Effects of in-plane extension on transverse thermal conductivity of a carbon–carbon 8-harness satin weave composite

Abstract

A finite element-based technique for coupled thermo-mechanical analysis of woven Ceramic Matrix Composites sheets is presented for the prediction of the degradation of transverse thermal transport behaviour with in-plane extension. The thermal conductivity–strain characteristics have been determined, at the tow level, from the properties of the constituent elements, and then extended to tows and composite. The non-linear thermal conductivity-extension behaviour of the tow has been discretised by multi-linear curves, and implemented in a user-defined subroutine in ABAQUS to model the behaviour of the homogenised orthotropic unidirectional tow and its matrix. By using this approach, an 8-Harness Satin weave HITCO C/C composite unit cell has then been analysed. The variation of through-thickness thermal conductivity degradation with in-plane extension has been predicted and compared with the results of experiments. Very good agreement has been achieved. Two classes of behaviour have been experimentally observed: one that exhibits a brittle response, and another that shows a quasi-ductile behaviour. Both classes of behaviour have been predicted and shown to relate, respectively, to strain localisation and instantaneous pull-out deactivation, without localisation being invoked. These responses are reflected directly in the predicted and experimental rates of decay of transverse thermal conductivity with axial extension. It is advocated that the reduction in transverse thermal conductivity with extension and damage can be used as a Structural Integrity Monitor for CMC operational components.

Introduction

The low density and good mechanical and thermal properties of Ceramic Matrix Composites (CMCs) at high temperatures make them favoured materials for use in rocket nozzles, thermal protection systems and gas turbine engines (Boyle et al., 2014; Class, 2008; Marshall and Cox, 2008; Raether, 2013; Wood, 2013). The driver for use in such applications is the potential for higher thermal efficiencies and lower emissions (Evans and Naslain, 1995). However, due to high costs and long lead times, the development of new components and engineering structures in these materials cannot be achieved by prototype development alone. Cox and Yang (2006) and McGlockton et al. (2003) have identified as an essential requirement the ability to predict, at the concept design stage, both the mechanical and thermal properties, coupled with the optimal manufacturing route, of CMC material test coupons and engineering components. Hence, accurate computer-based numerical simulation is an essential part of the materials selection process for CMC component design and manufacture. To achieve optimal design of engineering components by computer simulations, highly efficient yet computationally accurate methods of analysis are required which take into account the complex three-dimensional tow geometry (Blacklock et al., 2012a; Rinaldi et al., 2012) and the interactive damage mechanisms in woven CMCs that evolve with applied strain.

As demonstrated in the experiments of Sheikh et al. (2009) a major difficulty with CMCs is that their thermal conductivity degrades with composite strain. The effect is not only dependent on the properties of the constituent materials, but also on the damage mechanisms that accompany the straining process. The damage that is inevitably introduced in the manufacturing process by either applied mechanical loads or thermal stresses due to time-varying temperature gradients, subsequently forms micro and macro-cracks that become filled with air; these air pockets become the major barrier to thermal transport, and degrade the composites thermal conductivity (Lu and Hutchinson, 1996). This degradation, which is enhanced by in-service mechanical strain, influences temperature distributions in operating structures, resulting in increased thermal gradients and unwanted thermal stresses. The knock-on effect is that operating temperatures have to be decreased to achieve the required operating life spans, and consequently thermal efficiencies are compromised. It is therefore desirable to be able to analyse these effects at the early stages of design, and hence optimise materials selection in design and for optimal manufacturing route. Additionally, the use of finite element predictions in conjunction with periodic in-service measurements of transverse thermal conductivity has the potential for use as a structural health monitor, which perhaps can be used to decide when to replace a used component. A barrier to the complete analysis of CMC engineering structures is the inability to model the stress–strain and thermal conductivity–strain responses of CMCs, in particular its evolution during operation to the point of unserviceability, and the work reported here seeks to help rectify this.

Some of the approaches that used finite elements to model thermal behaviour of CMCs were two dimensional (Gao et al., 2014; Klett et al., 1999; Lu and Hutchinson, 1996). The limitation of these approaches was their simplicity, as the weave-related complexities of real composites (Blacklock et al., 2012a; in press; Cox et al., 2016) were neglected. Sheikh et al. (2001) presented a complex weave model of a plain weave CMC, and their three-dimensional model was a step towards the analysis of complex composite architectures. It included the effect of directionality in thermal transport by the introduction of the individual properties of fibre and matrix. However, initial manufacturing porosity was not modelled. Del Puglia et al. (2004a, 2004b, 2005) later overcame this deficiency with the inclusion of porosity at the macro-unit cell level for the same (DLR-XT) material.

Zhang and Hayhurst (2009, 2010) studied the mechanical behaviour of two CMCs; a Nicalon-CAS 0°/90° laminate (Tang and Hayhurst, 2011; Tang et al., 2011a) and a DLR-XT plain weave composite (Tang et al., 2011b). This was further extended by Hayhurst (2013) and Zhang and Hayhurst (2011) to include the degradation of transverse thermal conductivity for the same two materials. In both studies, good predictions of the experimental data of Sheikh et al. (2009) were made from tow properties obtained by using the tow model of Hayhurst et al. (1991) and Tang et al. (2009) The work of Zhang and Hayhurst (2010, 2011) was then extended to the more complex 8-Harness satin weave of a carbon fibre–amorphous carbon matrix composite (Zhang and Hayhurst, 2014). In the latter investigation, it was shown how the stress–strain composite response could be predicted from tow behaviour determined using the model due to Tang et al. (2009) and constituent materials properties of Blacklock and Hayhurst (2012b).

It is worth noting here that the work of Tang et al. (2009) linked the microscale of: fibres, interfaces, and matrices with damage in the form of delamination and porosity. To do this, the work of Lord Rayleigh was used (Rayleigh, 1892),who noted in his paper that a range of geometrical defects, porosity, and more generally damage influenced both thermal and electrical properties through the same field equations, hence providing a cross-disciplinary link. Also, Lubineau et al. (2013) and Selvakumaran and Lubineau (2014) have noted the similarity between the equations for electrical and thermal dissipation, and provided rigorous homogenisation at the micro-scale for electrical properties of composites. This is clearly an area where multi-physical interactions are apparent, and which have potential to provide cross-disciplinary insight.

This paper addresses the influence of applied in-plane uni-axial straining on the through-thickness thermal conductivity of a HITCO C/C 8-satin weave composite. The method adopted by Tang et al.(2009) for modelling the composite using assemblies of tows, i.e. a collection of thousands of fibres embedded in at least one matrix, has been embodied in the approach. Non-linear thermo-mechanical tow material properties have been modelled by a multi-linear elastic discretisation. The tow behaviour has been analysed using orthotropic finite elements. In this way, the multi-axial stress–strain response and the degradation of thermal conductivity with strain of the 8-Harness Satin weave composite have been analysed.

An idealised unit cell of the composite is shown schematically in Figure 1. The HITCO C/C 8-Harness Satin weave has T300 carbon fibre tows woven together to form a satin woven cloth or lamina (Tang et al., 2011b). Nine laminae have been stacked together and the material infiltrated to form a sheet of the composite. Further details of the material and its manufacturing route are given in Tang et al. (2011b). Sheikh et al. (2009) have reported uni-axial experimental data on the material for variation of axial stress and transverse (through-thickness) thermal conductivity with axial strain, and the results have been used to judge the fidelity of the analytical approach.

Figure 1.

Schematic of the unit cell of the 8-Harness Satin Weave HITCO material.

The next section introduces the thermal behaviour of a HITCO composite tow.

Thermal behaviour of a unidirectional CMC tow

As outlined in the introduction, finite element analysis has been carried out using tow stress–strain and thermal conductivity–strain behaviour as modelled by Hayhurst et al. (1991) and Tang et al. (2009). They considered the schematic representation of a tow, Figure 2, where the solid ellipses represent fibre cross sections, and the vertical lines in the 1–2 planes denote matrix cracks which form at relatively low levels of the stress (σ₃₃)_∞. Matrix cracking characteristically occurs at a uniform spacing w (Hayhurst et al., 1991), and triggers other phenomena such as wake debonding, fibre failure and pull-out, which all degrade the transverse thermal conductivity. The spacing w is defined in Figure 2(b) and also in the cross-sectional diagram of the block shown in Figure 3.

Figure 2.

(a) Schematic representation of the division of the composite tow into blocks of length equal to the matrix crack spacing w; fibres are denoted by solid ellipses, and (b) representation of a single block.

Figure 3.

Schematic diagram of a section of a fibre–matrix block of length w extracted from between adjacent matrix cracks. The interface is shown with (a) damage as a result of wake debonding, and (b) further interfacial damage due to transverse loading.

To understand the degradation of the thermal properties of the 8-Harness satin weave HITCO composite, it is necessary to perform a coupled mechanical strain-thermal conductivity analysis, for the unit cell shown in Figure 1, which utilises exactly the same definition of the boundary value problem (Hayhurst, 2013) for both mechanical and thermal analyses. For details of the mechanical part of the analysis of the HITCO 8-Harness woven CMC, the reader is directed to the paper by Zhang and Hayhurst (2014). In the next sections, various aspects of the controlling physical mechanisms are addressed in preparation for the presentation of the coupled mechanical strain–thermal conductivity analysis.

Tow mechanical behavior

For a unidirectional tow under longitudinal uni-axial loading, the material exhibits ductile behaviour, c.f. Figure 4(a), which is demonstrated as the gentle decreasing stress–strain curve following the peak (Blacklock and Hayhurst, 2011). Here the composite stress (σ₃₃)_∞ is the applied stress, and the composite strain (ɛ₃₃) _∞ is the gauge length strain, denoted by the subscript (∞); the components of stress and strain are parallel to the fibre tows, denoted by the subscript (33). This is due to wake debonding of the fibre-matrix interface that occurs gradually, and creates a partially intact fibre–matrix interface, which allows a failed fibre to pull out against a frictional stress along the wake debonding interface. This occurs within individual blocks, where a block is defined by a single fibre and its associated matrix that is contained between two adjacent matrix cracks, c.f. Figures 2 and 3. This phenomenon has been discussed in detail by Tang et al. (2009).

Figure 4.

Material properties for a CMC composite: (a) stress–strain curve of a unidirectional tow under longitudinal loading and variation of normalised number of wake debonded blocks N/N_T with composite strain (ɛ₃₃)_∞, and (b) stress–strain curve of a unidirectional tow (Blacklock and Hayhurst, 2011).

The variation within a tow of the normalised number of wake debonded blocks, N/N_T, with composite strain, (ɛ₃₃)_∞, where N is the number of failed blocks, and N_T is the total number of blocks in the tow segment shown in Figure 2, has been discussed by Blacklock and Hayhurst (2011) and is shown in Figure 4(a) c.f. right-hand ordinate. When a unidirectional tow is subjected to multi-axial loading, a positive transverse stress or a shear stress, c.f. Figure 3(b), advances wake debonding, and also degrades the partially intact interface to a non-contacting interface. Hence, the frictional stress between fibre and matrix reduces to almost zero, and the pullout mechanism is weakened.

Even though the strength of the pull-out mechanism is reduced, it cannot explain the catastrophic fibre failure observed in the experiments reported by Sheikh et al. (2009). A further mechanism called dynamic fibre failure by instantaneous pullout deactivation has been proposed by Blacklock and Hayhust (2011), that is believed to be the cause of brittle failure. In this mechanism, it is postulated that one half of all blocks in a tow (N/N_T = 0.5, at the strain ɛ_wd shown in Figure 4(a)) simultaneously undergoes instantaneous wake debonding and fibre pull-out deactivation. Ductile CMCs are designed so that the average wake debonding strain, ɛ_wd, is slightly less than the peak composite strain, and hence at this point, the fibres are stressed to a high fraction of the average fibre failure stress. When fibre failure occurs between matrix cracks, a shear stress wave propagates away from both matrix cracks at the extremities of the block, and to maintain equilibrium, a tension stress wave is induced in the fibre. The tension wave moves at a higher velocity than the shear wave, and the two fibre tension waves meet at the centre of the block, and reflect as a fibre tension wave with twice the amplitude of the incident wave. This doubling of the tensile stresses in the fibres causes complete fibre failure. Axial stress redistributes to adjacent fibres causing more widespread failure of fibres and tow. By this mechanism, the original ductile CMC stress–strain curve switches to the degraded curve, c.f. Figure 4(b), and this is for a tow subjected to a modicum of multi-axial loading.

Tow longitudinal thermal conductivity

Tang et al. (2009) have postulated that longitudinal thermal conductivity is controlled by the air gaps introduced on matrix cracking, with regular crack separation distance w, c.f. Figure 2(b), and by fibre failure. They derived equations, based on the thermo-mechanical properties of the constituent materials, which can be numerically integrated to produce variations of the local tow longitudinal thermal conductivity $k_{\mathrm{Long}}^{\ell }$ , with local tow strain ${\varepsilon }_{33}^{\ell }$ . For convenience, the derivations of Tang et al. (2009) have been summarised in the Tow stress–strain response and Tow thermal behaviour sections of Appendix 1 for the mechanical and thermal behaviour, respectively. It is worth reiterating that it is the physics modelled by these equations that permits the scaling of mechanisms operational at the fibre scale, through finite element analysis, to the macro-scale of engineering components. The relevant equations for variation of $k_{\mathrm{Long}}^{\ell }/k_{\mathrm{Long}}^{\ell -\mathrm{Initial}}$ with the tow damage variable ω are given by equations (8) and (9), and for the variation of $k_{\mathrm{Long}}^{\ell }/k_{\mathrm{Long}}^{\ell -\mathrm{Initial}}$ with ${\varepsilon }_{33}^{\ell }$ are given by coupled numerical integration of equations (4) and (6). Using these relations and value of the initial longitudinal thermal conductivity $k_{\mathrm{Long}}^{\mathrm{Initial}}$ , the variation of $k_{\mathrm{Long}}^{\ell }$ with ${\varepsilon }_{33}^{\ell }$ has been obtained from the material parameters of the composite base materials. These are given in Figure 5(a) for the HITCO 8-Harness woven CMC. The data in Figure 5(a) has been used as discretised multi-linear input data for the finite element model. The XX crosses denote the points used to define the linear segments of the linearised curve to be used in the analyses reported herein.

Figure 5.

Multi-linear representations of the variation of tow thermal conductivities with local axial tow strain ${\varepsilon }_{33}^{\ell }$ : (a) longitudinal thermal conductivity $k_{\mathrm{Long}}^{\ell }$ , and (b) transverse thermal conductivity $k_{\mathrm{Tran}}^{\ell }$ .

Tow transverse thermal conductivity

Tang et al. (2009) have shown that transverse thermal conductivity is controlled by the process of wake debonding which produces a cylindrical air gap at the interface between fibre and matrix in a block of material associated with a single fibre located between two adjacent matrix cracks as shown in Figure 3, due to Blacklock and Hayhurst (2011). The poor thermal conductivity of the cylindrical air gap prevents transverse heat flow, in the 2-direction, through the fibre and some of the matrix. The degradation of transverse thermal conductivity, $k_{\mathrm{Tran}}^{\ell }$ , with tow strain, ${\varepsilon }_{33}^{\ell }$ , is related to the ratio, N/N_T. The interaction between composite stress (σ₃₃)_∞ and composite strain (ɛ₃₃)_∞, and N/N_T is shown in Figure 4(a) due to Blacklock and Hayhurst (2011).

For convenience, the equations derived by Tang et al. (2009) for the variation of transverse thermal conductivity with strain are summarised in the Tow transverse thermal conductivity section of Appendix 1, and the calculated values are given in this section. The degradation of $k_{\mathrm{Tran}}^{\ell }$ with N/N_T is illustrated in Figure 10 of Tang et al. (2011b), and the corresponding graph of $k_{\mathrm{Tran}}^{\ell }$ against ${\varepsilon }_{33}^{\ell }$ for the HITCO 8-Harness satin weave tow is given in Figure 5(b) of this paper, which again has been used as multi-linear input data for the finite element model. The XX crosses denote the points used to define the linear segments of the linearised curve to be used in the analyses reported herein.

Effects of shear strains on thermal conductivity in the longitudinal and through-thickness plane

In the study of another material, a combined SiC-amorphous carbon matrix plain weave carbon fibre DLR-XT composite, the variation was determined of the transverse thermal conductivity response with in-plane composite strain, and to achieve this, Zhang and Hayhurst (2010) introduced a shear failure damage mechanism that deactivated thermal transport normal to the fibres/tows in the plane orthogonal to the local shear. However, for the mechanical loading of the HITCO composite studied here, this mechanism has been found by Zhang and Hayhurst (2014) to be inoperative. Instead, Zhang and Hayhurst (2014) found it necessary to introduce a continuum damage degradation in which the local transverse and shear moduli were controlled by the respective components of strain. This was necessary for the modelling to predict the observed shapes of the measured stress–strain curves. Their model admits micro-cracking on the local scale, of the order of the fibre radius, and since the moduli are coupled to strain it permits a gradual monotonic decrease of tow transverse thermal conductivity. The thermal transport in the composite through-thickness direction, orthogonal to the tow, has been assumed to be unchanged by shear or transverse stresses similar to those shown in Figure 3(b), and the local values of transverse thermal conductivity $k_{\mathrm{Tran}}^{\ell }$ are given by the data of Figure 5(b).

Potential transverse and piezothermal effects

Before proceeding to the next section on finite element analysis, it is worth reflecting on potential deficiencies of the material model used herein, particularly with respect to multi-axial stressing.

First, consider a plate of the woven composite loaded in uni-axial tension within the plane of the plate parallel to the 0° tow direction, and simultaneously in compression of equal magnitude in the plane of the plate at 90° to the tension loading. The approach formulated here assumes that influence on the through-thickness thermal conductivity, k₂₂, due to damage on the fibre–matrix interface initiated by the tension loading is unaffected by the orthogonal compression loading. Also, the assumption would apply equally to a compression loading applied normal to the surface of plate. For either of these two compression loadings, it would be expected that fibre–matrix debonding generated by the tensile loading would, in some measure, be closed up by the orthogonal compression. Recall the work of Liu and Hutchinson (1996) which predicted that a debonded air gap of the order of 0.1 micron, for the HITCO T300 carbon fibres of diameter 7 of 15 micron, would be sufficient to prevent local thermal conduction. A first-order approximation suggests that a compressive strain of the order of 1% would be necessary to nullify such effects. The work reported here has, to a first order of approximation, neglected such an effect, but it would be of interest to know those conditions and circumstances in which the effects would be significant.

Second, it is worth addressing if piezothermal effects could be present; by this one refers to the influence on thermal conductivity of pressure loadings, or in quantitative terms, the first stress invariant σ_ii = σ₁₁+ σ₂₂+ σ₃₃. In the situation addressed in the previous paragraph, if σ₁₁ = σ₂₂ = −X and σ₃₃ = +X, then one could expect the through-thickness thermal conductivity k₂₂ to be higher than for pure in-plane axial loading σ₃₃ = +X, with σ₁₁= σ₂₂ = 0, due to the compressive loadings partially closing up the deboned cracks. For tri-axial loadings in which σ₁₁ = σ₂₂ = σ₂₂ = +X, one would expect the through-thickness thermal conductivity k₂₂ to be lower than for pure in-plane axial loading, due to synergy of debonding in the three orthogonal directions. Since, to the author’s knowledge, no experimental evidence is available to allow quantification of such piezothermal effects, they warrant investigation. The conduct of experiments on tri-axial stressing of materials is notoriously difficult (Hayhurst and Felce, 1986), and it would be more expedient to address this challenge using theoretical/computational approaches.

Formulation of the finite element approach for a single tow

This paper reports the development of a coupled mechanical strain–thermal finite element model of an 8-Harness satin weave composite that has been based on the stress–displacement model of Zhang and Hayhurst (2010) for the mechanical loading of 0°/90° Nicalon-CAS and the plain weave DLR-XT composites. In this previous work, a unidirectional tow (or an assemblage of tows that form a lamina) was chosen to be the basic constituent in the finite element model, and hence an entire tow or a lamina, which consists of thousands of fibres embedded in matrix, was represented by a single 8-node solid finite element shown in Figure 6. The material properties were assumed to be multi-linear elastic and the discretised non-linear stress–strain and Poisson’s ratios-strain curves were used, c.f. Figures 10 and 14 of Zhang and Hayhurst (2010). The corresponding thermal analysis has been reported by Zhang and Hayhurst (2011).

Figure 6.

Homogenisation of a unidirectional tow or a unidirectional lamina to a single orthotropic block.

The equivalent mechanical strain analysis for the 8-Harness satin weave HITCO material, Figure 1, has been reported by Zhang and Hayhurst (2014), and the finite element package ABAQUS (SIMULIA, 2008) with a user-defined subroutine USDFLD will be used to carry out the thermal analyses of the 8-Harness satin weave HITCO material. This method has the benefit of being able to model a tow by a single finite element with orthotropic properties. Again, the coupled thermal-stress model reported here has been formulated using strain-dependent thermal material properties in the manner set out by Zhang and Hayhurst (2010, 2011) for the 0°/90° Nicalon-CAS and the plain weave DLR-XT composites. The formulation of the analysis of the coupled mechanical–thermal behaviour of the HITCO material is summarised as follows.

Heat flow in a homogenised unidirectional tow

A heterogeneous unidirectional tow or lamina has been homogenised to a single block shown in Figure 6, which has the same overall dimensions and orthotropic thermal material properties as the actual material. For the three-dimensional steady-state heat conduction problem under investigation, there are three independent tow thermal conductivities: k₁, k₂, and k₃, defined relative to the local tow axes given in Figure 6.

For anisotropic continua, the rate equation for heat conduction is given by Fourier’s law (Fourier, 1822), which has been reformulated for composite materials by White and Knutsson (1982) and Argyris et al. (1995)

$\{q1q2q3\}=-[k11k12k13k21k22k23k31k32k33]\{\partial T/\partial x1\partial T/\partial x2\partial T/\partial x3\}$

(1)

where q_i is the heat flux, $k\mathrm{ij}$ is the thermal conductivity and $\partial T/\partial xi$ is the thermal gradient (i, j = 1, 2, 3). It can be shown that the conductivity tensor $k\mathrm{ij}$ is symmetric with $k\mathrm{ij}=k\mathrm{ji}$ , and for an orthotropic material, the coupling thermal conductivities terms, i ≠ j, are zero and hence equation (1) becomes

$\{q1q2q3\}=-[k_{11}^{\ell }000k_{22}^{\ell }000k_{33}^{\ell }]\{\partial T/\partial x1\partial T/\partial x2\partial T/\partial x3\}$

(2)

In local coordinates, $k_{33}^{\ell }$ is the thermal conductivity along the tow, $k_{22}^{\ell }$ is the out-of-plane (through-thickness) transverse tow thermal conductivity, and $k_{11}^{\ell }$ is the in-plane transverse tow conductivity.

Determination of finite element heat flux

To perform the analysis at the finite element level, Figure 6, the tow non-linear thermal conductivities were discretised as the multi-linear curves given in Figure 5 for the HITCO tows. The mechanical straining in the 3-direction was imposed in terms of the displacement boundary conditions as a succession of small strain increments, and over each strain increment, the combined mechanical strain, steady-state thermal conductivity problem was solved for a unity thermal gradient between the upper and lower surfaces of the element, or of an assemblage of elements in the case of a partial composite unit cell. For each strain increment, values of the fields of element thermal conductivities were evaluated by ABAQUS from Figure 5 using the mechanical strain fields at the start of the increment, and these values were used by ABAQUS to compute the values of heat flux (Watts m⁻²) at nodes over the lower surfaces of the element, assemblages of elements or partial composite unit cell, at the end of the current strain increment.

Finite element evaluation of transverse thermal conductivity

The local values of heat flux (Watts m⁻²) discussed in the previous section were integrated over the lower surface, of either a single finite element, or of an assemblage of elements in the case of a partial composite unit cell, to determine the total heat flux. The integrated value is numerically equal to the transverse thermal conductivity of the unit cell, since a unity thermal gradient boundary condition was imposed. Such values of transverse thermal conductivity were determined at the end of each successive strain increment.

Implementation of Subroutine USDFLD in ABAQUS (SIMULIA, 2008)

The present thermo-mechanical model has been implemented using the finite element package, ABAQUS/standard and a user-defined subroutine, USDFLD. The discretised strain-dependent tow thermal conductivities of Figure 5 have been defined as functions of the strain field variables in the input file. For each increment, the subroutine USDFLD was used to read the local strains at each integration point and then to define the tow thermal conductivity properties using these strains. Automatic incrementation algorithms and the increment redefinition variable, PNEWDT, defined in ABAQUS, were continuously updated to ensure that values of the field variables at the end of the relevant strain increments were located exactly at the discretised points.

Numerical convergence studies

In the user subroutine USDFLD, the material properties were calculated using the strain values at the start of an increment, and they then remain unchanged throughout that increment. Hence, this may introduce a small error dependent upon the rate of change of material properties with strain, and on the magnitude of the strain increment. To reduce the error to acceptable levels and achieve the required solution convergence, the increment size was maintained less than a set value of 0.01. The latter value was established by carrying out repeated runs until the values of the composite global thermal conductivities were repeatable to within 0.1%.

The definition of the partial unit cell is addressed in the next section.

Definition of partial unit cell for HITCO 8-harness satin weave

A plan view of the entire unit cell is shown schematically in Figure 7(a), and this view replicates the isometric view of Figure 1. To reduce the number of finite elements used to model the HITCO 8-Harness Satin weave, and also to alleviate numerical difficulties that can arise with modelling localised unloading within the repeated woven zones, eight in total, of the entire unit cell of Figure 1, a basic repeatable unit has been sought. The minimal or partial unit cell selected is that associated with the square region of the weave that is defined by the outer broken line in Figure 7(b). For purposes of finite element analysis, this can be reduced further to the quadrant, shown in Figure 7(c). In this figure, the points X_1a and X_3a are on the centre lines of orthogonal tows. The dimensions defined in Figure 7(b) and (c) have been determined on a volumetric basis, such that the volume of the partial unit cell square in Figure 7(b), of unit thickness, is (2X_2b)(2X_3b), which in turn is one eighth of the volume of the entire unit cell of Figure 7(a), i.e. (4a 4a)/8. Given that X_2b = X_3b, it can then be shown that 2X_2b = 2X_3b =  $\sqrt{2}$ a. The finite element analysis has been carried out on the partial unit cell of Figure 7(c) using periodic boundary conditions. This approach for the definition of a partial unit cell is that used by Zhang and Hayhurst (2014) in the context of mechanical analysis without thermal loading.

Figure 7.

Definition of partial unit cell. (a) Plan view of entire unit cell shown schematically in Figure 1. (b) Weave segment identified in (a) by broken line square boundary. (c) Definition of a single quadrant of the partial unit cell shown in (b).

Formulation of the finite element partial unit cell model

Figure 8(a) shows two 0° tows, dark shading, and two 90° tows, light shading, and Figure 8(b) shows the assembled tows that make up the entire partial unit cell. An important objective of this study is to develop an approach which is capable of analysing large-scale composites and engineering components. This makes it a very difficult task to create a finite element model, which captures woven features of a tow, yet uses a minimum number of finite elements. The mesh and geometry of the HITCO 8-Harness satin weave partial cell shown in Figure 8 are believed to be some of the optimal solutions. The details of the finite element analysis approach for this partial unit cell are given in the paper by Zhang and Hayhurst (2014) for mechanical analysis without thermal loading, and the reader is referred to that source. Yang and Cox (2003, 2010) concluded that strains averaged over a gauge volume whose linear dimensions are equal to or exceed half the cross-sectional dimensions of a tow, are reasonably mesh independent. For the present HITCO 8-Harness satin weave partial unit cell model, the majority of the elements (93.3% by volume) are approximately equal to this gauge volume, and in addition, their strain distributions are rather uniform. Hence, it is reasonable to use the local strain at an integration point to predict different failure modes of the material. The fidelity of the model will be assessed by using the stress–strain experimental results of Sheikh et al. (2009).

Figure 8.

Finite element mesh of a single quadrant of the partial unit cell shown in Figure 4(c): (a) Exploded view showing 0° tows, dark shading, and 90° tows, light shading. (b) Assemblage of tows, shown in (a), that form one quarter of a partial unit cell.

Boundary conditions, tow waviness and material properties

Addressed in this section is the finite element analysis of the thermal conductivity–strain response of a HITCO 8-Harness Satin Weave partial unit cell, which is equivalent to the full unit cell shown in Figure 1, under uni-axial straining along the axes of the 0° fibres. The geometry and boundary conditions are defined, the analysis of fibre waviness on the overall out-of-plane thermal conductivity of the laminate is discussed, and the material properties of a unidirectional HITCO tow are now summarised.

Geometry and boundary condition

Tang et al. (2011b) have quoted HITCO tow dimensions as 13.450 × 1.681 × 0.138 mm³; however for the work reported here, these dimensions have been independently measured from larger scale micro-graphs, and the more accurate dimensions are 12.912 × 1.614 × 0.138 mm³. A partial unit cell of the HITCO 8-Harness Satin Weave laminate was used that consisted of nine laminae. Sheikh et al. (2009) give the laminate thickness as 2.490 mm, therefore the tow thickness is 0.138 mm (=2.490/(9 × 2)).

The geometry of a single woven tow, shown as the lower centre illustration in Figure 8(a), is given in Figure 9. By reference to Figure 8(a), and by virtue of symmetry, the dimension of the non-woven tows, shown as the top centre illustration in Figure 8(a), may be deduced.

Figure 9.

Geometry of a single woven tow, shown as the lower illustration in Figure 5(a). Tow waviness is shown by the broken thick line, and defined by the waviness angle ξ = ±2.44°. The weave angle is ζ = 9.4°. All linear dimensions are given in mm.

Uniform displacement boundary conditions were applied to the unit cell to simulate uni-axial straining along the 0° fibre direction (direction 3 in Figure 8). Steady-state heat conduction was modelled by application of a unity thermal gradient between the top and bottom faces of the unit cell, and all the other faces were lagged.

Modelling of tow waviness

During manufacture of the HITCO 8-Harness satin weave composites, some degree of waviness in the alignment of the fibres or tows has been introduced. The weave angle and the degree of waviness have been obtained from the micrographs presented by Sheikh et al. (2009). To the knowledge of the authors, this is the only information available in the literature, and measurements have been taken from that source. The value of the weave angle is ζ = 9.4°, c.f. Figure 9, and the average waviness angle is ξ = ±2.44°. The latter value has been used in the finite element model to evaluate the effects of waviness on stiffness reduction. Both the in-plane and out-of-plane waviness are modelled in the finite element analysis by assigning an associated local material orientation to each individual element. Figure 9 shows the fibre directions and their waviness angle of ξ = ±2.44°. To assess the effects of waviness on composite behaviour, stress–strain curves have been predicted for zero waviness angle ξ = 0° (Zhang and Hayhurst, 2014).

HITCO Material properties

This section addresses the specification of both mechanical and thermal material data.

HITCO: Tow mechanical properties

The material property data for the HITCO material has been described by Zhang and Hayhurst (2014) in Sections 6 and 7 of their paper. The variation with local strain,   ${\varepsilon }_{\mathrm{ij}}^{\ell }$ (i, j = 2–3), of five properties are given. These are longitudinal tow stress ${\sigma }_{33}^{\ell }$ ; tow Poisson’s ratios ${\nu }_{31}^{\ell }$ and ${\nu }_{32}^{\ell }$ ; transverse tow stresses ${\sigma }_{22}^{\ell }$ ; and tow shear stresses ${\sigma }_{23}^{\ell }$ . The characteristic variations of these parameters, with ${\varepsilon }_{33}^{\ell }$ , ${\varepsilon }_{22}^{\ell }$ and ${\varepsilon }_{23}^{\ell }$ , are provided in multi-linear discretised form, c.f. Figure 11, and equations (1) and (2) of Zhang and Hayhurst (2014).

Figure 10.

Comparison of experimental stress–strain curves for the HITCO 8-Harness Satin weave C/C laminate with curves predicted for ξ = 0° and ξ = ±2.44° levels of waviness and the transverse tow stress–strain curve given by the transverse modulus strain reduction model. Failure has been modelled by deformation localisation in the partial unit cell close to the weave section.

Figure 11.

Comparison of an experimental stress–strain curve for the HITCO 8-Harness Satin weave C/C laminate with curves predicted for ξ = 0° and ξ = ±2.44° levels of waviness and the transverse tow stress–strain curve given by the coupled damage stiffness reduction model. Failure has been modelled by dynamic fibre failure by instantaneous pullout deactivation.

HITCO: Longitudinal tow thermal properties

The variation of longitudinal thermal conductivity, $k_{\mathrm{Long}}^{\ell }$ , with tow strain, ${\varepsilon }_{33}^{\ell }$ , is a function of ω, the ratio of number of failed fibres to total fibres as given by equation (8) (equation (3.2) of Tang et al., 2009). This equation can be computed with numerical integration of equations (4) and (6) which are equations (2.11) and (2.12) of Tang et al. (2009) to yield Figure 3(a), with an initial value for the HITCO material of $k_{\mathrm{Long}}^{\mathrm{Initial}}$ = $k_{33}^{\ell }$  = 45.400 Wm⁻¹K⁻¹. The initial value $k_{\mathrm{Long}}^{\mathrm{Initial}}$ is determined in the Tow longitudinal thermal conductivity section of Appendix 1 from the basic physical properties of fibres and matrix. Figure 3(a) has been discretised as a multi-linear curve using 22 data points.

HITCO: Transverse tow thermal properties

The first step in the evaluation of the variation of $k_{\mathrm{Tran}}^{\ell }$ with ${\varepsilon }_{33}^{\ell }$ is to obtain the variation of ${\sigma }_{33}^{\ell }$ with ${\varepsilon }_{33}^{\ell }$ . This is achieved by integration of equation (7) (equation (3.1) of Tang et al., 2009) together with equation (3) (equation (11) of Hayhurst et al., 1991) with respect to ω. Proceeding in this way with substitution of $\Sigma \infty ={\sigma }_{33}^{\ell }/\sigma \mathrm{ff}$ , where $\sigma \mathrm{ff}$ is the average fibre failure stress, and of $\lambda \infty ={\varepsilon }_{33}^{\ell }/\varepsilon \mathrm{ff}$ , where $\varepsilon \mathrm{ff}=\sigma \mathrm{ff}/EI$ and E_I is the initial composite Young’s modulus, into equation (3.18) of Tang et al. (2009), the curve of Figure 5(b) can be predicted from an initial value of $k_{\mathrm{Tran}}^{\mathrm{Initial}}=k_{11}^{\ell }$ = $k_{22}^{\ell }$ =9.110 Wm⁻¹K⁻¹. This initial value $k_{\mathrm{Tran}}^{\mathrm{Initial}}$ is determined in the Tow Transverse thermal conductivity section of Appendix 1 from the basic physical properties of fibres and matrix. N.B. the fact that $k_{\mathrm{Long}}^{\mathrm{Initial}}\approx 5k_{\mathrm{Tran}}^{\mathrm{Initial}}$ is a consequence of the superior longitudinal properties of the T300 fibres. Again, 22 data points have been used to discretise the multi-linear curve of Figure 3(b).

Predictions for the HITCO 8-harness satin weave partial unit cell

First, predictions of the tow mechanical properties are addressed, followed by the transverse thermal properties for the two unit cell failure criteria examined by Zhang and Hayhurst (2014).

Composite mechanical response

The predicted variation of the applied global stress $(\sigma 33)\infty$ with global strain $(\varepsilon 33)\infty$ for the HITCO 8-Harness Satin weave has been obtained by Zhang and Hayhurst (2014) for the tow transverse modulus strain reduction model with two tow failure criteria: (i) deformation by localisation in the partial unit cell close to the weave section, c.f. Figure 10, and (ii) dynamic fibre failure by instantaneous pullout deactivation, c.f. Figure 11. In both figures, two levels of waviness have been examined: one for zero tow waviness ξ = 0°, and the other for the measured waviness of ξ = ±2.44°.

Note that curves Exp. 1 and Exp. 2, determined by experimentation, and presented in Figure 10, show quasi-ductile behaviour which corresponds with the prediction for the deformation localisation model. Alternatively, the experimentally derived curve, Exp. 4, of Figure 11, shows brittle behaviour that corresponds with the dynamic fibre failure mechanism of Blacklock and Hayhurst (2011). Figures 10 and 11 are those reported by Zhang and Hayhurst (2014) as Figures 14 and 15, respectively, and they are reproduced here for the convenience of the reader.

Composite transverse thermal response

For the materials tested by Sheikh et al. (2009), no results were obtained for global in-plane composite thermal conductivity. For this reason, no attempt has been made to predict in-plane thermal conductivities.

The variation of predicted composite transverse thermal conductivity, k₂₂, with composite strain (ɛ₃₃)_∞ in the 3-direction, where the direction is defined in Figure 6, is compared with the results of experiment in Figures 12 and 13. Comparisons are made for the two failure criteria: (i) failure by deformation localisation, and (ii) failure by dynamic fibre failure with instantaneous pullout deactivation, and each is now discussed in turn.

Figure 12.

Comparison of experimental thermal conductivity–strain curves for the HITCO 8-Harness Satin weave C/C laminate with predicted curves for the transverse tow stress–strain curve given by the transverse modulus strain reduction model. Failure has been modelled by deformation localisation in the partial unit cell close to the weave section for two levels of waviness levels ξ = 0° and ξ = ±2.44°.

Figure 13.

Comparison of experimental thermal conductivity–strain curves for the HITCO 8-Harness Satin weave C/C laminate with predicted curves for the transverse tow stress–strain curve given by the transverse modulus strain reduction model. Failure has been modelled by dynamic fibre failure by instantaneous pullout deactivation for two levels of waviness levels ξ = 0° and ξ = ±2.44°.

Figure 14.

Use of an assumed linear decrease in thermal conductivity k₂₂ (red solid line) as structural health monitor. Point E denotes deviation from linearity by 10%; further decrease in k₂₂ would render transverse thermal conductivity and mechanical strength of a structural member not fit for purpose.

Failure by deformation localisation in the partial unit cell close to the weave section

Predictions of the variation of k₂₂ with (ɛ₃₃)_∞ for the mechanism in which failure takes place by deformation localisation are shown in Figure 12, for two levels of tow waviness ξ = 0° and ξ = ±2.44° The predicted curves have a sudden drop in the level of k₂₂ which takes place at the strain level (ɛ₃₃)_∞ = 0.08; this is due to the strain associated with cracking of the amorphous carbon matrix.

Careful examination of the two curves in Figure 12 for waviness levels ξ = 0° and ξ = ±2.44° shows that the k₂₂−(ɛ₃₃)_∞ data for the larger value of waviness ξ = ±2.44° has consistently higher values of k₂₂. The reason for this is that the waviness provides a greater through-thickness inclination of fibres that allows the higher longitudinal thermal conductivity of the fibres, c.f. Figure 5, to be exploited in the through-thickness direction.

A further significant difference between the curves for ξ = 0° and ξ = ±2.44° may be seen between the strain (ɛ₃₃)_∞ = 0.29–0.35% where the curve for ξ = ±2.44° does not decrease as rapidly as that for ξ = 0°. The reason for this may be observed in Figure 10 where it can be seen that the stress–strain curve is higher for ξ = 0°, and also the failure strain is smaller. Both factors mean that the degree of fibre failure and the associated wake debonding is higher, and since the latter mechanism is mainly responsible for degradation of transverse thermal properties, the curve for ξ = 0° is lower in Figure 12. When judging the quality of the correspondence between predictions and experiment in Figure 12, it must be borne in mind that the prediction is for a single partial unit cell, and that the experiment records the behaviour of an assemblage of hundreds of unit cells positioned in the nine through-thickness laminae 2-direction, and staggered in the plane of the composite, both the 1- and 3-directions. For these reasons, one would expect the experimental curve to be below the predicted curve, since a failed unit cell will prevent heat flow through all other cells behind it. The overall thermal conductivity is therefore significantly reduced by this local unit cell failure mechanism.

Failure by dynamic fibre failure by instantaneous pullout deactivation

Predictions of the variation of k₂₂ with (ɛ₃₃)_∞ for the mechanism in which failure takes place by dynamic fibre failure by instantaneous pullout deactivation are shown in Figure 13, for two levels of tow waviness ξ = 0° and ξ = ±2.44°. Again, the predicted curves have a sudden drop in the level of k₂₂ which takes place that the strain level (ɛ₃₃)_∞ = 0.08 which is due to the strain associated with cracking of the amorphous carbon matrix.

Two global stress–strain curves are shown in Figure 11 for waviness levels ξ = 0° and ξ = ±2.44°, and the k₂₂ − (ɛ₃₃)_∞ curve of Figure 13, for the higher value of ξ = ±2.44°, consistently shows higher values of k₂₂. The same explanation as that given for Figure 12 applies in this case also.

The stress–strain curve for this failure mechanism is shown in Figure 11, with catastrophic failure taking place when the local fibre strain attains the value of ${\varepsilon }_{33}^{\ell }$ ≈ $\varepsilon \mathrm{wd}$ = 0.3%, where $\varepsilon \mathrm{wd}$ is defined by Blacklock and Hayhurst (2011) and discussed at the beginning of the Tow mechanical behavior section. This occurs when the composite strain (ɛ₃₃)_∞< 0.3%. The agreement between the predicted and the experimental curve is close.

Discussion of results

The results of the paper are now discussed under the following sections: Mechanical Response and Thermal Response.

Mechanical response

The discussion reported in this section is drawn from that reported by Zhang and Hayhurst (2014), and is reproduced here for the convenience of the reader.

For both fracture criteria and the two levels of waviness ξ = 0° and ξ = ±2.44° considered, the effect of increased waviness is to increase composite flexibility, or lower the respective moduli, with larger strains being achieved for the same stress. In the case of the deformation localisation failure criteria, waviness increases failure strains, but for dynamic fibre failure by instantaneous pullout deactivation, the failure stains for the two waviness values are almost the same.

To achieve accurate predictions of stress–strain behaviour, it has been found necessary to model transverse tension and shear behaviour of the tows by a transverse modulus strain reduction model (Zhang and Hayhurst, 2010). This is in contrast to the modelling of the behaviour of a plain weave carbon fibre – double SiC matrix composite DLR-XT of Zhang and Hayhurst (2010), that required the matrix to be modelled as an elastic–brittle medium. The fidelity of this model, and in particular its physical significance, requires further investigation.

The ratio of the levels of failure strain by instantaneous pullout deactivation and failure strain by localisation, ≈3.92/3.56 =1.10, is not huge, and for most datasets this difference would almost be hidden in the scatter on the data. If the different levels were deemed to be meaningful, then the practicalities of component design would dictate that the designer should be conservative by taking the lower value of failure strain by instantaneous pullout deactivation =3.92%.

Thermal response

For the two levels of waviness considered ξ = 0° and ξ = ±2.44°, the transverse thermal conductivity is slightly higher for the larger value of waviness. This is due to heat flow from the upper lamina to the lower lamina exploiting the higher axial thermal conductivity of the T300 fibre. Note Figure 5 shows that initially the ratio of longitudinal to transverse thermal conductivity $k_{\mathrm{Long}}^{\ell }$ / $k_{\mathrm{Tran}}^{\ell }$  ≈ 5. Although this component is present, it is not dominant, and the 8-Harness Satin weave behaves almost as a 0°/90° composite. The physical mechanism responsible for thermal property degradation is wake debonding.

For the case of zero waviness, ξ = 0°, there are only two orthogonal tows that are inclined and these have a shallow weave angle of ζ = 9.4°. It is these fibres/tows that have a temperature difference across their ends, and hence can allow heat to flow along their length. All other tows, which form the majority, are parallel to the plane of the laminae, and heat flow is through the composite thickness by flux normal to the fibres, without the possibility of exploiting $k_{\mathrm{Long}}^{\ell }$ / $k_{\mathrm{Tran}}^{\ell }$  ≈ 5.

The latter is altered only slightly when waviness ξ = ±2.44°, but this angle is not significant and is responsible for only minor changes.

Damage and reduction in transverse thermal conductivity as a structural health monitor

As an example, Figure 12 has been redrawn as Figure 14. Marked on the figure is the solid line AB that charts the linear decrease of the tow transverse thermal conductivity $k_{\mathrm{Tran}}^{\ell }$ =k₂₂ (Wm⁻¹K⁻¹) with the remote composite tensile strain (ɛ₃₃)_∞. It shows a decrement between A and B of Δk₂₂ = 0.6 (Wm⁻¹ K⁻¹) produced by a strain of (ɛ₃₃)_∞ = 0.22 (%), or a linear decrement rate of 2.73 (Wm⁻¹ K⁻¹ %⁻¹). The data from the two experiments indicate considerable noise/scatter; however, the trend of the data and that of the finite element predictions shows a deviation from linearity at point B, at a strain level of approximately (ɛ₃₃)_∞ = 0.22 (%); thereafter, the curve rapidly falling to point C. Although the falling curve from B to C is steep, it is stable and repeatable. This is thought to be due to the weave architecture which has relatively long straight segments that run parallel to the woven tows, and so provide compatibility of axial displacements and corresponding stability. If one draws the broken line DE with a 10% downward offset from the initial measured value of k₂₂ = 8.7 (Wm⁻¹K⁻¹) i.e. to 7.83 (Wm⁻¹K⁻¹), then the line DE falls substantially to below the experimental error, and the point of intersection with the finite element predictions at point E can be arbitrarily defined as the point at which the integrity of the structural member under examination can be called into question, and it is suggested that this point can be used to define the limits of reduction in mechanical strength and loss of thermal conductivity. At point E it can be observed from Figure 10 that the ratio of stress to the failure stress is 0.90. In practice, this level may be deemed to be too high and a strain level of (ɛ₃₃)_∞ = 0.15(%) might be selected on Figure 14, i.e. point F on the solid line AB. This would correspond to a ratio of stress to failure stress of 0.6, a value more typically employed in engineering design. If the latter procedure were to be followed, then an experimental technique for measurement of the tow transverse thermal conductivity should be used that would guarantee absence of scatter.

In practice, a high-temperature component in, say, an aero-engine would be examined on a regular service schedule (Sun et al., 2006), with measurements taken of the tow transverse thermal conductivity k₂₂, and after assessment using the procedure proposed above, a decision would be reached on whether the component should be returned to service or replaced by a new one.

Conclusions

For the HITCO 8-Harness Satin weave, good agreement has been achieved between experimental data and the predicted variations of composite transverse thermal conductivity with in-plane composite strain.

The finite element method employed has traceability to the thermo-mechanical properties of the constituent materials, i.e. the basic data for the fibres and matrices used in the tow model of Tang et al. (2009) Therefore, the macroscopic unit cell properties have been predicted from properties that relate to the micro-scale of the fibre and matrix.

The combined weave angle ζ = 9.4° and waviness angle ξ = ±2.44°, of ζ+ ξ = 11.84°, through which heat can flow along tows from the upper to the lower layers of a unit cell, is small. Furthermore, the number of weave segments in a unit cell that can exploit this heat flux mechanism with the higher thermal conductivity parallel to fibres $k_{\mathrm{Long}}^{\ell }$ {Ratio $k_{\mathrm{Long}}^{\ell }$ / $k_{\mathrm{Tran}}^{\ell }$  ≈ 5} is small, since the vast majority of tows are in-plane and forced to use the lower transverse thermal conductivity $k_{\mathrm{Tran}}^{\ell }$ to achieve transverse unit cell heat flux; hence, the 8-Harness Satin weave is a close approximation to a 0°/90° composite.

It is the wake debonding mechanism, modelled at the tow level by the equations of Tang et al. (2009) that is responsible for the degradation of transverse thermal conductivity.

A motivation for the research was to ascertain whether the use of monitoring the drop-off in transverse thermal conductivity could be used as a means of deciding when a component had reached the end of its serviceable life. For something like 75% of the strain history of a component–structure, the drop-off in $k_{\mathrm{Tran}}^{\ell }$ is little more than 10%; however, for this composite lay-up, most probably due to the 8-Harness Satin weave with straight tows running in parallel with woven tows, the drop-off beyond appears stable and there is potential to use this approach. A procedure is proposed for periodic examination of, say, an aero-space component that might be used to decide when to replace an in-service component. However, for this to be viable, an accurate scatter-free transverse thermal conductivity measurement device would be needed.