XXIV ICTAM, 21-26 August 2016, Montreal, Canada

INSTABILITIES OF A COMPRESSED HYPER ELASTIC PRISM:COMPETITION BETWEEN WRINKLES AND CREASES

Claire Lestringant ⇤1, Corrado Maurini1, Arnaud Lazarus1, and Basile Audoly2

1UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France

2Laboratoire de Mecanique des Solides, CNRS, Ecole polytechnique, Palaiseau, France

Summary We study the stability of a long hyper elastic prism whose cross-section is an isosceles triangle, subjected to axial compression.Experiments reveal extended buckling modes (wrinkles) when the ridge angle � is smaller than ⇡ 90�, and localized modes (creases) when� is larger than ⇡ 90�. The compression of a neo-Hookean half-space, which is known to produce creases, appears as a particular case ofour system (� = 180�). With this aim to explain the competition between the extended vs. localized buckling modes, we carry out a linearstability analysis based on a neo-Hookean (hyperelastic) model using the finite-element method. The first buckling mode and the associatedcritical strain ✏c are obtained as a function of the angle � at the apex. The simulations reproduce the different types of behaviors dependingon the ridge angle � as seen in the experiments.

We investigate the buckling of a prism having a triangular basis made of a soft elastic polymer (Vynilpolysiloxane - VPS),when subjected to axial compression. In our experiments, the prism is a ridge attached to a parallepipedic block. We usedVPS casting in a mold realized by rapid prototyping (laser cutter) to produce the samples. The block is then subjected to acompressive load P which forces the prism to shrink longitudinally, see Figure 1a.

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Experiments FEM : anti symmetric wrinkling FEM : symmetric wrinkling Biot’s wrinkling of a half space Thin plate model

Figure 1: (a): Geometry of undeformed sample and wrinkling mode under axial compressive load P : photo of the experimentwith ridge angle � = 20� and side length l = 13 mm. (b): sketch of the undeformed cross-section and boundary conditionsused in the simulation. (c): Buckling strain as a function of the ridge angle �. Numerical results are shown using red dots forthe anti symmetric wrinkling (ASW) mode and blue dots for the symmetric wrinkling (SW) mode. Prediction of a thin platemodel with varying thickness (dashed red line), and Biot strain corresponding the surface wrinkling instability of the lateralfaces (dashed blue line). Experimental results are shown with green dots.

For a sufficiently compressive load P , the prism buckles. The buckling mode observed in the experiments depends onthe ridge angle �: it is an extended mode (wrinkles, as in Figure 1a) when the ridge angle is smaller than ⇡ 90�, and alocalized mode (creases) when � is larger than ⇡ 90�. The well-studied compression of a neo-Hookean half-plane appears asa particular case of our system (� = 180�): our experiments reveal a transition from localized creases to extended wrinkleswhen the ridge angle � is decreased.

With the aim to explain these observations, we have set up a linear stability analysis of the compressed prism using thefinite-element method. The stability analysis is formulated in an original manner, namely as a polynomial eigenvalue problemin the triangular cross-sectional domain, with the axial wavenumber q entering as the eigenvalue. This formulation allows toaddress the case of an infinitely long prism with optimal numerical efficiency.

In the simulation, the first buckling mode and the associated critical strain ✏c are obtained as a function of the ridge angle�. The critical strain ✏c is small for slender cross-sections (✏c ! 0 for � ! 0). A stability analysis based on the von Karmanequations for thin plates (with varying thickness) accurately captures the dependence of ✏c on � for � ! 0 as ✏c ⇠ �2 (seedashed red line in Figure 1b, which is asymptotically consistent with the FEM results).

⇤Corresponding author. Email: claire.lestringant@gmail.com

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� = 120�DE✓ ✓DEr ✓DEr r

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Figure 2: (a): Sketch of the anti symmetric wrinkling (ASW) buckling mode predicted by the simulation for values of �below ⇡ 105�. (b): Sketch of the symmetric wrinkling (SW) buckling mode predicted by the simulation for values of � above⇡ 105� (above). Sketch of the symmetric creasing (SC) buckling mode obtained in the experiments for values of � above⇡ 90� (below). (c): Visualization of the first buckling mode in the cross-sectional plane: incremental radial strain DErr,shear strain DEr✓ and hoop strain DE✓✓ for a ASFW mode (top row, � = 90�) and a SSW mode (bottom row, � = 120�).Here, r and ✓ are the polar coordinates in the plane of the cross-section. Note that the modes have a harmonic dependence onthe axial variable, not shown.

The buckling mode predicted by the plate model is an anti symmetric (flexural) wrinkling (ASW) mode, making the ridgebend out of the plane of symmetry of the prism (see Figure 2a). A mode of this type is indeed predicted by the FEM analysis,for all ridge angles below � ⇡ 105� (red dots in Figure 1 and top row in Figure 2c).

When � reaches ⇡ 105�, the critical strain ✏c of the edge mode becomes larger than the value 0.55 corresponding to thesurface wrinkling instability in a neo-Hookean half-plane [1]. Consistently, when � is larger than ⇡ 105� the first bucklingmode predicted by the FEM analysis is a symmetric mode living on the surface of the lateral faces of the prism (see blue dotsand blue dashed line in Figure 1, and bottom row in Figure 2). For this mode, our linear analysis can only predict a wrinklingmode having an harmonic dependence on the axial coordinate z, but it is known that modes of this type appear concurrentlywith all possible wavelengths, and that the non-linear coupling between them gives rise to a creasing mode [2], [3]. In view ofthis, the wrinkles living on the faces of the prism predicted by our linear analysis (SW modes) are consistent with the creasingmodes (SC) observed in the experiments. Buckling strain for creasing instability in a neo-Hookean half-plane ✏C is known tobe lower than buckling strain for wrinkling: ✏C < ✏B [3], this probably explains why edge mode maintains up to bigger � insimulations (105�) than in experiments (90�).

An elastic prism under axial compression is an interesting system that features a competition between creasing modes forlarge ridge angles, like Biot’s Neo-Hookean half-space, and wrinkling modes for small ridge angles. We have observed thiscompetition in experiments, and explained it by a buckling analysis.

References

[1] Biot M.: Mechanics of Incremental Deformations. Wiley, 1965.[2] Hohlfeld E., Mahadevan L.: Unfolding the sulcus. Physical Review Letters, 2011.[3] Cao Y., Hutchinson J. W.: From wrinkles to creases in elastomers: the instability and imperfection-sensitivity of wrinkling. Proceedings of the Royal

Society A: Mathematical, Physical and Engineering Sciences, 2011.