隨著半導體元件持續朝微米與奈米尺度發展，微奈米尺度的機械性質量測對於晶圓結構的可靠度評估至關重要，而微柱壓縮試驗已成為評估此類微結構力學行為的主流測量方法之一。然而在微觀尺度下，試片與壓板之間的摩擦力會影響材料之受力狀態，導致材料內部產生不均勻應力與應變分佈，且此摩擦作用受試片深寬比之影響，進而改變材料之應力-應變曲線，因此，需建立可同時校正摩擦與深寬比效應之方法。
本研究以有限元素法建立單軸壓縮模型，探討摩擦與深寬比效應對壓縮行為之影響。模擬結果顯示，摩擦會影響接觸區域的應力狀態，且具有沿高度衰減之特性，基於觀察到的應力分佈變化，本研究系統性地建立校正摩擦與深寬比效應之方法，並定義有效摩擦影響深度，使不同深寬比與摩擦條件下之應力-應變曲線得以有效校正並且趨勢具有一致性，進一步結合 von Mises 降伏準則與應力路徑分析，建立摩擦與表觀降伏應力之關聯性。此外，透過晶體塑性有限元素分析顯示摩擦會改變滑移啟動與塑性變形的分佈。此修正模型基於力學理論可延伸應用於微柱壓縮試驗。本研究建立壓縮試驗中深寬比效應與摩擦效應之應力校正方法，增加壓縮試驗中所獲得的材料力學特性之準確性。

This study aims to establish a correction method for friction and aspect ratio effects in micropillar compression tests. As semiconductor devices continue to scale down to the micro- and nanoscale, accurate measurement of mechanical properties has become increasingly important, and micropillar compression testing has been widely used. However, at the microscale, friction between the specimen and compression platen can induce nonuniform stress and strain distributions, with its influence depending on the specimen aspect ratio, which can lead to deviations in the measured stress–strain curve. Therefore, a finite element uniaxial compression model was developed to investigate the effects of friction and aspect ratio. Based on the observed stress decay from the contact region along the specimen height, an effective friction-affected depth was defined, and a correction method was proposed to account for both effects, enabling consistent stress–strain trends under different friction conditions. By further combining stress-path analysis with crystal plasticity finite element simulations, the relationships among friction, apparent yield stress, slip activation, and plastic deformation distribution were clarified. The proposed correction model improves the accuracy of mechanical properties obtained from micropillar compression tests.

[1] J.R. Barber and M. Ciavarella, "Contact mechanics," International Journal of Solids and Structures, vol. 37, no. 1, pp. 29-43, 2000.
[2] A. Zmitrowicz, "Contact stresses: a short survey of models and methods of computations," Archive of Applied Mechanics, vol. 80, no. 12, pp. 1407-1428, 2010.
[3] A.C. Fischer-Cripps, Elastic Contact, in Introduction to Contact Mechanics. Boston, MA: Springer US, pp. 101-114, 2007.
[4] S. Zhang, D. Li, and Y. Liu, "Friction Behavior of Rough Surfaces on the Basis of Contact Mechanics: A Review and Prospects," Micromachines, vol. 13, no. 11,
[5] P.J. Blau, "The significance and use of the friction coefficient," Tribology International, vol. 34, no. 9, pp. 585-591, 2001.
[6] E. Pennestrì, V. Rossi, P. Salvini, and P.P. Valentini, "Review and comparison of dry friction force models," Nonlinear dynamics, vol. 83, no. 4, pp. 1785-1801, 2016.
[7] K. Dean, "Computer Simulation of Stick-Slip Friction in Mechanical Dynamic Systems," Journal of Dynamic Systems, Measurement, and Control, vol. 107, no. 1, pp. 100-103, 1985.
[8] X. Tan, "Comparisons of friction models in bulk metal forming," Tribology International, vol. 35, no. 6, pp. 385-393, 2002.
[9] V. Van Geffen, "A study of friction models and friction compensation," ed: Technische Universiteit Eindhoven Eindhoven, 2009.
[10] J. Yadav, S. Kumar Kurre, and R. Deorari, "An insight into facets of different dry friction models used for simulation," Materials Today: Proceedings, vol. 69, pp. 276-281, 2022.
[11] G.T. Rooney and P. Deravi, "Coulomb friction in mechanism sliding joints," Mechanism and Machine Theory, vol. 17, no. 3, pp. 207-211, 1982.
[12] A. Anvari, "Friction Coefficient Variation with Sliding Velocity in Copper with Copper Contact," Periodica Polytechnica Mechanical Engineering, vol. 60, 2016.
[13] D. Zhu, J. Wang, and Q. Jane Wang, "On the Stribeck Curves for Lubricated Counterformal Contacts of Rough Surfaces," Journal of Tribology, vol. 137, no. 2, 2015.
[14] H. Olsson, K.J. Åström, C.C. De Wit, M. Gäfvert, and P. Lischinsky, "Friction models and friction compensation," Eur. J. Control, vol. 4, no. 3, pp. 176-195, 1998.
[15] D. Shutin and Y. Kazakov, "Theoretical and Numerical Investigation of Reduction of Viscous Friction in Circular and Non-Circular Journal Bearings Using Active Lubrication," Lubricants, vol. 11, no. 5,
[16] R. Rathee, "Numerical modeling and simulation of friction models for mechanical systems: A brief review," Materials Today: Proceedings, 2023.
[17] L. Borello and M.D. Dalla Vedova, "Dry friction discontinuous computational algorithms," International Journal of Engineering and Innovative Technology (IJEIT), vol. 3, no. 8, pp. 1-8, 2014.
[18] P.R. Dahl, "A solid friction model," The Aerospace Corporation, El Segundo, CA, Technical Report TOR-0158(3107-18)-1, 1968.
[19] C.C. De Wit, H. Olsson, K.J. Astrom, and P. Lischinsky, "A new model for control of systems with friction," IEEE Transactions on automatic control, vol. 40, no. 3, pp. 419-425, 1995.
[20] P. Dupont, B. Armstrong, and V. Hayward, "Elasto-plastic friction model: contact compliance and stiction," in Proceedings of the 2000 American control conference. ACC (IEEE cat. no. 00CH36334), 2000, vol. 2: IEEE, pp. 1072-1077,
[21] J. Swevers, F. Al-Bender, C.G. Ganseman, and T. Projogo, "An integrated friction model structure with improved presliding behavior for accurate friction compensation," IEEE Transactions on automatic control, vol. 45, no. 4, pp. 675-686, 2002.
[22] R. Ebrahimi and A. Najafizadeh, "A new method for evaluation of friction in bulk metal forming," Journal of Materials Processing Technology, vol. 152, no. 2, pp. 136-143, 2004.
[23] M. Rastegaev, "New method of homogeneous upsetting of specimens for determining the flow stress and the coefficients of inner friction," Zavodszkaja Lab, vol. 6, pp. 354-355, 1940.
[24] K. Pöhlandt and W. Nester, "Methods of determining Stress‐Strain Curves by upsetting cylindrical test‐pieces," Materialwissenschaft und Werkstofftechnik, vol. 13, no. 4, pp. 116-119, 1982.
[25] A.J. Holzer and R.H. Brown, "Specimen Geometry and Friction in Dynamic Compression," CIRP Annals, vol. 29, no. 1, pp. 135-139, 1980.
[26] E. Parteder and R. Bünten, "Determination of flow curves by means of a compression test under sticking friction conditions using an iterative finite-element procedure," Journal of Materials Processing Technology, vol. 74, no. 1, pp. 227-233, 1998.
[27] E.D.H. Davies and S.C. Hunter, "The dynamic compression testing of solids by the method of the split Hopkinson pressure bar," Journal of the Mechanics and Physics of Solids, vol. 11, no. 3, pp. 155-179, 1963.
[28] M. Pankow, C. Attard, and A. Waas, "Specimen size and shape effect in split Hopkinson pressure bar testing," The Journal of Strain Analysis for Engineering Design, vol. 44, no. 8, pp. 689-698, 2009.
[29] A. Hrennikoff, "Solution of Problems of Elasticity by the Framework Method," Journal of Applied Mechanics, vol. 8, no. 4, pp. A169-A175, 1941.
[30] R. Courant, "Variational methods for the solution of problems of equilibrium and vibrations," Lecture notes in pure and applied mathematics, pp. 1-1, 1994.
[31] M.J. Turner, R.W. Clough, H.C. Martin, and L. Topp, "Stiffness and deflection analysis of complex structures," journal of the Aeronautical Sciences, vol. 23, no. 9, pp. 805-823, 1956.
[32] W.K. Liu, S. Li, and H.S. Park, "Eighty years of the finite element method: Birth, evolution, and future," Archives of Computational Methods in Engineering, vol. 29, no. 6, pp. 4431-4453, 2022.
[33] T. Robinson, H. Ou, and C.G. Armstrong, "Study on ring compression test using physical modelling and FE simulation," Journal of Materials Processing Technology, vol. 153-154, pp. 54-59, 2004.
[34] F. Fereshteh-Saniee and F. Fatehi-Sichani, "An investigation on determination of flow curves at room temperature and under forming conditions," Journal of Materials Processing Technology, vol. 177, no. 1, pp. 478-482, 2006.
[35] S. Solhjoo and S. Khoddam, "Evaluation of barreling and friction in uniaxial compression test: A kinematic analysis," International Journal of Mechanical Sciences, vol. 156, pp. 486-493, 2019.
[36] S. Khoddam, M. Fardi, and S. Solhjoo, "A verified solution of friction factor in compression test based on its sample's shape changes," International Journal of Mechanical Sciences, vol. 193, p. 106175, 2021.
[37] J.-Y. Kim and J.R. Greer, "Tensile and compressive behavior of gold and molybdenum single crystals at the nano-scale," Acta Materialia, vol. 57, no. 17, pp. 5245-5253, 2009.
[38] J.-Y. Kim, D. Jang, and J.R. Greer, "Insight into the deformation behavior of niobium single crystals under uniaxial compression and tension at the nanoscale," Scripta Materialia, vol. 61, no. 3, pp. 300-303, 2009.
[39] A.T. Jennings, M.J. Burek, and J.R. Greer, "Microstructure versus Size: Mechanical Properties of Electroplated Single Crystalline Cu Nanopillars," Physical Review Letters, vol. 104, no. 13, p. 135503, 2010.
[40] J.-Y. Kim, D. Jang, and J.R. Greer, "Tensile and compressive behavior of tungsten, molybdenum, tantalum and niobium at the nanoscale," Acta Materialia, vol. 58, no. 7, pp. 2355-2363, 2010.
[41] A.T. Jennings and J.R. and Greer, "Tensile deformation of electroplated copper nanopillars," Philosophical Magazine, vol. 91, no. 7-9, pp. 1108-1120, 2011.
[42] M. Liu, A.K. Tieu, C. Lu, H. Zhu, and G. Deng, "A crystal plasticity study of the effect of friction on the evolution of texture and mechanical behaviour in the nano-indentation of an aluminium single crystal," Computational Materials Science, vol. 81, pp. 30-38, 2014.
[43] D. Raabe, D. Ma, and F. Roters, "Effects of initial orientation, sample geometry and friction on anisotropy and crystallographic orientation changes in single crystal microcompression deformation: A crystal plasticity finite element study," Acta Materialia, vol. 55, no. 13, pp. 4567-4583, 2007.
[44] L.Y. Si, C. Lu, N.N. Huynh, A.K. Tieu, and X.H. Liu, "Simulation of rolling behaviour of cubic oriented al single crystal with crystal plasticity FEM," Journal of Materials Processing Technology, vol. 201, no. 1, pp. 79-84, 2008.
[45] N. Khlystov, D. Lizardo, K. Matsushita, and J. Zheng, "Uniaxial tension and compression testing of materials," 2013.
[46] Y. Wang, D. Raabe, C. Klüber, and F. Roters, "Orientation dependence of nanoindentation pile-up patterns and of nanoindentation microtextures in copper single crystals," Acta materialia, vol. 52, no. 8, pp. 2229-2238, 2004.
[47] N. Zaafarani, D. Raabe, R.N. Singh, F. Roters, and S. Zaefferer, "Three-dimensional investigation of the texture and microstructure below a nanoindent in a Cu single crystal using 3D EBSD and crystal plasticity finite element simulations," Acta Materialia, vol. 54, no. 7, pp. 1863-1876, 2006.
[48] X. Tan, "Evaluation of friction in upsetting," Production Engineering, vol. 5, no. 2, pp. 141-149, 2011.
[49] B. Roebuck, J.D. Lord, M. Brooks, M.S. Loveday, C.M. Sellars, and R.W. Evans, "Measurement of flow stress in hot axisymmetric compression tests," Materials at High Temperatures, vol. 23, no. 2, pp. 59-83, 2006.
[50] G.E. Dieter and D. Bacon, Mechanical metallurgy. McGraw-hill New York, 1976.
[51] P. Christiansen, P. Martins, and N. Bay, "Friction compensation in the upsetting of cylindrical test specimens," Experimental Mechanics, vol. 56, no. 7, pp. 1271-1279, 2016.