在瞬態接觸或瞬間負載作用下，潤滑壓力、油膜厚度與固體變形會在短時間內快速變化並相互耦合。傳統衝擊彈液動潤滑研究多假設等向性材料與牛頓流體，較難完整描述材料方向性、非牛頓流動及塑性變形的影響。
本研究建立橫向等向性材料於點接觸衝擊條件下之暫態彈塑性液動潤滑模型，結合修正雷諾方程式、油膜厚度方程式與球體運動方程式。材料方向性以橫向彈性模數E_{x}、縱向彈性模數E_{z}及剪切模數G_{xz}描述，並導入Hill’48降伏準則，以考量材料不同方向降伏強度對塑性區域、應力重分配及殘留變形的影響；流體則以流動指數n描述剪切稀化、牛頓與剪切增稠行為。
模擬結果發現材料方向性會影響壓力建立、油膜變化與基材應力分布，其中當縱向彈性模數E_{z}增加時，接觸區域的整體剛性提升，使中心壓力峰值明顯上升，且油膜厚度隨之增加；相較之下，E_{z}較低時，材料在衝擊時較容易產生垂直方向變形，導致壓力分布較為分散且油膜厚度降低。流動指數n會改變擠壓與回彈階段的流動阻力，其中當n<1時呈現剪切稀化行為，使流體黏度降低、流動性增加，導致壓力峰值下降且油膜較薄；當n>1時則呈現剪切增稠行為，使流體黏度提高、流動受阻，進而提升壓力峰值並增加油膜厚度，而Hill’48降伏準則所描述的方向性降伏行為，則會進一步影響塑性區域的發展、壓力分布、負載－位移關係及回彈反應。
未來可納入表面粗糙度、材料內含物、分層或塗層結構，以及熱效應與黏溫關係，以探討更貼近實際接觸條件下的壓力、油膜與塑性區域變化；亦可比較其他異向性降伏準則，並搭配油膜厚度量測、衝擊負載實驗或其他數值方法進行交叉驗證，以提升模型的可靠性與工程應用價值。

Under transient contact or impact loading, lubrication pressure, film thickness, and substrate deformation evolve rapidly and interact strongly. Conventional impact elastohydrodynamic lubrication models often assume isotropic materials and Newtonian fluids, limiting their ability to describe material anisotropy, non-Newtonian flow, and plastic deformation.
This study develops a transient elastoplastic model for a rigid ball impacting a lubricated transversely isotropic substrate. It couples the modified Reynolds, film-thickness, ball-motion, and solid-mechanics equations. Material behavior is characterized by E_{x}, E_{z}, G_{xz}, and the Hill’48 yield criterion, while the flow index n represents shear-thinning, Newtonian, and shear-thickening lubricants.
Results show that material anisotropy affects pressure development, film-thickness evolution, and substrate stress. Increasing E_{z} stiffens the contact region, raising the peak central pressure and film thickness, whereas lower E_{z} permits greater normal deformation and produces a more dispersed pressure distribution and thinner film. When n<1, shear thinning reduces the apparent viscosity, pressure peak, and film thickness; when n>1, shear thickening produces the opposite trend. Direction-dependent yielding also alters plastic-zone development, load–displacement response, and rebound behavior.
Future work may incorporate surface roughness, inclusions, layered or coated structures, and thermal effects. Comparing other anisotropic yield criteria and validating the model against experiments or alternative numerical methods could improve its reliability and engineering applicability.

[1] Hutchings, I. M., 1979, “Energy Absorbed by Elastic Waves During Plastic Impact,” J. Phys. D: Appl. Phys., 12(11), pp. 1819–1824.
[2] Thornton, C., 1997, “Coefficient of Restitution for Collinear Collisions of Elastic-Perfectly Plastic Spheres,” J. Appl. Mech., 64(2), pp. 383–386.
[3] Stronge, W. J., 2018, Impact Mechanics, 2nd ed., Cambridge University Press, Cambridge.
[4] Wu, C.-Y., Li, L.-Y., and Thornton, C., 2005, “Energy Dissipation During Normal Impact of Elastic and Elastic–Plastic Spheres,” Int. J. Impact Eng., 32(1–4), pp. 593–604.
[5] Hirai, N., Tosha, K., and Rouhaud, E., 2005, “Finite Element Analysis of Shot Peening—On the Form of a Single Dent,” Proceedings of the 9th International Conference on Shot Peening (ICSP9), pp. 82–87.
[6] 許仲勛, 2021, “衝擊負載下材料之硬化及彈塑性液動潤滑理論,” M.S. thesis, Department of Materials Science and Engineering, National Cheng Kung University, Tainan, Taiwan.
[7] Tower, B., 1883, “First Report on Friction Experiments,” Proc. Inst. Mech. Eng., 34(1), pp. 632–659.
[8] Petrov, N. P., 1883, “Friction in Machines and the Effect of the Lubricant,” Inzh. Zh. St-Peterb.
[9] Reynolds, O., 1886, “On the Theory of Lubrication and Its Application to Mr. Beauchamp Tower’s Experiments, Including an Experimental Determination of the Viscosity of Olive Oil,” Philos. Trans. R. Soc. Lond., 177, pp. 157–234.
[10] Martin, H. M., 1916, “Lubrication of Gear Teeth,” Engineering, 102, pp. 119–121.
[11] Grubin, A. N., 1949, “Fundamentals of the Hydrodynamic Theory of Lubrication of Heavily Loaded Cylindrical Surfaces,” Investigation of the Contact of Machine Components, Central Scientific Research Institute for Technology and Mechanical Engineering, Moscow, pp. 115–166.
[12] impact-PVEHL, 2026, “衝擊液動潤滑下黏彈性材料之彈/塑性變形研究,” Project No. to be determined, Aug. 1, 2026–Jul. 31, 2029.
[13] Liu, H. C., Zhang, B. B., Bader, N., Venner, C. H., and Poll, G., 2020, “Simplified Traction Prediction for Highly Loaded Rolling/Sliding EHL Contacts,” Tribol. Int., 148, p. 106335.
[14] Yang, P., and Wen, S., 1991, “Pure Squeeze Action in an Isothermal Elastohydrodynamically Lubricated Spherical Conjunction: Part 1. Theory and Dynamic Load Results,” Wear, 142(1), pp. 1–16.
[15] Larsson, R., and Höglund, E., 1994, “Elastohydrodynamic Lubrication at Pure Squeeze Motion,” Wear, 179(1–2), pp. 39–43.
[16] Cameron, A., and Gohar, R., 1966, “Theoretical and Experimental Studies of the Oil Film in Lubricated Point Contact,” Proc. R. Soc. Lond. A, 291(1427), pp. 520–536.
[17] Christensen, H., 1970, “Elastohydrodynamic Theory of Spherical Bodies in Normal Approach,” J. Lubr. Technol., 92(1), pp. 145–154.
[18] Safa, M. M. A., and Gohar, R., 1986, “Pressure Distribution Under a Ball Impacting a Thin Lubricant Layer,” J. Tribol., 108(3), pp. 372–376.
[19] Larsson, R., and Höglund, E., 1995, “Numerical Simulation of a Ball Impacting and Rebounding a Lubricated Surface,” J. Tribol., 117(1), pp. 94–102.
[20] Elkouh, A. F., Nigro, N. J., and Liou, Y. S., 1982, “Non-Newtonian Squeeze Film Between Two Plane Annuli,” J. Lubr. Technol., 104(2), pp. 275–278.
[21] Singh, C., Nailwal, T. S., and Sinha, P., 1982, “Elastohydrostatic Lubrication of Circular Plate Thrust Bearing With Power Law Lubricants,” J. Lubr. Technol., 104(2), pp. 243–247.
[22] Chu, H.-M., Li, W.-L., and Chen, M.-D., 2005, “Elastohydrodynamic Lubrication of Circular Contacts at Pure Squeeze Motion With Non-Newtonian Lubricants,” Tribol. Int., 39(9), pp. 897–905.
[23] Chu, H.-M., Chen, J.-L., Hsu, H.-C., and Li, W.-L., 2008, “Elastohydrodynamic Lubrication of Circular Contacts at Impact Loading With Generalized Newtonian Lubricants,” Tribol. Lett., 29(1), pp. 1–9.
[24] Mary, C., Philippon, D., Devaux, N., Fillot, N., Laurent, D., Bair, S., and Vergne, P., 2016, “Bridging High Pressure Rheology and Film-Forming Capacity of Polymer-Based Oil Solutions in EHL,” Tribol. Int., 93, pp. 502–510.
[25] Chu, L.-M., Cai-Wan, C.-J., and Chang, Y.-P., 2019, “Transient Elastohydrodynamic Lubrication of Artificial Hip Joints With Non-Newtonian Synovial Fluids,” J. Chin. Soc. Mech. Eng., 40(4), pp. 375–383.
[26] Su, J., Song, H., Ke, L., and Aizikovich, S. M., 2021, “The Size-Dependent Elastohydrodynamic Lubrication Contact of a Coated Half-Plane With Non-Newtonian Fluid,” Appl. Math. Mech. Engl. Ed., 42(7), pp. 915–930.
[27] Pei, X., and Yang, W., 2025, “Mixed Elastohydrodynamic Lubrication of Transversely Isotropic Coatings in Impact Motion,” Int. J. Mech. Sci., 287, Art. no. 109944.
[28] Willis, J. R., 1966, “Hertzian Contact of Anisotropic Bodies,” J. Mech. Phys. Solids, 14(3), pp. 163–176.
[29] Green, A. E., and Zerna, W., 1968, Theoretical Elasticity, 2nd ed., Oxford University Press, Oxford.
[30] Turner, J. R., 1980, “Contact on a Transversely Isotropic Half-Space, or Between Two Transversely Isotropic Bodies,” Int. J. Solids Struct., 16(5), pp. 409–419.
[31] Spilker, R. L., Donzelli, P. S., and Mow, V. C., 1992, “A Transversely Isotropic Biphasic Finite Element Model of the Meniscus,” J. Biomech., 25(9), pp. 1027–1045.
[32] Swanson, S. R., 2004, “Hertzian Contact of Orthotropic Materials,” Int. J. Solids Struct., 41(7), pp. 1945–1959.
[33] Li, L., and Hao, C.-T., 2011, “Constraints on Anisotropic Parameters in Transversely Isotropic Media and the Extensions to Orthorhombic Media,” Chin. J. Geophys., 54(6), pp. 798–809.
[34] Chen, C.-Y., Tseng, Y.-F., Chu, L.-M., and Li, W.-L., 2013, “Soft EHL for Transversely Isotropic Materials,” Tribol. Int., 67, pp. 240–253.
[35] Chu, L.-M., Chen, C.-Y., Tee, C.-K., Chen, Q.-D., and Li, W.-L., 2014, “Elastohydrodynamic Lubrication Analysis for Transversely Isotropic Coating Layer,” J. Tribol., 136(3), Art. no. 031502.
[36] Wang, Z., and Zhang, Y., 2019, “An Efficient Numerical Model of Elastohydrodynamic Lubrication for Transversely Isotropic Materials,” J. Tribol., 141(1), Art. no. 091501.
[37] Maier, E., Lengmüller, M., and Lohner, T., 2022, “Effect of Transversely Isotropic Elasticity on Elastohydrodynamic Lubrication of Point Contacts,” Polymers, 14(17), p. 3507.
[38] Nguyen, L. T. P., and Li, W.-L., 2024, “Impact Elastohydrodynamic Lubrication Analysis of Transversely Isotropic Materials in Point Contact,” J. Tribol., 146(2), Art. no. 024102.
[39] Ren, N., Zhu, D., Chen, W. W., and Wang, Q. J., 2010, “Plasto-Elastohydrodynamic Lubrication (PEHL) in Point Contacts,” J. Tribol., 132(3), Art. no. 031501.
[40] Hertelé, S., De Waele, W., and Denys, R., 2011, “A Generic Stress–Strain Model for Metallic Materials With Two-Stage Strain Hardening Behaviour,” Int. J. Non-Linear Mech., 46(3), pp. 519–531.
[41] Song, Z., and Komvopoulos, K., 2013, “Elastic–Plastic Spherical Indentation: Deformation Regimes, Evolution of Plasticity, and Hardening Effect,” Mech. Mater., 61, pp. 91–100.
[42] Wang, Z., Zong, Z., Qiao, L., and Li, W., 2018, “Elastoplastic Model for Transversely Isotropic Rocks,” Int. J. Geomech., 18(2), Art. no. 04017149.
[43] Hultqvist, T., Vrček, A., Johannesson, T., Marklund, P., and Larsson, R., 2020, “Transient Plasto-Elastohydrodynamic Lubrication Concerning Surface Features With Application to Split Roller Bearings,” Proc. Inst. Mech. Eng. J, J. Eng. Tribol., 234(12), pp. 1928–1942.
[44] Zheng, Y., Wang, C., Pu, C., Gong, J., and Meng, F., 2021, “Plasto-Elastohydrodynamic Lubrication Performance of Greased Ellipsoid Contact,” Lubr. Sci., 33(3), pp. 171–187.
[45] Bai, X., Dong, Q., Zheng, H., and Zhou, K., 2022, “Plasto-Elastohydrodynamic Lubrication of Heterogeneous Materials in Impact Motion,” Int. J. Mech. Sci., 236, Art. no. 107762.
[46] 董其融, 2023, “連續衝擊下產生加工硬化之彈塑性液動潤滑分析,” M.S. thesis, Department of Materials Science and Engineering, National Cheng Kung University, Tainan, Taiwan.
[47] Chen, H.-Y., and Li, W.-L., 2025, “On the Study of Impact Plasto-Elastohydrodynamic Lubrication of Point Contact Problems Lubricated With Non-Newtonian Lubricant,” J. Tribol., 147(3), Art. no. 034101.
[48] Tresca, H., 1869, Mémoires sur l’écoulement des corps solides, Imprimerie Impériale, Paris, France.
[49] von Mises, R., 1913, “Mechanik der festen Körper im plastisch-deformablen Zustand,” Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., 1913(4), pp. 582–592.
[50] Hill, R., 1948, “A Theory of the Yielding and Plastic Flow of Anisotropic Metals,” Proc. R. Soc. Lond. A, 193(1033), pp. 281–297.
[51] Hill, R., 1979, “Theoretical Plasticity of Textured Aggregates,” Math. Proc. Camb. Philos. Soc., 85(1), pp. 179–191.
[52] Hill, R., 1990, “Constitutive Modelling of Orthotropic Plasticity in Sheet Metals,” J. Mech. Phys. Solids, 38(3), pp. 405–417.
[53] Banabic, D., 1999, “Limit Strains in the Sheet Metals by Using the New Hill’s Yield Criterion (1993),” J. Mater. Process. Technol., 92, pp. 429–432.
[54] Caddell, R. M., Raghava, R. S., and Atkins, A. G., 1973, “A Yield Criterion for Anisotropic and Pressure Dependent Solids Such as Oriented Polymers,” J. Mater. Sci., 8(11), pp. 1641–1646.
[55] Deshpande, V. S., Fleck, N. A., and Ashby, M. F., 2001, “Effective Properties of the Octet-Truss Lattice Material,” J. Mech. Phys. Solids, 49(8), pp. 1747–1769.
[56] Roelands, C. J. A., 1966, “Correlational Aspects of the Viscosity-Temperature-Pressure Relationship of Lubricating Oils,” Ph.D. thesis, Technische Hogeschool te Delft, Delft, The Netherlands.
[57] Dowson, D., and Higginson, G. R., 1966, Elasto-Hydrodynamic Lubrication: The Fundamentals of Roller and Gear Lubrication, Pergamon Press, Oxford, pp. 88–92.
[58] Profito, F. J., Giacopini, M., Zachariadis, D. C., and Dini, D., 2015, “A General Finite Volume Method for the Solution of the Reynolds Lubrication Equation With a Mass-Conserving Cavitation Model,” Tribol. Lett., 60(1), Art. no. 18.
[59] Kaneta, M., Nishikawa, H., Mizui, M., and Guo, F., 2011, “Impact Elastohydrodynamics in Point Contacts,” Proc. Inst. Mech. Eng. J, J. Eng. Tribol., 225(1), pp. 1–12.
[60] Ding, H., Chen, W., and Zhang, L., 2006, Elasticity of Transversely Isotropic Materials, Springer, Dordrecht, The Netherlands.
[61] Pozharskii, D. A., 2018, “Contact of Transversely Isotropic Bodies in the Hertz Theory,” J. Appl. Mech. Tech. Phys., 59(3), pp. 491–497.
[62] Johnson, K. L., 1985, Contact Mechanics, Cambridge University Press, Cambridge, U.K.