由於過去在積體電路產業長期累積的應用經驗以及其本身優良的機械性質，矽材已經成為了微機電(Microelectromechanical Systems, MEMS)結構中相當廣泛使用的材料。由於這些微結構會受到長期的機械驅動或振動，將會導致疲勞現象的產生，進而影響到產品的精確性與可靠度，因此對長時處於振動狀態下的MEMS結構相關研究就顯得相當重要。截至目前已有些研究團隊對此現象進行討論，然而要從微結構試片上直接測得之應力值是相當困難的，通常會使用有限單元法來模擬獲得。
　　然而透過模擬所得到的應力值容易變動，歸因於有許多的有限元素法之設定如元素大小與類型、求解方式（節點解或元素解）或邊界條件等等，任何參數的小變動都可能引起求解上不等幅度的誤差，造成得到錯誤的應力值與判斷。因此這些參數都需要經過仔細且專業的選擇。
　　在本研究中會模擬多個矽質微結構，這些結構會根據過去與矽質疲勞性質有關之文獻，並在內顯式有限單元法軟體ANSYS®中建造模型，之所以要重新建造是因為模型需要處於相同的模擬環境，如上文所述。同時經過了重新建造模型與修正以後，在應力－壽命曲線(S-N Curve)上會有更合理與符合物理行為的曲線。
　　為了將這些文獻資料進行比較，上述之模擬設定都必須相近且正確，得到了可靠的最大應力值後各自對應相對的壽命大小，如此形成新的應力－壽命曲線。再根據這些曲線，可以得到一組壽命預估公式。
　　在另一方面，本研究將使用外顯式有限單元法軟體ANSYS®/LS-DYNA來印證不同的負載控制模式(力控制與位移控制)是否會導致頻率效應。
　　本研究提供完整的研究方法(Methodology)，即矽質微結構在彎矩疲勞負載下，如何提出壽命預估公式的研究流程以及設定，並以一組
過去文獻的研究數據為例，進行重新模擬及修正，而得到了較理想的結果。同時外顯式處理法的動態模擬研究中也發現到力控制模式的頻率效應。

　　Due to long-term accumulation of application experience in integration circuit (IC) industry and the excellent mechanical behavior, silicon has become a widely-used material in the microelectromechanical systems (MEMS). Since MEMS devices are exposed to long-period mechanical actuation or vibration, the high cycle fatigue characteristics of silicon become very important. There have been several research teams working in this topic, while the stresses on the MEMS structure are difficult to measure directly from actual specimen in experiments. The finite element analysis (FEA) is usually used to simulate the experiment to get the stress.
　　However, the simulation might have large error if some parameters like element mesh size, element type, solution type (nodal or element solution), boundary conditions and many others are not precisely chosen. The difference will cause magnitude of stress to vary, which may lead to wrong results and judgments. So these parameters should be carefully chosen.
　　In this study, silicon-based micro structures will be constructed according to several literatures which had experimental data of fatigue of silicon in implicit-method FEM software, ANSYS®. The reason why the models should be reconstructed is the setting as mentioned. After modification, the simulation agrees more to physical behavior and has better results than previous reference.
　　To bring these data together and compare, the parameters must be similar and chosen precisely for all the models to make the stress reliable. Then, each simulated stress was coupled to corresponding experimental number of life cycles from reference to form a curve plotted by stress versus number of cycles (S-N curve). And according to the S-N curves, a life-prediction formula is derived.
　　On the other hand, an explicit-method FEM software, ANSYS®/LS-DYNA is used in this research to verify if the frequency effect due to the different loading control types (force-control or displacement-control) exists or not.
　　This research provides a complete methodology to derive a life prediction equation, which is for silicon micro structures under bending fatigue tests. And taking the data from a literature as example, the simulation will be re-conducted and modified. The new results are more ideal. On the other hand, frequency effect of force- control type is discovered by transient explicit analysis.

摘要 I
Abstract　　　III
目錄　　　V
表目錄 VII
圖目錄 VIII
第一章 緒論 1
第二章 基礎理論 11
2.1 有限元素法理論 11
2.2 有限元素暫態分析法 14
2.2.1 外顯式時間處理法 15
2.2.1 零能量模式 17
2.3 有限元素法之接觸理論 18
2.3.1 拉格朗日乘子法 19
2.3.2 罰函數法 20
2.3.3 加強型拉格朗日法 21
2.4 有限元素等效應力推估法 21
2.5 彎矩樑原理 23
2.6 高週疲勞理論 23
第三章 內隱式處理法之各模型修正與等效應力推估 30
3.1 Hung與Hocheng之模型建構、修正與應力推估 30
3.2 應力結果之比較討論與壽命預估公式之建立 38
3.3 Baumert等人之模型修正與應力推估 41
第四章 外顯式處理法之頻率效應驗證 51
4.1 有限元素模型尺寸與建構 51
4.2 模型之邊界條件與負載設定 53
4.3 外顯式處理法模擬研究流程 55
4.4 外顯式處理法模擬結果與討論 56
第五章 結論與未來展望 63
參考文獻 65

[1] C. Liu, Founations of MEMS, 2nd ed., Pearson Education, 2012.
[2] 江國寧, 微電子系統封裝基礎理論與應用技術, 滄海書局, 2006.
[3] T. Y. Hung, C. C. Wang, and K. N. Chiang, "Bonding Wire Life Prediction Model of the Power Module under Power Cycling Test," Proc. EuroSimE 2013, Wroclaw, Poland, April 15-17, 2013.
[4] L. F. Coffin Jr., “A study of the effects of cyclic thermal stresses on a ductile metal,” Transactions of the ASME, vol. 76, pp. 931-950, 1954.
[5] SAE International, “Technical Report on Low Cycle Fatigue Properties Ferrous and Non-Ferrous of Materials,” 2002.
[6] D. M. Tanner, W. M. Miller, K. A. Peterson, M. T. Dugger, W. P. Eaton, L. W. Irwin, D. C. Senft, N. F. Smith, P. Tangyunyong and S. L. Miller, “Frequency dependence of the lifetime of a surface micromachined microengine driving a load,” Microelectronics Reliability, vol. 39, pp. 401-414, 1999
[7] E.K. Baumert, P.-O. Baumert and O.N. P ierron, “Investigation of the low-cycle fatigue mechanism for micron-scale monocrystalline silicon films,” Acta Materialia, vol. 58, pp. 2854–2863, 2010
[8] J. Bagdahn and W . N. Sharpe Jr., “Fatigue of polycrystalline silicon under long-term cyclic loading,” Sensors aand Actuators A, vol. 103, pp. 9-15, 2003
[9] T. Namazu and Y. Isono, “High-cycle fatigue damage evaluation for micro-nanoscale single crystal silicon under bending and tensile stressing,” Prof. The 17th Int. IEEE Micro Electro Mech. Syst. Conf., Maastricht, Netherlands, pp. 149-152, Jan. 2004.
[10] T. Namazu and Y. Isono, “Fatigue life prediction criterion for micro–nanoscale single-crystal silicon structures,” Journal of Microelectromechanical Systems, vol. 18, no. 1, Feb, 2009.
[11] J. N. Hung and H. Hocheng, “Frequency effects and life prediction of polysilicon microcantilever beams in bending fatigue,” J. Micro/Nanolith. MEMS MOEMS, vol. 11, no. 2, 2012.
[12] R. D. Cook, D. S. Malkus, M. E. Plesha and R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed., Wiley, 2002.

[13] J. M. Gere and B. J. Goodno, Mechanics of Materials, 8th ed., Nelson Education, 2013.
[14] O. H. Basquin, The exponential law of endurance tests, Proceedings of ASTM, vol.10, pp.625-630, 1910.
[15] ANSYS®, “ANSYS Elements Reference - 4.185 SOLID185 3-D Structural Solid,”http://www.ansys.stuba.sk/html/elem_55/chapter4/E
S4-185.htm.
[16] O. N. Pierron, C.C. Abnet and C.L. Muhlstein, “Methodology for low- and high-cycle fatigue characterization with kHz-frequency resonators,” Sensors and Actuators A: Physical, vol. 28, pp.140-150, 2006.
[17] P.-O. Baumert and O.N. Pierron, “Low-cycle fatigue testing of silicon resonators,” Proc. of SPIE, Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS and Nanodevices VII, vol. 7206 , 2009
[18] 藤田博之, 微機電系統技術入門, 全華圖書, 2004.
[19] H. K. Liu, B.J. Lee and P. P. Liu, “Low cycle fatigue of single crystal silicon thin films,” Sensors and Actuators A, vol.140, pp.257-265, 2007.
[20] K. Komai, K. Minoshima and S. Inoue, “Fracture and fatigue behavior of single crystal silicon microelements and nanoscopic AFM damage evaluation,” Microsystem Technologies, vol.5, iss.1, pp.30-37, Oct. 1998
[21] H. Kahn, R. Ballarini, R. L. Mullen and A. H. Heuer, “Electrostatically actuated failure of microfabricated polysilicon fracture mechanics specimens,” Proc. Mathematical, Physical and Engineering Sciences, vol.455, no.1990, pp.3807-3823, Oct. 1999.
[22] C. D. White, R. Xu, X. Sun, and K. Komvopoulos, “Dynamic MEMS devices for multi-axial fatigue and elastic modulus measurement,” Proc. SPIE 4980, Reliability, Testing, and Characterization of MEMS/MOEMS II, Jan. 2003
[23] ANSYS®, “ANSYS Elements Reference - 4.164 SOLID164 Explicit 3-D Structural Solid,” http://www.ansys.stuba.sk/html/elem_55/chapter
4/ES4-164.htm.
[24] ASM International,“Elements of Metallurgy and Engineering Alloys,” p.243-264, 2008, http://www.asminternational.org/documents/10192/
1849770/05224G_Chapter14.pdf