Title Page
Abstract
Contents
1. Introduction 19
1.1. Background and objectives 19
1.2. Research outline 27
2. Constraint-based design for kirigami unit cell tessellation 29
2.1. Introduction 29
2.2. Overview of a constraint-based design 31
2.3. Geometric characteristics of 3D surfaces 35
2.4. Design of 2D domain using conformal mapping 39
2.4.1. Boundary conditions and their effect on the 2D domain 43
2.4.2. Reducing distortion using cone points 45
2.5. Projection of 2D uniform kirigami unit cells 48
2.6. Symmetry and deployment conditions 51
2.7. Physical-hinge model in the planar configuration 55
3. Deployment analysis of kirigami surfaces 57
3.1. Introduction 57
3.2. Bistable mechanism of uniform polygonal unit cells 59
3.3. Finite element modeling of kirigami surfaces 62
3.4. Prediction of the deployed 3D shape and stability 64
3.5. Fabrication of 3d-printed kirigami surfaces 69
3.6. Experimental verification of predicted shape and stability 71
4. Indentation test for deployed kirigami surfaces 73
4.1. Introduction 73
4.2. Establishment of finite element analysis procedure 74
4.3. Experimental setup 76
4.4. Mechanical responses of deployed kirigami surfaces 77
4.5. Investigating the effect of hinge width and surface thickness 79
5. Kirigami surfaces with spatially varying hinges 82
5.1. Introduction 82
5.2. Design of four ring-like regions 84
5.3. Stability test and analysis of deployed kirigami surfaces 85
5.4. Modulating the nonlinear stiffness profiles 91
5.5. Broadening the range of controllable stiffness 94
5.6. Egg protection experiments 114
6. Extension of the design approach 117
6.1. Introduction 117
6.2. Unit cell effects on the kirigami surface 118
6.3. Designing n-fold symmetric surfaces 124
6.4. Application to other target surfaces 126
7. Conclusion 129
A. Appendix 130
A1. Material properties of a 3d-printed rubber-like material 130
A2. Investigation of strain energy and strain energy density 132
A3. Angular variation of polygonal units under vertical indentation 141
Bibliography 152
국문초록 158
Table 3.1. Coordinate differences of four lines sequentially obtained from the left side in the front-view profiles (x-z view) along the mid-line of deployed kirigami... 67
Table 3.2. Total strain energy and total strain energy density in the deployed state (second stable state) for the design with uniform variations in HR or surface thickness. 68
Table 4.1. Initial stiffness, peak force, and corresponding indentation depth for six different designs with uniform variations of HR or surface thickness under... 80
Table 5.1. Effect of design parameters of kirigami surfaces on the deployed 3D shape, stability, and stiffness. 83
Table 5.2. Coordinate differences of four lines sequentially obtained from the left side in the front-view profiles (x-z view) along the mid-line of deployed kirigami... 88
Table 5.3. Strain energy in the deployed state (second stable state) for the design with two different hinge widths in four ring-like regions. 89
Table 5.4. Strain energy density in the deployed state (second stable state) for the design with two different hinge widths in four ring-like regions. 90
Table 5.5. Initial stiffness, peak force, and corresponding indentation depth for seven designs with two different hinge widths under indentation test. 93
Table 5.6. Strain energy in the deployed state (second stable state) for the design with three different hinge widths in four ring-like regions. 99
Table 5.7. Strain energy density in the deployed state (second stable state) for the design with three different hinge widths in four ring-like regions. 101
Table 5.8. Strain energy in the deployed state (second stable state) for the design with four different hinge widths in four ring-like regions. 103
Table 5.9. Strain energy density in the deployed state (second stable state) for the design with four different hinge widths in four ring-like regions. 104
Table 5.10. Initial stiffness, peak force, and corresponding indentation depth for twelve designs with three different hinge widths under indentation test. 106
Table 5.11. Initial stiffness, peak force, and corresponding indentation depth for six designs with four different hinge widths under indentation test. 108
Table 5.12. Coordinate differences of four lines sequentially obtained from the left side in the front-view profiles (x-z view) along the mid-line of deployed kirigami... 110
Table 5.13. Coordinate differences of four lines sequentially obtained from the left side in the front-view profiles (x-z view) along the mid-line of deployed... 112
Table A3.1. Angles of four lines sequentially obtained from the left side in the front-view profiles (x-z view) along the mid-line of the deployed kirigami surfaces with... 142
Table A3.2. Angles of four lines sequentially obtained from the left side in the front-view profiles (x-z view) along the mid-line of the deployed kirigami surfaces with... 145
Table A3.3. Angles of four lines sequentially obtained from the left side in the front-view profiles (x-z view) along the mid-line of the deployed kirigami surfaces with... 147
Table A3.4. Angles of four lines sequentially obtained from the left side in the fron-tview profiles (x-z view) along the mid-line of the deployed kirigami surfaces with... 149
Figure 1.1. Illustration of the design process for bistable kirigami surfaces with controllable stiffness. (Design) Given a 3D target surface, a planar non-uniform... 26
Figure 2.1. The tessellations and geometric parameters of the triangular unit cell comprising the hexagonal kirigami before and after deployment. In the expanded... 31
Figure 2.2. Illustration of the design process: (i) selecting a symmetry condition of a target surface, (ii) calculating the 2D uniform kirigami assuming idealized... 34
Figure 2.3. Illustration of the Gaussian curvature (K) for various 3D surfaces. For a developable surface with zero Gaussian curvature, one of the two principal... 37
Figure 2.4. Non-developable surface discretized with triangular meshes (left) and the geometric parameters of the local region (right). Brown region represents... 37
Figure 2.5. Illustration of the reflection and rotational symmetry. The reflection model is singularly symmetric with no rotational symmetry. A rotationally... 38
Figure 2.6. Conformal mapping of a triangulated 3D surface. a) The target 3D surface and the corresponding 2D domain. b) Geometric parameters of the... 39
Figure 2.7. Illustration of the edge flipping method. If the sum of the opposite angle is greater than 180° (i.e., θi + θj〉 𝜋), the corresponding edge is flipped.[이미지참조] 42
Figure 2.8. Relationship between the target 3D surface and the kirigami unit cell. a) The 2D domain with distributions of area ratio for the dome-like target surface.... 42
Figure 2.9. Unique 2D domain (bottom) corresponding to various target surfaces (top) with arbitrary triangular meshes applied with free boundary conditions. The... 44
Figure 2.10. Design of various 2D domains (bottom) for the same target surface (top) by applying different fixed boundary conditions. The sum of the included... 44
Figure 2.11. Effect of a cone point on the 2D domain (right) corresponding to the mask-like target surface (left).The cone point angle of 4π/3 and straight line... 47
Figure 2.12. Effect of multiple cone points (red dots) on the 2D domain (bottom) corresponding to the target 3D surface (top). Colors represent the ratio of 3D... 47
Figure 2.13. Illustration of projecting vertex in the 2D domain (Left) onto the 3D target surface (Right). 50
Figure 2.14. Planar kirigami unit cell for geometric parameters of the edge lengths and included angles of polygonal units. 52
Figure 2.15. Geometric parameters of kirigami unit cells in the projected configuration for designing the constraints. 54
Figure 2.16. Iteration-maximum constraint violation curve with snapshots of updated deployed kirigami surfaces. Colors represent the distance between each... 54
Figure 2.17. Design process for creating the real model by replacing point hinges to physical hinges. ts and th represent slit and hinge width lengths, respectively.[이미지참조] 56
Figure 3.1. Deployment process of bistable kirigami unit cells. a) Predicted force-strain and b) strain energy-strain curves for the design with w/s of 0.5951... 60
Figure 3.2. Predicted energy barrier distributions for different geometries of polygonal unit cells. The black dashed line represents the transition between the... 61
Figure 3.3. Finite element model with the loading and boundary conditions for the deployment analysis and test. The reflection symmetry condition and... 63
Figure 3.4. Mesh resolution of the kirigami unit cell and the 3D solid elements for discretizing the FE model. 63
Figure 3.5. Deployed and undeployed configurations of the idealized point-hinge model by matching the polygonal units located at the cutting center. 65
Figure 3.6. FE analysis procedure to predict the deployed configuration and check its stability. The hinge width (HR) plays a dominant role in determining... 66
Figure 3.7. Front-view profiles along the mid-line of deployed kirigami surfaces with different hinge widths (HR). For all designs, surface thickness of 3.8 mm is used. 66
Figure 3.8. Front-view profiles along the mid-line of deployed kirigami surfaces with different surface thickness. For all designs, HR of 0.03 is used. 67
Figure 3.9. Post-processing of 3d-printed model in the fabrication. 70
Figure 3.10. Experimental deployment test using bistable and monostable 3d-printed models. Scale bar, 20 mm. 72
Figure 3.11. Front-view profiles of the target surface (black line), the idealized point-hinge model (blue line), the FE model (gray line), and the experimental... 72
Figure 4.1. Illustration of the FE analysis procedure for indentation test and analysis. 75
Figure 4.2. Predicted force-displacement curves for variations in friction coefficient and gravity under vertical indentation. For all tests, HR of 0.03 is used. 75
Figure 4.3. Experimental setup for the indentation test. 76
Figure 4.4. Snapshots of a stably deployed kirigami surface under vertical indentation predicted using FE analysis (top) and observed in the experiment... 78
Figure 4.5. Predicted and measured force-displacement curves. In the experimental curve, the shaded area corresponds to the standard deviation... 78
Figure 4.6. Effect of HR (surface thickness of 3.8mm) and surface thickness (HR of 0.03) on the force-displacement curve of stably deployed surfaces under... 80
Figure 4.7. Predicted force-displacement curves of two different deployed surfaces with similar initial stiffness under indentation load. 81
Figure 4.8. Predicted front-view profiles along the mid-line of the stably deployed surface with HR of 0.04 under indentation load. 81
Figure 5.1. Four ring-like regions for assigning different hinge widths. 84
Figure 5.2. Effect of the hinge width in Region 4 on the deployed 3D shapes. After deployment, partially folded structures are captured. In all cases, HR⁽¹⁾ of... 86
Figure 5.3. Effect of the hinge width in Region 4 on the deployed 3D shapes. After deployment, fully deployed (bistable) structures are captured. In all cases,... 87
Figure 5.4. Predicted front-view profiles of deployed kirigami surfaces with two different hinges. In all cases, HR⁽¹⁾ of 0.03 and HR⁽²⁾ of 0.05 are used. 87
Figure 5.5. Predicted and measured force-displacement curves for three representative hinge variations under indentation test. In the experimental curve,... 92
Figure 5.6. Predicted force-displacement curves for designs with two different hinges under indentation test. In all cases, HR⁽¹⁾ of 0.03 and HR⁽²⁾ of 0.05 are used. 92
Figure 5.7. Predicted force-displacement curves for HR⁽²⁾ variations (HR⁽¹⁾ of 0.03) under indentation test. For all curves, the same legends are used. 96
Figure 5.8. Predicted deployed 3D shapes of the stably deployed surfaces with three different hinges under deployment analysis. In all cases, HR⁽¹⁾ of 0.03, HR⁽²⁾... 97
Figure 5.9. Predicted deployed 3D shapes of the stably deployed surfaces with four different hinges under deployment analysis. In all cases, HR⁽¹⁾ of 0.03, HR⁽²⁾... 98
Figure 5.10. Predicted force-displacement curves for the stably deployed surfaces with three different hinges under indentation test. In all cases, HR⁽¹⁾ of... 105
Figure 5.11. Predicted force-displacement curves for the stably deployed surfaces with four different hinges under indentation test. In all cases, HR⁽¹⁾ of 0.03, HR⁽²⁾... 105
Figure 5.12. Front-view profiles of stably deployed kirigami surfaces with three different hinges. In all cases, HR⁽¹⁾ of 0.03, HR⁽²⁾ of 0.04, and HR⁽³⁾ of 0.05 are used. 109
Figure 5.13. Front-view profiles of stably deployed kirigami surfaces with four different hinges. In all cases, HR⁽¹⁾ of 0.03, HR⁽²⁾ of 0.04, HR⁽³⁾ of 0.05, and HR⁽⁴⁾... 109
Figure 5.14. Predicted force-displacement curves for HR⁽¹⁾ variations (HR⁽²⁾ of 0.03) in Region 4 under indentation test. 113
Figure 5.15. Predicted range of controllable stiffness profiles. 113
Figure 5.16. Egg protection experiments using two deployed bistable kirigami surfaces with different hinge variations. In all cases, HR⁽¹⁾ of 0.03 and HR⁽²⁾ of... 115
Figure 5.17. Geometry and dimensions of the support for egg protection experiments. 115
Figure 5.18. Experimental setup for egg protection experiments. 116
Figure 6.1. Predicted undeployed and deployed configurations of idealized point-hinge models using five different w/s in the constraint-based design. 119
Figure 6.2. Predicted deployed 3D shape and front-view profiles of stably deployed kirigami surfaces using four different w/s under deployment analysis. 120
Figure 6.3. Numerical snapshots of deployed bistable kirigami surface using two different w/s under indentation test. Colors represent the predicted distribution... 120
Figure 6.4. Predicted force-displacement curves for w/s variations under indentation test. 121
Figure 6.5. Predicted undeployed and deployed configurations of idealized point-hinge models using 25x6 and 36x6 kirigami unit cells in the constraint-based design. 121
Figure 6.6. Predicted deployed 3D shape and front-view profiles of stably deployed kirigami surfaces designed with different number of unit cells under... 122
Figure 6.7. Numerical snapshots of deployed bistable kirigami surface designed with different number of kirigami unit cells under indentation test. Colors... 122
Figure 6.8. Predicted force-displacement curves for different number of unit cells under indentation test. For all tests, HR of 0.03 is used. 123
Figure 6.9. Various bistable kirigami surfaces deployable to the same dome-like target configuration designed using different symmetry conditions and unit cell types. 125
Figure 6.10. An example of designing bistable kirigami surfaces using a saddle-like target surface. 127
Figure 6.11. An example of designing bistable kirigami surfaces using a hyperboloid-like target surface. 127
Figure 6.12. 3d-printed saddle-like surface. 128
Figure 6.13. 3d-printed hyperboloid-like surface. 128
Figure A1.1. Snapshots of three specimens for uniaxial tensile testing before (left) and after (right) fracture. ASTM D412 C type dogbone samples are used for all tests. 131
Figure A1.2. Stress-strain curves of the dogbone samples for 0°, 45° and 90° build orientations. 131
Figure A2.1. Strain energy- and strain energy density-displacement curves for stably deployed surfaces of different hinge widths (HR) under indentation test.... 133
Figure A2.2. Strain energy- and strain energy density-displacement curves for stably deployed surfaces of different surface thickness under... 133
Figure A2.3. Strain energy-displacement curves for four distinct regions of stably deployed surfaces with two different hinge widths under indentation test. For all... 135
Figure A2.4. Strain energy density-displacement curves for four distinct regions of stably deployed surfaces with two different hinge widths under indentation test.... 136
Figure A2.5. Strain energy-displacement curves for four distinct regions of stably deployed surfaces with three different hinge widths under indentation test. For all... 137
Figure A2.6. Strain energy density-displacement curves for four distinct regions of stably deployed surfaces with three different hinge widths under indentation... 138
Figure A2.7. Strain energy-displacement curves for four distinct regions of stably deployed surfaces with four different hinge widths under indentation test. For all... 139
Figure A2.8. Strain energy density-displacement curves for four distinct regions of stably deployed surfaces with four different hinge widths under indentation... 140
Figure A3.1. Illustration of four different angles (ω₁, ω₂, ω₃, and ω₄) with front-view profiles (x-z view) along the mid-line of the stably deployed surface. 142
Figure A3.2. Angle variations of four lines sequentially obtained from the left side in the front-view profiles (x-z view) along the mid-line of the stably deployed... 143
Figure A3.3. Angle variations of four lines sequentially obtained from the left side in the front-view profiles (x-z view) along the mid-line of the stably deployed... 144
Figure A3.4. Angle variations of four lines sequentially obtained from the left side in the front-view profiles (x-z view) along the mid-line of the stably deployed... 146
Figure A3.5. Angle variations of four lines sequentially obtained from the left side in the front-view profiles (x-z view) along the mid-line of the stably deployed... 150
Figure A3.6. Angle variations of four lines sequentially obtained from the left side in the front-view profiles (x-z view) along the mid-line of the stably deployed... 151