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[Kenta Goto](https://orcid.org/0000-0002-0102-0658), [Ikumu Watanabe](https://orcid.org/0000-0002-7693-1675), [Takahito Ohmura](https://orcid.org/0000-0001-7528-566X)

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[Inverse estimation approach for elastoplastic properties using the load-displacement curve and pile-up topography of a single Berkovich indentation](https://mdr.nims.go.jp/datasets/273ca8ed-b919-402e-9405-3adbcb7abd0f)

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Inverse estimation approach for elastoplastic properties using the load-displacement curve and pile-up topography of a single Berkovich indentationMaterials and Design 194 (2020) 108925Contents lists available at ScienceDirectMaterials and Designj ourna l homepage: www.e lsev ie r .com/ locate /matdesInverse estimation approach for elastoplastic properties usingthe load-displacement curve and pile-up topography of a singleBerkovich indentationKenta Goto a,⁎, Ikumu Watanabe b, Takahito Ohmura ba International Center for Young Scientists, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba 305-0047, Japanb Research Center for Structural Materials, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba 305-0047, JapanH I G H L I G H T S G R A P H I C A L A B S T R A C T• Elastoplastic properties were derivedfrom the mechanical response of an in-dentation test• An algorithm based on equivalent en-ergy model and limit-analysis was pro-posed• The coefficients for the algorithm weredetermined using finite element analy-sis• The inverse estimation of elastoplasticproperties based on the proposedmodel was highly correlated with thetensile tests⁎ Corresponding author.E-mail address: GOTO.Kenta@nims.go.jp (K. Goto).https://doi.org/10.1016/j.matdes.2020.1089250264-1275/© 2020 The Authors. Published by Elsevier Ltda b s t r a c ta r t i c l e i n f oArticle history:Received 12 February 2020Received in revised form 22 June 2020Accepted 24 June 2020Available online 29 June 2020Keywords:Elastic-plastic materialFinite elementsMechanical testingIndentationInverse analysisAn approach for the inverse estimation of the elastoplastic properties from a single indentation with a Berkovichindenter was developed. The relationship between the load-displacement and stress-strain curves was derivedbased on the equivalent energy principle, while an approximate equation for pile-up height was determinedusing elastic and plastic limits. The approach proposed in this study estimates the yield stress and strain-hardening exponent from hardness and pile-up height obtained from a single indentation based on these funda-mental equations. The coefficients in the equations were determined in a parametric study using finite elementanalyses. The accuracy of the inverse estimation technique was confirmed using aluminum alloy and stainlesssteel samples and reference tensile testing.© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).1. IntroductionIndentation testing is widely implemented in the field of engi-neering due to its simplicity. It is used to evaluate the mechanical. This is an open access article underproperties of a material, particularly hardness. An indentation test si-multaneously records the load (P) and displacement (h) (Fig. 1a),allowing the calculation of elastic modulus and hardness [1], residualstress [2,3], creep [4,5], phase transition [6], and the onset of plasticdeformation [7].In a bulk material, the reduced elastic modulus (Er) and indentationhardness (Hit) are calculated from the load-displacement curve accord-ing to the Oliver-Pharr (OP) method [8] as follows:the CC BY license (http://creativecommons.org/licenses/by/4.0/).http://crossmark.crossref.org/dialog/?doi=10.1016/j.matdes.2020.108925&domain=pdfhttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/https://doi.org/10.1016/j.matdes.2020.108925mailto:GOTO.Kenta@nims.go.jphttps://doi.org/10.1016/j.matdes.2020.108925http://creativecommons.org/licenses/by/4.0/http://www.sciencedirect.com/science/journal/www.elsevier.com/locate/matdesPmaxLoad, PDisplacement, hhmaxP = Ch2Wp WehchfSa)θcaElastoplasticElasticCorehpileVtb)Strain, ε ,sse rtSσ σyεyσeqεequeqc)Fig. 1. Schematic of the (a) load-displacement curve, (b) stress-strain curve and(c) deformation under a conical indenter based on Johnson's cavity model.2 K. Goto et al. / Materials and Design 194 (2020) 108925Er ¼ Sffiffiffiπp2ffiffiffiffiffiffiffiffiffiffiA hcð Þq , ð1ÞHit ¼ PmaxA hcð Þ, ð2Þ1Er¼ 1−ν2iEiþ 1−ν2sEs¼ 1−ν2iEiþ 1M, ð3Þwhere S is contact stiffness, hc is contact depth, Pmax is the maximumload and A(hc) is contact area at h = hc. E and ν are Young's modulusand Poisson's ratio, respectively, where subscripts i and s representthe indenter and sample, respectively. The authors define indentationmodulus (M) as Es/(1-νs2).The stress-strain curve obtained from a uniaxial tensile test (Fig. 1b)provides the essential information regarding mechanical properties re-quired in the design and evaluation of structuralmaterials. The simplestapproximation of the stress-strain curve uses Young's modulus (E), astrain hardening coefficient (K), and a strain hardening exponent (n),as given in Eqs. (4) and (5):σ ¼ Eε at σ ≤ σy, ð4Þσ ¼ Kεn at σ ≥ σy, ð5Þwhere σ and ε represent stress and strain, respectively, and σy is yieldstress. Eqs. (4) and (5) are continuous at the yield point (σ = σy), andthus K = Enσy1-n. Therefore, Eq. (5) is replaced by Eq. (6), as follows:σ ¼ Enσ1−ny εn: ð6ÞMany techniques have been proposed to inversely estimate thestress-strain relationship from the results of an indentation test, andthese techniques use either spherical or pyramidal indenters. Inverseestimation techniques using spherical indenters [9–12] generate a con-tinuous stress-strain curve from a single indentation as the strain fieldunder the indenter changes with increasing indentation depth. How-ever, the indenter tip radius must be adjusted according to the areabeing evaluated. Pyramidal indenters such as conical, Berkovich, andVickers indenters similarly have a geometry independent on depth.Most of the techniques using pyramidal indenters determine the elasticmodulus using Eqs. (1) and (3). The dual indenter technique [13–22] isa common inverse estimation technique applied to pyramidal in-denters. It estimates the yield stress and strain hardening exponentfrom two load-displacement curves, each obtained using indenterswith different tip angles, as the yield stress and strain hardening expo-nent of an unknown sample cannot be determined from a single load-displacement curve [23–25]. Numerous studies [15,17–19,26] haveused dimensionless polynomial functions based on Π theory, wherethe coefficients for the functions were determined using finite elementanalysis. A technique proposed by Ogasawara et al. [16] was based onlimit analysis that considered the elastic and plastic limits. Chen andCai [13] proposed a relationship between the load-displacement curveand elastoplastic properties based on an equivalent energy principleand Johnson's cavity model [27]. Furthermore, the effectiveness ofthese dual indenter techniques has been compared by Guelorget et al.[28] and Kang et al. [29].The stress-strain relationship can be estimated from a single inden-tation result by analyzing not only a load-displacement curve but alsothe three-dimensional pile-up topography after indentation using con-focal microscopy or atomic force microscopy (AFM). Goto et al. [30]demonstrated that the residual pile-up height is related to the yieldstress and strain hardening exponent as well as the load-displacementcurve in finite element analysis, where the residual pile-up height wasstrongly dependent on the strain hardening exponent. The sensitivityInput Er, Hit & HpCalculate M from Eq. (3)Assign trial εyOutput M, σy & nCalculate n from Eq. (22) Calculate C(Calc.) from Eqs. (21)R < Threshold?R = |C(Calc.)-C(Exp.)|Update εyσy = MεyYesNoCalculate C(Exp.) from Eq. (25)3K. Goto et al. / Materials and Design 194 (2020) 108925of the pile-up topography was further investigated by Meng et al. [31].The load-displacement curve and pile-up topography from a single in-dentation test were used to estimate stress-strain curves, as shown inGoto et al. [30],Meng et al. [32], and Bolzon et al. [33]. These studies per-formed a large number of indentation simulation iterations to obtain asimulation that accurately reflects the experimental result. However,these iterations of finite element analysis are time-consuming.This study aimed to establish a relationship between elastoplasticproperties and indentation results for an inverse estimation based onthe load-displacement curve and pile-up topography of a single inden-tation. A Berkovich indenter was chosen for its frequent use in previousstudies involving instrumented indentation. The study focused onmodeling and formulation, and determined the modeling coefficientsusing a finite element analysis parametric study. The proposed modelwas validated by conducting further inverse estimations.2. Inverse estimation model2.1. Load-displacement curveThe relationship between the load-displacement curve andelastoplastic properties was established using the equivalent energymodel [13]. An indentation causes elastoplastic deformation of the sam-plematerial beneath the indenter. The total strain energy (Ut) in the de-formed region (Vt) is obtained by integrating strain energy density (u)at each point (Fig. 1c),Ut ¼ZVtudV : ð7ÞThe equivalent strain energy (ueq) was defined as a volumetricallyaveraged value according to Eq. (8) as follows:ueq ¼RVtudVVt: ð8Þueq was decomposed at the yield point into two components, namelythe elastic (ueq,e) and elastoplastic (ueq,ep) components (Fig. 1b). Accord-ing to the power law in the elastoplastic region (Eq. (5)), ueq is repre-sented in terms of E, K, and n, as follows:ueq,e ¼Z εy0σdε ¼ Eε2y2¼ Kε1þny2, ð9Þueq,ep ¼Z εeqεyσdε ¼ K1þ nε1þneq −ε1þny� �, ð10Þ∴ueq ¼ ueq,e þ ueq,ep ¼ K1þ nε1þneq −1−n2ε1þny� �≈Kε1þneq1þ n, ð11Þwhere εy and εeq are the yield strain (= σy/E) and strain at u = ueq, re-spectively. It was assumed that εy is substantially smaller than εeq:1−n2εyεeq� �1þn≪ 1: ð12ÞAtkins and Tabor [34] proposed representative plastic strain (εr),which was the strain attributed to plastic deformation induced byindentation. The representative plastic strain has been determinedin previous studies [14,35], most of which use Eq. (13) to approxi-mate the relationship between εr and indenter tip angle (θ), as fol-lows:εr ¼ Be1 cot θ, ð13Þwhere Be1 is constant. It may be assumed that εeq has a similar relation-ship:εeq ¼ ke1 cot θ, ð14Þwhere ke1 is also a constant. The equivalent energy model assumes thatthe form of the indenter is conical, but the contact area of a conical in-denter is equivalent to a Berkovich indenter when θ = 70.3°.Deformed volume (Vt) may be estimated using Johnson's cavitymodel [13], which assumes that the deformed region is hemisphericaland consists of a core, elastoplastic, and elastic regions (Fig. 1c).Vt ¼ 23πc3 ¼ 23π f e1 εy ,n,θð Þa� �3¼ 23π f e1 εy ,n,θð Þhmax tan θ� �3, ð15Þwhere a and c are the radii of the core and elastic regions, respectively.fe1 = c/a is a function of the elastoplastic properties and the tip angle,but the elastic modulus has a minor effect on fe1 (see Section 3). Thus,the total strain energy Ut was derived from Eqs. (8), (11), (14), and(15) to give Eq. (16), as follows:Ut ¼ ueqVt ¼ 23πK1þ nke1 cot θð Þ1þn f e1 εy ,n,θð Þhmax tan θ� �3: ð16ÞThe total strain energy is equal to the total indentation work (Wt),where Wt = We + Wp, as follows:Ut ¼ Wt : ð17ÞWt is calculated by integrating the indentation load (P) with the dis-placement (h) during loading:Wt ¼Z hmax0Pdh ¼ 13Ch3max: ð18ÞHere, the load-displacement curve was approximated by a paraboliccurve:Fig. 2. Algorithm for the inverse estimation of Er, Hit, and Hp.x yza)IndenterSamplexzc)24.348.6xyb)24.3Fig. 3. Analytical model of a Berkovich indenter shown from the (a) perspective, (b) top and (c) side views.4 K. Goto et al. / Materials and Design 194 (2020) 108925P ¼ Ch2: ð19ÞThe curvature of the loading curve (C) was correlated to theelastoplastic parametersM, σy (=M × εy) and n using Eqs. (16)–(18):23πK1þ nke1 cot θð Þ1þn f e1 εy ,n,θð Þhmax tan θ� �3¼ 13Ch3max, ð20Þ∴C ¼2πk1þne1 f e1 εy ,n,θð Þ� �3Mε1−ny tan 2−nθ1þ n: ð21ÞYoung's modulus (E) was replaced by the indentation modulus (M)because the model was under plain-strain condition.2.2. Pile-up topographyPile-up occurs due to isovolumetric plastic flow parallel to theindenter surface when the strain is concentrated directly belowthe indenter. Therefore, the maximum pile-up height occurs whenεy = n = 0. Under elastic deformation, i.e., εy = ∞ or n = 1, theresidual pile-up height (hpile) must be 0 because the indentedsurface will return to its original form (i.e. no indentation)upon removal of the indenter. Goto et al. [30] demonstratedthat the residual pile-up height decreases with an increase in εyand n.The normalized residual pile-up height (Hp) is given in Eq. (22), asfollows:Hp ¼ hpile=hmax ¼ kp1 exp −kp2εy−kp3n� �, ð22Þwhere kp1, kp2, and kp3 are constants.Table 1The input elastoplastic properties for the finite element analyses in the parametric study(120 cases).Property ValueM [GPa] 40, 120, 200, 280εy 5 × 10−4, 2.5 × 10−3, 5 × 10−3, 7.5 × 10−3, 1 × 10−2n 0.0, 0.1, 0.2, 0.3, 0.4, 0.52.3. Extension to hardnessThe OP method evaluates the indentation hardness (Hit) in additionto the reduced elasticmodulus, as given in Eq. (2). Commercial softwarepackages calculate not C but the hardness value (Hit) from the load-displacement results. Therefore, Hit is preferable to C for the calculationof the elastoplastic properties.The OP method calculates contact depth according to Eq. (23):hc ¼ hmax−ePmaxS, ð23Þwhere e depends on indenter geometry and e= 0.75 for pyramidal in-denters. The ideal Berkovich indenter has a relationship described byEq. (24):A hð Þ ¼ C0h2, ð24Þwhere C0=24.56. Eqs. (2), (19), (23), and (24) is combined to calculateC from Er and Hit according to Eq. (25), as follows:C ¼ HitC0hchmax� �2¼ HitC02Er2Er þ εHitffiffiffiffiffiffiffiffiffiC0πp !2: ð25ÞAlthough the OP method is widely accepted for the calculation ofelastic modulus and hardness, it causes the error of contact area whenpile-up occurs [36,37] (See 3.4. Discussion). The error is remarkablefor materials with small strain hardening exponent. If different algo-rithm is used for the calculation of hardness, Eq. (25) should be modi-fied according to the used algorithm.2.4. Inverse estimation algorithmThe elastoplastic properties, including indentation modulus (M),yield stress (σy), and strain hardening exponent (n), are estimatedfrom the indentation results, specifically the reduced elastic modulus(Er), indentation hardness (Hit), and normalized pile-up height (Hp).The inverse estimation algorithm is based on the Er,Hit andHp results(Fig. 2).M is calculated from Er, and the assigned trial yield strain (εy) isused to calculate n fromHp. The εy value is optimized according to resid-ual error between the calculated and experimental C values. Iterationsare discontinued once a sufficiently low error is achieved, and σy andb)-0.2-0.4-0.6-0.8-1.20.1 0.2 0.3 0.4 0.5n [-][-] 0.20εy [-]5.0 10-42.5 10-35.0 10-37.5 10-31.0 10-20-1.04003002001000 100 200 300 400C (Eq. 28) [-]C (FEA) [-]n [-]0.00.10.20.30.40.5c)1.00.80.60.40.20 0.002 0.004 0.006 0.008 0.01εy [-]ln( ke1f e13) [-]M [GPa]40120200280a)Fig. 4.Determination of the coefficients for C, specifically (a) the relationship between ke1fe13 with εy at n=0, whichwas independent of elasticmodulus; (b) the relationship between theleft term in Eq. (29) and n; and (c) the high correlation between the C values calculated using Eq. (28) and finite element analyses (FEA).Table 2The coefficients determined using finite element analyses.Coefficient ke1 ke2 ke3 kp1 kp2 kp3Value 0.1702 42.94 0.8881 0.4162 130.4 7.4695K. Goto et al. / Materials and Design 194 (2020) 108925n are determined according to the trial εy. The C valuemay alternativelybe used instead of Hit for this estimation.3. Determination of coefficients using finite element analysis3.1. Analytical modelA finite element model of a Berkovich indentation is illustrated inFig. 3, generated using a commercial finite element analysis package(ABAQUS 6.14). The dimensions of the sample sections were48.6 × 24.3 × 24.3, and the smallest element sizewas 0.3. The simulatedsample was meshed with an 8-node hexahedral element (C3D8), andthe Berkovich indenter was assumed to be rigid and was meshed with3- and 4-node rigid elements. The model included a total of 15,805nodes and 14,543 elements. Bucaille et al. [14] found that the friction be-tween the indenter and sample had a minor effect at θ N 60°, but an in-crease in the friction coefficient caused an increase in the indentationload and a decrease in the pile-up height. A friction coefficient of 0.2was selected for the present study as well as Chen and Cai [13]. The in-denter was displaced by 1 in the loading step and returned to its initialposition in the subsequent unloading step. The indentation load andpile-up height were multiplied by a factor of 1.2 and 1.3, respectively,whichwas determined by applying forward analysis to an experimentaltensile test.A wide variety of input elastoplastic properties were included in thesample set to consider a diverse range of metals [38] (Table 1).3.2. Coefficients of the load-displacement curveA strain hardening exponent of n=0allows the adaption of Eq. (21)to Eq. (26), as follows:ke1 f e1 εy ,0,θð Þ� �3¼ C2πMεy tan 2θ: ð26ÞThe tip angle of the conical indenter (θ) is 70.3° to provide the samecontact area as a Berkovich indenter. The ln{ke1(fe1(εy,0,70.3°))3} term isdependent on εy at n = 0 (Fig. 4a), indicative of the linear relationshipwith εy, which was independent of the reduced elastic modulus (M).This was approximated by Eq. (27):ln ke1 f e1 εy ,0,70:3�ð Þ� �3  ¼ −ke2εy þ ke3, ð27Þwhere ke2 and ke3 are constants.Assuming fe1 is independent of n, Eq. (28) is obtained by substitutingEq. (27) into Eq. (21), as follows:C ¼ 2πkne1 exp −ke2εy þ ke3� �Mε1−ny tan 2−n70:3�1þ n: ð28Þ0.30.20.10 0.002 0.004 0.006 0.008 0.01εy [-]Hp[ -]M [GPa]40120200280a)0.40.30.20.10 0.1 0.2 0.3 0.4 0.5n [-]Hp[ -]M [GPa]40120200280b)0.40.40.20.10 0.1 0.2 0.4Hp (Eq. 22) [-]Hp(FEA) [ -]n [-]0.00.10.20.30.40.5c)0.30.3Fig. 5. The relationship betweenHp and (a) εy at afixedn value of 0 and (b) n at afixed εy value of 5 × 10−4, and (c) the high correlation between theHp values calculated using Eq. (22) andfinite element analysis (FEA).8004002000 5 10 1560025Strain ε [%] ssertSσ]aPM[EstimationTensile testEstimation with Er*20εcA5052SUS304Fig. 6. The true stress-strain curves of the aluminum alloy (A5052) and stainless steel(SUS304) estimated from the indentation results (Er, Hit, and Hp) and measured in thetensile tests. The estimated curves of stainless steel are given until ε reached 20%, as theupper limit of strain (εc) for an estimated curve is 20% at θ = 70.3°. The estimationresults for a reduced Er (Er⁎ = Er/1.3) are given in gray.6 K. Goto et al. / Materials and Design 194 (2020) 108925Eq. (28) can be rearranged to give Eq. (29):lnC 1þ nð Þ2πexp −ke2εy þ ke3� �Mε1−ny tan 2−n70:3� ¼ nlnke1: ð29ÞEq. (29) was validated (Fig. 4b) and ke1 was determined using theleast square method. The values of ke1, ke2 and ke3 are given in Table 2.The C values calculated using Eq. (28)were plotted against those ob-tained from the finite element analyses (Fig. 4c) to demonstrate thatEq. (28) provided an accurate estimation. The maximum error betweenthe C values calculated using Eq. (28) and the finite element analyseswas 36 GPa.3.3. Coefficients of pile-up topographyThe Hp value exponentially decreased with increasing εy at a fixed nvalue of 0 (Fig. 5a) and increasing n at a fixed εy value of 5 × 10−4(Fig. 5b), but was independent of M, as expected from the findings inSection 2. The coefficients in Eq. (22) were determined using the leastsquare method (Table 2).The Hp values calculated using Eq. (22) were plotted against thoseobtained from the finite element analyses (Fig. 5c) to reveal thatEq. (22) expressed the dependence of Hp across a wide range ofelastoplastic properties. Themaximum error between theHp values cal-culated using Eq. (22) and the finite element analyses was 0.02.1 Large difference is seen in the stress-strain curves estimated by various dual indentermethods as shown in Guelorget et al. [28].7K. Goto et al. / Materials and Design 194 (2020) 1089253.4. DiscussionThe Be1 coefficient for the representative plastic strain (εr) in Eq. (13)depends on the definition of εr, andwas found to be 0.0319 and 0.105 byOgasawara et al. [35] and Bucaille et al. [14], respectively. However, εeqincludes elastic strain in addition to plastic strain, and thus the coeffi-cient ke1 in Eq. (14) is larger than Be1. The ke1 value in this studywas sim-ilar to the findings of Chen et al. [13].The ratio of c to a (fe1) determines the volume affecting the load-displacement curve. fe1 was calculated from Eq. (27) to give Eq. (30),as follows:f e1 ¼ 1ke1exp −ke2εy þ ke3� �  13, ð30Þwhere fe1 has a maximum value of 2.409 at εy = 5 × 10−4 and a min-imum of 2.103 at εy = 1 × 10−2. The elastoplastic boundaries of theregion deformed by indentation were dependent on the materialand ranged between 3a and 6a [39,40]. These values were larger thanfe1, indicating that the deformed region of small plastic strain did notaffect the indentation results. Sudharshan Phani and Oliver [41] inves-tigated the effect of indentation spacing on Er and Hit using aBerkovich indenter and found that the elastoplastic regions of neigh-boring indentations were not required to obtain complete separationfor reliable results. A similar observation was made in the currentstudy.Pile-up occurs during indentation and increases the contact area, butwas not accounted for in the OPmethod. A parametric study usingfiniteelement analysis by Cheng and Cheng [36] and Bolshakov and Pharr [37]demonstrated that the OP method underestimates the contact area byas much as 60%. Cheng and Cheng [42] used indentation work valuesfrom finite element analysis to determine a non-dimensional functionrelated to Er/Hit. By combining this function with Eqs. (1) and (2), amodified contact area was calculated using the contact stiffness, maxi-mum load and indentation work values. Saha et al. [43] and Kese et al.[44,45] proposed geometrical modification approaches that assumedthe pile-up ridges were arc [43] and semi-ellipse [44] [45] shaped,where the ridge shape was calculated from pile-up height afterunloading. This can contribute to errors, as the pile-up behavior differsbetween loading and unloading. The modification of the OP method isstill controversial despite the proposal of numerous approaches, andthe present study chose to calculate Er using the unmodifiedOPmethod.The estimation model assumes adiabatic heating and strain-ratesensitivity are negligibly small. A slight increase of hardnesswith a load-ing rate was reported in previous studies [46,47]. The strain-rate sensi-tivity remarkably appears at ultra-high strain rate [48], causingoverestimation of the strain energy in the equivalent energy model.Therefore, the estimation should be performed using an indentation re-sult at small strain rate, and onemust pay attention to an estimation re-sult when a material has high strain-rate sensitivity.4. Inverse estimation of engineering materialsStress-strain curves estimated from indentation results were vali-dated using tensile testing on an aluminum‑magnesium alloy(A5052) and austenitic stainless steel (SUS304) as sample materials.The indentation samples were mechanically polished andelectropolished to minimize surface roughness and residual strain. In-dentations were performed at Pmax = 1.5 N and dP/dt = 100 mN/susing a Bruker TI-900 nanoindenter, followed by the measurementof pile-up topography using a Keyence VK-9700 laser microscope.The tensile tests were performed using a Shimadzu AG-X universaltesting instrument with constant crosshead speed of 0.6 mm/s. Theelastoplastic properties were calculated as mean values from indenta-tion tests performed in triplicate. Er and Hit values were calculatedfrom the load-displacement curves using the OP method. The hpilevalue was the mean maximum height, calculated from the cross-section through a vertex and center of the side of an indentationmark, and used to calculate Hp.True stress-strain curves in Fig. 6 were based on the proposedmethod estimates and the tensile test measurements. The aluminumalloy fractured at ε = 6.5%, thus these curves were only plotteduntil the fracture strain. Liu et al. [49] found that the upper limit ofstrain (εc) for the estimated curves was 20% at θ = 70.3°. Therefore,the stainless steel was only estimated until ε reached 20%. The estima-tion results were acceptably1 accurate with reference to tensile tests,although a small error was observed. These estimation errors were at-tributed to the stainless steel exhibiting bi-linear behavior and not thepowered law upon which the model was based.As mentioned in 3.4. Discussion, the OP method underestimatescontact area by 60% when pile-up occurs, corresponding to overesti-mation of Er by 30% [37]. In the present work, the estimated stress-strain curves possibly include the error due to the overestimation ofEr. To evaluate the error, the estimation was performed using 1.3times reduced Er (Er⁎ = Er/1.3). The estimated stress-strain curvesfor Er⁎ were comparable to that for Er though σy increased and n de-creased, as shown in Fig. 6. A 60% underestimation of contact areais the worst case scenario, and the error will be negligible inmost materials, particularly in those that exhibited large strainhardening.5. ConclusionsAn inverse estimation algorithm was proposed to estimate a stress-strain curve from a single indentation test. The relationship between theelastoplastic properties and indentation results was evaluated using theequivalent energy method and exponential approximation. A paramet-ric study of indentation was performed using finite element analysis todetermine the appropriate coefficients for the proposed equation. Theproposed equation provided an accurate reproduction of the simulationresults and was validated using an aluminum alloy and stainless steel.The stress-strain curves estimated from the indentation tests were cor-relatedwith the tensile tests. This technique allows for the estimation ofelastoplastic properties at various indentation size without exchangingthe indenter and is a promising alternative for the rapid evaluation oflocal mechanical properties.Data availabilityThe data required to reproduce these findings are available from thecorresponding author upon request.Declaration of Competing InterestThe authors declare that they have no known competing financialinterests or personal relationships that could have appeared to influ-ence the work reported in this paper.AcknowledgmentsThis work was financially supported by the Amada Foundation forMetal Work Technology in Japan (No. AF-2018035-C2). The authorswould like to thank Dr. Susumu Takamori, Ms. Eri Nakagawa, and Ms.Yuki Yamamoto for assistance with tensile tests, indentation, and meshverification, respectively.8 K. Goto et al. / Materials and Design 194 (2020) 108925References[1] W. Oliver, G. 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Inverse estimation approach for elastoplastic properties using the load-�displacement curve and pile-�up topography of a si... 1. Introduction 2. Inverse estimation model 2.1. Load-displacement curve 2.2. Pile-up topography 2.3. Extension to hardness 2.4. Inverse estimation algorithm 3. Determination of coefficients using finite element analysis 3.1. Analytical model 3.2. Coefficients of the load-displacement curve 3.3. Coefficients of pile-up topography 3.4. Discussion 4. Inverse estimation of engineering materials 5. Conclusions Data availability Declaration of Competing Interest Acknowledgments References