《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_sic-g-8

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