A number of assumptions are built into the everyday operation of mecha - PDF document

A number of assumptions are built into the everyday operation of mechanicaltesting devices, and nanoindentation devices in particular, even when the materials beingtested are “simple” (e.g. homogeneous, time-independent) engineering metals andglasses. Key issues associated specifically with nanoindentation testing will be discussedhere in detail within the context of time-independent materials (or materials with limitedtime-dependent responses); the issue of time-dependence will be addressed in Chapter 4.A critical assumption in indentation testing is that the instrument-tip combinationhave been calibrated correctly to allow for the measurement of quantitative materialproperty information from the indentation test data. Another question arises as towhether the assumptions in the Oliver-Pharr analysis, built into the instrument softwaretion analysis of stiff engineering materials,are valid for compliant materials. Finally, the frequently-reported relationship betweenelastic modulus and contact hardness will be critically examined. Nanoindentation is a “garbage in, garbage out” mechanical testing modality,particularly in the modern age of semi-automated instruments. As with any mechanicaltest, the raw load-displacement data is always available to the user, but materialproperties such as elastic modulus can only be obtained following instrument calibrationand advanced data analysis. In addition to the intrinsic machine compliance present in allmechanical testing devices, in indentation testing the diamond (or tungsten, alumina, orother stiff material) tip contacting the sample must be carefully calibrated and frequentlyon the specific details of instrument calibration or issues such as error propagation due topresented illustrating the calibrated modulus-depth data for several materials, on a fused silica standard sample at differentpeak load levels, typically in a logarithmic series (e.g. 0.1, 0.3, 1, 3, 10, 30, 100, 300mN). As in standard Oliver-Pharr data analysis, three parameters (maxmaxobtained directly from the raw load-displacement (data. From these parameters, themaxmax [3-2]. The reduced elastic modulus ( the known fused silica value for a diamond indenter, 69.6 GPa) and 4ER2 4ER21 [3-3] = compliance, the inverse of the stiffness. Based on this calculation of areafrom Eqn. 3-3 and the independent calcuare fit to a polynomial of the form Eqn. 3-1. The elastic modulus is then re-calculated forall indentation tests using the area function obtained from the polynomial fit. This technique is simple and straightforward, although in practice it is prone toerrors resulting from both the curve-fitting and the enforced numerical value of theelastic modulus. Fine-tuning of the area function is usually required, in which the areauntil the elastic modulus for one or more materials is constant with depth (but notnecessarily of a “forced”, known numerical value). Verification (and in some cases, fine-tuning or adjustments) of area functioncalibration is easily performed using continuous stiffness measurements (section 2.1.6.1), assuming the frame compliance was correctly set); the following section will examineframe compliance effects in area function calibration.A problem arises in this analysis in that the measured sample stiffness may beinfluenced by the compliance of the materials testing machine itself. This is a well-known issue in materials testing, but takes on extra importance in nanoindentationdevices compared to relatively stiff test-frames used for tensile testing at larger load- andlength-scales. Thus the calculated area function my have an artifact due to this machineframe compliance, , which must be calibrated separately. The level of importance ofthe frame compliance calibration actually varies due to both hardware and softwarethe major indentation device manufacturers,Hysitron and MTS. The Hysitron transducer is inherently more compliant than that usedin the MTS testing system, (approximately one order of magnitude: compare complianceliance)different designs of the support springs. Therefore, accurate compliance calibration isintrinsically more important in quantitative indentation analysis using the Hysitroninstrument. In addition to this magnitude difference, the MTS transducer is compliancecalibrated at the time of installation and this value is inserted in a protected location inthe software. (There is a user-level “frame compliance correction” that can beintroduced, and this may be a positive or negative value since it is merely a smallcorrection to a compliance value already set in the software, but this correction is seldomlarge and is frequently not necessary.) The frame compliance value in the Hysitronsystems is a user-level software level input which could be changed from day to day, testto test, and instrument user to instrument user. Because of these differences betweeninstrument manufacturers, the discussion of frame compliance calibration from this pointforward will be conducted exclusively in the framework of the Hysitron indentation sample 4ER2 4ER21 [3-7]The tip area function and frame compliance can then be calculated through an iterativeThe Hysitron software is employed for the current study, in which a set of real,experimentally obtained, fused silica indentation traces are analyzed and a calibration isdetermined as objectively as possible. In order to obtain this calibration, a new techniquethis investigation, the frame compliance is purposefully changed over a range of values,meant to be both larger and smaller than the expected range of machine compliance seenfor the instrument. A new area function is calculated to correspond to each framecompliance value that was set. The original fused silica data is analyzed with eachdifferent calibration pair (machine compliance and area function). The overall objectiveof this process is two-fold: (a) to examine the effects of machine compliance on differentaspects of indentation analysis, and (b) to arrive at a correct value of the machinecompliance and thus a correct calibration for the instrument by eliminating the incorrectcalibration pairs through objective measures. and 14 mN. The frame compliance was varied from 0 to 4 nm/mN and the effects offrame compliance on the calibration parameters was examined. Figure 3-4: Calculated contact area (The combination of the data from Figures 3-3 used for fitting the calibration area function, as shown in Figure 3-5. An approximatearea function fit is demonstrated here for each frame compliance level with quadraticpolynomial fits (although no experimentalist would consider calculating an area functionfrom data at only four different peak load levels!) Using these fits, the simple two-parameter paraboloid area function (Eqn 3-5) parameters are themselvesstrong functions of frame compliance, as is obvious from the large differences in thequadratic fits in Figure 4. The direct dependence of these area function parameters on the frame compliance 012342.0x104.0x106.0x108.0x101.0x101.2x101.4x101.6x101.8x102.0x102.2x10 Calculated Contact Area, (nmFrame Compliance, (mN nm max = 14 mN max = 10 mN max = 5 mN max = 2 mN contact hardness value is only constant and correct for a single value of the framecompliance (here ~ 2.5 nm mN). An alternative (better) second measure of the correctarea function-frame compliance calibration is a constant and numerically correct elasticmodulus value for tests conducted on a second material.It is also interesting to note the strong dependence of the first term area function on the frame compliance (Fig. 3-6). This number appears to provide a apredicted value for an ideal Berkovich indentation tip), something has gone wrong withthe frame compliance calibration. This internal check of the are function leading termhas worked well in the current system, in that thenm/mN, the value in the chosen “correct” calibration pair (based on the contact hardness01234 012341800200022002400260028003000 First () area coefficient, Frame Complaince, Second () area coefficient, materials will be discussed in the following section: whether a different calibrationprotocol is required, and if there are issues with the fundamental contact mechanics usedin Oliver-Pharr analysis that lead to systematic overestimates of the elastic modulus ofcompliant materials. A question is raised when indentation testing is being performed on materialssubstantially more compliant than those typically used as calibration standards. Forexample, fused silica and aluminum both have elastic modulus values near 70 GPa, whilemost polymeric materials have modulus values around 5 GPa or (much) less, dependingon the material and glass transition temperature. Bone and dentin have modulus valuesin the range from 10-30 GPa, which has resulted in discussion of potential alternativecalibration standards for indentation of compliant materials [Chang et al, 2003b]. Thereis no widely accepted polymer standard for indenter tip calibration; a polymer would beextremely difficult to use as a standard due to the intrinsic time-dependence of thematerial and the effects of this time-dependence on indentation response (to be discussedAs discussed above, the area function used to describe the tip-sample contact istypically described with the form of Eqn. 2-8, of which Eqn. 3-4 is a truncated form. Forcomparison of my own MTS-machine function I will include a third term here, such that[3-8]This sum for the contact area is plotted in Figure 3-8 for approximate values of thecalibration constants employed in the current work (along with each term separately. There are both physical and mechanical factors that could be implicated in thisdiscrepancy between indentation and homogeneous modulus measurements:physical aging of the polymer surface due to oxidation or other chemical mechanismscould be associated with changes in the surface mechanical properties, and those,rather than the bulk values, are the properties which are measured by nanoindentationtime-dependence in the polymer mechanical response could be affectingmeasurements (this will be addressed in detail in Chapter 4)errors in the estimation of contact depth, , from displacement, been previously discussed in indentation analysis of elastic-plastic systems,particularly in the context of work-hardening metals [Cheng and Cheng, 2004])0100020003000400050006000 Tensile ModulusElastic Modulus, (GPa)Displacement, (GPa) amorphous Selenium [Shimizu et al, 1999] in which the authors concluded thatexperimentally the value of is approximately unity. There are two ways that a material with ʳķˀ could cause problems inquantitative nanoindentation measurements. First, calibration of the tip area functionrequires a “well-behaved” material with a calibrations are undertaken using stiff glass and ceramic materials as the calibrationstandards. The use of a calibration material with an unexpected , as in many soft metalsand polymers, would result in numerical errors in the area function when attempting tothen use this area function to measure properties in a material with the correct Attempts to “force” the area function to conform to a uniaxial tension elastic modulus[Chang et al, 2003b] are intrinsically error-ridden, due to time-dependence in the polymerand the outstanding lack of certainty associated with the overestimation of polymermodulus values frequently observed. This topic will also be further discussed in section3.2.2 below, in which the data published previously with a forced polymer-based areafunction are corrected and found to be in agreement with the existing literature andindependent measurements using the same sample and a different indentation instrument.Since calibrations are typically carried out with stiff glass and ceramic materials,this issue would help explain why the Oliver-Pharr modulus values for many polymers,even after accounting for time-dependent effects, are overestimated in nanoindentation.An alternative model for time-dependent indentation data analysis, in which be 1, results in modulus values closer to the known modulus for polymers such as PL-1(Figure 4-15). This model (and these PL-1 polymer data) will be discussed later, withinthe context of time-dependence in Chapter 4. It is interesting to note, however, that thedifference in modulus values obtained at different depths remains. is equally applicable to viscoelastic problems, and this hasbeen confirmed by recent finite element analysis on conical indentation in a viscoelasticmedium [Cheng and Cheng, 2004]. However, problems with Having critically examined indentation calibration and issues for compliantmaterials, the technique will be applied (using Oliver-Pharr analysis) to elastic-plasticindentation testing and analysis on mineralized tissues, mainly bone. A brief overview ofa large healing bone study, from which samples will be used in several different portionsHealing bone samples adjacent to a dental implant interface were the focus ofplant interface were the focus ofal, 2003b]. In the study, the 4th premolars of six 2-year-old Sinclair miniswine wereunilaterally removed surgically and a titanium dental implant was inserted in the alveolarridge seven months after the extraction. The implants were shielded from bite forces andleft in place for one month (two animals), two months (two animals) or four months (twoanimals) prior to animal sacrifice. Three bone samples adjacent to the screw-implant(Figure 3-10) were harvested embedded in polymer resin (PL-1, Vishay Micro-Measurements, Raleigh, NC). The samples were sectioned 1-2 mm thick, andm using standard metallographic techniques, to prepare thesurfaces for indentation testing. The samples were stored frozen until indentation testingwas performed, and returned to the freezer between studies.Different groupings of these same samples will be examined in portions of thei.In section 3.2.2, one-month healing samples indented with two different instrumentswill be compared. The original (Hysitron) study uses samples 1-3 of both one monthsanimals and was published previously [Chang et al, 2003b]; the original result appearsin Figure 3-11. The data were re-analyzed following calibration and compared withnew experiments using the MTS instrument. The new MTS study uses one sample Figure 3-11: Plane strain modulus (for one month healing [from Chang et al, 2003b]It became of practical interest to verify that there was no possibility for machine-based and calibration-based differences in indentation elastic-plastic mechanicaltion instruments. The mineralized tissuenanoindentation studies (performed by myself) reported in the current work were doneusing an MTS Nanoindenter XP instrument. Indentation tests on the same series of bonesamples were done previously with a Hysitron Triboscope and preliminary results for onemonth bone healing were published [Chang et al, 2003b]. In the published study, a novelpolymer-based indenter tip calibration was used along with the Hysitron instrument. Values for the indentation modulus and contact hardness in porcine healing bone(reported in [Cheng et al, 2003b]) were at the extreme low end of the range of values (as was done in the original Cheng et al study) would result in dramatic overestimation ofthe contact area, which in turn would result in underestimated numerical values for ' and when testing unknown samples. protocol and modulus “forcing” to fused silica values (as per section 3.1.1 above) theHysitron data for bone with 1-month healing was found to have an average modulus = 0.4 GPa, much closer to previously published values forbone [Turner et al, 1999; Zysset et al, 1999]. The old (e.g. From Fig. 3-11 as originally new (recalculated after the recalibration) modulus values for the Hysitronexperiment 1-month healing data are showcompared to 12 GPa), again in better agreement with other experiments publishednts publishedFigure 3-12: Plot of 1-month healing bone plane strain modulus (The recalibrated modulus data from Figure 3-12 is shown as a function of distancefrom the bone-implant interface in Figure 3-13. The data from the new MTS experiments051015202530 Re-calibrated Hysitron Modulus, ' (GPa)Original Hysitron Modulus, ' (GPa) times of 1-, 2-, and 4-months post-implant(These original indentation experiments were performed previously by C-C Andy Liu butare presented here following this crucial re-calibration for comparison with the new MTSexperiments done on the same samples later in this chapter.) Modulus data showedsubstantial scatter (Figure 3-14) and a subtle trend towards increasing modulus with) from the bone-implant interface. The contact hardness datashowed a similarly large degree of variability and a similar, perhaps even subtler, trendwith respect to distance from the implant interface (plot not shown). Figure 3-14: Plot of modulus (for Hysitron indentation tests at 1, 2, aThese data demonstrate a slight increase in both plane strain modulus and contact0300600900120015001800 1 month 2 month 4 monthPlane Strain Modulus, ' (GPa)Distance, Figure 3-15: Frequency histograms for modulus ( 051015202530 Frequency051015202530 4 monthsModulus, ' (GPa)051015202530 2 months1 month Figure 3-16:Raw indentation load-displacement () responses for fused silicaThe raw load-displacement (10, and 100 mN are shown in Figure 3-16. Two things are immediately apparent fromeffect of geometric similarity of the Berkoviand demonstrate little variability at any load level. This is reflected in the modulus datadeconvoluted from the traces using Oliver-Pharr analysis; the modulus data are constantin value and show little variability in terms of total range of values (Figure 3-17). Thediminishing range of modulus values with increasing indentation peak load level isidentical so the broader lines at 1 mN reflect small variations in the responses. Thesedata have been plotted with an elastic modulus range of one order of magnitude, forcomparisons of the variability in responses for materials with different baseline elasticmodulus values in the following sections. 0100200300 0300600900100 03060900.00.20.40.60.81.0 Displacement, Load, (mN) Mineralized composite biological tissues were next examined within thisframework of indentation length-scale variability. Porcine bone samples were used, asTooth cross-sections were made by sectioning extracted 3 molars in the transverse planeto expose dentin and enamel, and wet-polishing the surface in a manner similar to thatused for the bone samples discussed in section 3.2 above. All bone and tooth sampleswere stored frozen until testing and were tested in ambient conditions (Elastic plane strain modulus-indentation peak load (maxFigure 3-19 for bone and in Figure 3-20 for the mineralized tooth tissues. Interestingly,considering the similarities in composition for bone and dentin, there was a substantialdifference in the degree of variability seen in the indentation responses. The enamel waseven less variable than the dentin at small depths. Figure 3-19: Variability in bone indentation modulus (levels (max0.1110100 Plane Strain Modulus, ' (GPa)Peak Load, max (mN) are comparable in variability to the homogeneous materials at both small (Table 3-2) andThere was a dramatic variability in the bone data for variability, both at smallbone variability can therefore be considered “real” and not due to artifacts of machine (bycomparison with homogeneous materials) or of sample preparation (by comparison withsimilarly-prepared tooth tissues). max = 0.1 or 1 mN) indentation modulus variability (datafrom plots 3-17 to 3-20)maxmax100% Fused Silica1 mN87 nm69.394.646.7Mineral apatite1 mN71 nm166.2826.8416.1Enamel1 mN90 nm117.0611.7510.0Bone0.1 mN131 nm14.328.2357.5Dentin0.1 mN62 nm27.764.3315.6 Having demonstrated in many different ways that the indentation responses ofbone are variable, especially at small loads, a new technique was developed to map outmaterial properties across a sample and allow for correlation of properties withmicrostructural features. The indentation experiments were performed at large peakmax = 100 mN) to minimize any variability effects due to surface preparation,individual discrete elements such as mineral particles, and to try to obtain a near-continuum mechanical response for each location. The mapping experiments wereperformed on a tooth cross section, as a control to establish the technique, as well as threebone samples (one each for one, two, and four months healing time as discussed insection 3.2.1). The bone and tooth samples underwent identical sample preparation,The dentin-enamel junction is an abrupt transitional region between two tissueswith dissimilar compositions and elastic modulus values. Therefore, this DEJ region wasused as a control to establish a technique of quantitative elastic modulus mapping viaindentation testing. Crucial to this mapping technique is the registration of optical andmechanical information on the same composite image. of single tooth cross-sectional sample m in thex-direction (approximately perpendicular to the DEJ) and 100-200 m in the y-directionmax = 100 mN and = 3 mN s). Optical images (160 x 120 pixels, corresponding toa sample region 600 m by 400 m) of the indented regions were captured in situ Figure 3-24 Plane strain elastic modulus (dulus value from one indentation testcomposite image of the DEJ (Fig. 3-22) 11000010900010800010700010600010500037000365003600035500350003450034000 Y-position (X-position ( 10.0025.0040.0055.0070.0085.00100.0115.0130.0 Figure 3-26: Raw indentation load-displacement (The procedure for generating the optical-mechanical maps was identical to thatdone for the DEJ (section 3.4.1 above): optical images of the indented regions captured were used to generate a composite image of the indented region. Indentationelastic modulus (obtained via Oliver-Pharr analysis) was mapped onto the optical images.For each experimental specimen (one each at one, two four months bone healing time)three images are presented in the following figures: (a) the composite optical image ofthe specimen, (b) the plane strain elastic modulus map, and (c) the combined optical-mechanical map. and mechanical information) on the polymeric embedding resin instead of the bone, thesymbol has been changed from a square to a circle and these points have been excludedfrom further data analysis. (These circles all correspond to modulus values of 7.5 GPa orless, associated with the polymer with a modulus of 4-5 GPa.) In some locations, nosquare symbol is presented on the image; in this case, the indentation test failed to runand that location and no mechanical information is available. These images will be further discussed en masse following presentation of thethree sets of images.01000200030004000100 BoneLoad, (mN)Displacement, (nm) Figure 3- 27(c): Overlay image of the combined optical and mechanical informationcontained in parts (a) and (b)112 Figure 3- 28(c): Overlay image of the combined optical and mechanical informationcontained in parts (a) and (b)114 Figure 3-29(c): Overlay image of the combined optical and mechanical informationcontained in parts (a) and (b)116 region clearly differentiable from the surrounding bone, typically darker in the opticalimage. Overall, modulus data from directly adjacent tests were frequently comparable,m indent spacing.However, there was no blindingly obvious trend in modulus with distance from theimplant interface in any of the samples.A histogram is presented in Figure 3-30 for the modulus information contained inthe three images 3-27 to 3-29. In contrast to the histograms presented earlier (Figure 3-15) for a larger group of samples at each time-point (including these three samples), thedata from the current investigation were not well-described by a normal distribution. different 1-month dry bone samples showed an average modulus of 17.9 GPa, in goodagreement with the current average 17.2 GPa at 1-month healing. However, and alsoquite interestingly, the mean modulus values for the current study demonstrated anopposite trend to that seen in the previous Hysitron study on these same samples (Figure3-15). In the previous study, the 1-month data are depressed compared to the 2- and 4-month data. In the current study, the 1-month data are, on average, elevated compared tothe 2- and 4-month data. However, since the bottom line continues to be that there islarge variability in the indentation modulus of bone, it is difficult to discern if this trendis real or is merely a sampling error. The focus of this chapter has been on elastic-plastic analysis in indentationtesting. The Oliver-Pharr analysis for conical-pyramidal indentation in elastic-plasticsolids is extremely popular, especially since the analysis is built directly in tocommercially available indentation testing instruments (including both of those describedin section 2.1.6 of this work). For large scale analysis of samples, such as the propertymapping performed here in section 3.4, the use of an automated instrument and the built-in elastic-plastic analysis results in extremely quick and effective materials propertydetermination compared to traditional testing modalities. The probe-based technique,allowing for individual measurements to be made at 100 m centers, presents interestingopportunities for gaining information about the local structure-properties relationships ininhomogeneous materials, such as mineralized biological tissues.In order to gain quantitative information from indentation testing, the tip areacalibration is extremely important. However, interestingly enough, the calibration forperforming tests on compliant materials is a bit more forgiving than that required forstiffer materials, as the total displacements are large relative to any blunting in the tip, ...). A perfect tip assumptionissues that arise in the tip calibration are somewhat machine-manufacturer dependent, butit was demonstrated in this work that with appropriate calibration (and an approximationfor the difference between wet and dry samples) the indentation results from differentmachines are in fact comparable (section 3.2.2). Variability in indentation responses can be assessed by examining point-to-pointwell as directly assessing location-dependent properties through a mapping technique(section 3.4). Overall, these examinations demonstrate that there is dramatic variability