94 K. A. Mac Donald and G. Ravichandran crystalline metals and ceramics. When applying these measures to polymers or other amorphous materials like glasses, the theories behind these measures no longer have the same meaning. However, the concept of fracture toughness as a measure of the energy needed to create new surfaces within the material is still applicable. 15.2 Background Linear elastic fracture mechanics (LEFM) theory predicts a parabolic crack opening profile described as δ(r) = 8K √2πEλ √r (15.1) where λ = 1 for plane stress, 1 1−ν2 for plane strain, (15.2) E is Young’s modulus, ν is Poisson’s ratio, r is the horizontal distance along the crack surface away from the tip, and K is the stress intensity factor [17]. If we defineδt as the crack opening where perpendicular lines from the vertex of the parabola (crack tip) intersect with the parabolic profile (crack edges), we can define rt =δt/2 as indicated in Fig. 15.1 [18, 19]. This allows us to determine Kin terms of elastic constants and a measure of the crack tip opening displacement (CTOD, δt): K= √πδt Eλ 4 . (15.3) For small scale yielding, the energy release rate is then G= K 2 Eλ = πδt Eλ 16 . (15.4) Many soft materials, including the gel used in this study, could be considered hyperelastic, which means there is a nonlinear region near the crack tip in which the parabolic crack profile would be altered. Geubelle and Knauss [20] estimate the size of this nonlinear region to be Fig. 15.1 Parabolic crack opening profile with crack tip opening displacement, δ_t, determined by the intersection of perpendicular lines from the crack tip with the crack edges
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