Residual stress and failure analyses of polymer matrix composites considering thermal cycling and temperature effects based on classical laminate plate theory

Abstract

The effects of temperature and thermal cycling on the residual stress and failure behavior of different polymer matrix composites have been investigated in this paper. A new algorithm within the framework of the classical laminate plate theory (CLPT) has been presented to calculate the residual stresses. The modified Tsai-Wu failure criterion has been employed to study the failure behavior of different stacking sequences. Numerical results show that the residual stress and failure index of the composites decrease with the increase of the temperature. It has also been established that thermal cycling condition leads to reduction of the residual stresses and increment of the failure index.

Keywords

Residual stress failure analysis thermal cycling polymer matrix composites modified Tsai-Wu failure criterion

Introduction

Spacecrafts, satellites, and other aeronautical vehicles are exposed to space environment during long-term missions. Structural parts can be submitted to cyclic mechanical loading and temperature variations (between −50℃ and about 130℃) during the supersonic flight.¹ Structural materials that are exposed to this environment can suffer degradation over time and their mechanical behavior such as vibration^2,3 or buckling reaction⁴ should be studied. Fatigue failure is a progressive form of local damage and can be produced by thermal cycling. Therefore, the investigation of thermal cycling effects would be necessary for future space studies.

González et al.⁵ presented an extended finite element analysis to simulate damage on carbon fiber-reinforced polymer as a consequence of thermal fatigue between 50℃ and 150℃ under atmospheres with different oxygen content. They realized that damage level in the oxidative environment is higher than the one that was in the inert environment. Jedidi et al.⁶ studied the drying effect of supersonic flight at high temperature (130℃), on the durability of the material. They also proposed different accelerated cycles adapted to the new situation of supersonic flights and focused on the cyclical drying of the material. Zhang et al.⁷ presented an experimental and finite element method (FEM) study of thermal cycling-induced microcracking in carbon/epoxy triaxial braided composites. Transverse microcrack morphology was investigated using X-ray computed tomography. The differing performance of two kinds of composites was discovered and analyzed. Simulation results exhibited a decrease in strength and stiffness with increasing crack density. They also investigated thermal cycling effects (ranging between 55℃ and 120℃) on two different composites and observed that the variation of the mass and the volume of composites were less than 1.0% after 160 cycles. Some other researches have been done on thermal fatigue effects and its simulation as well.^8–10

One of the most obvious effects of thermal cycling is appearing of thermal residual stresses^11–13 which is because of the mismatch in shrinkage between the fiber and the matrix. Residual stresses can cause several defects in composite laminates such as delamination and warpage or lead to fatigue failure.^14,15 Safarabadi¹⁶ proposed a procedure to integrate micro and macro thermal residual stresses in composite laminates. He employed the classical lamination theory (CLT) to predict macroscopic thermal residual stress of each layer in laminated composites. Available experimental data from hole-drilling method showed that the proposed combination approach yields more precise residual stresses predictions in comparison with the state that the CLT is used only. Studies have shown that it is difficult to determine the thermal residual stresses which are experienced by the matrix in laminated composite structures by experimental tests.^17,18 So, development of new models and algorithms in order to evaluate the residual stresses under thermal cycling is essential.

In this study, the effects of temperature and number of thermal cycling on residual stresses for different stacking sequences of the PMCs have been considered during thermal cycling. A new algorithm has presented to calculate the residual stresses under thermal cycling condition. The algorithm contains four boxes:

a) Thermal and mechanical property evaluation as a function of number of cycles and temperature

b) Geometry and stacking sequence consideration

c) Calculation of residual stresses using CLPT

d) Modified Tsai-Wu failure criterion method to obtain the failure index.

This analysis has considered stiffness and strength as a function of thermal cycles while multi variable polynomial approximation has been used to obtain the CTE as a function of temperature and thermal cycles. Residual stresses under different temperatures and thermal cycles for different stacking sequences have been analyzed. Finally, the failure behavior of different composites has been studied.

Experimental conditions

Thermal cycling

Researchers usually use different thermal cycling profiles to study the behavior of aerospace vehicles such as satellite components which depends on their case study or thermal cycling test apparatus.^19,20 For this study, one thermal cycle¹⁹ is used as a change in temperature from −70℃ to + 100℃ and back to −70℃. This profile has a transition rate of 3–5℃ per minute; a dwell period at the temperature extremes is about 15 min (Figure 1).

Figure 1.

Thermal cycling profile.

Materials

Graphite/epoxy composite (model: HFG CU-125NS) is considered to study the residual stresses during thermal cycling. The mechanical properties of this composite are degraded when the material is exposed to thermal cycling conditions.¹⁹ Up to a certain period of exposure, the composite may retain its strength and stiffness above its allowable limits. However, as time passes by, the strength and stiffness may become so low. The mechanical and thermal properties of the unexposed composite at room temperature are given in Table 1. The curing temperature of this laminate is 130℃²¹ and free stress temperature has been considered as 145℃. It should be mentioned that curing temperature is the particular temperature which is utilized during the curing process. Whereas the free stress temperature is when there are no residual stresses. Studies have proved that the free stress temperature is often 15℃ more than the curing temperature.^22,23

Table 1.

Mechanical and thermal properties of graphite/epoxy composites at 25℃.¹⁹

Material properties	HFG CU-125NS
Tensile modulus (GPa)	144.30
Shear modulus (GPa)	6.44
Poisson's ratio	0.31
Longitudinal CTE (1/℃)	−3.6 × 10⁻⁶
Transverse CTE (1/℃)	31.5 × 10⁻⁶

Problem formulation

Residual stress calculation

Residual stresses in macroscopic level can be estimated by CLPT.¹⁴ This theory has been employed to calculate the residual stresses during thermal cycling process of the carbon/epoxy composite. In the PMCs, fibers have much lower CTE in comparison to the matrix. Also, the matrix has more contraction during the curing process. Therefore, in cross-ply laminates, a compressive residual stress in longitudinal direction and a tensile residual stress in transverse direction are generated. Thermal forces and moment resultants are defined as $N^{T} = [N_{1}^{T} N_{2}^{T} N_{6}^{T}]$ and $M^{T} = [M_{1}^{T} M_{2}^{T} M_{6}^{T}]$ , respectively. For laminated composite materials, thermal loads in off-axis plane are defined as follows^14,22

\begin{matrix} {N_{1}^{T}, M_{1}^{T}} = \sum_{k = 1}^{n} (Q_{11}^{(k)} α_{1}^{(k)} + Q_{12}^{(k)} α_{2}^{(k)} + Q_{16}^{(k)} α_{6}^{(k)}) Δ T^{(k)} \\ . (t_{k}, t_{k} z_{k}) \\ {N_{2}^{T}, M_{2}^{T}} = \sum_{k = 1}^{n} (Q_{21}^{(k)} α_{1}^{(k)} + Q_{22}^{(k)} α_{2}^{(k)} + Q_{26}^{(k)} α_{6}^{(k)}) Δ T^{(k)} \\ . (t_{k}, t_{k} z_{k}) \\ {N_{6}^{T}, M_{6}^{T}} = \sum_{k = 1}^{n} (Q_{61}^{(k)} α_{1}^{(k)} + Q_{62}^{(k)} α_{2}^{(k)} + Q_{66}^{(k)} α_{6}^{(k)}) Δ T^{(k)} \\ . (t_{k}, t_{k} z_{k}) \end{matrix}

(1)

where n is the total number of layers,

t_{k}

and

z_{k}

are the ply thickness and distance to the mid-plane, respectively. Also, k is the lamina number;

Q^{(k)}

is the stiffness matrix in off-axis coordinate system, ΔT is the temperature difference between the instance and the stress-free temperatures and α^(k) is the vector of CTE in off-axis coordinate system. The strain and the curvature of mid-plane are determined as follows²³

\begin{matrix} {\begin{matrix} \overset{\cdot}{ɛ} \\ \overset{\cdot}{k} \end{matrix}} = [\begin{matrix} A \\ B \end{matrix} \begin{matrix} B \\ D \end{matrix}]^{- 1} {\begin{matrix} N^{T} \\ M^{T} \end{matrix}} \end{matrix}

(2)

where

\overset{\cdot}{ɛ}

and

\overset{\cdot}{k}

are normal strain and curvature of the mid-plane, respectively. Figure 2 shows the geometry of a polymer composite plate where A, B, and D are the extensional, coupling, and bending stiffness matrixes, respectively, and defined as²⁴

{A_{ij}, B_{ij}, D_{ij}} = \sum_{k = 1}^{n} \int_{z_{k}}^{z_{k + 1}} Q_{ij}^{(k)} (1, Z, Z^{2}) d z (i, j = 1, 2, 6)

(3)

Figure 2.

A multi-layer polymer composite plate.

In order to calculate the residual strain of the kth layer, equation (4) is proposed. After that the residual stress of the kth layer of PMCs can be obtained as follows²³

\begin{matrix} ɛ_{R}^{k} = \overset{\cdot}{ɛ} + z^{k} \overset{\cdot}{k} - α^{k} Δ T \Rightarrow σ_{R}^{k} = Q^{k} ɛ_{R}^{k} \end{matrix}

(4)

in which $σ_{R}$ and $ɛ_{R}$ are the vector of residual stresses and residual strains, respectively.

Mechanical properties

Strength and stiffness variations in composites can be predicted by specific equations as¹⁹

\begin{matrix} {\begin{matrix} M_{p} = A_{p} exp [- B_{p} N] p = 1, 2, \dots, 5 \\ W_{q} = C_{q} exp [- D_{q} N] q = 1, 2, \dots, 5 \end{matrix} \end{matrix} .

(5)

where

M_{p}

and

W_{q}

are the stiffness and strength properties, respectively. N is the number of thermal cycles. Also,

A_{p}

B_{p}

C_{q}

and

D_{q}

are exposure constants. The parameters

A_{p}

and

C_{q}

are defined as unexposed baseline values of stiffness and strength, and the parameters

B_{p}

and

D_{q}

are defined as loss rates or extents of degradation. So, if

B_{p}

and

D_{q}

have large values, strength or stiffness drops severely with an increase in the number of thermal cycles. The exposure parameters are presented in Table 2.

Table 2.

Exposure constants $A_{p}$ ,… $D_{q}$ .¹⁹

Mechanical property	Stiffness (GPa)			Strength (MPa)
Mechanical property	p	$A_{p}$	$B_{p}$	q	$C_{q}$	$D_{q}$
Longitudinal tensile modulus	1	144.30	0.0004	1	1809.40	0.0017
Longitudinal compressive modulus	2	132.68	0.0003	2	1119.80	0.0015
Transverse tensile modulus	3	9.09	0.0004	3	35.99	0.0029
Transverse compressive modulus	4	8.88	0.0005	4	130.31	0.0010
Shear modulus in the 1–2 plane	5	6.44	0.0015	5	79.95	0.0016

Thermal properties

The CTE can play an important role in determination of residual tensile stresses, hence an accurate prediction of CTEs during thermal cycling would be important. Available experimental results¹⁹ have shown that the longitudinal and transverse CTE of composites change as a function of temperature and thermal cycles.

There are a number of experiments available which have been designed based on the Taguchi method and that have analyzed the influence of various parameters such as the lay-up, number of thermal cycles and weight fraction of fibers on the mechanical properties.^8–10 The mechanical properties of specimens are obtained using the tensile test before and after the thermal cycling load. Also, a regression analysis was developed for modeling the mechanical properties using the effective factors. Finally, sensitivity analysis of the effective factors was studied to describe the relationship between these parameters.¹⁰

By a multi-variable polynomial approximation,²⁵ the CTE in longitudinal and transverse directions using the data that published by other researchers¹⁹ can be defined as

\begin{matrix} α_{j} (T, N) = a_{j} + b_{j} N + c_{j} T + d_{j} TN + e_{j} T^{2} \\ + f_{j} T^{2} N + g_{j} T^{3} + h_{j} T^{3} N \end{matrix}

(6)

where T is the instant temperature and

a_{j}, b_{j}, \dots, h_{j}

are the CTE constants. These constants are calculated by MATLAB software and presented in Table 3. For each curve, 50–100 points are used and the error limited to 1%. As the transverse CTE has the smooth variation, quadratic form of temperature exhibits adequate accuracy, so the cubic part of the function is ignored for simplicity.¹⁹

Table 3.

CTE constants.

CTE Direction	j	$a_{j}$	$b_{j}$	$c_{j}$	$d_{j}$ (10⁻³)	$e_{j}$ (10⁻³)	$f_{j}$ (10⁻⁶)	$g_{j}$ (10⁻⁶)	$h_{j}$ (10⁻⁶)
Longitudinal	L	−12.24	0.048	0.466	−2.1	−5.7	23.3	22.6	−0.09
Transverse	T	20.19	−0.039	0.543	−0.1	−2.5	−2	0	0

Figure 3 shows variation of longitudinal and transverse CTE in terms of temperature for various number of thermal cycles. As it is shown in this figure, the CTEs of composites decrease with increase of thermal cycles. The reduction in CTE is caused by matrix loss due to thermal cycles. Since the fiber has a lower CTE than the matrix, the increase in fiber volume fraction leads to a reduction in the CTEs of the composite.¹⁹

Figure 3.

(a) Variation of longitudinal CTE of the graphite/epoxy composite under thermal cycling conditions. (b) Variation of transverse CTE of the graphite/epoxy composite under thermal cycling conditions.

Modified failure criteria

In order to predict the failure cycle of composites induced by thermal cycling, different modified failure methods are used. In this study, Tsai–Wu failure criterion is employed to analyze the failure of composites under thermal cycling condition. This criterion is one of the most commonly used criteria, and also often a better prediction method than some phenomenological criteria.

From equation (5), five fundamental strengths can be expressed as a function of number of thermal cycles as

\begin{matrix} {\begin{matrix} W_{1} = X^{t} = C_{1} e^{- D_{1} N} \\ W_{2} = X^{c} = C_{2} e^{- D_{2} N} \\ W_{3} = Y^{t} = C_{3} e^{- D_{3} N} \\ W_{4} = Y^{c} = C_{4} e^{- D_{4} N} \\ W_{5} = S = C_{5} e^{- D_{5} N} \end{matrix} \end{matrix}

(7)

where

X^{t}

and

Y^{t}

are longitudinal and transverse tensile strength,

X^{c}

and

Y^{c}

are longitudinal and transverse compressive strength, and S is shear strength in the 1–2 plane. The Tsai–Wu method proposes a tensor polynomial failure criterion which is considered to be the general theory of strength for orthotropic materials.²⁶ For the case of plane stress, the Tsai–Wu failure criterion is reduced to^27,28

\begin{matrix} F (σ_{1}, σ_{2}, σ_{6})_{Tsai - Wu} = F_{1} σ_{1} + F_{2} σ_{2} + F_{6} σ_{6} + F_{11} σ_{1}^{2} \\ + F_{22} σ_{2}^{2} + F_{66} σ_{6}^{2} + 2 F_{12} σ_{1} σ_{2} \end{matrix}

(8)

where

\begin{matrix} {\begin{matrix} F_{1} = (\frac{1}{X^{t}} - \frac{1}{X^{c}}) \\ F_{2} = (\frac{1}{Y^{t}} - \frac{1}{Y^{c}}) \\ F_{6} = 0 \\ F_{11} = \frac{1}{X^{t} X^{c}} \\ F_{22} = \frac{1}{Y^{t} Y^{c}} \\ F_{66} = \frac{1}{S^{2}} \\ F_{12} = - \frac{1}{2} \sqrt{F_{11} F_{22}} \end{matrix} \end{matrix}

(9)

From equations (8) and (9), the modified Tsai–Wu failure criterion can be presented as follows

\begin{matrix} F (σ_{1}, σ_{2}, σ_{6}, N)_{Modified} \\ = (\frac{1}{C_{1} e^{- D_{1} N}} - \frac{1}{C_{2} e^{- D_{2} N}}) σ_{1} \\ + (\frac{1}{C_{3} e^{- D_{3} N}} - \frac{1}{C_{4} e^{- D_{4} N}}) σ_{2} - \sqrt{\frac{e^{(D_{1} + D_{2} + D_{3} + D_{4}) N}}{C_{1} C_{2} C_{3} C_{4}}} σ_{1} σ_{2} \\ + \frac{σ_{1}^{2}}{C_{1} C_{2} e^{- (D_{1} + D_{2}) N}} + \frac{σ_{2}^{2}}{C_{3} C_{4} e^{- (D_{3} + D_{4}) N}} + \frac{σ_{6}^{2}}{C_{5} e^{- 2 D_{5} N}} = 1 \end{matrix}

(10)

Solution procedure

The algorithm of residual stress calculation is presented in Figure 4. As you see in this figure, the procedure of calculation consists of four boxes (Thermal cycling, Geometry and layup design, Residual stresses calculation, and the modified Tsai-Wu failure criterion (Failure calculation)).

Figure 4.

The algorithm of residual stresses and failure calculation of PMCs under thermal cycling condition.

In thermal cycling box, thermal profile function and curing properties of composite (such as free stress temperature and $T_{g}$ ) are determined. Also, the mechanical and thermal properties degradations are obtained during thermal cycles consequently. Geometry, dimension, number of layers and stacking sequence of PMCs could be specified in the geometry and layup design box. Then the extensional, coupling, and bending stiffness matrixes are calculated for desired composite.

Subsequently, residual stress of the lamina at different temperatures and number of thermal cycles are obtained in residual stress calculation box. Finally, due to degradation of strength properties of PMCs, failure index of the lamina is calculated by using modified Tsai-Wu failure criterion in the fourth box.

A subroutine has been developed in MATLAB commercial software to implement the residual stress and failure calculation procedure which was described above (Figure 4).

Results and discussion

Validation

In this step, our study is validated with an available research¹⁹ that predicted the failure cycle of HFG CU-125NS graphite/epoxy composites exposed to the LEO environmental factors with a stacking sequence $[0 / 90 / \pm 45]_{s}$ . They considered two different loadings as follows

\begin{matrix} {\begin{matrix} (1) N_{1} = 200000 \frac{N}{m}, N_{2} = N_{6} = 0 \\ (2) N_{1} = 210000 \frac{N}{m}, N_{2} = 100000 \frac{N}{m}, N_{6} = 0 \end{matrix} \end{matrix}

Macro stresses of symmetric $[0 / 90 / \pm 45]_{s}$ laminates were analyzed by using presented algorithm and the failure index of the loading cases were obtained. The results of the algorithm have been validated with available results¹⁹ (Figure 5). As shown in this figure, the presented algorithm has a good agreement with available results. The maximum error of the presented approach was less than 2%.

Figure 5.

On-axis stress validation in each ply of the $[0 / 90 / \pm 45]_{s}$ laminate under different loading cases.

Residual stress analysis

Symmetric composite laminates are appropriate choices as the case of study in researches and experiments, therefore three different types of symmetric laminate are employed to investigate the effects of thermal cycling on the residual stress and failure index of the PMCs (Table 4).

Table 4.

Different stacking sequences as the case studies.

Type of laminate	Stacking sequences
Symmetric cross-ply	$[0_{2} / 90_{2}]_{s}$
Symmetric quasi-isotropic	$[0 / 90 / \pm 45]_{s}$
Symmetric 0/45 ply	$[0_{2} / \pm 45]_{s}$

Comparison of residual stresses

Residual stresses of each layer of different laminates (at room temperature) are compared and shown in Figure 6. Part (a) shows the normal residual stresses of each layer. As shown in this figure, $[0_{2} / \pm 45]_{s}$ laminate has the minimum normal residual stress (less than 15 MPa) while $[0_{2} / 90_{2}]_{s}$ has the maximum value (34.5 MPa). Also, it is observed that the $[0 / 90 / \pm 45]_{s}$ laminate has no normal residual stress in the second and third layers.

Figure 6

(a) Normal residual stresses of different stacking sequences at room temperature. (b) Transverse residual stresses of different stacking sequences at room temperature. (c) Shear residual stresses of different stacking sequences at room temperature.

In part (b), transverse residual stresses of different stacking sequences are compared. As shown in this figure, the $[0_{2} / 90_{2}]_{s}$ laminate has the greatest amount of transverse residual stress (34.5 MPa) and all layers of this laminate are similar. The transverse residual stress of the $[0 / 90 / \pm 45]_{s}$ laminate is equal to the $[0_{2} / 90_{2}]_{s}$ laminate in the first and fourth layers while, in the second and third layers it is zero. Also, it is observed that the amount of transverse residual stress of the $[0_{2} / \pm 45]_{s}$ laminate is the same in all layers (less than 15 MPa).

Part (c) shows shear residual stresses of different stacking sequences. As shown in this figure, the $[0_{2} / 90_{2}]_{s}$ laminate has no shear residual stress. Also, it is apperceived that shear residual stress of the $[0_{2} / \pm 45]_{s}$ laminate is greater than the $[0 / 90 / \pm 45]_{s}$ laminate, although they both have no residual stress in the first layer.

The influence of temperature on the residual stresses

Different layers of three types of laminate were analyzed in the previous part and it was clarified that the first layer of the $[0_{2} / 90_{2}]_{s}$ laminate, the second layer of the $[0 / 90 / \pm 45]_{s}$ laminate and the third layer of the $[0_{2} / \pm 45]_{s}$ laminate are the best choices to study.

Figure 7 depicts off-axis residual stresses of each stacking sequences in terms of temperature. It is necessary to notice that in this step, there is no thermal cycling effect and residual stresses have been compared before composites are subjected to thermal cycling conditions.

Figure 7.

(a) Residual stresses of the first ply of the $[0_{2} / 90_{2}]_{s}$ .laminate. (b) Residual stress of the second ply of the $[0 / 90 / \pm 45]_{s}$ laminate. (c) Residual stresses of the third ply of the $[0_{2} / \pm 45]_{s}$ laminate.

All these three figures illustrate that with increase of temperature, the value of residual stresses decreases.

The influence of thermal cycles on the residual stresses

In continuation of previous studies, the influence of thermal cycling on the residual stress of polymer matrix composites is investigated. For this purpose, the number of thermal cycles is considered between N = 0 and N = 200 cycles under the similar thermal cycling profile. The most critical layer of each stacking sequence is chosen and residual stresses are calculated and shown in Figure 8.

Figure 8.

(a) Normal residual stress of the first ply of the $[0_{2} / 90_{2}]_{s}$ laminate. (b) Shear residual stress of the second ply of the $[0 / 90 / \pm 45]_{s}$ laminate. (c) Shear residual stress of the third ply of the $[0_{2} / \pm 45]_{s}$ laminate.

As shown in parts (a) and (b), normal and shear residual stresses are similar in the first ply of the $[0_{2} / 90_{2}]_{s}$ laminate and the second ply of the $[0 / 90 / \pm 45]_{s}$ laminate, respectively. Numerical results indicate that with increase of the number of thermal cycles, residual stresses of the composites decrease. It is observed that when the number of thermal cycles is low, the maximum residual stress occurred at room temperature (almost 25℃) while the maximum residual stress at high thermal cycles occurred at almost 55℃.

Failure analysis

A parametric study is conducted in this section to analyze the failure of different composite laminates, with a particular focus on the effects of the temperature and thermal cycles.

Comparison of different laminates

The failure index of different composite laminates was calculated using the modified Tsai-Wu failure criterion (equation (10)) and is presented in graphical form in Figure 9. As shown in this figure, the failure indexes of both cross-ply and quasi-isotropic laminates are similar. The minimum and maximum failure indices of these laminates are 0.28 and 0.44, respectively. The failure index of the $[0_{2} / \pm 45]_{s}$ laminate is lower than the two other laminates. The maximum and minimum failure indices of these laminates are 0.41 and 0.26, respectively.

Figure 9.

Failure index of different laminates.

Investigation of the failure surface

Figure 10 shows the comparison of the failure surface and normal residual stress of the $[0_{2} / 90_{2}]_{s}$ laminate. It should be noticed that the failure surface has been calculated using the modified Tsai-Wu failure criterion method and residual stresses have been considered at 60℃. This figure confirms previous studies which mentioned that residual stress decreases with the increase of the number of thermal cycles. It also shows that the failure surface of the composite decreases with the increase of thermal cycles and the failure index increases consequently.

Figure 10.

Comparison of the failure surface and normal on-axis stress of the first ply of the $[0_{2} / 90_{2}]_{s}$ laminate.

The influence of temperature and thermal cycles on the failure index

Figure 11 shows the variation of failure index as a function of temperature and thermal cycles for different composite laminates. As shown in these figures, failure index decreases with the increase of temperature. It is also concluded that the failure index of composites increases linearly as the number of thermal cycles increases. The lowest amount of the failure index is observed when there is no thermal cycle (N = 0).

Figure 11.

(a) Failure index of the $[0_{2} / 90_{2}]_{s}$ and $[0 / 90 / \pm 45]_{s}$ laminates in terms of the temperature for various number of thermal cycles. (b) Failure index of the $[0_{2} / \pm 45]_{s}$ laminate in terms of the temperature for various number of thermal cycles.

Conclusion

The residual stress and failure index of three different composite laminates have been investigated in this paper within the framework of the classical laminate plate theory. A novel algorithm was presented to calculate the residual stress of composites under thermal cycling conditions. Different layers of each composite laminate were checked and the most critical ones were chosen to study. Then the modified Tsai-Wu failure criterion was employed to analyze the failure of different symmetric composites subjected to thermal cycles. Based on the investigation of the effects of the temperature and thermal cycles, the following conclusions can be made.

Residual stress

Residual stresses of the composites decrease with the increase of the number of thermal cycles. The value of residual stress at each step of the thermal cycle process highly depends on the initial residual stresses and layups. The numerical results showed that each cycle leads to almost 0.28% reduction of residual stress in the cross-ply stacking sequence of $[0_{2} / 90_{2}]_{s}$ at temperatures higher than 40℃ (56% reduction after 200 cycles). It was also proved that as the temperature raises, the residual stress decreases.

Failure index

Failure index decreases with the increase of the temperature. The $[0_{2} / \pm 45]_{s}$ laminate had the least amount of failure index which was 0.26 (at 100℃). It was also concluded that the failure index of composites increases linearly as the number of thermal cycles increases. The lowest amount of the failure index was observed when there was no thermal cycle (N = 0).

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful to the University of Kashan for supporting this work (grant no. 785401/05).

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