A Numerical Study on the Nanoindentation Response of a Particle Embedded in a Matrix

Transcription

1 A Numerical Study on the Nanoindentation Response of a Particle Embedded in a Matrix Teck Fei Low B Mechatronics Eng (Hons) Thesis submitted in accordance with the requirements for Master of Engineering Science (Research) Department of Mechanical and Aerospace Engineering Monash University February 2014

2 Copyright Notices Notice 1 Under the Copyright Act 1968, this thesis must be used only under the normal conditions of scholarly fair dealing. In particular no results or conclusions should be extracted from it, nor should it be copied or closely paraphrased in whole or in part without the written consent of the author. Proper written acknowledgement should be made for any assistance obtained from this thesis. Notice 2 I certify that I have made all reasonable efforts to secure copyright permissions for third-party content included in this thesis and have not knowingly added copyright content to my work without the owner's permission.

3 Statement of Authorship I hereby declare that the thesis titled A Numerical Study on the Nanoindentation Response of a Particle Embedded in a Matrix is submitted with accordance to requirements for Master of Engineering Science (Research) in the Department of Mechanical and Aerospace Engineering of Monash University. The thesis: Contains no material which has been accepted for the award to the candidate of any other degree or diploma at any university or equivalent institution, except where due reference is made in the text of examinable outcome. To the best of this candidate s knowledge contains no materials published or written by another person except where due reference is made in the text of the examinable outcome. Name: Signature: Date: i

4 Acknowledgements I would like to express my utmost gratitude to Dr. Wenyi Yan for allowing me to undertake this research and for his guidance throughout the project. I would like to sincerely thank the Department of Mechanical and Aerospace Engineering, Monash University and National Computational Infrastructure (NCI) National Facility for providing the necessary facilities required to conduct the research. I would also like to acknowledge the assistance in finite element modelling received from Mr. Jerome Pun. ii

5 Abstract An indentation test is an experimental method whereby an indenter is pressed into a specimen in order to create a contact impression. Based on experimental input and the mechanical response of a specimen, an indentation test can be used to measure material properties of a specimen. A nanoindentation test is an instrumented indentation test where the indentation depth is at the submicron scale, utilising the interpretation of recorded load-displacement data. Nanoindentation tests have proven to be popular commercially due to their non-destructive nature, with the Oliver-Pharr method being one of the most prominent analysis methods applied. In this research, nanoindentation tests on a semi-spherical particle embedded in a semi-infinite matrix were simulated using the finite element method. Dimensional analysis was utilised to conduct a numerical study of the nanoindentation response obtained from finite element simulations. The Oliver-Pharr method to estimate contact area from a nanoindentation test is unsuitable for the cases with pile-up deformation. In an indentation of a hard particle embedded in a soft matrix, the mismatch of material properties will cause the particle to indent itself into the matrix, which creates a secondary indentation effect. This makes it difficult to investigate the pile-up and sink-in behaviour for hard particle reinforced composites because the definition of pile-up for monolithic materials is no longer valid. In this research, a feasible method to quantify pile-up and sink-in for particle-matrix systems is proposed. Using a reference height, it was found that particle pile-up and sink-in can be quantified when the maximum indentation depth is less than 15% of the particle s radius. iii

6 The influence of particle and matrix material property on pile-up and sink-in was explored. Within the maximum indentation depth, which is less than 15% of the particle s radius, it was found that only the mechanical properties of the particle represented by particle s elastic modulus, particle s yield strength, and particle s workhardening exponent, have a significant influence on the degree of pile-up and sink-in from a nanoindentation test. In a nanoindentation test of a particle embedded in a matrix, the matrix material properties may influence the measured indentation hardness of the particle. A numerical study was carried out to identify the influence of particle and matrix material properties on the measured hardness. Results reveal that a mismatch in material properties is more sensitive on the measured hardness of a hard particle embedded in a soft matrix than a soft particle embedded in a hard matrix. Finally, a parametric study was carried out to identify a particle-dominated depth, within which the particle s hardness obtained from a nanoindentation test can be measured with confidence. Simulation results from the finite element analysis of a variety of material properties determined that when the maximum indentation depth is less than 13.5% of the particle s radius, the measured hardness is in good agreement with the particle s true hardness, with a maximum error of 5%. This finding can be applied in practice as a guideline to measure the hardness of a particle embedded in a matrix from a nanoindentation test and provide the theoretical basis to develop a particle-embedded method to measure the hardness of individual particles. iv

7 Publications Low, T. F., Yan, W. and Pun, C. L. (2012), Pile-Up and Sink-In in Nanoindentation of Stiff Particle Reinforced Composites, Proceeding of 4 th International Conference on Computational Methods, , 25 th -28 th November 2012, Gold Coast, Australia. (Published) Low, T. F. and Yan, W. (2014), Theoretical study on Nanoindentation Hardness Measurement of a Particle Embedded in a Matrix, to be submitted to Philosophical Magazine. v

8 Table of Contents Statement of Authorship... i Acknowledgements... ii Abstract... iii Publications... v List of Figures... ix List of Tables... xii Nomenclature... xiii 1.0 Introduction Problem Statement Research Objectives Literature Review Nanoindentation History of Indentation Testing Oliver-Pharr Method Pile-up and Sink-in Indentation Hardness Particle Reinforced Composites Nanoindentation of Particle Reinforced Composites Secondary Indentation Summary Pile-up and Sink-in in Nanoindentation on a Particle Embedded in a Matrix Methodology Finite Element Modelling vi

9 3.1.2 Dimensional Analysis Reference Height Method Results and Discussion Influence of Young s Modulus and Yield Strength on Secondary Indentation Accuracy of Reference Height Method to Determine Pile-up and Sinkin Influence of Material Properties on Pile-up and Sink-in Summary Measured Hardness in Nanoindentation on a Particle Embedded in a Matrix Methodology Finite Element Modelling Dimensional Analysis Results and Discussion Influence of Material Properties on Measured Hardness Influence of Indentation Depth on Measured Hardness Particle-Dominated Depth for Measuring Particle s Hardness Summary Conclusions and Recommendations Conclusions Recommendations References Appendix A: Rigid Indenter Assumption Appendix B: Semi-Infinite Matrix Assumption vii

10 Appendix C: Mesh Convergence viii

11 List of Figures Figure 1: Schematic representation of an instrumented indentation testing system... 6 Figure 2: Illustration of an indentation load-displacement curve Figure 3: Illustration of sink-in and pile-up deformation Figure 4: Illustration of different particle shapes Figure 5: Illustration of particle roundness and sphericity Figure 6: Load-displacement curve for soft and hard single phase material and particlematrix system Figure 7: Illustration of the method to determine the particle-dominated depth normalised by particle radius Figure 8: Relationship between normalised hardness and normalised contact radius for indentation on a disc shaped particle Figure 9: Illustration of an axisymmetric indentation profile of a hard particle reinforced composite Figure 10: Schematic illustration of the indentation model Figure 11: Axisymmetric finite element model Figure 12: Mismatch of Young s modulus and yield strength on secondary indentation Figure 13: Influence of Y/E on secondary indentation Figure 14: Influence of h max /R on d p with different ratios of Y p /E m Figure 15: Influence of h max /R on d p with different ratios of E p /E m Figure 16: Influence of E p /E m on pile-up and sink-in Figure 17: Influence of Y p /E m on pile-up and sink-in Figure 18: Influence of Y m /E m on pile-up and sink-in ix

12 Figure 19: Influence of v p on pile-up and sink-in Figure 20: Influence of v m on pile-up and sink-in Figure 21: Influence of n p on pile-up and sink-in Figure 22: Influence of n m on pile-up and sink-in Figure 23: Influence of h max /R on pile-up and sink-in Figure 24: Influence of Y m /Y p on H/Y p Figure 25: Influence of E p /Y p on H/Y p Figure 26: Influence of E m /Y p on H/Y p Figure 27: Influence of v p on H/Y p Figure 28: Influence of v m on H/Y p Figure 29: Influence of n p on H/Y p Figure 30: Influence of n m on H/Y p Figure 31: Relationship between H/H p and h max /R for different ratios of v p and v m Figure 32: Relationship between H/H p and h max /R for different ratios of E p /Y p, E m /Y p and Y m /Y p Figure 33: Relationship between H/H p and h max /R for different ratios of n p and Y m /Y p.. 73 Figure 34: Relationship between H/H p and h max /R for different ratios of n p Figure 35: Relationship between H/H p and h max /R for different ratios of n m and Y m /Y p. 74 Figure 36: Second mode of particle-dominated depth Figure 37: Relationship between particle-dominated depth and Y m /Y p for cases of Y m /Y p > Figure 38: Relationship between particle-dominated depth and n p Figure 39: Relationship between particle-dominated depth and Y m /Y p for different ratios of E p /Y p x

13 Figure 40: Relationship between particle-dominated depth and Y m /Y p for different ratios of E m /Y p Figure B1: P-h curve comparing different ratios of matrix size to h max with Sneddon s elastic contact solution Figure B2: Indentation load error at h=h max for different ratios of matrix size to h max 103 xi

14 List of Tables Table 1: Range of material properties studied Table 2: Material Properties for graphs in Figure Table 3: Material Properties for graphs in Figure Table 4: Material Properties for graphs in Figure Table 5: Material Properties for graphs in Figure Table 6: Material Properties for graphs in Figure Table 7: Material Properties for graphs in Figure Table 8: Material Properties for graphs in Figure Table 9: Material Properties for graphs in Figure Table 10: Range of material properties studied Table 11: Material Properties for graphs in Figure Table 12: Material Properties for graphs in Figure Table 13: Material Properties for graphs in Figure Table 14: Material Properties for graphs in Figure Table 15: Material Properties for graphs in Figure Table 16: Material Properties for graphs in Figure Table 17: Material Properties for graphs in Figure Table A1: Comparison between deformable indenter and rigid indenter assumption. 100 Table C1: Number of elements and simulation output xii

15 Nomenclature A a c C m d p E E i E m E p E r E m p H Projected contact area Contact radius Machine compliance Comparison between reference height and Leggoe s method Elastic modulus Indenter s elastic modulus Matrix s elastic modulus Particle s elastic modulus Reduced elastic modulus Measured Particle s elastic modulus Indentation hardness H/H p Normalised hardness H p h h c h f h max Particle s hardness Indentation depth Contact depth Final indentation depth Maximum indentation depth h max /R Normalised indentation depth h max sp h p Maximum indentation depth of single-phase material Particle s indentation depth h pd /R Particle-dominated depth h re h s Reference height Sink-in depth xiii

16 h sec n n m n p P P max p S R v v i v m v p Y Y 0 Y m Y p ε 0 ε axial ε p ε trans θ Secondary indentation depth Work-hardening exponent Matrix s work-hardening exponent Particle s work-hardening exponent Load Maximum Load Mean pressure Initial unloading stiffness Radius of particle Poisson s ratio Indenter s Poisson s ratio Matrix s Poisson s ratio Particle s Poisson s ratio Yield strength Representative stress Matrix s yield strength Particle s yield strength Representative strain Axial strain Plastic strain Transverse strain Indenter s included half-angle xiv

17 1.0 Introduction The nanoindentation test is an experimental method to probe the material properties of specimens, whereby a diamond indenter is usually pressed on to the surface of a specimen with depths ranging in submicron scale. Its load-displacement data is recorded to extract material properties such as indentation hardness and Young s Modulus (Pethica et al., 1983, Loubet et al., 1984, Doerner and Nix, 1986, Oliver and Pharr, 1992). Apart from its simplicity and convenience, the nanoindentation test has become a popular method to study the mechanical behaviour of small-sized materials due to the development of different theoretical methods that deliver highly accurate results. One of the most popular methods in nanoindentation tests is the Oliver-Pharr method. Oliver and Pharr (1992) developed a method to analyse the load-displacement data and to estimate the Young s modulus and hardness of a specimen from one complete cycle of loading and unloading of a nanoindentation test. The attractiveness of the Oliver- Pharr method is that direct observations and measurements of contact area are not required to evaluate hardness and Young s Modulus (Oliver and Pharr, 1992 and 2004, Cheng and Cheng, 1997 and 2004, Bolshakov and Pharr, 1998). In a particle-reinforced composite, nanoindentation on the particle could provide insight into properties of the reinforcement (Leggoe, 2004). However, Yan et al. (2011 and 2012) found that the Oliver-Pharr method cannot accurately determine the Young s Modulus of the particle in a matrix-influenced depth. This is due to the fact that Oliver- Pharr method was originally developed for monolithic materials. Durst et al. (2004) also 1

18 found that matrix material properties can influence the measured hardness of a particle in a particle-matrix system. Another issue with the Oliver-Pharr method is its inability to deal with indentation profiles that exhibit pile-up deformation. Pile-up is a phenomenon in which there is a material build-up around the contact impression. Much research has been carried out to understand the relationship between the material properties and pile-up phenomenon in bulk material and thin film coatings (Field and Swain, 1993 and 1995, Mencik and Swain, 1995, Bahr and Gerberich, 1996, Bolshakov et al., 1997, Bolshakov and Pharr, 1998, Cheng and Cheng, 1998b, Hay et al., 1998, Tsui and Pharr, 1999, Chen and Vlassak, 2001, Saha and Nix, 2002, Taljat and Pharr, 2004, Oliver and Pharr, 2004, Zhou et al., 2008). Despite this, there is still limited understanding of how material properties of particle-reinforced composites may contribute to pile-up deformation. The mismatch in material properties of the particle and the matrix may cause the particle itself to indent the matrix. This proves the definition of pile-up for monolithic materials invalid. 1.1 Problem Statement Durst et al. (2004) studied the influence of particle shape and determined that the particle s hardness can be measured when the contact radius normalised with the radius of the particle is not more than 0.7. However, it should be noted that the studies carried out by Durst et al. (2004) focussed more on the aspect ratio of particle shape and did not cover a wide range of material properties. Studies carried out were based on two cases: a hard particle in a soft matrix, where the yield strength of the particle was two times 2

19 greater than the yield strength of its matrix and a soft particle in a hard matrix, where the yield strength of the particle was half of the yield strength of its matrix. It means that only the influence of a mismatch in yield strength of particle and matrix on measured hardness has been identified in the two case studies. The limited number of material properties tested calls into question the validity of the rule established by Durst et al. (2004). It is difficult to extend this rule to real life situations as the mismatch in material properties can be much larger than the cases studied by Durst et al. (2004). In a nanoindentation test, the contact radius can only be probed after the conclusion of the loading cycle in a nanoindentation test. Yan et al. (2011 and 2012) established a particle-dominated depth based on the maximum indentation depth normalised by the radius of the particle for measuring the elastic modulus of the particle. The attractiveness of this particle-dominated depth is that the maximum indentation depth can be used as an input for a nanoindentation test and the radius of a particle is a constant. However, a particle-dominated depth for measuring hardness has yet to be found. Influence of material properties on pile-up and sink-in in single-phase material have been studied by many researchers. However, for particle reinforced composites, few studies have been carried out to understand the influence of matrix material properties on pile-up and sink-in. While Yan et al. (2011 and 2012) were able to determine the existence of a particle-dominated depth where the particle s elastic modulus can be measured, the issue of material pile-up can still provide complications to measurements when the Oliver-Pharr method is applied. 3

20 For indentation tests on a particle embedded in a matrix, a form of secondary indentation where the particle indents the matrix may occur. Indentation tests that exhibit secondary indentation cause the definition of pile-up for monolithic material to be unsuitable. Leggoe (2004) defined this secondary indentation as the difference between the maximum indentation depths measured from a particle-matrix system and that of a particle bulk material at a given load. Theoretically, applying Leggoe s method would allow the determination of particle s indentation depth. Practically, it is difficult to have a pure particle material indentation specimen with the same size as the composite sample. The possibility of carrying out the physical indentation test on the monolithic particle material specimen with the composite specimen size can be excluded. The computer-based virtual indentation test cannot be carried out either, because the properties of the particle material are generally unknown beforehand. Therefore, it is extremely difficult in practice to apply Leggoe s method to determine particle s indentation depth and to examine pile-up or sink-in. 1.2 Research Objectives The advantage of nanoindentation testing is the ability to conduct submicron scale indentations. As miniaturisation technology continues to develop, manufactured parts and components gradually become smaller. The need for smaller-sized indentation testing as a means of non-destructive testing will grow in importance. Furthermore, indentation testing on composites has always assumed that the specimen behaves in the same way as a bulk material. This is partly true for cases where the area of indent is many times larger than the particle size and multiple particles are indented. In these 4

21 cases, results reflect the composite s material properties as a whole. Nanoindentation testing may have an area of indent which is smaller relative to particle size. This allows the probing of indentation response of individual particles embedded in a matrix. This procedure will provide insight into a particle s mechanical behaviour when embedded in a matrix, which would aid in material selection of composites. In this thesis, research has been carried out by parametric study and dimensional analysis with the aid of finite element modelling software ABAQUS to achieve the following objectives: 1) Establish a method to quantify pile-up and sink-in in nanoindentation tests on a particle embedded in a matrix that takes into account secondary indentation. 2) Determine the influence of particle and matrix material properties on pile-up and sink-in in nanoindentation tests on a particle embedded in a matrix. 3) Determine the influence of particle and matrix material properties on measured hardness for nanoindentation tests on a particle embedded in a matrix. 4) Establish a particle-dominated indentation depth at which the particle s hardness can be measured with confidence for nanoindentation tests on a particle embedded in a matrix. 5

22 2.0 Literature Review 2.1 Nanoindentation Nanoindentation is an instrumented indentation method whereby the indentation depth ranges in the submicron scale. During a nanoindentation test, a force is applied via the force actuator to drive the indenter into the specimen and the displacement of the indenter is continuously recorded using the displacement sensor, as shown in Figure 1. The load-displacement data can then be used to estimate material properties using various methods such as the Oliver-Pharr Method. Figure 1: Schematic representation of an instrumented indentation testing system (Hay and Pharr, 2000) 6

23 2.1.1 History of Indentation Testing Indentation testing methods are the preferred methods in industry as compared to other material testing methods such as tensile test. This is because of its cheap, quick, nondestructive nature and flexibility to be applied to all factory outputs (Petrenko, 1930, Walley, 2012). According to Lysaght (1936), indentation testing methods became a popular material testing method with the emergence of mass production within industries which require every item to be quality tested, i.e., automotive industry. Indentation tests can be derived from the field of contact mechanics which begun with Heinrich Hertz s publication in 1882 on how the optical properties of multiple stacked lenses can change the force holding them together (Hertz, 1882). Hertz studied the behaviour of two spherical elastic solids with different radii and elastic constant in contact under loading and the theory developed widely became known as the Hertzian contact theory, which is based upon classical theory of elasticity and continuum mechanics (Oliver and Pharr, 1992). The first widely accepted and standardised indentation testing method was the Brinell s hardness test proposed by Johan August Brinell in 1900 (Wahlberg, 1901) whereby a steel ball is pressed into the surface of a specimen at a pre-determined load. The hardness is then calculated as a ratio of load to the area of indentation impression. Many other indentation testing methods were proposed following the Brinell s hardness test, such as Vicker s hardness test, Knoop hardness test, Rockwell hardness test and many others (Wahlberg, 1901, Rockwell and Rockwell, 1919, Smith and Sandland, 1922, Rockwell, 1924, Knoop et al., 1939). The steel ball indenter was replaced by diamond 7

24 indenters due to the inability of steel ball indenters to accurately determine the hardness of specimens that are very hard (Walley, 2012). In micro-indentation testing, pyramidal indenters such as the four-sided pyramid Vickers indenter and the tree-sided pyramid Berkovich indenter are preferred over spherical indenters due to the difficulty in machining high-quality spheres made from hard, rigid materials (Smith and Sandland, 1922, Berkovich, 1950, Newey et al., 1982, Hay and Pharr, 2000). Boussinesq (1885) first studied the state of stress of an elastic half space indented by a rigid punch that can be described as a body of revolution of a smooth function. Love (1939), Harding and Sneddon (1945), and Sneddon (1948) pursued the Boussinesq s problem to derive a closed-form analytical solution on the relationship between load, displacement and contact area for any punch on a semi-infinite elastic half space by a rigid indenter of simple geometry. Sneddon (1965) discovered that the loaddisplacement relationship for a punch of simple geometry can be described by a powerlaw function. Tabor (1970) studied the hardness of solids punched by a hardened spherical indenter and discovered that material recovery around the indentation impression during the unloading process is predominantly elastic. This proved to be a breakthrough in indentation testing as indentation models which include plasticity are much more complex since it consists of non-linear equations and a number of material parameters such as yield strength and work-hardening coefficients. The importance of this observation is that elastic contact analysis could be applied to the unloading portion of the indentation process to extract material properties since closed-form analytical 8

25 solutions for elastic contact by simple geometries already exists (Oliver and Pharr, 1992 and 2004, Tsui et al., 1996, Bolshakov et al., 1996, Woirgard and Dargenton, 1997, Cheng and Cheng, 1997, Fischer-Cripps, 2006). One of the major contributors in the use of load-depth sensing equipment in indentation tests to probe material properties of a specimen comes from workers from the Baikov Institute of Metallurgy in Moscow such as Bulychev, Alekhin and Shorshorov in the early 1970 s (Bulychev et al., 1975 and 1976, Oliver and Pharr, 1992). Based on elastic contact analysis, they discovered that the initial contact stiffness can be used to measure elastic modulus provided the contact area can be accurately determined. As the resolution of load-depth sensing equipment increased, the need to estimate the contact area without the need for direct observations gained importance, since optical imaging of the contact area on a submicron scale can be highly inaccurate. Doerner and Nix (1986) developed a method to estimate contact area without the need for direct observation, based on the assumption that the contact area remains constant during the initial stage of unloading. This behaviour is similar to a punch on a semi-infinite elastic half-space by a rigid indenter with flat geometry such as a cylindrical punch. Doerner and Nix used an empirical method to extrapolate the initial linear portion of the unloading curve and used the extrapolated depth together with the indenter area function to estimate the contact area (Oliver and Pharr, 1992). Oliver and Pharr (1992) later found that Doerner and Nix s assumption of linear unloading was incorrect and proposed a method to estimate contact area without the need of direct observation 9

26 (Pharr et al., 1992, Cheng and Cheng, 1997, Bolshakov and Pharr, 1998, Oliver and Pharr, 2004, Fischer-Cripps, 2006) Oliver-Pharr Method Figure 2: Illustration of an indentation load-displacement curve (Oliver and Pharr, 2004) The Oliver-Pharr method is a method to probe the material properties of a specimen through one complete cycle of loading and unloading in a nanoindentation test. The load-displacement curve recorded from the nanoindentation test can be used to estimate the elastic modulus and indentation hardness of the specimen. Figure 2 is an illustration of a load-displacement curve obtained from a nanoindentation test where P max is maximum indentation load, h max is maximum indentation depth, h f is the final depth of contact impression after unloading and S is the initial unloading stiffness. Oliver and 10

27 Pharr (1992) found that the unloading curve is usually well represented by the powerlaw relation: ( ) (1) where α and m are power-law fitting constants. The initial unloading stiffness is the slope of the unloading curve during the initial stages of unloading given by (Oliver and Pharr, 1992 and 2004, Cheng and Cheng, 1997 and 2004, Bolshakov and Pharr, 1998, Pharr and Bolshakov, 2002, Fischer-Cripps, 2006): ( ) ( ) (2) Oliver and Pharr (1992) developed an expression to estimate the contact area without the need for direct observation and measurement of the contact area. It was assumed that the contact periphery sinks-in (see Figure 3), as described by models for indentation of a flat elastic half-space by rigid punches of simple geometry (Sneddon, 1948 and 1965). In these sink-in situations, the contact depth, h c, which is the distance between the indenter tip and contact periphery in contact with the indenter in the loading direction at h=h max is given by: (3) where ε is a constant that depends on the geometry of the indenter, i.e., ε=0.72 for a conical punch (Sneddon, 1965). The projected contact area can be then derived by evaluating the indenter shape function at contact depth, h c, given by: The hardness can be estimated once the contact area is found: ( ) (4) (5) 11

28 The hardness measured is based on contact area under load and may differ from traditional hardness measured from the area of residual hardness impression. However, this is only important for materials with extremely small values of E/H which exhibits significant elastic recovery during unloading (Meyer, 1908, Bolshakov and Pharr, 1998, Cheng and Cheng, 2004). Measurements for elastic modulus rely on the theory of elastic contacts. The elastic modulus can be measured from its relationship to contact area and unloading stiffness (Bulychev et al., 1975 and 1976): (6) where β is a correction factor and E r is the reduced elastic modulus which accounts for elastic deformation to occur in both specimen and indenter. Initially, Bulychev et al. (1976) was able to show that Eq. (6) is valid for cylindrical and spherical indentation. Later, Pharr et al. (1992) was able to determine that Eq. (6) can be used for all indenters of simple geometry. Oliver and Pharr (2004) found that β lies between and through experiments and finite element calculations. They therefore proposed that is a good choice with a potential error of approximately ±0.05. E r is given by Hertzian contact theory (Hertz, 1882): (7) where v and E represents the Poisson s ratio and elastic modulus of the specimen respectively while v i and E i represents the Poisson s ratio and elastic modulus of the indenter respectively. Although, Eq. (7) was initially derived for indentation for elastic solids, Cheng and Cheng (1997) used the infinitesimal theory of continuum mechanics 12

29 to show that Eq. (7) applies equally well for elastic-plastic contact using indenters with axisymmetric smooth profiles Pile-up and Sink-in Pile-up deformation is a case where materials of the specimen pile up around the contact periphery with h c >h max while the opposite is true for sink-in deformation, as shown in Figure 3 (Field and Swain, 1995, Bolshakov et al., 1997, Hay et al., 1998, Cheng and Cheng, 1998b, Taljat and Pharr, 2004). Pile-up deformation usually occurs in materials that exhibit large plastic deformation, e.g., most metals, while sink-in deformation occurs in highly elastic material, e.g., polymers. Pile-up phenomenon is caused by the pattern of plastic flow, whereby displaced material flows up around the indenter to form a raised portion close to the indenter surface (Tabor, 1970). P P A h s A h max h c h max h c (a) (b) Figure 3: Illustration of sink-in (a) and pile-up (b) deformation The assumption in the Oliver-Pharr method is that contact periphery sinks in as described by Sneddon s elastic contact analysis (Oliver and Pharr, 2004). The assumption is acceptable as Tabor (1970) found that the unloading stage of the indentation test is mostly elastic deformation. However, the Oliver-Pharr method could 13

30 not account for large plastic deformation that occurs during the loading stage which leads to pile-up deformation (Cheng and Cheng, 1998b, Bolshakov and Pharr, 1998). Since ε, P max and S are positive values, Eq. (3) cannot accurately predict the contact depth when pile-up occurs, i.e, when h c >h max. Contact area could be underestimated by as much as 50% in pile-up cases (Field and Swain, 1993 and 1995, Cheng and Cheng, 1998b, Bolshakov and Pharr, 1998, Oliver and Pharr, 2004) Indentation Hardness Hardness can be defined as a substance s resistance towards indentation (O Neill, 1934, Cheng and Cheng, 2004, Mukhopadhyay and Paufler, 2006). The relationship between hardness and fundamental material properties has long been a subject of study for researchers. Cheng and Cheng (2004) defined hardness as a function of a material s Young s modulus, E, yield strength, Y, and work-hardening exponent, n. Hill (1950) developed an expression for mean pressure, p, based on expanding a cylindrical or spherical cavity in an elastic-plastic material given by: ( ( ) ) (8) Based on Hill s equation, other researchers such as Marsh (1964), Hirst and Howse (1969) and Johnson (1970) proposed modified versions of the expanding cavity model. Johnson (1970 and 1985) argued that the mode of deformation is of radial expansion caused by hydrostatic stress beneath the indenter and must accommodate the volume of material displaced. Johnson (1970) proposed a modified expanding cavity model based on Poisson s ratio, v=0.5 given by: ( ) (9) 14

31 where θ is the half-angle of a conical indenter. However, the expanding cavity model cannot be applied to material that has high ratios of Y/E and it breaks down when probing harder materials with pronounced strain-hardening characteristics, i.e., most ceramics (Tabor, 1986, Fischer-Cripps and Lawn, 1996, Lawn, 1996, Fischer-Cripps 1997). Tabor (1970) developed a relationship between hardness and yield strength based on slip-line field theory given by: (10) where c is a constant with a value between 2.6 to 3 and Y 0 is a representative stress at a representative strain, ɛ 0 (ɛ 0 =0.08 for Vicker s indenter). Cheng and Cheng (2004) studied Tabor s idea of representative strain using finite element modelling suggested a modification to Tabor s equation: when when (11) where the representative strain, ɛ 0 =0.1. Cheng and Cheng (2004) found that for Y/E<0.02, H/Y 0 is about 2.4 to 2.8 which is consistent with Tabor s equation. However, H/Y 0 approaches 1.7 for Y/E> Particle Reinforced Composites A composite is a structural material that consists of two or more constituents that are combined at a macroscopic level and are not soluble with each other. Composites can be largely divided into 4 main categories depending on its matrix constituent which are 15

32 polymer matrix composite, metal matrix composite, ceramic matrix composite and carbon-carbon composites (Kaw, 2006). A particle-reinforced composite consists of particle reinforcements embedded in a matrix such as alloys and ceramics. Particulate composites often provide an advantageous blend of properties of the individual materials, which include (Nielsen, 1967): (a) Stiff particles to increase rigidity of material (b) Regulate the coefficient of thermal expansion and thermal shrinkage of the material (c) Improve heat resistance (d) Reduce creep (e) Increase the strength of polymer or other matrix materials (f) Modify the permeability behaviour to gases and liquids (g) Improve electrical properties (h) Modify rheological properties (i) Lower cost of material Some examples of practical reasons in using particulate composites include adding rubber particles to rigid and brittle polymer matrices to improve impact resistance, adding carbon black to elastomers to improve stiffness and metal particulates added to polymer matrices to improve electrical conductivity (Genetti et al., 1998, Biwa et al., 2001, Sohn et al., 2003, Pal, 2005). Particles may vary in shape, roundness and sphericity as shown in Figures 4 and 5. 16

33 Figure 4: Illustration of different particle shapes (Department of Civil and Geological Engineering, University of Saskatchewan, n.d.) Figure 5: Illustration of particle roundness and sphericity (Department of Civil and Geological Engineering, University of Saskatchewan, n.d.) Nanoindentation of Particle Reinforced Composites Nanoindentation of composites had been studied for various reasons such as composite interfacial shear strength and effects of environmental aging (Haeberle et al., 2001). There is significant interest in evaluating mechanical properties in particulate composites such as pharmaceutical solids (Braem et al., 1987 and 1989, Masterson and Cao, 2008, Salerno et al., 2012), i.e., dental resin. Nanoindentation of the reinforcement 17

34 in a particle-reinforced composite could provide insight into the properties of the reinforcement. Particle properties are dominant at small indentation depths while matrix influence increases at larger indentation depths when indentations are carried out on a particle embedded in a matrix (Durst et al., 2004). Figure 6 shows the load-displacement curve for soft and hard single-phase material and particle-matrix system. The dark triangle data represents the P-h curve for a single-phase hard particle and the bright triangle data represents the P-h curve for a single-phase soft particle. The dark circle data represents the P-h curve for a hard particle in soft matrix system and the bright circle data represents the P-h curve for a soft particle in hard matrix system. Figure 6: Load-displacement curve for soft and hard single phase material and particlematrix system (Durst et al., 2004) It is known that the Oliver-Pharr method, which is widely used to measure material properties in monolithic materials, cannot account for the influence of matrix on particle 18

35 properties in a particle-matrix system. However, Yan et al. (2011 and 2012) found that Oliver-Pharr method can still be used to measure the particle s material properties with sufficient accuracy in a particle-dominated indentation depth provided that the real contact area is used. Figure 7 illustrates the method used by Yan et al. (2012) to determine the particle-dominated depth where h m is the maximum indentation depth and E m p is the measured particle elastic modulus given by: ( ) (12) where v p is the poisson s ratio of the particle. Figure 7: Illustration of the method to determine the particle-dominated depth normalised by particle radius (Yan et al., 2012) Durst et al. (2004) numerically studied the influence of particle shape on normalised hardness, H FEM /H p, which is the ratio of measured hardness, H FEM, and particle s hardness, H p, found that the particle s hardness can be measured with confidence when the normalised contact radius, a c /r p <0.7, where a c is the radius of contact and r p is the 19

36 radius of the particle, as shown in Figure 8. Durst et al. (2004) also determined that the influence of matrix is more prominent for a hard particle when the particle boundary parallel to loading direction is smaller and for a soft particle when the particle boundary normal to loading direction is smaller due to plastic zone development during indentation. Figure 8: Relationship between normalised hardness and normalised contact radius for indentation on a disc shaped particle (Durst et al., 2004) Secondary Indentation In a soft particle-reinforced composite, where the hardness of the particle is lower than the hardness of its matrix, plastic deformation is more prominent in the horizontal direction. When indenting the particle, the harder matrix constraints plastic flow causes particle material to pile-up around the particle-matrix boundary (Durst et al, 2004). However, during a nanoindentation test on the reinforcement of a hard particle 20

37 reinforced composite, the mismatch in material properties of the particle and matrix creates a secondary indentation effect, where the hard particle itself indents into the soft matrix, as shown in Figure 9 (Durst et al., 2004, Leggoe, 2004, Leisen et al., 2012). Leggoe (2004) defined this secondary indentation, h sec, as the difference between the maximum indentation depth measured for the real indentation on the finite particle embedded in the matrix, h max, and that for the imaged indentation on a monolithic specimen consisting of the pure particle material, h sp max, at a same given load: (13) The indentation depth of the particle embedded in the matrix, h p, can be defined as the difference between h max and h sec given by: This definition suggests that the indentation depth of the particle at any given load is equal to h max sp and is independent of the size of the particle. (14) Indenter Particle Matrix Figure 9: Illustration of an axisymmetric indentation profile of a hard particle reinforced composite 21

38 2.3 Summary Hardness is a useful measurement to determine a substance s resistance towards indentation. In an indentation test on a particle embedded in a matrix, the mismatch in material properties may cause the matrix to influence particle readings. Yan et al. (2011 and 2012) studied the influence of matrix on measured elastic modulus and determined a particle-dominated depth at which Oliver-Pharr method could be utilised to determine particle s elastic modulus provided the true contact area is known. Durst et al. (2004) studied the influence of particle s aspect ratio on hardness measurements and determined that particle s hardness can be measured with confidence when the normalised contact radius, a c /r p <0.7. However, research carried out by Durst et al. (2004) focuses on particle s aspect ratio, not the influence of particle and matrix material properties on measured hardness. Also, a particle-dominated depth based on indentation depth to measure particle s hardness has yet to be established. The influence of material properties on the degree of pile-up and sink-in in indentation tests on bulk material has been studied by a variety of researchers throughout the years. However, there is less exposure for research on the influence of particle and matrix material properties on pile-up and sink-in in particle-matrix systems. For indentation on a particle embedded in a matrix, the particle may indent the matrix during indentation testing rendering the definition of pile-up and sink-in for monolithic material to be a poor representation of the degree of pile-up or sink-in experienced by the particle. 22

39 3.0 Pile-up and Sink-in in Nanoindentation on a Particle Embedded in a Matrix There has been a lot of research on the influence of material properties on pile-up and sink-in in single-phase material. However, for particle-reinforced composites, very little study has been carried out to understand the influence of matrix material properties on pile-up and sink-in. While Yan et al. (2011 and 2012) were able to determine the existence of a particle-dominated depth where the particle s elastic modulus can be measured, the issue of material pile-up can still provide complications to its measurement when the Oliver-Pharr method is applied. For indentation on a particle embedded in a matrix, the particle may indent the matrix during indentation testing. This causes the definition of pile-up and sink-in for monolithic material to be a poor representation of the degree of pile-up or sink-in experienced by the particle. A definition for pile-up and sink-in can be formulated if h p is known: (Pile-up) (Sink-in) (15) Applying Leggoe s method to check for pile-up or sink-in, h p, i.e., h sp max must be obtained. Generally, it can be obtained from either an additional physical indentation test on the monolithic particle material specimen with the same size as the composite, or from a virtual indentation test via computational simulation. Practically, it is difficult to have a pure particle material indentation specimen with the same size as the composite sample. The possibility to carry out the physical indentation test on the monolithic particle material specimen with the composite specimen size can be excluded. Because 23

40 the properties of the particle material are generally unknown in advance, the computerbased virtual indentation test cannot be carried out either. Therefore, it is highly implausible in practice to apply Leggoe s method to determine h p and to examine pileup or sink-in. In this chapter, a viable method to obtain pile-up and sink-in values for nanoindentation tests on a particle embedded in a matrix is presented. The influence of particle and matrix s material properties on pile-up and sink-in are also identified. The research completed is based on a simplified model of a conical indentation on a semi-spherical particle embedded into a semi-infinite matrix. 3.1 Methodology Conical indenter P θ R Semi-spherical particle Semi-infinite matrix Figure 10: Schematic illustration of the indentation model A semi-spherical particle embedded in the surface of a semi-infinite matrix was investigated in this work, as shown in Figure 10. This model corresponds to idealised 24

41 particle-reinforced composites with particles discretely distributed in a matrix or inclusions discretely distributed in inhomogeneous materials. In the indentation test, a conical indenter with a half-angle of 70.3, which is representative of a Berkovich or Vickers indenter, was pressed into the centre of an idealised semi-spherical particle. As illustrated in Figure 10, P is the applied load, R is the radius of the particle and θ is the half-angle of the conical indenter. In this research, the nanoindentation test was simulated via one cycle of loading and unloading using depth controlled indentation, whereby the indenter was pressed into the specimen until a specified h max was achieved. Bucaille et al. (2003) studied the influence of friction on indentation of elastic-plastic materials and found that friction has a negligible influence when the indenter s half-angle is more than 50. Therefore, frictionless contact between indenter and specimen was used in the simulations Finite Element Modelling Finite element modelling was used to simulate nanoindentation on the reinforcement of a particle reinforced composite. Finite element analysis has been successfully used by many researchers to probe micro and nanoindentation response of single-phase or dualphase material, i.e., Bhattacharya and Nix (1988), Laursen and Simo (1992), Bolshakov et al. (1996), Cheng and Cheng (1997, 1998a, 1998b, 1998c, 1998d, and 2004), Bolshakov and Pharr (1998), Myers et al. (1998), Knapp et al. (1999), Chen and Vlassak (2001), Dao et al. (2001), Chollacoop et al. (2003), Oliver and Pharr (2004), Durst et al. (2004), Leggoe (2004), Yan et al. (2011 and 2012). 25

42 (a) (b) Indenter Matrix Particle (c) Figure 11: Axisymmetric finite element model (a) model overview, (b) near particlematrix boundary and (c) fine-mesh region The indenter was assumed as a rigid body and it was constrained to vertical motion only. Studies have been carried out to verify the rigid body assumption of the indenter, as shown in Appendix A, where results from rigid indenter assumption was compared with results obtained from indenter with diamond properties. Results in Appendix A indicate that a difference of less than 1% comparing various output the ratio of 26

43 E/E i <0.6557, which is the ratio of specimen s Young s modulus compared to indenter s Young s modulus. Therefore, the assumption of rigid indenter is valid as long as E/E i < In this finite element model, the semi-spherical particle has a radius, R, of 4μm. The bottom of the specimen was constrained in both vertical and horizontal directions. Abaqus was utilised to simulate the nanoindentation test using an axissymmetric model, which consists of node axissymmetric elements (CAX4). Figure 11 illustrates the finite element model. The size of matrix is 160μm x 160μm in order to simulate the semi-infinite matrix. The size of matrix was selected based upon a study of elastic solids comparing simulation results to Sneddon s elastic solution equation at different aspect ratios of indentation depth to matrix size, as shown in Appendix B. The size of finemesh region is 3.2μm x 1.2μm while the fine-mesh size is 10nm x 10nm selected through fine mesh convergence study, as shown in Appendix C. Both particle and matrix are treated as deformable body and they are described by the classical von Mises elastic-plastic constitutive material model with the consideration of isotropic hardening throughout the analysis Dimensional Analysis Dimensional analysis is an important tool for developing mathematical models of physical phenomena. The physical laws and formulas do not depend on the unit system as the laws of nature establish a link between the quantities (Barenblatt, 1996, Misic et al., 2010). The functions that express physical laws must process a certain mathematical 27

44 property, called generalised homogeneity which allows for the number of arguments in mathematical expressions to be reduced (Cheng and Cheng, 2004). The key theorem in dimensional analysis is the Buckingham -theorem (Buckingham, 1914, 1915a and 1915b). Dimensional analysis had been successfully applied to analyse indentation response by researchers such as Evans and Charles (1976), Cheng and Cheng (1998a, 1998b, 1998c, 1998d and 2004), Tunvisut et al. (2001), Dao et al. (2001), Chollacoop et al. (2003), Yan et al. (2011 and 2012) and Kan et al. (2013). The contact depth, h c, and the particle indentation depth, h p, (see Figure 9) as a function of all its independent parameters are given by: ( ) ( ) (16) where E p, E m, Y p, Y m, v p, v m, n p and n m are the Young s modulus, yield strength, Poisson s ratio, and work-hardening exponent of the particle and matrix respectively. By applying Buckingham Π-theorem expressed in terms of R and E m, the following expressions were obtained: ( ) (17) where h c /h p, E p /E m, Y p /E m, Y m /E m, v p, v m, n p, n m, h max /R and θ are all dimensionless. Data were collected to analyse the relationship between h c /h p and E p /E m, Y p /E m, Y m /E m, v p, v m, n p, n m and h max /R. In this research, E m was set constant at 100GPa. Studies were carried out on both elastic perfectly plastic material and work hardened materials. The range of material property studied is given in Table 1. 28

45 Table 1: Range of material properties studied Minimum Value Dimensionless Parameter Maximum Value 0.2 E p /E m Y p /E m Y m /E m v p v m n p n m h max /R Reference Height Method Figure 9 illustrates the axisymmetric indentation profile from the indentation problem where a reference height h re, measured from the surface of the particle matrix boundary to the indenter tip in the vertical direction was used to estimate particle indentation depth. The basic assumption of this method is that h max /R is small so that the particlematrix boundary is sufficiently far away from the impression whereby the amount of vertical compression is small enough so that h p h re. In practice, imaging techniques such as atomic force microscopy (AFM) and scanning electron microscope (SEM) may be used to measure the vertical displacement at the particle-matrix boundary. Therefore, it is practically possible to apply the reference height method to check for pile-up in a nanoindentation testing. Simulations have been carried out to compare the results for h c /h p from the proposed reference height method and from the definition provided by 29

46 Leggoe (2004), see Eqs. (13) and (14). The difference between h c /h p values obtained from reference height method and Leggoe s method in percentage can be quantified by: ( ) ( ) ( ) (18) where (h c /h p ) RF and (h c /h p ) Leggoe are the h c /h p values obtained from the reference height method and Leggoe s method to determine h sec respectively. 3.2 Results and Discussion Influence of Young s Modulus and Yield Strength on Secondary Indentation Mismatch in material properties may cause secondary indentation whereby the particle indents the matrix, as shown in Figure 9. For a hard particle embedded in a soft matrix, secondary indentation occurs when indentation is large enough depending on the material properties of the particle and matrix. To better understand this phenomenon, a study was carried out to determine how material properties may influence secondary indentation. Secondary indentation was identified through observation of the particlematrix boundary on the surface of the specimen. The mismatches of material properties studied are Young s modulus and yield strength, as shown in Figure

47 (a) 5Y p =Y m (b) Y p =5Y m Particle Matrix Particle Matrix (c) 5E p =E m (d) E p =5E m Particle Matrix Particle Matrix Figure 12: Mismatch of Young s modulus and yield strength on secondary indentation where E p =100GPa, Y p =200MPa, v p =v m =0.3, n p =n m =0, h max /R=0.2 and E p =E m for (a) and (b), Y p =Y m for (c) and (d) Secondary indentation occurs when Y p >Y m, as shown in Figure 12(b), where the indentation on the matrix is visible. When Y p <Y m, as shown in Figure 12(a), there is a pile-up of particle material at the particle-matrix boundary instead. Since the yield strength of a material affects the hardness, see Eq. (11), the results reflect the fundamental knowledge that when a harder material is pressed onto a softer material, the harder material will deform the softer material. Results obtained in Figures 12(c) and 12(d) suggest that the mismatch in elastic modulus does not affect secondary indentation as there is no large distortion of material around the particle-matrix boundary. 31

48 (a) (b) Particle Matrix Particle Matrix (c) (d) Particle Matrix Particle Matrix Figure 13: Influence of Y/E on secondary indentation where E p =E m =70GPa, v p =v m =0.3, n p =n m =0, h max /R=0.25, Y p =4Y m where (a) Y p =400MPa, (b) Y p =800MPa, (c) Y p =1.6GPa and (d) Y p =2GPa Although observation from Figure 12 suggests that elastic modulus does not play a role in secondary indentation, the ratio of Y/E of a material may influence secondary indentation, as shown in Figure 13. All 4 figures in Figure 13 display the surface of particle-matrix boundary where E p =E m and Y p =4Y m with different ratios of Y p /E p (Y p /E m = for Figure 13(a), Y p /E m = for Figure 13(b), Y p /E m = Figure 13(c) and Y p /E m = for Figure 13(d)). In Figures 13(a) and 13(b), there is a small amount of secondary indentation. However, in Figures 13(c) and 13(d), there is no clear indication that secondary indentation occurred. This shows that the ratio of Y/E does influence secondary indentation since the ratio of Y/E influences the hardness of a material, see Eq. (11). 32

49 3.2.2 Accuracy of Reference Height Method to Determine Pile-up and Sink-in Figure 14: Influence of h max /R on d p with different ratios of Y p /E m where E p /E m =1.5, Y m /E m = , v p =v m =0.3 and n p =n m =0 Figures 14 and 15 show the influence of h max /R on d p with different ratios of Y p /E m and E p /E m respectively. From the simulation results, the percentage difference of the two methods does not exceed 10% when h max /R Furthermore, the results obtained indicate that both Leggoe s method and reference height method are in much better agreement when h max /R as there is little change in d p. However, when h max /R>0.125, the slope of the curves becomes very steep and the difference between Leggoe s method and reference height method is much larger. Therefore, the assumption that the reference height method is a viable method to determine pile-up and sink-in in particle-matrix system when h max /R is small is acceptable. Referring to the results obtained from Figures 14 and 15, the reference height method was applied to 33

50 determine pile-up and sink-in for nanoindentation of particle-matrix systems when h max /R 0.15 in the following subsection. Figure 15: Influence of h max /R on d p with different ratios of E p /E m where Y p /E m =0.01, Y m /E m = , v p =v m =0.3 and n p =n m = Influence of Material Properties on Pile-up and Sink-in Figure 16 displays the influence of E p /E m on h c /h p where 0.2 E p /E m 5. In all the cases illustrated below, increase in E p /E m causes h c /h p to increase. Observations are also in good agreement with the understanding of pile-up for monolithic material where smaller values of Y/E have higher degree of pile-up deformation (Cheng and Cheng, 2004). Influence of h max /R on pile-up may differ depending on the ratio of E p /E m, as shown in Figure 16(a). At E p /E m =1, the material is single-phase and both curves should theoretically intersect since pile-up and sink-in for single-phase material is not 34

51 influenced by indentation depth. It can also be observed that when E p /E m <1, increase in h max /R increases pile-up and the opposite is true when E p /E m >1. However, it should be noted that the influence of h max /R and v p on h c /h p is very small, as shown in Figures 16(a) and 16(d). The influence of matrix material properties is negligible as can be seen from Figures 16(c), 16(e) and 16(g). Table 2: Material Properties for graphs in Figure 16 h max /R Y p /E m Y m /E m v p v m n p n m (a) (b) (c) (d) (e) (f) (g)

52 Figure 16: Influence of E p /E m on pile-up and sink-in (Material properties listed in Table 2) Analysing the results obtained, it can be seen that apart from Figure 16(f) when n p =0.3, influence of material property on pile-up reduces when h c /h p approaches a value of roughly 1.3. Chen and Vlassak (2001) found that the degree of pile-up or sink-in ranges from 0.64 to Taking into account numerical error and estimation error, the upper- 36

53 limit of pile-up/sink-in value proposed by Chen and Vlassak (2001) is in good agreement with pile-up/sink-in value obtained from the simulation of the particle-matrix system. Table 3: Material Properties for graphs in Figure 17 h max /R E p /E m Y m /E m v p v m n p n m (a) (b) (c) (d) (e) (f) (g)

54 Figure 17: Influence of Y p /E m on pile-up and sink-in (Material properties listed in Table 3) The relationship between Y p /E m and h c /h p was analysed within the range of Y p /E m 0.01, as shown in Figure 17. Simulation results reveal that h c /h p reduces with increasing Y p /E m. Results are consistent with observations made by Cheng and Cheng (2004) on single-phase materials, in which the increase in Y/E reduces pile-up 38

55 deformation. It can also be observed that the influence of h max /R and v p on h c /h p is very small, as shown in Figures 17(a) and 17(d), while the influence matrix material properties on pile-up is negligible, as shown in Figures 17(c), 17(e) and 17(g). Applying a curve fitting, the relationship between Y p /E m and h c /h p can be very well quantified by an exponential decay curve where R for all results obtained, as shown in Figure 17. Table 4: Material Properties for graphs in Figure 18 h max /R E p /E m Y p /E m v p v m n p n m (a) (b) (c) (d) (e) (f) (g)

56 Figure 18: Influence of Y m /E m on pile-up and sink-in (Material properties listed in Table 4) Figure 18 shows the relationship between Y m /E m and h c /h p where Y m /E m Simulation results reveal that Y m /E m does not influence h c /h p regardless of other dimensionless parameters. Although a change in h c /h p can be observed in Figures 18(a) and 18(f) when Y m /E m is very small, the difference in h c /h p when Y m /E m = and 40

57 Y m /E m =0.01 for Figure 18(a) when h max /R=0.15 is only 3.33% and for Figure 18(f) when n p =0.3 is only 3.14%. It is more likely that the anomaly is due numerical error as opposed to the influence of Y m /E m on pile-up and sink-in. Analysing Figure 18, it can be seen that only E p /E m, Y p /E m and n p have significant influence on h c /h p value, as shown in Figures 18(b), 18(c) and 18(f). Table 5: Material Properties for graphs in Figure 19 h max /R E p /E m Y p /E m Y m /E m v m n p n m (a) (b) (c) (d) (e) (f) (g)

58 Figure 19: Influence of v p on pile-up and sink-in (Material properties listed in Table 5) As shown in Figure 19, h c /h p increases with increasing values of v p within the range of 0.01 v p Poisson s ratio is defined by the following equation: v d d trans (19) axial 42

59 where ε trans and ε axial are the transverse and axial strains respectively when loading is applied in axial direction only. A larger Poisson s ratio value will therefore increase transverse deformation and cause more material to pile-up around the contact impression. A linear relationship between v p and h c /h p exists as shown by the linear curve fitting in Figure 19 where R for all results obtained. However, analysis of the effect of v p on pile-up deformation revealed that the influence is very small where the maximum difference between h c /h p values obtained when v p =0.01 and v p =0.49 for any 1 curve in Figure 19 is 5.75%. Results from Figure 20 displays the influence of v m on h c /h p. Results indicate that the influence of matrix s Poisson s ratio on pile-up is negligible. Table 6: Material Properties for graphs in Figure 20 h max /R E p /E m Y p /E m Y m /E m v p n p n m (a) (b) (c) (d) (e) (f) (g)

60 Figure 20: Influence of v m on pile-up and sink-in (Material properties listed in Table 6) 44

61 Table 7: Material Properties for graphs in Figure 21 h max /R E p /E m Y p /E m Y m /E m V p v m n m (a) (b) (c) (d) (e) (f) (g)

62 Figure 21: Influence of n p on pile-up and sink-in (Material properties listed in Table 7) The relationship between n p and h c /h p is shown in Figure 21 ranging from 0 n p 0.5. The influence of particle s work-hardening exponent on pile-up and sink-in is similar to the influence of particle s yield strength, where an increase in n p reduces h c /h p, as shown in Figure 17. Based on Hollomon s equation, the relationship between yield strength and work-hardening exponent is given by (Hollomon, 1945): (20) where K is the strength index and ɛ p is plastic strain. According to Eq. (20), a larger work-hardening exponent will result in higher yield strength at a given plastic strain. Since nanoindentation is based on deformation, which will cause significant plastic straining, similarity of the curve displayed by particle s work-hardening exponent and particle s yield strength influence on pile-up and sink-in can be explained by the 46

63 influence of work-hardening exponent on yield strength. Also similar to Y p /E m, the relationship between n p and h c /h p can be explained through an exponential decay curve shown in Figure 21 where exponential decay curve fitting with confidence level of 95% produced R for all the results obtained. Based on Figure 21(b), an observation can be made regarding the reduced influence of E p /E m on pile-up and sink-in when n p is large. At n p =0.5, both curves where E p /E m =1 and E p /E m =2 provide the same value of h c /h p. As shown in Figure 22, the influence of matrix work-hardening on pile-up and sink-in is negligible in the same way that matrix s yield strength has negligible influence on pile-up and sink-in. Table 8: Material Properties for graphs in Figure 22 h max /R E p /E m Y p /E m Y m /E m V p v m n p (a) (b) (c) (d) (e) (f) (g)

64 Figure 22: Influence of n m on pile-up and sink-in (Material properties listed in Table 8) 48

65 Simulation results showing the relationship between h max /R and h c /h p are displayed in Figure 23 within the range of 0.05 h max /R Based on the results obtained, the influence of h max /R on h c /h p seems to be negligible. However, this may be due to the range of h max /R that was studied. The limitations of reference height method which is based on the assumption of a small h max /R value means that the influence it has on pileup and sink-in cannot be accurately determined when h max /R is large. Referring to Figures 23(a) and 23(f) when E p /E m =2 and n p =0.3 respectively, a slight change in gradient of the curve can be seen when h max /R>0.1. However, the change is very small and may be subjected to numerical and estimation error. Therefore, it is difficult to conclude the influence of h max /R on h c /h p within the range of h max /R studied. Table 9: Material Properties for graphs in Figure 23 E p /E m Y p /E m Y m /E m v p v m n p n m (a) (b) (c) (d) (e) (f) (g)

66 Figure 23: Influence of h max /R on pile-up and sink-in (Material properties listed in Table 9) 50

67 3.3 Summary The influence of elastic modulus and yield strength on secondary indentation had been studied in this chapter and it was concluded that elastic modulus to yield strength s ratio, Y/E, of both the particle and the matrix influences the amount of secondary indentation experienced by the particle-matrix system. The reference height, h re, is a proposed method to approximately quantify the particle indentation depth, h p, based on the assumption that the normalised indentation depth, h max /R, is small enough that h p h re. Studies were carried out to determine the feasibility of using the reference height method to determine the degree of pile-up and sink-in from a nanoindentation test on the particle of a particle-matrix system. Comparisons were made with pile-up and sink-in value obtained using Leggoe s method to quantify h p. By applying a tolerance margin of 10%, results have shown that the reference height method can be used to evaluate pile-up and sink-in when h max /R The influence of particle and matrix material properties on the degree of pile-up and sink-in from a nanoindentation test on the particle of a particle-matrix system reveals that the mechanical properties of the particle represented by particle s elastic modulus, E p, particle s yield strength, Y p, and particle s work-hardening exponent, n p, have the greatest influence on pile-up and sink-in when h max /R 0.15: 1) Increase in Young s modulus of particle, E p, causes a larger degree of pile-up deformation, h c /h p. 51

68 2) Increase in yield strength of particle, Y p, causes a decrease in the degree of pileup deformation, h c /h p, which can be fitted by an exponential decay curve. 3) Increase in work-hardening exponent of particle, n p, causes a decrease in the degree of pile-up deformation, h c /h p, which can be fitted by an exponential decay curve. 4) Particle s Poisson s ratio, v p, has a linear relationship with the degree of pile-up deformation, h c /h p. However, the influence of v p on h c /h p is negligible. 5) Matrix material properties has negligible influence on pile-up and sink-in when the normalised indentation depth, h max /R

69 4.0 Measured Hardness in Nanoindentation on a Particle Embedded in a Matrix Durst et al. (2004) studied the influence of particle shape and determined that the particle s hardness can be measured when a c /r p <0.7. However, it should be noted that the study carried out by Durst et al. (2004) focussed on aspect ratio of particle size and did not cover a wide range of material properties. The conclusion was based on the study of indentation on a hard particle embedded in a soft matrix and a soft particle embedded in a hard matrix (E=200GPa and Y=800MPa for hard material and E=200GPa and Y=400MPa for soft material). As a result, only the influence of a mismatch in yield strength of particle and matrix on measured hardness has been partly identified. Also, the limited number of material properties tested calls into question the validity of the rule of a c /r p <0.7. It is difficult to extend this rule to practical situations as the mismatch in material properties can be much larger than the cases studied by Durst et al. (2004). In a nanoindentation test, a c can only be probed after the conclusion of the loading cycle in a nanoindentation test. Yan et al. (2011 and 2012) established a particle-dominated depth based on normalised indentation depth, h max /R for measuring elastic modulus. The attractiveness of this particle-dominated depth is that the maximum indentation depth, h max can be used as a guideline to measure the elastic modulus of a particle embedded in a matrix from a nanoindentation test. However, a particle-dominated depth for measuring hardness has yet to be found. 53

70 In this chapter, the influence of particle and matrix material properties on measured hardness was identified and a particle-dominated depth at which accurate measurements of particle s hardness can be obtained is documented. Research carried out is based on a simplified model of a conical indentation on a semi-spherical particle embedded in the surface of a semi-infinite matrix. 4.1 Methodology The model consists of a semi-spherical particle embedded in the surface of a semiinfinite matrix and a conical indenter with a half-angle of This is representative of a Berkovich indenter and was pressed into the centre of an idealised semi-spherical particle. Figure 10 represents the schematic illustration of the indentation model used in this research where P is the applied load, R is the radius of the particle and θ is the halfangle of the conical indenter. In this finite element model, the semi-spherical particle has a radius, R, of 4μm. In this research, nanoindentation test was simulated via one cycle of loading and unloading using depth controlled indentation whereby the indenter was pressed into the specimen until a specified h max was achieved. Bucaille et al. (2003) studied the influence of friction on indentation of elastic-plastic materials and found that friction has a negligible influence when the indenter s half-angle is more than 50. Therefore, frictionless contact between indenter and specimen was used in the simulations. 54

71 4.1.1 Finite Element Modelling To study the influence of material properties on measured hardness, the same finite element model presented in Chapter 3 was utilised. For details, refer to Chapter Indentation hardness, H, can be measured from Eq. (5). In order to avoid pile-up and sink-in complications which may produce significant reading errors if Oliver-Pharr method was used, the contact area, A, was calculated based on the radius of contact, a c, which was obtained directly from the finite element simulations. Since the finite element model is axisymmetric, the projected contact area, A, is given by: (23) Dimensional Analysis Indentation hardness, H, which is the particle s hardness measured from a nanoindentation test of a particle embedded in a matrix as a function of all the independent parameters is given by: ( ) (21) where E p, E m, Y p, Y m, v p, v m, n p and n m are the Young s modulus, yield strength, Poisson s ratio, and work-hardening exponent of the particle and matrix respectively. By applying Buckingham Π-theorem expressed in terms of R and Y p, the following expression can be obtained: ( ) (22) where H/Y p, E p /Y p, E m /Y p, Y m /Y p, v p, v m, n p, n m, h max /R and θ are all dimensionless. Data from numerical simulations were collected to analyse the relationship between H/Y p and 55

72 E p /Y p, E m /Y p, Y m /Y p, v p, v m, n p, n m and h max /R. In this research, Y p was set constant at 200MPa. Studies were carried out on both elastic perfectly plastic material and work hardened materials. The range of material property studied is given in Table 10. Table 10: Range of material properties studied Minimum Value Dimensionless Parameter Maximum Value 0.1 E p /Y p E m /Y p Y m /Y p v p v m n p n m h max /R 0.25 The representative strain method used by other researchers is based on an empirical study of Tabor s relationship of hardness and yield strength which originally could not account for work-hardening. One issue is that researchers could not agree on a specific value for representative strain. Therefore, the representative strain method was not used in this research. 56

73 4.2 Results and Discussion Influence of Material Properties on Measured Hardness Figure 24 shows the influence of Y m /Y p on H/Y p for 0.1 Y m /Y p 5 under different given conditions. H p is the particle s hardness as measured from indentation tests of a bulk material with particle properties, i.e., true hardness value of the particle material. H p /Y p is the ratio of particle s hardness and yield strength. It can be observed that the influence of Y m /Y p on measured hardness is large when Y m /Y p <1 while the influence of Y m /Y p on H/Y p is small when Y m /Y p >1. Results obtained shows that H/Y p increases with Y m /Y p. However, the aforementioned statement is not true for cases where Y p >>Y m. As can be seen in Figures 24(a) to 24(e) when Y m /Y p <0.2, H/Y p increases with decreasing Y m /Y p. An observation can be obtained from Figures 24(f) and 24(g) when particle and matrix materials are work-hardened. Previous observations indicates that H/Y p increases with decreasing Y m /Y p when Y m /Y p <0.2. In Figure 24(f) when n p =0.3, this phenomenon occurs when Y m /Y p is larger than 0.2. On the contrary, in Figure 24(g) when n m =0.3, this phenomenon does not occur even when Y m /Y p =0.1. In Figure 24(f), the hardness of particle compared to matrix would have been much larger due to work-hardening of the particle while the opposite is true for Figure 24(g) due to work-hardening of the matrix. The greater mismatch in hardness in Figure 24(f) may have caused H/Y p to increase with decreasing Y m /Y p much earlier compared to the elastic-perfectly plastic examples. Results suggest that particle properties are more dominant if hardness of particle is much greater than matrix. 57

74 Table 11: Material Properties for graphs in Figure 24 h max /R E p /Y p E m /Y p v p v m n p n m (a) (b) (c) (d) (e) (f) (g)

75 Figure 24: Influence of Y m /Y p on H/Y p (Material Properties listed in Table 11) Table 12: Material Properties for graphs in Figure 25 h max /R Y m /Y p E m /Y p v p v m n p n m (a) (b) (c) (d) (e) (f) (g)

76 Figure 25 shows the influence of E p /Y p on H/Y p within the range of 100 E p /Y p Results obtained show that the influence of E p /Y p on H/Y p is relatively small when E p /Y p 500 and n p =0. When E p /Y p <500 and n p =0, there is an increase in H/Y p value. Even though exact reasons are not fully understood, the repeatability of this phenomenon shows that the nature of the curve is not due to numerical errors. The only exceptions are cases with a work-hardened particle when n p =0.1 and n p =0.3, as shown in Figure 25(f). As observed in Figures 24(f) and 25(f), the influence of other material properties on H/Y p are significantly different between work-hardened particles and elastic-perfectly plastic particles. 60

77 Figure 25: Influence of E p /Y p on H/Y p (Material Properties listed in Table 12) Table 13: Material Properties for graphs in Figure 26 h max /R Y m /Y p E p /Y p v p v m n p n m (a) (b) (c) (d) (e) (f) (g)

78 Figure 26: Influence of E m /Y p on H/Y p (Material Properties listed in Table 13) 62

79 Figure 26 shows the influence of E m /Y p on H/Y p within the range of 100 E p /Y p Results obtained show that the influence of matrix s elastic modulus is small. The largest difference in H/Y p between any 2 data points in a curve shown in Figure 26 is 3.84% when n p =0.3. Table 14: Material Properties for graphs in Figure 27 h max /R Y m /Y p E p /Y p E m /Y p v m n p n m (a) (b) (c) (d) (e) (f) (g)

80 Figure 27: Influence of v p on H/Y p (Material Properties listed in Table 14) Figure 27 displays the relationship between H/Y p and v p for 0.01 v p Comparing all the curves obtained in Figure 27, the largest difference in H/Y p when v p changed from 0.01 to 0.49 is only 1.2%, which occurs when n p =0.3 (See Figure 27(f)). Referring to the results obtained, the influence of v p is negligible as the change in H/Y p corresponding to a change v p is very small. 64

81 Figure 28 displays the influence of v m on H/Y p. Results obtained from Figure 28 shows that matrix s Poisson s ratio does not influence the measured hardness. Observing graphs obtained in Figures 27 and 28, in line with conventional wisdom, it can be concluded that the influence on Poisson s ratio of hardness is negligible. Table 15: Material Properties for graphs in Figure 28 h max /R Y m /Y p E p /Y p E m /Y p v p n p n m (a) (b) (c) (d) (e) (f) (g)

82 Figure 28: Influence of v m on H/Y p (Material Properties listed in Table 15) Figure 29 displays the influence of n p on measured hardness. Results have shown that increasing particle s work-hardening exponent increases measured hardness. The influence of n p on measured hardness or particle s hardness is highly significant. When n p =0.5, H/Y p is between 15 and 25 for all cases shown in Figure 29. Comparing it with other graphs obtained in this section, H/Y p is usually between 2.2 and 2.7 when the 66

83 particle is elastic-perfectly plastic. Also, unlike previous figures which displayed a large difference in H/Y p value for different ratios of Y m /Y p, Figure 29(b) displays very similar results comparing Y m /Y p =1 and Y m /Y p =

84 Figure 29: Influence of n p on H/Y p (Material Properties listed in Table 16) Table 16: Material Properties for graphs in Figure 29 h max /R Y m /Y p E p /Y p E m /Y p v p v m n m (a) (b) (c) (d) (e) (f) (g) Figure 30 shows the relationship between n m and H/Y p. As n m increases, H/Y p increases as well. Also, when comparing n m =0.4 and n m =0.5, some of the curves recorded a drop in H/Y p value. It is highly likely that this phenomenon was caused by numerical error. Nevertheless, the influence of n m on measured hardness is small except when Y m /Y p =0.5, as shown in Figure 30(b), and n p =0.3, as shown in Figure 30(g). Referring 68

85 to Figures 24(g) and 30(b), it is evident that n m has a large influence on measured hardness when Y m /Y p <1. Table 17: Material Properties for graphs in Figure 30 h max /R Y m /Y p E p /Y p E m /Y p v p v m n p (a) (b) (c) (d) (e) (f) (g)

86 Figure 30: Influence of n m on H/Y p (Material Properties listed in Table 17) Influence of Indentation Depth on Measured Hardness Hardness measured from indentation testing of a single-phase material is independent of indentation depth. However, due to the dual-phase nature of particle-matrix systems, matrix material properties can influence measurements obtained from nanoindentation tests on the particle. Therefore, it is important to determine the influence of the mismatch in material properties between particle and matrix on measured hardness and how this influence may correlate with indentation depth. For the purpose of this study, the measured hardness, H, was normalised with the particle s hardness obtained from a finite element simulation of a nanoindentation test on a single-phase material with 70

87 particle properties, H p. The relationship between normalised hardness, H/H p, and normalised indentation depth, h max /R, was investigated. Figure 31 shows the influence of v p and v m on the relationship between normalised hardness, H/H p, and normalised indentation depth, h max /R. From the graphs obtained, it is clear that Poisson s ratio has a negligible influence on normalised hardness regardless of indentation depth. Therefore, the analysis can be simplified to exclude Poisson s ratio when determining the particle-dominated depth. Figure 31: Relationship between H/H p and h max /R for different ratios of v p and v m when Y m /Y p =1, E p /Y p =E m /Y p =500 and n p =n m =0, and (a) v m =0.3 and (b) v p =0.3 Figure 32 displays the influence of elastic modulus and yield strength on the relationship between normalised hardness and normalised indentation depth. When h max /R=0.1, all 4 curves in Figure 32 has normalised hardness, H/H p 1. As h max /R increases, H/H p diverges away from 1. The dashed lines in the graph are boundaries used to determine the particle-dominated depth at which the measured hardness is within 5% difference of particle s true hardness. Observing the trend of the curves in Figure 32, there is evidence to suggest that when particle and matrix are elastic- 71

88 perfectly plastic, the particle-dominated depth for nanoindentation testing to measure hardness is smaller when Y p >Y m. Figure 32: Relationship between H/H p and h max /R for different ratios of E p /Y p, E m /Y p and Y m /Y p when v p = v m =0.3 and n p =n m =0 Referring to Figure 33, H/H p 1 as n p increases for Y m /Y p =0.2. However, when Y m >Y p, the influence of n p on normalised hardness is not proportional. In Figure 33 for Y m /Y p =5, the curve representing n p =0.5 is between those of n p =0.1 and n p =0.3. Figure 34 studies the influence of n p on normalised indentation depth when Y m /Y p =5. Based on the results obtained in Figure 34, influence of matrix material properties is the most prominent when n p =0.3. Figure 35 displays the influence of n m on accuracy of measured hardness. When Y m /Y p =0.2, H/H p 1 when n m increases. For Y p >Y m, work-hardening of matrix increases 72

89 the accuracy of measured hardness. For Y m /Y p =5, curves for n m =0.1, n m =0.3 and n m =0.5 overlapped each other on the graph shown in Figure 35. Therefore, it can be concluded that n m has no influence on H/H p when Y m >Y p. Figure 33: Relationship between H/H p and h max /R for different ratios of n p and Y m /Y p when E p /Y p =500, E m /Y p =1000, v p =v m =0.3 and n m =0 73

90 Figure 34: Relationship between H/H p and h max /R for different ratios of n p when Y m /Y p =5, E p /Y p =100, E m /Y p =2500, v p =v m =0.3 and n m =0 Figure 35: Relationship between H/H p and h max /R for different ratios of n m and Y m /Y p when E p /Y p =500, E m /Y p =1000, v p =v m =0.3 and n p =0 74

91 4.2.3 Particle-Dominated Depth for Measuring Particle s Hardness As shown in Chapter 4.2.2, a mismatch between material properties of particle and matrix influences accuracy of measured hardness to determine particle s true hardness. At larger indentation depths relative to particle s radius, the measured hardness of particle may be significantly different from the particle s true hardness value. The purpose of this section of the research is to establish a particle-dominated indentation depth whereby the particle s hardness can be measured with sufficient accuracy in a way that the influence of matrix can be considered negligible. Particle-dominated depth, h pd /R, is an indentation depth normalised with particle radius within which indentation test to measure particle s hardness is viable. For the purpose of determining the particle-dominated depth, there are 2 modes to be considered. The first mode is when the indentation depth at which the influence of matrix properties on measured hardness can no longer be considered negligible. Applying a tolerance error of 5%, the normalised hardness, H/H p, is between 0.95 for hard particle in soft matrix and 1.05 for soft particle in hard matrix if the normalised indentation depth is less than the particle-dominated depth. Referring to Figures 32 to 35, the particle-dominated depth, h pd /R, can be determined from normalised indentation depth, h max /R, at the point which a curve intersects H/H p =0.95 or H/H p =1.05. The second mode to be considered is when the indenter establishes direct contact with matrix. In these cases before direct contact, matrix properties have a negligible influence on measured hardness, which occurs when particle s hardness is much greater than matrix s hardness and the presence of large secondary indentation. However, as 75

92 direct contact is established, matrix properties will have a direct influence on measurements. In this mode, particle-dominated depth was determined by the indentation depth at which indenter establishes direct contact with matrix. Figure 36 is an example where the second mode of particle-dominated depth occurred. In the example of Figure 36, the indentation depth, h, is 1.303µm and the normalised hardness, H/H p, is Although the normalised hardness is between 0.95 and 1.05, increasing indentation depth causes indenter and matrix to have direct contact, causing the matrix to exert direct influence on measurements. Therefore, the particle-dominated depth, h pd /R= Indenter Matrix Particle Figure 36: Second mode of particle-dominated depth where Y m /Y p =0.1, E p /Y p =2500, E m /Y p =500, v p =v m =0.3 and n p =n m =0 For the purpose of studying the influence of material properties on particle-dominated depth, v p and v m are considered to have negligible influence. This conclusion is 76