other part. Of course, linearly independent components of u and the surface traction vector, fi = σij nj, can alternatively be prescribed at a point of the boundary. Here, nj is the jth component of the outward unit normal to the boundary.
As is often done, we use the Voigt notation to express σ and e as six-dimensional vectors and write equation [1.2] as
In equation [1.4], C is a 6 x 6 symmetric matrix of elasticities of the layer material, and (σ1, σ2, σ3, σ4, σ5, σ6) = (σ11, σ22, σ12, σ13, σ23, σ33). A similar notation is used for e.
We use a mixed formulation and take sk = (u1k, u2k, u3k, σ4k, σ5k, σ6k, e1k, e1k, e3k) as unknowns at a point in each plate layer. In order to solve for sk, we define the following residuals on the kth layer that only involve first-order derivatives for the elements of sk:
[1.5a]
In equations [1.5b] and [1.5c], the coefficients Aijk represent the elasticities of the kth layer material, with respect to the layer material principal axes rotated with respect to the analysis coordinate system (see Figure 1.1). The
We note that Rka = 0, a = 1, 2, …, 9 are the nine equations for the nine unknowns in sk. Recall that there are three boundary conditions prescribed at each bounding face of the lamina. Generally, the top and the bottom faces of a lamina have prescribed tractions,
Equation [1.6]3 is equivalent to a null in-plane normal stress at the edge x = 0. Since in-plane stresses are not directly computed in the present formulation, we write this boundary condition residual in terms of the variables in s as
[1.7]
which also uses the constitutive relation in equation [1.4]. Thus, for each one of the six bounding faces, we will have three residuals that we denote by R10 through R27.
We define the functional, J, in terms of the residuals, with contributions from each material layer as
We note that the repeated index a goes from 1 to 9, and the second surface integral is for the six bounding surfaces of a lamina, and in it, Rbk (b = 10, 11, …, 27) are residuals for the boundary conditions of the kth layer. In equation [1.8], Ω denotes either the in-plane domain of the plate or one of its edge surfaces, hk is the thickness of the kth layer. In evaluating integrals with respect to x3, elasticities of the individual layer are considered.
Each element of the nine-dimensional vector sk is expressed as the product of complete Lagrange polynomials of degrees N1, N2 and N3 in x1, x2 and x3 defined as follows:
Note that equation [1.9] has 9 x N1 x N2 x N3 unknowns, Saijk, for each layer. In equation [1.10], the basis function ψi is written in natural coordinates, ξ, and in terms of the Pth-order Lagrange polynomial LP(ξ) and its derivative, indicated by the prime symbol. The quantity ξi is a root of the equation Pn(ξ) = 0, where Pn is a Legendre polynomial of order n. Basis functions given in [1.10] are associated with Gauss–Lobatto points. Substitution from equation [1.10] into equation [1.9], the result into equation [1.8],
