0 2 2 2 2 2 4 4 + + = ¶ ¶ + ¶ ¶ b W d W W x b x for 0) ( 0 - < £x A (9b) The relevant boundary conditions are: ( 0) 0 = = x W , ( 0) 0 ' = = x W , a x =- =- ) (' L Wu , ) 0 ( '' = - = L Wu x . (10) The governing Eqn. 9 together with the boundary conditions (10) and the continuity conditions at the end of the bridging zone ( 0A =- x ) will determine the deflection profile of the beam. Note that the bridging zone length (A0) will be dictated by the critical crack opening displacement ( c cw hW º ) required for failure of the bridging ligament. The crack energy release rate ( Total G ), as determined through the total energy balance is: ÷ ø ö ç è æ ¶ ¶ - ¶ ¶ - ¶ ¶ = a U a U a U B 1 G k s ext Total (11) where Uext is the work done by the applied load, Us is the strain energy, Uk is the kinetic energy, B is the uniform width of the DCB specimen, and a is the crack length. For steady state crack extension a = vt, where v is the crack velocity and t is time. For the DCB specimen loaded with a flying wedge this reduces to: 2 2 L 3 u 3 Total hv W 6 Eh G r a x a x - ¶ ¶ = =- (12) In addition, by application of the dynamic J-integral, the energy released at the crack tip is related to the bending moment M by [8]: ( ) 2 0 2 2 2 0 3 12 12 ÷ ÷ ø ö ç ç è æ ¶ ¶ = = = = x x x Eh W M Eh GTip (13) Comparing GTotal and GTip for the displacement fields derived for the linear bridging law, one finds that - = ò D = c 0 b 2 w Tip Total p dw G G G (14) where DGb represents the work that must be done against the bridging ligaments along the bridged zone. This result is identical to that for the quasi-static case for small scale bridging conditions. Since there is no rate dependence to the bridging law, it is not surprising that the small scale bridging limit relationship is obeyed. Since we limit our analysis to small scale bridging, tow failure must occur in the wake of the crack. Small scale bridging is ensured provided the displacement profile monotonically increases within the bridging zone from the crack tip and the pull-out required for tow failure is less than the maximum crack opening displacement within the bridging zone. This condition determines a criterion for the maximum allowable bridging zone length, max A , which is obtained by solving W( A ) / 0 max ¶ = ¶ = - x x . Therefore, if max 0A A£ , then ( W( A )) W W max critical c º = - £ x , and hence small scale bridging condition is ensured. Detailed calculations of the deflection profile, the crack energy release rate and the maximum allowable bridging length can be computed with the formalism presented above for both the Dugdale
RkJQdWJsaXNoZXIy MjM0NDE=