Key Engineering Materials Vols. 251-252 (2003) pp 339-344 Online available since 2003/Oct/15 at www.scientific.net Journal (to bePublications, inserted by the publisher) © (2003)Citation Trans Tech Switzerland doi:10.4028/www.scientific.net/KEM.251-252.339 Copyright by Trans Tech Publications
Influence of Pores on Effective Elastic Properties of Unidirectional Carbon/Carbon Composites I. Tsukrov1, R. Piat2 , J. Novak 1 and E. Schnack2 1
Department of Mechanical Engineering, University of New Hampshire, Durham, NH 03824 USA
2
Institute of Solid Mechanics, University of Karlsruhe, Kaiserstr.12, D-76128 Karlsruhe, Germany,
[email protected]
Keywords: effective elastic properties, modeling, composites, polarized light microscopy, microstructure.
Abstract. In this paper, we discuss the effect of porosity on the effective elastic properties of unidirectional carbon / carbon composites (carbon fibers in pyrolytic carbon matrix) densified by chemical vapor infiltration (CVI). The resulting composite consists of carbon fibers embedded in a porous matrix of pyrolytic carbon (Pyro-C). Such materials 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 in [1]. In dependence on the CVI conditions pores, with geometry, forms between fibers with pyro-C coating [2,3]. It is well known in the composite material theory that elastic moduli of unidirectional composites in longitudinal direction can be predicted with good accuracy by the so-called rule of mixtures [4]. Modeling of the effective elastic properties in the plane perpendicular to the fiber direction (transverse elastic moduli) is a more challenging task that requires not only knowledge of the transverse elastic constants for all the constituents but also adequate information on phases’ distribution and detailed geometry. These pores are analyzed using numerical conformal mapping procedure [5,6], 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 and compared with experimental results. Introduction Carbon/carbon (C/C) composites are manufactured by driving carbon-hydrate gas vapor through the highly porous bundles of carbon fibers at certain temperatures and pressures (CVI). During this process carbon polycrystals build up on the fiber cores filling up the large spaces between fibers with pyrolytic carbon material. As a result, large voids become partly densified decreasing the overall porosity. This significantly improves the material’s mechanical characteristics [7]. In this paper, we focus on the influence of resulting porosity on the overall elastic properties of C/C composite. It is well known in the composite material theory that elastic moduli of unidirectional composites in longitudinal direction can be predicted with good accuracy by the so-called rule of mixtures (see, for example[4]). Modeling of the effective elastic properties in the plane perpendicular to the fiber direction (transverse elastic moduli) is a more challenging task that requires not only knowledge of the transverse elastic constants for all the constituents but also adequate information on phases’ distribution and detailed geometry. To analyze the transverse effective elastic properties of unidirectional porous C/C composites, we employ the near-field micromechanical modeling procedure as described in [5,6]. It is assumed that All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 129.13.72.198-27/05/11,14:58:01)
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pores are sufficiently long in the direction of fibers so that the plane strain model can be used. Also, for the purpose of this study, the pyrolytic carbon material with randomly distributed fibers is treated as an equivalent transversely-isotropic matrix. Its Young’s modulus and Poisson’s ratio in the plane of isotropy are denoted as E M and n M , correspondingly. The homogenization procedure to predict the overall elastic properties of the pyrolytic carbon based on its nanostructure is presented in [8]. Analysis of microstructure A typical micrograph obtained by polarized light microscopy [2] is shown in Fig. 1a. We observe gray circular regions representing fibers surrounded by the pyrolytic carbon matrix, while large black regions of irregular shape indicate pores. This micrograph, along with other images of the same material, was processed and analyzed using ColorPoint 2.0 software [9]. Fig.1b shows the processed image with the pores placed in the equivalent matrix that consists of fibers and pyrolytic carbon. The average porosity obtained from the analysis of the micrographs for composite shown in Fig.1a is p = 0.11 .
b/
a/
Fig. 1. Example of unidirectional carbon / Pyro-C composite microstructure a/ Polarized light micrograph b/ Processed image (white – pores, gray - equivalent matrix consisting of PyroC and fibers) As can be seen from Fig. 1b, the pores in this composite material have very irregular shapes that cannot be accurately approximated by circles, ellipses or right polygons for which analytical elasticity solutions are available. Thus, a special procedure is needed to analyze these irregular pore shapes and their contribution to the effective elastic properties. Contribution of pores into the effective elastic properties To evaluate the effect of irregularly shaped pores on the effective elastic moduli of CVI densified C/C composites we employ the numerical conformal mapping (NCM) procedure recently developed by [6]. Contributions of the pores are described by the fourth-rank cavity compliance contribution tensors. The effective elastic compliance of the composite is represented as
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H (N )
S = SM +
341
(1)
N
where S M is the compliance of the equivalent matrix material, and H ( N ) is the hole compliance contribution tensor of the pore shape of N-th type. For each pore shape, the components of the corresponding H -tensor are expressed in terms of the hole area Ahole and shape factors hi (i = 1,K 6) : H 1111 = H 1211
Ahole h1 AE M
A = hole h5 AE M
H 2222 = H 1222
Ahole h2 AE M
H 1122 =
A = hole h6 AE M
H 1212
Ahole h4 AE M
A = hole h3 2 AE M
(2)
The remaining components of H are found from symmetry conditions H ijkl = H jilk = H klij . Hole shape factors constitute the proper geometric parameters that define contribution of pores of a certain shape into the effective elastic properties. To find these factors, the elasticity problem for a hole in the infinite plane subjected to the remotely applied stress field must be solved. This is done by conformal mapping of the region outside of the hole onto the unit disk and construction of the complex stress potentials satisfying the traction-free boundary conditions [10]. The conformal mapping function is found by numerical evaluation of Schwarz-Christoffel integrals. To obtain the hole shape factors, numerical elasticity solutions are compared with analytical expressions for components of H -tensor. The entire procedure is described in [6]. Assuming that pores of each shape type are randomly distributed and randomly oriented in the composite with porosity p N , the overall material is isotropic in the plane perpendicular to fibers. In this plane, the expressions for effective Young’s modulus and Poisson’s ratio are EM
E= 1+
pN N
uM -
pN N
u=
3 (h1 + h2 ) + 1 (h3 + h4 ) 8 4
1+
pN N
=
EM 1+ p N AN
(3)
N N
1 (h1 + h2 ) + 3 h4 - 1 h3 u M - p N BN 8 4 4 N N = 3 1+ p N AN (h1 + h2 ) + 1 (h3 + h4 ) N 8 4 N
(4)
As seen from Eq.3 and Eq.4, only two combinations of h-factors, AN and B N , describe contribution of each pore shape into the effective elastic properties. To analyze the transverse elastic moduli of the unidirectional C/C composite, we select several typical pore shapes as shown in Table 1. Note that some of the shapes presented in Table 1 are obtained not from the processed image in Fig. 1b, but from other micrographs of the same material. Application of NCM procedure yields the numerical values of h-coefficients that are also given in the Table. These values vary greatly for different pores - they depend not only on the shape of the pore but also on its orientation with respect to coordinate axes. However, comparison of calculated values of AN and B N for each pore shape shows surprising closeness of these parameters for different geometries. We attribute this fact to the identical manufacturing process during which cavities were formed. Assuming that all defects are presented with the same concentration, the above formulae can be simplified as
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E=
EM , 1+ A p
u=
uM - B p
(5)
1+ A p
where A and B are the averages of AN and BN .
h1
h2
h3
h4
h5
h6
dA
B
dB
6.263 3.399 6.839 -1.104 -0.237 -0.185 5.057
16.3%
-1.330
14.5%
2.960 5.042 4.725 -1.210 -0.705 -0.084 3.880
-10.7%
-1.088
-6.3%
3.628 6.366 5.861 -1.243 -1.186 -1.233 4.902
12.8%
-1.149
-1.1%
5.862 2.840 5.846 -1.191
4.427
1.9%
-1.267
9.0%
3.201 6.111 4.804 -1.461 -0.567 -0.353 4.328
-0.4%
-1.132
-2.5%
4.522 3.283 4.792 -1.124
3.844
-11.6%
-1.066
-8.3%
3.025 4.879 5.167 -1.063 -0.164 -0.276 3.990
-8.2%
-1.101
-5.2%
4.209 4.560 5.433 -1.199 -0.243 -0.164 4.347
0.0%
-1.162
0.0%
A
x2 x1
x2
x1 x2
x1
x2 x1
0.554
0.378
x2 x1
x2 0.607
0.607
x1 x2 x1
AVERAGE
Table 1. Typical carbon/carbon composite cavity shapes Fig. 2 presents variation of the overall Young’s modulus with porosity for various hole shapes. The results for equilateral triangles and circles are also shown for comparison. As can be seen, the curves for individual pyrolytic carbon pore shapes are located in a close vicinity of the “average pyrolytic carbon” curve. Even the hole with maximum deviation of A and B from the corresponding average values (“Table 1 - Pyro-C pore 1”) exhibits behavior that is is much closer to the average response than that of the holes of regular (triangular or circular) shapes.
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1,00 Table 1 - Pyro-C pore 1 Table 1 - Pyro-C pore 2 Average - Pyro-C pores Circular hole
0,75 E/E M
Triangular hole
0,50
0,25 0
0,05
0,1
0,15
0,2
0,25
porosity
Fig. 2. Variation of Young’s modulus with porosity for various hole’s shapes Conclusions The numerical conformal mapping method has been applied to analyze contribution of pores into the effective elastic properties of C/C composites densified by chemical vapor infiltration. For typical pore shapes, the complete set of components of the cavity compliance tensor has been found. It has been observed that numerical parameters describing the contribution of various pyrolytic carbon pores are relatively close. We attribute this to the identical manufacturing procedure during which they were formed. Since the effects of selected regular holes are noticeably different, it is not possible to approximate the real pores in the composite by regular shapes without considerable loss of accuracy.
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 (DGF) through the grant to Collaborative Research Center 551 “Carbon from the gas phase: elementary reactions, structures, materials”. References [1] R. Piat and E. Schnack: Carbon, submitted for publication. [2] R. Ermel :Private communications. [3] W. Benzinger and KJ. Hüttinger: Carbon. Vol. 37, 6 (1999), pp.941-946. [4] R.M. Christensen: Mechanics of composite materials ( Krieger Publishing Co., Malabar, USA 1991). [5] M. Kachanov, I. Tsukrov and B. Shafiro: Appl. Mech. Rev. Vol. 47, 1 (1994), pp.151-S174. [6] I. Tsukrov and J. Novak :Int. J. Sol. Struct. (2002). Vol. 39 (2002), pp.1539-1555.
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[7] B. Reznik, M. Guellali, D. Gerthsen, R. Oberacker and M.J. Hoffmann: Materials Letters Vol. 52, (2002), pp. 14-19. [8] R. Piat, B. Reznik, E. Schnack and D. Gerthsen : Carbon, accepted for publication. [9] P.G. Patri: ColorPoint 2.0 software. http://www.patrilab.com. [10] N.I. Muskhelishvili: Some Basic Problems of Mathematical Theory of Elasticity (Noordhoff, Groningen 1963).
Advances in Fracture and Damage Mechanics III doi:10.4028/www.scientific.net/KEM.251-252 Influence of Pores on Effective Elastic Properties of Unidirectional Carbon/Carbon Composites doi:10.4028/www.scientific.net/KEM.251-252.339
