Dynamic fracture: an example of convergence towards a discontinuous quasistatic solution

Abstract

Considering a one-dimensional problem of debonding of a thin film in the context of Griffith's theory, we show that the dynamical solution converges, when the speed of loading goes down to 0, to a quasi-static solution including an unstable phase of propagation. In particular, the jump of the debonding induced by this instability is governed by a principle of conservation of the total quasi-static energy, the kinetic energy being negligible.

Numerical investigation of dynamic brittle fracture via gradient damage models

“…13 the dynamic jump process is continuous (the crack propagates at a finite speed bounded by the shear wave speed) compared to the quasi-static one where the jump occurs necessarily in a discontinuous fashion between two iterations. We verify the conclusions drawn in [35] that the kinetic energy K plays only a transient role in this problem, as it attains a finite value during the jump and becomes again negligible after. The dynamic potential energy P = E + S after the jump is slightly bigger that its value before the jump, due to the fact that the loading speed k = 0.001 is small but not zero.…”

“…If this effect is ignored, the crack length after the jump is recorded in Table 3 for each case. From the static energy release rate evolution, we see that the crack length l m after the jump predicted in the first-order quasi-static numerical model is governed by G(l m ) = G c (l m ) from which authors of [35] …”

“…This antiplane tearing example is physically similar to the 1-d film peeling problem which can be studied using the classical Griffith's theory of dynamic fracture. According to [35] and [2], the crack speed, with respect to the loading displacement U = kt or to the physical time t, as a function of the loading velocity k is given by…”

“…On the one hand, it can be regarded as a numerical issue as the effective implementation of the quasi-static model is solely based on the first-order stability condition (28). For this particular problem based on quasi-static energy conservation we could predict a correct quasi-static crack evolution toward which the dynamic solution converges when the loading speed becomes small, see [35]. On the other hand, from a theoretic point of view, it is already known in [38] that there may not exist an energyconserving evolution which also respects the stability criterion at every time.…”

“…During the jump, the crack propagates at a speed comparable to the material speed of sound which, according to [35], is given by Table 4. A good agreement can be found between the numerical and the theoretic ones.…”