Modelling of concentration-dependent moisture diffusion in hybrid fibre-reinforced polymer composites

Abstract

Hybrid fibre-reinforced polymer composites have extensive applications due to their high strength, cost effectiveness, improved product performance, low maintenance and design flexibility. However, moisture absorbed by composite components plays a detrimental role in both the integrity and durability of hybrid structure because it can degrade the mechanical properties and induce interfacial delamination failures. In this study, the moisture diffusion characteristics in two-phase hybrid composites using moisture concentration-dependent diffusion method have been investigated. The two phases are unidirectional S-glass fibre-reinforced epoxy matrix and unidirectional graphite fibre-reinforced epoxy matrix. In the moisture concentration-dependent diffusion method, the diffusion coefficients are not only dependent on the environmental temperature but also dependent on the nodal moisture concentration due to the internal swelling stress built during the diffusion process. A user-defined subroutine was developed to implement this method into commercial finite element code. Three-dimensional finite element models were developed to investigate the moisture diffusion in hybrid composites. A normalization approach was also integrated in the model to remove the moisture concentration discontinuity at the interface of different material components. The moisture diffusion in the three-layer hybrid composite exposed to 45℃/84% relative humidity for 70 days was simulated and validated by comparing the simulation results with experimental findings. The developed model was extended to simulate the moisture diffusion behaviour in an adhesive-bonded four-layer thick hybrid composite exposed to 45℃/84% relative humidity for 1.5 years. The results indicated that thin adhesive layers (0.12-mm thick) did not significantly affect the overall moisture uptake as compared with thick adhesive layers (0.76-mm thick).

Keywords

Moisture concentration-dependent diffusion adhesive hybrid composite finite element model

Introduction

Hybrid fibre-reinforced polymer composites have been widely utilized in aerospace and marine structural applications where high strength and design flexibility are required. The combined properties of different components in hybrid composites are the weighed sum of the individual component’s properties so that some desirable balance between the inherent advantages and disadvantages can be achieved.¹ Hybrid fibre-reinforced polymer composites can generally be divided into two types: (a) polymeric matrix reinforced by several different types of fibres and (b) laminates of different types of fibre-reinforced composites. It is well known that hybrid composites are susceptible to the hygrothermal environment,^2,3 especially at elevated temperatures. Moisture penetrating from surfaces plays a detrimental role in both the integrity and durability of composite structures because it can degrade the mechanical properties^4–6 and induce interfacial delamination failures.⁷

Numerous diffusion models have been proposed to study moisture diffusion into various composites under different external hygrothermal conditioning. One-dimensional Fick’s law is the most frequently used one by researchers^3,8,9 to investigate moisture diffusion behaviour into single-fibre-reinforced composites. Gopalan et al.¹⁰ observed that the absorption curve in a mixed fibre-reinforced composite obeys Fick’s law. However, classical Fick’s law is not always adequate when explaining all the moisture diffusion behaviour in polymers or polymer composites. Gurtin and Yatomi¹¹ suggested using a two-stage Fickian process to explain the derivation from theoretical Fickian curve for fibre-reinforced composites. Bao and Yee⁷ proposed a dual-diffusivity model for hybrid composites to fit observed weight gain curves. Weitsman¹² developed a coupled damage and moisture transport non-Fickian model to describe moisture diffusion in transversely isotropic fibre-reinforced polymer composites. This model, however, was mathematically complex.

Some anomalies in moisture diffusion can be explained by the coupling between moisture transportation and local stress state. Both graphite and glass fibres are generally considered as impermeable. Compared with polymeric resin matrix, neither graphite fibres nor glass fibres are significantly affected by the presence of moisture or temperature changes. As moisture penetration proceeds or/and the environmental temperature elevates, the fibres will inhibit the matrix from free-swelling or thermal expansion. Consequently, the residual stresses will build up at the fibre/matrix interface. Some researchers have indicated a significant influence of internal (or external) stress on moisture diffusion behaviour. Whitney and Browning¹³ observed that the absorption curve of graphite/epoxy laminates deviates from the theoretical Fickian curve and proposed a stress-dependent diffusion method. In this method, the decrease in diffusivity corresponding to the swelling of the laminates relieves the tensile residual stress. Other researchers^14,15 also observed that the moisture diffusion process in carbon–epoxy composite is either accelerated under external tensile stresses or retarded under external compressive stresses. Crank¹⁶ suggested that the swelling internal stresses in a polymer sheet influence the diffusion coefficients. However, most of work available in the literature deals with the stress-dependent diffusion mechanism in homogeneous composites.

In this study, a moisture concentration-dependent diffusion model is proposed to investigate moisture diffusion behaviour in multi-layer unidirectional hybrid composites. The moisture model previously developed for the composite laminate by the authors¹⁷ is extended for hybrid composites. To guarantee the continuity of the nodal concentration at the interface of different material phases, normalized concentration is incorporated into the modelling. In the moisture concentration-dependent diffusion model proposed in this study, the diffusion coefficients are not only dependent on temperature but also depend on the nodal moisture concentration at every material point. Compared with the coupled damage and moisture transport non-Fickian model developed by Weitsman,¹² this model provides a significant simplification for this type of stress-dependent diffusion problems, and thus it is easy to be implemented in common finite element commercial codes using user-defined subroutines. In this study, the damage resulting from moisture diffusion is not considered and is limited to temperature and moisture concentration-dependent diffusion only. For model validation, the simulation results were compared with experimental findings of moisture absorption in three-layer unidirectional hybrid composites exposed to 45℃/84% RH for 70 days.

Moisture Diffusion Modelling

Mathematical background

The moisture diffusion behaviour in a simple orthotropic composite plate is governed by Fick’s second law:⁸

\frac{\partial c}{\partial t} = \frac{\partial}{\partial x} (D_{11} \frac{\partial c}{\partial x}) + \frac{\partial}{\partial y} (D_{22} \frac{\partial c}{\partial y}) + \frac{\partial}{\partial z} (D_{33} \frac{\partial c}{\partial z})

(1)

where

c (x, y, z, t)

is the moisture concentration, and

D_{ii}

(i = 1, 2, 3) are the diffusion coefficients along three principal axes (length direction, width direction and thickness direction, respectively). However, to take the moisture diffusion contribution from edges into account, the edge correction factor⁸ is usually used as shown below:

\bar{D} = D_{33} (1 + \frac{h}{l} \sqrt{\frac{D_{11}}{D_{33}}} + \frac{h}{w} \sqrt{\frac{D_{22}}{D_{33}}}) 2

(2)

where

\bar{D}

is the effective diffusivity through thickness, and

l, w, h

are the length, width and thickness of the plate, respectively. The diffusivity is generally considered to be dependent only on temperature, as expressed in the Arrhenius type equation:⁴

D = D_{0} e (- E_{d} / (RT))

(3)

where D₀ is the diffusivity constant,

E_{d}

is the diffusion activation energy, and T is the temperature in Kelvin. However, in the moisture concentration-dependent diffusion model, the diffusion coefficients are also dependent on the moisture concentration on every material node during the diffusion process. As the temperature conditioning in two cases studied in this work is constant during moisture diffusion process, the thermal expansion induced internal stresses due to elevated temperature are not taken into account in this study. Another important parameter is the equilibrium moisture content

M_{m}

, which has the same physical meaning as solubility. This parameter indicates the saturated moisture concentration under certain moist conditioning and temperature. This value is typically a constant if water immersion conditioning is applied, or in an exponential relationship with relative humidity if the humid air conditioning is applied, as follows:⁸

M_{m} = constant (liquidimmersion)

(4)

M_{m} = aRH b (humidair)

(5)

where

a

and b are both experimentally determined constants.

Normalization approach

A few similarities exist between Fick’s law and Fourier’s law, which govern the mass diffusion and heat transfer, respectively. The governing equation for three-dimensional heat transfer in orthotropic materials is given by [8]

ρ C_{p} \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} (K_{x} \frac{\partial T}{\partial x}) + \frac{\partial}{\partial y} (K_{y} \frac{\partial T}{\partial y}) + \frac{\partial}{\partial z} (K_{z} \frac{\partial T}{\partial z})

(6)

where ρ is the material density, C_p is the specific heat capacity, and T is the temperature. The thermal diffusivity

K / (ρ C_{p})

is the change rate of the temperature. For most materials, the thermal diffusivity is several orders higher than the mass diffusivity, which means that the material can reach thermal equilibrium state much faster than moisture equilibrium state.

The difference between heat transfer and mass diffusion is the continuity of primary variables at the interface for layered multi-material system. For heat transfer, the temperature is always continuous at the interface between different materials. While for moisture diffusion, the moisture concentration is discontinuous at the interface of different materials because different materials have different saturated moisture concentration. The moisture concentration discontinuity at the interface for bi-materials system can be expressed as

C_{1} \neq C_{2}

(7)

Material 1 has a higher saturated moisture concentration (solubility) than material 2 (Figure 1a). In both unsaturated and saturated conditions, the moisture concentration at the interface of a layered bi-material system is not continuous. To remove the concentration discontinuity at the interface, a new term “normalized concentration” was introduced¹⁸ and expressed as

φ = C / S

(8)

where C is the moisture concentration and S is the solubility. S primarily depends on material type and conditioning approach. After this new term is introduced, the moisture concentration discontinuity at the interface is removed. The continuity of moisture concentration at interface nodes is expressed as

φ = \frac{C_{1}}{S_{1}} = \frac{C_{2}}{S_{2}} or \frac{C_{1}}{C_{2}} = \frac{S_{1}}{S_{2}} = constant

(9)

Figure 1.

(a) Discontinuity of moisture concentration at the interface for a bi-material system. (b) Continuity of normalized concentration at the interface for a bi-material system.

The normalized concentration φ is continuous at the interface nodes in both unsaturated and saturated conditions (Figure 1b). Essentially, $C_{1} / C_{2} = constant$ is the necessary condition for the continuity of normalized concentration at interfacial nodes. The necessity of this condition could be further proved by Henry’s law.¹⁹

Finite element modelling

The three-dimensional Fickian equation with normalized concentration φ, can be expressed as

\frac{\partial φ}{\partial t} = \frac{\partial}{\partial x} (D_{11} \frac{\partial φ}{\partial x}) + \frac{\partial}{\partial y} (D_{22} \frac{\partial φ}{\partial y}) + \frac{\partial}{\partial z} (D_{33} \frac{\partial φ}{\partial z})

(10)

The finite element equations are given by

[K] {φ} + [M] {\overset{\cdot}{φ}} = {F}

(11)

[M] = \int [N] T [N] d Ω

(12)

[K] = \int [B] T [D] [B] d Ω

(13)

{F} = \int q [N] T d Ω

(14)

where

[K]

is the moisture diffusivity matrix,

[M]

is the moisture velocity matrix,

[N]

is the shape function,

{F}

is the moisture flow vector,

{φ}

is the nodal normalized moisture content and

{\overset{\cdot}{φ}}

is the change rate of the nodal normalized moisture concentration. The diffusivity matrix [D] is given by

[D] = [D_{11} 000 D_{22} 000 D_{33}]

(15)

The matrix of derivatives of shape functions $[B]$ is given by

[B] = [\frac{\partial [N]}{\partial x} \frac{\partial [N]}{\partial y} \frac{\partial [N]}{\partial z}]

(16)

Numerical Simulation

To validate the moisture concentration-dependent diffusion model for layered hybrid composites, a case study was conducted and the results were compared with experimental findings from the literature.^20,21 A detailed manufacturing process is presented in the same literature. All experimental specimens were made from unidirectional S-glass fibre-reinforced epoxy polymer GFRP prepreg 3 M SP250-S29 and unidirectional high modulus carbon fibre-reinforced epoxy polymer CFRP prepreg Cyanamid T152/751/135. In Case 1, unidirectional three-layer hybrid composite specimens were layered up with 4 plies of GFRP prepregs on both the top and bottom and 8 plies of CFRP prepregs in the middle (Figure 2). The dimensions after curing were 2.76 in. × 2.76 in. × 0.13 in. (70 mm × 70 mm × 3.2 mm). The specimens were conditioned at 45℃ and 84% RH for 70 days. The moisture weight gain of multi-layer hybrid structure was calculated with

M_{total} = \frac{W_{w 1} + W_{w 2} + \dots + W_{wn}}{W_{d 1} + W_{d 2} + \dots W_{dn}}

(17)

where

W_{wi}

and

W_{di}

(i = 1, 2 , … n) are the weight of absorbed moisture and initial weight for the nth layer component, respectively.

Figure 2.

Geometry of three-layer hybrid plate (Case 1).

In Case 1, both the length and width of the plate are much larger than the thickness (the aspect ratio was 21.88), the moisture diffusion from four edge sides can be ignored. Hence, this case can be modelled as a one-dimensional diffusion problem along the thickness direction, which significantly reduces the computational cost. A mesh convergence study was conducted in this one-dimensional model. Four different mesh sizes, with 7, 11, 32, and 64 elements, in the thickness direction were investigated. Differences were evident in 7 and 11 elements, both of which had higher moisture concentration and normalized moisture concentration compared with the other two cases (Figure 3a and b). Finite element models with mesh sizes of 32 and 64 elements showed the results overlapping over each other, implying convergence of results. The moisture content value jump in Figure 3(a) indicated the discontinuity of moisture concentration at the interfaces of CFRP and GFRP laminates, whereas the normalized moisture concentration is always continuous at the interfaces (Figure 3b). The convergence study was also conducted for later three-dimensional cases. In both one-dimensional and three-dimensional cases, the initial time increment is 0.01 h and maximum time increment is 60 h. The solution convergence with time is adaptively controlled by an iteration algorithm in ABAQUS.

Figure 3.

(a) Mesh convergence of moisture concentration (Case 1). (b) Mesh convergence of normalized moisture concentration (Case 1).

The saturated moisture content and various material properties for both fibre-reinforced composites were obtained from previous studies,^20–22 as listed in Table 1. Another important diffusion parameter was the diffusion coefficient along the thickness direction. Unlike the traditional Arrhenius relationship for diffusivities used in finite difference code,²¹ a moisture concentration-dependent diffusion method is incorporated in finite element modelling to explain the moisture weight gain for layered hybrid structures. In this method, as moisture penetrates into composites, the fibres restrain the matrix from free-swelling. Thus, the swelling stress builds up gradually, resulting in the decrease of diffusion coefficients. Consequently, the diffusion coefficients are not only dependent on temperature but also dependent on the nodal moisture concentration at each time increment. The moisture concentration-dependent diffusion coefficients are expressed as:

D_{z} = D_{0} Ψ (C) M (h) e (- E_{d} / (RT))

(18)

where

Ψ (C)

is the pattern function representing the dependence of diffusivity on nodal moisture concentration.

M (h)

is an experimentally determined thickness factor which is similar to the edge correction factor, representing the dependence of diffusivity on the specimen thickness. In this study, the initial effective diffusivities were obtained from the initial slopes of moisture absorption curves. The pattern functions were obtained by trial-and-error methods. The resultant effective diffusivities of CFRP and GFRP in Case 1 are illustrated in Figure 4(a). For both CFRP and GFRP composites, when nodal moisture concentration increases, the diffusivities continuously decrease, and when the moisture concentration approaches solubility value

M_{m}

, the diffusivities gradually drift to a constant value. The concentration-dependent diffusivity curves in Figure 4(a) are continuous fifth-order polynomial curves. The corresponding normalized pattern functions are given by

Ψ_{CFRP} = (2.011 \times 10 - 20) C 5 - (8.179 \times 10 - 16) C 4 + (1.190 \times 10 - 11) C 3 - (6.443 \times 10 - 8) C 2 - (2.650 \times 10 - 5) C + 1

(19)

Ψ_{GFRP} = (9.219 \times 10 - 20) C 5 - (2.878 \times 10 - 15) C 4 + (3.197 \times 10 - 11) C 3 - (1.338 \times 10 - 7) C 2 + (5.339 \times 10 - 6) C + 1

(20)

Figure 4.

(a) Effective diffusivities of CFRP and GFRP (Case 1). (b) Comparison between simulation results and experimental findings (Case 1).

Table 1.

Diffusion properties for CFRP and GFRP.

CFRP	GFRP
M_∞ at 84% RH	1.23%	0.86%
Density after curing (g/cm³)	1.54	1.90
Fibre volume fraction (%)	56	54

The pattern function for CFRP is different from that of GFRP. The initial effective diffusivities for CFRP and GFRP were $2.708 \times 10 - 6 mm 2 / s$ and $1.075 \times 10 - 6 mm 2 / s$ , respectively. Figure 4(b) illustrates that the simulation results from day 3 to day 49 overestimate the moisture uptake. But overall, the simulation results reasonably match with experimental findings.

The moisture concentration-dependent method was implemented using a user-defined subroutine USDFLD in ABAQUS version 6.10. Figure 5 illustrates the flowchart of subroutine USDFLD. At the beginning of every time step, the normalized moisture concentration φ and moisture concentration c are calculated at all integration points. The user-defined subroutine USDFLD checks the new φ and c at all material points, and the moisture diffusivity matrix is updated according to these values. Then the updated moisture diffusivity matrix is incorporated in new assembly equation, which is iteratively solved to get new normalized moisture concentration and moisture concentration for next time step.

Figure 5.

Flowchart of user-defined subroutine USDFLD.

The moisture concentration-dependent diffusion method had been validated by comparing simulation results with experimental findings in Case 1 (Figure 4b). This case study was extended to Case 2. In Case 2, four-layer unidirectional hybrid laminates with or without adhesive layers were conditioned at 45℃ and 84% RH for 1.5 years, and the effect of adhesives on the moisture diffusion behaviour was investigated. The laminate configuration, with and without adhesive layers, is illustrated in Figure 6. Two different adhesive thicknesses (0.12 and 0.76 mm) were considered in this case.

Figure 6.

Hybrid laminate configuration without (left) and with (right) adhesive layers (Case 2).

In this case, the thickness of multi-layer hybrid composite structure was considerable, and thus the moisture contribution from four edges must be taken into account. The laminate configuration under investigation was symmetric with respect to both geometry and boundary conditions along three principle axes. To save computational cost, 1/8th of the geometry was modelled for hybrid laminates (Figure 7a and b). The three outer surfaces of the laminate configuration, with and without adhesive layers, were subjected to saturated boundary conditions.

Figure 7.

(a) 1/8th model of four-layer symmetric hybrid composites with adhesive (Case 2). (b) 1/8th model of four-layer symmetric hybrid composites without adhesive (Case 2).

As Case 1 and Case 2 are under the same temperature and relative humidity conditions, the same normalized pattern functions for both CFRP and GFRP in Case 1 applied to Case 2. In Case 2, the through-thickness diffusivities for CFRP and GFRP were calculated by dividing the effective diffusivities in Case 1 with edge correction factor. The edge correction factor in Case 1 was 1.191, which was determined using equation (2). The longitudinal diffusivities were derived using the following equations:⁸

D_{11} = (1 - v_{f}) D_{r}

(21)

D_{22} = (1 - 2 \sqrt{v_{f} / π}) D_{r}

(22)

where

D_{11}

is the longitudinal diffusivity,

D_{22}

is the transverse diffusivity, and D_r is diffusivity in resin matrix. Because both the composite components are unidirectional, the diffusivity along thickness direction

D_{33}

in each component is the same as the transverse diffusivity. The resultant diffusion coefficients are illustrated in Figure 8(a) and (b) for CFRP and GFRP components, respectively, at 45℃/84% RH. The FM-300 adhesive layers were modelled as a homogenous material. The parameters in the Arrhenius equation were derived from the diffusivities taken at two temperatures.²³ The resultant Arrhenius-type diffusivity equation for FM-300 is given as

D_{FM - 300} = 9.2166 e (- 5523.2831 / T)

(23)

Figure 8.

(a) Effective diffusivity of CFRP (Case 2). (b) Effective diffusivity of GFRP (Case 2). (c) Moisture weight gain curves with and without adhesive layers (Case 2).

Substituting T = 318.15 K in equation (23), the diffusivity of FM-300 was calculated as $2.6604 \times 10 - 7 mm 2 / s$ . Similarly, the parameters in the solubility equation for FM-300 were derived from the equilibrium moisture content taken at two different levels of relative humidity.²³ The resultant solubility equation for FM-300 is given as

M_{m} = 3 . 3225 RH 1.3402

(24)

Substituting RH = 84% in equation (24), the equilibrium moisture content of FM-300 is calculated as 2.63%. In Figure 8(a), the ratio of longitudinal diffusivity $D_{11}$ to transverse diffusivity ( $D_{22}$ or $D_{33}$ ) for CFRP was 2.8278 as per equations (21) and (22) and the fibre volume fractions listed in Table 1. The ratio of longitudinal diffusivity to transverse diffusivity for GFRP was 2.693 as per the same equations. The initial effective longitudinal diffusivity for CFRP was $6.43 \times 10 - 6 mm 2 / s$ and the initial effective longitudinal diffusivity for GFRP was $2.431 \times 10 - 6 mm 2 / s$ . Figure 8(c) illustrates the overall moisture uptake curves of three different hybrid composite structures (with three 0.12 mm adhesive layers, with three 0.76 mm adhesive layers and without adhesive) exposed to 45℃/84% RH for 1.5 years. The results indicated that three 0.12-mm thick adhesive layers did not significantly influence the overall moisture uptake as compared with without-adhesive case. At the end of 1.5 years’ exposure, the overall moisture weight gain for hybrid structure without adhesive was 0.64%. For hybrid structure with 0.12 mm thickness adhesive, the moisture weight gain was 0.65%. For hybrid structure with 0.76 mm adhesive, the adhesive’s effect on moisture weight gain is negligible for the first 81 days, but after that, the difference of average moisture uptake between without-adhesive case and 0.76 mm adhesive case gradually increased. At the end of exposure, the moisture uptake percentage for hybrid structure with 0.76-mm thickness adhesive was 0.70%, which was 9.38% higher than without-adhesive case. The moisture concentration and normalized concentration distribution of three different hybrid composite structures after 1.5 years’ exposure are shown in Figures 9 to 11. Both discontinuity of the moisture concentration and continuity of the normalized concentration at the interfaces of different components can be clearly observed from these contours.

Figure 9.

Moisture concentration and normalized concentration contour after 1.5 years’ exposure.

Figure 10.

Moisture concentration and normalized concentration contour after 1.5 years’ exposure (0.12 mm adhesive).

Figure 11.

Moisture concentration and normalized concentration contour after 1.5 years’ exposure (0.76-mm adhesive).

To better demonstrate how adhesive layers affect the moisture diffusion behaviour among three different hybrid composites (0.12-mm adhesive layers, 0.76-mm adhesive layers and without adhesive layers), two path lines are selected to compare moisture concentration values among three different hybrid composites at the end of 1.5 years’ exposure. The location of selected two path lines is shown in Figure 7(a) and (b). Path line 1 is located on one of inner symmetric surfaces, and 0.26 mm from the top surface. Path line 2 is the axis line along the thickness direction. When comparing the moisture concentration values of three different hybrid composites along path line 2, only the nodes which belong to CFRP and GFRP layers in each type are considered (the adhesive nodes are ignored for with-adhesive laminates). Figure 12 compares moisture concentration along path line 1 for three hybrid structures at the end of 1.5 years’ exposure. The results showed that for nodes that are close to the outer surfaces, moisture concentration for hybrid structure with thicker adhesive layers is higher than the two other types of laminates. As the nodes gradually approach to the centre point when the path depth is larger than around 14 mm, the moisture concentration for hybrid structure with 0.76 mm adhesive layers is the lowest among three types of laminates. This is because, at early stages, the longitudinal and transverse diffusivities of CFRP and GFRP are higher than the diffusivity of adhesive layers. After 81 days’ conditioning, the diffusivities in partial saturated regions of CFRP and GFRP components gradually decrease due to the residual stresses, whereas the diffusivity of adhesive layers is constant and also its solubility is higher than that of CFRP and GFRP layers. As a result, the adhesive nodes near the side surfaces can absorb moisture more quickly from the longitudinal and transverse directions at later stages than CFRP and GFRP components. The higher moisture concentration in the adhesive layers compared with surrounding CFRP and GFRP laminate can be observed from Figures 10 and 11. As adhesive nodes near the side surfaces have higher moisture concentration than that of surrounding CFRP and CFRP laminate, those nodes play a role of accelerating the moisture diffusion to the surrounding CFRP and CFRP nodes. Although in the centre region, the number of saturated adhesive nodes is not as many as the side adhesive nodes, and thus the adhesive nodes near the centre will not be able to play the acceleration role as the side adhesive nodes do. Also in the centre region, most of CFRP and CFRP are not fully saturated; the diffusivity of CFRP and CFRP components is still higher than the diffusivity of adhesive layers. This is the reason that moisture concentration of hybrid structure with thicker adhesive layers along path line 2 is lower than the other two structures (as shown in Figure 13). However, as time elapses, more and more adhesive nodes will gradually get saturated and its acceleration role will be more evident (as shown in Figure 8c).

Figure 12.

Comparison of moisture concentration along path line 1 among three different hybrid structures after 1.5 years’ exposure.

Figure 13.

Comparison of moisture concentration along path line 2 among three different hybrid structures after 1.5 years’ exposure.

Conclusions

A moisture concentration-dependent method was proposed and implemented using user-defined subroutine USDFLD in commercial finite element code to simulate moisture diffusion behaviour in multi-layer unidirectional fibre-reinforced hybrid composite structures. The moisture concentration-dependent method assumes that the fibres restrain the matrix from free swelling. As a result, the diffusion coefficients gradually decrease due to swelling stress built inside the material during the diffusion process, and then drift to a constant value when moisture concentration approaches equilibrium moisture content. The concentration-dependent diffusivity curves are continuous fifth-order polynomial curves. The curve pattern function for CFRP component was different from that of GFRP. Finite element model for a three-layer hybrid composite structure was developed, and the simulation results were validated with experimental findings. This model was extended to simulate the moisture diffusion behaviour in adhesive-bonded four-layer hybrid symmetric composite laminates. The results indicated that thinner adhesive layers (0.12-mm thick) did not significantly affect the overall moisture uptake. Thicker adhesive layers (0.76-mm thick) noticeably accelerated the overall moisture uptake after 81 days’ conditioning. This is because, the diffusivities in partial saturated regions of CFRP and GFRP components gradually decrease due to the residual stresses, while the diffusivity of adhesive layers is constant and also its solubility is higher than that of CFRP and GFRP layers. As a result, the adhesive nodes near the side surfaces can absorb moisture more quickly from the longitudinal and transverse directions at later stages than CFRP and GFRP components. The dependency of adhesive’s diffusion coefficients on moisture concentration will be investigated in the future.

Footnotes

Acknowledgements

This research is supported by the Bell Helicopter Textron, Inc., Fort Worth, TX and partially supported by the National University Transportation Center (NUTC) at Missouri University of Science and Technology.

Conflict of interest

None declared.

References

Gururaja

Rao

ANH

. A review on recent applications and future prospectus of hybrid composites. Int J Soft Comput Eng 2012; 1: 352–355.

Springer

. Environmental effects on composite materials, Westport, CT: Technomic Publishing Co, 1981.

Loos

Springer

. Moisture absorption of graphite-epoxy composites immersed in liquids and in humid air. J Compos Mater 1979; 13: 131–147.

Shen

Springer

. Effects of moisture and temperature on the tensile strength of composite materials. J Compos Mater 1977; 11: 2–16.

Browning

. The mechanism of elevated temperature property losses in high performance structural epoxy resin matrix material after exposure to high humidity environments. Polym Eng Sci 1978; 18: 16–24.

Tsai

Bosze

Barjasteh

. Influence of hygrothermal environment on thermal and mechanical properties of carbon fiber/fiberglass hybrid composites. Compos Sci Technol 2009; 69: 432–437.

Bao

Yee

. Moisture diffusion and hygrothermal aging in bismaleimide matrix carbon fiber composites. II. Woven and hybrid composites. Compos Sci Technol 2002; 62: 2111–2119.

Shen

Springer

. Moisture absorption and desorption of composite materials. J Compos Mater 1976; 10: 2–20.

Loos

Springer

. Moisture absorption of polyester-E glass composites. J Compos Mater 1980; 14: 142–154.

10.

Gopalan

Rao

RMVGK

Murthy

MVV

. Diffusion studies on advanced fibre hybrid composites. J Reinforced Plast Compos 1986; 5: 51–61.

11.

Gurtin

Yatomi

. On a model for two phase diffusion in composite materials. J Compos Mater 1979; 13: 126–130.

12.

Weitsman

. Coupled damage and moisture-transport in fiber-reinforced, polymeric composites. Int J Solids Struct 1987; 23: 1003–1025.

13.

Whitney JM and Browning CE. Some anomalies associated with moisture diffusion in epoxy matrix composite materials. Advanced Composite Materials-Environmental Effects. ASTM STP 658. West Conshohocken, PA, USA: American Society for Testing and Materials, 1978, pp.43–60.

14.

Gillat O and Broutman LJ. Effect of an external stress on moisture diffusion and degradation in a graphite-reinforced epoxy laminate. In: Vinson JR (ed) Advanced Composite Materials-Environmental Effects, ASTM STP 658. West Conshohocken, PA, USA: American Society for Testing and Materials, 1978, pp.61–83.

15.

Wan

Wang

Huang

. Hygrothermal aging behaviour of VARTMed three-dimensional braided carbon-epoxy composites under external stresses. Compos Part A: Appl Sci Manufact 2005; 36: 1102–1109.

16.

Crank

. A theoretical investigation of the influence of molecular relaxation and internal stress on diffusion in polymers. J Polym Sci 1953; 11: 151–168.

17.

Roe N, Huo Z, Chandrashekhara K, et al. Advanced moisture modeling of polymer composites. J Reinforced Plast Compos 2013; 32(7): 437–449.

18.

Galloway

Miles

. Moisture absorption and desorption predictions for plastic ball grid array packages. IEEE Trans Compon Packag Manufact Technol Part A 1997; 20: 274–279.

19.

Wong EH, Teo YC, and Lim TB. Moisture Diffusion and vapour pressure modeling of IC packaging. In: Proceedings of the 48th Electronic Components and Technology Conference, Seattle, WA, 25–28 May 1998, pp.1372–1378.

20.

Zaffaroni G. Two-dimensional moisture diffusion in hybrid composite components. In: Gates TS and Zureick A (eds) High Temperature and Environmental Effects on Polymeric Composites. ASTM STP 1302. West Conshohocken, PA, USA: American Society for Testing and Materials, 1997. pp.97–109.

21.

Cappelletti

Rivolta

Zaffaroni

. Environmental effects on mechanical properties of thick composite structural elements. J Compos Technol Res 1995; 17: 107–114.

22.

Renzo

. Advanced Composite Materials: Products and Manufacturers, Park Ridge, NJ: Noyes Data Corp, 1988.

23.

Deiasi RJ and Schulte RL. Experimental Determination of the Effects of Moisture on Composite-to-Composite Adhesive Joints. NASA-CR-3387, 1981, pp.1–63.