Page 129 - Engineered Interfaces in Fiber Reinforced Composites

112 Engineered interfaces in jiber reinforced composites

Based on the Coulomb friction law, which governs the frictional stress transfer in
the debonded interface, and combining Eqs. (4.12) and (4.18) yield the MAS at the
interface (r = u)

(4.61)

Therefore, combination of Eqs. (4.50), (4.52), (4.53) and (4.60) yields a differential
equation for the FAS at the debonded interface

(4.62)

where the coefficients PI, 9 and P3 are given in Appendix C. The general solution of
Eq. (4.62) is obtained for the partially debonded interface in the region
((L - C) <z<L), which is subjected to the boundary conditions:

$(L) = 0, $(L - e) = . (4.63)
Thus,

where the coefficients Ql, Q2, Q3, Q4, ml and m2 are given in Appendix C. Eqs. (4.63)
and (4.65) hold for the positive axial direction (Le. the right-hand part of the fiber in
Fig. 4.6. The corresponding solutions valid for the negative axial direction are
obtained by symmetry of the FAS and anti-symmetry of the IFSS with respect to the
fiber center at z = 0.
Determination of the crack tip debond stress, q, at a debond length, l, is
contingent to the condition that the fiber axial strain is equivalent to the matrix axial
strain at the boundary between the bonded and debonded regions (i.e.
au;(z)/az = au;(a,z)/az at z = k(L - e)). Within the debonded region, the matrix
axial strain at the interface is greater than the fiber axial strain due to the relative slip
between fiber and matrix. Therefore, combining Eqs. (4.8), (4.9) and (4.61) at the
boundary, op is obtained from

(4.66)

where R1 and R2 are given in Appendix C.