On the role of kinetic energy during unstable propagation in a heterogeneous peeling test

Abstract

We study the dynamic debonding of a onedimensional inextensible film, subject to a monotonic loading and under the hypothesis that the toughness of the glue can take only two values. We first consider the case of a single defect of small length in the glue where the toughness is lower than in the remaining part. The dynamic solution is obtained in a closed form and we prove that it does not converge to the expected quasistatic one when the loading speed tends to zero. The gap is due to a kinetic energy which appears when the debonding propagates across the defect at a velocity which is of the same order as the sound velocity. The kinetic energy becomes negligible again only when the debonding has reached a critical distance beyond the defect. The case of many defects is then considered and solved using an exact numerical solution of the wave equation and the Griffith law of propagation. The numerical results highlight the effects of the time evolution of the kinetic energy which induce alternate phases of rapid and slow debonding, these oscillations depending essentially on the volume fraction of the highest toughness.Keywords brittle fracture · crack propagation · Griffith's criterion · dynamic fracture · quasistatic evolution · shock waves · nonlinear stability PACS 62.20.mj · 62.50.Ef · 43.25.Cb · 68.35.Ja · 83.60.Uv

A Note on the Convergence of Singularly Perturbed Second Order Potential-Type Equations

“…A similar phenomenon was observed in [8] where the authors give an example of non-convergence, in the context of a one-dimensional peeling test without viscosity: more precisely, their dynamic solutions converge to a limit that does not fulfill first order stability. Our non-convergence result in Sect.…”

On the 1d wave equation in time-dependent domains and the problem of debond initiation

“…Due to this simplification, we obtain a one-dimensional model, see Figure 3. This model was studied in [14,23] in particular cases (where the displacement's derivatives are piecewise constant). General results were given in [11,24,25] in the case where a portion of the film is already debonded at the initial time.…”

On the Approximation of Quasistatic Evolutions for the Debonding of a Thin Film via Vanishing Inertia and Viscosity

“…A more drastic scenario may even appear in the case of a discontinuous toughness, covered however by Theorem 1.5. The failure of the quasistatic approximation in this framework was observed in [10] and [15] where the authors considered explicit examples of piecewise constant toughness κ; they noticed that on discontinuity points of κ the limit solution does not fulfill Griffith's criterion, which has to be replaced by a suitable energy balance. This is in line with Proposition 4.7 (which however should be proven without assuming continuity of κ), where the appearance of the measure µ in (4.10) takes into account this feature.…”