Elasticity Solutions for Sandwich Arches considering Permeation Effect of Adhesive

In this work, analytical solution of simply supported sandwich arches considering permeation effect of adhesives is presented./e permeation layer is described by the functionally graded material, exponentially graded in the radial direction. /e stresses and deformations of each layer are based on the two-dimensional (2D) elasticity theory in the polar coordinate. /e governing equations of the arch are solved by the layer-wise method, which turns the differential equations with variable coefficients into constant coefficients./e solution can be obtained efficiently by means of the recursive matrix method, especially for the arch with many layers. /e present solution agrees well with the finite element solution with a fine mesh, while the finite element method is time consuming inmesh division and calculation./e one-dimensional (1D) solution based on the Euler–Bernoulli theory is close to the present one; however, the error increases as the arch becomes thick. /e effect of permeation layer thickness on the stresses is studied. It is indicated that the stress distributions tend to be smooth along the radial direction as the permeation layer thickness increases.


Introduction
Owing to various advantages such as corrosion resistance, antifatigue, and high specific strength and stiffness, composite structures are increasingly used in different branches of engineering. In addition, their mechanical behavior can be optimized by ingeniously choosing the material type and composite pattern. A typical application is the sandwich structures [1][2][3][4][5], usually composed of two stiff face layers and a relatively soft core layer, bonded together by the adhesive. It is known that the interfacial damage exists widely in sandwich structures due to the discontinuity of material property at the interface. In practice, the adhesive will permeate into adjacent macrovoid materials such as glass fiber, balsa wood, and aluminum honeycomb, which leads to a refined sandwich model, as shown in Figure 1. e permeation layer is the mixture of the face (or core) layer and the adhesive. is makes the material property smoothly vary along the radial direction, like of functionally graded material (FGM). Such a problem deserves to be deeply investigated. For sandwich plates considering permeation effect, analytical solution has been proposed by Huo et al. [6]. In addition to sandwich plates, sandwich arches have also received a lot of attention due to their advantage of artistic appearance and excellent load-carrying capacity. e present study aims at extending their work to investigate the sandwich arch with permeation effect.
A review of the literature indicates that many analytical models have been proposed for the mechanical behaviors of sandwich structures. Galuppi and Royer-Carfagni [7] studied the bending behavior of the sandwich arch formed by two elastic facial layers and a thin polymeric interlayer. In their report, the deformation of the arch was based on the Euler-Bernoulli assumption, which was suitable for thin arches with moderate curvature. e stress distribution in a sandwich plate with the FG face was studied by Raissi et al. [8] by the use of the layer-wise method based on the firstorder shear deformation theory. e vibration of sandwich microplates with nanocomposite face layers under magnetic and electric fields was studied by Mohammadimehr et al. [9] by virtue of the sinusoidal shear deformation theory. Livani et al. [10] investigated the supersonic flutter of doubly curved sandwich panels with variable thickness subjected to thermal load. In their study, the governing equations were based on a new higher-order shear deformation theory, in which the vertical and in-plane displacement components were assumed as quadratic and cubic functions, respectively. Bouchafa et al. [11] presented a new refined hyperbolic shear deformation theory with only four unknown functions to analyze the thermoelastic bending of FG sandwich plates. Bending, vibration, and buckling analysis of FG sandwich plates were analyzed by Nguyen et al. [12] according to the refined higher-order shear deformation theory. By using a higher-order sandwich theory, Rahmani et al. [13] tackled the free vibration of the composite sandwich shell with a flexible core.
In addition to the aforementioned analytical models, some numerical methods are efficient for the analysis of sandwich structures. By means of the differential quadrature finite element method, Liu et al. [14] investigated bending of the sandwich shell consisting of two homogenous face sheets and a FG core. Natarajan and Manickam [15] presented a QUAD-8 shear flexible element method based on higherorder shear deformation theory to study the bending and free flexural vibration of FG sandwich plates. Dynamic behavior of corrugated-core sandwich plates was analyzed by Peng et al. [16] via a mesh-free Galerkin method, in which the domain of the plate was discretized by scattered nodes, and no elements were required. By virtue of the isogeometric method, ai et al. [17] studied the free vibration and buckling analysis of sandwich plates. is method has the advantage to exactly represent domains and achieve approximation with arbitrarily high smoothness. Mantari et al. [18] developed a layer-wise finite element formulation-based trigonometric layer-wise shear deformation theory for the bending of thick sandwich plates. e analytical solution based on the elasticity theory is important because it can be used as a benchmark to assess other simplified solutions [19]. Based on the elasticity theory, Pagano [20] proposed an analytical solution for the bending of simply supported composite and sandwich plates. e elasticity solution of the cylindrical sandwich shell subjected to radial and axial loads was presented by Kardomateas [21]. Zenkour [22] studied the bending problem of cross-ply laminated plates by the use of the elasticity equations linked with the state space method. Kashtalyan and Menshykova [23] developed elasticity solution for sandwich plates with a FG core.
However, in the above investigations, the permeation effect of the adhesive between layers is neglected. In this paper, 2D elasticity solution is presented for a refined sandwich arch model considering the permeation effect of the adhesive. In the present work, the permeation layer is described as FGM, exponentially graded along the radial direction. e general solution of stresses and deformations with unknown coefficients is obtained by means of the layerwise method. e coefficients are determined via the recursive matrix method, which is efficient for arches with many layers. e solution obtained is compared with the finite element solution and 1D solution based on the Euler-Bernoulli theory, respectively.

Basic Equations.
Without loss of generality, we consider a sandwich arch with internal radius R 1 , external radius R 2 , thickness H, and angle β, composed of p FG layers with each thickness h i , as shown in Figure 2. e adjacent layers are perfectly bonded, i.e., the deformations are continuous along the radial direction. A polar coordinate system r − θ is established. e arch is simply supported at two edges and is subjected to radial load q(θ) acting on the external surface. e permeation layer is the material consisting in the mixture of face (or core) layer material and the adhesive material. Here, we assume its elastic modulus to be exponentially graded along the radial direction, i.e., E i (r) � E i 0 e k i r , in which E i 0 represents the elastic modulus at the internal surface of the i-th layer and k i denotes the gradient index. Poisson's ratio μ i is constant in each layer. e homogeneous layers, i.e., face, core, and adhesive layers, can be included in the model by letting According to the 2D elasticity theory in the polar coordinate [24], the constitutive relations are given by Core layer

Face layer
Face layer Permeation layer Adhesive layer 2 Advances in Polymer Technology in which σ i θ and σ i r are the circumferential and radial normal stresses, respectively; τ i rθ is the shear stress; ε i θ and ε i r are the circumferential and radial normal strains, respectively; and c i rθ is the shear strain. e geometrical equations are in which u i θ and u i r are the deformation components in the circumferential and radial directions, respectively. e equilibrium equations are given by e simply supported boundary conditions of the arch are e load condition on the external and internal surfaces of the arch can be expressed by

General Solution for a Sublayer. By combining equations
(1)-(3), two partial differential equations with respect to u i θ and u i r are derived: Here, the above equations are solved by the layer-wise method [25]. According to this method, each arch layer is divided into λ sublayers each with very small thickness so that the variable r in equation (6) can be taken as a constant. e radius, the middle plane of the j-th (j � 1,2, . . ., pλ) sublayer, is denoted by r ⌢ j . e distances from the arch center O to the internal and external surfaces of the j-th sublayer are, respectively, represented by d 0 j and d 1 j . en, we expand the deformation components of the j-th sublayer into Fourier series: where α m � (mπ/β). Substituting equation (7) into equation (6), we obtain two ordinary differential equations: where [j/λ] means the integer part of j/λ. e solution of equation (8) is a j mn e s j mn r , in which a j mn are unknown coefficients, and s j mn are the four roots of the following quartic equation: and ξ j mn can be expressed by s j mn as follows: Substituting equation (9) into equation (7), the general solutions of the deformation components with unknown coefficients are obtained: a j mn e s j mn r sin α m θ , ξ j mn a j mn e s j mn r cos α m θ , j � 1, 2, . . . , pλ.
Substitution of equation (12) into equations (1) and (2) gives the general solutions of the stress components By substituting equations (12) and (13) into equation (14), Θ j m (r), R j m (r), Y j m (r), and Z j m (r) can be further rearranged into the matrix form as follows: in which By replacing r in the above equation with d 0 j and d 1 j , respectively, one has By eliminating Γ j m in the two equations of equation (17), we obtain e out-of-plane variables are continuous between the adjacent sublayers, i.e., By combining equation (18) with equation (19), the relation between the internal and external surfaces of the arch is obtained: We define in which S 11 m (t), S 12 m (t), S 21 m (t), and S 22 m (t) are the 2 × 2 submatrixes. Equation (20) can be written as en, equation (22) is further decomposed into two matrix equations as follows: e load function q(θ) in equation (5) can be expanded into Fourier series Advances in Polymer Technology 5 Substituting the load condition into equation (23), one has By reusing equations (18) and (19), Ψ j m (d 1 j ) is obtained: Finally, the coefficients are determined: Substituting the coefficients back into equations (12) and (13), the solution of stress and deformation components is obtained.
It can be observed that, as the layer number increases, only the computation effort in equation (23) slightly increases. erefore, the present method is highly efficient for arches with many layers. Moreover, other boundary conditions can also be applied in the present model. For example, the clamped condition can be equivalent to the simply supported one acted by the unknown horizontal reaction, which can be further determined by the zero displacement condition at the clamped end [26].

Example and Discussion
In the following examples, the infinite series in the solution are truncated into the finite term, i.e., m � 1,2, . . ., N.

Convergence Study.
Consider a simply supported sandwich arch with R 1 � 1000 mm, R 2 � 1240 mm, and β � π/4. e face, core, and adhesive layers are made of carbon fiber, polyurethane foam, and epoxy, respectively. e thickness, elastic modulus, and Poisson's ratio of each arch layer are listed in Table 1.
Since the governing equations of equation (6) are solved by the layer-wise method, the effect of the divided number on the solution accuracy should be analyzed first. We assume the arch is subjected to radial load q(θ) � sin(π/β) N/mm. For sinusoidal load, one series term is sufficient, i.e., N � 1. Table 2 gives the present solution of stresses and deformations in the face layer and permeation layer with different divided number λ, respectively. It can be found that the solution tends to be convergent with the increase of λ. Moreover, the solution in the face layer converges faster than that in the permeation layer. e solution is accurate with at least three significant figures when λ � 10. erefore, the divided number is taken as λ � 10 for the following calculations.
en, the convergence property associated with the series terms for the solution is studied. e arch is acted by uniform load q(θ) � 1 N/mm. e present solution of stresses and deformations with different series number N is given in Table 3. It can be found that the present solution is rapidly convergent and has high convergence precision with four significant digits when N � 9.
us, the number of series terms is fixed at N � 9 unless stated.

Comparison Study.
We still consider the simply supported sandwich arch in Section 3.1; however, the acting load is q(θ) � 1 N/mm and β is the variable. e present solution is compared with the finite element (FE) solution obtained by ANSYS. In the FE modeling, the PLANE-182 element is used to simulate all the layers. Due to the symmetry of the arch, only half part of the arch is modeled, as shown in Figure 3. e FE mesh is created by dividing the arch length into 50 elements, while the thicknesses of face, adhesive, core, and permeation layers is divided into 4, 1, 20, and 10 elements, respectively. Figure 4 shows the comparison of stresses and deformations between the present solution and FE solution with different arch angle β. It can be found that the FE solution is in good agreement with the present one; however, the FE solution with the fine mesh is time  Table 2: e present solution of stresses and deformations in the face layer (i � 1) and permeation layer (i � 2) with different sublayer divided number λ, respectively.     By letting k i � 0 in the basic equations, the present solution can be used to predict stresses and deformations for the fully homogeneous layered arch. Consider a sandwich arch composed two face layers sandwiching a relatively soft thin interlayer, which was studied by Galuppi and Royer-Carfagni [7] based on the 1D Euler-Bernoulli theory. e geometric and material parameters are fixed at q(θ) � 0.5 N/mm 2 , E 1 � E 3 � 70000 MPa, E 2 � 2.6 MPa, μ 1 � μ 2 � μ 3 � 0.3, R 1 � 1000 mm, β � π/4, h 1 � h 3 , and h 2 � 2 mm. We define three variables: σ m � σ 1 θ at θ � 0.5β, r � R 1 ; τ m � τ 1 rθ at θ � 0, r � R 1 + h 1 ; and u rm � u 1 r at θ � 0.5β, r � R 1 .
e comparison of σ m , τ m , and u rm between the present, FE, and 1D solutions with different layer