ar X iv : c on d - m at / 9 50 70 56 v 1 1 5 Ju l 1 99 5 cond - mat / 9507 xxx Roughness of randomly crumpled elastic sheets

We study the roughness of randomly crumpled elastic sheets. Based on analytical and numerical calculations, we find that they are self affine with a roughness exponent equal to one. Such crumpling occurs e.g. when wet paper dries. We also discuss the case of correlated crumpling, which occurs in connection with flattening of randomly folded paper. PACS: 61.43.Hv, 62.20.Dc, 61.43.Bn, 68.60.Bs Typeset using REVTEX ∗Permanent address: Institutt for fysikk, Norges tekniske høgskole, N–7034 Trondheim, Norway 1 Take a sheet of paper. Wet it with water and let it dry. The sheet will crumple. A closer look at the dried surface does not reveal any preferred length scale in the crumpling. Rather, smaller crumples appear inside the larger ones, and so on to smaller and smaller scales, finally to reach the scale of the fibers making up the paper. Surfaces having this appearance are in fact abundant in nature. For example, fracture surfaces show roughness on all scales, as do surfaces that have been corroded [1–4]. Recently, mathematical tools have been developed to describe such surfaces in terms of their scaling properties [5,6]. We will in this letter analyse the roughness of dried paper surfaces in light of these scaling properties based on a theoretical model originating from the theory of linear elasticity of thin plates [7,8]. We also note the connection between this problem and the problem of unfolding a randomly wrapped paper [9]: Wrap a sheet of paper randomly into a tight ball. Then unfold it and try to flatten it as much as possible. It will stay rough by much the same mechanism that makes dried paper rough. However, the foldings of the paper have introduced long-range correlations between the local crumpling of the paper. These longrange correlations may change the roughness of the sheets. Experimental measurements of this roughness show, however, that it is within the experimental uncertainty equal to the roughness due to uncorrelated crumpling. We generalize the one-dimensional model of reference [9] to a two-dimensional elastic sheet. However, the roughness found in this model is larger than the one found in the random crumpling case and in the experiments. The scaling properties alluded to above, are more precisely described through the concept of self affinity. We choose for a given surface the (x, y) plane to be the mean plane, and the z axis to be the normal direction. The surface is self affine if it is (statistically) invariant under the rescaling transformation 1The baking of traditional Norwegian flatbread or Indian papadam produces a much stronger crumpling, but of the same kind.