利用相對成熟的半導體產業所發展出的材料與製程技術，矽質微結構是微機電系統(Micro-Electro-Mechanical Systems, MEMS)主要使用的材料之一。在實際使用上，微機電系統中的機械結構會不斷受到驅動或振動。如此長期往復施載作用下，可能會使材料微結構發生改變，甚至產生疲勞破壞的現象，使得微機電系統之準確度受到影響，最終發生失效。因此，對於矽質微結構受到高週波疲勞(High Cycle Fatigue)下產生破壞並失效的研究是必要的。時至今日也有許多針對這個議題的研究與實驗成果已經發表。
　　在研究過程中，為了探討疲勞現象的產生與影響，通常會採用疲勞／破壞理論搭配有限元素法(Finite Element Method, FEM)，求得受試結構之應力值及其疲勞／破壞壽命。然而受到有限元素法的特性影響，如網格的劃分、元素類型、求解方法以及負載和邊界條件設定等，上述之因素皆會影響所求得之數值。因此在有限元素模型的建立必須經過專業的檢視，此外固定一個穩定求得應力值的方法也是迫切需要的。
　　本研究會根據過去文獻中，與矽質微結構疲勞以及材料性質相關之研究，模擬文獻中之受試微結構，利用內隱式有限元素法軟體ANSYS®建立模型。利用有限元素法重建受試結構與環境，驗證實驗所得到之受力狀況與應力值。針對有應力集中結構之設計，透過模型的建立以及破裂力學為基礎的理論，尋找能相對穩定獲得應力值之方法，以便之後的研究者能以較少的數值誤差進行應力－壽命曲線(S-N Curve)的探討。
　　此外在文獻中也可以發現，以力控制模式在不同頻率下進行實驗，會對矽質微結構受疲勞破壞下的壽命產生影響。因此本研究也會使用外顯式有限元素法軟體ANSYS®/LS-DYNA，驗證在不同頻率以及負載控制模式下，受試結構所承受的應力值與變形狀況是否受到影響。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。另外，由於不同頻率施加力控制負載可能因為過衝導致壓縮的負載狀況，在本研究中也將都對前述之模型對其影響進行探討。




關鍵字：矽、微結構、高週波疲勞、有限元素法、頻率效應









 

　　Taking advantage of the materials and manufacturing technology developed by semiconductor industry, silicon-based microstructure is one of the main material of Micro-Electro-Mechanical Systems (MEMS). In practice, the mechanical structure in MEMS is actuated or bears vibration constantly. Under long term repeated loading, the microstructure of material might deform, even have fatigue fracture, which influence the accuracy of MEMS, and ended up lead to the failure. Therefore, the research of high cycle fatigue fracture and failure for silicon-based microstructure is necessary. To date, there are some published studies and experiment results on this topic.
　　In research, to investigate the reason and influence of fatigue, researchers usually use fatigue failure theory and FEM for the stress value and the fatigue failure life of the tested structure. However, because of the characteristic of FEM, such as meshing, element types, solution types, as well as loading and boundary conditions setting, these factors will all affect the solved values. Hence, the builds of finite element models should be examined, a way to get stress value steadily is also needed.
　　In this thesis, I will use implicit finite element method software ANSYS® to build model of tested microstructure in literature basing on the
research about silicon-based structure fatigue and material properties. Rebuilding the structure and environment, I will use the model to verify the loading condition and stress value. Furthermore, for design with stress concentration structure, the model and fracture mechanism are used to try to find a way that is able to derive stress value steadily, making it easier to study S-N curve with less variation.
　　In literature, we can also find that the life of structure varies as tested at different frequency under force control mode. To figure out how difference of frequencies and control modes affect the stress value and deformed shape of tested structure, the explicit finite element software ANSYS®/LS-DYNA will also be adopted in this thesis. In addition, since force-controlled loading at different frequencies might induce overshoot and cause compressive loading condition, the effect on the model will also be studied in this research.




Keywords: Silicon, Microstructure, High cycle fatigue (HCF), Finite element method, Frequency effect
 

摘要 I
Abstract III
目錄 V
表目錄 VII
圖目錄 VIII
第一章 緒論 1
1.1 研究動機 1
1.2 文獻回顧 2
1.3 研究目標 15
第二章 基礎理論 17
2.1 有限元素法理論[14] 17
2.1.1 線彈性有限元素法 17
2.1.2 有限元素暫態分析法 20
2.1.3 外顯式時間處理法 21
2.1.4 零能量模式 23
2.1.5 有限元素法等效應力推估 24
2.2 高週疲勞理論 25
2.2.1 高週疲勞之壽命預估理論 28
2.3 靜電式微致動器 30
2.3.1 梳狀致動器 31
2.4 斷裂力學 32
2.4.1 應變能釋放率[19] 33
2.4.2 虛擬裂紋與閉合技術[20] 33
第三章 研究方法 35
3.1 以斷裂力學為基礎之應力推估 35
3.1.1 Langfelder等人[4]的模型結構、尺寸與材料參數 36
3.1.2 應力收斂測試 37
3.1.3 以虛擬裂紋與閉合技術推估應力 39
3.2 應力幅度對疲勞失效的影響 43
3.2.1 應力與疲勞壽命關係 43
3.2.2 應力幅度－壽命圖 46
3.3 頻率效應 47
3.3.1 模型建立 48
3.3.2 力控制模式模擬結果 50
3.3.2 位移控制模式模擬結果 53
3.4 多晶矽微結構受高週波疲勞之壽命預估 56
第四章 研究成果 60
第五章 未來工作 62
參考文獻 63

參考文獻
[1] C. Liu, Foundations of MEMS. 2nd ed., New Jersey: Pearson Education, 2012.
[2] M. Tronconi. “MEMS and Sensors are the Key Enablers of Internet of Things”, SEMI MEMS Tech Seminar, Cornaredo, Italy, Sept. 26, 2013.
[3] W. Merlijn van Spengen. “MEMS Reliability from a Failure Mechanisms Perspective”, Microelectronics Reliability, vol. 43, no. 7, pp. 1049-1060, 2003.
[4] G. Langfelder, A. Longonia, F. Zaragaa, A. Coriglianob, A. Ghisib and A. Merassic. “A New on-Chip Test Structure for Real Time Fatigue Analysis in Polysilicon MEMS”, Microelectronics Reliability, vol. 18, no. 1, pp. 120-126, 2009.
[5] H. Kahn, R. Ballarini, and A. H. Heuer. “Dynamic Fatigue of Silicon”, Current Opinion in Solid State and Materials Science, vol. 8, no. 1, pp. 71-76, 2004.
[6] C. L. Muhlstein, S. B. Brown, and R. O. Ritchie. “High-Cycle Fatigue of Single-Crystal Silicon Thin Films”, Journal of Microelectromechanical Systems, vol. 10, no. 4, pp. 593-600, 2001.
[7] T. Namazu and Y. Isono. “Fatigue Life Prediction Criterion for Micro-Nanoscale Single-Crystal Silicon Structures”, Journal of Microelectromechanical Systems, vol. 18, no. 1, pp. 129-137, 2009.
[8] J. N. Hung and H. Hocheng. “Frequency Effects and Life Prediction of Polysilicon Microcantilever Beams in Bending Fatigue”, Journal Micro/Nanolithography MEMS and MOEMS, vol. 11, no. 2, pp. 021206-1, 2012.
[9] 林聖達, “彎曲循環負載下多晶矽微結構之高週波疲勞壽命預估及負載頻率效應研究.” 碩士論文, 國立清華大學動力機械所, 2017.
[10] C. L. Muhlstein, E. A. Stach, and R. O. Ritchie. “A Reaction-Layer Mechanism for the Delayed Failure of Micron-Scale Polycrystalline Silicon Structural Films Subjected to High-Cycle Fatigue Loading”, Acta Materialia, vol. 50, no. 14, pp. 3579-3595, 2002.
[11] C. L. Muhlstein, S. B. Brown, and R. O. Ritchie. “High-Cycle Fatigue and Durability of Polycrystalline Silicon Thin Film in Ambient Air”, Sensors and Actuators A: Physical, vol. 94, no. 3, pp. 177-188, 2001.
[12] D. H. Alsem, B. L. Boyce, E. A. Stach, and R. O. Ritchie. “Effect of Post-release Sidewall Morphology on the Fracture and Fatigue Properties of Polycrystalline Silicon Structural Films”, Sensors and Actuators A: Physical, vol. 147, no. 2, pp. 553-560, 2008.
[13] H. Kahn, R. Ballarini, and A. H. Heuer. “Electrostatically Actuated Failure of Microfabricated Polysilicon Fracture Mechanics Specimens”, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 455, no. 1990, pp. 3807-3823, 1999.
[14] R. D. Cook, D. S. Malkus, M. E. Plesha and R. J. Witt. Concepts and Applications of Finite Element Analysis. 4th ed., New York: Wiley, 2002.
[15] T. Nicholas, High cycle fatigue. 1st ed., Amsterdam: Elsevier, 2006.
[16] 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.
[17] O. H. Basquin, “The Exponential Law of Endurance Tests”, Proceedings of American Society of Testing Materials, vol. 10, pp. 625-630, 1910.
[18] 方維倫等, “微機電系統技術與應用”, 初版, 新竹市: 精密儀器發展中心, pp. 611-650, 2003.
[19] E. F. Rybicki, and M. F. Kanninen. “A Finite Element Calculation of Stress Intensity Factors by a Modified Crack Closure Integral”, Engineering Fracture Mechanics, vol. 9, no. 4, pp. 931-938, 1977.
[20] R. Krueger. “Virtual crack closure technique: history, approach, and applications”, Applied Mechanics Reviews, vol. 57, no. 2,pp. 109-143, 2004.
[21] ANSYS®, “ANSYS Elements Reference - 4.185 SOLID185 3-D Structural Solid,” http://www.ansys.stuba.sk/html/elem_55/chapter4/
ES4-185.htm.
[22] 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.