Predicting delayed instabilities in viscoelastic solids

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Science Advances  04 Sep 2020:
Vol. 6, no. 36, eabb2948
DOI: 10.1126/sciadv.abb2948
  • Fig. 1 The viscoelastic reference length evolution.

    At the resting state, all three length measures on the body, its measured length g (marked red), its instantaneous reference length g¯ (marked gray), and its rest reference length g¯0 (marked black) are all equal. When subjected to a constant displacement extension, the instantaneous reference length evolves away from the rest length and toward the presently assumed length, thus resulting in stress relaxation. It asymptotically approaches the stationary state g¯stat=βg+(1β)g¯0, in which the initial stress is reduced by a factor of 1 − β. When released, the unconstrained system immediately adopts its favored instantaneous reference length, which, in turn, gradually creeps toward the rest lengths.

  • Fig. 2 Schematic representation of the metrics collinearity.

    The minimization of the metric g (marked by a full black circle) is constrained and performed with respect to the subset of metrics that correspond to realizable configurations (thick black line). These metrics are in, particular, orientation preserving and Euclidean. Given an instantaneous reference metric, g¯ (marked by a full gray circle), the realized metric will correspond to the closest point from the set of admissible metrics to g¯ according to the distance function given by the instantaneous elastic energy Eq. 7. Starting from rest, g¯ evolves from g¯0 (marked by a full red circle) toward the g, which remains the closest admissible metric to g¯ due to collinearity of the three metrics. As g remains stationary, the evolution of g¯ will preserve the collinearity, asymptotically approaching g¯stat (marked by an open circle), which is also collinear. We stress that throughout this evolution, g remains unchanged; thus, no variation of the configuration will be observed despite the stress relaxation.

  • Fig. 3 Experimental verification of the viscoelastic stability diagram.

    (A) Straight and inverted conical poppers. Photo credit: Erez Y. Urbach, Weizmann Institute. (B) The two axes span the dimensionless geometrical properties of the truncated conical poppers. The background colors represent the theoretically predicted regions of each of the phases. Each marker corresponds to a different popper; different shaped (and colored) markers indicate the different phases observed in experiment. (C) Numerically calculated flipping time as a function of the normalized thickness of the conical popper for immediate release and long holding time. The different poppers were simulated by varying their thicknesses and constant radii rmin = 10 mm, rmax = 25 mm. The material properties taken were β = 0.1, and the memory kernel was assumed to be exponential with τ = 0.1 s, Young’s modulus E = 2.5 MPa, and Poisson’s ratio v = 0.47. Varying the kernel may lead to varying rate of divergence of the flipping time between the stable and acquired stability region, yet the location of this divergence will remain unchanged. The divergence of flipping times was addressed in (8), and more recently, the rate of divergence was studied in (18).

Supplementary Materials

  • Supplementary Materials

    Predicting delayed instabilities in viscoelastic solids

    Erez Y. Urbach and Efi Efrati

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