Evaluation of the stiffness of composite materials with hygrothermal conditions
Khaled AMARA^{1*}, Mokhtar BOUAZZA^{2}, Kamel ANTAR^{3 }and Abdelkader MEGUENI^{3}
^{1 }Department of Civil Engineering, University centre of Ain Témouchent, 46000, Algeria
^{2 }Department of Civil Engineering, University of Bechar, Bechar 08000, Algeria.
^{3 }Laboratory of Materials and Hydrology (LMH), University of Sidi Bel Abbes, 2200, Algeria.
Emails: amara3176@yahoo.fr; bouazza_mokhtar@yahoo.fr; mech2003@hotmail.com
^{* }Corresponding author, Phone: +213663248283
Abstract
The change in longitudinal modulus as a result of transverse ply cracking in simple crossply laminate is examined theoretically with the taking into consideration the decrease of the mechanical properties of material by the variation of temperature and moisture. The results show that the hygrothermal environment has a significant effect on the relative reduction of longitudinal Young’s modulus at the higher crack density, in contrast with the complete reduction of longitudinal Young’s modulus in which the sensitivity of the hygrothermal effects become weaker but significant at the lower crack density.
Keywords
Hygrothermal effect; Stiffness; CrossPly laminate; Crack density
Introduction
The variation of temperature and moisture reduces the elastic moduli and degrades the strength of the laminated material [1–8].
Also the environmental conditions in terms of temperature and moisture induce the hygrothermal stresses within the plies [9–13]. Furthermore, hygrothermal aging in the absence of external loads can lead to spontaneous formation of matrix micro cracks in laminates [14].
On the other hand, when crossply laminates are subjected to tensile loading, the first observed damage modes in crossply laminates is matrix cracking which cause the reduction of longitudinal Young’s modulus of laminates. These cracks are in general not critical for final failure. In order to design structure properly it is necessary to understand the damage process and to predict the level of degradation of mechanical properties as a result of transverse ply cracking in crossply laminate when this latter is exposed to hygrothermal ageing before applying the mechanical loading.
In the present study, the hygrothermal stresses [811] and the waterinduced micro cracks [12] are not taken into consideration. But the material properties are assumed to be functions of temperature and moisture. Both ambient temperature and moisture are assumed to have a uniform distribution. The plate is fully saturated such that the variation of temperature and moisture are independent of time and position.
The changes of longitudinal Young’s modulus of laminate due to the number of cracks formed in transverse layer are determined here, using two approaches: shear lag model [15–17] and Hashin’s model [18].
These latter, are modified to take into account the changes of the elastic moduli which are related to environmental conditions.
Material and method
It is well known in many studies [36] that the material Properties are function of temperature and moisture. In terms of a micromechanical model of laminate, the material properties may be written as [710]

(1) 

(2) 

(3) 

(4) 
In the above equations and are the fiber and matrix volume fractions and are related by:

(5) 
And E_{f}, G_{f}, and v_{f} are Young's modulus, shear modulus and Poisson's ratio of fibers, respectively, E_{m}, G_{m} and v_{m} are the properties corresponding to the matrix.
E_{L}, E_{T}, G_{LT} are the longitudinal modulus, transversal modulus and shear modulus, respectively.
It is proved that the longitudinal modulus E_{m} of matrix is a function of temperature and humidity, therefore, E_{L}, G_{LT} and E_{T} are also a function of temperature and humidity. The longitudinal modulus of the damaged crosslaminated in the x direction is given by [19]:

(6) 
where: E_{x}, E_{x0} are respectively the longitudinal modulus after damage and before damage.
, and α is a known function, dependent on elastic properties and geometry of 0° and 90° layer:

(7) 
where R(l) is the stress perturbation function

(8) 
G_{TT} out of plane shear modulus, t_{90} half thickness of the 90° layer and t_{0} thickness of the 0° layer
The properties of composite laminates studied AS4/3502 are given in the following Table1
Table 1: The properties of the composite laminate.
Material 
E_{L} (GPa) 
G_{LT} (GPa) 
E_{T} (GPa) 
G_{TT} (GPa) 
_{ LT} 
_{ TL} 
h_{0} (mm) 
AS4/3502 [12] 
144.8 
9.58 
4.79 
4.2 
0.31 
0.4 
0.147 
Results and discussion
Comparison of predictions with experimental data
Figure 1 and Figure 2 also show the phenomenon of reduction of the stiffness that is aggravated by the increased thickness of the layer 90°, i.e. the reduction of stiffness becomes larger when the thickness of the layer 90 augment, he is able to understand that increasing the thickness of the layer of 0° has a strong constraint on transverse cracking in the layer at 90° because the more the load is carried by the 0°.
Hygrothermal effect on the reduced elastic properties
The study, here, has been focused on the stiffness reduction due to transverse ply cracking in simple hygrothermal aged angleply laminate [0_{n}/90_{m}] _{S}.
For that, several numerical examples were presented. Graphite/epoxy composite material was selected in the present examples.
However, the analysis is equally applicable to other types of composite material.
For these examples the thickness of each ply is 0.125 mm and the material properties adopted are: E_{f }= 230 GPa, G_{f} = 9GPa, v_{f} = 0.203, v_{m} = 0.34 and E_{m} = (3.510.003T0.142 C) GPa, in which T = T_{0} + ΔT and T_{0} = 25°C (room temperature), and C = C_{0}+ΔC, C_{0 }= 0wt. % H_{2}O.
Three sets of environmental conditions are considered, referred to as 1, 2 and 3.
For environmental case 1, T=25°C, so that both ΔT and ΔC are zero.
For environmental case 2, ΔT=50°C and ΔC=0.5%, and for environmental case 3, ΔT=100°C and ΔC=1%. The environmental of case 1 will be regarded as the reference case. We calculate here, the loss of stiffness in the damaged laminate, which is already subjected to the hygrothermal ageing of type 1, 2 or 3.
The loss of stiffness in the laminate as a result of crack density is evaluated compared to the initial stiffness of the uncracked laminate.
We note that this initial stiffness of the uncracked laminate is function of temperature and moisture. Consequently, Eq. (6) becomes:

(9) 
The index (i) represents the considered case of the environmental conditions (case 1, 2 or 3).
Figures 3 and 4 illustrate the variation of the reduction of the longitudinal Young's modulus for various types of environmental conditions considered. The modulus reduction depends in addition to the crack density on the moisture and temperature. In fact, the longitudinal Young’s modulus is reduced with decreases in moisture and temperature especially when the crack density becomes higher.
Figure 2. Stiffness reduction with different crack densities for [0/90_{2}]_{s} and [0/90_{3}]_{s} AS4/3502 laminate
Figure 3. Hygrothermal effect on the longitudinal Young’s modulus degradation for [0/90_{2}]_{ s} AS4/3502 laminate
Figure 4. Stiffness reduction with different crack densitiesfor [0/90_{2}]_{s} AS4/3502 laminate under different temperature
Conclusions
The material properties were considered to be dependent on temperature and moisture, which are given explicitly in terms of the fibre and matrix properties and the fibrevolume ratio.
The results show that the longitudinal Young’s modulus degradation due to the transverse cracking is also reduced with decreasing moisture concentration and temperature, especially when the crack density takes higher values. However, this reduction is not very important.
In order to assess the effects of temperature and moisture on the reduction of longitudinal Young’s modulus in crossply laminates containing transverse cracking. The material properties are considered to be dependent on temperature and moisture, which are given explicitly in terms of the fiber and matrix properties and fiber volume ratio. The solution methodology is general in nature and may be applicable to the analysis of other types of environmental conditions, e.g. including the hygrothermal coupling in the governing equations.
The results presented herein show that the relative longitudinal Young’s modulus is reduced with decreases in moisture and temperature especially when the crack density becomes higher.
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