Y-F. Liu et al./ Mechanics of Materials 29(1998)111-121 where A is an incremental range and o is the discretized stress at a grid point X-I(Fig. 5). At the crack tip, on+I=o(1), whose value is dictated by qs.(1)and (2) to satisfy the crack tip condition 0.0L 1) with the following auxiliary integral variables T=ssin 8, S=x/cos (8) Fig. 5. Discretizing bridging stress at the bridging zone as a linear function of position Eq(4) may be transformed to 4( 丌E tion of /(X) into Eq. (5)results in an expression of 4(1-v2)cx arccos do Id( X )as a function of o. The obtained expression is again substituted into Eq. (9)to yield the following E 0}mnd(9)a(x)-+-“2 TE。6V1-x2=B(a)(10) The debond length, I(X, ) at X, is calculated as a function of o according to Eq(6). Then, substitu- The remaining terms in Eq. (9)is also expressed as a function of o, leading to the following equa tIon 4(1-v2)cr rarccos x do E Discretize o(x):01,02,,ON X sin6sin6d6≡A as a function ofσ Integration with regards to variables, 0 and , ar carried out by the trapezoidal rule For each discretized grid between co/csX<I it follows from Eqs. (10)and(11)that A=B() Substitute COD eqution Eq. (12)is a series of nonlinear equations with N unknowns, which is solved to obtain o by Newtons Calculate unknows a with Newton's iteration method iterative method. In actual calculation, many initial values of o were tested and unique convergent solutions were achieved. Once o is obtained other nd /(x) unknowns are easily determined(Fig. 4) 2. 4. Comparison with Marshall and Cox's results Calculation of KI To verify the above procedure for solving bridg ng problems, results of distributed bridging str for the same problem were compared to that of Fig. 4. Flow chart to solve the distributed spring model that Marshall and Cox (1987). Results shown in Fig. 6 considers debonding toughness as well as frictio were for a bridging law of p v8 with a partialY.-F. Liu et al.rMechanics of Materials 29 1998 111–121 ( ) 115 where D is an incremental range and s is the i discretized stress at a grid point X Ž . Fig. 5 . At the iy1 crack tip, sNq1ss Ž . 1 , whose value is dictated by Eqs. 1 and 2 to satisfy the crack tip condition Ž. Ž. uŽ . 1 . With the following auxiliary integral variables, tss sin u , ssxrcos f Ž . 8 Eq. 4 may be transformed to: Ž . 2 4 1Ž . yn c ' 2 u x Ž . s s 1yx p Ec 2 4 1Ž . yn cx arccos x df y H 2 p Ec 0 cos f = pr2 x H p sin u sin udu . 9Ž . ž / 0 cos f The debond length, l XŽ ., at X is calculated as a i i function of s according to Eq. 6 . Then, substitu- Ž . i Fig. 4. Flow chart to solve the distributed spring model that considers debonding toughness as well as friction. Fig. 5. Discretizing bridging stress at the bridging zone as a linear function of position. tion of l XŽ . Ž. into Eq. 5 results in an expression of i u XŽ . as a function of s . The obtained expression is i i again substituted into Eq. 9 to yield the following Ž . relation in normalized coordinates: 2 4 1Ž . yn c 2 u XŽ . Ž. Ž. i i ii y s(1yX 'B s . 10 p Ec The remaining terms in Eq. 9 is also expressed Ž . as a function of s , leading to the following equa- i tion: 4 1yn2 Ž . cX arccos X df H 2 p Ec 0 cos f pr2 X =H s sin u sin udu'A s . 11 Ž . ij j 0 ž / cos f Integration with regards to variables, u and f, are carried out by the trapezoidal rule. For each discretized grid between c rcFX-1, 0 it follows from Eqs. 10 and 11 that Ž. Ž. Aij j i i s sB Ž. Ž. s . 12 Eq. 12 is a series of nonlinear equations with Ž . N unknowns, which is solved to obtain s by Newton’s i iterative method. In actual calculation, many initial values of s were tested and unique convergent i solutions were achieved. Once s is obtained, other i unknowns are easily determined Fig. 4 . Ž . 2.4. Comparison with Marshall and Cox’s results To verify the above procedure for solving bridg￾ing problems, results of distributed bridging stress for the same problem were compared to that of Marshall and Cox 1987 . Results shown in Fig. 6 Ž . were for a bridging law of p;'d with a partial