本論文針對奈米壓痕試驗的理論建模與應用於多層複合薄膜材料性質及介面行為做深入的分析研究，其主要內容可分為三個主題。第一主題(第二章節)為建立對於硬脆材料的奈米壓痕試驗理論機械模型，研究其在不同負/卸載率過程下的接觸行為。系統的統馭方程式分別為壓頭尖端與試件壓痕深度個別的微分方程式組合而成，並利用壓痕深度冪次的關係表示。在系統的統馭方程式裡，其壓痕深度的冪次不論是在負載或是卸載過程中，皆為壓深的函式關係。本文中應用實數型基因演算法來估測系統之彈簧與阻尼係數，進而計算出如石英(Quartz)與矽(Silicon)等硬材的接觸投影面積，其計算值與應用Oliver和Pharr [1]等人所建立接觸面積函數的結果相當接近。藉由本理論模型的研究發現，奈米壓痕試驗中材料的相位延遲效應會隨著負/卸載率增加而變大。而本理論模型可適用於奈米壓痕試驗中任何負載率的變化，無須限制條件，但其接觸投影面積會隨著負載率的增加而變小。
根據上述的建模方法，第二個主題(第三章節)為建立一個在奈米壓痕試驗下廣義描述多層複合薄膜試件(包含底材)接觸行為下的機械模型，並且成功地以本研究中奈米壓痕分析理論對於多層複合薄膜試件，估算其接觸投影面積、複合硬度及複合模數等值。而系統的統馭方程式，分別為壓頭尖端與各層不同接觸力下的試件壓痕深度方程式聯立組合而成。因此，多層複合薄膜試件的複合硬度與複合模數對於底材效應的影響，就可以本研究理論隨著不同的壓深變化作出評估。以本研究方法應於碳(上層)/非晶矽(下層)/矽(底材)複合薄膜試件，其材料過渡效應與壓痕試驗的pop-in行為就可作出有效的預測。在本論文研究裡，負載過程中壓痕試驗的pop-in行為會在相當接近碳/非晶矽複合模的總厚度深度下發生。由負載-深度的關係曲線下可發現，壓痕試驗的pop-in行為可能與複合薄膜已脫離矽基材而產生相對的彎曲變形有關。
本論文的第三主題(第四章節)為利用薄膜理論建立在奈米壓痕試驗下，複合膜內應力 、正向力 及複合厚度t*的函式關係。將以Berkovich壓頭的奈米壓痕效應等效為兩端固定的矩型平板撓度模型，並合理地計算出其平均壓應力 與發生第一模態彎曲變形理論。當複合膜之內應力 與發生彎曲變形之壓應力( )buckling的關係式相等時，可計算出複合膜在彎曲變形時的壓痕深度。而本理論模型預測的複合膜彎曲變形之壓痕深度，將與實驗結果中壓痕試驗發生的pop-in位置相當接近，即使在不同的碳膜厚度試件亦是如此。本理論模型的建立成功地顯示出預測壓痕試驗的pop-in位置的特性，以及pop-in行為主要來自於複合薄膜在壓應力下發生的彎曲變形。整合以上三個主題，本論文可應用於各類型的多層薄膜試件，不論是機械性質的檢測或是薄膜介面脫層失效的預測，可提供一個相當具有效率的分析方法。

The present studies are theoretical modeling developed and applied to evaluate the mechanical and interfacial properties for the multilayer composite film produced in nanoindentation tests. The present studies can consist of three subjects. In the first subject (in the Chapter 2), a new mechanical model is developed in the present study for hard materials to investigate the behavior arising during the loading/unloading process of a nanoindentation test. Two governing differential equations are derived for the depth solution of the indenter tip and the depth solution formed at the separation point expressed in a power form. The exponent value in either the loading process or unloading process is considered to be a variable as a function of the indentation depth in the governing differential equation. All coefficients shown in these governing differential equations associated with the spring and damping behavior are determined by the real-coded genetic algorithm. Quartz and silicon were used as the examples of hard materials, and the contact projected area predicted by the present model is quite close to the solution predicted by the area function of Oliver and Pharr [1]. The phase lagging behavior demonstrated in the nanoindentation test at two different loading/unloading rates was investigated, and it is enhanced by increasing the loading/unloading rate. No restriction for hard materials in the loading rate is needed in the nanoindentation test if the present model is employed. The contact area is decreased by increasing the loading rate.
In the second subject (in the Chapter 3) and according the above modeling, a general mechanical model employed to describe the contact behaviors and deformations arising at all layers (including the substrate), is successfully developed in the present study for multilayer specimens in order to evaluate the contact projected area by a theoretical model, and thus the hardness and reduced modulus, using nano-indentation tests. The governing differential equations for the depth solutions of the indenter tip formed at all layers of the specimen under their contact load are developed individually. The influence of the material properties of the substrate on a multilayer specimen’s hardness and reduced modulus at various indentation depths can thus be evaluated. Using the present analysis in the C (top layer)/a-Si (buffer layer)/Si(substrate) specimen, the depths corresponding to the transition and pop-in behaviors can be predicted effectively. In the present study, the indentation depth corresponding to the pop-in arising in the loading process is found to be quite close to the C/a-Si composite film thickness. This load-depth behavior gives a clue that the occurrence of pop-in is perhaps related to the buckling of the composite film which had already delaminated from the silicon substrate.
In the third subject (in the Chapter 4), the membrane theory was first applied to develop the internal compression stress arising at the nanoindentation as a function of the normal load and the composite film thickness t*. A deflection model for a rectangular plate fixed at its two end and sides was developed equivalent to the indentation behavior using a Berkovich indenter. This model is convenient for us to determine the mean compression stress and thus its value occurring the first-mode buckling. The equality of and ( )buckling allows to determine the indentation depth of having buckling in the composite film. This indentation depth of buckling predicted by the present model is quite close to the pop-in depth obtained from experimental results, regardless of the change in the C-film thickness. This characteristic reveals that the present model is developed successfully to predict the pop-in depth of a specimen; and the pop-in is created due to the buckling of the composite film under a compression stress. From the above studies in multilayer specimen, the mechanical properties and delamination of the composite film can be predicted more efficiently.

[1]Oliver, W. C. and Pharr, G. M., 1992, “An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments”, Journal of Material Research, 7(4), pp. 1564-1583.
[2]Sakai, M., 1993, “Energy Principle Of The Indentation Induced Inelastic Surface Deformation And Hardness Of Brittle Materials”, Acta Materialia, 41(6), pp. 1751-1758.
[3]Field, J.S. and Swain, M.V., 1995, “Determining the Mechanical Properties of Small Volumes of Material from Submicrometer Spherical Indentations”, Journal of Materials Research, 10(1), pp. 101-112.
[4]Malzbender, J., 2003, “Comment on Hardness Definitions”, Journal of the European Ceramic Society, 23(9), pp.1355-1359.
[5]Suresh, S. and Giannakopoulos, A. E., 1999, “Determination of Elastoplastic Properties by Sharp Indentation”, Scripta Materialia, 40(10), pp. 1191-1198.
[6]Fischer-Cripps, A. C., 2000, “A Review of Analysis Methods for Submicron Indentation Test”, Vaccum, 58(4), pp. 569-585.
[7]Lucas, B. N., Oliver, W. C. and Swindman, J. E., 1998, “The Dynamics of Frequency-Specific, Depth-sensing Indentation Testing”, MRS Symp. Proc., 522, pp. 3-14.
[8]Shimizu, S., Yanagimoto, T. and Sakai, M., 1999, “Pyramidal Indentation Load-Depth Curve of Viscoelastic Materials”, Journal of Materials Research, 14(10), pp. 4075-4086.
[9]M. Sakai , S. Shimizu , N. Miyajima , Y. Tanabe , E. Yasuda, 2001, “Viscoelastic Indentation on Iodine-Treated Coal Tar Pitch”, Carbon, 39(4), pp. 605-614.
[10]Ma, X., Yoshida, F., and Shinbata, K., 2003, “On the Loading Curve in Microindentation of Viscoplastic Solder Alloy”, Materials Science and Engineering A, 344(1-2), pp. 296-299.
[11]Oyen, M.L. and Cook, R.F., 2003, “Load-displacement behavior during sharp indentation of viscous-elastic-plastic materials,” Journal of Material Research, 18(1), pp. 139-150.
[12]Lucas, B.N., Oliver, W.C., Pharr, G.M. and Loubet, J.L., 1997, “Time dependent deformation during indentation testing,” MRS Symp. Proc., 436, pp. 233-238.
[13]Nagn, A.H.W. and Tang, B., 2002, “Viscoelastic Effects During Unloading in Depth-Sensing Indentation”, Journal of Materials Research, 17(10), pp. 2604-2610.
[14]Feng, G. and Nagn, A.H.W., 2002, “Effects of creep and thermal drift on modulus measurement using depth-sensing indentation”, Journal of Materials Research, 17(3), pp. 660-668.
[15]Bhattacharya, A. K.; Nix, W. D.; 1998, “Finite element simulation of Indentation Experiments”, International Journal of Solids Structures, 24(12), pp. 1287-1298.
[16]Kovalev, A., Shulha, H., Lemieux, M., Myshkin, N. and Tsukruk, V.V., 2004, “Nanomechanical probing of layered nanoscale polymer films with atomic force microscopy”, Journal of Materials Research, 19(3), pp. 716-728.
[17]Ovecoglu, M.L., Doerner M F and Nix, W.D., 1987, “Elastic interactions of screw dislocation in thin-films on substrate”, Acta Meteallurgica, 35(12), pp. 2947-2957.
[18]Xu, Z.H. and Rowcliffe, D., 2004, “Finite element analysis of substrate effects on indentation behaviour of thin films”, Thin Solid Films, 447, pp. 399-405.
[19]Gao, H.J., Chiu, C.H. and Lee, J., 1992, “Elastic contact versus indentation modeling of multilayered materials”, International Journal of Solids Structures, 29(20), pp. 2471-2492.
[20]Clifford, C.A. and Seah, M.P., 2006, “Modelling of nanomechanical nanoindentation measurements using an AFM or nanoindenter for compliant layers on stiffer substrates”, Nanotechnology, 17(21), pp. 5283-5292.
[21]Mencik, J., Munz, D., Quandt, E. and Weppelmann, E.R., 1997, “Determination of elastic modulus of thin layers using nanoindentation”, Journal of Materials Research, 12(9), pp. 2475-2484.
[22]Perriot, A. and Bathel, E., 2004, “Elastic contact to a coated half-space: Effective elastic modulus and real penetration”, Journal of Materials Research, 19(2), pp. 600-608.
[23]Jung, Y.G., Lawn, B.R., Martyniuk, M., Huang, H. and Hu, X.Z., 2004, “Evaluation of elastic modulus and hardness of thin films by nanoindentation”, Journal of Materials Research, 19(10), pp. 3076-3080.
[24]Schwarzer, N., Richter, F. and Hecht, G., 1999, “The elastic field in a coated half-space under Hertzian pressure distribution”, Surface and Coatings Technology, 114(2-3), pp. 292-304.
[25]Burnett, P.J and Rickerby, D.S., 1987, “The mechanical-properties of wear-resistant cotings. 1. modeling of hardness behavior”, Thin Solid Films, 148, pp. 41-50.
[26]Bull, S.J. and Rickerby, D.S., 1989, “Evaluation of coatings”, British Ceramic Transactions and Journal, 88(5), pp. 177-183.
[27]Fabes, B.D., Oliver, W.C. Mckee, R.A. and Walker, F.J., 1992, “The determination of film hardness from the composite response of film and substrate to nanometer scale indentations”, Journal of Materials Research, 7(11), pp. 3056-3064.
[28]N.G. Chechenin, J. Bottiger, J.P. Krog, 1995, “Nanoindentation of amorphous aluminum-oxidfilms. 1. The influence of the substrate on the plastic properties”, Thin Solid Films, 261, pp. 219-227.
[29]Volinsky, A.A., Moody, N.R. and Gerberich, W.W., 2002, “Interfacial toughness measurements for thin films on substrates”, Acta Materialia, 50, pp. 441-466.
[30]Suresh, S. and Giannakopoulos, A. E.; 1999, “Determination of Elastoplastic Properties by Sharp Indentation”, Scripta Materialia, 40(10), pp. 1191-1198.
[31]Taljat, B., Zacharia, T. and Kosel, F.; 1998, “New Analytical Procedure to Determine StressStrain Curve from Spherical Indentation Data”, International Journal of Solids Structures, 35(33), pp. 4411-4426.
[32]Field, J.S. and Swain, M.V., 1993, “A Simple Predictive Model for Spherical Indentation,” Journal of Materials Research, 8(2), pp. 297-306.
[33]Fischer-Cripps, A.C., 2002, Nanoindentation, Springer
[34]Page, T. F., Pharr, G. M., Hay, J. C., Oliver, W. C., Lucas, B. N., Herbert, E. and Riester, L., 1998, ”Nanoindentation Characterization of Coated Systems: P/S2 - A New Approach Using the Continuous Stiffness Technique”, MRS Symp. Proc., 522, pp. 53-64.
[35]Wei, P.J. and Lin, J.F., 2005, “A new method developed to evaluate both the hardness and elastic modulus of a coating-substrate system”, Surface and Coatings Technology, 200, pp. 2489-2496.
[36]Li, X.D. and Bhushan, B., 2003, “Fatigue studies of nanoscale structures for MEMS/NEMS applications using nanoindentation techniques”, Surface and Coatings Technology, 163-164, pp. 521-526.
[37]Marshall, D.B. and Evans, A.G., 1984, “Measurement of adherence residually stressed thin films by indentation: I. Mechanics of interface delamination”, Journal of Applied Physics, 56(10), pp. 2632-2638.
[38]Rosenfeld, L.G., Ritter, J.E., Lardner, T.J. and Lin, M.R., 1990, “Use of the microindentation technology for determining the interfacial fracture energy”, Journal of Applied Physics, 67(7), pp. 3291-3296.
[39]Zhang, Y. W. and Yang, S., 2004, “Analysis of nanoindentation creep for polymeric materials”, Journal of Applied Physics, 95(7), pp. 3655-3666.
[40]Li, J. C. M. and Chu, S. N. G., 1979, “Impression Fatigue”, Scripta Materialia; 13(11), pp. 1021-1026.
[41]Li, J. C. M., 2002, “Impression creep and other localized tests”, Materials Science and Engineering A, 322, pp. 23-42.
[42]Kaszynski, P., Ghorbel, E. and Marquis, D., 1998, “An experimental study of ratcheting during indentation of 316L stainless steel”, ASME J. of Engineering Material and Technology, 120, 218-223.
[43]Xu, B.X. and Yue, Z. F., 2006, “Study of the ratcheting by the indentation fatigue method with a flat cylindrical indenter: Part I.Experimental study”, Journal of Materials Research, 21(7), pp. 1793-1797.
[44]Cheng, Y. T. and Cheng, C. M., 1999, “Scaling Relationships in Conical Indentation of Elastic-Perfectly Plastic Solids”, International Journal of Solids and Structures, 36, pp. 1231-1243.
[45]Wei, P. J. and Lin, J. F., 2007, “A New Model Developed for Brittle and Ductile Materials to Evaluate Mechanical Properties of a Lump Specimen in the Use of Indentation Test,” ASME J. of Engineering Material and Technology, 129, pp. 284-292.
[46]Sakai, M., Shimizu, S. and Ishikawa, T., 1999, “The Indentation Load-Depth Curve of Ceramics”, Journal of Materials Research, 14(10), pp.1471-1484.
[47]Haupt, R. L., and Haupt, S. E., 1998, Practical Genetic Algorithms, Wiley Interscience publication.
[48]Man, K. F., Tang, K. S. Tang, and Kwong, S., 1996, “Genetic Algorithms: Concepts and Applications”, IEEE Transactions on Industrial Electronics, 43(5), pp. 519-534.
[49]Goldberg D. E., 1989, Genetic Algorithms in Search, Optimization and Machine Learning, Reading, MA: Addison-Wesley.
[50]Adewuya A. A., 1996, New methods in genetic search with real-valued chromosomes, Master’s thesis, Cambridge: Massachusetts Institute of Technology.
[51]Cheng, Y. T. and Zheng, Z. M.; 1998, “Further analysis of indentation loading curves: Effects of tip rounding on mechanical property measurements”, Journal of Materials Research, 13(4), pp.1059-1064.
[52]Malzbender, J., de With, G. and den Toonder, J., 2000, “The P–h2 relationship in indentation”, Journal of Materials Research, 15(5), pp.1209-1212.
[53]Lou, J., Shrotriya, P., Buchheit T., Yang, D. and Soboyejo, W.O., 2002, “Nanoindentation study of plasticity length scale effects in LIGA Ni micro- electromechanical systems structures”, Journal of Materials Research, 18(3), pp. 719-728.
[54]Li, Z. Y.; Cheng, Y. T.; Yang, H. T.; Chandrasekar, S.; 2002, “On two indentation hardness definitions”, Surface and Coatings Technology, 154, pp.124-130.
[55]Johnson, K. L.; 1985, Contact mechanics; Cambridge University Press.
[56]Han, C.F. and Lin, J.F., 2008, “A New Model Developed to Evaluate the Contact Area Arising during Nano-Indentation Tests with Pile-up Behavior of Metal Materials", IEEE Transactions on Nanotechnology, 7(3), pp. 256-265,
[57]Wei, P.J. and Lin, J.F., 2008, “Determination for Elasticity and Plasticity from Time-Dependent Nanoindentations”, Materials Science and Engineering A, 496(1-2), pp. 90-97.
[58]Wei, P.J., Shen, W.X. and Lin, J.F., 2008, “Analysis and Modeling for Time-Dependent Behavior of Polymers Exhibited in Nanoindentation Tests”, Journal of Non-Crystalline Solids, 354(33), pp. 3911-3918.
[59]Mohammed T. Attaf, 2004, “Modeling the Nanoindentation of Elastoplastic Materials With Nonlinear Adaptive Springs (NASs)”, IEEE Transactions on Nanotechnology, 3(4), pp. 451-461.
[60]Chung, C.K. and Wu, B.H., 2006, “Effect of amorphous Si layer on the reaction of carbon and silicon in the C/Si multilayer by high vacuum annealing”, Thin Solid Films, 515, pp.1985–1991.
[61]Haq, A.J., Munroe, P.R., Hoffman, M., Martin, P.J. and Bendavid, A., 2007, “Deformation behaviour of DLC coatings on (111) silicon substrates”, Thin Solid Films, 516, pp. 267-271.
[62]Hainsworth, S. V.; Chandler, H. W.; and Page, T. F.; 1996, “Analysis of nano -indentation load-displacement loading curves”, Journal of Materials Research,11(8), pp. 1987-1995.
[63]Hibbeler, R.C., 1997, Mechanics of Materials, Prentice Hall International, Inc.