Mechanics of Advanced Materials and Structures, 12: 43–54, 2005 c Taylor & Francis Inc. Copyright ISSN: 1537-6494 print / 1537-6532 online DOI: 10.1080/15376490490492034
Micromechanical Modeling of Porous Carbon/Carbon Composites I. Tsukrov Department of Mechanical Engineering, University of New Hampshire, Durham, New Hampshire, USA
R. Piat Institute of Solid Mechanics, University of Karlsruhe, Karlsruhe, Germany
J. Novak Department of Mechanical Engineering, University of New Hampshire, Durham, New Hampshire, USA
E. Schnack Institute of Solid Mechanics, University of Karlsruhe, Karlsruhe, Germany
direct numerical simulations using, for example, finite element discretization [9]. Whereas analytical approaches are currently limited to very few regular geometries of inhomogeneities, numerical simulations can be used for all kinds of shapes. However, the later technique requires substantial computational power and are not universal. Our approach to calculating the effective elastic properties of porous composites combines numerical and analytical techniques: the elasticity problem for each type of irregularly shaped inhomogeneity is solved numerically, and this solution is used in the analytical procedure of micromechanical modeling. The proposed procedure is applied to predict the effective elastic moduli of unidirectional carbon/carbon (C/C) composites fabricated by chemical vapor infiltration (CVI) of carbon fiber preform. The technology of CVI is one of the methods used for obtaining materials with advanced mechanical and thermal properties. For the considered composite, it consists of synthesis of carbon particles from hydrocarbon gas (methane/hydrogen mixture) and their deposition on carbon fibers preliminarily placed in that environment. The process runs under a temperature of 1100◦ C and total pressures of 20–30 kPa until the carbon particles deposit and form a porous carbon matrix filling space between the fibers. Pores in this matrix have highly irregular shapes as shown in Figure 1. Further details on the infiltration procedure and sample properties are described by Reznik et al. [10] and Benzinger and H¨uttinger [11, 12]. The aim of the presented research is to analyze contributions of fibers and pores into the effective elastic properties of the unidirectional C/C composite with consideration of their geometry, distribution, and volume fraction in the material. It is well known
A procedure to model fiber-reinforced composites containing pores of irregular shapes is presented. Closed-form expressions for contributions of fibers and pores into effective elastic moduli are provided. The procedure is applied to predict the transverse elastic properties of unidirectional carbon/carbon composites (carbon fibers in pyrolytic carbon matrix) densified by chemical vapor infiltration. Infiltration treatment results in the formation of irregularly shaped pores randomly oriented in the plane perpendicular to the direction of fiber (transverse plane). These pores are analyzed using a numerical conformal mapping technique, and their contribution to the effective elastic properties is expressed in terms of the cavity compliance contribution tensor. Components of this tensor are found for a variety of typical pore shapes.
1. INTRODUCTION In this paper, we develop a micromechanical modeling procedure to predict the effective elastic properties of fiber-reinforced composites containing pores of irregular shapes. Our approach is based on the concept of a compliance contribution tensor as described, for example, by Kachanov et al. [1] and Sevostianov and Kachanov [2]. Other approaches include evaluation of upper and lower bounds (see [3–7] and later publications), analytical calculations using available elasticity solutions [8], and
Received 20 March 2003; accepted 13 January 2004. Address correspondence to I. Tsukrov, Department of Mechanical Engineering, University of New Hampshire, Durham, NH 03824, USA. E-mail:
[email protected]
43
44
I. TSUKROV ET AL.
This paper is organized as follows. Section 2 describes a micromechanical procedure to model materials with irregularly shaped inhomogeneities (inclusions or pores). Microstructure of the CVI-densified C/C composite and assumptions introduced to model this material are discussed in Section 3. In Section 4, the analytical formulas to evaluate contribution of fibers to the effective elastic moduli are derived. Section 5 employs a numerical conformal procedure to account for contribution of irregularly shaped pores. An example of application of the approach to evaluation of the in situ mechanical properties of the pyrolytic carbon is presented in Section 6.
FIG. 1.
Typical micrograph of C/Pyro-C composite material.
in composite material theory that elastic moduli of unidirectional composites in the longitudinal direction can be predicted with good accuracy by the so-called rule of mixtures (see, for example, [13]). In this study, we focus on transverse effective properties. We choose a two-step approach as demonstrated in Figure 2. First, the contribution of fibers is accounted for by substituting Pyro-C matrix and carbon fibers with an equivalent matrix material (denoted by a subscript FPC). Then the irregular pores (now embedded in the equivalent matrix FPC) are analyzed using a numerical conformal mapping technique [14]. A similar multiplestep approach of consecutive homogenization was successfully used by Roerden and Herakovich [15] to model the porous hybrid fiber-reinforced metal–matrix composites. The three-phase fiber material (alumina binder + pores + alumina filaments) was homogenized in the first two steps, and then the overall properties of the composite (fiber + metal matrix) were determined. Thus, multiple-step homogenization proves to be an efficient approach to micromechanical modeling of complex composite materials.
2. MICROMECHANICAL MODELING OF MATERIALS WITH INHOMOGENEITIES OF VARIOUS SHAPES Our approach to calculating the effective elastic properties of statistically homogeneous porous composite is based on the concept of the representative volume element (RVE)—a statistically representative portion of the microstructure for which various macroscopic parameters (for example, stress and strain) can be defined. We assume linear elasticity, perfectly bonded inclusions, and traction-free cavities. The effective elastic compliance S must satisfy the basic relation ε = S : σ,
[1]
where the colon denotes contraction over two indices, and ε and σ are the second-rank macroscopic strain and stress tensors. Traditionally, these tensors are defined as average values (over the RVE) of the corresponding fields. Since these fields are not clearly defined inside of the cavities (strain) and rigid inclusions (stress), we represent the macroscopic stress and strain in terms of the surface displacements and tractions (using the divergence theorem, as suggested by Hill [5]) as 1 1 ε=− (u ⊗ n + n ⊗ u) dγ, σ=− t ⊗ x dγ, 2V γ V γ [2]
FIG. 2. Two-step micromechanical modeling procedure.
POROUS CARBON/CARBON COMPOSITES MODELING
FIG. 4. FIG. 3.
45
Inhomogeneity of irregular shape.
Representative volume element.
hold: where V is the RVE with boundary γ; u, n, t, and x are the displacement, outward unit normal, traction, and position vectors of the boundary points, respectively (see Figure 3). To characterize the contribution of inhomogeneities to the effective elastic compliance, we introduce the inclusion compliance contribution tensor HRVE as S = SM + HRVE ,
[3]
M
where S is the compliance tensor of matrix material. The compliance contribution tensor (H tensor) has been used by Kachanov et al. [1] and Tsukrov and Novak [14] to analyze solids with various two-dimensional (2D) and three-dimensional (3D) holes and by Sevostianov and Kachanov [2, 16] to model solids with elastic ellipsoidal inclusions. The approach is based on the results for one inclusion. We represent the total strain and stress tensors (as defined by Eqs. (2)) in a reference volume V˜ containing an inclusion of volume VI (Figure 4) as sums: ε = ε + ε M
σ = σ + σ, M
[4]
H : σ = ε − SM : σ.
[7]
In the local coordinate system x1 , x2 , x3 with unit vectors e1 , e2 , and e3 (Figure 4), the compliance contribution tensor for an arbitrary inclusion has the following structure: H= Hi jkl ei ⊗ e j ⊗ ek ⊗ el , [8] where Hi jkl = H jilk = Hkli j . To find the components of the H tensor for a particular shape, we assume that the total stress in the reference volume is equal to the applied remote stress and evaluate the additional strain and stress tensors. Let us consider a 2D inhomogeneity under uniaxial tension P inclined at an angle θ to the x1 axis. The stress tensor for such a loading is σ = P[cos2 θ e1 ⊗ e1 + sin2 θ e2 ⊗ e2 + sin θ cos θ(e1 ⊗ e1 + e2 ⊗ e2 )].
[9]
Contraction of this tensor with H given by Eq. (8) produces the following expressions: P [(H1111 + H1122 ) + (H1111 − H1122 ) cos 2θ 2 + 2H1211 sin 2θ] P = [(H2222 + H1122 ) − (H2222 − H1122 ) cos 2θ 2 + 2H1222 sin 2θ] P = [(H1211 + H1222 ) + (H1211 − H1222 ) cos 2θ 2 + 2H1212 sin 2θ]. [10]
(H : σ)11 =
where εM =
1 V˜
ε(x) dA VM
σM =
1 V˜
σ(x) dA
[5]
VM
and inclusion contributions ε and σ are derived from the divergence theorem as 1 1 ε = − (u ⊗ n + n ⊗ u) d σ = − t ⊗ x d. 2V˜ V˜ [6] In the preceding equation, n is the unit normal to the inclusion boundary directed inward the inclusion. The contribution of the inclusion to the overall compliance of V˜ is given by tensor H, and the following relation must
(H : σ)22
(H : σ)12
Now the components of H are obtained by comparing these expressions with the corresponding elasticity predictions given by Eqs. (6) and (7) for various values of angle θ. Thus calculated values of the H tensor for individual inclusions are used to derive the overall properties of the composite as described below.
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I. TSUKROV ET AL.
We first consider the approximation of noninteracting inhomogeneities. This approximation is rigorous at small concentrations of inclusions (the so-called dilute limit) and can also be used as a basic building block for various first-order micromechanical schemes (see, for example, [17]). In this approximation, it is assumed for each inclusion that the reference volume V˜ is equal to RVE and the stress field is not disturbed by the presence of other inclusions. Then the overall response of the material can be characterized by the sum of H tensors as S = SM + HNI ,
[11]
where tensor HNI = H(k) is the noninteraction approximation of tensor HRVE and H(k) is the compliance contribution tensor of the kth inhomogeneity. For interacting inhomogeneities, predictions of the more advanced approximate micromechanical schemes are obtained from HNI [18]. In this paper, we use the Mori–Tanaka approximation [19, 20]. Each inclusion is assumed to be subjected to the remote stress equal to the average stress in the matrix phase. The compliance contribution tensor HMT and the effective compliance S are then derived as functions of the inclusion volume fraction Vf as HMT = HNI : [(1 − Vf )(Sf − SM ) + HNI ]−1 : (SI − SM ), S = SM + HMT .
[12]
3. MICROSTRUCTURE OF CVI DENSIFIED CARBON/CARBON COMPOSITE CVI densification results in the composite that consists of carbon fibers embedded in the porous matrix of pyrolytic carbon (Pyro-C). Such composites tend to have very complex microstructure and, to predict their effective mechanical properties, it is necessary to analyze them on different lengths scales. The four-level hierarchical material model has been proposed by Piat and Schnack [21]. On the microscale, the Pyro-C matrix has layered structure with layers arranged around fibers. Each layer has different mechanical properties in the axial, radial, and circumferential directions (i.e., is anisotropic). These layers are formed by different modifications of pyrolytic carbon, which is known to exhibit a broad variety of microstructures (isotropic, low-, medium-, or high-textured; see [22]). Number of layers, their thickness, order, and structure are determined by the deposition parameters. CVI conditions also influence the geometry, size, and concentration of pores that form between fibers with pyrolytic coating. In the case of random composite felts, the total porosity of 15–20% and the open porosity of approximately 10.5% have been observed by Benzinger and H¨uttinger [11, 12]. For the detailed analysis, we have chosen a specimen of the unidirectional C/C composite manufactured using the procedure described in [10–12]. Typical micrographs, obtained by optical light microscopy (R. Ermel, personal communication, 2002), are shown in Figures 5a and 6a in two different resolutions. We
FIG. 5. Microstructure of CVI densified C/C composite (on 500-µm length scale); (a) typical microstructure images; (b) processed images: Gray—pores, Black—fibers and Pyro-C.
47
POROUS CARBON/CARBON COMPOSITES MODELING
FIG. 6. Microstructure of CVI densified C/C composite (on 100-µm length scale); (a) typical microstructure images; (b) processed images: Black–pores, Gray—fibers and Pyro-C; (c) processed images: White—fibers, Gray—pores and Pyro-C.
observe the bright circular regions representing fibers (clearly visible in Figure 6) surrounded by the Pyro-C matrix, whereas large black regions of irregular shape indicate pores (cavities). The diameter of the fibers is 7 µm, and the typical pore dimensions are on the order of tens of microns. The micrograph images of the composite were processed and analyzed using ColorPoint 2.0 software (www.patrilab.com). Figures 5b, 6b, and 6c show distribution of pores and fibers. Images presented in Figures 5 and 6 and 13 other 200- and 500-µm scale images have been analyzed to obtain fiber volume fraction Vf and porosity Vp of the composite. The results are summarized in Table 1. These estimates of porosity can be compared with the measurements of open porosity reported (open) in [24]. According to their analysis, Vp = 8.8%. Thus, our estimates are consistent with the observation of Reznik et al. [24] that the total porosity is considerably greater than the open porosity. As can be seen in Figures 5 and 6, pores in this composite material are randomly distributed in the transverse plane without any preferential orientation. Their shapes are highly irregular and cannot be accurately approximated by circles, ellipses, or right polygons for which the analytical elasticity solutions are
available. It is also observed that the location of the fibers in the transverse plane is random, and so the overall elastic properties of the composite are transversely isotropic. To analyze the transverse elastic moduli of such a composite, we employ the near-field micromechanical modeling procedure, which is based on the concept of the compliance contribution tensor as described in Section 2. It is assumed that pores are sufficiently long in the direction of the fibers, so that the plane-strain model can be used. Also, the pyrolytic carbon matrix material, which can be high-textured and anisotropic on the nanoscale, is modeled as homogeneous and isotropic on the length scale of fibers and pores. The procedure to predict the homogenized elastic properties of the pyrolytic carbon based on its nanostructure and texture degree is presented by Piat et al. [25]. The typical pore size is about 5 to10 times greater than that of the fiber cross section. We propose that pores can be treated as being surrounded by an equivalent elastic material (FPC) consisting of Pyro-C and carbon fibers (Figure 2). Carbon fibers used for reinforcement of the C/C composite have circular cross section, and their stiffness in longitudinal and transverse directions is known. The contribution of fibers into the effective elastic moduli is investigated in Section 4.
TABLE 1 Fiber volume fraction and porosity of C/Pyro-C composites
4. CONTRIBUTION OF FIBERS INTO THE EFFECTIVE ELASTIC PROPERTIES Micromechanical modeling of the linear elastic material reinImage 1 Image 2 Image 3, . . . ,17 Average forced by unidirectional fibers of circular cross section is based (%) (%) (%) on the plane-strain elasticity solution for a circular inclusion in Porosity, V p 32.5 28.0 29.0 the infinite 2D solid. According to Hardiman [26], the stress in Fiber volume 18.0 18.0 ··· 18.0 such an inclusion subjected to remotely applied uniaxial tension fraction, Vf P (Figure 7) is uniform: Pyro-C volume 50.0 54.0 53.0 fraction, VPC σ = P(c e ⊗ e + c e ⊗ e ), [13] f
1 1
1
2 2
2
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I. TSUKROV ET AL.
where Vf is the volume fraction of fibers in the equivalent matrix FPC. If the stiffness of the fibers is much higher than that of pyrolytic carbon, they can be treated as absolutely rigid inclusions; the corresponding formulas are provided by Tsukrov [27]. For interacting fibers, we use Mori–Tanaka scheme (12), and the following expressions for the components of fiber compliance contribution tensor are obtained: FIG. 7. Circular inclusion (fiber) in an infinite solid (Pyro-C) subjected to remotely applied tension P.
where 1+β+γ β , 2 β − γ(1 − 2β + 2γ) β 1 − β + 3γ c2 = − 2 β − γ(1 − 2β + 2γ)
c1 =
[15b] H1122 = H2211 d1 (c1 + c2 ) d2 (c1 − c2 ) = Vf − 1 − Vf (1 − c1 − c2 ) 1 − Vf (1 − c1 + c2 ) (f)
and β=
E f 1 − ν2PC , E PC 1 − ν2f
γ=
1 1 β . − 4 1 − νPC 1 − νf
d2 (c1 − c2 ) , 1 − Vf (1 − c1 + c2 )
where 1 − νf − 2ν2f E PC − 1 − νPC − 2ν2PC E f d1 = 2E PC E f d2 =
[14]
where S(PC) is the average compliance of the pyrolytic carbon and H(f) is the fibers’ compliance contribution tensor. To evaluate H(f) , we use solution (13) to find traction t = σf ·n on the boundary of circular inclusion. Substituting in Eqs. (6) and taking into account that ε = S(f) : σ (where S(f) is the compliance of fiber material), we can utilize Eqs. (7) and (10) to find the H tensor of one inclusion. This yields the following expressions for components of H(f) in the assumption of noninteracting fibers: 1 − ν2f 1 − ν2PC (f) (f) H1111 = H2222 = Vf c1 − Ef E PC 1 + νPC 1 + νf c2 + νPC − νf E PC Ef 1 + νPC 1 + νf (f) (f) H1122 c1 [15a] = H2211 = Vf νPC − νf E PC Ef 1 − ν2PC 1 − ν2f + c2 − Ef E PC (f) (f) (f) (f) H1212 = H2121 = H1221 = H2112 1 1 + νPC 1 + νf (c1 − c2 ). = Vf − 2 E PC Ef
(f)
(f) (f) (f) (f) H1212 = H2121 = H1221 = H2112 = Vf
In Eq. (13), the transverse elastic properties of fibers and Pyro-C material have been assumed isotropic with Young’s moduli and Poisson’s ratios E f , νf and E PC , νPC , correspondingly. Following Eq. (3), we represent the elastic compliance of the carbon-fiber-reinforced matrix as a sum of two terms: S(FPC) = S(PC) + H(f) ,
(f) (f) H1111 = H2222 d1 (c1 + c2 ) d2 (c1 − c2 ) + = Vf 1 − Vf (1 − c1 − c2 ) 1 − Vf (1 − c1 + c2 )
(1 + νf )E PC − (1 + νPC )E f . 2E PC E f
The effective elastic properties of the fiber-reinforced Pyro-C matrix in the transverse plane are isotropic. In the assumption of noninteracting fibers, the corresponding Young modulus and Poisson ratio are found from Eqs. (14) and (15a) as E f E PC E f 1 + 2νPC + ν2PC − E f 1 + 2νPC + ν2PC 2 D − E PC 1 + 2νf + ν2f Vf c1 + E f 2 + νPC − ν2PC
−E PC 2 + νf − ν2f Vf c2 [16a] 1
= E f νPC (1 + νPC ) − [E f νPC (1 + νPC ) D − E PC νf (1 + νf )]Vf c1 + E f 1 − ν2PC
− E PC 1 − ν2f Vf c2
E FPC =
νFPC
where D = [E PC (1 + νf ) − E f (1 + νPC )]Vf c1 + [E f (1 + νPC ) − E PC (1 + νf )]Vf c2 + E f + νPC E f . Similarly, in the case of the Mori–Tanaka approximation for interacting fibers, the corresponding Young modulus and Poisson
49
POROUS CARBON/CARBON COMPOSITES MODELING
ratio are found from Eqs. (14) and (15b) as E FPC =
A1 − 2A2 , (−A1 + A2 )2
νFPC =
A2 , −A1 + A2
[16b]
where coefficients A1 and A2 are expressed as Vf d2 (c1 − c2 ) 1 − ν2PC Vf d1 (c1 + c2 ) + + E PC 1 − Vf (1 − c1 − c2 ) 1 − Vf (1 − c1 + c2 ) Vf d2 (c1 − c2 ) νPC + ν2PC Vf d1 (c1 + c2 ) A2 = − − . + E PC 1 − Vf (1 − c1 − c2 ) 1−Vf (1 − c1 + c2 )
A1 =
To illustrate these results, let us consider the T300 carbon fibers manufactured by Amoco with transverse properties E f = 14.7 GPa and νf = 0.47 [28] embedded into the stiff pyrolytic graphite having E PC = 38.6 GPa and νPC = 0.16. Variation of transverse Young’s modulus of the effective matrix with fiber concentration Vf is shown in Figure 8 for both noninteraction approximation and the Mori–Tanaka scheme. As can be seen, for this contrast of constituent stiffnesses (E f /E PC = 0.38), the deviation in the predicted effective elastic modulus is less then 7%, even at relatively high fiber volume fraction of Vf = 0.4. 5. CONTRIBUTION OF PORES INTO THE EFFECTIVE ELASTIC PROPERTIES For the purpose of this study, we consider pores as sufficiently long and being placed into the equivalent transversely isotropic elastic matrix. The elastic moduli of this matrix in the trans-
FIG. 8.
verse plane (Young’s modulus E FPC and Poisson’s ratio νFPC ) were derived in Section 4. Some preliminary results on the contribution of pores into elastic moduli of C/C composites have been reported by Novak et al. [29]. To evaluate the contribution of the irregularly shaped pores into the effective elastic properties of the CVI-densified C/C composites, we employ the numerical conformal mapping (NCM) procedure [14]. The procedure is based on the complex variable approach [30]. It uses the conformal mapping of the exterior of a pore onto the interior of a unit circle with the mapping function found by numerical evaluation of the Schwarz–Christoffel integral. Let us consider a hole of arbitrary shape in the complex plane z = x1 + i x2 with the origin inside, as in Figure 4. The solution requires mapping of the interior of the unit circle in canonical plane ζ onto the exterior of the hole in the z plane by the analytical mapping function ω(ζ). For arbitrary polygons, this can be done by evaluation of the Schwarz–Christoffel integral:
ω(ζ) =
ζ z1
k
ζ 1− zk
1−βk
1 ζ2
dζ,
[17]
where βk π are the interior angles of the polygon and ζk are the prevertices (points on the unit circle in the canonical plane that correspond to the vertices of the polygon z k ). We approximate the boundary of the hole via N -sided polygon. The accuracy of this approximation depends on the number of vertices of the polygon. In the case of regular polygons, the Schwarz–Christoffel integral can be evaluated analytically [31]. For more complicated shapes, this integral must be evaluated
Young’s modulus of the equivalent matrix (FPC) consisting of Pyro-C and carbon fibers as function of fiber concentration Vf .
50
I. TSUKROV ET AL.
numerically. (We used the MATLAB Schwarz–Christoffel toolbox developed by Driscoll [32].) The numerical character of this mapping function causes very low accuracy in calculations of the higher order derivatives needed for the evaluation of strains and stresses. To avoid this, we obtain the mapping function in closed form by slight modification in the shape of the approximating polygon. The procedure is described in the next paraghraph. The integrand is expanded in truncated Laurent series with the center at the origin ζ = 0: M ζ 1−βk 1 ∼ 1− ajζj. [18] = 2 ζ ζ k k j = −2 (Note that the integrand in Eq. (17) is analytic and single-valued in the entire domain of the unit disk except at singular points: prevertexes ζk and the origin ζ = 0. Also, coefficient a−1 is set to zero to satisfy the single-valuedness of the resulting mapping function.) Having constructed the mapping function ω(ζ), we are able to utilize the Kolosov–Muskhelishvili approach to solve the elasticity problem for an irregular 2D hole in an infinite plate loaded by remotely applied uniform tension. According to Muskhelishvili [30], the stresses and displacements can be expressed in terms of two complex stress functions ϕ(ζ) and ψ(ζ), as follows: 3 − νFPC 1 + νFPC ω(ζ) u + iv = ϕ(ζ) − ϕ (ζ) + ψ(ζ) E FPC E FPC ω (ζ) ϕ (ζ) σxx + σyy = 4Re ω (ζ) [19] σyy − σxx + 2iτxy ϕ (ζ) ω (ζ) ψ (ζ) =2 (ζ) ω(ζ) + − ϕ . ω2 (ζ) ω (ζ) ω3 (ζ) We represent functions ϕ(ζ) and ψ(ζ) as Taylor series and find the coefficients from the traction-free boundary conditions. By requesting that these boundary conditions are satisfied at a discrete set of boundary points, we obtain the system of linear equations for unknown Taylor coefficients. After this system is solved, functions ϕ(ζ) and ψ(ζ) are known, and the displacement of the hole boundary is calculated using Eq. (19). With the aforementioned procedure, 60–250 boundary points proved to be sufficient to obtain accurate results for most of the tested shapes. With boundary displacements known, we can calculate the components of additional strain tensor for any direction θ of uniaxial tension by numerically evaluating integrals (6). Repeating this procedure for various angles we find the components of ε and σ as numerical functions of θ. The components of the H tensor are obtained by comparing these functions with Eq. (10). We select a sufficient number of θ values and substitute them into Eq. (10) together with the corresponding values
of ε and σ. This produces a system of linear equations for components of the H tensor. Note that to improve the accuracy one may choose more values of θ (we used 10) and obtain an overdetermined system of linear equations. The components of H are then found by solving this system using the least-squares method. To analyze the unidirectional C/Pyro-C composite described in Section 2, we selected several typical pore shapes as shown in Table 2. (Note that some of the presented shapes are obtained from the micrographs that have not been included in Figures 5 and 6.) The components of the fourth rank hole compliance contribution tensor H(N ) for each individual pore were found using the NCM procedure. In Table 2, they are expressed in terms of the hole shape factors h i (i = 1, . . . , 6), related to the hole area Ahole and Young’s modulus of the matrix, E M , as follows: (N ) H1111 =
Ahole h AE M 1
(N ) H2222 =
Ahole h AE M 2
(N ) H1211 =
Ahole h AE M 5
(N ) H1222 =
Ahole h AE M 6
(N ) H1122 = (N ) H1212 =
Ahole h AE M 4 Ahole h . 2AE M 3
[20]
The remaining components of H are found from symmetry con) (N ) (N ) ditions Hi(N conjkl = H jikl = Hkli j . The overall hole compliance (N ) tribution tensor is obtained by direct summation, H( p) = H , and the effective compliance of composite is S = S(FPC) + H( p) . Thus, the effective elastic moduli are expressed in terms of components of the hole compliance contribution tensor. Assuming that pores of each shape type are randomly distributed and randomly oriented in the plane perpendicular to the direction of fibers, the overall material is transversely isotropic. The expressions for transverse effective Young’s modulus and Poisson’s ratio are E=
1+
E FPC
(N ) 3 (h N Vp 8 1
+ h 2 ) + 14 (h 3 + h 4 )
[21] N
νFPC − N Vp(N ) 18 (h 1 + h 2 ) + 34 h 4 − 14 h 3 N ν= , [22] 1 + N Vp(N ) 38 (h 1 + h 2 ) + 14 (h 3 + h 4 ) N where (h 1 , . . . , h 4 ) N and Vp(N ) are the shape factors and porosity of pores of the N th type. As can be seen from Table 2, the numerical values of h coefficients vary greatly for different pore shapes—they depend not only on the pore shape but also on its orientation with respect to coordinate axes. However, for a sufficiently large number of pores of each shape randomly distributed and oriented in the material, only two combinations of these coefficients describe contribution of each defect shape: 3 (h 1 + h 2 ) + 8 1 B = (h 1 + h 2 ) + 8 A=
1 (h 3 + h 4 ) 4 3 1 h4 − h3. 4 4
[23]
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POROUS CARBON/CARBON COMPOSITES MODELING
TABLE 2 Hole shape factors of typical Pyro-C pores (δA and δB quantify deviation of A and B from average values)
Average
A
δA (%)
B
0.178
4.015
−6.4
−1.146
0.1
−0.705
−0.084
3.880
−9.5
−1.088
−4.9
−1.243
−1.186
−1.233
4.902
14.3
−1.149
0.4
6.839
−1.104
−0.237
−0.185
5.057
17.9
−1.330
16.2
6.111
4.804
−1.461
−0.567
−0.353
4.328
−0.9
−1.132
−1.0
4.522
3.283
4.792
−1.124
0.607
0.607
3.844
−10.4
−1.066
−6.9
3.025
4.879
5.167
−1.063
−0.164
−0.276
3.990
−6.9
−1.101
−3.8
3.783
4.930
5.295
−1.213
−0.253
−0.192
4.288
0.0
−1.144
0.0
h1
h2
h3
h4
h5
h6
2.881
5.433
4.875
−1.288
0.479
2.960
5.042
4.725
−1.210
3.628
6.366
5.861
6.263
3.399
3.201
The analysis of calculated values of A and B for each shape (Table 2) shows surprising closeness of these parameters for different geometries of pores present in Pyro-C. We attribute this fact to the identical manufacturing process during which they were formed. Comparison of A and B with the corresponding parameters of selected regular shapes (Table 3) shows that no regular shape can be easily chosen to adequately model contribution of PyroC pores to the effective properties, even in the case when pores are randomly oriented and overall properties are isotropic. To demonstrate this phenomenon, Figure 9 presents variation of the overall Young’s modulus with porosity for various pore shapes. It is observed that the curves for various typical Pyro-C pores
δB (%)
are located much closer to the “averaged pore contribution” than the curves for presented regular holes. Thus, the effective transverse elastic moduli of the unidirectional porous C/C composite described in Section 2 are E=
E FPC , 1 + Vp A
ν=
νFPC − Vp B , 1 + Vp A
[24]
where A and B are the average values for pore contributions (last row of Table 2); porosity Vp and fiber volume fraction Vf can be determined by the analysis of micrograph images (as in Table 1); and E FPC , νFPC are expressed in terms of mechanical properties of fibers and Pyro-C by Eqs. (16).
52
I. TSUKROV ET AL.
FIG. 9.
Variation of Young’s modulus with porosity for various pore geometries (shape factors of typical pores 1 and 2 are taken from Table 2).
6. EVALUATION OF THE MECHANICAL PROPERTIES OF PYROLYTIC CARBON BASED ON THE MICROMECHANICAL ANALYSIS Mechanical properties of the pyrolytic carbon in C/C composite strongly depend on the deposition conditions (such as temperature and pressure) and cannot be determined from the experiments on the bulk material. We apply the procedure pre-
sented in the preceding sections to determine the elastic moduli of the in situ Pyro-C matrix from the tests performed on the entire composite. A series of uniaxial tension as well as three- and fourpoint bending tests were conducted with the composite described in Section 3 (see [23]). The results for stress–strain dependence in longitudinal and transverse directions are shown in Figure 10. The corresponding effective elastic properties of the composite
FIG. 10. Results of uniaxial tension tests performed on C/C composite (adopted from [23]).
53
POROUS CARBON/CARBON COMPOSITES MODELING
TABLE 3 Shape factors of selected regular holes h1
h2
h3
h4
h5
h6
A
B
6.916
6.880
7.962
−1.008
0.032
−0.027
6.912
−1.022
3.000
3.000
4.000
−1.000
0.000
0.000
3.000
−1.000
4.187
4.121
8.766
−0.241
0.030
−0.026
5.247
−1.334
1.200
20.99
12.09
−0.999
0.000
0.000
are estimated as E l = 53.6 GPa, νl = 0.33, E t = 4.4 GPa, and νt = 0.29, where subscripts l and t refer to the longitudinal and transverse directions, correspondingly. The mechanical properties of the fibers are known from R. Ermel (personal communication, 2002) and Donnet et al. [28] as E f,l = 190 GPa, νf,l = 0.26, E f,t = 14.7 GPa, and νf,t = 0.47. The fiber volume fraction and porosity were found in Section 3 as Vf = 0.18 and Vp = 0.29. In the longitudinal direction, the elastic stiffness of a unidirectional composite can be estimated with a reasonable accuracy by the rule of mixtures. The Young modulus of the Pyro-C in this direction is then found as E PC,l =
E l − Vf · E f,l 53.6 − 0.18 · 190 = = 36.6 GPa. 1 − Vp − Vf 1 − 0.29 − 0.18 [25]
This value is in reasonably good agreement with the experimental measurements E PC,l = 38.6 GPa of Papadakis and Bernstein [33]. One must remember, however, that the properties of the pyrolytic carbon vary considerably with the change of the manufacturing parameters, so the closeness of our estimates with their ultrasonic measurements might be accidental. In the transverse direction, the procedure is considerably more complicated. It is based on the results of Sections 2–5 and follows the steps presented in Figure 2. First, we have to determine the Young modulus and Poisson ratio of the equivalent matrix (FPC) consisting of Pyro-C and carbon fibers. Using Eqs. (24) and the average pore characteristics (Table 2), we obtain E FPC = 9.9 GPa and νFPC = 0.25. Then the transverse proper-
11.10
−0.998
ties of Pyro-C are calculated from Eq. (16b) as E PC,t = 8.5 GPa and νPC,t = 0.16. As can be seen, the elastic stiffnesses of Pyro-C in the longitudinal and transverse directions differ considerably. This can be expected, because the presence of fibers during CVI densification creates preferential directions in pyrolytic carbon—an observation also supported by the microscopic studies of Reznik et al. [10]. 7. CONCLUSIONS The proposed micromechanical modeling procedure for unidirectional composites involves calculation of the fourth-rank compliance contribution tensors for fibers and pores. For circular elastic fibers, the components of the H tensor can be obtained in closed form, and this yields the explicit representations for elastic moduli of the fiber-reinforced matrix material. The irregular shapes of pores are analyzed using a numerical conformal mapping procedure, and the hole shape factors produced by numerical analysis enter the expressions for the effective elastic constants of the considered composite. It has been observed that numerical parameters describing the contribution of various pyrolytic carbon pores are relatively close. We attribute this phenomenon to the identical manufacturing procedure during which they were formed. The corresponding parameters of selected regular holes are noticeably different. Thus, there is no obvious way to approximate the real pores in the composite by regular shapes without considerable loss of accuracy.
54
I. TSUKROV ET AL.
The CVI densification of the unidirectional C/C composite results in the formation of pyrolytic carbon that has different mechanical properties in the longitudinal and transverse directions. Thus, not only the resulting composite but also the Pyro-C matrix cannot be modeled as isotropic material. ACKNOWLEDGMENTS The authors would like to thank Dipl.-Ing. R. Ermel for providing the micrographs of the unidirectional C/C composite. We are also grateful to Prof. K. J. Huttinger and Dr. B. Reznik for useful discussions. This research was partially supported by the German Research Foundation through the grant to Collaborative Research Center 551, “Carbon from the Gas Phase: Elementary Reactions, Structures, Materials.” REFERENCES 1. M. Kachanov, I. Tsukrov, and B. Shafiro, Effective Properties of Solids with Cavities of Various Shapes, Appl. Mech. Rev., vol. 47, no. 1, pp. S151–S174, 1994. 2. I. Sevastianov and M. Kachanov, Explicit Cross-Property Correlations for Anisotropic Two-Phase Composite Materials, J. Mech. Phys. Solid, vol. 50, pp. 253–282, 2002. 3. Z. Hashin and S. Shtrikman, On Some Variational Principles in Anisotropic and Non-Homogeneous Elasticity, J. Mech. Phys. Solid, vol. 10, pp. 335– 342, 1962. 4. Z. Hashin and S. Shtrikman, A Variational Approach to the Theory of the Elastic Behavior of Multiphase Materials, J. Mech. Phys. Solid, vol. 11, pp. 127–140, 1963. 5. R. Hill, Elastic Properties of Reinforced Solids: Some Theoretical Approaches, J. Mech. Phys. Solid, vol. 11, pp. 357–372, 1963. 6. L. J. Walpole, On Bounds for the Overall Elastic Moduli of Inhomogeneous Systems—I, J. Mech. Phys. Solid, vol. 14, pp. 151–162, 1966. 7. L. J. Walpole, On Bounds for the Overall Elastic Moduli of Inhomogeneous Systems—II, J. Mech. Phys. Solid, vol. 14, pp. 289–301, 1966. 8. J. D. Eshelby, The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems, Proc. R. Soc. Lond. A, vol. 241, pp. 376– 396, 1957. 9. E. J. Garboczi and A.R. Day, An Algorithm for Computing the Effective Linear Elastic Properties of Heterogeneous Materials: Three-Dimensional Results for Composites with Equal Phase Poisson Ratios, J. Mech. Phys. Solid, vol. 43, no. 9, pp. 1349–1362, 1995. 10. B. Reznik, D. Gerthsen, and K. J. H¨utinger, Micro- and Nanostructure of the Carbon Matrix of Infiltrated Carbon Fibre Felts, Carbon, vol. 39, no. 2, pp. 215–229, 2001. 11. W. Benzinger and K. J. H¨uttinger, Chemistry and Kinetics of Chemical Vapor Infiltration of Pyrocarbon—V. Infiltration of Carbon Fiber Felt, Carbon, vol. 37, no. 6, pp. 941–946, 1999.
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