Y.-C. Chiang/ Engineering Fracture Mechanics 6.5(2000)15-28 ,、+一E aVm EmEr/avmt VEcts 4VFETEc The debonding length ld is then given by aVmEmo avmEm Ersd Eτ2 For the case of a purely frictional interface (i.e, Sd=0), the debond g length la equals the frictional slipping length and it is expressed as lasavmEmo The expression of Eq. (24)is consistent with the length at which the fiber stress at the crack plane transfers back to the matrix in the ACK model. It also can be seen from Eq .(23) that the inclusion of nterface bonding will decreases the debonding length To initiate the crack-wake debonding process (i.e, la>0), the interface debonding energy Sd should satisfy aLmemar VERE 4. Matrix cracking stress The energy relationship to evaluate the steady-state matrix cracking stress is expressed as [5] [=-9(+xR后广()2m veLd where Sm is the critical strain energy release rate of the matrix, Gm is the matrix shear modulus and the radius of the matrix R can be expressed as a/vf. The contribution of the shear energy term in Eq (26)was neglected in the ACK model. It was verified that this negligence is well accepted for the slipping length larger than a few fiber radii [5]. Following the ACK model, the contribution of shear energy is neglected in the present analysis. Substituting the fiber and matrix stresses of Eqs. (9),(10)and (4),(5)and the debonding length of Eq (23)into Eq(26), the energy balance equation leads to the form of A1()2+A2(0)a+A3(0)=0 (28a)l 2 d ÿ aVmEms VfEcts ld ‡ aVmEmEf Ect2 s " aVmEms2 4V 2 f EfEc ! ÿ zd # ˆ 0 …22† The debonding length ld is then given by ld ˆ aVmEms 2VfEcts ÿ  aVmEmEfzd Ect2 s s …23† For the case of a purely frictional interface (i.e., zd ˆ 0), the debonding length ld equals the frictional slipping length and it is expressed as ld ˆ aVmEms 2VfEcts …24† The expression of Eq. (24) is consistent with the length at which the ®ber stress at the crack plane transfers back to the matrix in the ACK model. It also can be seen from Eq. (23) that the inclusion of interface bonding will decreases the debonding length. To initiate the crack-wake debonding process (i.e., ld > 0), the interface debonding energy zd should satisfy zd < aVmEms2 4V 2 f EfEc …25† 4. Matrix cracking stress The energy relationship to evaluate the steady-state matrix cracking stress is expressed as [5] 1 2 …1 ÿ1 Vf Ef ÿ sU f ÿ sD f
2 ‡ Vm Em ÿ sU m ÿ sD m
2  dz ‡ 1 2pR2Gm …ld ÿld …R a ats r 2 2pr dr dz ˆ Vmzm ‡  4Vfld a  zd …26† where zm is the critical strain energy release rate of the matrix, Gm is the matrix shear modulus and the radius of the matrix R can be expressed as a=V 1=2 f : The contribution of the shear energy term in Eq. (26) was neglected in the ACK model. It was veri®ed that this negligence is well accepted for the slipping length larger than a few ®ber radii [5]. Following the ACK model, the contribution of shear energy is neglected in the present analysis. Substituting the ®ber and matrix stresses of Eqs. (9), (10) and (4), (5) and the debonding length of Eq. (23) into Eq. (26), the energy balance equation leads to the form of A1…s†s2 ‡ A2…s†s ‡ A3…s† ˆ 0 …27† where A1…s† ˆ VmEm VfEfEc ld …28a† Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28 21