Modern Trends in Structural and Solid Mechanics 1. Группа авторов

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СКАЧАТЬ other part. Of course, linearly independent components of u and the surface traction vector, f_i = σ_ij n_j, can alternatively be prescribed at a point of the boundary. Here, n_j 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

[1.4]

In equation [1.4], C is a 6 x 6 symmetric matrix of elasticities of the layer material, and (σ_1, σ_2, σ_3, σ_4, σ_5, σ₆) = (σ_11, σ_22, σ_12, σ_13, σ_23, σ₃₃). A similar notation is used for e.

      We use a mixed formulation and take s^k = (u₁^k, u₂^k, u₃^k, σ₄^k, σ₅^k, σ₆^k, e₁^k, e₁^k, e₃^k) as unknowns at a point in each plate layer. In order to solve for sk, we define the following residuals on the k^th layer that only involve first-order derivatives for the elements of sk:

      [1.5a]

[1.5b]

[1.5c]

In equations [1.5b] and [1.5c], the coefficients Aijk represent the elasticities of the k^th layer material, with respect to the layer material principal axes rotated with respect to the analysis coordinate system (see Figure 1.1). The quantities are found by expressing the in-plane stresses and transverse strains in terms of the transverse stresses and in-plane strains (i.e. the layerwise continuous variables) from equations [1.2] and [1.3] (see Moleiro et al. 2011). It should be highlighted that in-plane strains have been incorporated into the mixed formulation, in order to recast the governing equations into first-order form, which allows the use of C⁰ basis functions for the interpolation of unknown variables, discussed later. While this increases the number of unknowns to solve for, the advantage is that continuity of surface tractions and displacements at layer interfaces can now be enforced.

      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, , on them, and edges x = 0, a and y = 0, b have a combination of fi and ui, where and are the known function of the in-plane coordinates. For example, at a clamped edge, , and at a free edge, . At a simply supported edge x = 0, we set

[1.6]

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

[1.8]

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 k^th 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:

[1.9]

[1.10]

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 P_n(ξ) = 0, where P_n 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], СКАЧАТЬ