The Moisture Diffusion Equation for Moisture Absorption of Multiphase Symmetrical Sandwich Structures

: When hydrophilic materials (such as natural ﬁber, epoxy resin or concrete) compose sandwich structures, the moisture absorption from hydrothermal environments may signiﬁcantly affect their mechanical properties. Although some experimental works were carried out, few mathematical efforts have been made to describe the moisture diffusion of multiphase symmetrical sandwich structures thus far. In this paper, the moisture diffusion equation was developed to effectively predict the moisture diffusion behavior of multiphase symmetrical sandwich structures as the function of aging time. Both ﬁnite element analysis (FEA) and experimental works were carried out to validate the accuracy of the analytical method, and the analytical results show a good agreement with FEA and experimental data. The effect of the interface condition on the concentration at the interfaces was discussed; the difference between concentration and normalized concentration was illustrated; the correct interface condition, which is a continuous normalized concentration condition, was explained for the moisture diffusion behavior of sandwich structures.


Introduction
Sandwich structures have been widely used in daily life and engineering, such as automotive, sport equipment, building and aerospace, since they present good energy absorption properties, lightweight characteristics, and good designability [1][2][3][4]. Despite these advantages, if the materials used in the sandwich structures are hydrophilic materials, they have the drawback of high hydrophilicity when they are exposed to hydrothermal environments [5][6][7]. The moisture in the environment will diffuse into the sandwich structures and then affect the mechanical property of the materials until the failure of the structures [8][9][10][11]. The analysis of the moisture diffusion behavior of sandwich structures is important for their long-term performances and future applications [12][13][14]. Thus, an effective method should be developed to predict their moisture diffusion behavior.
Some experimental and numerical works have been carried out to study the moisture diffusion behavior of sandwich structures. For example, Saidane et al. [15] investigated the moisture absorption of flax/glass composite sandwich structures. It is found that if glass fiber layers increase, the water uptake and speed of diffusion will obviously reduce. Nurge et al. [16] investigated the moisture diffusion of a graphite/epoxy composite sandwich coupon with a foam core. They developed a finite-difference method by applying a mass-conserving approach to accurately predict the moisture uptake, and a good agreement was achieved with the experimental results. Katzman et al. [17] studied the moisture diffusion behavior of polymer core materials in sandwich structures. A similar finite-difference method was developed to predict the moisture uptake as a function of time. It is found that this method agreed well with the experimental results. Jalghaf [18] developed the numerical method to solve similar equations, and a comparative study of explicit and stable time integration schemes was carried out.
Although the experimental works have been done on the moisture diffusion behaviors of sandwich structures [19][20][21][22], few analytical works have been done to describe their moisture diffusion behaviors, especially for sandwich structures which have a multiphase structure. Yu and Zhou [23] studied two-phase moisture diffusion equations for the moisture diffusion behaviors of flax/glass fiber reinforced composites. The analytical calculations show a good agreement with FEA and experiment results. Joshi et al. [24] solved the moisture diffusion problem of sandwich structures. They used continuous moisture concentration conditions, which are treated as the simplest interface conditions. The form of interface condition between different phases can significantly affect the final results of the moisture absorption [25,26].
Comparative studies of the heat conduction and moisture diffusion problem were presented in some of the literature [27][28][29][30]. It should be noticed that when we solve the heat conduction problem, the temperature at the interface of sandwich structure is continuous, however, the concentration at the interface is discontinuous. This important difference is pointed by some studies [25,31]. According to the correct one, the interface conditions should be modified to continuous normalized concentration conditions, otherwise the analytical solution will be totally different from the real situation. Bao [31] discussed the moisture absorption of composite materials, and the moisture diffusivity models were developed according to continuous normalized concentration condition. They also indicated that the relative moisture concentration should correspond to temperature rather than absolute concentration.
From the literature review, the studies on moisture diffusion in two-phase symmetrical sandwich structure have been developed. However, research on moisture diffusion of multiphase symmetrical sandwich structures, which are important materials in engineering, is seldom found in references. In this paper, the moisture diffusion equation is developed to solve moisture diffusion behaviors of multiphase symmetrical sandwich structures. Firstly, the moisture diffusion problem was solved using continuous normalized concentration interface conditions. Then, both FEA and experimental works were introduced and carried out. Finally, the analytical results were compared with FEA and experiment results to validate the proposed analytical method. Moreover, the effects of different interface conditions on the moisture diffusion behavior of sandwich structures were also discussed.

Method
The diagram of multiphase symmetrical sandwich structure is shown in Figure 1a, and the moisture is applied at the upper and bottom surfaces of sandwich structure. To investigate moisture diffusion along the thickness direction, this moisture diffusion problem is simplified as shown in Figure 1b, where the white layer, grey layer and blue layer are defined as Phase 1, Phase 2 and Phase 3. The saturated moisture concentration C ∞ is defined as C ∞ = M ∞ /V, where M ∞ is saturated moisture uptake and V is volume of the sample [32]. The saturated moisture concentrations and diffusivities of phase 1-3 are C 1∞ , D 1 , C 2∞ , D 2 , C 3∞ and D 3 , respectively. The thicknesses of phases 1-3 are a 1 -a 2 , a 2 -a 3 and a 3 , respectively, and C 1 (x,t), C 2 (x,t) and C 3 (x,t) represent the moisture concentrations in phases 1-3, where t is the aging time.
The moisture diffusion equations of phases 1-3 are:   The moisture diffusion equations of phases 1-3 are: The boundary conditions are: The initial conditions are: The interface conditions are: The boundary conditions are: The initial conditions are: The interface conditions are: The Laplace transform of Equations (1)-(3) are: where q i = (p/D i ) 1/2 , i = 1,2,3, p is complex frequency. The solutions of Equations (11)-(13) are: where σ = (kD 2 /D 1 ), Thus, the solutions of concentrations in phase 1-3 are: where d 4 , d 5 and d 6 are shown in Appendix A. Then, the moisture absorptions of phase 1-3 are obtained by integral along the thickness direction: where d 7 and d 8 are shown in Appendix A. Thus, the total moisture uptake of sandwich structure is:

Experiment
Unidirectional glass fiber fabric was provided by Nanjing glass fiber research institute, unidirectional flax fiber fabric and jute fiber plane weave fabric were processed and manufactured by Nanjing Haituo composite material Co., Ltd., as shown in Figure 2a-c. The fiber fabric density and fiber density are given in Table 1. Non-hybrid and hybrid composite materials were formed by mold pressing, flax, glass and jute fiber fabrics. They were arranged on thick steel plates, and fabrics were manufactured according to the same fiber direction but according to a different layer sequence. The manufacturing process is illustrated in Figure 2d. Then, the composites were cured at room temperature after vac-uum pumping experiment. Finally, the composite plates were cut into moisture absorption specimens by a cutting machine; the specimen is shown in Figure 2e. There were three test pieces of each composite type. The layer sequence and thickness of flax/glass/jute fiber-reinforced sandwich structures are shown in Table 2. manufactured by Nanjing Haituo composite material Co., Ltd., as shown in Figure 2a-c. The fiber fabric density and fiber density are given in Table 1. Non-hybrid and hybrid composite materials were formed by mold pressing, flax, glass and jute fiber fabrics. They were arranged on thick steel plates, and fabrics were manufactured according to the same fiber direction but according to a different layer sequence. The manufacturing process is illustrated in Figure 2d. Then, the composites were cured at room temperature after vacuum pumping experiment. Finally, the composite plates were cut into moisture absorption specimens by a cutting machine; the specimen is shown in Figure 2e. There were three test pieces of each composite type. The layer sequence and thickness of flax/glass/jute fiber-reinforced sandwich structures are shown in Table 2.   The moisture absorption test pieces of sandwich structural composites were firstly put into a 60 °C constant temperature drying oven for 24 h, then the specimens were taken out and the four lateral sides of the moisture absorption test pieces were coated with waterproof material to ensure that the moisture diffuses along the thickness direction during the moisture absorption test. Next, the moisture absorption of specimens was tested by an electronic balance (Mettler Toledo al-104) and we recorded the initial mass W0 of the dried   The moisture absorption test pieces of sandwich structural composites were firstly put into a 60 • C constant temperature drying oven for 24 h, then the specimens were taken out and the four lateral sides of the moisture absorption test pieces were coated with waterproof material to ensure that the moisture diffuses along the thickness direction during the moisture absorption test. Next, the moisture absorption of specimens was tested by an electronic balance (Mettler Toledo al-104) and we recorded the initial mass W 0 of the dried test piece. Then, the specimens were put into a constant temperature environmental box with 60 • C and 100% relative humidity. The specimens were taken out at a certain time, wiped with the test paper, and their weight gains W t were weighed and recorded until the moisture absorption basically does not increase. The moisture absorption M t of the material is expressed by the following formula: The diffusivity can be calculated as below: where h is the specimen thickness, D is the diffusivity and M ∞ is its maximum moisture uptake in equilibrium state. k is the slope of the linear part of M t versus the t 0.5 curve [25].

Finite Element Analysis
The moisture diffusion of multiphase symmetrical sandwich structures through the thickness direction were solved by the analytical method. To validate the analytical model, the sandwich structure including phase 1-3 was developed in the commercial software Abaqus 6.11. The model established in Abaqus is shown in Figure 3. The mass diffusion method in Abaqus is Fick's second law (∂ψ/∂t = D∂ 2 ψ/∂x 2 ), which predicts how diffusion causes the concentration to change with time, where ψ represents normalized concentration. A mass diffusion option was used when the materials and steps in Abaqus were set up. Because Abaqus does not have the element type for moisture diffusion in family option, we used heat transfer element instead. The number of mesh element is 400, the element type is a 4-node linear heat transfer quadrilateral. The element type is shown in Figure 4. The boundary condition of analytical method is normalized concentration = 1. This boundary condition is also used in moisture diffusion experiment since water or humid = 100% environment was treated as normalized concentration = 1 at the surface of specimens in experiment. Thus, normalized concentration = 1 was applied at the upper and bottom surfaces.  (26) where h is the specimen thickness, D is the diffusivity and M∞ is its maximum moisture uptake in equilibrium state. k is the slope of the linear part of Mt versus the t 0.5 curve [25].

Finite Element Analysis
The moisture diffusion of multiphase symmetrical sandwich structures through the thickness direction were solved by the analytical method. To validate the analytical model, the sandwich structure including phase 1-3 was developed in the commercial software Abaqus 6.11. The model established in Abaqus is shown in Figure 3. The mass diffusion method in Abaqus is Fick's second law (∂ψ/∂t = D∂ 2 ψ/∂x 2 ), which predicts how diffusion causes the concentration to change with time, where ψ represents normalized concentration. A mass diffusion option was used when the materials and steps in Abaqus were set up. Because Abaqus does not have the element type for moisture diffusion in family option, we used heat transfer element instead. The number of mesh element is 400, the element type is a 4-node linear heat transfer quadrilateral. The element type is shown in Figure 4. The boundary condition of analytical method is normalized concentration = 1. This boundary condition is also used in moisture diffusion experiment since water or humid = 100% environment was treated as normalized concentration = 1 at the surface of specimens in experiment. Thus, normalized concentration = 1 was applied at the upper and bottom surfaces.   The parameters including thickness (mm), the diffusivity (mm 2 /hour) and equilibrium moisture content (%) in this structure are defined, and four cases are shown in Table  3. Table 3. The parameters in FEA model. The parameters including thickness (mm), the diffusivity (mm 2 /h) and equilibrium moisture content (%) in this structure are defined, and four cases are shown in Table 3.  Figure 5 illustrates the moisture uptake comparison of cases 1-4 between analytical results and FEA calculation. Four cases of moisture absorption for multiphase symmetrical sandwich structure were calculated by Abaqus, the thicknesses, diffusivities, and equilibrium moisture contents of phases 1-3 were changed to verify the analytical method by Equation (24). In reference [25], we know that the initial moisture uptake is a straight line for Fickian diffusion if we use t 0.5 as the x-axis. To conveniently observe whether the moisture is Fickian diffusion, t 0.5 or t 0.5 /h is often used as x-axis [25,26,31]. Here, we use t 0.5 /h as x-axis. The unit of thickness h is mm and the unit of time t is hour. Compared with the FEA calculation, the analytical results show a good agreement.

The Effect of Interface Condition
The interface condition will play a very important role for the final results of moisture absorption. By using analytical methods here, we will explain the details of concentration at the interface. Thus, the normalized concentration continuity condition can more accurately describe the concentration distribution sandwich structure.
To better understand concentration C and normalized concentration ψ, the explanation is shown as follow. The concentration C is defined as C = M/V, where M is moisture uptake and V is volume of the sample. The normalized concentration ψ = C/C ∞ . Obviously, the saturated moisture concentration C ∞ and V may be different from each other for different phases, but the normalized concentration ψ will finally reach 1. Thus, the normalized concentration is continuous, while concentration is discontinuous.  Table 4. It can be found from Figure 9

The Effect of Interface Condition
The interface condition will play a very important role for the final results of m absorption. By using analytical methods here, we will explain the details of concen at the interface. For example, the concentration and normalized concentration di tions at the interface of case 1 are shown in Figures 6 and 7  The reason is attributed to the fact that the glass fiber is non-hygroscopic while flax fiber is highly hydrophilic. Figure 9b shows the weight gain curve of [FGJ]s and [FGGF]s structures. From the figure, it can be seen that the moisture diffusion behaviors of these sandwich structures no longer fit Fick's law: their weight gains rapidly increase at first, then slowly increase until saturation. This phenomenon occurs because there are "FGGF" and "FGJ" structures in the ply, and the moisture diffuses faster in the flax fiber layer on the surface and inside, while the glass fiber layer in the middle diffuses slower.
FOR PEER REVIEW 9 of 15 0.997). Thus, the normalized concentration continuity condition can more accurately describe the concentration distribution sandwich structure.
To better understand concentration C and normalized concentration ψ, the explanation is shown as follow. The concentration C is defined as C = M/V, where M is moisture uptake and V is volume of the sample. The normalized concentration ψ = C/C∞. Obviously, the saturated moisture concentration C∞ and V may be different from each other for different phases, but the normalized concentration ψ will finally reach 1. Thus, the normalized concentration is continuous, while concentration is discontinuous.   (c) (d)     "FGJ" structures in the ply, and the moisture diffuses faster in the flax fiber layer on surface and inside, while the glass fiber layer in the middle diffuses slower.

Comparison between Experimental and Analytical Results
The root mean square error (RMSE) between analytical and experimental result shown in Table 5. From   The root mean square error (RMSE) between analytical and experimental results is shown in Table 5. From is less than 0.130. The errors between analytical and experimental results are small enough to prove obtained analytical results.

Conclusions
This paper presents a solution of moisture diffusion equations for multiphase symmetrical sandwich structures. Both the FEA and experimental works have been carried out to validate the analytical results, and the main conclusions are listed below:

1.
The analytical solution of moisture diffusion equation was given to predict the moisture absorption of multiphase symmetrical sandwich structures; the diffusivities and saturated moisture concentrations of different phases in sandwich structure can be obtained according to moisture diffusion experiments of non-hybrid fiber reinforced composites. Compared with FEA and other methods, the analytical solution of moisture uptake or concentration can be used as basic variables to further analyze the stress or strength of multiphase symmetrical sandwich structures when they are exposed to hydrothermal environments. The analytical solution is more convenient and intuitive.

2.
FEA results were obtained using a mass diffusion method in Abaqus, the validation of analytical method by FEA was carried out for four cases, and the results show a good agreement. 3.
The interface condition of different phases in sandwich structure was discussed. The concentration and normalized concentration at the interface were compared by FEA results. The fact was obtained that the normalized concentration is continuous at the interface. Thus, the correct interface conditions are continuous normalized concentration conditions. 4.