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 initially formulated that utilizes weakest-link statistical concepts to analyse and relate the individual contributions of matrix, intact/bridging 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 validated against the experimentally recorded load–displacement behaviour of a notched SiC-fibre-reinforced glass–ceramic matrix composite 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 combination 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 lengthto-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 insensitivity and their ability to effectively redistribute stresses around notches, voids and cracks. These prominent properties stem from the energy dissipation mechanisms activated 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 composite, 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 during 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 characterization of CFCC performance is the bridging law, a phenomenological 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, materialintrinsic fracture descriptor that is independent of the overall composite dimensions or geometry. The determination of the bridging law for various composite systems has been the aim of several research efforts in the last decade. 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 aforementioned studies, the current lack of confidence in CFCC applicability – especially with respect to the growing concern about high production costs – clearly demonstrates the necessity to comprehend the fundamentals of interactions between the composite phases using simple, straightforward 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 composite performance. In this study, a straightforward methodology is presented 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 contributions 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 correlation 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 behaviour of a CFCC is discussed against the results of a parametric analysis of the role of material parameters. The model can securely predict the load–displacement behaviour of any kind of double-edge notch (DEN) configuration, while it can be extended for the prediction of the fracture behaviour of compact tension (CT) and singleedge-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 displacement-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 procedure 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. Further loading leads to micro-cracks growing to fibre-bridged macro-cracks, while concurrent phenomena such as interface 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 manner 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 crackexposed length equal to the instant crack opening displacement. 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 without debonding and supports material brittleness instead of decreasing it. Besides its statistical origin, this ‘‘preferable fibre failure location’’ phenomenon is additionally promoted 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 controlled 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 linear elastic behaviour up to catastrophic failure. On the other hand, the ceramic matrix of a CFCC will fail gradually 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 mechanisms 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 matrix 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 composite, 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 established as the most suitable fibre failure distribution function as it couples a first-order growth term with an exponential decay term that can realistically simulate the sequential phenomena of elastic deformation and incremental 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 statistical parameter that may vary among fibres in the composite. 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 composite 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 volume 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 failure, 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 displacement 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 fracture behaviour of the composite depends on: macroscopical parameters such as specimen cross-sectional area Acomp and the gauge length over which displacement 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, respectively; and micromechanical parameters associated with the interface, such as the mean pull-out length, lp and the interfacial shear stress, s. In displacement-controlled testing, the total displacement 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