Figure 5

Dynamic responses of Zener elements. (a) “Short” and “long” Zener elements of natural length l and (L − l)/2, respectively. Here, we set the origin of the x-coordinate at the crack tip. (b,c) Representative behavior of dynamic responses of Zener elements for conventional elastomers (λ = 10³ and N = 10⁹). We show relaxational responses of Zener elements during a constant-velocity crack propagation, in contrast to vibrational responses in Fig. 3. The four curves are contours for Ψ_soft = 100, 3 and Ψ_hard = 3, 100 as a function of the initially applied energy density w and the distance from the crack tip χ ≡ x/x ₀. The conditions \({{\rm{\Psi }}}_{{\rm{soft}}}\gg 1\) and \({{\rm{\Psi }}}_{{\rm{hard}}}\gg 1\) correspond to soft- and hard-elastic regimes, respectively (see the text for details). Explicit forms of Ψ_soft, Ψ_hard, and x ₀ are given in Methods. Red dashed lines correspond to the velocity jump (w _jump) and black dashed lines correspond to the minimum (w ₀) and maximum (w _∞) values of w for with constant-velocity propagation.