DEVELOPMENT AND APPLICATION OF COMPOSITE MATERIAL LAMINATION THEORY FOR PRINTED CIRCUIT BOARDS by Kun-Yen Wang A thesis submitted to the Graduate Faculty of Auburn University in partial fulfillment of the requirements for the Degree of Master of Science Auburn, Alabama August 4, 2012 Copyright 2012 by Kun-Yen Wang Approved by Jeffrey C. Suhling, Chair, Quina Distinguished Professor of Mechanical Engineering Polapragada K. Raju, Thomas Walter Professor of Mechanical Engineering Winfred A. Foster, Professor of Aerospace Engineering ii Abstract Layers incorporating carbon fiber materials have been proposed to enhance the reliability of high performance printed circuit board production configurations. The carbon fiber layers feature high stiffness and high thermal conductivity, as well as near zero thermal expansion coefficients. In this research, theoretical approaches based on lamination theory have been developed for estimating the global/bulk coefficients of thermal expansion and mechanical properties of printed circuit boards incorporating carbon fiber layers based on the layer properties and stack-up configuration. Different types of carbon fiber laminates have been studied and tested. The coefficients of thermal expansion of individual carbon fiber PCB layers (plies) have been measured using strain gages with a temperature controlled chamber. The elastic moduli, Poisson?s Ratios, and shear modulus of individual carbon fiber plies have also been evaluated. An indirect method was developed and applied to find the shear modulus and Poisson?s ratio. The approach was applied to some example PCB configurations, and compared to the results of finite element analyses. iii Table of Contents Abstract .........................................................................................................................................ii I. INTRODUCTION ....................................................................................................................... 1 Literature Review ............................................................................................................. 1 II. LAMINATION THEORY FOR ORTHOTROPIC FIBER-REINFORCED COMPOSITES ...................... 6 Introduction ..................................................................................................................... 6 Modeling of Orthotropic Material Thermal Expansion ................................................... 7 Stress-Strain Relations and Material Properties for a Single Orthotropic Layer ............. 8 Analysis of Laminated Plates ......................................................................................... 13 III. STRAIN GAGE CALIBRATION AND MEASUREMENT EXPERIMENTS ....................................... 26 Introduction ................................................................................................................... 26 Example Application and Verification of the Strain Gage Measurement ...................... 28 IV. MEASUREMENT OF COEFFICIENTS OF THERMAL EXPANSION ............................................. 35 Introduction ................................................................................................................... 35 Description of Specimens and Environment Conditions ............................................... 35 Experiment Setup and Calibration ................................................................................. 37 Experimental Procedure and Results ............................................................................. 39 V. MEASUREMENT OF ELASTIC MODULI ................................................................................... 54 Introduction ................................................................................................................... 54 Description of Specimens and Testing Conditions ........................................................ 54 iv Results and Conclusion .................................................................................................. 55 VI. INDIRECT METHOD FOR EVALUATION OF SHEAR MODULUS AND POISSON?S RATIO ........ 80 Introduction ................................................................................................................... 80 Description of Specimens and Testing Conditions ........................................................ 81 Results and Conclusion .................................................................................................. 81 VII. APPLICATIONS TO STABLCOR PRINTED CIRCUIT BOARDS ................................................... 94 Description of Simulation Conditions ............................................................................ 94 Results and Conclusion .................................................................................................. 95 VIII. SUMMARY AND CONCLUSIONS ........................................................................................ 100 REFERENCES ............................................................................................................................. 101 APPENDIX A - ELASTIC MODULUS VERSUS OFF-AXIS ANGLE ................................................... 103 APPENDIX B - MATLAB CODE ................................................................................................... 177 v List of Figures Figure 1. Typical Flip-chip Assemblies ............................................................................................ 3 Figure 2. Single Fiber-Reinforced Layer .......................................................................................... 9 Figure 3. Single Fiber-Reinforced Layer with Fiber Angle ? ............................................................ 9 Figure 4. Various Angle Layers Stack Up Illustration .................................................................... 14 Figure 5. Illustration of N-ply Laminate Composition ................................................................... 15 Figure 6. Positive Direction for the Force Resultants on the Laminate ........................................ 20 Figure 7. Positive Directions for the Moment Resultants on the Laminate ................................. 20 Figure 8. Typical Thin Film Metal Foil Strain Gage ........................................................................ 31 Figure 9. Tension Testing Using a Strain Gage .............................................................................. 31 Figure 10. Compression Testing Using a Strain Gage ................................................................... 31 Figure 11. Strain Gage Calibration: Cantilever Beam Surface Strain Testing Setup ..................... 32 Figure 12. Surface Strain: Theoretical Prediction and Experimental Results (D = 8.25 in) .......... 34 Figure 13. Surface Strain: Theoretical Prediction and Experimental Results (D = 7.5 in) ............ 34 Figure 14. Environmental Chamber (Delta Design 9028) ............................................................. 40 Figure 15. Vishay Strain Gage Signal Processing Equipment ........................................................ 40 Figure 16. Sample of CTE Measurement Calculations .................................................................. 41 Figure 17. Testing Results for Aluminum CTE ............................................................................... 42 Figure 18. Testing Results for Copper CTE .................................................................................... 42 Figure 19. CTE Testing Specimens in Temperature Controlled Chamber..................................... 43 vi Figure 20. ST10F-0.005 Thermal Strain versus Temperature ....................................................... 43 Figure 21. ST10P-0.006 Thermal Strain versus Temperature ....................................................... 44 Figure 22. ST10F-0.006 Thermal Strain versus Temperature ....................................................... 44 Figure 23. ST10F-0.005 H/H Thermal Strain versus Temperature ................................................ 45 Figure 24. ST10P-0.006 H/H Thermal Strain versus Temperature ............................................... 45 Figure 25. ST10F-0.006 H/H Thermal Strain versus Temperature ................................................ 46 Figure 26. ST10F-0.006 1/1 Thermal Strain versus Temperature................................................. 46 Figure 27. ST10P-0.008 H/H Thermal Strain versus Temperature ............................................... 47 Figure 28. ST10F-0.008 H/H Thermal Strain versus Temperature ................................................ 47 Figure 29. ST10P-0.008 Thermal Strain versus Temperature ....................................................... 48 Figure 30. ST10F-0.008 Thermal Strain versus Temperature ....................................................... 48 Figure 31. ST10F-0.008 1/1 Thermal Strain versus Temperature................................................. 49 Figure 32. ST325P-0.010 Thermal Strain versus Temperature ..................................................... 49 Figure 33. ST325P-0.010 1/1 Thermal Strain versus Temperature .............................................. 50 Figure 34. ST325P-180 H/H Thermal Strain versus Temperature ................................................ 50 Figure 35. ST325P-0.010 H/H Thermal Strain versus Temperature ............................................. 51 Figure 36. ST325P-180 Thermal Strain versus Temperature ........................................................ 51 Figure 37. ST325P-0.008 H/H Thermal Strain versus Temperature ............................................. 52 Figure 38. Uniaxial Specimen in Mechanical Testing Machine..................................................... 56 Figure 39. Close-up Photograph of Uniaxial Specimen ................................................................ 56 Figure 40. ST10F-0.006 Stress versus Strain (1-direction) ............................................................ 57 Figure 41. ST10F-0.006 H/H Stress versus Strain (1-direction) .................................................... 57 vii Figure 42. ST10F-0.006 1/1 Stress versus Strain (1-direction) ..................................................... 58 Figure 43. ST10P-0.006 Stress versus Strain (1-direction) ............................................................ 58 Figure 44. ST10P-0.006 H/H Stress versus Strain (1-direction) .................................................... 59 Figure 45. ST10P-0.008 Stress versus Strain (1-direction) ............................................................ 59 Figure 46. ST10F-0.008 H/H Stress versus Strain (1-direction) .................................................... 60 Figure 47. ST10F-0.008 1/1 Stress versus Strain (1-direction) ..................................................... 60 Figure 48. ST10F-0.008 Stress versus Strain (1-direction) ............................................................ 61 Figure 49. ST10P-0.008 H/H Stress versus Strain (1-direction) .................................................... 61 Figure 50. ST325P-0.008 Stress versus Strain (1-direction) .......................................................... 62 Figure 51. ST325P-0.010 240 Stress versus Strain (1-direction) ................................................... 62 Figure 52. ST325P-0.010 H/H 255 Stress versus Strain (1-direction) ........................................... 63 Figure 53. ST325P-0.008 H/H Stress versus Strain (1-direction) .................................................. 63 Figure 54. ST325P-0.010 255 Stress versus Strain (1-direction) ................................................... 64 Figure 55. ST325P-0.010 1/1 240 Stress versus Strain (1-direction) ............................................ 64 Figure 56. ST325P-180 Stress versus Strain (1-direction) ............................................................. 65 Figure 57. ST10F-0.005 Stress versus Strain (1-direction) ............................................................ 65 Figure 58. ST325P-180 H/H Stress versus Strain (1-direction) ..................................................... 66 Figure 59. ST10F-0.005 H/H Stress versus Strain (1-direction) .................................................... 66 Figure 60. ST10F-0.005 H/H Stress versus Strain (2-direction) .................................................... 67 Figure 61. ST10P-0.006 Stress versus Strain (2-direction) ............................................................ 67 Figure 62. ST10F-0.008 Stress versus Strain (2-direction) ............................................................ 68 Figure 63. ST10F-0.006 H/H Stress versus Strain (2-direction) .................................................... 68 viii Figure 64. ST10P-0.006 H/H Stress versus Strain (2-direction) .................................................... 69 Figure 65. ST10F-0.005 H/H Stress versus Strain (2-direction) .................................................... 69 Figure 66. ST10P-0.008 Stress versus Strain (2-direction) ............................................................ 70 Figure 67. ST10F-0.008 H/H Stress versus Strain (2-direction) .................................................... 70 Figure 68. ST10F-0.006 Stress versus Strain (2-direction) ............................................................ 71 Figure 69. ST10P-0.008 H/H Stress versus Strain (2-direction) .................................................... 71 Figure 70. ST10F-0.008 1/1 Stress versus Strain (2-direction) ..................................................... 72 Figure 71. ST10F-0.006 1/1 Stress versus Strain (2-direction) ..................................................... 72 Figure 72. ST325P-180 Stress versus Strain (2-direction) ............................................................. 73 Figure 73. ST325P-0.010 240 Stress versus Strain (2-direction) ................................................... 73 Figure 74. ST325P-0.008 Stress versus Strain (2-direction) .......................................................... 74 Figure 75. ST325P-180 H/H Stress versus Strain (2-direction) ..................................................... 74 Figure 76. ST325P-0.008 H/H Stress versus Strain (2-direction) .................................................. 75 Figure 77. ST325P-0.010 255 Stress versus Strain (2-direction) ................................................... 75 Figure 78. ST325P-0.010 255 H/H Stress versus Strain (2-direction) ........................................... 76 Figure 79. ST325P-0.010 240 1/1 Stress versus Strain (2-direction) ............................................ 76 Figure 80. Example of Specimen with Mounted Strain Gages ..................................................... 79 Figure 81. Off-Axis Tensile Specimens .......................................................................................... 83 Figure 82. Sample Marked with Off-Axis Guide Lines .................................................................. 84 Figure 83. Hand Shear Used in Preparing Off-Axis Test Specimens ............................................. 84 Figure 84. ST10F-0.006 H/H 0-Degree Elastic Modulus ................................................................ 85 Figure 85. ST10F-0.006 H/H 90-Degree Elastic Modulus .............................................................. 85 ix Figure 86. ST10F-0.006 H/H 15-Degree Elastic Modulus .............................................................. 86 Figure 87. ST10F-0.006 H/H 30-Degree Elastic Modulus .............................................................. 86 Figure 88. ST10F-0.006 H/H 45-Degree Elastic Modulus .............................................................. 87 Figure 89. ST10F-0.006 H/H 60-Degree Elastic Modulus .............................................................. 87 Figure 90. ST10F-0.006 H/H 75-Degree Elastic Modulus .............................................................. 88 Figure 91. ST10F-0.006 H/H Elastic Modulus versus Off-Axis Angle ............................................ 88 Figure 92. ST10P-0.006 0-Degree Elastic Modulus ....................................................................... 89 Figure 93. ST10P-0.006 90-Degree Elastic Modulus ..................................................................... 89 Figure 94. ST10P-0.006 15-Degree Elastic Modulus ..................................................................... 90 Figure 95. ST10P-0.006 30-Degree Elastic Modulus ..................................................................... 90 Figure 96. ST10P-0.006 45-Degree Elastic Modulus ..................................................................... 91 Figure 97. ST10P-0.006 60-Degree Elastic Modulus ..................................................................... 91 Figure 98. ST10P-0.006 75-Degree Elastic Modulus ..................................................................... 92 Figure 99. ST10P-0.006 Elastic Modulus versus Off-Axis Angle .................................................... 92 Figure 100. Single STABLCOR Layer PCB Model............................................................................ 97 Figure 101. Single STABLCOR Layer Model Mesh with Solid Elements ........................................ 97 Figure 102. Three STABLCOR Layer PCB Model ............................................................................ 98 Figure 103. Three STABLCOR Layer Model Mesh with Solid Elements ........................................ 98 Figure 104. ST10F-0.005 0-Degree Elastic Modulus ................................................................... 105 Figure 105. ST10F-0.005 90-Degree Elastic Modulus ................................................................. 105 Figure 106. ST10F-0.005 15-Degree Elastic Modulus ................................................................. 106 Figure 107. ST10F-0.005 30-Degree Elastic Modulus ................................................................. 106 x Figure 108. ST10F-0.005 45-Degree Elastic Modulus ................................................................. 107 Figure 109. ST10F-0.005 60-Degree Elastic Modulus ................................................................. 107 Figure 110. ST10F-0.005 75-Degree Elastic Modulus ................................................................. 108 Figure 111. ST10F-0.005 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ................... 108 Figure 112. ST10F-0.005 H/H 0-Degree Elastic Modulus ............................................................ 109 Figure 113. ST10F-0.005 H/H 90-Degree Elastic Modulus .......................................................... 109 Figure 114. ST10F-0.005 H/H 15-Degree Elastic Modulus .......................................................... 110 Figure 115. ST10F-0.005 H/H 30-Degree Elastic Modulus .......................................................... 110 Figure 116. ST10F-0.005 H/H 45-Degree Elastic Modulus .......................................................... 111 Figure 117. ST10F-0.005 H/H 60-Degree Elastic Modulus .......................................................... 111 Figure 118. ST10F-0.005 H/H 75-Degree Elastic Modulus .......................................................... 112 Figure 119. ST10F-0.005 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ........... 112 Figure 120. ST10F-0.006 0-Degree Elastic Modulus ................................................................... 113 Figure 121. ST10F-0.006 90-Degree Elastic Modulus ................................................................. 113 Figure 122. ST10F-0.006 15-Degree Elastic Modulus ................................................................. 114 Figure 123. ST10F-0.006 30-Degree Elastic Modulus ................................................................. 114 Figure 124. ST10F-0.006 45-Degree Elastic Modulus ................................................................. 115 Figure 125. ST10F-0.006 60-Degree Elastic Modulus ................................................................. 115 Figure 126. ST10F-0.006 75-Degree Elastic Modulus ................................................................. 116 Figure 127. ST10F-0.006 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ................... 116 Figure 128. ST10F-0.006 1/1 0-Degree Elastic Modulus ............................................................. 117 Figure 129. ST10F-0.006 1/1 90-Degree Elastic Modulus ........................................................... 117 xi Figure 130. ST10F-0.006 1/1 15-Degree Elastic Modulus ........................................................... 118 Figure 131. ST10F-0.006 1/1 30-Degree Elastic Modulus ........................................................... 118 Figure 132. ST10F-0.006 1/1 45-Degree Elastic Modulus ........................................................... 119 Figure 133. ST10F-0.006 1/1 60-Degree Elastic Modulus ........................................................... 119 Figure 134. ST10F-0.006 1/1 75-Degree Elastic Modulus ........................................................... 120 Figure 135. ST10F-0.006 1/1 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ............ 120 Figure 136. ST10P-0.006 H/H 0-Degree Elastic Modulus ........................................................... 121 Figure 137. ST10P-0.006 H/H 90-Degree Elastic Modulus ......................................................... 121 Figure 138. ST10P-0.006 H/H 15-Degree Elastic Modulus ......................................................... 122 Figure 139. ST10P-0.006 H/H 30-Degree Elastic Modulus ......................................................... 122 Figure 140. ST10P-0.006 H/H 45-Degree Elastic Modulus ......................................................... 123 Figure 141. ST10P-0.006 H/H 60-Degree Elastic Modulus ......................................................... 123 Figure 142. ST10P-0.006 H/H 75-Degree Elastic Modulus ......................................................... 124 Figure 143. ST10P-0.006 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ........... 124 Figure 144. ST10F-0.008 0-Degree Elastic Modulus ................................................................... 125 Figure 145. ST10F-0.008 90-Degree Elastic Modulus ................................................................. 125 Figure 146. ST10F-0.008 15-Degree Elastic Modulus ................................................................. 126 Figure 147. ST10F-0.008 30-Degree Elastic Modulus ................................................................. 126 Figure 148. ST10F-0.008 45-Degree Elastic Modulus ................................................................. 127 Figure 149. ST10F-0.008 60-Degree Elastic Modulus ................................................................. 127 Figure 150. ST10F-0.008 75-Degree Elastic Modulus ................................................................. 128 Figure 151. ST10F-0.008 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ................... 128 xii Figure 152. ST10P-0.008 0-Degree Elastic Modulus ................................................................... 129 Figure 153. ST10P-0.008 90-Degree Elastic Modulus ................................................................. 129 Figure 154. ST10P-0.008 15-Degree Elastic Modulus ................................................................. 130 Figure 155. ST10P-0.008 30-Degree Elastic Modulus ................................................................. 130 Figure 156. ST10P-0.008 45-Degree Elastic Modulus ................................................................. 131 Figure 157. ST10P-0.008 60-Degree Elastic Modulus ................................................................. 131 Figure 158. ST10P-0.008 75-Degree Elastic Modulus ................................................................. 132 Figure 159. ST10P-0.008 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ................... 132 Figure 160. ST10F-0.008 H/H 0-Degree Elastic Modulus ............................................................ 133 Figure 161. ST10F-0.008 H/H 90-Degree Elastic Modulus .......................................................... 133 Figure 162. ST10F-0.008 H/H 15-Degree Elastic Modulus .......................................................... 134 Figure 163. ST10F-0.008 H/H 30-Degree Elastic Modulus .......................................................... 134 Figure 164. ST10F-0.008 H/H 45-Degree Elastic Modulus .......................................................... 135 Figure 165. ST10F-0.008 H/H 60-Degree Elastic Modulus .......................................................... 135 Figure 166. ST10F-0.008 H/H 75-Degree Elastic Modulus .......................................................... 136 Figure 167. ST10F-0.008 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ........... 136 Figure 168. ST10P-0.008 H/H 0-Degree Elastic Modulus ........................................................... 137 Figure 169. ST10P-0.008 H/H 90-Degree Elastic Modulus ......................................................... 137 Figure 170. ST10P-0.008 H/H 15-Degree Elastic Modulus ......................................................... 138 Figure 171. ST10P-0.008 H/H 30-Degree Elastic Modulus ......................................................... 138 Figure 172. ST10P-0.008 H/H 45-Degree Elastic Modulus ......................................................... 139 Figure 173. ST10P-0.008 H/H 60-Degree Elastic Modulus ......................................................... 139 xiii Figure 174. ST10P-0.008 H/H 75-Degree Elastic Modulus ......................................................... 140 Figure 175. ST10P-0.008 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ........... 140 Figure 176. ST10F-0.008 1/1 0-Degree Elastic Modulus ............................................................. 141 Figure 177. ST10F-0.008 1/1 90-Degree Elastic Modulus ........................................................... 141 Figure 178. ST10F-0.008 1/1 15-Degree Elastic Modulus ........................................................... 142 Figure 179. ST10F-0.008 1/1 30-Degree Elastic Modulus ........................................................... 142 Figure 180. ST10F-0.008 1/1 45-Degree Elastic Modulus ........................................................... 143 Figure 181. ST10F-0.008 1/1 60-Degree Elastic Modulus ........................................................... 143 Figure 182. ST10F-0.008 1/1 75-Degree Elastic Modulus ........................................................... 144 Figure 183. ST10F-0.008 1/1 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ............ 144 Figure 184. ST325P-180 0-Degree Elastic Modulus ................................................................... 145 Figure 185. ST325P-180 90-Degree Elastic Modulus ................................................................. 145 Figure 186. ST325P-180 15-Degree Elastic Modulus ................................................................. 146 Figure 187. ST325P-180 30-Degree Elastic Modulus ................................................................. 146 Figure 188. ST325P-180 45-Degree Elastic Modulus ................................................................. 147 Figure 189. ST325P-180 60-Degree Elastic Modulus ................................................................. 147 Figure 190. ST325P-180 75-Degree Elastic Modulus ................................................................. 148 Figure 191. ST325P-180 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot .................... 148 Figure 192. ST325P-180 H/H 0-Degree Elastic Modulus ............................................................ 149 Figure 193. ST325P-180 H/H 90-Degree Elastic Modulus .......................................................... 149 Figure 194. ST325P-180 H/H 15-Degree Elastic Modulus .......................................................... 150 Figure 195. ST325P-180 H/H 30-Degree Elastic Modulus .......................................................... 150 xiv Figure 196. ST325P-180 H/H 45-Degree Elastic Modulus .......................................................... 151 Figure 197. ST325P-180 H/H 60-Degree Elastic Modulus .......................................................... 151 Figure 198. ST325P-180 H/H 75-Degree Elastic Modulus .......................................................... 152 Figure 199. ST325P-180 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ............ 152 Figure 200. ST325P-0.008 0-Degree Elastic Modulus ................................................................. 153 Figure 201. ST325P-0.008 90-Degree Elastic Modulus ............................................................... 153 Figure 202. ST325P-0.008 15-Degree Elastic Modulus ............................................................... 154 Figure 203. ST325P-0.008 30-Degree Elastic Modulus ............................................................... 154 Figure 204. ST325P-0.008 45-Degree Elastic Modulus ............................................................... 155 Figure 205. ST325P-0.008 60-Degree Elastic Modulus ............................................................... 155 Figure 206. ST325P-0.008 75-Degree Elastic Modulus ............................................................... 156 Figure 207. ST325P-0.008 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ................. 156 Figure 208. ST325P-0.008 H/H 0-Degree Elastic Modulus ......................................................... 157 Figure 209. ST325P-0.008 H/H 90-Degree Elastic Modulus ....................................................... 157 Figure 210. ST325P-0.008 H/H 15-Degree Elastic Modulus ....................................................... 158 Figure 211. ST325P-0.008 H/H 30-Degree Elastic Modulus ....................................................... 158 Figure 212. ST325P-0.008 H/H 45-Degree Elastic Modulus ....................................................... 159 Figure 213. ST325P-0.008 H/H 60-Degree Elastic Modulus ....................................................... 159 Figure 214. ST325P-0.008 H/H 75-Degree Elastic Modulus ....................................................... 160 Figure 215. ST325P-0.008 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ......... 160 Figure 216. ST325P-0.010 240 0-Degree Elastic Modulus .......................................................... 161 Figure 217. ST325P-0.010 240 90-Degree Elastic Modulus ........................................................ 161 xv Figure 218. ST325P-0.010 240 15-Degree Elastic Modulus ........................................................ 162 Figure 219. ST325P-0.010 240 30-Degree Elastic Modulus ........................................................ 162 Figure 220. ST325P-0.010 240 45-Degree Elastic Modulus ........................................................ 163 Figure 221. ST325P-0.010 240 60-Degree Elastic Modulus ........................................................ 163 Figure 222. ST325P-0.010 240 75-Degree Elastic Modulus ........................................................ 164 Figure 223. ST325P-0.010 240 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot .......... 164 Figure 224. ST325P-0.010 255 0-Degree Elastic Modulus .......................................................... 165 Figure 225. ST325P-0.010 255 90-Degree Elastic Modulus ........................................................ 165 Figure 226. ST325P-0.010 255 15-Degree Elastic Modulus ........................................................ 166 Figure 227. ST325P-0.010 255 30-Degree Elastic Modulus ........................................................ 166 Figure 228. ST325P-0.010 255 45-Degree Elastic Modulus ........................................................ 167 Figure 229. ST325P-0.010 255 60-Degree Elastic Modulus ........................................................ 167 Figure 230. ST325P-0.010 255 75-Degree Elastic Modulus ........................................................ 168 Figure 231. ST325P-0.010 255 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot .......... 168 Figure 232. ST325P-0.010 255 H/H 0-Degree Elastic Modulus .................................................. 169 Figure 233. ST325P-0.010 255 H/H 90-Degree Elastic Modulus ................................................ 169 Figure 234. ST325P-0.010 255 H/H 15-Degree Elastic Modulus ................................................ 170 Figure 235. ST325P-0.010 255 H/H 30-Degree Elastic Modulus ................................................ 170 Figure 236. ST325P-0.010 255 H/H 45-Degree Elastic Modulus ................................................ 171 Figure 237. ST325P-0.010 255 H/H 60-Degree Elastic Modulus ................................................ 171 Figure 238. ST325P-0.010 255 H/H 75-Degree Elastic Modulus ................................................ 172 Figure 239. ST325P-0.010 255 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot .. 172 xvi Figure 240. ST325P-0.010 240 1/1 0-Degree Elastic Modulus ................................................... 173 Figure 241. ST325P-0.010 240 1/1 90-Degree Elastic Modulus ................................................. 173 Figure 242. ST325P-0.010 240 1/1 15-Degree Elastic Modulus ................................................. 174 Figure 243. ST325P-0.010 240 1/1 30-Degree Elastic Modulus ................................................. 174 Figure 244. ST325P-0.010 240 1/1 45-Degree Elastic Modulus ................................................. 175 Figure 245. ST325P-0.010 240 1/1 60-Degree Elastic Modulus ................................................. 175 Figure 246. ST325P-0.010 240 1/1 75-Degree Elastic Modulus ................................................. 176 Figure 247. ST325P-0.010 240 1/1 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ... 176 xvii List of Tables Table 1. Strain Gage Calibration: Surface Strain Theoretical and Experimental Comparison Results (D=8.25 inch) .............................................................................................. 33 Table 2. Strain Gage Calibration: Surface Strain Theoretical and Experimental Comparison Results (D=7.5 inch) ................................................................................................ 33 Table 3. STABLCOR Layer Material Table .................................................................................... 36 Table 4. Coefficients of Thermal Expansion for STABLCOR Ply Materials .................................. 53 Table 5. Elastic Modulus Results Obtain by Crosshead Displacement ....................................... 77 Table 6. Elastic Modulus Measurement Comparisons for Stress Evaluated Using Crosshead Displacements and Strain Gages ..................................................................... 78 Table 7. Shear Moduli and Poisson?s Ratios for STABLCOR Ply Materials ................................. 93 Table 8. Lamination Theory and Finite Element Method Model Material Constants ................ 96 Table 9. Laminate Layups ? Single STABLCOR Model ................................................................. 96 Table 10. Laminate Layups ? Tri-STABLCOR Model .................................................................... 96 Table 11. Single STABLCOR Layer Model CTE ............................................................................. 99 Table 12. Tri-STABLCOR Layer Model CTE .................................................................................. 99 Table 13. Curve Fit Shear Modulus and Poisson?s Ratio Chart with Page Number .................. 104 1 I. INTRODUCTION Literature Review For modern electronics, such as smart phones or other consumer electronics, improvement goals include much smaller scale, increased power, tougher durability, and more cost effective. IBM corporation invented the C4 (Controlled Collapse Chip Connection) IC attachment method [1]. This breakthrough in electronic packaging technology is also known as the flip-chip packaging. Moreover, flip-chip assembly was taken up by Delco Electronics in the 1970s, and is commonly used in automobile applications. Undoubtedly, flip chip assembly results in smaller products than traditional chip carrier-based systems. Since the chip has been directly mounted on the PCB (Printed Circuit Board) substrate, the area and height used are much smaller. In addition, the close distance between the flip chip and the substrate provides multiple benefits including lower inductance, capability of transferring higher-speed signals, and good heat conduction. However, due to the short distance, the connections are extremely stiff and coefficient of thermal expansion mismatch becomes a major concern. Figure 1 shows typical flip chip assemblies. Historically, ceramic substrates have ideal for flip chip attachment, since most ceramics have a similar CTE (Coefficient of Thermal Expansion) to the silicon chip. Large CTE mis-matches between the silicon chip and its substrate will cause cyclic stresses to be developed in the solder balls when the assembly is subjected to thermal cycling, and solder joint fatigue becomes the major failure mode. Use of ceramic substrates can eliminate this concern. However, due to the much higher production costs and 2 increased weight, ceramic packages are mostly limited to military and high performance applications. FR-4 (Flame Resistant) glass-reinforced epoxy laminate materials have been commonly used for PCBs (Printed Circuit Boards) in nearly all fields of electronic devices. FR-4 is a composite material composed of woven fiberglass with an epoxy resin binder that is flame resistant. FR-4 materials have several benefits when serving as an electronic substrate such as low cost, low density, good strength to weight ratios, insensitive to humidity changes, and good electrical insulation and manufacturing characteristics. All of the above characteristics result in wide usage of FR-4 in many applications. Due to the high CTE (16~20 ppm/oC) offered by conventional FR-4 glass fiber materials, a large CTE mis-match occurs between the silicon (2.6 ppm/oC) chip and the FR-4 substrate in flip chip on laminate applications [2]. This CTE mis-match leads to development of large stresses in the solder balls located between the flip chip and PCB. As a result, this can cause poor durability and short life-cycle of the device [3-5]. A general formulation referred to as classical lamination theory exists for mechanical analysis of composite materials. It can be used to calculate the deformations, strains, and stresses in a fiber-reinforced laminate such as a PCB that is subjected to mechanical loads and temperature changes. In addition, it can be used for predicting the average mechanical properties and CTEs of the laminate structure. The input properties to the theory include the mechanical properties and CTEs of each layer, as well as details of the stacking configuration including the order, orientations, and thicknesses of the various layers. 3 Figure 1. Typical Flip-chip Assemblies (a) Ceramic Substrate (b) FR-4 Substrate (c) STABLCOR Substrate 4 Lamination theory was originally proposed for fiber-reinforced composite materials, and was based on a linear elastic orthotropic constitutive model and the typical assumptions of classical small deflection plate theory [6-7]. Lamination theory has often been applied to PCBs. For instance, Nakagawa and Yokoyama [8] used composite lamination theory in order to estimate deformation of the PCB during the reflow process. They also used the theory to optimize the stack up of the laminate to reduce PCB deformation. Tan and Ume [9] also used lamination theory to study PCB warpage during soldering processes and under normal operations. In current high power electronic circuit design [10], it is common to have single or multiple thermal conductivity layers in PCB substrates. The ideal thermally conductive layers are metal plates or carbon-material plates. Metal materials such as copper-invar-copper, aluminum, and steel have been used as cores in PCBs to enhance thermal performance [11]. However, the main issue with these materials is undesirable weight. Jensen [12], patented a support board fabricated from graphite filament or aramid filament reinforced thermoset resin, which has a thermal coefficient of expansion less than 2 micro inch/inch/oF. Leibowitz [13] showed that a multilayer combination between graphite and dielectric material, such as a polytetrafluoroethylene (PTFE) and woven glass laminate can provide both low dielectric constant and good mechanical strength with low or negative coefficient of thermal expansion. Moreover, he also claimed one of the principal advantages of the use of carbon fiber reinforced layers in the PCB structure is that they serve as a good conductors of heat, which normally flows from the mounted components through the copper layers of the board, through mounting bolts, and then to a housing or other heat sink. 5 Vasoya [14] also stated that PCB laminates made of continuous carbon fiber layers have the advantages on both high thermal conductivity and low in-plane CTE. STABLCOR? [15] (previously Thermount) is a trademarked carbon fiber based material set. Characteristic of STABLCOR materials include high thermal conductivity, low CTEs, and high elastic moduli. They are also of relatively low density compared to metals such as copper. STABLCOR? [16-17] laminates have been used by various companies in the electronics industry such as Boeing, Ambient Energy, Infineon, Delphi, JPL, FLIR System, Los Alamos, Curtiss Wright, BAE System, Aehr Test System, Maxwell Technologies, Samsung, General Dynamics, and Raytheon. They are commonly used to replace certain layers in conventional PCBs such as copper-invar-copper, heavy copper, aluminum heat spreaders, thermal interface materials, and Thermount? materials [18]. STABLCOR? laminates also have additional benefits such as reduced moisture absorption, increasing rigidity without increasing weight, substantially increasing thermal conductivity, lowered assembly temperatures, and smaller z-axis expansion. By considering all the advantages and benefits of STABLCOR?, carbon fiber based PCB ply materials have the potential to offer significantly lower CTE values as well as other improvements. In order to tailor the PCB into a desired CTE value and other related attributes, the choice of layer material properties and the laminate configuration are critical. For analysis purposes, composite lamination theory provides a strong theoretical background. Moreover, it will be shown in this thesis that lamination theory can be used to design and calculate performance for various laminate configurations with different types of STABLCOR? ply materials and provide predictions of the in-plane CTEs of the resulting PCB laminate. 6 II. LAMINATION THEORY FOR ORTHOTROPIC FIBER-REINFORCED COMPOSITES Introduction The fiber-reinforced composite materials considered in this work for PCB applications consist of glass and/or carbon fiber fabrics hardened with an epoxy resin. Macroscopically, a fiber-reinforced epoxy layer is an orthotropic material, where the elastic moduli are directionally dependent and are greatly affected by the angle to the fiber direction. The angle between the force and fiber direction determines the stiffness of the laminate. The fibers in such layers are long and continuous as opposed to whiskers. Several fiber-reinforced epoxy layers are typically stacked and bonded to form a composite material laminate. The mechanical response of a laminate can be modeled using the general purpose lamination theory presented in this chapter. The layers in the composite laminate are usually designed to be symmetrical in relation to the thickness of the board. This is especially true for laminates with carbon fiber layers since they have very low CTE and very high tensile modulus, and thus dominate the other glass reinforced layers [14]. All of the fiber-reinforced laminates used in this research were constructed from bi- directional 0o/90o woven layers. In this case, the principal material directions are 0o (x1- direction) and 90o (x2-direction) and layer is classified as an orthotropic material. For orthotropic materials, like isotropic materials, application of normal stresses in a principal direction results in extension in the direction of the stress and contraction perpendicular to the stress. The strain in one principal material direction could be different from the strain in another principal material direction, even under the same normal stress. In other words, 7 different elastic moduli exist for the various principal material directions in the orthotropic material. Moreover, due to the different stiffness properties in the two principal material directions, the contractions will also be different for the two principal directions. Thus, the Poisson?s Ratios of an orthotropic material will typically vary with different pairs of principal material directions. Finally, the magnitude of the in-plane shearing deformation in an orthotropic layer is also independent from the elastic moduli and Poisson?s Ratios. As discussed below, there are four unique material properties needed to describe the in-plane mechanical behavior of orthotropic materials. Modeling of Orthotropic Material Thermal Expansion The relationship between the free thermal expansion and temperature change of a material is often modeled using a linear relationship TT ????? (1) where ?T is the thermal-induced strain, ? is the coefficient of thermal expansion (CTE) of the material, and ?T is the change in temperature. Thin orthotropic materials have two principal material directions in the plane of the layer. The x1-direction is the primary principal material direction, which is normally the stiffest direction. Moreover, the x2-direction is the in-plane direction perpendicular to the x1-direction. The thermal strains upon these two directions are normally different. Thus, eq. (1) is typically generalized as TT ??? 11)( ?? (2) 8 TT ??? 22)( ?? (3) where (?1)T and (?2)T are the thermal strains in the principal material directions, and ?1 and ?2 are the coefficients of thermal expansion in the principal material directions. During a temperature change, no thermally induced shear strains are produced in the coordinate system aligned along the directions of material symmetry. Stress-Strain Relations and Material Properties for a Single Orthotropic Layer A typical fiber reinforced layer is shown in Figure 2. The principal material directions of a constituent ply or lamina are denoted by 1, 2, and 3. They will often not be aligned with the chosen coordinate directions, denoted by x, y, and z. This is illustrated in Figure 3 for the case of rotation of angle ? about z-axis. Linear elastic stress-strain relations are often used to model an individual layer. For the case of plane stress in the 1-2-3 coordinate system, the plane stress version of linear elastic stress-strain relations for an orthotropic lamina exposed to a thermal/temperature change have the form ?? ? ? ? ?? ? ? ? ?? ?? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? 12 22 11 66 2212 1211 12 2 1 00 0 0 ? ?? ?? ? ? ? T T Q QQ QQ (4) where ?1, ?2, ?12 are the in-plane normal and shear stresses, and ?1, ?2, ?12 are the in-plane normal and shear strains. The coefficients in eq. (4) are the reduced stiffness?s of the material that can be evaluated from the material properties of the layer using the following relations: 9 Figure 2. Single Fiber-Reinforced Layer Figure 3. Single Fiber-Reinforced Layer with Fiber Angle ? 1x 3x 2x F i b e r D i r e c t i o n 1x z,x 3 2x F i b e r D i r e c t i o n ? x y 10 ? ? ? ? ? ? ? ? 0 11 1 1 2616 1266 2112 121 2112 212 2112 2112 2 22 2112 1 11 ?? ? ? ? ? ?? ? ? ? ? QQ GQ EE QQ E Q E Q ?? ? ?? ? ?? ?? (5) where E1 and E2 are the elastic moduli, 12? and 21? are the Poisson?s Ratios, and G12 is the shear modulus. Furthermore, it should be noted that the strains (?1, ?2, ?12) in eq. (4) are the total strains including both mechanical and thermal (CTE) induced contributions. From the basic theory of mechanics of materials, the transformation relationships for the in-plane stresses and strains are ? ?? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? xy y x T ? ? ? ? ? ? ? 12 2 1 (6) and ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ? ? ? ? ? 2 2 12 2 1 xy y x T ? ? ? ? ? ? ? (7) The transformation matrix ??? ??T in eqs. (6,7) is given by ? ?? ? ? ??? ? ? ? ?? ? ? ? ?? ?? 22 22 22 2 2 nmmnmn mnmn mnnm T ? (8) where ?cos?m , ?sin?n , and positive ? are defined as shown in Figure 2. The inverse of this matrix is easily evaluated to be 11 ? ?? ? ? ? ? ?? ??? ???? ? ? ? ?? ? ? ? ?? ??? T nmmnmn mnmn mnnm T 22 22 22 1 2 2 (9) Equation (7) can be used to characterize the transformation of the coefficients of thermal expansion by applying it to the thermal expansion strains. In this case, the strain transformations have the form ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? T xy Ty Tx T T T 2 0 2 1 ? ? ? ?? ? (10) or ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? T T T TT T xy y x 2 0 2 1 ? ? ? ?? ? (11) where x? , y? , and xy? are the in-plane coefficients of thermal expansion for the x-y-z coordinate system. Inversion of eq. (11) leads to ? ?? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 2 2 1 ? ? ? ? ? ? T xy y x (12) It is important to note that xy? is often not zero. In other words, shear strains can be generated in an off-axis coordinate system when the temperature of the lamina changes. 12 Substitution of eqs. (6-9,12) into eq. (4) gives the plane stress version of the off-axis stress-strain relations in the x-y-z coordinate system as ?? ? ? ? ?? ? ? ? ?? ?? ?? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? T T T QQQ QQQ QQQ xyxy yy xx xy y x ?? ?? ?? ? ? ? 662616 262212 161211 (13) where ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?4466226612221166 3 662212 3 66121126 3 662212 3 66121116 4 22 22 6612 4 1122 44 12 44 66221112 4 22 22 6612 4 1111 22 2 2 2 2 22 4 22 nmQnmQQQQQ nmQQQmnQQQQ mnQQQnmQQQQ mQnmQQnQQ nmQnmQQQQ nQnmQQmQQ ?????? ?????? ?????? ???? ????? ???? (14) are the transformed reduced stiffness coefficients. Expressions for the transformed coefficient of thermal expansion in eq. (13) are obtained by expanding eq. (12) to ? ?mn nm nm xy y x 21 2 1 2 2 2 2 2 1 ??? ??? ??? ?? ?? ?? (15) 13 Analysis of Laminated Plates As discussed previously, a typical fiber reinforced laminate is often constructed by stacking and bonding multiple orthotropic and/or isotropic layers together. An example is shown in Figure 4. In this section, the general theory for the thermal-mechanical behavior of laminated fiber-reinforced materials is presented. This formulation is usually known as composite material lamination theory [6]. In this development, each layer is considered as linear elastic and the laminate experiences only small out-of-plane deformations. A general illustration of a laminate composed of N lamina is shown in Figure 5. The layers of the laminate are numbered as 1, 2, 3,?,N from top to bottom. Each orthotropic ply is also associated with an angle ? which represents the orientation of the 1-direction for that ply. The angle between the 1-direction of the kth layer and the x-axis is denoted k? , where subscript k is the notation for the kth ply of the laminate. The z-axis is taken to be perpendicular to the plate with the positive downward direction and z = 0 at the middle surface. A basic assumption of lamination theory is that the out-of-plane stresses are negligible in comparison to the in-plane stresses. Therefore, the plane stress version of orthotropic Hooke?s law can be used for representing the linear elastic layers of the plate. Using eq. (13), the stress-strain relations for the kth layer are kxyxy yy xx kkxy y x T T T QQQ QQQ QQQ ?? ? ? ? ?? ? ? ? ?? ?? ?? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ?? ?? ?? ? ? ? 662616 262212 161211 (16) where the subscript k on the reduced stiffness matrix implies that the material properties and angle ? appropriate for kth layer should be used to evaluate the ijQ coefficients. 14 Figure 4. Various Angle Layers Stack Up Illustration 15 Figure 5. Illustration of N-ply Laminate Composition 16 The linearized strain-displacement relations for a body experiencing small strains and deformations are ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? x v y u y w z v x w z u z w y v x u xy yz xz z y x 2 1 2 1 2 1 ? ? ? ? ? ? (17) where u, v, w are the displacements in the x, y, z directions. The other standard assumption used in lamination theory is that the change in thickness dimension is relatively small and negligible. Using the third equation of eq. (17), rewritten as eq. (18). 0???? zw z? (18) Another classic assumption is that ?Plane sections remain plane after loading?. This implies that the out-of-plane shear strains xz? and yz? are approximately equal to zero. From the fourth and fifth sub-equations in eq. (17), this assumption becomes 021 021 ??? ? ? ??? ? ? ?? ? ?? ?? ? ?? ? ? ? ?? ? ?? y w z v x w z u yz xz ? ? (19) 17 If eqs. (18,19) are assumed to be equalities and integrated, the following results are obtained for the laminated plate displacement fields: ? ? ? ? ? ?yxww y wzyxVv x wzyxUu , , , 0 0 ? ? ??? ? ??? (20) where ? ?yxU ,0 , ? ?yxV ,0 are the middle surface in-plane displacement fields. Substituting eq. (20) into eqs. (17) gives the general form for the in-plane strain fields of the laminated plate ? ? ? ? ? ? ? ? yx w z x yxV y yxU y w z y yxV x w z x yxU xy x x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 00 2 2 0 2 2 0 2 ,, , , ? ? ? (21) The form of these relations is usually simplified by introducing additional notations for the middle surface strains and curvatures. The three middle surface strains (z = 0) are defined as ? ? ? ? ? ? ? ? x yxV y yxU y yxV x yxU xy y x ? ?? ? ?? ? ?? ? ?? ,, , , 000 00 00 ? ? ? (22) and the three middle surface curvatures are defined as 18 yx w y w x w x y x ?? ? ?? ? ? ?? ? ? ?? 2 2 2 2 2 2? ? ? (23) The six expressions showed in eqs. (22, 23) are a set of equivalent strain-displacement relations for the mid-surface strains and curvatures of the laminated plate. Using these definitions, eq. (21) can be rewritten in matrix form as ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ???? ? ? ?? ? ? ? xy y x xy y x xy y x z ? ? ? ? ? ? ? ? ? 0 0 0 (24) From eqs. (20, 24) it is clear that the in-plane displacement and strain fields vary linearly through the thickness of the laminated composite material. Substituting the results given in eq. (24) into eq. (16) gives the in-plane stress-strain relations for the laminated composite material plate in matrix form kxy y x xy y x xy y x kkxy y x Tz QQQ QQQ QQQ ?? ??? ?? ??? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 0 0 662616 262212 161211 (25) From eq. (25), it can be seen that the in-plane stresses vary linearly in any given layer of the laminated plate. However, the behavior is piecewise linear with discontinuities when graphing the stresses through the entire thickness of laminate. This is due to the fact that the values of the reduced stiffnesses ijQ can vary discontinuously from one layer to the other. From the development of classical plate theory [6], internal forces and moments per unit length can be defined and used as auxiliary variables which are related to the stresses but 19 independent of the thickness coordinate z. The in-plane forces per unit length or in-plane force resultants are defined as integrals of the in-plane stresses through the thickness of the plate as yx h h xyxy h h yy h h xx NdzN dzN dzN ?? ? ? ? ? ? ? ? ? ? ? ? 2 2 2 2 2 2 ? ? ? (26) where h is the thickness of the plate. The shear forces per unit length or the shear force resultants are similarly defined as the integrals of the out-of-plane shear stresses through the thickness of the plate ? ? ? ? ? ? ? ? 2 2 2 2 h h yzy h h xzx dzQ dzQ ? ? (27) The moments per unit length or the moment resultants are defined as the integrals of the moment of the in-plane stresses through the thickness of the plate as yx h h xyxy h h yy h h xx MzdzM zdzM zdzM ?? ? ? ? ? ? ? ? ? ? ? ? 2 2 2 2 2 2 ? ? ? (28) In these relations, both the force and moment resultants depend only on coordinates x and y. On the plate elements illustrated in Figures 6 and 7, the positive directions of all of the force and moment resultants are shown, which are consistent with the positive directions of the stresses integrated in eqs. (26-28). 20 Figure 6. Positive Direction for the Force Resultants on the Laminate Figure 7. Positive Directions for the Moment Resultants on the Laminate 21 For laminated plate calculations, eq. (26) is applied to evaluate the in-plane force resultants by integrating through each layer. Furthermore, the results from each individual layer are been added to treat as a bulk material. dzdz N N N k N k z z xy y xh h xy y x xy y x k k ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? 1 2 2 1 ? ? ? ? ? ? (29) where kz and 1?kz are the distances to the ply interfaces of the kth ply (see Figure 3). By substituting eq. (25) into eq. (29), the following expression results: dzTz QQQ QQQ QQQ N N N N k z z kxy y x xy y x xy y x kxy y x k k 1 0 0 0 662616 262212 161211 1 ? ? ? ? ?? ?? ? ? ?? ?? ? ? ?? ??? ?? ??? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? ? ? ? ? ? ? ? ? ? (30) The integrands in eq. (30) contain only terms that are constant and linear in z. These can be easily integrated to give ?? ??? ?? ???? ?? ??? ?? ??? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? T xy T y T x xy y x xy y x kxy y x N N N BBB BBB BBB AAA AAA AAA N N N ? ? ? ? ? ? 662616 262212 161211 0 0 0 662616 262212 161211 (31) or in concise form ? ? ? ?? ? ? ?? ? ? ?TNBAN ??? ?? 0 (32) where ? ? ? ? ? ? ? ? ? ?6,2,1, 2 1 1 2 1 2 1 1 ? ?? ?? ? ? ? ? ? ? ji zzQB zzQA N k kkkijij N k kkkijij (33) 22 Here, ijA are the extensional stiffnesses, and ijB are the coupling stiffnesses. The thermal stress term in eqs. (31, 32) is calculated from T d z QQQ QQQ QQQ N N N kxy y x k N k z zT xy T y T x k k ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? ? ? ? ? ? ? ? 1 662616 262212 161211 1 (34) By using a similar procedure, the moment resultants can also be expressed in terms of the mid-surface strains and curvatures. The integrals for each single layer can be obtained by breaking up the stacked total thickness integral. z d zz d z M M M k N k z z xy y xh h xy y x xy y x k k ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? 1 2 2 1 ? ? ? ? ? ? (35) Combining eq. (25) and (35), leads to dzTzzz QQQ QQQ QQQ M M M N k z z kxy y x xy y x xy y x kxy y x k k ? ? ? ? ?? ?? ? ? ?? ?? ? ? ?? ??? ?? ??? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? 1 2 0 0 0 662616 262212 161211 1 ? ? ? ? ? ? ? ? ? (36) Integrating eq. (36) then gives: ?? ??? ?? ???? ?? ??? ?? ??? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? T xy T y T x xy y x xy y x kxy y x M M M DDD DDD DDD BBB BBB BBB M M M ? ? ? ? ? ? 662616 262212 161211 0 0 0 662616 262212 161211 (37) or in concise form ? ? ? ?? ? ? ?? ? ? ?TMDBM ??? ?? 0 (38) The coupling stiffnesses ijB in eqs. (37, 38) have already been defined in eq. (33). The bending stiffnesses ijD are defined as 23 ? ? ? ? ? ?6,2,1, 3 1 1 3 1 3 ? ?? ? ? ? ji zzQD N k kkkijij (39) And the thermally-induced moment term can be expressed as T z d z QQQ QQQ QQQ M M M kxy y x k N k z zT xy T y T x k k ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? ? ? ? ? ? ? ? 1 662616 262212 161211 1 (40) By combining eq. (31) and (37), the following 6 ? 6 matrix equation can be established: ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? T xy T y T x T xy T y T x xy y x xy y x xy y x xy y x M M M N N N DDD DDD DDD BBB BBB BBB BBB BBB BBB AAA AAA AAA M M M N N N 0 0 0 0 0 0 662616 262212 161211 662616 262212 161211 662616 262212 161211 662616 262212 161211 ? ? ? ? ? ? (41) This relation is the well-known governing equation of composite material lamination theory and can be rewritten in concise form as ?? ? ?? ?? ?? ? ?? ?? ???????????? T T MNDB BAMN ?? 0 (42) For the case of a symmetric laminate ( ijB = 0) and no mechanical loading ( xN , yN , and xyN are 0), eq. (31) can be rewritten as ?? ??? ?? ???? ?? ??? ?? ??? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? T xy T y T x xy y x N N N AAA AAA AAA 0 0 0 662616 262212 161211 0 0 0 ? ? ? (43) or 24 ?? ??? ?? ??? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? 0 0 0 662616 262212 161211 xy y x T xy T y T x AAA AAA AAA N N N ? ? ? (44) This result can be inverted to give ?? ??? ?? ??? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? ??? ??? ??? T xy T y T x xy y x N N N AAA AAA AAA 1 33 1 32 1 31 1 23 1 22 1 21 1 13 1 12 1 11 0 0 0 ? ? ? (45) If the thermal loading consists of a uniform temperature change, eq. (34) can be rewritten as Th QQQ QQQ QQQ N N N k kxy y x k N kT xy T y T x ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? ? ? ? ? ? 1 662616 262212 161211 (46) where kh is the thickness of the kth layer Substitution of eq. (46) into eq. (45) yields Th QQQ QQQ QQQ AAA AAA AAA k kxy y x k N k xy y x ? ?? ?? ? ? ?? ?? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ??? ?? ??? ? ? ??? ??? ??? ? ? ? ? ? ? 1 662616 262212 161211 1 33 1 32 1 31 1 23 1 22 1 21 1 13 1 12 1 11 0 0 0 (47) The effective (average) CTE of a laminate can be obtained by rewriting eq. (1) as: TT?? ?? ?? (48) Thus, eq. (47) can be rearranged to yield expressions for the effective (average) expansion coefficients of the laminate: 25 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? ??? ??? k kxy y x k N k xy y x xy y x h QQQ QQQ QQQ AAA AAA AAA T T T ? ? ? ? ? ? ? ? ? 1 662616 262212 161211 1 33 1 32 1 31 1 23 1 22 1 21 1 13 1 12 1 11 0 0 0 0 0 0 (49) By using eq. (49), one can determine the surface (effective) CTEs for a laminated PCB, based on the relationship between thermal induced strain and temperature change. PCBs are typically constructed with symmetric layer configurations with all layers having fiber directions either at o0?? or o90?? [14]. In this case, the shear CTE xy? will be zero for all layers. There are seven material properties for each layer that need to be known in order to use eq. (49). The necessary material properties are 1? (the coefficient of thermal expansion along first principal material direction), 2? (the coefficient of thermal expansion along second principal material direction), E1 (elastic modulus along the first principal material direction), E2 (elastic modulus along the second principal material direction), 12? and 21? (Poisson?s Ratios along first and second principal material directions), and G12 (shear modulus). These properties have been evaluated for several PCB layers as discussed in subsequent chapters. 26 III. STRAIN GAGE CALIBRATION AND MEASUREMENT EXPERIMENTS Introduction Most typical strain gages contain a long, thin conductive strip in a serpentine pattern of parallel lines as shown in Figure 8. By this particular formation, even a small amount of stress in the direction of the orientation of the parallel lines results in a multiplicatively larger strain over the effective length of the conductor. Strain gage measurements have been used to provide accurate, reliable, and repeatable test results on isotropic materials for the past 70 years. They were invented by Edward E. Simmons and Arthur C. Ruge in 1938. Strain gages typically contain two parts, a metallic foil pattern and an insulating flexible backing film on which the foil pattern is bonded. The strain gage backing film can be bonded to the target object with various kinds of adhesives, such as cyanoacrylate or epoxy adhesives. Due to the bonding, the metallic foil pattern on the strain gage will experience the same deformations as the specimen it is bonded to. Once the metallic foil is deformed, its electrical resistance will be changed and correlated to the degree of deformation. The changing resistance can be measured using a circuit such as a Wheatstone Bridge. The main strain gage performance characteristics are the electrical conductance of the metal foil material and conductor?s geometry. In the tension case, the strain testing is limited within the elastic limit of the metal foil such that it does not break or yield as shown in Figure 9. The end-to-end electrical resistance will be increased. In the compressive case, the conductor will tend to become shorter and broader if not buckled. It will result in decreasing the overall end-to-end resistance as shown in Figure 10. In either case, the change of resistance will alter 27 the output voltage when current is passing through the metal foil. The altered output voltage can be observed by voltage meter. Strain gages measure only local deformations under the bonded region. The primary equation relating the resistance change to the strain is ? ?? GFRR ?? (50) where R? is the change in resistance caused by strain, R is the resistance of the undeformed gage, ? is the strain, and GF is the gage factor. For metallic foil gages, the gage factor is usually around 2.0. When using a Wheatstone Bridge in a quarter bridge configuration, the output voltage from the bridge is: 4 ???? GFBVV (51) where BV is the bridge excitation voltage. The excitation voltage is usually applied directly into the wires which are connected between the strain reader and strain gage terminals. Moreover, the typical input voltages are from 2V to 12V with output readings in millivolts. In this study, the excitation voltage has been set at 2V for all experiments, since higher excitation voltages tend to heat up the strain gage and raise the temperature of the metal foil, leading to measurement errors. Strain gages are attached to the specimen with special adhesives. The type of adhesive depends on both the required lifetime of the measurement system and the specimen material type. For instance, cyanoacrylic adhesive is often used as the preferable adhesive with short term testing (usually weeks) and measurements done under normal room temperature. However, for long lasting installation and wide range temperature operation, an epoxy 28 adhesive will be the more ideal solution. Epoxy adhesives need to be cured with a high temperature (80 oC ? 200 oC) curing profile in order to have an optimal bond to the specimen. Temperature variation during testing will cause a multitude of effects on strain gage measurements. In order to compensate for thermal errors, two identical strain gages need to be mounted on two different objects made from the same material. The first strain gage needs to be mounted on the target specimen while the other strain gage (dummy gage) should be mounted on an unloaded material. Both materials should experience the same temperature change. Using two arms of a Wheatstone Bridge, the thermally induced errors in each gage will be subtracted and compensate for each other in the bridge output voltage. Example Application and Verification of the Strain Gage Measurement As a trial experiment to explore the accuracy of strain gages, an axial gage was mounted on a cantilever aluminum bar (with one clamped boundary). A concentrated load was applied at the top of the beam using a micrometer as shown in Figure 11. The strain gage was mounted near the clamped end of the bar, and the load was applied at the free end of the bar. The strain gage used in this experiment was the general purpose strain gage CEA- 250UN-350 manufactured by Vishay Measurements Group [22], and the measurement instrument was a Vishay Micro-Measurements Model 3800 Wide Range Strain Indicator. This instrument has a resolution of 1 ?? at any gage factor from 0.0500 to 50.00. Moreover, the excitation voltage was changeable from 1 ? 15 Vdc with 0.02% error. The test cantilever bar was made of aluminum and had a thickness t = 0.248 inches, width b = 1.0 inches, length L = 10 inches, and the elastic modulus E = 10 x 106 psi. The strain 29 gage measurements were recorded for every 0.025 inch change of the micrometer displacement. Since the micrometer displacements were increased linearly and the beam did not yield, the resulting strain measurements should also show a linear dependence on the applied beam deflection. The classical beam theory solution for the out-of-plane deflection distribution of the cantilevered beam considered as ? ? ??? ? ??? ? ? ????????????? L xLxEIPLxv 2161 33 (50) where v is the beam deflection, P is the applied concentrated load, E is the modulus of elasticity, I is the moment of inertia, x is the distance measured from the fix end, and L is the length of the beam. This expression can be rewritten in terms of the prescribed end deflection ? ?Lv?? which is applied by the micrometer. Plugging x = L into eq. (52) and solving for the applied load gives 33LEIP ?? (51) By substituting eq. (53) into eq. (52), the theoretical out-of-plane deflection distribution is found in terms of the end displacement applied by the micrometer ? ? ??? ? ??? ? ? ????????????? L xLxxv 21613 3? (52) The strain at the gage location can be calculated from beam theory as ? ? ??? 23 2 3 33 ????????????? ?? LtDL dLtx (53) 30 where x? is the strain at the gage location,? is the deflection of beam at micrometer loading point, t is the beam thickness, L is the beam length, and d is the x-position of the strain gage. Theoretical and experimental results for the surface strain at the gage location are listed in Table 1 and 2 for 25.8?D and 5.7?D inch. The experimental results were found to agree well with theoretical calculations, and the errors were found to be lower than 3%. The normalized plots of the out-of-plane displacement versus longitudinal position matched extremely well, which are shown in Figure 12 and 13. Based on the accurate results of these example tests, the strain gage measuring method is considered as an effective and accurate method for measuring specimen strain. 31 Figure 8. Typical Thin Film Metal Foil Strain Gage Figure 9. Tension Testing Using a Strain Gage Figure 10. Compression Testing Using a Strain Gage E n d - to - en d T er m i n a l s Str a i n Sen s i ti v e P a tter n / M eta l Fo i l Hi g h er E n d - to - en d Res i s ta n c e T en s i o n : M eta l Fo i l A r ea N a r r o w s , Res i s ta n c e In c r ea s es L o w er E n d - to - en d Res i s ta n c e C o m p r es s i o n : M eta l Fo i l A r ea T h i c k en s , Res i s ta n c e D ec r ea s es 32 Figure 11. Strain Gage Calibration: Cantilever Beam Surface Strain Testing Setup L D t ? d 33 Table 1. Strain Gage Calibration: Surface Strain Theoretical and Experimental Comparison Results (D = 8.25 inch) Table 2. Strain Gage Calibration: Surface Strain Theoretical and Experimental Comparison Results (D = 7.5 inch) D = 8.25 inch N (increment of 0.025 inch) ? (inch) x? (?? )(Theoretical) x? (?? )(Experimental) %Error 1 0.025 77 79 2.9 2 0.050 153 157 2.3 3 0.075 230 236 2.5 4 0.100 307 314 2.3 5 0.125 384 393 2.4 6 0.150 460 471 2.3 7 0.175 537 547 1.8 8 0.200 614 626 1.9 9 0.225 691 704 1.9 D = 7.5 inch N (increment of 0.025 inch) ? (inch) x? (?? )(Theoretical) x? (?? )(Experimental) %Error 1 0.025 70 72 3.1 2 0.050 140 143 2.4 3 0.075 209 214 2.2 4 0.100 279 285 2.1 5 0.125 349 356 2.0 6 0.150 419 426 1.8 7 0.175 488 496 1.6 8 0.200 558 567 1.6 9 0.225 628 637 1.5 34 Surface S train Theor etical P rediction and Ex perimental Results D= 8.25 in Defl ection of B eam (in) 0.0 0.1 0.2 0.3 0.4 0.5 S urface S t rain ? x ( ? m) 0 200 400 600 800 1000 1200 1400 The or etical Pred ic tion Exper imen tal Re sults Figure 12. Surface Strain: Theoretical Prediction and Experimental Results (D = 8.25 in) Surface S train Theor etical P rediction and Ex perimental Results D= 7.5 in Defl ection of B eam (in) 0.0 0.1 0.2 0.3 0.4 0.5 S urface S t rain ? x ( ? m) 0 200 400 600 800 1000 1200 1400 The or etical Pred ic tion Exper imen tal Re sults Figure 13. Surface Strain: Theoretical Prediction and Experimental Results (D = 7.5 in) 35 IV. MEASUREMENT OF COEFFICIENTS OF THERMAL EXPANSION Introduction In this chapter, experimental measurements of the thermal expansion coefficients 1? and 2? along the first and second principal directions of several different single-ply carbon fiber reinforced layers are presented. These results will be used in modeling of the material properties of the carbon fiber reinforced printed circuit boards later in this thesis. The CTEs for the two principal directions were measured by strain gages. Two sets of samples were prepared for each ply material and three tests were performed for each sample. The same testing procedure was used for all of the different carbon fiber ply materials tested. The carbon fiber ply materials tested in this study were STABLCOR carbon fiber reinforced epoxy laminates layers that were specially designed to implement in high-end printed circuit boards. The STABLCOR laminates offer higher stiffness properties and better heat conduction than traditional FR-4 glass-epoxy materials. Moreover, STABLCOR laminates also have better strength to weight ratios, which make them attractive for high performance applications. Description of Specimens and Environment Conditions In this study, 18 different kinds of STABLCOR carbon-fiber epoxy ply materials were evaluated. The basic material designations and thicknesses are listed in Table 3. The copper weight column in the table describes the amount of copper plating on the specimen. In this category, ?NONE? corresponds to no copper plating layer, ?H/H? corresponds to a half ounce 36 Table 3. STABLCOR Layer Material Table Sample No. Material Description Product Category Copper Weight Core Thickness 1 ST10F-0.005 ST10 NONE 0.005" (0.127mm) 2 ST10F-0.006 ST10 NONE 0.006" (0.152mm) 3 ST10P-0.006 ST10 NONE 0.006" (0.152mm) 4 ST10F-0.006 H/H ST10 1/2 oz over 1/2 oz 0.006" (0.152mm) 5 ST10F-0.006 1/1 ST10 1 oz over 1 oz 0.006" (0.152mm) 6 ST10P-0.006 H/H ST10 1/2 oz over 1/2 oz 0.006" (0.152mm) 7 ST10F-0.008 ST10 NONE 0.008" (0.203mm) 8 ST10P-0.008 ST10 NONE 0.008" (0.203mm) 9 ST10F-0.008 H/H ST10 1/2 oz over 1/2 oz 0.008" (0.203mm) 10 ST10P-0.008 H/H ST10 1/2 oz over 1/2 oz 0.008" (0.203mm) 11 ST10F-0.008 1/1 ST10 1 oz over 1 oz 0.008" (0.203mm) 12 ST325P-180 ST325 NONE 0.007" (0.180mm) 13 ST325P-180 H/H ST325 1/2 oz over 1/2 oz 0.007" (0.180mm) 14 ST325P-0.008 ST325 NONE 0.008" (0.203mm) 15 ST325P-0.008 H/H ST325 1/2 oz over 1/2 oz 0.008" (0.203mm) 16 ST325P-0.010 ST325 NONE 0.010" (0.255mm) 17 ST325P-0.010 H/H ST325 1/2 oz over 1/2 oz 0.010" (0.255mm) 18 ST325P-0.010 1/1 ST325 1 oz over 1 oz 0.009" (0.240mm) 37 copper per square foot plating layer on both sides, and ?1/1? corresponds to a one ounce per square foot copper plating layer on both sides. The CTE specimens consisted of 1? x 1? square samples cut along the first and second principal material directions. Two strain gages were mounted on each of the test specimens, with one strain gage aligned along each principal material direction. All tests were conducted in a Delta Design 9028 temperature controlled chamber (Figure 14), which was controlled by a LabView based data acquisition program. The temperature controlled chamber heats by hot air and cools with liquid nitrogen. The tests were run from T = 30 oC to T = 140 oC, with measurements made on 10 oC increments. Experiment Setup and Calibration All strain gage readings were processed in real time by a Wheatstone Bridge based Vishay Measurement Group 2120 strain indicator with a model MG2000 Analog to Digital Converter (Figure 15). This system allowed the Wheatstone Bridge voltage signals from the strain gages to be conditioned and transformed into digital readings. A LabView program recorded both the temperature and strain gage readings. The testing equipment has the ability to track and record 8 channels simultaneously. The excitation voltages on all channels were calibrated with a HP 3478A digital multimeter with errors not larger than ?0.001 VDC. Initial testing for system calibration purposes was performed with copper and aluminum samples with known CTEs. The well established procedure for strain gage based CTE measurement requires strain gages to be mounted on both the specimen material of interest and a reference material of known CTE. Both materials are then subjected to controlled 38 temperature change. Due to thermally induced strain gage errors, the strain gage readings must be processed to extract the correct CTE values using the following expression [22]: 0)()()()( TT RSRSAPPRAPPS ?????? ????? (50) where R? is the CTE of a known reference material, S? is the unknown CTE of the test specimen, APPS)(? is the thermal output (apparent strain) of the specimen strain gage, APPR)(? is the thermal output (apparent strain) of the reference material strain gage, T is the testing temperature, and 0T is the initial (zero strain) temperature (normally room temperature). Equation (56) can be rewritten in compact form as RAPPRAPPSS T ???? ???? )()( (51) where ?TTT ??? is the temperature change from the initial reference temperature. The reference material is a critical element of the CTE experiments. In this work, the reference material was titanium silicate from Corning Glass Company. This material has an extremely low thermal expansion coefficient that has been well characterized over a wide temperature range. Vishay WK series strain gages were used in all of the CTE tests, and Vishay M-Bond 610 adhesive was used to mount them to the specimen and reference materials. The tests were performed with an initial temperature of 30 oC with temperature steps of 10 oC up to a maximum temperature of 140 oC. At each measurement temperature, the materials were allowed to stabilize for 20 minutes before readings were taken. Sample data for copper and aluminum test specimens are shown in Figure 16. Graphs of the copper and aluminum CTE data are shown in Figures 17 and 18. From the testing results, the average CTEs in Figure 16 39 were 24.15 ppm/oC for the aluminum specimen and 17.99 ppm/oC for the copper specimen. These values agree well with standard handbook values for these materials. Experimental Procedure and Results The coefficients of thermal expansion of the 18 different STABLCOR ply material types were evaluated using the procedure outlined in the previous pages. Specimens were placed at the center of the temperature controlled chamber with free unconstrainted edges as shown in Figure 19. Two samples were prepared for each STABLCOR material and each sample was tested 3 times. Example thermal strain vs. temperature raw data for the various materials are shown in Figures 20-37. All of the graphs show an extremely linear dependence of the thermal strain with temperature. The CTE values were calculated using eq. (57) and the procedure outlined in Figure 16. Tabulated results for the CTE values of the STABLCOR ply materials are listed in Table 4. 40 Figure 14. Environmental Chamber (Delta Design 9028) Figure 15. Vishay Strain Gage Signal Processing Equipment 41 Figure 16. Sample of CTE Measurement Calculations : T h e r m a l o u t p u t ( a p p a r e n t s t r a in ) o f g a g e o n s p e c im e n m a t e r ia l : T h e r m a l o u t p u t ( a p p a r e n t s t r a in ) o f g a g e o n r e fe r e n c e m a t e r ia l : T e m p e r a t u r e c h a n g e fr o m in it ia l r e fe r e n c e t e m p e r a t u r e o f 3 0 o C : C T E o f s p e c im e n m a t e r ia l : C T E o f r e fe r e n c e m a t e r ia l : C o r r e c t e d T h e r m a l I n d u c e d St r a in S a m pl e C T E M e a s ur e m e nt C a l cul a t i ons T e m p De lt a T C u Sp e c im e n St r a in G a g e R e a d in g A l Sp e c im e n St r a in G a g e Re a d in g R e fe r e n c e M a t e r ia l St r a in G a g e R e a d in g C u C o r r e c t e d S t r a in G a g e Re a d i n g A l C o r r e c t e d St r a in G a g e R e a d in g R e fe r e n c e M a t e r ia l C T E 30 0 - 1 2 . 2 3 - 2 6 . 0 4 6 - 7 . 0 1 2 - 5 . 2 1 8 - 1 9 . 0 3 4 - 0 . 9 40 10 - 8 0 . 4 1 9 - 3 0 . 4 0 3 - 2 6 4 . 6 2 8 1 8 4 . 2 0 9 2 3 4 . 2 2 5 - 1 50 20 - 1 4 6 . 2 4 3 - 4 4 . 7 2 6 - 5 0 2 . 2 4 3 356 4 5 7 . 5 1 7 - 0 . 9 60 30 - 2 2 6 . 7 3 2 - 5 7 . 7 4 7 - 7 5 3 . 4 0 1 5 2 6 . 6 6 9 6 9 5 . 6 5 4 - 0 . 8 70 40 - 3 0 4 . 4 9 7 - 7 0 . 1 9 3 - 1 0 0 7 . 1 9 3 7 0 2 . 6 9 6 937 - 0 . 6 80 50 - 3 8 0 . 1 9 5 - 8 6 . 4 7 4 - 1 2 4 6 . 6 4 4 8 6 6 . 4 4 9 1 1 6 0 . 1 7 - 0 . 4 90 60 - 4 5 6 . 9 6 2 - 9 0 . 0 3 - 1 4 9 9 . 4 7 5 1 0 4 2 . 5 1 3 1 4 0 9 . 4 4 5 0 . 2 100 70 - 5 3 7 . 1 0 5 - 9 6 . 9 1 1 - 1 7 5 3 . 8 6 8 1 2 1 6 . 7 6 3 1 6 5 6 . 9 5 7 0 . 7 110 80 - 5 9 0 . 6 1 8 - 1 0 0 . 6 9 1 - 2 0 0 4 . 5 1 8 1 4 1 3 . 9 1 9 0 3 . 8 2 7 1 . 3 120 90 - 6 6 7 . 0 5 2 - 9 3 . 2 3 4 - 2 2 5 0 . 7 0 3 1 5 8 3 . 6 5 1 2 1 5 7 . 4 6 9 1 . 8 130 100 - 7 2 6 . 1 4 3 - 7 9 . 6 3 8 - 2 4 9 1 . 6 5 2 1 7 6 5 . 5 0 9 2 4 1 2 . 0 1 4 2 . 4 140 110 - 7 7 5 . 2 8 7 - 6 2 . 2 2 6 - 2 7 2 9 . 7 4 6 1 9 5 4 . 4 5 9 2 6 6 7 . 5 2 3 . 1 Su m 660 - 4 9 0 3 . 4 8 3 - 8 3 8 . 3 1 9 - 1 6 5 1 1 . 0 8 3 Cu Al A v g C T E C T E 1 7 .9 9 2 4 .1 5 0 .4 0 8 R RGOTSGOT S T ? ?? ? ? ? ? ? )( )/(/)/(/ S? )/(/ SGOT? )/(/ RGOT? T? C O R R? R?S? )/(/ SGOT? C O R R? R? S? )/(/ SGOT? )/(/ RGOT? T? C O R R? R? T 42 Te m pe rat ure 0 C 0 20 40 60 80 100 120 140 160 M icro S t rain 0 500 1000 1500 2000 2500 3000 Al u mi n u m CT E ?? = 2 3 .7 4 (p p m/ o C) Figure 17. Testing Results for Aluminum CTE Co p p e r C TE Te m pe rat ure 0 C 0 20 40 60 80 100 120 140 160 M icro S t rain 0 500 1000 1500 2000 2500 3000 ?? = 1 7 .5 9 (p p m/ o C) Figure 18. Testing Results for Copper CTE 43 Figure 19. CTE Testing Specimens in Temperature Controlled Chamber Figure 20. ST10F-0.005 Thermal Strain versus Temperature S T1 0F -0. 005 T em pe rat ure ?C 0 20 40 60 80 100 120 140 160 M icro S t rain 0 500 1000 1500 2000 1 -d ir e ct io n 2 -d ir e ct io n ? 1 = 10.91 (ppm/ ?C) ? ? = 10.87 (ppm/ ?C) 44 Figure 21. ST10P-0.006 Thermal Strain versus Temperature Figure 22. ST10F-0.006 Thermal Strain versus Temperature ST10P-0 .006 Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =10.6 (ppm /?C) ? ? =11.9 (ppm /?C) ST10F- 0.00 6 Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =7.13 (ppm /?C) ? ? =8.37 (ppm /?C) 45 Figure 23. ST10F-0.005 H/H Thermal Strain versus Temperature Figure 24. ST10P-0.006 H/H Thermal Strain versus Temperature ST10F- 0.00 5 H/H Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? ? =12.85 (ppm /?C) ? ? =12.10 (ppm /?C) ST10P-0 .006 H/H Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =9.60 (ppm /?C) ? 2 =8.91 (ppm /?C) 46 Figure 25. ST10F-0.006 H/H Thermal Strain versus Temperature Figure 26. ST10F-0.006 1/1 Thermal Strain versus Temperature ST10F- 0.00 6 H/H Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =10.92 (ppm /?C) ? 2 =11.55 (ppm /?C) ST10F- 0.00 6 1/1 Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =12.02 (ppm /?C) ? 2 =13.95 (ppm /?C) 47 Figure 27. ST10P-0.008 H/H Thermal Strain versus Temperature Figure 28. ST10F-0.008 H/H Thermal Strain versus Temperature ST10P-0 .008 H/H Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =6.53 (ppm /?C) ? 2 =7.40 (ppm /?C) ST10F- 0.00 8 H/H Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =9.62 (ppm /?C) ? 2 =9.68 (ppm /?C) 48 Figure 29. ST10P-0.008 Thermal Strain versus Temperature Figure 30. ST10F-0.008 Thermal Strain versus Temperature ST10P-0 .008 Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =9.57 (ppm /?C) ? 2 =10.91 (ppm /?C) ST10F- 0.00 8 Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =8.64 (ppm /?C) ? ? =9.55 (ppm /?C) 49 Figure 31. ST10F-0.008 1/1 Thermal Strain versus Temperature Figure 32. ST325P-0.010 Thermal Strain versus Temperature ST10F- 0.00 8 1/1 Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =9.80 (ppm /?C) ? 2 =11.44 (ppm /?C) S T3 25P -0. 010 T em pe rat ure ?C 0 20 40 60 80 100 120 140 160 M icro S t rain 0 500 1000 1500 2000 1 -d ir e ct io n 2 -d ir e ct io n ? 1 = 4.17 (ppm/?C ) ? ? = 6.01 (ppm/?C ) 50 Figure 33. ST325P-0.010 1/1 Thermal Strain versus Temperature Figure 34. ST325P-180 H/H Thermal Strain versus Temperature ST325 P-0.01 0 1/1 Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =3.37 (ppm /?C) ? y =6.66 (ppm /?C)ST325 P-180 H/H Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =5.16 (ppm /?C) ? 2 =5.94 (ppm /?C) 51 Figure 35. ST325P-0.010 H/H Thermal Strain versus Temperature Figure 36. ST325P-180 Thermal Strain versus Temperature ST325 P-0.01 0 H/H Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =3.08 (ppm /?C) ? 2 =4.13 (ppm /?C) ST325 P-180 Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =9.80 (ppm /?C) ? 2 =11.44 (ppm /?C) 52 Figure 37. ST325P-0.008 H/H Thermal Strain versus Temperature ST325 P-0.00 8 H/H Temper ature ?C 0 20 40 60 80 100 120 140 160 Micr o S train 0 500 1000 1500 2000 1- directi on 2- directi on ? 1 =3.13 (ppm /?C) ?? =4.74 (ppm /?C) 53 Table 4. Coefficients of Thermal Expansion for STABLCOR Ply Materials Sample No. Material Description 1? CTE (ppm/ oC) Avg (Standard dev) 2? CTE (ppm/ oC) Avg (Standard dev) 1 ST10F-0.005 10.79 (0.27) 10.87 (0.28) 2 ST10F-0.006 7.22 (0.37) 8.15 (0.34) 3 ST10P-0.006 10.22 (0.33) 11.97 (0.32) 4 ST10F-0.006 H/H 11.05 (0.26) 11.55 (0.23) 5 ST10F-0.006 1/1 12.08 (0.26) 14.02 (0.27) 6 ST10P-0.006 H/H 9.75 (0.30) 9.05 (0.33) 7 ST10F-0.008 8.79 (0.33) 9.46 (0.34) 8 ST10P-0.008 9.58 (0.42) 11.03 (0.44) 9 ST10F-0.008 H/H 9.71 (0.28) 9.8 (0.28) 10 ST10P-0.008 H/H 6.57 (0.30) 7.33 (0.31) 11 ST10F-0.008 1/1 9.74 (0.27) 11.33 (0.38) 12 ST325P-180 6.7 (0.30) 7.07 (0.34) 13 ST325P-180 H/H 5.04 (0.29) 5.85 (0.33) 14 ST325P-0.008 4.58 (0.31) 6.17 (0.33) 15 ST325P-0.008 H/H 3.14 (0.29) 4.77 (0.30) 16 ST325P-0.010 4.1 (0.42) 6.04 (0.44) 17 ST325P-0.010 H/H 3.03 (0.26) 4.08 (0.28) 18 ST325P-0.010 1/1 3.42 (0.24) 6.71 (0.26) 54 V. MEASUREMENT OF ELASTIC MODULI Introduction Lamination theory requires input material properties for each fiber-reinforced layer including the elastic moduli 1E and 2E along the first and second principal material directions, the Poisson?s ratios 12? and 21? , and the shear modulus 12G . In this chapter, the elastic moduli of 18 different STABLCOR carbon fiber ply materials have been measured by mechanical testing. Description of Specimens and Testing Conditions The uniaxial test specimens for mechanical testing were cut out from single ply samples. The nominal dimensions were 5 mm in width and 60 mm in length. Three specimens were prepared for each material in each of the principal material directions, and three tests were performed on each specimen and the results were averaged. All tests were performed at 25oC with crosshead speed of 0.0006 mm/s. With the gage length of 60 mm, this resulted in a strain rate of 15 sec101 ?????? . Photographs of uniaxial specimens in the mechanical testing machine are shown in Figures 38 and 39. This setup recorded both axial force and crosshead displacement, from which the axial stress and axial strain could be calculated. Ten data points were recorded per second, and the maximum load limit was set to 50 N. To further refine the specimen strain measurements, axial strain gages were also applied to the specimens (type CEA-13-125UN-350 from Vishay). However, it was found that the gages only yielded accurate measurements for the fine weave carbon fiber specimens. For 55 larger weave materials, the gages were too small and picked up local mismatches in the deformations of the carbon fibers and epoxy matrix. Results and Conclusion Typical stress-strain curves recorded for the 18 carbon fiber ply materials are shown in Figures 40 ? 79. In these results, strains were calculated using the measured crosshead displacement, and Least-Squares linear regression fits were used to fit the data points. The regression fits are shown as red lines on the graphs. The slopes of the regression lines are the elastic moduli of the specimens. The average elastic moduli for all of the ply materials are tabulated in Table 5. Table 6 compares the results of the 1-direction elastic modulus measurements obtained from strains evaluated with the crosshead displacements with modulus measurements obtained from strains found with strain gages. An example specimen with strain gages mounted is show in Figure 80. In summary, good correlation was only obtained for the two methods when the carbon fiber ply material had a very small/fine weave (samples 1 ? 5). For the other specimen materials, the strain gages were too small compared to the weave and recorded local effects (e.g. only measured strains in the epoxy or in the fibers, instead of the average strains in the fiber-reinforced composite). 56 Figure 38. Uniaxial Specimen in Mechanical Testing Machine Figure 39. Close-up Photograph of Uniaxial Specimen 57 Figure 40. ST10F-0.006 Stress versus Strain (1-direction) Figure 41. ST10F-0.006 H/H Stress versus Strain (1-direction) ST 10 F- 0. 00 6 S t rain , ? 0 .0 0 0 0 0 .0 0 0 5 0 .0 0 1 0 0 .0 0 1 5 0 .0 0 2 0 0 .0 0 2 5 0 .0 0 3 0 S t ress, ? (MP a) 0 20 40 60 80 100 120 Str e ss v s Str a in Str e ss-Str a in cu rv e fit E= 38 . 6 G Pa ST 1 0 F- 0 .0 0 6 H /H St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 6 1 .4 GPa 58 Figure 42. ST10F-0.006 1/1 Stress versus Strain (1-direction) Figure 43. ST10P-0.006 Stress versus Strain (1-direction) ST 1 0 F- 0 .0 0 6 1 /1 St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 5 6 .4 GPa ST 1 0 P- 0 .0 0 6 St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 4 2 .3 GPa 59 Figure 44. ST10P-0.006 H/H Stress versus Strain (1-direction) Figure 45. ST10P-0.008 Stress versus Strain (1-direction) ST 1 0 P- 0 .0 0 6 H /H St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 4 6 .8 GPa ST 1 0 P- 0 .0 0 8 St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 2 5 .1 GPa 60 Figure 46. ST10F-0.008 H/H Stress versus Strain (1-direction) Figure 47. ST10F-0.008 1/1 Stress versus Strain (1-direction) ST 1 0 F- 0 .0 0 8 H /H St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 4 8 .0 GPa ST 1 0 F- 0 .0 0 8 1 /1 St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 5 0 .9 GPa 61 Figure 48. ST10F-0.008 Stress versus Strain (1-direction) Figure 49. ST10P-0.008 H/H Stress versus Strain (1-direction) ST 1 0 F- 0 .0 0 8 St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 C o l 1 v s C o l 2 x co l u m n v s y co l u m n E= 2 7 .8 GPa ST 1 0 P-0.0 0 8 H/H S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 49 .2 G Pa 62 Figure 50. ST325P-0.008 Stress versus Strain (1-direction) Figure 51. ST325P-0.010 240 Stress versus Strain (1-direction) ST 3 2 5 P- 0 .0 0 8 St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 7 3 .0 GPa ST 3 2 5 -0 .0 1 0 2 4 0 St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 7 1 .6 GPa 63 Figure 52. ST325P-0.010 H/H 255 Stress versus Strain (1-direction) Figure 53. ST325P-0.008 H/H Stress versus Strain (1-direction) ST 3 2 5 -0 .0 1 0 H /H 2 5 5 St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 9 5 .5 GPa ST 3 2 5 P- 0 .0 0 8 H /H St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 8 3 .3 GPa 64 Figure 54. ST325P-0.010 255 Stress versus Strain (1-direction) Figure 55. ST325P-0.010 1/1 240 Stress versus Strain (1-direction) ST 3 2 5 -0 .0 1 0 2 5 5 St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 7 7 .0 GPa ST 3 2 5 P- 0 .0 1 0 1 /1 2 4 0 St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 9 7 .0 GPa 65 Figure 56. ST325P-180 Stress versus Strain (1-direction) Figure 57. ST10F-0.005 Stress versus Strain (1-direction) ST 3 2 5 P- 1 8 0 St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 4 8 .4 GPa ST 1 0 F- 0 .0 0 5 St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 1 9 .4 GPa 66 Figure 58. ST325P-180 H/H Stress versus Strain (1-direction) Figure 59. ST10F-0.005 H/H Stress versus Strain (1-direction) ST 3 2 5 P- 1 8 0 H /H St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 5 7 .1 GPa ST 1 0 F- 0 .0 0 5 H /H St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 4 0 .5 GPa 67 Figure 60. ST10F-0.005 H/H Stress versus Strain (2-direction) Figure 61. ST10P-0.006 Stress versus Strain (2-direction) ST 1 0 F-0.0 0 5 Y S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 28 .3 G Pa ST 1 0 P-0.0 0 6 Y S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 20 .5 G Pa 68 Figure 62. ST10F-0.008 Stress versus Strain (2-direction) Figure 63. ST10F-0.006 H/H Stress versus Strain (2-direction) ST 1 0 F-0.0 0 8 Y S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 43 .3 G Pa ST 10F-0.006 H/H Y St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 63.9 G Pa 69 Figure 64. ST10P-0.006 H/H Stress versus Strain (2-direction) Figure 65. ST10F-0.005 H/H Stress versus Strain (2-direction) ST 10P -0.006 H/H Y St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 41.1 G Pa ST 1 0 F- 0 .0 0 5 H /H Y St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i tE= 3 9 .9 GPa 70 Figure 66. ST10P-0.008 Stress versus Strain (2-direction) Figure 67. ST10F-0.008 H/H Stress versus Strain (2-direction) ST 1 0 P-0.0 0 8 Y S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 44 .5 G Pa ST 10 F- 0. 00 8 H/ H Y S t rain , ? 0 .0 0 0 0 0 .0 0 0 5 0 .0 0 1 0 0 .0 0 1 5 0 .0 0 2 0 0 .0 0 2 5 0 .0 0 3 0 S t ress, ? (MP a) 0 20 40 60 80 100 120 Str e ss v s Str a in Str e ss-Str a in cu rv e fit E= 66 . 6 G Pa 71 Figure 68. ST10F-0.006 Stress versus Strain (2-direction) Figure 69. ST10P-0.008 H/H Stress versus Strain (2-direction) ST 1 0 F-0.0 0 6 Y S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 36 .7 G Pa ST 10P -0.008 H/H Y St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 39.6 G Pa 72 Figure 70. ST10F-0.008 1/1 Stress versus Strain (2-direction) Figure 71. ST10F-0.006 1/1 Stress versus Strain (2-direction) ST 1 0 F-0.0 0 8 1/1 Y S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 57 G Pa ST 10 F- 0. 00 6 1/ 1 Y S t rain , ? 0 .0 0 0 0 0 .0 0 0 5 0 .0 0 1 0 0 .0 0 1 5 0 .0 0 2 0 0 .0 0 2 5 0 .0 0 3 0 S t ress, ? (MP a) 0 20 40 60 80 100 120 Str e ss v s Str a in Str e ss-Str a in cu rv e fit E= 57 . 4 G Pa 73 Figure 72. ST325P-180 Stress versus Strain (2-direction) Figure 73. ST325P-0.010 240 Stress versus Strain (2-direction) ST 3 2 5 P-18 0 Y S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 62 .9 G Pa ST 3 2 5 P-0.0 1 0 24 0 Y S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 73 .7 G Pa 74 Figure 74. ST325P-0.008 Stress versus Strain (2-direction) Figure 75. ST325P-180 H/H Stress versus Strain (2-direction) ST 325P -0.008 Y St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 75.4 G Pa ST 3 2 5 P-18 0 H/H Y S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 63 .1 G Pa 75 Figure 76. ST325P-0.008 H/H Stress versus Strain (2-direction) Figure 77. ST325P-0.010 255 Stress versus Strain (2-direction) ST 325P -0.008 H/H Y St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f itE= 120 GPa ST 3 2 5 P- 0 .0 1 0 2 5 5 Y St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 5 7 .7 GPa 76 Figure 78. ST325P-0.010 255 H/H Stress versus Strain (2-direction) Figure 79. ST325P-0.010 240 1/1 Stress versus Strain (2-direction) ST 325P -0.010 255 H/H Y St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 109.2 GPa ST 3 2 5 P-0.0 1 0 24 0 1/1 Y S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 10 5 .5 G Pa 77 Table 5. Elastic Modulus Results Obtain by Crosshead Displacement Sample No. Material Description E1 (GPa) Avg (Standard Dev) E2 (GPa) Avg (Standard Dev) 1 ST10F-0.006 38.5 (0.7) 36.7 (0.5) 2 ST10P-0.006 42.0 (0.6) 20.4 (0.7) 3 ST10F-0.006 H/H 61.3 (0.7) 63.7 (0.7) 4 ST10F-0.006 1/1 56.3 (0.7) 57.2 (0.8) 5 ST10P-0.006 H/H 46.6 (0.7) 41.3 (0.9) 6 ST10F-0.008 27.7 (0.8) 43.4 (0.7) 7 ST10P-0.008 25.2 (0.5) 44.5 (0.6) 8 ST10F-0.008 H/H 48.1 (0.5) 66.4 (0.8) 9 ST10P-0.008 H/H 49.4 (0.8) 39.6 (0.7) 10 ST10F-0.008 1/1 50.8 (0.8) 57.2 (0.7) 11 ST325P-180 48.2 (0.7) 63 (0.7) 12 ST325P-180 H/H 57.1 (0.8) 63 (0.7) 13 ST325P-0.008 73.2 (0.6) 75.6 (0.6) 14 ST325P-0.008 H/H 83.5 (0.8) 120.4 (1.0) 15 ST325P-0.010 240 71.4 (0.6) 73.4 (0.6) 16 ST325P-0.010 255 77.2 (0.7) 57.6 (0.7) 17 ST325P-0.010 H/H 95.7 (1.0) 109.4 (1.1) 18 ST325P-0.010 1/1 97.2 (0.9) 105.5 (0.9) 78 Table 6. Elastic Modulus Measurement Comparisons for Stress Evaluated Using Crosshead Displacements and Strain Gages Sample No. Material Description E1 (GPa) From Crosshead Displacements E1 (GPa) From Strain Gage Measurements 1 ST10F-0.006 38.5 35.5 2 ST10P-0.006 42.0 39.6 3 ST10F-0.006 H/H 61.3 60.8 4 ST10F-0.006 1/1 56.3 63.6 5 ST10P-0.006 H/H 46.6 45.5 6 ST10F-0.008 27.7 46.6 7 ST10P-0.008 25.2 34.3 8 ST10F-0.008 H/H 48.1 63.1 9 ST10P-0.008 H/H 49.4 58.8 10 ST10F-0.008 1/1 50.8 60.8 11 ST325P-180 48.2 63.5 12 ST325P-180 H/H 57.1 72.6 13 ST325P-0.008 73.2 71.6 14 ST325P-0.008 H/H 83.5 107.2 15 ST325P-0.010 240 71.4 33.1 16 ST325P-0.010 255 77.2 49.3 17 ST325P-0.010 H/H 95.7 161.4 18 ST325P-0.010 1/1 97.2 76.5 79 Figure 80. Example of Specimen with Mounted Strain Gages 80 VI. INDIRECT METHOD FOR EVALUATION OF SHEAR MODULUS AND POISSON?S RATIO Introduction In order to apply lamination theory, both the shear modulus 12G and Poisson?s Ratio 12? of the constituent layers must be known. However, these two material properties are very difficult to obtain directly. As discussed in the previous chapter, it was not possible to obtain accurate strain gage measurements for many of the carbon fiber ply materials because they employed fairly coarse fiber weaving. Thus, Poisson?s ratios could not be found using the conventional approach where axial and transverse strains are monitored with strain gages during uniaxial testing. Thus, in this work an indirect method was employed to calculate the shear modulus 12G and Poisson?s ratio 12? from off-axis elastic modulus measurements. The theoretical expression for the variation of the in-plane elastic modulus with angle in a fiber-reinforced orthotropic layer is [6] ? ? 2 422 1 12 121 4 s i nc o ss i n21c o s1 EEGEE x ?????? ??????? ??? (52) where ? is the angle from the fiber direction (see Figure 81), ???xE is the off-axis elastic modulus, 1E is the elastic modulus in the first direction of material symmetry (1-direction), 2E is the elastic modulus in the second direction of material symmetry (2-direction), 12? is the Poisson?s ratio, and 12G is the shear modulus. Experimentally, the values of ???xE can be measured for several different angles including ??0? ( 1E ) and ??90? ( 2E ). If five or more 81 angles are tested, an indirect procedure can then be used to calculate 12G and 12? via nonlinear regression fitting of eq. (58) to the experimental xE vs. ? data. Description of Specimens and Testing Conditions Off-axis uniaxial samples were cut with dimensions of 3 mm in width and 60 mm in length. The material sheets were first marked with off-axis guide lines and then the specimens were cut by a hand shear as shown in Figures 82-83. Seven different specimen orientations were prepared for each ply material including ? = 0o, 15 o, 30 o, 45 o, 60 o, 75 o, and 90 o. The elastic moduli of the off-axis testing coupons were then obtained using mechanical testing as discussed in the previous chapter. The crosshead speed was again taken to be 0.0006 mm/s, resulting in a strain rate of 15 sec101 ?????? . All tests were conducted at 25 oC. Results and Conclusion Example off-axis stress-strain curves and elastic modulus data for the ST10F-0.006 H/H ply material are shown in Figures 84 ? 85 (? = 0o and 90o) and Figures 86 ? 90 (? = 15o, 30o, 45o, 60o, and 75o). The seven modulus values were regression fitted using eq. (58) to calculate 12G = 13.7 GPa and 12? = 0.27 as shown in Figure 91. The Matlab toolbox was used to perform the regression analysis. The fit of the model to the data is excellent, and an R-Squared value of 0.994 was obtained. Analogous measurements were made for the ST10P-0.006 material as shown in Figures 92 ? 98, and the regression fit is illustrated in Figure 99. In this case, the calculated values were 12G = 3.59 GPa and 12? = 0.21, with an R-Squared value of 0.993. Similar procedures were 82 followed for all 18 different STABLCOR carbon fiber ply materials. The measured 12G and 12? data are tabulated in Table 7, and the complete data sets are given in Appendix A. 83 Figure 81. Off-Axis Tensile Specimens 1x z,x 3 2x F i be r D i r e ct i on ? x y C ut O ut O f f - Ax i s T e ns i l e S pe ci m e ns f or V a r i ous Ang l e s 84 Figure 82. Sample Marked with Off-Axis Guide Lines Figure 83. Hand Shear Used in Preparing Off-Axis Test Specimens 85 Figure 84. ST10F-0.006 H/H 0-Degree Elastic Modulus Figure 85. ST10F-0.006 H/H 90-Degree Elastic Modulus ST 1 0 F- 0 .0 0 6 H /H 0 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 6 0 .6 GPa E 1 o0?? ST 10F-0.006 H/H 90 DE GRE E St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 61.8 G Pa E 2 o90?? 86 Figure 86. ST10F-0.006 H/H 15-Degree Elastic Modulus Figure 87. ST10F-0.006 H/H 30-Degree Elastic Modulus ST 1 0 F- 0 .0 0 6 H /H 1 5 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 5 3 .6 GPa o15?? ST 1 0 F- 0 .0 0 6 H /H 3 0 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 4 5 .1 GPa o30?? 87 Figure 88. ST10F-0.006 H/H 45-Degree Elastic Modulus Figure 89. ST10F-0.006 H/H 60-Degree Elastic Modulus ST 1 0 F- 0 .0 0 6 H /H 4 5 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 4 2 .3 GPa o45?? ST 1 0 F-0.0 0 6 H/H 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 44 .8 G Pa o60?? 88 Figure 90. ST10F-0.006 H/H 75-Degree Elastic Modulus Figure 91. ST10F-0.006 H/H Elastic Modulus versus Off-Axis Angle ST 1 0 F- 0 .0 0 6 H /H 7 5 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 5 4 .2 GPa o75?? 0 6 0 . 6 15 5 3 . 6 30 4 5 . 1 45 4 2 . 3 60 4 4 . 8 75 5 4 . 2 90 6 1 . 8 Of f-Ax is Ela st ic Mo d u lu s Fit ST 1 0 F-0 .00 6 H/ H A ng le( R ad) 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 1. 2 1. 4 1. 6 1. 8 E lastic M odulu s 30 40 50 60 70 E l as t i c v s A ngl e( R ad) C ur v e Fi t ? ( GP a ) E 7.13 27.0 12 12 ? ? G ? G o o d n es s o f f i t : R - s q u a r ed : 0 .9 9 4 A d j u s t ed R - s q u a r ed : 0 .9 9 2 89 Figure 92. ST10P-0.006 0-Degree Elastic Modulus Figure 93. ST10P-0.006 90-Degree Elastic Modulus ST 1 0 P- 0 .0 0 6 0 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 3 6 .6 GPa E 1 o0?? ST 1 0 P- 0 .0 0 6 9 0 D EG R EE X D ata 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 Y D ata 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 3 7 .2 GP a E 2 o90?? 90 Figure 94. ST10P-0.006 15-Degree Elastic Modulus Figure 95. ST10P-0.006 30-Degree Elastic Modulus ST 1 0 P- 0 .0 0 6 1 5 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 2 4 .0 GPa o15?? ST 1 0 P- 0 .0 0 6 3 0 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 1 6 .0 GP a o30?? 91 Figure 96. ST10P-0.006 45-Degree Elastic Modulus Figure 97. ST10P-0.006 60-Degree Elastic Modulus ST 1 0 P- 0 .0 0 6 4 5 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 1 2 .3 GP a o45?? ST 1 0 P- 0 .0 0 6 6 0 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 1 3 .8 GP a o60?? 92 Figure 98. ST10P-0.006 75-Degree Elastic Modulus Figure 99. ST10P-0.006 Elastic Modulus versus Off-Axis Angle ST 1 0 P- 0 .0 0 6 7 5 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 2 0 .6 GP a o75?? 0 3 6 . 6 15 2 4 . 0 30 1 6 . 0 45 1 2 . 3 60 1 3 . 8 75 2 0 . 6 90 3 7 . 2 ? ( GP a ) E Of f-A x is Elastic Mo d u lu s F it ST 1 0 P-0.0 0 6 A ng le( R ad) 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 1. 2 1. 4 1. 6 1. 8 E lastic M odulus 0 10 20 30 40 E l asti c v s A ng l e( Ra d) Cu r v e Fi t 59.3 21.0 12 12 ? ? G ? G o o d n es s o f f i t : R - s q u a r ed : 0 .9 9 3 A d j u s t ed R - s q u a r ed : 0 .9 9 2 93 Table 7. Shear Moduli and Poisson?s Ratios for STABLCOR Ply Materials Sample No. Material Description G12 (GPa) Poisson?s ratio, 12? 1 ST10F-0.005 9.08 0.30 2 ST10F-0.006 4.31 0.33 3 ST10P-0.006 3.59 0.21 4 ST10F-0.006 H/H 13.70 0.27 5 ST10F-0.006 1/1 15.70 0.30 6 ST10P-0.006 H/H 11.80 0.29 7 ST10F-0.008 5.17 0.22 8 ST10P-0.008 3.93 0.30 9 ST10F-0.008 H/H 9.32 0.30 10 ST10P-0.008 H/H 6.76 0.33 11 ST10F-0.008 1/1 10.29 0.29 12 ST325P-180 2.07 0.30 13 ST325P-180 H/H 3.98 0.30 14 ST325P-0.008 2.51 0.30 15 ST325P-0.008 H/H 7.22 0.29 16 ST325P-0.010 255 2.31 0.29 17 ST325P-0.010 H/H 6.22 0.30 18 ST325P-0.010 1/1 6.48 0.30 94 VII. APPLICATIONS TO STABLCOR PRINTED CIRCUIT BOARDS Description of Simulation Conditions The lamination theory expressions in eq. (49) have been coded into a Matlab program (see Appendix B) with material properties as the input data. The developed Matlab program allows the user to input layer number (k), layer thickness (zk), individual layer CTEs in the 1- direction ( 1? ) and 2-direction ( 2? ), elastic moduli in the 1 and 2 directions of each layer ( 1E , 2E ), shear modulus ( 12G ), Poisson?s ratios ( 12? , 21? ), and temperature difference ( T? ). Moreover, the program is able to calculate thermally induced normal stress, thermally induced normal strain, and laminate CTE along both axes. The results from the lamination theory Matlab program were compared to the finite element analysis solutions generated by ANSYS. The ANSYS program is able to create 3- dimensional models with various element attributes. In this work, 20 node solid elements (solid 186) were used, which provided providing non-linear deformation and large displacement simulations. All layers were considered perfectly bonded as a single bulk material. In this study, analysis was performed using typical FR-4 material properties [29-30]. Furthermore, the material properties of ST325P-0.010 carbon-epoxy plies have been used for the STABLCOR carbon fiber layers in the PCBs. Table 8 shows the material properties which were used in both the lamination theory Matlab calculations as well as in the ANSYS finite element analysis simulations. 95 Two laminate configurations were analyzed as described in Tables 9 and 10. The first model consisted of a single STABLCOR layer sandwiched between two FR-4 layers. Figure 100 shows the dimensions of the single layer STABLCOR model, and Figure 101 shows the corresponding finite element mesh. The second model consisted of three STABLCOR layers and four FR-4 layers. The dimensions are shown in Figure 102 and the finite element mesh in Figure 103. To simulate free expansion in the finite element models, temperature changes of 25 oC and 50 oC were applied as loads. The initial temperature was taken to be 25 oC (room temperature). Results and Conclusion The results from both lamination theory and ANSYS finite element method have shown reasonable agreement. Eq. (49) was used for lamination theory CTE calculation, which was coded in Matlab program shown in Appendix B. Comparism of the lamination theory calculations and FEA predictions for the laminate CTE values are shown in Tables 11 and 12. Correlation is very good with percent differences about 10%. Both lamination theory and the FEA simulations showed that the CTEs in x-direction were lower than in the y-direction. The differences in results for the 2 analysis methods are mainly due to the thermally induced expansions in the z-direction. The 3-dimension model used by the ANSYS simulation includes z-axis effects, while they are neglected in lamination theory. 96 Table 8. Lamination Theory and Finite Element Method Model Material Constants Layer Properties ST325P-0.010 FR-4 [28] 1E 77 GPa 12 GPa 2E 57.7 GPa 12 GPa 3E 7.7 GPa 1.2 GPa 12G 2.31 GPa 3.1 GPa 12? 0.29 0.27 21? 0.29 0.27 CTE ( 1? ) 4.1 (ppm/ oC) 12 (ppm/ oC) CTE ( 2? ) 6.04 (ppm/ oC) 12 (ppm/ oC) Thickness 0.255 mm 1.0 mm Table 9. Laminate Layup ? Single Layer STABLCOR Model Layer Material Orientation Angle of 1- direction with x-direction 1 FR-4 0o 2 ST325P-0.010 255 0o 3 FR-4 0o Table 10. Laminate Layup ? Three Layer STABLCOR Model Layer Material Orientation Angle of 1- direction with x-direction 1 FR-4 0o 2 ST325P-0.010 255 0o 3 FR-4 0o 4 ST325P-0.010 255 0o 5 FR-4 0o 6 ST325P-0.010 255 0o 7 FR-4 0o 97 Figure 100. Single STABLCOR Layer PCB Model Figure 101. Single STABLCOR Layer Model Mesh with Solid Elements 98 Figure 102. Three STABLCOR Layer PCB Model Figure 103. Three STABLCOR Layer Model Mesh with Solid Elements 99 Table 11. Single STABLCOR Layer Model CTE Lamination Theory [eq. (49)] ANSYS FEM Simulation 0x? 8.62 ppm/ oC 8.15 ppm/oC 0y? 10.83 ppm/ oC 9.76 ppm/oC Table 12. Three STABLCOR Layer Model CTE Lamination Theory [eq. (49)] ANSYS FEM Simulation 0x? 7.86 ppm/ oC 7.20 ppm/oC 0y? 10.46 ppm/ oC 9.14 ppm/oC 100 VIII. SUMMARY AND CONCLUSIONS Lamination theory is able to calculate the overall CTEs of the designed PCB. Lamination theory is able to calculate the mechanical properties of the PCB with different layer configuration. The CTEs of each STABLCOR layer can be measured by strain gages with correct procedures. The On-axis and off-axis elastic modulie can be measured by crosshead displacement. STABLCOR laminates could be edited into off-axis testing coupons for elastic modulus measurements. Off-axis elastic modulus regression fit can be used to calculate the shear modulus and Poisson?s ratio of STABLCOR layer. The calculation results show the carbon fiber laminates were able to lower the overall CTEs of the PCB. However, the stack-up ratios between STABLCOR and FR-4 layers have critical effects to the overall CTEs. Since the silicon chip has an extremely low CTE, the ideal PCB needs to have a CTE range from 3 ppm/oC to 4 ppm/oC. The calculated PCB CTEs were still not preferable with the stack-up configurations used in this study. Moreover, to set the CTE even lower than current models, higher ratio of STABLCOR to FR-4 may be needed. The calculated CTEs from lamination theory have good correspond with the simulation results from finite element method. 101 REFERENCES 1. Andrews, J. A., ?Flip Chip Package and Method of Making,? Patent number: 5352926, 1993. 2. Dally, J. W., Lall, P., and Suhling, J. C., Mechanical Design of Electronic Systems, College House Enterprises, 2008. 3. Engelmaier, W., and Attarwala, A. I., ?Surface-mount Attachment Reliability of Clip- leaded Ceramic Chip Carriers on FR-4 Circuit Boards,? 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Chamis, C.C., and Sinclair J.H., ?Ten-deg Off-Axis Test for Shear Properties in Fiber Composites,? Experimental Mechanics, Vol. 17(9), pp. 339-346, 1977. 28. Saliklis, E. P., and Falk, R. H., ?Correlating Off-Axis Tension Tests to Shear Modulus of Wood-Based Panels,? Journal of Structural Engineering, Vol. 126(5), pp. 621-625, 2000. 29. Kim, Y. K., ?Viscoelastic Effect of FR-4 Material on Packaging Stress Development,? IEEE Transactions on Advanced Packaging, Vol. 30(3), pp. 411-420, 2007. 30. Lau, J. H., and Lee, R. S. W., ?Effects of Build-Up Printed Circuit Board Thickness on the Solder Joint Reliability of a Wafer Level Chip Scale Package,? IEEE Transactions on Components and Packaging Technologies, Vol. 25(1), pp. 3-14, 2002. 103 APPENDIX A - ELASTIC MODULUS VERSUS OFF-AXIS ANGLE 104 Table 13. Curve Fit Shear Moduli and Poisson?s Ratios Sample No. Material Description G12 (GPa) Poisson?s ratio, 12? Graph Page No. 1 ST10F-0.005 9.08 0.30 105-108 2 ST10F-0.005 H/H 17.76 0.30 109-112 3 ST10F-0.006 4.31 0.33 113-116 4 ST10P-0.006 3.59 0.21 85-88 5 ST10F-0.006 H/H 13.70 0.27 89-92 6 ST10F-0.006 1/1 15.70 0.30 117-120 7 ST10P-0.006 H/H 11.80 0.29 121-124 8 ST10F-0.008 5.17 0.22 125-128 9 ST10P-0.008 3.93 0.30 129-132 10 ST10F-0.008 H/H 9.32 0.30 133-136 11 ST10P-0.008 H/H 6.76 0.33 137-140 12 ST10F-0.008 1/1 10.29 0.29 141-144 13 ST325P-180 2.07 0.30 145-148 14 ST325P-180 H/H 3.98 0.30 149-152 15 ST325P-0.008 2.51 0.30 153-156 16 ST325P-0.008 H/H 7.22 0.29 157-160 17 ST325P-0.010 240 2.67 0.23 161-164 18 ST325P-0.010 255 2.31 0.29 165-168 19 ST325P-0.010 H/H 6.22 0.30 169-172 20 ST325P-0.010 1/1 6.48 0.30 173-176 105 Figure 104. ST10F-0.005 0-Degree Elastic Modulus Figure 105. ST10F-0.005 90-Degree Elastic Modulus ST 1 0 F-0.0 0 5 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 90 .0 G Pa ?? 0? 1 E ST 1 0 F-0.0 0 5 90 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 91 .2 G Pa ?? 90? 2 E 106 Figure 106. ST10F-0.005 15-Degree Elastic Modulus Figure 107. ST10F-0.005 30-Degree Elastic Modulus ST 1 0 F-0.0 0 5 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 61 .3 G Pa ?? 15? ST 1 0 F-0.0 0 5 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 41 .3 G Pa ?? 30? 107 Figure 108. ST10F-0.005 45-Degree Elastic Modulus Figure 109. ST10F-0.005 60-Degree Elastic Modulus ST 1 0 F-0.0 0 5 45 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 29 .7 G Pa ?? 45? ST 1 0 F-0.0 0 5 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 39 .8 G Pa ?? 60? 108 Figure 110. ST10F-0.005 75-Degree Elastic Modulus Figure 111. ST10F-0.005 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 1 0 F-0.0 0 5 75 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 59 .5 G Pa ?? 75? 0 9 0 . 0 15 6 1 . 3 30 4 1 . 3 45 2 9 . 7 60 3 9 . 8 75 5 9 . 5 90 9 1 . 2 ? ( GP a ) E G o o d n es s o f f i t : R - s q u a r e d : 0 .9 9 2 5 A d j u s t e d R - s q u a r e d : 0 .9 9 1 Of f-Axis El astic Modulus Fit ST10F -0.005 Angle(R ad) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Elastic Modulus 20 30 40 50 60 70 80 90 100 El as t ic v s Angle(R ad) C ur v e Fi t 08.9 30.0 12 12 ? ? G ? 109 Figure 112. ST10F-0.005 H/H 0-Degree Elastic Modulus Figure 113. ST10F-0.005 H/H 90-Degree Elastic Modulus ST 1 0 F-0.0 0 5 H/H 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 11 8 .7 G Pa ?? 0? 1 E ST 1 0 F-0.0 0 5 H/H 90 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 12 0 .2 G Pa ?? 90? 2 E 110 Figure 114. ST10F-0.005 H/H 15-Degree Elastic Modulus Figure 115. ST10F-0.005 H/H 30-Degree Elastic Modulus ST 1 0 F-0.0 0 5 H/H 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 91 .2 G Pa ?? 15? ST 1 0 F-0.0 0 5 H/H 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 67 .6 G Pa ?? 30? 111 Figure 116. ST10F-0.005 H/H 45-Degree Elastic Modulus Figure 117. ST10F-0.005 H/H 60-Degree Elastic Modulus ST 1 0 F-0.0 0 5 H/H 45 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 64 .1 G Pa ?? 45? ST 1 0 F-0.0 0 5 H/H 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 67 .5 G Pa ?? 60? 112 Figure 118. ST10F-0.005 H/H 75-Degree Elastic Modulus Figure 119. ST10F-0.005 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 1 0 F-0.0 0 5 H/H 75 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 90 .1 G Pa ?? 75? 0 1 1 8 . 7 15 9 1 . 2 30 6 7 . 6 45 6 4 . 1 60 6 7 . 5 75 9 0 . 1 90 1 2 0 . 2 ? ( GP a ) E G o o d n es s o f f i t : R - s q u a r ed : 0 . 9 8 0 2 A d j u s t ed R - s q u a r ed : 0 .9 7 6 3 Of f-Axis El astic Modulus Fit ST10F -0.005 H/H Angle 0 20 40 60 80 100 Elastic Modulus 50 60 70 80 90 100 110 120 130 El as t ic v s Angle(R ad) C ur v e Fi t 76.17 30.0 12 12 ? ? G ? 113 Figure 120. ST10F-0.006 0-Degree Elastic Modulus Figure 121. ST10F-0.006 90-Degree Elastic Modulus ST 1 0 F- 0 .0 0 6 0 D EG R EE St r ai n, ? 0.000 0.002 0.004 0.006 0.008 0.010 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 3 8 .7 GPa E 1 o0?? ST 10 F- 0. 00 6 90 DEG REE S t rain , ? 0 .0 0 0 0 .0 0 2 0 .0 0 4 0 .0 0 6 0 .0 0 8 0 .0 1 0 S t ress, ? (MP a) 0 20 40 60 80 100 120 Str e ss v s Str a in Str e ss-Str a in cu rv e fit E= 36 . 7 G Pa E 2 o90?? 114 Figure 122. ST10F-0.006 15-Degree Elastic Modulus Figure 123. ST10F-0.006 30-Degree Elastic Modulus ST 1 0 F-0.0 0 6 15 DEG REE S t r ain , ? 0. 000 0. 002 0. 004 0. 006 0. 008 0. 010 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 33 .3 G Pa o15?? ST 1 0 F- 0 .0 0 6 3 0 D EG R EE St r ai n, ? 0.000 0.002 0.004 0.006 0.008 0.010 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 1 7 .8 GPa o30?? 115 Figure 124. ST10F-0.006 45-Degree Elastic Modulus Figure 125. ST10F-0.006 60-Degree Elastic Modulus ST 1 0 F- 0 .0 0 6 4 5 D EG R EE St r ai n, ? 0.000 0.002 0.004 0.006 0.008 0.010 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 1 4 .4 GPa o45?? ST 1 0 F- 0 .0 0 6 6 0 D EG R EE St r ai n, ? 0.000 0.002 0.004 0.006 0.008 0.010 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 1 7 .7 GPa o60?? 116 Figure 126. ST10F-0.006 75-Degree Elastic Modulus Figure 127. ST10F-0.006 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 1 0 F- 0 .0 0 6 7 5 D EG R EE St r ai n, ? 0.000 0.002 0.004 0.006 0.008 0.010 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 2 7 .2 GPa o75?? 0 3 8 . 7 15 2 7 . 8 30 1 7 . 8 45 1 4 . 4 60 1 7 . 7 75 2 7 . 2 90 3 6 . 7 ? )GP a(E Off -Ax is El a stic M o d u lu s Fit ST 1 0 F-0 .0 0 6 Ang le( R ad) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Elastic M odu lus 0 10 20 30 40 E l a sti c v s A n g l e ( R a d ) C u r v e Fi t G o o d n es s o f f i t : R - s q u a r ed : 0 . 9 9 9 A d j u s t ed R - s q u a r ed : 0 .9 9 9 31.4 33.0 12 12 ? ? G ? 117 Figure 128. ST10F-0.006 1/1 0-Degree Elastic Modulus Figure 129. ST10F-0.006 1/1 90-Degree Elastic Modulus ST 1 0 F-0.0 0 6 1/1 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 65 .8 G Pa ?? 0? 1 E ST 1 0 F-0.0 0 6 1/1 90 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 62 .7 G Pa ?? 90? 2 E 118 Figure 130. ST10F-0.006 1/1 15-Degree Elastic Modulus Figure 131. ST10F-0.006 1/1 30-Degree Elastic Modulus ST 1 0 F-0.0 0 6 1/1 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 59 .6 G Pa ?? 15? ST 1 0 F-0.0 0 6 1/1 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 48 .7 G Pa ?? 30? 119 Figure 132. ST10F-0.006 1/1 45-Degree Elastic Modulus Figure 133. ST10F-0.006 1/1 60-Degree Elastic Modulus ST 1 0 F-0.0 0 6 1/1 45 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 45 .2 G Pa ?? 45? ST 1 0 F-0.0 0 6 1/1 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 49 .2 G Pa ?? 60? 120 Figure 134. ST10F-0.006 1/1 75-Degree Elastic Modulus Figure 135. ST10F-0.006 1/1 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 1 0 F-0.0 0 6 1/1 75 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 58 .2 G Pa ?? 75? 0 6 5 . 8 15 5 9 . 6 30 4 8 . 7 45 4 5 . 2 60 4 9 . 2 75 5 8 . 2 90 6 2 . 7 ? ( GP a ) E Of f-Axis Elas tic Modulus Fi t ST 10F-0.006 1/1 A ngle( Rad) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 E las tic Modulus 30 40 50 60 70 Elast ic v s Ang le( Ra d) Cu rv e Fit G o o d n es s o f f i t : R - s q u a r e d : 0 .9 9 3 9 A d j u s t e d R - s q u a r e d : 0 .9 9 2 7 7.15 30.0 12 12 ? ? G ? 121 Figure 136. ST10P-0.006 H/H 0-Degree Elastic Modulus Figure 137. ST10P-0.006 H/H 90-Degree Elastic Modulus ST 1 0 P-0.0 0 6 H/H 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 54 .9 G Pa ?? 0? 1 E ST 1 0 P-0.0 0 6 H/H 90 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 56 .2 G Pa ?? 90? 2 E 122 Figure 138. ST10P-0.006 H/H 15-Degree Elastic Modulus Figure 139. ST10P-0.006 H/H 30-Degree Elastic Modulus ST 1 0 P-0.0 0 6 H/H 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 47 .7 G Pa ?? 15? ST 1 0 P-0.0 0 6 H/H 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 40 .2 G Pa ?? 30? 123 Figure 140. ST10P-0.006 H/H 45-Degree Elastic Modulus Figure 141. ST10P-0.006 H/H 60-Degree Elastic Modulus ST 1 0 P-0.0 0 6 H/H 45 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 36 .6 G Pa ?? 45? ST 1 0 P-0.0 0 6 H/H 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 40 .7 G Pa ?? 60? 124 Figure 142. ST10P-0.006 H/H 75-Degree Elastic Modulus Figure 143. ST10P-0.006 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 1 0 P-0.0 0 6 H/H 75 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 48 .2 G Pa ?? 75? 0 5 4 . 9 15 4 7 . 7 30 4 0 . 2 45 3 6 . 6 60 4 0 . 7 75 4 8 . 2 90 5 6 . 2 ? ( GP a ) E G o o d n es s o f f i t : R - s q u a r ed : 0 .9 8 9 A d j u s t ed R - s q u a r ed : 0 . 9 8 7 O ff -Axis Elast ic M odu lus Fit ST10P -0. 006 H/ H A ngle (Rad) 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 E lastic M odu lus 35 40 45 50 55 60 Ela st ic v s An g le (R a d ) C u rv e Fit 8.11 29.0 12 12 ? ? G ? 125 Figure 144. ST10F-0.008 0-Degree Elastic Modulus Figure 145. ST10F-0.008 90-Degree Elastic Modulus ST 1 0 F-0.0 0 8 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 27 G Pa ?? 0? 1 E ST 10F-0.008 90 DE GRE E St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 27.3 G Pa ?? 90? 2 E 126 Figure 146. ST10F-0.008 15-Degree Elastic Modulus Figure 147. ST10F-0.008 30-Degree Elastic Modulus ST 1 0 F-0.0 0 8 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 24 .8 G Pa ?? 15? ST 1 0 F-0.0 0 8 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 15 .3 G Pa ?? 30? 127 Figure 148. ST10F-0.008 45-Degree Elastic Modulus Figure 149. ST10F-0.008 60-Degree Elastic Modulus ST 1 0 F-0.0 0 8 45 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 12 .2 G Pa ?? 45? ST 1 0 F-0.0 0 8 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 15 .8 G Pa ?? 60? 128 Figure 150. ST10F-0.008 75-Degree Elastic Modulus Figure 151. ST10F-0.008 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 1 0 F-0.0 0 8 75 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 25 .3 G Pa ?? 75? 0 27 15 2 4 . 8 30 1 5 . 3 45 1 2 . 2 60 1 5 . 8 75 2 5 . 3 90 2 7 . 3 ? ( GP a ) E G o o d n e s s o f f i t : R - s q u a r e d : 0 .9 9 6 A d j u s t e d R - s q u a r e d : 0 .9 9 5 O ff -Axis Elast ic M odu lus Fit ST10F- 0. 008 A ngle (Rad) 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 E lastic M odu lus 10 12 14 16 18 20 22 24 26 28 30 Ela st ic v s An g le (R a d ) C u rv e Fit 17.5 22.0 12 12 ? ? G ? 129 Figure 152. ST10P-0.008 0-Degree Elastic Modulus Figure 153. ST10P-0.008 90-Degree Elastic Modulus ST 1 0 P-0.0 0 8 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 26 .1 G Pa ?? 0? 1 E ST 1 0 P-0.0 0 8 90 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 25 .9 G Pa ?? 90? 2 E 130 Figure 154. ST10P-0.008 15-Degree Elastic Modulus Figure 155. ST10P-0.008 30-Degree Elastic Modulus ST 1 0 P-0.0 0 8 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 20 .7 G Pa ?? 15? ST 10P -0.008 30 DEGRE E St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 15.4 G Pa ?? 30? 131 Figure 156. ST10P-0.008 45-Degree Elastic Modulus Figure 157. ST10P-0.008 60-Degree Elastic Modulus ST 1 0 P-0.0 0 8 45 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 12 .8 G Pa ?? 45? ST 1 0 P-0.0 0 8 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 15 .1 G Pa ?? 60? 132 Figure 158. ST10P-0.008 75-Degree Elastic Modulus Figure 159. ST10P-0.008 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 1 0 P-0.0 0 8 75 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 20 .5 G Pa ?? 75? 0 2 6 . 1 15 2 0 . 7 30 1 5 . 4 45 1 2 . 8 60 1 5 . 1 75 2 0 . 5 90 2 5 . 9 ? ( GP a ) E G o o d n e s s o f f i t : R - s q u a r ed : 0 .9 9 6 A d j u s t e d R - s q u a r e d : 0 .9 9 5 O ff -Axis Elast ic M odu lus Fit ST10P -0. 008 A ngle (Rad) 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 E lastic M odu lus 12 14 16 18 20 22 24 26 28 Ela st ic v s An g le (R a d ) C u rv e Fit 93.3 30.0 12 12 ? ? G ? 133 Figure 160. ST10F-0.008 H/H 0-Degree Elastic Modulus Figure 161. ST10F-0.008 H/H 90-Degree Elastic Modulus ST 1 0 F-0.0 0 8 H/H 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 62 .9 G Pa ?? 0? 1 E ST 1 0 F-0.0 0 8 H/H 90 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 63 .6 G Pa ?? 90? 2 E 134 Figure 162. ST10F-0.008 H/H 15-Degree Elastic Modulus Figure 163. ST10F-0.008 H/H 30-Degree Elastic Modulus ST 1 0 F-0.0 0 8 H/H 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 47 .9 G Pa ?? 15? ST 1 0 F-0.0 0 8 H/H 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 36 .3 G Pa ?? 30? 135 Figure 164. ST10F-0.008 H/H 45-Degree Elastic Modulus Figure 165. ST10F-0.008 H/H 60-Degree Elastic Modulus ST 1 0 F-0.0 0 8 H/H 45 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 31 .8 G Pa ?? 45? ST 1 0 F-0.0 0 8 H/H 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 36 .1 G Pa ?? 60? 136 Figure 166. ST10F-0.008 H/H 75-Degree Elastic Modulus Figure 167. ST10F-0.008 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 1 0 F-0.0 0 8 H/H 75 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 49 .5 G Pa ?? 75? 0 6 2 . 9 15 4 7 . 9 30 3 6 . 3 45 3 1 . 8 60 3 6 . 1 75 4 9 . 5 90 6 3 . 6 ? ( GP a ) E G o o d n es s o f f i t : R - s q u a r ed : 0 .9 9 4 A d j u s t ed R - s q u a r ed : 0 . 9 9 3 O ff -Axis Elast ic M odu lus Fit ST10F- 0. 008 H/ H A ngle (Rad) 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 E lastic M odu lus 30 35 40 45 50 55 60 65 70 Ela st ic v s An g le (R a d ) C u rv e Fit 32.9 30.0 12 12 ? ? G ? 137 Figure 168. ST10P-0.008 H/H 0-Degree Elastic Modulus Figure 169. ST10P-0.008 H/H 90-Degree Elastic Modulus ST 10P -0.008 H/H 0 DE GRE E St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 48.7 G Pa ?? 0? 1 E ST 10P -0.008 H/H 90 DEGRE E St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 49.1 G Pa ?? 90? 2 E 138 Figure 170. ST10P-0.008 H/H 15-Degree Elastic Modulus Figure 171. ST10P-0.008 H/H 30-Degree Elastic Modulus ST 10P -0.008 H/H 15 DEGRE E St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 36.3 G Pa ?? 15? ST 10P -0.008 H/H 30 DEGRE E St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 26.2 G Pa ?? 30? 139 Figure 172. ST10P-0.008 H/H 45-Degree Elastic Modulus Figure 173. ST10P-0.008 H/H 60-Degree Elastic Modulus ST 10P -0.008 H/H 45 DEGRE E St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 23.7 G Pa ?? 45? ST 10P -0.008 H/H 60 DEGRE E St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 27.4 G Pa ?? 60? 140 Figure 174. ST10P-0.008 H/H 75-Degree Elastic Modulus Figure 175. ST10P-0.008 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 10P -0.008 H/H 75 DEGRE E St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 37.4 G Pa ?? 75? 0 4 8 . 7 15 3 6 . 3 30 2 6 . 2 45 2 3 . 7 60 2 7 . 4 75 3 7 . 4 90 4 9 . 1 ? ( GP a ) E Of f-Axis Elas tic Modulus Fi t ST 10P-0.008 H/H A ngle( Rad) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 E las tic Modulus 20 25 30 35 40 45 50 55 Elast ic v s Ang le( Ra d) Cu rv e Fit G o o d n es s o f f i t : R - s q u a r e: 0 .9 9 2 A d j u s t ed R - s q u a r e: 0 .9 9 0 76.6 33.0 12 12 ? ? G ? 141 Figure 176. ST10F-0.008 1/1 0-Degree Elastic Modulus Figure 177. ST10F-0.008 1/1 90-Degree Elastic Modulus ST 1 0 F- 0 .0 0 8 1 /1 0 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 7 4 .0 GPa ?? 0? 1 E ST 1 0 F- 0 .0 0 8 1 /1 9 0 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 7 2 .0 GPa ?? 90? 2 E 142 Figure 178. ST10F-0.008 1/1 15-Degree Elastic Modulus Figure 179. ST10F-0.008 1/1 30-Degree Elastic Modulus ST 1 0 F- 0 .0 0 8 1 /1 1 5 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 5 8 .1 GPa ?? 15? ST 1 0 F- 0 .0 0 8 1 /1 3 0 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t ?? 30? E= 3 8 .3 GPa 143 Figure 180. ST10F-0.008 1/1 45-Degree Elastic Modulus Figure 181. ST10F-0.008 1/1 60-Degree Elastic Modulus ST 1 0 F- 0 .0 0 8 1 /1 4 5 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t ?? 45? E= 3 4 .1 GPa ST 1 0 F- 0 .0 0 8 1 /1 6 0 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t ?? 60? E= 3 9 .1 GPa 144 Figure 182. ST10F-0.008 1/1 75-Degree Elastic Modulus Figure 183. ST10F-0.008 1/1 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 1 0 F- 0 .0 0 8 1 /1 7 5 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 5 8 .8 GPa ?? 75? 0 7 4 . 0 15 5 8 . 1 30 3 8 . 3 45 3 4 . 1 60 3 9 . 1 75 5 8 . 8 90 7 2 . 0 ? ( GP a ) E G o o d n es s o f f i t : R - s q u a r e : 0 .9 9 5 A d j u s t ed R - s q u a r e: 0 . 9 9 4 Of f-Axis El astic Modulus Fit ST10F -0.008 1/1 Angle(R ad) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Elastic Modulus 30 40 50 60 70 80 El as t ic v s Angle(R ad) C ur v e Fi t 29.10 29.0 12 12 ? ? G ? 145 Figure 184. ST325P-180 0-Degree Elastic Modulus Figure 185. ST325P-180 90-Degree Elastic Modulus ST 3 2 5 P-18 0 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 49 .4 G Pa ?? 0? 1 E ST 3 2 5 P-18 0 90 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 51 .1 G Pa ?? 90? 2 E 146 Figure 186. ST325P-180 15-Degree Elastic Modulus Figure 187. ST325P-180 30-Degree Elastic Modulus ST 3 2 5 P-18 0 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 22 .0 G Pa ?? 15? ST 3 2 5 P-18 0 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 9.3 G Pa ?? 30? 147 Figure 188. ST325P-180 45-Degree Elastic Modulus Figure 189. ST325P-180 60-Degree Elastic Modulus ST 3 2 5 P-18 0 45 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 7.4 G Pa ?? 45? ST 3 2 5 P-18 0 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 10 .0 G Pa ?? 60? 148 Figure 190. ST325P-180 75-Degree Elastic Modulus Figure 191. ST325P-180 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 3 2 5 P-18 0 75 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 21 .4 G Pa ?? 75? 0 4 9 . 4 15 2 2 . 0 30 9 . 3 45 7 . 4 60 1 0 . 0 75 2 1 . 4 90 5 1 . 1 ? ( GP a ) E G o o d n e s s o f f i t : R - s q u a r e : 0 .9 9 9 A d j u s t e d R - s q u a r e : 0 .9 9 9 O ff -Axis Elast ic M odu lus Fit ST325 P-18 0 A ngle 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 E lastic M odu lus 0 10 20 30 40 50 60 Ela st ic v s An g le (R a d ) C u rv e Fit 07.2 30.0 12 12 ? ? G ? 149 Figure 192. ST325P-180 H/H 0-Degree Elastic Modulus Figure 193. ST325P-180 H/H 90-Degree Elastic Modulus ST 3 2 5 P-18 0 H/H 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 79 .3 G Pa ?? 0? 1 E ST 3 2 5 P-18 0 H/H 90 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 81 .1 G Pa ?? 90? 2 E 150 Figure 194. ST325P-180 H/H 15-Degree Elastic Modulus Figure 195. ST325P-180 H/H 30-Degree Elastic Modulus ST 3 2 5 P-18 0 H/H 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 36 .1 G Pa ?? 15? ST 3 2 5 P-18 0 H/H 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 20 .7 G Pa ?? 30? 151 Figure 196. ST325P-180 H/H 45-Degree Elastic Modulus Figure 197. ST325P-180 H/H 60-Degree Elastic Modulus ST 3 2 5 P-18 0 H/H 45 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 16 .7 G Pa ?? 45? ST 3 2 5 P-18 0 H/H 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 21 .8 G Pa ?? 60? 152 Figure 198. ST325P-180 H/H 75-Degree Elastic Modulus Figure 199. ST325P-180 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 3 2 5 P-18 0 H/H 75 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 35 .8 G Pa ?? 75? 0 7 9 . 3 15 3 6 . 1 30 2 0 . 7 45 1 6 . 7 60 2 1 . 8 75 3 5 . 8 90 8 1 . 1 ? ( GP a ) E G o o d n es s o f f i t : R - s q u a r e : 0 .9 9 3 8 A d j u s t e d R - s q u a r e : 0 .9 9 2 6 O ff -Axis Elast ic M odu lus Fit ST325 P-18 0 H/H A ngle 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 E lastic M odu lus 10 20 30 40 50 60 70 80 90 Ela st ic v s An g le (R a d ) C u rv e Fit 98.3 30.0 12 12 ? ? G ? 153 Figure 200. ST325P-0.008 0-Degree Elastic Modulus Figure 201. ST325P-0.008 90-Degree Elastic Modulus ST 3 2 5 P-0.0 0 8 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 68 .7 G Pa ?? 0? 1 E ST 3 2 5 P-0.0 0 8 90 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t ?? 90? 2 E E= 72 .7 G Pa 154 Figure 202. ST325P-0.008 15-Degree Elastic Modulus Figure 203. ST325P-0.008 30-Degree Elastic Modulus ST 3 2 5 P-0.0 0 8 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 26 .5 G Pa ?? 15? ST 3 2 5 P-0.0 0 8 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 14 .0 G Pa ?? 30? 155 Figure 204. ST325P-0.008 45-Degree Elastic Modulus Figure 205. ST325P-0.008 60-Degree Elastic Modulus ST 3 2 5 P-0.0 0 8 45 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 11 .0 G Pa ?? 45? ST 3 2 5 P-0.0 0 8 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 14 .7 G Pa ?? 60? 156 Figure 206. ST325P-0.008 75-Degree Elastic Modulus Figure 207. ST325P-0.008 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 3 2 5 P-0.0 0 8 75 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 25 .2 G Pa ?? 75? 0 6 8 . 7 15 2 6 . 5 30 1 4 . 0 45 1 1 . 0 60 1 4 . 7 75 2 5 . 2 90 7 2 . 7 ? ( GP a ) E G o o d n es s o f f i t : R - s q u a r e : 0 .9 9 6 A d j u s t ed R - s q u a r e: 0 . 9 9 5 O ff -Axis Elast ic M odu lus Fit ST325 P-0. 008 A ngle (Rad) 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 E lastic M odu lus 0 10 20 30 40 50 60 70 80 Ela st ic v s An g le (R a d ) C u rv e Fit 51.2 30.0 12 12 ? ? G ? 157 Figure 208. ST325P-0.008 H/H 0-Degree Elastic Modulus Figure 209. ST325P-0.008 H/H 90-Degree Elastic Modulus ST 3 2 5 P-0.0 0 8 H/H 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 88 .2 G Pa ?? 0? 1 E ST 3 2 5 P-0.0 0 8 H/H 90 DEG REE S t r ess, ? ( M P a) 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 90 .7 G Pa ?? 90? 2 E 158 Figure 210. ST325P-0.008 H/H 15-Degree Elastic Modulus Figure 211. ST325P-0.008 H/H 30-Degree Elastic Modulus ST 3 2 5 P-0.0 0 8 H/H 15 DEG REE S t r ess, ? ( M P a) 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 58 .2 G Pa ?? 15? ST 3 2 5 P-0.0 0 8 H/H 30 DEG REE S t r ess, ? ( M P a) 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 29 .4 G Pa ?? 30? 159 Figure 212. ST325P-0.008 H/H 45-Degree Elastic Modulus Figure 213. ST325P-0.008 H/H 60-Degree Elastic Modulus ST 3 2 5 P-0.0 0 8 H/H 45 DEG REE S t r ess, ? ( M P a) 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 23 .8 G Pa ?? 45? ST 3 2 5 P-0.0 0 8 H/H 60 DEG REE S t r ess, ? ( M P a) 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 30 .8 G Pa ?? 60? 160 Figure 214. ST325P-0.008 H/H 75-Degree Elastic Modulus Figure 215. ST325P-0.008 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 3 2 5 P-0.0 0 8 H/H 75 DEG REE S t r ess, ? ( M P a) 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 57 .6 G Pa ?? 75? 0 8 8 . 2 15 5 8 . 2 30 2 9 . 4 45 2 3 . 8 60 3 0 . 8 75 5 7 . 6 90 9 0 . 7 ? ( GP a ) E G o o d n e s s o f f i t : R - s q u a r e : 0 .9 9 5 A d j u s t e d R - s q u a r e : 0 .9 9 4 Off-Ax is El as tic Mod ul us Fi t ST 32 5P -0.00 8 H /H An g le ( R a d ) 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 1. 2 1. 4 1. 6 1. 8 Ela s tic Mo d u lu s 20 30 40 50 60 70 80 90 100 E las t ic v s A ngle(Rad) Curv e Fi t 22.7 29.0 12 12 ? ? G ? 161 Figure 216. ST325P-0.010 240 0-Degree Elastic Modulus Figure 217. ST325P-0.010 240 90-Degree Elastic Modulus ST 3 2 5 P-0.0 1 0 24 0 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 45 .7 G Pa ?? 0? 1 E ST 3 2 5 P-0.0 1 0 24 0 90 DEG REE S t r ess, ? ( M P a) 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 46 .8 G Pa ?? 90? 2 E 162 Figure 218. ST325P-0.010 240 15-Degree Elastic Modulus Figure 219. ST325P-0.010 240 30-Degree Elastic Modulus ST 3 2 5 P-0.0 1 0 24 0 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 23 .8 G Pa ?? 15? ST 3 2 5 P-0.0 1 0 24 0 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 10 .9 G Pa ?? 30? 163 Figure 220. ST325P-0.010 240 45-Degree Elastic Modulus Figure 221. ST325P-0.010 240 60-Degree Elastic Modulus ST 3 2 5 P-0.0 1 0 24 0 45 DEG REE S t r ess, ? ( M P a) 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 8.6 G Pa ?? 45? ST 3 2 5 P-0.0 1 0 24 0 60 DEG REE S t r ess, ? ( M P a) 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 10 .5 G Pa ?? 60? 164 Figure 222. ST325P-0.010 240 75-Degree Elastic Modulus Figure 223. ST325P-0.010 240 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 3 2 5 P-0.0 1 0 24 0 75 DEG REE S t r ess, ? ( M P a) 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 24 .8 G Pa ?? 75? 0 4 5 . 7 15 2 3 . 8 30 1 0 . 9 45 8 . 6 60 1 0 . 5 75 2 4 . 8 90 4 6 . 8 ? ( GP a ) E G o o d n es s o f f i t : R - s q u a r e : 0 .9 9 6 A d j u s t ed R - s q u a r e: 0 .9 9 6 Of f-Axis Elas tic Modulus Fi t ST 325P-0.010 240 A ngle( Rad) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 E las tic Modulus 0 10 20 30 40 50 Elast ic v s Ang le( Ra d) Cu rv e Fit 67.2 23.0 12 12 ? ? G ? 165 Figure 224. ST325P-0.010 255 0-Degree Elastic Modulus Figure 225. ST325P-0.010 255 90-Degree Elastic Modulus ST 325P -0.010 255 0 DE GRE E St r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 St r es s , ? (MPa ) 0 20 40 60 80 100 120 S t ress v s S t rain S t ress -St rain c urv e f it E= 72.2 G Pa ?? 0? 1 E ST 3 2 5 P-0.0 1 0 25 5 90 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 72 .4 G Pa ?? 90? 2 E 166 Figure 226. ST325P-0.010 255 15-Degree Elastic Modulus Figure 227. ST325P-0.010 255 30-Degree Elastic Modulus ST 3 2 5 P-0.0 1 0 25 5 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 25 .7 G Pa ?? 15? ST 3 2 5 P-0.0 1 0 25 5 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 11 .6 G Pa ?? 30? 167 Figure 228. ST325P-0.010 255 45-Degree Elastic Modulus Figure 229. ST325P-0.010 255 60-Degree Elastic Modulus ST 3 2 5 P-0.0 1 0 25 5 45 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 8.2 G Pa ?? 45? ST 3 2 5 P-0.0 1 0 25 5 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 11 .4 G Pa ?? 60? 168 Figure 230. ST325P-0.010 255 75-Degree Elastic Modulus Figure 231. ST325P-0.010 255 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 3 2 5 P-0.0 1 0 25 5 75 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 26 .3 G Pa ?? 75? 0 7 2 . 2 15 2 5 . 7 30 1 1 . 6 45 8 . 2 60 1 1 . 4 75 2 6 . 3 90 7 2 . 4 ? ( GP a ) E G o o d n es s o f f i t : R - s q u a r e : 0 .9 9 9 A d j u s t ed R - s q u a r e: 0 . 9 9 9 O ff -Axis Elastic M odulus Fit ST325P-0 .01 0 2 55 A ngle(Rad) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 E lasti c M odulus 0 20 40 60 80 100 120 Elast ic v s Angle (R ad ) C ur v e Fit 31.2 29.0 12 12 ? ? G ? 169 Figure 232. ST325P-0.010 255 H/H 0-Degree Elastic Modulus Figure 233. ST325P-0.010 255 H/H 90-Degree Elastic Modulus ST 3 2 5 P-0.0 1 0 25 5 H/H 0 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 94 .3 G Pa ?? 0? 1 E ST 3 2 5 P-0.0 1 0 25 5 H/H 90 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 95 .7 G Pa ?? 90? 2 E 170 Figure 234. ST325P-0.010 255 H/H 15-Degree Elastic Modulus Figure 235. ST325P-0.010 255 H/H 30-Degree Elastic Modulus ST 3 2 5 P-0.0 1 0 25 5 H/H 15 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 55 .5 G Pa ?? 15? ST 3 2 5 P-0.0 1 0 25 5 H/H 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 26 .5 G Pa ?? 30? 171 Figure 236. ST325P-0.010 255 H/H 45-Degree Elastic Modulus Figure 237. ST325P-0.010 255 H/H 60-Degree Elastic Modulus ST 3 2 5 P-0.0 1 0 25 5 H/H 45 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 21 .5 G Pa ?? 45? ST 3 2 5 P-0.0 1 0 25 5 H/H 60 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 25 .5 G Pa ?? 60? 172 Figure 238. ST325P-0.010 255 H/H 75-Degree Elastic Modulus Figure 239. ST325P-0.010 255 H/H Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 3 2 5 P-0.0 1 0 25 5 H/H 75 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 56 .7 G Pa ?? 75? 0 9 4 . 3 15 5 5 . 5 30 2 6 . 5 45 2 1 . 5 60 2 5 . 5 75 5 6 . 7 90 9 5 . 7 ? ( GP a ) E G o o d n e s s o f f i t : R - s q u a r e : 0 .9 9 9 A d j u s t e d R - s q u a r e : 0 .9 9 9 Off- Ax is El asti c Modu lu s Fi t ST 325P -0.010 255 H /H Angle(R ad) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Elastic Modulus 0 20 40 60 80 100 120 El as t ic v s Angle(R ad) C ur v e Fi t 22.6 30.0 12 12 ? ? G ? 173 Figure 240. ST325P-0.010 240 1/1 0-Degree Elastic Modulus Figure 241. ST325P-0.010 240 1/1 90-Degree Elastic Modulus ST 3 2 5 P- 0 .0 1 0 2 4 0 1 /1 0 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t ?? 0? 1 E E= 1 1 2 .4 GPa ST 3 2 5 P- 0 .0 1 0 2 4 0 1 /1 9 0 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 1 1 6 .9 GPa ?? 90? 2 E 174 Figure 242. ST325P-0.010 240 1/1 15-Degree Elastic Modulus Figure 243. ST325P-0.010 240 1/1 30-Degree Elastic Modulus ST 3 2 5 P- 0 .0 1 0 2 4 0 1 /1 1 5 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t ?? 15? E= 5 6 .1 GPa ST 3 2 5 P-0.0 1 0 24 0 1/1 30 DEG REE S t r ain , ? 0. 0000 0. 0005 0. 0010 0. 0015 0. 0020 0. 0025 0. 0030 S t r ess, ? ( M P a) 0 20 40 60 80 100 120 S t r es s v s S t r ai n S t r es s - S t r ai n c ur v e f i t E= 32 .8 G Pa ?? 30? 175 Figure 244. ST325P-0.010 240 1/1 45-Degree Elastic Modulus Figure 245. ST325P-0.010 240 1/1 60-Degree Elastic Modulus ST 3 2 5 P- 0 .0 1 0 2 4 0 1 /1 4 5 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 2 8 .5 GPa ?? 45? ST 3 2 5 P- 0 .0 1 0 2 4 0 1 /1 6 0 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 3 3 .4 GPa ?? 60? 176 Figure 246. ST325P-0.010 240 1/1 75-Degree Elastic Modulus Figure 247. ST325P-0.010 240 1/1 Elastic Modulus versus Off-Axis Angle Curve Fitting Plot ST 3 2 5 P- 0 .0 1 0 2 4 0 1 /1 7 5 D EG R EE St r ai n, ? 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 St r es s , ? (M Pa) 0 20 40 60 80 100 120 S t r e ss v s S t r a i n S t r e ss- S t r a i n cu r v e f i t E= 5 4 .3 GPa ?? 75? 0 1 1 2 . 4 15 5 6 . 1 30 3 2 . 8 45 2 8 . 5 60 3 3 . 4 75 5 4 . 3 90 1 1 6 . 9 ? ( GP a ) E G o o d n es s o f f i t : R - s q u a r e : 0 . 9 9 1 4 A d j u s t ed R - s q u a r e: 0 .9 8 9 6 Off- Ax is El asti c Modu lu s Fi t ST 325P -0.010 240 1/1 Angle(R ad) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Elastic Modulus 20 40 60 80 100 120 El as t ic v s Angle(R ad) C ur v e Fi t 48.6 30.0 12 12 ? ? G ? 177 APPENDIX B - MATLAB CODE 178 Matlab Code for Single-layer STABLCOR Model: clear all; close all; clc; %The program is able to calculate model construct with different numbers of %layers. %Enter E1 Elastic Moduli Grain Direction property for each layer(Top to Bottom) E1=[12E9 77E9 12E9]; %Enter E2 Elastic Moduli perpendicular to Grain Direction property for each layer(Top to Bottom) E2=[12E9 57.7E9 12E9]; %Enter G12 Shear Modulus property for each layer(Top to Bottom) G12=[3.1E9 2.31E9 3.1E9]; %Enter v12 property for each layer(Top to Bottom) V12=[0.27 0.29 0.27]; %Enter v21 property for each layer(Top to Bottom) V21=[0.27 0.29 0.27]; %Enter Thickness property for each layer(Top to Bottom) t=[0.1 0.0255 0.1]; %Alpha X, Alpha Y, Alpha XY AL1=[12E-6 4.1E-6 12E-6]; AL2=[12E-6 6.04E-6 12E-6]; AL12=[0 0 0]; ALA=[AL1; AL2; AL12]; %Temperature (Delta T) DT=25, ALT=ALA*DT; Q11=(E1./(1-(V12.*V21))); Q22=(E2./(1-(V12.*V21))); Q12=((V12.*E2)/(1-V12.*V21)); Q21=((V12.*E2)/(1-V12.*V21)); Q66=G12; DD=0; %Thermal Loads for i= 1 : length(t); Q=[Q11(:,i) Q12 0; Q12 Q22(:,i) 0; 0 0 Q66(:,i)]; D=Q*ALT(:,i)*t(:,i); %ALT DD=DD+D; end A11=sum(Q11.*t); A22=sum(Q22.*t); A12=sum(Q12.*t); A21=sum(Q21.*t); A66=sum(Q66.*t); AM=[A11 A12 0; A21 A22 0; 0 0 A66], AI=inv(AM); AI, C=AI*DD; D=C/DT; Thermalstrain=C, CTE=D, 179 Matlab Code For Tri-layer STABLCOR Model: clear all; close all; clc; %The program is able to calculate model construct with different numbers of %layers. %Enter E1 Elastic Moduli Grain Direction property for each layer(Top to Bottom) E1=[12E9 77E9 12E9 77E9 12E9 77E9 12E9]; %Enter E2 Elastic Moduli perpendicular to Grain Direction property for each layer(Top to Bottom) E2=[12E9 57.7E9 12E9 57.7E9 12E9 57.7E9 12E9]; %Enter G12 Shear Modulus property for each layer(Top to Bottom) G12=[3.1E9 2.31E9 3.1E9 2.31E9 3.1E9 2.31E9 3.1E9]; %Enter v12 property for each layer(Top to Bottom) V12=[0.27 0.29 0.27 0.29 0.27 0.29 0.27]; %Enter v21 property for each layer(Top to Bottom) V21=[0.27 0.29 0.27 0.29 0.27 0.29 0.27]; %Enter Thickness property for each layer(Top to Bottom) t=[0.1 0.0255 0.1 0.0255 0.1 0.0255 0.1]; %Alpha X, Alpha Y, Alpha XY AL1=[12E-6 4.1E-6 12E-6 4.1E-6 12E-6 4.1E-6 12E-6]; AL2=[12E-6 6.04E-6 12E-6 6.04E-6 12E-6 6.04E-6 12E-6]; AL12=[0 0 0 0 0 0 0]; ALA=[AL1; AL2; AL12]; %Temperature (Delta T) DT=25, ALT=ALA*DT; Q11=(E1./(1-(V12.*V21))); Q22=(E2./(1-(V12.*V21))); Q12=((V12.*E2)/(1-V12.*V21)); Q21=((V12.*E2)/(1-V12.*V21)); Q66=G12; DD=0; %Thermal Loads for i= 1 : length(t); Q=[Q11(:,i) Q12 0; Q12 Q22(:,i) 0; 0 0 Q66(:,i)]; D=Q*ALT(:,i)*t(:,i); %ALT DD=DD+D; end A11=sum(Q11.*t); A22=sum(Q22.*t); A12=sum(Q12.*t); A21=sum(Q21.*t); A66=sum(Q66.*t); AM=[A11 A12 0; A21 A22 0; 0 0 A66], AI=inv(AM); AI, C=AI*DD; D=C/DT; Thermalstrain=C, CTE=D,