International Journal of Mechanical Sciences 211 (2021) 106773

Available online 28 August 20210020-7403/© 2021 Elsevier Ltd. All rights reserved.

Evaluation of equi-biaxial residual stress from spherical indentation imprint

Chanyoung Jeong , Yewon Hwang , Naksoo Kim , Cheolsoo Lee , Hyungyil Lee *

Department of Mechanical Engineering, Sogang University, Seoul 04107, Republic of Korea

A R T I C L E I N F O

Keywords: Residual stresses Residual imprint Spherical indentation Finite element analysis (FEA) Inverse analysis Digital image correlation (DIC)

A B S T R A C T

A novel approach is proposed for evaluating equi-biaxial residual stresses (RS) from residual imprint displace-ment fields after spherical indentation. A unique region, where displacements are free from frictional effect, is first identified based on finite element analysis. Regression functions are proposed to relate imprint data with RS for a broad range of material properties. Digital image correlation (DIC) is employed for measuring the radial displacement fields. Experimental comparison of RS values from the present method and strain gauge showed the (maximum, mean) errors of only (10, 5) %. As the determined RS are accurate and reliable, the methodology can find more industrial applications compared with prior techniques.

1. Introduction

Residual stresses (RS) are induced deliberately (e.g., shot peening) or unavoidably (e.g., welding), and it is well known that RS affect me-chanical performance [1,2], such as fatigue life [3,4], corrosion [5], wear resistance [6,7], remarkably in a positive or negative way. For instance, in semiconductor devices, RS can lead to an abrupt decline in their service life, but under the pertinent internal RS, the speed of the device can markedly increase [8,9]. Therefore, a proper measurement of RS is essential to acquire a reliable engineering design, manufacturing of components and devices.

To measure RS, destructive and nondestructive methods, such as hole drilling and X-ray diffraction, are widely used [3,6,10–13]. How-ever, the destructive methods limits the wide application in industry despite their advantage of not requiring any reference sample to mea-sure RS quantitatively; and nondestructive methods have the problems associated with ease of measurement and flexibility for component scaling, thus require preparation of stress-free reference sample for comparison [9,14]. Consequently, micro / nano-indentation techniques were developed for its simplicity of use, nondestructive nature and suitability at various scales [15,16], and RS are analyzed via load-penetration depth (P–h) curve, imprint, or both with indentation simulation [13,17–21].

General observations reveal that tensile stress lowers indentation load-displacement (P–h) curve while compressive stress raises the curve [14], which allows the measurement of RS using P–h curves [22–28]. Hosseinzadeh and Mahmoudi [22] selected several parameters from loading part of P–h curve and derived a correlation function of

parameters, including material properties and RS. RS were then deter-mined with suggested equations. Moharrami and Sanayei [29] compared the accuracies of RS estimating methods using P–h curves from several studies [19,30–32], and developed the most accurate method, among them, by modifying the algorithm. However, for its inherent experimental errors [33,34], RS measurement based on P–h curve has trouble in giving fully reliable RS values. For RS calculation using P–h curves, indentation data of stress-free sample is also needed as Refs. [29,32], which sometimes causes a huge inconvenience. Rickhey et al. [35] established a relationship between stress-free P–h curve and material properties, compensating the defect of requirement of refer-ence sample when evaluating RS, but demands too many calculation for the whole evaluation. Therefore, a new method with (i) less influence from the machine and (ii) no need for reference data from stress-free sample is required.

Considering the weakness of RS evaluation using P–h curves, many inverse computation methods were proposed to estimate RS by using imprint shape, such as pile-up / sink-in or contact area [36–39]. Indentation forms residual imprint on the surface of metallic materials after unloading. It can remove experimental error from P–h curve, and using imprint needs no entire loading history [40,41]. Carlsson and Larsson [38] derived a single formula of RS by the ratio between actual and titular contact area to estimate RS. Lee et al. [37] selected normalized parameters, and with analytical regressions of data ob-tained, suggested a method for RS determination. Using imprint for estimating RS is an accurate and efficient method; however, the methods [38,42–45] included additional calculation of contact area after the analysis of imprint. Thus, sharp indenters were used for the easier calculation of contact area. However, Swadener et al. [46] stated that

* Corresponding author. E-mail address: hylee@sogang.ac.kr (H. Lee).

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International Journal of Mechanical Sciences

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https://doi.org/10.1016/j.ijmecsci.2021.106773 Received 15 April 2021; Received in revised form 16 August 2021; Accepted 25 August 2021

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the RS is more clearly evaluated using spherical indenters than sharp indenters.

Considering the simplicity of RS evaluating process, some studies directly used parameters of residual imprint, such as vertical displace-ment (e.g., pile-up / sink-in) [9,15,20] and radial in-plane displacement [47]. Xiao et al. [15] compared four models for calculating RS [14,30, 32,38,39] based on the sharp indentation. The study analyzed the fac-tors influencing the amount of pile-up / sink-in, which depends on material properties, regarding that the change of the amount of pile-up around the indentation is the key to understand the nonlinear effect of RS on indentation response. However, unlike the fixed friction coeffi-cient they used, it can vary depending on the state of material surface and indenter; and the frictional effect on the pile-up height becomes significant at larger penetration depths [48]. Thus, a certain region of pile-up / sink-in data that shows no frictional effect should be defined

before using it to evaluate RS. On the other hand, Chen et al. [47] suggested RS measurement via digital image correlation (DIC), for analyzing radial displacement, and sharp indentation test. The study showed acceptable prediction; however, used sharp indenter, which is less sensitive to RS. They showed only fitted evaluation of RS and determinant factor defined for a single material. Even though there were various attempts to evaluate RS using different parameters of residual imprint, the approaches have various drawbacks, and as a solution, novel perspective of using radial displacement for RS determination is required.

In this work, we propose a novel method for evaluating equi-biaxial RS from radial displacement of residual imprint with high accuracy and reliability. Finite element simulations of a single spherical indentation for analyzing a relationship between RS, material properties and resid-ual imprint are performed. A unique region of radial displacement free from frictional influence is defined, and with numerical regressions of the data obtained, mapping functions are generated for evaluating program of RS. The suggested method is validated first numerically; then experimentally by comparing RS from imprints with those from the strain gauge on the cruciform specimens under bending. The experiment residual imprints are measured with DIC instruments, and reference data from the specimen without RS is not needed.

Nomenclature

D diameter of indenter, mm E elastic modulus, GPa h indentation depth, mm n strain hardening exponent P indentation load, N Pmax maximum indentation load, N r max radius at maximum radial displacement, mm r rep, u rep representative radius and displacements, mm ur radial displacement, mm ur | max maximum radial displacement, mm

uz vertical displacement, mm εe, εp elastic and plastic strains εt true strain ( = εe + εp) εο yield strain μ friction coefficient v poisson’s ratio σ true stress, MPa σο yield strength, MPa σ eval evaluated stress from spherical indentation, MPa σr equi-biaxial residual stress, MPa σR1, σR2 stress in direction 1 and 2, MPa σref reference stress, MPa

Table 1 Material properties and test conditions of FE analysis.

material property symbol value

Young’s modulus [GPa] E 200 yield strength [MPa] σ o 200, 250, 400, 600, 800 (5 cases)

strain hardening exponent n 5, 10, 15 (3 cases) friction coefficient μ 0, 0.1, 0.2, 0.3 maximum load [N] P max 300

equi-biaxial residual stress σ r / σ o − 1, − 0.5, 0, 0.5, 1 (5 cases)

Fig. 1. Axisymmetric FE model for spherical indentation with CAX4 elements. Bottom surface is fixed in z-direction and the contact region beneath the indenter consists of elements with minimum size of 2.5 μm for accurate measurement of material deformation.

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2. Methodology

2.1. Finite element model for spherical indentation and material

This section describes the FE model for spherical indentation and material with the properties listed in Table 1 that comprehensively cover the property range of general metallic materials [42,49]. The purpose of FE analysis is to obtain relationship between RS and residual imprint, especially radial displacement ur . The relationship is the key part in RS estimation. For material properties and RS in Table 1, data of ur were collected through FE analysis of spherical indentation, which is detailed below.

An FE model of a single spherical indentation is illustrated as Fig. 1. Assuming that material is isotropic, the specimen is designed with 8470 axisymmetric continuum elements (CAX4) of different sizes. Axisym-metric boundary conditions are imposed to the nodes on axis. To obtain accurate measurement of imprint, the contact region beneath the indenter is consisted of elements with minimum size of 2.5 μm. Instead of multi-point constraints (MPC), trapezoidal elements are used in the element size transition regions to inhibit discontinuity of stress and strain. In the indentation tests, the indenter is fabricated from tungsten carbide with material properties of Young’s modulus Ei = 537 GPa and Poisson’s ratio νi = 0.24. Considering much stiffer material properties of the indenter compared with those of the specimen, the indenter with a diameter D = 1 mm is modeled with analytical rigid. Contact condition is set assuming surface-to-surface contact between the surfaces of specimen and material with the Coulomb type contact friction; μ implies friction coefficient. If too high indentation load is applied, it can be more influential to imprint than RS is; thus left imprint that cannot be distinguished by RS or vice versa. Therefore, the proper maximum indentation load P max of 300 N is applied on the indenter. Equi-biaxial RS are applied by Abaqus’ predefined field option. The list of ma-terial properties and values of test conditions, employed in the present study, is shown in Table 1.

Eq. (1) is the Hollomon’s piecewise power law model [20,37,50,51], in which a stress-strain curve is expressed by (σo, n) for given E. σo and εo ( = σo / E) are the elastic limit strength and strain. Note that σo is distinguished from the 0.2 % yield strength. Total strain εt ( = εe + εp) implies the sum of elastic and plastic strains, and n is the strain hard-ening exponent. While many indentation studies generally used Ramberg-Osgood stress-strain relation, Eq. (1) provides clear distinction between linear elastic and plastic regions by the elastic limit and consequently gives yield strength.

εt

εo=

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

σσo

for σ ≤ σo

(σσo

)n

for σ ≥ σo ; 1 < n ≤ ∞

(1)

2.2. Experiments

To verify the proposed method, indentation experiments are per-formed, and since the material properties of materials used in the indentation tests are required for the RS estimation, tensile tests are done. The test specimens are fabricated from the industrial metallic materials, SM45C and SS400. To avoid disparity in material properties, the tensile and indentation specimens are machined from the same sheet, and are annealed to remove the RS. The experimental results of indentation tests are compared with those obtained using conventional methods such as indentation tests using P–h curves and sharp indenter [13,52].

2.2.1. Tensile tests for plastic properties As the proposed method uses material properties to estimate RS,

tensile tests are done in the present study. Sheet-shape tensile specimens

are fabricated according to ASTM E8 standard, i.e. gauge length of 25 mm, width of 12.5 mm and thickness of 2 mm, from SM45C and SS400. Universal testing machine UTM (Instron 3367) was used for the tensile test, with a static load cell (Instron catalog 3 2530–444, capacity ± 30 kN) simultaneously measuring applied load. Displacement is evaluated by a static strain gauge within gauge length, and the load-displacement curve is converted into true stress-strain curve. By using the power-law regressions of Eq. (1), the material properties, E, σo and n are obtained. For each material, 5 tensile tests are performed and acquired properties are provided in Table 2. Determined material properties are average values after excluding upper and lower outliers; Poisson’s ratio is v =0.3.

2.2.2. Spherical indentation tests Adopting the set-up design of the device used by Rickhey et al. [35],

cruciform specimens of SM45C and SS400 as depicted in Fig. 2 are employed. The cruciform specimens, fabricated from the same sheet with the tensile specimens, are annealed to remove RS and the indenting surface is gradually grounded with 600-800-2400 grit sandpaper. As DIC results exhibit dependency on the surface pattern quality, certain pattern in appropriate size is necessary for image analysis, thus the surface was not further polished after grounding to use the slight scratches as patterns of the surface. As can be seen in Fig. 4, the sizes of remaining scratch width are much smaller than the imprint, thus the scratches can be considered not to affect resulting imprints. In addition, regarding that grounding with sandpaper might generate RS after annealing, every specimen is indented without stress induced. After the specimen is confirmed to be or be close to stress-free state, they are used for stress-induced indentation. Located between upper and lower jig, ends of the specimen are deflected with adjustment screws, inducing biaxial RS. Supposing that two orthogonally bended directions of spec-imen are direction 1 and 2, two strain gauges are attached to the top surface of specimen, oriented in direction 1 and 2, and measured elastic strains are calculated to reference stress σ ref using beam theory. The place of strain gauge is not always the same as in Fig. 2; thus slightly different stress might result from the locations of strain gauge slightly different from the indenting points.

The micro-indentation tests are implemented, using the same UTM with the tensile tests, under the compression mode, with the static load cell (Instron catalog # 2580, capacity ± 1 kN (Fig. 3). Spherical indenter with diameter of D = 1 mm, fabricated from tungsten carbide (WC), was used in the maximum indentation load Pmax = 300 N, as in FE analysis. After the contact, the displacement rate is controlled as 50 μm/min.

After indentation, surface image of residual imprint is obtained, using metallurgical microscope with FOV of 2.3 £ 1.7 mm and a magnification of 50 x. Radial displacement field ur on the surface of specimen is calculated using commercial DIC software (GOM correlate, GOM Corporation, Germany). For higher accuracy of ur measurement, surface images are taken before and after indentation at the same po-sition by employing jig fixture, designed in particular. When employing this method to actual industrial application, indentation would be done

Table 2 Material properties from tensile tests.

specimen number E [GPa] σo [MPa] n

SM45C 1 188 354 7.27 2 186 358 7.34 3 212 349 7.31 4 224 352 7.28 5 205 350 7.29

avg. 202 352 7.29 SS400 1 218 257 6.89

2 205 266 6.86 3 212 260 6.92 4 213 257 6.91 5 222 259 6.87

avg. 214 259 6.89

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on the material that has RS. In addition, the proposed method is based on the database obtained from FEA of material model having RS before and after indentation. Thus, for consistency between FEA, experimental validation and application, the residual imprints from indentation tests are measured, keeping stress-induced state by custom-made instru-mented spherical micro-indentation system. In addition, images were taken like Fig. 4 for each indentation for accurate measurements of ur in stress inducing directions.

3. Development of numerical method

Residual imprint forms as a result of spherical indentation after unloading due to permanent deformation of material. Imprint can be classified into vertical and radial in-plane displacement uz and ur. With ur data collected through FE analysis of spherical indentation for ma-terial properties and RS in Table 1, relationship between ur and RS is examined. For the ease of constructing the numerical function for RS evaluation, data of ur and RS are converted into the normalized terms,

(10 3 ur) / D and (σ r / σ o).

3.1. Relationship between radial displacement and residual stress

Fig. 5a shows the effect of RS on ur. On the specimen surface, ur is distributed differently depending on (σr / σo), which means ur is sensitive to the RS and in reverse, RS can be evaluated from the ur data. However, ur shows non-negligible friction influence, particularly at the region of contact and near the residual imprint (Fig. 5b–d) [49]. If friction-sensitive data is applied to the inverse analysis, it can cause error due to the deviation between experimental and numerical friction coefficients. In other words, to establish RS estimating method in which data from experiments can be simply applied, an undisputed relation-ship between ur and (σr / σo), a friction-free region of ur, where frictional effect can be neglected, is needed. Since ur has an arbitrary range after a certain point (Fig. 5b–d) that can be considered to be friction-free, nu-merical definition of that range of ur is made as detailed below.

Fig. 2. Scheme of custom-made equi-biaxial RS inducing experimental setup which provides both tensile and compressive stresses; all lengths in (mm) unit.

Fig. 3. Custom-made instrumented spherical micro-indentation system inducing equi-biaxial (a) tensile (b) compressive stresses; Spherical indenter with diameter of D = 1 mm and maximum testing load P max = 300 N are used as test conditions.

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3.1.1. Concept of representative displacement Hwang et al. [49] employed the concept of representative region to

extract representative displacement values that are considered to be free from friction effects. The point with minimum sum of the difference between ur for μ = 0, 0.1, 0.2, 0.3 was considered as a representative radial displacement point (rrep, urep), as shown in Fig. 5b–d. The repre-sentative point is regarded as the initiation point of the region without frictional effect, so for mapping function, the region after the point is made use of.

For the material of (E, σ o, n) = (200 GPa, 200 MPa, 5), when (σr / σo) = 0, 1, − 1 are applied, the average and maximum values of rrep / rmax are 2.3 and 2.7. In the work of Hwang et al. [49], the values of distance ratio rrep / rmax were measured as rrep / rmax = 1.8–2.6 without RS and for μ = 0.1. In addition, throughout similar ranges of material properties used for FEA, the maximum value of rrep / rmax was from material of (E, σ o, n) = (200 GPa, 200 MPa, 5), and the value of rrep / rmax decreased as the value of σo, n increased [49]. Therefore, since the average and maximum values of rrep / rmax, we obtained, have a sufficient gap, regarding the range of rrep / rmax from the study of Hwang et al. [49], we take 2.7 as the maximum value of rrep / rmax for the ranges of material properties on Table 1. Thus, rrep = 2.7 rmax is selected because for the cases with rrep / rmax < 2.7, the values are already in the region with little frictional effect.

3.1.2. Function development from radial displacement Based on the displacement data without frictional effect, a mapping

function is constructed to predict the (σr / σo) from ur in the range without frictional effect. Considering the field of view (FOV) of the microscope during the experiments and that after certain point, ur is almost 0, ur in the range of 1 ≤ r / rrep ≤ 2 is selected for use. The entire data of ur in the defined range are regressed (Fig. 6) for the considered

material properties and (σr / σo) in Table 1. As the value of friction co-efficient is insignificant in the range, μ = 0.1 is set for regression anal-ysis, since defined rrep = 2.7 rmax was established from the FEA data with μ = 0.1. The mapping function is established as

103 ur/

D = f u(r/D , σr /σo , n , σo /E)

=A 0/10+ exp (A 1 − A2(r/D))

; A i = f ui (σr / σo , n , σo /E) for i= 0, 1, 2

f ui (σr /σo , n , σo /E)=Bijk (σr /σo)

jn k

Bijk =Cijkl(103 σo

/E) l for j,k= 0, 1, 2, 3, 4 ; l= 0, 1, 2, 3, 4

(2)

The regression coefficients Cijkl are given in Appendix as Table A1.

3.2. Inverse analysis method

The Newton-Raphson (NR) method is a sensitive method to obtain unique polynomial solution and therefore, we use NR method to calculate the (σr / σo) that satisfies Eq. (2). To apply the mapping function to the NR method, a structure of equations is established as

F[(σr / σo)

i]=

∑

k

f u( rk

/D , (σr / σo)

i, n , σo

/E)− ur | k

= 0 (3)

(rk, ur | k) is the kth data in the range of 1 ≤ r / rrep ≤ 2 for regression, distributed in Section 3.1.1. The algorithm for determination of (σr / σo) is depicted in Fig. 7. When a single indentation test with Pmax = 300 N is done, ur is given and rrep, rrep are computed, so is the regression range. Regarding that we know the material properties (E, σo, n) of specimen, and rk / D, ur | k are the input data. At first iteration, the initial value of (σr / σo) i is assumed, and by using the Eq. (3), iterative computation continues to revise (σr / σo) i. The final solution of (σr / σo) is determined

Fig. 4. Surface image of specimen for measuring 1-direction ur (a) before (b) after indentation 2-direction ur (c) before (d) after indentation For each indentation, two forms of images, that are focused on 1, 2-direction of an imprint, were taken for accurate measurements of ur in stress inducing directions.

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Fig. 5. Distribution of ur for various (σ r / σ o) and friction coefficient μ when (E, σ o, n) = (200 GPa, 200 MPa, 5). First, different distribution of ur due to (σ r / σ o) is confirmed for (a) μ = 0.1. Second, different distribution of ur due to μ is confirmed for (σ r / σ o) = (b) 0 (c) 1 (d) − 1.

Fig. 6. Regression curves of (10 3 ur / D) vs. (r / D) of FE results with material properties of n = 5 (a) σ o = 200 (b) σ o = 800 MPa.

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when the maximum error e becomes less than the tolerance.

4. Results and discussion

Evaluation of the proposed method via FEA first, and then indenta-tion tests are performed. Using FEA results, numerical validations of algorithm and regression coefficients are verified before applying the method to the actual experiments, confirming proper estimating func-tion of the method. Before indentation tests, sensitivity of our inverse analysis algorithm is examined, regarding perturbation of indentation response, to investigate how sensitive the results from the method are to

the changes in (σ r / σ o). With acceptable sensitivity, the proposed method is applied to the imprint from experimental tests of single spherical indentation. Obtained results are also compared with those from different approach [13,52] for further reliable accuracy.

4.1. Numerical validation of the proposed method

To certify the reliability of proposed method, numerical validations of both algorithm and regression coefficients were performed. First, imprints from the indentation simulations for the 75 (= 5 £ 3 £ 5) combinations of (σo, n, σ r / σ o) in Table 1 were used. From each imprint, pairs of (r, ur) within the inverse range from Section 3.1.2 were obtained and put into RS determining algorithm. Estimated RS results of corre-sponding FE input values converged within e tol = 10 − 7, maximum error of 10 % and an average error of 1.8 % (Table 4, Fig. 8). As an additional verification of the method, comparison with computed equi-biaxial RS from the method of Lee et al. [37] is shown in Table 5 for (E, σo, n) =(200 GPa, 400 MPa, 10). The errors from the present study is smaller than the ones from Lee et al. [37].

Due to the high accuracy of estimation, considering that suggested algorithm is reliable, the same processes were done for 24 new combi-nations of (σ o, n, σ r / σ o) in Table 3, which were not used in obtaining regression coefficients. RS estimations of new combinations are shown as Table 6 and Fig. 9, with maximum error of 10% and an average error of 4.5%. Thus, both algorithm and regression coefficients of the sug-gested method were validated with high accuracy of RS estimation. Therefore, our new method, including regression coefficients and RS determining algorithm, was confirmed numerically that RS can be estimated properly.

4.2. Sensitivity analysis

The perturbation of indentation response is an issue for experimental estimation [23]. Due to inevitable experimental circumstances such as wear of indenter and measurement noise, small errors may accumulate

Fig. 7. Flow chart for determination of (σ r / σ o).

Fig. 8. Comparison of determined RS and corresponding FE input RS values for validation of algorithm for materials of (10 3 σo) / E = (1, 1.25, 2, 3, 4) and n = (a) 5 (b) 10 (c) 15.

Table 3 Material properties in additional FE analysis for the validation of regression coefficients.

material property symbol value

Young’s modulus [GPa] E 200 yield strength [MPa] σ o 300, 500, 700 strain hardening exponent n 7, 12 equi-biaxial residual stress (σ r / σ o) − 0.7, − 0.2, 0.2, 0.7

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in residual imprints from indentation tests, which differs from that of FE results. It is thus essential to examine the sensitivity of RS to the vari-ations in radial displacement ur. Perturbations of ± 2.5 % or ± 5 % are applied to radial displacement ur from the imprints of FEA for 6 cases in Table 7. All 75 combinations of (σo, n, σ r / σ o) in Table 1 are considered in the sensitivity test for each case. The average of absolute error values and standard deviations of the estimated RS are shown in Table 7.

The sensitivity of the present method, including the inverse analysis, shows almost one-to-one linearity. As the perturbation in ur doubles from ± 2.5 % to ± 5 % for the cases 1–2, errors are about twice, and so are the cases 3–6. The altered values in (σ r / σ o) due to the perturbation in ur are smaller than those in the studies [53,54], despite the larger perturbation. Dao et al. [54] stated that the sensitivity of an inverse analysis is acceptable when errors doubles for doubled perturbation. Zhao et al. [53] performed sensitivity tests with 2% perturbation, and mentioned that the value corresponds to the extreme case of error. Consequently, the sensitivity of the proposed inverse analysis algorithm is appropriate in engineering practice.

4.3. Experimental validation by spherical indentation tests

As an evaluation of the proposed method, a single spherical micro- indentation tests on the metal specimens (SS400, SM45C) are per-formed, in which biaxial tensile or compressive RS are induced, using a customized experimental setup for indentation tests described in

Table 4 Derived RS from RS determining algorithm for the validation of algorithm.

(10 3 σo) / E n (σ r / σ o) FEA (σ r / σ o) computed error (%)

1 5 − 1 − 1.002 − 0.2 − 0.5 − 0.519 − 3.7

0 − 0.009 −

0.5 0.487 − 2.5 1 1.008 0.8

10 − 1 − 0.982 1.8 − 0.5 − 0.479 4.2

0 − 0.015 −

0.5 0.505 1.0 1 0.995 − 0.5

15 − 1 − 0.993 0.7 − 0.5 − 0.519 − 3.9

0 0.030 −

0.5 0.475 − 5.1 1 0.998 − 0.2

1.25 5 − 1 − 0.997 0.3 − 0.5 − 0.491 1.8

0 0.058 −

0.5 0.497 − 0.6 1 0.998 − 0.2

10 − 1 − 1.006 − 0.6 − 0.5 − 0.504 − 0.8

0 − 0.010 −

0.5 0.503 0.5 1 0.996 − 0.4

15 − 1 − 1.002 − 0.2 − 0.5 − 0.513 − 2.5

0 0.044 −

0.5 0.473 − 5.3 1 0.998 − 0.2

2 5 − 1 − 0.997 0.3 − 0.5 − 0.511 − 2.1

0 0.021 −

0.5 0.532 6.4 1 1.007 0.7

10 − 1 − 0.997 0.3 − 0.5 − 0.477 4.5

0 0.009 −

0.5 0.513 2.5 1 0.998 − 0.2

15 − 1 − 0.995 0.5 − 0.5 − 0.497 0.6

0 − 0.022 −

0.5 0.502 0.4 1 0.994 − 0.6

3 5 − 1 − 0.996 0.4 − 0.5 − 0.496 0.7

0 − 0.013 −

0.5 0.519 3.8 1 0.980 − 2.0

10 − 1 − 1.002 − 0.2 − 0.5 − 0.504 − 0.8

0 − 0.007 −

0.5 0.486 2.7 1 0.999 − 0.1

15 − 1 − 1.002 − 0.2 − 0.5 − 0.519 − 3.8

0 0.044 −

0.5 0.494 − 1.3 1 1.016 1.6

4 5 − 1 − 0.999 0.1 − 0.5 − 0.490 3.6

0 0.017 −

0.5 0.490 − 9.4 1 0.987 − 3.7

10 − 1 − 1.007 − 0.7 − 0.5 − 0.482 3.6

0 − 0.040 −

0.5 0.453 − 9.4 1 0.963 3.7

15 − 1 − 1.004 − 0.4 − 0.5 − 0.506 − 1.2

0 − 0.079 −

0.5 0.454 − 9.1 1 1.010 1.0

avg. of absolute values of error 1.8

Table 5 Comparison of computed equi-biaxial RS for (E, σ o, n) = (200 GPa, 400 MPa, 10).

(σ r / σ o) FEA

(σ r / σ o) computed

error [%]

avg. of absolute values of error

Lee et al. (2010) [37]

− 0.5 − 0.473 5.5 6.4 − 1 − 0.955 4.5 0.5 0.470 6.0

1 0.903 9.8 present study − 0.5 − 0.477 4.5 4.5

− 1 − 0.997 0.3 0.5 0.453 − 9.4

1 0.963 3.7

Table 6 Derived RS from RS determining algorithm for the validation of regression coefficient.

(10 3 σo) / E n (σ r / σ o) FEA (σ r / σ o) computed error (%)

1.5 7 − 0.7 − 0.730 − 4.1 − 0.2 − 0.199 0.7

0.2 0.201 0.6 0.7 0.751 6.8

12 − 0.7 − 0.728 − 3.9 − 0.2 − 0.192 4.2

0.2 0.207 3.6 0.7 0.687 1.8

2.5 7 − 0.7 − 0.730 − 4.1 − 0.2 − 0.185 8.1

0.2 0.215 7.0 0.7 0.688 1.8

12 − 0.7 − 0.714 − 2.0 − 0.2 − 0.219 8.7

0.2 0.211 5.2 0.7 0.713 1.8

3.5 7 − 0.7 − 0.759 − 7.8 − 0.2 − 0.194 3.1

0.2 0.211 7.6 0.7 0.713 5.8

12 − 0.7 − 0.733 − 4.5 − 0.2 − 0.208 − 3.6

0.2 0.194 3.0 0.7 0.756 7.4

avg. of absolute values of error 4.5

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Section 2.2.2. Material properties (E, σo, n) are obtained by the tensile tests with UTM (Instron 3367), as mentioned in Section 2.2.1. Inden-tation tests were conducted by the same UTM for the specimens with two strain gauges bonded to the top surface, oriented in two orthogonal bending directions 1 and 2. During the tests, elastic strains are measured by the strain gauges simultaneously, and are calculated to reference bending stress σref using the beam theory. The tests were done with spherical indenter with diameter of D = 1 mm, made from tungsten carbide (WC), under the maximum indentation load P max = 300 N. After the indentation test, images of residual imprints and radial displacement field ur were obtained by metallurgical microscope and DIC software, and are employed in the algorithm in Fig. 7.

To confirm the reliability of the DIC analysis, radial displacement fields ur from the indentation tests are compared to those from FEA for the same conditions. Fig. 10 is the comparison of ur from SM45C indentation test and FEA with (σ R1 / σo) = 0.64, (σ R2 / σo) = 0.62 as in shown Table 8. Material properties from tensile tests in Table 2 and the values of (σ R1 / σo) and (σ R2 / σo) measured by the strain gauge are applied to FE analysis. The results can differ by the setting of ur range variation on both DIC program and FEA, but DIC measurement mostly showed similar results. However, ur from DIC includes measurement noise as Fig. 11, which cannot be applied to the inverse algorithm directly. As the inverse program estimates the value of RS by the itera-tive computation of the sum of errors between the assumed RS and calculated RS (using experimentally measured ur data), the final RS evaluation cannot be accurate with much noise in measured ur data. Thus, entire measured ur data is first regressed as an exponential decay form considering the form of Eq. (2), and then applied to the RS esti-mating algorithm.

For each biaxial RS case induced on the surface of cruciform spec-imen, three indentation tests are performed, and biaxial RS are deter-mined following the algorithm. An average of three estimations is selected as the final estimation of the corresponding biaxial RS case. Note that because all the experimental setting, including the degree of screw adjustment for inducing RS, is same for three indentation tests of each biaxial RS case, the estimated values were similar to each other.

Fig. 9. Comparison of determined RS and corresponding FE input RS values for validation of algorithm for materials of (10 3 σo) / E = (1.5, 2.5, 3.5) and n = (a) 7 (b) 12.

Table 7 Sensitivity of the estimated RS to the perturbation in ur.

case # Perturbation [%] in ur (σ r / σ o) average [%] SD

0 1.8 2.1 1 ± 2.5 3.1 2.5 2 ± 5 5.6 4.5 3 + 2.5 3.3 2.8 4 + 5 5.2 4.7 5 – 2.5 3.0 2.4 6 – 5 5.9 4.2

Fig. 10. Comparison of radial displacement field ur from indentation tests and FEA (SM45C). Displacement fields are those of (σ R1 / σo) = 0.64, (σ R2 / σo) = 0.62 as in Table 8 and ur from the experiment is calculated by DIC program in the patterned region.

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There are only slight differences in indentation RS for identical RS cases, for slight different indentation points inside the test area.

Comparison of evalueated RS to the σ ref are shown as Table 8 and Fig. 12. σ R1 is stress induced in direction 1, and σ R2 in direction 2. The RS values from the imprint experiment show a good agreement with the stresses from strain gauges, within error of 10%, average (absolute value) error of (4.5, 4.7)% for whole (σ R1, σ R2) and (4.3, 4.9, 4.6)% for (SS400, SM45C, whole experiment). In brief, the proposed algorithm provides reliable RS values in the material. As a verification of the effectiveness of the present indentation method, accuracies of σ R1 & σ R2 of equi-biaxial RS are compared to the previous studies using conven-tional indentation methods [13,52] in Table 9 . All the errors are calculated to the average value of absolute error values. Although the studies tested different materials, the accuracy of present method seems to be quite acceptable, indicating that the proposed method provides reliable RS estimation.

5. Summary and concluding remarks

A new equi-biaxial RS evaluating method by using the residual imprint from spherical indentation test was proposed. Based on the FEA results from various combinations of material properties and equi- biaxial RS, a mapping function between ur and σο / Е, n, (σ r / σ o)

was established. Combined with inverse analysis algorithm, the method for RS estimating system was made and via the numerical / experimental validations of the method, the following conclusions are drawn.

(1) Unlike other studies for estimating RS, the proposed method employs spherical indentation, which provides more information on RS than sharp indentation does. Also, instead of using P-h curve, the use of residual indentation imprint alleviates the (i) effect of machine compliance and (ii) need for stress-free refer-ence sample. This makes our method more convenient compared with others.

(2) The inverse range of radial displacement ur, correlating with RS and material properties with no friction effect, was defined. Ob-tained region of imprints from experiments can applied to the inverse algorithm established from the FEA data free from fric-tional influence.

(3) The proposed method was both numerically and experimentally validated. The accuracy of inverse algorithm with obtained regression coefficients for estimating RS was confirmed by applying imprints from FEA of materials under RS. With RS

Table 8 Measured stresses via strain gauge and indentation test.

material strain gauge indentation error in σ R1 / σο [%] error in σ R2 / σο [%] σ R1 /σο σ R2 / σο σ R1 / σο σ R2 / σο

SS400(σο = 259 MPa) 0.72 0.71 0.75 0.74 4.2 4.2 0.56 0.58 0.57 0.56 1.8 − 3.4 0.33 0.33 0.31 0.30 − 6.1 − 9.1 0.00 0.00 0.02 0.01 − −

− 0.21 − 0.23 − 0.23 − 0.24 − 9.5 − 4.3 − 0.47 − 0.45 − 0.46 − 0.47 2.1 − 4.4 − 0.81 − 0.79 − 0.82 − 0.80 − 1.2 − 1.3

avg. of absolute value of error 4.1 4.5

SM45C(σο = 352 MPa) 0.83 0.81 0.80 0.79 − 3.6 − 2.5 0.48 0.47 0.51 0.50 6.3 6.4 0.18 0.20 0.19 0.21 5.6 5.0 0.00 0.00 0.04 0.02 − −

− 0.37 − 0.35 − 0.36 − 0.38 2.7 − 8.6 − 0.55 − 0.55 − 0.59 − 0.58 − 7.3 − 5.5 − 0.73 − 0.75 − 0.76 − 0.74 − 4.1 1.3

avg. of absolute values of error 4.9 4.9 avg. of absolute values of error for overall data 4.5 4.7

Fig. 11. ur from FEA and raw experimental data of SM45C with of (σ R1 / σo) =0.64, (σ R2 / σo) = 0.62. Raw experimental data is regressed into the fine exponential decay form, and then applied to inverse algorithm for RS evaluation.

Fig. 12. RS from spherical indentation (σ eval) vs. strain gauge (σ ref) Values from indentation tests were obtained through the evaluating algorithm and those from strain gauge are calculated using the beam theory.

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values from the data used in regression, inverse algorithm was validated, and with RS values from the data not used in regres-sion, obtained regression coefficients were validated.

(4) Before experimental validation, sensitivity of proposed inverse analysis method and the reliability of digital image correlation (DIC) program for ur measurement were verified and shown to be acceptable, which were appropriate in engineering practice.

(5) Spherical indentation tests were conducted on the cruciform specimen for two kinds of metal materials under various equi- biaxial RS. The specimen surface was analyzed by DIC program. The obtained RS values from the imprint method were in acceptable agreements with the stresses from the strain gauges within error gap of 10 % and average errors of (4.3, 4.9, 4.6) % for (SS400, SM45C, whole experiment).

(6) Comparisons with prior studies, using conventional indentation methods, further verified the accuracy and sensitivity of our

inverse analysis algorithm. The results indicate that the suggested method not only has strength on convenience but also on the accuracy of RS estimation.

The form of RS, estimated in this study, is equi-biaxial only, but as residual imprint after indentation reveals the directionality of RS [55], it can provide a framework for investigation of non-equi-biaxial RS. RS evaluations considering material anisotropy [41,56] and severe surface roughness [57] are also future studies. As the form and range of RS extends, regression of radial displacement and derivation of estimating equation, like Eq. (2), may be much complicated and take certain effort and time, but it could be substituted by artificial neural network (ANN). Since similar approach of using residual imprint for material plastic properties evaluation has been investigated by Hwang et al. [49], future works will be focused on simultaneous evaluation of material properties and RS using residual imprint with ANN. The extended form of esti-mating method would be an essential development to certify the safety and reliability of the designed structures.

CRediT authorship contribution statement

Chanyoung Jeong: Conceptualization, Methodology, Writing – re-view & editing, Formal analysis. Yewon Hwang: Data curation. Naksoo Kim: Investigation. Cheolsoo Lee: Project administration. Hyungyil Lee: Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Table A1 Regression coefficients Cijkl for 10 3 ur / D in Eq. (2).

i ¼ 0 j = 0 j = 1 j = 2

k 0 1 2 0 1 2 0 1 2

l = 0 − 17.5 3.49 − 0.095 − 63.7 8.32 − 0.183 − 15.9 − 0.32 − 0.069 l = 1 38.0 − 6.23 0.157 125. − 14.9 0.291 32.1 0.99 0.111 l = 2 − 24.1 4.03 − 0.101 − 83.7 9.59 − 0.173 − 24.0 − 0.76 − 0.063 l = 3 6.27 − 1.10 0.028 23.1 − 2.59 0.044 7.38 0.23 0.015 l = 4 − 0.59 0.11 − 0.003 − 2.25 0.25 − 0.004 − 0.79 − 0.03 − 0.001

i ¼ 1 j = 0 j = 1 j = 2

k 0 1 2 0 1 2 0 1 2

l = 0 − 33.9 7.64 − 0.248 − 8.12 0.89 0.016 − 31.7 − 4.09 0.066 l = 1 64.8 − 13.0 0.426 23.9 − 3.32 − 0.036 67.13 7.92 − 0.130 l = 2 − 42.5 8.54 − 0.292 − 20.7 2.69 0.042 − 46.7 − 5.52 0.105 l = 3 11.8 − 2.41 0.085 6.67 − 0.80 − 0.016 13.1 1.61 − 0.035 l = 4 − 1.16 0.24 − 0.009 − 0.71 0.08 0.002 − 1.29 − 0.16 0.004

i ¼ 2 j = 0 j = 1 j = 2

k 0 1 2 0 1 2 0 1 2

l = 0 20.3 − 2.74 0.163 − 66.4 7.22 − 0.057 − 37.2 4.35 − 0.292 l = 1 − 35.2 6.15 − 0.323 139. − 15.8 0.087 80.2 − 7.74 0.526 l = 2 24.7 − 3.81 0.181 − 99.7 10.9 − 0.020 − 56.1 4.59 − 0.309 l = 3 − 6.90 0.91 − 0.039 29.0 − 3.01 − 0.004 15.8 − 1.10 0.074 l = 4 0.67 − 0.07 0.003 − 2.92 0.29 0.001 − 1.56 0.09 − 0.006

i ¼ 0 i ¼ 1 i ¼ 2 j = 3 j = 4 j = 3 j = 4 j = 3 j = 4

k 0 1 0 0 1 0 0 1 0

l = 0 53.7 − 4.88 24.0 8.63 − 1.05 56.0 59.5 − 5.51 15.8 l = 1 − 104.0 9.53 − 46.6 − 24.0 3.17 − 113.0 − 126.0 12.3 − 40.0 l = 2 69.6 − 6.38 32.1 21.4 − 2.80 77.6 92.5 − 9.16 29.9 l = 3 − 19.3 1.76 − 9.22 − 7.11 0.91 − 21.8 − 27.5 2.71 − 8.72 l = 4 1.87 − 0.17 0.94 0.78 − 0.10 2.15 2.8 − 0.28 0.88

Table 9 Comparison of accuracies in estimation of (σ R1 = σ R2) but measured as (σ R1 ∕= σ R2).

material average of absolute error values [%]

average of total estimation error [%]

Lu et al. [13] (σ R1

= σ R2) Al 2024 18.0 15 Al 7075 4.5 Copper C11000

16.1

Ti Grade 5 22.0 Pham and Kim

[52] (σ R1 ∕= σ R2)

SS400 11.2 12 12.2

SM490 12.5 11.7

Present study (σ R1

∕= σ R2) SS400 5.5 4.8

5.0 SM45C 4.1

4.4

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Acknowledgments

This work was supported by the Sogang University Research Fund (No. 202019062.01), and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF- 2021R1A2C2011210).

Appendix A. Regression coefficients Cijkl

See Table A1

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