Instrumented indentation testing (IIT) is a technique for

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1 FEATURE Nanomechanical Characterization of Materials by Nanoindentation Series INTRODUCTION TO INSTRUMENTED INDENTATION TESTING by J. Hay Instrumented indentation testing (IIT) is a technique for measuring the mechanical properties of materials. It is a development of traditional hardness tests such as Brinell, Rockwell, Vickers, and Knoop. IIT is similar to traditional hardness testing in that a hard indenter, usually diamond, is pressed into contact with the test material. However, traditional hardness testing yields only one measure of deformation at one applied force, whereas during an IIT test, force and penetration are measured for the entire time that the indenter is in contact with the material. All of the advantages of IIT derive from this continuous measurement of force and displacement. IIT is particularly well suited for testing small volumes of material such as thin films, particles, or other small features. It is most commonly used to measure Young s modulus (E) and hardness (H). 1 2 The Young s modulus for a material is the relationship between stress and strain when deformation is elastic. If an engineer knows the Young s modulus for his design material, then he can predict strain for a known stress, and vice versa. In metals, hardness depends directly on the flow stress of the material at the strain caused by the indentation. In other words, hardness is an indirect but simple measure of flow stress; within a class of metals, the metal with the higher hardness will also have the higher flow stress. In addition to Young s modulus and hardness, IIT has also been used to measure complex modulus in polymers and biomaterials, 3,4 yield stress and creep in metals, 5,6 and fracture toughness in glasses and ceramics. 7 This article is the first in an ET feature series on the technique. Thus, this article covers basic testing and analysis procedures; future articles will address advanced applications. Readers who wish to gain a comprehensive understanding of this technology should begin by reading the landmark article published in 1992 by Warren Oliver and George Pharr 2 ; as of 2009, this article had over 4700 citations according to the ISI Web of Knowledge. Next, review articles 8 11 and a few textbooks provide a strong foundation for understanding current research. Finally, in 2004 and 2009, the Journal of Materials Research published special-focus issues Editor s Note: This article is part of the ET Feature Series on Nanomechanical Characterization of Materials by Nanoindentation (instrumented indentation testing) which lends itself to testing small volumes of material. This series features articles describing the main experimental technique(s), background theory and data reduction, testing of polymers, and low-k thin films. Additionally, most recent developments in these techniques will be addressed throughout the series. Series editor: Ibrahim Miskioglu, Michigan Technological University. J. Hay is an applications engineer in the Agilent Technologies Nanotechnology Measurements Operation, Oak Ridge, TN. on IIT These will be excellent reference volumes for years to come. Terminology is often a significant barrier to understanding a new technique. Therefore, a summary of terms that are most often used in IIT is provided in Table 1. Figure 1 illustrates some of these terms. TYPICAL TEST CHRONOLOGY Figure 2 shows a typical force time history for an IIT test; numbers on the chart correspond with these test segments: 0. The indenter approaches the test surface until contact is sensed. Contact is sensed as an increase in stiffness, relative to the stiffness of the mechanism supporting the indenter column. Approach rate and detection limit are likely user-specified inputs. 1. The indenter is pressed into contact with the test material until the maximum force or penetration is achieved. The mechanism for pressing may be either force-controlled or displacement-controlled. Both the rate and limit are likely user-specified inputs. 2. The force on the indenter is held constant for a dwell time at the peak force. Dwell time is likely a userspecified input. 3. The indenter is withdrawn from the sample at a rate that is comparable to the pressing rate until the force becomes a small percentage of the peak force, usually 10%. 4. The force on the indenter is held constant for a dwell time. The purpose of this test segment is to acquire enough data to determine how much of the measured displacement should be attributed to thermal expansion and contraction of the equipment and/or test material, called thermal drift. If thermal drift is expected to be small relative to the overall penetration of the test, this segment may be omitted. 5. The indenter is withdrawn from the sample completely. ANALYSIS STAGE 1: GENERATE A FORCE DISPLACEMENT CURVE Analysis of IIT data may be divided into two stages. The first stage is rather system-dependent, but it ends with the generation of a force displacement curve, which is shown in Fig. 3. The origin of this plot is the point at which the indenter first contacted the test surface. As the applied force increases, the displacement increases as well, until the peak test force is achieved. Then, as the contact force decreases, some of the displacement is generally recovered, though usually not all. If the contact were completely doi: /j x 66 EXPERIMENTAL TECHNIQUES November/December , Society for Experimental Mechanics

2 Table 1 Summary of common terms SYMBOL DIMENSIONS TERM RELEVANCE P F Contact force Force exerted by the indenter on the sample and vice versa. Letter F is also used sometimes to represent this quantity. h L Displacement Penetration of the indenter into the test material, relative to the position at which the indenter first contacted the surface. S FL 1 Contact stiffness Elastic stiffness of the contact; relationship between F and h when deformation produced by relative motion between indenter and sample is entirely elastic. h c L Contact depth Depth over which the material makes contact with the indenter tip; usually h c is less than h due to the elastic deformation of the surface outside the contact. a L Contact radius Normal distance from the indenter axis to the edge of contact for axisymmetric indenters; if the indenter is a pyramid, then an equivalent contact radius is calculated as a = (A/π) 1/2. A L 2 Contact area Projected contact area, that is surface contact area projected onto a plane normal to the direction of indentation. E r FL 2 Reduced elastic modulus Effectively, this is the plane-strain modulus of the test material, although strictly, this is only true in the case of a perfectly rigid indenter and an isotropic test material. E FL 2 Young s modulus Young s modulus of the test material; generally, E provides the link between stress and strain for a material undergoing elastic deformation, σ = Eε. ν Poisson s ratio Poisson s ratio of the test material; this value must be known or estimated in order to calculate E. Generally, ν is an elastic property of the material that provides the relationship between elastic strains in two directions. In a tensile test of a cylindrical specimen that produces radial strain ε r and axial strain ε l, ν = ε r /ε l. E i FL 2 Young s modulus of the indenter Young s modulus of the material comprising the indenter; for diamond, this value is 1140 GPa. ν i Poisson s ratio of the indenter Poisson s ratio of the material comprising the indenter; for diamond, this value is H FL 2 Hardness Mean pressure of the contact; H = P/A. In metals, the hardness measured for a plastic contact is directly related to the flow stress at the strain caused by the indentation. Fig. 1: h h c a Schematic of an indentation test ψ plastic, then the unloading curve would be exactly vertical. If the contact were completely elastic, then the unloading curve would coincide with the loading curve. The process of generating a force displacement curve generally comprises these two steps: 1. Raw measurements of force, displacement, and time are tared at the point of contact. The point of contact is not necessarily the same point at which segment 0 of the test chronology terminated. With pre- and post-contact data available, it is usually possible to refine the determination of the contact point. P z A r 2. Channels of P and h are calculated as a function of raw measurements, machine influence, and environment. For example, the calculation of P should include a term to account for the force exerted by the mechanism supporting the indenter shaft. The calculation of h should include terms to account for machine compliance and thermal drift. The manufacturer of the indentation equipment should provide the details of these calculations to the user. ANALYSIS STAGE 2: CALCULATE MATERIAL PROPERTIES The second stage of analysis is the calculation of material properties from the load displacement curve shown in Fig. 3. Here, calculations are presented in an order that is intuitive, but exactly opposite to the order in which calculations are performed in practice. That is, calculations of the desired properties, H and E, are presented first, and then supporting calculations are presented as they are required. But when data are analyzed in practice, Eq. 7 is executed first, and Eqs. 1 and 2 are executed last. Hardness is calculated as H = P/A (1) November/December 2009 EXPERIMENTAL TECHNIQUES 67

3 Force [mn] Time [sec] Fig. 2: Typical force time history for an instrumented indentation test; the origin of this plot is the point at which the indenter first touches the test surface P, Force on surface [mn] h, Displacement into Surface [nm] Although calculation of Young s modulus (Eq. 2) requires knowing the Poisson s ratio of the sample (ν), the sensitivity is weak. Sensitivity analysis reveals that a generous uncertainty of 40% in the Poisson s ratio manifests as only a 5% uncertainty in the Young s modulus. If the Poisson s ratio of the test sample is unknown, a value should be chosen according to the following guidelines: 0.2 for glasses and ceramics, 0.3 for metals, and 0.45 for polymers. Contact area, which is used in Eqs. 1 and 3, is calculated as a function of contact depth A = f (h c ) (4) Fig. 3: Typical force displacement curve for an instrumented indentation test; this plot is the output of Analysis Stage 1 and the input to Analysis Stage 2. Numbers on the plot correspond to the test segments identified in Fig. 2; data in segments 2 and 4 are plotted, but are not visible as a function of displacement Young s modulus is calculated from the reduced modulus, E r,as [ E = (1 ν 2 1 ) 1 ] 1 ν2 i (2) E r E i and the reduced modulus is calculated as 1,2 π S E r = (3) 2 A The exact form of this function, called the area function, depends on the geometry of the indenter and the scale of the test. The most common indenter is a Berkovich diamond; Berkovich is the geometry and diamond is the material. The Berkovich geometry is a three-sided pyramid that has about the same aspect ratio as a Vickers pyramid (which is four-sided). Figure 4 shows a residual impression left by a Berkovich indenter. A Berkovich diamond indenter is ideal for most testing because: Its mechanical properties are well known. It is not easily damaged. It is relatively easy to manufacture well. It induces plasticity at very small loads, thereby allowing a meaningful measure of hardness. Its relatively large included angle minimizes the influence of friction. The similarity to the Vickers indenter allows the calculation of a meaningful Vickers Hardness Number (VHN); the relationship is VHN (kg/mm 2 ) = 92.7 H(GPa). 68 EXPERIMENTAL TECHNIQUES November/December 2009

4 Table 2 Area functions for various indenter geometries TIP TYPE AREA FUNCTION COMMENTS Fig. 4: Residual impression in nickel, made by a Berkovich diamond indenter For an ideal Berkovich indenter, the area function is A = 24.56h 2 c (5a) If the total indentation depth is greater than 2 microns, then Eq. 5a may be used to calculate A, becausethedifference between ideal and real geometry is negligible at this scale. If the total indentation depth is less than 2 microns, then we must account for rounding at the apex of the indenter with a second term, making the area function A = 24.56h 2 c + Ch c (5b) In the second term, C is a constant that is determined empirically by indenting a known material, usually fused silica. For most Berkovich indenters manufactured today, C has a value of 150 nm or less. The manufacturer of the indentation equipment should provide a process for determining the value of C; this process is often called the area-function calibration. For other indenter geometries, other forms of the area function must be used. Table 2 provides a summary of these functions. In order to use Eq. 5, we must know the contact depth. For pyramidal, conical, and spherical indenters, the contact depth is calculated as h c = h 0.75P/S (6) Contact stiffness, S, which is used in Eqs. 3 and 6, is the relationship between force and displacement when the indenter is first retracted from the sample; that is, when the material response is entirely elastic. Logistically, S is determined by fitting the force displacement data acquired Ideal Berkovich A = 24.56h 2 c Used when h c > 2microns Real Berkovich A = 24.56h 2 c + Ch c C is determined by indenting a known material and is about 150 nm. Real cube-corner A = 2.60h 2 c + Ch c C is determined by indenting a known material and is about 150 nm. Sphere A = 2πRh c R is tip radius; value may be known or determined by indenting a known material. Cone A = π tan 2 ψh 2 c ψ is the half-included angle of the cone as illustrated in Fig. 1. Sphere-tipped cone A = π tan 2 ψh 2 c + 2πRh c Superposition of the area functions for a sphere and a cone. Flat-ended cylinder A = πa 2 A is the punch radius; A is constant (independent of indentation depth). during unloading to an expression of the form P = B(h h f ) m, (7a) where P and h are ordered pairs of force displacement data, and B, h f,andm are best-fit constants. Typically, not all the force displacement data acquired during unload are used in this fit, but only data above a certain force level, often 50% of the maximum force. Once B, h f,andm have been determined, Eq. 7a is analytically differentiated with respect to displacement, and then evaluated at the maximum displacement, giving S = dp dh = Bm(h max h f ) m 1. h=h max (7b) So, beginning with Eq. 7 and working our way back, we have everything that we need to calculate Young s modulus (E) and hardness (H) from an instrumented indentation test. Figure 5 is a comparison between Young s modulus measured by IIT versus Young s modulus measured by other means. The agreement is quite good. (The means used to determine nominal Young s modulus depend on the material. Polymers were tested by dynamic mechanical analysis; metals, glasses, and ceramics were tested by tensile test or ultrasound.) Equation 6 is the crux of IIT. It was first proposed by Warren Oliver and George Pharr 2 ; when people refer to November/December 2009 EXPERIMENTAL TECHNIQUES 69

5 Indentation Young's Modulus, GPa Cu (polycrystalline) Mg (polycrystalline) Mo (polycrystalline) Stainless steel (polycrystalline) Sn (polycrystalline) Ti (polycrystalline) SiO2 (amorphous) brass (polycrystalline) Pyrex (amorphous) BK-7 (amorphous) Delrin (amorphous) Polycarbonate Unity Nominal Young's Modulus, GPa Fig. 5: Young s modulus measured by IIT versus nominal Young s modulus. Nominal values were determined by tensile test, ultrasound, or dynamic mechanical analysis the Oliver-Pharr model, Eq. 6 is what they are referencing. It is valuable because it provides a means for calculating contact depth, and thus contact area, directly from the force and displacement data. This ability is what distinguishes IIT from other hardness tests in which contact area is determined by imaging the residual impression in some way. Equation 6 works well for large indents, but it is especially advantageous when making nanometer-scale indents, because it renders imaging the residual impression unnecessary. Not only does Eq. 6 provide for more accurate results at the nanometer scale, it also simplifies testing at all scales. Because there is no need to image the residual impression, hundreds of tests can be performed at a time with no user interaction. Although the derivation of Eq. 6 is rather complex, 1,2 the expression itself is intuitive, as illustrated in Fig. 6. Soft metals approach the limit of purely plastic deformation; they exhibit minimal elastic recovery during unloading. That is, the unloading curve is nearly vertical, making S very large. In Eq. 6, this drives the second term to zero, and so the contact depth is nearly equal to the total depth. This matches our intuition for soft metals, because we do not expect the sample surface outside the contact area to deflect elastically as a result of the test. Hard ceramics approach the limit of purely elastic behavior, which means that the slope of the unloading curve approaches the slope of the loading curve. In this case, the second term makes a significant contribution to the right-hand side of Eq. 6, and so the contact depth is less that the total depth. This matches our intuition for ceramics, because we expect the surface outside the contact area to deflect elastically as a result of the test. There are some scenarios in which Eq. 6 does not accurately predict contact depth. When people say, the Oliver-Pharr model doesn t work in this situation, they are saying that the calculation provided by Eq. 6 either over- or underestimates contact depth to an extent that is unacceptable. If the true contact depth is greater than that which is predicted by Eq. 6, we call it pile-up, because this happens when material piles up adjacent to the indenter. Pile-up tends to occur (1) in soft metals that have no further capacity for hardening (i.e. work-hardened copper) and (2) soft films on hard substrates. If pile-up were to occur unbeknownst to the user, then the contact area calculated by Eqs. 5 and 6 would be too small; Eqs. 1 and 3 reveal that (1) hardness and Young s modulus would be too big and (2) the adverse effect would be twice as great for hardness as for Young s modulus, because hardness goes as 1/A, whereas modulus goes as 1/ A. Thus, if the hardness and Young s modulus are larger than expected for a certain material, then one should suspect pile-up. If pile-up has indeed occurred, then the sides of the residual impression made with a Berkovich indenter will bow out as illustrated in Fig. 7a. It can also happen that the true contact depth is less than that predicted by Eq. 6; this is called excessive sink-in. 70 EXPERIMENTAL TECHNIQUES November/December 2009

6 (a) (b) P P h c = h h h c < h h Fig. 6: Schematic illustration of the calculation of contact depth, h c = h 0.75P/S, for (a) plastic indentation into a soft metal and (b) elastic indentation into a hard ceramic (a) (b) (c) (d) Fig. 7: Schematics of Berkovich residual impressions manifesting various degrees of pile-up and sink-in (a) pile-up, (b) no pile-up or sink-in, (c) sink-in, (d) excessive sink-in. If the residual impression looks like (b) or (c), the Oliver-Pharr model works well. If it looks like (a) or (d), the Oliver-Pharr model will underestimate or overestimate the true contact depth, respectively The modifier excessive is used, because Eq. 6 does predict a degree of sink-in. Excessive sink-in is most likely to occur when testing hard films on soft substrates the greater the mismatch in hardness, the greater the likelihood of excessive sink-in. 17 Excessive sink-in has the opposite effect of pile-up: the contact area calculated by Eqs. 5 and 6 is too big, so the calculated hardness and Young s modulus are too small. When excessive sink-in has occurred, the sides of the residual impression will bow in excessively as illustrated in Fig. 7d. When faced with either pile-up or excessive sink-in, the user has limited choices: (1) decide that the error is acceptable, (2) modify Eq. 6 using analytic or computational modeling, or (3) measure the contact area directly using some form of microscopy, and use this directly measured area for A in Eqs. 1 and 3. GOOD EXPERIMENTAL PRACTICE Environmental Control. To take full advantage of the fine displacement resolution available in most IIT testing systems, the testing environment should be chosen and prepared with care. Uncertainties and errors in measured displacements arise from two separate environmental sources: (1) vibration and (2) variations in temperature that cause thermal expansion and contraction of the sample and testing system. To minimize vibration, testing systems should be located on quiet, solid foundations (ground floors) and mounted on vibration-isolation systems. Thermal stability can be provided by enclosing the testing apparatus in an insulated cabinet to thermally buffer it from its surroundings and by controlling room temperature to within ±0.5 C. Surface Preparation. Because contact area is deduced from contact depth on the presumption that the test surface is flat, the surface roughness must be small relative to the indentation depth. For metallographic specimens, a good guide for surface preparation is ASTM E One can normally determine whether roughness is an issue by performing multiple tests in an area and examining the scatter in measured properties. For a homogeneous material with minimal roughness, scatter of less than a few percent can be expected with a good testing system and technique. Testing Procedure. To avoid interference, successive indentations should be separated by at least times the maximum depth when using a Berkovich or Vickers indenter. For other indenter geometries, the distance between indents should be about 10 times the maximum contact radius. The importance of frequently testing a standard material cannot be overemphasized. Optically polished fused silica is a good choice for such a standard, because it is smooth, isotropic, homogenous, and does not degrade over time. It is good practice to perform 5 10 indents on the standard material every time the instrument is used. If the measured properties of the standard do not match expected values, the user is immediately alerted to problems in the testing equipment and/or procedure. Testing Standards. Two standards govern IIT: ISO and ASTM E Copies of these standards may November/December 2009 EXPERIMENTAL TECHNIQUES 71

7 be purchased directly from either ISO or ASTM. Both of these standards prescribe (1) procedures for instrument verification and (2) a test procedure. When purchasing an IIT system, one should look for a system that conforms to at least one of these standards. CONCLUSIONS IIT is used to measure mechanical properties at the nanometer scale. It is an essential tool for evaluating films, coatings, and surface layers which are used to improve mechanical performance and longevity. The contact-mechanics theory behind IIT is complex, but commercial systems simplify the implementation. In practice, IIT is one of the simplest and fastest types of mechanical testing, because sample preparation is relatively easy, and hundreds of tests can be performed on a single sample. References 1. Sneddon, I.N., The Relation between Load and Penetration in the Axisymmetric Boussinesq Problem for a Punch of Arbitrary Profile, International Journal of Engineering Science 3:47 56 (1965). 2. Oliver, W.C., and Pharr, G.M., An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments, Journal of Materials Research 7(6): (1992). 3. Herbert, E.G., Oliver, W.C., and Pharr, G.M., Nanoindentation and the Dynamic Characterization of Viscoelastic Solids, Journal of Physics D: Applied Physics 41:1 9 (2008). 4. Madsen, E.L., Hobson, M.A., Frank, G.R., Shi, H., Jiang, J., Hall, T.J., Varghese, T., Doyley, M.M., Weaver, J.B., Anthropomorphic Breast Phantoms for Testing Elastography Systems, Ultrasound in Medicine and Biology 32(6): (2006). 5. Herbert, E.G., Oliver, W.C., and Pharr, G.M., On the Measurement of Yield Strength by Spherical Indentation, Philosophical Magazine 86:5521 (2006). 6. Lucas, B.N. and Oliver, W.C., Indentation Power-Law Creep of High-Purity Indium, Metallurgical and Materials Transactions A, 30: (1999). 7. Pharr, G.M., Harding, D.S., and Oliver, W.C., Measurement of Fracture Toughness in Thin Films and Small Volumes Using Nanoindentation Methods, Mechanical Properties and Deformation Behavior of Materials Having Ultra-Fine Microstructures, Kluwer Academic Publishers, Norwell, MA, pp (1993). 8. Hay, J.L., and Pharr, G.M., Instrumented Indentation Testing, Kuhn, H., and Medlin, D., (eds), ASM Handbook Volume 8: Mechanical Testing and Evaluation, 10th Edition, ASM International, Materials Park, OH, pp (2000). 9. VanLandingham, M.R., Review of Instrumented Indentation, Journal of Research of the National Institute of Standards and Technology, 108(4): (2003). 10. Oliver, W.C., and Pharr, G.M., Measurement of Hardness and Elastic Modulus by Instrumented Indentation: advances in Understanding and Refinements to Methodology, Journal of Materials Research 19(1):3 20 (2004). 11. Golovin, Y., Nanoindentation and Mechanical Properties of Solids in Submicrovolumes, Thin Near-Surface Layers, and Films: AReview, Physics of the Solid State, 50(2) pp (2008). 12. Tabor, D., Hardness of Metals, Clarendon Press, Oxford, UK (1951). 13. Johnson, K., Contact Mechanics, Cambridge University Press, Cambridge, UK (1985). 14. Fischer-Cripps, A., Nanoindentation, Springer-Verlag, New York, NY (2004). 15. Pike, G.E., Faupel, F., Yamamoto, H., (eds), Journal of Materials Research 19(1):(2004). 16. Pharr, G.M., Sundararajan, G., Hutchings, I.M., Sakai, M. Cheng, Y-T., Moody, N.R., Swain, M.V., (eds), Journal of Materials Research: Focus Issue on Indentation Methods in Advanced Materials Research 24(3):(2009). 17. Hay, J.C., and Pharr, G.M., Critical Issues in Measuring the Mechanical Properties of Hard Films on Soft Substrates by Nanoindentation Techniques, Thin Films - Stresses and Mechanical Properties VII (Materials Research Society Symposium Proceedings) San Francisco, California, Vol 505, pp (1998). 18. Standard Guide for Preparation of Metallographic Specimens, E3 01, ASTM International, West Conshohocken, PA (2007). 19. Metallic Materials Instrumented Indentation Test for Hardness and Materials Parameters, ISO 14577, International Organization for Standardization, Geneva, Switzerland (2002). 20. Standard Practice for Instrumented Indentation Testing, E , ASTM International, West Conshohocken, PA (2007). 72 EXPERIMENTAL TECHNIQUES November/December 2009