本研究主要係建立有限元素模型和模擬結構鋼材在循環載重下之彈塑性行為。在循環載重下，材料會呈現循環完全彈性、循環彈性安定性、循環塑性安定性或棘輪的行為模式。本論文主要容分為三部份，首先，建立線性等向性硬化和線性運動硬化組合模型，以簡單懸臂樑受反覆載重作用為例，分析結果與ANSYS比較，以驗證有限元素分析程式之準確性。其次，採用Chaboche非線性運動硬化模型，探討金屬材料在單軸循環拉伸壓縮作用下之塑性行為。由實驗觀察發現，在非對稱應力控制條件下，材料會發生循環棘輪行為，並隨著循環載重週期累加。有限元素分析結果顯示，Chaboche模型能有效分析材料之棘輪反應，並與實驗結果相符。最後，建立彈塑性接觸有限元素模型，分析接觸表面受循環滾滑動接觸作用之彈塑性反應。本研究採用Prager模型和Chaboche模型模擬軌道鋼和CS1026碳鋼之金屬材料於循環載重週期之塑性變形。研究結果發現，Prager模型對於安定性分析能得到良好的預測結果，但無法有效模擬材料棘輪反應，而Chaboche能有效預測材料之棘輪行為。另外，由安定性極限分析結果發現，接觸應力於安定性極限範圍內，因塑性變形、殘留應力和應變硬化之影響，材料內部降伏強度會隨載重週期增加而提升，並抑制材料進一步塑性變形的發生。當接觸應力超過安定性極限，殘留應力並不影響材料塑性流動的累積，在循環載重過程，塑性應變會呈現循環棘輪反應，而棘輪應變率隨著週期數的增加遞減。

The purpose of this research is to develop a finite element model to study the elastic-plastic deformation of structural steel under cyclic loading. The finite element analysis is applied to investigate the elastic-plastic behavior of structural steel with cyclic elastic-entirely, cyclic elastic shakedown, cyclic plastic shakedown or ratcheting under cyclic loading. First, the constitutive law for linear isotropic hardening and kinematic hardening model with Von-Mises yield surface and associated flow rule is developed. Numerical solutions of a cantilever beam under reversed of loading are compared with ANSYS solutions and there is a good agreement between two solutions. Next, the constitutive model of nonlinear kinematic hardening rule proposed by Chaboche is used to investigate the ratcheting response of CS1026 carbon steel with uniaxial loading history for an unsymmetrical stress controlled system. Finally, the finite element analysis is applied to investigate the elastic-plastic response of rail steel and CS1026 carbon steel under rolling and sliding line contact. Numerical results show that the Prager’s model can give a good prediction on the shakedown limit and the Charboche model can effectively simulate the ratcheting behavior of materials under cyclic contact loading. The shakedown limits are also calculated for various combination of contact loading to assess the plastic flow that occurs when the shakedown limit is exceeded. When the contact loading exceeds the shakedown limit, the plastic strain can accumulate with increasing number of contact cycles.

目錄
誌謝 Ⅰ
中文摘要 Ⅱ
英文摘要 Ⅲ
目錄 Ⅳ
表目錄 Ⅷ
圖目錄 Ⅸ

第一章 緒論 1
1.1研究動機與目的 1
1.2文獻回顧 2
1.3論文內容 4

第二章 彈塑性有限元素分析 6
2.1緒論 6
2.2彈塑性力學理論 6
2.2.1Von-Mises降伏準則 8
2.2.2塑性流法則 9
2.2.3硬化法則 10
2.2加載卸載判定 14
2.3塑性組合律模型 15
2.4非線性有限元素分析 17
2.4.1分析流程概述 18
2.4.2有限元素分析 20
2.4.3彈性應力和應變增量運算 24
2.4.4初始降伏應力計算 24
2.4.5應力塑性修正分析 26
2.5平面組合律模型 28
2.6結果與討論 29
2.6.1有限元素模型 29
2.6.2懸臂樑平面應力問題分析 29
2.6.3懸臂樑平面應變問題分析 31
2.6.4結論 32

第三章 結構鋼材單軸循環載重棘輪效應之有限元素分析 59
3.1緒論 58
3.2金屬材料棘輪效應(ratcheting) 59
3.3棘輪組合律模型 61
3.3.1塑性力學方程組 61
3.3.2Prager模型 61
3.3.3AF模型 62
3.3.4Chaboche 模型 62
3.3.5Chaboche 修正模型 63
3.3.6Ohno -Wang 模型 64
3.3.7Jiang -Sehitoglu模型 65
3.3.8Guionnet模型 65
3.3.9Bower模型 66
3.3.10Bari-Hassan模型 66
3.3.11其他模型 67
3.4數值分析 68
3.4.1Chaboche組合律分析 68
3.4.2塑性方程式運算分析 70
3.5單軸循環拉伸和壓縮模型分析 71
3.5.1Chaboche模型材料參數分析 71
3.5.2有限元素模型 72
3.5.3邊界條件應力分析 73
3.6結果與討論 74
3.6.1單軸應力控制之棘輪效應 74
3.6.2固定平均應力對棘輪效應之影響 74
3.6.3固定應力振幅對棘輪效應之影響 75
3.6.4非等比例應力控制對棘輪效應之影響 75
3.6.5等應力應變分析 76
3.6.6結論 76

第四章 彈塑性滾滑動接觸有限元素分析 100
4.1緒論 99
4.2二維圓柱接觸理論 101
4.2.1赫茲接觸理論 101
4.2.2彈塑性接觸理論 103
4.3滾滑動接觸分析 104
4.3.1循環應力應變反應 104
4.3.2安定性(shakedown)理論 104
4.3.3安定性極限(shakedown limit)分析 105
4.4有限元素分析 108
4.4.1二維圓柱接觸模型分析 108
4.4.2安定性極限模型分析 109
4.4.3滾滑動接觸模型分析 110
4.5結果與討論 111
4.5.1二維圓柱之彈塑性接觸應力和接觸區域分析 111
4.5.2赫茲接觸應力安定性極限分析 111
4.5.3線性硬化模型滾滑動接觸分析 112
4.5.4Chaboche模型滾滑動接觸分析 113
4.5.5結論 114

第五章 綜合結論與建議 149
5.1研究結論 149
5.2研究建議 151

參考文獻 153
附錄 161

參考文獻
1.Prager, W. and Providence,R.I., “A new method of analyzing stresses and strains in work hardening plastic solid.”, J. Appl.Mech., 1956, 493-496.
2.Ziegler,H., “A modification of Prager’s hardening rule.”, Quarterly of Applied Mathematics, 17, 1959, 55-65.
3.R.Hill, “The mathematical theory of plasticity”, Oxford, 1950.
4.Prager, W., and Hodge, P.G., “Theory of perfectly plastic solids.”, John Wiley & Sons, 1951.
5.Mendelson , A., “Plasticity: Theory and application.”, Krieger, 1968.
6.Chaboche, J.L., “Mechanics of solid materials.”, Cambridge University Press, 1990.
7.Khan , A.S., and Huang, S., “ Continuum theory of plasticity”, Wiley. 1995.
8.Bairstow, L., “The elastic limits of iron and steel under cyclic variation of stress”, Philosophical Transactions of the Royal Society of London, Series A, 210, 1911, 35-55.
9.Broom, T. and Ham, R.K., “The hardening and softening of metals by cyclic stressing.”, Philosophical Transactions of the Royal Society of London, Series A, 242, 1957, 166-179.
10.Benham, P.P., “Axial load and strain cycling fatigue of copper at low endurance.”, Journal of Institutive of metals, 89, 1961, 328-338.
11.Coffin, L.F., JR., “The influence of mean stress on the mechanical hysteresis loop shift of 1100 Aluminium.”, ASME Journal of Basic Engineering, 86, 1964, 673-678.
12.Ruiz, Z., “High-strain fatigue of stainless-steel cylinders: Experimental results and their application to pressure-vessel design.”, Journal of Strain Analysis, 2(4), 1967, 290-297.
13.Hassan, T., and Kyriakieds, S., “Ratcheting in cyclic plasticity, Part I: Uniaxial behavior.”, International Journal of Plasticity, 8,1992, 91-116.
14.Hassan, T., and Kyriakieds, S., “Ratcheting of cyclically hardening and softening materials, Part I: Uniaxial behavior.”, International Journal of Plasticity, 10(2),1994, 149-184.
15.Hassan, T., and Kyriakieds, S., “Ratcheting of cyclically hardening and softening materials, Part II: Multiaxial behavior.”, International Journal of Plasticity, 10(2),1994, 185-212.
16.Hassan, T., Corona, E., and Kyriakieds, S., “Ratcheting in cyclic plasticity, Part II: Multiaxial behavior.”, International Journal of Plasticity, 8,1992, 117-146.
17.Corona, E., Hassan, T., and Kyriakides, S., “On the performance of kinematic hardening rules in predicting a class of biaxial ratcheting histories.”, International Journal of Plasticity, 12,1996, 117-145.
18.Yanyao, J. and Sehitoglu, H., “Cyclic ratcheting of 1070 steel under multiaxial stress state.”, International Journal of Plasticity, 10(5),1994, 579-608.
19.Yanyao, J. and Sehitoglu, H., “Multiaxial cyclic ratcheting under multiple step loading.”, International Journal of Plasticity, 10(8),1994, 849-870.
20.GuoZheng K., Qing G., Lixun C., and Yafang S., “Experimental study on uniaxial and nonproportionally multiaxial ratcheting of SS304 stainless steel at room and high temperatures.”, Nuclear engineering and design, 216, 2002, 13-26.
21.Scavuzzo, R.J., Lam, P.C., and Gau, J.S., “Experiment studies of ratcheting of pressurized pipe.”, Journal of Pressure Vessel Technology, 113,1991,210-218.
22.Bower, A.F., “Cyclic hardening properties of hard-drwan copper and rail steel.”, Journal of Mechanics and Physics of Solids, 54(4), 1989, 455-470.
23.Chaboche, J.L.and Nouailhas, D. “Constitutive modeling of ratcheting effects─Part I: Experimental facts and properties of the classical models.”, ASME Journal of Engineering Materials and Technology, 111, 1989, 384-392.
24.GuoZheng K, and Gao, Q., “Uniaxial and non-proportionally multiaxial ratcheting of U71Mn rail steel: experiments and simulations.”, Mechanics of Materials, 34, 2002, 809-820.
25.Mizuno, M., Mima, Y., Abdel-Karim, M., and Ohno, N., “Uniaxial ratcheting of 316FR steel at room temperature─ Part I: Experiments.”, ASME Journal of Engineering Materials and Technology, 122, 2000, 29-34.
26.GuoZheng K., Yujie, L., and Zhao, L., “Experimental study on ratcheting fatigue interaction of SS304 steel in uniaxial cyclic stressing.”, Material Science & Engineering A, 435-436, 2006, 396-404.
27.Chaboche, J.L., “Time-independent constitutive theories for cyclic plasticity.”, International of Journal Plasticity, 2(2), 1986, 149-188.
28.Chaboche, J.L., “Constitutive equations for cyclic plasticity and cyclic viscoplasticity.”, International of Journal Plasticity, 5, 1989, 247-302.
29.Nobutada O., and J.D., Wang, “Kinematic hardening rules with critical state of dynamic recovery, part I: Formulation and basic features for ratcheting behavior.”, International Journal of Plasticity, 9, 1993, 375-390.
30.Nobutada O., and J.D., Wang, “Kinematic hardening rules with critical state of dynamic recovery, part II: Application to experiments of ratcheting behavior.”, International Journal of Plasticity, 9, 1993, 391-403.
31.Yanyao, J., and Sehitoglu, H., “Modeling of cyclic ratcheting plasticity, part I: Development of constitutive relations.”, ASME Journal of Applied Mechanics, 63, 1996, 720-725.
32.Yanyao, J., and Sehitoglu, H., “Modeling of cyclic ratcheting plasticity, part II: Comparison of model simulations with experiments.”, ASME Journal of Applied Mechanics, 63, 1996, 726-733.
33.Bower, A.F., “Cyclic hardening properties of hard-drwan copper and rail steel.”, Journal of Mechanics and Physics of Solids, 54(4), 1989, 455-470.
34.Xihui, X. and Fernand E. “A constitutive model with capability to simulate complex multiaxial ratcheting behavior of material.”, International of Journal Plasticity, 13(1/2), 1997, 127-142.
35.Bari, S., and Hassan, T. “Anatomy of coupled constitutive models for ratcheting simulation”, International Journal of Plasticity, 16, 2000, 381-409.
36.Bari, S., and Hassan, T., “An advancement in cyclic plasticity modeling for multiaxial ratcheting simulation.”, international journal of plasticity, 18, 2002, 873-894.
37.GuoZheng K., Nobutada O., and Akira N., “Constitutive modeling of strain range dependent cyclic hardening.” International Journal of Plasticity, 19, 2003, 1801-1819.
38.Drucker, D. C. and Palgen, L., On stress-strain relations suitable for cyclic and other loadings, Journal of Applied Mechanics, 48, 1981, 479-485.
39.Marcal, P.V. and King, I.P., “ A stiffness method for elastic-plastic problem.”, Int. J. Mech. Sci., 7, 1965, 229-238.
40.Marcal, P.V. and King, I.P., “Elastic-plastic analysis of two-dimensional stress system by the finite element method.”, Int. J. Mech. Sci., 9, 1967, 143-155.
41.Yamada, Y. and Yoshimura, N., “Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite element method.”, International Journal of Mechanical Sciences, 10, 1967, 343-354.
42.Zienkiewicz, O.C., Valliappan, S. and King, I.P., “Elasto-plastic solutions of engineering problems initial stress, finite element approach.”, International Journal for Numerical Methods in Engineering, 1, 1969, 75-100.
43.Owen, D.R.J., Hinton, E., “Finite element in plasticity: Theory and practice.”, Prinergi Press Limited, 1980.
44.Nyssen, C., “An efficient and accurate iterative method, allowing large incremental step, to solve elasto-plastic problems.”, Computer & Structural, 13, 1981, 63-71.
45.Sloan, S.W., “Substepping schemes for the numerical integration of elastoplastic stress-strain relations.”, International Journal for Numerical Methods in Engineering, 24, 1987, 893-911.
46.Ortiz, M., Popov, E.P., “Accuracy and stability of integration algorithm for elastoplastic constitutive relations.”, International Journal for Numerical Methods in Engineering, 21, 1985, 1561-1576.
47.Chaboche, J.L., “On some modification of kinematic hardening to improve the description of ratcheting effects.”, International of Journal Plasticity, Vol. 7, pp. 661-678, 1991.
48.Yaguchi, M., and Takahashi, Y., “Ratchetting of viscoplastic material with cyclic softening, Part I: experiments on modified 9Cr-1Mo steel.”, International Journal of Plasticity, 21, 2005, 43-65.
49.Halama, R., “Ratcheting simulations of CS1026 steel using FEM.”, In: Book of extended abstracts: Session 9:"Computational and experimental analysis of strength", International Scientific Conference held on the occasion of the 55th anniversary of founding the Faculty of Mechanical Engineering VŠB-TU Ostrava, Ostrava 7.-9.9.2005. Ed. K. Frydrýšek, 2005.
50.Suh. N.P., “The delamination theory of wear.”, Wear, 25, 1973, 111-124.
51.Merwin, J.E., and Johnson, K.L., “An analysis of plastic deformation in rolling contact.”, Proceeding Institution of Mechanical Engineers, 177, 1963, 676-685.
52.Johnson, K.L., “Contact mechanics.”, Cambridge University Press, Cambridge, England, 1985.
53.Johnson, K.L., “A graphical to shakedown in rolling contact.”, Apllied Stress Analysis, Elsevier Apllied Science, 1990, 263-274.
54.Bower, A.F., and Johnson, K.L., “Plastic flow and shakedown of the rail surface in repeated wheel-rail contact.”, Wear, 144, 1991, 1-17.
55.Johnson, K.L., “Contact mechanics and the wear of metals.”, Wear, 190, 1995, 162-170.
56.Kapoor, A, and William, J.A., “Shakedown limit in sliding contact on a surface-hardened half-space.”, Wear, 172, 1994, 197-206.
57.Hill, D.A., and Ashelby, D.W., “The influence of residual stresses on contact-load-bearing capacity.”, Wear, 75, 1982, 221-240.
58.William, J.A., “The influence of repeated loading, residual stresses and shakedown on the behavior of tribology contact.”, Tribology International, 38, 2005, 786-797.
59.Hearle, A.D, Johnson, K.L., “Cumulative plastic flow in rolling and sliding contact.”, Journal of Applied Mechanic, 54, 1987, 1-7.
60.Bower, A.F., and Johnson, K.L. “The influence of strain hardening on cumulative plastic deformation in rolling and sliding line contact.”, Journal of Mechanics and Physics of Solids, 37, 1989, 471-493.
61.Yanyao, J. and Sehitoglu, H., “An analytical approach to elastic-plastic stress analysis of rolling contact.”, Journal of Tribology, 116,1994, 577-587.
62.Yanyao, J. and Sehitoglu, H., “Rolling contact stress analysis with the application of a new plasticity model.”, Wear, 191,1996, 35-44.
63.Bhargava, V., Hahn, G.T., and Rubin, C.A., “An elastic-plastic finite element model of rolling contact. Part 1, Analysis of single contacts.”, Journal of Applied Mechanics, 52, 1985,67-74.
64.Bhargava, V., Hahn, G.T., and Rubin, C.A., “An elastic-plastic finite element model of rolling contact. Part 2, Analysis of repeated contacts.”, Journal of Applied Mechanics, 52, 1985, 75-82.
65.Bhargava, V., Hahn, G.T., and Rubin, C.A., “Analysis of rolling contact with kinemtaic hardening for rail steel properties.”, Wear, 122, 1988, 267-283.
66.Hahn, G.T., Bhargava, V., Rubin, C.A., Chen, Q., and Kim, K., “Analysis of the rolling contact residual stresses and cyclic plastic deformation of SAE 52100 steel ball bearings.”, Journal of Tribology, 109, 1987, 619-626.
67.Kumar, A.M. , Rubin, C.A., and Hahn, G.T., “Analysis of the effects of rigid and elastic boundary conditions in the elastic-plastic finite element analysis of rolling contact.”, Computational Mechanics, 6, 1990, 309-314.
68.Kumar, A.M., Hahn, G.T., Bhargava, V., and Rubin, C., “Elasto-plastic finite element analyses of two-dimensional rolling and sliding contact deformation of bearing steel.”, Journal of Tribology, 111, 1989, 309-314.
69.Howell, M., Hahn, G.T., Rubin, C., and McDowell, D.L., “Finite element analysis of rolling contact for nonlinear kinematic hardening bearing steel.”, Journal of Tribology, 117, 1995, 729-736.
70.McDowell, D.L., and Moyar, G.J. “Effect of non-linear kinematic hardening on plastic deformation and residual stresses in rolling line contact.” Wear, 144, 1991, 19-37.
71.Yanyao, J., Jianjun, C., and Biqiang, X., “Elastic-plastic finite element stress analysis of two dimensional rolling contact.”, Hydraulic Failure Analysis: Fluids, Components, and Systems Effect, ASTM 1339, 2001, 59-74.
72.Ringsberg, J.W., “Cyclic ratcheting and railure of a pearlitic rail steel.”, Fatigue & fracture of engineering materials & structures, 23, 2000, 747-758.
73.Dan Van, K., and Maitournam, M.H., “Steady-state flow in classical elastoplasticity: Applications to repeated rolling and sliding contact.”, Journal of Mechanics and Physics of Solids, 41(11), 1993, 1691-1710.
74.Yu, M., Moran, B., and Keer, L.M., “A direct analysis of two-dimensional elastic-plastic rolling contact.”, Journal of Tribology, 115, 1993, 115-227.
75.Kogut, L., Etsion, I. “Elastic-plastic contact analysis of a sphere and a rigid flat.”, Journal of Applied Mechanics, 69, 2002, 657-662.
76.Robert, L. J., and Itzhak, G. “A finite element study of elasto-plastic hemispherical contact against a rigid flat.”, Journal of Tribology, 127, 2005, 343-354.
77.Vincent, B, Daniel N., Shuanbiao, L., Q. Jane, W., and Leon, M. K., “Contact analyses for bodies with frictional heating and plastic behavior.”, Journal of Tribology, 127, 2005, 355-364.
78.Axelsson, K. and Samuelsson, A., “Finite element analysis of elastic-plastic material displaying mixed hardening.”, International Journal for Numerical Methods in Engineering, 14, 1979, 211-225.
79.Zwirlein, O., and Schlicht, H., “Rolling contact fatigue mechanisms—Accelerated testing versus field performance.”, Rolling Contact Fatigue Testing of Bearing Steels, ASTM 771, J. J. C. Hoo, Ed., American Society for Testing and Materials, 1982, 358-379.
80.Tomasello, C. M., Burrier, H. I., Knepper, R. A., Balliett, S., and Maloney, J. L., “Progress in the evaluation of CSS-42L: A high performance bearing alloy.”, Bearing Steel Technology, ASTM STP 1419, Beswick, E.d., American Society for Testing and Materials International, West Conshohocken, PA, 2002, 375-385.