Relaxation Constant in the Folding of Thin Viscoelastic Sheets

Abstract

If one folds a thin viscoelastic sheet under an applied force, a line of plastic deformation is formed which shapes the sheet into an angle. We determine the parameters that define this angle experimentally and show that no matter how much load one applies or what is the thickness of the sheet, it is impossible to make angles less than a certain minimum angle in a definite time. Moreover, it is shown that regardless of whether the sheet is released freely afterward or kept under the load, a robust logarithmic relaxation follows the first deformation immediately. The slope of this logarithmic process is the same in both cases and depends neither on the applied force nor on the thickness of the sheet, which indicates it is directly a probe of molecular mobility of the material. We measure this intrinsic relaxation constant ~0.01 and ~5.7 for Mylar and paper sheets, respectively. We also suggest that the observed minimum angle of folding can be defined as a characteristic index for the plasticity of different materials.Folding, as a seemingly trivial way of making multilayer stacks or three-dimensional objects from thin sheets, is an everyday experience. While simple at first glance, folding problems arise in a variety of living organisms such as biological membranes, insect wings, and the cortical brain structure; as well as in artistic and technological applications ranging from decorative art and fashion to space solar panels. For instances, the shape of viral shells are determined by the energies of folding; 1, 2 the cerebral cortex expansion occurs with increasing degrees of folding of the cortical surface; 3 creasing properties of fabrics like crease-resistance and recovery are very important for designing skirts, suits, trousers, and ladies' dresses; and engineers seek to design series of solar panels which can change between folded stowed and planar configurations. 4 Other examples which require a better understanding of folding and have recently attracted a great deal of attention are origami designed structures 5-9 and crumpled sheets, 10-12 consisting respectively of ordered and random networks of creases created in a sheet. Mechanical properties of such systems are determined by not only the network but also by the response of each of the building block creases which can be considered as an elastic hinge of specific stiffness connecting flexible panels. 13 Furthermore, next-generation soft robots 14, 15 and wearable electronics 16 include thin 2D elastomeric parts that undergo continual bending and folding during use. Understanding the structural and geometrical changes, force production, as well as non-linear and time-

Predicting frictional aging from bulk relaxation measurements

“…However, in both cases, after a brief onset ( t < 1 s) the force relaxation is logarithmic over many decades in time: where C and B are constants. Previously, other molecular and/or athermal disordered systems including creased sheets 26 , 27 , crumpled papers 28 , 29 , and foams 29 were also reported to display such logarithmic mechanical response to deformation. Moreover, enthalpy and specific volume measurements during the aging of polymers show a similar logarithmic behavior 30 .…”

“…Moreover, enthalpy and specific volume measurements during the aging of polymers show a similar logarithmic behavior 30 . In the case of creased polymeric (Mylar and paper) sheets, which contain lines of localized plastic deformations, analysis of the relationship between C and B revealed that the logarithmic process is governed by a constant that doesn’t change with the sheet thickness or the applied load 27 . Similarly, for our polymer spheres, B is proportional to C for the whole range of the squeezing forces, from 0.1 N to above 50 N (Fig.…”

Simulation of crumpled sheets via alternating quasistatic and dynamic representations

“…However, crumpling often proceeds in an unpredictable manner: As a thin sheet is confined, stresses spontaneously localize to produce a complex network of creases in the sheet [8,9]. To elucidate the mechanisms underlying this complex process, crumpled structures have been investigated at different length scales, from the energy and geometry of individual ridges [10,11,12,13] and vertices [14,15], to the mechanical response of single folds [16,17,18,19], to the collective properties of disordered ridge networks, such as slow relaxation [20,21,22,23,24] and non-monotonic aging [25]. However, more remains to be understood about these rich phenomena.…”

Non-monotonic Dynamics in the Onset of Frictional Slip

“…This implies that, although the two transient dynamics occur on very different time scales (2-3 s vs 25,000 s), they are similar when plotted as a function of sliding distance where t 0 → 0 is a singular point in time from which the interfacial system evolves by aging and rejuvenation. Such a singular behavior arises in all disordered systems demonstrating logarithmic aging, ranging from creased/crumpled paper [25,26] and granular piles [27] to static frictional interfaces [7,12,28]. The consequence is that, if t 0 → 0 , the first term of Eq.…”