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Reversibility of crumpling on compressed thin sheets ⋆ Reversibility of crumpling
| Content Provider | Semantic Scholar |
|---|---|
| Author | Pocheau, Alain Roman, Benoît |
| Copyright Year | 2014 |
| Abstract | Compressing thin sheets usually yields the formation of singularities which focus curvature and stretching on points or lines. In particular, following the common experience of crumpled paper where a paper sheet is crushed in a paper ball, one might guess that elastic singularities should be the rule beyond some compression level. In contrast, we show here that, somewhat surprisingly, compressing a sheet between cylinders make singularities spontaneously disappear at large compression. This “stress defocusing” phenomenon is qualitatively explained from scale-invariance and further linked to a criterion based on a balance between stretching and curvature energies on defocused states. This criterion is made quantitative using the scalings relevant to sheet elasticity and compared to experiment. These results are synthesized in a phase diagram completed with plastic transitions and buckling saturation. They provide a renewed vision of elastic singularities as a thermodynamic condensed phase where stress is focused, in competition with a regular diluted phase where stress is defocused. The physical differences between phases is emphasized by determining experimentally the mechanical response when stress is focused or defocused and by recovering the corresponding scaling laws. In this phase diagram, different compression routes may be followed by constraining differently the two principal curvatures of a sheet. As evidenced here, this may provide an efficient way of compressing a sheet that avoids the occurrence of plastic damages by inducing a spontaneous regularization of geometry and stress. Systems driven far from equilibrium usually first undergo a homogeneous increase of their energy density which is thus equally distributed in space and time. However, beyond some distance to equilibrium, most of them undergo a spontaneous focalization of energy on localized areas corresponding to singularities, defects or so-called coherent structures. Examples include vorticity concentration in turbulent fluids, rogue waves occurrence on sea surface, shock wave formation in thermodynamic systems, dielectric breakdown in media submitted to electrostatic field, fracture in stressed solids, etc. In all these phenomena, the energy density has spontaneously turned from a homogeneous distribution to a highly localized concentration. The origin of this “energy-focusing” phenomenon, which stands as an emblematic example of selforganization, still rises important questions and issues. In particular, as the increase of the distance to equilibrium goes together with a rise of the energy density, one might wonder whether energy-focusing is correlated to energy ⋆ Contribution to the Topical Issue “Irreversible Dynamics: A topical issue dedicated to Paul Manneville” edited by Patrice Le Gal and Laurette S. Tuckerman. a e-mail: alain.pocheau@irphe.univ-mrs.fr density, for instance by being a preferred state beyond an energy density threshold. If so, driving a system far from equilibrium would imply triggering energy-focusing by the irremediable occurrence of singularities, defects or coherent structures. This will, however, not be the case in the system described below. Here, we consider the “energy-focusing” issue in the context of elasticity by addressing the ultimate fate of compressed thin sheets. Such sheets stand as an efficient mean to separate domains, treat surfaces or confine volumes in the form of thin envelops, thin layers or thin films. Examples include graphene sheets [1], epitaxial deposit at sub-micrometric scales [2], membranes at micrometric scales [3], packaging at sub-millimeter scales [4], metallurgical structures at millimeters scales, geological layers at even larger scales [5], the scale meaning here the thickness of the object. Their common feature is to display a small dimension, their thickness, in comparison to their length and width. Following it, most of their properties can be recovered by treating them as 2d surfaces involving flexural effects. However, in many instances, sheets may be submitted to geometrical constraints that force them to fit into a reduced space. They then have to adapt their form to restrictive conditions, something they may do smoothly Page 2 of 15 Eur. Phys. J. E (2014) 37: 28 or sharply, i.e. with small or large curvatures as compared to their inverse thickness. In the latter case, they escape the 2d surface assumption at the locations of large curvature. They then form at these places surface singularities in which a large elastic energy is focused with many important implications. In particular, the sheet properties are usually altered there regarding electronic properties, robustness or even elasticity, with a possible transition to plasticity at the core of singularities. In this context, the general issue regarding the origin of energy-focusing turns out addressing whether singularity formation results here from the global rise of the elastic energy density of the sheet by compression or from another origin. To address this issue in the present context, we shall benefit from two major rules for self-organization. First, as there is no intrinsic scale in elasticity, scale-invariance and scaling arguments apply. Second, as elasticity is nondissipative in its elastic regime, energy landscapes can be used to infer the preferred states, including those involving elastic singularities. In particular, the occurrence of a stress-focused state may be understood as the fact that it has became energetically preferred as compared to a stress-distributed state. Applying both these rules should then largely help elucidating stress focusing, but with possible surprises. In particular, the popular example of crumpled paper where the compaction of a sheet in a ball generates scars (fig. 2 right), usually yields the common intuition that singularities and stress focusing should irremediably persist when increasing compaction. On the opposite, we shall find here that, surprisingly, the two above rules deny this belief, in the sense that scalings imply that singular states should no longer be preferred at large compaction, if they previously were: stress should thus defocus at large compaction. To highlight the possibility of energy defocusing when rising the mean energy density by compaction, we shall exploit the existence of two curvature radii to specify the geometry of a sheet surface. Fixing one of them here will then provide a compression route definitely different from that of usual paper crumpling. In particular, stress defocusing will actually be evidenced on it in the linear regime of elasticity. This will show the possibility of reversibility of singularity formation in elasticity. When stress defocusing applies, the ultimate state of a compressed sheet should be smooth and regular. Its actual existence will be evidenced on a dedicated experiment and the apparent paradox regarding the common experience of crumpled paper will be clarified [6]. This will enable us to identify singularities as a thermodynamic condensed phase surrounded in phase space by the regular diluted phase corresponding to regular geometries and defocused stress. In particular, the persistence of singularities on some actual compression routes will be shown to refer to plasticity instead of elasticity. In this regard, the popular demonstration of paper crumpling with hands will appear as a misleading example of linear elasticity since the singularities that form should disappear at large compression but actually do not because of plasticity only. Altogether, this study will thus provide a modified vision of the nature of elastic singularities and of sheet adaptation to compression. In particular, on compression routes, singularities, instead of being the rule beyond some compression level, will actually appear as a transient state governed by a balance between two forms of energy density, irrespective of the absolute amount of mean energy density. In the following, we first emphasize in sect. 1 the relevance of an intermediate compression route between Elastica and crumpled paper to address singularity occurrence. We then report in sect. 2 the experimental compression of an elastic sheet between cylinders and the resulting evidence of stress defocusing. Energy arguments for stress focusing or defocusing are then addressed in sect. 3 together with scaling arguments so as to end up with a quantitative criterion for defocusing. Taking a more general viewpoint, the surprising phenomenon of stress defocusing under compression is linked in sect. 4 to the natural stress defocusing under decompression by using scale-invariance and an additional scale symmetry. Limiting phenomena during sheet compression as the buckling saturation and the plastic transition are determined in sect. 5 before synthesizing the results in sect. 6 in a phase diagram. Forces during compression are then determined experimentally in sect. 7 in both stress-focused or -defocused phases together with their relevant scalings. This is followed by a conclusion on the implications of this study for the nature of singularities in elasticity. 1 On singularities in plate elasticity: from Elastica to crumpled paper The deformation of thin sheets can be decomposed into two parts: one related to the stretching of a sheet viewed as a 2d surface and one related to the sheet’s curvature [7– 9]. The energy associated with the former is proportional to the sheet thickness h and the latter to its cube, h. As this thickness is the smallest scale on thin sheets, stretch appears as the dominant stress both on singularities and on the non-singular states on which they appear. Accordingly, singularity formation actually corresponds to focusing that stretch on singularities, leaving unstretched but possibly curved domains in between. Viewed this way, the role of singularities may be considered as to relax the elastic stress in the remaining sheet parts: the bending stress for |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://blog.espci.fr/benoitroman/files/2018/04/Pocheau14.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
