Internal Stresses Lead to Net Forces and Torques on Extended Elastic Bodies

Hillel Aharoni, John M. Kolinski, Michael Moshe, Idan Meirzada, and Eran Sharon

Phys. Rev. Lett. 117, 124101 – Published 16 September 2016

Abstract

A geometrically frustrated elastic body will develop residual stresses arising from the mismatch between the intrinsic geometry of the body and the geometry of the ambient space. We analyze these stresses for an ambient space with gradients in its intrinsic curvature, and show that residual stresses generate effective forces and torques on the center of mass of the body. We analytically calculate these forces in two dimensions, and experimentally demonstrate their action by the migration of a non-Euclidean gel disc in a curved Hele-Shaw cell. An extension of our analysis to higher dimensions shows that these forces are also generated in three dimensions, but are negligible compared to gravity.

Received 23 May 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.124101

Physics Subject Headings (PhySH)

Classical mechanics Continuum mechanics Mechanical deformation

Nonlinear Dynamics

Authors & Affiliations

Hillel Aharoni^1,2, John M. Kolinski¹, Michael Moshe^1,3,4, Idan Meirzada¹, and Eran Sharon^1,*

¹The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
²Department of Physics & Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
³Department of Physics, Soft matter program, Syracuse University, Syracuse, New York 13244, USA
⁴Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

^*erans@vms.huji.ac.il

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Vol. 117, Iss. 12 — 16 September 2016

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Figure 1
Experimental schematic and demonstration of migration of a 2D frustrated gel. (a) A picture of the confining plates of the cell. Arrows indicate periodic forcing of the gap at 5 Hz using a shaker. The exaggerated gap shows how the plates form the curved Hele-Shaw cell. Scalebar: 2 cm). The cell’s Gaussian curvature $K$ is plotted as a function of radius (inset). (b) An oblique schematic view of the setup: arrows indicate the gel, the glass surface and the steel surface. This setup is immersed in a temperature-regulated water bath; by controlling the temperature, we can manipulate the intrinsic curvature of the gel. (c) A gel with positive spontaneous curvature migrates from the lower portion of the cell, where $K (r) < 0$ , to the upper portion, where $K (r) > 0$ . The gel is shown in its final, stable position, where it has ceased migration. The dots trace out the trajectory of the gel, where the color corresponds to time from start (blue) to finish (red). Scale bar: 5 cm (d) Several trajectories for gels with different values of reference curvature are plotted against time, with background color indicating the local value $K (r)$ . The gels with spontaneous negative curvature (blue curves) both migrate to a point near the lowest value of the local curvature. Two examples of a gel with positive spontaneous curvature (red lines) show their propagation against gravity to regions of $K (r) > 0$ . A flat gel (dark gray) migrates to the zero in the bell. Each gel is approximately 5 cm in diameter, and thus the precision with which the gels migrate is approximately 2 cm.
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Figure 2
Frustration driven migration of a gel with temperature-dependent intrinsic curvature. (a) The NIPA gel is cast such that it will change the value of its spontaneous curvature in response to changes in the temperature of the bath: here, the gel changes its spontaneous curvature from positive at high-temperatures to approximately flat at lower temperatures. (b) The gel is embedded in the cell at $r = 5 cm$ , where the local curvature is negative, while the water bath is at $45 ° C$ . The gel migrates upward, as shown in the red curves of Fig. 1. At $t = 150 s$ , we begin the cooling cycle, and the water bath temperature drops to approximately $35 ° C$ over 2 min, and the gel’s spontaneous curvature is now near zero; as the bath cools, the gel simultaneously migrates down the bell to more closely match this curvature, where it remains for the remainder of the experiment.
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