# A Simple Method for Measuring Plastic Properties of Power Hardening Metals via the Indentation Curve with a Large Depth

Authors: Ying Kan; Yiwen Wu; Liping Ren; Huaining Chen
Journal: Advances in Materials Science and Engineering
 (2020)

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## Abstract

The spherical indentation technique provides an easy way to evaluate the integrity of in-service structures because it is nondestructive. In this study, a simple method was proposed to measure mechanical properties such as the yield strength, the ultimate tensile strength, and the strain hardening exponent from the indentation curve at a large indentation depth, which is 0.4 times of the indenter radius. Based on finite element analyses, a simple function was proposed to relate representative stress to indentation data. Besides, representative strains at different indentation depths were identified according to the load-depth curves from simulations. The calculated plastic properties from the developed method were compared well with experimental results.


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## Body

## 1. Introduction

Mechanical properties of materials such as ultimate tensile strength (UTS) and yield strength are considered to be important for evaluating the integrity of structures [1, 2]. However, conventional tensile tests are destructive and need preparation of specimens. In order to obtain mechanical properties of local domains and in-service components, the instrument indentation test (IIT) was proposed [3–6].Compared with sharp indenter, spherical indenter is more practical since it can measure elastoplastic properties by just one indentation [7–10]. Many studies have been carried out with the purpose of extracting mechanical properties from spherical indentation parameters [11–13]. Some researchers directly relate the indentation parameters to the tensile strain-stress points. In the direct method, the representative stress was defined as the averaged true stress under the indenter, while the representative strain was defined by using tangent function or sine function [14], which leads to different accuracies of the calculation results. Jeon et al. [15] found that, compared with the tangent function, the sine function results in a too large work hardening exponent for materials that obey Hollomon hardening law. Besides, both the sine function and the tangent function contain the parameter of contact depth, which is difficult to be measured in practical. Other researchers extract the elastoplastic properties from a closed-form expression of the indention load [16] or work [17] as a function of parameters of the Hollomon hardening law. Cao and Lu [16] derived the indention load as a function of representative stress. However, the expression contains dozens of coefficients, which need to be fitted and lack physical meanings. Ogasawara et al. [18] established a relatively simple equation of the loading work and the representative stress. However, the strain hardening exponent in the equation is in contradiction with the definition of the representative stress.Indentation curves with a shallow depth cannot guarantee the uniqueness of the solution since different materials may generate identical indentation curves when the indentation depth is small [19]. Chen et al. [20] suggest that a deep indentation with the depth-radius ratio (h/r) of 0.3 may distinguish the indentation curves. Lee et al. [17] demonstrate that the load-depth curves of different materials clearly separated from each other when the h/r is 0.4, which may be the largest depth ratio in literatures.In the current study, a simple method was proposed to obtain the yield strength, the ultimate tensile strength, and the strain hardening exponent from an indentation curve with a large indentation depth. Based on finite element analyses, the developed representative stress function is independent with the strain hardening exponent. Besides, the representative strains are determined at different depth-radius ratios with a maximum value of 0.4. The effectiveness of the method is verified on the Q235 steel and the 2219Al alloy by indentation tests and tensile tests.


## 2. Materials and Experiment

One of the experimental materials is a Q235 low carbon steel with the typical compositions of 0.17C-0.26Si-0.46Mn (wt%). The other material is an annealed 2219Al alloy with the chemical compositions of 5.80Cu-0.20Mn-0.20Si-0.30Fe-0.10Zn-0.02Mg (wt%).The samples used in indentation tests and tensile tests were cut from the same blank. The tensile specimens were machined into cylindrical bars with a gage section of 5 mm in diameter and 30 mm in length. The indentation specimens were cut into cuboids with the dimensions of 30 mm × 30 mm × 20 mm. The uniaxial tests were performed on an Instron 5582 universal testing machine at room temperature. The indentation tests were performed on a Shimadu AG-X testing machine, whose load and displacement resolution are 0.2 N and 0.2μm, respectively. A constant loading rate of 0.01 mm/s was applied for all indentation tests. The indenter is a tungsten carbide spherical with a radius of 0.25 mm. Four indentation and tensile tests were carried out for each material.


## 3. Finite Element Modelling

Finite element simulations were performed using ABAQUS software. An axisymmetric 2D model was constructed to simulate the indentation load-depth curves of materials. The Mises yield criterion and isotropic hardening law were used for the materials in simulations. The model contains more than 50,000 4-node bilinear axisymmetric quadrilateral elements (CAX4R in ABAQUS). Fine meshes of size 2μm were used on the region beneath the indenter, which was modeled as an elastic material with Young’s modulus 534 GPa and Poisson ratio 0.3.The maximum indentation depth-radius ratio is 0.4, which is deep enough to obtain unique solution of material properties in reverse analysis [17]. 720 simulations were carried out for different combinations of material properties within the range of E from 70 GPa to 210 GPa, E/σy from 0.001 to 0.04, and n from 0 to 0.5. Figure 1 shows the geometry, mesh, and boundary conditions of the finite element model. The dimensions of the blank are 12 mm in radius and 8 mm in height, which are large enough to get rid of the size effect of the samples.Figure 1
Geometry, mesh, and boundary conditions of the finite element model.


## 4. Verification of the Finite Element Model

The mechanical properties obtained from tensile tests are listed in Table1. Figure 2 illustrates the experimental true stress-strain curves of Q235 steel and 2219Al, which were used for the simulation. As can be seen in Figure 2, the 2219Al alloy and Q235 steel have different behaviors between the elastic and plastic transitions. 2219Al shows a smooth curve, while the Q235 steel curve contains a passage of Lüders strain.Table 1
Mechanical properties of Q235 steel and 2219Al.
Material
E (GPa)
R
p
0.2
/
R
L (MPa)
UTS (MPa)
Elongation (%)
n
2219Al
70
83.9/—
218.0
27.1
0.297
Q235
210
—/242.9
438.2
36.6
0.239Figure 2
Experimental stress-strain curves of Q235 steel and 2219Al alloy.The friction coefficientf is an important parameter in indentation simulation. Numerous studies have discussed the influence of friction coefficient on indentation curves based on simulation results. However, the reported research studies rarely compared the simulation load-depth curve (p-h curve) with that of experiment when the friction coefficient is fixed. In the present work, the coulomb’s friction law is assumed between the contact surfaces. Figure 3 shows the experimental and numerical indentation curves of 2219Al alloy and Q235 steel when the friction coefficients in numerical simulation are 0.2, 0.3, and 0.4, which are reasonable values in experiment. It can be seen that simulated curves with various f (0.2 ≤ f ≤ 0.4) are compared well with experimental curves. Therefore, the friction coefficient was set as 0.3 for simulations in this paper.Figure 3
Comparison between experimental and numerical indentation curves of 2219Al alloy and Q235 steel: (a) the integral curve; (b) partial enlarged view.
(a)
(b)


## 5. Measuring Method

Ogasawara et al. [18] proposed a dimensionless function to calculate the representative stress as follows:(1)Wh3σr1+sn=1meE¯/σr1+sn+1mp−1,where W is the indentation work during loading; h is the indentation depth; σr is the representative stress; n is the strain hardening exponent; s is the sphere factor, which represents the work hardening effect; and me and mp are coefficients. The values of W, σr, s, me, and mp are influenced by h/r for a certain material. E¯ is the plain strain modulus, which is as follows [18]:(2)E¯=E1−υ2,where υ is the Poisson ratio and E is Young’s modulus of the test material. It should be noted that equation (2) did not contain the term that reflects the effect of indenter properties. Thus, the reduced elastic modulus E∗ was introduced in this paper to replace E¯. The E∗is given as follows [16]:(3)E∗=1−υ2E+1−υi2Ei−1,where υ is the Poisson ratio and Ei is Young’s modulus of the indenter material.According to the principle of the representative stress-strain method [7], representative stress function should be independent with n. Therefore, the introduction of n in equation (1) is in contradiction with the principle of this method. The specific form of the representative stress function depends on the load-depth data from simulations. In the current study, it was found that the W/h3σr−E∗/σr curves are in coincidence with each other under different n values when a representative strain εr is given. Figure 4 shows the relationship between W/h3σr and E∗/σr on the condition of E 70 GPa, εr 0.055, and h/r 0.4. It can be seen from Figure 4 that the W/h3σr−E∗/σr curve is independent of n. Thus, equation (1) can be replaced by the following function:(4)Wh3σr=1meE∗/σr+1mp−1.Figure 4
Relationship betweenW/h3σr and E∗/σr.Equation (4) is independent of n and more simplified than equation (1). According to the simulation data, the values of me and mp under different indentation depths and E were fitted by the following equations:(5)me=AEd×rhBEd,0.1≤h/r≤0.4,mp=MEd×rh+NEd,0.1≤h/r≤0.4.Based on the indentation data obtained from finite element simulations, the coefficientsAEd, BEd, MEd, and NEd can be fitted as(6)AEd=0.71031+0.24281Ed,BEd=0.46839−0.1232Ed,MEd=−4.78397−36.44447Ed+158.40876Ed2−365.36863Ed3,NEd=12.48498+9.87187Ed−42.81818Ed2+90.62353Ed3,where Ed is a dimensionless coefficient which is defined as Ed=E/E1, in which E1 is 106 MPa.In this paper, the representative strainεr was defined to be equal to the plastic strain εp. Correspondingly, based on the Hollomon equation, the relationship between the representative stress and strain is as follows [18]:(7)σr=σyEσynεr+σrEn,where σy is the theoretical yield strength of Hollomon equation. According to the simulation data that was shown in Figure 5, the εr under different h/r can be fitted as(8)εr=0.00607+0.16048hr−0.09524hr2,0.1≤h/r≤0.4.Figure 5
Simulation data ofεr.The reverse analysis means calculation of mechanical properties from ap-h curve. In the inverse analysis, the W and h were obtained from the p-h curve, while the r and E were taken as known conditions in the current study. Then, the σr and εr can be calculated using equations (4) and (8), respectively. The n was calculated by using equation (7). According to the relationship between engineering and true stress and strain [21], 0.2% offset yield stress Rp0.2 is calculated by using the following equation:(9)Rp0.2=σ0.2e0.02+σ0..2/E,in which σ0.2 was solved by the following equation:(10)σ0.2=σyEσyn0.002+σ0.2En.In this study, the Lüders strainεL is calculated using the following equation for materials with Lüders elongations [22]:(11)lnK/σL=εL−n⋅lnεL,where [22](12)K=σεpn.The true stress at Lüders strain is given as(13)σL=σyEσynεLn.The engineering stress at Lüders strain is given as(14)RL=σLeεL+σL/E.The ultimate tensile strength UTS was calculated using the following equation [23]:(15)UTS=σy×nE2.718σyn.


## 6. Results and Discussion

In order to verify the measuring method, indentation curves of Q235 steel, 2219Al, and 3 imaginary materials were employed to extract mechanical properties. The experimental stress-strain curves of Q235 steel and 2219Al do not match equation (7) precisely. The 3 imaginary materials are ideal power law hardening materials. The input mechanical properties of the 3 imaginary materials are shown in Table 2. The numerical simulations of indentation were carried out to obtain P-h curves of the 3 imaginary materials. The indentation works W under different depths are collected in Table 3.Table 2
Mechanical properties of 3 imaginary materials.
Material
E (GPa)
σ
y(MPa)
R
p
0.2 (MPa)
n
Material 1
70
83.0
106.6
0.230
Material 2
195
241.0
302.0
0.214
Material 3
210
242.0
280.8
0.140Table 3
W (N·mm) of 2219Al, Q235 steel, and the 3 imaginary materials.
h/r
W of 2219Al
W of Q235
W of mat. 1
W of mat. 2
W of mat. 3
0.10
0.26906
0.58524
0.27781
0.75228
0.64412
0.15
0.62710
1.38302
0.66173
1.79128
1.50282
0.20
1.15406
2.54280
1.21757
3.29364
2.72737
0.25
1.84569
4.06635
1.94695
5.26291
4.30728
0.30
2.69975
5.93597
2.84725
7.69149
6.23573
0.35
3.71410
8.14890
3.91624
10.5712
8.49717
0.40
4.88300
10.6800
5.14757
13.8849
11.0735Figure6 is the comparison between calculated stress-strain points and input curves or experimental curves. It can be seen from Figure 6 that the calculated data points are compared well with the input curve for the 3 imaginary materials. However, the stress-strain points of Q235 and 2219Al show deviations from the experimental tensile curves. It should be noted that the combination of E, σy, and n is not unique for real materials that do not strictly obey the power law equation. Figure 7 shows comparison between experimental tensile and fitted curves of 2219Al. It can be seen that the same stress-strain curve can be fitted as several combinations such as E 70 GPa, σy52.8 MPa, and n 0.296 and E 23 GPa, σy83.9 GPa, and n 0.296 when E or σy was fixed. In this study, the σy was an unknown parameter, while the E was taken as a known parameter. Consequently, the σy can only be fitted as 52.8 MPa, which is much smaller than the yield strength (83.9 MPa) of 2219Al. This phenomenon was also reported in literatures. Byun et al. [23] introduced an offset constant to calculate the yield strength from the indentation test. However, the proposed constant was determined from extra tensile tests versus indentation tests on the same material. It is well known that the 0.2% offset yield strength Rp0.2 was taken as the yield strength of the material without Lüders strain in conventional tensile tests. Thus, instead of the σy in the Hollomon equation, the Rp0.2 extracted from indentation stress-strain curve should be taken as the yield strength and compared with that of the tensile test. As shown in Figure 7, the Rp0.2 are 83.9 MPa and 80.6 MPa on the experimental curve and fit curve, respectively. Nevertheless, the Rp0.2 may deviate from the yield strength of materials with Lüders strains, which make the yield strain unknown. In general, the yield strength is directly obtained from the stress-strain curve in the tensile test for the material with a yield point. In this work, the measuring method is based on the Hollomon equation, which cannot describe the passage of Lüders strain. Thus, the stress at the Lüders strain εL is proposed to be the yield strength of this kind of materials.Figure 6
Comparison between input or tensile curves and stress-strain points calculated by the reverse analysis: (a) 3 imaginary materials; (b) 2219Al and Q235 steel.
(a)
(b)Figure 7
Comparison between experimental tensile and fitted curves of 2219Al.Tables4 and 5 summarize the extracted material properties by using the developed method and Ogasawara’s method, respectively. It was shown that the errors of mechanical properties calculated by using the developed method are smaller than these calculated by using Ogasawara’s method. It also can be seen in Table 4 that the errors of Rp0.2 and RL of Q235 are −17.8% and 21.4%. The corresponding strains ε0.2and εL can be regarded as the lower and upper limits of the real Lüders strain, which is usually larger than ε0.2. In the present work, the calculated εL 0.0147 is larger than the real εL 0.0098, which leads to an overestimated RL of Q235 steel. Thus, an average value (247.3 MPa) of Rp0.2 and RL will be a conservative estimate of the yield strength in indentation tests for materials with Lüders strain. It should be noted that the average yield strength is still an conservative value when the calculated εL is smaller than the real εL [24].Table 4
Plastic properties obtained from the developed method.
Material
σ
y (MPa)
R
p
0.2
/
R
L (MPa)
ε
0.2
/
ε
L (10−3)
n
UTS (MPa)
ErrorRp0.2RL/n/UTS (%)
2219Al
73.6
97.6/—
3.4/—
0.241
214.4
16.4/−18.9/−1.6
Q235
138.8
199.6/295.0
2.9/14.7
0.243
457.4
−17.8(21.4)/1.6/4.4
Material 1
81.0
105.5/—
3.5/—
0.239
227.9
−1.0/3.9/—
Material 2
227.2
290.4/—
3.5/—
0.224
589.8
−3.6/4.6/—
Material 3
229.0
272.4/—
3.3/—
0.157
427.1
−3.0/12.1/—Table 5
Plastic properties obtained from Ogasawara’s method.
Material
σ
y (MPa)
R
p
0.2
/
R
L (MPa)
ε
0.2
/
ε
L(10−3)
n
UTS (MPa)
ErrorRp0.2(RL)/n/UTS (%)
2219Al
78.1
101.5/—
3.4/—
0.222
202.6
20.9/−25.5/−7.0
Q235
129.1
191.3/306.8
2.9/14.7
0.253
459.7
−21.3(26.3)/6.0/4.9
Material 1
72.7
97.8/—
3.4/—
0.251
224.1
−8.2/9.1/—
Material 2
195.7
264.2/—
3.4/—
0.249
602.1
−12.3/16.3/—
Material 3
200.0
249.4/—
3.2/—
0.183
435.9
−11.1/30.7/—In order to investigate the sensitivity to the possible experimental errors for indentation testing, a 3% error band was artificially added to the indentation curves of Q235 steel and 2219 aluminium alloy (as shown in Figure8). Table 6 shows the plastic properties obtained from the indentation curves with 3% error. It can be seen that, compared with properties in Table 4, the maximum error is −4.9%.Figure 8
Indentation curves with error bands.Table 6
Plastic properties obtained from the indentation curves with 3% error.
Material
Error in indentation curve (%)
R
p
0.2
/
R
L (MPa)
n
UTS (MPa)
Error compared with values in Table4Rp0.2(RL)/n/UTS (%)
2219Al
+3
101.8/—
0.250
223.4
4.3/4.0/−4.2
2219Al
−3
93.9/—
0.229
206.5
−3.8/−4.7/−3.7
Q235
+3
208.1/308.8
0.250
472.9
4.3(4.7)/3.1/3.4
Q235
−3
191.1/285.8
0.231
439.5
−4.2(3.1)/−4.9/−3.9


## 7. Conclusions

In this study, a new energy-based approach was proposed to extract the stress-strain curve from the spherical indentation curve. The relationships between indentation stress-strain curve and mechanical properties were derived and discussed. The conclusions can be summarized as follows:(1)
A new representative stress function was proposed for the spherical indentation test. Compared with the original function, the new function is more reasonable since it is independent with the strain hardening exponent. Besides, it can be applied to a maximum depth of 0.4 times of the indenter radius. The corresponding representative strain function was also given.(2)
A new method for extracting the yield strength from indentation stress-strain curve was proposed. Instead of the parameterσy in the Hollomon equation, the Rp0.2 extracted from the indentation stress-strain curve was identified as the yield strength for materials without Lüders strains. It is suggested to take the average value of Rp0.2 and the stress at Lüders strains RL as the yield strength for materials with Lüders strains.(3)
The mechanical properties calculated by the developed method are more precise than those calculated by Ogasawara’s method. Based on the developed method, the maximum errors of yield strength, strain hardening exponent, and ultimate tensile strength are 21.4%, 18.4%, and 4.38%, respectively.



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