# Wrinkling Patterns and Stress Analysis of Tensile Membrane with Rigid Elements
## 4 Discussion  
In this section, we use the finite-element method to analyze the eigenvalue buckling of the tension membrane with four square rigid elements, study the influence of the position of the rigid element on the critical buckling tension in the tension membrane, and analyze the influence mechanism of the position of the rigid element on the wrinkle pattern of the tension membrane. We selected a rigid element with a side length of 10 mm and a thickness of 1 mm and changed the position \(R_{c}\) of the rigid element from 10 mm to 60 mm. The eigenvalue buckling analysis was performed on the membrane to obtain the critical buckling tension and buckling mode of the membrane. The results are shown in Figure 5.  
As can be seen in Figure 5, the tensile membrane containing rigid elements has two distinct buckling modes. In mode 1, there are both wrinkling that passes through the rigid element and wrinkles that are distributed over both sides of the rigid element. The wrinkles in mode 2 are distributed over the upper side of the rigid element, which is similar to the wrinkle distribution of the uniform membrane. Mode 3 is the wrinkle distribution of the uniform membrane structure. As can be seen from Figure 5, the critical buckling tension in mode 1 is smaller than that of mode 3 when the rigid element is in the region of \(R_{c}\leq 45\) mm, which indicates that the rigid element reduces the stability of the membrane  
Figure 4: Experimental results of wrinkle patterns of rigid elements on the membrane at different positions: (**a**–**d**) indicate that the positions \(R_{c}\) of the rigid elements are 30 mm, 40 mm, 50 mm, and 60 mm, respectively. Figure 5: The critical buckling tension of the membrane when the rigid element is at different positions. in the region close to the membrane boundary. The critical buckling tension in mode 1 increases from the increase in position \(R_{c}\) of the rigid element, which means that under the same tension, the wrinkle amplitude decreases with the increase in position \(R_{c}\) of the rigid element. The buckling mode of the membrane changes from mode 1 to mode 2 when the position of the rigid element \(R_{c}\geq 50\) mm, the wrinkles on both sides with the rigid element disappear, and the wrinkle distribution of the membrane is similar to that of mode 3. Therefore, the rigid element has no effect on the wrinkle distribution of the membrane in this area. However, when the position of the rigid element \(R_{c}\geq 50\), the critical buckling tension in mode 2 is greater than that of mode 3, and the introduction of the rigid element increases the stability of the membrane. It can be seen from the above analysis that with the change in the position of the rigid element, the superposition of the rigid element and the tensile stress places the membrane in different stress states, which leads to the transformation of the instability mechanism of the membrane, thus showing different types of wrinkle pattern. Therefore, the size of the rigid element also affects the stress distribution and buckling mode of the membrane. We can change the wrinkle mode and buckling mode of the membrane by designing the position and size of the rigid element. According to the eigenvalue buckling analysis, the relationship between the critical buckling mode of the membrane and the size and position of the rigid element is shown in Figure 6. When the rigid element is in the blue area, there are both wrinkles passing through the rigid element and wrinkles distributed over both sides of the rigid element. When the rigid element is in the black line and the red area, the membrane has a wrinkle distribution similar to that of the uniform membrane, and the stability of the heterogeneous membrane in this region is greater than that of the uniform membrane.