GB2540821A

GB2540821A - Method for measuring mechanical properties of materials using depth sensing indentation

Info

Publication number: GB2540821A
Application number: GB1513480.2A
Authority: GB
Inventors: Khaliq Shah Abdul; Renevier Nathalie
Original assignee: University of Central Lancashire
Current assignee: University of Lancashire
Priority date: 2015-07-30
Filing date: 2015-07-30
Publication date: 2017-02-01
Anticipated expiration: 2035-07-30
Also published as: GB201513480D0; WO2017017409A1; GB2540821B

Abstract

Apparatus for use in determining a property of a sample material tested by a depth sensing indentation system, comprises a memory storing data points on a load-displacement curve for a test of the sample generated by the depth-sensing indentation system, the data including an unloading curve of the measured displacement, ℎ, of the probe of the indentation system as the previously applied indentation load, P, thereon is released; a full elastic point, FEP, determination module configured to determine, based on the unloading curve data, a point on the unloading curve having the highest load at which the measured acceleration of the indenter displacement, optionally using d2h/dP2 as dP/dt as constant, becomes a constant indicating that the indenter and sample thereafter move together at the same speed, said point taken to be the full elastic point, FEP, at which any delayed elasticity in the sample in response to the indentation ceases; and a material property determination module configured to determine, based on the unloading curve data at the FEP, one or more material properties of the sample.

Description

Method for measuring mechanical properties of materials using depth sensing indentation [0001] This invention relates to apparatus and a method for use in determining a property of a sample material tested by a depth-sensing indentation system.

BACKGROUND

[0002] Nanoindentation has been industrialized for investigating the mechanics of materials at small scales. The technique, referred to more generally as a type of depthsensing indentation (a term which can also include testing using larger indentations), is used for characterisation of a sample material and also to gain an understanding of the material physics by measuring the deformation processes.

[0003] As shown in Figure 1, depth-sensing indentation equipment comprises a support platform 1 comprising a material sample 2 to be tested, an indenter 3, formed as a probe typically having a well-characterised shape and well-understood material properties, provided adjacent the material sample 2, and an actuator 4 configured to apply a load to the indenter 3 whereby to cause the indenter 3 to deform the material sample 2. The actuator 4 applies a monotonically increasing and subsequently decreasing load to the indenter 3. The applied load is measured using a load sensor (not shown). The displacement of the indenter 3 as a result of the loading by the actuator 4 is measured using a displacement sensor (not shown). In particular, the unloading portion of the load-displacement data may be referred to as an unloading curve.

[0004] The deformation processes exhibited by the material during the indentation test of loading and unloading are due to different phenomena, such as geometrical and materials based variations in the material. These phenomena affect the load-displacement curve.

The technique of nanoindentation testing has been made possible, firstly, by the development of equipment that can accurately place small indentations while recording load and displacement with precision and secondly, by mathematical models by which the load displacement data can be interpreted to obtain hardness, modulus (see Oliver WC, Pharr GM. An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments. J Mater Res. 1992;7(06):1564-83), and other mechanical properties. Traditionally the unloading curve is assumed to be totally elastic, and almost all nanoindenter equipment manufacturers have adopted the Oliver and Pharr procedure for analysing the data, as shown in Figure 2.

[0005] By the Oliver and Pharr method, the raw load-displacement data acquired by the nanoindenter initially has to be corrected for a zero point correction, thermal drift and load frame compliance. After these corrections have been made, the gradient of the load against penetration depth is calculated as this is defined as being equal to the measured stiffness, S, at that load-displacement point. The calculated stiffness at maximum load is then used in the calculation of a plastic depth (hc) by the following Equation (1). Figure 3 is an illustration of a diagram showing the indentation of a surface. A material 12 is indented by an indenter tip 14. In a first region of the indentation 16, the indenter 14 and the surface of the material 12 are in contact. In a second region of the indentation 18, the surface of the material 12 separates from the indenter 14. The outermost point of the indentation still in contact with the indenter 14 defines a circle of contact 20. The height of the circle of contact 20 from the base of the indentation is termed the plastic depth. The height of the upper surface of the material 12 from the base of the indentation is termed the maximum displacement.

(1) [0006] Here, hmax is the maximum displacement, Pmax is the maximum load. The factor ε in equation (1) is due to Oliver and Pharr adapting the original equation for plastic depth described by Doerner and Nix (see Doerner MF, Nix WD. A Method for Interpreting the Data from Depth-Sensing Indentation Instruments. J Mater Res. 1986; 1(04):601-9), from one describing an elastic unloading of a cone, to an unloading of an indenter of a parabolic shape. When ε=1 we have the Doerner and Nix equation. For a Bercovich indenter ε is typically 0.75, but to account for any variations of the tip for a specific indenter, ε can be characterised and calculated using Equation (2). (see Pharr GM, Bolshakov A. Understanding Nanoindentation Unloading Curves. J Mater Res. 2002;17(10):2660-71).

(2) [0007] Where m is the exponent of the unloading curve when fitted to power law function of the type

(3) [0008] An area function, A(hc), of the indenter is typically described as a function of the plastic depth. Equation (4) presents a 5th order polynomial to describe the area function and is widely used. The area function is the contact area (area within the circle of contact 20) of the indenter 14 as a function of the plastic depth, hc.

(4) [0009] Thus the area of contact, A(hc), can be established once the plastic depth is known. To relate the stiffness to the modulus, much work was first established by researchers at the Baikov Institute of Metallurgy in Moscow during the 1970's (for a review see Bulychev S., Alekhin V., Shorshorov M., Ternovskii A., Shnyrev G. Determining Young’s modulus from the indentor penetration diagram. Zavod Lab. 1975;41 (9):1137—40). The contact was modelled using an analytical model for a contact between a rigid indenter with a homogeneous isotropic elastic half space. The relationship presented by Sneddon (see Sneddon IN. Boussinesq’s problem for a rigid cone. Math Proc Camb Philos Soc. 1948;44(04):492-507) was thus used to obtain the expression.

15) [0010] Here, S is the measured stiffness (in Oliver and Pharr, the stiffness at maximum load), A is the area function calculated based on the plastic depth inferred by equation 3, and Er is the reduced modulus, which is the modulus resulting from the combination of the modulus of the sample and the modulus of the indenter.

[0011] One of the basic assumptions of the above-described approach for recovering the modulus of a sample is that deformation upon unloading is purely elastic. This equation originally was derived for a conical indenter, but holds equally well for any indenter that can be described as a body of revolution of a smooth function (see Pharr G m., Oliver W c., Brotzen F r. On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation. J Mater Res. 1992;7(03):613-7) and for pyramidal indenters (see King RB. Elastic analysis of some punch problems for a layered medium.

Int J Solids Struct. 1987;23(12): 1657-64). Due to deviations from the assumptions used in Sneddon’s elastic derivation a correction factor β is added in equation 5. King proposed β to be unity for axisymmetric indenters, for pyramidal indenters β is close to unity, β = 1.012 for square-based indenter(Vickers) and β = 1.034 for a triangular punch (Berkovich).

[0012] For the calculation of the Hardness the following equation is used.

(6) [0013] Here, H is the hardness, A is the area function calculated based on the plastic depth inferred by equation 3, and Pmax is the maximum load applied in the test.

[0014] In Figure 4 a number of plots are shown highlighting the various responses of different materials. From this it can be seen the initial unloading is different for each plot. This can be either because of the material itself or the experimental conditions for a given indenter/sample setup. It should be mentioned here that initially Doerner and Nix noticed the unloading curve to be linear for most materials when indented with a flat punch, i.e. materials which displayed unload curves similar to Figure 4a. However Oliver and Pharr concluded that the initial unloading curve wasn’t linear but in fact a power law fit for the majority of materials i.e. the typical elastic-plastic unloading response as shown in Figure 4b&c. By using the power law fit to the unloading data good results were found when determining the modulus and comparing values to macro scale.

[0015] Apart for correction factors to account for indenter geometry there were concerns with the analysis when testing materials that exhibited creep behaviour. These kind of materials display unloading curves similar to Figure 4c&d. Feng (see G. Feng AHWN. Effects of Creep and Thermal Drift on Modulus Measurement Using Depth-sensing Indentation. J Mater Res. 2002; 17(03):660-8) proposed a method for correcting the creep effects in the modulus measurement. The solution is based on ideal viscoelasticity -. By Feng, the apparent compliance of the material measured in the Oliver and Pharr method is recognised as being due to the elasticity and creep components of viscoelasticity. In developing the Oliver and Pharr method, Feng also made use of a multi-loading sequence to eliminate any reverse plasticity by the testing procedure itself, and has to be performed at conditions when the “nose-out effect” is present. Feng developed the method of Oliver and Pharr by further correcting the stiffness for viscous effects. The solution is based on ideal viscoelastic material and any delayed elasticity due to plastic deformations are accounted for within the test procedure not the solution, therefore the testing conditions are limited. A “nose-out effect” occurs in viscous materials around the period of maximum displacement, and occurs when the point of maximum displacement occurs during the unloading stage i.e., when the load has been reduced from the maximum load, yet the material continues to deform to a greater displacement. By Feng’s method, the assumption is that the initial unloading curve is fully elastic after multiple unloading. Creep is then corrected and the whole unloading curve is used in the materials properties analysis.

[0016] The reason for the assumption of fully elastic unloading in Oliver and Pharr’s (and Feng’s) method is mostly due to Tabor (see Tabor D. A Simple Theory of Static and Dynamic Hardness. Proc R Soc Lond Ser Math Phys Sci. 1948 Feb 4; 192(1029):247-74) who showed that, by using the entire unloading curve, the total amount of recovered displacement can be accurately related to the elastic modulus along with the size of the contact impression, for both spherical and conical indenters, only if the indentation was loaded and unloaded a number of times before the load displacement behaviour became perfectly reversible i.e. elastic. A limited amount of plasticity sometimes occurs in each of the first few loading and unloading sequences. Tabor also concluded that the effects of non-rigid indenters on the load-displacement behaviour can be accounted by defining a reduced modulus, Er, by the following equation

(7) [0017] Here, Er is the reduced modulus as determinable by equation 5, and Ej is the modulus of the indenter, whereas E (or also Es) is the modulus of the sample. In this way, the sample modulus can be determined by the Oliver and Pharr method, and, for materials exhibiting creep, the Feng method. Furthermore, v is the Poisson’s ratio of the material and νέ is the Poisson’s ratio of the indenter.

[0018] Nevertheless, further developments in nanoindentation testing to improve the testing results and process would be beneficial.

[0019] It is in this context that the present disclosure has been devised.

BRIEF SUMMARY OF THE DISCLOSURE

[0020] In view of the above, the present disclosure recaps on the herein identified limitations of Oliver and Pharr’s, and Feng’s procedures to determine material properties from the unloading data using nanoindentation. That is, while their results are valid around their assumptions of elastic behaviour and methods, utilizing the assumption that the initial unloading curve is fully elastic after multiple unloading, in reality different materials display a spectrum of viscous behaviour.

[0021] The method disclosed herein utilizes this concept and has recognised that, to compare results to macro scale, the effects of actual local deformation should be eliminated because practically for any materials a power law fit doesn’t always exist, depending on the experimental conditions, at the very initial unloading (see Figure 4) using monotonic loading. When it comes to examining viscous materials both previous methods (Oliver & Pharr and Feng), even under multi-loading sequence testing, show an increase in variability in the results if tested when a nose-out is present. This is hereby recognised as being a direct consequence of contact between the sample and indenter being in nonequilibrium, as the contact area is changing, present in the early stages of the unloading. This is hereby recognised as being a result of relative movement between the indenter and the sample. The present disclosure recognises that this non-equilibrium of contact needs to be eliminated in the model to obtain a pure viscoelastic response of the sample material.

[0022] In accordance with one aspect of the present disclosure, there is provided apparatus for use in determining a property of a sample material tested by a depth-sensing indentation system. The apparatus comprises a memory storing data representative of points on a load-displacement curve for a test of the sample generated by the depthsensing indentation system. The data includes an unloading curve of the measured displacement, h, of the probe of the depth-sensing indentation system as the previously applied indentation load, P, thereon is released. The apparatus further comprises a full elastic point, FEP, determination module configured to determine, based on the unloading curve data, a point on the unloading curve having the highest load at which the measured acceleration of the indenter displacement (optionally using ^ instead of ^7 , as ^ is constant), becomes a constant indicating that the indenter and sample thereafter move together at the same speed, said point taken to be the full elastic point, FEP, at which any delayed elasticity in the sample in response to the indentation ceases. The apparatus further comprises a material property determination module configured to determine, based on the unloading curve data at the full elastic point, FEP, one or more material properties of the sample.

[0023] Thus a method is provided for measuring elastic and viscoelastic properties of any material by depth-sensing indentation. This method is as good as Oliver and Pharr’s method and for viscous material, as good as Oliver and Pharr’s method with Feng’s creep correction. Traditionally for viscous material a number of cycles are needed to obtain the material properties, whereas this method just needs one single cycle, as the data at the FEP is taken to provide a pure elastic or viscoelastic response of the material, even on the first cycle, thus reducing the time of study considerably. This method enables one to reliably study the properties of materials including viscous materials at all conditions using nanoindentation, all performed in a single loading-unloading cycle. It can then be used to determine other mechanical properties such as hardness by traditional adapted means. The fits of the data are extremely accurate, over wide experimental conditions and lead to reduced variability associated with the analysis procedure and higher repeatability of results, giving lower measurement errors.

[0024] The FEP determination module may be configured to fit a linear regression to a plot of the indentation load, P, against the first derivative of the measured displacement of dh the probe with respect to the indentation load, —, for the unloading curve data, and to

dP dh identify the point on the unloading curve having the highest load above which — deviates from the fitted linear regression.

[0025] The FEP determination module may be configured to fit a second order polynomial regression to a plot of the indentation load, P, against the measured displacement, h, for the unloading curve data, and to identify the point on the unloading curve having the highest load above which the measured displacement, h, deviates from the fitted second order polynomial regression.

[0026] To eliminate initial nose out data for the FEP determination where forward plasticity or viscosity in the sample, the FEP determination module may be configured to fit a fourth order polynomial regression to a plot of the indentation load, P, against the measured displacement, h, for the unloading curve data, to identify, as a nose out point, the point on the unloading curve having the highest load above which the measured displacement, h, deviates from the fitted fourth order polynomial regression, and to use only the unloading curve data having loads below the nose out point when determining the FEP.

[0027] To eliminate initial nose out data for the FEP determination where forward plasticity or viscosity in the sample, the FEP determination module may be configured to identify, as a nose out point, the point on the unloading curve having the maximum measured displacement, hmax, and to use only the unloading curve data having loads below the nose out point when determining the FEP.

[0028] The FEP determination module may be configured to evaluate a viscofactor for the unloading curve of the sample material by the equation:

wherein Pmax is the maximum load applied in the test cycle and Pmaxdisplacement >s the load applied when the probe is at its maximum displacement, hmax. The FEP determination module may be configured to only eliminate initial nose out data from the FEP determination if the evaluated viscofactor for the unloading curve of the sample material is greater than zero.

[0029] To identify the point on the unloading curve having the highest load above which the measured stiffness or displacement, h, deviates from the fitted linear regression or second or fourth order polynomial regression, the FEP determination module may be configured to identify the highest load at which the difference between the unloading curve and the fitted linear regression or second or fourth order polynomial regression matches the mean difference therebetween.

[0030] To fit the linear or second or fourth order polynomial regression models to the unloading curve data, the FEP determination module may be configured to perform a least squares analysis to fit a regression model by minimising the errors between model to the unloading curve data.

[0031] The FEP determination module may be configured to use only the unloading curve data having within a given percentage of the maximum load, Pmax, when determining the FEP.

[0032] By determining a second order fit, the load, P, can be expressed in the form:

[0033] Where, h is the displacement and a, b and c are constants to be determined according to the second order polynomial fit.

[0034] The material property determination module may be configured to determine the apparent stiffness SFEP of the sample at the full elastic point based on the relation:

wherein, hFEP is the modelled probe displacement h at the full elastic point.

[0035] The material property determination module may be configured to determine the plastic-corrected stiffness S from the apparent stiffness SFEP based on the relation (wherein the stiffness S is the plastic corrected stiffness at the FEP):

where |P| is the unloading rate and hpiastic at fep's the plastic rate at which plastic deformation occurs during the period of delayed elasticity before the full elastic point.

[0036] To determine the plastic rate, hplastic at FEP, the material property determination module may be configured to determine a plastic displacement hplastic at fep based on the following relation:

wherein hmax is the maximum displacement, hmax fltted is what the maximum displacement of a regression model fitted to the unloading curve data would be up to the maximum load, hmax_load is the displacement at the maximum load, and hFEP is the displacement at the full elastic point. The material property determination module may be further configured to determine the plastic rate, hpiastic, based on the relation:

where tmax ioad-FEP >s the time taken from the maximum load to the load at the full elastic point.

[0037] To determine whether or not a nose-out is present in the data, the material property determination module may be configured to evaluate a viscofactor for the unloading curve of the sample material by the relation:

wherein Pmax is the maximum load applied in the test cycle and Pmax displacementis the load applied when the probe is at its maximum displacement, hmax. The material property determination module may be further configured to deem there to be a nose-out in the unloading curve if the evaluated viscofactor for the unloading curve of the sample material is greater than zero, and to otherwise deem there to be no nose-out in the unloading curve.

[0038] The material property determination module may be configured to determine the plastic depth of the probe contact at the full elastic point, hc, to correct for pile up effects, by the relation:

where hc (elastic method)'s the plastic depth calculated according to the relation for elastic materials, hc (elastic method)'s the plastic depth calculated according to the relation for the elastic plastic perfectly plastic materials, ε is a factor compensating for the shape of the indenter probe, c1 and c2 are both 1.2 for elastic plastic perfectly plastic materials, hFEP is the measured probe depth at the full elastic point, SFEP is the measured apparent or plastic corrected stiffness of the sample at the full elastic point.

[0039] To determine hardness H of the sample, the material property determination module may be configured to determine a projected area i4p(hc) of the indentation of the probe at the plastic depth hc at the full elastic point. The material property determination module may be further configured to determine the hardness H at the full elastic point by the relation:

[0040] To determine reduced modulus Er of the sample and indenter, the material property determination module may be configured to determine a plastic depth hc of the probe contact at the full elastic point. The material property determination module may be further configured to determine a projected area Ap (K) of the indentation of the probe at the plastic depth hc at the full elastic point. The material property determination module may be further configured to determine the reduced modulus Er of the sample and indenter at the full elastic point by the relation:

[0041] The material property determination module may be further configured to determine the sample modulus E$ by correcting the determined reduced modulus Er to remove the contribution of the indenter modulus Fj.

[0042] The FEP determination module and the material property determination module may be configured to respectively determine the FEP and the one or more material properties using only data from a single indentation cycle on the sample, preferably the first indentation cycle performed on the sample.

[0043] The apparatus may further comprise one or more processors and computer readable medium carrying instructions which when executed by one or more of the processors cause the apparatus to instantiate the FEP module and/or the material property determination module.

[0044] Viewed from another aspect, the present disclosure provides a depth-sensing indentation system comprising a sample stage, a probe for indenting the surface of a sample arranged on the stage, an actuator for applying a load to the probe to indent the surface of a sample arranged on the stage, a load sensor for sensing a load on the probe, a displacement sensor for sensing a displacement of the probe, a controller configured to operate the actuator, load sensor and displacement sensor to perform a depth-sensing indentation test on a sample arranged on the stage by collecting data representative of points on a load-displacement curve including an loading curve of the measured displacement, h, of the probe as the indentation load, P, is applied and an unloading curve of the measured displacement, h, of the probe as the previously applied indentation load, P, thereon is released, and apparatus for use in determining a property of a sample material tested by a depth-sensing indentation system as claimed in any preceding claim, the apparatus being configured to store in the memory thereof the data representative of points on a load-displacement curve produced by the controller.

[0045] The indentation test may be a nanoindentation test or a microindentation test.

[0046] Viewed from one aspect, the present invention provides a computer readable medium carrying instructions which when executed by one or more processors of apparatus or depth-sensing indentation system as described in relation to the above aspect of the disclosure, cause the apparatus or depth-sensing indentation system to instantiate the FEP module and/or the material property determination module.

BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Embodiments of the invention are further described hereinafter, by way of example only, with reference to the accompanying drawings, in which:

Figure 1 is an illustration of a schematic of a depth-sensing indentation equipment;

Figure 2 is an illustration of a schematic flowchart of Oliver and Pharr’s procedure for the calculation of Hardness and Reduced Modulus;

Figure 3 is an illustration of a material deformed by an intenter tip;

Figure 4 is an illustration of four typical nanoindentation unloading curves with a) linear fit, b), c) & d) y = a*(x-b)c power law fits;

Figure 5 is an illustration of a schematic flow diagram of an embodiment of the disclosed method for determining a property, in particular the Hardness and Reduced Modulus, of a sample material tested by a depth-sensing/nanoindentation system;

Figure 6 is an illustration of graphs showing the second derivative of displacement with respect to load against load for three separate materials;

Figure 7 is an illustration of a graph showing a stiffness of experimental data against load, plotted with a linear fit to the experimental data;

Figure 8 is an illustration of graphs showing a measured modulus of PET and PEN determined in a multi loading indentation test using a selection of different methods;

Figure 9 is an illustration of graphs showing stiffness against load determined with a) no creep correction, b) creep correction and c) plastic correction;

Figure 10 is an illustration of a graph showing normalised stiffness against normalised contact depth for PET using an elastic model or an elasto-perfectly plastic model;

Figure 11 is an illustration of graphs showing the dependency of normalised contact depth and modulus with depth for three different load rates, hold periods and unloading rates;

Figure 12 is an illustration of graphs showing a determined modulus of non-viscous materials using a single cycle for Alumina, Aluminium, Brass, Copper, High speed steel EN24, Mild steel and Cast iron for six different methods;

Figure 13 is an illustration of graphs showing a determined modulus of viscous materials using a single cycle for Nylon, Perspex, Rubber, Silver steel, PET and PEN for eight different methods;

Figure 14 is an illustration of a graph showing the fit of a second order polynomial to a subset of unloading data for an indentation test of a non viscous material;

Figure 15 is an illustration of a graph showing the difference between the measured data and the second order polynomial for the data shown in Figure 14;

Figure 16 is an illustration of a graph showing the fit of a fourth order polynomial to a subset of unloading data for an indentation test of a viscous material;

Figure 17 is an illustration of a graph showing the fit of a second order polynomial to a further subset of the subset of unloading data shown in Figure 16; and

Figure 18 is an illustration of a graph showing the difference between the measured data and the second order polynomial for the data shown in Figure 16.

DETAILED DESCRIPTION

[0048] A method is described for determining the mechanical properties of viscous and non-viscous materials. Uniquely implemented by acquiring initial load-displacement data from nanoindentation hardware and processed by software modules, overall comprising of logical components that carry out steps according to the flow diagram shown in Figure 5.

[0049] The method 100 requires unloading curve data to be obtained by nanoindentation hardware 102, substantially as described in relation to Figure 1. As described previously, the nanoindentation hardware 102 comprises a sample stage 104, a force actuator/load sensor 106 and a displacement sensor 108. When operated with a material sample, the nanoindentation hardware 102 produces raw load-displacement data 110 giving the variation of displacement during loading and unloading of a force on the material sample through the indenter tip. This data is stored in a memory 112 which may be located onboard the nanoindentation hardware 102, or external to the nanoindentation hardware 102. Subsequently, logical modules 120, are used to analyse the raw load-displacement data 110 stored in the memory 112. The logical modules 120 may be implemented in data processing apparatus comprising one or more data processing units, such as general-purpose processors, operating under software control to instantiate in memory the logical modules. While the example described herein indicates a software-driven implementation of components of the apparatus by a more general-purpose processor such as a CPU core based on program logic stored in a memory, in alternative embodiments, certain components or modules may be partly embedded as pre-configured electronic systems or embedded controllers and circuits embodied as programmable logic devices, using, for example, application-specific integrated circuits (ASICs) or Field-programmable gate arrays (FPGAs), which may be partly configured by embedded software or firmware.

[0050] The software modules 120 comprises a load frame compliance correction module 122, providing an input to a zero point correction module 124, providing an input to a thermal drift correction module 126. The load-displacement data 110, modified by none, or one, or more, of the load frame compliance correction module 122, the zero point correction module 124, and the thermal drift correction module 126, is passed into an FEP determination module 130 and into a material property determination module 140.

[0051] The FEP determination module 130 comprises an elimination of “nose-out” step 132 (if necessary), which provide the data for input into the data separation at FEP determination step 134.

[0052] The material properties determination module 140 performs a determine stiffness step 142. The stiffness is used to calculate a plastic correction 144. The plastic correction 144 is used to determine a plastic depth 146 and partially to determine the modulus at FEP 150. The plastic depth 146 is used to determine a contact area 148, itself used for determining, in conjunction with the plastic correction 144, the modulus at FEP 150. The contact area is also used to determine the hardness at FEP. The operation of the various steps and modules described above will be explained in more detail below.

[0053] Specific implementations of the logical modules for analysing the raw load-displacement data of a nanoindentation tester, in particular the FEP Determination Module 130 and material properties determination module 140 that are provided in the apparatus that is the subject of this disclosure, will be explained in more detail below. It will be appreciated that the logical modules 120 may be implemented in a processor physically connected to the nanoindentation hardware 102, or instead may be performed on a hardware device separate from the nanoindentation hardware 102. Indeed, the unloading curve data may be transmitted by any suitable means to a data processing apparatus implementing the logical modules 120 at a location remote from the nanoindentation tester. In addition, the processing of the raw load-displacement data by the logical modules may occur at a time later than, and possibly significantly later than, the time of capture of the data by the nanoindentation hardware. Furthermore, although Figure 5 illustrates several modules performing different functions as part of the software modules 120, it will be appreciated that some of these modules may not be needed for certain materials or in certain testing regimes, or certain modules or functions thereof may be merged. The particular selection of software modules required will be apparent to the person skilled in the art on consideration of the disclosure of this document in combination with their own knowledge of nanoindentation testing methods.

[0054] FEP Determination Module 130 [0055] In the FEP Determination Module 130,_the unloading curve data is analysed to eliminate all data until delayed elasticity ceases. This is referred to as the full elastic point (FEP).. To determine it, first, in step 132, the top 60% (or 80%) of the unloading curve data is taken and then split at the maximum displacement (i.e. at nose tip). This can be done by closely fitting a fourth order polynomial to the data, using, for example, a least squares method. The highest load where the unloading curve data diverges from the fourth order polynomial fit is taken and data above this point is discarded, to discard data above the nose out point, for the next step of identifying the FEP. This is particularly important when analysing viscous materials where a nose out is present. A viscofactor (as described herein) can be used to determine whether or not a “nose out” is present.

[0056] Next in step 134, another regression model is fitted to the unloading curve. Typically, the polynomial fit is a second order polynomial fit. The fourth order polynomial fit is used to remove data that may otherwise unacceptably skew the result of the second order polynomial fit, adversely affecting the accuracy of the determined material properties.. After performing the polynomial fit, and comparing the fitted data to the experimental, the highest load at which the two datasets first start to differ is the FEP. The reason for this is that, the FEP determined in this way corresponds to the highest load at which the measured acceleration of the indenter displacement (optionally using — instead of as ^ is constant)becomes a constant indicating that the indenter and sample thereafter move together at the same speed, said point taken to be the full elastic point, FEP, at which any delayed elasticity in the sample in response to the indentation ceases.

[0057] As will be apparent herein, the unloading curve data (at the FEP) or, more preferably the fitted second order polynomial regression model (at the FEP) is then used in the subsequent material properties determination module 140 to determine the material stiffness, and with that, hardness and modulus.

[0058] Figure 14 is an illustration of a graph showing the fit of a second order polynomial to a subset of unloading data for an indentation test of a non viscous material. This corresponds to the data separation at FEP 134 in the flowchart shown in Figure 5. In particular, the raw data has already had a fourth order polynomial fit performed, which removed some of the data past the “nose out” seen. As can be seen from Figure 14, the fit between the second order polynomial and the data is particularly good, and the datasets are visually indistinguishable. Figure 15 is an illustration of a graph showing the difference between the measured data and the second order polynomial for the data shown in Figure 14. The mean difference is less than 0.005mN. The Full Elastic Point (FEP) is determined as the first point (highest load) in the unloading curve at which the difference between the measured displacement of the indenter and the second order polynomial fit is below the mean difference. This is equivalent to the point at which the acceleration of the indenter displacement with respect to load becomes a constant, indicating that the indenter and sample thereafter move together at the same speed. In Figure 15, the FEP is determined at a displacement of approximately 1290nm, which corresponds to a load of approximately 7.5mN.

[0059] Figure 16 is an illustration of a graph showing the fit of a fourth order polynomial to a subset of unloading data for an indentation test of a viscous material. The procedure used to determine the data for this plot is substantially as described in relation to Figures 14 and 15 above. It can be seen that the fourth order polynomial is a reasonably good match to the unloading data one the initial displacement “nose-out” has been completely finished. The fourth order polynomial fit is used to determine where the “nose-out” is. By taking the difference between the unloading dataset and the determined fourth order polynomial fit, and looking for the point at which the unloading data first falls within a threshold difference, the data before this point (and relating to the “nose-out”) can be ignored.

[0060] Figure 17 is an illustration of a graph showing the fit of a second order polynomial to a further subset of the subset of unloading data shown in Figure 16. It can be seen that the data and second order polynomial match each other less well when compared to the data of Figures 14 and 15. In particular, the model overestimates the load upto a displacement of approximately 1.385nm, underestimates the load between approximately 1.385nm to approximately 1.485nm, overestimates the load between approximately 1.485nm to approximately 1.555nm and finally underestimates the load beyond approximately 1.555nm. Figure 17 is an illustration of a graph showing the difference between the measured data and the second order polynomial for the data shown in Figure 16. It can be seen in Figure 17 that the differences between the measured displacement data and the second order polynomial are larger than for the non-viscous material. The FEP is again determined as the first point (highest displacement) in the unloading curve at which the difference between the measured displacement of the indenter and the second order polynomial fit is below the mean difference. This is equivalent to the point at which the acceleration of the indenter displacement with respect to load becomes a constant, indicating that the indenter and sample thereafter move together at the same speed. In Figure 17, the FEP is determined at a displacement of approximately 1.56nm, which corresponds to a load of approximately 17.3mN.

[0061] Material Properties Determination Module 140 [0062] As mentioned above, and as shown in Figure 5, the material properties determination module 140 takes in the load-displacement curve data from step 126 and the determined FEP from FEP determination module 130 (this may include the regression model fit (i.e. the second order polynomial details) and/or the FEP data point). The material properties determination module 140 then performs a determine stiffness step 142. The stiffness SFEp is determined at the FEP from the unloading curve data at the FEP or, more preferably, from the second order polynomial regression model fitted to the data by the FEP determination module 130.

[0063] By determining a second order polynomial fit, the load, P, can be expressed in the regression model in the form:

rip

The stiffness — at a modelled load-displacement point is thus given by the differentiation: dh

[0064] Where, h is the displacement and a, b and c are constants to be determined according to the second order polynomial fit.

[0065] The material property determination module 140 may be configured to, in step 142, determine the apparent stiffness SFEP of the sample at the full elastic point based on the relation:

wherein, hFEP is the modelled probe displacement h at the full elastic point. Alternatively, the unloading curve data at the FEP may be used to determine the stiffness, SFep of the sample at the full elastic point by evaluating the value of — from the data.

[0066] The stiffness at the FEP is then used in step 144 to calculate a plastic correction to the stiffness, to recover the component S of the stiffness measured at the FEP (SFep) that due only to the elastic component, (i.e. by correcting SFep for the plastic component). This will now be explained in more detail.

[0067] Plastic Correction 144 [0068] When determining mechanical properties from the split data of the unloading curve the assumptions have to be made clear. Under actual localised deformation, as will be apparent from this disclosure, for hardness and modulus calculations, compliance is taken at FEP. The history of the indentation can be ignored before this point. Only at the FEP is the following equation for stiffness valid, whereas it is not valid before or after the FEP.

[0069] A fit cannot be extrapolated to max load due to non-conformity and non-elasticity.

[0070] In previous methods (Oliver & Pharr and Feng) the deformation was always seen as fully-elastic-localised (based on hertzian localised deformation) in which two assumptions were made: [0071] 1) Delayed elasticity at the onset of unloading is ignored. What actually happens is that local deformation causes densification/plasticity around the indenter due to diffusion and geometrical necessary dislocations, and upon unloading there is reverse plasticity (referred to as actual localised deformation in this document).

[0072] 2) The material behaviour is fully elastic or viscoelastic. In a fully-elastic-localised deformation the overall elasticity can also be seen as a viscous and elastic component i.e. viscoelasticity, thus a broader term to call this and will be referred to as localised-reversible deformation in this document [0073] In their methods assumption (2) was taken in to account however for assumption (1) it was with multiple loading the delayed elasticity was ignored. Under monotonic loading the initial conformity and delayed elastic response have a marked effect on the measured stiffness especially for viscous material; by eliminating them first from the unloading data the stiffness uniquely represents the material’s elastic or viscoelastic behaviour. When using a multi-loading test the plasticity reduces from cycle to cycle and the final response indicated in the past to be due to the elasticity of the material. By adding another cycle more plasticity/viscous deformation is introduced into the system and temperature would also increase. This would also affect the mechanical properties measured. Apart from this the material density around the material reaches a limit after successive cycles and the elasticity measured would presumably be due to a composite effect of the material around the indenter and the subsurface properties. By measuring the elasticity at the first cycle, at FEP, in accordance with the present disclosure these issues can be eliminated.

[0074] In previous methods viscous effect has always been an issue. Feng corrects the stiffness for viscous effects. The solution is based on ideal viscoelastic material and any delayed elasticity due to plastic deformations is accounted for within the test procedure not the solution. Therefore, the testing conditions are limited. In Feng’s analysis Equation (9) is usually interpreted as Equation (8).

(8) (9) [0075] In the method in accordance with the present disclosure, under the assumptions of localized-reversible deformation the apparent stiffness measured from extrapolating the data from the FEP to maximum load is taken to be due to the elasticity and creep components. S is the effective stiffness measured from the unloading curve in Equation (9). When comparing Equation (9) to Equation (8) the elastic component is related to 1/2aEr, thus Su is the stiffness due the elastic component. Whereas, the (^) term relates to the viscous behaviour. Typically the absolute value of unload rate is taken that’s why the elastic and creep components are added, otherwise the creep component subtracts. A further note should be taken that the penetration rate at the end of hold is typically measured at steady state, thus the hold period should be sufficient to eliminate primary creep. It should be mentioned that the Feng’s correction is applied at the turn over point, so to be applicable for the new method two assumptions are made: [0076] 1) The turn over point occurs within a range from max load to the load at FEP. There is no instantaneous elastic response when unloading from max load. Thus it is reasonable to assume the FEP as the turn over point.

[0077] 2) By determining the true viscoelastic response and also assuming localised-reversible deformation the penetration rate at FEP is determined. To achieve this, instead of using the penetration rate at the end of hold (hh), the rate at which plastic deformation occurs ([hpiastic) is used (which is due to delayed elasticity), this will be referred to as the “plastic correction”.

[0078] To calculate this plastic rate the plastic displacement can be determined use the following equation.

(10) [0079] Two cases exist, no nose-out and the nose-out. In both cases the unloading data up to the FEP is compared with a fully viscoelastic or elastic response determined by a fit to the data after the FEP. For the nose-out case the surface initially dips in and after the nose-out the surface dips out, these displacements are determined by the second bracketed expression for the nose-out equation, after it is subtracted from the first bracketed expression (the displacements for ideal viscoelastic or elastic response). Once the plastic displacement is determined, the plastic rate is calculated by dividing by the time taken from max load to the FEP load. As the turn over point occurs at FEP, the plasticity has already occurred due to delayed elasticity; at this point the stiffness doesn’t need correcting in terms of creep. However it is due to this occurred plastic rate that the stiffness at FEP needs to be corrected for. This method also allows for hold periods where primary creep is present, and since the penetration at end of hold isn’t needed a hold period isn’t required. However, a hold period can still be applied in order to stabilise the plastic deformation during the loading.

[0080] To determine whether or not a nose-out is present in the data, the material property determination module may be configured to evaluate a viscofactor for the unloading curve of the sample material by the relation:

[0081] wherein Pmax is the maximum load applied in the test cycle and Pmaxdisplacementis the load applied when the probe is at its maximum displacement, hmax. The material property determination module may be further configured to deem there to be a nose-out in the unloading curve if the evaluated viscofactor for the unloading curve of the sample material is greater than zero, and to otherwise deem there to be no nose-out in the unloading curve.

[0082] The material property determination module 140 thus in step 144 is configured to determine the plastic-corrected stiffness S from the apparent stiffness SFEP based on the relation (wherein the stiffness S is the plastic corrected stiffness at the FEP):

where \P\ is the unloading rate and hpiasticat FEP is the plastic rate at which plastic deformation occurs during the period of delayed elasticity before the full elastic point.

[0083] To determine the plastic rate, hpiasticat FEP’ the material property determination module 140 may be configured in step 144 to determine a plastic displacement hpiastic at FEP based on the following relation:

wherein hmax is the maximum displacement, hmaxfltted is what the maximum displacement of a regression model fitted to the unloading curve data would be up to the maximum load, hmax_load is the displacement at the maximum load, and hFEP is the displacement at the full elastic point. The material property determination module 140 may be further configured in step 144 to determine the plastic rate, hpiastic, based on the relation:

where tmax ioad-FEP's the time taken from the maximum load to the load at the full elastic point.

[0084] Referring again to Figure 5, in order to determine the hardness and modulus of the material, the area function of the contact between the indenter and sample must be determined. This is determined at the plastic depth (hc, as explained with reference to Figure 3), and so, the material properties determination module 140 in step 146 is configured to use the plastic corrected stiffness S evaluated in step 144 and the load-displacement curve data to determine a determine the plastic depth 146. This will now be explained in more details [0085] Plastic depth determination 146 [0086] Finding accurate contact depth is essential in nanoindentation tests. The two methods most commonly used are the elastic case (See Oliver and Pharr) and the elastic-perfectly plastic case (Bee et al). In Fujisawa and Swain’s work (see Fujisawa N, Swain MV. Effect of unloading strain rate on the elastic modulus of a viscoelastic solid determined by nanoindentation. J Mater Res. 2006;21 (03):708-14), for quasi-static unloading, these two models were used to determine the normalised contact depth against the normalised stiffness, in which the crossover between the two curves was used to determine if either to use the elastic model equation or the elastic plastic model equation. Previously it has been shown that the modulus of a polymer near the surface can be a number of times higher than the bulk modulus (see Tweedie CA, Constantinides G,

Lehman KE, Brill DJ, Blackman GS, Van Vliet KJ. Enhanced Stiffness of Amorphous Polymer Surfaces under Confinement of Localized Contact Loads. Adv Mater. 2007 Sep 17; 19(18):2540-6) however in Fujisawa and Swain’s work they hypothesised that the modulus of the amorphous polymer is dependent only on the unloading strain rate and is independent of the indentation depth. Their findings established this for PMMA polymer and the reason given was when considering the depth there is no clear correlation based on the statistical analysis, which doesn’t regard the depth dependency. However to achieve this, the overestimation of the contact depth was eliminated by extra unloading tests to establish quasi static test conditions. Thus they were able to single out the dependency of modulus on the strain rate. For the new method determined at the FEP the equations used to determine the plastic depth are valid since quasi-static behaviour occurs for even highly viscoelastic materials. Apart from the “plastic correction” in the method, the pile-up effects can be substantial in highly viscous materials and must be accounted for in the measurement of the plastic depth. Equation (1) is based on elastic unloading, however for viscous materials Bee et al approach is widely adopted which leads to Equation (11) below.

(11) [0087] Bee et al found Ci=1.2 & C2=1.2 for elastic-plastic perfectly plastic materials. In this method a similar procedure is adapted as highlighted in Fujisawa & Swain, to determine which method to use. Thus, in accordance with the presently disclosed method, the material properties determination module 140 is configured to, in step 146, determine the plastic depth, hc(S), based on either the elastic method (calculated using Equation (1)) or the elastic-plastic perfectly plastic method (calculated using Equation Error! Reference source not found.) using the logic shown in Equation 12. In Fujisawa & Swain’s work the selection procedure is valid for tests done at one particular strain-rate test condition. For testing done at a variety of different conditions the following logic is used.

(12) [0088] Here, all parameters, including the displacement h, the load P, and the stiffness S, are determined at FEP.

[0089] Referring again to Figure 5, the plastic depth hc determined in step 146 is used in step 148 to determine a contact area Ap (hc) of the indentation of the probe at the plastic depth hc based on equation (4) for the indenter.

[0090] The contact area Ap (hc) determined in step 148 is itself used for determining, in conjunction with the plastic correction 144, the modulus in step 150. That is, in step 150, the material properties determination module 140 is configured to determine reduced modulus Er of the sample and indenter, by the relation:

[0091] The material property determination module may be further configured to determine the sample modulus E$ by correcting the determined reduced modulus Er to remove the contribution of the indenter modulus E[ based on equation (7).

[0092] The contact area is also used to determine the hardness H by the relation:

[0093] In contrast, traditionally the hardness is determined by first finding the contact area at max load Pmax (in hardness tests at micro scale the contact area is typically determined by optically inspecting the residual impression), and once the area is found the hardness can be determined by Pmax/A.

[0094] This residual impression includes the reserve effects during unloading i.e. reverse plasticity and elasticity, and hardness is termed the true hardness. When determining the hardness using nanoindentation by establishing the contact area by indirect method then only for an ideal rigid plastic material the hardness equals the true hardness. The apparent hardness, which is the hardness determined using the Oliver and Pharr method, do not account for the reversal effects during unloading. However as it is been shown that the area determined isn’t correct when there is substantial delayed elasticity and creep when using the tradition procedure, and it is only at FEP the contact area can be determined. Determining the hardness using the contact area at FEP, results in a contact area accounting for the initial reverse plasticity. As Tabor stated that the residual impression doesn’t change much upon unloading for elastic-plastic material i.e. only the depth recovers, it can be stated that the reversal effect due to elasticity has negligible effect on the residual impression. Thus in accordance with the presently disclosed method, the hardness found at FEP using the contact area and load at FEP is equivalent to the true hardness.

[0095] The reasoning behind the method as detailed above in relation to the FEP determination module 130 of determining the FEP is given below. Figure 6a, b & c shows the second derivative of displacement with respect to load against load for three separate materials. This second derivative will also be directly proportional to the acceleration of the indenter since the unloading rate is constant. Thus these graphs can be seen as acceleration against load. For PET (see Figure 6a) there is no acceleration change i.e. thus the contact is in equilibrium. The fluctuations seen in the data are due to signal noise. It can be seen that for Rubber and Nylon (see Figure 6b & c) there is a negative acceleration in region 1, which is changing with load. This is due to non-conformity of contact i.e. the area is changing. From Figure 6b the non-conformity can be seen to occur at lower load and doesn't occur at higher loads even the viscous effects are large.

[0096] All materials which deform plastically display delayed elasticity upon unloading (in actual localized deformation). This is due to plasticity (forwards or reversed) or viscous plasticity during unloading i.e. upon unloading the material is attaining internal equilibrium before elastically responding (see Tabor). When a “nose-out effect’ is present plasticity is occurring alongside creep, the surface is moving inwards, results in a difference in the relative motion of the indenter and sample. Also there could be some adhesion or even plunging effects.

[0097] The main step in the analysis is to eliminate all data until delayed elasticity ceases i.e. FEP. In viscous materials after contact conformity the acceleration is still changing due to delayed elastic response, this is actually the minimum load in the region 2 shown in Figure 6c, and can be determined when the first derivative of displacement with respect to load (i.e. 1/stiffness) starts responding negatively proportional to the load, as seen in

Figure 7. The FEP is determined when the acceleration become a constant i.e. the indenter and sample move together at the same speed. At this point, the material responds fully elastic or viscoelastic without the influence of plasticity, and is unique to the test conditions. The contact can be said to be in “fully conformed elastic or visco-elastic equilibrium”. After this point, upon further unloading, the direction of creep would depend on the system and its boundary conditions. After the FEP a second order polynomial (or a power-law) can be fitted to the curve. A second order polynomial is more accurate and an exact match since 1/stiffness against load is a linear fit.

[0098] Experimental results [0099] Figure 8 shows six different methods of determining the modulus plotted against each cycle of a multi-loading test. Each bar is averaged over 10 tests showing the error bars. The solid line indicates the value of new method (determined at FEP) at the first cycle and the dotted line is the value of Feng’s creep correction method with a 3rd order power law determined at the fourth cycle. A number of observations can be made from these plots. The new method (determined at max load) and Oliver and Pharr method displays the highest variability in the first cycle; this can be related to the high viscous behaviour and plasticity at the end of hold. Also the variability of the new method when plastic corrected shows the least variability i.e. elimination of plasticity at end of hold. This high variability is definitely related to primary creep for this experimental condition, as the number of cycle's increase the variability is reduced considerably. Determining the modulus from the first cycle using the new method with creep corrected will be inaccurate in this circumstance. By finding the modulus at FEP for the new method the results give close reading to the forth cycle determined by creep correction OP method. The modulus difference is about 1Gpa for PET and less than 1Gpa for PEN. This modulus represents the true value of the response of the material upon unloading from max load.

[00100] It is hypothesised that this reduction in modulus in successive cycles is not due to the reverse plasticity but due to the increase in effective contact area and also heating generated when indenter penetrates into the surface (discussed in more detail below).

Thus the modulus can be efficiently acquired at the first cycle. By using the new method at FEP and the plastic correction the diverse effect of creep is eliminated. Referring to Figure 9 the effect of the plastic correction on the stiffness can be seen, which is compared to the creep corrected and non-creep corrected data. Although the range of stiffness values for the plastic correction isn’t as narrow as the creep correction, the stiffness’s are a true representation at the FEP. The modulus is determined efficiently. Acquiring this modulus from different experiments at different loads represents the viscoelastic modulus with different viscous behaviour. Even with Feng’s creep correction the relationship with load is similar but the modulus is slightly lower, roughly 1Gpa, that is in regimes where unloading is quasi-static. However for non-quasi-static conditions, when creep behaviour are high the method isn’t valid because the penetration rate at end of hold is too high due to primary creep being present. It is by the plastic method, at FEP where quasi-static conditions occur and where the penetration can be acquired with some accuracy which enables the determination of the true viscoelastic behaviour related to the max load. From Figure 9 it can be seen that that the stiffness determined by the same experiments are not the same at separate experiments, thus the fitted modulus-load curves determined from the unloading data can displays an opposite behaviour to what is measured experimentally. This is a direct consequence of the behaviour of the determined contact area during unloading.

[00101] The loading curve and unloading curves show different power-law exponents.

This relates to the fact that the unloading geometry is effected by the plastic zone (see Fischer-Cripps AC. Nanoindentation. 3rd Edition. Springer; 2011. 280 p). It is further noted that the contact angle changes with depth (see Fischer-Cripps). Also it has been stated that finite element has revealed an elastic zone is beneath the indenter which depends on the value E/Y (see Fischer-Cripps). This indicates that there is strong evidence that the plasticity is the cause. Johnson’s expanding cavity model is based on a hemispherical hydrostatic core of a radius equal to the contact circle; this is by far the most popular model when the plasticity has reached the surface. This is a representation of the event at maximum load in a nanoindentation test with a conical indenter. It is noted that plasticity caused during loading produces geometrical necessary dislocation around the indenter (see Fischer-Cripps). As the density of geometrical necessary dislocations increase with decreasing depth (see Fischer-Cripps) it is believed that at the early loading, plasticity occurs the most. There is evidence that at low loads plasticity can be present even when the contact is believed to be elastic (see Fischer-Cripps AC, Karvankova P, Veprek S. On the measurement of hardness of super-hard coatings. Surf Coat Technol. 2006 May 8;200( 18-19):5645-54). This clearly strengthens the material around the indenter, and it is hypothesised that the effective geometry changes i.e. a combination of the indenter geometry and the hardened/dense material around the indenter tip. In Oliver and Pharr method a parabolic indenter is modelled to represent the unloading curve even though indented with a Bercovich indenter. Thus modelling parabolic and spherical shapes more represent the shape of the indenter when comparing experimental data and shape of the effective indenter geometry can be either depending on the study sample. On further loading as the plasticity develops beneath the indenter a second phenomenon is also believed to occur. Its effect is also hypothesised to change the effective geometry. During unloading these two phenomena are reversed. The initial delayed elasticity is due to reversing of the plasticity beneath the indenter whereas the hardened/dense material around the indenter tip reverses at low loads. Thus at FEP the contact area can be distinguished by the area function, since there are no reverse effects. Before and after the FEP the contact area is not accurately determined and must not be used to determine the modulus and hardness.

[00102] Increasing creep and plasticity decreases the measured modulus of a material since the deformation (strain) is enhanced by these processes. This is why the stiffness decreases when these processes are active during unloading. For multi-loading tests, Figure 8 shows that with each cycle the modulus decreases, however the plasticity is decreasing, so what is causing this decrease. It can be said that the effect of plasticity on the geometry of the indenter increases the contact area and this in turn decreases the modulus. It is not the modulus that changes but how it is determined by using an equation which is not valid under these circumstances. On further cycles extra heat is generated, which in turn increases the viscosity and reduces the modulus further. This implies that modulus should be determined at the first cycle.

[00103] Examining the modulus as a function of load, it is seen that experimentally from indent to indent the modulus decreases with increasing load i.e. more viscous effects results in low modulus. However, the modulus determined from the unloading curve has an opposite effect with load. An explanation for this is that at max load the penetration rate causes forwards plasticity/creep. Following reverse plasticity at initial unloading which causes the delayed elasticity, this then dominates opposing the forwards plasticity/creep. When delayed elasticity ceases at FEP, only the viscous component of the viscoelasticity is present. The only conclusions that can be made is first the effective contact after unloading is always reducing but at different rates. This is why in Oliver and Pharr’s work at initial loading for plastic contact on an tungsten sample the stiffness-time plot shows a second peak on each of the cycles. This also implies that the continuous stiffness method can be used to determine FEP. Secondly the area after FEP can be said to be more than the elastic case and this effect increases relative to the decreasing load, as at lower loads the density of geometrical necessary dislocation increase.

[00104] Despite the fact Fujisawa and Swain’s work was at very low loads, PET polymer under various conditions was studied revealing that there is a dependency on the depth as well as other testing conditions related to the strain rate. As these effects are coupled they cannot be separated. For the experiment considered, the plastic depths were accurately determined by the methodology presented previously. Unloading from maximum load the non-instantaneous elastic response is due to the stored energy within the system. The release of this energy upon unloading is unique to the system and boundary conditions. At full load, for viscous materials the contact depth can be overestimated, due to the forwards plasticity/viscous behaviour upon unloading. However at the full elastic point the contact depth which is determined is solely due to the elastic or viscoelastic response, providing quasi-static conditions. Thus, there is much confidence in using the equations which were initially developed for elastic contact and valid when these quasi-static conditions are met. On the other hand the underestimation due to the pileup and sink-in is corrected by the Bee at el(see Bee S, Tonck A, Georges J-M, Georges E, Loubet J-L. Improvements in the indentation method with a surface force apparatus. Philos Mag A. 1996 Nov;74(5):1061-72) method. Therefore both of these conditions i.e. viscous effects, and pile-up & sink-in can be corrected for. Thereby the modulus of the material can be determined for a number of testing conditions, where the load rate, unload rate, hold and the maximum load dictate the maximum displacements and the strain rates.

[00105] The equations to determine contact depths can be a rearranged in terms of normalised contact depth and normalised stiffness as presented by Fujisawa and Swain (see Figure 10). This equation can be used to fit the normalised contact depth and the normalised stiffness experimental data to the equation itself. When performing these fits one can deduce the true values of a and ε for the range of test conditions for that particular material/indenter contact. The fits to the data are reasonable with little variation. For the elastic case the value of a value was found to be 1 & the value of ε roughly 0.72. As for the elastic-perfectly plastic case a value was 1.2 whereas the value of ε was 0.91. These values can be iterated to give more accurate values of the contact depth if required. It should be pointed out that the contact depth was determined at the FEP therefore the point of determination as seen on the plot is lower than 2.25, as found by Fujisawa and Swain. Figure 11 shows the dependency of the normalised contact depth and the modulus with the experimental parameters. First examining Figure 11a, b & cthe calculated contact depth has negligible effect with the hold period and load rate. There is a clear dependency with depth and the unload rate, decreasing unload rate increases the gradient of the fitted data for a particular test condition. As mention above, this is due to the combined effect of the strain rate (dependent upon the experimental conditions) and depth related properties of the material. So for all the experimental condition even when at the lowest load rate, which gave high viscous behaviour accurate contact depths were achieved. Looking at Figure 11d, e & f the modulus is also determined with some degree of variability. As found for contact depths, its dependency with depth is similarly represented; this is because one is determined from the other. At low load rate where the viscous behaviour was high the modulus values are about 1Gpa lower, especially at the higher depths. This is reflected by the fact the appearance of the “nose-out” on the load-displacement graph and the determined stiffness, which again is used to determine the modulus. During loading, low load rates cause more plastic deformation/creep since the material is forced by the indenter over a longer period. This in turn introduces more internal energy in the material which is released upon unloading. In the case of PET this initial energy causes a phenomenon where high viscous behaviour is detected during unloading. So the modulus is low because the material behaves in this way and the analysis is perfectly valid. As for other methods accurate modulus values would have not been established at these conditions.

[00106] Monotonic loading-unloading experiments were conducted to determine the relative validity of the new method to existing methods. Thirteen different materials Alumina, Aluminium, Brass, Cast Iron, Copper, High speed steel EN24, Mild steel, Nylon, Perspex, Rubber, Silver steel, PET and PEN were tested using a micro material nanoindenter; these materials are split into two separate categories, viscous and non-viscous. These experiments were performed at a maximum load of 100mN with 5 sec dwell period, load rate and unload rate being 10mN/s. For viscous materials a number of influencing factors were compared such as the effect of applying creep correction, the effect of applying plastic correction to new method and also the effect of applying a sink-in correction.

[00107] The analysis was validated at micro scale, by comparing to a Doerner and Nix linear method and the traditional Oliver and Pharr power law method. The new method was determined at both the FEP and maximum load, with and without the plastic correction. Figure 12 shows the determined modulus for all these method. The dotted lines indicate the lower and upper limits of modulus as determined in literature by different means. It is seen that the new method determined at the FEP give closest results to the traditional Oliver and Pharr method, although the values are slightly higher, they are more accurate when compared to the literature values.

[00108] The analysis was validated at micro scale, by comparing the new method, at both maximum load and FEP, to determine the stiffness from the unloading curve, to the traditional Oliver and Pharr using third order power law. Figure 13 shows the determined modulus for all these method. Again the dotted lines indicate the lower and upper limits of modulus as determined in literature. The first observation that can be seen is when applying a sink-in correction the modulus is substantially reduced. The second observation is the creep correction increases the modulus values for the new method. The third observation is the same for the non-viscous material that the modulus at max load is greater than the modulus at FEP for the new method. Forth observation is the variation associated with the new method at FEP is less than the other methods. It can be seen that the new method is effective in determining the modulus at FEP with less variability and the values are closer and slightly less than the Oliver and Pharr method. When comparing to the multi-loading testing in Figure 8 the first cycle gives high variation for creep corrected methods. The reason for this is the high creep phenomena at that particular experimental condition. The variation in the creep correction in Figure 13 is low due to the penetration rate at end of hold being only due to the secondary creep. Under these circumstances the modulus values determined by the new method are as accurate at the Oliver and Pharr method. However when high creep is present, the Oliver and Pharr method overestimate the modulus with high variation.

[00109] A more reliable and accurate method is given to determine elastic, viscoelastic modulus and hardness for viscous and non-viscous materials. It has been found that at the FEP an elastic-viscoelastic contact is established, and it is claimed that before and after this point the determined contact area is not accurate, as the equations used to determine it are not representative of the system, so must not be used in determining the modulus and hardness due to the reversal effects of plasticity. Also it is necessary that the first cycle is considered, as further cycles change the contact geometry and boundary conditions such as temperature, internal energy etc.

[00110] For all the 13 different materials studied by using just a single loading-unloading cycle, the method determined values equally well as previous methods, exhibiting closer values to the ones found in literature. In employing this method time and cost of experimentation can be reduced, and studies can also be conducted at non quasi-static test conditions, making it suitable for polymer characterisation.

[00111] In determining the modulus of PET using this established method a clear dependency with depth is revealed as well as for other testing conditions related to the strain rate. As these effects are coupled they cannot be separated, thus all input variable which affect the modulus of viscous materials need to be studied together and statistically determined, to identify their effects.

[00112] From the disclosure provided herein, it can be seen that determining the reduced modulus and hardness of a material using the method disclosed herein, where a full elastic point is determined and used as the point for determining material properties, produces results which are close to the results determined from the Oliver and Pharr method, but requiring only a single loading-unloading cycle. Furthermore, as can be seen from Figure 8, 12 and 13, the error bars on the resulting modulus estimate are minimal when compared with the error bars seen using other methodologies (such as Oliver and Pharr) and only a single cycle.

[00113] Throughout the description and claims of this specification, the words “comprise” and “contain” and variations of them mean “including but not limited to”, and they are not intended to (and do not) exclude other moieties, additives, components, integers or steps. Throughout the description and claims of this specification, the singular encompasses the plural unless the context otherwise requires. In particular, where the indefinite article is used, the specification is to be understood as contemplating plurality as well as singularity, unless the context requires otherwise.

[00114] Features, integers, characteristics, compounds, chemical moieties or groups described in conjunction with a particular aspect, embodiment or example of the invention are to be understood to be applicable to any other aspect, embodiment or example described herein unless incompatible therewith. All of the features disclosed in this specification (including any accompanying claims, abstract and drawings), and/or all of the steps of any method or process so disclosed, may be combined in any combination, except combinations where at least some of such features and/or steps are mutually exclusive. The invention is not restricted to the details of any foregoing embodiments.

The invention extends to any novel one, or any novel combination, of the features disclosed in this specification (including any accompanying claims, abstract and drawings), or to any novel one, or any novel combination, of the steps of any method or process so disclosed.

[00115] The reader's attention is directed to all papers and documents which are filed concurrently with or previous to this specification in connection with this application and which are open to public inspection with this specification, and the contents of all such papers and documents are incorporated herein by reference.

Claims

1. Apparatus for use in determining a property of a sample material tested by a depthsensing indentation system, comprising: a memory storing data representative of points on a load-displacement curve for a test of the sample generated by the depth-sensing indentation system, the data including an unloading curve of the measured displacement, h, of the probe of the depth-sensing indentation system as the previously applied indentation load, P, thereon is released; a full elastic point, FEP, determination module configured to determine, based on the unloading curve data, a point on the unloading curve having the highest load at which d2h dP the measured acceleration of the indenter displacement, optionally using —- as — is dPz dt constant, becomes a constant indicating that the indenter and sample thereafter move together at the same speed, said point taken to be the full elastic point, FEP, at which any delayed elasticity in the sample in response to the indentation ceases; and a material property determination module configured to determine, based on the unloading curve data at the full elastic point, FEP, one or more material properties of the sample.

2. Apparatus as claimed in claim 1, wherein, to determine the FEP, the FEP determination module is configured: to fit a linear regression to a plot of the indentation load, P, against the first derivative of the measured displacement of the probe with respect to the indentation dh load, — for the unloading curve data; and to identify the point on the unloading curve having the highest load above which dh — deviates from the fitted linear regression. dP a

3. Apparatus as claimed in claim 1, wherein, to determine the FEP, the FEP determination module is configured: to fit a second order polynomial regression to a plot of the indentation load, P, against the measured displacement, h, for the unloading curve data; and to identify the point on the unloading curve having the highest load above which the measured displacement, h, deviates from the fitted second order polynomial regression.

4. Apparatus as claimed in claim 1, 2 or 3, wherein, to eliminate initial nose out data from the FEP determination where forward plasticity or viscosity in the sample, the FEP determination module is configured: to fit a fourth order polynomial regression to a plot of the indentation load, P, against the measured displacement, h, for the unloading curve data; to identify, as a nose out point, the point on the unloading curve having the highest load above which the measured displacement, h, deviates from the fitted fourth order polynomial regression; and to use only the unloading curve data having loads below the nose out point when determining the FEP.

5. Apparatus as claimed in claim 1, 2 or 3, wherein, to eliminate initial nose out data from the FEP determination where forward plasticity or viscosity in the sample, the FEP determination module is configured: to identify, as a nose out point, the point on the unloading curve having the maximum measured displacement, hmax; and to use only the unloading curve data having loads below the nose out point when determining the FEP.

6. Apparatus as claimed in claim 4, wherein the FEP determination module is configured: to evaluate a viscofactor for the unloading curve of the sample material by the equation:

wherein Pmax is the maximum load applied in the test cycle and Pmaxdispiacement is the load applied when the probe is at its maximum displacement, hmax, and wherein the FEP determination module is configured to only eliminate initial nose out data from the FEP determination if the evaluated viscofactor for the unloading curve of the sample material is greater than zero.

7. Apparatus as claimed in any of claims 2 to 5, wherein, to identify the point on the unloading curve having the highest load above which the measured stiffness or displacement, h, deviates from the fitted linear regression or second or fourth order polynomial regression, as the FEP determination module is configured to identify the highest load at which the difference between the unloading curve and the fitted linear regression or second or fourth order polynomial regression matches the mean difference therebetween.

8. Apparatus as claimed in any of claims 2 to 6, wherein, to fit the linear or second or fourth order polynomial regression models to the unloading curve data, the FEP determination module is configured to perform a least squares analysis to fit a regression model by minimising the errors between model to the unloading curve data.

9. Apparatus as claimed in any preceding claim, wherein the FEP determination module is configured: to use only the unloading curve data having within a given percentage of the maximum load, Pmax, when determining the FEP.

10. Apparatus as claimed in any preceding claim, wherein the material property determination module is configured to: determine the apparent stiffness SFEP of the sample at the full elastic point based on the relation:

wherein, a and b are constants to be determined according to a second order polynomial fit, and hFEP is the modelled probe displacement h at the full elastic point.

11. Apparatus as claimed in any preceding claim, wherein the material property determination module is configured to: determine the plastic-corrected stiffness S at FEP from the apparent stiffness SFEP based on the relation:

• φ where |P| is the unloading rate and hpiastiC at FEP is the plastic rate at which plastic deformation occurs during the period of delayed elasticity before the full elastic point.

12. Apparatus as claimed in claim 11, wherein, to determine the plastic rate, hpiasticat fep< the material property determination module is configured to: determine a plastic displacement hpiasticat FEP based on the following relation:

wherein hmax is the maximum displacement, hmax_fftted is what the maximum displacement of a regression model fitted to the unloading curve data would be up to the maximum load, hmax_Ioad is the displacement at the maximum load, and hFEP is the displacement at the full elastic point; and determine the plastic rate, hpiastic, based on the relation:

where tmax ioad-FEP is the time taken from the maximum load to the load at the full elastic point.

13. Apparatus as claimed in claim 12, wherein, to determine whether or not a nose-out is present in the data, the material property determination module is configured to: evaluate a viscofactor for the unloading curve of the sample material by the relation:

wherein Pmax is the maximum load applied in the test cycle and Pmaxdispiacement is the load applied when the probe is at its maximum displacement, hmax, and wherein the material property determination module is configured to deem there to be a nose-out in the unloading curve if the evaluated viscofactor for the unloading curve of the sample material is greater than zero, and to otherwise deem there to be no nose-out in the unloading curve.

14. Apparatus as claimed in any preceding claim, wherein the material property determination module is configured to determine the plastic depth of the probe contact at the full elastic point, hc, to correct for pile up effects, by the relation:

Wherein hc (eiastic method) is the plastic depth calculated according to the relation for elastic materials, hc (eiastic method)is the plastic depth calculated according to the relation for the elastic plastic perfectly plastic materials, ε is a factor compensating for the shape of the indenter probe, c1 and c2 are both 1.2 for elastic plastic perfectly plastic materials, hFEP is the measured probe depth at the full elastic point, SFEP is the measured apparent or plastic corrected stiffness of the sample at the full elastic point.

15. Apparatus as claimed in any preceding claim, wherein, to determine hardness H of the sample, the material property determination module is configured to: determine a projected area Ap(hc) of the indentation of the probe at the plastic depth hc at the full elastic point; and determine the hardness H at the full elastic point by the relation:

16. Apparatus as claimed in any preceding claim, wherein, to determine reduced modulus Er of the sample and indenter, the material property determination module is configured to: determine a plastic depth hc of the probe contact at the full elastic point; determine a projected area Ap (.he) of the indentation of the probe at the plastic depth hc at the full elastic point; and determine the reduced modulus Er of the sample and indenter at the full elastic point by the relation:

17. Apparatus as claimed in any preceding claim, wherein the material property determination module is further configured to determine the sample modulus Es by correcting the determined reduced modulus Er to remove the contribution of the indenter modulus Ei.

18. Apparatus as claimed in any preceding claim, wherein the FEP determination module and the material property determination module are configured to respectively determine the FEP and the one or more material properties using only data from a single indentation cycle on the sample, preferably the first indentation cycle performed on the sample.

19. Apparatus as claimed in any preceding claim, further comprising one or more processors and computer readable medium carrying instructions which when executed by one or more of the processors cause the apparatus to instantiate the FEP module and/or the material property determination module.

20. A depth-sensing indentation system comprising: a sample stage; a probe for indenting the surface of a sample arranged on the stage; an actuator for applying a load to the probe to indent the surface of a sample arranged on the stage; a load sensor for sensing a load on the probe; a displacement sensor for sensing a displacement of the probe; a controller configured to operate the actuator, load sensor and displacement sensor to perform a depth-sensing indentation test on a sample arranged on the stage by collecting data representative of points on a load-displacement curve including an loading curve of the measured displacement, h, of the probe as the indentation load, P, is applied and an unloading curve of the measured displacement, h, of the probe as the previously applied indentation load, P, thereon is released; and apparatus for use in determining a property of a sample material tested by a depthsensing indentation system as claimed in any preceding claim, the apparatus being configured to store in the memory thereof the data representative of points on a load-displacement curve produced by the controller.

21. Apparatus as claimed in any of claims 1 to 19 or a depth-sensing indentation system as claimed in claim 20, wherein the indentation test is a nanoindentation test or a microindentation test.

22. Computer readable medium carrying instructions which when executed by one or more processors of apparatus depth-sensing indentation system as claimed in any preceding claim cause the apparatus or depth-sensing indentation system to instantiate the FEP module and/or the material property determination module.

23. Apparatus for use in determining a property of a sample material tested by a depthsensing indentation system, substantially as hereinbefore described with reference to the drawings.

24. A depth-sensing indentation system substantially as hereinbefore described with reference to the drawings.