Monte-Carlo simulation to the tensile mechanical behaviors of unidirectional composites at low temperature

Cryogenics 41 (2001) 683±691

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Monte-Carlo simulation to the tensile mechanical behaviors of unidirectional composites at low temperature X.F. Wang, J.H. Zhao

*

Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, People's Republic of China Received 7 August 2000; accepted 20 September 2001

Abstract This paper presents an analytical approach which combines the modi®ed shear-lag model and the Monte-Carlo simulation technique to simulate numerically the mechanical behaviors including the failure processes, tensile sti€ness and strength, etc. for unidirectional composites at room and low temperatures. Comparisons of the simulated results to the experimental data at 296 and 77 K are made. The results show that the tensile moduli and strengths at low temperatures are generally larger than those at room temperatures for carbon or glass ®bre unidirectional composites, and they depend mainly on the properties of constituent materials and the status of interface at low temperature. Ó 2001 Published by Elsevier Science Ltd. Keywords: Unidirectional composites; Cryogenic mechanical behaviors; Monte-Carlo simulation

1. Introduction Many cases of the composites applied to aerospace and superconducting engineering deal with the e€ect of cryogenic environments [1]. Most materials making up the structures operating at cryogenic temperatures need good mechanical and thermal properties. The ®bre-reinforced composites are generally superior to the metal in this point. The straps for suspending nonmetallic cryostat or nuclear magnetic resonance equipment and for reinforcing of cryogenic vessel are often made by unidirectional ®bre composites. The straps, of course, experience tension, and must have good tensile sti€ness and strength at low temperatures. In most cases, material screening and initial design data for using at low temperature are obtained by testing a unidirectional ®bre-reinforced plate at the same conditions. However, a new test is necessary as we hope to change the design, and it is obviously of a high cost. The Monte-Carlo simulation technique coupled with an analytical model is a probable route with low cost for obtaining some initial design data and understanding the mechanical behavior of unidirectional composites at low temperatures. Usually, the Monte-Carlo simulation technique is coupled with the classical shear-lag model *

Corresponding author. E-mail address: [email protected] (J.H. Zhao).

[2,3]. It is clear that the classical shear-lag model cannot be used to take into account the breaking of matrix and handle the coupled e€ects of loading and rising (or reducing) temperatures. While Abdelmohsen [4] conducted the simulation of tensile strength of unidirectional composites at low temperature, he incorporated only the e€ect of cooling into the ®bre failure strength distribution by an increase in the scale parameter value instead of taking into account the coupled e€ects of loading and temperature. In this study, the modi®ed shear-lag model [7] was extended to simulate the failure of unidirectional composites at low temperature. The process of failure was simulated as the cumulative fracture containing a series of micro-break events, for example, matrix break, interface debonding and random fracture of ®bres. In addition, the in¯uence of the matrix and interface strengths as well as Weibull scale parameter of ®bre strength distribution on the failure strength of unidirectional composites at low temperature was studied, and comparisons with some experimental results were made too.

2. Analytical method 2.1. Analytical model A body of unidirectional composites which is composed of m strips of the ®bre and m 1 strips of the

0011-2275/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd. PII: S 0 0 1 1 - 2 2 7 5 ( 0 1 ) 0 0 1 4 9 - 7

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X.F. Wang, J.H. Zhao / Cryogenics 41 (2001) 683±691

matrix is shown in Fig. 1. According to the modi®ed shear-lag model, both ®bres and matrix are regarded as carrying tensile stress and also acting as medium for stress transfer. In addition, it is assumed that the ®bre and matrix are homogeneous linear elastic, and the strengths of the interface and the matrix are de®nitive, but the ®bre strength is described statistically by the two parameter Weibull distribution F …X †, as expressed in the following equation: (  b ) L X F …X † ˆ 1 exp ; …1† L0 r 0 where F …X † is the probability that the ®bre strength is 6 X , while r0 and b are the Weibull scale and shape parameters, respectively. Note that L0 is the original gage length at which the single ®lament tension test and estimation of Weibull parameters are conducted, while L is the extrapolated ®bre length of interest. 2.2. Governing equation and boundary conditions It is assumed that loading is only applied at the end of the ®bres, but the ®bres and matrix all experience the homogeneous change DT of temperature. In Fig. 2, ui and ui stand for the displacements at the centerline of the ®bre and matrix strips, respectively. The normal stresses in the ®bres and matrix are denoted by ri and ri , respectively. si 1 is the shear stress at the left interface of the ith ®bre strip and si as the one at the right interface. Consider the force equilibrium for a micro-segment of any strips, and use ri ˆ Ef

dui dx

Ef af DT

and

ri ˆ Em

dui dx

Em am DT :

We can obtain a set of governing equations as follows:

Fig. 2. Representative segments of analytical model.

d2 ui ‡ h…si si 1 † ˆ 0; dx2 d2 u1 Ef Af 2 ‡ hs1 ˆ 0; dx d2 um Ef A f ‡ hsm 1 ˆ 0 dx2

Ef A f

i ˆ 2; 3; . . . ; m

1; …2†

for the ®bre strips and d2 ui ‡ h…si si † ˆ 0; i ˆ 2; 3; . . . ; m dx2 d 2 u Em Am 21 ‡ h…s1 s1 † ˆ 0; dx d2 um 1 Em A m ‡ h…sm 1 sm 1 † ˆ 0 dx2

Em Am

2; …3†

for the matrix strips. Boundary conditions are: ui …x ˆ 0† ˆ 0;

Ef

dui …x ˆ L† ˆ p ‡ Ef af DT dx

…4†

for the ®bre, i ˆ 1; 2; . . . ; m. ui …x ˆ 0† ˆ 0

and

dui …x ˆ L† ˆ am DT dx

…5†

for the matrix, i ˆ 1; 2; . . . ; m 1, where Af , Ef and Am , Em are the cross-sectional areas and Young's moduli of the ®bre and matrix, respectively. af , am are the coecient of thermal expansion for the ®bre and matrix along the ®bre direction, respectively. h is the thickness of the unidirectional composites, DT is the change of temperature, and p is the tensile stress applied on the end of the ®bre. According to [7], we have si ˆ f2Gf Gm =…df Gm ‡ Gf dm †g…ui

Fig. 1. Analytical model of unidirectional composites.

ui †;

…6†

where df , dm and Gf , Gm are the diameter and shear moduli of the ®bre and matrix, respectively. When considering the ®bre-reinforced polymer matrix composites, we have generally Gf  Gm , hence

X.F. Wang, J.H. Zhao / Cryogenics 41 (2001) 683±691

si ˆ

2Gm  …u dm i

u†:

…7†

si ˆ

2Gm …ui‡1 dm

ui †:

…8†

Introducing the marks ( ui ; i ˆ 1; 3; . . . ; 2m Vi ˆ ui ; i ˆ 2; 4; . . . ; 2m

1;

…9†

2;

then Eqs. (2) and (3) reduce to the following form: d2 Vi ‡ Si ˆ 0; i ˆ 1; 2; . . . ; 2m 1; dx2 where ( h …si si 1 †; i ˆ 1; 3; . . . ; 2m Ef Af Si ˆ h …si si †; i ˆ 2; 4; . . . ; 2m Em Am

…10†

1; 2;

…11†

and the boundary conditions (4) and (5) become dVi ˆ Zi ; Vi jxˆ0 ˆ 0; dx xˆL i ˆ 1; 2; . . . ; 2m where ( Zi ˆ

p Ef

1;

‡ af DT ;

am DT ;

1;

i ˆ 2; 4; . . . ; 2m

2:

dVi Vi;j Vi;j ˆ dx Dx

1

…13†

2

…Dx†

…Xi;j 6 ri;j †;

1

…Xi;j > ri;j †;

…16†

where em ; ei;j are the failure strain and the current strain of the matrix segment, …i ˆ 2; 4; . . . ; 2m 2; j ˆ 1; 2; . . . ; n†, respectively. (2) When the shear stress of the interface reaches the interfacial shear strength sm on the segment of the interface, the interface debonding will occur, and the interfacial strength will reduce to the sliding friction ss between both debonded interfaces instead of sm , as shown in Fig. 3. Thus, we have Vi;j † ‡ Pi

1;j …Vi 1;j

Vi;j †g

Dss f…1 Pi;j †sgn…Vi‡1;j Vi;j † Gm ‡ …1 Pi 1;j †sgn…Vi 1;j Vi;j †g;

in which ( 0 Pi;j ˆ 1

…14†

( Wi ˆ

…18†

…sm 6 jsi;j j†; (

sgn…n† ˆ

;

2Vi;j ‡ Vi;j‡1

0

where Xi;j ; ri;j are the tensile strength and the current normal stress of the ®bre segment, …i ˆ 1; 3; . . . ; 2m 1; j ˆ 1; 2; . . . ; n†, respectively, and  0 …em 6 ei;j †; …17† Hi;j ˆ H …em ei;j † ˆ 1 …em > ei;j †;

…12†

i ˆ 1; 3; . . . ; 2m

ri;j † ˆ

‡ Wi

Usually, a way of solving Eq. (10) is the ®nite di€erence method. As shown in Fig. 1, we divide every strip of the ®bres and matrix into n segments with the length of Dx which is taken as the ine€ective length of the ®bre. The following di€erence forms are adopted:

1

(

Si;j ˆ Wi fPi;j …Vi‡1;j

2.3. Solution ± ®nite di€erence method

d2 Vi Vi;j ˆ dx2

in which Hi;j ˆ H …Xi;j

Similarly

685

…19†

…sm > jsi;j j†; 0

…n P 0†;

1

…n < 0†;

Gm h=EAD; 



Gm h=E A D;

…20†

i ˆ 1; 3; . . . ; 2m

1;

i ˆ 2; 4; . . . ; 2m

2;

D ˆ dm =2: Substituting Eqs. (15) and (18) into Eq. (10), and incorporating the boundary conditions (12), we have Vi;j ˆ

1 ‰UL ‡ UR ‡ C2 Dx2 …UU ‡ UD†Š; C1 ‡ C2 C3 Dx2 …22†

;

where Vi;j …i ˆ 1; 3; . . . ; 2m 1; j ˆ 1; 2; . . . ; n† is the displacement of the nodal point j on the ith strip of the ®bre, and Vi;j …i ˆ 2; 4; . . . ; 2m 2; j ˆ 1; 2; . . . ; n† is the displacement of the nodal point j on the ith strip of the matrix. The following two points need to be noted in the later solutions: (1) When the break occurs on a segment of the ®bre or matrix [2], we have d2 Vi 4fHi;j‡1 …Vi;j‡1 Vi;j † Hi;j …Vi;j Vi;j 1 †g ˆ ; dx2 …2 ‡ Hi;j ‡ Hi;j‡1 †…Dx†2

…21†

…15† Fig. 3. Stress/strain curve of interface.

686

X.F. Wang, J.H. Zhao / Cryogenics 41 (2001) 683±691

where  Hi;j‡1 ‡ Hi;j ; j ˆ 1; 2; . . . ; n 2; C1 ˆ Hi;j ; j ˆ n 1;  …2 ‡ Hi;j ‡ Hi;j‡1 †Wi =4; j ˆ 1; 2; . . . ; n C2 ˆ …2 ‡ Hi;j †Wi =4; j ˆ n 1; 8 i ˆ 1; > < Pi;j ; C3 ˆ Pi;j ‡ Pi 1;j ; i ˆ 2; 3; . . . ; 2m 2; > : Pi 1;j ; i ˆ 2m 1;  Hi;j Vi;j 1 ; j ˆ 2; 3; . . . ; n; UL ˆ 0; j ˆ 1;  Hi;j‡1 Vi;j‡1 ; j ˆ 1; 2; . . . ; n 2; UR ˆ Hi;j‡1 DxZi ; j ˆ n 1; UU 8 Pi;j Vi‡1;j ‡ Dss …1 Pi;j † > > < sgn…Vi‡1;j Vi;j †=Gm ; ˆ > > : 0;

i ˆ 1; 2; . . . ; 2m i ˆ 2m

2;

2;

1:

Using the successive over-relaxation method [5], we have n o q q 1 q 1 ‰Vi;j Š ˆ ‰Vi;j Š ‡ x Vi;j0 ‰Vi;j Š ; …23† where q is the time of iteration, x is the relaxation factor, and the calculations showed that it is the best convergent at x ˆ 1:95. When Vi;j is obtained from Eq. (23), the current stresses on the segment of the ®bre, interface and the current strain on the segment of the matrix can be calculated from the following expression: Vi;j 1 for the fibre or matrix; Dx Vi;j Vi;j 1 ei;j ˆ for the matrix; Dx Vi‡1;j Vi;j si;j ˆ Gm for the interface: D ri;j ˆ E

Vi;j

…24† …25† …26†

2. Find an initial value pi of the stress acting on the end of the ®bre through several tests of calculations. It should be noted that under pi , the break of the weakest ®bre segment only occurs but there are no break events on the rest segments of unidirectional composites. The subsequent calculations are started from the load corresponding pi in increment Dp. 3. For a given applied stress p P pi , the displacement, the normal stress and strain on the segments of the ®bre and matrix and the shear stresses on the segments of the interface can be calculated from Eqs. (22), (23), (24), (25) and (26), respectively. 4. Make the judgements of breakage for the segments of the ®bre, matrix and interface on the basis of the stress or strain obtained from step 3 and failure criterion. It should be noted that the maximum stress criterion is adopted for the ®bre and interface in simulation, while the maximum strain criterion for the matrix. If there are no additional break events, the applied stress r and corresponding strain e on the end of the unidirectional composites in the current calculation can be calculated according to the following expressions: r ˆ pVf ; eˆ

…28† 2X m 1

1 …2m

1†L

Vi;n ;

…29†

iˆ1

where Vf is the volume fraction of the ®bre. Subsequently, turn the calculation to step 5. If there are additional and possibly more break events than one, the e€ect of the broken segments on governing equation (22) needs to be considered. In order to make the break events of the segments to be successive, the less loading increment would be chosen. A modi®ed cmin method is used to make the break events of segments to be successive at a loading increment. Let ri;j ; ri;j

cm i;j ˆ

em ; ei;j

cfi;j ˆ

According to the above statement, the failure strains of the matrix and interface are considered to be de®nitive, while the tensile strength of the ®bre obeys the Weibull distribution in the following simulations: 1. Taking the length Dx of the ®bre segment as the length L in Eq. (1), the strength of the ®bre segment …i; j† with the length of Dx is 1=b  L0 1=b Xi;j ˆ r0 ln…1 z† ˆ rDx ‰ ln…1 z†Š ; …27† Dx

where em is the failure strain of the matrix. Take cmin ˆ minfcfi;j ; cii;j ; cm i;j g, as cmin > 1, the corresponding segments of composites do not break and as cmin 6 1, the breaks of the corresponding segments will occur. In the latter case, change Hi;j and Pi;j (Eqs. (17) and (19)), and Eq. (22) will be changed too. Then, solve the changed equation (22) again under holding of the applied stress p, until cmin > 1 and turn the calculation to the next step. 5. Increase the applied stress p in increment Dp, then repeat steps 3 and 4 until the composite fractures fail. Thus the strength rc and the stress±strain curve of unidirectional composite can be obtained.

in which rDx ˆ …L0 =Dx†1=b r0 acts as the scale parameter and z is a random number taken from the uniform distribution on the interval ‰0; 1Š.

cii;j ˆ

sm ; si;j

2.4. Simulation procedure

…30†

X.F. Wang, J.H. Zhao / Cryogenics 41 (2001) 683±691 Table 1 comparison of simulated results for test procedure

Table 2 Material constants used in the present simulation

Simulated strength rc a rc b a b

6.47 GPa 6.62 GPa

Result of [3]. Result of the present work.

3. Results of numerical simulations and discussion 3.1. Test of calculation program and selection of some parameters The numerical simulation of the strength at room temperature was conducted for unidirectional CFRP to test the described simulation procedure above. The related parameters were chosen to be the same with the ones listed in Table 1 of [3]. The calculations showed that the simulated strengths depend on the number m of the ®bres and the length Dx of the segment in the model of unidirectional composites. As Dx ˆ 10df [3], variation of the simulated strength with m is shown in Fig. 4. It can be seen that the simulated results approach almost a stationary value as m ˆ 50. Thus, in the following numerical simulations, m and Dx would be taken as 50 or more and 10df , respectively. It should be noted that the simulated strengths would change with di€erent runs. As m ˆ 50 and n ˆ 50, it showed that the coecient of variation of 20 runs is lower than 1%. Table 1 shows a comparison of the calculated result in this work to the one in [3]. The comparison shows that the result of this work is somewhat larger than that of [3]. It may be the e€ect of the matrix which was taken into account in this work. Thus, we can consider that the developed analytical model and the simulation procedure are more appropriate for the strength simulation relating to the temperature of unidirectional composites.

Fig. 4. The relation of simulated strength rc and the number m of ®bres.

687

E (GPa) em (%) sm (MPa) ss (MPa) b rDx (MPa) Gm (MPa) aL …10 6 =K† aT …10 6 =K† A …mm2 † Vf …%† dm …mm† h …mm† Dx …mm† L0 …mm†

E-glass [12]

Carbon [13,14]

70 ± 59.5 59.5/2 9.4 1950 ± 4.8 4.8 2.25  10 65 0.00425 0.015 0.0866 8

230 ± 59.5 59.5/2 6.98 4320 ± 1.1 48 3.844  10 60 0.0026 0.0062 0.062 8

4

Matrix [8] 3.2 6

5

± ± ± ± 1130 48 48 ± ± ± ± ± ±

3.2. Numerical simulations of mechanical behavior for unidirectional composites For examining the e€ect of temperature, the numerical simulations were carried out for the cases of room and low temperatures, respectively. The related parameters at room temperature (about 296 K) are given in Table 2. But the following parameters: Em , sm , ss would be changed at low temperature: Em ˆ 6000 MPa;

em ˆ 4%

for the matrix [8], sm ˆ 170 MPa;

ss ˆ sm =2

for E-glass ®bre unidirectional composites [10], sm ˆ 150 MPa;

ss ˆ sm =2

for carbon ®bre unidirectional composites [10]. Generally, the ®bre shape parameter b is assumed to be independent of temperature. As for Weibull scale parameter rDx , some experimental data at a few temperature points can be found in [11] for glass ®bre. A ®tted curve on basis of these data is shown in Fig. 5. Extending to low temperature along the ®tted curve, we

Fig. 5. Weibull scale parameters rDx of glass ®bre versus temperature T.

688

X.F. Wang, J.H. Zhao / Cryogenics 41 (2001) 683±691

(a)

(b)

(c)

Fig. 6. Fracture process of carbon/epoxy unidirectional composite at 296 K: (a) 0.923 GPa, (b) 1.402 GPa and (c) 1.459 GPa (N, breakage of ®bre; , breakage of matrix; , debonding of interface).

Fig. 7. Fracture process of carbon/epoxy unidirectional composite at 77 K: (a) 1.227 GPa, (b) 1.842 GPa and (c) 1.859 GPa (N, breakage of ®bre; , breakage of matrix; , debonding of interface).

expect rDx to be about 2350 MPa at 77 K for glass ®bre. Similarly, rDx to be 5220 MPa at 77 K for carbon ®bre. 3.2.1. Failure processes Figs. 6 and 7 show the examples of the failure processes at 296 and 77 K, respectively, for a carbon ®bre unidirectional composites. Fig. 8 is an example of the failure process at 77 K for a glass ®bre unidirectional composite. We see that a common char-

acteristic of the failure processes is that ®rst break occurs on the weakest ®bre segment, and then the matrix break and the debonding (interface break) occur near the breakpoint of the broken ®bre, as shown in case (a) of those Figs. With increasing of the applied stress, more break events in the ®bre, matrix and interface occur as in case (b). In case (b), the failure processes progress further and leads quickly to the ®nal failure as in case (c), i.e. the unidirectional

Fig. 8. Fracture process of E-glass/epoxy unidirectional composite at 77 K: (a) 1.068 GPa, (b) 1.392 GPa and (c) 1.433 GPa (N, breakage of ®bre; , breakage of matrix; , debonding of interface).

X.F. Wang, J.H. Zhao / Cryogenics 41 (2001) 683±691

689

perature and the jagged appearance of the fracture surface is more obvious at room temperature for carbon ®bre unidirectional composites. 3. While the micro-break events of the matrix and interface are more at low temperature, the jagged appearance of the fracture surface is not obvious for glass ®bre unidirectional composites.

Fig. 9. Simulated stress/strain curve under controlling load for carbon/ epoxy unidirectional composite at 296 and 77 K.

Fig. 10. Recorded stress/strain curve under controlling displacement for carbon/epoxy unidirectional composite at 296 and 77 K [9].

composites completely fail. The patterns of the failure processes tell us: 1. During most part of the loading period, the microbreak events are less, and increase sharply only when the loading approaches the critical case, as shown in case (c). 2. The micro-break events of the matrix and interface at room temperature are more than the ones at low tem-

3.2.2. Simulating of stress±strain relations Fig. 9 shows the stress±strain relations obtained by the numerical simulation for carbon ®bre unidirectional composites. Fig. 10 shows the curves of stress±strain, which were recorded experimentally by authors [9] for carbon ®bre unidirectional composites. Owing to the unknown properties and geometry parameters of constituent materials for the tested composites, it is not possible to compare quantitatively the curves in Fig. 9 with the ones in Fig. 10. But its tendency demonstrates enough the feasibility of the simulation procedure developed in this work. From Figs. 9 and 10, it is seen that the slope of stress±strain curve at 77 K is larger than the one at 296 K, and the relations are all linear for both the cases. The numerical calculations illustrate that the effect of the moduli of constituent materials, especially the ®bre, on the slope of stress±strain curve is not neglected. Table 3 gives some of the simulated results and experimental data for comparison. We see that the simulated results are well in agreement with the experimental data, and the moduli at 77 K are generally larger than the ones at 296 K for both unidirectional composites. 3.2.3. Simulation of strengths Table 4 shows the simulated results of the failure strength. It is clear that the simulated results are well in agreement with the experimental data and the strengths at 77 K are always larger than those at 296 K for both unidirectional composites. 3.2.4. Variables for in¯uencing the properties of unidirectional composites at low temperature From the simulated results of Young's modulus and the strength for unidirectional composites in this work and some experimental data in [6,9], we can see that these parameters at low temperature are generally larger than those at room temperature. Thus, exploring the cause of this phenomenon is interesting. First, we considered only the e€ect of reducing temperature, i.e. whether the case of room or low temperature, the

Table 3 Simulated results of Young's moduli for unidirectional composites Carbon/epoxy T (K) E (GPa)

296 138 (133)

The values in parentheses are the experimental data [6].

E-glass/®bre 77 148 (145)

296 47 (43)

77 51 (45)

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X.F. Wang, J.H. Zhao / Cryogenics 41 (2001) 683±691

Table 4 Comparisons of strength rc (GPa)

Experimental data [6] Experimental data [9] Simulated results

Carbon/epoxy

E-glass/®bre

296 K

77 K

296 K

77 K

1.46 1.463 1.45

1.88 1.787 1.86

1.10 ± 1.08

1.40 ± 1.43

parameters listed in Table 2 were all adopted as needful data in simulations. The result shows the strength to be 1.465 at 77 K, while 1.459 at 296 K for the case of carbon ®bre, and 1.091 at 77 K, while 1.080 at 296 K for the case of glass ®bre. It illustrates that only reducing of temperature is not an essential cause of mechanical properties trending upward at low temperature. However, the property change of the constituent materials which resulted from reducing of temperature will in¯uence more the properties of unidirectional composites. Figs. 11±13 show the e€ects of the ®bre Fig. 13. E€ect of interface shear strength sm on rc of glass/epoxy unidirectional composite at 77 K.

Fig. 11. E€ect of scale parameter rDx on rc of carbon/epoxy unidirectional composite at 77 K.

scale parameter rDx , Young's modulus Em of the matrix and the shear strength sm of interface on the failure strength of unidirectional composites. We can see that: (1) the e€ect of the matrix Young's modulus on the strength rc of unidirectional composites is not notable, while the failure strain em of the matrix needs to be considered; (2) rc increases with increasing of the rDx and sm . Also, the calculations show that the increasing of rc with a rate of about 30% at 77 K is mainly due to the contributions of increasing rDx and sm with decreasing of temperature in which the contribution of rDx is notable for the case of carbon ®bre, while sm for the case of glass ®bre.

4. Conclusions

Fig. 12. E€ect of Young's modulus Em of matrix on rc of glass/epoxy unidirectional composite for variant em at 77 K.

An analytical method and simulating produce developed in this work could be used to simulate well the failure process including the ®bre break, the matrix crack and the interface debonding, and predict Young's modulus and the tensile strength for unidirectional composites at room and low temperatures. Simulated results showed a good agreement with experimental data for glass/epoxy and carbon/epoxy unidirectional composites. Also, variables for in¯uencing the properties of unidirectional composites at low temperature were explored. It indicated that the essential causes of making the mechanical behavior of unidirectional composite improve at low temperature

X.F. Wang, J.H. Zhao / Cryogenics 41 (2001) 683±691

are increasing the ®bre scale parameter and shear strength in the interface with cooling for glass/epoxy and carbon/epoxy, while a direct e€ect of reducing only on the temperature is less. Acknowledgements The ®nance support for this subject was provided by the grant (19732001) from the National Science Foundation of China for which the authors are grateful. References [1] Schutz JB. Properties of composite materials for cryogenic applications. Cryogenics 1998;38:3±12. [2] Kong POH. A Monte Carlo study of the strength of unidirectional ®ber-reinforced composites. J Compos Mater 1979;13:311±27. [3] Goda K, Leigh PS. Reliability approach to the tensile strength of unidirectional CFRP composites by Monte-Carlo simulation in a shear-lag model. Compos Sci Technol 1994;50:457±68. [4] Abdelmohsen HH. Simulation of tensile strength of anisotropic ®bre-reinforced composites at low temperature. Cryogenics 1991;31:399±404.

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[5] Ferzigr JH. Numerical methods for engineering application. New York: Wiley; 1981. [6] Reed RP, Golda M. Cryogenic properties of unidirectional composites. Cryogenics 1994;33:909±27. [7] Shojiro O, Karl S, Piet WMP. Strain concentration factors ®bers and matrix in unidirectional composites. Compos Sci Technol 1991;41:237±56. [8] Hartwig G. Status and future of ®bre composites. In: Reed RP, Fichett FR, editors. Advances in cryogenic engineering (materials), vol. 40B. New York: Plenum Press; 1994. p. 961±75. [9] Wang XF. The research on the mechanical behavior of ®brereinforced composites under low temperature. Ph.D. Thesis, University of Science and Technology of China; 2001. [10] Hartwig G, Knaak S. Fibre-epoxy composites at low temperatures. Cryogenics 1984:639±47. [11] Wang Z. The research for the distortion, damage and fracture process of unidirectional composites under tensile impact loading. The paper for Doctor degree of USTC; 1996 (in Chinese). [12] Rosen BW. Mechanics of composite strengthening. In: Fibre composite materials. Cleveland, OH: American Society for Metals; 1965. p. 19±24. [13] Kimpara I, Watanabe I, Okatsu K, Ueda T. A simulation of failure process of ®bre-reinforced materials. In: The 7th Symposium of Composite Materials, Japan; 1974. p. 169 (in Japanese). [14] Tsai SW, Hahn HT. Introduction to composite material. New York: Technomic Publishing; 1980.