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Simulation of tensile strength of anisotropic fibre-reinforced composites at low temperature
Simulation of tensile strength of anisotropic fibre-reinforced composites at low temperature H.H. Abdelmohsen* Applied Superconductivity Wl 537O6, USA
Center,
University
of
Wisconsin-Madison,
Madison,
Received 23 May 1990; revised 15 November 1990 This article presents a simulation scheme to predict the effect of fibre anisotropy on composite tensile failure strength at room and low temperatures. The simulation model combines the shear lag equation with the chain of bundles probability model to describe the composite failure behaviour. The effect of fibre anisotropy on thermal stresses developed in composite constituents due to cooling to low temperatures is considered. The failure strength of composites composed of fibres with different degrees of anisotropy is obtained and comparisons are made with experiments. Simulated results for E-glass-, graphite- and Kevlar-epoxy type composites showed close agreement with the experiments.
Keywords: fibre-reinforced composites; simulation models; tensile strength
Analytical studies for predicting the mode and the load at failure for unidirectional fibre-reinforced composites, on the basis of the properties of the fibre and the matrix, are based in a broad sense on two distinct models. In the first model, it is assumed that the distribution of strain across the specimen is uniform and that all the fibres break at the same strain and at the same cross-section ~. The second model takes into account the effect of the statistical distribution of fibre strength 2-~°. It is possible to split the failure models developed on statistical bases in relation to fibre-reinforced composites into two categories: the weakest link model and fracture models. The weakest link model also known as the series model is based on the assumption that the whole structure fails if the weakest link fails. The first application of this model to the study of material strength was by Weibull 8. The series model does not apply to composite materials, since the unbroken fibres continue to carry loads after the weakest fibres break. Turning to the fracture models, two major models are presented in the literature: the cumulative fracture model and the fracture propagation model. In the first model, the matrix is assumed not to contribute directly to the tensile strength of the composites, although it provides a means to transfer the load in shear to the fibres. The specimen is divided into layers (bundles) of a length defined as an ineffective length ~0. When the specimen is loaded, the fibres are assumed to be stressed uniformly and as the load increases the fibres in each
*Present address: 4 EI-Douha Street No. 7, Zizinia, Alexandria, Egypt 0011 - 2 2 7 5 / 9 1 / 0 6 0 3 9 9 06 ,c~ 1991 B u t t e r w o r t h - H e i n e m a n n
bundle start to break randomly and stresses are redistributed uniformly among the unbroken fibres in each bundle. When a sufficient number of fibres in a bundle fail, the specimen fails. Since the stress redistribution is assumed to be uniform along all the unbroken fibres, no stress concentration factors were employed. It is clear that the cumulative fracture model or the parallel model does not account for the stresses developed between the fibres. This disadvantage is overcome by the fracture propagation model. In the fracture propagation (series-parallel) model, which is an alternative to the weakest link model and cumulative fracture model described above, the composite is modelled as a chain of n links in series and each link is a bundle of m fibres in parallel. The use of a series-parallel model is an attempt to account for the interaction between the fibres when the composite is loaded to failure. Neither of the above models consider residual stresses generated in the fibre and the matrix due to the cooling process. Most mechanical and thermal properties of fibre composites are of a tensorial type due to fibre arrangement, intrinsic fibre anisotropy and f i b r e - m a t r i x interracial bonds. The effect of these parameters on composite behaviour is well summarized by Hartwig ~'~2. Fibreglass possesses isotropic mechanical and thermal behaviour which results in high strength in all directions. Good f i b r e - m a t r i x interfacial bonding is anticipated when using a glass f i b r e - m a t r i x composite at low temperature, since glass fibre contracts less than polymers. Carbon fibres are both mechanically and thermally anisotropic which results in a low transverse strength and stiffness compared to that in the longitudinal direction. Anisotropy is more pronounced in
Ltd
Cryogenics 1991 Vol 31 June
399
Simulation of tensile strength: H.H. Abdelmohsen Kevlar fibres, which consist of stretched aramide molecules with strong covalent bonding in the fibre direction only. This produces very low transverse shear strengths and stiffness. Since Kevlar fibres contract more than polymers in the transverse direction, weak bonding and adhesion occurs between the fibres and the polymers. An elastic solution is utilized by Vedula et al. to obtain thermal residual stresses for composites with anisotropic fibres 13'14. However, their analysis is restricted to fibres with isotropic elastic behaviour in all directions. A more general solution is given by Avery and Herakovich ~5 where the fibres possess anisotropic elastic and thermal properties. The present paper is intended to explore analytically and numerically the relationship between fibre anisotropy and the tensile failure strength of unidirectional fibre-reinforced composites at low temperatures.
each fibre by itself. Equation (2) takes a slightly different form if it is applied to the first or to the last fibre in the bundle.
Analysis of thermal stresses For a long fibre in an isotropic elastic matrix, as shown in Figure 1, the governing differential equation under a condition of plane strain with uniform axial strain, cx, has the following form 15
Cr r [[ 02w Or 2 + ~1f0f w rrl-Coo
-
w r: -
1 - -
r
+-
(Co,
1 r
-
Cx)
,
(crj - coj)
Ar
(4)
Micromechanic model The model is based on two basic assumptions: the fibre strength is described statistically by the two parameter Weibull distribution and the shear lag equation governs the fibre displacement field. While the matrix does n o t contribute directly to the composite strength, it does provide a means to transfer the load in shear to the fibres. The first assumption states that, if F(o*) is the probability that the fibre strength is _<0", then
Equation (4) is based on the assumption that both stresses and strains are independent of 0 and that no shear-extension coupling exists. C0 are stiffness coefficients, % is the coefficient of thermal expansion in the j direction and AT is the uniform temperature change. Cij values are functions of the fibre elastic constants 16 u, v and w are the axial, hoop and radial displacements, respectively. The general solution to Equation (4) is w f ( r ) = Air xl + A f r x2 + Glexrln r + GzATrln r
F(o*) = 1 - exp [ - (~)~,Sac]
(5a)
(1) for a transversely isotropic fibre and
where: 71and/3 are Weibull shape and scale parameters, respectively; and Ax is the fibre segment size. The second assumption considers the unidirectional composite lamina as a thin sheet consisting of m fibres spaced uniformly parallel to the x axis. If the lamina is loaded in the x direction, the force equilibrium equation or the shear lag equation in a non-dimensional form is --
(5b)
w f ( r ) = A~r ×1 + Afr x2 + Hl¢x r + H2ATr
+ (ui-i - 2ui + ui+O = 0
do 2
where p = [EfAfS/Gmhm] l/2x
(2)
where: u~ is the displacement of the ith fibre; EfAf is the fibre tensile stiffness; Gmh m is the matrix shear stiffness; and S is the spacing between fibres. S is assumed to be constant and uniform. For a composite pulled out in simple tension, the boundary conditions are
r(w)
ui(O) = 0 du i
dp
-
oJE,
(3)
where oc is the applied stress on the lamina. Equation (2) guarantees the equilibrium of the whole composite lamina, whereas Equation (3) ensures the equilibrium of
400
Cryogenics 1991 Vol 31 June
Matrix Fibre Figure 1 Composite geometry
Simulation of tensile strength: H.H. Abdelmohsen for a transversely orthotropic fibre, where Gl-
H, -
Cox - Crx 2Coo
,
G2-
(Cox - Crx )
( Cri
H2 -
( C r - Coo)'
-
-
Coi)O[i
2Coo ( Cri - Coi)OLi
(6)
(C~r -- Coo)
and /g"*
\1/2
The repeated indices used in Equations (4) and (6) indicate summation over x, r and 0. For the isotropic matrix the solution has the form
wm(r)
=
(7)
A["r + A~' F
f and m in Equations (5) and (7) denote the fibre and matrix, respectively. Equations (5) and (7) contain five constants to be evaluated using the following boundary conditions: 1 fibre radial displacement is bounded at r = 0; 2 continuity of radial displacement at the f i b r e - matrix interface; 3 continuity of radial stress at the fibre-matrix interface; 4 traction free at the outer boundary of the matrix; and 5 traction free at the composite boundary for pure thermal loading. Once the coefficients that appear in Equations (5) and (7) are known, stresses could be obtained by differentiating the expression for radial displacement. Details of the derivation are shown in Appendix A.
Solution scheme The composite lamina is divided into m x n mesh points, where m is the total number of fibres and n is the total number of mesh points in the x direction (i.e. the number of bundles). The segment size, 2tx (fibre length/n), also known as the ineffective length t°,
Table 1
Numerical examples The analysis outlined in the previous three sections is applied here to predict the tensile failure strength of glass-, graphite- and Kevlar-epoxy composites at room and liquid helium temperatures. Numerical values of fibre and matrix mechanical and thermal properties used in the simulation are listed in Table 1. Fibre Weibull parameters were reported by Fariborz ~'~°. To assess the convergence of the numerical scheme, different numbers of fibres and mesh points (number of bundles ~7~8) are tested until convergence occurs at
Mechanical and thermal constants
Elastic modulus, E~ (GPa) Elastic modulus, ET ~ (GPa) Poisson's ratio, PLTb Poisson's ratio, PTLb Coefficient of thermal expansion, ~L b (10 Coefficient of thermal expansion, ozTb (10 /3 (GPa) Of (/~m)
depends on the mechanical properties of the fibre and the matrix, the size of the composite and the geometry of the specimen. Fariborz 9'~° suggested a conservative value of 6 - 10 fibre diameters. For the m x n mesh points, a random number for a*, corresponding to each segment strength, is generated using the distribution defined in Equation (1). The load is applied in increments, Aa, starting with an initial value, Oo, and Equation (2) is solved with the boundary conditions given in Equation (3) using the successive over-relaxation method, as described in Appendix B. Once the solution of Equation (2) is known at each load increment, the stress in each fibre segment, o = Er dui4/dP (where i = 1, 2 . . . . . m and j = 1, 2, . . . . n), is checked against its strength, o*. Two cases exist; either a* _> o or a* < ~. The first case indicates that this segment is in a state of constant stress and the displacement is a single valued function. On the other hand, the second case implies that the fibre has been broken at this segment and a multivalued displacement function exists in this segment. As long as the first case prevails, Equation (2) holds. If the first case is violated and the second case exists, Equation (2) has to be modified to allow the displacement to assume multivalues. Appendix B presents the solution to Equation (2) in a finite difference form for the two cases presented above. For each increment of loading, the broken fibres and the locations of such breaks are defined. The composite has failed if one cleavage of breaks is formed. The existence of one cleavage causes the solution of Equation (2) to diverge since the boundary conditions on both sides of the broken region (at the position of the cleavage and at x = lamina length) are of the Neumann type. The solution scheme is depicted in Appendix C and numerical results are given in the next section.
6/K) 6/K)
E-Glass
Graphite
Kevlar
Matrix
72.5 72.5 0.22 0.22 4.8 4.8 8.2 1.96 8.0
240.0 32.0 a 0.30 a 0.04 a 1.1 48 a 7.0 2.30 8.0
131.0 4.13 0.35 0.01a 2 48 a 8.2 1.96 11.9
2.16 2.16 0.40 0.40 48 48 -
Assumed values, based on analysis given by Hartwig TM 12 bL and T denote the longitudinal and transverse direction, respectively
Cryogenics 1991 Vol 31 June
401
Simulation of tensile strength: H.H. Abdelmohsen Table 2 Comparison between simulated composite failure strength and experimental results (in parentheses) at room and low temperatures
Room temperature Liquid helium temperature
E-Glass-epoxy
Graphite-epoxy
Kevlar-epoxy
0.93 (1.05) 1.20 (1.25)
0.91 (0.84) 0.79 (0.80)
1.24 (1.18) 1.18 (1.15)
80 x 80 mesh points. Thermal stresses developed in the composite constituents due to cooling are evaluated using the analysis given above. Due to cooling, glass fibres exhibit longitudinal compressive stress of 0.46 GPa. Graphite and Kevlar fibres develop the same type of stresses but an order of magnitude lower. In all three composites, tensile stress exists in the matrix longitudinal direction (x in Figure 1). These stresses create thermal residual strain in the matrix equal to 0.5% for a glass fibre type composite and 0.3% for graphite fibre and Kevlar fibre type composites. Compressive stresses developed in the fibre due to cooling are incorporated into the fibre failure strength distribution by adding their values to the fibre scale parameter. This causes the fibre strength distribution to shift by such a stress value on Weibull graph paper 17. The fibre shape parameter is assumed to be temperature independent. Table 2 shows the simulated tensile failure strengths compared with results from the experiments ~9'2° on the three composites at room and low temperatures. Results are based on a fibre volume fraction equal to 50% and 20 computer simulations. As shown there is remarkable agreement between the model results and the experimental data. The argument given in the introduction is strongly supported by the results reported in Table 2. Fibre anisotropic mechanical and thermal properties control thermal stresses generated in the composite components due to cooling. Basically, fibres will exhibit compressive stresses and the matrix will be subjected to tensile stresses in the longitudinal direction. The resulting fibre compressive stresses tend to increase the tensile strength and this could be reflected by the increase in the scale parameter value. Matrix tensile thermal stresses reduce the matrix free strain ~2.
2.0
0.0 -1,0
-
-2.0 -
t-
.J
-3.0 -
R"
-4.0 -5.0 -0.40
I
_5 I i-
/
-1.0
-30
-4.0 -5.0 -0.6
I -0.4
i
I
-0.2 Ln 5"
0.0
0.2
Figure 3 Weibull distribution of graphite-epoxy composite. PR = probability of failure, S = failure strength
2.0 1.0 L
"
"
•
'
i--1
0.0 I -~
-1.0
I
~-
-2.0
d
-5.0 -4.0 0.0
I 0.1
I 0.2 Ln 5"
I 0.3
Figure 4 Weibull distribution of Kevlar-epoxy PR = probability of failure, S = failure strength
0.4 composite.
L I
I --0.20
I
I 0.00 Ln 5"
I
I 0.20
Figure 2 Weibull distribution of glass-epoxy PR = probability of failure, S = failure strength
402
20// 0.0
However, this has no effect on the composite tensile strength since the matrix is assumed to be merely a means to transfer loads in shear to the fibres and not to contribute to composite strength ~7. Figures 2, 3 and 4 depict tensile strength distribution for the three composites at room and liquid helium temperatures. As shown, Weibull distribution describes the tensile strength of glass- and graphite-epoxy composites fairly well, while there is poor agreement for the Kevlar-epoxy composite.
1.0
I c. I
2.0
Cryogenics
1991
Vol
31 June
[
0.40 composite.
Conclusions
This article presents a simulation scheme to predict the effect of fibre anisotropy on composite tensile failure strength at room and low temperatures. Results show
Simulation of tensile strength: H.H. Abdelmohsen that composite tensile strength tends to improve upon cooling for composites with isotropic fibre properties (mechanical and thermal). Composites with anisotropic fibres may lose part of their strength due to cooling. Simulated results for E-glass-, graphite- and Kevlarepoxy type composites agree well with expeirmental data.
References 1
2
3 4 5 6
7
8 9
10
11 12
13
14
15
16 17
18
19
20 21
Jech, R.W., McDaniles, D.L. and Weston, J.W. Fiber reinforced metallic composites Proc 6th Sagamore Ordnance Materials Research Conference (1959) Rosen, R.W. Mechanics of Composite Strengthening: Fiber Composite Materials American Society of Metals, Metals Park, Ohio (1965) Zweben, C. Tensile failure analysis of fibrous composites AIAA J (1968) 6 2325 Coleman, B.D. A stochastic process model for mechanical breakdown Trans Soc Rheology (1957) 1 153 Gucer, D. and Gurland, J. Comparison of the statistics of the two fracture modes J Mech Phys Sol (1962) 10 365 Zweben, C. and Rosen, B.W. A statistical theory of material strength with application to composite materials, AIAA Paper No 69-123, presented at AIAA 7th Aerospace Sciences Meeting, New York, USA (1969) McKee, R.W. and Sines, G. A statistical model for the tensile fracture of parallel fiber composites, ASME Paper No. 68-WAIRP-7, presented at ASME Winter Annual Meeting, New York, USA (1968) Oh, K.P. A Monte Carlo study of the strength of unidirectional fiberreinforced composites J Composite Mat (1979) 13 31 l Fariborz, S.J., Tang, C.L. and Harlow, D.G. The tensile behavior of intraply hybrid composites I: Model and simulation J Composite Mat (1985) 19 334 Fariborz, S.J. and Harlow, D.G. The tensile'behavior of interply hybrid composites II: Micromecbanical model J Composite Mat (1987) 21 856 Hartwig, G. and Knaak, S. Fiber-epoxy composites at low temperatures Cryogenics (1984) 24 639 Hartwig, G. Low temperature ductile matrices for advanced fiber composites, in: Mechanics of Composite Materials (Eds Dvorak, G.J. and Laws, N ) Society of Mechanical Engineers (1988) Vedula, M., Pangborn, R.N. and Queeney, R.A. Fiber anisotropic thermal expansion and residual thermal stress in a graphite/aluminium composite Composites (1988) 19 55 Vednla, M., Pangborn, R.N. and Qneeney, R.A. Modification of residual thermal stress in a metal-matrix composite with the use of a tailored interfacial region Composites (1988) 19 133 Avery, W.B. and Herakovlch, C.T. Effect of fiber anisotropy on thermal stresses in fibrous composites J Appl Mechanics (1986) 53 751 Jones, R.M. Mechanics of Composite Materials McGraw-Hill (1975) Abdelmohsen, H.H. Prediction of tensile strength of unidirectional fiber reinforced composites at low temperature, paper presented at ICMC, Los Angeles, California, USA (1989) Abdelmohsen, H.H. Effect of fiber anisotropy on tensile strength of unidirectional fiber reinforced composite at low temperature, paper presented at ICMC, Los Angeles, California, USA (1989) Kasen, M.B. Mechanical and thermal properties of filamentaryreinforced structural composites at cryogenic temperatures 2: Advanced composites Cryogenics (1975) 24 701 Kasen, M.B. Composites. in: Materials at Low Temperature (Eds Reed, R P and Clark. A F ) American Society of Metals (1983) Ferziger, J.H. Numerical Methods fi~r Engineering Application John Wiley (1981)
w'(r)=A Ir+
-+Biqrlnr+B2ATrlnr r
(AI)
win(r) = A3r r
Using Equation (A1) the stresses can be expressed as
Oif = A l ( C l ; 4- C~r ) 4- A2(Cl; -- e l ; )
1 r~
!
f
f
+ C,.~q - C o a j A T
o m = A 3 ( C ~ + Cimr) --~ A 4 ( C ~ -- Cim)
1 F-
+ Ci"~e, - C~'ot]nAT
(A2)
where i and j are indices that refer to x, r or 0 where x, r and 0 are the longitudinal, the radial and the tangential directions, respectively, r = 0 is located on the fibre longitudinal axis. Repeated indices indicate summation over the repeated index, f and m denote the fibre and the matrix, respectively. In Equations (A1) and (A2), the A terms are coefficients to be determined from the boundary conditions, e~ is the strain in the x direction, AT is the temperature drop, the C terms are fibre and matrix stiffness coefficients which are related to their mechanical properties ~6, the B terms are functions of the C terms and the thermal properties, and the c~ terms are the coefficients of thermal expansion. Satisfying the boundary conditions at the f i b r e - m a t r i x interface for pure thermal loadings and using the fact that the solution has to be bounded everywhere, the following system of equations is formed to obtain the unknown coefficients and c~ [A] Ix] = [B]
(A3)
where the elements of each matrix are A(1, 1) = a A(1, 2 ) = - a A(1, 3) = - 1 / a
A(1, 4) =
Blaln(a )
A(2, 1) = C~o + C~r A(2, 2) = - ( C ~
+ C~lr)
A(2, 3) = - ( C ~ - Crr)/a" m
A(2, 4) = C~, - C~m A(3, 1) = 0 A(3, 2) = C~ + Crn) A(3, 3) = ( C ~
-
Cnr'r)/b 2
A(3, 4) = C~',
Appendix A For fibres with transversely isotropic mechanical and thermal properties, embedded in an isotropic elastic matrix, the fibre and the matrix displacements in the r direction are given byl5
A(4, 1) = (C~o + C f r ) a : / 2 A(4. 2) = C.r(b m 2 - aZ)/2 A ( 4 , 3) = 0 A(4, 4) = [C",',(b 2 - a z) + C ,f~ a - ] / 2
Cryogenics 1991 Vol 31 June
403
Simulation of tensile strength: H.H. Abdelmohsen B(1) = - B 2 A T a In(a)
where o~ is the relaxation factor with a value between 1 and 2 (Reference 21). The use of the Christopherson method, as demonstrated by Oh s, allows Equation (B1) for a broken fibre to have the following two forms, if evaluated to the right or to the left of the break, respectively
B(2) = C[; oLfA T - C~ o~'~A T a(3) = C~c~TAT B(4) = [Cxfjot~a2 + C~ot]~(b 2 - a b ] A T / 2 x(1)=A~ x(2) = a 3
u(i, j ) =
x(3) = A 4 x(4) =ex Finally, Bl = (Cox - C,-,)/Coo/2 and B 2 = (Cri - Coi)Oti/ Coo~2.2a = Df is the fibre diameter and 2b - 2a = S is the spacing between fibres. Once the unknown coefficients are evaluated by solving Equation (A3), the displacements and the stresses are determined from Equations (AI) and (A2).
o~(3Ap2[u(i -- 1, j ) + u(i + 1, j ) ] + 4u(i, j + 1)) 2(3Ap2 + 2)
J
+(1
-
o~)u(i, j)
(B2)
u(i, j ) =
Appendix B Equation (1) in a finite difference form using the successive over-relaxation algorithim is 21
wf3ap2[u(i -
1, j ) + u(i + 1, j ) ] - 4u(i, j - 1))
--25 +(1 - w)u(i, j )
u(i, j ) = o~{([u(i - 1, j ) + u(i + 1, j ) ] A p 2
3 (B3)
+ u(i, j - 1) +u(i, j + 1))/2(1 + ApZ)} +(1
404
-
oo)u(i, j)
Cryogenics 1991 Vol 31 June
(B1)
The above equations have different forms for the first and the last fibre in a bundle.
Center,
University
of
Wisconsin-Madison,
Madison,
Received 23 May 1990; revised 15 November 1990 This article presents a simulation scheme to predict the effect of fibre anisotropy on composite tensile failure strength at room and low temperatures. The simulation model combines the shear lag equation with the chain of bundles probability model to describe the composite failure behaviour. The effect of fibre anisotropy on thermal stresses developed in composite constituents due to cooling to low temperatures is considered. The failure strength of composites composed of fibres with different degrees of anisotropy is obtained and comparisons are made with experiments. Simulated results for E-glass-, graphite- and Kevlar-epoxy type composites showed close agreement with the experiments.
Keywords: fibre-reinforced composites; simulation models; tensile strength
Analytical studies for predicting the mode and the load at failure for unidirectional fibre-reinforced composites, on the basis of the properties of the fibre and the matrix, are based in a broad sense on two distinct models. In the first model, it is assumed that the distribution of strain across the specimen is uniform and that all the fibres break at the same strain and at the same cross-section ~. The second model takes into account the effect of the statistical distribution of fibre strength 2-~°. It is possible to split the failure models developed on statistical bases in relation to fibre-reinforced composites into two categories: the weakest link model and fracture models. The weakest link model also known as the series model is based on the assumption that the whole structure fails if the weakest link fails. The first application of this model to the study of material strength was by Weibull 8. The series model does not apply to composite materials, since the unbroken fibres continue to carry loads after the weakest fibres break. Turning to the fracture models, two major models are presented in the literature: the cumulative fracture model and the fracture propagation model. In the first model, the matrix is assumed not to contribute directly to the tensile strength of the composites, although it provides a means to transfer the load in shear to the fibres. The specimen is divided into layers (bundles) of a length defined as an ineffective length ~0. When the specimen is loaded, the fibres are assumed to be stressed uniformly and as the load increases the fibres in each
*Present address: 4 EI-Douha Street No. 7, Zizinia, Alexandria, Egypt 0011 - 2 2 7 5 / 9 1 / 0 6 0 3 9 9 06 ,c~ 1991 B u t t e r w o r t h - H e i n e m a n n
bundle start to break randomly and stresses are redistributed uniformly among the unbroken fibres in each bundle. When a sufficient number of fibres in a bundle fail, the specimen fails. Since the stress redistribution is assumed to be uniform along all the unbroken fibres, no stress concentration factors were employed. It is clear that the cumulative fracture model or the parallel model does not account for the stresses developed between the fibres. This disadvantage is overcome by the fracture propagation model. In the fracture propagation (series-parallel) model, which is an alternative to the weakest link model and cumulative fracture model described above, the composite is modelled as a chain of n links in series and each link is a bundle of m fibres in parallel. The use of a series-parallel model is an attempt to account for the interaction between the fibres when the composite is loaded to failure. Neither of the above models consider residual stresses generated in the fibre and the matrix due to the cooling process. Most mechanical and thermal properties of fibre composites are of a tensorial type due to fibre arrangement, intrinsic fibre anisotropy and f i b r e - m a t r i x interracial bonds. The effect of these parameters on composite behaviour is well summarized by Hartwig ~'~2. Fibreglass possesses isotropic mechanical and thermal behaviour which results in high strength in all directions. Good f i b r e - m a t r i x interfacial bonding is anticipated when using a glass f i b r e - m a t r i x composite at low temperature, since glass fibre contracts less than polymers. Carbon fibres are both mechanically and thermally anisotropic which results in a low transverse strength and stiffness compared to that in the longitudinal direction. Anisotropy is more pronounced in
Ltd
Cryogenics 1991 Vol 31 June
399
Simulation of tensile strength: H.H. Abdelmohsen Kevlar fibres, which consist of stretched aramide molecules with strong covalent bonding in the fibre direction only. This produces very low transverse shear strengths and stiffness. Since Kevlar fibres contract more than polymers in the transverse direction, weak bonding and adhesion occurs between the fibres and the polymers. An elastic solution is utilized by Vedula et al. to obtain thermal residual stresses for composites with anisotropic fibres 13'14. However, their analysis is restricted to fibres with isotropic elastic behaviour in all directions. A more general solution is given by Avery and Herakovich ~5 where the fibres possess anisotropic elastic and thermal properties. The present paper is intended to explore analytically and numerically the relationship between fibre anisotropy and the tensile failure strength of unidirectional fibre-reinforced composites at low temperatures.
each fibre by itself. Equation (2) takes a slightly different form if it is applied to the first or to the last fibre in the bundle.
Analysis of thermal stresses For a long fibre in an isotropic elastic matrix, as shown in Figure 1, the governing differential equation under a condition of plane strain with uniform axial strain, cx, has the following form 15
Cr r [[ 02w Or 2 + ~1f0f w rrl-Coo
-
w r: -
1 - -
r
+-
(Co,
1 r
-
Cx)
,
(crj - coj)
Ar
(4)
Micromechanic model The model is based on two basic assumptions: the fibre strength is described statistically by the two parameter Weibull distribution and the shear lag equation governs the fibre displacement field. While the matrix does n o t contribute directly to the composite strength, it does provide a means to transfer the load in shear to the fibres. The first assumption states that, if F(o*) is the probability that the fibre strength is _<0", then
Equation (4) is based on the assumption that both stresses and strains are independent of 0 and that no shear-extension coupling exists. C0 are stiffness coefficients, % is the coefficient of thermal expansion in the j direction and AT is the uniform temperature change. Cij values are functions of the fibre elastic constants 16 u, v and w are the axial, hoop and radial displacements, respectively. The general solution to Equation (4) is w f ( r ) = Air xl + A f r x2 + Glexrln r + GzATrln r
F(o*) = 1 - exp [ - (~)~,Sac]
(5a)
(1) for a transversely isotropic fibre and
where: 71and/3 are Weibull shape and scale parameters, respectively; and Ax is the fibre segment size. The second assumption considers the unidirectional composite lamina as a thin sheet consisting of m fibres spaced uniformly parallel to the x axis. If the lamina is loaded in the x direction, the force equilibrium equation or the shear lag equation in a non-dimensional form is --
(5b)
w f ( r ) = A~r ×1 + Afr x2 + Hl¢x r + H2ATr
+ (ui-i - 2ui + ui+O = 0
do 2
where p = [EfAfS/Gmhm] l/2x
(2)
where: u~ is the displacement of the ith fibre; EfAf is the fibre tensile stiffness; Gmh m is the matrix shear stiffness; and S is the spacing between fibres. S is assumed to be constant and uniform. For a composite pulled out in simple tension, the boundary conditions are
r(w)
ui(O) = 0 du i
dp
-
oJE,
(3)
where oc is the applied stress on the lamina. Equation (2) guarantees the equilibrium of the whole composite lamina, whereas Equation (3) ensures the equilibrium of
400
Cryogenics 1991 Vol 31 June
Matrix Fibre Figure 1 Composite geometry
Simulation of tensile strength: H.H. Abdelmohsen for a transversely orthotropic fibre, where Gl-
H, -
Cox - Crx 2Coo
,
G2-
(Cox - Crx )
( Cri
H2 -
( C r - Coo)'
-
-
Coi)O[i
2Coo ( Cri - Coi)OLi
(6)
(C~r -- Coo)
and /g"*
\1/2
The repeated indices used in Equations (4) and (6) indicate summation over x, r and 0. For the isotropic matrix the solution has the form
wm(r)
=
(7)
A["r + A~' F
f and m in Equations (5) and (7) denote the fibre and matrix, respectively. Equations (5) and (7) contain five constants to be evaluated using the following boundary conditions: 1 fibre radial displacement is bounded at r = 0; 2 continuity of radial displacement at the f i b r e - matrix interface; 3 continuity of radial stress at the fibre-matrix interface; 4 traction free at the outer boundary of the matrix; and 5 traction free at the composite boundary for pure thermal loading. Once the coefficients that appear in Equations (5) and (7) are known, stresses could be obtained by differentiating the expression for radial displacement. Details of the derivation are shown in Appendix A.
Solution scheme The composite lamina is divided into m x n mesh points, where m is the total number of fibres and n is the total number of mesh points in the x direction (i.e. the number of bundles). The segment size, 2tx (fibre length/n), also known as the ineffective length t°,
Table 1
Numerical examples The analysis outlined in the previous three sections is applied here to predict the tensile failure strength of glass-, graphite- and Kevlar-epoxy composites at room and liquid helium temperatures. Numerical values of fibre and matrix mechanical and thermal properties used in the simulation are listed in Table 1. Fibre Weibull parameters were reported by Fariborz ~'~°. To assess the convergence of the numerical scheme, different numbers of fibres and mesh points (number of bundles ~7~8) are tested until convergence occurs at
Mechanical and thermal constants
Elastic modulus, E~ (GPa) Elastic modulus, ET ~ (GPa) Poisson's ratio, PLTb Poisson's ratio, PTLb Coefficient of thermal expansion, ~L b (10 Coefficient of thermal expansion, ozTb (10 /3 (GPa) Of (/~m)
depends on the mechanical properties of the fibre and the matrix, the size of the composite and the geometry of the specimen. Fariborz 9'~° suggested a conservative value of 6 - 10 fibre diameters. For the m x n mesh points, a random number for a*, corresponding to each segment strength, is generated using the distribution defined in Equation (1). The load is applied in increments, Aa, starting with an initial value, Oo, and Equation (2) is solved with the boundary conditions given in Equation (3) using the successive over-relaxation method, as described in Appendix B. Once the solution of Equation (2) is known at each load increment, the stress in each fibre segment, o = Er dui4/dP (where i = 1, 2 . . . . . m and j = 1, 2, . . . . n), is checked against its strength, o*. Two cases exist; either a* _> o or a* < ~. The first case indicates that this segment is in a state of constant stress and the displacement is a single valued function. On the other hand, the second case implies that the fibre has been broken at this segment and a multivalued displacement function exists in this segment. As long as the first case prevails, Equation (2) holds. If the first case is violated and the second case exists, Equation (2) has to be modified to allow the displacement to assume multivalues. Appendix B presents the solution to Equation (2) in a finite difference form for the two cases presented above. For each increment of loading, the broken fibres and the locations of such breaks are defined. The composite has failed if one cleavage of breaks is formed. The existence of one cleavage causes the solution of Equation (2) to diverge since the boundary conditions on both sides of the broken region (at the position of the cleavage and at x = lamina length) are of the Neumann type. The solution scheme is depicted in Appendix C and numerical results are given in the next section.
6/K) 6/K)
E-Glass
Graphite
Kevlar
Matrix
72.5 72.5 0.22 0.22 4.8 4.8 8.2 1.96 8.0
240.0 32.0 a 0.30 a 0.04 a 1.1 48 a 7.0 2.30 8.0
131.0 4.13 0.35 0.01a 2 48 a 8.2 1.96 11.9
2.16 2.16 0.40 0.40 48 48 -
Assumed values, based on analysis given by Hartwig TM 12 bL and T denote the longitudinal and transverse direction, respectively
Cryogenics 1991 Vol 31 June
401
Simulation of tensile strength: H.H. Abdelmohsen Table 2 Comparison between simulated composite failure strength and experimental results (in parentheses) at room and low temperatures
Room temperature Liquid helium temperature
E-Glass-epoxy
Graphite-epoxy
Kevlar-epoxy
0.93 (1.05) 1.20 (1.25)
0.91 (0.84) 0.79 (0.80)
1.24 (1.18) 1.18 (1.15)
80 x 80 mesh points. Thermal stresses developed in the composite constituents due to cooling are evaluated using the analysis given above. Due to cooling, glass fibres exhibit longitudinal compressive stress of 0.46 GPa. Graphite and Kevlar fibres develop the same type of stresses but an order of magnitude lower. In all three composites, tensile stress exists in the matrix longitudinal direction (x in Figure 1). These stresses create thermal residual strain in the matrix equal to 0.5% for a glass fibre type composite and 0.3% for graphite fibre and Kevlar fibre type composites. Compressive stresses developed in the fibre due to cooling are incorporated into the fibre failure strength distribution by adding their values to the fibre scale parameter. This causes the fibre strength distribution to shift by such a stress value on Weibull graph paper 17. The fibre shape parameter is assumed to be temperature independent. Table 2 shows the simulated tensile failure strengths compared with results from the experiments ~9'2° on the three composites at room and low temperatures. Results are based on a fibre volume fraction equal to 50% and 20 computer simulations. As shown there is remarkable agreement between the model results and the experimental data. The argument given in the introduction is strongly supported by the results reported in Table 2. Fibre anisotropic mechanical and thermal properties control thermal stresses generated in the composite components due to cooling. Basically, fibres will exhibit compressive stresses and the matrix will be subjected to tensile stresses in the longitudinal direction. The resulting fibre compressive stresses tend to increase the tensile strength and this could be reflected by the increase in the scale parameter value. Matrix tensile thermal stresses reduce the matrix free strain ~2.
2.0
0.0 -1,0
-
-2.0 -
t-
.J
-3.0 -
R"
-4.0 -5.0 -0.40
I
_5 I i-
/
-1.0
-30
-4.0 -5.0 -0.6
I -0.4
i
I
-0.2 Ln 5"
0.0
0.2
Figure 3 Weibull distribution of graphite-epoxy composite. PR = probability of failure, S = failure strength
2.0 1.0 L
"
"
•
'
i--1
0.0 I -~
-1.0
I
~-
-2.0
d
-5.0 -4.0 0.0
I 0.1
I 0.2 Ln 5"
I 0.3
Figure 4 Weibull distribution of Kevlar-epoxy PR = probability of failure, S = failure strength
0.4 composite.
L I
I --0.20
I
I 0.00 Ln 5"
I
I 0.20
Figure 2 Weibull distribution of glass-epoxy PR = probability of failure, S = failure strength
402
20// 0.0
However, this has no effect on the composite tensile strength since the matrix is assumed to be merely a means to transfer loads in shear to the fibres and not to contribute to composite strength ~7. Figures 2, 3 and 4 depict tensile strength distribution for the three composites at room and liquid helium temperatures. As shown, Weibull distribution describes the tensile strength of glass- and graphite-epoxy composites fairly well, while there is poor agreement for the Kevlar-epoxy composite.
1.0
I c. I
2.0
Cryogenics
1991
Vol
31 June
[
0.40 composite.
Conclusions
This article presents a simulation scheme to predict the effect of fibre anisotropy on composite tensile failure strength at room and low temperatures. Results show
Simulation of tensile strength: H.H. Abdelmohsen that composite tensile strength tends to improve upon cooling for composites with isotropic fibre properties (mechanical and thermal). Composites with anisotropic fibres may lose part of their strength due to cooling. Simulated results for E-glass-, graphite- and Kevlarepoxy type composites agree well with expeirmental data.
References 1
2
3 4 5 6
7
8 9
10
11 12
13
14
15
16 17
18
19
20 21
Jech, R.W., McDaniles, D.L. and Weston, J.W. Fiber reinforced metallic composites Proc 6th Sagamore Ordnance Materials Research Conference (1959) Rosen, R.W. Mechanics of Composite Strengthening: Fiber Composite Materials American Society of Metals, Metals Park, Ohio (1965) Zweben, C. Tensile failure analysis of fibrous composites AIAA J (1968) 6 2325 Coleman, B.D. A stochastic process model for mechanical breakdown Trans Soc Rheology (1957) 1 153 Gucer, D. and Gurland, J. Comparison of the statistics of the two fracture modes J Mech Phys Sol (1962) 10 365 Zweben, C. and Rosen, B.W. A statistical theory of material strength with application to composite materials, AIAA Paper No 69-123, presented at AIAA 7th Aerospace Sciences Meeting, New York, USA (1969) McKee, R.W. and Sines, G. A statistical model for the tensile fracture of parallel fiber composites, ASME Paper No. 68-WAIRP-7, presented at ASME Winter Annual Meeting, New York, USA (1968) Oh, K.P. A Monte Carlo study of the strength of unidirectional fiberreinforced composites J Composite Mat (1979) 13 31 l Fariborz, S.J., Tang, C.L. and Harlow, D.G. The tensile behavior of intraply hybrid composites I: Model and simulation J Composite Mat (1985) 19 334 Fariborz, S.J. and Harlow, D.G. The tensile'behavior of interply hybrid composites II: Micromecbanical model J Composite Mat (1987) 21 856 Hartwig, G. and Knaak, S. Fiber-epoxy composites at low temperatures Cryogenics (1984) 24 639 Hartwig, G. Low temperature ductile matrices for advanced fiber composites, in: Mechanics of Composite Materials (Eds Dvorak, G.J. and Laws, N ) Society of Mechanical Engineers (1988) Vedula, M., Pangborn, R.N. and Queeney, R.A. Fiber anisotropic thermal expansion and residual thermal stress in a graphite/aluminium composite Composites (1988) 19 55 Vednla, M., Pangborn, R.N. and Qneeney, R.A. Modification of residual thermal stress in a metal-matrix composite with the use of a tailored interfacial region Composites (1988) 19 133 Avery, W.B. and Herakovlch, C.T. Effect of fiber anisotropy on thermal stresses in fibrous composites J Appl Mechanics (1986) 53 751 Jones, R.M. Mechanics of Composite Materials McGraw-Hill (1975) Abdelmohsen, H.H. Prediction of tensile strength of unidirectional fiber reinforced composites at low temperature, paper presented at ICMC, Los Angeles, California, USA (1989) Abdelmohsen, H.H. Effect of fiber anisotropy on tensile strength of unidirectional fiber reinforced composite at low temperature, paper presented at ICMC, Los Angeles, California, USA (1989) Kasen, M.B. Mechanical and thermal properties of filamentaryreinforced structural composites at cryogenic temperatures 2: Advanced composites Cryogenics (1975) 24 701 Kasen, M.B. Composites. in: Materials at Low Temperature (Eds Reed, R P and Clark. A F ) American Society of Metals (1983) Ferziger, J.H. Numerical Methods fi~r Engineering Application John Wiley (1981)
w'(r)=A Ir+
-+Biqrlnr+B2ATrlnr r
(AI)
win(r) = A3r r
Using Equation (A1) the stresses can be expressed as
Oif = A l ( C l ; 4- C~r ) 4- A2(Cl; -- e l ; )
1 r~
!
f
f
+ C,.~q - C o a j A T
o m = A 3 ( C ~ + Cimr) --~ A 4 ( C ~ -- Cim)
1 F-
+ Ci"~e, - C~'ot]nAT
(A2)
where i and j are indices that refer to x, r or 0 where x, r and 0 are the longitudinal, the radial and the tangential directions, respectively, r = 0 is located on the fibre longitudinal axis. Repeated indices indicate summation over the repeated index, f and m denote the fibre and the matrix, respectively. In Equations (A1) and (A2), the A terms are coefficients to be determined from the boundary conditions, e~ is the strain in the x direction, AT is the temperature drop, the C terms are fibre and matrix stiffness coefficients which are related to their mechanical properties ~6, the B terms are functions of the C terms and the thermal properties, and the c~ terms are the coefficients of thermal expansion. Satisfying the boundary conditions at the f i b r e - m a t r i x interface for pure thermal loadings and using the fact that the solution has to be bounded everywhere, the following system of equations is formed to obtain the unknown coefficients and c~ [A] Ix] = [B]
(A3)
where the elements of each matrix are A(1, 1) = a A(1, 2 ) = - a A(1, 3) = - 1 / a
A(1, 4) =
Blaln(a )
A(2, 1) = C~o + C~r A(2, 2) = - ( C ~
+ C~lr)
A(2, 3) = - ( C ~ - Crr)/a" m
A(2, 4) = C~, - C~m A(3, 1) = 0 A(3, 2) = C~ + Crn) A(3, 3) = ( C ~
-
Cnr'r)/b 2
A(3, 4) = C~',
Appendix A For fibres with transversely isotropic mechanical and thermal properties, embedded in an isotropic elastic matrix, the fibre and the matrix displacements in the r direction are given byl5
A(4, 1) = (C~o + C f r ) a : / 2 A(4. 2) = C.r(b m 2 - aZ)/2 A ( 4 , 3) = 0 A(4, 4) = [C",',(b 2 - a z) + C ,f~ a - ] / 2
Cryogenics 1991 Vol 31 June
403
Simulation of tensile strength: H.H. Abdelmohsen B(1) = - B 2 A T a In(a)
where o~ is the relaxation factor with a value between 1 and 2 (Reference 21). The use of the Christopherson method, as demonstrated by Oh s, allows Equation (B1) for a broken fibre to have the following two forms, if evaluated to the right or to the left of the break, respectively
B(2) = C[; oLfA T - C~ o~'~A T a(3) = C~c~TAT B(4) = [Cxfjot~a2 + C~ot]~(b 2 - a b ] A T / 2 x(1)=A~ x(2) = a 3
u(i, j ) =
x(3) = A 4 x(4) =ex Finally, Bl = (Cox - C,-,)/Coo/2 and B 2 = (Cri - Coi)Oti/ Coo~2.2a = Df is the fibre diameter and 2b - 2a = S is the spacing between fibres. Once the unknown coefficients are evaluated by solving Equation (A3), the displacements and the stresses are determined from Equations (AI) and (A2).
o~(3Ap2[u(i -- 1, j ) + u(i + 1, j ) ] + 4u(i, j + 1)) 2(3Ap2 + 2)
J
+(1
-
o~)u(i, j)
(B2)
u(i, j ) =
Appendix B Equation (1) in a finite difference form using the successive over-relaxation algorithim is 21
wf3ap2[u(i -
1, j ) + u(i + 1, j ) ] - 4u(i, j - 1))
--25 +(1 - w)u(i, j )
u(i, j ) = o~{([u(i - 1, j ) + u(i + 1, j ) ] A p 2
3 (B3)
+ u(i, j - 1) +u(i, j + 1))/2(1 + ApZ)} +(1
404
-
oo)u(i, j)
Cryogenics 1991 Vol 31 June
(B1)
The above equations have different forms for the first and the last fibre in a bundle.