Entropic rigidity of a crumpling network in a randomly folded thin sheet

Abstract

We have studied experimentally and theoretically the response of randomly folded hyperelastic and elasto-plastic sheets on the uniaxial compression loading and the statistical properties of crumpling networks. The results of these studies reveal that the mechanical behavior of randomly folded sheets in the one-dimensional stress state is governed by the shape dependence of crumpling network entropy. Following to the original ideas by Edwards for granular materials, we derive an explicit force-compression relationship which precisely fit the experimental data for randomly folded matter.Experimental data also indicate that the entropic rigidity modulus scale as the power of the mass density of folded ball with the universal scaling exponent.

Edwards's statistical mechanics of crumpling networks in crushed self-avoiding sheets with finite bending rigidity

“…Notice that Eq. (7) is consistent with the results of experimental studies [17,21,22,40,[48][49][50] and molecular dynamics simulations [29][30][31]37], On the other hand, it is easy to understand that the maximum ridge length is always less than or equal to the folded ball diameter. Furthermore, it is a straightforward matter to deduce that in the limit of p -» 1 the width of all ridges is equal to Wmin = ji h and so /max -> /m in -Consequently, in this limit, the total number of ridges is expected to be equal to…”

“…It was also recognized that both fractal dimensions are independent of the sheet elastic properties [20][21][22][23][24][25][26][27][28][29][30], but may change due to plastic deformations of the sheet material [26,28,31]. Consequently, although the crumpling processes appear quite haphazard, the crumpling behavior is well defined in a statistical sense and rather well reproducible in experiments [13][14][15][16][17][18][19][20]32], Moreover, almost all thin materials display nearly the same scale invariant crumpling behavior [22], This has allowed the authors of Ref. [28] [13] for randomly folded aluminum foils.…”

“…Consequently, the crumpling behavior of self-avoiding sheets with finite bending rigidity is governed by an evolving network of crumpling ridges. This is reflected in anomalously large resistance of folded sheets to hydrostatic compression [12][13][14][15], whereas their resistance to shear and axial loads is quite low [15][16][17][18][19][20].…”

“…Hence, one might expect that the mechanical response of randomly crumpled sheets can be explained within a framework of the Edwards's statistical mechanics of crumpling networks (see Refs. [17,19,41,42]). Although the Edwards's statistical mechanics was originally developed to deal with granular matter and spin glasses [42,43], its applicability to cellular systems and crumpling networks with jam m ed states has been already discussed in Refs.…”

“…Although the Edwards's statistical mechanics was originally developed to deal with granular matter and spin glasses [42,43], its applicability to cellular systems and crumpling networks with jam m ed states has been already discussed in Refs. [ 17,18,41,42,44].…”