臺灣博碩士論文加值系統

為利用銲點試片(Solder Joint)尋求受損傷錫球之應力應變行為，本文首先定義材料測試系統之勁度，其值可由穩態循環負荷－位移遲滯曲線量取線性卸(加)載範圍之斜率求得。在前述線性反應下，銲點材料呈線彈性反應其楊氏及剪力模數與塊狀(Bulk)材料相同，故於定循環比例位移(Φ)下可決定銲點試片(彈性)勁度。然後將材料測試系統以彈簧串聯觀念可計算出夾具勁度值，在測試過程(Φ)中設定夾具勁度不變，進而可得銲點試片真實位移，及其負荷－位移遲滯曲線。
本文利用含9顆Sn/3.5Ag/0.75Cu錫球之銲點試片於定溫下不含損傷之單剪實驗為基準找出內涵時間黏塑性理論之核心函數及內涵時間內之材料函數，其值由Φ=90°算起向Φ=0°遞增，但隨非彈性應變振幅增大而變小。依此等向性含損傷內涵時間黏塑性理論在不同循環比例位移路徑下負荷－位移遲滯曲線計算結果與實驗數據比較，兩者相當吻合。依Lee等人提出之循環損傷演化方程式，及非彈性應變振幅隨循環圈數疊加只略微增大，可推導出損傷因子與循環圈數之關係： ，其中參數A由Φ=90°算起向Φ=0°遞增，同時隨等效非彈性應變範圍增大而上升，n則與A趨勢相反。在各種循環比例角度下由D－N及負荷－位移遲滯曲線兩種數據，發現臨界損傷因子0.4<Dc<0.5，並與循環比例位移角度無關。

In order to find stress-strain-damage behavior of solder ball by using solder joint specimen, this thesis defined, at first, the stiffness of material testing system, whose value could be determined from the slope of linear range of loading or unloading of the load-displacement hysteresis loop. Within this range, the solder joints behaved linear elastic with Young’s modulus and shear modulus of bulk material. As a result, the (elastic) stiffness of solder joint specimen under proportional displacement path (Φ) could be determined. Employing the series combination of spring for material testing system, the stiffness of grips could be calculated, whose value was fixed under its corresponding proportional displacement path. Consequently, the load-displacement hysteresis loop of solder joint specimen could be constructed.
In this paper, isothermal undamage simple shear test data of solder joint specimen having 9 Sn/3.5Ag/0.75Cu balls were used as a base to determine the kernel function of endochronic viscoplasticity and the material function in the intrinsic time measure whose values were increased from Φ=90° to 0° but decreased as increased of the effective inelastic strain amplitude. Based on these, the isotropic endochronic viscoplasticity with cyclic damage under various proportional displacement path was used to compute load-displacement hysteresis loops. The computational results were in excellent agreement with data. According to Lee’s evolution equation of cyclic damage and the small increase in effective inelastic strain amplitude with increasing cyclic number, the relationship between damage factor D and the cyclic number N could be derived: , here was increased both from Φ=90° to 0° and the increase of effective inelastic strain amplitude. But the trend of was reversed. Usage of data of D vs. N curves and load-displacement hysteresis loops of various Φ, critical values of damage factor could be determined, 0.4<Dc<0.5 and its value was indepent of Φ.

摘要.....................................................Ⅰ
Abstract.................................................Ⅱ
致謝.....................................................Ⅲ
目錄.....................................................Ⅳ
表錄.....................................................Ⅶ
圖目錄...................................................Ⅷ
符號說明...............................................ⅩⅢ
第一章 緒論...............................................1
1-1 前言.................................................1
1-2 研究動機.............................................2
1-3 文獻回顧.............................................2
1-3-1 內涵時間黏塑性理論之文獻回顧......................2
1-3-2 損傷力學之文獻回顧................................5
第二章 含循環損傷內涵時間黏塑性本構方程式.................7
2-1 由熱力學架構探討本構方程式...........................7
2-2 循環損傷演化律之方程式..............................13
2-3 含循環損傷內涵時間黏塑性本構方程式之增量式..........15
2-3-1 循環穩態下增量式本構方程式.......................15
2-3-2 f0與材料參數之決定...............................16
2-3-3 拉-扭應變下增量式本構方程式......................21
第三章 Sn/3.5Ag/0.75Cu銲點試片循環比例位移路徑下之實驗及
負荷-位移修正法...................................27
3-1 實驗方法............................................27
3-2 負荷-位移修正法.....................................28
第四章 Sn/3.5Ag/0.75Cu銲點試片循環比例位移路徑下內涵時間黏
塑性理論之計算結果與實驗比較......................35
4-1 以單剪試驗為基準決定內涵時間黏塑性理論之材料參數....35
4-1-1 定溫下以位移振幅(da=10μm)為基準之核心函數........36
(Ⅰ) 材料參數α之決定..................................38
(Ⅱ) 材料參數K及ρ0之決定..............................38
(Ⅲ) 指數遞減函數近似核心函數ρ(Z)之決定...............39
4-1-2 材料函數κ之決定..................................42
4-2 不同循環比例位移路徑下增量式內涵時間黏塑性本構方程式之
計算................................................45
4-2-1 材料函數κ之決定..................................45
4-2-2 循環軸向位移(Φ=0°)之計算結果與實驗之比較.........46
4-2-3 材料函數κ(27°)、κ(45°)、κ(63°)之決定.............47
4-2-4 循環比例位移角度(Φ= )之計算結果與實驗之比較......49
第五章 Sn/3.5Ag/0.75Cu銲點試片循環比例位移路徑下含循環損傷
內涵時間黏塑性理論之計算結果與實驗比較............51
5-1 循環損傷塑性應變範圍之討論..........................51
5-2 循環損傷因子之決定..................................52
5-3 不同循環比例位移路徑下含循環損傷內涵時間黏塑性本構方程
式之增量式計算......................................55
5-3-1 含循環損傷剪向位移(Φ=90°)之計算結果與實驗之比較..55
5-3-2 含循環損傷軸向位移(Φ=0°)之計算結果與實驗之比較...57
5-3-3 含循環損傷比例位移角度Φ(Φ=45°)之計算結果與實驗之比
較...............................................58
5-4 臨界循環損傷因子Dc之決定............................60
第六章 結論與未來方向....................................62
6-1 結論................................................62
6-2 未來發展及研究方向..................................64
附表.....................................................65
附圖.....................................................72
參考文獻................................................125

[1] Valanis, K. C., “A Theory of Viscoplasticity Without a Yield Surface, Part I. General Theory”, Archives of Mechanics, pp.517-533, 1971.
[2] Valanis, K. C., ”An Energy-Probability Theory of Fracture (An endochronic theory)”, J. de Me’canique, Vol.14, pp.843-862, 1975.
[3] Valanis, K. C., “Fundamental Consequences of a New Intrinsic Time Measure: Plasticity as a Limit of The Endochronic Theory”, Archives of Mechanics, Vol.32, pp.171-191, 1980.
[4] Valanis, K. C., ”A Probabilistic Endochronic Theory of Fracture”, in Defects and Fracture, eds. G. C. Sih and H. Zorski, pp.179-198, 1982.
[5] Valanis, K.C. and Fan, J., “A Numerical Algorithm for Endochronic Plasticity and Comparison With Experiment”, Computers and Structures, Vol.19, pp.717-729, 1984.
[6] Lee, C. F., “A Systematic Method of Determining Material Functions in the Endochronic Plasticity”, J. Chinese Society of Mechanical Engineers, Vol.8, pp.419-430, 1987.
[7] Lee, C. F. and Hsiao, S. T., “Endochronic Plastic Behavior of Brass under Uniaxial-Torsional Loading History”, 13th National Conference on Theoretical and Applied Mechanics, Taichung, Taiwan, Dec.15, 1989.
[8] Lee, C. F., “Numerical Method of The Incremental Endochronic Plasticity”, The Chinese J. of Mechanics, Vol.8, No.4, pp.377-396, 1992.
[9] Lee, C. F. and Shieh, T. J., “Theory of Endochronic Cyclic Viscoplasticity of Eutectic Tin/Lead Solder Alloy”, J. of Mechanics, Vol.22, No.3, pp.181-191, 2006.
[10]Lee, C. F. and Chen, Y. C., “Thermodynamic Formulation of Endochronic Cyclic Viscoplasticity with Damage-Application to Eutectic Sn/Pb Solder Alloy”, will be Published in the J. of Mechanics, Sept., 2007.
[11]Kachanov, L. M., “Introduction to Continuum Damage Mechanics”, Kluwer Academic Publishers, 1986.
[12]Rabotnov, Y. N., “Creep Problems in Structural Members”, North Holland, 1969.
[13]Lemaitre, J., “A Course on Damage Mechanics”, Springer-Verlag, Germany, 1992.
[14]Budiansky, B. and O’Connell, R. J., “Elastic Moduli of a Cracked Solder”, International J. of Plasticity, Vol.12, pp.81-97, 1976.
[15]Stolkarts, V., Keer, L.M. and Fine, M.E., “Damage Evolution Governed by Microcrack Nucleation with Application to the Fatigue of 63Sn-37Pb Solder”, J. of Mechanics and Physics of Solid, Vol.47, pp.2451-2468, 1999.
[16]Park, T. S. and Lee, S. B., “Isothermal Low Cycle Fatigue Test of Sn/3.5Ag/0.75Cu and 63Sn/37Pb Solder Joint under Mixed-Mode Loading Cases”, Electronic Components and Technology Conference, pp.979-984, 2002.
[17]Lehman, L.P., Kinyanjui, R.K., Wang, J., Xing, Y., Zavalij, L., Borgesen, P. and Cotts, E. J., “Microstructure and Damage Evolution in Sn-Ag-Cu Solder Joints”, Electronic Components and Technology Conference, 2005.
[18]Frear, D. R., Jones, W. B. and Kinsman, K. R., “Solder Mechanics：A State of the Assessment”, TMS. Pub. Co. USA, 1991.