本文內涵時間黏塑性理論核心函數 及應變率敏感函數 ，依Wei等人對Sn/3.9Ag/0.6Cu銲錫塊材於溫度範圍 間不同應變率之循環穩態應力-應變遲滯曲線實驗數據而建立；然後再進行銲錫於定應變率 及熱循環下之力學行為的計算，在In-phase條件下，循環應力-應變計算結果與實驗數據相當吻合，Out-phase之計算結果，也與Wei等人之Damage-Coupled Constitutive Model之結果大致上趨勢一致。
本文對循環損傷因子以循環最大應力的下降率來定義，結合定應變振幅之實驗數據，及Percolation理論來決定臨界循環損傷因子 。以含損傷內涵時間黏塑性理論，模擬Sn/3.8Ag/0.7Cu銲錫於定應變振幅0.8%之循環損傷之應力-應變滯遲曲線，與實驗比較相當貼合。並利用Lee and Chen所提議之內涵損傷演化方程式，配合Zeng等人對Sn/3.8Ag/0.7Cu銲錫材料於298 下循環應力-應變曲線，推導出單軸拉伸下循環損傷與非彈性應變範圍之關係式。在臨界循環損傷因子0.3時，得到Coffin-Mason修正式，有效的預測該銲錫之疲勞初始壽命。

In this paper, the kernel function and the strain rate sensitive function in the Endochronic viscoplasticity were established by using Sn/3.9Ag/0.6Cu experimental cyclically steady hysteresis loops of Wei et. al., in the temperature range (298K to 373K) and strain rateThen, the mechanical behavior of solder under thermal cycle(298K to 373K) and strain rate was computed. The results and the experimental data were in very good agreement under In-phase condition, and had almost the same trend with that of the Damage-Coupled Constitutive Model of Wei et. al. under Out-phase condition.
To define the cyclic damage factor, the reducing rate of maximum cyclic stress was used. The critical cyclic damage could be found by combining experimental vs. data and the Percolation theory. The Endochronic viscoplasticity with damage was used to simulate Sn/3.8Ag/0.7Cu cyclic stress-strain hysteresis loops with damage under strain amplitude 0.8% provided by Zeng et. al. in temperature 298K. The results were in very good agreement with data. Employed the evolution equation of intrinsic damage proposed by Lee and Chen and the computed cyclic stress-inelastic strain relation, modified Coffin-Manson relationship was derived and used to predict the data of fatigue initiation life very effectively.

[1] Kachanov, L. M., “Introduction to Continuum Damage Mechanics,” Kluwer Academic Publishers, 1986.
[2] Rabotnov, Y. N., “Creep Problems in Structural Members,” North Holland, 1969.
[3] Lemaitre, J., “A Course on Damage Mechanics,” Springer-Verlag, Germany, 1992.
[4] Budiansky, B. and O’Connell, R. J., “Elastic Moduli of a Cracked Solid ,” Int. J. Solids Struct., Vol. 12, pp.81-97, 1976.
[5] Stolkarts, V., Keer, L. M., Fine, and M. E., “Damage Evolution Governed by Microcrack Nucleation with Application to the Fatigue of 63Sn-37Pb Solder,” J. Mech. Phys. Solids Packaging, Vol. 47, pp.2451-2468, 1999.
[6] Wei, Y., Lau, K. J., Vianco, P., and Fang, H. E., “Behavior of Lead-Free Solder under Thermomechanical Loading,” ASME J. Electronic Packaging., Vol. 126, pp. 367-373, 2004.
[7] Zeng, Q., Wang, Z., Xian, A., and Shang, J., “Low Cycle Fatigue Behavior of Sn-3.8Ag-0.7Cu Lead-Free Solder,” Chinese J. Material Research., Vol. 18, pp. 11-17, 2004.
[8] Shang, J. K., Zeng, Q. L., Zhang, L., and Zhu, Q. S., “Mechanical Fatigue of Sn-rich Pb-free Solder Alloys,” J. of Mater. Sci : Mater. in Electron., Vol. 18(1-3), pp. 211-227, 2007.
[9] Valanis, K. C., “A Theory of Viscoplasticity Without a Yield Surface Part І. General Theory,” Archives of Mechanics, pp.517-533, 1971.
[10] 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.
[11] Valanis, K.C. and Lee, C.F., “Endochronic Theory of Cyclic Plasticity With Applications,” ASME J. of Applied Mechanics, Vol. 51, pp. 367-374, 1984.
[12] Lee, C. F., “A Systematic Method of Determining Material Functions in the Endochronic Plasticity,” J. the Chinese Society of Mechanical Engineers, Vol. 8, pp. 419-430, 1987.
[13] 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.
[14] Lee, C. F., “Recent Finite Element Applications of the Incremental Endochronic Plasticity,” International Journal of Plasticity, Vol. 11, No.7, pp.843-865, 1995.
[15] Lee, C. F. and Shieh, T. J., “Theory of Endochronic Cyclic Viscoplasticity of Eutectic Tin/Lead Solder Alloy,” J. of Mech.,Vol. 22, No.3, pp.181-191, 2006.
[16] 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 Mech., Sept., 2007.
[17] Qian, Z., Abhijit, D., and Peter, H., “Viscoplastic Constitutive Properties and Energy-Partitioning Model of Lead-Free Sn/3.9Ag/0.6Cu,” Electronic Components and Technology Conference, pp.1862-1868, 2003.