a combined inverse finite element elastoplastic modelling method to simulate the size effect in nanoindentation and characterise materials from the nano to micro scale

Accepted Manuscript A combined inverse finite element – elastoplastic modelling method to simulate the size-effect in nanoindentation and characterise materials from the nano to micro-scale X Chen , I A Ashcroft , R D Wildman , C J Tuck PII: DOI: Reference: S0020-7683(16)30327-4 10.1016/j.ijsolstr.2016.11.004 SAS 9356 To appear in: International Journal of Solids and Structures Received date: Revised date: Accepted date: 15 March 2016 30 October 2016 November 2016 Please cite this article as: X Chen , I A Ashcroft , R D Wildman , C J Tuck , A combined inverse finite element – elastoplastic modelling method to simulate the size-effect in nanoindentation and characterise materials from the nano to micro-scale, International Journal of Solids and Structures (2016), doi: 10.1016/j.ijsolstr.2016.11.004 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain ACCEPTED MANUSCRIPT A combined inverse finite element – elastoplastic modelling method to simulate the size-effect in nanoindentation and characterise materials from the nano to micro-scale X Chen, I A Ashcroft1, R D Wildman, C J Tuck Abstract CR IP T Faculty of Engineering, The University of Nottingham, NG7 2RD, UK PT ED M AN US Material properties such as hardness can be dependent on the size of the indentation load when that load is small, a phenomenon known as the indentation size effect (ISE) In this work an inverse finite element method (IFEM) is used to investigate the ISE, with reference to experiments with a Berkovich indenter and an aluminium test material It was found that the yield stress is highly dependent on indentation depth and in order to simulate this, an elastoplastic constitutive relation in which yielding varies with indentation depth/load was developed It is shown that whereas Young’s modulus and Poisson’s ratio are not influenced by the length scale over the range tested, the amplitude portion of yield stress, which is independent of hardening and corresponds to the initial stress for a bulk material, changes radically at small indentation depths Using the proposed material model and material parameters extracted using IFEM, the indentation depth-time and loaddepth plots can be predicted at different loads with excellent agreement to experiment; the relative residual achieved between FE modelling displacement and experiment less than 0.32% An improved method of determining hardness from nanoindentation test data is also presented, which shows goof agreement with that determined using the IFEM Introduction AC CE It is generally recognised that material properties, especially plasticity, relate to length scale, with materials exhibiting increased resistance to deformation at smaller length scales (Al-Rub and Voyiadjis, 2004); this has been shown by microbending and microtorsion experiments, leading to the development of a model involving strain gradient plasticity (Fleck at al, 1994), and the discovery that hardness decreases with indentation depth, until at large indentation depths a depth-independent bulk material hardness is found This phenomenon of hardness being depth dependent is referred to as the indentation size-effect (ISE) (Gane and Bowden, 1968; Wendelin et al, 2013; Page et al, 1992) The ISE has been explained in metals using dislocation theory, with the dislocation density under the tip of an indenter being dependent on indentation depth From this interpretation, the ISE has been successfully modelled by Nix and Gao (1998) by presenting the dislocation density as being inversely proportional to indentation depth Based on the model of Nix and Gao (NG), various improvements Corresponding author: ian.ashcroft@nottingham.ac.uk ACCEPTED MANUSCRIPT CR IP T have been made For instance, an improved model considered the size of the plastic zone (Durst et al, 2005, 2006) and a general shape/size-effect law for nanoindentation was presented by Pugno (2007) Voyiadjis and Peters (2010) incorporated a material hardening effect (equivalent plastic strain) into the NG model to develop an analytical formulation for the indentation hardness calculation, although the equivalent plastic strain was computed with finite element analysis (FEA) Gerberich et al (2002) considered indentation with depth less than a few hundred nanometres where strain gradient plasticity is insufficient and developed an analytical model to simulate ISE Similar advances in analytical modelling of the ISE have been reported in (Alderighi et al 2009; Huang et al, 2006; Jeng and C Tan, 2006; Wang and Lu, 2002; Elmustafa and Stone, 2003; Al-Rub and Faruk, 2012; Al-Rub, 2007; Pharr, Herbert and Gao, 2010; Durst, Găoken and Pharr, 2008; Qiao, Starink and Gao, 2010; Lucas, Gall and Riedo, 2008; Xu and Li, 2006) AN US ISE has also been observed in polymers For example, Samadi-Dooki et al, (2016) conducted nanoindentation tests with a polycarbonate material, with an ISE being seen at different loading rate Voyiadjis and Malekmotiei (2016) carried out indentation testing of glassy polymers keeping the ratio of loading rate against load constant during the indentation Again, higher hardness was observed at smaller indentation depth Alhough the ISE has been recorded for polymers, the underlying mechanisms are likely to be different CE PT ED M ISE has also been studied using FEA A crystal plasticity FE model to simulate ISE was developed by Liu et al (2015) Yield stress was considered to be strain rate dependent and a function of hardening modulus This method can be used to predict hardness, although only the FE simulated depth-load curve was shown and no comparison with experiments was made in the paper Faghihi and Voyiadjis (2011) developed a viscoplastic constitutive material model in order to simulate the uniaxial/multiaxial deformation of metals at low and high strain rates and temperatures, considering the motion of dislocations The model was used to predict hardness, achieving good agreement with experimental data Bittencourt (2013) presented a FEA method based on crystal plasticity theory, considering strain rate influence and crystal slip to simulate ISE The simulated ISE was in good agreement with the NG model Similarly, Guha et al (2013) developed a FEA method with high order strain gradient theory within the framework of large deformation and elastic-viscoplasticity to simulate the ISE Similar investigations of the FE modelling of ISE or computation of indentation hardness have been reported in (Harsono, et al, 2011; Salehi and Salehi, 2014; Celentano, et al, 2012; Gomez and Basaran, 2006; Swaddiwudhipong, 2012; Oliver and G.M Pharr, 2004) AC Although various material constitutive relations have been used in the FE simulation of ISE, only the simulated hardnesses were compared with test data in some cases (Faghihi and Voyiadjis, 2011), or only the modelled depth-load curves in others (Harsono et al, 2011; Celentano et al, 2012; Swaddiwudhipong, 2012) When both have been shown, good agreement with experimental data was not seen (Gomez and Basaran, 2006) In some works, no comparison with experimental data was made, (Liu et al, 2015; Bittencourt, 2013; Guha et al, 2013; Salehi and Salehi, 2014) A further drawback to the application of all these FEA based methods is that they require the coding of material subroutine scripts based on the corresponding theories In this work, an inverse FE method (IFEM) is used to characterise the property parameters of an indentation sample material according to the reference data obtained from experiments This ACCEPTED MANUSCRIPT approach has been shown to be a powerful tool for identifying complex material relationships, and has been demonstrated in applications as diverse as 3D printing (Chen et al, 2015) and biomaterials evaluation (Abyaneh et al, 2013), high strain rate deformation (Hernandez, et al, 2011), in simultaneous identification of boundary conditions and material properties (Dennis, et al, 2011) and in the characterisation of tissue (Sangpradit et al, 2009; Kauer et al, 2002) Experimental study M AN US CR IP T In this paper it is proposed that the results of an IFEM can be used to empirically determine a suitable material model and model parameters to accurately simulate the ISE As nanoindentation is a pseudo-static process, with low strain rates, and the test materials mainly show strain-hardening behaviour, an elastoplastic material constitutive relation is used in the IFEM in this work However, the yield rule developed is different from traditional elastoplastic constitutive relations, with yield stress directly expressed as a function of indentation depth/load With this material constitutive relation, an excellent fit between experiment and FEA will be demonstrated Based on the characterized material property, the modelled load-depth curves and hardness are compared with experimental data, with good agreement at all the indentation loads tested The proposed method enables the ISE to be explicitly interpreted as the change of yield stress of a material at different length-scales/depths Moreover, elastoplasticity can be directly applied in most finite element packages; hence, the ISE can be incorporated into mechanical design using the proposed approach without the need to code a subroutine In addition, the paper also presents an improved method for the evaluation of hardness from nanoindentation test data for the more accurate characterisation of material hardness PT ED An industrial aluminium alloy was used as the sample material in this study The material composition was analysed using a scanning electron microscope TM3030 (Hitachi High-Technologies Corporation), with results shown in Table I The aluminium material was machined into a cylinder, with diameter and height of 30 mm The test surface was ground flat and polished to a one micron diamond paste finish CE Table I Sample material composition Element Aluminium Silver Silicon Manganese AC [norm.at.%] 98.8537 0.2772 0.4992 0.3700 Indentation tests were conducted using a NanoTest 600 (Micro Materials Ltd., Wrexham, UK) In this work, load control was used with a linear load and unload pattern, as shown in figure 1a A suitable load-hold stage at maximum load should be applied if creep occurs under the applied loading conditions, such as in Chen et al (2015) However, at room temperature, creep is insignificant for the aluminium alloy tested, which can therefore be considered as a rate independent, elastoplastic material (Rathinam et al 2009; Liu et al 2015) Figure 1b shows a typical depth-time curve for an ACCEPTED MANUSCRIPT indentation with 400 mN of maximum load It can be seen that deformation during loading is nonlinear, whereas unloading appears linear, at least initially, with an unrecovered deformation which is indicative of plasticity CR IP T A indentation array was made in the sample surface and then repeated at least three times The separation between indentations in an array was 50 while the distance between arrays was 100 A range of maximum loads were used in each array, varying between and 400 mN Typical experimental results are shown in figure where (a) shows the load-depth curves for an array of indents and (b) shows the calculated hardness as a function of depth from the same data, using the analysis method presented by Oliver and Pharr (2004) An obvious ISE can be observed, which is most severe below about 50 mN, but is still apparent at 400 mN, indicating the bulk hardness has not been achieved It can be seen that the loading curves for different maximum loads almost lie on the same curve; however, there are some small differences that can be attributed to spatial variation in indentation response due to random factors, such as defects, inclusions and grain size This is reflected in some scatter in the hardness values in figure 2b AN US As both the load and unload time are fixed, the loading and unloading rates depend on the maximum load of an indentation In this experiment, the loading/unloading rate changes from 0.02 to mN/s The change of loading and unloading rate may affect the experimental results of some materials (e.g Voyiadjis and Malekmotiei, 2016) but is not expected to be of significance for the current test materials, as explained earlier AC CE PT ED M It should also be noted that mechanical polishing can harden the sample test surface, and the oxide layer on metal samples may also affect the surface hardness of the sample These effects can be significant at very low loads; however, the maximum indentation load in this test is 400 mN corresponding to indentation depth around and, hence, can be ignored In the case of very small indentation loads, with penetration in the nanoscale, results can also be affected by local anisotropy, grain size, orientation alignment and boundaries Figure (a) Load-control pattern in tests (b) Indentation depth-time curve for maximum load of 400 mN ACCEPTED MANUSCRIPT Inverse FE modelling and constitutive relation CE PT ED M AN US CR IP T 3.1 Introduction of IFEM The inverse FE method (IFEM) is predicated on the basis that a single combination of material properties and boundary conditions will result in the observed experimental conditions Proceeding from this assumption, it is possible to iterate the possible material parameters for the given boundary conditions and the given constitutive relation until the properties that lead to results closest to the experimental observations are obtained IFEM, therefore, provides the possibility of combining the output of the nanoindentation test with an FE model to inversely obtain the material properties, despite the complex stress state in the material induced by the indenter (Chen et al, 2015) In the modelling process, a key step is selecting a material constitutive model that is able to capture the main mechanical responses of the material against an actuation of indentation; only a correct material constitutive relation can lead to a good fit between the FE simulated result and the tested, or in other words, if a sufficiently good fit can be achieved by an assumed material constitutive relation by IFEM, then the material constitutive relation selected is close to being correct This is essential for the method to establish meaningful material property parameters and a good fit between FEA modelling and experiment AC Figure Test result of an indentation array with linear load variation from to 400 mN along the array (a) The load-depth curve obtained for indents with different indentation loads and (b) the variation of evaluated hardness with indentation loads 3.2 Preliminary investigation A preliminary investigation into the applicability of the proposed IFEM was carried out using experimental depth-time curves with a number of different maximum loads as reference data and assuming that the yield stress obeys the commonly used exponential hardening law, given by (1) ACCEPTED MANUSCRIPT where is initial and hardening independent yield stress, is equivalent plastic strain, and are material constants In this equation is assumed to be independent of depth and indentation load Using this material constitutive relation, a FE model (details of which will be discussed in later sections) to simulate the indentation process was developed The IFEM was then used to update the material parameters in equation until the error between the FE simulated indentation displacement and experimental reference data was minimised AN US CR IP T Typical results from this preliminary investigation are shown in figure Representative examples of the stress-strain curves constructed with the optimised values of the material parameters in equation from the IFEM at different maximum loads, p, are shown in figure 3a It can be seen that higher yield stresses are seen with smaller indentations, indicating the sensitivity of yielding to length scale and the fact that a single set of material parameters cannot be used in equation to characterise the indentation response at different loads Figures 3b-3c compares the experimental indentation time-depth plots (‘Test’) with simulated plots using the IFEM optimised values of equation (FF) It can be seen that even when the optimal material parameters for a particular load are used, the fit of the FE simulated time-depth plot to experimental data is not good, particularly at low loads, although the fitting is similar to that seen in the published literature (Harsono et al, 2011; Celentano et al, 2012; Swaddiwudhipong, 2012) The residual error between simulated and experimental curves (Res) can also be seen in the plots, which is contrasted against a plot of zero residual error (Zero) AC CE PT ED M This preliminary investigation has proven, then, that yielding in the indentation process is depth/load dependent and cannot be accurately predicted with standard elastoplasticity An elastoplastic model capable of modelling the ISE is proposed in the next section AN US CR IP T ACCEPTED MANUSCRIPT ED M Figure Preliminary IFEM result for depth-independent yielding, showing (a) stress-strain plots with IFEM optimised parameters for equation 1at various value of maximum load, p (mN), (b) – (d) best fit between experimental (Test) and simulated (FF) depth-time plots at (b) p = 6.4 mN, (c) p = 50 mN and (d) p = 400 mN, where “Res” indicates the residual error between FE and experiment, and ”Zero” indicates zero residual error PT 3.3 A proposed indentation elastoplastic constitutive relation AC CE From the observations of the dependence of material strength to length scale (Al-Rub and Voyiadjis, 2004) and the ISE observed in tests on metal materials, several theories have been developed, such as a viscoplastic theory (Faghihi and Voyiadjis, 2011), high order gradient crystal plasticity (Bittencourt, 2013), higher order strain gradient theory (Guha et al, 2013), micropolar theory (Salehi and Salehi, 2014) and gradient-enhanced plasticity (Swaddiwudhipong, 2012) With the various theories emphasizing different aspects of the observed size-effect, in this paper, a new constitutive relation to account for the ISE is proposed based on traditional elastoplastic theory In traditional isotropic elastoplastic theory, the yield stress is assumed to be dependent on the equivalent plastic strain, dictated by a hardening rule The integration procedure for an elasto-plastic material is usually divided into two steps In the first step, a trial equivalent von Mises stress is computed by ignoring any possible plastic flow during the considered time increment; if the computed trial von Mises stress is lower than the given initial yield stress, or a yield stress determined in a previous step, then the trial stress is the actual von Mises stress; otherwise, in the second integration step, the equivalent plastic strain increment is computed based on a hypothesis that yielding occurs on the yield surface, with flow in the direction of the normal to the yield surface ACCEPTED MANUSCRIPT (Dunne and Petrinic, 2005) Hence, the yield condition is that the trial von Mises stress should be higher than the current yield stress at any material point In order to develop a constitutive model accounting for the ISE based on traditional elasto-plasticity, it is assumed that: ii) iii) The current yield stress consists of two parts, a part that is independent of strain hardening (corresponding to the initial yield stress in traditional elastoplasticity) and a part that is due to strain hardening Only the part of yield stress independent of strain hardening is indentation load/depth dependent, presented by ( ) or ( ) for as long as the trial equivalent stress is higher than the current yield stress, plastic flow occurs on the yield surface and flows in the direction normal to the yield surface CR IP T i) This can be referred to as indentation elastoplasticity As an example, with an exponential hardening law and ignoring any strain rate effect, the constitutive relation can be expressed as: ( ) AN US , , (2) (3) M where is the current yield stress, ( ) is the part yield stress that is indentation load (and hence indentation size) dependent, is a material constant, is the material hardening index, is the von Mises equivalent plastic strain and is the trial von Mises stress The second part of the yield stress in equation is the strain hardening function The material constants and may also be indentation load/size dependent; however, for simplicity, they are assumed to be indentation load independent in this work AC CE PT ED Equations and have a similar form to traditional elastoplasticity, but with the ( ) expressed as a function of indentation load Equation shows the yielding condition, which will differ from traditional elastoplasticity when combined with equation as the current yield stress , can now vary with indentation load If a high current yield stress is achieved at some point, i.e at the start of an indentation, this current yield stress would be of significance in traditional theory and only when the trial equivalent stress at this point is higher than the remembered yield stress would plastic flow occur again at this point However, in the presented constitutive relation, the current yield stress, as computed from equation 2, is used such that as long as equation is satisfied, plastic flow occurs This means that a material yielding at a high yield stress at some instance, may yield at a lower yield stress at later; or vice versa, depending only on whether equations and are satisfied Hence, the plastic flow which occurs at a high yield stress at the start of an indentation does not prevent plastic flow at a lower yield stress at the end of the indentation; which is why the ISE seen during indentations is a challenge to model with traditional elastoplastic theory (see figure 5) Other aspects of the proposed constitutive relation, such as the plastic flow direction and integration procedures, are the same as traditional elastoplasticity (Dunne and Petrinic, 2005) As the yield condition in the proposed material constitutive relation is different from traditional elastoplasticity, it needs to be coded in a material subroutine script for use within a commercial FE package In this work, ABAQUS (Dassault Systèmes Simulia Corp.) FE software was used and a user’s ACCEPTED MANUSCRIPT material subroutine (UMAT) was coded for use in ABAQUS standard Compared with traditional elastoplasticity, only the yield conditions and yield stress computations are different, therefore, the integration procedures and method of determining material stiffness presented in ABAQUS manual can be used AN US Results of the inverse finite element modelling CR IP T It is worth noting that the above material model is mainly suitable for multi-crystalline alloys demonstrating hardening behaviour with a grain size much smaller than indentation contact diameter or the equivalent that can be treated as isotropic For the indentation with single crystal material, the material constitutive relation presented in Gao et al (2015) can be referred to, where the ISE is found to be related to sample preparation It can also be noted that as the proposed model is not explicitly based on the physics of the deformation it could potentially be applied to any material showing similar phenomenological deformation behaviour to that exhibitted by the test material in the paper 4.1 FE model CE PT ED M As IFEM is a time-consuming and computationally expensive modelling process, the Berkovich indenter was represented in the FEA by a conical indenter with an equivalent half angle of 70.3° (Salehi and Salehi, 2014; Celentano et al, 2012; Gomez and Basaran, 2006; Swaddiwudhipong, 2012) while the finite sharpness of the tip was represented by a circular curve with diameter of 100 nm This enables a computationally efficient axisymmetrical 2D model to be used The diamond indenter was simplified to a rigid body and contact between the indenter and the sample surface was assumed to be frictionless The FE sample geometry was a 2000 diameter cylinder, 1000 in height as computational experiments demonstrated that with these dimensions, the solution of the FE analysis was insensitive to further increase in dimensions Figure shows the FE geometry, boundary conditions and mesh Convergence tests were conducted to determine an adequately refined mesh, which is particularly important in the region of the indenter tip (Liu et al, 2015), as shown in the inserts in Figure 4(b) ( ) In an initial study, ( ) was discretized to allow for the minimisation of the residual between FE and experiment in the IFEM Twelve discrete values of ( ) plus A and m (equation 2), were selected as variables and varied systematically using the approach presented in Chen et al (2015) to minimise the residual between an experimental depth-time reference curve and the corresponding FE simulated curve by using a gradient based nonlinear least square technique Linear interpolation was used to compute the value of ( ) between the discrete variables The greater the number of discrete variable used, the more accurately variations in ( ) can potentially be represented, however, the increased number variables also increases the computational cost of the IFEM Trials indicated that 14 discrete variables was a reasonable compromise value for this problem AC 4.2 Discrete variable modelling of CR IP T ACCEPTED MANUSCRIPT AN US Figure The used FE model and the mesh applied (a) Geometry and boundary conditions of the FE model (b) The applied mesh in the FE model √ ∑( ) ( CE PT ED M The results of the IFEM based on the representation of ( ) as described above are shown in figure From Figure 5a, it can be seen that ( ) is highest at small indentation depths and decreases rapidly with increasing indentation size initially before attaining an almost constant value at high indentation depths Figure 5b shows that yield stress increases with equivalent plastic strain, as in traditional strain hardening, and also with indentation depth Figures 5c and 5d compare the experimental reference curves with corresponding FE simulated curves using the optimal parameters from the IFEM using the discrete variable approach Visually, a good correspondence can be seen, however, in order to quantitatively to assess the fit between experiment and FE modelling, a relative residual (Rres) was computed according to equation in Chen at al (2015): ) , (4) AC where and are the indentation depths from experiment and FE simulation respectively at time The relative residual is included in Figure 5c, showing a reasonably constant value of approximately 0.41% This demonstrates that the material constitutive relation expressed by equations and can be used to capture the mechanical response of aluminium during indentation much more accurately than with a value of which is independent of load/depth, as seen by comparison with Figure 3d, where the relative residual is 2.67% 10 ED M AN US CR IP T ACCEPTED MANUSCRIPT AC CE PT Figure IFEM result with discrete first part of yield stress (see equation 2) as variables for the minimisation of residual between experiment and FE modelling (a) Shows the change of the first part of yield stress along with indentation depth and (b) illustrates the corresponding yield stress distribution along both equivalent plastic strain and the indentation depth, where d is the depth (c) and (d) Comparison between experiment and FE modelling in depth-time and in load-depth domains respectively 11 ED M AN US CR IP T ACCEPTED MANUSCRIPT CE PT Figure 6: IFEM result with ( ) represented by a combined hyperbolic and polynomial function (a) Variation of ( ) with indentation depth, (b) yield stress as a function of equivalent plastic strain and indentation depth, where d is the depth (c) and (d) show the comparison between test reference curve and FE modelled curve in both depth-time and load-depth domains AC 4.3 Continuous function modelling of ( ) Although the previous section showed that the discrete variable approach to representing ( ) was successful, the shape of the curve in Figure 5a indicated that representation by continuous segments of curves may enable the computation cost to be reduced whilst retaining at least as high a fitting accuracy After investigation of various curve types, a hyperbolic function was chosen to represent ( ) at small indentation loads ( ) and a polynomial function was used at high loads ( ) This can be written as ( ) 12 ( ), (5) ACCEPTED MANUSCRIPT where , , and by IFEM For the case of the variation of ( ) in equation are the search variables in the minimisation of the residual , a polynomial function with an order of three was selected to model ( ) ( ) (6) ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅ ; ̅ ̅ (̅ ( ̅ ) ) PT ED M (̅ ( ̅ ( (7) and P2 to P3 respectively From the AN US ̅̅̅̅̅̅) are the slope of the segments of to where ̅̅̅̅̅̅ conditions in equation 7, the constants can be determined as: CE CR IP T Three points were selected to construct the polynomial function, with coordinates as follows: ( ̅ ) and ( ̅ ); ( ̅ ), where and is the maximum indentation load The boundary conditions to determine the constants in equation are: ( ( ( ̅ ) ( ) (̅ ̅) ( ) ) (8) ) ) ) ̅ ) ̅ AC Using equations 5-8, the search vector for minimisation of the residual can be stated as ( ̅ ̅ ) (9) The results from the solution of equation by IFEM are shown in table II These parameters, including Young’s modulus and Poisson’s ratio, can be applied to simulate the indentation process using the coded subroutine 13 ACCEPTED MANUSCRIPT Table II Solution variables for minimisation of residual ( ) ( ) ( ( ( ) ) ) CR IP T ̅ ( ̅ ( ( ) ) ) M AN US Compared with the discrete method in section 4.1, the computation is more efficient as there are fewer search variables The results of the IFEM are presented in figure Comparison with figure shows that the distributions of ( ) and yield stress as functions of indentation load/depth are similar with different amplitudes due to the different fitting accuracy Figures 6c and 6d show an excellent fit between experiment and simulation, with an Rres value of 0.32% that is even lower than that obtained with the discrete variable representation of ( ) This demonstrates that the material constitutive relation, expressed by equations to 3, captures the basic mechanical response of the sample material to the indentation load and that equations to accurately represent the variation of ( ) with indentation load More generally, it can be seen that the ISE can be accurately modelled by simply substituting a load dependent yield stress into traditional elastoplasticity (next section) Figures and show that the yield stress is strongly indentation depth dependent, with ( ) at small indentation depth much higher than the bulk material’s initial yield stress, but the elastic constants (Young’s modulus of 70 GPa and Poisson’s ratio of 0.35) can be considered as independent of indentation depth ED 4.4 FE modelling of hardness AC CE PT Hardness is commonly calculated as the average indentation pressure, which is the force divided by the projected area of indentation Thus hardness modelling is equivalent to the modelling of contact area during an indentation (Spary et al, 2006) Many investigations have been performed to simulate indentation hardness by FE modelling (Faghihi and Voyiadjis, 2011; Bittencourt, 2013; Guha et al, 2013; Harsono et al, 2011; Salehi and Salehi, 2014; Celentano et al, 2012; Gomez and Basaran, 2006; Swaddiwudhipong, 2012) or by using analytical models (Nix and Gao 1998; Durst et al, 2005; Pugno, 2007) In order to accurately model the contact area in this work, a fine mesh around the indenter area, as shown in figure 4b, and the proposed indentation elastoplastic constitutive model with parameters determined by the IFEM were used Hardness was then determined from the simulated contact area This method is referred to as H-IFEM Hardness can also be computed from experimental load-depth curves using the method proposed by Oliver and Pharr (2004); this is referred to as H-OP In this work a new method to compute hardness from experimental load-depth curves is presented (see next section), this is referred to as H-new when applied to experimental data and H-FE when applied to FE simulated load-depth data Hardness computed by the different methods is shown in figure 7a Compared with the hardness evaluated from the experimental loaddepths curves, the simulated hardness was higher This can be shown to be caused by the errors involved in the traditional analytical evaluation method, as will be analysed in the following section 14 AN US CR IP T ACCEPTED MANUSCRIPT Figure Comparison of indentation hardness (a) and load-depth (b) between experiment and FE modelling for an indentation array with indentation load from to 400 mN using the material properties extracted from the test reference curve at 400 mN load, as shown by figure Proposed method for the evaluation of indentation hardness PT ED M In standard nanoindentation (or depth sensing indentation) hardness is evaluated from an experimental load-depth curve based on an assumption that sink-in is proportional to the indentation load and inversely proportional to the material stiffness (Oliver and Pharr, 2004) This may be one of the reasons that rarely are both hardness and load-depth response predicted well by FEA Traditionally, the contact area is evaluated first by calculating the contact depth as CE (10) AC where is the measurable maximum depth, is the maximum indentation load, is the slope of the load-depth curve at the start of unloading and is a constant (Oliver and Pharr, 2004) When is obtained from equation 10, the contact area and the projection of contact impression can be computed from the geometry of the indenter used In equation 10, the part is referred to as the sink-in displacement in the literature (Oliver and Pharr, 2004) It is the vertical sink-in displacement of the sample surface that is at the final stage of contact with the indenter when the indentation load arrives at its maximum The assumption that sink-in is linearly proportional with may be true for an indentation where the contact area is constant For most indentation tests, however, the contact area is an increasing function of indentation load, which may result in the sink-in to be nonlinear with respect to assumption inherent in equation 10 may result in errors in the evaluation of hardness 15 Hence, the AN US CR IP T ACCEPTED MANUSCRIPT Figure 8: Comparison of FE simulated and fitted nonlinear sink-in displacements ( ) , (11) PT ED M The FE simulated sink-in with a conical indenter with an aluminium sample is shown in figure with the continuous line Considering the distribution shown in figure 8, it is assumed that the sink-in, , is governed by a power law relationship with indentation load, as shown by equation 11: CE where and are constants determined by nonlinear regression analysis, is the reduced modulus of indentation, determined by the elastic parameters of both sample and indenter (Oliver and Pharr, 2004) AC (12) where and are Young’s modulus and Poisson’s ratio of sample material, and and are Young’s modulus and Poisson’s ration of indenter The constants of and are shown to be and with dimension of for the sample material, determined by nonlinear regression analysis The comparison of FE simulated sink-in and the fitted equation are shown in figure and it can be seen that a good fit is obtained with these constants Revisiting equation 10, with the proposed description of the sink-in (equation 13), the contact depth can be expressed as: ( 16 ) (13) ACCEPTED MANUSCRIPT Although equation 13 can be used to predict the contact depth, the reduced modulus is involved It is not a directly measurable parameter in an indentation test and therefore, equation 13 cannot be directly used to calculate the contact depth of A general relation between slope (see equation 10) and the reduced modulus exists (Oliver and Pharr, 2004): √ (14) is the projection of contact area of the indentation For a conical indenter, CR IP T where , , (15) AN US where is the radius of the projection of contact impression From the geometry of the conical indenter, (16) where is the half angle of the conical indenter On insertion of equations 14-16 into equation 13, the following expression is obtained: M ( ) ( It can be seen from equation 17 that sink-in, expressed as ( ) ( ) because the sink-in displacement is also dependent on ED proportional to ( ) (17) ) in this case is not itself Now all the CE PT parameters in equation 17 are directly measurable in an indentation test, except the contact depth ; i.e it is a nonlinear algebraic equation involving only one unknown, corresponding to the nonlinear value of the sink-in displacement shown by figure The contact depth of the indentation can, therefore, be determined from equation 17 using Newton’s iterative method AC It can be seen then that equation 17, can be used instead of equation 10 to determine hardness from an indentation test Different computational methods for hardness introduced in the previous section were used to calculate the hardness All these data are shown in figure 7a with different symbols and can be compared with each other As expected, it can be seen that the hardness obtained based on equation 10 (Oliver and Pharr, 2004) is lower than that using equation 17 This is because when sufficient nonlinearity of sink-in is considered, the sink-in is higher than the linear assumption based on equation 10 Thus smaller contact depth is obtained when the total maximum depth of is given, resulting in a higher evaluated hardness It is observable in figure 7a that the hardness evaluated by the new method from tested load-depth curves, shown by a continuous line in the figure, is in good agreement with the hardness from FE modelling, either by H-IFEM or by H-FE methods Sufficient non-linearity of the sink-in should also be considered In this case, the contact area based on equation 17 is almost the 17 ACCEPTED MANUSCRIPT same as that by direct FE modelling, leading to the agreement between H-IFEM and H-FE On the other hand, the agreement between experimental load-depth curves and the FE simulated loaddepth curves, as shown by figures 5c, 5d, 6c, 6d and 7b, it is natural that the hardness evaluated from these curves based on equation 17 should be in good agreement with experiment CR IP T It can be seen by comparing figure 7a and figure 6a (or figure 5a) and the analysis in this work that the so called indentation size effect is equivalent to the resistance of the material to deformation being indentation depth/load dependent, especially for the part of the yield stress that is independent of hardening Excellent agreement of both load-depth curves between FE and test, and hardness between test and FE modelling has been achieved, showing that the indentation elastoplasticity constitutive relation presented, as shown by equations and 3, captures the main mechanical behaviour of aluminium metal during nanoindentation AN US Conclusions M A method for determining material properties from indentation tests, whilst accounting for the indentation size effect has been proposed that utilizes IFEM and a novel indentation elastoplastic constitutive relation It is demonstrated that this method can be used to characterize the material properties of aluminium from nano to micron length scales By using the determined material properties, load- or time-depth curves and hardness variations (size-effect) can be accurately predicted, in excellent agreement with experimental results AC CE PT ED The average relative residual of time-depth curves between the experimental and FE modelling was approximately 0.32%, showing that the material constitutive relation used captures the main mechanical behaviour of the material The yield stress changes significantly from the micro-length scale to bulk material, and this change can be captured by means of a load (or depth) dependent ( ) , while the hardening components does not initial yield stress in the elastoplasticity model, change greatly with length scale It is further shown that the 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Size effects in nanoindentation: an experimental and analytical study Acta Mech 211, 131–153 Voyiadjis G Z and Malekmotiei L., 2016 Variation of the Strain Rate during CSM Nanoindentation of Glassy Polymers and Its Implication on Indentation Size Effect J Polymer Sci Part B: Polymer Phys DOI: 10.1002/polb.24127 AC CE PT ED M Xu Z H and Li X D., 2006 Sample size effect on nanoindentation of micro-/nanostructures Acta Materialia 54 1699–1703 21 ...ACCEPTED MANUSCRIPT A combined inverse finite element – elastoplastic modelling method to simulate the size- effect in nanoindentation and characterise materials from the nano to micro- scale. .. and model parameters to accurately simulate the ISE As nanoindentation is a pseudo-static process, with low strain rates, and the test materials mainly show strain-hardening behaviour, an elastoplastic. .. hybrid approach to determining cornea mechanical properties in vivo using a combination of nano- indentation and inverse finite element analysis, journal of the mechanical behavior of biomedical materials