本文以含損傷內涵時間黏塑性理論整合Kanchanomai等人以Sn-3.5Ag銲錫於定溫 頻率( )及應變範圍( )下單軸循環拉伸之循環應力-應變曲線、應變壽命曲線及應力下降圖實驗數據。因銲錫試件在實驗之前即存在表面裂紋，導致一開始其循環拉應力振幅會隨循環圈數的增加而有急劇下降的現象。文中以 及 為基準決定核心函數 ， 及其應變率敏感函數 ，並於 、 及 下，計算含表面裂紋效應下初始循環之應力-應變曲線。此結果，與實驗數據相當吻合。配合應力下降圖調整 以扣除表面裂紋效應得到 modified循環應力-非彈性應變曲線 ，式中 、 及 為一定值。
依應力下降曲線圖定義臨界循環損傷因子 及其壽命 在扣除因表面裂紋效應所對應之循環圈數 後可得 及 。文中依循環損傷現象，提議損傷程度隨 上升及 下降而擴大，推導出 modified損傷乘冪公式。結合Endochronic viscoplasticity及 為定值下推導出 modified Lee-Coffin Manson( )壽命預估公式: 。根據在 、 及 下進行應力下降之實驗數據中以 及 決定乘冪指數 及 。頻率由0.001Hz~1Hz下之疲勞壽命 以 為基準利用 進行數據之整理，決定 後，以 vs. 作圖，在以 分界下，可得兩段曲線， ，斜率C=0.72； ，C=0.97。此斜率與一般 modified CM經驗公式之斜率為0.89，在 小時，大致相同，對壽命預估值也大致相同。當 越大時， 預估差值越大，但本文公式之預估值較為保守。

In this paper, the damage-coupled Endochronic viscoplasticity was used to integrate the experimental data-Cyclic Stress-Strain Curves, the Strain-Life Curves and the Load Drop Curves of uniaxial cyclic tests of Sn-3.5Ag solder alloy at 293K under frequency [0.001~1.0Hz] and strain range [0.5%~2%]. Before experiment, specimens had already have surface cracks, these lead to a sudden drop of stress amplitude within several cycles. Based on and , the kernel function was determined as ( ) and the strain rate sensitive function as . Under at and , the cyclic stress - strain curves calculated under the effect of surface cracks, were in good agreement with the experimental data. According to the Load Drop Curves without effects of surface cracks adjust and then obtain the modified cyclic stress – inelastic strain curve .
Let represent the cycles affected by the surface cracks ,the critical cyclic damage factor and initiation fatigue life defined in Load Drop Curves, were adjusted into and . From the phenomenon of cyclic damage, the damage degree was proposed to depend positively on and nonpositively on , and then derived modified damage formula. Using the Endochronic viscoplasticity and constant value of , the modified Lee-Coffin-Manson ( ) life prediction formula: can be derived. According to 、 , and under the Load Drop experimental data, the damage exponent and . Based on , the experimental data of the fatigue life( 0.001Hz ~ 1Hz) with , can be potted in vs. paper. Divided at , two straight lines can be drawn in which the slope when and when . These lines were then compared with the straight line( ) of modified Coffin-Manson emipical formula. When , both lines can predict data quite well, but when line predicts lower values of life. These conservative life prediction are more reasonable due to theoretical bases in the derivation of equation.

[1] Solomon, H. D., “Low Cycle Fatigue of Sn96 Solder with Reference to Eutectic Solder and a High Pb Solder,” Journal of Electronic Packaging, Vol. 113, pp. 102-108,1991.
[2] Valanis, K. C., “A Theory of Viscoplasticity Without a Yield Surface Part І. General Theory, ” Archives of Mechanics, pp.517-533,1971.
[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] Lee, C. F. and Shieh, T. J., “Theory of Endochronic Cyclic Viscoplasticity of Eutectic Tin/Lead Solder Alloy,” Journal of Mechanics,Vol. 22, No.3, pp.181-191, 2006.
[5] Lee, C. F. and Chen, Y. C., “Thermodynamic Formulation of Endochronic Cyclic Viscoplasticity with Damage-Application to Eutectic Sn/Pb Solder Alloy”, Journal of Mechanics, Vol. 23, No.4, pp. 433-444, 2007.
[6] Kanchanomai C., Miyashita Y., and Mutoh Y., “Low-Cycle Fatigue Behavior and Mechanisms of a Lead-Free Solders 96.5Sn/3.5Ag,” Journal of Electronic Materials, Vol. 31, pp. 142-151, 2002.
[7] Kanchanomai C., Miyashita Y. and Mutoh Y.“Influence of Frequcncy on Low Cycle Fatigue Behavior of Pb-free Solder 96.5Sn-3.5Ag,”Materials Science and Engineering, A345, pp.90-98, 2003.
[8] 曾信傑，“電子構裝用無鉛銲錫之低週疲勞行為研究”，碩士論文，國立中央大學機械工程學系，2003。
[9] Nozaki, M., Sakane, M., Tsukada Y., and Nishimura H., “Creep- Fatigue Life Evaluation for Sn-3.5Ag Solder,” ASME Journal of Engineering Materials and Technology, 128, pp. 142-150, 2006.
[10] Takahashi T., Hioki S., Shohji I. and Kamiya O.,“Low-Cycle Fatigue Life Definition for Sn-3.5Ag and Sn-0.7Cu Lead-free Solders Using Surface Deformation,” 溶接學會論文集(日本)，第24卷，第3號，pp.253-258, 2006.
[11] Kanchanomai C. and Mutoh Y.,“Fatigue Crack Initiation and Growth in Solders Alloys,” Fatigue & Fracture of Engineering Materials & Structures, Vol. 30, No. 5, pp. 443-457, 2006.
[12] Lee, C. F., “Numerical Method of The Incremental Endochronic Plasticity”, The Chinese Journal of Mechanics, Vol.8, No.4, pp. 377-396, 1992.
[13] Kanchanomai C., Miyashita Y. and Mutoh Y.,“Low-cycle Fatigue Behavior of Sn-Ag, Sn-Ag-Cu, and Sn-Ag-Cu-Bi Lead-Free Solders,”Journal of Electronic Materials, Vol. 31, No. 5, pp. 456-465, 2002.
[14] Sheng J. K., Zeng Q. L., Zhang L. and Zhu Q. S., “Mechanical Fatigue of Sn-rich Pb-free Solder Alloys,” Journal of Material Science : Mater. In Electron., Vol. 18, pp. 211-227, 2007.
[15] 李志豪，Sn/Au/Cu循環熱-力行為及疲勞初始壽命預估-含損傷內涵時間黏塑性理論之應用，碩士論文-國立成功大學工程科學系，2007。
[16] 黃達雄，Sn/3.5Ag銲錫試件其疲勞初始壽命之預估-含損傷內涵時間黏塑性理論之應用，碩士論文-國立成功大學工程科學系，2009。
[17] Kanchanomai C. and Mutoh Y.,“Low-Cycle Fatigue Prediction Model for Pb-Free Solder 96.5Sn-3.5Ag,”Journal of Electronic Materials, Vol. 31, No.4, pp. 456-465, 2004.
[18] Kanchanomai C., Miyashita Y. and Mutoh Y.“Low Cycle Fatigue and Fatigue Crack Growth Behavior of Sn-Ag Eutectic Solder,”Soldering and Surface Mount Technology, Vol. 14, No. 3, pp.30-36, 2002.