Composites Science and Technology 61 (2001) 1729–1734 www.elsevier.com/locate/compscitech
Effective transverse elastic moduli of a composite reinforced with multilayered hollow-cored fibers L. Stagni* Dipartimento di Ingegneria Meccanica e Industriale, Universita` degli Studi di Roma Tre, Via della Vasca Navale 79, 00146 Rome, Italy, and Istituto Nazionale per la Fisica della Materia, Rome, Italy Received 16 January 2001; accepted 8 May 2001
Abstract The transverse elastic moduli of a unidirectional composite material reinforced with identical multilayered hollow-cored fibers are evaluated by the use of a multiphase generalized self-consistent model, on the basis of a previously established plane-elasticity solution to the problem of a multilayered hollow-cored circular inclusion under uniform remote stresses. The analysis leads to a pair of algebraic equations, which can easily be numerically inverted. Numerical examples for composites reinforced with thinly-coated hollow-cored fibers, and with osteon-like fibers (modelling cortical bone tissue), are presented. In particular, it is found that the effective transverse Poisson ratio becomes negative beyond certain porosity levels, and that the parity of the number of fiber layers is a significant parameter. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Coating; B. Modelling; B. Porosity; C. Elastic properties; Negative Poisson ratio
1. Introduction In order to derive the relationships between the effective elastic moduli of a composite and the properties of the constituents (or phases), appropriate models of fibers or particles are of crucial importance. The models should be such that even when uniform fibers or inclusions are used, the material should be modeled as one with non-homogeneous reinforcements. Actually, a transition zone between the matrix and the other phases often originates as a result, for example, of chemical reactions, plastic flow, or internal strains [1,2]. On the other hand, composites with coated or multilayered reinforcements are widely employed nowadays [3,4]. Other aspects to be taken into account are the presence of voids or pores, which often influence to a large extent the overall mechanical behaviour [5,6], and the growing use of hollow-cored fibers [7,8]. The evaluation of the transverse elastic moduli of unidirectional fiber-reinforced composites, that represents
* Tel.: +39-06-55173308; fax: +39-06-5593732. E-mail address:
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one of the major problems in micromechanics, has been attempted by various authors. Lately, Ju and Zhang [9] estimated the effective transverse elastic moduli of composites with randomly located aligned circular fibers, Chinh [10] estimated the transverse shear modulus of a quite general class of unidirectional composites, and Wang et al. [11] applied an elasticity solution for an inhomogeneity in a finite plane region to evaluate the effective elastic constants of a unidirectional composite. However, no voids or pores within the reinforcements have been considered. Recently, an elastic analysis of a multilayered hollowcored cylindrical inclusion under uniform remote stress has been proposed [12]. In the present paper, these results are used, together with a multiphase generalized self-consistent model [13,14], to evaluate the transverse elastic moduli of a unidirectional composite material with multilayered hollow-cored fibers (circular cylindrical inclusions). The self-consistent scheme takes the form of a (n+1)-phase model, where the representative volume of the composite is a concentric circular hollow cylinder, constituted by the (n1)-layered fiber, surrounded by an outer ring (n-th layer) with the same properties of the matrix; this cylinder is supposed to be embedded in an infinite medium, or (n+1)-th phase, whose elastic moduli
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00071-9
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are equal to the effective moduli of the composite. The analysis leads to a pair of coupled algebraic equations, which can be easily numerically inverted, for the effective transverse shear modulus and Poisson’s ratio. These results are illustrated by two numerical examples. The first example, dealing with a composite reinforced by hollow-cored, simply coated fibers, assesses the effect of a thin coating layer. The second example deals with fibers composed of layers of uniform thickness, but of alternate hardness, as it happens in secondary osteons of cortical bone tissue [15–17]. The porosity of the material, defined as the volume fraction of fiber hollow core, is taken as the main parameter. Actually, though distribution, size, and shape of voids are known to play a significant role in determining the overall elastic behavior [6,18,19], the hollow core may be dimensioned, as a first approximation, so to include also the effect of voids dispersed in the material.
2. Analysis 2.1. Average strains and stresses in an n-layered cylindrical inclusion Consider a finite region S of an infinite elastic plane Oxy (x, y being Cartesian coordinates, and r, polar coordinates) bounded by the contour l. If u ¼ u ðx; yÞ ¼ u~ ðr; Þ;
If S is a circular region of radius R, then S=R2, dl=Rd, nx=cos, ny=sin, and Eqs. (1) and (2) yield 1 1 RefIx g; Im Iy ; "yy ¼ R R 1 Re Iy þ ImfIx g ¼ 2R
"xx ¼ "xy
ð3Þ
where ð 2 UðR; Þcos d;
Ix ¼ 0 ð 2
ð4Þ UðR; Þsin d:
Iy ¼ 0
Next, let R=rN be the outer radius of a hollow-cored, N-layered circular inclusion under uniform remote 1 stresses ð; ¼ x; yÞ (see Fig. 1). According to the available solution of the related elasticity problem [12], the elastic potentials for R 5 rN, i.e. within the infinite layer LN+1, are given by ’Nþ1 ðzÞ ¼ G1 r0 ANþ1 þ CNþ1 1 ; ð5Þ 1 þ FNþ1 3 Nþ1 ðzÞ ¼ G1 r0 DNþ1 þ ENþ1 where
¼ z=r0 ¼ ej ; ¼ r=r0 ;
ð; ¼ x; yÞ
denotes the elastic displacement field, then the average strains within region S are given by ð 1 @u @u " ¼ þ dS 2S S @x @x ð 1 u n þ u n dl ¼ ð1Þ 2S l
r0 is the hollow core radius, G1 is the shear modulus of the first (innermost) layer L1, and AN+1, EN+1 (real), and CN+1, DN+1, FN+1 (complex), are expansion coefficients.
where S is the area of the region S, dl is the element of l, and (nx, ny) is the outward unit normal to l [20]. By defining the complex variable z=x +jy=re j 2 (j =1), and representing the elastic field through the Muskhelishvili complex potentials ’(z) and (z) [21], we can write Uðr; Þ ¼ u~ x ðr; Þ þ ju~ y ðr; Þ ¼
i 1 h ’ðzÞ z’0 ðzÞ ðzÞ z¼re j 2G
ð2Þ
where =3–4 for plane strain, =(3– )/(1+ ) for plane stress, G is the shear modulus and the Poisson’s ratio, a prime denotes complex differentiation, and a bar the complex conjugate.
Fig. 1. n-Layered circular inclusion with a hollow core in infinite elastic plane.
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In particular, AN+1 and DN+1 are given by the remote stresses alone 1 ¼ G1 2ANþ1 Re DNþ1 ;
xx 1 ¼ G1 2ANþ1 þ Re DNþ1 ;
yy 1 ¼ G1 Im DNþ1 ;
xy
ð6Þ
1 xy ¼ xy ;
and the following relationships hold 1Þ 0Þ ANþ1 þ EðNþ1 ; ENþ1 ¼ EðNþ1
1Þ CNþ1 ¼ CðNþ1 D Nþ1
! N X 2 ¼ 2 !Nþ1 i Pi;N N ;
1Þ CðNþ1
N X ¼ 2!Nþ1 i Qi;N þ 3i QiþN;N 2N
ð8Þ
For any applied remote uniform stress field, this implies
i¼1
i ¼ 4i 4i1 :
að1Þ ð 1ÞN ¼ 0;
The matrix elements Pj,N and Qj+N,N can be evaluated 0Þ with the formulae reported in the Appendix, and EðNþ1 is a quantity associated to the eigenstrains. Combining Eqs. (2)–(5) and (7) yields 8 Nþ1 ð1Þ ð1Þ > < "xx ¼ 2N a ANþ1 þ b Re DNþ1 þ "0 ¼ "xx þ "0 Nþ1 ð1Þ ð1Þ "yy ¼ 2N a ANþ1 b Re DNþ1 þ "0 ¼ "yy þ "0 > : "xy ¼ 2Nþ1 bð1Þ Im DNþ1 ¼ "xy N ð9Þ
where Nþ1 ¼ G1 =GNþ1 , and a ð1 Þ ¼ b
ð1 Þ
¼
ð1 Þ N ð Nþ1 1Þ 1 N ENþ1 ð1Þ N þ 1 N Nþ1 CNþ1 :
1 xy ¼ xy
whence, taking advantage of Eqs. (6) and (11), it follows 8 ð1Þ ð1 Þ a > > 2 1 N ANþ1 þ b þ N Re DNþ1 ¼ 0 < ð1Þ a ANþ1 bð1Þ þ N Re DNþ1 ¼ 0 ð12Þ 2 N 1 > > : ð1Þ b þ N Im DNþ1 ¼ 0
i¼1
i ¼ 2i 2i1 ;
1 yy ¼ yy
ð7Þ
where 1Þ EðNþ1
same as the macroscopic strain (stress) imposed to the composite medium at infinite. As a consequence, we can assume that the N-th layer of the composite inclusion represents the matrix, while the surrounding infinite medium LN+1 is the ‘‘effective’’ composite material, whose transverse elastic moduli are given by N+1 and N+1, (which, then, will be hereafter termed and ). Thus, we can write
b ð1 Þ þ N ¼ 0
namely, by exploiting Eq. (10), 1Þ EðNþ1 ¼ 0;
1Þ CðNþ1 ¼ 0:
ð13Þ
By comparing Eqs. (8) and (13), we deduce the following set of equations for the unknown and 8 N P > > < ð þ 1Þ i Pi;N ¼ 2 i¼1
N P > > : ð þ 1Þ i Qi;N þ 3i QiþN;N ¼ 1
ð14Þ
i¼1
ð10Þ
ð0 Þ The quantity "0 ¼ 22 N Nþ1 ENþ1 is the average eigenstrain, and " are the variations of the average strains due to the applied remote load. Finally, relying on Hooke’s law, Eqs. (9) allow us to write the variations of the average stresses within the multilayered inclusion as follows ð1Þ 8 1 2a ð1 Þ > ¼ G A þ b Re D 1 Nþ1 Nþ1 > xx N Nþ1 1 < ð1Þ 2a 1 ð1 Þ ð11Þ ¼ G A b Re D 1 Nþ1 Nþ1 yy N Nþ1 1 > > : ð1 Þ xy ¼ G1 1 N b Im DNþ1
It is easily found (see Appendix) that the quantities Pi,N contain the unknown alone, while Qi,N and Qi+N,N contain both and . Thus, Eq. (14) implies that the effective transverse elastic constants and can be worked out by finding the roots of the function fðÞ ¼ ½ ðÞ þ 1
N X i Qi;N ½; ðÞ þ 3i QiþN;N ½; ðÞ 1 i¼1
where 2
ð Þ ¼
N P
1:
i Pi;N ðÞ
i¼1
2.2. Effective moduli According to the self-consistent scheme [13], the average strain (stress) in the composite inclusion is the
On physical grounds, we expect f () to possess one and only one real root, a condition which has been found to be fulfilled in all the computations performed.
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Fig. 2 shows the effective transverse elastic moduli (Young modulus Et in units of the matrix modulus Em, and Poisson’s ratio t) of a composite reinforced with hollow-cored fibers coated with a single thin layer (N=3), for plane strain. The relative coating thickness, ð2 1 Þ=1 , is 0.025, the fiber volume fraction 22 =23 is
0.7, Poisson’s ratios of fiber, matrix and coating are 0.3, 0.3 and 0.4, respectively. The effective moduli are plotted against porosity, defined as the hollow-core volume fraction 20 =23 , for different values of matrix and coating compliance (m ¼ G1 =G3 and c ¼ G1 =G2 , respectively). By inspecting the graphs, the following noteworthy features are deduced: (i) the effective transverse Young modulus of a composite with fibers harder than the matrix (m=2), increases with increasing porosity (see Fig. 2a); (ii) the effective transverse Poisson’s ratio of a
Fig. 2. Effective transverse Young modulus (a) and Poisson’s ratio (b) versus porosity of a composite reinforced with simply-coated hollow cored fibers, for two values of matrix (m) and coating (c) compliance. Coating thickness is 2.5% fiber radius, fiber volume fraction is 70%, Poisson’s ratios of inner fiber, matrix and coating are 0.3, 0.3 and 0.4, respectively.
Fig. 3. Effective transverse Young modulus (a) and Poisson’s ratio (b) versus porosity of a composite reinforced with hollow cored alternate multilayered fibers. Each fiber is made of 4 evenly spaced layers having alternate compliances. Curves for two values of m=G1/Gm (matrix compliance) and of 2 =G1/G2 are plotted. Fiber volume fraction is 70%, and uniform Poisson’s ratio is 0.3.
3. Numerical examples 3.1. Simply-coated hollow fiber
L. Stagni / Composites Science and Technology 61 (2001) 1729–1734
composite with fibers softer than the matrix (m=0.5, see Fig. 2b), becomes negative beyond a certain porosity (about 10% for c =10); (iii) both Et and t increase with increasing coating hardness. 3.2. Osteon-like fibers Let us consider now a composite reinforced with fibers whose structure resembles that of secondary osteons, which are typical constituents of the cortical bone tissue. These structures may be idealized as hollow-cored multilayered cylinders [15] with evenly spaced layers, and either uniform or alternate distribution of elastic constants (alternate means that all the odd layers of the fiber have identical elastic moduli G1, 1, and all the even layers have elastic moduli G2, 2). Fig. 3 refers to a four-layer-fiber composite (N=5), and shows Et and t versus porosity 20 =2N , for different values of matrix (m) and even-layer (2) compliance. Uniform Poisson’s ratio equal to 0.3, plane strain conditions, and 70% fiber volume fraction are assumed. It is seen that the modulus mismatch between odd and even layers plays a major role in determining the effective moduli. Transverse Young modulus increasing with porosity (beyond a certain porosity) is found for a matrix softer than both odd and even layers (top curve of Fig. 3a). Negative transverse Poisson’s ratio also occurs (see Fig. 3b). The most interesting result appears to be that fibers with even layers harder than the odd ones (2=0.5 in the graphs) yield always effective moduli larger than those found in the opposite case (2=2). This different behavior of odd and even layers is best displayed by
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plotting the effective modulus versus the number of layers, as in Fig. 4. It is seen that a composite with even fibers presents an effective transverse modulus always larger than that of a composite with odd fibers, and that the two values approach each other for N!1. Such a parity effect has been also found for the elastic behavior of a single osteon loaded in axial tension [17].
4. Conclusions We have derived formulae for evaluating the transverse elastic constants of a multilayered hollow fiber composite. Application to relevant configurations shows that the hollow core volume fraction (porosity) controls crucial properties of such kind of composites, mainly their transverse Poisson’s ratio t. Actually, all other properties given, t becomes negative if a certain porosity value is exceeded, an interesting feature in view of the importance that materials with negative Poisson’s ratio (also called auxetic materials) are assuming in technology [22–26]. Moreover, under certain conditions, increasing porosity results in an increase of the transverse Young modulus. These results may be employed, together with the findings of other researches, to create high-strength auxetic composites. In particular, Wei and Edwards [27] worked out the effective moduli of a twophase disordered composite for homogeneous auxetic inclusions of various shapes, and found auxeticity windows and maximum effective Young modulus as a function of the fiber volume fraction. While their approach may minimize stress concentration problems, the present results may lead to lighter auxetic materials without the need of employing auxetic components. Finally, the investigation of the effects related to the fiber layer sequence may introduce other surprising results, such as the importance of the parity of the number of layers, rather than the number itself.
Acknowledgements This work was supported by the Ministero della Ricerca Scientifica e Tecnologica, Italy.
Appendix The matrix elements Pi,N appearing in Eq. (8) belong to the inverse of the NN matrix [p] defined as [12] Fig. 4. Effective transverse Young modulus versus porosity of a composite reinforced with hollow cored alternate multilayered fibers, for six values of the number (N1) of fiber layers, displaying the parity effect (m=0.8, 2=2, fiber volume fraction is 70%, and uniform Poisson’s ratio is 0.3).
8 pi;k ¼ ðiþ1 i Þk for k
> < for k>i þ1 pi;k ¼0 ði; k¼1; 2; . . . ; NÞ 2i þ2i1 þiþ1 i pi;i ¼i i 1 > > 2 : pi;iþ1 ¼ !iþ1
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and the elements Qi+N,N belong to the inverse of the (2N)(2N) matrix [q] defined as 8 qi;k ¼ ðiþ1 iþ1 i i Þk > > > < qi;kþN ¼ 3ðiþ1 iþ1 i i Þk > qiþN;k ¼ ðiþ1 i Þ k 2i k > > : qiþN;kþN ¼ ðiþ1 i Þ 4k 32i k for k < i
[7] [8]
[9]
ði; k ¼ 1; 2; . . . ; NÞ
8 qi;i ¼ iþ1 iþ1 i i > > > > > qi;iþ1 ¼ 2!iþ1 > > > > > qi;iþN ¼ 3 iþ1 iþ1 i þ i i 4i1 þ i 4i > > > >q 2 > > < i;iþNþ1 ¼ 6!iþ1 i qiþN;i ¼ ðiþ1 i Þ2i1 i > > > qiþN;iþ1 ¼ 0 > > > > > qiþN;iþN ¼ iþ1 6i þ 32i 4i1 46i1 > > > > 6 2 4 6 > > > þi i i 3i i1 þ 4i1 > : qiþN;iþNþ1 ¼ 2!iþ1 4i
[10] [11]
[12] [13] [14]
[15]
ði ¼ 1; 2; . . . ; NÞ [16]
qi;k ¼ qi;kþN ¼ qiþN;k ¼ qiþN;kþN ¼ 0 for k > i þ 1 ði; k ¼ 1; 2; . . . ; NÞ
[17] [18]
where i ¼
G1 ; Gi
[19]
!i ¼ 12 i ð i þ 1Þ2i1
i ¼ 2i 2i1 ;
i ¼ 4i 4i1 ;
[20]
i ¼ 6i 6i1 :
[21]
[22]
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