15 Using Crack Geometry to Determine Fracture Properties 95 rNL = 1 3π J μ (15.5) where μis the shear modulus, which can be related to Eand ν for a linear elastic material, and J is Rice’s J-integral, which corresponds to the energy release rate as J =Gfor a linear elastic material [17, 19]. Thus, using Eq 15.4, rNL is found to depend only on the Poisson’s ratio (compressibility) and δt: rNL = 1+ν 24 λδt . (15.6) For an incompressible hyperelastic gel, ν =0.5, so for plane stress rNL =δt/16, and for plane strain rNL =δt/12. In both cases, the nonlinear region is significantly smaller than the crack tip opening displacement, so assuming LEFM holds in determining the CTOD is appropriate. 15.3 Summary of Experiment and Results Due to the high water content of many hydrogels (often >90%) and their very low toughness, it can be challenging to conduct traditional fracture tests on these materials. We also conduct in situ confocal microscope imaging on the submillimeter scale to understand both crack geometry and the damage processes occurring in the polymer network, introducing further experimental challenges. For these reasons, we use a fracking-inspired method to induce stable, slow crack propagation in a thin, confined disk of hydrogel. The experimental methods are described in detail in [21]. A second-order polynomial (parabola) fits the crack shapes observed in the experiments very well, as shown in Fig. 15.2. Details of the calculated fracture properties of this gel using this method to induce crack propagation can be found in [21]. For a propagating crack in this gel, which has a Young’s Modulus of approximately 78 kPa, the mean CTOD (δt) is 339μm Fig. 15.2 A parabola fit to a confocal microscope image of an internal plane of the gel indicating the measured CTOD (mean value, 339μm) and estimated size of the nonlinear region (mean value, 28 μm)
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