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Micromechanical strength formulae of unidirectional composites
August 1999
Materials Letters 40 Ž1999. 164–169 www.elsevier.comrlocatermatlet
Micromechanical strength formulae of unidirectional composites Zheng Ming Huang
)
Department of Mechanical and Production Engineering, National UniÕersity of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 27 November 1998; received in revised form 12 March 1999; accepted 15 March 1999
Abstract When isotropic fibers are unidirectionally arranged in another isotropic matrix material, the resulting composite has different mechanical properties from those of both the fibers and the matrix. Its elastic stiffness can be accurately estimated with some simple micromechanical formulae such as rule of mixture formulae, based on the properties of the constituent fiber and matrix materials as well as fiber volume fraction only. However, little is known about the ultimate strengths of the composite. This letter presents a set of concise micromechanical formulae for the ultimate tensile strengths of the unidirectional composite under individual uniaxial loads. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Micromechanics; Unidirectional composite; Failure criterion; Strength formulae; Stress state
1. Introduction Micromechanics has found much success in prediction of the stiffnessrcompliance of unidirectional fibrous composites w1x. On the other hand, little is known about the ultimate tensile strength, another important mechanical property, of the composite materials. Most strength theories for fibrous composites have been developed based on macromechanical considerations similar to those for isotropic materials, including the maximum stressrstrain criteria and the Tsai–Wu theory w2,3x. Unlike in isotropic cases, extensive experiments including biaxial tests, which may be difficult or expensive to conduct in some cases, have to be performed to determine the strength parameters involved in these macromechanical theories whenever they are applied to any composite. Even with the same constituent materials, different )
Corresponding author
composite having different fiber volume fraction still requires similar tests. Another drawback with these theories is that it cannot predict the failure mode of the composite. They are unable to indicate which of the constituent phases initiates the failure of the composite and meanwhile how rich the strength of the other phase is. Several micromechanics based attempts to the ultimate strength of fibrous composites have been made w4–6x. However, they were focused mainly on the microstructural failure mechanisms and were developed by pre-assuming a priori deformation modes. The overall failure of a composite was not treated in terms of the average responses of its constituent phases. The purpose of this paper is to present a set of simple micromechanical formulae for the ultimate tensile strengths of the unidirectional fibrous composites under individual uniaxial loads. The failure prediction of the composites is based on understanding for the stress states in the constituent materials
00167-577Xr99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 5 7 7 X Ž 9 9 . 0 0 0 6 9 - 5
Z.M. Huangr Materials Letters 40 (1999) 164–169
165
without any pre-assumed deformation mode. Only the material properties of the constituent fiber and matrix as well as fiber volume fraction are required to apply these formulae.
2. Elastic stress states Suppose that both the fiber and matrix are isotropic materials. Let us use E, n , and G to represent the Young’s Želastic. modulus, Poisson’s ratio, and shear modulus of a constituent material, respectively. Let w Si j x to be the compliance matrix of the material. Suffixes ‘f’ and ‘m’ will refer to the fiber and matrix, whereas a quantity without any suffix will refer to the composite. Thus, Vf denotes the fiber volume fraction in the composite, E f represents the Young’s modulus of the fiber, w Simj x is the compliance matrix of the matrix material, etc. It has been shown in Ref. w7x that the elastic stress state in the matrix material can be correlated with that in the fiber by a bridging matrix through
si m 4 s w a i j x sj f 4
Ž 1.
where si m 4 s s 11m , s 22m , s 12m 4T, si 4 s s 11f , s 22f , s 12f 4T, and a11
a12 a22 0
w ai j x s 0
0 0 a 33
Fig. 1. Predicted and measured Žw10x. transverse modulus of a glassrepoxy composite Ž Ef s 73.1 GPa, Em s 3.45 GPa, n f s 0.22, and nm s 0.35..
which are sufficiently accurate w1x. Eq. Ž3.3. is the result of a precise elastic solution for the overall in-plane shear modulus Ž G 12 . of the composite w8x. The accuracy of Eq. Ž3.2. can be seen from a comparison between predicted and experimental results for the transverse modulus of a composite shown in Fig. 1. Let us then consider an arbitrary overall stress state, Ž s 11 , s 22 , s 12 ., applied to the composite. From the micromechanical relationship, si 4 s Vf si f 4 q Ž1 q Vf . si m 4 , and by virtue of Eq. Ž1., the stress shares in the fiber and matrix can be explicitly determined. They are given as:
Ž 2.
s 11f s a e1f s 11 q a e4f s 22 , s 22f s a e2f s 22 , s 12f s a e3f s 12 , Ž 4.
The non-zero elements in Eq. Ž2. are defined as w7x
s 11m s a e1m s 11 q a e4m s 22 , s 22m s a e2m s 22 , s 12m s a e3m s 12 , Ž 5.
0
a11 s E mrE f , a 22 s a 33 s
1 2 1 2
ž ž
1q 1q
Ž 3.1 . m
E
E G
f
m
Gf
/ /
,
Ž 3.2 .
a e1f s ,
Ž 3.3 . a e2f s
and a12 s
where
f m S12 y S12 f m S11 y S11
Ž a11 y a22 . .
Ž 3.4 .
Using Eqs. Ž3.1. and Ž3.4., the resulting overall longitudinal Young’s modulus and Poisson’s ratio Ž E11 and n 12 . of the composite are exactly the same as those given by the rule of mixture formulae w7x,
a e3f s f a e4 s
Ef Vf E f q Ž 1 y Vf . E m
,
Ef Vf E f q 0.5 Ž 1 y Vf . Ž E m q E f . Gf Vf G f q 0.5 Ž 1 y Vf . Ž G m q G f .
Ž 6.1 . ,
Ž 6.2 .
,
Ž 6.3 .
Ž 1yVf . E f Ž n f E m y n m E f .
w Vf E
f
q Ž 1yVf . E m x w 2Vf E f q Ž 1yVf . Ž E f q E m . x
,
Ž 6.4 .
Z.M. Huangr Materials Letters 40 (1999) 164–169
166
a e1m s a e2m s a e3m s
Em f
Vf E q Ž 1 y Vf . E
m
,
0.5 Ž E f q E m . Vf E f q 0.5 Ž 1 y Vf . Ž E m q E f . 0.5 Ž G f q G m . Vf G f q 0.5 Ž 1 y Vf . Ž G m q G f .
m a e4 sy
Ž 7.1 . ,
Ž 7.2 .
,
Ž 7.3 .
Vf E f Ž n f E m y n m E f .
w Vf E
f
q Ž 1yVf . E m x w 2Vf E f q Ž 1yVf . Ž E f q E m . x
.
Ž 7.4 .
3. Strength formulae Most matrix materials are capable of undergoing significant inelastic deformation before failure. To tailor the strength problem, the plastic deformation of the constituent materials must be taken into account. For simplicity, let us assume that the fibers are elastic until rupture and the matrix is a bilinear elastic–plastic material. Let E T , s Y , and su to denote the hardeningmodulus Žtangential modulus in plastic region., yield stress, and ultimate tensile strength of the material, respectively. These parameters are obtainable through a uniaxial tensile test. The compliance matrix of the matrix material, w Simj x, will depend on current load level. As the bridging matrix, w a i j x, correlates the stress states generated in the constituent fiber and matrix materials, its elements can only depend on the material properties as well as on packing geometries of the constituents. When any constituent material deforms from an elastic region to a plastic one, the packing geometries Ži.e., fiber volume fraction, relative arrangement of fibers in the matrix, fiber cross-sectional shape, etc.. do not change or only vary by negligibly small amount. Therefore, only the elastic moduli in Eqs. Ž3.1., Ž3.2. and Ž3.3., which define the independent elements of the bridging matrix, need to be changed when the matrix undergoes plastic deformation. Namely, E m and G m in Eqs. Ž3.1., Ž3.2. and Ž3.3. should be replaced by ETm and G Tm Žs E Tm r3.. On the other hand, the upper triangular part of the bridging matrix, w a i j x given by Eq. Ž2., will be fully occupied and the off-diagonal elements will be deter-
mined using the symmetric condition of the resulting overall compliance matrix of the composite, i.e., S ji s S i j, where w Si j x s Ž Vf w Sif j x q Vm w Simj xw a i j x. = Ž Vf w I x q Vm w a i j x.y1 Žwhich is obtained by using following relations: d si 4 s Vf d si f 4 q Vm d s im 4 , d ´ i 4 s Vf d ´ if 4 q Vm d ´ im 4 , d ´ im 4 s w Sif j x d sj f 4 , d ´ im 4 s w Simj x d sj m 4 , d ´ i 4 s w Si j x d sj 4 and Eq. Ž1.x. The stress calculation has to be carried out incrementally. If, however, the overall applied stress is only uniaxial, we can easily generalize Eqs. Ž4. and Ž5. to determine the total stresses generated in the fiber and matrix. Let us focus on an ultimate tensile strength of the composite under a uniaxial load only. In this letter, the ultimate tensile strength of the composite is defined as the overall applied stress under which the maximum normal stress Ži.e., the first principal stress, s I . of a constituent material attains its ultimate value, su . Based on this definition, the ultimate strength of the composite under longitudinal tension Ž s 11 ., transverse tension Ž s 22 ., or in-plane shear Ž s 12 ., respectively, is obtained as follows. Ža. Longitudinal tensile strength due to a longitudinal tensile load Ž s 11 . only u a 11 s min
with
½
f f suf y Ž a e1 y a p1 . s110 , sum y Ž ae1m y a p1m . a 110 m f a p1 a p1
s 110 s min
½
s Ym suf , , a e1m a e1f
5
5
Ž 8.1 .
Žb. Transverse tensile strength due to a transverse tensile load Ž s 22 . only u a 22 s min
½
f f suf y Ž a e2 y a p2 . s 220 , sum y Ž ae2m y a p2m . a 220 m f a p2 a p2
with s 220 s min
½
s Ym suf , , a e2m a e2f
5
5
Ž 8.2 .
Žc. In-plane shear strength due to an in-plane shear load Ž s 12 . only u a 12 s min
½
f f suf y Ž a e3 y a p3 . s120 , sum y Ž ae3m y a p3m . a 120 m f a p3 a p3
with s 120 s min
½
s Ym
'3 ae3m
,
suf a e3f
5
,
5
Ž 8.3 .
Z.M. Huangr Materials Letters 40 (1999) 164–169
ponents in the fiber and matrix, these components are believed to be small compared with their transverse counterparts and have been ignored in deriving Eq. Ž8.2.. Let us apply the above strength formulae to several unidirectional fibrous laminae. The first lamina is a unidirectional SiC-fiber and titanium ŽTi. matrix composite w9x. The SiC-fiber is an isotropic elastic material until rupture, with E f s 400 GPa and n f s 0.25. The measured ultimate tensile strength of the extracted fiber specimens, however, varied from 2520 MPa to 4540 MPa. In this study, a fiber ultimate strength of suf s 3480 MPa, which was measured using fiber samples extracted from a composite panel whose tensile stress–strain curve was plotted in Fig. 6 of Ref. w9x, is used. The titanium matrix is a bilinear elastic–plastic material Žsee Fig. 2 of Ref. w9x., having properties: E m s 110 GPa, n m s 0.33, ETm s 2.16 GPa, s Ym s 850 MPa, and sum s 1000 MPa. Substituting the fiber and matrix properties into Eq. Ž8.1., the calculated longitudinal tensile strengths vs. fiber volume fractions are plotted in Fig. 2. The experimental data taken from Table 5 of Ref. w9x are also shown in the figure. Correlation between the predicted and measured results is very good. The other laminae to be studied are continuous alumina fiber reinforced pure aluminium matrix composites, which were recently investigated by Bushby w11x. The alumina fibers, Altex and Nextel 610, are considered as isotropic linear elastic until rupture and the pure aluminium as isotropic elastic– plastic. The fiber properties, based on manufacturers’ data given in Ref. w11x, are listed in Table 1. However, except for Young’s modulus and Poisson’s ratio, the other matrix parameters are not available from Ref. w11x. They are taken from Ref. w12x. With
Fig. 2. Predicted and measured Žw5x. longitudinal tensile strength of SiCrTi composite vs. fiber volume fraction. The used material properties are: E f s 400 GPa, n f s 0.25, suf s 3480 MPa, E m s 110 GPa, n m s 0.33, E Tm s1.62 GPa, s Ym s850 MPa, and sum s1000 MPa.
In Eqs. Ž8.1., Ž8.2. and Ž8.3., f a p1 s
m a p1 s
f a p2 s
m a p2 s
f a p3 s
m a p3 s
Ef Vf E f q Ž 1 y Vf . E Tm ETm Vf E f q Ž 1 y Vf . E Tm
,
Ž 9.1 .
,
Ž 9.2 .
Ef Vf E f q 0.5 Ž 1 y Vf . Ž E Tm q E f . 0.5 Ž E f q E Tm . Vf E f q 0.5 Ž 1 y Vf . Ž E Tm q E f .
,
Ž 9.3 .
,
Ž 9.4 .
3G f 3Vf G f q 0.5 Ž 1 y Vf . Ž E Tm " 3G f . 0.5 Ž 3G f q ETm . 3Vf G f q 0.5 Ž 1 y Vf . Ž E Tm q 3G f .
,
Ž 9.5 .
.
Ž 9.6 .
167
It should be noted that although a transversely applied load also generates longitudinal stress com-
Table 1 Constituent properties Altex w11x Nextel 610 w11x Matrix w11,12x
E ŽGPa.
n
E T ŽMPa.
s Y ŽMPa.
su ŽMPa.
Reduction by cold rolling Ž%.
210 373 70 70 70 70
0.2 0.2 0.3 0.3 0.3 0.3
– – 70 5 13.4 46
– – 10 57 75 91
1800 2500 45 59 77 96
– – 0 10 20 40
Z.M. Huangr Materials Letters 40 (1999) 164–169
168
Table 2 Measured uniaxial tensile behavior of alumina fiberraluminium matrix laminae Fiber used
Altex Vf s 0.4 Altex Vf s 0.6 Nextel 610 Vf s 0.7
Ultimate strength ŽMPa.
Ultimate strainŽ%.
Longitudinal
Transverse
Longitudinal
Transverse
761 " 61 930 " 120 1254 " 48
84 " 25 109 " 10 138 " 12
0.85 " 0.06 0.69 " 0.10 0.50 " 0.04
0.50 " 0.1 0.33 " 0.05 0.28 " 0.08
increased ratio of reduction by cold rolling, the yield and tensile strengths of the pure aluminium elevate. Hence, four different groups of the matrix parameters, corresponding to different reduction ratios, have been tried. The used matrix properties are given in Table 1. Measured w11x longitudinal and transverse failure strengths and strains of three composites with fiber volume fractions of 0.4 and 0.6 for the Altex composites and 0.7 for the Nextel composite are summarised in Table 2. Using the constituent parameters given in Table 1, the composite longitudinal and transverse tensile strengths have been calculated from Eqs. Ž8.1. and Ž8.2., whereas the ultimate strains of the composites can be obtained based on d ´ i 4 s Vf w Sif j x d si f 4 q Vm w Simj x d si m 4 Žnote: the matrix compliance matrix w Simj x after yielding is speci-
fied using the Plandtl–Reuss theory w13x.. Results are shown in Table 3. The calculated longitudinal tensile strengths are all accurate for the Altex composites regardless of any group of matrix parameters used. This is because the longitudinal strength of the composite is controlled by the fiber strength. For the Nextel composite, correlation between the predicted and measured longitudinal strengths is less satisfactory if the fiber tensile strength, 2500 MPa, provided by the manufacturer, is used. However, the correlation can be much improved if the Altex fiber strength, 1800 MPa, is also taken as the Nextel fiber strength, as indicated in Table 3. Whereas the longitudinal tensile strength of the composite is controlled by the fiber strength, the transverse failure strength of the composite is seen to
Table 3 Predicted ultimate values of the composites upon different parameters of matrix plasticity Fiber used
Altex Vf s 0.4 Ž suf s 1800 MPa.
Altex Vf s 0.6 Ž suf s 1800 MPa.
Nextel Vf s 0.7 Ž suf s 1800 MPa.
Nextel Vf s 0.7 Ž s Mf s 2500 MPa.
Matrix properties
Failure strength
Failure strain
sum ŽMPa.
s Ym ŽMPa.
E Tm ŽMPa.
Longitudinal
Transverse
Longitudinal
Transverse
45 59 77 96 45 59 77 96 45 59 77 96 45 59 77 96
10 57 75 91 10 57 75 91 10 57 75 91 10 57 75 91
70 5 13.4 46 70 5 13.4 46 70 5 13.4 46 70 5 13.4 46
726.5 754.3 765.2 774.2 1084.4 1103.0 1110.2 1116.6 1263.3 1277.4 1282.8 1287.6 1753.2 1767.3 1772.7 1777.5
60.9 70.8 92.4 115.2 68.8 76.7 100.1 124.8 74.1 87.3 113.9 142.0 74.1 87.3 113.9 142.0
0.8573 0.8572 0.8573 0.8572 0.8573 0.8573 0.8575 0.8572 0.4827 0.4827 0.4827 0.4827 0.6703 0.6703 0.6703 0.6703
18.803 0.0661 0.0862 0.1075 11.828 0.0579 0.0756 0.0942 9.7273 0.0426 0.0556 0.0693 9.7273 0.0426 0.0556 0.0693
Z.M. Huangr Materials Letters 40 (1999) 164–169
depend heavily on the matrix strength used. Compared with the experimental evidences, it is evident that the matrix tensile strength must be considered as higher than 45 MPa, because the predicted ultimate strains of the composites based upon this strength are too higher than the measured data. Using the other three groups of the matrix properties, all the predicted transverse tensile strengths of the composites agree well with the experimental data. Amongst, the predicted results using the matrix properties having 20% reduction by cold rolling are in closest agreement with the measured strengths. It is to be noted that the actual tensile strengths of the in situ constituent materials are somewhat different from those specified by the manufacturers in general. They should be obtained using the in situ constituent materials under same condition as in fabricating the composites.
4. Conclusion Simple formulae, as concise as rule of mixture formulae for composite stiffness, are presented in this paper to calculate the ultimate tensile strengths of a unidirectional fibrous lamina subjected to various uniaxial load conditions. The formulae only use the gross uniaxial tensile parameters of the constituent fiber and matrix materials together with fiber volume fraction as input data. These parameters are easily obtainable through individual tests. The proposed formulae have been applied to calculate the ultimate tensile strengths of several unidirectional fibrous composites. The calculated results are in good agreement with the available experimental data.
169
References w1x R.L. McCullough, Micro-models for composite materialscontinuous fiber composites, in: J.M. Whitney, R.L. McCullough ŽEds.., Micromechanical Materials Modeling, Delaware Composites Design Encyclopedia, Vol. 2, Technomic Publishing, Lancaster, 1990. w2x P. Labossiere, K.W. Neal, Macroscopic failure criteria for fiber-reinforced composite materials, Solid Mech. Its Arch. 12 Ž1987. 439–450. w3x J. Echaabi, F. Trochu, R. Gauvin, Review of failure criteria of fibrous composite materials, Polym. Compos. 17 Ž6. Ž1996. 786–798. w4x A.M. Skudra, Micromechanics of failure of reinforced plastics, in: G.C. Sih, A.M. Skudra ŽEds.., Failure Mechanics of Composites, North-Holland, Amsterdam, 1985, pp. 1–69. w5x W.A. Curtin, Ultimate strengths of fiber-reinforced ceramics and metals, Composites 24 Ž2. Ž1993. 98–102. w6x M.J. Pindera, Micromechanical aspects of yielding and failure criteria of composites, in: J.P. Boehler ŽEd.., Failure Criteria of Structured Media, Balkema, Rotterdam, Netherlands, 1993, pp. 121–141. w7x Z.M. Huang et al., A New Modeling Procedure for Estimating the Elastic Moduli of Knitted Fabric Composites, Proc. 5th Japan Int. SAMPE Symposium, Tokyo, 1997, pp. 719– 724. w8x M.W. Hyer, Stress Analysis of Fiber-Reinforced Composite Materials, WCB, McGraw-Hill, Boston, 1997, pp. 120–124. w9x D.B. Gundel, F.E. Wawner, Experimental and theoretical assessment of the longitudinal tensile strength of unidirectional SiC-fiberrtitanium-matrix composites, Comp. Sci. Technol. 57 Ž1997. 471–481. w10x S.W. Tsai, H.T. Hahn, Introduction to Composite Materials, Technomic Publishing, Lancaster, Basel, 1980, pp. 377–431. w11x R.S. Bushby, Evaluation of continuous alumina fiber reinforced composites based upon pure aluminium, Mater. Sci. Technol. 14 Ž1998. 877–886. w12x ASM Handbook, ASM International, The Materials Information Society, 2, 1990, p. 65. w13x D.F. Adams, Elastoplastic behavior of composites, in: G.P. Sendeckyj ŽEd.., Mechanics of Composite Materials, Academic Press, New York, 1974, pp. 169–208.
Materials Letters 40 Ž1999. 164–169 www.elsevier.comrlocatermatlet
Micromechanical strength formulae of unidirectional composites Zheng Ming Huang
)
Department of Mechanical and Production Engineering, National UniÕersity of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 27 November 1998; received in revised form 12 March 1999; accepted 15 March 1999
Abstract When isotropic fibers are unidirectionally arranged in another isotropic matrix material, the resulting composite has different mechanical properties from those of both the fibers and the matrix. Its elastic stiffness can be accurately estimated with some simple micromechanical formulae such as rule of mixture formulae, based on the properties of the constituent fiber and matrix materials as well as fiber volume fraction only. However, little is known about the ultimate strengths of the composite. This letter presents a set of concise micromechanical formulae for the ultimate tensile strengths of the unidirectional composite under individual uniaxial loads. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Micromechanics; Unidirectional composite; Failure criterion; Strength formulae; Stress state
1. Introduction Micromechanics has found much success in prediction of the stiffnessrcompliance of unidirectional fibrous composites w1x. On the other hand, little is known about the ultimate tensile strength, another important mechanical property, of the composite materials. Most strength theories for fibrous composites have been developed based on macromechanical considerations similar to those for isotropic materials, including the maximum stressrstrain criteria and the Tsai–Wu theory w2,3x. Unlike in isotropic cases, extensive experiments including biaxial tests, which may be difficult or expensive to conduct in some cases, have to be performed to determine the strength parameters involved in these macromechanical theories whenever they are applied to any composite. Even with the same constituent materials, different )
Corresponding author
composite having different fiber volume fraction still requires similar tests. Another drawback with these theories is that it cannot predict the failure mode of the composite. They are unable to indicate which of the constituent phases initiates the failure of the composite and meanwhile how rich the strength of the other phase is. Several micromechanics based attempts to the ultimate strength of fibrous composites have been made w4–6x. However, they were focused mainly on the microstructural failure mechanisms and were developed by pre-assuming a priori deformation modes. The overall failure of a composite was not treated in terms of the average responses of its constituent phases. The purpose of this paper is to present a set of simple micromechanical formulae for the ultimate tensile strengths of the unidirectional fibrous composites under individual uniaxial loads. The failure prediction of the composites is based on understanding for the stress states in the constituent materials
00167-577Xr99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 5 7 7 X Ž 9 9 . 0 0 0 6 9 - 5
Z.M. Huangr Materials Letters 40 (1999) 164–169
165
without any pre-assumed deformation mode. Only the material properties of the constituent fiber and matrix as well as fiber volume fraction are required to apply these formulae.
2. Elastic stress states Suppose that both the fiber and matrix are isotropic materials. Let us use E, n , and G to represent the Young’s Želastic. modulus, Poisson’s ratio, and shear modulus of a constituent material, respectively. Let w Si j x to be the compliance matrix of the material. Suffixes ‘f’ and ‘m’ will refer to the fiber and matrix, whereas a quantity without any suffix will refer to the composite. Thus, Vf denotes the fiber volume fraction in the composite, E f represents the Young’s modulus of the fiber, w Simj x is the compliance matrix of the matrix material, etc. It has been shown in Ref. w7x that the elastic stress state in the matrix material can be correlated with that in the fiber by a bridging matrix through
si m 4 s w a i j x sj f 4
Ž 1.
where si m 4 s s 11m , s 22m , s 12m 4T, si 4 s s 11f , s 22f , s 12f 4T, and a11
a12 a22 0
w ai j x s 0
0 0 a 33
Fig. 1. Predicted and measured Žw10x. transverse modulus of a glassrepoxy composite Ž Ef s 73.1 GPa, Em s 3.45 GPa, n f s 0.22, and nm s 0.35..
which are sufficiently accurate w1x. Eq. Ž3.3. is the result of a precise elastic solution for the overall in-plane shear modulus Ž G 12 . of the composite w8x. The accuracy of Eq. Ž3.2. can be seen from a comparison between predicted and experimental results for the transverse modulus of a composite shown in Fig. 1. Let us then consider an arbitrary overall stress state, Ž s 11 , s 22 , s 12 ., applied to the composite. From the micromechanical relationship, si 4 s Vf si f 4 q Ž1 q Vf . si m 4 , and by virtue of Eq. Ž1., the stress shares in the fiber and matrix can be explicitly determined. They are given as:
Ž 2.
s 11f s a e1f s 11 q a e4f s 22 , s 22f s a e2f s 22 , s 12f s a e3f s 12 , Ž 4.
The non-zero elements in Eq. Ž2. are defined as w7x
s 11m s a e1m s 11 q a e4m s 22 , s 22m s a e2m s 22 , s 12m s a e3m s 12 , Ž 5.
0
a11 s E mrE f , a 22 s a 33 s
1 2 1 2
ž ž
1q 1q
Ž 3.1 . m
E
E G
f
m
Gf
/ /
,
Ž 3.2 .
a e1f s ,
Ž 3.3 . a e2f s
and a12 s
where
f m S12 y S12 f m S11 y S11
Ž a11 y a22 . .
Ž 3.4 .
Using Eqs. Ž3.1. and Ž3.4., the resulting overall longitudinal Young’s modulus and Poisson’s ratio Ž E11 and n 12 . of the composite are exactly the same as those given by the rule of mixture formulae w7x,
a e3f s f a e4 s
Ef Vf E f q Ž 1 y Vf . E m
,
Ef Vf E f q 0.5 Ž 1 y Vf . Ž E m q E f . Gf Vf G f q 0.5 Ž 1 y Vf . Ž G m q G f .
Ž 6.1 . ,
Ž 6.2 .
,
Ž 6.3 .
Ž 1yVf . E f Ž n f E m y n m E f .
w Vf E
f
q Ž 1yVf . E m x w 2Vf E f q Ž 1yVf . Ž E f q E m . x
,
Ž 6.4 .
Z.M. Huangr Materials Letters 40 (1999) 164–169
166
a e1m s a e2m s a e3m s
Em f
Vf E q Ž 1 y Vf . E
m
,
0.5 Ž E f q E m . Vf E f q 0.5 Ž 1 y Vf . Ž E m q E f . 0.5 Ž G f q G m . Vf G f q 0.5 Ž 1 y Vf . Ž G m q G f .
m a e4 sy
Ž 7.1 . ,
Ž 7.2 .
,
Ž 7.3 .
Vf E f Ž n f E m y n m E f .
w Vf E
f
q Ž 1yVf . E m x w 2Vf E f q Ž 1yVf . Ž E f q E m . x
.
Ž 7.4 .
3. Strength formulae Most matrix materials are capable of undergoing significant inelastic deformation before failure. To tailor the strength problem, the plastic deformation of the constituent materials must be taken into account. For simplicity, let us assume that the fibers are elastic until rupture and the matrix is a bilinear elastic–plastic material. Let E T , s Y , and su to denote the hardeningmodulus Žtangential modulus in plastic region., yield stress, and ultimate tensile strength of the material, respectively. These parameters are obtainable through a uniaxial tensile test. The compliance matrix of the matrix material, w Simj x, will depend on current load level. As the bridging matrix, w a i j x, correlates the stress states generated in the constituent fiber and matrix materials, its elements can only depend on the material properties as well as on packing geometries of the constituents. When any constituent material deforms from an elastic region to a plastic one, the packing geometries Ži.e., fiber volume fraction, relative arrangement of fibers in the matrix, fiber cross-sectional shape, etc.. do not change or only vary by negligibly small amount. Therefore, only the elastic moduli in Eqs. Ž3.1., Ž3.2. and Ž3.3., which define the independent elements of the bridging matrix, need to be changed when the matrix undergoes plastic deformation. Namely, E m and G m in Eqs. Ž3.1., Ž3.2. and Ž3.3. should be replaced by ETm and G Tm Žs E Tm r3.. On the other hand, the upper triangular part of the bridging matrix, w a i j x given by Eq. Ž2., will be fully occupied and the off-diagonal elements will be deter-
mined using the symmetric condition of the resulting overall compliance matrix of the composite, i.e., S ji s S i j, where w Si j x s Ž Vf w Sif j x q Vm w Simj xw a i j x. = Ž Vf w I x q Vm w a i j x.y1 Žwhich is obtained by using following relations: d si 4 s Vf d si f 4 q Vm d s im 4 , d ´ i 4 s Vf d ´ if 4 q Vm d ´ im 4 , d ´ im 4 s w Sif j x d sj f 4 , d ´ im 4 s w Simj x d sj m 4 , d ´ i 4 s w Si j x d sj 4 and Eq. Ž1.x. The stress calculation has to be carried out incrementally. If, however, the overall applied stress is only uniaxial, we can easily generalize Eqs. Ž4. and Ž5. to determine the total stresses generated in the fiber and matrix. Let us focus on an ultimate tensile strength of the composite under a uniaxial load only. In this letter, the ultimate tensile strength of the composite is defined as the overall applied stress under which the maximum normal stress Ži.e., the first principal stress, s I . of a constituent material attains its ultimate value, su . Based on this definition, the ultimate strength of the composite under longitudinal tension Ž s 11 ., transverse tension Ž s 22 ., or in-plane shear Ž s 12 ., respectively, is obtained as follows. Ža. Longitudinal tensile strength due to a longitudinal tensile load Ž s 11 . only u a 11 s min
with
½
f f suf y Ž a e1 y a p1 . s110 , sum y Ž ae1m y a p1m . a 110 m f a p1 a p1
s 110 s min
½
s Ym suf , , a e1m a e1f
5
5
Ž 8.1 .
Žb. Transverse tensile strength due to a transverse tensile load Ž s 22 . only u a 22 s min
½
f f suf y Ž a e2 y a p2 . s 220 , sum y Ž ae2m y a p2m . a 220 m f a p2 a p2
with s 220 s min
½
s Ym suf , , a e2m a e2f
5
5
Ž 8.2 .
Žc. In-plane shear strength due to an in-plane shear load Ž s 12 . only u a 12 s min
½
f f suf y Ž a e3 y a p3 . s120 , sum y Ž ae3m y a p3m . a 120 m f a p3 a p3
with s 120 s min
½
s Ym
'3 ae3m
,
suf a e3f
5
,
5
Ž 8.3 .
Z.M. Huangr Materials Letters 40 (1999) 164–169
ponents in the fiber and matrix, these components are believed to be small compared with their transverse counterparts and have been ignored in deriving Eq. Ž8.2.. Let us apply the above strength formulae to several unidirectional fibrous laminae. The first lamina is a unidirectional SiC-fiber and titanium ŽTi. matrix composite w9x. The SiC-fiber is an isotropic elastic material until rupture, with E f s 400 GPa and n f s 0.25. The measured ultimate tensile strength of the extracted fiber specimens, however, varied from 2520 MPa to 4540 MPa. In this study, a fiber ultimate strength of suf s 3480 MPa, which was measured using fiber samples extracted from a composite panel whose tensile stress–strain curve was plotted in Fig. 6 of Ref. w9x, is used. The titanium matrix is a bilinear elastic–plastic material Žsee Fig. 2 of Ref. w9x., having properties: E m s 110 GPa, n m s 0.33, ETm s 2.16 GPa, s Ym s 850 MPa, and sum s 1000 MPa. Substituting the fiber and matrix properties into Eq. Ž8.1., the calculated longitudinal tensile strengths vs. fiber volume fractions are plotted in Fig. 2. The experimental data taken from Table 5 of Ref. w9x are also shown in the figure. Correlation between the predicted and measured results is very good. The other laminae to be studied are continuous alumina fiber reinforced pure aluminium matrix composites, which were recently investigated by Bushby w11x. The alumina fibers, Altex and Nextel 610, are considered as isotropic linear elastic until rupture and the pure aluminium as isotropic elastic– plastic. The fiber properties, based on manufacturers’ data given in Ref. w11x, are listed in Table 1. However, except for Young’s modulus and Poisson’s ratio, the other matrix parameters are not available from Ref. w11x. They are taken from Ref. w12x. With
Fig. 2. Predicted and measured Žw5x. longitudinal tensile strength of SiCrTi composite vs. fiber volume fraction. The used material properties are: E f s 400 GPa, n f s 0.25, suf s 3480 MPa, E m s 110 GPa, n m s 0.33, E Tm s1.62 GPa, s Ym s850 MPa, and sum s1000 MPa.
In Eqs. Ž8.1., Ž8.2. and Ž8.3., f a p1 s
m a p1 s
f a p2 s
m a p2 s
f a p3 s
m a p3 s
Ef Vf E f q Ž 1 y Vf . E Tm ETm Vf E f q Ž 1 y Vf . E Tm
,
Ž 9.1 .
,
Ž 9.2 .
Ef Vf E f q 0.5 Ž 1 y Vf . Ž E Tm q E f . 0.5 Ž E f q E Tm . Vf E f q 0.5 Ž 1 y Vf . Ž E Tm q E f .
,
Ž 9.3 .
,
Ž 9.4 .
3G f 3Vf G f q 0.5 Ž 1 y Vf . Ž E Tm " 3G f . 0.5 Ž 3G f q ETm . 3Vf G f q 0.5 Ž 1 y Vf . Ž E Tm q 3G f .
,
Ž 9.5 .
.
Ž 9.6 .
167
It should be noted that although a transversely applied load also generates longitudinal stress com-
Table 1 Constituent properties Altex w11x Nextel 610 w11x Matrix w11,12x
E ŽGPa.
n
E T ŽMPa.
s Y ŽMPa.
su ŽMPa.
Reduction by cold rolling Ž%.
210 373 70 70 70 70
0.2 0.2 0.3 0.3 0.3 0.3
– – 70 5 13.4 46
– – 10 57 75 91
1800 2500 45 59 77 96
– – 0 10 20 40
Z.M. Huangr Materials Letters 40 (1999) 164–169
168
Table 2 Measured uniaxial tensile behavior of alumina fiberraluminium matrix laminae Fiber used
Altex Vf s 0.4 Altex Vf s 0.6 Nextel 610 Vf s 0.7
Ultimate strength ŽMPa.
Ultimate strainŽ%.
Longitudinal
Transverse
Longitudinal
Transverse
761 " 61 930 " 120 1254 " 48
84 " 25 109 " 10 138 " 12
0.85 " 0.06 0.69 " 0.10 0.50 " 0.04
0.50 " 0.1 0.33 " 0.05 0.28 " 0.08
increased ratio of reduction by cold rolling, the yield and tensile strengths of the pure aluminium elevate. Hence, four different groups of the matrix parameters, corresponding to different reduction ratios, have been tried. The used matrix properties are given in Table 1. Measured w11x longitudinal and transverse failure strengths and strains of three composites with fiber volume fractions of 0.4 and 0.6 for the Altex composites and 0.7 for the Nextel composite are summarised in Table 2. Using the constituent parameters given in Table 1, the composite longitudinal and transverse tensile strengths have been calculated from Eqs. Ž8.1. and Ž8.2., whereas the ultimate strains of the composites can be obtained based on d ´ i 4 s Vf w Sif j x d si f 4 q Vm w Simj x d si m 4 Žnote: the matrix compliance matrix w Simj x after yielding is speci-
fied using the Plandtl–Reuss theory w13x.. Results are shown in Table 3. The calculated longitudinal tensile strengths are all accurate for the Altex composites regardless of any group of matrix parameters used. This is because the longitudinal strength of the composite is controlled by the fiber strength. For the Nextel composite, correlation between the predicted and measured longitudinal strengths is less satisfactory if the fiber tensile strength, 2500 MPa, provided by the manufacturer, is used. However, the correlation can be much improved if the Altex fiber strength, 1800 MPa, is also taken as the Nextel fiber strength, as indicated in Table 3. Whereas the longitudinal tensile strength of the composite is controlled by the fiber strength, the transverse failure strength of the composite is seen to
Table 3 Predicted ultimate values of the composites upon different parameters of matrix plasticity Fiber used
Altex Vf s 0.4 Ž suf s 1800 MPa.
Altex Vf s 0.6 Ž suf s 1800 MPa.
Nextel Vf s 0.7 Ž suf s 1800 MPa.
Nextel Vf s 0.7 Ž s Mf s 2500 MPa.
Matrix properties
Failure strength
Failure strain
sum ŽMPa.
s Ym ŽMPa.
E Tm ŽMPa.
Longitudinal
Transverse
Longitudinal
Transverse
45 59 77 96 45 59 77 96 45 59 77 96 45 59 77 96
10 57 75 91 10 57 75 91 10 57 75 91 10 57 75 91
70 5 13.4 46 70 5 13.4 46 70 5 13.4 46 70 5 13.4 46
726.5 754.3 765.2 774.2 1084.4 1103.0 1110.2 1116.6 1263.3 1277.4 1282.8 1287.6 1753.2 1767.3 1772.7 1777.5
60.9 70.8 92.4 115.2 68.8 76.7 100.1 124.8 74.1 87.3 113.9 142.0 74.1 87.3 113.9 142.0
0.8573 0.8572 0.8573 0.8572 0.8573 0.8573 0.8575 0.8572 0.4827 0.4827 0.4827 0.4827 0.6703 0.6703 0.6703 0.6703
18.803 0.0661 0.0862 0.1075 11.828 0.0579 0.0756 0.0942 9.7273 0.0426 0.0556 0.0693 9.7273 0.0426 0.0556 0.0693
Z.M. Huangr Materials Letters 40 (1999) 164–169
depend heavily on the matrix strength used. Compared with the experimental evidences, it is evident that the matrix tensile strength must be considered as higher than 45 MPa, because the predicted ultimate strains of the composites based upon this strength are too higher than the measured data. Using the other three groups of the matrix properties, all the predicted transverse tensile strengths of the composites agree well with the experimental data. Amongst, the predicted results using the matrix properties having 20% reduction by cold rolling are in closest agreement with the measured strengths. It is to be noted that the actual tensile strengths of the in situ constituent materials are somewhat different from those specified by the manufacturers in general. They should be obtained using the in situ constituent materials under same condition as in fabricating the composites.
4. Conclusion Simple formulae, as concise as rule of mixture formulae for composite stiffness, are presented in this paper to calculate the ultimate tensile strengths of a unidirectional fibrous lamina subjected to various uniaxial load conditions. The formulae only use the gross uniaxial tensile parameters of the constituent fiber and matrix materials together with fiber volume fraction as input data. These parameters are easily obtainable through individual tests. The proposed formulae have been applied to calculate the ultimate tensile strengths of several unidirectional fibrous composites. The calculated results are in good agreement with the available experimental data.
169
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