Читать книгу Modern Trends in Structural and Solid Mechanics 1 - Группа авторов - Страница 16

We formulate the problem for a rectangular a x b x h lamina that can be easily extended to a laminate made of p layers. We employ rectangular Cartesian coordinates (x₁, x₂, x₃), shown in Figure 1.1, to describe the position of a material particle in the unstressed reference configuration. In the absence of body forces, infinitesimal deformations of the lamina are governed by the following equilibrium equations [1.1], stress–strain relations [1.2] and strain–displacement relations [1.3]:

[1.1]

[1.2]

[1.3]

Figure 1.1. Geometry and coordinate-axes of a laminated plate

In equations [1.1]–[1.3], σ is the stress tensor, e is the infinitesimal strain tensor, u is the displacement vector and C is the matrix of elasticities. Here, a layer is modeled as a transversely isotropic material with the axis of transverse isotropy along the fiber. A repeated index implies summation over the range of the index. The substitution from equation [1.3] into [1.2], and the result into equation [1.1] gives three second-order partial differential equations for the three displacement components that are to be solved under the prescribed boundary conditions of surface tractions on one part of the boundary and displacements on the 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], and the numerical evaluation of the integral by using the Gauss–Lobatto quadrature rule of order Pn in the three directions, gives J as a function of Saijk. We deduce the needed linear algebraic equations by setting

[1.11]

We realize that expressions for the residuals have different units. When equation [1.11] is written as KA = F, it is likely that the use of non-dimensional variables throughout the chapter will improve the condition number of the matrix K and reduce error. However, we have not tried this. A feature of the equations derived from [1.11] using basis functions of type [1.10] is that they are insensitive to shear-locking effects, which means that reduced integration is not needed in the thickness direction.

We note that the above formulation holds for a laminate, when the continuity of variables u₁, u₂, u₃, σ₄, σ₅, σ₆, e₁, e₂, e₃ is enforced by adding the appropriate residuals in equation [1.8] or using a layerwise theory. Here, we use a layerwise theory, where the contribution from each layer is included in the summation in equation [1.8] and the continuity of the variables in s at each layer interface is ensured.