Initial delay in Griffith growth under proportional loading ($G(s,t)=t^{2}G(s,1)$)

For linear antiplane elasticity with body forces and Dirichlet data proportional to time, uniqueness gives $u(s,t)=tu(s,1)$ at any fixed crack length $s$. The minimized elastic-plus-load potential and its energy release rate scale quadratically. The local Griffith fracture criterion laws are $\dot{\ell }\ge 0$, $G(\ell ,t)\le G_c$ and $\dot{\ell }(G_c-G)=0$, with positive toughness $G_c$.

If $G(s,1)$ is finite and continuous near the initial length, it is bounded there by $M$. Continuity of the crack path keeps it in that neighborhood for a positive interval. Choose the interval short enough that $t^{2}M<G_c$; complementarity then forces zero growth throughout it. This proves a positive initial waiting time without assuming the crack is stationary in advance.