《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_A micromechanical bridging law model for CFCCs

Availableonlineatwww.sciencedirect.co ScienceDirect Acta materialia ELSEVIER Acta Materialia 55(2007)83-92 www.actamat-journals.com A micromechanical bridging law model for CFCCs Konstantinos G. Dassios * Vassilis Kostopoulos marc Steen c oundation of Research and gy Hellas, Institute of Chemi nical Engineering and High Temperature Cher mical processes Stadiou Street. Platani. Patras GR 26504, Achaia. Greece Department of Mechanical Engineering and Aeronautics, Unicersity of Patras, Rio GR 26.500, Greece Istitute for Energy, Joint Research Centre, European Commission, Westerduinweg 4, 1755 LE Petten, The Netherlands Received 2 June 2006: received in revised form 17 July 2006: accepted 17 July 2006 Available online 7 November 2006 Abstract In the present work, a methodology is presented for the assessment of bridging laws for continuous fibre-reinforced ceramic matrix composites based on material properties as well as micromechanics of fibre deformation and failure. A load-displacement model is ini- ally formulated that utilizes weakest-link statistical concepts to analyse and relate the individual contributions of matrix, intact/bridg ing and failed/pull-out fibres during the composite fracture process. The total and individual contributions to the bridging law and crack growth resistance of the material are determined by identifying the non-elastic part of displacement as crack opening. The model is val- dated against the experimentally recorded load-displacement behaviour of a notched Sic-fibre-reinforced glass-ceramic matrix compos- ite tested under monotonic tension. The output parameters of the converged regression procedure remain within a small scattering range from the corresponding mean values that compare favourably with known material properties. a parametric analysis of the effect of fibre volume fraction, Weibull modulus of fibres and interfacial shear stress in overall composite performance is presented in view of the ability of the model to serve as an a priori fracture prediction tool. o 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Ceramic matrix composites; Fiber reinforced composites; Fracture 1. Introduction to-diameter ratios has become established as the strongest and toughest of all types of ceramic composites. CFCCs The brittle nature of ceramics has hindered the use of are used today as structural components in commercial hese materials in high-temperature applications with and military aircraft and high-speed vehicles(e.g. braking creased mechanical demands for many decades. The systems, stabilizers), in aerospace applications (e.g development, in the 1970s, of ceramic matrix composite mechanical parts and thermal barriers for space shuttles, (CMC) materials, consisting of a bulk ceramic phase, the rocket nozzles), and in incinerators, heat exchangers and matrix, reinforced by an embedded phase in the form of many other high-temperature applications. Among the particulates, whiskers or fibres, unleashed the potential of admirable characteristics of CFCCs are their increased these materials to equip a structure with a unique combina- crack growth resistance and improved damage tolerance tion of properties to withstand simultaneous thermal and compared with monolithic ceramics, their notch insensitive A specially designated class of CMCs that evolved Ity and their ability to effectively redistribute stresses mechanical loading and pushed up the limits. during the 1990s, continuous fibre-reinforced ceramic erties stem from the energy dissipation mechanisms acti- matrix composites(CFCCs), with fibres of high length- vated during CFCC fracture that temper the fatal work of crack propagation at the crack front (crack process nor. Tel: +302610965277: fax: +302610965228. zone)by consuming a part of the externally applied energy. lassos(@iceht forthgr(K.G. Dassios ). The two major energy dissipation mechanisms in CFCO 1359-6454/$30.00 O 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:l0.1016 actant200607.039

A micromechanical bridging law model for CFCCs Konstantinos G. Dassios a,*, Vassilis Kostopoulos b , Marc Steen c a Foundation of Research and Technology Hellas, Institute of Chemical Engineering and High Temperature Chemical Processes, Stadiou Street, Platani, Patras GR 26504, Achaia, Greece b Department of Mechanical Engineering and Aeronautics, University of Patras, Rio GR 26500, Greece c Institute for Energy, Joint Research Centre, European Commission, Westerduinweg 4, 1755 LE Petten, The Netherlands Received 2 June 2006; received in revised form 17 July 2006; accepted 17 July 2006 Available online 7 November 2006 Abstract In the present work, a methodology is presented for the assessment of bridging laws for continuous fibre-reinforced ceramic matrix composites based on material properties as well as micromechanics of fibre deformation and failure. A load–displacement model is ini￾tially formulated that utilizes weakest-link statistical concepts to analyse and relate the individual contributions of matrix, intact/bridg￾ing and failed/pull-out fibres during the composite fracture process. The total and individual contributions to the bridging law and crack growth resistance of the material are determined by identifying the non-elastic part of displacement as crack opening. The model is val￾idated against the experimentally recorded load–displacement behaviour of a notched SiC-fibre-reinforced glass–ceramic matrix compos￾ite tested under monotonic tension. The output parameters of the converged regression procedure remain within a small scattering range from the corresponding mean values that compare favourably with known material properties. A parametric analysis of the effect of fibre volume fraction, Weibull modulus of fibres and interfacial shear stress in overall composite performance is presented in view of the ability of the model to serve as an a priori fracture prediction tool. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Ceramic matrix composites; Fiber reinforced composites; Fracture 1. Introduction The brittle nature of ceramics has hindered the use of these materials in high-temperature applications with increased mechanical demands for many decades. The development, in the 1970s, of ceramic matrix composite (CMC) materials, consisting of a bulk ceramic phase, the matrix, reinforced by an embedded phase in the form of particulates, whiskers or fibres, unleashed the potential of these materials to equip a structure with a unique combina￾tion of properties to withstand simultaneous thermal and mechanical loading and pushed up the limits. A specially designated class of CMCs that evolved during the 1990s, continuous fibre-reinforced ceramic matrix composites (CFCCs), with fibres of high length￾to-diameter ratios, has become established as the strongest and toughest of all types of ceramic composites. CFCCs are used today as structural components in commercial and military aircraft and high-speed vehicles (e.g. braking systems, stabilizers), in aerospace applications (e.g. mechanical parts and thermal barriers for space shuttles, rocket nozzles), and in incinerators, heat exchangers and many other high-temperature applications. Among the admirable characteristics of CFCCs are their increased crack growth resistance and improved damage tolerance compared with monolithic ceramics, their notch insensitiv￾ity and their ability to effectively redistribute stresses around notches, voids and cracks. These prominent prop￾erties stem from the energy dissipation mechanisms acti￾vated during CFCC fracture that temper the fatal work of crack propagation at the crack front (crack process zone) by consuming a part of the externally applied energy. The two major energy dissipation mechanisms in CFCCs 1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.07.039 * Corresponding author. Tel.: +30 2610 965277; fax: +30 2610 965228. E-mail address: kdassios@iceht.forth.gr (K.G. Dassios). www.actamat-journals.com Acta Materialia 55 (2007) 83–92

K.G. Dassios et al. Acta Materialia 55(2007 )83-92 are, in order of importance, matrix-crack bridging by intact the fracture mechanism associated with each contribution fibres and pull-out of failed fibres. Both mechanisms are and rational conditions to simulate the physics of their cor- direct results of the interactions of a third phase in the com- relation during composite fracture, a model fracture law is posite, the interface region(interphase) between the matrix initially developed that assesses the load-displacement and the fibres, that, despite having a thickness of only a few behaviour of the composite. The bridging law is then nanometres, occupies a vast surface area in the composite obtained by a simple subtraction of the matrix contribution and commands its macroscopical fracture behaviour. and the conversion of the load-displacement law to the Externally applied stress is transmitted from the matrix stress-crack opening displacement form. The efficiency of to the bridging fibres through the interface zone while the model relies on the simplicity of concepts used in its pull-out is a result of frictional sliding of the outer surface derivation, the minimal number of assumptions it invokes of failed fibres across the debonded interface and its utilization of measurable parameters with explicit The fracture behaviour of most contemporary CFCCs physical significance, such as the properties of the matrix, exhibits large-scale bridging(LSB), a phenomenon ere fib res and the interphase. The validity of the derived model the dimensions of the bridging zone formed by fibres is tested against the experimentally recorded fracture bridging the separating crack flanks of a macro-crack dur- behaviour of a Sic- fibre-reinforced glass-ceramic matrix to a characteristic specimen dimension (e.g. thickness). ability of the model to Pla quasi-static tensile loading. The ing material fracture are large enough to be comparable composite tested under a priori, the fracture b Under these conditions, conventional fracture mechanics iour of a CFCC is discussed against the results of a para- parameters such as the crack growth resistance of the metric analysis of the role of material parameters. The material are dependent on specimen dimension and con- model can securely predict the load-displacement behav iguration and cannot serve as material-intrinsic fracture iour of any kind of double-edge notch(DEN) configura descriptors [ 21. tion, while it can be extended for the prediction of the An alternative approach proposed for the characteriza- fracture behaviour of compact tension(CT) and single tion of CFCC performance is the bridging law, a phenom- edge-notch beam specime enological correlation between the stresses acting on fibres within the bridging zone(bridging stresses)and the crack 2 Model formulation pening displacement( COD)[3] Owing to its localized range of effect around the bridging zone, the law is The derivation of the model relies on the principle that believed capable of serving as a constitutive, material- the total load-displacement behaviour of a composite can intrinsic fracture descriptor that is independent of the be expressed as the sum of three terms corresponding to overall composite dimensions or geometry. The determi- the individual contributions of the matrix, intact/bridging nation of the bridging law for various composite systems fibres and failed/pull-out fibres. For the formulation of has been the aim of several research efforts in the last dec- the model, equal strain conditions are assumed to apply ade. Different law forms have been indirectly evaluated among the composite constituents, as in common displace sing either micromechanical modelling approaches [4]. ment-controlled testing. The development of the model energy-related approaches such as the J-integral [5, 6]. relies on the good understanding of the physics and ompliance-based techniques [7, 8]. weight function and mechanics of successive stages during CFCC fracture and stress intensity approaches [9] or weakest-link statistical the role of each of the composite's constituents(matrix, considerations [10-12]. While the calculated bridging laws fibre and interphase)in these stages. The fracture proce- have, in general, succeeded in assessing the fracture dure of a CFCC has been analysed in a previous study behaviour of individual CFCCs used in the foremen- [13] and is summarized in the following tioned studies the current lack of confidence in CFCc applicability -especially with respect to the growing con- 2. 1. Mechanical behaviour of a bridged crack cern about high production costs- clearly demonstrates the necessity to comprehend the fundamentals of interac During the initial loading stages the composite deforms tions between the composite phases using simple, straight- in a linear elastic manner until the first matrix micro-crack forward methods. The biggest challenge in the composites appears to trigger the formation of the bridging zone. Fur- world today is the linking of the physics and mechanics of ther loading leads to micro-cracks growing to fibre-bridged micro-and nanoscale interactions to the overall compos- macro-cracks, while concurrent phenomena such as inter- ite performance face debonding and crack deflection at the interface take In this study, a straightforward methodology is pre- place. Depending on specimen configuration, the bridging ted for the determination of bridging laws for composite zone may either span the whole width of the material, as materials exhibiting LSB and pull-out. The principle of the for example, in a dEn sample [13], or attain saturated methodology relies on the deconvolution of the fracture shape and dimensions and propagate in a self-similar man- behaviour of the composite into the individual contribu- ner through the material, as, for example, in a CT sample tions of the matrix, intact and failed fibres Using simple [8]. Intact fibres bridging the macro-crack may fail due to Weibull statistical concepts to mathematically describe surface flaws [10] at any position along their crack-exposed

are, in order of importance, matrix-crack bridging by intact fibres and pull-out of failed fibres. Both mechanisms are direct results of the interactions of a third phase in the com￾posite, the interface region (interphase) between the matrix and the fibres, that, despite having a thickness of only a few nanometres, occupies a vast surface area in the composite and commands its macroscopical fracture behaviour. Externally applied stress is transmitted from the matrix to the bridging fibres through the interface zone while pull-out is a result of frictional sliding of the outer surface of failed fibres across the debonded interface. The fracture behaviour of most contemporary CFCCs exhibits large-scale bridging (LSB), a phenomenon where the dimensions of the bridging zone formed by fibres bridging the separating crack flanks of a macro-crack dur￾ing material fracture are large enough to be comparable to a characteristic specimen dimension (e.g. thickness). Under these conditions, conventional fracture mechanics parameters such as the crack growth resistance of the material are dependent on specimen dimension and con- figuration and cannot serve as material-intrinsic fracture descriptors [1,2]. An alternative approach proposed for the characteriza￾tion of CFCC performance is the bridging law, a phenom￾enological correlation between the stresses acting on fibres within the bridging zone (bridging stresses) and the crack opening displacement (COD) [3]. Owing to its localized range of effect around the bridging zone, the law is believed capable of serving as a constitutive, material￾intrinsic fracture descriptor that is independent of the overall composite dimensions or geometry. The determi￾nation of the bridging law for various composite systems has been the aim of several research efforts in the last dec￾ade. Different law forms have been indirectly evaluated using either micromechanical modelling approaches [4], energy-related approaches such as the J-integral [5,6], compliance-based techniques [7,8], weight function and stress intensity approaches [9] or weakest-link statistical considerations [10–12]. While the calculated bridging laws have, in general, succeeded in assessing the fracture behaviour of individual CFCCs used in the aforemen￾tioned studies, the current lack of confidence in CFCC applicability – especially with respect to the growing con￾cern about high production costs – clearly demonstrates the necessity to comprehend the fundamentals of interac￾tions between the composite phases using simple, straight￾forward methods. The biggest challenge in the composites world today is the linking of the physics and mechanics of micro- and nanoscale interactions to the overall compos￾ite performance. In this study, a straightforward methodology is pre￾sented for the determination of bridging laws for composite materials exhibiting LSB and pull-out. The principle of the methodology relies on the deconvolution of the fracture behaviour of the composite into the individual contribu￾tions of the matrix, intact and failed fibres. Using simple Weibull statistical concepts to mathematically describe the fracture mechanism associated with each contribution and rational conditions to simulate the physics of their cor￾relation during composite fracture, a model fracture law is initially developed that assesses the load–displacement behaviour of the composite. The bridging law is then obtained by a simple subtraction of the matrix contribution and the conversion of the load–displacement law to the stress-crack opening displacement form. The efficiency of the model relies on the simplicity of concepts used in its derivation, the minimal number of assumptions it invokes and its utilization of measurable parameters with explicit physical significance, such as the properties of the matrix, fibres and the interphase. The validity of the derived model is tested against the experimentally recorded fracture behaviour of a SiC-fibre-reinforced glass–ceramic matrix composite tested under quasi-static tensile loading. The ability of the model to predict, a priori, the fracture behav￾iour of a CFCC is discussed against the results of a para￾metric analysis of the role of material parameters. The model can securely predict the load–displacement behav￾iour of any kind of double-edge notch (DEN) configura￾tion, while it can be extended for the prediction of the fracture behaviour of compact tension (CT) and single￾edge-notch beam specimens. 2. Model formulation The derivation of the model relies on the principle that the total load–displacement behaviour of a composite can be expressed as the sum of three terms corresponding to the individual contributions of the matrix, intact/bridging fibres and failed/pull-out fibres. For the formulation of the model, equal strain conditions are assumed to apply among the composite constituents, as in common displace￾ment-controlled testing. The development of the model relies on the good understanding of the physics and mechanics of successive stages during CFCC fracture and the role of each of the composite’s constituents (matrix, fibre and interphase) in these stages. The fracture proce￾dure of a CFCC has been analysed in a previous study [13] and is summarized in the following. 2.1. Mechanical behaviour of a bridged crack During the initial loading stages the composite deforms in a linear elastic manner until the first matrix micro-crack appears to trigger the formation of the bridging zone. Fur￾ther loading leads to micro-cracks growing to fibre-bridged macro-cracks, while concurrent phenomena such as inter￾face debonding and crack deflection at the interface take place. Depending on specimen configuration, the bridging zone may either span the whole width of the material, as, for example, in a DEN sample [13], or attain saturated shape and dimensions and propagate in a self-similar man￾ner through the material, as, for example, in a CT sample [8]. Intact fibres bridging the macro-crack may fail due to surface flaws [10] at any position along their crack-exposed 84 K.G. Dassios et al. / Acta Materialia 55 (2007) 83–92

K.G. Dassios et al. I Acta Materialia 55(2007)83-92 across the debonding length. Due to the regularly weak Pm(d) Acomp( 1-V Em, d. exp-dn lengths or along their length inside the matrix, mainly nature of the interface. the debond length on the onset of ment. Consequently, the probability of a fibre failing along The two-parameter Weibull model is commonly estab. the debond length is much larger than that of a fibre failing lished as the most suitable fibre failure distribution func- within the crack flanks [14]. The same concept can be used to rationalize the fact that strong interfaces are generally tion as it couples a first-order growth term with an undesirable in CFCCS, as the resultant strong bonding exponential decay term that can realistically simulate the between fibres and the matrix promotes fibre fracture with- sequential phenomena of elastic deformation and incre- out debonding and supports material brittleness instead of mental fibre failure under the global load-sharing principle decreasing it. Besides its statistical origin, this"preferable respectively [10, 15]. Analogously to Eq. (3), the intact fibre failure location"phenomenon is additionally pro- fibre's contribution to the load-displacement behaviour moted by sliding of the stretching intact fibre's surface of the composite can be expressed as failure by tampering with the fatal nature of its surface Pr(d)Acomp rEr dexp _/d) across the debonded interface, an action that triggers fibre flaws. Fibres failing along the debond length contribute to the pull-out mechanism where shear forces develop where Er is the youngs modulus of the fibres and mf and between the matrix and fibres due to frictional sliding of dr o their Weibull shape and location parameters the failed fibre's surface across the interface. On the onset of total fibre failure, the composite's behaviour is con 2. 4. Pull-out contribution trolled purely by the pull-out mechanism that endures until all fibres have pulled out completely, disengaging from the Contrary to the intact/bridging fibres contribution to the load-displacement behaviour of a composite, which is associated with axial forces, the origins of the pull-out 2.2 Matrix contribution mechanism are shear forces developing along the debonded interface. The shear force acting on a single fibre on the A mechanically loaded monolithic ceramic will fail in onset of pull-out can be converted to axial force TAp)by brittle manner, exhibiting minimal deviation from the lin- taking a simple force balance around the fibre ear elastic behaviour up to catastrophic failure. On the Tr(p)=(2rr Ip).T other hand, the ceramic matrix of a CFCC will fail gradu where Ip is the pull-out length, t is the interfacial shear ally and in a stepwise manner due to normally existing stress and rr is the fibre radius. The pull-out length is a sta open and closed porosity, the interactions with the fibres and the interface. Crack deflection along the interface. tistical parameter that may vary among fibres in the com- interfacial debonding and stepwise multi-failure of matrix posite. Although the exact pull-out length distribution of a composite may be measured post mortem directly on parts between fibres and fibre bundles are damage mecha- the composite, for the scope of this analysis, and without nisms that contribute to the ductility of the matrix fracture loss of generality, the mean pull-out length Ip can be used Weibull distribution function where the load-displacement in Eq. (5) contribution of the matrix is expressed b The number of failed fibres at any instance during com- posite fracture, Np, is reciprocal to the number of surviving Pm(d) d ( (bridging) fibres in the composite. Hence where Pm(d is the load carried by the matrix as a function of displacement d, Cm o is the elastic compliance of the m where No is the total number of fibres in the composite trix and mm and dm.o are the respective Weibull shape and Ps(d) is the survival probability of a fibre at a displacement location parameters of the matrix. The compliance Cm o is d and is given by related to the matrix properties through L Acomp(1-Vr)Em However, among the total number of failed fibres, only where L is the reference length over which displacement is those that are oriented non-perpendicularly to the loading measured, Acomp is the cross-sectional area of the compos- direction can contribute to pull-out. Denoting by No efr the ite, Vr is the fibre volume fraction and Em is the Youngs portion of the total number of fibres in the composite that modulus of the matrix. By substituting Eq (1)in Eq (2) are pull-out-effective, and by combination of Eqs.(6)and the following expression is obtained (7), the effective number of pull-out fibres is obtained as

lengths or along their length inside the matrix, mainly across the debonding length. Due to the regularly weak nature of the interface, the debond length on the onset of fibre failure is usually much larger than the fibre’s crack￾exposed length equal to the instant crack opening displace￾ment. Consequently, the probability of a fibre failing along the debond length is much larger than that of a fibre failing within the crack flanks [14]. The same concept can be used to rationalize the fact that strong interfaces are generally undesirable in CFCCs, as the resultant strong bonding between fibres and the matrix promotes fibre fracture with￾out debonding and supports material brittleness instead of decreasing it. Besides its statistical origin, this ‘‘preferable fibre failure location’’ phenomenon is additionally pro￾moted by sliding of the stretching intact fibre’s surface across the debonded interface, an action that triggers fibre failure by tampering with the fatal nature of its surface flaws. Fibres failing along the debond length contribute to the pull-out mechanism where shear forces develop between the matrix and fibres due to frictional sliding of the failed fibre’s surface across the interface. On the onset of total fibre failure, the composite’s behaviour is con￾trolled purely by the pull-out mechanism that endures until all fibres have pulled out completely, disengaging from the matrix. 2.2. Matrix contribution A mechanically loaded monolithic ceramic will fail in a brittle manner, exhibiting minimal deviation from the lin￾ear elastic behaviour up to catastrophic failure. On the other hand, the ceramic matrix of a CFCC will fail gradu￾ally and in a stepwise manner due to normally existing open and closed porosity, the interactions with the fibres and the interface. Crack deflection along the interface, interfacial debonding and stepwise multi-failure of matrix parts between fibres and fibre bundles are damage mecha￾nisms that contribute to the ductility of the matrix fracture behaviour. This gradual failure can be approximated by a Weibull distribution function where the load–displacement contribution of the matrix is expressed by: P mðdÞ ¼ 1 Cm;0 d exp d dm;0 mm   ð1Þ where Pm(d) is the load carried by the matrix as a function of displacement d, Cm,0 is the elastic compliance of the ma￾trix and mm and dm,0 are the respective Weibull shape and location parameters of the matrix. The compliance Cm,0 is related to the matrix properties through: Cm;0 ¼ L Acompð1 V fÞEm ð2Þ where L is the reference length over which displacement is measured, Acomp is the cross-sectional area of the compos￾ite, Vf is the fibre volume fraction and Em is the Young’s modulus of the matrix. By substituting Eq. (1) in Eq. (2) the following expression is obtained: P mðdÞ ¼ Acompð1 V fÞEm L d exp d dm;0 mm   ð3Þ 2.3. Intact/bridging fibres contribution The two-parameter Weibull model is commonly estab￾lished as the most suitable fibre failure distribution func￾tion as it couples a first-order growth term with an exponential decay term that can realistically simulate the sequential phenomena of elastic deformation and incre￾mental fibre failure under the global load-sharing principle, respectively [10,15]. Analogously to Eq. (3), the intact fibre’s contribution to the load–displacement behaviour of the composite can be expressed as: PfðdÞ ¼ AcompV fEf L d exp d df;0 mf   ð4Þ where Ef is the Young’s modulus of the fibres and mf and df,0 their Weibull shape and location parameters. 2.4. Pull-out contribution Contrary to the intact/bridging fibres contribution to the load–displacement behaviour of a composite, which is associated with axial forces, the origins of the pull-out mechanism are shear forces developing along the debonded interface. The shear force acting on a single fibre on the onset of pull-out can be converted to axial force Tf(lp) by taking a simple force balance around the fibre: T fðlpÞ¼ð2prf lpÞ s ð5Þ where lp is the pull-out length, s is the interfacial shear stress and rf is the fibre radius. The pull-out length is a sta￾tistical parameter that may vary among fibres in the com￾posite. Although the exact pull-out length distribution of a composite may be measured post mortem directly on the composite, for the scope of this analysis, and without loss of generality, the mean pull-out length lp can be used in Eq. (5). The number of failed fibres at any instance during com￾posite fracture, Np, is reciprocal to the number of surviving (bridging) fibres in the composite. Hence Np ¼ N0 ½1 psðdÞ ð6Þ where N0 is the total number of fibres in the composite. ps(d) is the survival probability of a fibre at a displacement d and is given by: psðdÞ ¼ exp d df;0 mf   ð7Þ However, among the total number of failed fibres, only those that are oriented non-perpendicularly to the loading direction can contribute to pull-out. Denoting by N0,eff the portion of the total number of fibres in the composite that are pull-out-effective, and by combination of Eqs. (6) and (7), the effective number of pull-out fibres is obtained as: K.G. Dassios et al. / Acta Materialia 55 (2007) 83–92 85

K.G. Dassios et al. Acta Materialia 55(2007)83-92 N p. eff N (8) where ve a is the pull-out-effective fibre volume fraction of the composite The force exerted on the composite at the onset of total No er can be evaluated through the pull-out-effective vol- fibre failure, T can be expressed as the product of Eqs. (5) ume of the composite given by and (9 2A T (10) Table I Designation of regression parameters Parameter Status Value/reference Gauge length, L F 25 mm Cross-sectional area. A Youngs modulus of matrix, Em UV Youngs modulus of fibres, Er UV Fibre radius, rr 7pm[l6.17] Fibre volume fraction, vr 0.35[18] Weibull modulus of fibres, ne Characteristic failure displacement of matrix, dm.o Characteristic failure displacement of fibres, dro Displacement at first fibre failure, dr Interfacial shear stress, t Mean pull-out length, Ip 60mm[16] UV in exp[-(d-d)/pl F: Fixed, UV: unconditionally varying, CV: conditionally varying Table 2 Initial and boundary conditions of the regression procedure Origin Material properties IC Em= 80 GPa gineering value for glass Er= 260 GPa Approximate engi ineering value for ceramic fibre Weibull parameters IC. 3 Approximate engineering value for brittle failure IC. 4 Approximate engineering value for ceramic fibre dm0=0.05mm Experimental load-displacement behaviour IC.6 Experimental load-displacement behaviour d=0.02 mm Experimental load-displacement behaviour t=5MPa Approximate engineering value for moderate Boundary conditions Rationalization The matrix contributi Ty than the valu observation that macro. bridging zone are fully spanning the width of the notched lig efore the maximum load is attained [13] The intact fibres contribution term, Eq(4),may Eq(16)does not intrinsically relate the confined not extend to displacements corresponding to ranges of effect of intact and failed fib pure pull-out distribution BC. 3 The value of dr is defined by the displacement Eq.(16) does not intrinsically relate first fibre where the intact fibre contribution term deviates failure to appearance of pull-out from linearity

Np;eff ¼ N0;eff 1 exp d df;0 mf     ð8Þ N0,eff can be evaluated through the pull-out-effective vol￾ume of the composite given by: N0;eff ¼ AcompV f;eff pr2 f ð9Þ where Vf,eff is the pull-out-effective fibre volume fraction of the composite. The force exerted on the composite at the onset of total fibre failure, T, can be expressed as the product of Eqs. (5) and (9): T ¼ 2AcompV f;efflps rf 1 exp d df;0 mf     ð10Þ Table 1 Designation of regression parameters Parameter Status Value/reference Macroscopical parameters Gauge length, L F 25 mm Cross-sectional area, Acomp F Specimen-dependent Material properties Young’s modulus of matrix, Em UV Young’s modulus of fibres, Ef UV Fibre radius, rf F 7 lm [16,17] Fibre volume fraction, Vf F 0.35 [18] Weibull parameters Weibull modulus of matrix, mm UV Weibull modulus of fibres, mf UV Characteristic failure displacement of matrix, dm,0 UV Characteristic failure displacement of fibres, df,0 UV Displacement at first fibre failure, d f CV (see Table 2) Interface-related parameters Interfacial shear stress, s UV Mean pull-out length, lp F in lps=rf 690 mm [16] UV in exp½ðd d fÞ=lp F: Fixed, UV: unconditionally varying, CV: conditionally varying. Table 2 Initial and boundary conditions of the regression procedure Condition Origin/rationalization Initial conditions Origin Material properties IC.1 Em = 80 GPa Approximate engineering value for glass IC.2 Ef = 260 GPa Approximate engineering value for ceramic fibre Weibull parameters IC.3 mm = 10 Approximate engineering value for brittle failure IC.4 mf = 6 Approximate engineering value for ceramic fibre IC.5 dm,0 = 0.05 mm Experimental load–displacement behaviour IC.6 df,0 = 0.15 mm Experimental load–displacement behaviour IC.7 d f ¼ 0:02 mm Experimental load–displacement behaviour Interface-related parameters IC.8 s = 5 MPa Approximate engineering value for moderate bonding Boundary conditions Rationalization BC.1 The matrix contribution term, Eq. (3), may not extend to displacements larger than the value corresponding to the maximum load Typical brittle fracture. Optical microscope observation that macro-crack and corresponding bridging zone are fully developed, spanning the width of the notched ligament, before the maximum load is attained [13] BC.2 The intact fibres contribution term, Eq. (4), may not extend to displacements corresponding to pure pull-out Eq. (16) does not intrinsically relate the confined ranges of effect of intact and failed fibre distribution BC.3 The value of d f is defined by the displacement where the intact fibre contribution term deviates from linearity Eq. (16) does not intrinsically relate first fibre failure to appearance of pull-out 86 K.G. Dassios et al. / Acta Materialia 55 (2007) 83–92

K.G. Dassios et al. Acta Materialia 55(2007)83-92 As the composite displaces further, failed fibres gradually 2.5. Total modelled behaviour disengage from the matrix and the pull-out length, over which shear forces are acting, decreases. This effect leads The theoretically expected load-displacement behaviour to a decrease, with increasing displacement, in the axial of the composite can be expressed as the sum of Eqs. (3), force exerted to the composite due to pull-out. It has been (4)and(12) shown that the decrease in load due to pull-out of failed fi- Pom(d)=Pm()+ Pr(d)+T. the decay constant being equal to the mean pull-out length, Based on the above, the total pull-out force as a function f specimen displacement, Ttor(d), can be written as - exp As demonstrated through Eq(13), the modelled fra ure behaviour of the composite depends on (11) macroscopical parameters such as specimen cross tional area Acomp and the gauge length over which Eq (11)generates non-zero values even for infinitesimally placement is measured,L, small displacements. However, in practice, the appearance the properties of the matrix( such as Em), of the fibres of the pull-out mechanism is triggered by the first fibre fail (such as Ef and re) and of the composite(such as the ure,an action occurring after the end of the fibres'elastic total and effective fibre volume fractions Vf and veeff displacement regime. In order to account for this effect, respectively) the total pull-out force Eq (11)can be shifted in displace- statistical parameters describing the characteristic ment by the displacement at which the intact/bridging fibre failure distribution Eq. (4)deviates from linearity, denoted distributions mm, mm.f for the matrix and fibres, respec df. This live micromechanical parameters associated with the inter- face, such as the mean pull-out length, Ip and the inter- d facial shear stress.t p In displacement-controlled testing, the total displace- ment of the material is the sum of a linear contribution Results of the regressions Specimen constants Specimen I Decimen 3 Specimen width, w(mm) 12 10 Specimen thickness, t(mm) Notch-to-width ratio, 2xo/W(- 0.6 0.4 Cross-section area, Aoomp=IW1-2ao/W)(mm Regression output values Material properties Youngs modulus of matrix, Em(GPa) 71.5 Youngs modulus of fibres, Er(GPa 221.0 213.8 Weibull parameters Weibull modulus of matrix, mm(-) 1.885 Weibull modulus of fibres, m(-) 2.540 Char failure displacement of matrix, dm,o(mm) 0.0351 Char failure displacement of fibres, dEo(mm) Displacement at first fibre failure, df(mm) 0.019 0.0190 Interface-related parameters Interfacial shear stress, t(MPa) 3.04 Mean pull-out length, Ip(mm) Computed parameters Value Composite elastic modulus(GPa 25.4 1213

As the composite displaces further, failed fibres gradually disengage from the matrix and the pull-out length, over which shear forces are acting, decreases. This effect leads to a decrease, with increasing displacement, in the axial force exerted to the composite due to pull-out. It has been shown that the decrease in load due to pull-out of failed fi- bres follows a first-order exponential decay function, with the decay constant being equal to the mean pull-out length, lp [10]. Based on the above, the total pull-out force as a function of specimen displacement, Ttot(d), can be written as: T totðdÞ ¼ 2AcompV f;efflps rf 1 exp d df;0 mf     exp d lp ! ð11Þ Eq. (11) generates non-zero values even for infinitesimally small displacements. However, in practice, the appearance of the pull-out mechanism is triggered by the first fibre fail￾ure, an action occurring after the end of the fibres’ elastic displacement regime. In order to account for this effect, the total pull-out force Eq. (11) can be shifted in displace￾ment by the displacement at which the intact/bridging fibre failure distribution Eq. (4) deviates from linearity, denoted by d f . This leads to: T tot ¼ 2AcompV f;efflps rf 1 exp d d f df;0 mf     exp d d f lp " # ð12Þ 2.5. Total modelled behaviour The theoretically expected load–displacement behaviour of the composite can be expressed as the sum of Eqs. (3), (4) and (12): PcompðdÞ ¼ P mðdÞ þ PfðdÞ þ T tot ¼ Acomp L d ð1 V fÞEm exp d dm;0  mm h i þ V fEf exp d df;0  mf h iþ þ 2 d lpV f;eff s rf 1 exp dd f df;0  mf n o h i exp dd f lp h i 0 @ 1 A ð13Þ As demonstrated through Eq. (13), the modelled frac￾ture behaviour of the composite depends on:  macroscopical parameters such as specimen cross-sec￾tional area Acomp and the gauge length over which dis￾placement is measured, L;  the properties of the matrix (such as Em), of the fibres (such as Ef and rf) and of the composite (such as the total and effective fibre volume fractions, Vf and Vf,eff, respectively);  statistical parameters describing the characteristic failure displacements dm,0, df,0 and shape of the Weibull distributions mm, mm,f for the matrix and fibres, respec￾tively; and  micromechanical parameters associated with the inter￾face, such as the mean pull-out length, lp and the inter￾facial shear stress, s. In displacement-controlled testing, the total displace￾ment of the material is the sum of a linear contribution Table 3 Results of the regressions Value Specimen constants Specimen 1 Specimen 2 Specimen 3 Specimen width, W (mm) 12 12 10 Specimen thickness, t (mm) 2 2 3 Notch-to-width ratio, 2a0/W (–) 0.6 0.4 0.4 Cross-section area, Acomp = tW(1 2a0/W) (mm2 ) 9.6 14.4 18 Varying parameters Regression output values Material properties Young’s modulus of matrix, Em (GPa) 73.9 63.2 71.5 Young’s modulus of fibres, Ef (GPa) 221.0 189.0 213.8 Weibull parameters Weibull modulus of matrix, mm (–) 1.885 1.895 1.889 Weibull modulus of fibres, mf (–) 2.540 2.525 2.520 Char. failure displacement of matrix, dm,0 (mm) 0.0357 0.0351 0.0390 Char. failure displacement of fibres, df,0 (mm) 0.0915 0.1040 0.0901 Displacement at first fibre failure, d f (mm) 0.0197 0.0190 0.0190 Interface-related parameters Interfacial shear stress, s (MPa) 2.86 3.04 3.21 Mean pull-out length, lp (mm) 0.506 0.615 0.588 Computed parameters Value Composite elastic modulus (GPa) 125.4 107.2 121.3 K.G. Dassios et al. / Acta Materialia 55 (2007) 83–92 87

os et al. /Acta materialia 55 corresponding to reversible deformation and a non-linear Given the large number of parameters utilized in Eq contribution corresponding to the irreversible mechanism (16), an infinite number of combinations of their values of fracture, which, in turn, is directly associated with crack can leave the expression invariant. In this regard, experi- opening and the corresponding formation of the bridging mental values were introduced for parameters entering as zone. In simple specimen configurations, such as a dou- multipliers or divisors in the same terms. Care was taken ble-edge- notched tension specimen, a direct relation can in introducing only those parameters that are easily mea be drawn between the crack opening, e, and the load car- surable while keeping the number of introduced constants ried by intact/bridging and failed/pull-out fibres within to an absolute minimum. In this rationale, unambiguously that crack opening, Pbre), by subtracting the elastic dis- determinable material properties such as elastic and Wei- placement from the total displacement of the system bull moduli were free to vary during the regression proce- e(d)=d-CoPcomp(d where Co is the compliance of the composite within the elastic regime. Pbr(e) can then be expressed as the sum of Eqs. (3)and(4) considered with respect to e, while the bridging law, Obr(e), can be obtained by dividing Pbr(e) by the cross-sectional area of the composite, A above leads to 1000 obr(e L.d. ( Er.exp|-(e (15) Displacement, d [mm] a Specimen 1(1=2.0 mm, W=12 mm, 200/W=0.6) It is important to note that only fibres that are oriented non-perpendicularly to the loading direction can contribute 2500 to bridging. Hence, only the effective fibre volume fraction Veeff is relevant for the intact-fibre term of the bridging law. In the present case, the model is applied in a cross-ply lam inate for the fibers that are parallel to the loading direction hence the calculation of Vf eff is trivial. In case of existing other reinforcing directions, VEer must be calculated before application of the model. 3. Application to composite The established model, Eq(13), was validated against Displacement, d [mml the experimentally recorded load-displacement behaviour b Specimen 2(1=12.0 mm, W=12 mm, 2ao/W=0. 4) of DEN SiC-fibre-reinforced glass-ceramic matrix compos- ite coupons tested under uniaxial monotonic tension. For the [0/90Js(where n=2 and 3)cross-ply laminate used in this study, the effective fibre volume fraction is given as Vee= Vd2 and Eq (13)attains the form 1000 +{1-sp[-(=)}ep The assessment of the original Pcomp(d) behaviour of the c Specimen 3(1=3.0 mm, W=10 mm, 2ao/W=0. 4) composite was performed computationally through an iter ative regress Fig. I. Assessment of the experimentally obtained load-displacement procedure where convergence was con- behaviour of the composite by the established model. The individual trolled by the x reduction criterion (computed for th contributions of matrix, intact/bridging and failed/pull-out fibres ar total fit denoted by indices M, I.F. and P respectively

corresponding to reversible deformation and a non-linear contribution corresponding to the irreversible mechanism of fracture, which, in turn, is directly associated with crack opening and the corresponding formation of the bridging zone. In simple specimen configurations, such as a dou￾ble-edge-notched tension specimen, a direct relation can be drawn between the crack opening, e, and the load car￾ried by intact/bridging and failed/pull-out fibres within that crack opening, Pbr(e), by subtracting the elastic dis￾placement from the total displacement of the system: eðdÞ ¼ d C0PcompðdÞ ð14Þ where C0 is the compliance of the composite within the elastic regime. Pbr(e) can then be expressed as the sum of Eqs. (3) and (4) considered with respect to e, while the bridging law, rbr(e), can be obtained by dividing Pbr(e) by the cross-sectional area of the composite, Acomp. The above leads to: rbrðeÞ ¼ V f;eff L d Ef exp e ef;0 mf   þ 2 d lps rf 1 exp e e f ef;0 mf     exp e e f lp " #! ð15Þ It is important to note that only fibres that are oriented non-perpendicularly to the loading direction can contribute to bridging. Hence, only the effective fibre volume fraction Vf,eff is relevant for the intact-fibre term of the bridging law. In the present case, the model is applied in a cross-ply lam￾inate for the fibers that are parallel to the loading direction, hence the calculation of Vf,eff is trivial. In case of existing other reinforcing directions, Vf,eff must be calculated before application of the model. 3. Application to composite The established model, Eq. (13), was validated against the experimentally recorded load–displacement behaviour of DEN SiC-fibre-reinforced glass–ceramic matrix compos￾ite coupons tested under uniaxial monotonic tension. For the [0/90]ns (where n = 2 and 3) cross-ply laminate used in this study, the effective fibre volume fraction is given as Vf,eff = Vf/2 and Eq. (13) attains the form: PcompðdÞ ¼ AcompV f L d ð1V fÞEm V f exp d dm;0  mm h i þ Ef exp d df;0  mf h iþ þ 1 d lps rf 1 exp dd f df;0  mf n o h i exp dd f lp h i 0 B@ 1 CA ð16Þ The assessment of the original Pcomp(d) behaviour of the composite was performed computationally through an iter￾ative regression procedure where convergence was con￾trolled by the x2 reduction criterion (computed for the total fit). Given the large number of parameters utilized in Eq. (16), an infinite number of combinations of their values can leave the expression invariant. In this regard, experi￾mental values were introduced for parameters entering as multipliers or divisors in the same terms. Care was taken in introducing only those parameters that are easily mea￾surable while keeping the number of introduced constants to an absolute minimum. In this rationale, unambiguously determinable material properties such as elastic and Wei￾bull moduli were free to vary during the regression proce- 0.0 0.2 0.4 0.6 0.8 0 500 1000 1500 2000 P I.F. M Displacement, d [mm] Load, P comp [N] Experimental data Model - Total Model - Individual contributions a. Specimen 1 (t=2.0 mm, W=12 mm, 2α0/W=0.6) 0.0 0.2 0.4 0.6 0.8 1.0 0 500 1000 1500 2000 2500 P I.F. M Experimental data Model - Total Model - Individual contributions Load, P comp [N N ] Displacement, d [mm] b. Specimen 2 (t=12.0 mm, W=12 mm, 2α0 /W=0.4) 0.0 0.2 0.4 0.6 0.8 0 1000 2000 3000 4000 P I.F. M Load, P comp [] Displacement, d [mm] Experimental data Model - Total Model - Individual contributions c. Specimen 3 (t=3.0 mm, W=10 mm, 2α0/W=0.4) Fig. 1. Assessment of the experimentally obtained load–displacement behaviour of the composite by the established model. The individual contributions of matrix, intact/bridging and failed/pull-out fibres are denoted by indices M, I.F. and P respectively. 88 K.G. Dassios et al. / Acta Materialia 55 (2007) 83–92

K.G. Dassios et al. Acta Materialia 55(2007)83-92 dure in order for their output values to serve in validating range of effect of the individual contributions. These condi the efficiency of the performed regression. Fibre radius, rf, tions are tabulated in Table 2. and experimental constants such as specimen cross-sec- Table 3 lists the results of three regressions performed tional area, Acomp, and the gauge length of the extensome- on the experimental load-displacement behaviours of ter used for recording displacement on the specimen, L, equal-in-number DEn specimens with different thicknesses were introduced per se. By examination of Eq(16) it is and notch-to-width ratios. The regressions are depicted in observed that the fibre volume fraction, Vf, and one of Ip graphical form in Fig. I or t must also be fixed. Although reference values for the As manifested through Fig. 1, as well as through the val interfacial shear stress of the specific composite exist [16], ues of output parameters in Table 3, the model is successful it was chosen that Ip be fixed and t be deduced due to in assessing the actual fracture behaviour of the composite the simplicity associated with the experimental measure- This remark is supported by the correlation coefficients of ment of pull-out length as opposed to the difficulties the regressions, which did not exceed a value of 0.997 in all encountered in measuring interfacial shear stress. The cases. The most important finding, though, is the fact that validity of this remark is supported by the large spread the values of output parameters remain practically con values for t available in the literature, ranging, for the stant among specimens of different thickness and notch pecific composite, between 2 and 5 MPa(relevant cofi- depths, and hence also of bridging zone dimensions. Table ent of variation 60%), in contrast to the unambiguous 4 summarizes the statistical properties of the output measurements of mean pull-out length, Ip, with values parameters along with reference values for known quant ranging between 670 and 710 um(relevant coefficient of ties. It is observed that parameter values vary within 10% variation 2%)[16]. As will be demonstrated in a later sec- among specimens of different dimensions. Considering ion of this study, a variation in the value of t by 2 MPa the role of interfacial shear stress, the accuracy associated (approximately equal to only half the standard deviation with the determination of this parameter as compared with from the mean of 3. 5 MPa within the 2-5 MPa range) its wide spread in the literature is of particular importance has a dramatic effect on the load-displacement behaviour Equally important is the value obtained for the fibres'Wei- of the composite. On the other hand, a variation in Ip by bull modulus that, although initially set to a much higher 30 um (equal to twice the standard deviation from the value (IC. 4, Table 2), exhibits smaller scatter from the mean 690 um within the 670-710 um range) has a minor mean output value of 2.5 that compares favourably with effect. One exception to the above was that the mean the reference pull-out length, Ip, was chosen to vary in the last exponen- The modelled bridging law and the corresponding indi- tial term of Eq.(16), exp[-(d-d)/p, when it enters vidual contributions of intact and pull-out fibres can be cal- independently. A list of all parameters used in the regres- culated for each specimen through Eqs. ( 14)and(15).At sion procedure is presented in Table I the same time, the crack growth resistance of the material Common to the aforementioned case, a large number of can be obtained by integration of the bridging law with combinations among the individual contributions of the respect to crack opening while the individual contributions matrix, intact and failed fibres can leave Eq (16)invariant. of intact and pull-out fibres to the resistance can be Analogously, the minimum possible number of initial and obtained as the integrals of the corresponding bridging boundary conditions with respect to the independent vari- law terms. The obtained bridging law and R-curves are pre- able, d, were introduced in order to realistically relate the sented in Fig. 2 I properties of regressions Mean,μ SD. a Variance, o/u() Material properties modulus of matrix, Em (GPa) 69.5 5.62 8.08 70[8] modulus of fibres, Er(GPa 207.9 200[18] Weibull modulus of matrix, mm(-) 1.889 0.0047 0.24 Weibull modulus of fibres, mr(-) 0.0100 0.39 23-27[17 Char failure displacement of matrix, dm.o(mm) 0.00210 Char failure displacement of fibres, dro(mm) Displacement at first fibre failure, dr 0.0193 0.00041 2.12 Interface-pelated Interfacial sh 0.174 2-5[6] 0.690[16] cted parameters mposite elastic modulus(GPa) 1179 9.54 23[18]

dure in order for their output values to serve in validating the efficiency of the performed regression. Fibre radius, rf, and experimental constants such as specimen cross-sec￾tional area, Acomp, and the gauge length of the extensome￾ter used for recording displacement on the specimen, L, were introduced per se. By examination of Eq. (16) it is observed that the fibre volume fraction, Vf, and one of lp or s must also be fixed. Although reference values for the interfacial shear stress of the specific composite exist [16], it was chosen that lp be fixed and s be deduced due to the simplicity associated with the experimental measure￾ment of pull-out length as opposed to the difficulties encountered in measuring interfacial shear stress. The validity of this remark is supported by the large spread of values for s available in the literature, ranging, for the specific composite, between 2 and 5 MPa (relevant coeffi- cient of variation 60%), in contrast to the unambiguous measurements of mean pull-out length, lp, with values ranging between 670 and 710 lm (relevant coefficient of variation 2%) [16]. As will be demonstrated in a later sec￾tion of this study, a variation in the value of s by 2 MPa (approximately equal to only half the standard deviation from the mean of 3.5 MPa within the 2–5 MPa range) has a dramatic effect on the load–displacement behaviour of the composite. On the other hand, a variation in lp by 30 lm (equal to twice the standard deviation from the mean 690 lm within the 670–710 lm range) has a minor effect. One exception to the above was that the mean pull-out length, lp, was chosen to vary in the last exponen￾tial term of Eq. (16), exp½ðd d fÞ=lp, when it enters independently. A list of all parameters used in the regres￾sion procedure is presented in Table 1. Common to the aforementioned case, a large number of combinations among the individual contributions of the matrix, intact and failed fibres can leave Eq. (16) invariant. Analogously, the minimum possible number of initial and boundary conditions with respect to the independent vari￾able, d, were introduced in order to realistically relate the range of effect of the individual contributions. These condi￾tions are tabulated in Table 2. Table 3 lists the results of three regressions performed on the experimental load–displacement behaviours of equal-in-number DEN specimens with different thicknesses and notch-to-width ratios. The regressions are depicted in graphical form in Fig. 1. As manifested through Fig. 1, as well as through the val￾ues of output parameters in Table 3, the model is successful in assessing the actual fracture behaviour of the composite. This remark is supported by the correlation coefficients of the regressions, which did not exceed a value of 0.997 in all cases. The most important finding, though, is the fact that the values of output parameters remain practically con￾stant among specimens of different thickness and notch depths, and hence also of bridging zone dimensions. Table 4 summarizes the statistical properties of the output parameters along with reference values for known quanti￾ties. It is observed that parameter values vary within 10% among specimens of different dimensions. Considering the role of interfacial shear stress, the accuracy associated with the determination of this parameter as compared with its wide spread in the literature is of particular importance. Equally important is the value obtained for the fibres’ Wei￾bull modulus that, although initially set to a much higher value (IC.4, Table 2), exhibits smaller scatter from the mean output value of 2.5 that compares favourably with the reference. The modelled bridging law and the corresponding indi￾vidual contributions of intact and pull-out fibres can be cal￾culated for each specimen through Eqs. (14) and (15). At the same time, the crack growth resistance of the material can be obtained by integration of the bridging law with respect to crack opening while the individual contributions of intact and pull-out fibres to the resistance can be obtained as the integrals of the corresponding bridging law terms. The obtained bridging law and R-curves are pre￾sented in Fig. 2. Table 4 Statistical properties of regressions Varying parameters Mean, l SD, r Variance, r/l (%) Reference Material properties Young’s modulus of matrix, Em (GPa) 69.5 5.62 8.08 70 [18] Young’s modulus of fibres, Ef (GPa) 207.9 16.82 8.09 200 [18] Weibull parameters Weibull modulus of matrix, mm (–) 1.889 0.0047 0.24 – Weibull modulus of fibres, mf (–) 2.528 0.0100 0.39 2.3–2.7 [17] Char. failure displacement of matrix, dm,0 (mm) 0.0366 0.00210 5.73 – Char. failure displacement of fibres, df,0 (mm) 0.0952 0.00768 8.06 – Displacement at first fibre failure, d f (mm) 0.0193 0.00041 2.12 – Interface-related parameters Interfacial shear stress, s (MPa) 3.04 0.174 5.72 2–5 [16] Mean pull-out length, lp (mm) 0.570 0.0569 9.98 0.690 [16] Extracted parameters Composite elastic modulus (GPa) 117.9 9.54 8.09 123 [18] K.G. Dassios et al. / Acta Materialia 55 (2007) 83–92 89

K.G. Dassios et al. Acta Materialia 55(2007)83-92 285副 Crack opening, e[mm] Crack opening, e[mm a b Specimen 1(t=12.0 mm, W=12 mm, 20o/W=0. 4) E385苏巴6 Crack opening, e[mm] Crack opening, e [mm] Specimen 2((=2.0 mm, W=12 mm, 2oo/W=0.6) Crack opening, e[mm] Crack opening, e[mm] Specimen 3(1=3.0 mm, W=10 mm, 20o/W=0. 4 Fig. 2. Experimental and modelled bridging laws (left column) and R-curves(right column). 4. Discussion used in the model have explicit physical meaning and are experimentally measurable quantities. It is also understood The agreement between analytical and experimental that use of the specific model expression is valid only for dings, as well as the accordance with the values of calcu- simple geometries where fracture is not confined by spec lated parameters in the literature, is representative of the men configuration characteristics, such as in a three-point potential of the model to assess the fracture behaviour of bending or in a compact tension specimen. In fact, in such le specific composite system in a realistic manner. cases the established law can be utilized once modified to On the other hand, the ability of the model to predict, a account for the specific needs of the geometry [14 .Never priori, the macromechanical fracture behaviour of any theless, an analysis of the effect of selected parameters on given composite (load-displacement behaviour, bridging the magnitude and shape of load-displacement behaviour, law and R-curve)can be limited by the lack of knowledge bridging law and r- ve of the material under the current of certain parameters' values. However, all parameters configuration is of particular interest

4. Discussion The agreement between analytical and experimental findings, as well as the accordance with the values of calcu￾lated parameters in the literature, is representative of the potential of the model to assess the fracture behaviour of the specific composite system in a realistic manner. On the other hand, the ability of the model to predict, a priori, the macromechanical fracture behaviour of any given composite (load–displacement behaviour, bridging law and R-curve) can be limited by the lack of knowledge of certain parameters’ values. However, all parameters used in the model have explicit physical meaning and are experimentally measurable quantities. It is also understood that use of the specific model expression is valid only for simple geometries where fracture is not confined by speci￾men configuration characteristics, such as in a three-point bending or in a compact tension specimen. In fact, in such cases the established law can be utilized once modified to account for the specific needs of the geometry [14]. Never￾theless, an analysis of the effect of selected parameters on the magnitude and shape of load–displacement behaviour, bridging law and R-curve of the material under the current configuration is of particular interest. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 50 100 150 200 P I.F. Bridging Stress, br [MPa] Crack opening , e [mm] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 10 20 30 40 50 60 P I.F. Crack opening , e [mm] Crack growth resistance, R [kJ.m-2 ] 1a. 1b. Specimen 1 (t=12.0 mm, W=12 mm, 2 0 /W=0.4) 0.0 0.2 0.4 0.6 0.8 0 50 100 150 200 P I.F. Bridging Stress, br [MPa] Crack opening , e [mm] 0.0 0.2 0.4 0.6 0.8 0 10 20 30 40 50 60 P I.F. Crack growth resistance, R [kJ.m-2 ] Crack opening , e [mm] 2a. 2b. Specimen 2 (t=2.0 mm, W=12 mm, 2 0 /W=0.6) 0.0 0.2 0.4 0.6 0.8 0 50 100 150 200 P I.F. Bridging Stress, br [MPa] Crack opening , e [mm] 0.0 0.2 0.4 0.6 0.8 0 10 20 30 40 50 60 P I.F. Crack growth resistance, R [kJ.m-2 ] Crack opening , e [mm] 3a. 3b. Specimen 3 (t=3.0 mm, W=10 mm, 2 0/W=0.4) Fig. 2. Experimental and modelled bridging laws (left column) and R-curves (right column). 90 K.G. Dassios et al. / Acta Materialia 55 (2007) 83–92

K.G. Dassios et al. Acta Materialia 55(2007)83-92 The effect of fibre volume fraction, Weibull modulus of state and as surface area providers for the development of fibres and interfacial shear stress on three fracture descrip- frictional forces in the failed state. By examination of tors are presented in Fig 3(1), (2)and (3), respectively. The Fig. 3(1), a vast change in the crack growth resistance values of varying parameters are included in the corre- capacity of the material is observed for a shift in volume sponding graph legends while the values of the remaining fraction by 17% while the specific parameter comes into parameters were in every case kept constant to the mean effect, as expected, only after the first matrix crack occurs. values obtained through the regression procedure (Table The effect of fibre Weibull modulus on the total fracture behaviour of the composite follows a more complex sce- The effect of fibre volume fraction in the fracture behav- nario. Fig. 3(2a) manifests that fibres of higher Weibull iour of the composite is presented in Fig 3(1)(left column modulus contribute to the attainment of higher maximum graphs). It is observed that lower values of Vr result in a loads. However, the sharper decrease in the intact fibre decrease in the maximum load attainable by the composite contribution associated with higher Weibull modulus indi- while, at the same time, causing a decrease in the pull-out cates a more massive, hence also more critical, composite contribution to the crack resistance of the material. This failure scenario. The effect of Weibull modulus on load dual role of fibre volume fraction stems from the ability vanishes after the composite fracture changes to being of the fibres to serve both as reinforcing media in the intact purely pull-out-dominated. It is interesting to note that, Displacement, d[mm] Crack opening, e(mm] ack opening, e [mm] Crack opening, e[mml Crack opening. e[mm) Crack opening, e(mm Crack opening. e[m Fig 3. Effect of fibre volume fraction(1), Weibull modulus of fibres(2)and interfacial shear stress (3)on load-displacement behaviour of the composite (a), bridging law(b) and crack growth resistance (c)

The effect of fibre volume fraction, Weibull modulus of fibres and interfacial shear stress on three fracture descrip￾tors are presented in Fig. 3(1), (2) and (3), respectively. The values of varying parameters are included in the corre￾sponding graph legends while the values of the remaining parameters were in every case kept constant to the mean values obtained through the regression procedure (Table 4). The effect of fibre volume fraction in the fracture behav￾iour of the composite is presented in Fig. 3(1) (left column graphs). It is observed that lower values of Vf result in a decrease in the maximum load attainable by the composite while, at the same time, causing a decrease in the pull-out contribution to the crack resistance of the material. This dual role of fibre volume fraction stems from the ability of the fibres to serve both as reinforcing media in the intact state and as surface area providers for the development of frictional forces in the failed state. By examination of Fig. 3(1), a vast change in the crack growth resistance capacity of the material is observed for a shift in volume fraction by 17% while the specific parameter comes into effect, as expected, only after the first matrix crack occurs. The effect of fibre Weibull modulus on the total fracture behaviour of the composite follows a more complex sce￾nario. Fig. 3(2a) manifests that fibres of higher Weibull modulus contribute to the attainment of higher maximum loads. However, the sharper decrease in the intact fibre contribution associated with higher Weibull modulus indi￾cates a more massive, hence also more critical, composite failure scenario. The effect of Weibull modulus on load vanishes after the composite fracture changes to being purely pull-out-dominated. It is interesting to note that, 0.0 0.2 0.4 0.6 0.8 1.0 0 1000 2000 3000 Load, Pcomp [ ] Displacement, d [mm] Vf = 0.33 Vf = 0.50 Vf = 0.66 0.0 0.2 0.4 0.6 0.8 1.0 0 500 1000 1500 2000 Load, Pcomp [ ] Displacement, d [mm] mf = 2 mf = 2.5 mf = 3.5 0.0 0.2 0.4 0.6 0.8 1.0 0 500 1000 1500 2000 Load, Pcomp [ ] Displacement, d [mm] = 2 MPa = 3 MPa = 4 MPa 1a. 2a. 3a. 0.0 0.2 0.4 0.6 0.8 1.0 0 50 100 150 200 250 Vf = 0.33 Vf = 0.50 Vf = 0.66 Bridging stress, br [MPa] Crack opening, e [mm] 0.0 0.2 0.4 0.6 0.8 1.0 0 50 100 150 200 mf = 2 mf = 2.5 mf = 3.5 Bridging stress, br [MPa] Crack opening, e [mm] 0.0 0.2 0.4 0.6 0.8 1.0 0 50 100 150 200 = 2 MPa = 3 MPa = 4 MPa Bridging stress, br [MPa] Crack opening, e [mm] 1b. 2b. 3b. 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 Vf = 0.33 Vf = 0.50 Vf Crack growth resistance, = 0.66 R [kJ.m-2 ] Crack opening, e [mm] 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 60 mf = 2 mf = 2.5 mf = 3.5 Crack growth resistance, R [kJ.m-2 ] Crack opening, e [mm] 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 60 70 = 2 MPa = 3 MPa = 4 MPa Crack growth resistance, R [kJ.m-2 ] Crack opening, e [mm] 1c. 2c. 3c. Fig. 3. Effect of fibre volume fraction (1), Weibull modulus of fibres (2) and interfacial shear stress (3) on load–displacement behaviour of the composite (a), bridging law (b) and crack growth resistance (c). K.G. Dassios et al. / Acta Materialia 55 (2007) 83–92 91

K.G. Dassios et al. /Acta materialia 55 while the effect of Weibull modulus is clear in the load-dis- individual, appropriately coupled contributions of matrix, placement curve within the fibre failure regime, the same intact/bridging and failed/pull-out fibres. The established does not hold for the R-curve, where a limited shift in crack law assesses the load-displacement behaviour while a sim- growth resistance is noted at displacements corresponding ple concept equating the non-linear part of displacement to pure pull-out. The observed shift stems from the depen- to crack opening is used to convert the original form into dence of the characteristic failure displacement upon the bridging law and, in turn, into crack growth resistance of Weibull modulus, with higher values of the latter providing the material. The law was successful in assessing the lower values for the former. The same dependence is experimentally established load-displacement behaviour responsible for the small effect of this parameter within of a notched Sic-fibre-reinforced ceramic matrix compos- the fibre failure regime of the R-curve: by increasing max- ite with LSB characteristics tested in monotonic tension. imum bridging stress while simultaneously decreasing the The output values of the regression parameters remained width of the intact fibre distribution, a higher Weibull practically constant among specimens of varying cross- modulus leaves the integral of this contribution practically sectional areas and notch lengths. The model was utilized ariant in analysing the effect of selected parameters in composite The interfacial shear stress appears to affect most dra- performance, as well as in linking the micromechanical matically the mechanical performance of the composite interactions with the macroscopic response of the mate ( Fig. 3(3)). The role of t, as demonstrated through Eq. rial to applied deformation (12), is analogous to that of the fibre volume fraction with the exception that the former comes into effect after first Acknowledgement fibre failure. Higher values of this parameter induce the development of more intense shear -hence also axial The authors gratefully acknowledge the valuable re- forces during frictional sliding of the intact or failed fibres marks of Dr. P. Hahner(Institute for Energy, Joint Re- surface along the debonded interface. It is of particula search Center of the European Commission, Petten, The interest to observe that, for the range of bibliographically Netherlands) to the present work available values of t for the composite of this study, 2-5 MPa, all three fracture descriptors, load-displacement, References bridging law and R-curve, change radically In this regard, the ability of the established model to [] Cox BN. Acta Mater 1991: 39: 1189 predict a priori the fracture behaviour of a specific CFCC [2] Fett 1.315 Munz D, Geraghty RD, White Kw. Eng Fract Mech biting LSB relie lies catalytically on the precision of the Foote RML, Mai Y-W, Cotterell B J Mech Phys Solids 1986: 34: introduced value for interfacial shear stress which. com- mon to all parameters entering the model, is an experimen- [4]Rausch G, Kuntz M, Gratwohl G. J Am Ceram Soc 2000: 84 tally measurable quantity of explicit physical significance It is still interesting to note, however, that the same param 5]Sorensen BF, Jacobsen TK. Compos Part A-Appl S 1998: 29: 1443 eters can also be evaluated through the assessment of the [6] Jacobsen TK, Sorensen BF Compos Part A-Appl S 2001: 32: 1 V Hu XZ, Mai Yw. J Mater Sci 1992: 27: 3502 experimental load-displacement behaviour of a single test [8] Pappas YZ, Kostopoulos V Eng Fract Mech 2001: 68: 155 specimen by the law proposed in this study, Eq(13) 9 Fett T, Munz D J Eur Ceram Soc 1995: 15:337 [10] Sutcu M. Acta Mater 1989: 37: 651 5. Conclusions [1l] Thouless MD, Evans AG. Acta Metall 1988: 36: 517. [12] Llorca J, Singh N J Am Ceram Soc 1991:74: 2882. a bridging model has been developed for the [Dassios kG, Galiotis C, Kostopoulos V, Steen M. Acta Mater of the fracture behaviour of continuous-[14] Dassios KG, PhD thesis. Department of Mechanical Engineering and fibre-reinforced ceramic matrix composites based on ity of Pat composite parameters, as well as on micromechanical [15] Dassios KG, Steen M, Filiou K Mat Sci Eng A Struct 2003:349:63 properties of fibres, matrix and the interface. The model [6 Benet P conci F. aoto sn 1, Reynaud P, Rouby D, Tallaron C formulation relies on the concept of convolution of the [17]Simon G, Bunsell AR J Mater Sci 1984: 19:3649 load-displacement behaviour of the composite into the [18] Drissi-Habti M. J Eur Ceramic Soc 1997; 17: 33

while the effect of Weibull modulus is clear in the load–dis￾placement curve within the fibre failure regime, the same does not hold for the R-curve, where a limited shift in crack growth resistance is noted at displacements corresponding to pure pull-out. The observed shift stems from the depen￾dence of the characteristic failure displacement upon the Weibull modulus, with higher values of the latter providing lower values for the former. The same dependence is responsible for the small effect of this parameter within the fibre failure regime of the R-curve: by increasing max￾imum bridging stress while simultaneously decreasing the width of the intact fibre distribution, a higher Weibull modulus leaves the integral of this contribution practically invariant. The interfacial shear stress appears to affect most dra￾matically the mechanical performance of the composite (Fig. 3(3)). The role of s, as demonstrated through Eq. (12), is analogous to that of the fibre volume fraction with the exception that the former comes into effect after first fibre failure. Higher values of this parameter induce the development of more intense shear – hence also axial – forces during frictional sliding of the intact or failed fibres surface along the debonded interface. It is of particular interest to observe that, for the range of bibliographically available values of s for the composite of this study, 2–5 MPa, all three fracture descriptors, load–displacement, bridging law and R-curve, change radically. In this regard, the ability of the established model to predict a priori the fracture behaviour of a specific CFCC exhibiting LSB relies catalytically on the precision of the introduced value for interfacial shear stress which, com￾mon to all parameters entering the model, is an experimen￾tally measurable quantity of explicit physical significance. It is still interesting to note, however, that the same param￾eters can also be evaluated through the assessment of the experimental load–displacement behaviour of a single test specimen by the law proposed in this study, Eq. (13). 5. Conclusions A bridging law model has been developed for the assessment of the fracture behaviour of continuous- fibre-reinforced ceramic matrix composites based on composite parameters, as well as on micromechanical properties of fibres, matrix and the interface. The model formulation relies on the concept of convolution of the load–displacement behaviour of the composite into the individual, appropriately coupled contributions of matrix, intact/bridging and failed/pull-out fibres. The established law assesses the load–displacement behaviour while a sim￾ple concept equating the non-linear part of displacement to crack opening is used to convert the original form into bridging law and, in turn, into crack growth resistance of the material. The law was successful in assessing the experimentally established load–displacement behaviour of a notched SiC-fibre-reinforced ceramic matrix compos￾ite with LSB characteristics tested in monotonic tension. The output values of the regression parameters remained practically constant among specimens of varying cross￾sectional areas and notch lengths. The model was utilized in analysing the effect of selected parameters in composite performance, as well as in linking the micromechanical interactions with the macroscopic response of the mate￾rial to applied deformation. Acknowledgement The authors gratefully acknowledge the valuable re￾marks of Dr. P. Hahner (Institute for Energy, Joint Re￾search Center of the European Commission, Petten, The Netherlands) to the present work. References [1] Cox BN. Acta Mater 1991;39:1189. [2] Fett T, Munz D, Geraghty RD, White KW. Eng Fract Mech 2000;66:375. [3] Foote RML, Mai Y-W, Cotterell B. J Mech Phys Solids 1986;34: 593. [4] Rausch G, Kuntz M, Gratwohl G. J Am Ceram Soc 2000;84: 2762. [5] Sorensen BF, Jacobsen TK. Compos Part A-Appl S 1998;29:1443. [6] Jacobsen TK, Sorensen BF. Compos Part A-Appl S 2001;32:1. [7] Hu XZ, Mai YW. J Mater Sci 1992;27:3502. [8] Pappas YZ, Kostopoulos V. Eng Fract Mech 2001;68:1557. [9] Fett T, Munz D. J Eur Ceram Soc 1995;15:337. [10] Sutcu M. Acta Mater 1989;37:651. [11] Thouless MD, Evans AG. Acta Metall 1988;36:517. [12] Llorca J, Singh N. J Am Ceram Soc 1991;74:2882. [13] Dassios KG, Galiotis C, Kostopoulos V, Steen M. Acta Mater 2003;51:5359. [14] Dassios KG, PhD thesis. Department of Mechanical Engineering and Aeronautics, University of Patras, Greece, 2003. [15] Dassios KG, Steen M, Filiou K. Mat Sci Eng A Struct 2003;349:63. [16] Brenet P, Concin F, Fantozzi G, Reynaud P, Rouby D, Tallaron C. Compos Sci Technol 1996;56:817. [17] Simon G, Bunsell AR. J Mater Sci 1984;19:3649. [18] Drissi-Habti M. J Eur Ceramic Soc 1997;17:33. 92 K.G. Dassios et al. / Acta Materialia 55 (2007) 83–92