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Material characterization by instrumented spherical indentation Minh-Quy Le ⇑ Department of Mechanics of Materials, School of Mechanical Engineering, Hanoi University of Science and Technology, No. 1, Dai Co Viet Road, Hanoi, Viet Nam International Center for Computational Materials Science, Hanoi University of Science and Technology, No. 1, Dai Co Viet Road, Hanoi, Viet Nam

a r t i c l e

i n f o

Article history: Received 12 March 2011 Received in revised form 11 October 2011 Available online 19 December 2011 Keywords: Elastic–plastic material properties Finite element analysis Hardness Instrumented indentation

a b s t r a c t It was illustrated by the author in the previous work that combinations between material properties and indentation parameters can be used as mixed parameters in dimensionless functions to capture the sharp indentation response of materials. These issues are further extended for spherical indentation in the present study. Instrumented spherical indentation was performed by a parametric ﬁnite element analysis for a wide range of materials with maximum indentation depth-indenter radius ratios rising from 0.01 to 0.3 to investigate several fundamental features within the frame work of limit analysis. Frictional effects are taken into account. Regarding dimensional analyses and using a Taylor series expansion, a new set of dimensionless functions is constructed for spherical indentation parameters and hardness associated to a 70.3° conical indenter. Based on formulated functions, a reverse analysis procedure is suggested to extract material properties and hardness from spherical indentation force-depth curves with respect to two different indentation depth-indenter radius ratios. Effects of indenter compliance on indentation parameters and reverse results are considered. The accuracy of the proposed method is studied and discussed by carrying out reverse and sensitivity analyses for 22 representative materials with rigid and deformable indenters. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Instrumented indentation tests are an important tool to explore various characteristics of materials on small scales, where it is difﬁcult to use other conventional testing techniques. Spherical indenter is one of the earliest studied through the Brinell hardness test and Hertz solution in 1900s (Johnson, 1985). Elastic modulus and hardness can be estimated from spherical indentation force-depth curves (Fischer-Cripps, 2001; Oliver and Pharr, 2004). Analogy between spherical and conical indentations is commonly considered (Johnson, 1985). Thus, it is possible to deduce material properties from the spherical indentation response at different depths. ⇑ Corresponding author. Address: Department of Mechanics of Materials, School of Mechanical Engineering, Hanoi University of Science and Technology, No. 1, Dai Co Viet Road, Hanoi, Viet Nam. Tel./fax: +84 4 38680047. E-mail address: [email protected] 0167-6636/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2011.10.008

Huber and Tsakmakis (1999a, 1999b) applied neural network and cyclic indentation tests to determine the stress– strain curve. Their method is complex since cyclic loading is needed. Nayebi et al. (2002) established an analytical framework to extract plastic properties of steels from spherical indentation loading data only. Ma et al. (2003) also used spherical indentation loading curves to estimate the plastic properties of power law and linear strain hardening materials. Their dimensionless functions were given in the form of ﬁgures; however no analytical framework was put forward. Lee et al. (2005, 2010) proposed numerical techniques which did not take full advantage of the varying rich information during loading. These methods are still based on the determination of the contact area and require a relatively complex algorithm. Collin et al. (2008) investigated a new method to estimate plastic properties of materials from cyclic spherical indentation. Their method was established only for steels with known elastic modulus, and also complex due to cyclic indentation. Recently, Jiang et al.

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M.-Q. Le / Mechanics of Materials 46 (2012) 42–56

(2009) modiﬁed the expanding cavity model by considering the piling-up effect to describe the relationship between indentation work and plastic properties. A method was then provided to determine plastic properties of materials from spherical indentation loading curves. More recently, Alcalá and Esqué-de los Ojos (2010) demonstrated correlations between hardness, yield strength, power-law strain hardening exponent, and Young’s modulus, and suggested a hardnessbased method to extract mechanical properties from spherical indentation at multiple depths. The concept of representative strain in sharp indentation (Atkins and Tabor 1965; Dao et al., 2001; Ogasawara et al., 2005) have been also extended to construct dimensionless relationships in instrumented spherical indentation (Cao and Lu, 2004; Zhao et al., 2006; Cao et al., 2007; Ogasawara et al., 2009) for material characterization. It should be emphasized that most of previous spherical indentation methods can only determine plastic properties of materials and require that the elastic modulus must be a priori known for the reverse analysis (Ahn and Kwon, 2001; Nayebi et al., 2002; Cao and Lu, 2004; Beghini et al, 2006; Kucharski and Mroz, 2007; Collin et al., 2008; Jiang et al., 2009), while others are still based on the determination of contact area (Lee et al., 2005; Kucharski and Mroz, 2007; Alcalá and Esqué-de los Ojos, 2010; Lee et al., 2010) or require cyclic indentation (Huber and Tsakmakis, 1999a; 1999b; Collin et al., 2008) or average contact pressures at multiple indentation depths (Alcalá and Esqué-de los Ojos, 2010), making the measurement process and reverse analysis complicated in practice. Moreover, studies on shallow indentation depths (Huber and Tsakmakis, 1999a; 1999b, Nayebi et al., 2002; Ma et al., 2003; Cao and Lu, 2004; Cao et al., 2007; Lee et al., 2005; Beghini et al, 2006; Collin et al., 2008; Alcalá and Esqué-de los Ojos, 2010) may not lead to unique solutions (Chen et al., 2007), and hence cause signiﬁcant errors in practice (Herbert et al., 2006). Recently, Le (2008, 2009, 2011) proposed a new approach to consider instrumented sharp indentation response. Regarding dimensional analysis and based on extensive ﬁnite element analysis (FEA), the author illustrated that combinations between material properties and indentation parameters can be used as mixed parameters in dimensionless functions for instrumented sharp indentation. He also suggested a dual sharp indenter method for material characterization. In the present work, this approach for sharp indentation is extended to spherical indentation to formulate a new set of dimensionless functions within the frame work of limit and dimensional analyses and with the aid of a Taylor series expansion. A simple method is hence suggested to estimate the elastic modulus, yield strength, strain hardening exponent and Vickers hardness of materials from spherical indentation force-depth curves.

simple elasto-plastic, true stress–true strain behavior is assumed to be:

r ¼ Ee; ðr 6 YÞ r ¼ Y EYe n ; ðr P YÞ;

ð1Þ

where E is the Young’s modulus, Y the initial compressive uniaxial yield stress, and n the strain hardening exponent. For the spherical indentation of elastic–plastic substrates as schematically shown in Fig. 1, the indentation force P during loading can be expressed as (Cao and Lu, 2004):

P ¼ PðE; m; Ei ; ;mi ; Y; n; h; RÞ;

ð2Þ

where E and m, and Ei and mi are the elastic modulus and Poisson’s ratio of the indented material and the indenter, respectively. h and R are the indentation depth, and the indenter radius, respectively. Using the reduced Young’s modulus (Johnson, 1985), Eq. (2) can be reduced to

P ¼ PðE ; Y; n; h; RÞ;

ð3Þ

where

E ¼

1 1 m2 1 m2i : þ E Ei

ð4Þ

Applying the P-theorem in dimensional analysis, Eq. (3) can be written under the following form: 2

P ¼ E h f1

Y h : ; n; E R

ð5Þ

Eq. (5) provides the total indentation work with the maximum indentation depth hm:

Wt ¼

Z

hm

0

2

Pdh ¼ E hm f2

E hm : ; n; Y R

ð6Þ

Conducting similar analysis, the indentation force P during unloading can be expressed as:

Pu ¼ Pu ðE ; Y; n; h; hm ; RÞ;

ð7Þ

and under a dimensional form 2

Pu ¼ E h f3

Y hm h : ; n; ; E R hm

ð8Þ

P Wt=Wp+We

P

R h

S

he=hm-hr Pm

Loading Unloading Wp

We h

2. Theoretical backgrounds and ﬁnite element model 2.1. Theoretical backgrounds Elastic-plastic behavior of many engineering solid materials can be modeled by a power law description. A

hr

(a)

hm

(b)

Fig. 1. Schematic representation of a spherical indentation: (a) axisymmetric model of the indenter and specimen, and (b) typical indentation load-depth curve and notations.

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M.-Q. Le / Mechanics of Materials 46 (2012) 42–56

Note that Pu = 0 when h = hr, thus from Eq. (8), we obtain

hr Y hm : ¼ f4 ; n; E hm R

ð9Þ

Combining Eqs. (8) and (9) yields the elastic indentation work

We ¼

Z

hm

hr

2

P u dh ¼ E hm f5

E hm ; n; : Y R

ð10Þ

Combining Eqs. (6) and (10) yields

Wt E ¼ f6 ; n; r : We Y

ð11Þ 3

where r = hm/R. If we note C ¼ W t =hm , Eq. (6) can be put into the alternative forms as follows

E E ¼ f7a ; n; r ; C Y

ð12aÞ

E E ¼ f7b ; n; r ; Y C

ð12bÞ

E E ; ;r : Y C

ð12cÞ

n ¼ f7c

Substituting E⁄/Y from Eq. (12b) into Eq. (11) yields

Wt E ¼ f8a ; n; r : We C

ð13Þ

Eq. (11) can be put into another alternative form for r1 = 0.1 as follows

E ¼ f9 Y

Wt We

;n :

ð14Þ

0:1

depth-indenter radius ratios are considered in the present study. The mesh was adapted separately for each geometry. Therefore, several meshes were used. A typical mesh comprises about approximately 5000 large strain fournode and eight-node axisymmetric elements, and 14800 nodes. Frictionless roller boundary conditions were applied along the centerline and bottom. Outside surfaces were taken as free surfaces. Residual stresses were not taken into account in the analysis. Displacement-controlled procedure was used in this work. The Coulomb’s friction law is used between contact surfaces. Since typical friction coefﬁcient between well polished metallic surfaces and diamond lies within 0.1 and 0.15 (Tabor, 1951), the friction coefﬁcient k is taken as 0.15. This value is also consistent with most literature in the area of indentation simulation. Nevertheless, friction coefﬁcient is also varied in Section 4.1 to explore its effects. A large number of different combinations of elasto-plastic properties with n ranging from 0 to 0.6 and Y/E ranging from 5.0E5 to 6.0E2 were used for maximum indentation depth-indenter radius ratio r = 0.1, 0.15, 0.2, 0.25, and 0.3. Additional FEA were also carried out for r = 0.01 and 0.03 to explore effects of indentation depths to the indentation response. The Poisson’s ratio is ﬁxed at 0.3. The wide range of model materials covers most of metals and engineering alloys. Model materials are assumed to obey Von Mises criterion. The material properties used in the computations are given in detail in (Le, 2009). Spherical indenters were modeled as rigid bodies in most of simulations in the present study. Elastic indenter is also considered in Section 4.2 and 5 to explore its effects on reverse results. Elastic indenters were meshed by around 500 and 700 four-node axisymmetric elements for r = 0.1 and 0.3, respectively.

Combining Eq. (11) and (14) yields

Wt ¼ f10 We

Wt We

; n; r :

ð15Þ

3. Dimensionless relationships for P–h curve

0:1

Substituting n from Eq. (12c) with r1 = 0.1 into Eq. (12a) yields

E E E ; ;r : ¼ f11 C C 0:1 Y

ð16Þ

2.2. Finite element model Since the indentation problem of a rigid sphere into half-space is axisymmetric (Fig. 1a) only one-half of the system is used in the modeling. Therefore, elastic-plastic indentation was simulated using the axisymmetric capacities of the ABAQUS ﬁnite element code. The specimen was modeled as a large cylinder. The radius and the height of the sample are equal or ﬁfty times larger than the contact radius at maximum indentation depth. These dimensions were found to be large enough to approximate a semi-inﬁnite half-space for indentation. This was evidenced by an insensitivity of calculated results to further increase in specimen size. Elements were ﬁnest in the central contact area and became gradually coarser outwards. At the maximum indentation depth, no less than 100 elements came into contact. Several values of maximum indentation

3.1. Dimensionless relationships at given strain hardening n Relationships between Wt/We and E⁄/C, and between (Wt/We)r1 and (Wt/We)r2 are here considered using limit analysis. Relationship between indentation force and depth for spherical indentation into an elastic material is given as follow (Johnson, 1985):

P¼

4 E 3

qﬃﬃﬃﬃﬃﬃﬃﬃ 3 h R:

ð17Þ

Rh 3 Note that W t ¼ 0 m Pdh and C ¼ W t =hm . (a) For elastic limit, we have Wt/We ? 1, E⁄/C ? ce, where ce is E⁄/C ratio in elastic contact and determined as below:

E 15 ce ¼ ¼ 8 C

rﬃﬃﬃﬃﬃﬃ hm : R

ð18Þ

(b) For rigid-perfectly-plastic materials, we have Wt/ We ? 1, E⁄/C ? 1. For the relationships between Wt/We and E⁄/C, elastic limit (Wt/We?1) is clearly seen in Fig. 2a at shallow indentation depths, r = 0.01 and 0.03, and with high yield

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M.-Q. Le / Mechanics of Materials 46 (2012) 42–56

4

Y/E=0.004

3

(a)

n =0.4 r=0.01 r=0.03

Y/E=0.003

7

Y/E=0.002

W t /W e 2

9 8

(W t /W e) r

(a)

Y/E=0.016

Y/E=0.007

6 5

Y/E=0.02

4

n =0.4 r=0.03 r=0.15 r=0.3

3

ce=0.1875

2

ce=0.3248

1

1

0.1

0.2

0.3

0.4

0.5

0.6

0

1

2 (W t /W e)0.01

E*/C 200

(b)

150

Y/E=5e-4 W t /W e

210

4

Y/E=5e-4

180

Y/E=2.5e-4

150 (W t /W e) 0.3

(b)

3

100

Y/E=0.001

n=0 n=0.2 n=0.5

50

r =0.1 0 0

10

20

30

Y/E=0.002

90 60

n=0 n=0.2 n=0.5

30 40

0 0

E*/C

20

40

60

100

80

(W t /W e)0.1

Fig. 2. Relationship between Wt/We and E⁄/C for different maximum indentation depth-indenter radius ratios: (a) r = 0.01 and 0.03, and (b) r = 0.1.

strength-elastic modulus ratios Y/E. Fig. 2b is pertained to the plastic limit for a deeper indentation depth r = 0.1, Wt/ We and E⁄/C become large when Y/E tends to zero. For the relationships between (Wt/We)r1 and (Wt/We)r2, elastic limit (Wt/We ? 1) is clearly shown in Fig. 3a when one of two indentation processes is taken at a shallow indentation depth, r1 = 0.01, and high yield strength-elastic modulus ratio Y/E is involved. Plastic limit is indicated in Fig. 3b when Wt/We becomes large with very low values of Y/E for relatively deeper indentation depths. It is found that a linear law can be taken as a ﬁrst order of approximation for the relationship between Wt/We and E⁄/C at relatively deep indentation depths. Hence, if f is noted as deviations between initial values and regressed values of E⁄/C and regarding Eq. (13), f can be expressed as a function of We/Wt, n, and r. Therefore, the relationship between Wt/We and E⁄/C can be written as follows:

Wt E We ¼ K w1 ðn; rÞ þ K w2 ðn; rÞ þ f ; n; r : We C Wt

Y/E=0.001

120

ð19Þ

Noting x = We/Wt, it should be emphasized that x falls between 0 and 1. x = 0 and 1 for rigid-perfectly-plastic and elastic materials, respectively. It is supposed that f is a continuous and smooth function of x, n, and r. If x is now considered as a variable, and n and r play the role of parameters, f can be expanded as a Taylor series of x as below (Polyanin and Manzhirov, 2007):

Fig. 3. Relationships for Wt/We with respect to two different maximum indentation depth-indenter radius ratios: (a) (Wt/We)r and (Wt/We)0.01, and (b) (Wt/We)0.3 and (Wt/We)0.1.

f 0 ðx0 ; n; rÞ f 00 ðx0 ; n; rÞ ðx x0 Þ þ 1! 2! f ðmÞ ðx0 ; n; rÞ 2 ðx x0 Þm ðx x0 Þ þ þ m! þ Rem ðx; n; rÞ;

f ðx; n; rÞ ¼ f ðx0 ; n; rÞ þ

ð20Þ Rem ðx; n; rÞ

where x0 is a certain value between 0 and 1, and is the remainder term in the Taylor formula. Substituting Eq. (20) into Eq. (19) and using different notations for coefﬁcients yield: m X Wt E ¼ Ae ðn; rÞ þ A0 ðn; rÞ þ Ai ðn; rÞðx x0 Þi We C i¼1

þ Rem ðx; n; rÞ:

ð21Þ

Since |x x0| < 1 due to x and x0 2 (0, 1), higher order terms in Eq. (21) will vanish. The second order term in the Taylor series in Eq. (20) is here used to assume a high accuracy in reverse analysis:

Wt E We ¼ Ase ðn; rÞ þ As0 ðn; rÞ þ As1 ðn; rÞ We C Wt 2 We þ As2 ðn; rÞ : Wt

ð22Þ

By using a least square ﬁtting procedure, coefﬁcients of Eq. (22) are formulated as cubic polynomial functions of n and

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M.-Q. Le / Mechanics of Materials 46 (2012) 42–56

parabolic polynomial functions of r and given in the appendix for r 2 [0.1, 0.3]. It is also found that linear functional forms can be used for the relationship between (Wt/We)r and (Wt/We)0.1 as a ﬁrst order of approximation at relatively deep indentation depths. Conducting a similar analysis, Eq. (15) expressing relationship between (Wt/We)r and (Wt/We)0.1 can be written as below:

Wt We

Wt ¼ F w ðn; rÞ þ F 0 ðn; rÞ W e 0:1 r m X þ F i ðn; rÞðx x0 Þi þ Rw m ðx; n; rÞ;

ð23Þ

i¼1

where x = (We/Wt)r, and Rw m ðx; n; rÞ is the remainder term in the Taylor formula. The second order term in Taylor series in Eq. (23) is taken to assume a high accuracy in reverse analysis:

Wt We

r

Wt ¼ F sw ðn; rÞ þ F s0 ðn; rÞ W e 0:1 2 We We þ F s2 ðn; rÞ : þ F s1 ðn; rÞ W t 0:1 W t 0:1

ð24Þ

By using a least square ﬁtting procedure, coefﬁcients of Eq. (24) are formulated as cubic polynomial functions of n and parabolic polynomial functions of r and given in the appendix for r 2 [0.15, 0.3]. 3.2. Dimensionless relationships at given Y/E Fig. 4 plots the evolution of E⁄/C0.3 versus E⁄/C0.1 at different values of Y/E. It is found that Eq. (16) can be conveniently written under the following form

E E ¼ B1 C 0:3 C 0:1

B2 ð25Þ

:

By using a least square ﬁtting procedure, coefﬁcients B1 and B2 are determined as functions of E⁄/Y and given in the appendix. 3.3. Relationships between spherical indentation parameters and Vickers hardness Within the frame work of limit and dimensional analyses and with the aid of a Taylor series expansion, Le (2011) 80

E */C 0.3

60 40

E E We pﬃﬃﬃﬃﬃﬃﬃ ¼ K e ðnÞ þ K 0 ðnÞ þ K 1 ðnÞ C Wt CH 2 We þ K 2 ðnÞ ; Wt E Wt We þ M 0 ðnÞ þ M1 ðnÞ ¼ M w ðnÞ H We Wt 2 We þ M 2 ðnÞ : Wt

ð26aÞ

ð26bÞ

Hardness H is here deﬁned as the ratio of the maximum indentation force to the contact area under load or the mean pressure supported by the material under load. It should be further emphasized that in Eq. (26) hardness H and indentation parameters are corresponding to frictionless and frictional indentation, respectively. Since 70.3° conical indenters correspond to commonly used Berkovich and Vickers indenters, hardness H may be here considered as Vickers hardness. Hardness associated to a 70.3° conical indenter was estimated by FEA for model materials in (Le, 2009). In order to estimate Vickers hardness from instrumented spherical indentation, parameters of P–h curves for 70.3° conical indenter in Eq. (26) are now replaced by spherical indentation parameters. Conducting a similar analysis as in Section 3.1, relationship between Vickers hardness and parameters of spherical indentation forcedepth curves can be expressed as below:

E E W pﬃﬃﬃﬃﬃﬃﬃ ¼ K se ðn; rÞ þ K s0 ðn; rÞ þ K s1 ðn; rÞ e C Wt CH 2 We þ K s2 ðn; rÞ : Wt E Wt We þ M s0 ðn; rÞ þ M s1 ðn; rÞ ¼ M sw ðn; rÞ H We Wt 2 We þ M s2 ðn; rÞ : Wt

ð27aÞ

ð27bÞ

pﬃﬃﬃﬃﬃﬃﬃ Fig. 5 depicts variations of E = CH versus E⁄/C and variations of E⁄/H versus Wt/We at different strain hardening exponents n for spherical indentation with maximum indentation depth-indenter radius ratios r = 0.3. By using a least square ﬁtting procedure, coefﬁcients of Eq. (27) are established as cubic polynomial functions of n and parabolic polynomial functions of r, and given in the appendix for r 2 [0.1, 0.3].

Y/E 4. Effects of friction and indenter compliance

5e-5 5e-4 0.001

20

0

established dimensionless relationships between hardness and parameters of P–h curves for 70.3° conical indenters as below:

0

10

20

4.1. Frictional effects 30

E */C 0.1 Fig. 4. Evolution of E⁄/C0.3 versus E⁄/C0.1 at different values of Y/E.

Frictional effects in instrumented spherical indentation have been previously investigated (Cao et al., 2007; Lee et al., 2010). In general, frictional effects increase with indentation depth, the yield strength-elastic modulus ratio

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M.-Q. Le / Mechanics of Materials 46 (2012) 42–56

240

(a) Difference inC & W t /We, %

(a)

E */(CH )

1/2

180

120

r =0.3 n=0 n=0.2 n=0.5

60

0

4

k=0.2; % in C k=0.1; % in Wt/We k=0.2; % in Wt/We

2

0

-2

0

8

16

24

32

40

48

56

64

0

E*/C

350 300 250

r =0.3

150

n=0 n=0.2 n=0.5

100 50 15

30

45

60

75

90

6 4

r =0.3

2 0 -2

k=0.1; % in C k=0.2; % in C k=0.1; % in Wt/We k=0.2; % in Wt/We

-4 -6 0

0 0

(b) Difference inC & W t /We, %

400

200

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Y/E

(b) 450

E*/H

k=0.1; % in C

r =0.1

105

W t /W e Fig. 5. Relationship between hardness H associated to the 70.3° conical indenter, reduced elastic modulus E⁄, and spherical indentation parameters for the maximum indentation depth-indenter radius ratio r = 0.3 at different strain hardening exponents n: (a) E⁄/(CH)1/2 versus E⁄/C, and (b) E⁄/H versus Wt/We.

Y/E, or the strain hardening exponent n. Therefore, to explore frictional effects with various friction coefﬁcients, it is sufﬁcient to take elastic-perfectly-plastic materials into consideration. Fig. 6 shows relative differences in C and the total indentation work-elastic indentation work ratio Wt/We between k = 0.1 and k = 0.15 and between k = 0.15 and k = 0.2 for elastic-perfectly-plastic materials with Y/E 2 [0.001, 0.04] at the maximum indentation depth-indenter radius ratio r = 0.1 and 0.3. Frictional effects decrease rapidly with the increasing Y/E. Friction has very slight inﬂuence on the loading curve for the maximum indentation depth-indenter radius ratio r = 0.1 since deviations in C fall within ±1%. However, at the maximum indentation depth-indenter radius ratio r = 0.3, deviations in C caused by the uncertainties of the friction coefﬁcient,0:1 6 k 6 0:2;appear from 4.2% to 2.5%. Deviations in the total indentation work-elastic indentation work ratio Wt/We rise from 1.1% to 3.5%, and 4.3% to 5.6% for the maximum indentation depth-indenter radius ratio r = 0.1 and 0.3, respectively.

4.2. Effects of indenter compliance Twenty two representative materials, which were previously investigated (Le, 2009, 2011), are considered for

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Y/E

Fig. 6. Frictional effects on indentation parameters for: (a) r = 0.1 and (b) r = 0.3.

effects of indenter compliance in this section and for the reverse analysis in the next section. Their mechanical properties are listed in Table 1. Poisson’s ratio is taken as 0.3 otherwise it is noted. Some of the ﬁrst 16 metals and alloys in Table 1 have been investigated as representative materials by other authors (Cao and Lu, 2004; Zhao et al., 2006; Cao et al., 2007). The last six materials are rare groups of mystical materials with ﬁxed Poisson’s ratios (material 17 and 18, m = 0.3) and with varied Poisson’s ratios (materials 19, 20, 21, and 22) for conical indenters (Chen et al., 2007). To study effects of elastic indenter deformation, FEA were performed to simulate spherical instrumented indentation tests with rigid and elastic indenters for 22 representative materials. For the elastic indenter, the material properties of diamond is applied with Ei = 1140 GPa and mi = 0.07. Main indentation parameters of the representative materials are tabulated in Table A1 in the appendix. It is found from Fig. 7 that differences in C (or Wt) and Wt/We between the cases of elastic diamond indenter and rigid indenter increase overall with the increasing of H/Ei and E/Ei, respectively. Effects of indenter compliance on C (or Wt) at relatively shallow indentation depth r = 0.1 is higher than that at deeper indentation depth r = 0.3. At r = 0.1, effects of indenter compliance on C can be neglected (variation in C < 2%) for soft materials with H/Ei < 0.0008 (lead, aluminum, gold, silver, zinc, and copper). At r = 0.3, effects of

48

M.-Q. Le / Mechanics of Materials 46 (2012) 42–56

Table 1 Mechanical properties of the representative materials. Materials

Notations

E (GPa)

Y (MPa)

n

Y/E

Copper Aluminum Gold Lead Silver Tungsten Iron Titanium 1 Steel 1 Nickel Steel 2 Titanium 2 Aluminum alloy Ti-6Al-4 V Zinc Silicon Material 17 Material 18 Material 19 (m = 0.35) Material 20 (m = 0.3 Material 21 (m = 0.25) Material22(m = 0.2)

Cu Al1 Au Pb Ag W Fe Ti1 S1 Ni S2 Ti2 Al2 Ti3 Zn Si M17 M18 M19 M20 M21 M22

128 70 79 16 83 411 180 120 210 207 210 110 70 110 9 107 103.75 100 116.9 120 122.2 123.9

10 20 38 10 60 550 300 230 500 800 900 600 500 830 300 6000 715.61 872.47 659.4 691.8 725.5 762.8

0.5 0.15 0.22 0.05 0.27 0.005 0.25 0.12 0.1 0.4 0.3 0.1 0.122 0.15 0.05 0.025 0.10663 0 0.2038 0.1913 0.1784 0.1653

7.813E05 2.857E04 4.810E04 6.250E04 7.229E04 1.338E03 1.667E03 1.917E03 2.381E03 3.865E03 4.286E03 5.455E03 7.143E03 7.545E03 3.333E02 5.607E02 6.897E03 8.725E03 5.641E03 5.765E03 5.937E03 6.157E03

(a) H/Ei: 3e-5

6.4e-4 . 4

Pb Al1 Au

Ag

Zn

0.0013 .0

Cu Ti1

W Al2

0.0020 S1

Ti2

0.0028 0.

0.0111

Fe M18 M17 M19 M20 M21 M22 Ti3

S2

Ni

Si

Difference in C , %

0 -2 -4 -6

r=0.1

-8

r=0.3

(b) E/Ei: 0.008 Zn

0.061

0.069

Pb Al1 Al2 Au

0.094 Ag M18 M17 Si

0.105

0.112

Ti2 Ti3 M19 Ti1 M20 M21 M22 Cu

0.182 Fe

Ni

0.361 S1

S2

W

Difference in W t /W e, %

0 -5 -10 -15 -20

r=0.1

r=0.3

-25 Fig. 7. Differences between indentation parameters performed by the elastic diamond indenter and the rigid indenter for 22 representative materials: (a) in C, and (b) in Wt/We. All deviations were computed as (Xdiamond Xrigid)/Xrigid, where X represents a parameter.

indenter compliance on C are slight for most of 22 representative materials with H/Ei < 0.003, while variation in C is higher than 2.5% for harder materials with H/Ei > 0.005 such that steel 2, nickel, and silicon.

Effects of indenter compliance on Wt/We are almost similar for two indentation depths, r = 0.1 and 0.3. Variation in Wt/We due to the indenter compliance is neglected for materials with very low E/Ei such as zinc and lead

M.-Q. Le / Mechanics of Materials 46 (2012) 42–56

(E=Ei 6 0:014), while this variation is higher than 4%, 10%, and 22% for E=Ei P 0:06, 0.15 and 0.36, respectively.

5. Reverse and sensitivity analyses 5.1. Reverse procedure Using Eqs. (22), (24), (25), and (27a), a simple reverse method is here suggested for material characterization from force-depth curves of one spherical indentation test with two cycles performed at two different depths r1 = 0.1 and r2 = 0.3. Indentation parameters (Wt/We)0.1, (Wt/We)0.3, C0.1 and C0.3 are used as input data. The strain hardening exponent n is ﬁrst estimated from Eq. (24). The reduced elastic modulus E⁄, the yield strength Y and Vickers hardness H are then evaluated from Eq. (22), (25), and (27a), respectively. It should be emphasized that the strain hardening exponent is taken as 0 or 0.5 when Eq. (24) gives a solution being negative or higher than 0.5, respectively. It should be also noted that the yield strength Y is simply estimated from Eq. (25) when the reduced elastic modulus is a priori known.

5.2. Numerical veriﬁcation Although indenter compliance affects signiﬁcantly on P–h curves for most of metals and engineering alloys as shown in Section 4.2, numerical veriﬁcation of recent reverse procedures has been carried out only for indentation data performed with rigid indenters (Cao and Lu, 2004; Zhao et al., 2006; Ogasawara et al., 2009; Zhang et al., 2009) and hence effects of indenter deformation on reverse results were not well considered. Indentation data, which were numerically generated by FEA with rigid and elastic diamond indenters as tabulated in Table A1, are used as input for the reverse analysis to extract the mechanical properties of representative materials. Reverse results are in general good as shown in Tables 2–5. Fig. 8 shows P–h curves performed with elastic diamond indenters for six mystical materials. It is found that mystical materials with varied Poisson’s ratios are the most severe since their P–h curves are visually identical. In fact, their corresponding indentation parameters (see Table A1) exhibit very low deviation (within ±0.8%), whenever indenters are modeled as rigid or elastic diamond. Hence, Poisson’s ratio has a strong effect in such cases. Errors in reverse results of material 22 are slightly higher than those of the other mystical materials due to the large difference between its Poisson’s ratio and those of model materials. Errors of reverse results in the present study are also summarized in Table 7 in comparison with several spherical indenter methods from the literature. In the present method, deviations Dn rise from 0.02 to 0.01 for the strain hardening exponent n whenever rigid or elastic models are used for indenters. These deviations are much lower than those reported in (Zhang et al., 2009) for the method suggested by Zhao et al. (2006), Dn 0.25 for high n and low Y/E.

49

When the indenter is assumed to be rigid, errors appear within (0.8%, 1.5%), (2.4%, 3.3%), and (5.3%, 6.6%) for the elastic modulus E, hardness H, and yield strength Y, respectively. It should be emphasized that errors in the yield strength of other spherical indenter methods could reach 40% (Zhao et al., 2006), 10% (Jiang et al., 2009), and 15% (Ogasawara et al., 2009), see Table 7 for details. When indenter deformation is taken into account, hardness exhibits errors from 4.2% to 1.4%. Errors in Y are higher than 10% for copper, gold, silver, titanium 1, material 22; and lower than 10% for other materials. Errors fall in general within ±1.7% for the elastic modulus E, excepting tungsten (6.7% of error in the elastic modulus) due to its comparable elastic modulus with that of the diamond indenter (E/Ei = 0.361) and subsequently high effects of indenter compliance as shown in Fig. 7b. Cao et al. (2007) reported that indenter compliance induced in their reverse results around 10% of error in the reduced elastic modulus E⁄ of tungsten (about 13.8% of error in the elastic modulus E). Overall, the present method provides lower errors in reverse results than several spherical indenter methods from the literature as summarized in Table 7, although deformable indenters are taken into account in the reverse analysis. It should be here emphasized that the range of tested materials (7.81E5 6 Y/E 6 0.056; 0 6 n 6 0.5) in the present study is larger than those conducted in (Cao et al., 2007; Zhang et al., 2009; Ogasawara et al., 2009), while high errors were generally indicated in their work for materials of extremely low or high Y/E. At least four materials (copper, aluminum, zinc and silicon) among 22 representative materials considered in the present work fall well outside the range of tested materials investigated in (Zhang et al., 2009). Further, their veriﬁcation was performed with numerical indentation data, which ignored indenter deformation. It can be seen that low errors are found for the strain hardening exponent n, elastic modulus E, and hardness H, even though indenter compliance is taken into account. Rodrıguez et al. (2011) also reported that variation in indentation parameters did not signiﬁcantly decrease the capability to estimate hardness and elastic modulus based on the ratio of the residual indentation depth to maximum indentation depth, hr/hm, while this indentation parameter rose up to 26% due to the indenter deformation. 5.3. Sensitivity analysis Although effects of indenter compliance on reverse results are considered in the previous section, friction causes several percents of indentation uncertainties as indicated in Section 4.1. Therefore, indentation data performed with elastic diamond indenters in Table A1 are perturbed. Two cases of perturbations of the input data are considered in this work as listed in Table 6. Sensitivity analysis was carried out for 22 representative materials. Maximum relative errors in E, H, and Y, and deviations Dn of the strain hardening exponent n in each case are plotted in Figs. 9–11. The strain hardening exponent n shows low to moderate sensitivities. Deviations of n vary from 0.08 to 0.11

50

M.-Q. Le / Mechanics of Materials 46 (2012) 42–56

Table 2 Reverse results of the strain hardening exponent n for the representative materials. Materials

Original data

Reverse results Rigid indenter

Copper Aluminum Gold Lead Silver Tungsten Iron Titanium 1 Steel 1 Nickel Steel 2 Titanium 2 Aluminum alloy Ti-6Al-4 V Zinc Silicon Material 17 Material 18 Material 19 Material 20 Material 21 Material 22

Elastic diamond indenter

n

n

Dn

n

Dn

0.5 0.15 0.22 0.05 0.27 0.005 0.25 0.12 0.1 0.4 0.3 0.1 0.122 0.15 0.05 0.025 0.10663 0 0.2038 0.1913 0.1784 0.1653

0.5059 0.1557 0.2241 0.0582 0.2712 0.0135 0.2501 0.1219 0.1005 0.4041 0.2991 0.1016 0.1275 0.1562 0.0281 0.0332 0.1108 0.0052 0.2110 0.1930 0.1752 0.1593

0.006 0.006 0.004 0.008 0.001 0.009 0.000 0.002 0.001 0.004 0.001 0.002 0.006 0.006 0.022 0.008 0.004 0.005 0.007 0.002 0.003 0.006

0.4996 0.1571 0.2270 0.0638 0.2712 0.0000 0.2333 0.1194 0.0877 0.3908 0.2849 0.0883 0.1265 0.1522 0.0344 0.0126 0.0982 0.0000 0.2029 0.1872 0.1706 0.1539

0.000 0.007 0.007 0.014 0.001 0.005 0.017 0.001 0.012 0.009 0.015 0.012 0.004 0.002 0.016 0.012 0.008 0.000 0.001 0.004 0.008 0.011

Table 3 Reverse results of the elastic modulus E for the representative materials. Materials

Original data

Reverse results Rigid indenter

Copper Aluminum Gold Lead Silver Tungsten Iron Titanium 1 Steel 1 Nickel Steel 2 Titanium 2 Aluminum alloy Ti-6Al-4 V Zinc Silicon Material 17 Material 18 Material 19 Material 20 Material 21 Material 22

Elastic diamond indenter

E (GPa)

E (GPa)

% error E

E (GPa)

% error E

128 70 79 16 83 411 180 120 210 207 210 110 70 110 9 107 103.75 100 116.9 120 122.2 123.9

129.96 70.19 79.30 16.18 83.31 417.36 179.97 119.18 208.26 209.72 210.55 109.25 69.84 109.94 8.97 107.41 103.39 100.48 116.59 119.66 121.72 123.33

1.5 0.3 0.4 1.1 0.4 1.5 0.0 0.7 0.8 1.3 0.3 0.7 0.2 0.1 0.4 0.4 0.4 0.5 0.3 0.3 0.4 0.5

130.15 70.73 79.94 16.23 83.71 438.67 179.41 120.22 211.61 208.59 210.38 109.14 69.85 109.98 8.89 106.73 103.10 100.89 116.45 119.61 121.68 123.18

1.7 1.0 1.2 1.4 0.8 6.7 0.3 0.2 0.8 0.8 0.2 0.8 0.2 0.0 1.2 0.2 0.6 0.9 0.4 0.3 0.4 0.6

in case 1, and from 0.12 to 0.16 in case 2. High sensitivity of n appears for materials of low strain hardening and high yield strength-elastic modulus ratio Y/E (zinc and silicon in the present study). Elastic modulus E exhibits relatively low sensitivity from 8% to 15% in case 1, from 10% to 17% in case 2. Sensitivities of hardness H fall in the ranges (10%, 8%) and

(12%, 11%) in cases 1 and 2, respectively. High error and sensitivity in elastic modulus and hardness appear in tungsten due to its comparable elastic modulus with that of the diamond indenter (E/Ei = 0.361). It should be emphasized that errors in Vickers hardness estimated by the contact area can rise to 15% due to the typical friction between the indenter and specimen (Mata and Alcalá, 2004).

51

M.-Q. Le / Mechanics of Materials 46 (2012) 42–56 Table 4 Reverse results of the yield strength Y for the representative materials. Materials

Original data

Reverse results Rigid indenter

Copper Aluminum Gold Lead Silver Tungsten Iron Titanium 1 Steel 1 Nickel Steel 2 Titanium 2 Aluminum alloy Ti-6Al-4 V Zinc Silicon Material 17 Material 18 Material 19 Material 20 Material 21 Material 22

Elastic diamond indenter

Y (MPa)

Y (MPa)

% error Y

Y (MPa)

% error Y

10 20 38 10 60 550 300 230 500 800 900 600 500 830 300 6000 715.61 872.47 659.4 691.8 725.5 762.8

10.32 21.18 37.93 10.45 58.40 579.98 288.26 222.60 493.55 800.71 917.44 614.25 512.88 853.84 306.18 5998.78 733.94 891.46 703.04 711.22 717.42 722.44

3.2 5.9 0.2 4.5 2.7 5.5 3.9 3.2 1.3 0.1 1.9 2.4 2.6 2.9 2.1 0.0 2.6 2.2 6.6 2.8 1.1 5.3

8.51 18.34 33.14 9.26 51.54 551.68 272.25 201.57 464.40 749.26 877.95 568.93 467.81 788.72 278.85 5483.47 679.99 823.05 652.08 659.34 664.04 669.24

14.9 8.3 12.8 7.4 14.1 0.3 9.2 12.4 7.1 6.3 2.5 5.2 6.4 5.0 7.1 8.6 5.0 5.7 1.1 4.7 8.5 12.3

Table 5 Reverse results of the hardness H for representative materials. Materials

Copper Aluminum Gold Lead Silver Tungsten Iron Titanium 1 Steel 1 Nickel Steel 2 Titanium 2 Aluminum alloy Ti-6Al-4 V Zinc Silicon Material 17 Material 18 Material 19 Material 20 Material 21 Material 22

FEA

Reverse results

Le (2009)

Rigid indenter

H (GPa)

H (GPa)

% error H

H (GPa)

% error H

0.8902 0.1313 0.3345 0.0345 0.6077 1.452 2.225 1.006 1.978 7.174 5.937 2.172 1.84 3.235 0.7262 12.68 2.548 2.295 3.169 3.187 3.197 3.210

0.8990 0.1337 0.3396 0.0350 0.6170 1.492 2.263 1.018 1.992 7.250 6.010 2.190 1.872 3.296 0.7498 12.741 2.586 2.325 3.244 3.211 3.169 3.133

1.0 1.8 1.5 1.3 1.5 2.8 1.7 1.2 0.7 1.0 1.2 0.8 1.8 1.9 3.3 0.5 1.5 1.3 2.4 0.7 0.9 2.4

0.8776 0.1319 0.3355 0.0347 0.6071 1.459 2.195 1.001 1.945 6.955 5.818 2.137 1.839 3.234 0.7365 12.169 2.526 2.291 3.171 3.146 3.109 3.074

1.4 0.5 0.3 0.5 0.1 0.4 1.3 0.5 1.7 3.1 2.0 1.6 0.0 0.0 1.4 4.0 0.9 0.2 0.1 1.3 2.8 4.2

Yield strength Y presents high sensitivities but still in reasonable accuracy for most of representative materials. Maximum variations of Y appear in the ranges (43%, 41%) and (55%, 64%) for other materials in cases 1 and 2, respectively. Errors in the yield strength are exceptionally high for copper, 98% in case 1 and 170% in case 2.

Elastic diamond indenter

High sensitivity for the yield strength in copper and for the strain hardening exponent in zinc and silicon may be explained by the boundary effects since their material properties fall in the neighborhood of the bounds of the range of model materials used in the simulation to formulate dimensionless functions in the present study.

52

M.-Q. Le / Mechanics of Materials 46 (2012) 42–56

0.006 M17 M18

0.005

2

P /(E iR )

M19

0.004

M20 M21

0.003

M22

0.002 0.001 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

h/R Fig. 8. Indentation force-depth curves performed with elastic diamond indenters for 6 mystical materials.

Table 6 Case study for sensitive analysis of the representative materials. Case study

Changes in the input data

(1) (2)

±2% error in C0.1, C0.3, (Wt/We)0.1 and (Wt/We)0.3. ±2% error in C0.1, C0.3, and (Wt/We)0.1, and ±4% errors in (Wt/We)0.3.

It is found that if the input data is all added by the same error, the reverse results are better, than for the case that only one input has errors. Indenter compliance induces relatively low errors because indentation data is all decreased due to the indenter compliance for two indentation depths, r = 0.1 and 0.3. Especially, effects of indenter compliance on Wt/We are almost similar for two indentation depths, r = 0.1 and 0.3. It was also noted by (Zhang et al., 2009) for spherical indenter methods and by (Swaddiwudhipong et al., 2005; Lan and Venkatesh, 2007; Le, 2011) for dual sharp indenters methods that several percents of indentation errors could cause large variations in reverse results of the yield strength especially for high n and low Y/E. For example, Zhang et al. (2009) indicated that variations in Y could reach 60% for their considered materials in the method by (Zhao et al., 2006) when indentation force corresponding to h = 0.13R was perturbed by ±3%. Features of mystical materials, which seem to share or closely share the same indentation force-depth curves as described earlier by (Chen et al., 2007) were cohesively related to characteristics of low sensitivities in the reduced elastic modulus E⁄ and hardness H as previously discussed in (Le, 2008, 2009) for sharp indentation. Le (2009) demonstrated that mystical materials exhibit not only fair differences in their elastic modulus as pointed out by (Chen et al., 2007) but also in their hardness. These two features are related to low variations of

Table 7 Comparison of the present study with other spherical indenter methods. Comparison Present study

Zhao et al. (2006) reported by Zhang et al. (2009).

Jiang et al. (2009) reported by Zhang et al. (2009).

Cao et al. (2007)

Ogasawara et al. (2009)

Estimation

E, Y, n, and H

E, Y and n

E, Y and n

E, Y and n

Tested range

22 materials, 7.81E5 6 Y/E 6 0.056, 0 6 n 6 0.5

0.0003 6 Y/E 6 0.03, 0 6 n 6 0.5

Y and n. E is a priori known 0.0003 6 Y/E 6 0.03, 0 6 n 6 0.5

10 materials, 0.001 < Y/E 6 0.03, 0 6 n 6 0.4

Errors

For rigid indenters: 0.8 6 %E 6 1.5%, 5.3% 6 %Y 6 6.6%, 0.022 6 Dn 6 0.009, 2.4% 6 %H 6 3.3%

Rigid indenters: For high n and low Y/ E:%Y 40%, Dn 0.25. Mostly:%Y < 20%, Dn < 0.15

Rigid indenters: For high n and low Y/ E:%Y 10%, Dn 0.05. Mostly:%Y < 5%, Dn < 0.02

9 materials, 2.8E4 < Y/ E 6 0.056, 0.005 6 n 6 0.27 Rigid indenters: .25% 6 %E 6 2.5%, 6.5% 6 %Y 6 2%, 0.01 6 Dn 6 0.02

For elastic indenters: 1.2 6 %E 6 6.7%, 14.9% 6 %Y 6 0.3%, 0.017 6 Dn 6 0.014, 4.2% 6 %H 6 1.4%

Rigid indenters: 5.8% 6 %E 6 0%, 15% 6 %Y 6 5.9%, 0.02 6 Dn 6 0.01

0.18

Case 1 Case 2

0.15

n

0.12 0.09 0.06 0.03 0.00 Cu Al1 Au

Pb

Ag

W

Fe

Ti1

S1

Ni

S2 Ti2 Al2 Ti3

Zn

Si M17 M18 M19 M20 M21 M22

Fig. 9. Sensitivity study on the strain hardening exponent n.

53

M.-Q. Le / Mechanics of Materials 46 (2012) 42–56

18

Case 1, % Error E Case 1, % Error H

% Error E & H

15

Case 2, % Error E Case 2, % Error H

12 9 6 3 0 Cu Al1 Au

Pb

Ag

W

Fe

Ti1

S1

Ni

S2

Ti2 Al2 Ti3

Zn

Si M17 M18 M19 M20 M21 M22

Fig. 10. Sensitivity study on the elastic modulus E and hardness H.

65

% Error Y

Case 1 Case 2 50

35

20 Cu Al1 Au

Pb

Ag

W

Fe

Ti1

S1

Ni

S2

Ti2 Al2 Ti3

Zn

Si M17 M18 M19 M20 M21 M22

Fig. 11. Sensitivity study on the yield strength Y.

elastic modulus and hardness due to uncertainties of input data (Le, 2008, 2009). The present study considers small deviations in corresponding indentation parameters of such materials as perturbation, hence reverse solution can be obtained in such severe cases without any special treatment.

6. Conclusions In the present work, several fundamental features in instrumented spherical indentation are investigated by conducting an extensive FEA for a wide range of maximum indentation depth-indenter radius ratios within the frame work of limit analysis. The main results are summarized as follows: The approach suggested by the author in the previous work for sharp indentation (Le, 2008, 2009, 2011) is successfully extended to spherical indentation. Based on dimensional analyses and a Taylor series expansion, a new set of dimensionless functions is constructed for spherical indentation parameters. Hardness H associated to a 70.3° conical indenter is also functionally related to characteristics of spherical indentation force-depth curves and material properties. Effects of friction and indenter compliance are studied. It is found that indenter compliance decreases the total indentation work Wt and the total indentation work-elastic indentation work ratio Wt/We. Differences

in C and Wt/We between the cases of elastic diamond indenter and rigid indenter increase overall with the increasing of H/Ei and E/Ei, respectively. Using formulated explicit equations, a reverse analysis procedure is proposed to extract material properties and hardness from spherical indentation force-depth curves with respect to two indentation depth-indenter radius ratios, r = 0.1 and 0.3. Reverse results are very good for 22 representative materials when the indenter is assumed to be rigid. When indenter deformation is taken into account, reverse results remain overall accurate for the strain hardening exponent n, hardness H and elastic modulus E. However, errors in elastic modulus is around 7% for materials, which exhibit a comparable elastic modulus with that of the indenter (E/Ei = 0.36). Indenter compliance reduces slightly the capacity to estimate the yield strength Y by the proposed reverse method, since this plastic property is underestimated from 5% to 15% for most of 22 representative materials. Comprehensive sensitivity analyses are carried out for indentation data performed by elastic diamond indenters. In practice, frictional effects, imperfection of indenter tips, indenter compliance, etc. . . are inevitable. In addition, real materials do not obey power law strain hardening exactly. Such mentioned factors cause uncertainties of input data and hence several percent of indentation error is very common in practice. Therefore, it may be practically difﬁcult to estimate accurately the strain hardening exponent and yield strength due to their moderate sensitivity. How-

54

M.-Q. Le / Mechanics of Materials 46 (2012) 42–56

ever, much less sensitivities were found for the elastic modulus and hardness, and their predicted values remain reasonable for a wide range of materials with certain perturbations of the input data.

As2 ¼ ð1156:3r 2 þ 558:6r 28:237Þn3 þ ð1486:2r 2 766:1r þ 31:424Þn2 þ ð594:71r 2 þ 342:27r 11:881Þn þ ð56:779r 2 44:645r þ 0:7991Þ: The coefﬁcients in Eq. (24) for r 2 [0.15, 0.3]:

F sw ¼ ð21:444r2 1:03r þ 0:1212Þn3 þ ð36:421r2

Acknowledgements This work was supported by Vietnam National Foundation for Science & Technology Development (NAFOSTED), Ministry of Science & Technology, under the basic research program.

þ 2:1388r 0:3977Þn2 þ ð20:0022r2 6:7223r þ 0:8265Þn þ ð2:7381r2 þ 5:3594r þ 0:4606Þ; F s0 ¼ ð206:031r 2 þ 115:31r 8:5921Þn3 þ ð232:094r 2 158:166r þ 12:5065Þn2 þ ð89:5883r 2 þ 82:4664r 7:1986Þn þ ð7:4885r 2 15:6244r

Appendix A.

þ 1:5014Þ; Table A1. The coefﬁcients in Eq. (22) for r 2 [0.1, 0.3]: 2

2

Ase ¼ ð1:4942r þ 1:673r þ 2:204Þn þ ð22:815r

F s1 ¼ ð666:08r 2 266:1r þ 14:736Þn3 þ ð887:2r 2 2

þ 391:85r 24:915Þn2 þ ð427:4r 2 215:76r

2

þ 12:527r 6:1263Þn þ ð37:763r 23:17r

þ 16:284Þn þ ð61:616r2 þ 39:25r 3:4Þ;

þ 7:0349Þ;

F s2 ¼ ð487r 2 þ 153:57r 4:22Þn3 þ ð708:45r 2 2

3

As0 ¼ ð363:96r þ 148:23r 4:3654Þn þ ð439:68r

2

247:586r þ 11:403Þn2 þ ð377:034r2 þ 153:52r

197:51r þ 1:6082Þn2 þ ð165:02r 2 þ 88:933r

10:269Þn þ ð63:216r 2 31:819r þ 2:595Þ:

2

þ 2:5137Þn þ ð13:873r 12:323r 1:7972Þ;

It is noted that y = ln(E⁄/Y). The coefﬁcients in Eq. (25):

B1 ¼ ð0:0000476y6 þ 0:0020358y5 0:0356359y4

As1 ¼ ð1243:4r 2 533:76r þ 23:416Þn3 þ ð1573:2r 2 þ 725:63r 21:235Þn2 þ ð627:4r 2 325:77r

þ 0:327875y3 1:68653y2 þ 4:682359y

þ 6:0425Þn þ ð60:715r 2 þ 42:393r þ 0:0598Þ;

5:106263Þ1 ;

Table A1 Indentation parameters of the representative materials. Materials

Cu Al1 Au Pb Ag W Fe Ti1 S1 Ni S2 Ti2 Al2 Ti3 Zn Si M17 M18 M19 M20 M21 M22

Rigid indenter

Elastic diamond indenter

C0.1 (GPa)

C0.3 (GPa)

(Wt/We)0.1

(Wt/We)0.3

C0.1 (GPa)

C0.3 (GPa)

(Wt/We)0.1

(Wt/We)0.3

19.701 4.7165 10.533 1.4321 17.472 63.554 62.142 33.657 66.448 151.100 139.269 63.043 49.099 82.415 13.221 183.790 69.882 69.403 80.614 80.712 80.578 80.503

8.4532 1.5992 3.7970 0.4502 6.5800 19.579 23.645 11.753 23.120 65.080 56.933 23.294 18.794 32.165 5.7972 86.800 26.443 25.262 31.507 31.578 31.556 31.560

22.022 69.612 32.383 57.349 19.261 33.615 11.467 15.754 13.957 4.4892 5.3094 6.9773 5.4071 4.9290 1.9780 1.4663 5.7182 5.7565 5.5406 5.5165 5.4925 5.4694

31.141 132.791 57.240 121.463 32.361 74.167 19.296 30.047 27.020 6.5333 8.3148 12.840 9.5136 8.4228 3.1462 2.1974 10.244 11.218 9.2173 9.2872 9.3580 9.4187

19.310 4.6383 10.336 1.4069 17.115 61.606 60.148 32.802 64.279 142.010 131.665 60.989 47.665 79.262 12.981 168.955 67.453 67.042 77.574 77.664 77.531 77.460

8.4270 1.6030 3.7977 0.4511 6.5682 19.502 23.429 11.721 22.959 63.214 55.503 23.094 18.649 31.733 5.805 82.892 26.163 25.030 31.097 31.160 31.139 31.142

20.057 66.643 30.760 57.390 18.233 25.313 10.064 14.525 12.064 3.9040 4.5949 6.4606 5.1769 4.5726 1.9808 1.3671 5.3176 5.3721 5.0964 5.0751 5.0550 5.0342

28.461 126.882 54.181 120.768 30.594 57.338 17.067 27.666 23.450 5.6853 7.1939 11.928 9.0743 7.7750 3.1450 2.0360 9.5400 10.488 8.4683 8.5175 8.5754 8.6338

M.-Q. Le / Mechanics of Materials 46 (2012) 42–56

B2 ¼ ð0:0000274y6 0:0009419y5 þ 0:0119572y4 3

2

0:062617y þ 0:041365y þ 0:815711y 1:565274Þ1 : The coefﬁcients in Eq. (27a) for r 2 [0.1, 0.3]:

K se ¼ ð101:91r 2 þ 61:63r 11:605Þn3 þ ð150:809r 2 90:82r þ 16:452Þn2 þ ð115:602r 2 þ 71:174r 13:141Þn þ ð62:847r2 40:854r þ 10:428Þ; K s0 ¼ ð29:387r 2 44:32r þ 6:0393Þn3 þ ð21:253r 2 þ 50:907r 12:5586Þn2 þ ð0:319r 2 17:907r þ 9:1070Þn þ ð0:5486r 2 þ 2:5991r 2:9447Þ; K s1 ¼ ð114:3r 2 þ 174:1r 3:840Þn3 þ ð12:574r2 136:79r þ 12Þn2 þ ð49:013r 2 þ 13:426r 7:4198Þn þ ð7:3477r 2 þ 2:0862r þ 1:9055Þ; K s2 ¼ ð50:235r 2 198:3r þ 6:0689Þn3 þ ð54:813r 2 þ 154:411r 9:2933Þn2 þ ð55:536r 2 21:461r þ 4:4668Þn þ ð2:9523r2 0:2248r 1:601Þ; The coefﬁcients in Eq. (27b) for r 2 [0.1, 0.3]:

M sw ¼ ð312:06r 2 þ 164:89r 25:033Þn3 þ ð442:66r2 233:67r þ 33:445Þn2 þ ð291:34r2 þ 165:91r 22:433Þn þ ð104:76r2 68:293r þ 15:259Þ; M s0 ¼ ð2616:7r 2 1097r þ 68:365Þn3 þ ð3362:2r 2 þ 1502:5r 112:31Þn2 þ ð1495:9r 2 742:72r þ 71:947Þn þ ð241:83r 2 þ 144:54r 19:591Þ; M s1 ¼ ð8504:9r 2 þ 3612:8r 98:18Þn3 þ ð10315:8r 2 4517:1r þ 165:14Þn2 þ ð4100r 2 þ 1897:8r 103:96Þn þ ð506:06r2 270:49r þ 24:853Þ; M s2 ¼ ð8387:4r 2 3995:1r þ 123:311Þn3 þ ð10035:7r 2 þ 4801:9r 164:132Þn2 þ ð3908r2 1901:2r þ 83:371Þn þ ð470:145r2 þ 250:48r 18:030Þ: References Ahn, J.H., Kwon, D., 2001. Derivation of plastic stress–strain relationship from ball indentations: examination of strain deﬁnition and pileup effect. Journal of Materials Research 16, 3170. Alcalá, J., Esqué-de los Ojos, D., 2010. Reassessing spherical indentation: contact regimes and mechanical property extractions. International Journal of Solids and Structures 47, 2714–2732. Atkins, A.G., Tabor, D., 1965. Plastic indentation in metals with cones. Journal of the Mechanics and Physics of Solids 13, 149–164. Beghini, M., Bertini, L., Fontanari, V., 2006. Evaluation of the stress–strain curve of metallic materials by spherical indentation. International Journal of Solids and Structures 43, 2441–2459. 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.

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