G N. Morscher, J D Cawley/ Journal of the European Ceramic Society 22(2002) 2777-2787 2783 maximum time and is the fraction of fibers that failed tested in rupture have several matrix cracks exposed to prior to the time of embrittlement. The number of the hot zone depending on the crack-density associated embrittled fiber failures can then be determined from with the applied stress condition. An increase in crack the product of the number of fibers per unit thickness, density increases the effective gage length of strongly Nx, and the integration of emb over the maximum bonded fibers, and thereby the fraction of fiber failures depth of oxidation embrittlement, To account for this, all that is required is to multiply femb by the number of cracks, Nc, which results in NcNemb(single crack) (23) NxCe-XGa)1-2m (18) where Nemb( single crack) This equation is valid up to the time that all of the is the solution to either Eq(18)or(19), depending on fibers throughout the entire matrix cross-section have the time and Nc can be found from been embrittled, femb. If no embrittled fibers failed prior to the time it takes to strongly bond all of the fibers in Nc=peg (24) through-thickness crack. the number of embrittled fiber failures would be equal to where Lg is the gage length, i.e. the length of specimen in the hot zone N盐bm=Mm+Nr(中1-a) 3. 2. Applying the model It was decided to model two regimes of rupture (m/n-im/n for t>temb (19) behavior rather than the entire rupture curve:(a)"oxi dation kinetics"controlled rupture and(b)"single fiber failure""controlled rupture. The first case is where many where N is the total number of fibers in the composite fibers break during the rupture condition due to higher cross-section applied stresses and high crack densities. The latter When a strongly bonded fiber fails at time tail, it will condition is the case where the first embrittled fiber to be assumed that all of the embrittled fibers fail in the break in a composite causes ultimate composite failure. cross-section of the matrix crack. Then, if the load shed The entire rupture curve could be modeled similar to onto the remaining pristine fibers cannot be carried by Lara-Curzio in an iterative fashion where a computer the remaining fibers, the composite will rupture. The program essentially continues to increase time in dis- ultimate stress-criterion 19 for most conditions will be crete steps and solves the above equations in order to for the case of a composite that is not crack-saturated determine if the equation offal> Cult is fulfilled for a given stress/crack-density condition. However, for the Cult ace/m+l; for 28<P-1 (20) case of several matrix cracks in a hot zone. after first fiber failure, each crack has to be treated independently where ooft. T) must be used as the reference stress in the and there exists an intermediate stress range where letermination of o. [Eq (4)]. If the matrix were saturated higher applied stresses will yield longer rupture times with matrix cracks Eq.(13)would be used. The stress than lower applied stress conditions. b This may actually remaining on pristine fibers can be determined from: be indicative of some of the scatter in rupture results; Fail (21) however. what is usually considered to be of greatest importance is the"run-outstress condition, i.e. case b For example, for a higher stress condition, first embrittled fiber failure occurs at a time less than at a lower stress condition. however 2 the shorter-time-first-embrittled-fiber-failure may not be at a condition where ultimate composite failure would occur, i.e. x is too small to satisfy oil >Cult, and one would have to"wait"until another fiber fails in that specific matrix crack for ultimate failure to occur. It is If affil>cult, then the composite fails possible that the time for the lower stress condition first embrittled fiber failure to be less than the second fiber failure of the higher stress One further consideration must be taken into condition and sufficient for ultimate composite failure. 23(It should be account. All of the analysis up to this point has been for noted that the determination of Emb in Ref 23 is inappropriate under the case of a single matrix crack. Most of the specimens this scenario, the analysis used here should be used instead.maximum time and t is the fraction of fibers that failed prior to the time of embrittlement. The number of embrittled fiber failures can then be determined from the product of the number of fibers per unit thickness, N x; and the integration of emb over the maximum depth of oxidation embrittlement, xmax: Nt<temb femb ¼ N x ðxmax 0 embdx ¼ N xC Cm frupture C2m=n ox x 2m ð Þ n þ1 max 1 1 2m n þ 1 2 6 4 3 7 5 ð18Þ This equation is valid up to the time that all of the fibers throughout the entire matrix cross-section have been embrittled, temb. If no embrittled fibers failed prior to the time it takes to strongly bond all of the fibers in a through-thickness crack, the number of embrittled fiber failures would be equal to: Nt>temb femb ¼ Nt<temb femb þ Nf t temb   ¼ Nt<temb femb þ Nf K Cm frupture t m=n t m=n emb  ; for t>temb ð19Þ where Nf is the total number of fibers in the composite cross-section. When a strongly bonded fiber fails at time tffail, it will be assumed that all of the embrittled fibers fail in the cross-section of the matrix crack. Then, if the load shed onto the remaining pristine fibers cannot be carried by the remaining fibers, the composite will rupture. The ultimate stress-criterion 19 for most conditions will be for the case of a composite that is not crack-saturated:
ult ¼
ce1=mþ1 ; for 2<1 c ð20Þ where so(t,T) must be used as the reference stress in the determination of sc [Eq. (4)]. If the matrix were saturated with matrix cracks Eq. (13) would be used. The stress remaining on pristine fibers can be determined from:
ffail ¼
f 1femb ð21Þ where femb ¼ 2xtffail b ð22Þ If sf fail>sult, then the composite fails. One further consideration must be taken into account. All of the analysis up to this point has been for the case of a single matrix crack. Most of the specimens tested in rupture have several matrix cracks exposed to the hot zone depending on the crack-density associated with the applied stress condition. An increase in crack density increases the effective gage length of strongly bonded fibers, and thereby the fraction of fiber failures. To account for this, all that is required is to multiply femb by the number of cracks, Nc, which results in: Nfemb ¼ NcNfembð Þ single crack ð23Þ where Nfembð Þ single crack is the solution to either Eq. (18) or (19), depending on the time, and Nc can be found from: Nc ¼ cLg ð24Þ where Lg is the gage length, i.e. the length of specimen in the hot zone. 3.2. Applying the model It was decided to model two regimes of rupture behavior rather than the entire rupture curve: (a) ‘‘oxi￾dation kinetics’’ controlled rupture and (b) ‘‘single fiber failure’’ controlled rupture. The first case is where many fibers break during the rupture condition due to higher applied stresses and high crack densities. The latter condition is the case where the first embrittled fiber to break in a composite causes ultimate composite failure. The entire rupture curve could be modeled similar to Lara-Curzio 8 in an iterative fashion where a computer program essentially continues to increase time in dis￾crete steps and solves the above equations in order to determine if the equation sf fail>sult is fulfilled for a given stress/crack-density condition.23 However, for the case of several matrix cracks in a hot zone, after first fiber failure, each crack has to be treated independently and there exists an intermediate stress range where higher applied stresses will yield longer rupture times than lower applied stress conditions.b This may actually be indicative of some of the scatter in rupture results; however, what is usually considered to be of greatest importance is the ‘‘run-out’’ stress condition, i.e. case b For example, for a higher stress condition, first embrittled fiber failure occurs at a time less than at a lower stress condition. However, the shorter-time-first-embrittled-fiber-failure may not be at a condition where ultimate composite failure would occur, i.e. x is too small to satisfy sffail>sult, and one would have to ‘‘wait’’ until another fiber fails in that specific matrix crack for ultimate failure to occur. It is possible that the time for the lower stress condition first embrittled fiber failure to be less than the second fiber failure of the higher stress condition and sufficient for ultimate composite failure.23 (It should be noted that the determination of Øemb in Ref.23 is inappropriate under this scenario, the analysis used here should be used instead.) G.N. Morscher, J.D. Cawley / Journal of the European Ceramic Society 22 (2002) 2777–2787 2783