Figure 4
From: Exactly solvable model for a velocity jump observed in crack propagation in viscoelastic solids
Reproduced velocity jump and simple characterization of the w-V curve. (a,b) Two representative plots of the crack propagation velocity V as a function of the initially applied energy density w. The cases (a) with and (b) without velocity jump are obtained for \(1 < \lambda \ll N\) and \(1\ll N\lesssim \lambda \), respectively. (These plots are obtained for (a) λ = 103, N = 109 and (b) λ → ∞, N = 109). Four characteristic velocity-scales and three energy-scales are indicated in (a), which are important for toughening. (c–e) V/V 0 vs. \(w/({w}_{0}N)={\varepsilon }^{2}/{\varepsilon }_{c}^{2}\), obtained from equation (1) on a log-log scale. The normalization factors for velocity and energy are \({V}_{0}\simeq l{E}_{0}/\eta \) and \({w}_{0}\equiv {E}_{0}l{\varepsilon }_{c}^{2}/(2L)\), respectively. The cases with velocity jump are demonstrated for various λ with a fixed N in (c) and for various N with a fixed λ in (d). The Kelvin-Voigt limit, λ → ∞, is shown for various N in (e) as an example of the case without velocity jump.
