Formulation of constitutive relations based on indentation test

FORMULATION OF CONSTITUTIVE RELATIONS BASED ON INDENTATION TESTS THO KEE KIAT (B. Eng. (Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 This work is dedicated to my parents, brother and sisters. i ACKNOWLEDGEMENTS The author wishes to express his sincere gratitude to his supervisor, Professor Somsak Swaddiwudhipong, for his help, guidance, encouragement and understanding throughout the course of this study. The author deeply appreciates his generosity with time for consultation despite his extremely busy schedule as the deputy head of department. The author is extremely grateful to the National University of Singapore for the financial support in the form of NUS Research Scholarship and President’s Graduate Fellowship. In addition, the author would like to thank Dr Hua Jun who is always available for advices and meaningful discussions. The author would also like to thank Mr Brandon Oei Nick Sern for sharing his views and advices on artificial neural network and support vector machine. The author is indebted to Associate Professor Lim Chwee Teck for kindly allowing the use of nanoindentation equipments at the Nano Biomechanics Lab, Assistant Professor Zeng Kaiyang for facilitating the sample preparation and nanoindentation experiments at IMRE, Mr Hairul Nizam Bin Ramli at Nano Biomechanics Lab and Ms Shen Lu at IMRE for their assistances in the nanoindentation experiments and the staff members of SVU and eITU for providing technical assistances on computing facilities. Last but not least, the author would like to thank his beloved parents, brother, sisters and girlfriend, Rwe Yun, for their constant love, care, support and encouragement. ii Table of Contents TABLE OF CONTENTS ACKNOWLEDGEMENTS ii SUMMARY . vi LIST OF TABLES viii LIST OF FIGURES . x NOMENCLATURE .xiv CHAPTER 1: INTRODUCTION . 1.1 Background 1.2 Literature Review 1.3 Objectives and Scope of Study 14 1.4 Organisation of Report 15 CHAPTER 2: FINITE ELEMENT ANALYSIS . 18 2.1 Overview 18 2.2 Conical Indenters 19 2.2.1 Boundary Conditions 20 2.2.2 Contact Interface 20 2.2.3 Far-Field Effects 21 2.2.4 Finite Element Mesh and Convergence Studies 22 2.3 Three-Sided Pyramidal Indenters 23 2.3.1 Boundary Conditions 24 2.3.2 Contact Interface 24 2.3.3 Far-Field Effects 25 2.3.4 Finite Element Mesh and Convergence Studies 26 2.4 Comparison between Finite Element Results and Experimental Data 27 CHAPTER 3: FUNDAMENTAL ASPECTS OF LOADDISPLACEMENT CURVES . 47 3.1 Overview 47 3.2 Material Model 47 3.3 Indentation Load-Displacement Curves 48 3.3.1 Loading Curve 49 3.3.2 Unloading Curve 53 iii Table of Contents 3.3.3 Relationship between Indentation Work and Total Work Done 54 3.3.4 Surfaces Describing Functions f1, f2 and f3 56 3.4 Forward and Reverse Analysis Algorithms 57 3.4.1 Forward Analysis Scheme 57 3.4.2 Reverse Analysis Scheme 58 CHAPTER 4: UNIQUENESS OF REVERSE ANALYSIS BASED ON SINGLE INDENTER . 68 4.1 Overview 68 4.2 Reverse Analysis Based on Single Indenter 68 4.3 Proof on Non-Uniqueness 74 4.3.1 Analytical Derivation 74 4.3.2 Computational Verification 78 4.3.3 Effect of hr on Sensitivity of Parameters a and b 79 4.3.4 Discussion on Uniqueness 79 CHAPTER 5: REVERSE ANALYSIS BASED ON DUAL INDENTERS 86 5.1 Overview 86 5.2 Generalized Dimensionless Functions 86 5.3 Reverse Analysis Algorithms 87 5.4 Uniqueness of Results from Reverse Analysis 88 5.5 Artificial Neural Network Model 92 5.5.1 Overview 92 5.5.2 Model Definition 93 5.5.3 Results and Discussion 96 5.6 Least Squares Support Vector Machine 98 5.6.1 Overview 98 5.6.2 Model Definition 99 5.6.3 Results and Discussion 101 CHAPTER 6: CONVENTIONAL MECHANISM-BASED STRAIN GRADIENT PLASTICITY MODEL 106 6.1 Overview 106 6.2 Review of Strain Gradient-Dependent Plasticity 107 6.2.1 Taylor Hardening Model 107 iv Table of Contents 6.2.2 Effective Plastic Strain Rate ε p 111 6.2.3 Constitutive Relation 112 6.3 C0 CMSG Solid Elements 115 6.3.1 Derivation 6.4 C0 CMSG Axisymmetric Elements 6.4.1 Derivation 115 119 119 6.5 C CMSG Plane Strain/Stress Elements 126 6.6 Numerical Examples on C0 CMSG Finite Elements 126 6.6.1 Bar under Body Force and Traction at Free End 126 6.6.2 Nanoindentation 128 6.6.3 Stress Distribution Near Crack Tip 131 CHAPTER 7: NANOINDENTATION EXPERIMENTS 139 7.1 Overview 139 7.2 Sample Preparation 139 7.3 Nanoindentation Experiments 140 7.4 Numerical Analysis of Nanoindentation Results 140 CHAPTER 8: CONCLUSION 154 8.1 Concluding Remarks 154 8.2 Further Research 157 REFERENCES 158 LIST OF PUBLICATIONS 166 APPENDIX A: Derivation of C0 CMSG Plane Strain/Stress Elements . 169 v Summary SUMMARY Extensive large strain large deformation finite element analyses are performed to investigate the response of elasto-plastic materials obeying power law strain-hardening during the loading and unloading process of instrumented indentation with conical and pyramidal indenters of different apex angles. The functional forms of the relationships between the characteristics of the load-indentation curve and the material properties are examined. Two simple algorithms are proposed for forward and reverse analyses based on a single indenter. By considering the load-displacement curve of Al6061, it is demonstrated that a one-to-one relationship between the elasto-plastic material properties and the load-displacement curve does not always exist and the material properties obtained from the load-displacement curve of a single indenter is nonunique. The curvature of the loading curve, the initial slope of the unloading curve and the ratio of the residual depth to maximum indentation depth are three main quantities that can be established from an indentation load-displacement curve. A relationship between these three quantities is analytically derived for four indenters with different geometries. The existence of an intrinsic relationship between these three quantities implies that only two independent quantities are obtainable from a single loaddisplacement curve and these are insufficient to uniquely solve for the three unknown material properties. A reverse analysis algorithm based on load-displacement curves obtained from dual indenters is presented. It is demonstrated that the proposed reverse analysis algorithm can uniquely recover the elasto-plastic material properties from the load-displacement curves of two indenters with different geometries. vi Summary Artificial neural network (ANN) and least squares support vector machines (LS-SVM) are robust and efficient tools to perform multi-dimensional surface regression. They enable the direct mapping of the characteristics of the load-displacement curves to the elasto-plastic material properties. Direct mapping via ANN and LS-SVM alleviate the need to adopt an iterative procedure in the reverse analysis. The proposed ANN and LS-SVM models can predict accurately the material properties when presented with new sets of load-indentation curves which are not used in the training and verification of the model. A series of C0 solid, axisymmetric and plane strain/stress finite elements for materials with strain gradient effects is established. The formulation is based on conventional mechanism-based strain gradient plasticity (CMSG) theory. The model is implemented in ABAQUS. These elements are adopted to study the plastic strain distribution in a bar subject to uni-axial tension and body force, the indentation size effect and the state of stress in the vicinity of the crack tip. Comparison with other analytical solutions and test results, besides showing good agreement also reflects the necessity of incorporating the effects of strain gradient plasticity when the material and characteristic length scales of non-uniform plastic deformation are of the same order at micron level. Nanoindentation experiments with indentation depths varying from 400nm to 3400nm are performed on Al7075 and copper. In the presence of indentation size effect, the strength of the material increases with decreasing indentation depth. The proposed C0 solid elements incorporating the CMSG plasticity theory is able to simulate this phenomenon rather accurately. vii List of Tables LIST OF TABLES Table 2.1: Characteristics of pyramidal indenters 23 Table 2.2: Different domain sizes for B-123 and B-142 indenters . 26 Table 3.1: Characteristics of load-displacement curves for 15 different material properties when indented using C-600 indenter 51 Table 3.2: Characteristics of load-displacement curves for 15 different material properties when indented using C-703 indenter 51 Table 3.3: Characteristics of load-displacement curves for 15 different material properties when indented using B-123 indenter 52 Table 3.4: Characteristics of load-displacement curves for 15 different material properties when indented using B-142 indenter 52 Table 3.5: Relationships between hr/hmax and WR/WT for B-123, B-142, C-600 and C703 indenters 56 Table 4.1: Typical combinations of material properties resulting in similar indentation curves . 70 Table 4.2: The quantities C, S, WR for different combinations of material properties 70 WT Table 4.3: Prediction of forward analysis algorithm based on material combinations derived by Capehard and Cheng (2003) 71 Table 4.4: Sensitivity of parameters a and b due to variation of hr 79 Table 5.1: Forward analysis on Al6061 89 Table 5.2: Forward analyses results for different combination of material properties. 90 Table 5.3: Summary of finite element results for Al7075, steel, iron and zinc 91 Table 5.4: Summary of reverse analysis results 91 Table 5.5: Characteristics of ANN-1 and ANN-2 96 Table 5.6: Summary of finite element results for Al6061, Al7075, steel and iron . 97 Table 5.7: Prediction from artificial neural network model . 98 Table 5.8: Characteristics of LS-SVM 101 viii List of Tables Table 5.9: Prediction from proposed LS-SVM model 102 Table 6.1: Material and loading data 127 ix Chapter 8: Conclusion A series of C0 solid, axisymmetric and plane strain/stress elements for materials with strain gradient effects has been established. The formulation is based on conventional mechanism-based strain gradient plasticity incorporating Taylor dislocation model through intrinsic material length scale. As only the constitutive condition is affected, higher order stress and hence higher order continuity requirements and/or additional nodal parameters with mixed formulation are no longer necessary. The model has been implemented in ABAQUS, a finite element package. These elements have been adopted to study the plastic strain distribution in a bar subject to uni-axial tension and body force, the load-displacement relationship of Berkovich indentation on electropolished nickel and the state of stress in the vicinity of the crack tip. Comparison with other analytical solutions and test results, besides showing good agreement also reflects the necessity of incorporating the effects of strain gradient plasticity in the formulation when the material and characteristic length scales of non-uniform plastic deformation are of the same order at micron level. A series of nanoindentation experiments with maximum indentation depths varying from 400 nm to 3400 nm are carried out on Al7075 and copper. In the presence of indentation size effect, the strength of the material increases with decreasing indentation depth. The curvature of the loading curve, which is a representation of the mean contact pressure, also increases with decreasing indentation depth. The proposed C0 solid elements incorporating the CMSG plasticity theory is able to simulate this phenomenon rather accurately. 156 Chapter 8: Conclusion 8.2 Further Research The proposed reverse analysis algorithm, ANN and LS-SVM models can be further developed to include the results from multiple indenters. The accuracy of the predicted elasto-plastic material properties is expected to increase as the characteristics of the load-displacement curves from more indenters are included in the analysis. The methods proposed in the present research focus on material characterization of bulk materials. Further research efforts should be directed at studying the feasibility of extending the proposed methods to cover material characterization of thin films on substrate. In the present study, the identification of the internal material length scale parameter, α involves a rather tedious iterative process. Further research could be carried out to explore the feasibility of building ANN or LS-SVM models that will automatically identify the value of α based on the indentation response at various indentation depths. 157 References REFERENCES 1. ABAQUS/Standard User’s Manual Version 6.3. Hibbit, Karlsson and Sorensen Inc., 2002. 2. Alkorta, J., Martinez-Esnaola, J.M., Gil Sevillano, J. Absense of one-to-one correspondence between elastoplastic properties and sharp-indentation loadpenetration data, J. Mater. Res., 20, pp.432-437. 2005. 3. Arsenlis, A., Parks, D.M. Cystallographic aspects of geometrically-necessary and statistically-stored dislocation density, Acta Mater., 47, pp.1597-1611. 1999. 4. Ashby, M.F. The deformation of plastically non-homogeneous alloys. Phil. Mag., 21, pp.399-424. 1970. 5. Bhattacharya, A.K., Nix,W.D. Finite element simulations of indentation experiments. Int. J. Solids Struct., 24, pp.881-891. 1988. 6. Bishop, J.F.W., Hill, R. A theory of plastic distortion of a polycrystalline aggregate under combined stresses. Phil. Mag., 42, pp.414-427. 1951. 7. Bucaille, J.L., Stauss, S., Felder, E. and Michler, J. Determination of Plastic Properties of Metals by Instrumented Indentation Using Different Sharp Indenters, Acta Mater., 51, pp.1663-1678. 2003. 8. Capehart, T.W. and Cheng, Y.T. Determining Constitutive Models from Conical Indentation: Sensitivity Analysis, J. Mater. Res., 18, pp.827-832. 2003. 9. Chen, S. H., Tao, C.J., Wang, T.C. A study of size-dependent microindentation, Acta Mech., 167, pp.57-72. 2004. 158 References 10. Cheng, Y.T. and Cheng C.M. Scaling Approach to Conical Indentation in Elastic-Plastic Solids with Work Hardening, J. Appl. Phys., 84, pp. 1284-1291. 1998a. 11. Cheng, Y.T. and Cheng, C.M. Can Stress-Strain Relationships be Obtained from Indentation Curves using Conical and Pyramid Indenters ?, J. Mater. Res., 14, pp.3493-3496. 1999b. 12. Cheng, Y.T. and Cheng, C.M. Relationships between Hardness, Elastic Modulus, and the Work of Indentation, Appl. Phys. Lett., 73, pp.614-616. 1998b. 13. Cheng, Y.T. and Cheng, C.M. Scaling Relationships in Conical Indentation of Elastic-Perfectly Plastic Solids, Int. J. Solids. Struct., 36, pp.1231-1243. 1999a. 14. Cheng, Y.T., Li, Z. and Cheng, C.M. Scaling Relationships for Indentation Measurements, Phil. Mag. A, 82, pp,1821-1829. 2002. 15. Chollacoop, N., Dao, M. and Suresh, S. Depth-Sensing Instrumented Indentation with Dual Sharp Indenters, Acta Mater., 51, pp.3713-3729. 2003. 16. Cristianini, N., Shawe-Taylor, J. An Introduction to Support Vector Machines. UK: Cambridge University Press. 2000. 17. Dao, M., Chollacoop, N., Van Vliet, K.J., Venkatesh, T.A. and Suresh, S. Computational Modeling of the Forward and Reverse Problems in Instrumented Sharp Indentation, Acta Mater., 49, pp.3899-3917. 2001. 18. Doerner, M.F. and Nix, W.D. A Method for Interpreting the Data from DepthSensing Indentation Instruments, J. Mater. Res, 1, pp.601-609. 1986. 19. Fischer-Cripps, A.C. Study of analysis methods of depth-sensing indentation test data for spherical indenters, J. Mater. Res., 16, pp.1579-1584. 2001. 20. Fleck, N.A., Hutchinson, J.W. A phenomenological theory for strain gradient effects in plasticity, J. Mech. Phys. Solids., 41, pp.1825-1857. 1993. 159 References 21. Fleck, N.A., Hutchinson, J.W. Strain gradient plasticity. Adv. Appl. Mech., 33, pp.295-361. 1997. 22. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W. Strain gradient plasticity: theory and experiment, Acta Metall. Mater., 42, pp.475-487. 1994. 23. Gao, H., Huang, Y., Nix, W.D., Hutchinson, J.W. Mechanism-based strain gradient plasticity-I. Theory. J. Mech. Phys. Solids., 47, pp.1239-1263. 1999. 24. Giannakopoulos, A.E. and Larsson, P.-L. Analysis of Pyramid Indentation of Pressure-Sensitive Hard Metals and Ceramics, Mech. Mater., 25, pp.1-35. 1997. 25. Giannakopoulos, A.E. and Suresh, S. Determination of Elastoplastic Properties by Instrumented Sharp Indentation, Scripta Mater., 40, pp.1191-1198. 1999. 26. Giannakopoulos, A.E., Larsson, P.-L. and Vestergaard, R. Analysis of Vickers Indentation, Int. J. Solids. Struct., 31, pp.2679-2708. 1994. 27. Huang, Y., Gao, H., Nix, W.D., Hutchinson, J.W. Mechanism-based strain gradient plasticity-II. Analysis. J. Mech. Phys. Solids., 48, pp.99-128. 2000a. 28. Huang, Y., Qu, S., Hwang, K.C., Li, M., Gao, H. A conventional theory of mechanism-based strain gradient plasticity, Int. J. Plasticity, 20, pp.753-782. 2004. 29. Huang, Y., Xue, Z., Gao, H., Nix, W.D., Xia, Z.C. A study of microindentation hardness tests by mechanism-based strain gradient plasticity, J. Mater. Res., 15, pp.1786-1796. 2000b. 30. Hutchinson, J.W. Bounds and self-consistent estimates for creep of polycrystalline materials. Proc. Roy. Soc. Lon. A, 348, pp.101-127. 1976. 31. Jiang, H., Huang, Y., Zhuang, Z, Hwang, K.C. Fracture in mechanism-based strain gradient plasticity, J. Mech. Phys. Solids., 49, pp.979-993. 2001. 160 References 32. Johnson, K.L. The Correlation of Indentation Experiments, J. Mech. Phys. Solids., 18, pp.115-126. 1970. 33. Kalpakjian S. and Schmid S. R. Manufacturing Processes for Engineering Materials. pp.33, New Jersey: Prentice Hall. 2003. 34. Kok, S., Beaudoin, A.J., Tortorelli, D.A. A polycrystal plasticity model based on the mechanical threshold, Int. J. Plasticity, 18, pp.715-741. 2002. 35. Larsson, P.-L., Giannakopoulos, A.E., Soderlund, E., Rowcliffe, D.J. and Vestergaard, R. Analysis of Berkovich Indentation, Int. J. Solids. Struct., 33, pp.221-248. 1996. 36. Loubet, J.L., Georges, J.M., Marchesini, O., Meille G. Vickers Indentation Curves of Magnesium Oxide (MgO), J. Tribol., 106, pp.43-48. 1984. 37. Luk, V.K., Forrestal, M.J. and Amos, D.E. Dynamic Spherical Cavity Expansion of Strain-Hardening Materials, J. Appl. Phys., 58, pp.1-6. 1991. 38. Ma, Q.J., Clarke, D.R. Size dependent hardness in silver single crystals, J. Mater. Res., 10, pp.853-863. 1995. 39. Marsh, D.M. Plastic Flow in Glass, Proc. Roy. Soc. A, 279, pp.420-435. 1964. 40. Martin, M. and Troyon, M. Fundamental Relations Used in Nanoindentation: Critical Examination Based on Experimental Measurements, J. Mater. Res., 17, pp.2227-2234. 2002. 41. MatWeb: http://www.matweb.com, 2003, by Automation Creations, Inc. 42. McElhaney, K.W., Vlassak, J.J., Nix, W.D. Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments, J. Mater. Res., 13, pp.1300-1306. 1998. 43. Mindlin, R.D. Second gradient of strain and surface tension in linear elasticity, Int. J. Solids. Struct., 1, pp.417-438. 1965. 161 References 44. Nix, W.D. Mechanical properties of thin films, Metall. Trans., 20A, pp.22172245. 1989. 45. Nix, W.D., Gao, H. Indentation size effect in crystalline materials: A law for strain gradient plasticity, J. Mech. Phys. Solids, 46, pp.411-425. 1998. 46. Nye, J. Some geometrical relations in dislocated crystals. Acta Metall. et Mater., 1, pp153-162. 1953. 47. Oliver, W.C. and Pharr, G.M. An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments, J. Mater. Res., 7, pp.1564-1583. 1992. 48. Pelckmans, K., Suykens, J.A.K. , Van Gestel, T., De Brabanter, J., Lukas, L., Hamers, B., De Moor, B., Vandewalle, J. LS-SVMlab: a Matlab/C toolbox for Least Squares Support Vector Machines Internal Report 02-44, ESAT-SISTA K.U.Leuven, Belgium. 2002. 49. Pethica, J. B., Hutchings, R. and Oliver, W. C. Hardness Measurement At Penetration Depths As Small As 20 nm, Phil. Mag. A, 48, pp.593-606. 1983. 50. Pharr, G.M. and Bolshakov, A. Understanding nanoindentation unloading curves, J. Mater. Res., 17, pp.2660-2671. 2002. 51. Qiu, X., Huang, Y., Wei, Y., Gao, H., Hwang, K.C. The flow theory of mechanism-based strain gradient plasticity, Mech. Mater., 35, pp.245-258. 2003. 52. Shi, M., Huang, Y., Jiang, H., Hwang, K.C., Li, M. The boundary-layer effect on the crack tip field in mechanism-based strain gradient plasticity, Int. J. Frac., 112, pp.23-41. 2001. 53. Sneddon, I.N. The Relation Between Load and Penetration in the Axisymmetric Boussinesq Problem for a Punch of Arbitrary Profile, Int. J. Eng. Sci., 3, pp.4757. 1965. 162 References 54. Stolken, J.S., Evans, A.G. A microbend test method for measuring the plasticity length scale, Acta Mater., 46, pp.5109-5115. 1998. 55. Suykens J.A.K. , Van Gestel, T., De Brabanter, J., De Moor, B., Vandewalle, J. Least Squares Support Vector Machines. Singapore:World Scientific, Singapore. 2002. 56. Suykens, J.A.K., Vandewalle, J. Least squares support vector machine classifiers, Neural Proc. Let., 9, pp.293-300. 1999. 57. Swaddiwudhipong, S., Hua, J., Tho, K.K., Liu, Z.S. C0 plane finite elements incorporating conventional mechanism-based strain gradient plasticity, 2005a. (Submitted for publication) 58. Swaddiwudhipong, S., Hua, J., Tho, K.K., Liu, Z.S. C0 solid elements for materials with strain gradient effects, Int. J. Numer. Meth. Eng., 64, pp.14001414. 2005b. 59. Swaddiwudhipong, S., Tho, K.K., Hua, J., Liu, Z.S. Mechanism-based strain gradient plasticity in C0 axisymmetric element, Int. J. Solids Struct. 2005c. (In Press) 60. Swaddiwudhipong, S., Tho, K.K., Liu, Z.S., Hua, J., Ooi, N.S.B. Material characterization via least squares support vector machines, Modelling Simul. Mater. Sci. Eng., 13, pp.993-1004. 2005d. 61. Swaddiwudhipong, S., Tho, K.K., Liu, Z.S., Zeng, K. Material characterization based on instrumented indentation, J. Metastable Nanocryst. Mater., 23, pp.359362. 2005e. 62. Swaddiwudhipong, S., Tho, K.K., Liu, Z.S., Zeng, K. Material characterization based on dual indenters, Int. J. Solids. Struct., 42, pp.69-83. 2005f. 163 References 63. Swadener, J.G., George, E.P., Pharr, G.M. The correlation of the indentation size effect measured with indenters of various shapes, J. Mech. Phys. Solids., 50, pp.681-694. 2002. 64. Tabor, D. A Simple Theory of Static and Dynamic Hardness, Proc. Roc. Soc. Lon. A, 192, pp. 247-274. 1948. 65. Taylor, G.I. The mechanism of plastic deformation of crystals. Part I.- theoretical. Proc. Roy. Soc. Lon. A, 145, pp.362-387. 1934. 66. Tho, K.K., Swaddiwudhipong, S., Liu, Z.S., Hua, J. Artificial neural network model for material characterization by indentation, Modelling Simul. Mater. Sci. Eng., 12, pp.1055–1062. 2004a. 67. Tho, K.K., Swaddiwudhipong, S., Liu, Z.S., Zeng, K. Simulation of instrumented indentation and material characterization, Mat. Sci. Eng. A., 390, pp.202-209. 2005. 68. Tho, K.K., Swaddiwudhipong, S., Liu, Z.S., Zeng, K., Hua, J. Uniqueness of reverse analysis from conical indentation tests, J. Mater. Res., 19, pp.2498-2502. 2004b. 69. Toupin, R.A. Elastic materials with couple stresses, Archive for Rational Mechanics and Analysis, 11, pp.385-414. 1962. 70. Venkatesh, T.A., Van Vliet, K.J., Giannakopoulos, A.E. and Suresh, S. Determination of Elasto-Plastic Properties by Instrumented Sharp Indentation: Guidelines for Property Extraction, Scripta Mater., 42, pp.833-839. 2000. 71. Wang, W., Huang, Y., Hsia, K.J., Hu, K.X., Chandra, A. A study of microbend test by strain gradient plasticity, Int. J. Plasticity, 19, pp.365-382. 2003. 72. Xu, Z.H. and Rowcliffe, David. Method to Determine the Plastic Properties of Bulk Materials by Nanoindentation, Phil. Mag. A, 82, pp.1893-1901. 2002. 164 References 73. Yu, H.S. Cavity Expansion Methods in Geomechanics. Pp.32-91, The Netherlands: Kluwer Academic Publishers. 2000. 74. Zeng, K. and Chiu, C. h. An Analysis of Load-Penetration Curves from Instrumented Indentation, Acta Mater., 49, pp.3539-3551. 2001. 75. Zeng, K. and Shen, Lu. A New Analysis of Nanoindentation Load-Displacement Curves, Phil. Mag. A, 82, pp.2223-2229. 2002. 165 List of Publications LIST OF PUBLICATIONS The following is a list of publications based on the work reported in this study. Journal Publications 1. Swaddiwudhipong, S., Tho, K.K., Hua, J., Liu, Z.S. Mechanism-based strain gradient plasticity in C0 axisymmetric element, Int. J. Solids Struct., 43, pp.1117-1130. 2006. 2. Swaddiwudhipong, S., Hua, J., Tho, K.K., Liu Z.S. Equivalency of Berkovich and conical load-indentation curves, Modelling Simul. Mater. Sci. Eng., 14, pp.71-82. 2006. 3. Swaddiwudhipong, S., Hua, J., Tho, K.K., Liu, Z.S. C0 solid elements for materials with strain gradient effects, Int. J. Numer. Meth. Eng., 64, pp.14001414. 2005. 4. Swaddiwudhipong, S., Tho, K.K., Liu, Z.S., Hua, J., Ooi, N.S.B. Material characterization via least squares support vector machines, Modelling Simul. Mater. Sci. Eng., 13, pp.993-1004. 2005. 5. Swaddiwudhipong, S., Tho, K.K., Liu, Z.S., Zeng, K. Material characterization based on dual indenters, Int. J. Solids. Struct., 42, pp.69-83. 2005. 6. Swaddiwudhipong, S., Tho, K.K., Liu, Z.S., Zeng, K. Material characterization based on instrumented indentation, J. Metastable Nanocryst. Mater., 23, pp.359-362. 2005. 7. Tho, K.K., Swaddiwudhipong, S., Hua, J., Liu, Z.S. Numerical simulation of indentation with size effect, Mat. Sci. Eng. A-Struct., 2006. (In Press) 166 List of Publications 8. Tho, K.K., Swaddiwudhipong, S., Liu, Z.S., Hua, J. Artificial neural network model for material characterization by indentation, Modelling Simul. Mater. Sci. Eng., 12, pp.1055–1062. 2004. 9. Tho, K.K., Swaddiwudhipong, S., Liu, Z.S., Zeng, K. Simulation of instrumented indentation and material characterization, Mat. Sci. Eng. AStruct., 390, pp.202-209. 2005. 10. Tho, K.K., Swaddiwudhipong, S., Liu, Z.S., Zeng, K., Hua, J. Uniqueness of reverse analysis from conical indentation tests, J. Mater. Res., 19, pp.24982502. 2004. Conference Publications 1. Liu, Z.S., Swaddiwudhipong, S., Tho, K.K., Zeng, K., Hua, J. Numerical simulation of material characterisation by nanoindentation approach. In. Proc. International Conference on Scientific and Engineering Computation, June 2004, Singapore. 2. Liu, Z.S., Swaddiwudhipong, S., Tho, K.K., Zeng, K., Hua, J. Uniqueness of material characterization of single conical indenter. In. Proc. The Third International Conference on Advances in Structural Engineering and Mechanics, September 2004, Seoul, Korea. 3. Swaddiwudhipong, S., Tho, K.K., Liu, Z.S., Hua, J. Material characterization using artificial neural network. In. Proc. Xth International Conference on Computing in Civil and Building Engineering, June 2004, Weimar, Germany. 4. Swaddiwudhipong, S., Tho, K.K., Liu, Z.S., Zeng, K. Material characterization based on instrumented sharp indentation. In. Proc. International Conference on Materials for Advanced Technologies, December 2003, Singapore. 167 List of Publications 5. Tho, K.K., Swaddiwudhipong, S., Liu, Z.S., Zeng, K. Material characterization through indentation process. In. Proc. The Third International Conference on Advances in Structural Engineering and Mechanics, September 2004, Seoul, Korea. 168 Appendix A APPENDIX A: Derivation of C0 CMSG Plane Strain/Stress Elements For a plane-strain or plane-stress isoparametric element with n nodes, the coordinates, x and y , and the corresponding displacement components, u and v , can be expressed as n n i =1 i =1 n n i =1 i =1 x = ∑ N i (g , h )xi , y = ∑ N i (g , h ) yi (A.1) u = ∑ N i ( g , h )ui , v = ∑ N i (g , h )vi (A.2) where g and h are the corresponding natural coordinates. Coordinate transformation can be performed through Jacobian matrix and its inverse expressed as J= ∂ ( x, y ) ⎡ x , g =⎢ ∂(g , h ) ⎣ y, g J −1 = ∂(g , h ) ⎡ g, x =⎢ ∂ ( x, y ) ⎣ h, x x,h ⎤ y ,h ⎥⎦ (A.3) g, y ⎤ h, y ⎥⎦ (A.4) The strain vector {ε }can be expressed as {ε } = [ε x ε y γ xy ] [B ] = [B1 ⎡ ∂u =⎢ ⎣ ∂x B2 " Bn ] ∂v ∂y T ∂u ∂v ⎤ + = [B ]{δ } ∂y ∂x ⎥⎦ (A.5) (A.6) 169 Appendix A {δ } = [δ1 δ ⎡ N i,x [Bi ] = ⎢⎢ ⎢ N i, y ⎣ " δn ] ⎤ ⎥ N i, y ⎥ , N i , x ⎥⎦ (A.7) ⎧u i ⎫ ⎬ , i = to n ⎩ vi ⎭ {δ i } = ⎨ (A.8) In Eq. (A.5), ε x , ε y and γ xy are the three plane strain components. The out-of-plane strain component, ε z , vanishes for plane-strain element. However, for plane-stress element, εz = (A.9) v (ε x + ε y ) 1− v where v is the Poisson ratio. Coordinate transformation can be carried out through Eq. (A.10). [N i,x ] [ Ni, y = N i, g ] (A.10) N i ,h J −1 The strain gradient is obtained through the derivatives of the strain vector and can be shown to be {ε }, x ⎡ ∂ 2u =⎢ ⎣ ∂x [Bi ], x ⎡ N i , xx ⎢ =⎢ ⎢ N i , xy ⎣ ∂ 2v ∂ x∂ y T ∂ 2u ∂ v ⎤ + ⎥ = [B ], x {δ } ∂x∂y ∂x ⎦ ⎤ ⎥ N i , xy ⎥ N i , xx ⎥⎦ (A.11) (A.12) The derivatives of strain vector with respect to y can be similarly derived. 170 Appendix A According to the chain rule of derivative, N i , xx = (A.13) ∂N i , x ∂g ∂N i , x ∂h + = N i , xg g , x + N i , xh h, x ∂g ∂x ∂h ∂x Hence, ⎡ N i , xx ⎢N ⎣ i , yx N i , xy ⎤ ⎡ N i , xg = N i , yy ⎥⎦ ⎢⎣ N i , yg N i , xh ⎤ ⎡ g , x N i , yh ⎥⎦ ⎢⎣ h, x g,y ⎤ h, y ⎥⎦ (A.14) Equation (6.121) can be expressed in compact form as ∂( N i,x , N i, y ) ∂ ( x, y ) Similarly, N i , gg = N ∂ ( N i , g , N i ,h ) ∂ ( g , h) = ∂ ( N i , x , N i , y ) ∂ ( g , h) ∂ ( g , h) i , gx = x ,g + N ∂ ( x, y ) i , gy = ∂( N i,x , N i, y ) ∂ ( g , h) J −1 y ,g ∂ ( N i , g , N i ,h ) ∂ ( x, y ) ∂ ( N i , g , N i , h ) = J ∂ ( x, y ) ∂ ( g , h) ∂ ( x, y ) (A.15) (A.16) (A.17) Note that ∂( N i,x , N i, y ) ∂ ( g , h) ⎛ ∂ ( N i , g , N i ,h ) ⎞ ⎟⎟ = ⎜⎜ ⎝ ∂ ( x, y ) ⎠ T (A.18) It can be shown that ∂( N i,x , N i, y ) ∂ ( x, y ) = J −T ∂ ( N i , g , N i ,h ) ∂ ( g , h) J −1 (A.19) The derived expressions for the strain and the strain gradient matrices can be conveniently implemented via the User subroutine provided for in ABAQUS (2002), an existing finite element program. 171 [...]... indentation depth of 15 micron, (d) Maximum indentation depth of 20 micron 151 Figure 7.8: Comparison between the numerical load-displacement curves of classical plasticity theory and CMSG plasticity theory for copper (a) Maximum indentation depth of 5 micron, (b) Maximum indentation depth of 10 micron, (c) Maximum indentation depth of 15 micron, (d) Maximum indentation depth of 20 micron ...List of Figures LIST OF FIGURES Figure 1.1: Indentation apparatus 17 Figure 1.2: Schematic representation of an instrumented indentation machine 17 Figure 2.1: Schematic drawing of a typical finite element model for indentation using conical indenter 28 Figure 2.2: Indentation load-displacement curves of various domain sizes Comparison between 150 micron x 150 micron, 200 micron... results in a unique set of elasto-plastic material properties is still a point of contention The interpretation of instrumented indentation results is further complicated by presence of indentation size effect when the indentation depth is in the order of tenths of nanometers up to several microns The strength of materials is observed to increase significantly with decreasing indentation depths Due to the... effect of pile-up and sink-in of the materials on the interpretation of an indentation load-displacement curve They noted that as a consequence of pile-up or sink-in, large differences may arise between the 6 Chapter 1: Introduction true contact area and the apparent contact area which is usually observed after indentation They pointed out that knowledge of the relationship between the indentation load... indenters are occasionally used In general, instrumented indentation experiments can be broadly classified into microindentation and nanoindentation The two classifications defers only in the scale at which they are carried out In microindentation experiments, the displacement of the indenter is in the order of tenths of micron while in nanoindentation experiments, the displacement of the indenter into... using ANSYS software They observed that the ratio of elastic depth of indentation to maximum depth of indentation is a function of the strain hardening exponent and the ratio of yield stress to Young’s modulus They further observed that pile-up or sink-in behaviour of the materials around the indenter is influenced by the strain hardening exponent and the ratio of elastic depth of indentation to the maximum... dimensional analysis and finite element analysis to propose several relationships that relate features of indentation loading and unloading curves to hardness, Young’s modulus and the work of indentation for conical indentations with various cone angles From the finite element results, they proposed three linear relationships between the ratio of hardness to Young’s modulus, the ratio of plastic work done... They concluded that the numerical results were in good agreement with experimental observations Based on their numerical results, universal relations between indentation load and indentation depth for the loading curve were proposed for elastic and elastoplastic materials Larsson et al (1996) performed a similar analysis for Berkovich indentation and concluded that Berkovich and Vickers indentation tests... the classical plasticity theory An alternative constitutive model taking into account of the material length scale is imperative in this case In general, the indentation size effect is only significant in nanoindentation experiments In microindentation experiments, the indentation depths are typically in excess of 10 microns and the effect of indentation size effect diminishes Therefore, the classical... maximum indentation depth They also suggested that the ratio of plastic energy to total energy is equal to the ratio of residual depth to maximum depth Cheng and Cheng (1999a) proposed several scaling relationships for conical indentation of elastic-perfectly plastic solids using dimensional analysis and finite element calculations These scaling relationships were used to reveal the general relationships . copper. (a) Maximum indentation depth of 5 micron, (b) Maximum indentation depth of 10 micron, (c) Maximum indentation depth of 15 micron, (d) Maximum indentation depth of 20 micron 153 Nomenclature. FORMULATION OF CONSTITUTIVE RELATIONS BASED ON INDENTATION TESTS THO KEE KIAT (B. Eng. (Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF. of classical plasticity theory and CMSG plasticity theory for Al7075. (a) Maximum indentation depth of 5 micron, (b) Maximum indentation depth of 10 micron, (c) Maximum indentation depth of