## Abstract

A physical model, previously developed by one of the authors, has been extended to cover thermal conductivity degradation owing to the uni-axial stress–strain response of aligned groups of fibres or tows found in ceramic matrix composites. Both the stress–strain and thermal models, together with their coupling, have been shown to predict known composite behaviour qualitatively. The degradation of longitudinal thermal properties is shown to be driven by strain-controlled fibre failure; while the degradation of transverse thermal properties is because of the growth of fibre–matrix interface wake-debonded cracks, coupled with strain-driven fibre failure.

## 1. Introduction

Ceramic matrix composites (CMCs; Kelly 1989) are candidate replacement materials for metallic superalloys used in high-temperature parts of aeroengines, heavy-duty gas turbines, nozzles in rocket engines and flight and combustion surfaces of hypersonic vehicles and devices. Increased operating temperatures from 900–1200^{°}C for coated metallic superalloys, to above 1300^{°}C, for CMCs, have the potential to achieve higher thermal efficiencies and lower emissions (Evans & Naslain 1995). In addition to favourable mechanical properties, these materials require good thermal properties for heat transfer in CMC engine components.

Cox & Yang (2006) have identified, as a necessary requirement, the ability to predict at the design stage both mechanical and thermal properties of a CMC composite material test sample and those of an engineering component of the same material. It is only by proceeding in this way that one can achieve optimal selection of: constituent materials; composite materials lay-up; weaving and manufacturing route; thermo-mechanical response of a component; prediction of component failure; and component life-cycle cost. Hence, there is a very strong driver to be able to model the life scenario using advanced computer techniques, and so avoid the unsatisfactory, costly and time-consuming current practice of prototype development. Probably the most demanding of these requirements is the ability to predict the degradation of thermal properties with composite strain, and this paper addresses the topic.

A deficiency of CMCs, when compared with metallic superalloys, is the degradation of thermal transport properties because of internal damage. The presence of damage and cracks can be introduced in an engineering component, either during manufacture or in service. First, damage is introduced in manufacture as a result of the different thermo-mechanical properties of the constituent materials, which, during cooling, introduce thermal gradients, thermal stresses, localized failure and hence damage. This manifests itself after cooling as micro-porosity. Second, damage is created in-service (Bruggeman 1935) by mechanical overloads, fatigue and time-dependent and environmental effects that produce matrix and fibre–matrix interfacial cracks. The air contained within the micro-porosity and cracks degrades the thermal transport properties and renders the component unserviceable; at that point, either component repair or replacement is necessary. The very strong coupling between mechanical behaviour and thermal properties is, at the present time, not well understood, and is not capable of being accurately predicted. Hence, the driver for this research is the need to describe and predict these effects at the design stage.

The modelling of CMCs requires a number of factors to be taken into account that influence thermal transport, such as composite architecture, properties of constituent materials and influence of defects (Del Puglia & Sheikh 2001). Some of the first approaches to thermal finite-element modelling were two-dimensional (Lu & Hutchinson 1996; Klett *et al*. 1999). The limitation of their approaches is simplicity, which does not reflect the complexities of real composites. Sheikh *et al.* (2001) have presented a complex weave model of a plain weave CMC. Their model is three-dimensional, and is a step towards the modelling of complex composite architectures. Their research included the effect of directionality in thermal transport by the introduction of the individual properties of fibre and matrix. However, the main deficiency was the absence of initial porosity.

Del Puglia *et al.* (2004*a*) focused on the identification and classification of initial/manufacturing porosity and introduced four different classes. A further paper by Del Puglia *et al.* (2004*b*) addressed how, for three classes of porosity, finite-element analysis techniques can be used to quantify the effect of each type on the spatial heat transport properties assessed at the level of a micro-unit cell of tow and matrix. The research of Del Puglia *et al.* (2004*a*,*b*) was further advanced (Del Puglia *et al.* 2005) for the same (DLR-XT) material, by inclusion of a fourth class of porosity at the macro-unit cell level. They proceeded to accurately predict the measured thermal properties of a 10-high laminate, plain weave DLR-XT composite from knowledge of the individual constituent materials properties and of the levels of manufacturing porosity. In this way, they established the link between constituent material properties and those of the unit cell.

The present paper builds on the work of Del Puglia *et al.* (2005) by considering a single tow of fibres and its associated matrices. The objective is to establish a simple tow model for both the stress–strain and thermal conductivity–strain responses, which is linked to the physical and mechanical properties of the constituent materials. Then, in subsequent papers, the tow model will be developed to cater for multi-axial loadings that include superimposed tension orthogonal to the tow and shear stresses in planes both parallel and transverse to the tow.

In the multi-axial mechanical model, the interactions will be included between damage mechanisms that are associated with the loading systems operating in three mutually perpendicular directions. For example, the following will be considered: matrix cracking; inter-laminae damage; and the effect of wake debonding on the degradation of Poisson’s ratio. Since there is a dearth of experimental data that can be used to isolate the influence of individual mechanisms, the computational modelling approach will be used to postulate mechanisms, predict their consequences and then assess their significance by assessment against the results of experiments that have been carried out on DLR-XT and HITCO woven composites.

Once an understanding has been obtained of the controlling damage mechanisms and their interactions, the multi-axial tow model will be used to construct a unit cell model. A follow-up paper by Tang *et al.* (2009) reports predictions, based on the tow model developed here, of the thermo-mechanical response of a simplified unit cell model of 0/90 and woven composites, which takes account of only the axial tow mechanical behaviour, coupled with longitudinal and transverse thermal transport. The latter investigation has been carried out to provide insight into the dominant damage mechanisms and their interactions.

The key to the achievement of the latter objective will be the use of a numerical technique that makes an approximation with acceptable accuracy possible, and yet has the computational efficiency required to model large assemblages of unit cells that might be encountered in structural components. Two approaches can meet these requirements: first, the binary method of finite-element analysis of Cox *et al*. (1994), Xu *et al.* (1995) and McGlockton *et al.* (2003), and second, the modelling of tows using a finite-element-based multi-linear elastic orthotropic materials approach. The finite-element-based multi-linear elastic orthotropic materials approach has the simplicity of considering only one material and avoids the complexity of having to define the effective medium properties used in the binary approach. Either of the two approaches allows a composite unit cell to be modelled by a small number, usually less than 20, of finite elements, compared with the tens of thousands of conventional finite elements used by Del Puglia *et al.* (2005). Inevitably, this reduction in element numbers is achieved with some loss of accuracy; however, it is these features that make possible the finite-element analysis of large components. The common link between a composite tow, modelled by multi-axial finite elements, unit cells and engineering components, is provided by the fibre tow.

The present paper does not address the implementation or use of the finite-element method. Instead, it focuses on the development of a simple, approximate, uni-axial tow model that describes the stress–strain response of a fibre tow and associated matrices and the dependency of axial and transverse tow thermal conductivities on axial tow strain. Two classes of fibre tow and associated matrices are considered here. They are: (i) C/C–SiC DLR-XT material (Krenkel 2005) and (ii) C–C HITCO material (R. Mathur 1991, personal communication). The former material has been produced as a 10-high plain weave laminate and the latter as a 9-high 8-harness satin weave laminate. The tow configurations associated with these materials are shown schematically in figure 1 for an idealized tow of rectangular cross section.

The paper reports the uni-axial thermo-mechanical tow theory and the identification of those damage mechanisms that are responsible for the degradation of tow thermal properties with strain. The next section addresses mechanical behaviour at the tow level.

## 2. Mechanical behaviour

### (a) Stress–strain response and length scale

#### (i) Stress–strain analysis of uni-directional tow model

When a uni-directional fibre-reinforced CMC is under remote tensile stress, , the schematic stress–strain response shown in figure 2 can be observed (Hayhurst *et al.* 1991).

The four characteristic stages in the stress–strain curve are now described.

AB: initially, the composite behaves as a virgin, undamaged, linear elastic material,

BC: periodic matrix cracking with intact fibre bridging (Evans & Marshall 1989) originates from pre-existing flaws, which can occur in manufacture (figure 3), and forms a continuum of matrix crack damage,

CD: weaker fibres fail and determine the maximum stress, and

DE: the majority of fibres fail and are pulled out against the frictional shear stress along the interface, hence giving the brittle material ‘toughness’.

Two assumptions are made in the *σ*−*ϵ* model (Hayhurst *et al.* 1991). First, the average nominal stress on the fibres (load/total initial fibre area) can be expressed as a function of the damage variable, *ω*=*r*/*n*, where *r* is the number of failed fibres and *n* is the total number of fibres. In this paper, * ω* is treated as a continuum damage parameter, and, in the same way as stress and strain, it has continuum properties. The nominal stress has two contributions: to extend the fibres and to pull out failed fibres from the matrix. Second, the strain of the composite is controlled by two aspects of the elasticity of the unfailed fibres: (i) the stress increment required to change the damage at constant elastic modulus and (ii) the change in strain, at constant stress, owing to the change in current modulus. This is expressed as
2.1
where the subscript ‘f’ refers to fibres. This relationship can be rewritten in terms of the global or composite stress, , and strain, Before matrix cracking , the effect of damage on Young’s modulus of the fibres is insignificant, and the second term on the right-hand side of equation (2.1) can be neglected; hence,

*σ*

_{f}and

*ϵ*

_{f}for the fibres can then be replaced by and . After matrix cracking , the majority of the load is carried both by fibres and by fibre pullout. The strain

*ϵ*

_{f}is set equal to , the stress

*σ*

_{f}becomes (Hayhurst

*et al.*1991) and equation (2.1) can be written as 2.2 where is the initial Young’s modulus of the entire composite and is given by

*E*

_{I}=

*V*

_{f}

*E*

_{f}+

*V*

_{im}

*E*

_{im},

*V*

_{f}and

*V*

_{im}denote the volume fraction of fibres and inner matrix, respectively,

*E*

_{f}and

*E*

_{im}denote Young’s moduli of the materials and

*σ*

_{ff}is the average Weibull (1939) fibre failure stress. The term

*E*(

*ω*) in equations (2.1) and (2.2) refers to the change in Young’s modulus caused by the failure of the fibres after matrix cracking.

A considerable amount of micro-structural evidence shows that, when reinforced ceramic composites are subjected to homogeneous states of uni-directional stress, failure of the matrix takes place at not by the propagation of a single dominant crack, but by the development of a field of matrix cracks with periodical crack spacing, *w*, shown in figure 3 (Aveston *et al.* 1971; Hashin 1980). Namely, a stable continuum damage field is established.

#### (ii) Effect of damage on remote composite elastic modulus

After matrix cracking, the elastic modulus is assumed to be *Θ**E*_{I}, where *Θ* is a parameter relating to the change of Young’s modulus due to matrix cracking at *σ*_{mc} and *Θ*=(*E*_{I}+Δ*E*_{I})/*E*_{I} (figure 2). It is assumed that once matrix cracking has occurred, all direct load-carrying capability of the matrix is lost, therefore Δ*E*_{I}=−*V*_{im}*E*_{im}. For subsequent values of *ω*, the value of Young’s modulus is *E*(*ω*)=*Θ**E*_{I}(1−*ω*); however, because of the interaction between matrix and fibres, this relationship is nonlinear and is assumed to be (Hayhurst *et al.* 1991)
2.3
where *B*≥1 is the elastic modulus decay index (Vavakin & Salganik 1975; Budiansky & O’Connell 1976), and for simplicity has been given the value of unity.

#### (iii) Contribution to overall axial force from unfailed fibres

Weibull (1939) statistics will be used to describe the failure of the brittle fibres. It is assumed that the strength, *σ*_{f}, of a particular fibre depends on its physical length, ℓ, a reference length, ℓ_{0}, the average failure strength of the fibres, *σ*_{ff}, and the Weibull index, *m*. Based on the statistical model of Hult & Travnicek (1983), and on the length scale theory by Jansson & Leckie (1992) and Ashby & Jones (2005), the probability of the fibre failure at a stress *σ*_{f} is
2.4
where the length scale ratio of ℓ_{0}/ℓ=1 has been used throughout this paper.

When fibres begin to fail, 0≤*r*≤*n*, stresses are redistributed among the unfailed fibres, and the stress on intact fibres is expressed in terms of the nominal fibre stress, *σ*_{0},
2.5
Combination of equations (2.4) and (2.5) yields
2.6
As fibre–matrix interfacial debonding will influence equation (2.5) more strongly than reflected by the linear term (1−*ω*), this term will be replaced by (1−*ω*)^{A}, where *A* is a constant to be determined by experiment (Hayhurst *et al.* 1991). Then, equation (2.6) becomes
2.7

#### (iv) Contribution to overall axial force from pullout of failed fibres

By considering a fibre that fails at *ω*_{po} and is then pulled out at the stress *σ*_{po}, the normalized pullout stress of a single fibre can be written as
2.8
where is the maximum normalized pullout stress and *D* is a parameter that controls the nonlinear decay of the pullout stress. Calibration of *D* is addressed in §2*b*. Integration of equation (2.8), at the current damage variable, *ω*_{po}, gives the contribution of all failed fibres as
2.9

#### (v) Overall axial force

The total stress on the fibres involves contributions from both the failed and unfailed fibres, and summation of these two terms yields (Hayhurst *et al.* 1991)
2.10
Differentiation of equation (2.10) with respect to *ω* gives
2.11
Combination of equations (2.2), (2.3) and (2.11) gives
2.12
Equation (2.12) describes the normalized stress–strain relationship for a uni-directional tow; this is shown in figure 4 for typical material properties: *A*=1.025, *B*=1, *D*=0.1375, *S*=0.103, *m*=4.8, *σ*_{ff}=1.748 GPa, *σ*_{mc}=108.5 MPa, *Θ*=0.937, *E*_{f}=294 GPa, *E*_{im}=62.56 GPa, *V*_{f}=0.494, *V*_{im}=0.156 and *V*_{om}=0.35. The value of *Θ* depends on whether matrix cracking has occurred. Prior to matrix cracking, it is unity, as the initial modulus is unchanged. After matrix cracking, it is calculated (for the parameters given earlier) by the equation *Θ*=(*E*_{I}+Δ*E*_{I})/*E*_{I}=0.937, previously discussed in §2*a*(ii). This change in slope takes place at the strain level λ_{mc}=0.062, and figure 4 shows it to be insignificant. Equation (2.12) allows the stress–strain tow response to be predicted from constituent material properties and some relatively insensitive empirical parameters. In the following sections, the calibration of *D* is addressed, and then the model will be used to study the degradation of thermal properties with normalized composite strain,

It is worth addressing how the stress–strain curve shown in figure 4 for the uni-axial tow might be influenced by multi-axial loading. It is expected that the application of either a small orthogonal tension stress or a small shear stress in a plane either parallel or perpendicular to the tow axis would attenuate the curve in figure 4. Two alternative outcomes might be expected. First, wake debonding, and the contribution of stress owing to fibre pullout, could be negated, as shown by Hayhurst *et al.* (1991) in fig. 7 of that paper. Second, this first mechanism could be enhanced by catastrophic fibre failure following wake debonding. The first alternative would produce a steadily decreasing stress–strain curve, and the second option would give a sudden drop in stress at constant strain. It is the latter that is often observed in experimental data (Sheikh *et al.* in press). These options have been explored by Zhang & Hayhurst (2009).

### (b) Calibration of fibre pullout stress decay index D

In this section, the mechanisms of matrix cracking, fibre failure and fibre pullout are each discussed in turn. An expression is derived for the variation of fibre pullout distance with strain and damage, which can then be used to calibrate *D*. Fibre pullout adjacent to a single matrix crack, of the type represented in figure 3, is shown schematically in figure 5. In the figure, *δ*_{f} represents the fibre pullout distance after failure, is the average failure location after all fibres fail (Thouless & Evans 1988), *U*_{0} is the equilibrium separation distance of adjacent cracked matrix surfaces (Marshall *et al.* 1985) and ξ is the remaining embedded length of failed fibre within the matrix.

#### (i) Matrix cracking separation distance

For steady-state crack propagation, the distance between two parallel matrix surfaces will reach an equilibrium distance 2*U*_{0} (Marshall *et al.* 1985) (cf. experiments carried out by Jansson & Leckie 1992). The expression for *U*_{0} is
2.13
where *D*_{f} is the fibre diameter and *τ*_{po} is the constant fibre pullout interfacial shear stress.

#### (ii) Fibre failure and pullout

Fibre strength varies statistically, resulting in different fibre failure stresses and locations. The average fibre failure location, when all the fibres have failed, has been determined by Thouless & Evans (1988) using a statistical approach,
2.14
where is an arbitrary normalizing area, defined such that is constant and *Γ*ma; is the complete gamma function where *x* ≥ 0 is an arbitrary variable (Thouless & Evans 1988). The term will be used to determine the variation of the failed fibre pullout distance with damage, *ω*, and normalized composite strain, .

A fibre having failed at *ω*_{po} sustains a normalized stress (*σ*_{po}/*σ*_{ff}), given by equation (2.8). The pullout process and the decay of the pullout stress are shown in figure 6. There are only two stresses acting along the axis of a failed fibre: the pullout stress *σ*_{po} and the equilibrating frictional stress around its surface, *τ*. The remaining embedded length of the fibre, ξ, can be derived by equating the frictional force with the pullout force given in equation (2.8). Assuming the frictional resistance stress, *τ*, to be constant (*τ*_{po}), the force equilibrium equation is
2.15
Rearrangement of equation (2.15) gives
2.16
It can be observed, from figure 5, that the pullout distance, *δ*_{f}, can be expressed by the subtraction of *U*_{0} and ξ from the average failure location of the fibre
2.17
Substitution of and ξ into equation (2.17) gives an expression for the fibre pullout distance,
2.18
When normalized pullout distance, *δ*_{f}/*D*_{f}, is plotted against normalized composite strain, , it becomes evident that the selection of the value of *D* is crucial to maintaining the physical sense of the equation. Namely, *D* influences the predicted normalized stress–strain curve and the value of *δ*_{f}/*D*_{f} when . It is therefore necessary to determine *D* such that *δ*_{f}/*D*_{f}=0 when (cf. figure 7).

The set of typical material properties, defined in §2*a*(iv), together with *τ*_{po}=10 MPa, *D*_{f}=5.5 μm, *A*_{0}=0.4 m^{2}, *ω*_{po}=0.622 and an optimized value of *D*=0.1375, have been used to determine the variation of *δ*_{f}/*D*_{f} with normalized composite strain; the results are presented in figure 7.

### (c) Discussion on mechanical response

The normalized stress–strain tow response and the normalized pullout distance-composite strain response have been predicted and are shown in figures 4 and 7 respectively. Both figures reflect the known macroscopic composite behaviour (Jansson & Leckie 1992; Sheikh *et al.* in press).

The fidelity of the model tow predictions can only be assessed after unit cell behaviour has been analysed for the woven laminates that have been used to test industrial samples of the composite. This will be addressed in subsequent papers (Tang *et al.* 2009; Zhang & Hayhurst 2009).

## 3. Thermal behaviour

### (a) Longitudinal thermal response of a uni-directional tow

The thermal response of a CMC under tension is usually measured in terms of the degradation of thermal conductivity, *k*, with respect to strain. The initial longitudinal thermal conductivity, of a typical tow that consists of an inner tow matrix and an outer composite matrix (figure 1*a*) can be represented by a simple one-dimensional parallel rule of mixtures model, and can be expressed by
3.1
where *k*_{f//} represents the thermal conductivity of fibres in the longitudinal direction, *k*_{im} and *k*_{om} are the thermal conductivities of the inner tow matrix and the outer composite matrix, respectively, *V*_{om} is the volume fraction of the outer matrix and *k*_{air} and *V*_{air} are the thermal conductivity and volume fraction of initial porosity present in the tow. For a material with a single inner tow matrix, the term *k*_{om}*V*_{om} is omitted, and since *k*_{air}≈0, the term *k*_{air}*V*_{air} is also neglected.

Micrographs (Del Puglia *et al.* 2004*a*) show that the outer matrix of a typical CMC is subjected to cracking in manufacture, which occurs on planes perpendicular to the tow axis. The consequence of matrix cracking is to create a thermal barrier across the cracks and thus prevent heat flow through the matrix. According to Lu & Hutchinson (1996), the air gap must be at least 0.1 μm to prevent heat flow effectively. Hence, *k*_{om} parallel to the tow axis becomes zero during manufacture. In addition, after cracking of the inner matrix because of applied strain, the right-hand side of equation (3.1) reduces to *k*_{f//}*V*_{f}.

As well as matrix cracking, fibre failure must also be taken into account. Once a fibre has failed and an air gap generated, it is assumed that it can no longer conduct heat along it owing to the distance between the fibre ends at the failure point being greater than 0.1 μm. Since the damage variable, *ω*, is directly related to the number of failed fibres, the following relationship can be obtained for the longitudinal thermal conductivity of a single tow:
3.2

Equations (2.11) and (2.12) can be coupled to yield an expression for this expression has been integrated numerically and, together with equation (3.2), produces the curve, shown in figure 8, for typical material data defined in §2*a*(v) with *k*_{f//}=40 W m^{−1}K^{−1} and *k*_{im}=10 W m^{−1}K^{−1}.

The effect of cracking within the inner matrix is significant and shows a sharp drop in conductivity at the matrix cracking strain, λ_{mc}=0.062. The curve proceeds at a near-constant thermal conductivity until the majority of fibres begin to fail at Between normalized strains of 0.3 and 1.0, approximately 80 per cent of thermal conductivity is lost. After this, the remaining conductivity reduces to zero as the remaining fibres fail.

There is no experimental data available to categorize the longitudinal thermal response of CMCs under in-plane tensile straining.

### (b) Transverse thermal response of a uni-directional tow

Thermal conductivity of a CMC in the transverse direction, *k*_{ T}, is usually much lower than that in the longitudinal direction owing to the orthotropic nature of the fibres. It has been reported by Taylor *et al.* (1993), Gibson (1994) and Taylor (2000) that the transverse thermal conductivity of fibres, *k*_{f⊥}, is typically one order of magnitude lower than the longitudinal value *k*_{f//}. Prediction of the transverse thermal properties of a uni-directional tow in its undamaged state is more complex than in the longitudinal direction, and this is because the uni-directional transverse thermal model must be set up as a combination of zones in parallel and series. Figure 9*a* shows transverse heat flow through a cross section of a tow, and figure 9*b* shows the division into parallel and serial elements. Zone A in figure 9*b* represents the direct heat flow path through the outer matrix to the side of the fibres. Since zone A only contains an outer matrix, it can be assumed that its transverse conductivity is that of the outer matrix, i.e. *k*_{TA}=*k*_{om}. Zone B consists of fibres and both inner and outer matrices. The initial value of the transverse thermal conductivity, , can be calculated using a series model comprising the two different regions. The first region consists of only the top and bottom layers of outer matrix, and can be assumed to have a thermal conductivity of the outer matrix, *k*_{om}. The second or central region contains both fibre and inner matrix material. The initial transverse thermal conductivity of this region, , can be calculated using the following equation:
3.3
where is the volume fraction of fibres in the central region of zone B. This expression, which can be found as eqn (12) in Mottram & Taylor (1987), has its origins in the work of Rayleigh (1892) and subsequently that of Bruggeman (1935). This development has been reviewed by Clayton (1971) and formalized by Mottram & Taylor (1987) and Whittaker & Taylor (1990). Equation (3.3) describes transverse heat conduction through a two-phase medium: the inner matrix material is treated as the continuous phase and the cylindrical fibres as the discontinuous phase. The shape factor *X*=1 relates to a regular array of cylindrical fibres. Rearrangement of equation (3.3) gives in terms of *k*_{im},*k*_{f⊥} and ,
3.4
where . The central region and outer matrix region in zone B in figure 9*b* can now be coupled to determine the overall transverse thermal conductivity of zone B, *k*_{TB}, using a simple series heat flow equation,
3.5
where represents volume, i.e. is the volume of outer matrix material within zone B. Consideration of heat flux through zones A and B using a simple parallel model yields the following expression for overall transverse thermal conductivity, *k*_{T}:
3.6
where *V*_{A} and *V*_{B} are volume fractions of zones A and B, respectively. For a material with a single matrix, *k*_{om} is removed from the derivation, and equation (3.6) simply becomes
3.7

#### (i) Principle damage mechanisms affecting transverse thermal conductivity

To predict the degradation of transverse thermal conductivity of a tow, it is necessary to consider the initiation and growth of damage by the predominant mechanisms active during axial loading. These damage mechanisms must be considered separately and then, if necessary, coupled to form a prediction of the overall degradation. Mechanisms that involve pockets of air between adjacent tows, categorized as Class D porosity by Del Puglia *et al.* (2005), have not been addressed here, as they do not relate to tow behaviour, are stable and unlikely to produce changes in transverse thermal conductivity with strain. Four inter-fibre micro-porosity damage mechanisms are considered here:

crack front and wake debonding,

spherical pores,

shrinkage debonded cracks, and

fibre failure and pullout.

All four of these strain-dependent damage mechanisms influence only the thermal conductivity of zone B, shown in figure 9*b*. The transverse thermal conductivity of zone A is assumed to be unaffected by strain. Hence, in subsequent sections, all thermal conductivities are calculated for zone B only.

The mechanism of crack front and wake debonding has been found to principally affect the transverse thermal conductance, and it will now be addressed. Each of the remaining failure mechanisms is discussed in detail in appendix A.

Crack front and wake debonding damage precedes fibre failure and takes place at both the matrix crack tip and, more significantly, in its wake. It is assumed to completely surround the fibre, thus preventing heat flow through that region. Wake debonding is triggered, similarly to shrinkage debonding, by the interfacial shear stress between the inner matrix and fibre. As shown in figure 10, the maximum levels of shear stress occur at the locations of the matrix cracks, *x*=0 and *x*=*w*; therefore, when the shear stresses at these locations reach the critical level, wake debonding will propagate from *x*=0 and *x*=*w* and move towards the centre (*x*=*w*/2) until saturation. This mechanism is the strongest, and most prevalent, and will be assumed to control transverse thermal conductivity. To quantify this, an assemblage of simple representative volume elements or blocks (figure 11*b*) taken from a single tow (figure 11*a*) are analysed.

#### (ii) Analysis of a fibre–matrix block between adjacent matrix cracks

The central region of zone B in figure 9 is shown schematically in figure 11. The solid ellipses denote cross sections of fibres passing through the tow. The cross-sectional arrangement of fibres in the tow is assumed constant along its length. The tow can be further broken down into a series of blocks, as shown in figure 11*a*, where the length of each block is the matrix crack spacing *w*. Each fibre passes through a series of blocks. A single block, shown in figure 11*b*, contains one fibre that will fail at an unknown damage level, * ω*. Fibres within a tow are assumed to be uniformly arranged into a number of layers, ψ, each containing an equal number of fibres, η. The number of matrix crack spacings, φ, is given by
3.8
where

*G*is the total length of the parallel segments of one tow contained within a composite unit cell. The total number of blocks,

*N*

_{block}, is therefore given by 3.9 The volume of air created by matrix cracking will not be considered in this analysis because the ratio between the matrix crack separation, 2

*U*

_{0}, and the matrix crack spacing,

*w*+2

*U*

_{0}, is less than 0.01 per cent, and therefore, the volume of air created at matrix cracking is negligible.

#### (iii) Wake debonding and transverse thermal conductivity

Wake debonding allows stresses to redistribute throughout the material and thus avoid high local stress concentrations. It is assumed to occur before fibre failure and occupies a region from the matrix crack to the fibre failure location, (figure 12). For a typical material, Jansson & Leckie (1992) have determined that , and this has been confirmed by calculations using the material parameters specified earlier. From this, it can be assumed that there is a single fibre failure location at the midpoint between every pair of matrix cracks within every fibre, and therefore within every block shown in figure 11.

Wake debonding surrounds the fibre, as shown in figure 12, and is assumed to have a perfect cylindrical form. Eqn (15) of Mottram & Taylor (1987) may be used to calculate the effect of wake debonding on in terms of the volume fraction of air, *V*_{p}, produced by debonding,
3.10
where is the thermal conductivity of the central region of zone B (cf. figure 9) as a result of wake debonding. The equation has been derived using equation (3.3) for the fibre–matrix system with zero porosity as the continuous phase and introducing the porosity *V*_{p} as the discontinuous phase. A shape factor of *X* = 0.1 has been chosen to accurately reflect the behaviour of cylindrical porosity segments with heat flow normal to the axes of the cylindrical fibres, and also to represent partial shielding of the heat flow to the material downstream of the air-filled cracks.

The volume fraction of air for wake debonding, *V*_{p}, has been calculated for a single block. It can be determined from the cylindrical air volume surrounding the fibre that has length *w*. Proceeding in this way, the volume fraction *V*_{p} for a single block is
3.11
where *b* and *c* are the width and height of the tow, *t*_{om} is the thickness of the outer matrix material, shown in figure 9 (*t*_{om} for a single-matrix material will be zero) and *d* is the average air gap produced by wake debonding (figure 12); a value of *d*=1.0 μm has been taken as being typical of CMC composites (Fareed 2005). This value is above the critical level of 0.1 μm determined by Lu & Hutchinson (1996) for zero heat conductance. It is therefore a reasonable assumption that heat cannot flow transversely through the region affected by wake debonding. For the whole tow, the total number of blocks in which wake debonding has occurred, *N*, at any given value of strain must be included into equation (3.11). In addition, there will be an increase in the volume of the block owing to the introduction of the porosity, and this can be taken into account by using the equation . Equation (3.11) for a single block then becomes the following for the entire tow:
3.12

A model, based upon the maximum interfacial shear stress, has been developed to calculate the total number of wake-debonded blocks, *N*. After matrix cracking, fibres carry the majority of the load; however, before wake debonding occurs, a small amount of the load is transferred to the matrix via the interfacial shear stress. Therefore, as the load in the fibres increases, so does this shear stress, until a critical value is reached at the location of the matrix crack, *x*=0 (cf. appendix B), and wake debonding takes place. It is therefore important to be able to calculate the maximum shear stress along the fibre–matrix interface at different levels of strain. An equation for the variation of shear stress, *τ*, at *x*=0 in an individual block, before the inception of wake debonding, with composite strain is derived in appendix A and given as
3.13
Figure 13 shows the variation of *τ*(*x*=0) with normalized composite strain, , in blocks that have not wake debonded. Prior to matrix cracking, , the shear stress is assumed to be zero. This shear stress can be used to calculate the total number of wake-debonded blocks using a statistical approach similar to that used for fibre failure in equation (2.4),
3.14
where *g* is the Weibull distribution index for interfacial shear failure and *τ*_{c} is the average critical shear stress for wake debonding. Initial values of *τ*_{c}=25 MPa and *g*=4.5 have been arbitrarily used for the illustrative calculations presented here. Substitution of equation (3.14) into equation (3.12) gives the volume fraction of porosity, *V*_{p}, in terms of shear stress,
3.15
where the constant *Φ*=(*b*−2*t*_{om})(*c*−2*t*_{om})/ηψπ(*D*_{f}*d*+*d*^{2}). The overall degradation of thermal conductivity can then be calculated by substitution of equation (3.15) into equation (3.10),
3.16

It is now possible to calculate the degradation of the transverse thermal conductivity of the central region of zone B (cf. figure 9) owing to wake debonding. The degradation of thermal conductivity of the whole tow can be determined by replacing the *k*_{TB.Cent.} term in equation (3.5) with and substitution into equation (3.6); further substitutions from equations (3.13) and (3.16) yield the following expression *for a dual-matrix material*:
3.17

where Substitution of equations (3.13) and (3.16) into equation (3.7) yields the following expression *for a single-matrix material*:
3.18
The results for the variation of normalized transverse thermal conductivity, with normalized composite strain, , predicted in this way for a dual-matrix material using equation (3.18), are shown in figure 14 for typical material properties given in previous sections and *t*_{om}=0.0247 mm,φ=56,*b*=0.803 mm,*c*=0.161 mm,*g*=4.5 and *d*=1 μm, with three values of *τ*_{c}. The effect of wake debonding, shown in figure 14, is small at low values of strain because of the relatively small number of blocks that contain wake debonding. The sharp drop in thermal conductivity at a normalized strain of around is caused by cracking of the inner matrix. This causes an increase in shear stress (cf. figure 13), which, in turn, results in wake debonding occurring in the weakest blocks. The curve then continues at a relatively constant value, owing to comparatively constant shear stresses. At a normalized strain of approximately figure 13 shows that there is a rapid rise in shear stress that causes the majority of wake debonding to occur; this results in the significant drop in thermal conductivity shown in figure 14. The value of thermal conductivity then levels off as all of the blocks contain wake debonding, and the only available heat flow paths are through matrix material (zones A and matrix section of B in figure 9*b*). This latter feature is responsible for the final plateau of figure 14.

#### (iv) Sensitivity of *k*_{T} to model parameters for a dual-matrix material

In the model for degradation of transverse thermal conductivity presented earlier, three parameters have been identified that control the shape of the curve for normalized transverse thermal conductivity with normalized composite strain; these are: (i) the critical shear stress for wake debonding, *τ*_{c}, (ii) the Weibull distribution index for the fibre–matrix interface, *g*, and (iii) the wake debonding air gap distance, *d*. Presented in the following is a sensitivity study for each of these parameters to establish their effect on the curve.

##### Critical shear stress for wake debonding, *τ*_{c}

The effect on the degradation of transverse thermal conductivity owing to the critical shear stress for wake debonding, *τ*_{c}, is shown in figure 14. It can be seen that, as *τ*_{c} decreases, there is a greater reduction in thermal conductivity at matrix cracking . For lower values of *τ*_{c}, a greater number of blocks will fail by wake debonding according to the Weibull distribution, and an earlier degradation of the transverse thermal conductivity will occur. The minimum value of transverse thermal conductivity is not affected by *τ*_{c}, as it is determined by the outer matrix and the remaining intact inner matrix material.

##### Weibull distribution index, *g*

The effect on normalized transverse thermal conductivity owing to the variation of the Weibull distribution index, *g*, is presented in figure 15 for *τ*_{c}=25 MPa and *d*=1 μm. As *g* increases, fewer blocks will wake debond at shear stresses that deviate from *τ*_{c}. Hence, for larger values of *g*, the thermal conductivity curve experiences a lower drop at matrix cracking, and a steeper decrease in thermal conductivity, when the majority of blocks wake debond . This is because wake debonding occurs over a smaller strain range than that for smaller values of *g*.

##### Wake debonding air gap distance, *d*

The effect on normalized transverse thermal conductivity owing to the variation of the average wake debonding air gap distance, *d*, is shown in figure 16 for *τ*_{c}=25 MPa and *g*=4.5. The major changes are at matrix cracking and once all blocks have wake debonded . The value of *d* determines the volume of porosity present in a wake-debonded block; therefore, as *d* increases, there will be more porosity present and the value of thermal conductivity will be lower. The shape of the curve is unaffected because *d* does not determine the number of blocks that contain wake debonding at any given value of normalized strain.

It is evident from this study that the curve for normalized transverse thermal conductivity against normalized strain is dependent on all three parameters, and no single parameter dominates. It is, however, clear that *d* controls the final value of the thermal conductivity. It can also be argued that *g* controls the steepness of the curve when the majority of degradation is taking place and that *τ*_{c} influences the level of strain at which the majority of degradation occurs. All three parameters have some effect on the level of degradation at matrix cracking. In order to fit this model to experimental data, it is important to consider the simultaneous effect of all three parameters.

#### (v) Influence of multi-axial stressing

It is worth addressing the question of how the normalized transverse thermal conductivity, curves against normalized composite strain, might be influenced by multi-axial loading. As discussed in §2*a*(iv), the most deleterious stress components are likely to be transverse tension and shear, either parallel or perpendicular to the tow axis. It has been argued, in that section, that their effect would be to negate the wake-debonding mechanism and effectively introduce cylindrical air gaps along the fibres. If this is correct, the gentle decays in with shown in figures 14–16 may be replaced by sudden drops of at almost constant strain. Data available on woven CMC composites by Sheikh *et al.* (in press) indicate that this could be the operative mechanism. For 0/90 and plain weave CMC composites, this is observed to be the case, but the 8-harness satin weave composite does not show this mechanism. Hence, the tow model would require appropriate enhancement for multi-axial loading.

## 4. Discussion and conclusions

A uni-directional model has been developed for assemblages of fibres or tows contained within matrices, which describes the uni-directional stress–strain response, the axial and transverse thermal conductivities and their dependencies on axial composite strain. The stress–strain model includes the statistics of fibre failure and fibre pullout on the stress–strain response.

The approximate thermal model is based on one-dimensional heat flow assumptions.

Longitudinal or axial tow thermal conductivity is considered to be degraded predominantly by fibre failure, and its variation with composite axial strain has been predicted by the model.

Transverse tow heat flow is considered to be degraded by four possible mechanisms: (i) wake debonding at matrix cracks and its growth with strain, (ii) growth of spherical pores associated with porosity created during manufacturing/processing on the interface between fibre and matrix, (iii) shrinkage debonding, and (iv) fibre failure and pullout.

*Wake debonding* is relatively dormant in the early stages of composite straining, but predominates as the majority of fibres fail in the region of peak composite stress. It is dominated by the statistical variation of the interfacial shear stress required to initiate the mechanism, here assumed to be described by a Weibull distribution. The shape of the curve of normalized transverse thermal conductivity against axial normalized composite strain is sensitive to the Weibull index, *g*, interfacial failure shear stress, *τ*_{c}, and the size of the air gap associated with wake debonding, *d*.

*Spherical pore and shrinkage debonding* mechanisms do not propagate sufficiently to degrade the transverse tow thermal conductivity.

*Fibre failure and pullout* occur after and within the same region as wake debonding and therefore its effect on transverse thermal conductivity is negated.

The model predicts the expected shape of curve for the variation of normalized transverse thermal conductivity with axial composite strain for a uni-directional tow.

To validate the uni-directional tow model, it is necessary to compare predictions with experimental results on uni-directional composites. Since such experimental data for CMCs are not usually available, it is necessary to use data for woven composite laminates. For the latter situations, the tows are not straight and are subjected to multi-axial loading. Hence, before the fidelity of the model predictions can be assessed, it is necessary to extend the tow model to multi-axial stress conditions. An approximate model (Tang *et al.* 2009) has been developed for this purpose and shows encouraging results, and an orthotropic multi-linear elastic model (Zhang & Hayhurst 2009) addresses the complexities of the interaction of several damage mechanisms.

## Appendix A. Failure mechanisms potentially affecting transverse thermal conductivity

In §3*b*(i), four damage mechanisms were cited, but only that owing to crack front and wake debonding was considered in detail. The remaining three mechanisms are now addressed.

Inter-fibre micro-porosity occurs within the inner matrix material between fibres within a tow. It is made up of a number of voids or spherical pores and of cracks known as shrinkage debonding (Del Puglia *et al.* 2004*a*). It is clear from the micrographs of the DLR-XT composite obtained by Del Puglia *et al.* (2004*a*) that inter-fibre micro-porosity is present in the virgin material as a result of processing. In order to cause a degradation of transverse thermal conductivity upon loading of the composite owing to spherical pores and shrinkage debonding, it is necessary for them to propagate. To assess whether this is likely, it is necessary to examine micrographs of the composite to determine the location, relative size and spatial distribution of the porosity. Each mechanism is now addressed.

*Spherical pores*. Examination of micrographs of DLR-XT reveals spherical pores, described as Class A1 porosity by Del Puglia *et al.* (2005), cf. the micrograph of a section perpendicular to fibres given in figure 17*a*, and the schematic of a fibre segment removed from between two adjacent matrix cracks, of spacing *w*, as shown in figure 17*b*. The spherical pores are adjacent to the fibres, randomly dispersed in the inner matrix and have diameter, *q*, which is a fraction of a micron. Propagation of spherical pores is strain controlled, and the growth of the porosity will be determined by Δ*q*=*f*(*ϵ*_{im},*q*), where Δ*q* is the change in spherical pore size and *ϵ*_{im} is the strain within the inner matrix. The strain field, *ϵ*_{im}, is strongly constrained by the adjacent stiff fibres; this, coupled with the small physical size of spherical pores, relative to the fibres, and their low concentration means that growth will be negligible. Hence, their effect on the degradation of transverse thermal conductivity can be safely neglected.

*Shrinkage debonding* (Del Puglia *et al.* 2004*a*) occurs at the interface between the inner matrix and fibres, as shown in figure 17. It has been termed Class A2 porosity by Del Puglia *et al.* (2005), and observed from micrographs to be randomly dispersed along the interface, with varying crack lengths. The shape of the thin shrinkage debonded interfacial cracks is typically circular/elliptical, with a characteristic dimension of approximately one-tenth of the fibre diameter. Propagation of shrinkage debonding occurs when the interfacial shear stress between the fibres and inner matrix reaches a critical level. When this level is achieved, the crack extends along the interface and reduces the transverse thermal conductivity accordingly. An expression can be derived (appendix B) for the shear stress at any position along the fibre–matrix interface between two matrix cracks. This model is dependent upon the stress within the fibres and the overall composite strain. Figure 10 shows that the shear stress significantly reduces with distance along the fibre, *x*, to zero at *x*=*w*/2. An assessment of the effects of shrinkage debonding on transverse thermal conductivity has been carried out based on the critical shear stress required to propagate an interfacial fibre–matrix crack.

Owing to the dilute concentration of shrinkage debonded cracks, the surrounding strain field is close to that found in the absence of the cracks, and the shrinkage debonded crack tip displacements are strongly constrained. Hence, the interfacial shear stress *τ* does not reach the critical value required for crack propagation, even at the fibre–matrix interface near the matrix crack (*x*=0), where the shear stresses are highest, cf. equation (B 12). Consequently, shrinkage debonded cracks are unlikely to propagate, will have a minimal effect on the overall transverse conductivity and can be neglected.

*Fibre failure and subsequent pullout* creates an air gap where the fibre has been pulled out. This air gap is assumed to have zero conductivity and could, therefore, have an effect on transverse thermal conductivity. However, fibre pullout takes place after wake debonding has occurred, and since wake debonding is assumed to surround the fibre, it thus prevents heat flow through it. The effect of fibre pullout is therefore already taken into account through wake debonding, and can be neglected.

## Appendix B. Determination of fibre–matrix interfacial shear stress

In this section, the derivation is given for shear stress across the fibre–matrix interface located between adjacent matrix cracks (figure 18). The distribution of shear stress along the fibre axis (*x* direction) has been investigated by many authors. Kelly (1989) assumed a constant distribution of shear stress for a weak fibre–matrix interface; whereas Kerans & Parthasarathy (1991) argued that the shear stress is zero at the free surfaces of the matrix, and then increases to a maximum value at small *x*. After this, it will decay to zero at a point at or before *w*/2, depending on Young’s moduli and volume fractions of fibre and matrix. In the analyses by Cox (1952) and Kelly & Macmillan (1986), the initial rise of shear stress at small *x* was neglected and a hyperbolic expression derived for the shear stress distribution. This relation will be used here.

After matrix cracking, the stress within a single fibre is given by , and the strain of the overall composite is determined by the strain within a single fibre. However, as the stress within the fibre along the fibre axis is not constant (Kelly & MacMillan 1986), owing to shear interaction with the matrix, an average value needs to be calculated. This average value is equal to the overall composite strain and is assumed to be given by
B 1
At a distance *x* from the free surface of the matrix, the stresses and the cross-sectional areas of matrix and fibre are, respectively, *σ*_{m}(*x*),*A*_{m}, *σ*_{f}(*x*) and *A*_{f}. Kelly & MacMillan (1986) assumed that
B 2
where *σ*_{m}(*x*) is the matrix stress, *H* is the constant determined by Young’s moduli and geometrical arrangement of fibres and matrix, *u* is the longitudinal displacement of the matrix and *v* is the displacement, linear in *x*, that the fibre would undergo at the same point if the matrix were absent. For the matrix,
B 3
where *e* is the uniform fibre strain in the absence of the matrix. Substitution of equation (B 3) into the differentiated form of equation (B 2) gives
B 4
Solution of this differentiated equation yields
B 5
where is the shear modulus of the fibre, *A*_{f} is the fibre cross-sectional area within a block and *R*_{b} is the half height of a single block, as shown in figure 18. Typical values of *G*_{f}=11.9 GPa and *R*_{b}=7 μm have been used for the calculations presented here. As the stress within the matrix is transferred from the fibre via the shear stress, the force equilibrium equation of the matrix gives
B 6
Substitution of equation (B 6) into (B 5) gives the expression for the shear stress,
B 7
this equation can be normalized as
B 8
Equation (B 8) is plotted in figure 10. The maximum shear stress occurs at the free surfaces of the matrix. For the force equilibrium of the single block, the following equation applies:
B 9
Substitution of *σ*_{f}(*x*) by *E*_{f}*ϵ*_{f}(*x*) and of *σ*_{m}(*x*) from equation (B 5) into equation (B 9) gives an equation in terms of *ϵ*_{f}(*x*). Substitution of this equation into equation (B 1), after integration, gives
B 10
Rearrangement of equation (B 10) yields the expression
B 11
Substitution of equation (B 11) into equation (B 7) yields the maximum shear stress at *x*=0,
B 12
Equation (B 12) gives the variation of the maximum shear stress with the remote composite stress and strain .

## Footnotes

- Received May 28, 2009.
- Accepted June 2, 2009.

- © 2009 The Royal Society