玻纖基板在封裝製程中扮演著連結電路板和晶片的重要角色，由於熱膨脹係數與矽晶之不匹配，進而造成IC元件的翹曲，可能會導致線路斷裂和元件失效，傳統上分析均假設基板為彈性或線黏彈性，然而當基板內部應力過大而開始產生非線性黏彈性行為時，分析上就會開始產生極大的誤差，為了增加IC產品之可靠度，建構非線性黏彈模型來分析玻纖基板是必然的趨勢。本文的重心著重注於模擬玻纖基板之非線性潛變與潛變恢復行為。
本文建構兩組玻纖基板黏彈性本構模型，此兩組模型為考慮高分子材料的非線性黏彈（與時間、溫度及應力相關）之本構模型；分別為時間-溫度-應力重疊法(time-temperature-stress-superposition; TTSSP)非線性黏彈模型和Schapery非線性黏彈本構模型。文中藉由潛變實驗量測玻纖基板之非線性黏彈行為，再利用潛變與潛變恢復實驗結果作曲線擬合求取模型參數，然後運用這兩組非線性黏彈模型模擬玻纖基板在循環負載應力下應變行為，最後與實驗結果做比對。從比對結果發現，TTSSP與Schapery非線黏彈本構模型在應力負載為30 MPa時模擬玻纖基板之非線性黏彈行為均有不錯的表現，雖然當應力達到60 MPa時，兩組模型均會開始產生些微的偏差，但是綜觀整體趨勢，本文所建構之兩組非線性黏彈模型，在描述玻纖基板之機械行為仍然有不錯的表現。

Organic substrate is a key constituent for connecting printed circuit board and silicon die in electronic packages. Package warpage typically occurs because of the thermal expansion mismatch among various packing constituents such as silicon die, molding compound and multilayer substrate. When package warpage is too large, it may lead to difficulties of electrical interconnection. It is typically assumed that substrate is either elastic or linear viscoelastic. However, the substrate may exhibit nonlinear viscoelastic behavior when subjected to higher stress loading. In order to improve the accuracy of warpage analysis, it is important to establish a nonlinear viscoelastic constitutive model of the substrate. This study focuses on modeling the nonlinear creep and creep recovery behaviors of an organic package substrate.
In this research, two nonlinear viscoelastic constitutive models were developed for describing the nonlinear viscoelastic behavior of the organic substrate: a time- temperature-stress-superposition model and the Schapery model. Creep and creep recovery experiments were first conducted to examine the nonlinear viscoelastic behavior of the substrate. The nonlinear parameters were then obtained by curve-fitting the creep and creep recovery experimental results. For evaluating the accuracy of these two models, numerical analyses were performed to simulate substrate response under cyclic stress condition, and compared to experimental results. It was observed that these two constitutive models exhibit good agreements with experimental results at 30 MPa-condition, but slightly over-predict strain response at 60 MPa.

[1]TSSOP Data Sheet, Amkor Inc., Korea, 2001.
[2]PBGA Data Sheet, Amkor Inc., Korea, 2011.
[3]fcCSP Data Sheet, Amkor Inc., Korea, 2010.
[4]R. B. R. van Silfhout, J. G. J. Beijer, K. Zhang and W. D. van Driel, “Modelling methodology for linear elastic compound modelling versus visco-elastic compound modelling,” Proceedings of the 6th International Conference. Thermal, Mechanical and Multi-Physics Simulation and Experiments in Micro-Electronics and Micro-Systems. EuroSimE2005, Berlin, Germany, pp. 483-489, 2005.
[5]R. A. Schapery, “On the characterization of nonlinear viscoelastic materials,” Polymer Engineering and Science, vol. 9, pp. 295–310, 1969.
[6]Y. C. Lou and R. A. Schapery, “Viscoelastic characterization of a nonlinear fiber-reinforced plastic,” Journal of Composite Materials, vol. 5, pp. 208-234, 1971.
[7]M. E. Tuttle and H. F. Brinson, “Prediction of the long-term creep compliance of general composite laminates, ” Experimental Mechanics, vol. 26, pp. 89-102, 1986.
[8]D. A. Dillard, M. R. Straight and H. F. Brinson, “The nonlinear viscoelastic characterization of graphite/epoxy composites,” Polymer Engineering and Science, vol. 27, pp. 116–123, 1987.
[9]S. Y. Zhang and X. Y. Xiang, “Creep characterization of a fiber reinforced plastic material,” Journal of Reinforced Plastics and Composites, vol. 11, pp. 1187-1194, 1992.
[10]G. C. Papanicolaoua, S. P. Zaoutsosa and A. H. Cardonb, “Prediction of the non-linear viscoelastic response of unidirectional fiber composites,” Composites Science and Technology, vol. 59, pp. 1311–1319, 1999.
[11]P. P. Provenzano, R. S. Lakes, D. T. Corr and R. Vanderby, “Application of nonlinear viscoelastic models to describe ligament behavior,” Biomechanics and Modeling in Mechanobiology, vol. 1, pp. 45-57, 2002.
[12]R. A. Schapery, “Nonlinear viscoelastic and viscoplastic constitutive equations based on thermodynamics,” Mechanics of Time-Dependent Materials, vol. 1, pp. 209-240, 1997.
[13]M. Megnis and J. Varna, “Nonlinear viscoelastic, viscoplastic characterization of unidirectional GF/EP composite,” Mechanics of Time-Dependent Materials, vol. 7, pp. 269-290, 2003.
[14]L. Nordin and J.Varna, “Nonlinear viscoelastic behavior of paper fiber composites,” Composites Science and Technology, vol. 65, pp. 1609–1625, 2005.
[15]J. Lai and A. Bakker, “3-D schapery representation for non-linear viscoelasticity and finite element implementation,” Computational mechanics, vol. 18, pp. 182-191, 1996.
[16]W. G. Knauss and I. J. Emri, “Non-linear viscoelasticity based on free volume consideration,” Computers and Structures, vol. 13, pp. 123–128, 1981.
[17]C. F. Popelar and K. M. Liechti, “A Distortion-Modified Free Volume Theory for Nonlinear Viscoelastic Behavior,” Mechanics of Time-Dependent Materials, vol. 7, pp. 89-141, 2003.
[18]R. Zhao, W. Luo, C. Wang and X. Tang, “Application of time-temperature-dtress superposition principle to nonlinear creep of poly(methyl methacrylate),” Key Engineering Materials, vol. 353-358, pp. 1386-1389, 2007.
[19]R. C. Dunne, An Integrated Process Modeling Methodology and Module for Sequential Multilayered High-Density Substrate Fabrication for Microelectronic Packages, Ph. D. Dissertation, Georgia Technology Institute, 2000.
[20]J. D. Ferry, Viscoealstic Properties of Polymer, 3rd edition, John Wiley and Sons Inc., New York, 1980.
[21]H. F. Brinson and L. C. Brinson, Polymer Engineering Science and Viscoelasticity An Introduction.
[22]W. Luo, T. Q. Yang and Q. An, “Time-temperature-stress equivalence and its application to nonlinear viscoelastic materials,” Acta Mechanica Solida Sinica, vol. 14, pp. 195-199, 2001.
[23]R. A. Schapery, “A theory of non-linear thermoviscoelasticity based on irreversible thermodynamics,” Proceedings of 5th U.S. National Congress of applied mechanics ASME, pp. 511-530, 1966.
[24]MATLAB, R2011a, MathWorks, Inc., 2011.
[25]D. R. Jones, C. D. Perttunen and B. E. Stuckman, “Lipschitzian optimization without the Lipschitz constant,” Journal of Optimization Theory, vol. 79, pp. 157-181, 1993.
[26]Z. Xia, X. Shen and F. Ellyin, “Cyclic deformation behavior of an epoxy polymer. Part I: experimental investigation,” Polymer Engineering and Science, vol. 44, pp. 2240–2246, 2004.
[27]Z. Xia, X. Shen and F. Ellyin, “Cyclic deformation behavior of an epoxy polymer. Part II: Predictions of viscoelastic constitutive models,” Polymer Engineering and Science, vol. 45, pp. 103–113, 2005.
[28]ASTM D3039/D3039M – 08, “Standard test method for tensile properties of polymer matrix composite materials1,” West Conshohocken, PA, 2008.
[29]Maple 16, Waterloo, ON, Canada: Maplesoft, 2012.