Reliability Engineering and System Safety 56 (1997) @ 1997El All rights reserved. Printed in Northern ELSEVIER PII:s0951·8320(97)00094-4 0951-8320/97/517 Ultimate strength properties of fiber reinforced composites* Francois Hild Pascal Feillard Laboratoire de Mecanique et Technologie, ENS Cachan/CNRS/Universite Paris VI, 6I avenue du President Wilson, F-94235 Cachan Cedex. france pull-out. Depending on the load redistribution around a broken fiber,two different regimes can be obtained. The results are applied to the prediction of ultimate strengths of Sic fiber-reinforced composites subjected to tension, pure flexure and three-point flexure. C 1997 Elsevier Science Limited 1 INTRODUCTION failure mechanism leads to the failure of the com- posite (e.g, a fiber adjacent to a previously broken fiber breaks because of the stress concentration In spite of the fact that Fiber-Reinforced Composites duced by the initially broken fiber [8] bundle pull-out (FRCS)are made of brittle fibers, the composites can (91). Because of localized load shedding from a broken exhibit a fracture resistance behavior [1, 2 ]. The origin fiber on to the nearest neighboring fibers, localized of the fracture resistance is the consequence of the breaks induce the final failure of the composite statistical failure strength of fibers (i. e, all the fibers do not break at the same load level and they can be The assessment of ultimate strengths of FRCs subject to multiple breakage)and of sliding which mainly based upon the assumption of a constant occurs between broken fibers and the surrounding [10-13] or ductile matrices [14-16] in the global load matrix. The non- linearity of a stress/strain curve close to the ultimate point is one of the indications of sharing regime although the same assumption can be used to describe bundle pull-out [9]. A constant multiple fragmentation of fibers [3]. In this paper we interfacial sliding stress is also assumed to study dictated by fiber breakage and fiber pull-out. The wer iple fragmentation taking place in polymeric-as multiple fragmentation of fibers depends upon the s ceramic-matrix composites [17-20]. To carr statistical nature of fiber breakage as well as the load out the analysis, an elementary cell can be considered whose size, the recovery length, is directly related to redistribution around a fiber break. Two extreme the longest fiber that can be pulled out and cause a regimes(or modes)can be exhibited as described in reduction in load carrying capacity [10,13]. In the case an earlier analysis by Gucer and Gurland [4]. The of polymeric matrices other models have been used first,referred to as a global load sharing regime [21j based upon more refined mechanical models of the interfacial behavior [22]. The latter studies deal equally shared by all the unbroken fibers. This with the distribution of fragment length and therefore hypothesis was also used to model dry fiber bundles one relevant length to consider is the average [5-7]. Global load sharing is an important assumption fragment length. This length can be used to derive that allows us to relate the fragmentation analysis of a ultimate strengths of FRCs, regardless of the details of single fiber composite to fiber breakage and fiber the mechanical behavior of the interface. From a pull-out in a multi-fiber composite On the other hand, design point of view, it is important not only to a local load sharing regime assumes that there exists a weakest zone in the material where the onset of a evaluate the ultimate tensile strength of FRCs but to model the stress/strain response of the composite so *This work was partially supported by the Advanced that the capacity of the components subjected to Research Projects Agency through the University Research multiple loadings can be estimated [23]. We focus our Initiative, ONR Contract NO0014-12-J-180 attention on determining the strength of rectangula
ELSEVIER Reliability Engineering and System Safety 56 (1997) 225-235 1997 Elsevier Science Limited All rights reserved. Printed in Northern Ireland Pll: S0951-8320197)00094-4 0951-8320/97/$17.00 Ultimate strength properties of fiberreinforced composites* Francois Hild & Pascal Feillard Laboratoire de M~canique et Technologie, ENS Cachan/CNRS/Universit~ Paris VI, 61 avenue du President Wilson, F-94235 Cachan Cedex, France The volume effect and stress heterogeneity effect (i.e., the effect of loading type) on the ultimate strength are analyzed for fiber-reinforced composites. The main failure mechanisms are assumed to be fiber breakage and fiber pull-out. Depending on the load redistribution around a broken fiber, two different regimes can be obtained. The results are applied to the prediction of ultimate strengths of SiC fiber-reinforced composites subjected to tension, pure flexure and three-point flexure. © 1997 Elsevier Science Limited. 1 INTRODUCTION In spite of the fact that Fiber-Reinforced Composites (FRCs) are made of brittle fibers, the composites can exhibit a fracture resistance behavior [1, 2]. The origin of the fracture resistance is the consequence of the statistical failure strength of fibers (i.e., all the fibers do not break at the same load level and they can be subject to multiple breakage) and of sliding which occurs between broken fibers and the surrounding matrix. The non-linearity of a stress/strain curve close to the ultimate point is one of the indications of multiple fragmentation of fibers [3]. In this paper we study the case where the ultimate strength of FRCs is dictated by fiber breakage and fiber pull-out. The multiple fragmentation of fibers depends upon the statistical nature of fiber breakage as well as the load redistribution around a fiber break. Two extreme regimes (or modes) can be exhibited as described in an earlier analysis by GUcer and Gurland [4]. The first, referred to as a global load sharing regime assumes that the load carried by a fiber at failure is equally shared by all the unbroken fibers. This hypothesis was also used to model dry fiber bundles [5-7]. Global load sharing is an important assumption that allows us to relate the fragmentation analysis of a single fiber composite to fiber breakage and fiber pull-out in a multi-fiber composite. On the other hand, a local load sharing regime assumes that there exists a weakest zone in the material where the onset of a *This work was partially supported by the Advanced Research Projects Agency through the University Research Initiative, ONR Contract N00014-12-J-1808. 225 failure mechanism leads to the failure of the composite (e.g., a fiber adjacent to a previously broken fiber breaks because of the stress concentration induced by the initially broken fiber [8] bundle pull-out [9]). Because of localized load shedding from a broken fiber on to the nearest neighboring fibers, localized breaks induce the final failure of the composite. The assessment of ultimate strengths of FRCs is mainly based upon the assumption of a constant interfacial sliding stress in the case of ceramic matrices [10-13] or ductile matrices [14-16] in the global load sharing regime although the same assumption can be used to describe bundle pull-out [9]. A constant interracial sliding stress is also assumed to study multiple fragmentation taking place in polymeric- as well as ceramic-matrix composites [17-20]. To carry out the analysis, an elementary cell can be considered whose size, the recovery length, is directly related to the longest fiber that can be pulled out and cause a reduction in load carrying capacity [10, 13]. In the case of polymeric matrices other models have been used [21] based upon more refined mechanical models of the interfacial behavior [22]. The latter studies deal with the distribution of fragment length and therefore one relevant length to consider is the average fragment length. This length can be used to derive ultimate strengths of FRCs, regardless of the details of the mechanical behavior of the interface. From a design point of view, it is important not only to evaluate the ultimate tensile strength of FRCs but to model the stress/strain response of the composite so that the capacity of the components subjected to multiple loadings can be estimated [23]. We focus our attention on determining the strength of rectangular
F. Hild, P. feillard beams made of unidirectional composites subjected to stress field along the fibers in the elementary cell a combination of tension and flexure so that F(T, z)by closed-form solutions can be obtained. In the first part f(2 it is shown that the ultimate tensile strength can be derived by considering an elementary cell of size equal L(T)J-LO to the average fragment length. The second part is where z is the current position along the fiber concerned with a constitutive law derived in tension direction, and f the fiber volume fraction. Since the and compression to analyze the pure flexural loadings: reference stress T is directly proportional to the discussed in the third part. In pure flexure, the failure macroscopic strain E, the ultimate tensile strength is mode is assumed to be identical to that in tension (i.e reached when no compressive mode is considered in the present analysis). The fourth part establishes the interaction between tensile and flexural loadings and the fifth is concerned with the study of the ultimate strength in where the reference stress corresponding to the three-point flexure. Experimental data are compared ultimate tensile strength is denoted by Tu. If we with predictions and the two load transfer regimes are assume that the interfacial shear resistance is modeled by a constant shear stress t, the average stress is given by 2 ULTIMATE TENSILE STRENGTH where LR is the so-called recovery length. The latter is In this section, expressions of the ultimate tensile related to the reference stress by strength of FRCs are derived. The key mechanism RT leading to final failure of the considered FRC is LR(T) (5) assumed to be fiber breakage. This mechanism is usually characterized by the fact that a fiber undergoes where R is the fiber radius. The recovery length refers multiple fragmentation until final fracture. The matrix to twice the longest fiber that can be pulled out and contribution is supposed to be negligible compared to cause a reduction of the load carrying capacity(Fig that of the fibers. Depending on the analyzed system, 1) different models are used to describe the behavior of It is worth noting that eqn(4)can be used in the the fiber/matrix interface. The simplest assumption is case of a constant interfacial shear stress when the to consider a constant interfacial shear stress t ratio LR(T)/L(T) is approximated by the cumulative [24, 25]. Other models have been proposed [26, 22]in failure probability PF(T)of a piece of composite of hich the debonding propagation is based upon a length LR(T)[13]. As long as the interaction between shear lag analysis limited by an interfacial shear strength and a Coulomb friction law which models the load transfer along the debond length. In both cases expressions of the average frag cally [ 17, 27] as w [19, 28]. The latter quantity will be ultimate tensile strength of FRCs In the first part a global load sharing regime is supposed to occur. By assuming that the fibers do not interact, a single fiber system is then representative of the whole composite behavior [18, 11, 16. Let us (T,z) consider an elementary cell of size equal to the average fragment length L. The average fragment length is defined as the ratio of the total composite 2T/R length L divided by the average number of fiber breaks N(T) in a single fiber Fiber breaks where T is the reference stress equal to the stress level Fig. 1. Distribution of the fiber stress field oF(T, z)for a given value of the reference stress T in the case of a constant in an unbroken fiber for the same macroscopic strain interfacial shear stress t. Two different fibers are E. Furthermore, the average stress o is related to the
226 F. Hild, P. Feillard beams made of unidirectional composites subjected to a combination of tension and flexure so that closed-form solutions can be obtained. In the first part it is shown that the ultimate tensile strength can be derived by considering an elementary cell of size equal to the average fragment length. The second part is concerned with a constitutive law derived in tension stress field along the fibers in the elementary cell O'F(T,z) by (L(T)/2 ~(T) = ~ J_£(r)/Ea~F(T,z)dz (2/ where z is the current position along the fiber direction, and f the fiber volume fraction. Since the and compression to analyze the pure flexural loa~gs:: reference stress T is directly proportional to the discussed in the third part. In pure flexure, the:faiqube ~ ~ ;: macroscopic:::, strain g, the ultimate tensile strength is mode is assumed to be identical to that in tension (i,e.i ....... reached when no compressive mode is considered in the present analysis). The fourth part establishes the interaction between tensile and flexural loadings and the fifth is concerned with the study of the ultimate strength in three-point flexure. Experimental data are compared with predictions and the two load transfer regimes are discussed. 2 ULTIMATE TENSILE STRENGTH In this section, expressions of the ultimate tensile strength of FRCs are derived. The key mechanism leading to final failure of the considered FRC is assumed to be fiber breakage. This mechanism is usually characterized by the fact that a fiber undergoes multiple fragmentation until final fracture. The matrix contribution is supposed to be negligible compared to that of the fibers. Depending on the analyzed system, different models are used to describe the behavior of the fiber/matrix interface. The simplest assumption is to consider a constant interracial shear stress [24, 25]. Other models have been proposed [26, 22] in which the debonding propagation is based upon a shear lag analysis limited by an interfacial shear strength and a Coulomb friction law which models the load transfer along the debond length. In both cases, expressions of the average fragment length are derived numerically [17,27] as well as analytically [19, 28]. The latter quantity will be used to assess the ultimate tensile strength of FRCs. In the first part a global load sharing regime is supposed to occur. By assuming that the fibers do not interact, a single fiber system is then representative of the whole composite behavior [18,11,16]. Let us consider an elementary cell of size equal to the average fragment length /~. The average fragment length is defined as the ratio of the total composite length L divided by the average number of fiber breaks N(T) in a single fiber L /:(T) = ~r(T ) (1) where T is the reference stress equal to the stress level in an unbroken fiber for the same macroscopic strain g. Furthermore, the average stress ~ is related to the O~'(T = T,) = 0 (3) aT where the reference stress corresponding to the ultimate tensile strength is denoted by T,. If we assume that the interfacial shear resistance is modeled by a constant shear stress ~', the average stress is given by LR(r)] t~(r) =fT[1 ~j (4) where LR is the so-called recovery length. The latter is related to the reference stress by RT LR(T) = (5) r where R is the fiber radius. The recovery length refers to twice the longest fiber that can be pulled out and cause a reduction of the load carrying capacity (Fig. 1). It is worth noting that eqn (4) can be used in the case of a constant interfacial shear stress when the ratio LR(T)/f_.(T) is approximated by the cumulative failure probability PF(T) of a piece of composite of length LR(T) [13]. As long as the interaction between i m m L R Id Z = Z ~ FC T, z) i = L R =1 Fig. 1. Distribution of the fiber stress field o'F(T,z) for a given value of the reference stress T in the case of a constant interracial shear stress r. Two different fibers are considered
trength properties of Frc's fiber breaks is negligible (i.e, LR(T)/L(T)<1), the failure stress, OFT is proportional to V-m, for any evolution of the average fragment length is given by composite volume V 1-ex where T( is the Euler function of the second kind where m is the Weibull modulus [29] modeling fiber (also called gamma function), S, is the Weibull scale failure, and S is the characteristic strength [17] arameter, Vs is the reference volume and m, is the defined as Weibull shape pa soLot\"T (7 localized failure Darameter modeling the onset of R If debonding propagation is based upon a shear lag analysis limited by an interfacial shear strength, and a So is the scale parameter and Le Coulomb friction law modeling the load transfer along corresponding gauge length of the Weibull law the debond length [26, 22], the exclusion zones where modeling fiber failure. Equation (6) neglects the no additional fiber breakage is possible are not as fragments of size less than or equal to LR(T)(i.e, it is easily determined as previously (since the recovery assumed that there are no over-lapping recove length is of statistical nature). However, the general regions ). This assumption is not very strong since the formalism of eqns(2)and(3)still applies. Instead of strain levels at the ultimate point are significantly one quantity modeling the interfacial behavior, i.e., t, lower than those at fiber breakage saturation [13]. a two quantities are used, viz. the friction coefficient u first order solution of the ratio LR(T)/L(T)is found defining the shear stress level t,=-Aorr in the debond zone, which is dependent upon the radial stresses orr, and the maximum shear strength td. The LR(T) Poisson effect as well as the residual stresses are accounted for in this model 31]. An expression of the and the corresponding ultimate strength ours is given average fragment length has to be derived numerically 31] for any combination of ta, u and initial residual Ours= fSe +2)m+2 fragments can be approximated by a Weibull distribution and the average fragment length is This expression is rigorously identical to that found therefore given by by Curtin [10]. Equation (9) shows that the haracteristic strength is the scaling stress needed to BL(T)=BLi(T)r1+ derive the ultimate tensile strength. Furthermore, eqn (9)is valid provided the composite length L is greater where L,(T)and m, (T)are the Weibull scale and than the recovery length at the ultimate LR(Tu), which shape parameters, respectively. The parameters model is proportional to the reference length 8 defined by the fragment length distribution, are identified from 9 the numerical simulations and depend upon the reference stress level T The parameter B is a function (10) of the elastic properties of the matrix and the fiber as well as the fiber volume fraction [32 The characteristic strength S can be reinterpreted as the average strength of a fiber of length 8.When (14) the previous condition is not met, the ultimate tensile ERIn strength is length-dependent and the dry fiber bundle strength [6, 7] is a good approximation of the ultimate where E is the Youngs modulus of the fibers, Gmis tensile strength[30, 13] the shear modulus of the matrix, 2R is the average distance between fibers. When fiber breakage (11) saturates, the parameters L,(T)and m, (r)saturate as SiC/LAS composite). The numerical computations are Equations(9)and(11)show that the volume effect is performed on a single filament representative of the different from that of purely brittle materials or FRCs composite. The results of Fig. 2 were obtained with an exhibiting a local load sharing regime for which the analysis of a fiber containing 500,000 elements of ultimate tensile strength, defined as the average length 20 um whose strength is randomly distributed
Strength properties of FRC's 227 fiber breaks is negligible (i.e., LR(T)/L(T)<< 1), the evolution of the average fragment length is given by [10, 13] LR(T) _ exp[ T m+l /,(r) - 1 where m is the Weibull modulus [29] modeling fiber failure, and S~ is the characteristic strength [17] defined as S~ = \--~----/ (7) where So is the scale parameter and Lo the corresponding gauge length of the Weibull law modeling fiber failure. ]Equation (6) neglects the fragments of size less than or equal to LR(T) (i.e., it is assumed that there are no over-lapping recovery regions). This assumption is not very strong since the strain levels at the ultimate point are significantly lower than those at fiber breakage saturation [13]. A first order solution of the ratio L~(T)/£(T) is found to be (rF+' and the corresponding ultimate strength ~vrs is given by [ 2 ] -,-h ffurs~--~° "-~-~-~]]o~ m + 1 (9) m+2" This expression is rigorously identical to that found by Curtin [10]. Equation (9) shows that the characteristic strength is the scaling stress needed to derive the ultimate tensile strength. Furthermore, eqn (9) is valid provided the composite length L is greater than the recovery length at the ultimate LR(T,), which is proportional to the reference length 6~ defined by [191 RSc 6~ = (10) The characteristic strength Sc can be reinterpreted as the average strength of a fiber of length 6c. When the previous condition is not met, the ultimate tensile strength is length-dependent and the dry fiber bundle strength [6, 7] is a good approximation of the ultimate tensile strength [30, 13] 8urs_~( Lo )~ (11) fSo " Equations (9) and (11) show that the volume effect is different from that of purely brittle materials or FRCs exhibiting a local load sharing regime for which the ultimate tensile strength, defined as the average failure stress, CTFr is proportional to V -1/ms for any composite volume V (+1) tYs= "SF 1 (12) s, where F(.) is the Euler function of the second kind (also called gamma function), S, is the Weibull scale parameter, V~ is the reference volume and ms is the Weibull shape parameter modeling the onset of localized failure. If debonding propagation is based upon a shear lag analysis limited by an interracial shear strength, and a Coulomb friction law modeling the load transfer along the debond length [26, 22], the exclusion zones where no additional fiber breakage is possible are not as easily determined as previously (since the recovery length is of statistical nature). However, the general formalism of eqns (2) and (3) still applies. Instead of one quantity modeling the interracial behavior, i.e., ~, two quantities are used, viz. the friction coefficient/x defining the shear stress level rI =- tzo,r in the debond zone, which is dependent upon the radial stresses Or,, and the maximum shear strength rd. The expression of the friction stress r s shows that the Poisson effect as well as the residual stresses are accounted for in this model [31]. An expression of the average fragment length has to be derived numerically [31] for any combination of rd, /Z and initial residual stresses in the composite. The distribution of fragments can be approximated by a distribution and the average fragment therefore given by /3/_~(T) =/3LI(T)F(1 Weibull length is where LI(T) and ml(T) are the Weibull scale and shape parameters, respectively. The parameters model the fragment length distribution, are identified from the numerical simulations and depend upon the reference stress level T. The parameter/3 is a function of the elastic properties of the matrix and the fiber as well as the fiber volume fraction [32] ,I 2Gm_ /3 = EyRZln(R ) (14) where E I is the Young's modulus of the fibers, Gm is the shear modulus of the matrix, 2/~ is the average distance between fibers. When fiber breakage saturates, the parameters Ll(T) and m~(T) saturate as well (see Fig. 2 in the case of a unidirectional SiC/LAS composite). The numerical computations are performed on a single filament representative of the composite. The results of Fig. 2 were obtained with an analysis of a fiber containing 500,000 elements of length 20/zm whose strength is randomly distributed + (13)
F. Hild, p. feillard 6 2 10 15 102 阝L 更φ 00.0020.0040.0060.008 10210110°102102103104105 Normalized reference stress, T/E Normalized composite length, B 2. Shat of the fiber fr Fig. 4. Normalized ultimate tensile strength of an E distribution as a function of normalized reference stress for Glass/Epoxy composi e as a function of normalized a SiC/LAS composite (f=0. 5, B=50mm, So composite length(B=10 mm S,=900 MPa, Ln=10 mm 0MPa,Ln=10mm,m=4,=016,t=50MPa) m=10,H=09,=80MPa,R=55m) strength of an Epoxy matrix reinforced by E Glass ccording to a Weibull law. Because of the statistical fibers is analyzed. Figure 4 shows the evolution of the distribution of fragment lengths, the debond lengths ultimate tensile strength as a function of the are also of statistical nature. a more detailed composite length. Each simulation of Fig. 4 is again presentation of the numerical analysis used to predict the result of 1.000 realizations for the same fiber the present results can be found in [31, 331 The fragmentation model can be utilized to model length. Three different regimes can be exhibited atrix cracking as well as fiber breakage. If matrix First, for very small composite BL500 Before saturation, however, the previous results do the onset of debonding signals final failure of the not apply and a constant shear strength hypothesis can composite(see Fig. 5). The number of breaks is not be a crude approximation of the actual interfacial on the order of unity as in the case of small composite behavior. In the following, the ultimate tensile 10 5 Simulation Equations(9-10) 10210l10010110210310410 0.01 Normalized composite length, BL Normalized composite length, L/8 ig. 5, Normalized debond length of an E glas Fig 3. Normalized ultimate tensile strength of a SiC/LAs composite as a fu f normalized composite composite as a function of normalized composite length (B=10 mm fS,=900 MPa, Lo=10mm, m=10,u=0.9 (f=05,S=2500MPa,m=4) R=55μm)
228 F. Hild, P. Feillard ff 104 ~ 103 N 102 ~ 101 N 10 ° O z 0 I ( = 0.002 m I ~L i , i , , , I , 0.004 0.006 5 4 3 2 ' ' 1 0.008 Normalized reference stress, T/Ef • ~, Ib ~ O Z Fig. 2. Shape and scale parameters of the fiber fragment distribution as a function of normalized reference stress for a SiC/LAS composite (f=0.5, /3 =50mm-', So= 1500 MPa, Lo = 10 mm, m = 4,/x = 0.16, ra = 50 MPa). 1 t ~ ti--~--~-- i 0.5 ....... ~ ........ ' ........ ' ........ ' ' ":'"" ........ ~ ...... 10 -2 10 -1 100 101 102 103 104 105 Normalized composite length, [~L Fig. 4. Normalized ultimate tensile strength of an E Glass/Epoxy composite as a function of normalized composite length (/3 = 10 mm ~, fSo = 900 MPa, L{) = 10 mm, m = 10,/z = 0.9, T,l = 80 MPa, R = 5-5 #zm). according to a Weibull law. Because of the statistical distribution of fragment lengths, the debond lengths are also of statistical nature. A more detailed presentation of the numerical analysis used to predict the present results can be found in [31, 33]. The fragmentation model can be utilized to model matrix cracking as well as fiber breakage. If matrix cracking saturates, the interracial shear stress rs does not evolve significantly when the reference stress increases. Under these circumstances, the results obtained by a model with a constant shear strength are expected to be close to those obtained by the above-discussed model. In particular the evolution of the ultimate tensile strength as a function of the composite length is identical as shown in Fig. 3 when r = zs. Each simulation of Fig. 3 is the result of 1,000 realizations for the same fiber length. Before saturation, however, the previous results do not apply and a constant shear strength hypothesis can be a crude approximation of the actual interfacial behavior. In the following, the ultimate tensile strength of an Epoxy matrix reinforced by E Glass fibers is analyzed. Figure 4 shows the evolution of the ultimate tensile strength as a function of the composite length. Each simulation of Fig. 4 is again the result of 1,000 realizations for the same fiber length. Three different regimes can be exhibited. First, for very small composite lengths (i.e., /3L 500) the onset of debonding signals final failure of the composite (see Fig. 5). The number of breaks is not on the order of unity as in the case of small composite 10 ...... , ........ , ........ , ,., • Simulation O Z 0.1 ......................... 0.01 0.1 1 Normalized composite length, L/8 c Fig. 3. Normalized ultimate tensile strength of a SiC/LAS composite as a function of normalized composite length (f = 0-5, Sc = 2500 MPa, m = 4). 100 80 o~ 60 -,% N 40 -,..~ ~ 20 E O Z 0 "i'il ' ' '~'~ ........ ~ ........ ' ....... I ........ , ........ ....... d ........ I .................. ~,~,~,.lill.,~ 10 -2 10 -1 100 10 ] 102 103 104 105 Normalized composite length, I3L Fig. 5. Normalized debond length of an E Glass/Epoxy composite as a function of normalized composite length (/3 = 10 mm -l, fSo = 900 MPa, L. = 10 mm, m = 10, #z = 0.9, z,i = 80 MPa, R = 5"5 #xm)
trength properties of frC's lengths. According to the Cox model [32], the onset of constitutive equation modeling gradual fiber breakage debonding is given by the condition [31] can be written only in the global load sharing regime BL(T) for which the composite behavior is length 2Ta=I.BR tanh( (15) independent In the following, it is assumed that the stress/strain behavior can be characterized by a series expansion of greater than 3/B, the previous condition can simplified the macroscopic stress o as a function of the reference to become 2ta a TuBR. This criterion is deterministic so that the ultimate tensile strength can be expressed material (i.e, fiber breakage, debonding and pull-out) (T)T tant BL(T n≥1 OUTs=fUll BL(Tu (16) where w, are the coefficients of the series expansion and B(n) is a linear combination of Equation(18)will be used later to predict the ultimate It is worth noting that the format of eqn(16)is flexural strengt identical to that of eqn (4). Furthermore, when If the ultimate point is reached when T=Tu= Ase BL(T)>100 the previous expression can be where A is a constant, the ultimate tensile strength simplified so that the ultimate tensile strength can be can be rewritten as approached by fTu oE=1+∑A)1 (19) In the present case, the ultimate tensile strength is For example, in the case of a constant interfacial equal to 0. 96 fSo. This value is in good agreement with the numerical simulations shown in Fig. 4. Third, for shear stress, the parameters in eqn(18 )are such that composites of intermediate lengths (i.e,0-5<Bl< symbol, B(n)=8mn(m+2), S is given by eqn(7)and 500)the onset of debonding is not fatal to the A=(2/(m+2) composite(see Fig. 5). On the other hand, there is no Another application can be complete debonding so none of the two previous found in [13] regimes is relevant. Moreover, the number of fiber The constitutive eqn (18)is only valid in the tensile part, and does not take into account matrix cracking breaks increases as the composite length increases. In since the initial Young's modulus is assumed to be the the case of the present simulations, the number or Young's modulus of the unbroken fibers fE. In the breaks for intermediate lengths was less than 10 compressive part we assume that the behavior is Lastly, the ultimate tensile strength decreases with the unaltered upon loading and is given by the behavior of total length to approach the length-independent the virgin material. Therefore the compressive regime when the length of the composite increases behavior is defined by the Youngs modulus of the composite E. No compressive failure mechanism is These simulations show that the features exhibited considered herein. a damage variable, D,, is by a model assuming a constant interfacial shear stress. introduced to measure the difference in Young's can be obtained by different models even when matrix modulus in tension(T/Sc <<1)and compression ng's cracking is not involved. However there is(are)some additional regime(s)in-between. Moreover the length D1=1 fE independent regime can be explained by different reasons as shown by the two ultimate strength studies within the global load sharing framework In the next sections. this constitutive law will be used to determine ultimate strengths in pure and three-point flexure 3 CONSTTTUTIVE EQUATION In this section the behavior of frcs is studied from 4 ULTIMATE FLEXURAL STRENGTH macroscopic point of view. Therefore we will only consider the average stress, or macroscopic stress o This section is devoted to the determination of the nd the corresponding macroscopic strain ultimate strength of rectangular beams under pure equivalently, the reference stress T/Er. The constitu- flexure by making use of the model derived in tension tive equations will be derived using the microscopic and compression. The beam is made of one layer approach of Section 2. It is worth remembering that a whose fibers are aligned along the z-direction. It
Strength properties of FRC's 229 lengths. According to the Cox model [32], the onset of debonding is given by the condition [31] 2Td = T,/3R tanh(~). (15) When the average fiber fragment length [.(T,) is greater than 3//3, the previous condition can simplified to become 2rd ~- T,/3R. This criterion is deterministic so that the ultimate tensile strength can be expressed as [ (16) It is worth noting thai: the format of eqn (16) is identical to that of eqn (4). Furthermore, when /3£(T,) > 100 the previous expression can be simplified so that the ultimate tensile strength can be approached by ~urs ~fT, (17) In the present case, the ultimate tensile strength is equal to 0.96 fSo. This value is in good agreement with the numerical simulations shown in Fig. 4. Third, for composites of intermediate lengths (i.e., 0.5 -1 For example, in the case of a constant interracial shear stress, the parameters in eqn (18) are such that ~On =-6,,,/2, where 6,,, denotes the Kronecker symbol,/3(n) = 6,,,(rn + 2), Sc is given by eqn (7) and A = (2/(m + 2)) "'-~+~. Another application can be found in [13]. The constitutive eqn (18) is only valid in the tensile part, and does not take into account matrix cracking since the initial Young's modulus is assumed to be the Young's modulus of the unbroken fibers fEy. In the compressive part we assume that the behavior is unaltered upon loading and is given by the behavior of the virgin material. Therefore the compressive behavior is defined by the Young's modulus of the composite /~. No compressive failure mechanism is considered herein. A damage variable, D1, is introduced to measure the difference in Young's modulus in tension (T/S~ << 1) and compression Dl=l fEI /~. (20) In the next sections, this constitutive law will be used to determine ultimate strengths in pure and three-point flexure. In this section the behavior of FRCs is studied from a macroscopic point of vie.w. Therefore we will only consider the average stress, or macroscopic stress and the corresponding macroscopic strain g, or, equivalently, the reference stress T/E I. The constitutive equations will be derived using the microscopic approach of Section 2. It :is worth remembering that a 4 ULTIMATE FLEXURAL STRENGTH This section is devoted to the determination of the ultimate strength of rectangular beams under pure flexure by making use of the model derived in tension and compression. The beam is made of one layer whose fibers are aligned along the z-direction. It is
Hild p. feillard TFF/GFT for the same volume loaded in pure flexure and in tension is given by h =[2m+1) where oFF is the average failure stress in pure flexure To illustrate the previous results, a constant interfacial Fig. 6. Definition of the beam geometry in the case of pure shear strength is assumed in the framework of global load sharing. By using the approximations to derive eqn( 8), the position of the neutral axis is given by assumed that there is a sufficient number of fibers in h the width of the beam so that a global load sharing )mn+1(m+4(27) h 1+√/(1-D1) hypothesis is made for each height y. The Bernoulli kinematic condition [ 34] leads to and the ultimate flexural strength oUFs becomes (21) h where g is the curvature, and y the height ordinate h)(m+ D,h measured from the neutral axis(see Fig. 6) (m+1)m+5) Using eqn(18), we obtain the variation of the normalized tensile stress (m+2)(m+4) In Fig. 7 the evolution of the normalized fexural Σ=Y+∑nyC (22) strength is plotted as a function of the Weibull parameter m and the damage parameter D,. The with lower the value of the Weibull parameter, the higher ∑=0ady=Emy the normalized flexural strength. On the other hand the higher the value of the damage parameter, the higher the normalized flexural strength. These results The position of the neutral axis hi(see Fig. 6)is have been obtained differently in [13 but are ver determined from the force balance equation. The close to those presented herein ultimate flexural strength is reached when o(h, Lastly, when the only difference in the Ours and the position of the neutral axis is given by tension/compression behavior is given by the Young,s h=1+(1-D)1+2∑A4m (23) modulus difference(m→+∞) modeled by the 2≥1B(m)+ le D,, eqns (27)and(28) show that ratio of the ultimate flexural strength to the ultimate he ultimate flexural ensile strength 2/(1+Ⅵ1-D1) fS. h)l1-D,h 5 INTERACTION BETWEEN FLEXURE AND TENSION +1+3SA(m n1B(n)+2 (24) In this section we will study stress states where we The ratio of the ultimate flexural strength to the combine tensile and flexural loads. The beam hose used ultimate tensile strength is then given by characteristics are the same previous section. We still assume a bernoulli +1+3丝Am) hypothesis [34]. The kinematic condition leads to a n≥1B(n)+2 linear strain field in the beam of height h such that ()=r(y) Equation(25)shows that the ultimate strength ratio with /Ours is independent of the composite length, h but depends upon the damage parameter D, in addition to the parameters n, B(n)and A. This result constitutes a second difference with a weakest link The variable y now denotes the height ordinate hypothesis for which the average strength ratio measured from the mid-plane of the beam, and the
230 F. HiM, P. Feillard Fig. 6. Definition of the beam geometry in the case of pure flexure. assumed that there is a sufficient number of fibers in the width of the beam so that a global load sharing hypothesis is made for each height y. The Bernoulli kinematic condition [34] leads to g = gty (21) where ~ is the curvature, and y the height ordinate measured from the neutral axis (see Fig. 6). Using eqn (18), we obtain the variation of the normalized tensile stress with = y + ~' q,, yt~-) (22) and Y = ~y. The position of the neutral axis h~ (see Fig. 6) is determined from the force balance equation. The ultimate flexural strength is reached when ~(h0 = #urs and the position of the neutral axis is given by h=l+ (1-D 0 1+2~ 13(n)+l l" (23) hi n_~l Using the moment equation, the ultimate flexural strength ~'UFS is given by ~UFS /hl\2f 1 / h ~3 fsca = 2~-~ ) ~1--~1~-~1- 1 ) qtnA t3¢~)- 1 t ~ + 1 + ,>_l~(n) +2 ]" 3~'~ .... (24) The ratio of the ultimate flexural strength to the ultimate tensile strength is then given by j t l ---z-ff, Y, - 1 + 1 + 3 ~ 1 l~ (n) + 2 . m 6"VTS 1 + ~] qJ,,A ~¢')-1 n~l (25) Equation (25) shows that the ultimate strength ratio ~UFS/~UVS is independent of the composite length, but depends upon the damage parameter D1 in addition to the parameters q,,, fl(n) and A. This result constitutes a second difference with a weakest link hypothesis for which the average strength ratio O'FF/O'FT for the same volume loaded in pure flexure and in tension is given by Or Fr = [2(ms + 1)]~ (26) ~FT where O'FF is the average failure stress in pure flexure. To illustrate the previous results, a constant interfacial shear strength is assumed in the framework of global load sharing. By using the approximations to derive eqn (8), the position of the neutral axis is given by h ~/ (~+ 1)(m +4)(27) h, 1+ (1 D1) +2)(m+3) and the ultimate flexural strength 8UFS becomes O'UFS /h1\2/ 2 \m--~+ f 1 / h fSc -- 2~--) ~-~--~) [1__---S~-~-- 1) 3 + (m + 2)(m + " In Fig. 7 the evolution of the normalized flexural strength is plotted as a function of the Weibull parameter m and the damage parameter D1. The lower the value of the Weibull parameter, the higher the normalized flexural strength. On the other hand, the higher the value of the damage parameter, the higher the normalized flexural strength. These results have been obtained differently in [13] but are very close to those presented herein. Lastly, when the only difference in the tension/compression behavior is given by the Young's modulus difference (m--+ +m) modeled by the damage variable D1, eqns (27) and (28) show that ratio of the ultimate flexural strength to the ultimate tensile strength approaches 2/(1 + V1 - D1). 5 INTERACTION BETWEEN FLEXURE AND TENSION In this section we will study stress states where we combine tensile and flexural loads. The beam characteristics are the same as those used in the previous section. We still assume a Bernoulli hypothesis [34]. The kinematic condition leads to a linear strain field in the beam of height h such that (see Fig. 8) g(y) = ~- Y(y) (29) Lr with l+a 1-a h h Y(Y)= 2 + h y -2-<Y---2" The variable y now denotes the height ordinate measured from the mid-plane of the beam, and the
Strength properties of FRC's =●--D=0.2 1.6 -, 1.5 D.=0 1.4 --▲--D,=0.6 b 0 5 Shape para Fig. 7. Evolution of the normalized flexural strength as a function of the Weibull parameter m and the damage parameter D, constant a, which is less than or equal to 1, measures positive, the beam is in a pure tensile mode. Therefore the ratio of E(h /2)to E(h/2) the results are independent of the variable D,. The resultant force equation yields e(-h/2 e(h/2) Gn=4{1-a2+∑业 (33) When a lies between 0 and 1 then the whole beam undergoes tensile stresses, whereas negative values of The moment equation enables us to derive the a lead to a mixed tensile/compressive mode. The flexural strength to be tensile strength is obtained from a force balance equation and is defined as 3(1+a)+26A r+(1 N[G(h/2) OUTS (31) NaN(1- β(m)+ where n denotes the resultant force and b the width of the beam. The flexural strength is obtained from a When a is negative, a part of the beam is in moment balance and is defined as compression. In that case the results depend on the damage variable D,. The resultant force equation ≈6M[(h/2)=urs (32) here m denotes the resultant moment. when a is 0=1-c2-20-D)+21B0m)+1)035 and the moment equation enables us to derive the M efSA Efe/T ψnA D, (36) To illustrate the previous results, a constant Fig.8. Definition of the beam geometry in the case of interfacial shear strength is assumed. By using the flexure combined with tension approximations discussed to derive eqn 8), the
Strength properties of FRC's 231 1.6 1.5 x o ~ 1.4 1.3 --I'~ 1.2 o 1.1 N 0w.l --e-- D =0.2 1 ',' , .... i ~ D 1 = 0.3 I ~,~l, ~ --m--D 1 = 0.4 D l = 0.5 I • ~"~:k- ' " --a--D =0.6 • ,~ 1 ' - " ,,. ..... ,b.'. ~ "" "A .... _ ~. ~.........~ ----~ ..... .__. "" " .... 0--; 1 0 .... ; .... 1'0 .... 1'5 .... 20 o Z Shape parameter, m Fig. 7. Evolution of the normalized flexural strength as a function of the Weibull parameter m and the damage parameter D~. constant a, which is less than or equal to 1, measures the ratio of g(- hi2) to g(h/2) g(- h]2) t~ (30) g(h/2) When a lies between 0 and 1 then the whole beam undergoes tensile stresses, whereas negative values of lead to a mixed tensile/compressive mode. The tensile strength is obtained from a force balance equation and is defined a,; N[#(h/2) = ~vrs] O'T =- bh (31) where N denotes the resultant force, and b the width of the beam. The flexural strength is obtained from a moment balance and is defined as 6M[~(h/2) = ~VTS] ~F = bh 2 (32) where M denotes the resultant moment. When a is N4 Y -~N Fig. 8. Definition of the beam geometry in the case of flexure combined with tension. positive, the beam is in a pure tensile mode. Therefore the results are independent of the variable D~. The resultant force equation yields fS~A {1- a 2 q4,A t3('0-1 } I~T -- i5; 2 + ~ (1 -- a ~(")+1) (33) n~l fl(n) + 1 The moment equation enables us to derive the flexural strength to be 3(1 + 2fScA ~F -- a'~r + 1 - a (1 - a) 2 O,A¢(")-'(1 - O, fl(n)+2)'~ X {1 - a3 + 3~_> 1 fl(n) + 2 j. (34) When a is negative, a part of the beam is in compression. In that case the results depend on the damage variable D1. The resultant force equation allows us to get the tensile strength fScA a: ~,A~(")2~ ~r-l_-a{~ 2(1-D1) +.~>, fl(n)+lJ (35) and the moment equation enables us to derive the flexural strength 3(1 - a) _ 2fScA -- --Or T .-[--- t?r 1 -- a (1 -- a) 2 ot a q&A¢(")- 11 --- 3~ .... (36) x[1 I_D+ ._~,t~(n)+eJ" To illustrate the previous results, a constant interfacial shear strength is assumed. By using the approximations discussed to derive eqn (8), the
F. Hild, p. feillard following results are obtained. When a is positive, the plastic material. In that case the interaction is given by resultant force equation leads to U fS.(2 Lastly, when the only difference in the 1-a(m+2 (m+2)(m+3)(37) tension/compression behavior is given by the Young's modulus difference(m→+∞) modeled by the and the moment equation damage variable DI, the interaction diagram is only a function of the parameter D,(see Fig 9). This last 3(1+a) 2fs 2 esult constitutes a lower bound in the interaction local load sharing regime nteraction 2)(m+4) has no meaning. However, one can study the combined effect of tension and pure flexure the When a is negative, the resultant force equation strain field defined in eqn(29)is used. The average llows us to get tensile strength failure stress ratio for the same volume is given by 2 (m+1)(m+4) 2(m+2)(m+3)2(1-D)(39) where or is the average failure stress under combined loading conditions, and Hm. is the Weibull stress and the moment equation enables us to derive the heterogeneity factor [35]defined as flexural strength +a) 2 a)(m,+1) when a≥ (m+1)(m+5)a3 when a <o (m+2(m+4)1-D」(40) (1-a)(m+1) In Fig. 9 an interaction diagram is plotted when m=4, and D,= 2/7 for a constant interfacial shea 6 ULTIMATE THREE-POINT FLEXURAL strength. The interaction diagram is also compared to STRENGTH a linear prediction: in that particular case a straight line describes very well the interaction line(see dotted The aim of this section is to derive the ultimate line in Fig 9). This is one difference when we compare strength of rectangular beams loaded in three-point the same hypotheses as those made in the previous sections. However, in three-point flexure the stress field profile along the fiber direction is no longer 14 uniform. Therefore, eqn (4) cannot be used m=4: Independent Throughout this section, the longitudinal stress field m→∞:ofD1(a20) o(T, 2), is assumed to be symmetric about z=0 and 苏1 maximum at z=0. Depending on the position of the 0.8 fiber along the y-axis, the stress field will vary. In the first part of the section, only a set of fibers is analyzed 0.6 for which the position along the y-axis is the same. A 0.4 global load sharing regime is assumed in a plane Dependent normal to the y-axis, for each value of y onD1(<0) Furthermore, the beam length is assumed to be sufficiently large with respect to the beam height and 00.20.40.60.8 width so that the shear stresses can be neglected For Normalized Tensile Strength,OT the sake of simplicity, the interfacial sliding is assumed constant. The contribution of the broke diagram(solid line) between fibers within the recovery length or pull-out stress extre flexural Foo, at z=0 is written as ashed line shows the interaction diagram (7) ap(t dP(L, LR(o)
232 F. Hild, P. Feillard following results are obtained. When a is positive, the resultant force equation leads to l_ m+3 ] Or-l-ol\m+2/ [ 2 (m+2)-~+3) and the moment equation 5F- 3(1 + a) _ 2f& {2__.~] ''-h+l O'T + (1 - a)2km + 2/ 3(1-0~m+4) ] x [1- 0/3 (m ; 2-)g 74) J" (37) (38) When a is negative, the resultant force equation allows us to get the tensile strength O'r (l---a) \m +2/ × [ (m +4) 1 (39) L2(m +2)(m +3) 2(I~-D,)I and the moment equation enables us to derive the flexural strength 3(1 + a) 2f& [2~ .,+--% OF 1-a (1---~)2 \m + 2/ 1--/ +2)(m+4) l-D1 ' In Fig. 9 an interaction diagram is plotted when m = 4, and D~ = 2/7 for a constant interracial shear strength. The interaction diagram is also compared to a linear prediction: in that particular case a straight line describes very well the interaction line (see dotted line in Fig. 9). This is one difference when we compare to the interaction diagram of an elastic-perfectly ,=" ~o e- "O t,q O Z 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 m = 4 I Independent i ~...".'.'.'.'.'.~ m...~ool I of Dl(t~->O ) : "~. ~ ~ i '-%. i f _%pen%nt f °nDl(~ 0 (42) 1 H,,, (1 - a)(m, + 1) when a < 0. (43) 6 ULTIMATE THREE-POINT FLEXURAL STRENGTH The aim of this section is to derive the ultimate strength of rectangular beams loaded in three-point flexure. The tensile and compressive behavior follows the same hypotheses as those made in the previous sections. However, in three-point flexure the stress field profile along the fiber direction is no longer uniform. Therefore, eqn (4) cannot be used. Throughout this section, the longitudinal stress field, cr(T,z), is assumed to be symmetric about z = 0 and maximum at z = 0. Depending on the position of the fiber along the y-axis, the stress field will vary. In the first part of the section, only a set of fibers is analyzed for which the position along the y-axis is the same. A global load sharing regime is assumed in a plane normal to the y-axis, for each value of y. Furthermore, the beam length is assumed to be sufficiently large with respect to the beam height and width so that the shear stresses can be neglected. For the sake of simplicity, the interfacial sliding resistance is assumed constant. The contribution of the broken fibers within the recovery length or pull-out stress, ~po, at z = 0 is written as ffpo(T)_ foT~p(t) dPF(t~n(t)) dt (44) f
Strength properties of frc's where p()denotes the average stress at z=0 when a Y(T)is the failure probability density associated with fiber breaks at a location z, and at a reference stress the length increase from LR(t) to Lr(T)+ lr(t)dt. level t, and PF(t, LR(i)) is the failure probability that a The first term is identical to that given in [36, 37, 18] fiber break within the recovery length LR(O)at or when using a Weibull law, the second term is due to below a reference stress t. In tension, the probability the load dependence of the recovery length LR(T). By density of fracture locations is uniform, therefore using eqns (46)and (47), the expressions of the op(0=t/2. On the other hand, when the stress field functions and are, respectively, given by along the fiber direction is not homogeneous, the previous result does not apply, and one needs to Φ(T,z)=-{1-P(T,LR(T) determine the average fracture location to estimate the pull-out stress. The average stress o applied to the composite at the plane z=0, is given by [10] [n(1-PPo(T, z)lI G=T{1-P2(T,L(7)+2(i()P(T,L(T)(45) v(T)=-{1-P(T,LR(T) LR(T) The first term of the right hand side of eqn (45) corresponds to the contribution of the unbroken fibers, and the second term is the pull-out contribution (1n(1-Pr(T, l2 )I (50) (stress opo(T)). The quantity (h(r))represents the average pull-out length, or average fracture location in When LR(T) is independent of T, LR(T)/2 when the reference stress is less than or PPo(T, LR(T)/2) vanishes, y(r) vanishes as well, and equal to T. The aim of the present calculation is to the results given in [37] still apply in the case of a evaluate the average fracture location(h(T))when the Weibull law. By dimensional analysis and by stress filed along the fiber direction is not inspection of eqn (47), the average (h(T))of the homogeneous. fracture location h(T)for a piece of composite of Oh and Finnie [36] derived the average fracture length LR(T) at a reference stress T is [38] location of a brittle material. In that case the Le(t》2 considered volume is constant. In the present case, the zΦ(t,z)dz+Lk()u()dr length to consider(i.e the recovery length) is varying with the stress level measured by T. Therefore the (h(T) (51) results need to be generalized. By recasting the z)dz +y(t cumulative failure probability PF(T, LR(T)) of a part of length LR(T), useful functions are exhibited to This expression of the average fracture location determine the average fracture location. Under the does not consider the fibers that originally broke weakest link assumption and the independent events outside the recovery length but were brought into it as hypothesis, the cumulative failure probabilit the load level increased. The approximation made PA(T, LR(T)) of a piece of fiber of length LR(T)is here is worst when the stress field is constant over the LR(T)2 however, it can be shown that this hypothesis is not P(r(T)=1-x[m(1-P(rx) very strong [13.E quation(4 5)is used for each height y to determine the global stress state of the composite in three-point flexure. By studying the most loade where Po(T, z)denotes the cumulative failure fibers in the tensile part, it can be shown that when probability of a single link of length Lo whose the specimen length is greater than 5 times the mid-section is located at z. The failure probability characteristic length 8, the gradient effect in density is obtained by differentiation of eqn(46)with three-point fiexure along the fiber direction is respect to T negligible when compared to the one due to load recovery in the vicinity of a fiber break [39 Under dPF(T, LR(T)) 2@(T, z)dz+2(T)(47) these circumstances the ultimate strength in three point flexure Gu3F is on the same order of magnitude o that eqn(46) can be rewritten as the ultimate flexural strength our TUBF E UES when L> 58. (52) PF(T, LRTD p(t, z )dz+y(oldr (48) This result constitutes a major difference with where p(t, z) corresponds to the failure probability materials for which a weakest link concept can be density for a reference stress varying between T and used. In that case the ratio of the ultimate strength in T+dT, of an element of length dz centered at z, and three-point flexure is greater than that in pure flexure
Strength properties of FRC's 233 where 6"p(t) denotes the average stress at z = 0 when a fiber breaks at a location z, and at a reference stress level t, and PF(t,LR(t)) is the failure probability that a fiber break within the recovery length LR(t) at or below a reference stress t. In tension, the probability density of fracture locations is uniform, therefore ~p(t) = t/2. On the other hand, when the stress field along the fiber direction is not homogeneous, the previous result does not apply, and one needs to determine the average fracture location to estimate the pull-out stress. The average stress # applied to the composite at the plane z = 0, is given by [10] = T{1 - PF(T, Ln(T))} + ~(h(T))PF(T, LR(T)). (45) f The first term of the right hand side of eqn (45) corresponds to the contribution of the unbroken fibers, and the second term is the pull-out contribution (stress O-po(T)). The quantity (h(T)) represents the average pull-out length, or average fracture location in LR(T)/2 when the reference stress is less than or equal to T. The aim of the present calculation is to evaluate the average fracture location (h(T)) when the stress filed along the; fiber direction is not homogeneous. Oh and Finnie [36] derived the average fracture location of a brittle material. In that case the considered volume is constant. In the present case, the length to consider (i.e., the recovery length) is varying with the stress level measured by T. Therefore the results need to be generalized. By recasting the cumulative failure probability PF(T,LR(T)) of a fiber part of length LR(T), useful functions are exhibited to determine the average fracture location. Under the weakest link assumption and the independent events hypothesis, the cumulative failure probability PF(T,LR(T)) of a piece of fiber of length LR(T) is expressed as [ '? rLR(T,/2 ] PF(T, LR(T)) = 1-exp ~,~ J. In{1 - PFo(r,z)}dz (46) where PFo(T,z) denotes the cumulative failure probability of a single link of length Lo whose mid-section is located at z. The failure probability density is obtained by differentiation of eqn (46) with respect to T dPF(T, Ln(T)) fL,,r)/2 -- JO 2~(T,z)dz + 2qJ(T) (47) dT so that eqn (46) can be rewritten as "o~T I(Lt~(t)/2 +tla(t)}dt (48) PF(T, LR(T)) = 2 ° dP(t,z)dz where ~(t,z) corresponds to the failure probability density for a reference stress varying between T and T +dT, of an element of length dz centered at z, and W(T) is the failure probability density associated with the length increase from LR(T) to LR(T) + L'n(T)dT. The first term is identical to that given in [36, 37, 18] when using a Weibull law, the second term is due to the load dependence of the recovery length LR(T). By using eqns (46) and (47), the expressions of the functions • and W are, respectively, given by 1 • (T,z) = - {1 - PF(T, LR(T))}-~o 0 X ~-~ [ln{1 - Ppo(T,z)}] (49) L'n( T) W(T) = - {1 - PF(T, Lk(T))} -~o When LR(T) is independent of T, or Peo(T, LR(T)/2) vanishes, U~(T) vanishes as well, and the results given in [37] still apply in the case of a Weibull law. By dimensional analysis and by inspection of eqn (47), the average (h(r)) of the fracture location h(T) for a piece of composite of length LR(T) at a reference stress T is [38] fr f fLR(t)/2 Jo tJo + i Lg(t)tIJ(t)}dt z.(t,z z (h(T)) (T f fL.U)/2 W(t) ]dt~ (51) Jo U0 This expression of the average fracture location does not consider the fibers that originally broke outside the recovery length but were brought into it as the load level increased. The approximation made here is worst when the stress field is constant over the whole length (i.e, pure tension). In that case, however, it can be shown that this hypothesis is not very strong [13]. Equation (45) is used for each height y to determine the global stress state of the composite in three-point flexure. By studying the most loaded fibers in the tensile part, it can be shown that when the specimen length is greater than 5 times the characteristic length 3c, the gradient effect in three-point flexure along the fiber direction is negligible when compared to the one due to load recovery in the vicinity of a fiber break [39]. Under these circumstances the ultimate strength in threepoint flexure O-U3F is on the same order of magnitude as the ultimate flexural strength 6"VFS O'U3F ~ ~UFS when L > 5~c. (52) This result constitutes a major difference with materials for which a weakest link concept can be used. In that case the ratio of the ultimate strength in three-point flexure is greater than that in pure flexure
234 Hild. P. feillard Table 1. Comparison between experimental (Exp. and predicted (Pred )ultim ate strengths expressed in MPa for three composites reinforced by SiC fibers Tension Pure Flexure Three-point Flexure Exp Pred Exp. SiC/LAS 455 SiC/Al 700 When the two volumes are identical, the average some similar features can be shown, in particular the failure stress ratio is given by increase in terms of normalized ultimate strength. This case corresponds to a situation where the 'ductility'is 2=(m,+1) (53) distributed within a large region of the structure the other hand, when the 'ductility'is confined within here dB is the average failure stress in three-point a very small area of the structure the increase in terms of the ultimate strength is expected to be less The previous results are applied to three composites important, though the gradients along the fiber reinforced by SiC fibers (Table 1). Two of the three direction could play a role and increase it again composites have ultimate strengths tollowing a global ultimate strength has the same order of magnitude as load sharing hypothesis(LAS and C matrices). This that in pure flexure within the framework of global result is related to the fact that matrix cracking has load sharing. On the other hand, in the local load saturated and that the interfacial shear strength is sharing regime, the ultimate strength in three-point small (on the order of a few MPa). On the other hand, the aluminum matrix leads to ultimate strengths flexure increases as compared with pure flexure when described by a local load sharing hypothesis(m, =7) the two volumes are identical. The three-point flexure experiments therefore allow to discriminate the two and to higher values of the interfacial shear strength load transfer regimes when compared with the pure flexure case. This result can be observed when ultimate strengths of LAS and C matrix composites 7 CONCLUSIONS global load sharing regime)are ompared with Al matrix composites(exhibiting local load sharing regime) subjected to pure tension A unified approach to the prediction of the ultimate pure flexure and three-point flexure tensile strength of FRCs has been proposed. The key quantity to consider is the average fragment length Expressions of the ultimate tensile strength are REFERENCES derived within this general framework and compared with other existing theories. In particular, in the global 1. Netravali, AN,Henstenburg,RB load sharing regime it is shown that the ultimate tensile strength is mostly length-independent. Con single-filament-composite test. Part I graphite fibers in epoxy Polymer Con versely, in the local load sharing regime, the ultimate 226-241 tensile strength is always length-dependent. A 2. Thouless, M D, Sbaizero, O, Sigl, L.S. and Evans description of the behavior of these FRCs up to the A.G., Effect of interface mechanical properties on ultimate tensile point has been derived in th in a siC-fiber- reinforced lithium aluminium glass-ceramic. Journal of the American Ceramic framework of Continuum mechanics A generalization to pure flexural modes is proposed to derive the ultimate flexural strength. In th and Curtin, W.A.. n8, S.M., Evans, A.G.,Mosher, P. load sharing regime, the increase in terms of posites reinforced with nicalon flexural strength is due to the conjunction phenomena: difference in elastic moduli in tension and 4. Gucer, D.E. and Gurlan in compression, and 'ductility' due to fiber breakage atistics of two fracture models. Journal of the and fiber pull-out. It has been shown that the Mechanics and Physics of solids, 1962, 10, 365-373 ductility contributes substantially to this increase. 5. Pierce, F.T., Tensile tests for cotton yarns V: The An interaction between tension and flexure is weakest link 'theorems on the strength of long and of composite specimens. Journal of the Textile institute, studied. Although some differences are noticed when 1926,17,T355-T368 compared to an elastic perfectly plas 6. Daniels, H.E., The statistical theory of the strength
234 F. Hild, P. Feillard Table 1. Comparison between experimental (Exp.) and predicted (Pred.) ultimate strengths expressed in MPa for three composites reinforced by SiC fibers Loading Tension Pure Flexure Three-point Flexure Material Exp. Pred. Exp. Pred. Exp. Pred. SiC/LAS 4° 790 850 1050 1080 1180 1090 SiC/C 41 345 340 455 430 455 435 SiC/A142 700 690 815 820 930 930 When the two volumes are identical, the average failure stress ratio is given by = (ms + 1)'1', (53) O'FF where ~F3 is the average failure stress in three-point flexure. The previous results are applied to three composites reinforced by SiC fibers (Table 1). Two of the three composites have ultimate strengths following a global load sharing hypothesis (LAS and C matrices). This result is related to the fact that matrix cracking has saturated and that the interracial shear strength is small (on the order of a few MPa). On the other hand, the aluminum matrix leads to ultimate strengths described by a local load sharing hypothesis (m, = 7). This matrix is more sensitive to localized fiber failures and to higher values of the interfacial shear strength. 7 CONCLUSIONS A unified approach to the prediction of the ultimate tensile strength of FRCs has been proposed. The key quantity to consider is the average fragment length. Expressions of the ultimate tensile strength are derived within this general framework and compared with other existing theories. In particular, in the global load sharing regime it is shown that the ultimate tensile strength is mostly length-independent. Conversely, in the local load sharing regime, the ultimate tensile strength is always length-dependent. A description of the behavior of these FRCs up to the ultimate tensile point has been derived in the framework of Continuum Mechanics. A generalization to pure flexural modes is proposed to derive the ultimate flexural strength. In the global load sharing regime, the increase in terms of ultimate flexural strength is due to the conjunction of two phenomena: difference in elastic moduli in tension and in compression, and 'ductility' due to fiber breakage and fiber pull-out. It has been shown that the 'ductility' contributes substantially to this increase. An interaction between tension and flexure is studied. Although some differences are noticed when compared to an elastic perfectly plastic behavior, some similar features can be shown, in particular the increase in terms of normalized ultimate strength. This case corresponds to a situation where the 'ductility' is distributed within a large region of the structure. On the other hand, when the 'ductility, is confined within a very small area of the structure, the increase in terms of the ultimate strength is expected to be less important, though the gradients along the fiber direction could play a role and increase it again. Lastly, in the case of three-point flexure, the ultimate strength has the same order of magnitude as that in pure flexure within the framework of global load sharing. On the other hand, in the local load sharing regime, the ultimate strength in three-point flexure increases as compared with pure flexure when the two volumes are identical. The three-point flexure experiments therefore allow to discriminate the two load transfer regimes when compared with the pure flexure case. This result can be observed when ultimate strengths of LAS and C matrix composites (exhibiting a global load sharing regime) are compared with A1 matrix composites (exhibiting a local load sharing regime) subjected to pure tension, pure flexure and three-point flexure. REFERENCES 1. Netravali, A.N., Henstenburg, R.B., Phoenix, S.L. and Schwartz, P., Interracial shear strength studies using the single-filament-composite test. Part I: Experiments on graphite fibers in epoxy. Polymer Composites, 1989, 10, 226-241. 2. Thouless, M.D., Sbaizero, O., Sigl, L.S. and Evans, A.G., Effect of interface mechanical properties on pullout in a SiC-fiber-reinforced lithium aluminium silicate glass-ceramic. Journal of the American Ceramic Society, 1989, 72, 525-532. 3. Heredia, F.E., Spearing, S.M., Evans, A.G., Mosher, P. and Curtin, W.A., Mechanical properties of carbon matrix composites reinforced with nicalon fibers. Journal of the American Ceramic Society, 1992, 75, 3017-3025. 4. GUcer, D.E. and Gurland, J., Comparison of the statistics of two fracture models. Journal of the Mechanics and Physics of Solids, 1962, 10, 365-373. 5. Pierce, F.T., Tensile tests for cotton yarns V: The 'weakest link' theorems on the strength of long and of composite specimens. Journal of the Textile Institute, 1926, 17, T355-T368. 6. Daniels, H.E., The statistical theory of the strength of