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A simple phenomenological approach to nanoindentation creep
Materials Science and Engineering A 385 (2004) 74–82
A simple phenomenological approach to nanoindentation creep A.C. Fischer-Cripps∗ CSIRO Division of Telecommunications and Industrial Physics, P.O. Box 218, Lindfield, NSW 2070, Australia Received 5 February 2004; received in revised form 19 April 2004
Abstract Nanoindentation is frequently used to measure elastic modulus and hardness of structural materials such as ceramics, metals and thin films. The assumption behind conventional nanoindentation analysis methods, where the unloading data is analysed, is that the material behaves in an elastic-plastic manner. However, many materials can also exhibit a visco-elastic and visco-plastic response which is commonly termed “creep”. In a nanoindentation test, this is usually observed as an increase in depth during a hold period at maximum load in the load-displacement data. Creep is not accommodated in conventional nanoindentation analysis methods. The present work shows how conventional linear spring and dashpot elements can be used to model the creep response of a wide range of materials using the hold period force-displacement data. The method shown can be readily incorporated into a computer program and can be used with any conventional nanoindentation test instrument using either spherical or sharp indenters. Crown Copyright © 2004 Published by Elsevier B.V. All rights reserved. Keywords: Nanoindentation; Indentation creep; Depth-sensing indentation
1. Introduction Nanoindentation has proven to be an effective and convenient method of determining the mechanical properties of solids, most notably elastic modulus and hardness. The most popular method relies on an analysis of the unloading load-displacement response which is assumed to be elastic, even if the contact is elastic-plastic [1]. The method relies on plasticity occurring instantaneously upon satisfaction of a constitutive criterion and that there are no time-dependent effects. Many materials, however, have a time-dependent behavior when placed under load and as a result, conventional nanoindentation test methods may not provide an adequate estimation of material properties of interest. There are two basic approaches to managing time-dependent behavior in nanoindentation testing. The first is the application of an oscillatory displacement or force, in which the transfer function between the load and displacement provides a method of calculating the storage and loss modulus of the material. In the second, the application of a step load or displacement and subsequent measurement of depth (creep) or force as a function of time (relaxation) is used to calculate visco-elastic
∗
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properties of the specimen material. It is the second method that is the subject of the present work. In a nanoindentation test, creep and plastic deformation in the conventional sense, i.e., that occurring due to shear-driven slip for example, should be regarded separately. Plasticity, in the sense of yield or hardness, is conveniently thought of as being an instantaneous event (although in practice, it can take time for yield processes to complete). In contrast, creep can occur over time in an otherwise elastic deformation as a result of the diffusion and motion of atoms or movement of dislocations in the indentation stress field, the extent of which depends very much on the temperature. If we ignore effects arising from the formation of cracks, permanent deformation in a material under indentation loading is thus seen as arising from a combination of instantaneous plasticity (which is not time-dependent) and creep (which is time-dependent). A material which undergoes elastic and non-time-dependent plastic deformation is called elasto-plastic. A material that deforms elastically but exhibits time-dependent behavior is called visco-elastic. A material in which time-dependent plastic deformation occurs is visco-plastic [1]. The term creep is often used to describe a delayed response to an applied stress or strain that may be a result of visco-elastic or visco-plastic deformation. In a nanoindentation test, the depth recorded at each load increment will be, in general, the addition of that
0921-5093/$ – see front matter Crown Copyright © 2004 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2004.04.070
A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82
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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.
Fig. 1. (a) Three-element Voigt spring and dashpot representation of a visco-elastic material (delayed elasticy), (b) Maxwell representation of a visco-elastic material (steady creep), (c) four-element combined Maxwell–Voigt model.
due to the elastic-plastic properties of the material and that occurring to due creep, either visco-elastic or visco-plastic. Time dependent properties of materials are conventionally analyzed in terms of mechanical models such as those shown in Fig. 1. The elastic response of such a model is quantified by what we call the storage modulus. The fluid-like response is quantified by the loss modulus. In rheology, the material behavior is towards the fluid end of the spectrum where any elastic response, or the storage modulus, is dominated by the shear modulus of the material. In solids, such as those usually tested in nanoindentation, the material behavior tends towards a predominantly elastic response. In an indentation test, the nature of the loading is a complex mixture of hydrostatic compression, tension, and shear. Unlike a fluid, the storage modulus in this case contains contributions from all three of these types of materials response. In the present work, we shall, in the interests of simplicity, assume that the storage modulus is a measure of the conventional (tensile/compressive) elastic modulus in recognition of the large component of hydrostatic stress in the indentation stress field. The fluid-like response we shall call “viscosity” although in practice, viscosity is usually frequency and temperature dependent and not single-valued. Radok [2], and Lee and Radok [3] have analyzed the visco-elastic contact problem using a correspondence principle in which elastic constants in the elastic equations of contact are replaced with time dependent operators [4]. For the case of a rigid spherical indenter in contact with a material represented by a three-element Voigt model as shown in Fig. 1(a), we can write that for a steady applied load Po , the depth of penetration increases with time is given by the well-known Hertz equation with the addition of a time-dependent exponential 3 Po 1 1 −tE∗2 /η (1) h32 (t) = √ + (1 − e ) 4 R E1∗ E2∗
The square-bracketed term in Eq. (1) represents the displacement-time response of the mechanical model to a step increase in load. A step increase in applied load results in an initial elastic displacement (t = 0) followed by a delayed increase in displacement to a maximum value at t = ∞ 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 and Poisson’s ratio of the specimen material (E∗ = E/(1 − ν2 )) and not the combined modulus of the indenter and specimen as is normally the case. The numerical factor 3/4 in Eq. (1) differs from that usually seen in reference literature due to our working with the elastic modulus E∗ rather than the shear modulus G. The justification for the change in variable being that in indentation loading, the greater proportion of the deformation is hydrostatic compression and the materials we consider are more likely to be dominated by solid-like properties than fluid-like behavior. A similar approach is appropriate for the case of a conical indenter in which we obtain, for the case of the three-element Voigt model 1 1 2 −tE∗2 /η h (t) = Po cot α + ∗ (1 − e ) (2) 2 E1∗ E2 For the case of a Maxwell model Fig. 1(b), the time-dependent depth of penetration for a spherical indenter is given by 3 Po 1 1 h32 (t) = √ + t (3) 4 R E1∗ η and for a conical indenter, we obtain 1 1 h2 (t) = Po cot α t + 2 E1∗ η
(4)
From the above information, it is relatively easy to construct equations for more complicated models by adding components in series or parallel as would be done in mechanical–electrical analogs conventionally applied to this type of modeling. For example, in the present work, we shall also consider a four-element Maxwell–Voigt combination as shown in Fig. 1(c) which gives 3 Po 1 1 1 −tE∗2 /η2 (5) h32 (t) = √ + (1 − e ) + t 4 R E1∗ E2∗ η1
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h2 (t) =
A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82
1 1 1 −tE∗2 /η2 + (1 − e ) + t Po cot α 2 E1∗ E2∗ η1
(6)
The equations given above are expressed in terms of the modulus E∗ which is the combination of elastic modulus and Poisson’s ratio of the specimen. In the present work, the elastic material properties of the specimen material are given in terms of E∗ as is done in conventional nanoindentation analysis and η is the viscosity term that quantifies the time-dependent property of the material. It should be remembered that Eqs. (1)–(6) assume a step increase in load to Po and are expressed here in a form to be easily fitted to experimentally obtained creep data to provide values for E∗ and η. Should modeling of an arbitrary time displacement response be required, a suitable superposition [5,6] can be employed but this is beyond the scope of the present work. There are several detailed theoretical studies of visco-elastic indentation creep available in the literature [7–12]. A popular motivation for such modeling is the behavior of materials at elevated temperatures [13,14]. Modeling of indentation creep for nanoindentation applications has also widely reported in the literature. Such treatments focus on either constant load (creep [15]) or constant displacement (relaxation [16]) or both [17–21]. Traditionally, intrinsic material properties are modeled in terms of spring and dashpot elements under indentation loading. For example, Feng and Ngan [22,23] applied a Maxwell two-element model to the creep displacement at maximum load in a conventional load-displacement response and determined an equivalent expression for the contact stiffness that included the creep rate expressed as a displacement over time. Their work illustrates and quantifies the forward going “nose” that appears in the unloading curve in indentation experiments in which creep is significant. Cheng et al. [17] applied a method of functional equations to the visco-elastic contact problem in conjunction with a step increase in load, or displacement, to a three-element Voigt model to provide equations for steady creep, or relaxation, for the case of a rigid spherical indenter in contact with both incompressible and compressible materials. In contrast, in a recent work, Oyen and Cook [24] presented a phenomenological approach which sought to include elasticity, viscosity, and plasticity in terms of modeling elements that represented the quadratic character of the contact equations rather than the intrinsic properties of the specimen material. Those workers found a very good agreement between the predicted and observed load-displacement curves for the bounding conditions of an elastic-plastic response (e.g. metals and ceramics) to visco-elastic deformation (e.g. elastomers). The previous works mentioned here are characterized by very formal constitutive equations and complex methods of solution. By way of contrast, in draft standard ISO14577 [25], creep is simply expressed as a change in depth (or load) over time for fixed load, or fixed displacement loading. The intention of the present work is to fill the gap between those more formal treatments discussed above and that spec-
ified in draft standard ISO14577. A simplified treatment of nanoindentation creep is presented and a readily accessible method of analysing Eqs. (1)–(4) in a manner suitable for automation within a computer program is provided.
2. Experimental Several materials were selected for study. The first, a 1 m thick film of high purity aluminum, a metal known to exhibit significant indentation creep. The second, a sample of fused silica, a material in which creep is not expected to be significant, and the third, an 100–150 m thick polyurethane acrylic copolymer film, a material expected to exhibit significant visco-elasticity the amount of which is dependent upon the additive molecules. Two acrylic co-polymer materials were tested, a baseline material and another with a cross-linker added. Tests were done on a UMIS nanoindentation instrument [26]. This instrument is characterized by a real-time electronic force-feedback control loop that ensures the indenter load is held constant regardless of the depth of penetration during the creep period. Conventional nanoindentation load-displacement curves were done along with step loading and hold periods at maximum load. Analysis of the unloading data in the usual manner yields values for modulus and hardness on the assumption of no time-dependent response. Fitting to Eqs. (1)–(6) for the types of models considered here was done using a least squares method. For the Maxwell model, this is relatively straight-forward, there being only two unknowns. For the three-element Voigt and four-element combination model, a non-linear least squares method was used the details of which are given in the Appendix A. In this method, starting values of E∗ and η are required. The method allows for the specification of a tolerance level for convergence and also the specification of relaxation factors to be applied for each variable to prevent instability in the computations. The procedure for materials that exhibit significant creep is relatively straight forward, convergence is rather rapid. For solid like materials, such as the aluminum and fused silica tested here, it is necessary to reduce the value of the relaxation factors progressively and to undertake many iterations. In all the experiments reported here, testing was done in under very closely controlled laboratory conditions in which the ambient temperature was held at 21 ± 0.1 ◦ C. Each test was performed after a long thermal soak period, and thermal drift was assessed before each experiment by monitoring the displacement output of the indentation instrument at the initial contact force. Thermal drift during all the experiments was deemed to be negligible and no correction of the final data was made during the analysis. The creep times selected for this study are comparatively short compared to larger scale indentation testing. This ensures that any thermal drift errors that might occur are kept to a minimum. For the aluminum specimen, a nominal 20 m radius diamond spherical indenter was used to perform standard
A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82 Table 1 Values of elastic modulus, E, hardness, H, and total penetration depth, ht , from the unloading response of conventional load-displacement curves on a 1 m Al film deposited on silicon Hold (s) 0 10 20 40 80
E (GPa) 109 97.8 112 105 102
H (GPa) 0.932 0.928 0.926 0.926 0.922
ht (nm) 73.8 74.7 76.7 77.4 79.9
load–unload indentation tests with varying hold periods at maximum load to determine the effect of creep on the computed values of modulus and hardness. The actual radius of the indenter at the penetration depth for each experiment was determined by calibration against a fused silica specimen and used in the modeling procedure. It was found that for the indenter used here, there was a substantial flattening of the radius at small penetration depths and a sharpening of the radius at higher depths. This is not unusual with sphero-conical indenters used in nanoindentation work. The results for modulus and hardness with varying creep times are shown in Table 1. A Poisson’s ratio of ν = 0.35 was assumed for aluminum to extract the specimen modulus E from the combined modulus E∗ . Further, tests with a step load followed by a hold period were performed and the resulting data analyzed using the Eqs. (1) and (3). A step load of 10 mN was applied to the same indenter and held constant for 20 s. The change in depth as a function of time is shown in Fig. 3. The depth changed from an initial value of 70.4–78.6 nm over the 20 s hold period. A non-linear least squares analysis of the hold period data for a three-element Voigt model according to Eq. (1)
Fig. 3. Creep response for hold period of 20 s at 10 mN on a 1 m Al film on silicon with a spherical indenter 20 m nominal radius (36 m actual radius). Data points show experimental results. The solid line shows the fitted response according to Eq. (1) (three-element Voigt model). The dotted line shows the response according to Eq. (3) (two-element Maxwell model).
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yielded E1 = 58.6 GPa, E2 = 1197 GPa and η = 1886 GPa s. A least squares fit to the two-element Maxwell model, gave E1 = 75.6 GPa and η = 28797 GPa s. The value of radius used for the fitting was adjusted according to the area function of the indenter and at the depth of penetration measured, translated into an actual radius of 36 m. The displacement response function using the values in Eq. (1) is shown as a full line in Fig. 3. The data shown in Fig. 3 was taken with some consideration for minimizing the dynamic response of the measurement instrument. The displacement data was taken after amplification from the depth sensor but before the normal filtering circuitry so that any time-related delays with the electronic filtering would not affect the results obtained. The data therefore contains a level of electronic noise that would ordinarily not be measured in a nanoindentation test. The application of the 10 mN load took approximately 1 s, the fastest rate available with the test instrument. For the fused silica specimen, a corner cube indenter was used in both a conventional load–unload indentation test and also in step loading and hold. Conventional analysis of the unloading yielded E = 74.2 GPa and H = 9.5 GPa at a depth of penetration of 464 nm for a maximum load of 10 mN. A value of ν = 0.17 was used to extract the specimen modulus from the combined modulus. These results are in reasonable agreement with the commonly accepted values of 72.5 and 9.2 GPa, respectively. The effective cone angle at this depth was found to be 57◦ from the calibrated area function of the indenter. For a step loading to 10 mN and hold period for 20 s, the resulting change in displacement as a function of time was analyzed using Eq. (2) using the non-linear least squares method. A significant number of iterations 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 comparison of the experimental and fitted data for the hold period 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 material 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 high level of indentation stress so as to induce a visco-elastic or visco-plastic response more readily than would be possible with a blunter type of indenter. With the cube corner indenter, reloading the impression in this material after some minutes from the initial testing showed a completely
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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)
Fig. 4. Creep response for hold period of 20 s at 10 mN on a fused silica with a corner cube indenter with an effective cone angle of 57◦ as determined from the indenter area function. The solid line shows the fitting according to Eq. (2) (three-element Voigt model) while the dotted line shows the fitting according to Eq. (4) (two-element Maxwell model).
flat creep response during a hold period indicating that the creep observed here is related to visco-plasticity rather than visco-elasticity. Load-displacement response and a creep test at step load of 1 mN for 10 s were performed on the two acrylic co-polymer materials to a maximum load of 1 mN with a 20 m radius spherical indenter. At the depths of penetration in this material, the actual radius of the indenter tip was estimated to be 13.5 m from the calibrated area function of the indenter. The load-displacement curves were performed at a constant strain rate of 20%/s and are shown in Fig. 5. The results are given in Table 2 along with the moduli and viscosity estimations from least squares fitting to the creep response from a step loading to 1 mN and hold over 10 s. The creep responses for the two materials are shown in Fig. 6 along with the fitted curves from Eqs. (1), (3) and (5). A value of ν = 0.4 was used to extract the specimen
Polymer (a)
Polymer (b)
Load/unload test to 1 mN at constant strain rate 20%/s E (GPa) H (GPa) ht (nm)
ht = 1680 nm
ht = 313 nm
0.154 0.0086 1680
1.209 0.0571 313
Creep test at step load to 1 mN, hold for 10 s
ht = 2000 nm
ht = 373 nm
Two-element Maxwell model E1 (GPa) η (GPa s)
0.0896 1.687
0.9414 41.25
Three-element Voigt model E1 (GPa) E2 (GPa) η (GPa s)
0.163 0.103 0.135
1.033 2.732 11.18
Four-element Maxwell–Voigt model E1 (GPa) E2 (GPa) η1 (GPa s) η2 (GPa s)
0.175 0.136 0.696 2.856
1.055 4.703 7.688 72.6
modulus E from the combined modulus E∗ . For these tests, the load was applied over a period of X and Y s. 3. Discussion As mentioned in Section 1, the depth recorded at each load increment in a nanoindentation test will be that arising from elastic-plastic, visco-elastic and visco-plastic deformations. In conventional nanoindentation testing, the instantaneous elastic and plastic deformations are usually considered. Elastic equations of contact are applied to the unloading data to find the depth of the circle of contact at full load
Fig. 5. Load–unload response 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.
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Fig. 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 a 13.5 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 or visco-plastic response (i.e. creep), then this analysis is invalid. In extreme cases, the unloading curve has a negative slope which, if the standard unloading analysis is applied, results in a negative elastic modulus [22,27]. When fitting creep curves to mechanical models, we must be sure that we distinguish between visco-elasticity and visco-plasticity. The models shown here are appropriate for visco-elastic deformation but may be shown to provide some information for visco-plastic deformation. If a step load is applied to an indenter in contact with a material and the resulting depth of penetration monitored, then a response similar to that shown in Fig. 2(a) or (b) may be obtained. Fitting the appropriate equation (Eqs. (1)–(4)) to this data yields values for the moduli and viscosity terms. The accuracy of the results so obtained depends upon the rapidity with which the step increase in load is applied and the time-dependent nature of the specimen material. In practice, an increase in load is applied over a finite time period within which, for elastic-plastic materials, plastic deformation can occur quite rapidly and this causes the initial step response in displacement to be greater than that predicted, particularly when a sharp indenter is used. The resulting value of modulus can be very much less than the nominal modulus of the material. For a step load with a spherical indenter (to reduce the possibility of time-independent plasticity) on a material with a significant viscous response, the resulting analysis is likely to result in a reasonable measurement of the visco-elastic properties of the material. The significance of this can be observed by comparing the values of modulus obtained from conventional unloading curve analysis to that obtained from the creep curve fitting. For aluminum and fused silica, the moduli E1 from the creep curve fitting procedures are very much less than expected because of the plastic deformation arising during the step loading. This is because the models assume an initial instantaneous elastic response (i.e. the models represent a visco-elastic deformation) whereas in these materials, there is an elastic-plastic response, and in the case of aluminum, a relatively strong
visco-plastic behavior. In the case of aluminum, the effect is aggravated due to piling-up of material around the edge of the contact area. In instrumented indentation tests, the piled up material serves to make the specimen appear harder and stiffer because conventional analysis techniques do not account for it. In the present work, the loading time for the step load was limited by the time-response of the electronics of the test instrument. An attempt was made to gauge the effect of the loading time for the case of the cube corner indenter on fused silica by performing a number of creep indentation tests while attempting to increase the loading time by altering the time response of the instrument circuitry. Faster loading times could be achieved, but it was found that the indenter penetrated the sample more deeply due to overshoot in the force servo-control feedback loop. Attempts to reduce the overshoot resulted in a more damped response which increased the time taken to achieve the maximum load which also, as discussed above, increased the penetration depth due to plastic deformation in the specimen. These attempts showed that for highly elastic materials, a creep analysis employing a step increase in load is likely to provide values of E1 only to within an order of magnitude. In this case, conventional nanoindentation would be more appropriate for the measurement of elastic properties. However, measurements of the time-dependent properties of such a material are not so affected by the non-instantaneous step application of load and the measurement of the viscosity terms would be expected to be reliable. In the extreme case of solid-like materials (e.g. the fused silica specimen tested here), the procedure may be useful in determining a quantitative account of the time-dependent nature of the deformation under extremely high contact pressures which would ordinarily be inaccessible in conventional tensile testing where fracture of the specimen occurs before plastic deformation. For visco-elastic property measurements, the creep analysis procedure presented here is more suitable for use on materials in which time-dependent behavior represents a significant contribution to the overall deformation (such as the
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polymer materials tested here). In the case of the polymer specimens, a relatively low load of 1 mN was found to provide a reasonably large penetration depth in each case and requirement for a more ideal step-like application thus more easily accomplished. For the case of the two polymeric materials tested here, the specimen with the added cross-linker shows a much stiffer elastic and more viscous response compared to the base material. This is expected as the cross linker serves to restrict the motion of molecular chains under the applied shear stress in the indentation stress field. In general, the response of a particular specimen material may be affected by indentation size effects arising from strain-gradient plasticity, the presence of oxide layers, surface roughness, etc. Such effects are beyond the scope of the present work but should be considered as possible sources of errors or variations in the data, especially in the case of crystalline solids. The load–unload curves shown in Fig. 5 deserve some comment. The polymeric materials tested here were selected on the expectation of a significant visco-elastic response (in contrast to an elastic-plastic response). The curves shown in Fig. 5 support this, especially for the cross-linked material. Although there is a readily identifiable plastic deformation as evidenced by the area enclosed between the loading and unloading curves, the elastic recovery of the material is quite substantial. This implies that the energy dissipation within these types of materials is substantially a result of viscous losses (in the sense of visco-elasticity) and not plastic deformation (in the sense of elasto-plasticity). Similar behavior in various materials has been previously reported in the literature [28,29] but has not in general been satisfactorily explained. If we recognize the limitations of the method, then the values for E2 and η, in the case of the three-element Voigt model, may still have validity for visco-plastic contact since, as shown in Fig. 2, they influence only the time-dependent character of the deformation and not the initial response to the step loading. A decision as to the nature of the specimen material with regard to its elasto-plastic, visco-plastic or visco-elastic character can thus in some cases be made upon the observed shape of the conventional load-displacement response. While the use of simple mechanical models such as those shown in Fig. 1 allow some comparison to be made between specimens, it should be noted that they offer very little in terms of a basic understanding of the physical mechanisms involved in the deformation. The mathematical models in Eqs. (1)–(6) are simply standard elastic equations with a time-dependent terms added to represent the fluid-like behavior of the material. They are models only with no real physical significance. They serve only to give some quantitative description of mechanical events. Time-dependent behavior often depends on the strain-hardening characteristics of the material which in turn, depend upon microstructural variables. Various constitutive laws [30] have been proposed that apply to many different types of materials and these should be investigated if a more detailed account of
the deformation is desired. The scope of the present paper is to present a simple phenomenological approach only. A comparison of the fitted creep response curves and the experimental data presented in Figs. 3, 4 and 6 demonstrate that the mechanical models used may not precisely match the response of the specimens, particularly in the case of the two-element Maxwell model. The three-element Voigt model provides a reasonable fit while the four-element model (Fig. 6) shows a very good fit. The physical significance of the elements within a model depends upon the microstructural characteristics of the specimen material. In the present case, no weighting scheme was used other than an adjustment to the values of the relaxation factors. However, the non-linear least square fitting procedure is general enough to allow the addition of more elements as desired. The fitted curves shown here represent a minimization of the sum of the squares of the differences between the fitted and actual data, but more control is possible through the use of the terms wi given in the Appendix A. Using these factors, the differences between the fitted and actual data can be weighted in favor of data at one or other ends of the range of values as desired. Some workers [27,31] have studied the effect of specimen creep on the values of modulus and hardness obtained using conventional methods of analysis of the unloading response. The general conclusion is that for materials which exhibit creep during an indentation test, the modulus so calculated from the unloading response is not reliable if the hold period at maximum load is too short due to bowing or “nose” of the unloading response to larger depth values resulting from creep. Briscoe et al. [31] introduced a 10 s hold period into their tests on polymeric specimens to eliminate the nose in the unloading data. Chudoba and Richter [27] found that the hold period at maximum load has to be long enough such that the creep rate has decayed to a value where the depth increase in 1 min is less than 1% of the indentation depth. According to Chudoba and Richter, allowing creep to proceed to relative completion and then obtaining the unloading data would provide a value of unloading stiffness dP/dh that would occur at the increased depth free from the effect of creep. Feng and Ngan [22] draw a similar conclusion and show how a value of dP/dh can be obtained using shorter hold periods if the unloading slope is corrected by a factor which is dependent upon the creep rate and the unloading load rate.
4. Conclusion The intention of the present work is to provide a simple and accessible method of obtaining a quantitative measure of the elastic and viscous properties of materials from indentation creep curves. The present work is not intended to offer a rigorous account of indentation creep or materials constitutive behavior. Three representative classes of materials were tested: a highly elastic ceramic, a soft metal and two
A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82
visco-elastic polymer systems. For highly elastic solid-like materials, the results using the creep method for the elastic modulus are very much less than that expected due to the sensitivity of the technique to the non-deal step-like initial application of load. The technique may however provide quantitative information about the plastic behavior of these materials which would not ordinarily be accessible through conventional tensile or compressive tests where fracture usually occurs before deformation. For visco-elastic materials, the technique is expected to be useful for the measurement of both viscous and elastic properties. Lower loads may be used to obtain reasonable penetration depths thus allowing a step-like application of load to be more easily accomplished. Depending on the material tested, the results can be considered a figure of merit, or absolute values of elasticity and viscosity. The theoretical work presented here is based on standard phenomenological models and the fitting procedure can be readily automated. Acknowledgements The author thanks Avi Bendavid for supplying the high purity aluminum film, Jeffrey T. Carter for supplying the polymer films used in this study, and A.H.W. Ngan and an anonymous referee for useful comments. Appendix A. Linear approximation, non-linear least squares Let Zi be a function that provides fitted values of a dependent variable yi at each value of an independent variable xi . Zi can be a function of many parameters a0 , a1 , . . . , ar . Zi = Zi (xi : a0 , a1 , a2 , . . . , aj , . . . , ar )
(A.1)
It is presumed that initial values or estimates of these parameters are known and that the desired outcome is an optimisation of the values of these parameters using the method of least squares. The true value of the parameter aj is found by adding an error term δaj to the initial value ajo . aj = ajo + δaj
yi − Zi at each data point i can be weighted by a factor wi to reflect the error associated in the observed values yi . Thus, the sum of the squares is expressed as X2 =
N wi [yi − Zi ]2 i=1
2 N r o δZ i = wi yi − Zio + δaj δaj i=1
j=1
2 N r o δZ i = wi (yi − Zio ) − δaj , δaj i=1
j=1
yi = yi − Zio ,
j=1
This is a linear equation in δaj and is thus amenable to multiple linear least squares analysis. Now, by least squares theory, we wish to minimise the sum of the squares of the differences (or residuals) between the observed values yi and the fitted values Zi . The differences
wi yi −
i=1
r δZo i
δaj
j=1
2 δaj
δX2 = 0, δ(δaj ) o N r δZio δZi 0= wi yi − δaj δaj δaj i=1
(A.6)
j=1
This expression can be expanded by considering a few examples of j. Letting j = 1, we obtain N
wi yi
i=1
= δa1
δZio δa1
N
wi
i=1
+ · · · + δar
If the errors δaj are small, then the function Zi can be expressed as a Taylor series expansion r δZio Zi = Zio + (A.4) δaj δaj
The weighting factor wi for the present application can be simply the magnitude of yi on the assumption that the error at each data point is inversely proportional to the magnitude of the data at that point. The objective is to minimise this sum with respect to the values of the error terms δaj , thus we set the derivative of X2 with respect to δaj to zero
(A.2)
(A.3)
X2 =
N
(A.5)
Thus, the function Zi becomes Zio = Zi (xi : a1o , a2o , a3o , . . . , aro )
81
δZio δa1
N
2 + δa2
wi
i=1
N
wi
i=1
δZio δZio δa1 δar
δZio δZio δa1 δa2
(A.7)
At j equal to some arbitrary value of k, we obtain: N
wi yi
i=1
= δa1
δZio δak
N
wi
i=1 N
+ δak
i=1
δZio δZio δak δa1
wi
δZio δak
2
+ δa2
N
wi
i=1
+ · · · + δar
N i=1
δZio δZio δak δa2
wi
+ ···
δZio δZio δak δar
(A.8)
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A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82
In matrix notation, the sums for each error term δa from 1–r is expressed [Yj ] = [Ajk ][Xj ], Y1 A11 Y 2 . Yk = . . . Ar1 Yr
.
.
.
. .
. Ajk
.
.
.
. .
.
.
.
A1r
X1
. X2 . Xk . . Arr Xr
(A.9)
where Yj =
N i=1
δZo wi yi i , δaj
Ajk = Akj =
N
wi
i=1
δZio δZio , δaj δak
Xj = δaj
(A.10)
and yi = yi − Zio
(A.11)
The solution is the matrix X that contains the error terms to be minimised. Thus [Xj ] = [Ajk ]−1 [Yj ]
(A.12)
When values of δaj are calculated, they are added to the initial values ajo to give the fitted values aj aj1 = ajo + Lδaj
(A.13)
The process may then be repeated until the error terms δaj become sufficiently small indicating that the parameters aj have converged to their optimum value. L in Eq. (A.13) is a relaxation factor that is applied to error terms to prevent instability during the initial phases of the refinement process.
References [1] A.C. Fischer-Cripps, Nanoindentation, Springer-Verlag, NY, 2002. [2] J.R.M. Radok, Q. Appl. Math. 15 (1957) 198–202.
[3] E.H. Lee, J.R.M. Radok, Trans. ASME Series E, J. Appl. Mech. 27 (1960) 438–444. [4] K.L. Johnson, Contact Mechanics, Cambridge University Press, 1985. [5] L. Cheng, X. Xia, W. Yu, L.E. Scriven, W.W. Gerberich, J. Polym. Sci. B 38 (2000) 10–22. [6] M. Sakai, Phil. Mag. A 82 (10) (2002) 1841–1849. [7] P.M. Sargent, M.F. Ashby, Mater. Sci. Technol. 8 (1992) 594–601. [8] R. Hill, B. Storåkers, A.B. Zdunek, Proc. R. Soc. Lond. A423 (1989) 301–330. [9] R. Hill, Proc. R. Soc. Lond. A436 (1992) 617–630. [10] B. Storåkers, P.-L. Larsson, J. Mech. Phys. Solids 42 (2) (1994) 307–332. [11] S.Dj. Mesarovic, N.A. Fleck, Proc. R. Soc. Lond. A455 (1999) 2707–2728. [12] Y.-T. Cheng, C.-M. Cheng, Phil. Mag. Lett. 81 (2001) 9–16. [13] T.R.G. Kutty, C. Ganguly, D.H. Sastry, Scripta Mater. 34 (12) (1996) 1833–1838. [14] M. Sakai, S. Shimizu, J. Non-Cryst. Solids 282 (2001) 236– 247. [15] W.B. Li, R. Warren, Acta Metall. Mater. 41 (10) (1993) 3065– 3069. [16] S. Shimizu, Y. Yanagimoto, M. Sakai, J. Mater. Res. 14 (10) (1999) 4075–4085. [17] L. Cheng, L.E. Scriven, W.W. Gerberich, Mater. Res. Symp. Proc. 522 (1998) 193–198. [18] A. Strojny, W.W. Gerberich, Mater. Res. Soc. Symp. Proc. 522 (1998) 159–164. [19] S.A. Syed Asif, J.B. Pethica, J. Adhes. 67 (1998) 153–165. [20] X. Xia, A. Stronjy, L.E. Scriven, W.W. Gerberich, A. Tsou, C.C. Anderson, Mater. Res. Symp. Proc. 522 (1988) 199–204. [21] 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. [22] G. Feng, A.H.W. Ngan, J. Mater. Res. 17 (3) (2002) 660–668. [23] A.H.W. Ngan, B. Tang, J. Mater. Res. 17 (10) (2002) 2604– 2610. [24] M.L. Oyen, R.F. Cook, J. Mater. Res. 18 (1) (2003) 139–150. [25] ISO14577, Metallic materials—instrumented indentation test for hardness and materials parameters, ISO Central Secretariat, 1 rue de Varembé, 1211 Geneva 20, Switzerland. [26] CSIRO Division of Telecommunications and Industrial Physics, P.O. Box 218, Lindfield, NSW 2007, Australia. [27] T. Chudoba, F. Richter, Surf. Coat. Tech. 148 (2001) 191–198. [28] M.V. Swain, J.S. Field, Phil. Mag. A 74 (5) (1996) 1085– 1096. [29] M.W. Seto, K. Robbie, D. Vick, M.J. Brett, L. Khun, J. Vac. Sci. Technol. B 17 (5) (1999) 2172–2177. [30] D. François, A. Pineau, A. Zaoui, Mechanical Behaviour of Materials, Kluwer Academic Publishers, The Netherlands, 1998. [31] B.J. Briscoe, L. Fiori, E. Pelillo, J. Phys. D 31 (1998) 2395– 2405.
A simple phenomenological approach to nanoindentation creep A.C. Fischer-Cripps∗ CSIRO Division of Telecommunications and Industrial Physics, P.O. Box 218, Lindfield, NSW 2070, Australia Received 5 February 2004; received in revised form 19 April 2004
Abstract Nanoindentation is frequently used to measure elastic modulus and hardness of structural materials such as ceramics, metals and thin films. The assumption behind conventional nanoindentation analysis methods, where the unloading data is analysed, is that the material behaves in an elastic-plastic manner. However, many materials can also exhibit a visco-elastic and visco-plastic response which is commonly termed “creep”. In a nanoindentation test, this is usually observed as an increase in depth during a hold period at maximum load in the load-displacement data. Creep is not accommodated in conventional nanoindentation analysis methods. The present work shows how conventional linear spring and dashpot elements can be used to model the creep response of a wide range of materials using the hold period force-displacement data. The method shown can be readily incorporated into a computer program and can be used with any conventional nanoindentation test instrument using either spherical or sharp indenters. Crown Copyright © 2004 Published by Elsevier B.V. All rights reserved. Keywords: Nanoindentation; Indentation creep; Depth-sensing indentation
1. Introduction Nanoindentation has proven to be an effective and convenient method of determining the mechanical properties of solids, most notably elastic modulus and hardness. The most popular method relies on an analysis of the unloading load-displacement response which is assumed to be elastic, even if the contact is elastic-plastic [1]. The method relies on plasticity occurring instantaneously upon satisfaction of a constitutive criterion and that there are no time-dependent effects. Many materials, however, have a time-dependent behavior when placed under load and as a result, conventional nanoindentation test methods may not provide an adequate estimation of material properties of interest. There are two basic approaches to managing time-dependent behavior in nanoindentation testing. The first is the application of an oscillatory displacement or force, in which the transfer function between the load and displacement provides a method of calculating the storage and loss modulus of the material. In the second, the application of a step load or displacement and subsequent measurement of depth (creep) or force as a function of time (relaxation) is used to calculate visco-elastic
∗
Tel.: +61 2 9413 7544; fax: +61 2 9413 7457. E-mail address: [email protected] (A.C. Fischer-Cripps).
properties of the specimen material. It is the second method that is the subject of the present work. In a nanoindentation test, creep and plastic deformation in the conventional sense, i.e., that occurring due to shear-driven slip for example, should be regarded separately. Plasticity, in the sense of yield or hardness, is conveniently thought of as being an instantaneous event (although in practice, it can take time for yield processes to complete). In contrast, creep can occur over time in an otherwise elastic deformation as a result of the diffusion and motion of atoms or movement of dislocations in the indentation stress field, the extent of which depends very much on the temperature. If we ignore effects arising from the formation of cracks, permanent deformation in a material under indentation loading is thus seen as arising from a combination of instantaneous plasticity (which is not time-dependent) and creep (which is time-dependent). A material which undergoes elastic and non-time-dependent plastic deformation is called elasto-plastic. A material that deforms elastically but exhibits time-dependent behavior is called visco-elastic. A material in which time-dependent plastic deformation occurs is visco-plastic [1]. The term creep is often used to describe a delayed response to an applied stress or strain that may be a result of visco-elastic or visco-plastic deformation. In a nanoindentation test, the depth recorded at each load increment will be, in general, the addition of that
0921-5093/$ – see front matter Crown Copyright © 2004 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2004.04.070
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.
Fig. 1. (a) Three-element Voigt spring and dashpot representation of a visco-elastic material (delayed elasticy), (b) Maxwell representation of a visco-elastic material (steady creep), (c) four-element combined Maxwell–Voigt model.
due to the elastic-plastic properties of the material and that occurring to due creep, either visco-elastic or visco-plastic. Time dependent properties of materials are conventionally analyzed in terms of mechanical models such as those shown in Fig. 1. The elastic response of such a model is quantified by what we call the storage modulus. The fluid-like response is quantified by the loss modulus. In rheology, the material behavior is towards the fluid end of the spectrum where any elastic response, or the storage modulus, is dominated by the shear modulus of the material. In solids, such as those usually tested in nanoindentation, the material behavior tends towards a predominantly elastic response. In an indentation test, the nature of the loading is a complex mixture of hydrostatic compression, tension, and shear. Unlike a fluid, the storage modulus in this case contains contributions from all three of these types of materials response. In the present work, we shall, in the interests of simplicity, assume that the storage modulus is a measure of the conventional (tensile/compressive) elastic modulus in recognition of the large component of hydrostatic stress in the indentation stress field. The fluid-like response we shall call “viscosity” although in practice, viscosity is usually frequency and temperature dependent and not single-valued. Radok [2], and Lee and Radok [3] have analyzed the visco-elastic contact problem using a correspondence principle in which elastic constants in the elastic equations of contact are replaced with time dependent operators [4]. For the case of a rigid spherical indenter in contact with a material represented by a three-element Voigt model as shown in Fig. 1(a), we can write that for a steady applied load Po , the depth of penetration increases with time is given by the well-known Hertz equation with the addition of a time-dependent exponential 3 Po 1 1 −tE∗2 /η (1) h32 (t) = √ + (1 − e ) 4 R E1∗ E2∗
The square-bracketed term in Eq. (1) represents the displacement-time response of the mechanical model to a step increase in load. A step increase in applied load results in an initial elastic displacement (t = 0) followed by a delayed increase in displacement to a maximum value at t = ∞ 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 and Poisson’s ratio of the specimen material (E∗ = E/(1 − ν2 )) and not the combined modulus of the indenter and specimen as is normally the case. The numerical factor 3/4 in Eq. (1) differs from that usually seen in reference literature due to our working with the elastic modulus E∗ rather than the shear modulus G. The justification for the change in variable being that in indentation loading, the greater proportion of the deformation is hydrostatic compression and the materials we consider are more likely to be dominated by solid-like properties than fluid-like behavior. A similar approach is appropriate for the case of a conical indenter in which we obtain, for the case of the three-element Voigt model 1 1 2 −tE∗2 /η h (t) = Po cot α + ∗ (1 − e ) (2) 2 E1∗ E2 For the case of a Maxwell model Fig. 1(b), the time-dependent depth of penetration for a spherical indenter is given by 3 Po 1 1 h32 (t) = √ + t (3) 4 R E1∗ η and for a conical indenter, we obtain 1 1 h2 (t) = Po cot α t + 2 E1∗ η
(4)
From the above information, it is relatively easy to construct equations for more complicated models by adding components in series or parallel as would be done in mechanical–electrical analogs conventionally applied to this type of modeling. For example, in the present work, we shall also consider a four-element Maxwell–Voigt combination as shown in Fig. 1(c) which gives 3 Po 1 1 1 −tE∗2 /η2 (5) h32 (t) = √ + (1 − e ) + t 4 R E1∗ E2∗ η1
76
h2 (t) =
A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82
1 1 1 −tE∗2 /η2 + (1 − e ) + t Po cot α 2 E1∗ E2∗ η1
(6)
The equations given above are expressed in terms of the modulus E∗ which is the combination of elastic modulus and Poisson’s ratio of the specimen. In the present work, the elastic material properties of the specimen material are given in terms of E∗ as is done in conventional nanoindentation analysis and η is the viscosity term that quantifies the time-dependent property of the material. It should be remembered that Eqs. (1)–(6) assume a step increase in load to Po and are expressed here in a form to be easily fitted to experimentally obtained creep data to provide values for E∗ and η. Should modeling of an arbitrary time displacement response be required, a suitable superposition [5,6] can be employed but this is beyond the scope of the present work. There are several detailed theoretical studies of visco-elastic indentation creep available in the literature [7–12]. A popular motivation for such modeling is the behavior of materials at elevated temperatures [13,14]. Modeling of indentation creep for nanoindentation applications has also widely reported in the literature. Such treatments focus on either constant load (creep [15]) or constant displacement (relaxation [16]) or both [17–21]. Traditionally, intrinsic material properties are modeled in terms of spring and dashpot elements under indentation loading. For example, Feng and Ngan [22,23] applied a Maxwell two-element model to the creep displacement at maximum load in a conventional load-displacement response and determined an equivalent expression for the contact stiffness that included the creep rate expressed as a displacement over time. Their work illustrates and quantifies the forward going “nose” that appears in the unloading curve in indentation experiments in which creep is significant. Cheng et al. [17] applied a method of functional equations to the visco-elastic contact problem in conjunction with a step increase in load, or displacement, to a three-element Voigt model to provide equations for steady creep, or relaxation, for the case of a rigid spherical indenter in contact with both incompressible and compressible materials. In contrast, in a recent work, Oyen and Cook [24] presented a phenomenological approach which sought to include elasticity, viscosity, and plasticity in terms of modeling elements that represented the quadratic character of the contact equations rather than the intrinsic properties of the specimen material. Those workers found a very good agreement between the predicted and observed load-displacement curves for the bounding conditions of an elastic-plastic response (e.g. metals and ceramics) to visco-elastic deformation (e.g. elastomers). The previous works mentioned here are characterized by very formal constitutive equations and complex methods of solution. By way of contrast, in draft standard ISO14577 [25], creep is simply expressed as a change in depth (or load) over time for fixed load, or fixed displacement loading. The intention of the present work is to fill the gap between those more formal treatments discussed above and that spec-
ified in draft standard ISO14577. A simplified treatment of nanoindentation creep is presented and a readily accessible method of analysing Eqs. (1)–(4) in a manner suitable for automation within a computer program is provided.
2. Experimental Several materials were selected for study. The first, a 1 m thick film of high purity aluminum, a metal known to exhibit significant indentation creep. The second, a sample of fused silica, a material in which creep is not expected to be significant, and the third, an 100–150 m thick polyurethane acrylic copolymer film, a material expected to exhibit significant visco-elasticity the amount of which is dependent upon the additive molecules. Two acrylic co-polymer materials were tested, a baseline material and another with a cross-linker added. Tests were done on a UMIS nanoindentation instrument [26]. This instrument is characterized by a real-time electronic force-feedback control loop that ensures the indenter load is held constant regardless of the depth of penetration during the creep period. Conventional nanoindentation load-displacement curves were done along with step loading and hold periods at maximum load. Analysis of the unloading data in the usual manner yields values for modulus and hardness on the assumption of no time-dependent response. Fitting to Eqs. (1)–(6) for the types of models considered here was done using a least squares method. For the Maxwell model, this is relatively straight-forward, there being only two unknowns. For the three-element Voigt and four-element combination model, a non-linear least squares method was used the details of which are given in the Appendix A. In this method, starting values of E∗ and η are required. The method allows for the specification of a tolerance level for convergence and also the specification of relaxation factors to be applied for each variable to prevent instability in the computations. The procedure for materials that exhibit significant creep is relatively straight forward, convergence is rather rapid. For solid like materials, such as the aluminum and fused silica tested here, it is necessary to reduce the value of the relaxation factors progressively and to undertake many iterations. In all the experiments reported here, testing was done in under very closely controlled laboratory conditions in which the ambient temperature was held at 21 ± 0.1 ◦ C. Each test was performed after a long thermal soak period, and thermal drift was assessed before each experiment by monitoring the displacement output of the indentation instrument at the initial contact force. Thermal drift during all the experiments was deemed to be negligible and no correction of the final data was made during the analysis. The creep times selected for this study are comparatively short compared to larger scale indentation testing. This ensures that any thermal drift errors that might occur are kept to a minimum. For the aluminum specimen, a nominal 20 m radius diamond spherical indenter was used to perform standard
A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82 Table 1 Values of elastic modulus, E, hardness, H, and total penetration depth, ht , from the unloading response of conventional load-displacement curves on a 1 m Al film deposited on silicon Hold (s) 0 10 20 40 80
E (GPa) 109 97.8 112 105 102
H (GPa) 0.932 0.928 0.926 0.926 0.922
ht (nm) 73.8 74.7 76.7 77.4 79.9
load–unload indentation tests with varying hold periods at maximum load to determine the effect of creep on the computed values of modulus and hardness. The actual radius of the indenter at the penetration depth for each experiment was determined by calibration against a fused silica specimen and used in the modeling procedure. It was found that for the indenter used here, there was a substantial flattening of the radius at small penetration depths and a sharpening of the radius at higher depths. This is not unusual with sphero-conical indenters used in nanoindentation work. The results for modulus and hardness with varying creep times are shown in Table 1. A Poisson’s ratio of ν = 0.35 was assumed for aluminum to extract the specimen modulus E from the combined modulus E∗ . Further, tests with a step load followed by a hold period were performed and the resulting data analyzed using the Eqs. (1) and (3). A step load of 10 mN was applied to the same indenter and held constant for 20 s. The change in depth as a function of time is shown in Fig. 3. The depth changed from an initial value of 70.4–78.6 nm over the 20 s hold period. A non-linear least squares analysis of the hold period data for a three-element Voigt model according to Eq. (1)
Fig. 3. Creep response for hold period of 20 s at 10 mN on a 1 m Al film on silicon with a spherical indenter 20 m nominal radius (36 m actual radius). Data points show experimental results. The solid line shows the fitted response according to Eq. (1) (three-element Voigt model). The dotted line shows the response according to Eq. (3) (two-element Maxwell model).
77
yielded E1 = 58.6 GPa, E2 = 1197 GPa and η = 1886 GPa s. A least squares fit to the two-element Maxwell model, gave E1 = 75.6 GPa and η = 28797 GPa s. The value of radius used for the fitting was adjusted according to the area function of the indenter and at the depth of penetration measured, translated into an actual radius of 36 m. The displacement response function using the values in Eq. (1) is shown as a full line in Fig. 3. The data shown in Fig. 3 was taken with some consideration for minimizing the dynamic response of the measurement instrument. The displacement data was taken after amplification from the depth sensor but before the normal filtering circuitry so that any time-related delays with the electronic filtering would not affect the results obtained. The data therefore contains a level of electronic noise that would ordinarily not be measured in a nanoindentation test. The application of the 10 mN load took approximately 1 s, the fastest rate available with the test instrument. For the fused silica specimen, a corner cube indenter was used in both a conventional load–unload indentation test and also in step loading and hold. Conventional analysis of the unloading yielded E = 74.2 GPa and H = 9.5 GPa at a depth of penetration of 464 nm for a maximum load of 10 mN. A value of ν = 0.17 was used to extract the specimen modulus from the combined modulus. These results are in reasonable agreement with the commonly accepted values of 72.5 and 9.2 GPa, respectively. The effective cone angle at this depth was found to be 57◦ from the calibrated area function of the indenter. For a step loading to 10 mN and hold period for 20 s, the resulting change in displacement as a function of time was analyzed using Eq. (2) using the non-linear least squares method. A significant number of iterations 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 comparison of the experimental and fitted data for the hold period 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 material 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 high level of indentation stress so as to induce a visco-elastic or visco-plastic response more readily than would be possible with a blunter type of indenter. With the cube corner indenter, reloading the impression in this material after some minutes from the initial testing showed a completely
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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)
Fig. 4. Creep response for hold period of 20 s at 10 mN on a fused silica with a corner cube indenter with an effective cone angle of 57◦ as determined from the indenter area function. The solid line shows the fitting according to Eq. (2) (three-element Voigt model) while the dotted line shows the fitting according to Eq. (4) (two-element Maxwell model).
flat creep response during a hold period indicating that the creep observed here is related to visco-plasticity rather than visco-elasticity. Load-displacement response and a creep test at step load of 1 mN for 10 s were performed on the two acrylic co-polymer materials to a maximum load of 1 mN with a 20 m radius spherical indenter. At the depths of penetration in this material, the actual radius of the indenter tip was estimated to be 13.5 m from the calibrated area function of the indenter. The load-displacement curves were performed at a constant strain rate of 20%/s and are shown in Fig. 5. The results are given in Table 2 along with the moduli and viscosity estimations from least squares fitting to the creep response from a step loading to 1 mN and hold over 10 s. The creep responses for the two materials are shown in Fig. 6 along with the fitted curves from Eqs. (1), (3) and (5). A value of ν = 0.4 was used to extract the specimen
Polymer (a)
Polymer (b)
Load/unload test to 1 mN at constant strain rate 20%/s E (GPa) H (GPa) ht (nm)
ht = 1680 nm
ht = 313 nm
0.154 0.0086 1680
1.209 0.0571 313
Creep test at step load to 1 mN, hold for 10 s
ht = 2000 nm
ht = 373 nm
Two-element Maxwell model E1 (GPa) η (GPa s)
0.0896 1.687
0.9414 41.25
Three-element Voigt model E1 (GPa) E2 (GPa) η (GPa s)
0.163 0.103 0.135
1.033 2.732 11.18
Four-element Maxwell–Voigt model E1 (GPa) E2 (GPa) η1 (GPa s) η2 (GPa s)
0.175 0.136 0.696 2.856
1.055 4.703 7.688 72.6
modulus E from the combined modulus E∗ . For these tests, the load was applied over a period of X and Y s. 3. Discussion As mentioned in Section 1, the depth recorded at each load increment in a nanoindentation test will be that arising from elastic-plastic, visco-elastic and visco-plastic deformations. In conventional nanoindentation testing, the instantaneous elastic and plastic deformations are usually considered. Elastic equations of contact are applied to the unloading data to find the depth of the circle of contact at full load
Fig. 5. Load–unload response 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.
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79
Fig. 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 a 13.5 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 or visco-plastic response (i.e. creep), then this analysis is invalid. In extreme cases, the unloading curve has a negative slope which, if the standard unloading analysis is applied, results in a negative elastic modulus [22,27]. When fitting creep curves to mechanical models, we must be sure that we distinguish between visco-elasticity and visco-plasticity. The models shown here are appropriate for visco-elastic deformation but may be shown to provide some information for visco-plastic deformation. If a step load is applied to an indenter in contact with a material and the resulting depth of penetration monitored, then a response similar to that shown in Fig. 2(a) or (b) may be obtained. Fitting the appropriate equation (Eqs. (1)–(4)) to this data yields values for the moduli and viscosity terms. The accuracy of the results so obtained depends upon the rapidity with which the step increase in load is applied and the time-dependent nature of the specimen material. In practice, an increase in load is applied over a finite time period within which, for elastic-plastic materials, plastic deformation can occur quite rapidly and this causes the initial step response in displacement to be greater than that predicted, particularly when a sharp indenter is used. The resulting value of modulus can be very much less than the nominal modulus of the material. For a step load with a spherical indenter (to reduce the possibility of time-independent plasticity) on a material with a significant viscous response, the resulting analysis is likely to result in a reasonable measurement of the visco-elastic properties of the material. The significance of this can be observed by comparing the values of modulus obtained from conventional unloading curve analysis to that obtained from the creep curve fitting. For aluminum and fused silica, the moduli E1 from the creep curve fitting procedures are very much less than expected because of the plastic deformation arising during the step loading. This is because the models assume an initial instantaneous elastic response (i.e. the models represent a visco-elastic deformation) whereas in these materials, there is an elastic-plastic response, and in the case of aluminum, a relatively strong
visco-plastic behavior. In the case of aluminum, the effect is aggravated due to piling-up of material around the edge of the contact area. In instrumented indentation tests, the piled up material serves to make the specimen appear harder and stiffer because conventional analysis techniques do not account for it. In the present work, the loading time for the step load was limited by the time-response of the electronics of the test instrument. An attempt was made to gauge the effect of the loading time for the case of the cube corner indenter on fused silica by performing a number of creep indentation tests while attempting to increase the loading time by altering the time response of the instrument circuitry. Faster loading times could be achieved, but it was found that the indenter penetrated the sample more deeply due to overshoot in the force servo-control feedback loop. Attempts to reduce the overshoot resulted in a more damped response which increased the time taken to achieve the maximum load which also, as discussed above, increased the penetration depth due to plastic deformation in the specimen. These attempts showed that for highly elastic materials, a creep analysis employing a step increase in load is likely to provide values of E1 only to within an order of magnitude. In this case, conventional nanoindentation would be more appropriate for the measurement of elastic properties. However, measurements of the time-dependent properties of such a material are not so affected by the non-instantaneous step application of load and the measurement of the viscosity terms would be expected to be reliable. In the extreme case of solid-like materials (e.g. the fused silica specimen tested here), the procedure may be useful in determining a quantitative account of the time-dependent nature of the deformation under extremely high contact pressures which would ordinarily be inaccessible in conventional tensile testing where fracture of the specimen occurs before plastic deformation. For visco-elastic property measurements, the creep analysis procedure presented here is more suitable for use on materials in which time-dependent behavior represents a significant contribution to the overall deformation (such as the
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polymer materials tested here). In the case of the polymer specimens, a relatively low load of 1 mN was found to provide a reasonably large penetration depth in each case and requirement for a more ideal step-like application thus more easily accomplished. For the case of the two polymeric materials tested here, the specimen with the added cross-linker shows a much stiffer elastic and more viscous response compared to the base material. This is expected as the cross linker serves to restrict the motion of molecular chains under the applied shear stress in the indentation stress field. In general, the response of a particular specimen material may be affected by indentation size effects arising from strain-gradient plasticity, the presence of oxide layers, surface roughness, etc. Such effects are beyond the scope of the present work but should be considered as possible sources of errors or variations in the data, especially in the case of crystalline solids. The load–unload curves shown in Fig. 5 deserve some comment. The polymeric materials tested here were selected on the expectation of a significant visco-elastic response (in contrast to an elastic-plastic response). The curves shown in Fig. 5 support this, especially for the cross-linked material. Although there is a readily identifiable plastic deformation as evidenced by the area enclosed between the loading and unloading curves, the elastic recovery of the material is quite substantial. This implies that the energy dissipation within these types of materials is substantially a result of viscous losses (in the sense of visco-elasticity) and not plastic deformation (in the sense of elasto-plasticity). Similar behavior in various materials has been previously reported in the literature [28,29] but has not in general been satisfactorily explained. If we recognize the limitations of the method, then the values for E2 and η, in the case of the three-element Voigt model, may still have validity for visco-plastic contact since, as shown in Fig. 2, they influence only the time-dependent character of the deformation and not the initial response to the step loading. A decision as to the nature of the specimen material with regard to its elasto-plastic, visco-plastic or visco-elastic character can thus in some cases be made upon the observed shape of the conventional load-displacement response. While the use of simple mechanical models such as those shown in Fig. 1 allow some comparison to be made between specimens, it should be noted that they offer very little in terms of a basic understanding of the physical mechanisms involved in the deformation. The mathematical models in Eqs. (1)–(6) are simply standard elastic equations with a time-dependent terms added to represent the fluid-like behavior of the material. They are models only with no real physical significance. They serve only to give some quantitative description of mechanical events. Time-dependent behavior often depends on the strain-hardening characteristics of the material which in turn, depend upon microstructural variables. Various constitutive laws [30] have been proposed that apply to many different types of materials and these should be investigated if a more detailed account of
the deformation is desired. The scope of the present paper is to present a simple phenomenological approach only. A comparison of the fitted creep response curves and the experimental data presented in Figs. 3, 4 and 6 demonstrate that the mechanical models used may not precisely match the response of the specimens, particularly in the case of the two-element Maxwell model. The three-element Voigt model provides a reasonable fit while the four-element model (Fig. 6) shows a very good fit. The physical significance of the elements within a model depends upon the microstructural characteristics of the specimen material. In the present case, no weighting scheme was used other than an adjustment to the values of the relaxation factors. However, the non-linear least square fitting procedure is general enough to allow the addition of more elements as desired. The fitted curves shown here represent a minimization of the sum of the squares of the differences between the fitted and actual data, but more control is possible through the use of the terms wi given in the Appendix A. Using these factors, the differences between the fitted and actual data can be weighted in favor of data at one or other ends of the range of values as desired. Some workers [27,31] have studied the effect of specimen creep on the values of modulus and hardness obtained using conventional methods of analysis of the unloading response. The general conclusion is that for materials which exhibit creep during an indentation test, the modulus so calculated from the unloading response is not reliable if the hold period at maximum load is too short due to bowing or “nose” of the unloading response to larger depth values resulting from creep. Briscoe et al. [31] introduced a 10 s hold period into their tests on polymeric specimens to eliminate the nose in the unloading data. Chudoba and Richter [27] found that the hold period at maximum load has to be long enough such that the creep rate has decayed to a value where the depth increase in 1 min is less than 1% of the indentation depth. According to Chudoba and Richter, allowing creep to proceed to relative completion and then obtaining the unloading data would provide a value of unloading stiffness dP/dh that would occur at the increased depth free from the effect of creep. Feng and Ngan [22] draw a similar conclusion and show how a value of dP/dh can be obtained using shorter hold periods if the unloading slope is corrected by a factor which is dependent upon the creep rate and the unloading load rate.
4. Conclusion The intention of the present work is to provide a simple and accessible method of obtaining a quantitative measure of the elastic and viscous properties of materials from indentation creep curves. The present work is not intended to offer a rigorous account of indentation creep or materials constitutive behavior. Three representative classes of materials were tested: a highly elastic ceramic, a soft metal and two
A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82
visco-elastic polymer systems. For highly elastic solid-like materials, the results using the creep method for the elastic modulus are very much less than that expected due to the sensitivity of the technique to the non-deal step-like initial application of load. The technique may however provide quantitative information about the plastic behavior of these materials which would not ordinarily be accessible through conventional tensile or compressive tests where fracture usually occurs before deformation. For visco-elastic materials, the technique is expected to be useful for the measurement of both viscous and elastic properties. Lower loads may be used to obtain reasonable penetration depths thus allowing a step-like application of load to be more easily accomplished. Depending on the material tested, the results can be considered a figure of merit, or absolute values of elasticity and viscosity. The theoretical work presented here is based on standard phenomenological models and the fitting procedure can be readily automated. Acknowledgements The author thanks Avi Bendavid for supplying the high purity aluminum film, Jeffrey T. Carter for supplying the polymer films used in this study, and A.H.W. Ngan and an anonymous referee for useful comments. Appendix A. Linear approximation, non-linear least squares Let Zi be a function that provides fitted values of a dependent variable yi at each value of an independent variable xi . Zi can be a function of many parameters a0 , a1 , . . . , ar . Zi = Zi (xi : a0 , a1 , a2 , . . . , aj , . . . , ar )
(A.1)
It is presumed that initial values or estimates of these parameters are known and that the desired outcome is an optimisation of the values of these parameters using the method of least squares. The true value of the parameter aj is found by adding an error term δaj to the initial value ajo . aj = ajo + δaj
yi − Zi at each data point i can be weighted by a factor wi to reflect the error associated in the observed values yi . Thus, the sum of the squares is expressed as X2 =
N wi [yi − Zi ]2 i=1
2 N r o δZ i = wi yi − Zio + δaj δaj i=1
j=1
2 N r o δZ i = wi (yi − Zio ) − δaj , δaj i=1
j=1
yi = yi − Zio ,
j=1
This is a linear equation in δaj and is thus amenable to multiple linear least squares analysis. Now, by least squares theory, we wish to minimise the sum of the squares of the differences (or residuals) between the observed values yi and the fitted values Zi . The differences
wi yi −
i=1
r δZo i
δaj
j=1
2 δaj
δX2 = 0, δ(δaj ) o N r δZio δZi 0= wi yi − δaj δaj δaj i=1
(A.6)
j=1
This expression can be expanded by considering a few examples of j. Letting j = 1, we obtain N
wi yi
i=1
= δa1
δZio δa1
N
wi
i=1
+ · · · + δar
If the errors δaj are small, then the function Zi can be expressed as a Taylor series expansion r δZio Zi = Zio + (A.4) δaj δaj
The weighting factor wi for the present application can be simply the magnitude of yi on the assumption that the error at each data point is inversely proportional to the magnitude of the data at that point. The objective is to minimise this sum with respect to the values of the error terms δaj , thus we set the derivative of X2 with respect to δaj to zero
(A.2)
(A.3)
X2 =
N
(A.5)
Thus, the function Zi becomes Zio = Zi (xi : a1o , a2o , a3o , . . . , aro )
81
δZio δa1
N
2 + δa2
wi
i=1
N
wi
i=1
δZio δZio δa1 δar
δZio δZio δa1 δa2
(A.7)
At j equal to some arbitrary value of k, we obtain: N
wi yi
i=1
= δa1
δZio δak
N
wi
i=1 N
+ δak
i=1
δZio δZio δak δa1
wi
δZio δak
2
+ δa2
N
wi
i=1
+ · · · + δar
N i=1
δZio δZio δak δa2
wi
+ ···
δZio δZio δak δar
(A.8)
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A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82
In matrix notation, the sums for each error term δa from 1–r is expressed [Yj ] = [Ajk ][Xj ], Y1 A11 Y 2 . Yk = . . . Ar1 Yr
.
.
.
. .
. Ajk
.
.
.
. .
.
.
.
A1r
X1
. X2 . Xk . . Arr Xr
(A.9)
where Yj =
N i=1
δZo wi yi i , δaj
Ajk = Akj =
N
wi
i=1
δZio δZio , δaj δak
Xj = δaj
(A.10)
and yi = yi − Zio
(A.11)
The solution is the matrix X that contains the error terms to be minimised. Thus [Xj ] = [Ajk ]−1 [Yj ]
(A.12)
When values of δaj are calculated, they are added to the initial values ajo to give the fitted values aj aj1 = ajo + Lδaj
(A.13)
The process may then be repeated until the error terms δaj become sufficiently small indicating that the parameters aj have converged to their optimum value. L in Eq. (A.13) is a relaxation factor that is applied to error terms to prevent instability during the initial phases of the refinement process.
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