applied fracture mechanics

Fractal Fracture Mechanics Applied to Materials Engineering 77 i L 0d U U U F (33)dLwhich can be rewritten asd dU( F UL) (34)dLdLsince Uiis constant. On the left hand side of equation (34), dF dL dULdL is the amountof energy that remains available to increase crack extension by an amount dL . On the righthand side of equation (34), dU dL is the surface energy that must be released to form therugged crack surfaces. This energy is the crack growth resistance.Deriving equation (30) with respect to the projected crack length L0, one hasddLdU (35)( F UL)0dL0Considering postulate II, one can apply the derivation chain rule and obtainConsidering the following cases:i. Fixed grips condition with F constant : sinced dL dUdL( F UL) (36)dL dL dL dL0 02 2L0 0U U m * L 2 E ' decreases withthe crack length, and using equations (10) and (25) in (36), one can deriveL2rOr, by using equations (17) and (26) in (35), one findsm* L 2 r.(37)E'22 2H2m*rL0H0 l 012H2 0.2 E' l 0L 0 ii. Condition of constant loading or stress, where necessarily F 2 UL(38), since2 2L0 0U U m * L 2 E ' increases with the work of external forces, and usingLequations (10) and (25) in (36), one can find2rm* L 2 r.(39)E'