A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82 75 Fig. 2. Displacement response for a step increase in load for (a) Voigt model, (b) Maxwell model. f(h) = h2 for cone, f(h) = h3/2 for sphere. The square-bracketed term in Eq. (1) represents the displacement-time response of the mechanical model to aFig. 1. (a) Three-element Voigt spring and dashpot representation of step increase in load. A step increase in applied load re-a visco-elastic material (delayed elasticy), (b) Maxwell representation sults in an initial elastic displacement (t = 0) followed by aof a visco-elastic material (steady creep), (c) four-element combined delayed increase in displacement to a maximum value at tMaxwell–Voigt model. = ∞ as shown in Fig. 2(a). It should be noted that Eq. (1) applies to the case of a rigid indenter in which case the symbol E∗ is the combination of the elastic modulus anddue to the elastic-plastic properties of the material and that Poisson’s ratio of the specimen material (E∗ = E/(1 − ν2 ))occurring to due creep, either visco-elastic or visco-plastic. and not the combined modulus of the indenter and speci- Time dependent properties of materials are conventionally men as is normally the case. The numerical factor 3/4 inanalyzed in terms of mechanical models such as those shown Eq. (1) differs from that usually seen in reference literaturein Fig. 1. The elastic response of such a model is quanti- due to our working with the elastic modulus E∗ rather thanﬁed by what we call the storage modulus. The ﬂuid-like re- the shear modulus G. The justiﬁcation for the change insponse is quantiﬁed by the loss modulus. In rheology, the variable being that in indentation loading, the greater pro-material behavior is towards the ﬂuid end of the spectrum portion of the deformation is hydrostatic compression andwhere any elastic response, or the storage modulus, is dom- the materials we consider are more likely to be dominatedinated by the shear modulus of the material. In solids, such by solid-like properties than ﬂuid-like behavior.as those usually tested in nanoindentation, the material be- A similar approach is appropriate for the case of a conicalhavior tends towards a predominantly elastic response. In indenter in which we obtain, for the case of the three-elementan indentation test, the nature of the loading is a complex Voigt modelmixture of hydrostatic compression, tension, and shear. Un- 1 1 −tE∗ /ηlike a ﬂuid, the storage modulus in this case contains contri- h2 (t) = Po cot α ∗ + E∗ (1 − e 2 ) (2)butions from all three of these types of materials response. 2 E1 2In the present work, we shall, in the interests of simplicity, For the case of a Maxwell model Fig. 1(b), theassume that the storage modulus is a measure of the con- time-dependent depth of penetration for a spherical indenterventional (tensile/compressive) elastic modulus in recogni- is given bytion of the large component of hydrostatic stress in the in-dentation stress ﬁeld. The ﬂuid-like response we shall call 3 Po 1 1 h32 (t) = √ ∗ + ηt (3)“viscosity” although in practice, viscosity is usually fre- 4 R E1quency and temperature dependent and not single-valued. and for a conical indenter, we obtain Radok , and Lee and Radok  have analyzed thevisco-elastic contact problem using a correspondence prin- 1 1 h2 (t) = Po cot α ∗ + ηt (4)ciple in which elastic constants in the elastic equations of 2 E1contact are replaced with time dependent operators . For From the above information, it is relatively easy to con-the case of a rigid spherical indenter in contact with a ma- struct equations for more complicated models by addingterial represented by a three-element Voigt model as shown components in series or parallel as would be done inin Fig. 1(a), we can write that for a steady applied load mechanical–electrical analogs conventionally applied to thisPo , the depth of penetration increases with time is given type of modeling. For example, in the present work, we shallby the well-known Hertz equation with the addition of a also consider a four-element Maxwell–Voigt combinationtime-dependent exponential as shown in Fig. 1(c) which gives 3 Po 1 1 −tE∗ /η 3 Po 1 1 −tE∗ /η2 1h32 (t) = √ ∗ + E∗ (1 − e 2 ) (1) h32 (t) = √ ∗ + E∗ (1 − e 2 )+ t (5) 4 R E1 2 4 R E1 2 η1
76 A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82 1 1 −tE∗ /η2 1 iﬁed in draft standard ISO14577. A simpliﬁed treatment ofh2 (t) = Po cot α ∗ + E∗ (1 − e 2 )+ t (6) nanoindentation creep is presented and a readily accessible 2 E1 2 η1 method of analysing Eqs. (1)–(4) in a manner suitable for The equations given above are expressed in terms of the automation within a computer program is provided.modulus E∗ which is the combination of elastic modulusand Poisson’s ratio of the specimen. In the present work,the elastic material properties of the specimen material are 2. Experimentalgiven in terms of E∗ as is done in conventional nanoinden-tation analysis and η is the viscosity term that quantiﬁes Several materials were selected for study. The ﬁrst, a 1 mthe time-dependent property of the material. It should be re- thick ﬁlm of high purity aluminum, a metal known to ex-membered that Eqs. (1)–(6) assume a step increase in load hibit signiﬁcant indentation creep. The second, a sample ofto Po and are expressed here in a form to be easily ﬁtted to fused silica, a material in which creep is not expected to beexperimentally obtained creep data to provide values for E∗ signiﬁcant, and the third, an 100–150 m thick polyurethaneand η. Should modeling of an arbitrary time displacement acrylic copolymer ﬁlm, a material expected to exhibit sig-response be required, a suitable superposition [5,6] can be niﬁcant visco-elasticity the amount of which is dependentemployed but this is beyond the scope of the present work. upon the additive molecules. Two acrylic co-polymer ma- There are several detailed theoretical studies of visco-elastic terials were tested, a baseline material and another with aindentation creep available in the literature [7–12]. A popu- cross-linker added. Tests were done on a UMIS nanoinden-lar motivation for such modeling is the behavior of materials tation instrument . This instrument is characterized by aat elevated temperatures [13,14]. Modeling of indentation real-time electronic force-feedback control loop that ensurescreep for nanoindentation applications has also widely re- the indenter load is held constant regardless of the depth ofported in the literature. Such treatments focus on either penetration during the creep period.constant load (creep ) or constant displacement (relax- Conventional nanoindentation load-displacement curvesation ) or both [17–21]. Traditionally, intrinsic material were done along with step loading and hold periods at max-properties are modeled in terms of spring and dashpot el- imum load. Analysis of the unloading data in the usual man-ements under indentation loading. For example, Feng and ner yields values for modulus and hardness on the assump-Ngan [22,23] applied a Maxwell two-element model to tion of no time-dependent response. Fitting to Eqs. (1)–(6)the creep displacement at maximum load in a conventional for the types of models considered here was done using aload-displacement response and determined an equivalent least squares method. For the Maxwell model, this is rel-expression for the contact stiffness that included the creep atively straight-forward, there being only two unknowns.rate expressed as a displacement over time. Their work For the three-element Voigt and four-element combinationillustrates and quantiﬁes the forward going “nose” that model, a non-linear least squares method was used the de-appears in the unloading curve in indentation experiments tails of which are given in the Appendix A. In this method,in which creep is signiﬁcant. Cheng et al.  applied a starting values of E∗ and η are required. The method allowsmethod of functional equations to the visco-elastic contact for the speciﬁcation of a tolerance level for convergence andproblem in conjunction with a step increase in load, or also the speciﬁcation of relaxation factors to be applied fordisplacement, to a three-element Voigt model to provide each variable to prevent instability in the computations. Theequations for steady creep, or relaxation, for the case of a procedure for materials that exhibit signiﬁcant creep is rela-rigid spherical indenter in contact with both incompress- tively straight forward, convergence is rather rapid. For solidible and compressible materials. In contrast, in a recent like materials, such as the aluminum and fused silica testedwork, Oyen and Cook  presented a phenomenological here, it is necessary to reduce the value of the relaxationapproach which sought to include elasticity, viscosity, and factors progressively and to undertake many iterations.plasticity in terms of modeling elements that represented In all the experiments reported here, testing was done inthe quadratic character of the contact equations rather than under very closely controlled laboratory conditions in whichthe intrinsic properties of the specimen material. Those the ambient temperature was held at 21 ± 0.1 ◦ C. Each testworkers found a very good agreement between the predicted was performed after a long thermal soak period, and thermaland observed load-displacement curves for the bounding drift was assessed before each experiment by monitoringconditions of an elastic-plastic response (e.g. metals and the displacement output of the indentation instrument at theceramics) to visco-elastic deformation (e.g. elastomers). initial contact force. Thermal drift during all the experiments The previous works mentioned here are characterized by was deemed to be negligible and no correction of the ﬁnalvery formal constitutive equations and complex methods of data was made during the analysis. The creep times selectedsolution. By way of contrast, in draft standard ISO14577 for this study are comparatively short compared to larger, creep is simply expressed as a change in depth (or scale indentation testing. This ensures that any thermal driftload) over time for ﬁxed load, or ﬁxed displacement loading. errors that might occur are kept to a minimum.The intention of the present work is to ﬁll the gap between For the aluminum specimen, a nominal 20 m radiusthose more formal treatments discussed above and that spec- diamond spherical indenter was used to perform standard
A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82 77Table 1 yielded E1 = 58.6 GPa, E2 = 1197 GPa and η = 1886 GPa s.Values of elastic modulus, E, hardness, H, and total penetration depth, A least squares ﬁt to the two-element Maxwell model, gaveht , from the unloading response of conventional load-displacement curveson a 1 m Al ﬁlm deposited on silicon E1 = 75.6 GPa and η = 28797 GPa s. The value of radius used for the ﬁtting was adjusted according to the area func-Hold (s) E (GPa) H (GPa) ht (nm) 0 109 0.932 73.8 tion of the indenter and at the depth of penetration measured,10 97.8 0.928 74.7 translated into an actual radius of 36 m. The displacement20 112 0.926 76.7 response function using the values in Eq. (1) is shown as a40 105 0.926 77.4 full line in Fig. 3.80 102 0.922 79.9 The data shown in Fig. 3 was taken with some consid- eration for minimizing the dynamic response of the mea- surement instrument. The displacement data was taken af-load–unload indentation tests with varying hold periods at ter ampliﬁcation from the depth sensor but before the nor-maximum load to determine the effect of creep on the com- mal ﬁltering circuitry so that any time-related delays withputed values of modulus and hardness. The actual radius of the electronic ﬁltering would not affect the results obtained.the indenter at the penetration depth for each experiment The data therefore contains a level of electronic noise thatwas determined by calibration against a fused silica speci- would ordinarily not be measured in a nanoindentation test.men and used in the modeling procedure. It was found that The application of the 10 mN load took approximately 1 s,for the indenter used here, there was a substantial ﬂatten- the fastest rate available with the test instrument.ing of the radius at small penetration depths and a sharpen- For the fused silica specimen, a corner cube indenter wasing of the radius at higher depths. This is not unusual with used in both a conventional load–unload indentation test andsphero-conical indenters used in nanoindentation work. The also in step loading and hold. Conventional analysis of theresults for modulus and hardness with varying creep times unloading yielded E = 74.2 GPa and H = 9.5 GPa at a depthare shown in Table 1. A Poisson’s ratio of ν = 0.35 was of penetration of 464 nm for a maximum load of 10 mN. Aassumed for aluminum to extract the specimen modulus E value of ν = 0.17 was used to extract the specimen modulusfrom the combined modulus E∗ . from the combined modulus. These results are in reason- Further, tests with a step load followed by a hold period able agreement with the commonly accepted values of 72.5were performed and the resulting data analyzed using the and 9.2 GPa, respectively. The effective cone angle at thisEqs. (1) and (3). A step load of 10 mN was applied to the depth was found to be 57◦ from the calibrated area functionsame indenter and held constant for 20 s. The change in depth of the indenter. For a step loading to 10 mN and hold pe-as a function of time is shown in Fig. 3. The depth changed riod for 20 s, the resulting change in displacement as a func-from an initial value of 70.4–78.6 nm over the 20 s hold pe- tion of time was analyzed using Eq. (2) using the non-linearriod. A non-linear least squares analysis of the hold period least squares method. A signiﬁcant number of iterationsdata for a three-element Voigt model according to Eq. (1) and adjustments to the relaxation factors was required to obtain convergence. The results of the ﬁtting yielded E1 = 36.33 GPa, E2 = 1.27 × 108 GPa and η = 8150 GPa s. Linear least squares ﬁtting for the two-element Maxwell model yielded E1 = 36 GPa and η = 52816 GPa s. A com- parison of the experimental and ﬁtted data for the hold pe- riod is shown in Fig. 4. The relatively large amount of scatter (≈ ±1 nm) in Fig. 4 arises from taking the depth readings before any ﬁltering to ensure the most representative mate- rial time-related response. As can be seen from the results above, there is a signiﬁcant difference in the value for E1 provided by the creep analysis compared to that obtained with the conventional unload analysis. This arises from the ﬁnite loading time of the indenter to the maximum load and more discussion on this is given below. The corner cube indenter was selected for this material since experience with a spherical indenter showed that it was difﬁcult to obtain any creep response in this material. A cube-corner indenter was thought to provide a very highFig. 3. Creep response for hold period of 20 s at 10 mN on a 1 m Al ﬁlm level of indentation stress so as to induce a visco-elasticon silicon with a spherical indenter 20 m nominal radius (36 m actualradius). Data points show experimental results. The solid line shows the or visco-plastic response more readily than would be pos-ﬁtted response according to Eq. (1) (three-element Voigt model). The sible with a blunter type of indenter. With the cube cor-dotted line shows the response according to Eq. (3) (two-element Maxwell ner indenter, reloading the impression in this material aftermodel). some minutes from the initial testing showed a completely
78 A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82 Table 2 Results from conventional nanoindentation analysis on the unloading portion of load–unload tests and creep tests at 1 mN for 10 s for indentation with a nominal 20 m radius spherical indenter on (a) acrylic co-polymer ﬁlm and (b) acrylic co-polymer ﬁlm with cross-linker additive (both ﬁlms were approximately 100–150 m) Polymer (a) Polymer (b) Load/unload test to 1 mN at constant ht = 1680 nm ht = 313 nm strain rate 20%/s E (GPa) 0.154 1.209 H (GPa) 0.0086 0.0571 ht (nm) 1680 313 Creep test at step load to 1 mN, ht = 2000 nm ht = 373 nm hold for 10 s Two-element Maxwell model E1 (GPa) 0.0896 0.9414 η (GPa s) 1.687 41.25Fig. 4. Creep response for hold period of 20 s at 10 mN on a fusedsilica with a corner cube indenter with an effective cone angle of 57◦ Three-element Voigt modelas determined from the indenter area function. The solid line shows the E1 (GPa) 0.163 1.033ﬁtting according to Eq. (2) (three-element Voigt model) while the dotted E2 (GPa) 0.103 2.732line shows the ﬁtting according to Eq. (4) (two-element Maxwell model). η (GPa s) 0.135 11.18 Four-element Maxwell–Voigt model E1 (GPa) 0.175 1.055ﬂat creep response during a hold period indicating that the E2 (GPa) 0.136 4.703creep observed here is related to visco-plasticity rather than η1 (GPa s) 0.696 7.688visco-elasticity. η2 (GPa s) 2.856 72.6 Load-displacement response and a creep test at stepload of 1 mN for 10 s were performed on the two acrylic modulus E from the combined modulus E∗ . For these tests,co-polymer materials to a maximum load of 1 mN with a the load was applied over a period of X and Y s.20 m radius spherical indenter. At the depths of penetra-tion in this material, the actual radius of the indenter tip wasestimated to be 13.5 m from the calibrated area function of 3. Discussionthe indenter. The load-displacement curves were performedat a constant strain rate of 20%/s and are shown in Fig. 5. As mentioned in Section 1, the depth recorded at eachThe results are given in Table 2 along with the moduli and load increment in a nanoindentation test will be that arisingviscosity estimations from least squares ﬁtting to the creep from elastic-plastic, visco-elastic and visco-plastic deforma-response from a step loading to 1 mN and hold over 10 s. tions. In conventional nanoindentation testing, the instanta-The creep responses for the two materials are shown in neous elastic and plastic deformations are usually consid-Fig. 6 along with the ﬁtted curves from Eqs. (1), (3) and ered. Elastic equations of contact are applied to the unload-(5). A value of ν = 0.4 was used to extract the specimen ing data to ﬁnd the depth of the circle of contact at full loadFig. 5. Load–unload response for indentation with a nominal 20 m radius spherical indenter on (a) acrylic co-polymer ﬁlm and (b) acrylic co-polymerﬁlm with cross-linker additive.
A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82 79Fig. 6. Creep response for hold period of 10 s at 1 mN on (a) acrylic co-polymer ﬁlm and (b) acrylic co-polymer ﬁlm with cross-linker additive with a13.5 m spherical indenter as determined from the indenter area function. Experimental data is shown as data points. Line plots indicate ﬁtting to two-,three- and four-element models.(under elastic-plastic conditions). If there is a visco-elastic visco-plastic behavior. In the case of aluminum, the effect isor visco-plastic response (i.e. creep), then this analysis is in- aggravated due to piling-up of material around the edge ofvalid. In extreme cases, the unloading curve has a negative the contact area. In instrumented indentation tests, the piledslope which, if the standard unloading analysis is applied, up material serves to make the specimen appear harder andresults in a negative elastic modulus [22,27]. stiffer because conventional analysis techniques do not ac- When ﬁtting creep curves to mechanical models, we must count for it.be sure that we distinguish between visco-elasticity and In the present work, the loading time for the step loadvisco-plasticity. The models shown here are appropriate for was limited by the time-response of the electronics of thevisco-elastic deformation but may be shown to provide some test instrument. An attempt was made to gauge the effectinformation for visco-plastic deformation. If a step load is of the loading time for the case of the cube corner indenterapplied to an indenter in contact with a material and the on fused silica by performing a number of creep indenta-resulting depth of penetration monitored, then a response tion tests while attempting to increase the loading time bysimilar to that shown in Fig. 2(a) or (b) may be obtained. altering the time response of the instrument circuitry. FasterFitting the appropriate equation (Eqs. (1)–(4)) to this data loading times could be achieved, but it was found that theyields values for the moduli and viscosity terms. The ac- indenter penetrated the sample more deeply due to over-curacy of the results so obtained depends upon the rapid- shoot in the force servo-control feedback loop. Attempts toity with which the step increase in load is applied and the reduce the overshoot resulted in a more damped responsetime-dependent nature of the specimen material. In practice, which increased the time taken to achieve the maximum loadan increase in load is applied over a ﬁnite time period within which also, as discussed above, increased the penetrationwhich, for elastic-plastic materials, plastic deformation can depth due to plastic deformation in the specimen. These at-occur quite rapidly and this causes the initial step response tempts showed that for highly elastic materials, a creep anal-in displacement to be greater than that predicted, particu- ysis employing a step increase in load is likely to providelarly when a sharp indenter is used. The resulting value of values of E1 only to within an order of magnitude. In thismodulus can be very much less than the nominal modulus case, conventional nanoindentation would be more appro-of the material. For a step load with a spherical indenter priate for the measurement of elastic properties. However,(to reduce the possibility of time-independent plasticity) on measurements of the time-dependent properties of such aa material with a signiﬁcant viscous response, the resulting material are not so affected by the non-instantaneous stepanalysis is likely to result in a reasonable measurement of application of load and the measurement of the viscositythe visco-elastic properties of the material. The signiﬁcance terms would be expected to be reliable. In the extreme caseof this can be observed by comparing the values of mod- of solid-like materials (e.g. the fused silica specimen testedulus obtained from conventional unloading curve analysis here), the procedure may be useful in determining a quantita-to that obtained from the creep curve ﬁtting. For aluminum tive account of the time-dependent nature of the deformationand fused silica, the moduli E1 from the creep curve ﬁtting under extremely high contact pressures which would ordi-procedures are very much less than expected because of the narily be inaccessible in conventional tensile testing whereplastic deformation arising during the step loading. This is fracture of the specimen occurs before plastic deformation.because the models assume an initial instantaneous elastic For visco-elastic property measurements, the creep analysisresponse (i.e. the models represent a visco-elastic deforma- procedure presented here is more suitable for use on ma-tion) whereas in these materials, there is an elastic-plastic terials in which time-dependent behavior represents a sig-response, and in the case of aluminum, a relatively strong niﬁcant contribution to the overall deformation (such as the
80 A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82polymer materials tested here). In the case of the polymer the deformation is desired. The scope of the present pa-specimens, a relatively low load of 1 mN was found to pro- per is to present a simple phenomenological approach only.vide a reasonably large penetration depth in each case and A comparison of the ﬁtted creep response curves and therequirement for a more ideal step-like application thus more experimental data presented in Figs. 3, 4 and 6 demon-easily accomplished. strate that the mechanical models used may not precisely For the case of the two polymeric materials tested here, the match the response of the specimens, particularly in thespecimen with the added cross-linker shows a much stiffer case of the two-element Maxwell model. The three-elementelastic and more viscous response compared to the base ma- Voigt model provides a reasonable ﬁt while the four-elementterial. This is expected as the cross linker serves to restrict model (Fig. 6) shows a very good ﬁt. The physical signif-the motion of molecular chains under the applied shear stress icance of the elements within a model depends upon thein the indentation stress ﬁeld. In general, the response of a microstructural characteristics of the specimen material.particular specimen material may be affected by indentation In the present case, no weighting scheme was used othersize effects arising from strain-gradient plasticity, the pres- than an adjustment to the values of the relaxation factors.ence of oxide layers, surface roughness, etc. Such effects are However, the non-linear least square ﬁtting procedure is gen-beyond the scope of the present work but should be consid- eral enough to allow the addition of more elements as de-ered as possible sources of errors or variations in the data, sired. The ﬁtted curves shown here represent a minimizationespecially in the case of crystalline solids. of the sum of the squares of the differences between the ﬁt- The load–unload curves shown in Fig. 5 deserve some ted and actual data, but more control is possible through thecomment. The polymeric materials tested here were selected use of the terms wi given in the Appendix A. Using theseon the expectation of a signiﬁcant visco-elastic response (in factors, the differences between the ﬁtted and actual datacontrast to an elastic-plastic response). The curves shown in can be weighted in favor of data at one or other ends of theFig. 5 support this, especially for the cross-linked material. range of values as desired.Although there is a readily identiﬁable plastic deformation Some workers [27,31] have studied the effect of specimenas evidenced by the area enclosed between the loading and creep on the values of modulus and hardness obtained usingunloading curves, the elastic recovery of the material is quite conventional methods of analysis of the unloading response.substantial. This implies that the energy dissipation within The general conclusion is that for materials which exhibitthese types of materials is substantially a result of viscous creep during an indentation test, the modulus so calculatedlosses (in the sense of visco-elasticity) and not plastic defor- from the unloading response is not reliable if the hold periodmation (in the sense of elasto-plasticity). Similar behavior at maximum load is too short due to bowing or “nose” ofin various materials has been previously reported in the lit- the unloading response to larger depth values resulting fromerature [28,29] but has not in general been satisfactorily ex- creep. Briscoe et al.  introduced a 10 s hold period intoplained. If we recognize the limitations of the method, then their tests on polymeric specimens to eliminate the nose inthe values for E2 and η, in the case of the three-element Voigt the unloading data. Chudoba and Richter  found that themodel, may still have validity for visco-plastic contact since, hold period at maximum load has to be long enough suchas shown in Fig. 2, they inﬂuence only the time-dependent that the creep rate has decayed to a value where the depthcharacter of the deformation and not the initial response to increase in 1 min is less than 1% of the indentation depth.the step loading. A decision as to the nature of the speci- According to Chudoba and Richter, allowing creep to pro-men material with regard to its elasto-plastic, visco-plastic or ceed to relative completion and then obtaining the unload-visco-elastic character can thus in some cases be made upon ing data would provide a value of unloading stiffness dP/dhthe observed shape of the conventional load-displacement that would occur at the increased depth free from the effectresponse. of creep. Feng and Ngan  draw a similar conclusion and While the use of simple mechanical models such as those show how a value of dP/dh can be obtained using shortershown in Fig. 1 allow some comparison to be made between hold periods if the unloading slope is corrected by a factorspecimens, it should be noted that they offer very little in which is dependent upon the creep rate and the unloadingterms of a basic understanding of the physical mechanisms load rate.involved in the deformation. The mathematical models inEqs. (1)–(6) are simply standard elastic equations with atime-dependent terms added to represent the ﬂuid-like be- 4. Conclusionhavior of the material. They are models only with no realphysical signiﬁcance. They serve only to give some quan- The intention of the present work is to provide a simpletitative description of mechanical events. Time-dependent and accessible method of obtaining a quantitative measurebehavior often depends on the strain-hardening characteris- of the elastic and viscous properties of materials from inden-tics of the material which in turn, depend upon microstruc- tation creep curves. The present work is not intended to of-tural variables. Various constitutive laws  have been pro- fer a rigorous account of indentation creep or materials con-posed that apply to many different types of materials and stitutive behavior. Three representative classes of materialsthese should be investigated if a more detailed account of were tested: a highly elastic ceramic, a soft metal and two
A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82 81visco-elastic polymer systems. For highly elastic solid-like yi − Zi at each data point i can be weighted by a factor wi tomaterials, the results using the creep method for the elas- reﬂect the error associated in the observed values yi . Thus,tic modulus are very much less than that expected due to the sum of the squares is expressed asthe sensitivity of the technique to the non-deal step-like ini- Ntial application of load. The technique may however provide X2 = wi [yi − Zi ]2quantitative information about the plastic behavior of these i=1materials which would not ordinarily be accessible through 2conventional tensile or compressive tests where fracture usu- N r δZi oally occurs before deformation. For visco-elastic materials, = wi yi − Zi + o δaj δajthe technique is expected to be useful for the measurement i=1 j=1of both viscous and elastic properties. Lower loads may be 2 N rused to obtain reasonable penetration depths thus allowing a δZi o = wi (yi − Zi ) − o δaj ,step-like application of load to be more easily accomplished. δaj i=1 j=1Depending on the material tested, the results can be consid- 2ered a ﬁgure of merit, or absolute values of elasticity and N r δZi oviscosity. The theoretical work presented here is based on yi = yi − Zi , o X2 = w i yi − δaj δajstandard phenomenological models and the ﬁtting procedure i=1 j=1can be readily automated. (A.5) The weighting factor wi for the present application can beAcknowledgements simply the magnitude of yi on the assumption that the error at each data point is inversely proportional to the magnitude The author thanks Avi Bendavid for supplying the high of the data at that point.purity aluminum ﬁlm, Jeffrey T. Carter for supplying the The objective is to minimise this sum with respect to thepolymer ﬁlms used in this study, and A.H.W. Ngan and an values of the error terms δaj , thus we set the derivative ofanonymous referee for useful comments. X2 with respect to δaj to zero δX2Appendix A. Linear approximation, non-linear least = 0,squares δ(δaj ) N r δZi o δZi o Let Zi be a function that provides ﬁtted values of a de- 0= wi yi − δaj (A.6) δaj δajpendent variable yi at each value of an independent variable i=1 j=1xi . Zi can be a function of many parameters a0 , a1 , . . . , ar . This expression can be expanded by considering a fewZi = Zi (xi : a0 , a1 , a2 , . . . , aj , . . . , ar ) (A.1) examples of j. Letting j = 1, we obtain It is presumed that initial values or estimates of these N δZi oparameters are known and that the desired outcome is an wi y i δa1optimisation of the values of these parameters using the i=1method of least squares. The true value of the parameter aj N 2 N δZi o δZi δZi o ois found by adding an error term δaj to the initial value aj . o = δa1 wi + δa2 wi δa1 δa1 δa2 i=1 i=1aj = aj + δaj o (A.2) N δZi o δZi o Thus, the function Zi becomes + · · · + δar wi (A.7) δa1 δar i=1Zi = Zi (xi : a1 , a2 , a3 , . . . , ar ) o o o o o (A.3) At j equal to some arbitrary value of k, we obtain: If the errors δaj are small, then the function Zi can be Nexpressed as a Taylor series expansion δZi o wi y i r δZi o δak i=1Zi = Zi o + δaj (A.4) δaj N δZi δZi o o N δZi δZi o o j=1 = δa1 wi + δa2 wi + ··· δak δa1 δak δa2 This is a linear equation in δaj and is thus amenable to i=1 i=1multiple linear least squares analysis. N 2 N δZi o δZi δZi o o Now, by least squares theory, we wish to minimise the sum + δak wi + · · · + δar wi δak δak δarof the squares of the differences (or residuals) between the i=1 i=1observed values yi and the ﬁtted values Zi . The differences (A.8)
82 A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82 In matrix notation, the sums for each error term δa from  E.H. Lee, J.R.M. Radok, Trans. ASME Series E, J. Appl. Mech. 271–r is expressed (1960) 438–444.  K.L. Johnson, Contact Mechanics, Cambridge University Press, 1985.  L. Cheng, X. Xia, W. Yu, L.E. Scriven, W.W. Gerberich, J. Polym.[Yj ] = [Ajk ][Xj ], Sci. B 38 (2000) 10–22. Y1 A11 . . . A1r X1  M. Sakai, Phil. Mag. A 82 (10) (2002) 1841–1849.  P.M. Sargent, M.F. Ashby, Mater. Sci. Technol. 8 (1992) 594–601.Y 2 . . . . . X2  R. Hill, B. Storåkers, A.B. Zdunek, Proc. R. Soc. Lond. A423 (1989) Yk = . . Ajk . . Xk (A.9) 301–330.  R. Hill, Proc. R. Soc. Lond. A436 (1992) 617–630. . . . . . . .  B. Storåkers, P.-L. Larsson, J. Mech. Phys. Solids 42 (2) (1994) 307–332. Yr Ar1 . . . Arr Xr  S.Dj. Mesarovic, N.A. Fleck, Proc. R. Soc. Lond. A455 (1999) 2707–2728.where  Y.-T. Cheng, C.-M. Cheng, Phil. Mag. Lett. 81 (2001) 9–16. N  T.R.G. Kutty, C. Ganguly, D.H. Sastry, Scripta Mater. 34 (12) (1996) δZi o 1833–1838.Yj = wi yi , δaj  M. Sakai, S. Shimizu, J. Non-Cryst. Solids 282 (2001) 236– i=1 247. N  W.B. Li, R. Warren, Acta Metall. Mater. 41 (10) (1993) 3065– δZi δZi o oAjk = Akj = wi , Xj = δaj (A.10) 3069. δaj δak  S. Shimizu, Y. Yanagimoto, M. Sakai, J. Mater. Res. 14 (10) (1999) i=1 4075–4085.and  L. Cheng, L.E. Scriven, W.W. Gerberich, Mater. Res. Symp. Proc. 522 (1998) 193–198. yi = yi − Zi o (A.11)  A. Strojny, W.W. Gerberich, Mater. Res. Soc. Symp. Proc. 522 (1998) 159–164. The solution is the matrix X that contains the error terms  S.A. Syed Asif, J.B. Pethica, J. Adhes. 67 (1998) 153–165.to be minimised. Thus  X. Xia, A. Stronjy, L.E. Scriven, W.W. Gerberich, A. Tsou, C.C. Anderson, Mater. Res. Symp. Proc. 522 (1988) 199–204.[Xj ] = [Ajk ]−1 [Yj ] (A.12)  K.B. Yoder, S. Ahuja, K.T. Dihn, D.A. Crowson, S.G. Corcoran, L. Cheng, W.W. Gerberich, Mater. Res. Symp. 522 (1998) 205–210. When values of δaj are calculated, they are added to the  G. Feng, A.H.W. Ngan, J. Mater. Res. 17 (3) (2002) 660–668.initial values aj to give the ﬁtted values aj o  A.H.W. Ngan, B. Tang, J. Mater. Res. 17 (10) (2002) 2604– 2610.aj = aj + Lδaj 1 o (A.13)  M.L. Oyen, R.F. Cook, J. Mater. Res. 18 (1) (2003) 139–150.  ISO14577, Metallic materials—instrumented indentation test for The process may then be repeated until the error terms δaj hardness and materials parameters, ISO Central Secretariat, 1 rue debecome sufﬁciently small indicating that the parameters aj Varembé, 1211 Geneva 20, Switzerland.  CSIRO Division of Telecommunications and Industrial Physics, P.O.have converged to their optimum value. L in Eq. (A.13) is a Box 218, Lindﬁeld, NSW 2007, Australia.relaxation factor that is applied to error terms to prevent in-  T. Chudoba, F. Richter, Surf. Coat. Tech. 148 (2001) 191–198.stability during the initial phases of the reﬁnement process.  M.V. Swain, J.S. Field, Phil. Mag. A 74 (5) (1996) 1085– 1096.  M.W. Seto, K. Robbie, D. Vick, M.J. Brett, L. Khun, J. Vac. Sci. Technol. B 17 (5) (1999) 2172–2177.References  D. François, A. Pineau, A. Zaoui, Mechanical Behaviour of Materials, Kluwer Academic Publishers, The Netherlands, 1998.  A.C. Fischer-Cripps, Nanoindentation, Springer-Verlag, NY, 2002.  B.J. Briscoe, L. Fiori, E. Pelillo, J. Phys. D 31 (1998) 2395–  J.R.M. Radok, Q. Appl. Math. 15 (1957) 198–202. 2405.