A simple framework of spherical indentation for measuring elastoplastic properties

Mechanics of Materials 41 (2009) 1025–1033

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A simple framework of spherical indentation for measuring elastoplastic properties Nagahisa Ogasawara a,*, Norimasa Chiba a, Xi Chen b,* a b

Department of Mechanical Engineering, National Defense Academy, Hashirimizu, Yokosuka 239-8686, Japan Nanomechanics Research Center, School of Engineering and Applied Sciences, MC 4709, Columbia University, New York, NY 10027-6699, USA

a r t i c l e

i n f o

Article history: Received 29 July 2008 Received in revised form 19 April 2009

a b s t r a c t Based on the indentation load–displacement curve, spherical indentation may deduce material elastoplastic properties from the measurements at several depths (which mimics the dual/plural sharp indentation method). The previous approaches, however, have very complex formulations and involve many fitting parameters that lack theoretical backgrounds; moreover, studies based on shallow indentation may not lead to unique solution. To close these gaps, we propose a simple framework of spherical indentation based on a new limit analysis-based representative strain analysis, which contains minimum number of fitting parameters. Two simple equations of the normalized loading work (at two different depths) are proposed, which can determine the material plastic properties accurately from the loading curve. In addition, by using either the established Fischer-Cripps method or an extra equation based on the contact stiffness, both the elastic and plastic properties are determined with reasonable accuracy. The simple framework may be useful for guiding the measurement of elastoplastic properties via spherical indentation. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Instrumented indentation requires minimum effort for sample preparation and it is widely used to probe the elastic and plastic properties of engineering materials (Cheng and Cheng, 2004; Gouldstone et al., 2007), small material structures (Cao and Chen, 2006; Chen et al., 2006), and biomaterials (Gordon et al., 2004; Ko et al., 2006). During the experiment, a rigid indenter penetrates normally into a homogeneous solid (Fig. 1(a)), where the indentation load, P, and displacement, d, are continuously recorded during loading and unloading (Fig. 1(b)). Denoting the specimen Young’s modulus by E and yield stress by ry, without losing generality, the uniaxial stress–strain (r–e) curve of a stress-free solid can be approximated in a power-law form (Fig. 1(c)):

* Corresponding authors. Tel.: +1 212 854 3787; fax: +1 212 854 6267 (X. Chen). E-mail addresses: [email protected] (N. Ogasawara), xichen@civil. columbia.edu (X. Chen). 0167-6636/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2009.04.010

r ¼ Ee for e 6 ry =E and r ¼ Ren for e P ry =E; ð1Þ where n is the work-hardening exponent and R  ry(E/ry)n is the work-hardening rate. For most ductile metals and alloys n is between 0.1 and 0.5. The Poisson’s ratio can be taken as m = 0.3 which is often regarded as a minor factor for indentation analyses (Cheng and Cheng, 2004). The material parameter set (E, R, n) is implicitly related to the indentation characteristics that are exhibited through the P–d curve. The main objective of the indentation analysis is to correlate the shape factors of the indentation load–displacement curve and indenter geometry with material properties, such that the elastoplastic properties can be measured from indentation experiments. Among various shapes of indenter tips, spherical indenter is one of the earliest studied through the Brinell hardness test and Hertz solution in 1900’s. Oliver and Pharr (2004) and Fischer-Cripps (2001) proposed algorithms to estimate the projected contact area and to derive the hardness and modulus for spherical indentation. Johnson (1985) showed that if the dimensionless strain of spherical

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N. Ogasawara et al. / Mechanics of Materials 41 (2009) 1025–1033

Fig. 1. (a) Schematics of spherical indentation on a homogeneous, isotropic semi-infinite bulk specimen; (b) typical indentation load–displacement curve obtained from an experiment; (c) uniaxial stress–strain curve of materials obeying the power-law hardening law Eq. (1), and the definition of the representative strain (which equals to the plastic strain).

indentation is chosen as Ea=ðrry Þ (where E ¼ E=ð1  m2 Þ is the plane strain modulus, a is the contact radius, and r is the indenter radius), the variation of the dimensionless strain with the normalized hardness H/ry (where H = P/ (pa2)) can be correlated to that of a conical indenter with half apex angle a, except that the dimensionless strain of the conical indenter is E tanðp=2  aÞ=ry . Subsequently, there exists an analogy between the geometrical factor d/r of spherical indentation and the indenter angle a in conical impression (Chen et al., 2005), and it is possible to deduce

the material elastic and plastic properties from the spherical indentation response at different depths (Cao and Lu, 2004; Cao et al., 2007; Zhao et al., 2006b), which mimics the effect of the dual/plural sharp indenter method (Bucaille et al., 2003; Chollacoop et al., 2003; Ogasawara et al., 2005, 2006b) yet it requires only one spherical indentation test. Chen et al. (2007) further showed that the P–d curve of a deep spherical indentation test (with d=r P 0:3) may has a one-to-one correspondence with (E, R, n) of power-law materials. Therefore, a properly designed spherical inden-

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N. Ogasawara et al. / Mechanics of Materials 41 (2009) 1025–1033

tation approach may represent a simple alternative to complement the sharp indenter method involving single/dual/ plural tips (Bucaille et al., 2003; Chollacoop et al., 2003; Ogasawara et al., 2006a, 2007), which is the focus of the present study. There have been several attempts to measure the elastoplastic properties based on the information of spherical indentation P–d curves. Lee et al. (Lee et al., 2005) proposed a numerical technique which involved quite a few fitting parameters and it did not take full advantage of the varying rich information during loading. Huber and Tsakmakis (1999) proposed the use of a neural network method to extract material elastoplastic properties from spherical indentation, which included many fitting parameters with a relatively complex algorithm. Cao and Lu (2004) applied the concept of representative strain to normalize the loading P–d curves of spherical indentation, and utilized data at two very shallow indentation depths. However, their formulation of the representative strain was shown to be inapplicable to a wide range of materials (Zhao et al., 2006b). Cao et al. improved their definition of the representative strain in a recent study (Cao et al., 2007), yet the formulation involved dozens of fitting parameters, and an advanced numerical analysis was required. Moreover, a prominent disadvantage of the previous shallow spherical indentation studies (Cao and Lu, 2004; Cao et al., 2007; Huber and Tsakmakis, 1999; Lee et al., 2005) is that they may not lead to unique solution (Chen et al., 2007), causing significant error in practice (Herbert et al., 2006). In addition, shallow nanoindentation experiment may interact with the strain gradient effect (Qu et al., 2006) and makes it difficult to measure the intrinsic material properties. It is arguable that for the reverse analysis of spherical indentation technique to be more robust, the indentation depth needs to be sufficiently deep (Liu et al., 2009). Zhao et al. (2006b) proposed that by utilizing the unloading contact stiffness, along with the normalized indentation loads obtained at two moderately deep indentation depths, it is possible to measure the material elastoplastic properties from a spherical indentation test. During penetration, the normalized loading curvatures were taken from d/r = 0.13 and 0.3, and the contact stiffness was measured at unloading where dmax/r = 0.3. This method, however, still involves quite a few fitting parameters and its complex form requires advanced numerical analyses, and users may make mistakes during the reverse analysis if they were not careful (Ogasawara et al., 2008). In this paper, we present a relatively simple framework for solving the material elastoplastic properties from spherical indentation, based on a new formulation of representative strain. The number of fitting parameters involved is reduced to minimum. For elastic-perfectly plastic solid, the Young’s modulus and yield strength can be derived in closed form. For materials exhibit hardening behavior, we propose several methods (depending on different scenarios) to measure the elastoplastic properties, which all exhibit good accuracy for ductile material examples. The studies in this paper may extend the usefulness of the spherical indentation technique and the relatively simple algorithms may receive broader perspectives.

2. Computation method 2.1. Representative strain of spherical indentation Similar to the work of (Zhao et al., 2006b), for spherical indentation we define the representative strain to be the plastic strain of a uniaxial stress–strain curve (Fig. 1(c)):

e ¼ ee þ ep  ee þ eR :

ð2Þ

The representative strain eR is a function of d/r. Correspondingly, the representative stress is:



rR ¼ R eR þ

rR n E

ð3Þ

:

For spherical indentation at two moderately deep indentation depths d1/r and d2/r, the corresponding indentation loads, representative strains, and representative stresses are P1 and P2, eR1 and eR2, and rR1 and rR2, respectively. Dimensional analysis leads to:

C1

rR1 C2

rR2

 

P1 d21 rR1 P2 d22 rR2

¼ f1 ¼ f2

E

rR1 E

rR2

! ;n ;

ð4Þ

! ;n ;

ð5Þ

where C is also referred to as the loading curvature. The general purpose for introducing the representative strain is to expect that (4) and (5) may be independent of n by optimizing the value of eR for a given d/r (Cao and Lu, 2004), such that these two equations would only involve the representative stresses as the apparent unknowns and thus simplify the reverse analysis. However, the complete independence of n may be difficult for spherical indentation (Zhao et al., 2006b). The overall purpose of this paper is still to keep the functional forms of (4) and (5) sufficiently simple to facilitate the reverse analysis. The optimized relationships are sought from numerical forward analyses.

2.2. Material and finite element method used in forward analysis Although there were theoretical studies concerning the spherical indentation, most of them focused on rigid plastic materials and shallow indentation regimes (Biwa and Storakers, 1995; Hill et al., 1989; Richmond et al., 1974); theoretical approach can be quite challenging for elastoplastic materials with strain hardening and deep penetration. Since indentation involves finite plastic deformation, the relationship between material properties and indentation responses are established through numerical simulations based on the finite element method (FEM) using ANSYS (ANSYS, 2003). The indenter is assumed rigid and the specimen is semi-infinite. A typical mesh for the axisymmetric indentation model comprises about 10,000 4-node elements with reduced integration, and mesh convergence tests are carried out. In the forward analysis, the material parameters are varied over a large range to cover essentially all engineering materials, with E=rR from 3 to 6000 and n from 0 to 0.5

N. Ogasawara et al. / Mechanics of Materials 41 (2009) 1025–1033

(a)

140

Wt 1

3

δ1 σ R1(1+ s1n )

120 100

60

0

(b)

d32

0

1

¼@ E rR2 ð1 þ s2 nÞ me2 rR2 ð1þs 2 nÞ

11 1 A þ ; mp2

2000

3000

4000

Wt 2

30 n = 0.0 n = 0.1 n = 0.2 n = 0.3 n = 0.4 n = 0.5 eq.(7)

25 20

δ2 /r = 0.3 εR2 = 0.0160

15 10

sphere factor, s2 = 1.34 0

1000

2000

3000

4000

E σ R 2 (1+ s2n )

(c)

1600

δ2 /r = 0.3 εR2 = 0.0160

1400

S 2 (1 + s 2n ) E (δ 2 − δ f ' )

1200

sphere factor, s2 = 1.34 n = 0.0 n = 0.1 n = 0.2 n = 0.3 n = 0.4 n = 0.5 eq.(15)

1000 800 600 400

ð6Þ

200 0

ð7Þ

5000

40

0

where me1 = 1.69, mp1 = 125.7, s1 = 0.94, eR1 =0.0095. Physically, me1 represents the limit of the behavior of elastic materials (when E=rR1 ! 0) and can be derived analytically below, and mp1 is related with the limit of rigid plastic materials when E=rR1 ! 1 (Ogasawara et al., 2006b). The sphere factor s1 represents the effect of work hardening. Comparing with other formulations for spherical indentation (Cao and Lu, 2004; Cao et al., 2007; Huber and Tsakmakis, 1999; Lee et al., 2005; Zhao et al., 2006b), the limit-based formulation (6) is very simple that involves minimum number of fitting parameters and with clearer physical meanings. A similar relationship holds at d2/r = 0.3 (Fig. 2(b)):

W t2

1000

35

11

1 1 A ¼@ þ ; E mp1 d31 rR1 ð1 þ s1 nÞ me1 rR1 ð1þs 1 nÞ

0

E σ R1(1+ s1n )

3.1. Formulation for loading

W t1

sphere factor, s1 = 0.94

20

3. Representative strain formulation

0

δ1/r = 0.1 εR1 = 0.0095

5

The work done by the indenter during loading is Rd Rd W t1 ¼ 0 1 Pdd and W t2 ¼ 0 2 Pdd, respectively, at d1/r = 0.1 and d2/r = 0.3. It is found that when W t1 =ðd31 rR1 Þ is plotted as a curve with respect to E=rR1 for a given n value, the spacing between such curves scales linearly with n. Therefore, inspired by the work of (Ogasawara et al., 2006b) on sharp indenters, the following simple relationship is found to work remarkably well for spherical indentation at d1/ r = 0.1 (Fig. 2(a)):

n = 0.0 n = 0.1 n = 0.2 n = 0.3 n = 0.4 n = 0.5 eq.(6)

80

40

3

(for larger n values, the range of E/ry is much larger than that of E=rR ). The Coulomb’s friction law is used between contact surfaces, and the friction coefficient is taken to be 0.15, which was obtained for metals and diamond by Bowden and Tabor (Bowden and Tabor, 1950). The Poisson’s ratio, which is regarded as a minor factor (Cheng and Cheng, 2004), is fixed at 0.3 which is a good approximation for most metals and alloys. The numerical indentation tests are displacement-controlled and data are obtained from various depths, d/ r = 0.10.3. Without losing generality, in Eqs. (4) and (5) we let d1/r = 0.1 and d2/r = 0.3, and the corresponding functional relationships will be used for reverse analysis. These two depths are sufficiently different and can lead to robust solutions of elastoplastic properties, see below. In addition, d1/r is sufficiently large to overcome the strain gradient effect (Qu et al., 2006) whereas d2/r is deep enough to ensue uniqueness of reverse analysis (Chen et al., 2007). If the material Young’s modulus is already known, only the loading curve is needed for measuring the plastic parameters; in that case, the maximum penetration does not have to be controlled precisely, as long as it is larger than 30% of the indenter tip radius. Whereas, if the goal is to explore the elastoplastic properties, unloading at dmax/r = 0.3 is required, see below.

δ 2 σ R 2 (1+ s2n )

1028

0

1000

2000

3000

4000

E σ R2 (1+ s2n ) Fig. 2. (a) Relationship between normalized indentation work and

E rR1 ð1þs1 nÞ

W t1 d31 rR1 ð1þs1 nÞ

for spherical indentation at d1/r = 0.1. The simple dimen-

sionless function (6) is also shown, which has no fitting parameter. (b) Relationship between normalized indentation work d3 r 2

W t2

R2 ð1þs2 nÞ

E and rR2 ð1þs 2 nÞ

for spherical indentation at d2/r = 0.3. The simple dimensionless function (7) is also shown, which has only one fitting parameter. (c) Relationship between normalized contact stiffness

S2 ð1þs2 nÞ Eðd2 df 0 Þ

E and rR2 ð1þs for spherical 2 nÞ

indentation at d2/r = 0.3. The dimensionless function (15) is also shown.

where me2 = 0.97, mp2 = 37.2, s2 = 1.34, eR2 = 0.016. Likewise, when d/r is varied between 0.1 and 0.3, dimensionless relationships similar to that in Eqs. (6) and (7) can

N. Ogasawara et al. / Mechanics of Materials 41 (2009) 1025–1033

be established and the generic formulation can be written as:

0

Wt d3 rR ð1 þ snÞ

1

¼@ E me rR ð1þsnÞ

11 1A þ : mp

ð8Þ

Both the representative strain and sphere factor depend on d/r, which can be fitted as:

eR ¼ 0:033  d=r þ 0:00616 2

s ¼ 5:31  ðd=rÞ þ 4:02  d=r þ 0:604:

ð9Þ ð10Þ

The elastic and plastic limits also depend on penetration. If the specimen is purely elastic, the Hertz solution (Johnson, 1985) dictates that

P¼

4 pffiffiffiffiffiffiffi E d3 r : 3

ð11Þ

By taking the limit E=rR ! 0 in Eq. (8), one can derive

me ¼

8 15

rffiffiffi r d

ð12Þ

which agrees very well with the results from FEM simulations as d/r is varied. On the other hand, mp may be derived from the theoretical solution of spherical indentation on rigid plastic material using slip line theory, where for d/r up to about 0.03, the relationship P = 25.38  rdry can be fitted from (Richmond et al., 1974). It then follows that

r mp ¼ 12:69  d

ð13Þ

when d/r = 0.1, the error of this formula with respect to mp1 is about 1%. Therefore the value of mp1, for shallower penetration, can be obtained from the slip line theory. However, with the increase of d/r, because the original solution (Richmond et al., 1974) was only intended for very shallow indentation, the error starts to increase and becomes about 12% for d/r = 0.3. Since the theoretical solution based on the slip line theory was not explicit and not focused on deep indentation, in order to pursue maximum accuracy and a simple/explicit formulation, based on extensive FEM analyses with varying d/r (between 0.1 and 0.3) the following functional form is fitted:

r mp ¼ 13:27   7:0: d

ð14Þ

Within the framework presented above, one can choose any two different values of d/r (between 0.1 and 0.3), and the corresponding normalized indentation works in the form of Eq. (8) represent the foundation of solving material plastic properties via the reverse analysis (see Section 4.1 for details). In the following we focus on the cases with d1/r = 0.1 and d2/r = 0.3, hence Eqs. (6) and (7) are employed, where among the elastic and plastic limits only mp2 needs to be fitted (according to Eq. (14)) and other coefficients can be derived analytically as described above. From Eqs. (6) and (7), the representative stresses rR1 and rR2 can be solved (which are related with R and n) if E is known.

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3.2. Formulation for unloading Unloading information is useful for determining elastic modulus concurrently with plastic properties, elaborated in Sections 4.2 and 4.3. The specimen is unloaded at dmax/r = d2/r = 0.3, and the contact stiffness is denoted as unloading 2 ð1þs2 nÞ , Fig. 1(c). It is found that when SEðd S2 ¼ dP=ddjd¼d2 2 df 0 Þ E is plotted as a function of rR2 ð1þs2 nÞ, the relationship is apparently independent of n, Fig. 2(c). Here, df’ is the reduced residual penetration, which is defined as the indentation depth near the end of unloading and at the moment P = 0.1Pmax = 0.1P2, see Fig. 1(b); the use of such a reduced residual penetration could avoid the uncertainties upon full unloading (Zhao et al., 2007). The values of rR2 (and the related eR2) and s2 are identical to that presented in Eq. (7). Furthermore, the relationship in Fig. 2(c) can be fitted as:

!2 E rR2 ð1 þ s2 nÞ ! E þ 1:33: þ 0:518  rR2 ð1 þ s2 nÞ

S2 ð1 þ s2 nÞ ¼ 2:29  105  Eðd2  df 0 Þ

ð15Þ

When this equation is combined with Eqs. (6) and (7), the elastic modulus E may be solved along with the plastic properties. 4. Measuring material properties The established new framework of spherical indentation is now employed to derive the material elastoplastic properties from the measured indentation load-depth curve. We consider the following scenarios in reverse analyses: in Section 4.1, we first assume E is known and solve the plastic properties; in Section 4.2, we use the developed method (Fischer-Cripps, 2001) to estimate the elastic modulus from contact stiffness, and then solve (R, n) following the method in Section 4.1; in Section 4.3, we solve the elastoplastic properties (E, R, n) concurrently. The performances of the different approaches are compared with each other, through the examples of ten representative materials. First, three ideal power-law materials, brass, gold, and aluminum are chosen whose constitutive relationships are assumed to obey Eq. (1) exactly. The Young’s modulus, yield stress and ultimate tensile strength of them are obtained from literature (Ashby and Jones, 1996), which are fitted and converted to power-law constitutive constants (E, R, n), tabulated in Table 1 and then used as the input data for the numerical indentation experiment to obtain the P–d curves. Next, a work-hardened copper and an annealed copper are chosen. These two coppers are real materials and their uniaxial stress–strain behaviors have been obtained from uniaxial tensile tests (Ogasawara et al., 2005), which are then directly employed as the input data for material properties in numerical FEM analyses of indentation test. Note that these two copper materials do not obey the power-law relationship exactly and thus only their Young’s moduli are given in Table 1. Finally, a group of 5 materials are taken from Alkorta et al. (2005), who have shown that there are combinations of

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N. Ogasawara et al. / Mechanics of Materials 41 (2009) 1025–1033

Table 1 Reference (input) values of elastoplastic properties of 10 materials. The two coppers are real materials whose stress–strain curves do not conform power-law Eq. (1) exactly, and thus their n and R are not listed; instead the original measured stress–strain curves (Ogasawara et al., 2005) are directly used as the input properties in numerical indentation experiments.

Al Au Br Annealed Cu W-hardened Cu mat 0.0 mat 0.1 mat 0.2 mat 0.3 mat 0.4

E (GPa)

n

R (MPa)

70.0 82.0 96.0 120.0 120.0 10.2 11.1 12.0 12.8 13.4

0.30 0.25 0.36 – – 0.00* 0.10* 0.20* 0.30* 0.40*

159.0 268.0 843.0 – – 10.2* 14.8* 21.6* 31.1* 45.2*

tionship used in Alkorta et al. (2005) was slightly different than Eq. (1), therefore, we still use their original relationship/parameters (Table 1) as the input material properties for forward analyses. However, later in order to examine the performance of our proposed techniques, these 5 materials are refitted into the form of Eq. (1) so as to obtain their effective power-law constants, and compared with the reverse analysis results in Table 2. 4.1. Determining material plastic properties From the indentation P–d loading curve, Wt1 and Wt2 are measured at d1/r and d2/r, respectively. When E is known, by solving (6) and (7) simultaneously, the representative stresses are:

*

The plastic properties of the five materials, mat 0.0–mat0.4, are taken directly from Alkorta et al. (2005). It is important to note that Alkorta et al. defined their power-law materials in a way that is different than Eq. (1). These material properties (along with the power-law formulation in Alkorta et al. 2005) are used in the forward analysis. Later, for comparison purposes, in Table 2 the stress–strain curves of the five materials are refitted to the form of Eq. (1) and then compared with our reverse analysis results.

materials that have identical load–displacement curves for Berkovich indentation – the proposed spherical indentation method can be further validated if it could successfully distinguish these 5 materials. Note that the power-law rela-

rR1

1 þ ns1 1 d3 ¼  þ 1 mp1 me1 E W t1

rR2 ¼

1 þ ns2 1 d3  þ 2 mp2 me2 E W t2

!1 ð16Þ !1 ð17Þ

:

Although n is unknown at this moment, by combining these two equations with



rR1 ¼ R eR1 þ rR2

rR1 n

ð18Þ

E  rR2 n ¼ R eR2 þ : E

ð19Þ

Table 2 Elastoplastic properties obtained from reverse analyses, and errors between the results and the input data shown in Table 1. The corresponding stress–strain curves are given in Figs. 3–5. E (GPa)

n

R (MPa)

Output

Error (%)

Output

Error (%)

Output

Error (%)

4.1 E known

Al Au Br Annealed Cu W-hardened Cu mat 0.0 mat 0.1 mat 0.2 mat 0.3 mat 0.4

– – – – – – – – – –

– – – – – – – – – –

0.28 0.24 0.34 0.47 0.09 0.00 0.11 0.20 0.29 0.37

4.2 2.3 6.4 – – 0.0 5.5 0.9 5.2 7.2

153.7 264.7 789.6 426.7 313.2 10.1 15.1 21.8 29.7 41.2

3.3 1.2 6.3 – – 0.0 2.2 0.9 4.7 8.8

4.2 E from F -C method

Al Au Br Annealed Cu W-hardened Cu mat 0.0 mat 0.1 mat 0.2 mat 0.3 mat 0.4

69.0 83.1 93.9 108.5 130.2 11.4 11.8 12.3 12.6 12.7

1.4 1.3 2.2 9.5 8.5 11.9 6.6 2.8 1.5 5.3

0.28 0.24 0.34 0.47 0.10 0.00 0.11 0.20 0.29 0.37

4.3 2.1 6.7 – – 0.0 7.3 1.2 5.2 7.4

153.6 264.9 787.7 426.1 316.3 10.1 15.2 21.9 29.6 41.1

3.4 1.1 6.6 – – 0.0 2.6 1.0 4.8 8.9

4.3 E solved Concurrently (Unknown)

Al Au Br Annealed Cu W-hardened Cu mat 0.0 mat 0.1 mat 0.2 mat 0.3 mat 0 4

69.9 79.5 90.4 113.1 114.2 10.2 11.1 11.9 12.5 12.8

0.2 3.0 5.9 5.7 4.8 0.3 0.3 0.5 2.4 4.0

0.28 0.24 0.34 0.48 0.09 0.00 0.11 0.21 0.29 0.38

5.7 3.4 6.4 – – 0.0 7.2 4.0 3.1 5.8

151.2 262.3 793.8 437.8 309.8 10.3 15.2 22.4 30.4 42.1

4.9 2.1 5.8 – – 0.0 2.9 3.3 2.3 6.7

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N. Ogasawara et al. / Mechanics of Materials 41 (2009) 1025–1033

The unknowns rR1, rR2, R and n can be readily solved. And ry = (En/R)1/(n-1) . For the ten material examples, once the plastic properties are obtained from the reverse analysis, the identified uniaxial stress–strain curves are drawn as thick solid lines in Fig. 3 and compared with the input properties (dash lines). The specific results and errors are also tabulated in Table 2. It can be seen that in general there is very good agreement between input and output parameters, and the present technique can effectively measure the plastic properties of ductile metals and alloys. Besides the ideal power-law materials (brass, gold, aluminum), the proposed method works quite well for the work-hardened and annealed coppers who do not obey the power-law behaviors exactly (Fig. 3a). In addition, thanks to the deep indentation method, the five materials proposed by (Alkorta et al., 2005) have distinct spherical indentation responses and can be easily and effectively distinguished by the proposed technique (Fig. 3b).

Al Au Au Brass Brass W-hardenedCu Cu W-hardened Annealed Cu Annealed Cu

stress (MPa)

500 400

W-hardened Cu Annealed Cu

200

Au

ð20Þ

where S2 is the contact stiffness at the onset of unloading at d2/r = 0.3, and a2 is the corresponding projected contact radius that can be estimated as:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R a2 ¼ ðd2 þ 3dS Þ; 2

ð21Þ

where dS = d2 - P2/S2 (see Fig. 1b). Subsequently, n and R are solved following the approach in Section 4.1. The comparisons between the uniaxial stress–strain curves obtained from the reverse analysis (thick solid lines) are compared with the true solution (dash lines) in Fig. 4 and with good agreement. From the results tabulated in Table 2, it can be seen that although there are some errors of E (up to about 10%) estimated

input EE from input fromF-C F-C

600

Al Al Au Au Brass Brass W-hardened W-hardenedCuCu Annealed Cu Annealed Cu

400

Brass

W-hardened Cu Annealed Cu

200

100

Au

Al

Al 0

(b) 30

0.05

0.1

0.15 strain

0.2

0.25

0.3

(b) 30

input E input E known known

15 10

0.05

0.1

0.15 strain

0.2

0.25

0.3

0.2

0.25

0.3

input EE from input fromF-C F-C mat 0.0 0.0 mat mat 0.1 0.1 mat mat 0.2 0.2 mat mat 0.3 0.3 mat mat 0.4 0.4 mat

25 stress (MPa)

20

0 0

mat 0.0 mat 0.0 mat 0.1 mat 0.1 mat 0.2 mat 0.2 mat 0.3 mat 0.3 mat 0.4 mat 0.4

25 stress (MPa)

S2 ; 2a2

(a)

Brass

300

20 15 10

5 0

E¼

input E known known input E

600

0

If the elastic modulus is not known, it can be obtained following (Fischer-Cripps, 2001):

stress (MPa)

(a)

4.2. Determining elastic and then plastic properties

5

0

0.05

0.1

0.15 strain

0.2

0.25

0.3

Fig. 3. Comparison between the uniaxial stress–strain curves based on input properties in Table 1 (dash lines), and reverse analysis results based on the method described in Section 4.1 with E known (thick solid lines). (a) Three ideal power-law materials and two real materials. (b) Five materials which have identical load–displacement curves for Berkovich indentation (Alkorta et al., 2005).

0

0

0.05

0.1

0.15 strain

Fig. 4. Comparison between the uniaxial stress–strain curves based on input properties in Table 1 (dash lines), and reverse analysis results based on method described in Section 4.2 where E is obtained from the FischerCripps method (thick solid lines). (a) Three ideal power-law materials and two real materials. (b) Five materials which have identical load– displacement curves for Berkovich indentation (Alkorta et al., 2005).

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by the Fischer-Cripps method (Fischer-Cripps, 2001), the plastic properties are still determined with reasonably good accuracy (comparing with the results in Section 4.1 where E is known exactly). The reason that the present technique is not very sensitive to the perturbation of E is me d31 E >> 1 and that according to Eqs. (16) and (17), if W t1 me d32 E >> 1, then the variation of E does not affect the soluW t2 tion of rR1 and rR2 in a significant way. This condition is roughly satisfied for ductile materials with E/rR2 > 400. In addition, for these ductile materials eR1E/rR1 > > 1 and eR2E/rR2 > > 1 are satisfied, which means that (according to Eqs. (18) and (19)) the perturbation of E does not strongly alter the plastic properties. Therefore, the proposed method works well for ductile metals and alloys with relatively large values of E/ry.

interesting that E derived this way is (on average) more accurate for these ductile materials than that from the Fischer-Cripps method (see last section). Again, the results of extracted plastic properties do not seem to be affected significantly by the perturbation of E . Although the reverse analysis still needs to be carried out numerically, we note that the present spherical indentation algorithm is much simpler than those proposed earlier (Cao and Lu, 2004; Cao et al., 2007; Huber and Tsakmakis, 1999; Lee et al., 2005; Zhao et al., 2006b) thanks to the more concise form of representative strainbased formulation. In addition, in case n = 0, for elasticperfectly plastic materials, by utilizing (16) and (17), the closed-form solution of material properties E and R = ry can be obtained:

4.3. Determining elastoplastic properties concurrently

E¼

By solving Eqs. (6), (7), and (15) concurrently, the elastoplastic properties (E, R, n) may be obtained numerically. The relevant results are given in Fig. 5 and Table 2 where the material properties identified from the reverse analysis show good agreement with the input values. It is quite

(a)

input EE unknown input unknown

600

Al Au Au Brass Brass W-hardened W-hardenedCu Cu Annealed Cu Annealed Cu

stress (MPa)

500 400

Brass

W-hardened Cu

300

Annealed Cu

200

Au

100

Al

0 0

(b)

0.1

0.15 strain

0.2

0.25

ðme2 mp1  me1 mp2 ÞW t1 W t2 mp1 mp2 ðme2 W t1 d32  me1 W t2 d31 Þ

ð23Þ

:

Compare with the Oliver–Pharr formula (Oliver and Pharr, 2004) for the same elastic-perfectly plastic material, this formulation does not require the measurement or estimation of the projected contact area, which is often difficult; in addition, unloading is not required in this particular case. For five representative elastic-perfectly plastic materials whose properties are varied as 25 < E/ry < 700 (which correspond to hard metals and ceramics that show nearly no strain hardening (Ashby, 1999)), the Young’s modulus and the yield stress are obtained from the explicit formulae (22) and (23), and compared with the input properties in Table 3. It is readily seen that this explicit method, which remarkably has only one fitting parameter, can obtain quite reasonable results for elastic-perfectly plastic materials. 5. Conclusion We propose a new framework of spherical indentation based on representative strain. The number of fitting parameters is significantly reduced comparing with previous spherical indentation studies, which enables a much simplified, and occasionally explicit, reverse analysis for obtaining the elastoplastic properties from indentation load-depth curves. In addition, most fitting parameters have clear physical meanings and backgrounds.

mat 0.0 mat 0.1 mat 0.2 mat 0.3 mat 0.4

20

ry ¼

ð22Þ

me1 me2 ðmp1 W t2 d31  mp2 W t1 d32 Þ

0.3

input E unknown

30 25

stress (MPa)

0.05

ð1  m2 Þðme1 mp2  me2 mp1 ÞW t1 W t2

15 10

Table 3 Comparison between the results based on the explicit method (Eqs. (22) and (23)) and the input data for several examples of elastic-perfectly plastic materials.

5 0

0

0.05

0.1

0.15 strain

0.2

0.25

0.3

Fig. 5. Comparison between the uniaxial stress–strain curves based on input properties in Table 1 (dash lines), and reverse analysis results based on method in Section 4.3 where E is unknown and solved concurrently (thick solid lines). (a) Three ideal power-law materials and two real materials. (b) Five materials which have identical load–displacement curves for Berkovich indentation (Alkorta et al., 2005).

E/cry

25 60 180 333 700

E (GPa)

ay (MPa)

Input

Output

Error (%)

Input

Output

Error (%)

50 90 90 10 210

59 95 81 9 218

17.9 5.2 10.0 9.0 3.9

2000 1500 500 30 300

2034 1564 520 30 298

1.7 4.2 4.0 1.4 0.7

N. Ogasawara et al. / Mechanics of Materials 41 (2009) 1025–1033

By practicing the proposed technique on ten representative materials, it is found that the elastic and plastic properties extracted from the reverse analyses are in good agreement with the true solutions. The method works not only very well for ideal power-law materials, but also for several real materials that do not obey power-law constitutive relationship exactly. In addition, the deep spherical indentation method can effectively distinguish special materials that are otherwise difficult to measure in sharp indentation. When the elastic modulus is known a priori, the proposed method can easily determine the plastic properties of the material using only information from the loading curve. Similarly, for an elastic-perfectly plastic material, the modulus and yield strength can be obtained explicitly from the loading P - d curve of spherical indentation. When the modulus is unknown, it can be either estimated from a previous study (Fischer-Cripps, 2001), or solved concurrently (using contact stiffness) with plastic properties. For ductile metals/alloys, the extracted plastic properties are not very sensitive to small errors of the determined elastic modulus. The findings in this paper contribute to understanding and simplifying spherical indentation analysis, and provide useful guideline of measuring elastoplastic properties using spherical indentation. The versatile framework may be extended to cases where residual stresses present (Zhao et al., 2006a), indentation on a thin film (Zhao et al., 2007), and materials with microstructure (Xiang et al., 2006). More systematic experimental verifications will be carried out in future. Acknowledgement The work is supported by NSF CMS-0407743 and NSF CMMI-CAREER-0643726. References Alkorta, J., Martinez-Esnaola, J.M., Gil Sevillano, J., 2005. Absence of oneto-one correspondence between elastoplastic properties and sharpindentation load-penetration data. J. Mater. Res. 20, 432–437. ANSYS, 2003. Ansys Release 8.0 Documentation. ANSYS Inc., Canonsburg, PA. Ashby, M.F., Jones, D.R.H., 1996. Engineering Materials I. Butterworth Heinemann. Ashby, M.F., 1999. Materials selection in mechanical design, second ed. Elsevier. Biwa, S., Storakers, B., 1995. An analysis of fully plastic Brinell indentation. J. Mech. Phys. Solids 43, 1303–1334. Bowden, F.P., Tabor, D., 1950. The Friction and Lubrications of Solids. Oxford University Press, Oxford. Bucaille, J.L., Stauss, S., Felder, E., Michler, J., 2003. Determination of plastic properties of metals by instrumented indentation using different sharp indenters. Acta Mater. 51, 1663–1678. Cao, G., Chen, X., 2006. Mechanisms of nanoindentation on single-walled carbon nanotubes: the effect of nanotube length. J. Mater. Res. 21, 1048–1070. Cao, Y.P., Lu, J., 2004. A new method to extract the plastic properties of metal materials from an instrumented spherical indentation loading curve. Acta Materialia 52, 4023–4032. Cao, Y.P., Qian, X., Huber, N., 2007. Spherical indentation into elastoplastic materials: indentation-response based definitions of the representative strain. Mater. Sci. Eng. A 454–455, 1–13. Chen, X., Hutchinson, J.W., Evans, A.G., 2005. The mechanics of indentation induced lateral cracking. J. Am. Ceram. Soc. 88, 1233– 1238.

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