Fractal features of a crumpling network in randomly folded thin matter and mechanics of sheet crushing

Abstract

We study the static and dynamic properties of networks of crumpled creases formed in hand crushed sheets of paper. The fractal dimensionalities of crumpling networks in the unfolded (flat) and folded configurations are determined. Some other noteworthy features of crumpling networks are established. The physical implications of these findings are discussed. Specifically, we state that self-avoiding interactions introduce a characteristic length scale of sheet crumpling. A framework to model the crumpling phenomena is suggested. Mechanics of sheet crushing under external confinement is developed. The effect of compaction geometry on the crushing mechanics is revealed.

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

“…[22], In this context, it is pertinent to point out that the fractal dimensions of ball configuration and the fractal dimension of a set of balls folded by the same forces are generally different due to elastic strain relaxation after the confinement force is withdrawn [23]. 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.…”

“…Another noteworthy feature of crumpling networks in randomly crushed thin sheets is their statistical scale invariance within a wide range of length scales [21,22], This gives rise to the fractal geometry of both a crumpled sheet configuration [22][23][24][25][26] and a set of balls folded from sheets of different sizes under the same confinement force [1,13,14,20,[27][28][29][30][31]. The relationships between the fractal dimensions of a crumpling network and sheet configuration were established in Ref.…”

“…As the compaction ratio K -L / R in creases, the packing density increases up to p = 1 at = ( n L / 6 h ) l/3. Experiments [12][13][14][15], theoretical con siderations [9][10][11][12][13]15,22,28,36], and numerical simulations [29][30][31]35,36,37] suggest that once K exceeds the threshold value K lh ( K lh & 1.7 [29,30]) the sheet packing density exhibits a power-law dependence on P. That is,…”

“…The crumpling behavior of thin sheets is strongly dependent on the confinement mode [15,22] and sheet geometry [22,34], One of the basic modes is the crumpling of a thin sheet subjected to isotropic confinement [22,[29][30][31]35], Under increasing hydrostatic pressure (P ) an initially flat square sheet is folded into an approximately spherical ball of diameter R with the packing density 6 hL? where h and L are the sheet thickness and edge size, respectively. As the compaction ratio K -L / R in creases, the packing density increases up to p = 1 at = ( n L / 6 h ) l/3.…”

“…The relationships between the fractal dimensions of a crumpling network and sheet configuration were established in Ref. [22], In this context, it is pertinent to point out that the fractal dimensions of ball configuration and the fractal dimension of a set of balls folded by the same forces are generally different due to elastic strain relaxation after the confinement force is withdrawn [23]. 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].…”