《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_Thermal shock fracture in unidirectional fibre-reinforced

Availableonlineatwww.sciencedirect.com 8CIENCE DIRECTO COMPOS SCIENCE TECHNOLO 8 ELSEVIER Composites Science and Technology 65(2005)1880-1890 www.elsevier.com/locate/compscitech Thermal shock fracture in unidirectional fibre-reinforced ceramic-matrix composites C. Kastritseas. P.A. Smith * J.A. Yeomans School of Engineering(H6), Unirersity of Surrey, Guildford GU2 7XH, Surrey, UK Received 26 October 2004: received in revised form 16 March 2005: accepted 9 April 2005 Available online 6 june 2005 Abstract The onset of multiple matrix cracking due to thermal shock in unidirectional fibre-reinforced ceramic-matrix composites is inves- tigated in this study. A simple, semi-empirical formula is developed that allows prediction of the critical quenching temperature dif- ferential, ATe, that initiates fracture, as a function of the processing temperature of the composite, the temperature of the quenching medium, and material properties measured at room temperature. The approach considers the anisotropic stress field generated dur o. the shock. Application of the formula to the cases of four different CMCs reinforced with Nicalon fibres revealed significant discrepancies with experimental results, which were attributed to a reduction in the effective value of the interfacial shear stress dur- ing the shock due to the bi-axial nature of the applied stress field. The phenomenon was modelled successfully using a modified Coulomb-type friction law and estimates were obtained for the value of the stress reduction factor, A, that characterises the heat ransfer conditions. a methodology based on this analysis is then proposed that enables reasonable estimates of AT for the class of CMCs under consideration c 2005 Elsevier Ltd. All rights reserved Keywords: A Ceramic-matrix composites(CMCs): B High-temperature properties; Interfacial strength; Matrix cracking 1. Introduction temperature gradients. The more serious condition devel- ops when a heated material comes into contact with a Ceramic-matrix composites(CMCs) reinforced with medium of much lower temperature, i.e. in the case of continuous ceramic fibres are an emerging class of engi- cold shock. In this case, tensile stresses develop at the sur- neering materials that demonstrate the excellent high face of the material that are balanced by a distribution of temperature properties of ceramics(e.g. low density, high compressive stresses at the interior. Such stresses result in strength and stiffness, good creep, oxidation, erosion, and the propagation of a range of flaws on the surface of wear resistance) but also aim to overcome their major monolithic ceramic components which can cause abrupt drawback, i.e. low fracture toughness. The mechanical and significant property degradation or even catastrophic properties of these materials were thoroughly investi- failure [2]. Fibre-reinforced CMCs have been found to gated in the past two decades and a good understanding possess superior resistance to thermal shock compared of their behaviour under various conditions exists [1] to their monolithic counterparts as catastrophic failure a less investigated aspect is their behaviour under con- is prevented, but they have been shown to exhibit damage ditions of thermal shock, i.e. in the presence of transient and property degradation even at moderate shocks [3]. In CMCs reinforced with unidirectional(UD)fibres, Corresponding author. Tel. +44 1483 689616: fax: +44 145 he main damage mode comprises matrix micro-cracks 87629 oriented perpendicular to the fibre orientation, similar E-mail address: P.Smith@surrey. ac uk(PA. Smith) to those observed in UD CMCs under tensile loading 0266-35 see front matter 2005 Elsevier Ltd. All rights reserved C.compscitech.2005.04.004

Thermal shock fracture in unidirectional fibre-reinforced ceramic–matrix composites C. Kastritseas, P.A. Smith *, J.A. Yeomans School of Engineering (H6), University of Surrey, Guildford GU2 7XH, Surrey, UK Received 26 October 2004; received in revised form 16 March 2005; accepted 9 April 2005 Available online 6 June 2005 Abstract The onset of multiple matrix cracking due to thermal shock in unidirectional fibre-reinforced ceramic–matrix composites is inves￾tigated in this study. A simple, semi-empirical formula is developed that allows prediction of the critical quenching temperature dif￾ferential, DTc, that initiates fracture, as a function of the processing temperature of the composite, the temperature of the quenching medium, and material properties measured at room temperature. The approach considers the anisotropic stress field generated dur￾ing the shock. Application of the formula to the cases of four different CMCs reinforced with Nicalon fibres revealed significant discrepancies with experimental results, which were attributed to a reduction in the effective value of the interfacial shear stress dur￾ing the shock due to the bi-axial nature of the applied stress field. The phenomenon was modelled successfully using a modified Coulomb-type friction law and estimates were obtained for the value of the stress reduction factor, A, that characterises the heat transfer conditions. A methodology based on this analysis is then proposed that enables reasonable estimates of DTc for the class of CMCs under consideration.
2005 Elsevier Ltd. All rights reserved. Keywords: A. Ceramic–matrix composites (CMCs); B. High-temperature properties; Interfacial strength; Matrix cracking 1. Introduction Ceramic–matrix composites (CMCs) reinforced with continuous ceramic fibres are an emerging class of engi￾neering materials that demonstrate the excellent high temperature properties of ceramics (e.g. low density, high strength and stiffness, good creep, oxidation, erosion, and wear resistance) but also aim to overcome their major drawback, i.e. low fracture toughness. The mechanical properties of these materials were thoroughly investi￾gated in the past two decades and a good understanding of their behaviour under various conditions exists [1]. A less investigated aspect is their behaviour under con￾ditions of thermal shock, i.e. in the presence of transient temperature gradients. The more serious condition devel￾ops when a heated material comes into contact with a medium of much lower temperature, i.e. in the case of cold shock. In this case, tensile stresses develop at the sur￾face of the material that are balanced by a distribution of compressive stresses at the interior. Such stresses result in the propagation of a range of flaws on the surface of monolithic ceramic components which can cause abrupt and significant property degradation or even catastrophic failure [2]. Fibre-reinforced CMCs have been found to possess superior resistance to thermal shock compared to their monolithic counterparts as catastrophic failure is prevented, but they have been shown to exhibit damage and property degradation even at moderate shocks [3]. In CMCs reinforced with unidirectional (UD) fibres, the main damage mode comprises matrix micro-cracks oriented perpendicular to the fibre orientation, similar to those observed in UD CMCs under tensile loading 0266-3538/$ - see front matter
2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.04.004 * Corresponding author. Tel.: +44 1483 689616; fax: +44 1483 876291. E-mail address: P.Smith@surrey.ac.uk (P.A. Smith). Composites Science and Technology 65 (2005) 1880–1890 www.elsevier.com/locate/compscitech COMPOSITES SCIENCE AND TECHNOLOGY

C. Kastritseas et al Composites Science and Technology 65(2005)1880-1890 [I]. Kagawa et al. [4] described perpendicular matrix where stifness (E), coefficient of thermal expansion cracking on the surface of a PyrexM-matrix composite CTE (a), and Poissons ratio (v)are either matrix einforced with Nicalon fibres after quenching test or volume-averaged properties [8]. The parameter 'A pecimens into room temperature water(a20C)from is termed the 'stress reduction factorand accounts for a temperature higher than 620C, i.e. the critical the finite value of the coefficient of heat transfer, h, be- quenching temperature differencefor the onset of crack- tween the material and the quenching medium. Residual ing was AT>600C. The same authors also reported thermal stresses were also taken into account. The de- =800C for a water-quenched UD rived formula for ATe in both cases had the same form Nicalon/LAS (lithium aluminosilicate), after observ ing a range of cracking phenomena, some of them AT 1-D perpendicular to the fibre axis, appearing on the material surface. Blissett et al. [5] performed water- where omu is the uniaxial matrix streng quench tests on a UD Nicalon CAS (calcium alumi posite stress to cause matrix cracking) gres is the nosilicate)and found that AT=400C. while residual thermal stress in the matrix along the axial Boccaccini et al.[6, 7] reported that AT. oC for a direction, but differed in the choice of models for these water-quenched UD Nicalon/DuranM. Such cracks two parameters. However, by using an expression de- (Fig. 1) are confined to the surface of the material and rived for monolithic materials both sets of authors ne while their number increases significantly at larger glect the effect of material anisotropy on the shocks(ATs)[5] their depth never exceeds two to three magnitude of shock-induced stresses. In addition, dis ibre diameters [4]. However, their appearance can affect crepancies between predicted and experimentally ob- gnificantly the performance of the material as they can erved values of ATc are explained by either varying act as crack initiation sites during possible subsequent e value of A or by assuming that micro-structural loading, and thus affect the properties of the materia hanges affect the matrix fracture energy or make the interior of the composite, especially the sen- This study describes an investigation into AT of Ul sitive fibre-matrix interfaces, susceptible to environmen CMCs that addresses these issues by taking into account I attack by allowing the ingress of oxygen and other the effect of material anisotropy on the magnitude of the corrosive agents shock-induced stresses as well as the significance of the This has led to attempts to devise models that bi-axial nature of these stresses on the behaviour of predict the onset of thermal shock damage. i.e. the UD CMCs Te. Both Blissett et al. [5] and Boccaccini [8] postu lated that the level of stress to cause matrix cracking should be the same whether applied mechanically or 2. Derivation of the predictive model thermally. They characterised the maximum value of the bi-axial thermal shock-induced stress at the surface 2. 1. The condition for cracking due to thermal shock of the material using the classic relationship [9] VYse We follow the approach of Blissett et al. [5]and Boc- (1) caccini [8] and postulate that multiple matrix cracking perpendicular to the fibres occurs when the thermally in- duced stresses along the fibre direction become equal to the stress required to cause matrix fracture. Thus, we can equate the thermal stresses in the matrix with the uniaxial matrix strength, i.e where othM describes the thermally induced stresses in the matrix along the direction of the fibres the ther mally induced stresses may comprise, apart from ther mal shock-induced stresses. residual thermal stresses usually present in CMCs due to differences in the coeffi- cient of thermal expansion between the matrix and the reinforcing fibres. Thus, Fig. I. Reflected light micrograph showing matrix cracking due where alm is the axial thermal shock-induced stress in to thermal shock at AT = 400C in UD Nicalon'M/cas (after he matrix. Accordingly, the critical condition for the Blissett et al. 5) onset of fracture(3)becomes through(4):

[1]. Kagawa et al. [4] described perpendicular matrix cracking on the surface of a PyrexTM–matrix composite reinforced with NicalonTM fibres after quenching test specimens into room temperature water (
20 C) from a temperature higher than 620 C, i.e. the
critical quenching temperature difference for the onset of crack￾ing was DTc > 600 C. The same authors also reported that DTc = 800 C for a water-quenched UD NicalonTM/LAS (lithium aluminosilicate), after observ￾ing a range of cracking phenomena, some of them perpendicular to the fibre axis, appearing on the material surface. Blissett et al. [5] performed water￾quench tests on a UD NicalonTM/CAS (calcium alumi￾nosilicate) and found that DTc = 400 C, while Boccaccini et al. [6,7] reported that DTc
585 C for a water-quenched UD Nicalon/DuranTM. Such cracks (Fig. 1) are confined to the surface of the material and while their number increases significantly at larger shocks (DTs) [5] their depth never exceeds two to three fibre diameters [4]. However, their appearance can affect significantly the performance of the material as they can act as crack initiation sites during possible subsequent loading, and thus affect the properties of the material, or make the interior of the composite, especially the sen￾sitive fibre–matrix interfaces, susceptible to environmen￾tal attack by allowing the ingress of oxygen and other corrosive agents. This has led to attempts to devise models that predict the onset of thermal shock damage, i.e. the DTc. Both Blissett et al. [5] and Boccaccini [8] postu￾lated that the level of stress to cause matrix cracking should be the same whether applied mechanically or thermally. They characterised the maximum value of the bi-axial thermal shock-induced stress at the surface of the material using the classic relationship [9]: rTS ¼ AEaDT ð1 mÞ ; ð1Þ where stiffness (E), coefficient of thermal expansion￾CTE (a), and Poissons ratio (m) are either matrix [5] or volume-averaged properties [8]. The parameter
A is termed the
stress reduction factor and accounts for the finite value of the coefficient of heat transfer, h, be￾tween the material and the quenching medium. Residual thermal stresses were also taken into account. The de￾rived formula for DTc in both cases had the same form: DT c ¼ 1 v AEa rmu rRES 1;M
; ð2Þ where rmu is the uniaxial matrix strength (or the com￾posite stress to cause matrix cracking) and rRES 1;M is the residual thermal stress in the matrix along the axial direction, but differed in the choice of models for these two parameters. However, by using an expression de￾rived for monolithic materials both sets of authors ne￾glect the effect of material anisotropy on the magnitude of shock-induced stresses. In addition, dis￾crepancies between predicted and experimentally ob￾served values of DTc are explained by either varying the value of A or by assuming that micro-structural changes affect the matrix fracture energy. This study describes an investigation into DTc of UD CMCs that addresses these issues by taking into account the effect of material anisotropy on the magnitude of the shock-induced stresses as well as the significance of the bi-axial nature of these stresses on the behaviour of UD CMCs. 2. Derivation of the predictive model 2.1. The condition for cracking due to thermal shock We follow the approach of Blissett et al. [5] and Boc￾caccini [8] and postulate that multiple matrix cracking perpendicular to the fibres occurs when the thermally in￾duced stresses along the fibre direction become equal to the stress required to cause matrix fracture. Thus, we can equate the thermal stresses in the matrix with the uniaxial matrix strength, i.e. rth 1;M ¼ rmu; ð3Þ where rth 1;M describes the thermally induced stresses in the matrix along the direction of the fibres. The ther￾mally induced stresses may comprise, apart from ther￾mal shock-induced stresses, residual thermal stresses usually present in CMCs due to differences in the coeffi- cient of thermal expansion between the matrix and the reinforcing fibres. Thus, rth 1;M ¼ rTS 1;M þ rRES 1;M ; ð4Þ where rTS 1;M is the axial thermal shock-induced stress in the matrix. Accordingly, the critical condition for the onset of fracture (3) becomes through (4): Fig. 1. Reflected light photomicrograph showing matrix cracking due to thermal shock at DTc = 400 C in UD NicalonTM/CAS (after Blissett et al. [5]). C. Kastritseas et al. / Composites Science and Technology 65 (2005) 1880–1890 1881

C. Kastritseas et al Composites Science and Technology 65(2005)1880-1890 The parameters included in(5)need to be described ana- f2+Eth 2=0 (12) lytically before the equation can be applied. This is the heme of the following paragraph From Eqs. (6), (7),(11) and(12), we have el=-cth=-m1(70-71)=x1(71-T0)=x1△T, 2. 2. The thermal shock-induced stress field (13) To obtain the shock-induced stress in the matrix, aiN the full thermo-elastic stress field developed at the surface 12=-ch2=-2(70-T1)=x(T1-10)=x2△T of the material during thermal shock needs to be charac- terised. Such an approach would require complex three- The elastic strains cause'thermal stresses' along the prin- dimensional transient stress analysis similar to that cipal axes of the material and can be written as performed by Wang and Chou [10, 11]. However, a simpli- fied approach can be followed if only the maximum values of the induced stresses are taken into account E, Consider the surface of a rectangular plate of UD CMC initially at temperature TI(Fig. 2). The composite V1201 consists of a matrix of volume fraction Vm with proper E2 El ties Em, am, Vm, which contains parallel fibres of volume Young,'s moduli along the principal material direction fraction Vr with properties Ef, af, ve. and the minor Poissons ratio are given by If the material is rapidly cooled from Ti to To and E1=EmVm+E(1-Vm), perfect heat transfer between the plate and the cooling medium is assumed, the surface immediate adopts the ErEm temperature To while the other parts of the plate E2(Er/m+ EmVe) remain at T,. This case corresponds to having a plate that can freely expand in the 3-direction (i.e. perpendicular E to the plane of Fig. 1), with suppressed expansion in the 1-and 2-directions. In the absence of displacement It has to be noted that more sophisticated approaches restrictions, the plate would expand along the 1-and 2-directions by thermal strains of could have been used to estimate transverse properties (E2 and a2), e.g.[12]. However, in fibre-reinforced ct=m1(70-T1) (6) CMCs the differences between such models and the sim- ple expressions adopted here are small. h2=m2(70-71) By substituting(13)and(14)in(15)and(16) respec The CTes along the principal material directions are gi- tively and solving first for dl and then for a2 we can ven obtain the thermal shock-induced stresses along the Ema.m/m+ Erarve) principal axes of the material as EmVm+ErVe) Gs=AO,△T a2=(1+vm )am Vm +(1+vraurVr-a1v12. (9)a5=AQ2△T, (21) In addition, the major Poisson's ratio is given by 1 Since thermal expansion in both directions is completely (1-v12V21) suppressed, elastic strains are created that compensate (v12E21+E2x2) (1-v12V21) The stress reduction factor A has also been included in (20)and(21). The thermal shock-induced stress in the matrix can be found by employing the iso-strain condition in the axial (1-)direction as ig. 2. A UD composite(fibres aligned along l-direction) subjected to hermal shock. Thermal shock-induced stresses are also show =吧=g① El Em Er

rTS 1;M þ rRES 1;M ¼ rmu. ð5Þ The parameters included in (5) need to be described ana￾lytically before the equation can be applied. This is the theme of the following paragraphs. 2.2. The thermal shock-induced stress field To obtain the shock-induced stress in the matrix, rTS 1;M, the full thermo-elastic stress field developed at the surface of the material during thermal shock needs to be charac￾terised. Such an approach would require complex three￾dimensional transient stress analysis similar to that performed by Wang and Chou [10,11]. However, a simpli- fied approach can be followed if only the maximum values of the induced stresses are taken into account. Consider the surface of a rectangular plate of UD CMC initially at temperature T1 (Fig. 2). The composite consists of a matrix of volume fraction Vm with proper￾ties Em, am, mm, which contains parallel fibres of volume fraction Vf with properties Ef, af, mf. If the material is rapidly cooled from T1 to T0 and perfect heat transfer between the plate and the cooling medium is assumed, the surface immediate adopts the temperature T0 while the other parts of the plate remain at T1. This case corresponds to having a plate that can freely expand in the 3-direction (i.e. perpendicular to the plane of Fig. 1), with suppressed expansion in the 1- and 2-directions. In the absence of displacement restrictions, the plate would expand along the 1- and 2-directions by thermal strains of eth;1 ¼ a1ðT 0 T 1Þ; ð6Þ eth;2 ¼ a2ðT 0 T 1Þ. ð7Þ The CTEs along the principal material directions are gi￾ven by a1 ¼ ð Þ EmamV m þ EfafV f ð Þ EmV m þ EfV f ; ð8Þ a2 ¼ ð1 þ mmÞamV m þ ð1 þ mfÞafV f a1m12. ð9Þ In addition, the major Poissons ratio is given by m12 ¼ mfV f þ mmV m. ð10Þ Since thermal expansion in both directions is completely suppressed, elastic strains are created that compensate the thermal strains, i.e. eel;1 þ eth;1 ¼ 0; ð11Þ eel;2 þ eth;2 ¼ 0. ð12Þ From Eqs. (6), (7), (11) and (12), we have eel;1 ¼ eth;1 ¼ a1ðT 0 T 1Þ ¼ a1ðT 1 T 0Þ ¼ a1DT ; ð13Þ eel;2 ¼ eth;2 ¼ a2ðT 0 T 1Þ ¼ a2ðT 1 T 0Þ ¼ a2DT . ð14Þ The elastic strains cause
thermal stresses along the prin￾cipal axes of the material and can be written as eel;1 ¼ rTS 1 E1 m21rTS 2 E2 ; ð15Þ eel;2 ¼ rTS 2 E2 m12rTS 1 E1 . ð16Þ Youngs moduli along the principal material directions and the minor Poissons ratio are given by E1 ¼ EmV m þ Efð1 V mÞ; ð17Þ E2 ¼ EfEm ðEfV m þ EmV fÞ ; ð18Þ m21 ¼ m12E2 E1 . ð19Þ It has to be noted that more sophisticated approaches could have been used to estimate transverse properties (E2 and a2), e.g. [12]. However, in fibre-reinforced CMCs the differences between such models and the sim￾ple expressions adopted here are small. By substituting (13) and (14) in (15) and (16) respec￾tively and solving first for rTS 1 and then for rTS 2 we can obtain the thermal shock-induced stresses along the principal axes of the material as rTS 1 ¼ AQ1DT ; ð20Þ rTS 2 ¼ AQ2DT ; ð21Þ where Q1 ¼ ð Þ E1a1 þ m21E2a2 ð Þ 1 m12m21 ; ð22Þ Q2 ¼ ð Þ m12E2a1 þ E2a2 ð Þ 1 m12m21 . ð23Þ The stress reduction factor, A, has also been included in (20) and (21). The thermal shock-induced stress in the matrix can be found by employing the iso-strain condition in the axial (1-) direction as e1 ¼ e m 1 ¼ e f 1 ¼ rTS 1 E1 ¼ rTS 1;M Em ¼ rTS 1;F Ef ; ð24Þ 1 2 TS σ 1 TS σ 2 Fig. 2. A UD composite (fibres aligned along 1-direction) subjected to thermal shock. Thermal shock-induced stresses are also shown. 1882 C. Kastritseas et al. / Composites Science and Technology 65 (2005) 1880–1890

C. Kastritseas et al Composites Science and Technology 65(2005)1880-1890 where aIF is the thermal shock-induced stress in the fi. 2. 4. The matrix cracking stress bres. Eq (24) yields through(20) The strength of the matrix in the direction parallel to Emrs AEmO1AT (25) the aligned fibres in UD CMCs has been the subject of numerous investigations [1]. Blissett et al. [5] used in At the onset of cracking AT= AT= Tmax - To, where heir work the classic model of Aveston et al. (ACK) Tmax is the temperature from which the material should [16] which has been shown to be valid for the assump- tion of "long initial flaws and, thus, provides a lower be quenched in a medium of temperature To for cracking bound estimate of the matrix strength [17]. By contrast, to initiate Boccaccini [8] chose the model of Pagano and Kim [18] 2.3. The residual stress field which is valid for initial damage in the form of localised cracks that do not interact and arrest when they encoun As stated above due to differences in Cte between ter the nearest fibre. The major difference between the the matrix and the reinforcing fibres, residual stresses two models has to do with the interfacial shear stress t. included in the aCK model but absent from that of are established in CMCs when they are cooled down from their high processing temperatures (e.g Pagano and Kim Micrographs, such as that of Fig. I suggest that the 'long crack assumption can be consid >1200C for most glass ceramic-matrix composites). ered valid for the initial thermal shock damage observed There attempts to the magnitude of these stresses and model the effect UD CMCs. In addition the interfacial condition has they have on mechanical properties. Usually, co-axial been shown to be the single most important factor that ylinder models subject to thermo-mechanical loading determines the properties of fibre-reinforced CMCs under various conditions. Thus. the aCK model was are utilised (e.g. [13] and include micro-mechanical determined to be more appropriate to describe the onset (e.g.[14 of thermal shock damage This gives matrix failure strain In this case, we use the model of Budiansky et al. [ 15] that states that residual stresses are governed by the mis- 12τ7 emEr fit strain, e, between the fibre and the matrix, which if Emu-EELrV the misfit arises solely from thermal expansion differ where ,m is the fracture surface energy, and r is the fibre radius. The stress in the matrix to initiate cracking is 9=(xm-x)△T hen given by The parameter ATF in(26) is the temperature difference between processing temperature(Tp) and the tempera- omu=Ememu=/oTmEmErv? ture of operation. For example, room temperature oper- Ej ation is at 20-22 oC. whereas in the case of therma where Im(=2,m) is the matrix fracture energy (cold)shock it is at the temperature to which the mate- rial has been heated prior to quenching (i.e. 2.5. Application of the critical condition ATF=Tp-T1) The axial residual stress in the matrix oLE, is then given by As all the parameters included in(5)have been deter mined, we can now proceed with the application of the (1+E1/Er)/r22 critical condition in order to determine the critical v quenching temperature difference, ATc. Substituting (25),(28)and(30)in(5), we find 1+E1/ErEmEr Vr(am-a) AT -m(1-到)E1(1-m2) (27)AEmQ△T 6tlmEmEfk +61△TF= (31 This can be written more simply as By solving(31) for ATc, we get the critical quenching temperature difference for the onset of matrix cracking ORES=OATF a function of△Tras At the onset of thermal shock cracking ATF=Tp E1/6tlmEmErV Tmax. The model predictions were found to be in close EMei FIrm (61△TF) agreement with the values obtained by the more omplex, analytical model of Powell et al. [14, which In (32), ATc=Tmax -To and ATF=Tp-Tmax. As is based on the analysis of co-axial isotropic cylinders Tma features on both sides of the equation we can

where rTS 1;F is the thermal shock-induced stress in the fi- bres. Eq. (24) yields through (20): rTS 1;M ¼ Em E1 rTS 1 ¼ AEmQ1DT E1 . ð25Þ At the onset of cracking DT = DTc = Tmax T0, where Tmax is the temperature from which the material should be quenched in a medium of temperature T0 for cracking to initiate. 2.3. The residual stress field As stated above, due to differences in CTE between the matrix and the reinforcing fibres, residual stresses are established in CMCs when they are cooled down from their high processing temperatures (e.g. >1200 C for most glass ceramic–matrix composites). There have been a number of attempts to quantify the magnitude of these stresses and model the effect they have on mechanical properties. Usually, co-axial cylinder models subject to thermo-mechanical loading are utilised (e.g. [13]) and include micro-mechanical analyses of stress transfer between fibre and matrix (e.g. [14]). In this case, we use the model of Budiansky et al. [15] that states that residual stresses are governed by the mis- fit strain, X, between the fibre and the matrix, which if the misfit arises solely from thermal expansion differ￾ences is given by X ¼ ðam afÞDT F. ð26Þ The parameter DTF in (26) is the temperature difference between processing temperature (Tp) and the tempera￾ture of operation. For example, room temperature oper￾ation is at 20–22 C, whereas in the case of thermal (cold) shock it is at the temperature to which the mate￾rial has been heated prior to quenching (i.e. DTF = Tp T1). The axial residual stress in the matrix, rRES 1;M , is then given by rRES 1;M ¼ ð Þ 1 þ E1=Ef EmEfV fX 2 1 ð12m12Þ 2ð1m12Þ 1 E1 Ef h i
E1ð1 m12Þ ¼ ð Þ 1 þ E1=Ef EmEfV fðam afÞ 2 1 ð12m12Þ 2ð1m12Þ 1 E1 Ef h i
E1ð1 m12Þ DT F. ð27Þ This can be written more simply as rRES 1;M ¼ H1DT F. ð28Þ At the onset of thermal shock cracking DTF = Tp Tmax. The model predictions were found to be in close agreement with the values obtained by the more complex, analytical model of Powell et al. [14], which is based on the analysis of co-axial isotropic cylinders. 2.4. The matrix cracking stress The strength of the matrix in the direction parallel to the aligned fibres in UD CMCs has been the subject of numerous investigations [1]. Blissett et al. [5] used in their work the classic model of Aveston et al. (ACK) [16] which has been shown to be valid for the assump￾tion of
long initial flaws and, thus, provides a lower bound estimate of the matrix strength [17]. By contrast, Boccaccini [8] chose the model of Pagano and Kim [18] which is valid for initial damage in the form of localised cracks that do not interact and arrest when they encoun￾ter the nearest fibre. The major difference between the two models has to do with the interfacial shear stress, s, included in the ACK model but absent from that of Pagano and Kim. Micrographs, such as that of Fig. 1, suggest that the
long crack assumption can be consid￾ered valid for the initial thermal shock damage observed on UD CMCs. In addition, the interfacial condition has been shown to be the single most important factor that determines the properties of fibre-reinforced CMCs under various conditions. Thus, the ACK model was determined to be more appropriate to describe the onset of thermal shock damage. This gives matrix failure strain as emu ¼ 12scmEmEfV 2 f E1E2 mrV m 1 3 ; ð29Þ where cm is the fracture surface energy, and r is the fibre radius. The stress in the matrix to initiate cracking is then given by rmu ¼ Ememu ¼ 6sCmEmEfV 2 f E1rV m 1 3 ; ð30Þ where Cm (=2cm) is the matrix fracture energy. 2.5. Application of the critical condition As all the parameters included in (5) have been deter￾mined, we can now proceed with the application of the critical condition in order to determine the critical quenching temperature difference, DTc. Substituting (25), (28) and (30) in (5), we find AEmQ1DT c E1 þ H1DT F ¼ 6sCmEmEfV 2 f E1rV m 1 3 . ð31Þ By solving (31) for DTc, we get the critical quenching temperature difference for the onset of matrix cracking as a function of DTF as DT c ¼ E1 AEmQ1 6sCmEmEfV 2 f E1rV m 1 3 ð Þ H1DT F " #. ð32Þ In (32), DTc = Tmax T0 and DTF = Tp Tmax. As Tmax features on both sides of the equation we can C. Kastritseas et al. / Composites Science and Technology 65 (2005) 1880–1890 1883

C. Kastritseas et al Composites Science and Technology 65(2005)1880-1890 proceed by substituting AT and AT in(32)and, subse- NicalonM/DuranM, and UD NicalonM/LAS quently, solving for Tmax. In this way, we can find the obtained by quenching heated specimens into room temperature from which the material should be temperature water. The data used in the calculation quenched into a medium of temperature To to initiate [4-6, 8, 19-21] are presented in Tables I and 2. multiple matrix cracking due to thermal shock as It should be noted that. in contrast to blissett et al. [5](A=0.5)and Boccaccini [8](A=0.6), the stress -日1Tn+mgtn reduction factor, A, is not assigned a single pre- (33) determined value but is allowed to vary between 0 and 0.66. This is mainly because h, on which the value of Finally, the critical temperature difference for the onset A depends, can not be readily determined during a of multiple matrix cracking is given by water quench test, and its value has been reported to vary between very low values and a maximum of 6rlmEmErI FIrm 0,Tp+=E 60 kW mK depending on temperature and material △Te= Emg-0, (34) surface finish. In addition, A is a function of the size of the component under investigation. However, it has been shown that the maximum value a can attain in It can be noted that(34)provides AT as a function of such tests is 0.66[3] Tp, To and a number of material properties. Room tem- The calculated and experimentally determined values perature values of these properties are usually employed of. can be seen in Table 2 as information of their change with increasing tempera ture is scarce. In the following paragraph, the predic- 2.7 Discussion tions of (34) are compared with experimental data for a range of UD CMCs. Before discussing the results from the application of 2.6. Comparison with experimental results (34), it is interesting to compare the present approach with the models of Blissett et al. [5] and Boccaccini [8] (Eq.(2)). To facilitate such a comparison, (32) can be The values of ATc predicted by Eq(34)are com- written, after substitution of @1 from(22), d this section with experimental results fo UD Nicalon/ CAS, UD Nicalon/Pyrex, UD AT A(Em/E1)(E141+V2 E2u2(mu -GRES (35) althe d oRe depend ing on the models chosen for their description, the main Table l difference is centred in the parameter (1-V12v21/ Material properties used in the calculation of AT. E101+ v2 E202)present in(35), which shows that, con- Material E(GPa) x (10 r(m-2) r(um) trary to the other models for ATc, the anisotropic prop Nicanor 0.2 erties of the material have been taken into consideration 90 3633 in the present approach. In addition, the appearance of the parameter(Em/En shows that the critical condition 0.27.5 LAS for the onset of thermal shock fracture(Eq. (5)has been applied in terms of matrix stresses. Blissett et al. [51 Table 2 The properties of the four CMCs under consideration used in the model and the experimentally determined and predicted(through(34) values of ATc. The values of t used were experimentally determined room temperature ones obtained from the literature Material ATF(°C t(MPa) △Tc(°C Nicalon/CAS Nicalon/dura 1000 1000 86 Minimum predicted values of ATc corresponding to 4= 20 mPa l b Average value of t, as reported values range from 2 to 20 [20.21 (28)and(39) are not valid for UD Nicalon/LAS as the es not arise solely from Az [31] Reported room temperature residual stress values are oRE =-50 MPa and oRES= 20 MPa [19, 31] Thus, O1 and O2 are back-calculated by applying (28)and(39)for

proceed by substituting DTc and DTF in (32) and, subse￾quently, solving for Tmax. In this way, we can find the temperature from which the material should be quenched into a medium of temperature T0 to initiate multiple matrix cracking due to thermal shock as T max ¼ 6sCmEmEf V 2 f E1rV m
1 3 H1T p þ AEmQ1T 0 E1 AEmQ1 E1 H1 . ð33Þ Finally, the critical temperature difference for the onset of multiple matrix cracking is given by DT c ¼ 6sCmEmEf V 2 f E1rV m
1 3 H1T p þ AEmQ1T 0 E1 AEmQ1 E1 H1 2 6 4 3 7 5 T 0. ð34Þ It can be noted that (34) provides DTc as a function of Tp, T0 and a number of material properties. Room tem￾perature values of these properties are usually employed as information of their change with increasing tempera￾ture is scarce. In the following paragraph, the predic￾tions of (34) are compared with experimental data for a range of UD CMCs. 2.6. Comparison with experimental results The values of DTc predicted by Eq. (34) are com￾pared in this section with experimental results for UD NicalonTM/CAS, UD NicalonTM/PyrexTM, UD NicalonTM/DuranTM, and UD NicalonTM/LAS obtained by quenching heated specimens into room temperature water. The data used in the calculation [4–6,8,19–21] are presented in Tables 1 and 2. It should be noted that, in contrast to Blissett et al. [5] (A = 0.5) and Boccaccini [8] (A = 0.6), the stress reduction factor, A, is not assigned a single pre￾determined value but is allowed to vary between 0 and 0.66. This is mainly because h, on which the value of A depends, can not be readily determined during a water quench test, and its value has been reported to vary between very low values and a maximum of 60 kW m2 K1 depending on temperature and material surface finish. In addition, A is a function of the size of the component under investigation. However, it has been shown that the maximum value A can attain in such tests is 0.66 [3]. The calculated and experimentally determined values of DTc can be seen in Table 2. 2.7. Discussion Before discussing the results from the application of (34), it is interesting to compare the present approach with the models of Blissett et al. [5] and Boccaccini [8] (Eq. (2)). To facilitate such a comparison, (32) can be written, after substitution of Q1 from (22), as DT c ¼ ð Þ 1 m12m21 A Eð Þ m=E1 ð Þ E1a1 þ m21E2a2 rmu rRES 1;M
. ð35Þ Although there are differences in rmu and rRES 1;M depend￾ing on the models chosen for their description, the main difference is centred in the parameter (1 m12m21/ E1a1 + m21E2a2) present in (35), which shows that, con￾trary to the other models for DTc, the anisotropic prop￾erties of the material have been taken into consideration in the present approach. In addition, the appearance of the parameter (Em/E1) shows that the critical condition for the onset of thermal shock fracture (Eq. (5)) has been applied in terms of matrix stresses. Blissett et al. [5] Table 1 Material properties used in the calculation of DTc Material E (GPa) a (106 K1 ) v C (J m2 ) r (lm) Nicalon 190 3.3 0.2 – 8 CAS 90 4.6 0.25 25 – Duran 63 3.3 0.2 7.5 – Pyrex 63 3.3 0.2 7.5 – LAS 83 0.9 0.3 30 – Table 2 The properties of the four CMCs under consideration used in the model and the experimentally determined and predicted (through (34)) values of DTc. The values of s used were experimentally determined room temperature ones obtained from the literature Material Vf DTF (C) s (MPa) DTc (C) Experimental Predicteda Nicalon/CAS 0.34 1200 15 400 >485 Nicalon/Duran 0.4 1000 14 585 >840 Nicalon/Pyrex 0.5 1000 10b >600 >896 Nicalon/LAS 0.4 1350c 2–3 800 >900 a Minimum predicted values of DTc, corresponding to A = 0.66. b Average value of s, as reported values range from 2 to 20 MPa [20,21]. c (28) and (39) are not valid for UD Nicalon/LAS as the misfit stress does not arise solely from Da [31]. Reported room temperature residual stress values are rRES 1;M ¼ 50 MPa and rRES 2 ¼ 20 MPa [19,31]. Thus, H1 and H2 are back-calculated by applying (28) and (39) for DTF = 1350 C. 1884 C. Kastritseas et al. / Composites Science and Technology 65 (2005) 1880–1890

C. Kastritseas et al Composites Science and Technology 65(2005)1880-1890 chose to characterise matrix stresses by applying(1) 3. Modelling changes in interfacial shear stress due to sing matrix properties, while Boccaccini [8] seems to thermal shock have applied (1) with volume-averaged properties in or der to characterise the composite stresses required to 3. 1. Introduction cause matrix fracture Concentrating now on the application of(34), the The enhanced toughness of fibre-reinforced CMCs results presented in Table 2 show that it significantly with non-oxide matrices compared to monolithic ceram overestimates the AT of all CMCs for all possible ics is mainly achieved through properly engineered fi- values of the stress reduction factor. a. while it can bre-matrix interfaces that allow fibres to debond be argued that the values of material properties ahead of advancing cracks and slide past the matrix included in (34) may change as a result of the [l]. One school of thought in the experimental analysis short-term high-temperature exposure to which test and description of interfacial behaviour in CMCs views specimens are subjected during a quench test, these the sliding stress at the fibre/matrix interface as a Cou changes are expected to be small and not enough to ac- lomb law of friction [22-26]. This means that the count for the discrepancies observed between experi- frictional stress is assumed to be directly proportional mental and predicted values of ATc. However, further to the radial clamping stress, which includes contribu consideration of the analysis presented in Section 2 tions mainly from radial residual thermal stresses and fi reveals that, because of the bi-axial nature of the ther- bre roughness-induced compressive stresses [21] mal shock-induced stresses. in addition to an axi In the case of thermal shock. the interface can be con thermal shock-induced stress, o, there is a transverse sidered to be under the influence of a tensile thermal thermal shock-induced stress, o,, given by(21), that shock-induced stress, given by (21), that reduces the can be thought of as acting perpendicular to the fi- magnitude of the radial clamping stress. The extent of bre/matrix interface during the shock. Hence, aS his reduction at each AT would depend on the respec- may reduce the value of the radial clamping stress at tive value of the thermal shock-induced stress. In the fol the interface on which the value of the interfacial shear lowing paragraphs, a Coulomb-type expression is first stress, t, is assumed to depend. This means that during derived for an interface under thermal shock and is then the shock the value of t is reduced from its room tem- applied to the cases of the thermally shocked CMCs perature value to a significantly lower one, which will mentioned in Paragraph 2.6 depend on the magnitude of the applied stress a,. As can be seen from(34), a reduction in t would cause 3.2. Model derivation a reduction in om. which would in turn lead to lower predicted values of ATe. In other words, by taking into Following the review of the relevant theory presented ccount the bi-axiality of the applied thermal stress by Drissi-Habti and Nakano [26] and ignoring Poisson field, it can be argued that the discrepancy between effects, we postulate that the value of interfacial shear experimental and predicted values of AT is caused tress,t, when the surface considered in Fig. 2 is sub- by the use in(34) of values of t measured at room jected to a thermal shock AT, is described by a temperature Coulomb-type relationship of the form: It has to be noted that a different value of t should be applicable during the shock even without taking into account the effect of o2. This is because the In(36), u is the coefficient of friction between fibre and clamping stress at the interface depends on the resid- matrix and o2 is the clamping stress at the fibre/matrix ual radial stress caused by differences in Cte between interface. The value of g, in the case of thermal shock matrix and fibres as the CMC is cooled from its pro- treatment is given by cessing temperature. Since this stress is proportional to△ TF and the△ TF of heated material(=Tp-Tm 02=d5+BA+d1 (37) is lower than that of material at room temperature (=Tp-To), the value of the clamping stress, and con- Eq.(37)states that the value of the clamping stress, 2, sequently of t, should be changed at the onset of depends on the difference between(a) the sum of the fracture radial residual thermal stress, oRES, and the fibre rough As it has been qualitatively argued that the applied ness-induced stress, oRA, and(b)the thermal shock thermal stresses can be thought of as affecting the radial induced radial stress, oIs. So,(36)becomes through clamping stress at the fibre/matrix interface, an attempt (37) is presented in the following section to model changes in t=-u(oRES +oR+orS) at various ATs using a modified Coulomb-type rela ionship usually employed to characterise the results of We continue to employ the model of Budiansky et al micro-indentation tests [15] for the calculation of a?, in contrast to

chose to characterise matrix stresses by applying (1) using matrix properties, while Boccaccini [8] seems to have applied (1) with volume-averaged properties in or￾der to characterise the composite stresses required to cause matrix fracture. Concentrating now on the application of (34), the results presented in Table 2 show that it significantly overestimates the DTc of all CMCs for all possible values of the stress reduction factor, A. While it can be argued that the values of material properties included in (34) may change as a result of the short-term high-temperature exposure to which test specimens are subjected during a quench test, these changes are expected to be small and not enough to ac￾count for the discrepancies observed between experi￾mental and predicted values of DTc. However, further consideration of the analysis presented in Section 2 reveals that, because of the bi-axial nature of the ther￾mal shock-induced stresses, in addition to an axial thermal shock-induced stress, rTS 1 , there is a transverse thermal shock-induced stress, rTS 2 , given by (21), that can be thought of as acting perpendicular to the fi- bre/matrix interface during the shock. Hence, rTS 2 may reduce the value of the radial clamping stress at the interface on which the value of the interfacial shear stress, s, is assumed to depend. This means that during the shock the value of s is reduced from its room tem￾perature value to a significantly lower one, which will depend on the magnitude of the applied stress rTS 2 . As can be seen from (34), a reduction in s would cause a reduction in rmu, which would in turn lead to lower predicted values of DTc. In other words, by taking into account the bi-axiality of the applied thermal stress field, it can be argued that the discrepancy between experimental and predicted values of DTc is caused by the use in (34) of values of s measured at room temperature. It has to be noted that a different value of s should be applicable during the shock even without taking into account the effect of rTS 2 . This is because the clamping stress at the interface depends on the resid￾ual radial stress caused by differences in CTE between matrix and fibres as the CMC is cooled from its pro￾cessing temperature. Since this stress is proportional to DTF and the DTF of heated material (=Tp Tmax) is lower than that of material at room temperature (=Tp T0), the value of the clamping stress, and con￾sequently of s, should be changed at the onset of fracture. As it has been qualitatively argued that the applied thermal stresses can be thought of as affecting the radial clamping stress at the fibre/matrix interface, an attempt is presented in the following section to model changes in s at various DTs using a modified Coulomb-type rela￾tionship usually employed to characterise the results of micro-indentation tests. 3. Modelling changes in interfacial shear stress due to thermal shock 3.1. Introduction The enhanced toughness of fibre-reinforced CMCs with non-oxide matrices compared to monolithic ceram￾ics is mainly achieved through properly engineered fi- bre–matrix interfaces that allow fibres to debond ahead of advancing cracks and slide past the matrix [1]. One school of thought in the experimental analysis and description of interfacial behaviour in CMCs views the sliding stress at the fibre/matrix interface as a Cou￾lomb law of friction [22–26]. This means that the frictional stress is assumed to be directly proportional to the radial clamping stress, which includes contribu￾tions mainly from radial residual thermal stresses and fi- bre roughness-induced compressive stresses [21]. In the case of thermal shock, the interface can be con￾sidered to be under the influence of a tensile thermal shock-induced stress, given by (21), that reduces the magnitude of the radial clamping stress. The extent of this reduction at each DT would depend on the respec￾tive value of the thermal shock-induced stress. In the fol￾lowing paragraphs, a Coulomb-type expression is first derived for an interface under thermal shock and is then applied to the cases of the thermally shocked CMCs mentioned in Paragraph 2.6. 3.2. Model derivation Following the review of the relevant theory presented by Drissi-Habti and Nakano [26] and ignoring Poisson effects, we postulate that the value of interfacial shear stress, s, when the surface considered in Fig. 2 is sub￾jected to a thermal shock DT, is described by a Coulomb-type relationship of the form: s ¼ lr2. ð36Þ In (36), l is the coefficient of friction between fibre and matrix and r2 is the clamping stress at the fibre/matrix interface. The value of r2 in the case of thermal shock treatment is given by r2 ¼ rRES 2 þ rRA 2 þ rTS 2 . ð37Þ Eq. (37) states that the value of the clamping stress, r2, depends on the difference between (a) the sum of the radial residual thermal stress, rRES 2 , and the fibre rough￾ness-induced stress, rRA 2 , and (b) the thermal shock￾induced radial stress, rTS 2 . So, (36) becomes through (37): s ¼ l rRES 2 þ rRA 2 þ rTS 2  . ð38Þ We continue to employ the model of Budiansky et al. [15] for the calculation of rRES 2 , in contrast to C. Kastritseas et al. / Composites Science and Technology 65 (2005) 1880–1890 1885

C. Kastritseas et al Composites Science and Technology 65(2005)1880-1890 Table 3 Calculated values of friction coefficie Material Nicalon/Duran Nicalon/Pyrex 0.0544 0.0144 Drissi-Habti and Nakano [26] and other workers [22- 25]. The model gives oRES as (62△TF)+ CAn +(AQ2△T (43) BBS=62△TrF where 3.3. Application of the modified Coulomb-type model Em(1-Ve(am-af) (43)is now applied for the UD CMCs mentioned 2[1-=(1-2)(1-m2) in Paragraph 2.6, using the data presented in Tables I and 2, at increasing values of AT. The stress reduction factor, A, is allowed to vary between 0 and 0.66, while The model produces similar results with that of Powell et al. [14] and agreement with the model employed in he roughness amplitude of the Nicalon fibre has been determined to be n 30 nm with the use of atomic the analysis of micro-indentation tests is fair The fibre roughness-induced compressive stress, oRA, force microscopy (AFM)[27, 28]. In addition, a number of values have been reported for A, mostly in the 1 is characterised by the following expression as [26] before a pending on whether the fibres have been coated before composite manufacture or a thin interfacial layer (41) has been formed during processing due to fibre/matrix reaction [21]. In this study, u values are obtained by applying (43)at room temperature (where a,S=0) using experimentally determined t values reported in Er(1+vm )+Em(l-ve) 42) the literature(Table 2). The values obtained, which are resented in Table 3, are in good agreement with pub- In(42), r is the fibre radius and Ar is the roughness lished results from micro-indentation studies (e.g amplitude of the fibre surface. Finally, (38)becomes Lara-Curzio and Ferber [29] determined for Nicalon through(21),(39)and CAS that u=0.05-0 01002003004005006007008 010020030040 0800 Quenching Temperature Difference (c) Quenching Temperature Difference cc) 30 UD Nicalon/Pyre UD NicalonILAS Quenching Temperature Difference (c) Quenching Temperature Difference (c) Fig 3. Interfacial shear stress as a function of AT given by (43)for (a) UD Nicalon"M/CAS.(b) UD Nicalon/Duran,(c) UD Nicalon/Pyrex, (d) UD Nicalon/LAS

Drissi-Habti and Nakano [26] and other workers [22– 25]. The model gives rRES 2 as rRES 2 ¼ H2DT F; ð39Þ where H2 ¼ Emð1 V fÞðam afÞ 2 1 ð12m12Þ 2ð1m12Þ 1 E1 Ef h i
ð1 m12Þ . ð40Þ The model produces similar results with that of Powell et al. [14] and agreement with the model employed in the analysis of micro-indentation tests is fair. The fibre roughness-induced compressive stress, rRA 2 , is characterised by the following expression as [26]: rRA 2 ¼ C Ar r ; ð41Þ where C ¼ EmEf Efð1 þ mmÞ þ Emð1 mfÞ . ð42Þ In (42), r is the fibre radius and Ar is the roughness amplitude of the fibre surface. Finally, (38) becomes through (21), (39) and (41): s ¼ l Hð Þþ 2DT F CAr r  þ ð Þ AQ2DT  . ð43Þ 3.3. Application of the modified Coulomb-type model Eq. (43) is now applied for the UD CMCs mentioned in Paragraph 2.6, using the data presented in Tables 1 and 2, at increasing values of DT. The stress reduction factor, A, is allowed to vary between 0 and 0.66, while the roughness amplitude of the NicalonTM fibre has been determined to be 30 nm with the use of atomic force microscopy (AFM) [27,28]. In addition, a number of values have been reported for l, mostly in the range 0–0.2, depending on whether the fibres have been coated before composite manufacture or a thin interfacial layer has been formed during processing due to fibre/matrix reaction [21]. In this study, l values are obtained by applying (43) at room temperature (where rTS 2 ¼ 0) using experimentally determined s values reported in the literature (Table 2). The values obtained, which are presented in Table 3, are in good agreement with pub￾lished results from micro-indentation studies (e.g. Lara-Curzio and Ferber [29] determined for Nicalon/ CAS that l = 0.05–0.06). Table 3 Calculated values of friction coefficient, l Material Nicalon/CAS Nicalon/Duran Nicalon/Pyrex Nicalon/LAS l 0.0544 0.062 0.087 0.0144 0 2 4 6 8 10 12 14 16 0 100 200 300 400 500 600 700 800 Quenching Temperature Difference (°C) Interfacial Shear Stress (MPa) UD Nicalon/CAS 0 2 4 6 8 10 12 14 16 0 100 200 300 400 500 600 700 800 Quenching Temperature Difference (°C) Interfacial Shear Stress (MPa) UD Nicalon/DURAN 0 2 4 6 8 10 0 100 200 300 400 500 600 700 800 Quenching Temperature Difference (°C) Interfacial Shear Stress (MPa) UD Nicalon/Pyrex 0 0.5 1 1.5 2 2.5 0 100 200 300 400 500 600 700 800 900 Quenching Temperature Difference (°C) Interfacial Shear Stress (MPa) UD Nicalon/LAS (a) (b) (c) (d) Fig. 3. Interfacial shear stress as a function of DT given by (43) for (a) UD NicalonTM/CAS, (b) UD NicalonTM/Duran, (c) UD Nicalon/Pyrex, (d) UD Nicalon/LAS. 1886 C. Kastritseas et al. / Composites Science and Technology 65 (2005) 1880–1890

C. Kastritseas et al. Composites Science and Technology 65(2005)1880-1890 The results of the application of (43)are presented in Fig 3(aHd) 3. 4. Discussion 0苏0 In Section 2 of this paper it was argued qualitatively that the interfacial shear stress, t, which governs the mechanical behaviour of CMCs, may be reduced from ts room temperature to a lower value during thermal shock. Fig 3(ahd) show that such reductions are pos- sible if the thermal shock stress component at is taken Quenching Temperature Difference (c) into consideration and modelled as acting perpendicular to the fibre/matrix interface. The extent of such reduc Fig 4. Change in t for increasing AT from(43)(dotted line)and from tions depends on the relative magnitudes of aland the residual clamping stress induced clamping stress, oRA, is independent of temper- As it can be seen, the values of t predicted by (43)and ature or temperature differential. In the case of Nicalon/CAS(Fig. 3(a)), o,S and GRES act in a (44)for the same ATc overlap for all CMCs under con sideration. This is better visualised if the values pre- synergistic way since the value of o2 is increasing while dicted by(44) for increasing AT are superimposed on GRES is decreasing for increasing AT. This causes large the graphs of Fig. 3(aHd). An example(for Nic- reductions in t even at low values of AT, which partly explains why this material has the lowest AT(the other alon/CAS) is shown in Fig. 4. Similar graphs can be obtained for the other three cmcs reason being that the matrix is under residual tension in the fibre direction). The Pyrex-and Duran'-matrix Table 4 and Fig 4 reveal that the proposed modi composites also exhibit large reductions in t with fied Coulomb-type model for the description of possi- increasing AT, albeit of a smaller rate than the CAs-ma ble changes in t during thermal shock not only predicts correctly the trends of the changes of t with trix system. This is because no thermal residual stresses AT but can also provide quantitative estimates for are present in these materials, which is evident in Fig. (b)and(c)as t becomes equal to its room temperature imental data the values of t=f(AT) that correlate well with exper value for A=0 irrespective of AT. In the LAs-matrix It has to be noted that more sophisticated approaches effect of o2s on t opposes that of o2s. This can be ob- Use of such models will share similar features with the served in Fig 3(d)as for A=0 t actually increases due to residual stress relaxation with increasing AT. present one(e.g. inclusion of a2 and may describe It is also of interest to investigate whether the reduc- he underlying phenomena more accurately, especially tions in t predicted by (43)correspond to the values of t be significant during thermal slo effect are shown to if phenomena such as the Poisson required for thermal shock cracking to initiate at the experimentally observed values of ATc. The values t should acquire so that (34) gives correct prediction can be obtained by solving (34)for t and applying the 4. Determination of the stress reduction factor A resulting equation at the experimentally determined val- ues of ATc(Table 2). Eq (34)becomes The previous analysis was performed by assuming that the stress reduction factor. A. varied from o to =(a)(a) 4g1△T+61△T 0.66 for reasons mentioned in Paragraph 2.6. As we have now established that the two equations describing hange in t with AT[i.e(43)and (44)] produce overlap- Similarly, (43)is applied for the known values of ATc. In ping results, estimates of A can be obtained. This is th equations, A is again varied from 0 to 0.66. The re- achieved by plotting t(obtained from both equations) Its are presented in Table 4. against A(=0-0. 66) for all ATs. The point where th relation of values of t given by (43 )and (44) for the ATc of each CMC Nicalon/CAS Nicalon/Duran Nicalon/Pyrex Nicalon/LAS (△T=400°C) (△Tc≈585°C (△Tc=800°C) T(MPa) from(44 0.4-11.3 5.6-13.5 1.44-14 1.64-10 0.5-24

The results of the application of (43) are presented in Fig. 3(a)–(d). 3.4. Discussion In Section 2 of this paper it was argued qualitatively that the interfacial shear stress, s, which governs the mechanical behaviour of CMCs, may be reduced from its room temperature to a lower value during thermal shock. Fig. 3(a)–(d) show that such reductions are pos￾sible if the thermal shock stress component rTS 2 is taken into consideration and modelled as acting perpendicular to the fibre/matrix interface. The extent of such reduc￾tions depends on the relative magnitudes of rTS 2 and the residual clamping stress, rRES 2 , since the roughness￾induced clamping stress, rRA 2 , is independent of temper￾ature or temperature differential. In the case of NicalonTM/CAS (Fig. 3(a)), rTS 2 and rRES 2 act in a synergistic way since the value of rTS 2 is increasing while rRES 2 is decreasing for increasing DT. This causes large reductions in s even at low values of DT, which partly explains why this material has the lowest DTc (the other reason being that the matrix is under residual tension in the fibre direction). The PyrexTM– and DuranTM–matrix composites also exhibit large reductions in s with increasing DT, albeit of a smaller rate than the CAS–ma￾trix system. This is because no thermal residual stresses are present in these materials, which is evident in Fig. 3(b) and (c) as s becomes equal to its room temperature value for A = 0 irrespective of DT. In the LAS–matrix system, changes in s for increasing DT are small as the effect of rTS 2 on s opposes that of rRES 2 . This can be ob￾served in Fig. 3(d) as for A = 0 s actually increases due to residual stress relaxation with increasing DT. It is also of interest to investigate whether the reduc￾tions in s predicted by (43) correspond to the values of s required for thermal shock cracking to initiate at the experimentally observed values of DTc. The values s should acquire so that (34) gives correct predictions can be obtained by solving (34) for s and applying the resulting equation at the experimentally determined val￾ues of DTc (Table 2). Eq. (34) becomes s ¼ E1rV m 6CmEmEfV 2 f  Em E1 AQ1DT c þ H1DT F  3 . ð44Þ Similarly, (43) is applied for the known values of DTc. In both equations, A is again varied from 0 to 0.66. The re￾sults are presented in Table 4. As it can be seen, the values of s predicted by (43) and (44) for the same DTc overlap for all CMCs under con￾sideration. This is better visualised if the values pre￾dicted by (44) for increasing DT are superimposed on the graphs of Fig. 3(a)–(d). An example (for Nic￾alonTM/CAS) is shown in Fig. 4. Similar graphs can be obtained for the other three CMCs. Table 4 and Fig. 4 reveal that the proposed modi- fied Coulomb-type model for the description of possi￾ble changes in s during thermal shock not only predicts correctly the trends of the changes of s with DT but can also provide quantitative estimates for the values of s = f(DT) that correlate well with exper￾imental data. It has to be noted that more sophisticated approaches for modelling s are available in the literature, e.g. [30]. Use of such models will share similar features with the present one (e.g. inclusion of rTS 2 ) and may describe the underlying phenomena more accurately, especially if phenomena such as the Poisson effect are shown to be significant during thermal shock. 4. Determination of the stress reduction factor
A The previous analysis was performed by assuming that the stress reduction factor, A, varied from 0 to 0.66 for reasons mentioned in Paragraph 2.6. As we have now established that the two equations describing change in s with DT [i.e. (43) and (44)] produce overlap￾ping results, estimates of A can be obtained. This is achieved by plotting s (obtained from both equations) against A (=0–0.66) for all DTs. The point where the Table 4 Correlation of values of s given by (43) and (44) for the DTc of each CMC Nicalon/CAS (DTc = 400 C) Nicalon/Duran (DTc
585 C) Nicalon/Pyrex (DTc > 600 C) Nicalon/LAS (DTc = 800 C) s (MPa) from (44) 0.4–11.3 0–5.2 0–3.1 0–1.4 from (43) 5.6–13.5 1.44–14 1.64–10 0.5–2.4 0 2 4 6 8 10 12 14 16 0 100 200 300 400 Quenching Temperature Difference ( C) Interfacial Shear Stress (MPa) UD Nicalon/CAS ˚ Fig. 4. Change in s for increasing DT from (43) (dotted line) and from (44) (continuous line). C. Kastritseas et al. / Composites Science and Technology 65 (2005) 1880–1890 1887

C. Kastritseas et al Composites Science and Technology 65(2005)1880-1890 0.50.60.7 Stress Reduction Factor ' A' Stress Reduction Factor 'A 100c 9苏 0.00.1 0.60.7 0.50.60.7 (c) Stress Reduction Factor ' A (d) Fig.5. Determination of the stress reduction factor, A, for water-quenching of (a) UD NicalonM/CAS. (b) UD NicalonM/Duran, (c)UD Nicalon/Pyrex, (d) UD Nicalon /LAS Table 5 Values of the stress reduction factor, 4, for four different UD CMCs determined by the method outlined in Section 4 Material Nicalon/CAS Nicalon/Duran Nicalon/LA 0°C) △T≈585° T>600°C 0.53 Table 6 Estimates of ATc for four different UD CMCs determined by the methods outlined in Section 5 Nicalon/Pyrex (△Tc=400° (△Tc≈585°) (△Tc>600°C) (△Tc=800°C AT.(° A=0.55 A=0.66 >300 >500 >550 >650 two branches corresponding to the experimentally deter- 5. The present approach as a predictive model for ATe mined ATc meet denotes the value of A for the particular material and quenching medium. The resulting graphs The above analysis can also be used to obtain for the composites under consideration are presented approximate estimates for the AT of UD CMCs. More in Fig. 5(aHd analytically, we can substitute the value of t from(43)in The values of A derived for each composite are pre-(34), then solve for Tmax and subsequently determine sented in Table 5 ATc(=Tmax - To). Two estimates can then be obtained Although these values can only be considered simple the first assuming an average value of the stress reduc estimates as they are subject to inaccuracies regarding tion factor Aav =0.55, which seems to be a valid material properties and limitations of the incorporated approximation at least for this class of CMCs, and the models, it is interesting to note that they all fall within second assuming the maximum possible value of A,i.e the range A=0.5-0.6. The same limits to the value of A=0.66. Both estimates can also be determined graph A can be set by combining the work of Blissett et al ically using the plots of Fig. 5(a)-d) [5] and Boccaccini [8] The values of ATc obtained are presented in Table 6

two branches corresponding to the experimentally deter￾mined DTc meet denotes the value of A for the particular material and quenching medium. The resulting graphs for the composites under consideration are presented in Fig. 5(a)–(d). The values of A derived for each composite are pre￾sented in Table 5. Although these values can only be considered simple estimates as they are subject to inaccuracies regarding material properties and limitations of the incorporated models, it is interesting to note that they all fall within the range A = 0.5–0.6. The same limits to the value of A can be set by combining the work of Blissett et al. [5] and Boccaccini [8]. 5. The present approach as a predictive model for DTc The above analysis can also be used to obtain approximate estimates for the DTc of UD CMCs. More analytically, we can substitute the value of s from (43) in (34), then solve for Tmax and subsequently determine DTc (=Tmax T0). Two estimates can then be obtained; the first assuming an average value of the stress reduc￾tion factor Aav = 0.55, which seems to be a valid approximation at least for this class of CMCs, and the second assuming the maximum possible value of A, i.e. A = 0.66. Both estimates can also be determined graph￾ically using the plots of Fig. 5(a)–(d). The values of DTc obtained are presented in Table 6. 0 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Stress Reduction Factor 'A' Interfacial Shear Stress (MPa) UD Nicalon/Pyrex 100 C 200 C 300 C 400 C 500 C 600 C 400 C 300 C 200 C Solid symbols correspond to (43) Open symbols correspond to (44) 0 2 4 6 8 10 12 14 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Stress Reduction Factor 'A' Interfacial Shear Stress (MPa) UD Nicalon/CAS 400 C 100 C 200 C 300 C 100 C 200 C 300 C Solid symbols correspond to (43) Open symbols correspond to (44) 0 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Stress Reduction Factor 'A' Interfacial Shear Stress (MPa) UD Nicalon/DURAN 100 C 200 C 300 C 400 C 500 C 600 C 400 C 300 C 200 C Solid symbols correspond to (43) Open symbols correspond to (44) 0.0 0.5 1.0 1.5 2.0 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Stress Reduction Factor 'A' Interfacial Shear Stress (MPa) UD Nicalon/LAS 200 C 300 C 400 C 500 C 600 C 700 C 800 C 700 C 600 C 500 C 400 C 300 C Solid symbols correspond to (43) Open symbols correspond to (44) (a) (b) (c) (d) Fig. 5. Determination of the stress reduction factor, A, for water-quenching of (a) UD NicalonTM/CAS, (b) UD NicalonTM/Duran, (c) UD NicalonTM/Pyrex, (d) UD NicalonTM/LAS. Table 5 Values of the stress reduction factor, A, for four different UD CMCs determined by the method outlined in Section 4 Material Nicalon/CAS (DTc = 400 C) Nicalon/Duran (DTc
585 C) Nicalon/Pyrex (DTc > 600 C) Nicalon/LAS (DTc = 800 C) A 0.53 0.57 600 C) Nicalon/LAS (DTc = 800 C) DTc (C) A = 0.55 370 607 671 780 A = 0.66 >300 >500 >550 >650 1888 C. Kastritseas et al. / Composites Science and Technology 65 (2005) 1880–1890

C. Kastritseas et al Composites Science and Technology 65(2005)1880-1890 l889 The estimates for A=0.55 give the best agreement References with experimental data as the error in the calculated val ues of AT ranges from 2.5% to 12%. However, the [1 Parthasarathy TA, Kerans RJ. Failure of ceramic composites. In: approximation of A=0.55 cannot be guaranteed to be Milne I, Ritchie RO, Karihaloo B, editors. Comprehensive reasonable in the case of other CMCs with appreciably 2 Munz D, Fett T. Ceramics: Mechanical properties, failure different thermal and mechanical properties. For behaviour, materials selection. Springer: 1999 A=0.66, only fair agreement with experimental data [3] Wang H, Singh RN. Thermal shock behaviour of ceramics and is achieved as the error ranges between 8% and 25% ceramic composites. Int Mat Rev 1994: 39: 228-44 However, the value of AT obtained is the most conser- 4 Kagawa Y, Kurosawa N, Kishi T. Thermal shock resistance of vative possible and can be utilised as a guideline indicat ing the worst case scenario, i.e. the resistance of the [5]Blissett MJ, Smith PA, Yeomans JA. Thermal shock behaviour of material to thermal shock under the most severe condi nidirectional silicon carbide fibre reinforced calcium aluminosil- tions possible nvestigation of cyclic thermal shock behaviour of fibre reinforced glass matrix composites using technique. Mat Sci Tech 1997: 13: 852-8 6. Concluding remarks [7 Boccaccini AR, Janczak-Rusch J, Pearce DH, Kern H. As nent of damage induced by thermal shock in SiC-fibre-reinfc A theoretical investigation of the conditions for the borosilicate glass onset of multiple matrix cracking due to thermal(cold) 05-12 shock in UD CMCs was conducted in this study. A [8] Boccaccini AR Predicting the thermal shock resistance of fiber reinforced brittle matrix composites. Scripta Mater model was developed to predict the critical quenching 9988:1211-7. temperature differential, ATe, that required as inputs 9] Kingery WD. Factors affecting thermal stress resistance of he processing temperature of the composite, the tem- ceramic materials. J Am Ceram Soc 1955: 38: 3-15. perature of the quenching medium, and material proper- [10 Wang HS, Chou T-W. Transient therm analysis of es measured pom temperature. The approach [u Wang H, Chou t.W. Transient thermal behaviour of a thermally and elastically orthotropic medium. AIAA J 1986: 24: 664-72 the shock, which is an improvement compared to avail- [12] Hashin Z. Analysis of properties of fiber composites with able models. Numerical predictions for water-quenche anisotropic constituents. J Appl Mech 1979: 46: 543-50 CAS, PyrexM-, DuranTM, and LAS-matrix compos [13] Kuntz M, Meier B, Grathwohl G. Residual stresses in fiber ites reinforced with Nicalon M fibres suggested that the reinforced ics due to thermal expansion mismatch. J Am Ceram Soc1993;76:2607-12 value of the effective interfacial shear stress may be sig [14 Powell KL, Smith PA, Yeomans JA. Aspects of residual thermal nificantly reduced during the shock compared to the stresses in continuous fibre-reinforced ceramic matrix composites. standard (room temperature)value due to the presence Compos Sci Technol 1993: 47: 359-67 of a thermal shock sress component that acts perpendic [15 Budiansky B, Hutchinson JW, Evans AG. Matrix fracture in ular to the fibre/matrix interface. The phenomenon was [16] Aveston JA. Cooper GA, Kelly A. Single and multiple fracture modelled successfully using a modified Coulomb-type In: The properties of fibre composites. Proceedings of Nationa friction law and satisfactory correlation with experimen- Physical Laboratory. London: IPC Science and Technology tal data was achieved. Subsequently, the combined mod- Ltd:1971.p.15-25 els were used to provide predictions for the stress [17 Marshall DB, Cox BN, Evans AG. The mechanics of m reduction factor that characterises the heat transfer con- cracking in brittle matrix fiber composites. Acta M 985;33:2013-21 dition during the shock. An average value of 4=0.55[18] Pagano NJ, Kim RY Progressive microcracking in unidirectional was determined to be a satisfactory approximation for brittle matrix composites. Mech Compos Mater Struct this type of CMCs that have similar mechanical and 1994:1:3-29 thermal properties. A methodology based on the as [19] Beyerle DS, Spearing SM, Zok FW, Evans AG. Damage and sumed average and the maximum value of the stress failure in unidirectional ceramic matrix composites. J Am Ceram Soc1992;75:2719-25 reduction factor provided satisfactory and conservative [20 Bleay SM, Scott VD. Microstructural property relationship in stomates of ATe respectively Pyrex glass composites reinforced with Nicalon fibres. J Mater So 21]Lewis MH Interfaces in ceramic matrix composites. In: Kelly A, Zweben C, editors. Comprehensive composite materials, vo Acknowledgement 4. Elsevier: 2000. [22]Kerans R, Parthasarathy TA. Theoretical analysis of the fiber CK acknowledges financial support from the Engi-(23) Jero PD, Kerans RI, Parthasarathy TA. Efect of interfacial neering and Physical Sciences Research Council (UK), oughness on the frictional stress measured using pushout tests. J and the 'Alexander s. Onassis' Foundation Am Ceram Soc 1991: 74: 2793-801

The estimates for A = 0.55 give the best agreement with experimental data as the error in the calculated val￾ues of DTc ranges from 2.5% to 12%. However, the approximation of A = 0.55 cannot be guaranteed to be reasonable in the case of other CMCs with appreciably different thermal and mechanical properties. For A = 0.66, only fair agreement with experimental data is achieved as the error ranges between 8% and 25%. However, the value of DTc obtained is the most conser￾vative possible and can be utilised as a guideline indicat￾ing the worst case scenario, i.e. the resistance of the material to thermal shock under the most severe condi￾tions possible. 6. Concluding remarks A theoretical investigation of the conditions for the onset of multiple matrix cracking due to thermal (cold) shock in UD CMCs was conducted in this study. A model was developed to predict the critical quenching temperature differential, DTc, that required as inputs the processing temperature of the composite, the tem￾perature of the quenching medium, and material proper￾ties measured at room temperature. The approach considers the anisotropic stress field generated during the shock, which is an improvement compared to avail￾able models. Numerical predictions for water-quenched CAS–, PyrexTM–, DuranTM–, and LAS–matrix compos￾ites reinforced with NicalonTM fibres suggested that the value of the effective interfacial shear stress may be sig￾nificantly reduced during the shock compared to the standard (room temperature) value due to the presence of a thermal shock sress component that acts perpendic￾ular to the fibre/matrix interface. The phenomenon was modelled successfully using a modified Coulomb-type friction law and satisfactory correlation with experimen￾tal data was achieved. Subsequently, the combined mod￾els were used to provide predictions for the stress reduction factor that characterises the heat transfer con￾dition during the shock. An average value of A = 0.55 was determined to be a satisfactory approximation for this type of CMCs that have similar mechanical and thermal properties. A methodology based on the as￾sumed average and the maximum value of the stress reduction factor provided satisfactory and conservative estimates of DTc respectively. Acknowledgement CK acknowledges financial support from the Engi￾neering and Physical Sciences Research Council (UK), and the
Alexander S. Onassis Foundation. References [1] Parthasarathy TA, Kerans RJ. Failure of ceramic composites. In: Milne I, Ritchie RO, Karihaloo B, editors. Comprehensive structural integrity, vol. 2. Elsevier; 2003. [2] Munz D, Fett T. Ceramics: Mechanical properties, failure behaviour, materials selection. Springer; 1999. [3] Wang H, Singh RN. Thermal shock behaviour of ceramics and ceramic composites. Int Mat Rev 1994;39:228–44. [4] Kagawa Y, Kurosawa N, Kishi T. Thermal shock resistance of SiC fibre-reinforced borosilicate glass and lithium aluminosilicate matrix composites. J Mater Sci 1993;28:735–41. [5] Blissett MJ, Smith PA, Yeomans JA. Thermal shock behaviour of unidirectional silicon carbide fibre reinforced calcium aluminosil￾icate. J Mater Sci 1997;32:317–25. [6] Boccaccini AR, Pearce DH, Janczak J, Beier W, Ponton CB. Investigation of cyclic thermal shock behaviour of fibre reinforced glass matrix composites using non-destructive forced resonance technique. Mat Sci Tech 1997;13:852–8. [7] Boccaccini AR, Janczak-Rusch J, Pearce DH, Kern H. Assess￾ment of damage induced by thermal shock in SiC–fibre-reinforced borosilicate glass composites. Compos Sci Technol 1999;59:105–12. [8] Boccaccini AR. Predicting the thermal shock resistance of fiber reinforced brittle matrix composites. Scripta Mater 1998;8:1211–7. [9] Kingery WD. Factors affecting thermal stress resistance of ceramic materials. J Am Ceram Soc 1955;38:3–15. [10] Wang HS, Chou T-W. Transient thermal stress analysis of a rectangular orthotropic slab. J Compos Mater 1985;19:424–41. [11] Wang H, Chou T-W. Transient thermal behaviour of a thermally and elastically orthotropic medium. AIAA J 1986;24:664–72. [12] Hashin Z. Analysis of properties of fiber composites with anisotropic constituents. J Appl Mech 1979;46:543–50. [13] Kuntz M, Meier B, Grathwohl G. Residual stresses in fiber￾reinforced ceramics due to thermal expansion mismatch. J Am Ceram Soc 1993;76:2607–12. [14] Powell KL, Smith PA, Yeomans JA. Aspects of residual thermal stresses in continuous-fibre-reinforced ceramic matrix composites. Compos Sci Technol 1993;47:359–67. [15] Budiansky B, Hutchinson JW, Evans AG. Matrix fracture in fiber-reinforced ceramics. J Mech Phys Solids 1986;33:167–89. [16] Aveston JA, Cooper GA, Kelly A. Single and multiple fracture. In: The properties of fibre composites. Proceedings of National Physical Laboratory. London: IPC Science and Technology Ltd.; 1971. p. 15–25. [17] Marshall DB, Cox BN, Evans AG. The mechanics of matrix cracking in brittle matrix fiber composites. Acta Metall 1985;33:2013–21. [18] Pagano NJ, Kim RY. Progressive microcracking in unidirectional brittle matrix composites. Mech Compos Mater Struct 1994;1:3–29. [19] Beyerle DS, Spearing SM, Zok FW, Evans AG. Damage and failure in unidirectional ceramic matrix composites. J Am Ceram Soc 1992;75:2719–25. [20] Bleay SM, Scott VD. Microstructural property relationship in Pyrex glass composites reinforced with Nicalon fibres. J Mater Sci 1991;26:2229–39. [21] Lewis MH. Interfaces in ceramic matrix composites. In: Kelly A, Zweben C, editors. Comprehensive composite materials, vol. 4. Elsevier; 2000. [22] Kerans RJ, Parthasarathy TA. Theoretical analysis of the fiber pullout and pushout tests. J Am Ceram Soc 1991;74:1585–96. [23] Jero PD, Kerans RJ, Parthasarathy TA. Effect of interfacial roughness on the frictional stress measured using pushout tests. J Am Ceram Soc 1991;74:2793–801. C. Kastritseas et al. / Composites Science and Technology 65 (2005) 1880–1890 1889