Received 5 April 2003; received in revised form 29 May 2003

Transcription

1 Materials Science and Engineering A360 (2003) 339/344 Connection between elastic moduli and thermal conductivities of anisotropic short fiber reinforced thermoplastics: theory and experimental verification Igor Sevostianov a, *, Mar Kachanov b a Department of Mechanical Engineering, New Mexico State University, P.O. Box 30001, Las Cruces, NM 88003, USA b Department of Mechanical Engineering, Tufts University, Medford, MA 02155, USA Received 5 April 2003; received in revised form 29 May 2003 Abstract Cross-property connections for two phase composites derived recently by the authors are specified for short fiber reinforced thermoplastics. They are verified by comparison with experimental data on glass fiber reinforced composites available in literature. The comparison demonstrates a good agreement for the entire set of nine orthotropic constants with the exception of C 33. The results mae it possible to estimate anisotropic elastic constants from the conductivity measurements. # 2003 Elsevier B.V. All rights reserved. Keywords: Polymer matrix composites; Short-fiber composite; Elastic properties; Conductivity; Cross-property connection 1. Introduction Correlations between the effects of microcracs, pores and various inhomogeneities on different physical properties*/elastic and conductive, for example*/constitute one of the most challenging problems in materials science. They started to attract attention over 40 years ago, in connection with microcracing in metals: the idea to estimate its effect on the mechanical properties via the resistivity data was discussed in Refs. [1,2] and was clearly formulated in the classical wor of Bristow [3]. Cross-property correlations are also implicitly utilized in other areas, lie a non-destructive evaluation (NDE) where the so-called eddy current method is used for NDE of steel structural elements [4]. A new approach, based on bounds, was developed in Ref. [5]. It was later substantially advanced in Refs. [6,7]. All the mentioned wors, with the exception of [7], deal with the case of isotropic materials. Further wor in * Corresponding author. Tel.: / ; fax: / address: igor@me.nmsu.edu (I. Sevostianov). Ref. [7] extended the bounds to transversely isotropic materials. Many realistic microstructures are strongly anisotropic, thus calling for the extension of cross-property correlations to anisotropic cases. In addition, the cases of high contrast between the phases are relevant to many applications: porous/microcraced materials (inclusions have zero stiffness) constitute one example; another example is fiber reinforced plastics (inclusions are much stiffer than the matrix). These practical needs of materials science call for the cross-property connections that, preferably, have the explicit form and that can be applied to strongly anisotropic microstructures with high contrast between the phases. This tas is quite challenging: not only the governing equations of elasticity and conductivity are different, but even the tensors that characterize the mentioned properties are of different rans (fourth ran tensor of elastic moduli vs. second ran tensor of conductivity), so that the cross-property connections should interrelate different numbers of independent components. In their earlier wor, the present authors established explicit elastic /conductive cross-property connections for anisotropic porous/microcraced materials [8], as /03/$ - see front matter # 2003 Elsevier B.V. All rights reserved. doi: /s (03)

2 340 I. Sevostianov, M. Kachanov / Materials Science and Engineering A360 (2003) 339/344 well as in the more general case of two phase anisotropic composites [9]. Their advantage is that, being explicit, they cover the general case of elastic orthotropy and, in addition, can be applied to the cases of high contrast between the phases. However, the connections were derived in the framewor of non-interaction approximation. Formally speaing, this limits their applicability to the cases of low concentration of inclusions. To extend the applicability of the derived connections to high concentrations of inclusions, we suggest and experimentally verify the following ey hypothesis: interactions between inclusions affect both groups of properties*/elastic and conductive*/in a similar way, so that the cross-property connections derived in the non-interaction approximation continue to hold (although this approximation may yield substantial errors for each of the properties separately). This idea was first suggested by Bristow [3] for a material with randomly oriented microcracs. We mention that this hypothesis was experimentally confirmed for metal foams, at volume concentration of pores 70/90% [10] and for the effects of pores/microcracs on the elastic/ conductive properties of composites [11] and metals [12]. In the present wor, we focus on short (glass) fiber reinforced thermoplastics. We first specialize the theory for such materials. Then we verify the cross-property connections by using the available experimental data on the full set of orthotropic elastic and conductive constants [13]. Note that two fiber materials are considered in Ref. [13]*/carbon and glass fiber reinforced composites; the first material is strongly anisotropic (as reported in Ref. [13]), the second one is isotropic. The cross-property connection has been derived for materials with the isotropic phases. Therefore, we restrict our analysis to the glass fiber reinforced composites only. We emphasize that the mentioned ey hypothesis is verified here on a material that is characterized by (1) a non-small concentration of inclusions (volume concentrations up to 26%), (2) strong anisotropy (the ratios of Young s moduli in two perpendicular directions up to 2.5) and (3) high elastic/ conductive contrast between the fibers and the matrix. This is, therefore, a rather challenging test. 2. Elastic properties For a linear elastic specimen of volume V containing one linear elastic inclusion, the macroscopic stress s ij (that is defined as (1/V)f S t i x j ds in terms of surface tractions t i on boundary S of V, with x j being a position vector drawn from a fixed interior point of V to points on S, and that is equal to the average over V stress) can be represented as a sum S ij C 0 ijl o l N ijl o l (1) where C 0 is the stiffness tensor of the matrix and o ij are uniform strains at infinity (i.e. the uniform strain field that would have existed at the site of inclusion in a homogeneous material, in absence of the inclusion). The second term is the change due to the inclusion. Tensor N is the stiffness contribution tensor of the inclusion. For a general ellipsoid, components N ijl are expressed in terms of elliptic integrals. They reduce to elementary functions for a spheroid and N G 0 V + V D + 1 II D+ 2 fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} J D + 3 (mmi Imm)D+ 4 (mim) isotropic terms D + 5 mmmm For the explicit expressions of D i, see [9,14]. They are specified for the particular spheroidal shapes relevant to our analysis in the text to follow. Remar: The volume V should be sufficiently large as compare to the inclusion size, in order to utilize Eshelby s results [14]. This is the usual assumption in the theory of the effective properties of composites. In the case of many inclusions contained in volume V, S ij C 0 ijl X N () ijl (2) o l (3) where the sum a N ( ijl ) represents the collective effect of all inclusions. In the non-interaction approximation, each individual tensor N ( ) is taen as the one for an isolated inclusion (without accounting for the influence of neighbors). Establishing the cross-property connections crucially depends on the possibility to express, with sufficient accuracy, the sum a N ( ijl ) entering Eq. (3) in terms of a second ran tensor (plus the unit tensor). Such a possibility is based on the following two facts. The possibility of approximate representation N of a spheroidal inclusion in terms of a second ran tensor mm, where m is the unit vector along the spheroid s axis (dyadic notations, lie mm or mi, denote tensors with components m i m j and m i d j ): N : ˆN G 0V + V D 1 II D 2 J fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} isotropic terms D 3 (mmi Imm)D 4 (mim) (4) where V * is the inclusion volume; G 0 is shear modulus of the matrix. This possibility was established in Ref. [15], in the case of pores and in Ref. [9] in the more general case of spheroidal inclusions, with D i factors explicitly ex-

3 I. Sevostianov, M. Kachanov / Materials Science and Engineering A360 (2003) 339/ pressed in terms of spheroid s aspect ratio and the elastic constants. The representation Eq. (4) is approximate one and its accuracy depends on three parameters: contrast in elastic properties of the matrix and the inclusions, geometry of the inclusions and Poisson s ratio of the matrix. Sevostianov and Kachanov [9] constructed the accuracy maps (accuracy of Eq. (3) as a function of the matrix/inclusion elastic constants and the inclusion s aspect ratio). The accuracy is particularly good at very low and very high Poisson s ratio of the matrix. The last case is relevant for polymer matrix composites. The possibility to establish a similar representation, in terms of a certain symmetric second ran tensor v, for a solid with many inclusions gives: X N () : X ˆN () G 0 cd 1 II cd 2 J d fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} 3 (vi Iv) isotropic terms d 4 (v J J v) where d i /f 0 D i (g)f(g)dg, f(g) is the distribution density of aspect ratios g (that, in the case of short fibers, is defined as the ratio length/diameter of a fiber), c is volume concentration of inclusions (note that c /trv) and 2J ijl /d i d jl /d il d j. In the case of inclusions of the same shapes, d i /D i. Remar: Representation Eq. (4) for one inclusion constitutes a necessary, but not a sufficient condition for Eq. (5) to hold (unless all the inclusions have identical shapes and thus identical D 3, D 4 ). Indeed, for mixtures of diverse shapes, coefficients D 3 and D 4 in Eq. (3) are different for inclusions of different aspect ratios, so that summation over inclusions may produce two, generally, non-coaxial, different second ran tensors (rather than one such tensor). Nevertheless, representation Eq. (5) was found to hold with good accuracy [8,9], with tensor v given by v X (5) v () 1 X (V + mm) () (6) V In the case of cracs, tensor v is reduced to crac density tensor, while in the case of spherical inclusions it gives volume concentration of inclusions multiplied by second ran unit tensor. Note, that the exact representation of the property contribution tensor should include both second ran tensor v/(1/v)a (V * mm) ( ) and fourth ran tensor (1/V)a (V * mmmm) ( ) [16]. We now consider a needle-lie inclusion (short fiber) and specify tensor N and coefficients d 1 4 of the approximate representation Eq. (5) for this shape. Tensor N entering Eq. (1) is given by N ijl n 1 u ij u l n 2 (u i u lj u il u j u ij u l ) n 3 (u ij m m l u l m i m j ) n 5 (u i m l m j u il m m j u j m l m i u jl m m i ) n 6 m i m j m m l (7) where u/d ij /m i m j (m is the unit vector along the fiber) coefficients n i are given by c n 1 1 2c ; n 2 2 ; G 0 (1 0 )c 1 4G 0 (2 0 )c 2 and where In the case of axial symmetry of the inclusion shape (spheroid, for example), the representation Eq. (3) implies the following restrictions on coefficients n i : (n 6 n 1 n 2 )=(22n 3 n 5 )0 (10) We now approximate N by tensor ˆN; with coefficients ˆn i replacing n i, where ˆn i are obtained by multiplying n i by either (1/d) or(1/d), and choosing d in such a way that Eq. (10) is satisfied exactly for ˆn i : ˆn 1 n 1 (1d sign n 1 ); ˆn 3 n 3 (1d sign n 3 ); ˆn 2 n 2 (1d sign n 2 ); ˆn 5 n 5 (1d sign n 5 ); where n 3 c 3 G 0 (1 0 )c 1 n 5 2c 5 8G 0 c 5 ; n 6 2c 6 G 0 (1 0 )(c 1 c 6 c 3 c 4 ) 2G 0 [G 0 (1 0 )c 1 ] 0 1 2(1 n 0 ) (l 0 G 0 )=(l 0 2G 0 ); c 1 (l + G + )(l 0 G 0 ); c 2 2(G + G 0 ) c 3 c 4 l + l 0 ; c 5 4(G + G 0 ); c 6 (l + 2G + )(l 0 2G 0 ) (9) d (n 6 n 1 n 2 )=(2 2n 3 n 5 ) (jn 6 jjn 1 jjn 2 j)=(2 2jn 3 jjnj 5 ) ˆn 6 n 6 (1d sign n 6 ) (11) (12) Then the error of this approximation is equal to jdj (see [9] for details). Coefficients D i for the approximate stiffness contribution tensor ˆN are as follows: D 1 ( ˆn 1 ˆn 2 )=2; D 2 ˆn 2 D 3 2ˆn 3 ˆn 2 2ˆn 1 ; D 4 ˆn 5 2ˆn 2 (13) (8)

4 342 I. Sevostianov, M. Kachanov / Materials Science and Engineering A360 (2003) 339/ Effective conductivity and the cross-property connections The change of heat flux vector DQ (per volume V) due to the inclusion is a linear function of the far-field temperature gradient 9T and hence can be written in the form: DQH C 9T (14) where H C is the conductivity contribution tensor of the inclusion (superscript C stands for conductivity, to distinguish H C from the elastic compliance contribution tensor H). In the case of a spheroid of conductivity * in a matrix of conductivity 0, H C V + V 0 [A 1 (I mm)a 2mm] (15) where coefficients A 1, A 2 are functions of the aspect ratio g, defined as the ratio of the length to the diameter of a fiber, and conductivities *, K 0 : A ( 0 )f 0 (g) ; ( A 2 0 ) 2 [3f 0 (g) 1] [ 2( 0 )f 0 (g)][ 0 ( 0 )f 0 (g)] f 0 (g) g2 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ln g p ffiffiffiffiffiffiffiffiffiffiffiffiffi g 2 1 p 2(g 2 1) 2g g 2 1 g ffiffiffiffiffiffiffiffiffiffiffiffiffi g 2 1 In the asymptotics of short fibers, (16) cross-property connection (a blunder in Ref. [9] is corrected in the formula below): C C 0 d1 a 2 2d 3 a 1 G 0 a 2 (a 2 3a 1 ) II d 2 a 2 2d 4 a 1 a 2 (a 2 3a 1 ) J [tr(k= 0 )3] d 3 [(K= 0 I)I I(K= 0 I)] a 2 d 4 a 2 [(K= 0 I) J J (K= 0 I)] (19) or, in components, (d 1 d 2 )a 2 2(d 3 d 4 )(a 2 2a 1 ) 1 d 2 )a 2 2(d 3 d 4 )(a 2 2a 1 ) 1 d 2 )a 2 2(d 3 d 4 )(a 2 2a 1 ) 0 3 (a 1 a 2 ) 33 0 C 1111 C G 0 a 2 (a 2 3a 1 ) 0 (d 1 d 2 )a 2 2(d 3 d 4 )a a 2 (a 2 3a 1 ) 0 0 C 2222 C G 0 a 2 (a 2 3a 1 ) 0 (d 1 d 2 )a 2 2(d 3 d 4 )a a 2 (a 2 3a 1 ) 0 0 C 3333 C G 0 a 2 (a 2 3a 1 ) 0 (d 1 d 2 )a 2 2(d 3 d 4 )a a 2 (a 2 3a 1 ) 0 0 C 1122 C d 1a 2 d 3 (a 1 a 2 ) G 0 2a 2 (a 2 3a 1 ) 0 0 d 1a 2 2d 3 a a 2 (a 2 3a 1 ) 0 C 1133 C d 1a 2 d 11 0 G 0 2a 2 (a 2 3a 1 ) 0 0 A 1 2( + 0 ) + 0 ; A 2 ( + 0 )2 0 ( + 0 ) (17) d 1a 2 2d 3 a a 2 (a 2 3a 1 ) 0 C 2233 C d 1 a 2 d 3 (a 1 a 2 ) G 0 2a 2 (a 2 3a 1 ) 0 0 In the case of many inclusions, considered in the noninteraction approximation, we have the following expression for the effective conductivity K 0 [(a 1 c1)i a 2 v] (18) where v is given by Eq. (6) and a i /f 0 A i (g)f(g)dg and f(g) is the shape distribution density. In the case of inclusions of the same shapes, a i /A i. The ey observation is that both Eq. (18) and a similar Eq. (5) for the effective elasticity are formulated in terms of the same tensor v. Therefore, v can be eliminated, yielding the explicit elastic /conductive d 1a 2 2d 3 a a 2 (a 2 3a 1 ) 0 C 1212 C d 2a 2 2d 11 0 G 0 4a 2 (a 2 3a 1 ) 0 0 d 2 a 2 2d 4 a a 2 (a 2 3a 1 ) 0 C 1313 C d 2a 2 d 4 (a 1 a 2 ) G 0 4a 2 (a 2 3a 1 ) 0 0 d 2a 2 2d 4 a a 2 (a 2 3a 1 ) 0 C 2323 C d 2a 2 d 22 0 G 0 4a 2 (a 2 3a 1 ) 0 0 d 2a 2 2d 4 a (20) 4a 2 (a 2 3a 1 ) 0

5 I. Sevostianov, M. Kachanov / Materials Science and Engineering A360 (2003) 339/ Experimental verification of the derived cross-property connections We compare the derived cross-property connections with the experimental data of [13] on the full set of orthotropic elastic and conductive constants of thermoplastics (polyphenilene sulfide, PPS) reinforced by short glass fibers (with the aspect ratio 16). The accuracy of the data was within 8%, for both elastic and conductive constants. As seen from Table 1, the elastic/conductive contrast between the fibers and the matrix was quite high. The fibers tended to be parallel, with a substantial scatter. The overall anisotropy (both elastic and conductive) was strongly pronounced. Moreover, the orientational distribution of fibers was such that the elastic anisotropy was fully orthotropic (nine independent constants) and not reducible to the transverse isotropy. Samples were taen from two different regions of a specimen, a surface layer close to the surface and a middle layer (these two regions had different orientational distribution of fibers). These factors*/strongly pronounced, fully orthotropic anisotropy, high contrast between the phases and strongly non-spherical shapes of inclusions*/made this set of data a challenging test for our theory. This can be seen from the accuracy maps constructed in Ref. [9] that present the maximal expected errors of the crossproperty connections in terms of the inclusions shapes and the inclusions/matrix properties. The weight fraction of the fibers was 30 and 40%, translating into 19 and 26% volume fractions (see Table 1 for densities of the constituents). At such volume fractions, the interaction effects are substantial, so that the non-interaction approximation may yield substantial errors, for each of the conductive and the elastic properties separately. Nevertheless, we expect, in accordance with our ey hypothesis that the connections between the conductive and the elastic properties that are derived in the non-interaction approximation, continue to hold. Table 2 compares predictions of the effective elastic stiffnesses, made via the cross-property connections (from three measured principal conductivities given in Table 3), with the experimentally measured stiffnesses Table 1 Elastic properties, thermal conductivities and densities of glass fibers and of poly(phenylene sulfide) matrix Glass fibers E (GPa) G (GPa) n K (mw (cm K) 1 ) r (g cm 3 ) Polymer matrix Table 2 Comparison of the effective elastic stiffnesses calculated using crossproperty connections (plane font) with experimental data (bold font) 30% fibers 30% fibers 40% fibers 40% fibers C / / / /13.0 C / / / /12.0 C / / / /21.2 C / / / /2.95 C / / / /3.45 C / / / /2.65 C / / / /6.9 C / / / /7.2 C / / / /7.1 Table 3 Experimentally measured thermal conductivities (in mw (cm K) 1 ) 30% fibers 30% fibers 40% fibers 40% fibers K K K (plain font vs. boldface font). The agreement between the theoretical predictions and the data is quite good for the entire set of nine orthotropic stiffnesses with only one exception*/coefficient C 33 for weight fraction of fibers 40%. In all the other cases the agreement is better than 12%. We emphasize that the agreement is good in spite of the fact that this material presents a challenging test for our theory. 5. Discussion and conclusions The cross-property connections for anisotropic composites, that express the anisotropic elastic constants in terms of conductivities, are verified, using the data of [13] on of thermoplastics reinforced by short glass fibers. The fibers had preferential orientations resulting in a strong anisotropy, and their volume fraction was up to 26%. This data set contains a full set of orthotropic elastic and conductive constants measured with accuracy within 8%. The goal was to verify the central hypothesis*/that the explicit cross-property connections for a full set of anisotropic constants, that were derived in the noninteraction approximation, continue to hold at higher volume fractions of inclusions (the reason being that the interactions affect both groups of properties in a similar way). The data set presented a challenging test for our theoretical predictions*/the anisotropy was strongly pronounced, the contrast between the phases was high and the inclusion shapes were strongly non-spherical.

6 344 I. Sevostianov, M. Kachanov / Materials Science and Engineering A360 (2003) 339/344 Agreement between the theoretical predictions of the elastic constants (inferred from the data on conductivities, via the cross-property connections) and the directly measured ones is found to be good for the entire set of nine orthotropic elastic constants with the exception of C 33. For C 33 the disagreement may rich up to 30%. This demonstrates that the explicit cross-property connections derived, originally, in the non-interaction approximation, may be used at higher volume concentrations of inclusions. We note, in this connection, that a similar finding was reported for several other material systems [10/12]. We note that, although the elastic and the conductive properties are sensitive to the distribution of orientations and lengths of inclusions (for example, to the extent of orientational scatter, in the case when there is a preferential orientation), see [17,18], the cross-property connection is not sensitive to the exact form of the distribution. This follows from the fact that the mentioned distribution affects the elastic and the conductive properties in a similar way. This insensitivity is one of the advantages of the cross-property connections. The obtained results also indicate that the accuracy maps constructed in Ref. [9], that present the maximal expected errors of the cross-property connections in terms of the inclusions shapes and the inclusions/matrix properties, may be overly conservative. They appear to overestimate the error of representation Eq. (5) on which the cross-property correlations are based. In particular, the error for PPS reinforced by glass fibers (value of in d Eq. (12)) is estimated as 50%, while, in reality, it is significantly lower. References [1] L.M. Clarenbrough, M.E. Hargreaves, G.W. West, Proc. Roy. Soc. A 232 (1955) 252. [2] A.N. Stroh, Phylos. Mag. 2 (1957) 1. [3] J.R. Bristow, Br. J. Appl. Phys. 11 (1960) 81. [4] P. McIntire (Ed.) Nondestructive Testing Handboo, Vol. 4: Electromagnetic Testing, second ed., American Society for Nondestructive Testing, London, [5] J.G. Berryman, G.W. Milton, J. Phys. D 21 (1988) 87. [6] L.V. Gibiansy, S. Torquato, Phylos. Trans. Roy. Soc. L A353 (1995) 243. [7] L.V. Gibiansy, S. Torquato, Proc. Roy. Soc. A 452 (1996) 253. [8] M. Kachanov,I.Sevostianov, B. Shafiro, J. Mech. Phys. Solids 49 (2001) 1. [9] I. Sevostianov, M. Kachanov, J. Mech. Phys. Solids 50 (2002) 253. [10] I. Sevostianov, J. Kováci, F. Simancí, Int. J. Fracture 114 (2002) L23. [11] I. Sevostianov, V. Verijeno, M. Kachanov, Compos. B 33 (2002) 205. [12] I. Sevostianov, M. Bogarapu, P. Tabaov, Int. J. Fracture 115 (2002) L15. [13] C.L. Choy, W.P. Leung, K.W. Kow, F.P. Lau, Polymer Compos. 13 (1992) 69. [14] J.D. Eshelby, Proc. Roy. Soc. A 241 (1957) 376. [15] B. Shafiro, M. Kachanov, J. Mech. Phys. Solids 47 (1999) 877. [16] M. Kachanov, Int. J. Fracture 97 (1999) 1. [17] G. Za, M. Harberer, C.B. Par, B. Benhabib, Rapid Prototyping J. 6 (2000) 107. [18] S.-Y. Fu, B. Laue, Comp. Sci. Technol. 56 (1996) 1179.