the zero-thickness interface element in this study is a 2-dimensional Lagrange shape function rather than that of a 3-dimensional although the interface element resembles the geometrical configuration of a solid element. The same concept has been successfully applied by Coutinho et al. [49] for a 6-node triangular zero-thickness interface element, where displacements were well represented. Note that the interface considered in this study is an orthotropic material with null normal stresses in x- and y-directions (��x = 0 and ��y = 0) and null in-plane shear stress in x-y plane (��xy = 0).

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