1718 R H. Jones, C.H. HenagerJr /Journal of the European Ceramic Sociery 25(2005)1717-1722 2. Suberitical crack growth in SiC/Sic Matrix Matrix 2.1. Mechanisms and model development Bonded Pacific Northwest National Laboratory (PNNL) wa mong the first to identify and study time-dependent bridg- ing in ceramic composites-'and we have proposed a crack growth mechanism map based on available experimental data as a function of temperature and oxygen partial pressure for continuous fiber composites with carbon interphases An approach to modeling dynamic time-dependent crack bridging has emerged from the work of Buckner and Rice crack based on the use of weight-functions to calculate crack opening opening displacements. Once a relationship . between V displacement crack-opening displacement and bridging tractions from crack-bridging elements is included, a governing integral Fig. 1. Schematic of frictionally bonded fiber acted upon by force Pr due to equation is obtained that relates the total crack opening, and debonding and sliding across a Mode I crack with opening ut. The lengths the bridging tractions, to the applied load. The solution of deb and l free are shown. this equation gives the force on the crack-bridges and the crack-opening displacement everywhere along the crack CG and Hi-Nicalon fibers as and applied it to a variety of time-dependent bridging cases for linear creep laws 12-14. Recently, a more appropriate Ef=Ao"rPexP(RT bridging relation for creeping fibers has been developed by Cox et al. that provides axial and radial stresses in creeping C(T)otP= Ef=C()o"ptP- fibers with linear and non-linear creep laws where A is a constant, t is time, Q is an activation energy for creeping fibers that also considers the case for interface re- creep, Ris the gas constant and T is temperature, n is the stress moval due to oxidation. We treat discrete fiber bridges as exponent, p is the time-temperature exponent, o is stress, Et opposed to a bridging force distribution because each fiber Is the fiber strain and C(T)is A exp(-POIRT). The unbonded has a different environmental exposure. A non-linear (in time length given by lfree creeps at the bridging stress applied to and stress)creep law is used to compute bridge extensions the fiber. The portion of fiber that is debonded creeps at a stress that varies over the debond length, which is found by 2.2. Enironmental effects: model and mechanism map integration. The creep extension of a bridge becomes 2. 1. Compliance of a frictionally bondedfiber(bridge △lfte=C(T) We introduce an expression for the compliance of a bridg- ing fiber, b, using the assumptions of frictional bonding with a weak, debonding interphase. The frictionally bonded fiber for the free length of fiber and involves an unbonded or free length, Ifree, and a debonded or frictionally bonded and sliding length, deb, for a fiber bridg- Aldeb =C(T)prP-1At0(2)dz=C(T)prP-A ng a crack of opening ur, as shown in Fig. I The free length of fiber, which is not subject to fric- tional forces, is assigned a compliance, pb, and the portion (1+E)(1+n) of the fiber subjected to a frictional sliding resistance along length Deb due to Pf (Fig. I)is assigned a non-linear compli for the debonded length when the axial fiber stress is given ance,b, such that bb=pbPb+Pb[15], where Pb is the by 2D normalized bridge force and 8b, is the bridge displace dependent debonding from fiber contraction due to Poisson o()=A 2rz ment. As suggested by Cox et al. one could include a time- effect and creep deformation, which provides an additional time-dependent fiber compliance term where t is the total creep time, At is the time step, is the axial distance along the fiber measured from the crack face. 2.2.2. Fiber relaxation (FR) due to creep t is the shear stress and 5 is given by Ef/(1-NEm. Time, t, The time-dependent extension of a fiber bridge at elevated is referenced to the inception of each bridge into the bridging temperatures obeys a power-law creep equation for Nicalon- zone and is tracked separately for each bridge1718 R.H. Jones, C.H. Henager Jr. / Journal of the European Ceramic Society 25 (2005) 1717–1722 2. Subcritical crack growth in SiC/SiC 2.1. Mechanisms and model development Pacific Northwest National Laboratory (PNNL) was among the first to identify and study time-dependent bridg￾ing in ceramic composites1–3 and we have proposed a crack growth mechanism map based on available experimental data as a function of temperature and oxygen partial pressure for continuous fiber composites with carbon interphases4. An approach to modeling dynamic time-dependent crack bridging has emerged from the work of Buckner ¨ 5 and Rice6 based on the use of weight-functions to calculate crack￾opening displacements7. Once a relationship8,9 between crack-opening displacement and bridging tractions from crack-bridging elements is included, a governing integral equation is obtained that relates the total crack opening, and the bridging tractions, to the applied load. The solution of this equation gives the force on the crack-bridges and the crack-opening displacement everywhere along the crack face5–7,10. Begley et al.11 first developed a dynamic model and applied it to a variety of time-dependent bridging cases for linear creep laws12–14. Recently, a more appropriate bridging relation for creeping fibers has been developed by Cox et al.9 that provides axial and radial stresses in creeping fibers with linear and non-linear creep laws. We have developed a similar bridging law for non-linear creeping fibers that also considers the case for interface re￾moval due to oxidation. We treat discrete fiber bridges as opposed to a bridging force distribution because each fiber has a different environmental exposure. A non-linear (in time and stress) creep law is used to compute bridge extensions. 2.2. Environmental effects: model and mechanism map 2.2.1. Compliance of a frictionally bonded fiber (bridge) We introduce an expression for the compliance of a bridg￾ing fiber,Φb, using the assumptions of frictional bonding with a weak, debonding interphase. The frictionally bonded fiber involves an unbonded or free length, lfree, and a debonded or frictionally bonded and sliding length, ldeb, for a fiber bridg￾ing a crack of opening ut, as shown in Fig. 1. The free length of fiber, which is not subject to fric￾tional forces, is assigned a compliance, Φl b, and the portion of the fiber subjected to a frictional sliding resistance along length ldeb due to Pf (Fig. 1) is assigned a non-linear compli￾ance, Φn b, such that δb = Φl bPb + Φn bP2 b [15], where Pb is the 2D normalized bridge force and δb, is the bridge displace￾ment. As suggested by Cox et al.9 one could include a time￾dependent debonding from fiber contraction due to Poisson effect and creep deformation, which provides an additional time-dependent fiber compliance term. 2.2.2. Fiber relaxation (FR) due to creep The time-dependent extension of a fiber bridge at elevated temperatures obeys a power-law creep equation for Nicalon￾Fig. 1. Schematic of frictionally bonded fiber acted upon by force Pf due to debonding and sliding across a Mode I crack with opening ut. The lengths ldeb and lfree are shown. CG and Hi-Nicalon fibers as: εf = Aσnt p exp
−pQ RT = C(T )σnt p ⇒ ε˙f = C(T )σnptp−1 (1) where A is a constant, t is time, Q is an activation energy for creep, R is the gas constant and T is temperature, n is the stress exponent, p is the time–temperature exponent, σ is stress, Ef is the fiber strain and C(T) is A exp(−pQ/RT). The unbonded length given by lfree creeps at the bridging stress applied to the fiber. The portion of fiber that is debonded creeps at a stress that varies over the debond length, which is found by integration. The creep extension of a bridge becomes: lfree = C(T )ptp−1t
Pb 2rf n lfree (2) for the free length of fiber and ldeb = C(T )ptp−1t ldeb 0 σn(z)dz = C(T )ptp−1t ×
Pb 2rf n ldeb
1 − 1 (1 + ξ) n(1 + n) (3) for the debonded length when the axial fiber stress is given by: σ(z) = Pb 2rf − 2τz rf (4) where t is the total creep time, t is the time step, z is the axial distance along the fiber measured from the crack face, τ is the shear stress and ξ is given by fEf/(1 − f)Em. Time, t, is referenced to the inception of each bridge into the bridging zone and is tracked separately for each bridge.