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A model for the transverse strain response of unidirectional ceramic matrix composites during tensile testing
Materials Science and Engineering A250 (1998) 222 – 230
A model for the transverse strain response of unidirectional ceramic matrix composites during tensile testing Eddy Vanswijgenhoven *, Omer Van Der Biest Department of Metallurgy and Materials Engineering, Katholieke Uni6ersiteit Leu6en, De Croylaan 2, B-3001 He6erlee, Belgium
Abstract A comprehensive micromechanical model relating the longitudinal stress and transverse strain of unidirectional fibre toughened ceramic matrix composites (CMCs) is presented. The model uses different cylindrical unit-cells to describe the composite throughout a tensile test and considers all relevant damage mechanisms. The proposed model takes into account the Poisson contraction of fibre and matrix, the relief of thermal residual stresses upon damage development, and the build-up of compressive radial stresses at the interface due to mismatch between fibre and matrix after debonding and sliding. Thus the modelled transverse strain response depends on a wide range of microstructural and micromechanical parameters. The approach is checked by comparing the experimentally observed and simulated response of a unidirectional SiC/CAS composite of which all constituent properties were determined experimentally. The agreement between experiment and theory is excellent. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Ceramic matrix composites; Interfacial debonding; Transverse strain
1. Introduction The relation between longitudinal stress s and transverse strain otr of unidirectional fibre toughened ceramic matrix composites has been studied experimentally by several authors [1 – 8]. Fig. 1 shows the transverse strain response of different ceramic matrix composites (CMCs) during tensile testing. Initially the transverse strain decreases linearly with applied stress. Matrix cracking generally causes the transverse strain of the composite to become positive. When the saturation crack density is reached, the transverse strain remains approximately constant up to failure. The transverse strain of some composite systems, however, continues to decrease throughout a tensile test. It has been suggested that the observed strain reversal is caused by fibres tending to broom out [1] and the relief of residual stress as the matrix cracks [2 – 8], axial cracking of the matrix [3,4], and the build-up of radial mismatch at debonded interfaces [6,7]. A comprehensive model for the transverse strain response of unidirectional fibre toughened composites in not available in literature. Trying to explain the * Corresponding author. 0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0921-5093(98)00595-4
observed behaviour Sørensen et al. presented a micromechanical study based on the finite element method (FEM) [6]. The overall response of the composite was deduced from the response of a representative volume element consisting of concentrically placed cylinders of fibre and surrounding matrix, bounded axially by a matrix crack and a symmetry plane. The interface between fibre and matrix was assumed to be a frictional sliding contact. In all cases the FEM model did not give the experimentally observed positive transverse strains. This was due to the disappearance of the fibre/matrix contact at sufficiently high stresses and the assumption of perfectly smooth fibres. It was suggested that the roughness effect, as also evidenced and modelled by other authors for different loading conditions [9–20], could override the Poisson contraction of the fibre and cause an increase in transverse strain upon fibre/matrix interface sliding during tensile testing. Indeed by assuming that stress transfer between fibre and matrix along a debonded interface follows Coulomb’s friction, Sørensen [7] proved that the loss of the interfacial contact by Poisson contraction of the fibre could be compensated by fibre roughness. However, his model only gave a description of a totally debonded composite material and did not fully describe the evolution of the
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transverse strain as a function of longitudinal stress. In addition to the model proposed by Sørensen [7], two other models relating longitudinal stress and transverse strain have been proposed [21,22]. These models do however neglect interfacial mismatch and are guaranteed to be significantly wrong. In this paper, a comprehensive model relating the longitudinal stress and transverse strain of unidirectional fibre toughened ceramic matrix composites is proposed. The model describes the transverse strain response of a model composite throughout an entire tensile test, taking into account all relevant damage. The mechanisms governing the response of the composite are the Poisson contraction of fibre and matrix, the relief of residual thermal stresses upon damage development, and the build-up of compressive radial stresses at the interface caused by interfacial mismatch after debonding and sliding. The proposed model is the first which describes the material response throughout an entire test and takes into account all mechanisms relevant to the transverse strain response. In addition its analytical and micromechanical character gives the material engineer a versatile and powerful tool to evaluate the impact of changes in constituent properties. The paper is organised as follows: a detailed description of the proposed model in Section 2 is followed by a comparison between the experimentally observed and theoretically predicted transverse strain response of unidirectional SiC/CAS in Section 3. Section 4 contains some concluding remarks.
2. Model description The proposed model focuses on the transverse strain response of a unidirectional fibre toughened CMC under uniaxial stressing parallel to the fibres. The model
Fig. 1. Experimentally observed relationships between the longitudinal stress and the transverse strain of different CMCs during tensile testing.
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Fig. 2. Schematic representation of the microstructure and the damage state in an idealised composite and an example of the description of the idealised composite by a unit-cell.
composite consists of a hexagonal array of perfectly aligned fibres in a matrix (Fig. 2). The thickness of the interphase between fibre and matrix is assumed to be negligible (it is a true interface), whereas the surface of the fibre is assumed to have a certain roughness. Before continuing with the description of the model, the most important assumptions made are briefly described and discussed. These assumptions concern: (1) the development of damage during tensile testing, (2) the interfacial mismatch after interface debonding and sliding, and (3) the stress transfer across debonded interfaces. Two major damage mechanisms are operative during tensile testing of unidirectional fibre toughened CMCs: multiple matrix cracking accompanied by interface debonding and fibre fracture accompanied by fibre pull-out [1–8,21–32]. The present model only considers the effect of matrix cracking and interface debonding on the transverse strain response. Fibre failure and pull-out are very important for modelling of the longitudinal strain response [22–32] but the effect on the transverse strain response is neglected in this paper as a first approximation because (1) the matrix is the continuous phase in the transverse direction, (2) before composite failure only a limited amount of fibres are broken in a representative material volume [25], and (3) composite failure is often a localised process [25]. The formation of matrix microcracks, which are often observed through acoustic emission or in-situ microscopy below the proportional limit, is not taken into
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account either, because this type of damage hardly influences the overall stress – strain response [2]. During macrocracking of the matrix, the number as well as the spatial distribution of the matrix cracks are related to the stress on the composite. Previous work on unidirectional fibre toughened CMCs has indicated that during tensile stressing matrix cracks develop into more or less periodic arrays, with a characteristic spacing [1–8,21–32]. The proposed model assumes the cracks to be equidistant and spanning the entire crosssection of the material while the dependence between the matrix crack density r and the stress s is assumed to be given by a three parameter Weibull function:
r(s)= rs 1−exp
s −s* s0
m
(1)
with rs the saturation matrix crack density, s* the stress at which matrix cracks first appear, s0 a scale parameter, and m the Weibull modulus governing matrix cracking [32]. These parameters are treated in this paper as independent constituent properties of the composite system which are determined experimentally. The authors realise however that they can be related to other, truly independent properties [23 – 26,32]. The second item to be discussed beforehand concerns the interfacial mismatch after interface debonding and sliding. At this point in the paper it is advisable to summarise the current knowledge on fibre roughness and interfacial mismatch. The fibres used in CMCs are known to exhibit a variation in diameter along their length [9–20]. Two types of diameter variation are to be distinguished: a variation over large distances (typically cm) and a variation over short distances (typically mm). In this paper only short distance variations are considered. A thorough study of the short distance topography of small diameter SiC fibres, as used in for example SiC/CAS, has not been reported. Preliminary work based on atomic force microscopy and the study of the push-out behaviour by Jero and reported by Kerans [15,16] estimated the short distance fibre roughness to be about 20 nm. This value is consistent with a roughness value of 17 nm reported by Martin et al. [19] and the value of 24 nm used by Sørensen [7] to explain the experimental behaviour of unidirectional SiC/CAS. The roughness wavelength was estimated to be about 1 mm by Jero et al. [12,13] by interpretation of the pushout and pushback behaviour of Nicalon fibres in an 1723 aluminosilicate glass. Later Kerans [15] corrected this value by taking into account the change of length of the loaded debonded fibre; he proposed a roughness period of about 0.3 mm. These roughness periods are of the same order of magnitude as the relative axial displacement between fibre and matrix at a debonded interface causing roughness to play a major role even early in the debonding process. From the above literature review, it is obvious that the descrip-
tion of the interfacial mismatch after interface debonding and sliding is a complex issue. Consequently, some simplifying assumptions are necessary to study the effect of the interfacial mismatch upon the otr response during tensile testing. First, it is assumed that there is no interphase between the fibre and the matrix; after processing, the fibre and the matrix are in direct contact. Second, the fibre is assumed to have a certain roughness profile. The proposed model is thus strictly only valid for composites with fibres that debond along a weak interface. It is however believed that the adopted approach will give reasonable approximations for other debonding scenarios [11]. Third, the fibre roughness is assumed to be random, non-periodic, and small compared with the fibre diameter and its effect on the transverse strain response is assumed to be described by an average roughness amplitude DRmax and an average roughness wavelength L [7,11,13–16]. Fourth, it is assumed that the average radial interfacial mismatch DR(s, x) at a distance x from the matrix crack along the debonded interface depends linearly on the relative axial displacement between fibre and matrix Du(s, x) when Du(s, x) is less than half of L [13,15,16]: DRf = DRmax ×
with Du(s, x)=
Du L if Du 5 L 2 2
& Ld(s) x
(2)
s deb s deb(x) f (x) − m dx Ef Em
&
(3)
if the fibre/matrix interface is not totally debonded or Du(s, x)=
1 (2r(s)) x
s deb s deb(x) f (x) − m dx Ef Em
(4)
if total debonding has occurred, Ld(s) the distance over deb which debonding occurs, s deb f (x) and s m (x) the axial stresses in the fibre and the matrix where the interface has debonded, and x the distance to the matrix crack. These parameters will be defined below. For relative axial displacements greater than half of the roughness wavelength, the interfacial mismatch is assumed to be at its maximum [13] (Fig. 3): L DRf = DRmax if Du ] . 2
(5)
The third topic to be discussed is the stress transfer across debonded interfaces. The fibre/matrix interphase is assumed to be sufficiently weak to allow debonding upon matrix cracking; this generally happens if Gi 5Gf/ 4 with Gi and Gf the fracture energy of the interphase and the fibre [26,33]. The matrix cracks are then bridged by fibres for which the interphase with the matrix has debonded over a certain distance. Interfacial slip occurs against an interfacial sliding stress t. Such sliding generally occurs in accordance with a friction
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law: t= t0 −msi, rwith m is the Coulomb friction coefficient of the interface and si, r the radial stress at the interface [6,26,34– 36]. In this paper, t is assumed to be constant, independent of the radial stress at the interface. The constant interfacial sliding stress assumption has been used with great success by several authors and it has been demonstrated that the use of an average interfacial sliding stress independent of the local radial stress at the interface has quite a small effect on the stress distributions in the fibre and the matrix [6,34,35]. By making the assumptions above, a composite consisting of a hexagonal distribution of fibres in a matrix can be described by combinations of three different unit-cells throughout a tensile test. For computational reasons cylindrical unit-cells are used [6,7,21,22,36,37] (Fig. 2). Each unit-cell consists of a fibre within a matrix with Rm = Rf/ Vf (Rf is the fibre radius, Rm is the radius of the unit-cells, and Vf is the fibre volume fraction) and represents a damage state characteristic for unidirectional fibre toughened CMCs. Before damage development, the composite is represented by a unit-cell with no damage. In the presence of matrix cracks and partially debonded interfaces, the composite is described by a corresponding unit-cell with a matrix crack and a partially debonded interface. After complete debonding, a unit-cell with a matrix crack and a totally debonded interface is used.
2.1. Unit-cell with no damage (unit-cell A) Fig. 4(a) schematically shows the unit-cell with no damage describing the undamaged composite. The fibre and the matrix are initially bonded and axial, radial, and circumferential thermal residual stresses are be-
Fig. 4. Schematic representation of unit-cell A with no damage (a), a unit-cell B with a matrix crack and a partially debonded interface (b), and a unit-cell C with a matrix crack and a totally debonded interface (c) together with the axial stress state in the fibre and the matrix.
lieved to be present within the fibre, the matrix and at the interface. The relationship between the longitudinal stress and the transverse strain of unit-cell A is assumed to be linear and was first given in this form by Hashin et al. [38,39]: s otr,A = − nc . Ec
Fig. 3. (a) Development of interfacial mismatch between fibre and matrix after interface debonding and sliding and (b) the proposed evolution of the interfacial mismatch DRf as a function of the difference in the axial displacement between the fibre and the matrix Du.
(6)
The composite Young’s modulus Ec and Poisson coefficient nc depend on the fibre volume fraction Vf and the Young’s moduli and the Poisson coefficients of fibre and matrix Ef, Em, nf, nm. The equations giving these relationships are given in [39] and are not reproduced here. The thermal residual stress state in the unit-cell plays an important role in the transverse strain response of
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the unit-cells with damage as will be demonstrated below. Most important are the axial residual stress in R the fibre S R f, z and the matrix S m, z and the radial residual stress at the fibre/matrix interface S R i,r. Several analytical models have been put forward to determine the elastic stress field in a set of two or more coaxial cylinders subject to thermomechanical loading [40–42]. In this paper the thermal stresses are calculated for a cylinder of fibre surrounded by a cylindrical matrix shell and assuming a radial stress free matrix surface. The thermal stresses depend on Vf, Rf, Ef, Em, nf, nm, the thermal expansion coefficients of the fibre and the matrix af, am, and the temperature difference between the test temperature and the stress free temperature DT (DT is negative). The equations used here for the calculation of the thermal residual stresses in unit-cell A are given in [42] and are not reproduced here.
2.2. Unit-cell with a matrix crack and a partially debonded interface (unit-cell B) Fig. 4(b) is a schematic representation of a unit-cell with a matrix crack and a partially debonded interface, and of the axial stress state in the fibre and the matrix. The length of the unit-cell B equals 1/r(s). The axial deb stress in the fibre s deb f (x) and the matrix s m (x) vary linearly with the distance to the matrix crack x according to the simple shear lag theory until they reach their value prior to cracking. The relationship between the stress on the unit-cell and the debond length Ld(s) is given by [43]: Ld(s)=
Rf s E f − s −S R f,z . 2t Vf Ec
(7)
The calculation of the transverse strain of this unitcell is done as follows. The transverse strain is first calculated as a function of the position in the cell x (x being the distance from the matrix crack) and then averaged over the entire cell: 1
&
otr,B(s)=
Ld(s)
o deb tr,B(s, x)dx +
0
&
2r(s)
o bon tr,B(s, x)dx
L d(s)
1/2 r(s)
DRf(s, x) nf s + + (am − af)DT Rf Vf Ef nm Vf nf t + −2 x Rf Em Vm Ef si,r(s, x) = . (9) 1+ Vf + nm Vm (1− nf) + Em Ef To ensure compatibility between unit-cells the outer surface of the cylinders must remain straight; i.e. the surface must have the same radial displacement along the cylinder length [6,7,21]. This is automatically true for the unit-cells with no damage. The straightness of the outer surface of unit-cells with a cracked matrix and a partially debonded interface is guaranteed as follows. If radial pressure continues to exist across debonded interfaces, then the effect of applying radial stresses on the part with a debonded interface does not differ from the effect of applying radial stresses on the part with a bonded interface. In both cases the unit-cell cylinder behaves as a linear elastic medium with the same elastic properties. Applying tensile and compressive radial stresses resulting in a constant transverse strain equal to the average transverse strain of the unit-cell with zero radial stresses at the outer surface ensures unit-cell compatibility. A similar procedure was followed by Sørensen et al. [6,7] and by He et al. [21]. The same approach is also true for the unit-cells with a matrix crack and a totally debonded interface. In the following, it is thus silently assumed that compressive radial stresses continue to exist across debonded interfaces. The proposed model is thus only valid for composite systems for which si, r as calculated above in Eq. (7) is negative. After calculation of the radial interfacial stress, the transverse strain in the part of the unit-cell with a debonded interface can be calculated. Elastic analysis results in the following equation for the transverse strain [7,44,45] (Fig. 5): −
.
(8)
The transverse strain response of the part of the unit-cell with a still bonded interface is similar to that of the unit-cell with no damage and is given by Eq. (4). The transverse strain in the part of the unit-cell with a debonded interface is calculated in two steps. First the radial stress at the debonded fibre/matrix interface is calculated as a function of s and x. Following the elastic analysis by Sørensen [7] and Timoshenko and Goodier [44] of a system consisting of a fibre cylinder surrounded by a hollow matrix cylinder with a radial interfacial mismatch the following formula for the radial stress at the interface is obtained [45]:
Fig. 5. Transverse expansion of an undamaged unit-cell at the stress free temperature, of an undamaged unit-cell at the test temperature in unloaded condition, and of a damaged unit-cell at the test temperature in loaded condition.
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o deb tr, B(s, x) =−
2Vf S R 2Vf si, r(s, x) s deb SR m (s, x) i, r m, z −nm + +nm Em Em Vm Vm Em Em (10)
The transverse strain in the part of the unit-cell with a debonded interface depends on a wide range of constituent properties. It increases, i.e. tends to become positive, for high compressive radial stresses at the interface, for small tensile axial stress in the matrix, for high tensile radial thermal residual stresses at the interface, and for high tensile axial thermal residual stresses in the matrix.
2.3. Unit-cell with a matrix crack and a totally debonded interface (unit-cell C) Unit-cell C is schematically represented in Fig. 4(c), together with the stress evolution in the fibre and matrix. The average transverse strain of this unit-cell is calculated in a manner similar to the transverse strain of the unit-cell with a partially debonded interface. The calculation is less complex since this unit-cell no longer consists of two parts with a different transverse response:
&
1 2r(s)
otr,C(s)=
otr,C(s, x)dx
0
1 2r(s)
(11)
Fig. 6. Experimentally observed relationship between longitudinal stress, longitudinal strain, and transverse strain during tensile testing of unidirectional SiC/CAS specimens investigated by Sørensen et al. [5 – 7].
creases steeply to a value of about 0.02%. At higher stresses, the transverse strain remains constant at this value. The constituent properties used for the simulation of the transverse strain response of the unidirectional SiC/ CAS under investigation are summarised in Table 1. Most of these properties have been determined experimentally by Sørensen et al. [5–7]. Only the choice of
with the transverse strain as in Eq. (10).
3. Comparison between the experimentally observed and simulated behaviour In this section, the experimentally observed and simulated relation between longitudinal stress and transverse strain of a unidirectional nicalon silicon carbide (SiC) fibre toughened calcium aluminosilicate (CAS) glass–ceramic matrix composite are compared. In the previous section it has been shown that the transverse strain response depends on a wide variety of constituent properties. All of these properties are relatively wellknown for the SiC/CAS system under investigation although some of them have not been determined with a high accuracy yet or are subject to inherent scatter [5 –7,15,16]. The experimentally observed relationship between longitudinal stress, longitudinal strain, and transverse strain during tensile testing of unidirectional SiC/CAS investigated by Sørensen et al. is given in Fig. 6 [5–7]. Initially the transverse strain decreases linearly with an increase in longitudinal stress (Poisson’s ratio of 0.24). Between 260 and 420 MPa, the transverse strain in-
Table 1 Constituent properties of unidirectional SiC/CAS used to simulate the transverse strain response in Fig. 10[5 – 7,15,16] Basic constituent properties Vf Rf Ef Em nf nm t DRmax L af am DT rs s* s0 m
0.35 7.5 mm 200 GPa 98 GPa 0.15 0.3 20 MPa 20 nm 0.5 mm 3×10−6 K−1 5×10−6 K−1 −1000 K 6 cracks mm−1 150 MPa 180 MPa 3
Derived properties Ec nc SR i, r SR f, z SR m, z
134 GPa 0.245 −91 MPa −232 MPa 125 MPa
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Fig. 7. A comparison between the experimentally observed relation between matrix crack density and longitudinal strain [5– 7] and the matrix crack density evolution used in the simulation.
the maximum interfacial mismatch DRmax, the interfacial mismatch wavelength L, and the relation between matrix crack density and stress r(s) are further highlighted in this paragraph. The experimentally determined evolution of the matrix crack density as observed on replicas is given in Fig. 7[5]. In this paper, the evolution of the matrix crack density is assumed to be given by the three parameter Weibull function of Eq. (1). The values used for s*, s0, and m are given by 150 MPa, 180 MPa, and 3, respectively(Fig. 7). The values for the maximum interfacial mismatch DRmax and the interfacial mismatch wavelength L used are 20 nm and 0.5 mm, respectively. This of the same order of magnitude as put forward by other authors [7,12,15,16]. The simulated damage development during tensile testing is schematically represented in Fig. 8. Up to a stress of 150 MPa, no damage develops and the behaviour is perfectly elastic. The composite is described by unit-cell A (Fig. 8(a)). The relationship between
Fig. 8. Schematic representation of the simulated damage development during tensile testing of unidirectional SiC/CAS; (a) no damage developing; (b) matrix cracking and interface debonding; (c) matrix cracking.
Fig. 9. Simulated evolution of the radial mismatch DRf as a function of the distance from the matrix crack x and the applied longitudinal stress s.
longitudinal stress and transverse strain is given by Eq. (6). Between 150 and 330 MPa, the composite is described by B unit-cells (Fig. 8(b)). The length of the B unit-cell equals the crack spacing. The relationship between longitudinal stress and transverse strain is given by Eqs. (7)–(9) and Eq. (10). At a longitudinal stress of 330 MPa, the interfaces become totally debonded. Above this stress level the composite is described by C unit-cells and the transverse strain is given by Eqs. (9) and (10) and Eq. (11) (Fig. 8(c)). Theoretically speaking matrix cracking should stop at this point since the axial stress in the matrix can no longer increase and the cracks are assumed to be equidistant. In reality cracking continues because spacings are not equal because time-dependent cracking occurs, and because some regions of the matrix are under high loads. In this paper the experimentally measured evolution of the matrix crack density is used as a first approximation. Because the present model does not take into account fibre failure, it cannot predict tensile failure. The simulation in this paper is arbitrarily stopped at the experimentally observed tensile strength of 540 MPa [5–7]. The incorporation of fibre failure is however the subject of ongoing investigations [45]. The evolution of the radial mismatch between the fibre and the matrix is given in Fig. 9 as a function of the distance from the matrix crack at some representative stress levels. Before damage development (e.g. 100 MPa), no radial mismatch exists. The radial mismatch at a debonded interface is calculated as follows. First the difference between the axial displacement between the fibre and the matrix; Du, is calculated. The radial mismatch is then calculated using Eqs. (2)–(4) and Eq. (5). At 200 MPa Du and consequently DRf become zero near the matrix crack. At 350 MPa the radial mismatch increases strongly near the matrix crack while it is nearly zero near the debond tip. At 500 MPa Du is
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larger than half of the fibre roughness wavelength L near the matrix crack; there the radial mismatch is at its maximum. The simulated relationship between longitudinal stress and transverse strain is given in Fig. 10. The calculations were performed using a simple spreadsheet program (Microsoft Excel). The agreement between the experimentally observed and theoretically observed relationship is quite good using the constituent properties of Table 1. The discrepancy is most pronounced at the onset of matrix cracking. A possible explanation might be the fact that the crack density measurements of Sørensen et al. [5] did not discriminate between matrix microcracks with no or very limited interfacial debonding and matrix macrocracks extending over large distances and accompanied by extensive debonding. Since microcracks typically form at low stresses, this might lead to an overestimation of the amount of debonding and sliding and consequently of the transverse strain at low stresses.
4. Conclusions A comprehensive micromechanical model describing the relation between longitudinal stress and transverse strain during tensile loading of unidirectional fibre toughened CMCs has been presented. The model describes a composite by different unit-cells, each representing a damage state characteristic for the material. It takes into account the Poisson contraction of fibre and matrix, the relief of thermal residual stresses upon damage development, and the build-up of radial misfit stresses at the fibre/matrix interface due to interfacial mismatch after debonding and sliding. Thus the simulated transverse strain response depends on a variety of constituent properties. For an experimentally determined set of these
Fig. 10. Comparison between the experimentally observed and the theoretically simulated relation between longitudinal stress and transverse strain of SiC/CAS; the constituent properties used as input are given in Table 1.
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constituent properties, the simulated and experimentally observed response of SiC/CAS are in excellent agreement. Compared with other models available in literature [7,21,22] the proposed model combines comprehensiveness (the response is predicted throughout a tensile test), completeness (all relevant phenomena are taken into account), ease of use (the calculations can be done using simple spreadsheet programs), and accuracy (the SiC/ CAS response is accurately predicted). In addition, the analytical and micromechanical character of the model gives the materials engineer a powerful and versatile tool to evaluate the impact of a change in constituent properties upon the transverse strain response. These results will however be reported elsewhere [45].
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Z. Hashin, B. Rosen, J. Appl. Mech. 31 (1964) 223. Z. Hashin, J. Appl. Mech. 50 (1983) 481. H. Oel, V. Frechette, J. Am. Ceram. Soc. 69 (1986) 342. B. Budiansky, J. Hutchinson, A. Evans, J. Mech. Phys. Solids 34 (1986) 167. M. Kuntz, B. Meier, G. Grathwohl, J. Am. Ceram. Soc. 76 (1993) 2607. A. Pryce, P. Smith, Acta Metall. Mater. 41 (1993) 1269. S. Timoshenko, J. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1951, p. 58. E. Vanswijgenhoven, O. Vand Der Biest, Acta Mater. 45 (1997) 3349.
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A model for the transverse strain response of unidirectional ceramic matrix composites during tensile testing Eddy Vanswijgenhoven *, Omer Van Der Biest Department of Metallurgy and Materials Engineering, Katholieke Uni6ersiteit Leu6en, De Croylaan 2, B-3001 He6erlee, Belgium
Abstract A comprehensive micromechanical model relating the longitudinal stress and transverse strain of unidirectional fibre toughened ceramic matrix composites (CMCs) is presented. The model uses different cylindrical unit-cells to describe the composite throughout a tensile test and considers all relevant damage mechanisms. The proposed model takes into account the Poisson contraction of fibre and matrix, the relief of thermal residual stresses upon damage development, and the build-up of compressive radial stresses at the interface due to mismatch between fibre and matrix after debonding and sliding. Thus the modelled transverse strain response depends on a wide range of microstructural and micromechanical parameters. The approach is checked by comparing the experimentally observed and simulated response of a unidirectional SiC/CAS composite of which all constituent properties were determined experimentally. The agreement between experiment and theory is excellent. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Ceramic matrix composites; Interfacial debonding; Transverse strain
1. Introduction The relation between longitudinal stress s and transverse strain otr of unidirectional fibre toughened ceramic matrix composites has been studied experimentally by several authors [1 – 8]. Fig. 1 shows the transverse strain response of different ceramic matrix composites (CMCs) during tensile testing. Initially the transverse strain decreases linearly with applied stress. Matrix cracking generally causes the transverse strain of the composite to become positive. When the saturation crack density is reached, the transverse strain remains approximately constant up to failure. The transverse strain of some composite systems, however, continues to decrease throughout a tensile test. It has been suggested that the observed strain reversal is caused by fibres tending to broom out [1] and the relief of residual stress as the matrix cracks [2 – 8], axial cracking of the matrix [3,4], and the build-up of radial mismatch at debonded interfaces [6,7]. A comprehensive model for the transverse strain response of unidirectional fibre toughened composites in not available in literature. Trying to explain the * Corresponding author. 0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0921-5093(98)00595-4
observed behaviour Sørensen et al. presented a micromechanical study based on the finite element method (FEM) [6]. The overall response of the composite was deduced from the response of a representative volume element consisting of concentrically placed cylinders of fibre and surrounding matrix, bounded axially by a matrix crack and a symmetry plane. The interface between fibre and matrix was assumed to be a frictional sliding contact. In all cases the FEM model did not give the experimentally observed positive transverse strains. This was due to the disappearance of the fibre/matrix contact at sufficiently high stresses and the assumption of perfectly smooth fibres. It was suggested that the roughness effect, as also evidenced and modelled by other authors for different loading conditions [9–20], could override the Poisson contraction of the fibre and cause an increase in transverse strain upon fibre/matrix interface sliding during tensile testing. Indeed by assuming that stress transfer between fibre and matrix along a debonded interface follows Coulomb’s friction, Sørensen [7] proved that the loss of the interfacial contact by Poisson contraction of the fibre could be compensated by fibre roughness. However, his model only gave a description of a totally debonded composite material and did not fully describe the evolution of the
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transverse strain as a function of longitudinal stress. In addition to the model proposed by Sørensen [7], two other models relating longitudinal stress and transverse strain have been proposed [21,22]. These models do however neglect interfacial mismatch and are guaranteed to be significantly wrong. In this paper, a comprehensive model relating the longitudinal stress and transverse strain of unidirectional fibre toughened ceramic matrix composites is proposed. The model describes the transverse strain response of a model composite throughout an entire tensile test, taking into account all relevant damage. The mechanisms governing the response of the composite are the Poisson contraction of fibre and matrix, the relief of residual thermal stresses upon damage development, and the build-up of compressive radial stresses at the interface caused by interfacial mismatch after debonding and sliding. The proposed model is the first which describes the material response throughout an entire test and takes into account all mechanisms relevant to the transverse strain response. In addition its analytical and micromechanical character gives the material engineer a versatile and powerful tool to evaluate the impact of changes in constituent properties. The paper is organised as follows: a detailed description of the proposed model in Section 2 is followed by a comparison between the experimentally observed and theoretically predicted transverse strain response of unidirectional SiC/CAS in Section 3. Section 4 contains some concluding remarks.
2. Model description The proposed model focuses on the transverse strain response of a unidirectional fibre toughened CMC under uniaxial stressing parallel to the fibres. The model
Fig. 1. Experimentally observed relationships between the longitudinal stress and the transverse strain of different CMCs during tensile testing.
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Fig. 2. Schematic representation of the microstructure and the damage state in an idealised composite and an example of the description of the idealised composite by a unit-cell.
composite consists of a hexagonal array of perfectly aligned fibres in a matrix (Fig. 2). The thickness of the interphase between fibre and matrix is assumed to be negligible (it is a true interface), whereas the surface of the fibre is assumed to have a certain roughness. Before continuing with the description of the model, the most important assumptions made are briefly described and discussed. These assumptions concern: (1) the development of damage during tensile testing, (2) the interfacial mismatch after interface debonding and sliding, and (3) the stress transfer across debonded interfaces. Two major damage mechanisms are operative during tensile testing of unidirectional fibre toughened CMCs: multiple matrix cracking accompanied by interface debonding and fibre fracture accompanied by fibre pull-out [1–8,21–32]. The present model only considers the effect of matrix cracking and interface debonding on the transverse strain response. Fibre failure and pull-out are very important for modelling of the longitudinal strain response [22–32] but the effect on the transverse strain response is neglected in this paper as a first approximation because (1) the matrix is the continuous phase in the transverse direction, (2) before composite failure only a limited amount of fibres are broken in a representative material volume [25], and (3) composite failure is often a localised process [25]. The formation of matrix microcracks, which are often observed through acoustic emission or in-situ microscopy below the proportional limit, is not taken into
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account either, because this type of damage hardly influences the overall stress – strain response [2]. During macrocracking of the matrix, the number as well as the spatial distribution of the matrix cracks are related to the stress on the composite. Previous work on unidirectional fibre toughened CMCs has indicated that during tensile stressing matrix cracks develop into more or less periodic arrays, with a characteristic spacing [1–8,21–32]. The proposed model assumes the cracks to be equidistant and spanning the entire crosssection of the material while the dependence between the matrix crack density r and the stress s is assumed to be given by a three parameter Weibull function:
r(s)= rs 1−exp
s −s* s0
m
(1)
with rs the saturation matrix crack density, s* the stress at which matrix cracks first appear, s0 a scale parameter, and m the Weibull modulus governing matrix cracking [32]. These parameters are treated in this paper as independent constituent properties of the composite system which are determined experimentally. The authors realise however that they can be related to other, truly independent properties [23 – 26,32]. The second item to be discussed beforehand concerns the interfacial mismatch after interface debonding and sliding. At this point in the paper it is advisable to summarise the current knowledge on fibre roughness and interfacial mismatch. The fibres used in CMCs are known to exhibit a variation in diameter along their length [9–20]. Two types of diameter variation are to be distinguished: a variation over large distances (typically cm) and a variation over short distances (typically mm). In this paper only short distance variations are considered. A thorough study of the short distance topography of small diameter SiC fibres, as used in for example SiC/CAS, has not been reported. Preliminary work based on atomic force microscopy and the study of the push-out behaviour by Jero and reported by Kerans [15,16] estimated the short distance fibre roughness to be about 20 nm. This value is consistent with a roughness value of 17 nm reported by Martin et al. [19] and the value of 24 nm used by Sørensen [7] to explain the experimental behaviour of unidirectional SiC/CAS. The roughness wavelength was estimated to be about 1 mm by Jero et al. [12,13] by interpretation of the pushout and pushback behaviour of Nicalon fibres in an 1723 aluminosilicate glass. Later Kerans [15] corrected this value by taking into account the change of length of the loaded debonded fibre; he proposed a roughness period of about 0.3 mm. These roughness periods are of the same order of magnitude as the relative axial displacement between fibre and matrix at a debonded interface causing roughness to play a major role even early in the debonding process. From the above literature review, it is obvious that the descrip-
tion of the interfacial mismatch after interface debonding and sliding is a complex issue. Consequently, some simplifying assumptions are necessary to study the effect of the interfacial mismatch upon the otr response during tensile testing. First, it is assumed that there is no interphase between the fibre and the matrix; after processing, the fibre and the matrix are in direct contact. Second, the fibre is assumed to have a certain roughness profile. The proposed model is thus strictly only valid for composites with fibres that debond along a weak interface. It is however believed that the adopted approach will give reasonable approximations for other debonding scenarios [11]. Third, the fibre roughness is assumed to be random, non-periodic, and small compared with the fibre diameter and its effect on the transverse strain response is assumed to be described by an average roughness amplitude DRmax and an average roughness wavelength L [7,11,13–16]. Fourth, it is assumed that the average radial interfacial mismatch DR(s, x) at a distance x from the matrix crack along the debonded interface depends linearly on the relative axial displacement between fibre and matrix Du(s, x) when Du(s, x) is less than half of L [13,15,16]: DRf = DRmax ×
with Du(s, x)=
Du L if Du 5 L 2 2
& Ld(s) x
(2)
s deb s deb(x) f (x) − m dx Ef Em
&
(3)
if the fibre/matrix interface is not totally debonded or Du(s, x)=
1 (2r(s)) x
s deb s deb(x) f (x) − m dx Ef Em
(4)
if total debonding has occurred, Ld(s) the distance over deb which debonding occurs, s deb f (x) and s m (x) the axial stresses in the fibre and the matrix where the interface has debonded, and x the distance to the matrix crack. These parameters will be defined below. For relative axial displacements greater than half of the roughness wavelength, the interfacial mismatch is assumed to be at its maximum [13] (Fig. 3): L DRf = DRmax if Du ] . 2
(5)
The third topic to be discussed is the stress transfer across debonded interfaces. The fibre/matrix interphase is assumed to be sufficiently weak to allow debonding upon matrix cracking; this generally happens if Gi 5Gf/ 4 with Gi and Gf the fracture energy of the interphase and the fibre [26,33]. The matrix cracks are then bridged by fibres for which the interphase with the matrix has debonded over a certain distance. Interfacial slip occurs against an interfacial sliding stress t. Such sliding generally occurs in accordance with a friction
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law: t= t0 −msi, rwith m is the Coulomb friction coefficient of the interface and si, r the radial stress at the interface [6,26,34– 36]. In this paper, t is assumed to be constant, independent of the radial stress at the interface. The constant interfacial sliding stress assumption has been used with great success by several authors and it has been demonstrated that the use of an average interfacial sliding stress independent of the local radial stress at the interface has quite a small effect on the stress distributions in the fibre and the matrix [6,34,35]. By making the assumptions above, a composite consisting of a hexagonal distribution of fibres in a matrix can be described by combinations of three different unit-cells throughout a tensile test. For computational reasons cylindrical unit-cells are used [6,7,21,22,36,37] (Fig. 2). Each unit-cell consists of a fibre within a matrix with Rm = Rf/ Vf (Rf is the fibre radius, Rm is the radius of the unit-cells, and Vf is the fibre volume fraction) and represents a damage state characteristic for unidirectional fibre toughened CMCs. Before damage development, the composite is represented by a unit-cell with no damage. In the presence of matrix cracks and partially debonded interfaces, the composite is described by a corresponding unit-cell with a matrix crack and a partially debonded interface. After complete debonding, a unit-cell with a matrix crack and a totally debonded interface is used.
2.1. Unit-cell with no damage (unit-cell A) Fig. 4(a) schematically shows the unit-cell with no damage describing the undamaged composite. The fibre and the matrix are initially bonded and axial, radial, and circumferential thermal residual stresses are be-
Fig. 4. Schematic representation of unit-cell A with no damage (a), a unit-cell B with a matrix crack and a partially debonded interface (b), and a unit-cell C with a matrix crack and a totally debonded interface (c) together with the axial stress state in the fibre and the matrix.
lieved to be present within the fibre, the matrix and at the interface. The relationship between the longitudinal stress and the transverse strain of unit-cell A is assumed to be linear and was first given in this form by Hashin et al. [38,39]: s otr,A = − nc . Ec
Fig. 3. (a) Development of interfacial mismatch between fibre and matrix after interface debonding and sliding and (b) the proposed evolution of the interfacial mismatch DRf as a function of the difference in the axial displacement between the fibre and the matrix Du.
(6)
The composite Young’s modulus Ec and Poisson coefficient nc depend on the fibre volume fraction Vf and the Young’s moduli and the Poisson coefficients of fibre and matrix Ef, Em, nf, nm. The equations giving these relationships are given in [39] and are not reproduced here. The thermal residual stress state in the unit-cell plays an important role in the transverse strain response of
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the unit-cells with damage as will be demonstrated below. Most important are the axial residual stress in R the fibre S R f, z and the matrix S m, z and the radial residual stress at the fibre/matrix interface S R i,r. Several analytical models have been put forward to determine the elastic stress field in a set of two or more coaxial cylinders subject to thermomechanical loading [40–42]. In this paper the thermal stresses are calculated for a cylinder of fibre surrounded by a cylindrical matrix shell and assuming a radial stress free matrix surface. The thermal stresses depend on Vf, Rf, Ef, Em, nf, nm, the thermal expansion coefficients of the fibre and the matrix af, am, and the temperature difference between the test temperature and the stress free temperature DT (DT is negative). The equations used here for the calculation of the thermal residual stresses in unit-cell A are given in [42] and are not reproduced here.
2.2. Unit-cell with a matrix crack and a partially debonded interface (unit-cell B) Fig. 4(b) is a schematic representation of a unit-cell with a matrix crack and a partially debonded interface, and of the axial stress state in the fibre and the matrix. The length of the unit-cell B equals 1/r(s). The axial deb stress in the fibre s deb f (x) and the matrix s m (x) vary linearly with the distance to the matrix crack x according to the simple shear lag theory until they reach their value prior to cracking. The relationship between the stress on the unit-cell and the debond length Ld(s) is given by [43]: Ld(s)=
Rf s E f − s −S R f,z . 2t Vf Ec
(7)
The calculation of the transverse strain of this unitcell is done as follows. The transverse strain is first calculated as a function of the position in the cell x (x being the distance from the matrix crack) and then averaged over the entire cell: 1
&
otr,B(s)=
Ld(s)
o deb tr,B(s, x)dx +
0
&
2r(s)
o bon tr,B(s, x)dx
L d(s)
1/2 r(s)
DRf(s, x) nf s + + (am − af)DT Rf Vf Ef nm Vf nf t + −2 x Rf Em Vm Ef si,r(s, x) = . (9) 1+ Vf + nm Vm (1− nf) + Em Ef To ensure compatibility between unit-cells the outer surface of the cylinders must remain straight; i.e. the surface must have the same radial displacement along the cylinder length [6,7,21]. This is automatically true for the unit-cells with no damage. The straightness of the outer surface of unit-cells with a cracked matrix and a partially debonded interface is guaranteed as follows. If radial pressure continues to exist across debonded interfaces, then the effect of applying radial stresses on the part with a debonded interface does not differ from the effect of applying radial stresses on the part with a bonded interface. In both cases the unit-cell cylinder behaves as a linear elastic medium with the same elastic properties. Applying tensile and compressive radial stresses resulting in a constant transverse strain equal to the average transverse strain of the unit-cell with zero radial stresses at the outer surface ensures unit-cell compatibility. A similar procedure was followed by Sørensen et al. [6,7] and by He et al. [21]. The same approach is also true for the unit-cells with a matrix crack and a totally debonded interface. In the following, it is thus silently assumed that compressive radial stresses continue to exist across debonded interfaces. The proposed model is thus only valid for composite systems for which si, r as calculated above in Eq. (7) is negative. After calculation of the radial interfacial stress, the transverse strain in the part of the unit-cell with a debonded interface can be calculated. Elastic analysis results in the following equation for the transverse strain [7,44,45] (Fig. 5): −
.
(8)
The transverse strain response of the part of the unit-cell with a still bonded interface is similar to that of the unit-cell with no damage and is given by Eq. (4). The transverse strain in the part of the unit-cell with a debonded interface is calculated in two steps. First the radial stress at the debonded fibre/matrix interface is calculated as a function of s and x. Following the elastic analysis by Sørensen [7] and Timoshenko and Goodier [44] of a system consisting of a fibre cylinder surrounded by a hollow matrix cylinder with a radial interfacial mismatch the following formula for the radial stress at the interface is obtained [45]:
Fig. 5. Transverse expansion of an undamaged unit-cell at the stress free temperature, of an undamaged unit-cell at the test temperature in unloaded condition, and of a damaged unit-cell at the test temperature in loaded condition.
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o deb tr, B(s, x) =−
2Vf S R 2Vf si, r(s, x) s deb SR m (s, x) i, r m, z −nm + +nm Em Em Vm Vm Em Em (10)
The transverse strain in the part of the unit-cell with a debonded interface depends on a wide range of constituent properties. It increases, i.e. tends to become positive, for high compressive radial stresses at the interface, for small tensile axial stress in the matrix, for high tensile radial thermal residual stresses at the interface, and for high tensile axial thermal residual stresses in the matrix.
2.3. Unit-cell with a matrix crack and a totally debonded interface (unit-cell C) Unit-cell C is schematically represented in Fig. 4(c), together with the stress evolution in the fibre and matrix. The average transverse strain of this unit-cell is calculated in a manner similar to the transverse strain of the unit-cell with a partially debonded interface. The calculation is less complex since this unit-cell no longer consists of two parts with a different transverse response:
&
1 2r(s)
otr,C(s)=
otr,C(s, x)dx
0
1 2r(s)
(11)
Fig. 6. Experimentally observed relationship between longitudinal stress, longitudinal strain, and transverse strain during tensile testing of unidirectional SiC/CAS specimens investigated by Sørensen et al. [5 – 7].
creases steeply to a value of about 0.02%. At higher stresses, the transverse strain remains constant at this value. The constituent properties used for the simulation of the transverse strain response of the unidirectional SiC/ CAS under investigation are summarised in Table 1. Most of these properties have been determined experimentally by Sørensen et al. [5–7]. Only the choice of
with the transverse strain as in Eq. (10).
3. Comparison between the experimentally observed and simulated behaviour In this section, the experimentally observed and simulated relation between longitudinal stress and transverse strain of a unidirectional nicalon silicon carbide (SiC) fibre toughened calcium aluminosilicate (CAS) glass–ceramic matrix composite are compared. In the previous section it has been shown that the transverse strain response depends on a wide variety of constituent properties. All of these properties are relatively wellknown for the SiC/CAS system under investigation although some of them have not been determined with a high accuracy yet or are subject to inherent scatter [5 –7,15,16]. The experimentally observed relationship between longitudinal stress, longitudinal strain, and transverse strain during tensile testing of unidirectional SiC/CAS investigated by Sørensen et al. is given in Fig. 6 [5–7]. Initially the transverse strain decreases linearly with an increase in longitudinal stress (Poisson’s ratio of 0.24). Between 260 and 420 MPa, the transverse strain in-
Table 1 Constituent properties of unidirectional SiC/CAS used to simulate the transverse strain response in Fig. 10[5 – 7,15,16] Basic constituent properties Vf Rf Ef Em nf nm t DRmax L af am DT rs s* s0 m
0.35 7.5 mm 200 GPa 98 GPa 0.15 0.3 20 MPa 20 nm 0.5 mm 3×10−6 K−1 5×10−6 K−1 −1000 K 6 cracks mm−1 150 MPa 180 MPa 3
Derived properties Ec nc SR i, r SR f, z SR m, z
134 GPa 0.245 −91 MPa −232 MPa 125 MPa
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Fig. 7. A comparison between the experimentally observed relation between matrix crack density and longitudinal strain [5– 7] and the matrix crack density evolution used in the simulation.
the maximum interfacial mismatch DRmax, the interfacial mismatch wavelength L, and the relation between matrix crack density and stress r(s) are further highlighted in this paragraph. The experimentally determined evolution of the matrix crack density as observed on replicas is given in Fig. 7[5]. In this paper, the evolution of the matrix crack density is assumed to be given by the three parameter Weibull function of Eq. (1). The values used for s*, s0, and m are given by 150 MPa, 180 MPa, and 3, respectively(Fig. 7). The values for the maximum interfacial mismatch DRmax and the interfacial mismatch wavelength L used are 20 nm and 0.5 mm, respectively. This of the same order of magnitude as put forward by other authors [7,12,15,16]. The simulated damage development during tensile testing is schematically represented in Fig. 8. Up to a stress of 150 MPa, no damage develops and the behaviour is perfectly elastic. The composite is described by unit-cell A (Fig. 8(a)). The relationship between
Fig. 8. Schematic representation of the simulated damage development during tensile testing of unidirectional SiC/CAS; (a) no damage developing; (b) matrix cracking and interface debonding; (c) matrix cracking.
Fig. 9. Simulated evolution of the radial mismatch DRf as a function of the distance from the matrix crack x and the applied longitudinal stress s.
longitudinal stress and transverse strain is given by Eq. (6). Between 150 and 330 MPa, the composite is described by B unit-cells (Fig. 8(b)). The length of the B unit-cell equals the crack spacing. The relationship between longitudinal stress and transverse strain is given by Eqs. (7)–(9) and Eq. (10). At a longitudinal stress of 330 MPa, the interfaces become totally debonded. Above this stress level the composite is described by C unit-cells and the transverse strain is given by Eqs. (9) and (10) and Eq. (11) (Fig. 8(c)). Theoretically speaking matrix cracking should stop at this point since the axial stress in the matrix can no longer increase and the cracks are assumed to be equidistant. In reality cracking continues because spacings are not equal because time-dependent cracking occurs, and because some regions of the matrix are under high loads. In this paper the experimentally measured evolution of the matrix crack density is used as a first approximation. Because the present model does not take into account fibre failure, it cannot predict tensile failure. The simulation in this paper is arbitrarily stopped at the experimentally observed tensile strength of 540 MPa [5–7]. The incorporation of fibre failure is however the subject of ongoing investigations [45]. The evolution of the radial mismatch between the fibre and the matrix is given in Fig. 9 as a function of the distance from the matrix crack at some representative stress levels. Before damage development (e.g. 100 MPa), no radial mismatch exists. The radial mismatch at a debonded interface is calculated as follows. First the difference between the axial displacement between the fibre and the matrix; Du, is calculated. The radial mismatch is then calculated using Eqs. (2)–(4) and Eq. (5). At 200 MPa Du and consequently DRf become zero near the matrix crack. At 350 MPa the radial mismatch increases strongly near the matrix crack while it is nearly zero near the debond tip. At 500 MPa Du is
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larger than half of the fibre roughness wavelength L near the matrix crack; there the radial mismatch is at its maximum. The simulated relationship between longitudinal stress and transverse strain is given in Fig. 10. The calculations were performed using a simple spreadsheet program (Microsoft Excel). The agreement between the experimentally observed and theoretically observed relationship is quite good using the constituent properties of Table 1. The discrepancy is most pronounced at the onset of matrix cracking. A possible explanation might be the fact that the crack density measurements of Sørensen et al. [5] did not discriminate between matrix microcracks with no or very limited interfacial debonding and matrix macrocracks extending over large distances and accompanied by extensive debonding. Since microcracks typically form at low stresses, this might lead to an overestimation of the amount of debonding and sliding and consequently of the transverse strain at low stresses.
4. Conclusions A comprehensive micromechanical model describing the relation between longitudinal stress and transverse strain during tensile loading of unidirectional fibre toughened CMCs has been presented. The model describes a composite by different unit-cells, each representing a damage state characteristic for the material. It takes into account the Poisson contraction of fibre and matrix, the relief of thermal residual stresses upon damage development, and the build-up of radial misfit stresses at the fibre/matrix interface due to interfacial mismatch after debonding and sliding. Thus the simulated transverse strain response depends on a variety of constituent properties. For an experimentally determined set of these
Fig. 10. Comparison between the experimentally observed and the theoretically simulated relation between longitudinal stress and transverse strain of SiC/CAS; the constituent properties used as input are given in Table 1.
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constituent properties, the simulated and experimentally observed response of SiC/CAS are in excellent agreement. Compared with other models available in literature [7,21,22] the proposed model combines comprehensiveness (the response is predicted throughout a tensile test), completeness (all relevant phenomena are taken into account), ease of use (the calculations can be done using simple spreadsheet programs), and accuracy (the SiC/ CAS response is accurately predicted). In addition, the analytical and micromechanical character of the model gives the materials engineer a powerful and versatile tool to evaluate the impact of a change in constituent properties upon the transverse strain response. These results will however be reported elsewhere [45].
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