2.1 Brittle Fracture 77 Consequently,the total energy available for the creation of a new surface area Bdx along the crack front can be written as B dW Gurdz 9eff,rdz. (2.11) 0 As follows from Figure 2.1,however,the real new elementary surface area dS=RABdz is greater than Bdr since dz RA B (2.12) coso(z)cos(z) In Equation 2.12,RA is the roughness of the fracture surface and drdz/(cos ocos) is the area of the hatched rectangle in Figure 2.1.Because GIci is the intrinsic resistance to crack growth,the total fracture energy must be dW GIcidS GIci RA B dt. (2.13) Combining Equations 2.11 and 2.13 and denoting ⊙ gef.r= B gef,rdz, 0 one obtains GuI≡GIc= RA GIei. (2.14) geff.r In general,,Gie≥GIei since geff,r≤1 and RA≥1.Therefore,.the nominally measured fracture toughness Gic is usually higher than the in- trinsic (real)matrix resistance GIci.According to the relation Gre/GIci (KIe/KIci)2,Equation 2.14 can be eventually rewritten to obtain Equation 2.10. Values of gefr and RA must be estimated by using numerical (or ap- proximate analytical)models of the real tortuous crack front combined with appropriate experimental methods for fracture surface roughness determina- tion.In Sections 2.1.2 and 2.1.3,the so-called pyramidal-and particle-induced models are presented.In the context of 2D crack models,the tortuosity is usu- ally described by a double-or even single-kink geometry and RA =1/cos is assumed.In the 2D single kink approximation at Equation 2.8,the crack front is assumed to be straight (RA =1).Consequently,Equation 2.10 takes the following form: KIci=cos2(0/2)KIc.2.1 Brittle Fracture 77 Consequently, the total energy available for the creation of a new surface area Bdx along the crack front can be written as dW = GuIdx B 0 geff ,rdz. (2.11) As follows from Figure 2.1, however, the real new elementary surface area dS = RABdx is greater than Bdx since RA = 1 B B 0 dz cos φ(z) cos ϑ(z) . (2.12) In Equation 2.12, RA is the roughness of the fracture surface and dxdz/(cos φ cos ϑ) is the area of the hatched rectangle in Figure 2.1. Because GIci is the intrinsic resistance to crack growth, the total fracture energy must be dW = GIcidS = GIci RA B dx. (2.13) Combining Equations 2.11 and 2.13 and denoting g¯eff ,r = 1 B B 0 geff ,rdz, one obtains GuI ≡ GIc = RA g¯eff ,r GIci. (2.14) In general, GIc ≥ GIci since ¯geff ,r ≤ 1 and RA ≥ 1. Therefore, the nominally measured fracture toughness GIc is usually higher than the in￾trinsic (real) matrix resistance GIci. According to the relation GIc/GIci = (KIc/KIci)2, Equation 2.14 can be eventually rewritten to obtain Equation 2.10. Values of ¯geff ,r and RA must be estimated by using numerical (or ap￾proximate analytical) models of the real tortuous crack front combined with appropriate experimental methods for fracture surface roughness determina￾tion. In Sections 2.1.2 and 2.1.3, the so-called pyramidal- and particle-induced models are presented. In the context of 2D crack models, the tortuosity is usu￾ally described by a double- or even single-kink geometry and RA = 1/ cos θ is assumed. In the 2D single kink approximation at Equation 2.8, the crack front is assumed to be straight (RA = 1). Consequently, Equation 2.10 takes the following form: KIci = cos2(θ/2)KIc.