In this section, the deployment simulation of the multi-element Miura membrane was carried out, in which the membrane crease was replaced by proxy model. A series of characteristics such as deployment rate, deployment error, maximum Mises stress, stretch ratio and wrinkle deformation were studied.
2.1. Membrane material and crease proxy model
PET (Polyethylene terephthalate) membrane has excellent mechanical properties with tensile strength and impact strength higher than ordinary membranes. And it is used more and more frequently in space tasks. The PET membrane is used for the study, with thickness of 100 μm, yield stress of 65 MPa, and Young's modulus of about 4800 MPa.15
The proxy model is used to replace membrane creases.14 This method mainly uses ABAQUS Hinge connectors to connect the membrane mesh nodes to replace the crease connections between the membrane surfaces. The crease width is defined based on the deploying simulation of Z-fold membrane, which was given in reference . The ABAQUS connector behavior simulates the elastic-plastic behaviors of the membrane creases. The connection method of the ABAQUS connector is shown in Fig. 1, and the elastic-plastic behavior at the crease is shown in Fig. 2. The same method is used to analyze the folding and deployment of the multi-element Miura membrane, in which the behavior at the crease is simulated only by the nonlinear elastic behavior of the ABAQUS connector.
2.2. Modeling of Miura membrane
The 210mm×210mm square PET membrane is used as the research object, and the Miura Origami method is used. The number of parallelogram elements is 5×5, as shown in Fig. 3. It is worth noting that the solid red line is the boundary of the membrane, and the part of the parallelogram element that exceeds the boundary will be removed. The dotted line creases are valley creases, the solid line creases are mountain creases. And the angle between the longitudinal crease and the y-axis direction is 15°. The ABAQUS connector is used to connect the mesh nodes in the membrane crease area, and the elastic-plastic behavior of the connector replaces the elastic-plastic properties of the crease. In addition, in order to avoid excessive stress concentration at the four corners of the membrane when the deployment force is too large, the four corners are removed and replaced by a linear loadable region that is 45° or 135° with the positive x-axis. The side length of this linear loadable region is 10mm, as shown in the partial enlarged area in Fig. 3.
2.3. Analysis of membrane folding and deploying process
2.3.1. The folding and deploying process of membrane
Taking the fully folded membrane as the research object, the deploying process of the membrane is discussed. It is known that the membrane still has a certain elastic deformation at the crease after it is fully folded, so the membrane will have a certain degree of free rebound after being relaxed. Assuming that the elastic-plastic properties of each crease of the membrane are consistent, the deployment characteristic curve is shown in Fig. 2. The area in the membrane that is about to produce creases is removed, and the mesh nodes are connected by hinge connectors. The elastic-plastic properties at the creases are used as the connection behavior of hinge connectors, so the proxy model is set up. Using the ABAQUS/Standard solver to perform quasi-static simulation, the membrane in a flat state will gradually folded under the action of the connector's connection behavior, until it reaches a stable state formed by the free rebound of the membrane without a proxy model instead of the crease after it is fully folded. The restriction of the membrane is shown in Fig. 4. Among them, U1, U2, U3 represent the displacement degrees of freedom in the X, Y, and Z directions, respectively. And UR1, UR2, and UR3 represent the rotation degrees of freedom in the X, Y, and Z directions, respectively. The membrane folding process is shown in Fig. 5, where the mesh size is 1mm and the simulation time is 1s.
It can be seen from Fig. 5(e) that when the membrane is in a release configuration, the stress concentration phenomenon mainly occurs at the intersection of the creases, and the maximum Mises stress is 22.41Mpa. In other places, such as the crease area far away from the intersection of the crease, there is no stress concentration phenomenon, and the stress value is small. In this release configuration, a linear force T parallel to the xoy plane is applied to the four linear loadable regions to gradually deploy the membrane. The deployment force T is gradually increased from 0N/mm to 10N/mm. The deploying process of membrane is shown in Fig. 6, and the specific application direction of the deployment force T is marked in Fig. 6(a).
2.3.2. Deployment result of membrane
(1) The deployment rate and deployment error
During the membrane deploying process, the projected area along the normal direction of the ideal membrane fully deployed plane (the xoy plane in Fig. 6) can be recorded as the effective area. The effective area is an important parameter of the space membrane front structure. Its ratio to the ideal deployment area is defined as the deployment rate of the membrane.14
In addition, the neutral surface of the membrane during the deploying process is analyzed, and the two planes parallel to the ideal deployment plane of the membrane are used to encompass the actual deployment neutral surface of the membrane. And the minimum distance between the two parallel planes is regarded as the deployment error.14 The deployment error during the membrane deploying process is marked in Fig. 7.
The deployment rate and the deployment error are used to describe the changes of the membrane in the effective area projection plane and the normal direction of the effective area projection plane. Since the simulation is quasi-static, there will be no vibration in the dynamic simulation, and each sample can reflect the equilibrium state of the membrane at this time.
Fig. 8(a) shows the change of the deployment rate during the deploying process of membrane. Since the membrane has elastic deformation at the crease after folded, it will produce free rebound behavior. The effective area of the membrane in the release configuration is 1364.97mm2, and the deployment rate is 3.1%. Subsequently, the deployment force T is applied to continue the deployment of the membrane. When 0N/mm<T≤0.16N/mm, the slope value of the deployment rate curve fluctuates around 3.98, but its rapid growth trend remains unchanged. When 0.16N/mm<T≤1N/mm, the slope value of the deployment rate curve gradually decreases and approaches zero. In this interval, the deployment rate increases to 96.26% at the end of the curve. And the effective area of the membrane is 42356.56mm2, which basically meets the application requirements.
When T=1N/mm, although the deployment rate of the membrane is 96.26%, and its effective area is already close to the state when the membrane is deployed, the deployment error is still very large, which is 15.68mm. When the condition of the deployment rate is satisfied, the deployment error during the membrane deploying process is shown in Fig. 8(b).
As shown in Fig. 8(b), the overall curve shows a downward trend, but the rate of decline gradually decreases. When T=6N/mm, the deployment error of the membrane is 1.79mm. Since the deployment error when the membrane fully folded is 42mm, the solution error of the deployment error at this moment is 4.26%. In addition, when 6N/mm<T≤10N/mm, the rate of reduction of the deployment error approaches zero. At the end of the curve, the deployment error is 1.25mm, and the average value of the longitudinal crease deployment angle is 69.94°, the average value of the transverse crease deployment angle is 71.37°. The rotation angle of the connector is still within the effective range of the proxy model instead of the membrane crease. Due to the elastic deformation of the membrane, although the creases have not yet reached the fully deployed state, the deployment rate of the membrane has reached 100%.
In summary, a certain degree of deployment force can make the membrane deployment rate reach the condition that the solution error is less than 5%, but if the solution error of the membrane deployment error is less than 5%, the above-mentioned deployment force needs to be increased to 5-6 times of its own.
(2) In-plane stretch ratio
The deployable structure must have a large enough size to meet the working requirements after deployment, but it still need to meet the storage requirements of small volume after folding. Therefore, it needs to be calculated for the stretch ratio. Considering that the effective area is an important parameter of the space membrane front structure, the ratio of the effective area of the membrane in the deployed configuration to the release configuration is taken as its in-plane stretch ratio. Obviously, when the membrane conforms to the small-volume folding, its in-plane stretch ratio is very important.16
In the formula, S1 is the effective area after the membrane structure is deployed, and S0 is the effective area when the membrane structure is released. The in-plane stretch ratio of the 5×5 elements membrane is 32.24, and the envelope volume in release configuration is 102.64cm3.
(3) Mises stress
It can be seen from the membrane deploying process in Fig. 6. When T≤1N/mm, although the four linear loadable regions of the membrane have the effect of deployment force T, the stress concentration area mainly exists the membrane crease area. With the gradual increase in the deployment force T, the maximum Mises stress point gradually approaches the intersection of the creases. When 1N/mm<T≤10N/mm, the stress concentration phenomenon in linear loadable regions of the membrane gradually intensifies. At the end of the deployment force growth, the maximum Mises stress exists in linear loadable regions, which is 91.98 MPa.
(4) Wrinkling phenomenon of deployed membrane
The thickness of the membrane structure is on the order of micrometers, its bending stiffness is very small. And it is prone to local instability when subjected to force, and then wrinkles.17 Affected by wrinkles, the membrane surface precision is reduced, and the working performance of the lightweight front reflection structure made of membrane materials is greatly reduced. There are two main types of wrinkles in membrane structure: one is material wrinkles, also called plastic wrinkles, which are irreversible wrinkles caused by plastic deformation of membrane materials during processing or in the folded configuration; the other is structural wrinkles, also called elastic wrinkles, it is a kind of buckling response behavior of the membrane under the action of external force. The structural wrinkle is completely reversible, and its amplitude changes with the force state.18
In this paper, the membrane is folded, plastic deformation occurs at the crease, so there are material wrinkles. In addition, after the membrane is deployed, wrinkles appear on the surface of the non-crease area, so there are structural wrinkles. In the case of deployment force T=10N/mm, the displacement of the 1/4 part of the membrane in the Z-axis direction is shown in Fig. 9.
Since the square membrane has a symmetrical structure, the material wrinkle of the structure is generally symmetrical about the symmetry axis shown in the Fig. 9. However, since the creases are asymmetric, the wrinkle shape of the membrane after the overlap of the material wrinkles and structural wrinkles is asymmetric. The A-A path passes through the intersection of the crease and is parallel to the membrane linear loadable region with distance of 48.9mm. Since the wrinkle shape of the membrane that changes along this path is more obvious, the coordinate axis is established from the bottom left to the top right as the positive direction, and the starting point of the membrane surface on the A-A path is used as the coordinate axis origin to analyze the out-of-plane deformation (the deformation in the Z-axis direction) of the membrane. The specific value of the deformation is shown in Fig. 10.
It can be seen from the Fig. 10 that the curve has three local maximum points and two local minimum points. Considering the accumulation of material wrinkles caused by many creases, the out-of-plane deformation of the membrane along the A-A path is all negative. The second local minimum point has the largest Z-axis negative out-of-plane deformation, which is 1.18mm. And the third local maximum point has the smallest Z-axis negative out-of-plane deformation, which is 0.23mm. Among them, the position of the largest negative out-of-plane deformation of the Z-axis is the position of the intersection of the creases, where has the largest amount of material wrinkles along the A-A path. Together with the effect of structural wrinkles, the current out-of-plane deformation of the membrane is formed.
In addition, the out-of-plane deformation curve of the membrane near the two end points of the A-A path does not fluctuate up and down and converges to a certain stable value. This is because the end points on both sides are close to the mountain crease area of the membrane, which causes the out-of-plane deformation near the end points on both sides showed a tendency to increase from the inside to the edge of the membrane.