Determination of Elastic-Plastic and Visco-Plastic Material Properties from Instrumented Indentation Curves JiJun Kang MEng Thesis submitted to The University of Nottingham for the degree of Doctor of Philosophy July 2013 For Dad and Mum i | Pa g e Abstract Instrumented indentation techniques at micro or nano-scales have become more popular for determining mechanical properties from small samples of material. These techniques can be used not only to obtain and to interpret the hardness of the material but also to provide information about the near surface mechanical properties and deformation behaviour of bulk solids and/or coating films. In particular, various approaches have been proposed to evaluate the elastic-plastic properties of power-law materials from the experimental loading-unloading curves. In order to obtain a unique set of elastic-plastic properties, many researchers have proposed to use more than one set of loading-unloading curves obtained from different indenter geometries. A combined Finite Element (FE) analysis and optimisation approach has been developed, using three types of indenters (namely, conical, Berkovich and Vickers), for determining the elastic-plastic material properties, using one set of ‘simulated’ target FE loading-unloading curves and one set of real-life experimental loading-unloading curves. The results obtained have demonstrated that excellent convergence can be achieved with the ‘simulated’ target FE loading-unloading curve, but less accurate results have been obtained with the real-life experimental loading-unloading curve. This combined technique has been extended to determine the elastic and visco-plastic material properties using only a single indentation ‘simulated’ loading-unloading curve based on a two-layer viscoplasticity model. A combined dimensional analysis and optimisation approach has also been developed and used to determine the elastic-plastic material properties from loading-unloading curves with single and dual indenters. The dimensional functions have been established based on a parametric study using FE analyses and the loading and linearised unloading portions of the indentation curves. It has been demonstrated that the elastic-plastic material properties cannot be uniquely determined by the test curves of a single indenter, but the unique or more accurate results can be obtained using the test curves from dual indenters. Since the characteristic loading-unloading responses of indenters can be approximated by the results of dimensional analysis, a simplified approach has been used to obtain the elastic- plastic mechanical properties from loading-unloading curves, using a similar optimisation ii | Pa g e procedure. It is assumed that the loading-unloading portions of the curves are empirically related to some of the material properties, which avoids the need for time consuming FE analysis in evaluating the load-deformation relationship in the optimisation process. This approach shows that issues of uniqueness may arise when using a single indenter and more accurate estimation of material properties with dual indenters can be obtained by reducing the bounds of the mechanical parameters. This thesis highlights the effects of using various indenter geometries with different face angles and tilted angles, which have not been covered previously. The elastic-plastic material parameters are estimated, for the first time, in a non-linear optimisation approach, fully integrated with FE analysis, using results from a single indentation curve. Furthermore, a linear and a power-law fitting scheme to obtain elastic-plastic material properties from loading-unloading indentation curves have been introduced based on dimensional analysis, since there are no mathematical formulas or functions that fit the unloading curve well. The optimisation techniques have been extended to cover time-dependent material properties based on a two-layer viscoplasticity model, has not been investigated before. iii | Pa g e Acknowledgement This thesis would not have been possible without the support of my supervisors at the University of Nottingham. I am greatly indebted to Professor Adib Becker and Wei Sun for their continuous support, guidance and encouragement during my PhD study. I would like to thank all my friend and colleagues in the Structural Integrity and Dynamics (SID) group, the Department of Mechanical Engineering, The University of Nottingham for sharing their experience and knowledge. They have also made my time as a student enjoyable and memorable. Thanks to my brother, JunHyuk Kang, for encouraging, understanding, patience and supporting me during this period of time to finish my Ph.D program. Finally yet most importantly, I would like to thank my parents, who unconditionally support me financially and mentally, so I could complete this long journey without concern. iv | Pa g e Nomenclature Projected area of the hardness impression √ π Radius of the circle of contact Ratio of the contact radius Depth of penetration C Independent of initial plastic strain Compliance of the loading instrument Total compliance Compliance of indenter material D The diameter of indenter (mm) d The diameter of indenter (mm) ⁄ The initial slope of the unloading curve Reduced modulus F( ) Objective function Dimensional functions h Spherical indenter at any point with radius r from the centre of contact Circle of contact Final depth of the contact impression after indenter removed Maximum displacement of indenter The hardness of Meyer’s law H’ Power law hardening with work hardening exponent Distance between the surface of specimen and the edge of contact at full load or Contact depth Hardness v | Pa g e A specific position K (Yield coefficient) Elastic modulus of the elastic plastic network Elastic modulus of the elastic-viscous network m Power law index or N Total number of points n Work-hardening exponent Work hardening exponent Norton creep parameters P Indenter load Hertzian pressure distribution Mean contact pressure Maximum indenter load Unloading Force The (experimental) force from target data The predicted total force R Relative radius of the two contacting bodies’ curvature Radius of a rigid indenter A vector in the n-dimensional space S Initial slope of unloading curve or ( ) UTS Ultimate tensile strength Total work done Work done during unloading Y Initial yield stress α The angle of indenter vi | Pa g e β Correction factor 1.034 for a Berkovich indenter and 1.024 for a Vickers indenter Geometric constant: 0.727 for conical and 0.75 for Berkovich and Vickers Y/E ⁄ A representative flow stress Stress in the elastic-viscous network Initial yield stress Stress in the elastic-plastic network Initial plastic strain Representative strain Optimisation variable set Total strain Elastic strain Plastic strain The elastic strain in the elastic-plastic network Elastic strain in the elastic-viscous network Distance of mutual approach ν Poisson’s ratio Abbreviations BHN Brinell hardness number CAX3 Three-Node Axisymmetric Triangular Continuum Elements CAX4R Four-Node Axisymmetric Quadrilateral Continuum Elements C3D4 Four-Node Linear Tetrahedron Continuum Elements FEA Finite element analysis .EXE Executable File .INP ABAQUS Input File LSQNONLIN Non-linear Least Square Function vii | Pa g e .M MATLAB Script File .OBD ABAQUS Output Data File PEEQ Equivalent Plastic Strain RF1 Reaction Force in X-direction (in Newton) RF2 Reaction Force in Y-direction (in Newton) RF3 Reaction Force in Z-direction (in Newton) R3D4 Four-Node Bilinear Quadrilateral Continuum Elements 2D Axisymmetric model 3D 3-Dimensional UTS Ultimate tensile strength U1 Displacement in X-direction (in Millimetre) U2 Displacement in Y-direction (in Millimetre) U3 Displacement in Z-direction (in Millimetre) SNRE The squared norm of the residual error viii | Pa g e List of Contents 1 Introduction……………………………………………………………………………..1 1.1 Background ........................................................................................................... 1 1.2 Research Aims and Objectives ............................................................................. 1 1.3 Thesis Outline ....................................................................................................... 2 2 Literature Review……………………………………………………………………….5 2.1 Basic theory of Indentation ................................................................................... 5 2.2 Type of indenters ................................................................................................ 14 2.3 Review of instrumented indentation measurement ............................................. 17 2.3.1 Scaling approach to indentation modelling ..................................................... 17 2.3.2 Numerical approach based on FE Analysis ..................................................... 20 2.3.3 The concept of representative strain ................................................................ 21 2.4 Summary and Knowledge gaps .......................................................................... 25 3 Research Methodology………………………………………………………………….27 3.1 Research Approach ............................................................................................. 27 3.2 Development of Methodology ............................................................................ 28 3.2.1 FE model development .................................................................................... 28 3.2.2 Optimisation model development ................................................................... 30 3.3 Conclusions ......................................................................................................... 32 4 Effects of indenter geometries on the prediction of material properties………………..33 4.1 Introduction ......................................................................................................... 33 4.2 Commonly Used Indenters ................................................................................. 33 4.3 Finite element modelling .................................................................................... 34 4.3.1 ABAQUS Software Package ........................................................................... 34 4.3.2 Types of Load application ............................................................................... 35 4.3.3 Indenter Geometry Definition ......................................................................... 36 4.4 FE Models ........................................................................................................... 37 4.4.1 Typical Indentation Behaviour ........................................................................ 38 4.4.2 Mesh Sensitivity .............................................................................................. 39 4.5 Comparison between Different Indenter Geometries ......................................... 40 ix | Pa g e