# Improvement of a Tunable Stiffness Organ-Grasping Device by Design of a Wavy-Shaped Beam Structure
## 2 Methods
### Improved Design of the Beams  
In this study, we approached the task to increase the stiffness gain of the device by increasing the stiffness in a high-stiffness state with suppressing the stiffness in a low-stiffness state. As written above, the wavy structure was also adapted in line with this strategy; however, optimal shape of the wave has not been investigated. We improve the beam design by two-step process; first investigate the shape of the wave to increase the stiffness in a high-stiffness state, second explore the arrangement of waves to decrease the stiffness in a low-stiffness state. The principal stress line is a concept that we use to maximize the stiffness of structures. Here we introduce the concept of principal stress line in two-dimensional material. Generally, a small square area around a point in the material receives normal and shear stress (Figure 4a). By rotating the square, we can find the orientation in which only normal stress is applied on the edges (Figure 4b). The normal stress in this orientation is called principal stress. The direction of the principal stress \(\theta\) is the solutions of the following equation:  
\[\tan 2\theta=\frac{2\tau_{xy}}{\sigma_{x}-\sigma_{y}}, \tag{3}\]  
where \(\sigma_{x}\) and \(\sigma_{y}\) are normal stress in the direction of \(x\) axis and \(y\) axis, respectively, and \(\tau_{xy}\) is shear stress. Four orientations that satisfies the equation correspond to the four normal stress on the edges. The angles gradually change according to the position and lead two sets of streamlines that are orthogonal each other. The lines are called principal stress lines (Figure 4b). The principal stress lines are relative to the deformation of curve in the material. The relationship is shown in Figure 4c,d. A curve that deviates from the principal stress lines receives normal and shear stress (Figure 4c). Let the angle between the curve and principal stress line \(\phi\) be small. From Mohr's stress circle, the normal stress \(\sigma_{x^{\prime}}\) and \(\sigma_{y^{\prime}}\) and the shear stress \(\tau_{x^{\prime}y^{\prime}}\) are as follows:  
\[\sigma_{x^{\prime}} =\frac{\sigma_{1}+\sigma_{2}}{2}+\frac{\sigma_{1}-\sigma_{2}}{2 }\cos 2\phi\approx\sigma_{1}, \tag{4}\] \[\sigma_{y^{\prime}} =\frac{\sigma_{1}+\sigma_{2}}{2}-\frac{\sigma_{1}-\sigma_{2}}{2 }\cos 2\phi\approx\sigma_{2}\] (5) \[\tau_{x^{\prime}y^{\prime}} =-\frac{\sigma_{1}-\sigma_{2}}{2}\sin 2\phi\approx-(\sigma_{1}- \sigma_{2})\phi, \tag{6}\]  
where \(\sigma_{1}\) and \(\sigma_{2}\) are the principal stress. On the other hand, a curve along a principal stress line receives only the normal stress \(\sigma_{1}\) and \(\sigma_{2}\) (Figure 4d). These suggest that the curve on the principal stress line gives the minimal deformation because of the elimination of shear stress. Figure 4: Illustration of principal stress lines. (**a**) Stress on a small square in the material. (**b**) Principal stress and principal stress lines. (**c**) Stress on a line deviated from principal stress lines. (**d**) Stress on a line aligned to principal stress lines. Black and white arrows represent normal and shear stress, respectively. Our strategy to form the shape of the wave is to make the edge of the wave follow the principal stress lines in a high-stiffness state. This strategy will suppress the shear stress across the edge of material and is expected to suppress the deformation. To design the wavy shape, we first derive the principal stress line of the device in a high-stiffness state. A part of the device in a high-stiffness state can be modeled as a cantilever, which is fixed on the position of handle and uniform load is applied on as Figure 5a,b. For simplicity, we supposed that the cantilever is made of homogeneous material in the following calculation.