The basic idea to unfold a 2D layer is to constrain each node of the upper border of the object to displace to a target curve (horizontal line in the example below). Each of these node is free to move along the curve and all the other nodes are set free. These constraints allow the system to converge until equilibrium is reached and the layer is unfolded.

We first tested the effect of the mechanical properties on 2D restoration of a simple fold.

Figure 1: (a) Mesh of a folded layer. (b) Ratio of the final area over the initial are (A_f/A_i) for a homogeneous case (E = 2.5 Pa, n = 0.25). (c) Ratio Af/Ai for a heterogeneous case where E varies from 6 Pa to 1 Pa from top to bottom. The thin black lines represent the neutral line. Red is contraction, blue is dilation.

We then tested the effect of the boundary conditions on 2D restoration of a simple fold.

Figure 2: (a) Mesh of a folded layer. Here slip is allowed between layers. (b) Ratio A_f/A_i for a model that is constrained only at the upper edge. (c) Ratio Af/Ai for a model that is constrained at both the upper and left edges. The thin black lines represent the neutral line. Red is contraction, blue is dilation.

The unfolding technique was tested on a folded sheet of paper. The goal was to test if a developable surface could be unfolded to its original shape. A 10*10 cm thin sheet of paper was first slightly crumpled, then its surface was digitized with a laser at the Stanford Computer Sciences Laboratories, which has a precision of 0.25 mm in the three dimensions (see Fig. 3a). The folded surface of the paper was then meshed with triangular elements (Fig. 3b).

Figure 3: a) Perspective view of a 10*10 cm crumpled sheet of paper digitized with a laser (resolution 0.25 mm). b) Mesh of the digitized surface

The unfolding process consisted of constraining each node of the folded surface to displace to a horizontal planar target surface. The X and Y coordinates of every node was set to be free to move in the XY plane in order to let the system converge freely to its unfolded shape. We used a finite plain strain deformation, a Young’s modulus of 10 GPa, and a Poisson’s ratio of 0.49. The restoration computed by the program (see Figs. 4) is in good agreement with the initial state. The initial squared shape of the sheet of paper is restored and the area is conserved. The minor variations of area of the unfolded state (i.e. 0.0005%) and its slightly undulating borders, result in part from the accuracy of the method used to digitize the surface, but most importantly from the meshing (coarse mesh) of the surface.

3. 3D unfolding of a non-developable surface and effect of material properties

The unfolding technique was tested on a hemisphere (see Fig. 5a), which is a non-developable surface. Here, we expect the triangular elements to contract or dilate in order to minimize the deformation caused by the unfolding process. The same procedure has been used and the surface was unfolded. Figure 5 shows the result for a homogeneous case. The contours represent the ratio of the final area over the initial are (A_f/A_i).The model shows that contraction (red) and dilation (blue) occur in the central region and in the peripheral region of the hemisphere respectively.

Figure 5: (a) 3D mesh of a hemisphere (folded surface). b) Contours of the deformation caused by the unfolding process. Contraction and dilation occur in the central region and in the peripheral region respectively. The area variation between the folded and unfolded states is 0.08 %.

Here we test the effect of the mechanical properties on the same non developable surface. Figure 6a shows the contours of the ratio A_f/A_i for a heterogeneous case (varying n) and Figure 6b for a heterogeneous case (varying E). Red is contraction, blue is dilation, and white is neutral line. Dashed line is the deformed boundary of mechanical properties.

4. 3D volume restoration

The basic idea to unfold a 3D volume is to constrain each node of the upper surface of the object to displace to a target surface. Each of these node is free to move along that surface and all the other nodes are set free in x, y and z directions. These constraints allow the system to converge until equilibrium is reached and the layer is unfolded. Here we test the effect of the layer thickness of a gentle fold on the stress perturbations. We show that a thicker fold (Fig. 7a) will produce larger stresses than a thinner fold (Fig. 7b).

Figure 7: Effects of layer thickness on 3D volume restoration.