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Small-scale indentation of a hemispherical inhomogeneity in an elastic half-space
European Journal of Mechanics A/Solids 53 (2015) 151e162
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Small-scale indentation of a hemispherical inhomogeneity in an elastic half-space Ivan I. Argatov, Federico J. Sabina* ticas Aplicadas y en Sistemas, Universidad Nacional Auto noma de M Instituto de Investigaciones en Matema exico, Apartado Postal 20-726, n Alvaro Obregon, 01000, M Delegacio exico, D. F., Mexico
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 December 2014 Accepted 8 April 2015 Available online 18 April 2015
The axisymmetric problem of small-scale frictionless indentation of an elastic hemispherical inhomogeneity embedded at the free surface of a semi-infinite elastic matrix is considered. It is assumed that the radius of contact area is relatively small compared with the radius of the inhomogeneity. The first-order asymptotic model for the incremental indentation stiffness is presented in terms of the coefficient of local compliance, which is evaluated based on the analytical solution for the surface Green's function. The influence of both Poisson's ratios on the corresponding indentation scaling factor, which reflects the effect of localized inhomogeneity, is studied in detail. © 2015 Elsevier Masson SAS. All rights reserved.
Keywords: Indentation testing Hemispherical inhomogeneity Asymptotic model
1. Introduction Indentation technique (Hardy et al., 1971; Follansbee and Sinclair, 1984; Hill et al., 1989) has been proved very useful in testing mechanical properties of small material samples and, in particular, thin films (Antunes et al., 2007; Hemmouche et al., 2013). By modeling the sample configuration as an elastic layer bonded to a rigid base or to an elastic substrate, one can study the corresponding thickness effect (Hayes et al., 1972; Argatov et al., 2013) or the substrate effect (Yu et al., 1990; Gao et al., 1992; Chen and Vlassak, 2001; Perriot and Barthel, 2004; Argatov and Sabina, 2014). In recent years, the AFM indentation tests have been applied for characterizing composite materials (Gregory and Spearing, 2005; Constantinides et al., 2006), for which new identification methods should be developed in order to take into account the effect of material inhomogeneity (Kabele et al., 2008). In particular, an important sample geometry is represented by a hemispherical inhomogeneity embedded at the free surface of an elastic half-space made of another material (Fig. 1a). Axiallysymmetric finite-element solutions of the indentation problems for an elastic hemispherical inhomogeneity were obtained by Batog et al. (2008) and Kabele et al. (2008) under simplifying assumptions
* Corresponding author. E-mail addresses: [email protected] (I.I. Argatov), [email protected]. mx (F.J. Sabina). http://dx.doi.org/10.1016/j.euromechsol.2015.04.003 0997-7538/© 2015 Elsevier Masson SAS. All rights reserved.
that the values of Poisson's ratio is assumed to be the same in the hemispherical inhomogeneity and the semi-infinite matrix. In the present paper, we develop the first-order asymptotic model for the incremental indentation stiffness, which in the axisymmetric case can be written as follows (Argatov, 2010; Argatov and Sabina, 2014):
dP 2aEeff x : dw 1 ε 2a0
(1)
p
Here, P is the contact force, w is the indenter displacement, ε ¼ a/l is a small parameter, a is the radius of contact area, l is the radius of inhomogeneity, Eeff is the effective elastic modulus defined through the formula
1 1 n 1 n20 þ ¼ : Eeff E0 2G
(2)
Recall that the effective modulus Eeff is used to account for the effect of elastic deformation of the indenter, whose Young's modulus and Poisson's ratio are denoted by E0 and n0, respectively. Formula (1) contains the so-called (Argatov, 2002) coefficient of local compliance a0, which bears information about the inhomogeneity geometry, the interface conditions between the inhomogeneity and matrix, and depends on Poisson's ratios n and n as well as on the inhomogeneity-matrix shear moduli ratio G ¼ G=G.
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2. The hemispherical inhomogeneity subjected to a concentrated force In this section, we briefly outline the analytical solution obtained by Tsuchida et al. (1990) for the three-dimensional mixed boundary value problem of a hemispherical inhomogeneity, u, embedded at the free surface of an elastic half-space, ℝ3þ ¼ fx ¼ ðx1 ; x2 ; x3 Þ : x3 > 0g, and subjected to a concentrated force, P. By the way, we correct misprints and shortcomings in the mentioned study. We assume that the inhomogeneity u is perfectly bonded to the semi-infinite medium, and the continuity and equilibrium conditions along the interface g ¼ vu∩ℝ3þ are formulated as follows:
uðxÞ ¼ uðxÞ;
s
ðnÞ
ðxÞ ¼ s
ðnÞ
ðxÞ;
x2g:
(3)
Here, u and sðnÞ , u and sðnÞ are the displacement and stress vectors in the inhomogeneity u and in the matrix ℝ3þ nu, respectively. Let us denote the systems of cylindrical and spherical coordinates by (r,q,z) and (R,q,f) respectively (see Fig. 1). Then, in the spherical coordinates, the boundary conditions (3) can be rewritten as
ðuR ÞR¼l ¼ ðuR ÞR¼l ; ðsR ÞR¼l ¼ ðsR ÞR¼l ; Note that tRq ¼ tRq ¼ 0.
in
uf
R¼l
¼ uf R¼l ;
axisymmetric
G¼
G G
(8)
as well as on Poisson's ratios n and n, where G and n are the shear modulus and Poisson's ratio of the semi-infinite matrix. 2.2. Boussinesq potentials In the case of axisymmetry about the x3-axis, the general solution to the elasticity equations in the matrix is given by the Boussinesq potentials
vF0 vF þ m R 3 ð3 4nÞF3 ; uq ¼ 0; vR vR sinf vF0 vF þ sinf m 3 þ ð3 4nÞF3 ; 2Guf ¼ R vm vm 2GuR ¼
where F0 and F3 are harmonic functions, i. e., V2 F0 ¼ V2 F3 ¼ 0,
(5) case,
uq ¼ uq ¼ 0
V2 ¼
and
v 1 v 1 v 2 v 2 R þ 1 m ; vR vm R2 vR R2 vm
m ¼ cosf:
The corresponding stress field is given by1
1 m2 vF3 vF3 2n ; sR ¼ þ mR 2ð1 nÞm vR vm R vR2 vR2 i 1 vF h 1 vF0 m vF0 vF 3 2 þ ð1 2nÞm 3 2n þ ð1 2nÞm2 ; sq ¼ R vR R vm vm vR R i 1 vF h 1 m2 v2 F0 m vF0 1 vF0 m 1 m2 v2 F3 vF 3 sf ¼ þ þ ; þ ð1 2nÞm 3 1 þ ð1 2nÞ 1 m2 2 2 2 vm 2 R R vm R vR vR R R vm vm ! 1 vF0 1 v2 F0 vF v2 F 3 m vF3 þ ð1 2nÞ 3 m þ 2ð1 nÞ tRf ¼ sinf 2 ; R vRvm R vm vR vRvm R vm v2 F0
(9)
(4)
tRf R¼l ¼ tRf R¼l : the
Here, G and n are the shear modulus and Poisson's ratio of the inhomogeneity, l is the radius of the inhomogeneity. The asymptotic constants a0,a1,… are called the coefficients of local compliance (Argatov, 2002). We emphasize that the local-compliance coefficients are dimensionless and depend on the inhomogeneitymatrix shear moduli ratio
v2 F 3
(10)
tfq ¼ tRq ¼ 0:
2.1. Green's function and asymptotic coefficients of local compliance At the coordinate center, O, the vector-function u should satisfy the following asymptotic condition:
The solution T of the Boussinesq problem is expressed in terms of the Boussinesq potentials as
uðxÞ ¼ PTðxÞ þ Oð1Þ;
p l2 0 F0 ¼ 0 ð1 2nÞlnðR þ x3 Þ; 2
x/O:
(6)
Here, T is the solution of the Boussinesq problem for an elastic half-space comprised of the same material properties as the inhomogeneity. In the case of a unit force P, the solution to the elastic problem (3), (6) represents Green's vector-function with a pole at the point O. The structure of the next terms in the asymptotic expansion (6) was studied in detail by Argatov (2002). It was shown that in asymptotic analysis of frictionless contact problems, the following asymptotic expansion plays an important role: ∞ n 2pG 1 1X an 2 2 x u3 ðx1 ; x2 ; 0Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ x : 1 2 2n ð1 nÞP x2 þ x2 l n¼0 l 1
2
p l2 1 0 ; F3 ¼ 0 2 R
(11)
where p0 is the equivalent pressure given by
p0 ¼
P ; pl2
(12)
and the quantities denoted by a bar refer to the inhomogeneity.
(7) 1 Formulas (10) coincide with the corresponding relations used by Tsuchida et al. (1990) except for the formula for sf.
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153
Fig. 1. Hemispherical inhomogeneity under a concentrated force in the spherical (a) and cylindrical (b) coordinates.
The potentials (11) yield the following displacements2:
1 uR ¼ ½4ð1 nÞcosf ð1 2nÞ; 2 p 0 l2
2RG 1 1 2n sinf ð3 4nÞ u ¼ : f 2 1 þ cosf p 0 l2
V
F0 ¼ ð1 2nÞp0
2RG
p l2 1 FI3 ¼ 0 F0 ; R 2
(13)
P ðmÞ ¼ 2ð1 nÞp0 An 2n ; R2nþ1 n¼0
∞ X ðmÞ P ð2n þ 1ÞAn 2nþ1 ; FII3 ¼ p0 2nþ2 R n¼0
FIII 0 ¼ ð1 2nÞp0
∞ X
Bn
n¼0
FIII 3
¼ p0
∞ X
ð2n þ 2ÞBn
n¼0
P2nþ1 ðmÞ ; R2nþ2
P2nþ2 ðmÞ R2nþ3
(14)
IV
∞ X
¼ p0
∞ X
for the inhomogeneity (R < l). Here, Pn(m) are the Legendre functions of the first kind, F0 is an unknown quantity, and An, Bn, An , Bn are series coefficients. 2.3. Resulting linear algebraic system
0
(15)
(16)
IV
V
Fi ¼ Fi þ Fi þ Fi ;
Fi ¼ FIi þ FIIi þ FIII i
ði ¼ 0; 3Þ
(19)
for the inhomogeneity and the matrix, respectively, one can express stresses and displacements in terms of the unknown coefficients An, Bn, An , Bn . In order to apply the boundary conditions along the interface (4) and (5), the Legendre polynomials of odd order are expanded in terms of the Legendre polynomials of even order using the following “half-range expansion” (Tsuchida et al., 1990):
P2kþ1 ðmÞ ¼
∞ X
ðnÞ
wk P2n ðmÞ;
0 P2kþ1 ðmÞ ¼
n¼0
∞ X
ðnÞ
0 wk P2n ðmÞ:
n¼0
0 ðmÞ is the derivative of the Legendre polynomial P (m) Here, P2n 2n of degree 2n, and we used the notation
An R2nþ2 P2nþ2 ðmÞ;
n¼0 IV F3
(18)
∞ X ¼ p0 ð2n þ 1ÞBn R2n P2n ðmÞ
Using the Boussinesq potentials
for the matrix (R > l), and
F0 ¼ 2ð1 nÞp0
V F3
n¼0
∞ X
FII0
Bn R2nþ1 P2nþ1 ðmÞ;
n¼0
Following Tsuchida et al. (1990), the required auxiliary Boussinesq potentials are taken in the form
p l2 FI0 ¼ 0 F0 ð1 2nÞlnðR þ x3 Þ; 2
∞ X
(17)
ð2n þ 2ÞAn R2nþ1 P2nþ1 ðmÞ;
n¼0
ðnÞ
wk ¼
ð4n þ 1Þð2k þ 1Þ P ð0ÞP2k ð0Þ: ð2k þ 1 2nÞð2k þ 2 þ 2nÞ 2n
Thus, the first boundary condition (4) implies3
i lF0 h An ð2n þ 1Þ A 2nð2n 1Þ ðnÞ ðnÞ a þ n1 ð2n þ 3 4nÞ ð1 2nÞd0 þ 4ð1 nÞw0 þ 2nþ2 4n þ 3 2n 2 l l2n ( 4n 1 ∞ i 1 X Bk ð2k þ 2Þ h ð2n þ 1Þð2n þ 2Þ ðnÞ ðnÞ a ð2n 2 þ 4nÞ þ þ w þ ð2k þ 3Þð2k þ 6 4nÞw An l2nþ1 2kþ2 k kþ1 2kþ3 ð4k þ 5Þ G 4n þ 3 l k¼0 ) ∞ i X 2n ð2k þ 1Þ h ðnÞ ðnÞ 2kð2k 3 þ 4nÞwk1 þ a2k1 wk a2n1 þ Bk l2k þ An1 l2n1 ð4n 1Þ ð4k þ 1Þ k¼0 i l h ðnÞ ðnÞ ð1 2nÞd0 þ 4ð1 nÞw0 ðn ¼ 0; 1; 2; …Þ; ¼ 2G
2 Formulas (13) differ from the corresponding relations used by Tsuchida et al. (1990) by the factor 1/2 on the right-hand sides instead of the factor 2 that appeared in Tsuchida et al. (1990).
(20)
3 Note that the term with An1 in Eq. (20) differs from the corresponding term in Eq. (2.15) derived by Tsuchida et al. (1990) by an additional factor of n.
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ðnÞ
where di
denotes the Kronecker delta, and
an ¼ ðn þ 1Þ2 2 þ 2n;
an ¼ ð1Þn
an ¼ ðn þ 1Þ2 2 þ 2n:
ð2n þ 1Þ! 22n1 ðn!Þ2
l2n1 Bn :
(24)
Note that according to (22), the coefficients An ; Bn ; An , and Bn have dimensions of L2nþ3,L2nþ4,L2n, and L12n, respectively.
From the second boundary condition (4) it follows that
"
# ∞ X 4n þ 1 4k þ 3 An 1 A ð2n 1Þ ðnÞ ðnÞ ð3 4nÞw0 þ ð1 2nÞ w a þ n1 ð2n 4 þ 4nÞ ð1 2nÞ þ 2nþ2 2n ð4n 1Þ 2nð2n þ 1Þ ð2k þ 1Þð2k þ 2Þ k 4n þ 3 2n l l k¼0 ∞ h i X Bk 1 ð2n þ 2Þ 1 a 1 ðnÞ ðnÞ a2kþ2 wk þ ð2k þ 2Þð2k 1 þ 4nÞwkþ1 An l2nþ1 ð2n þ 5 4nÞ An1 l2n1 2n1 þ 2kþ3 ð4k þ 5Þ ð4n þ 3Þ G G ð4n 1Þ l lF 0 2
k¼0
i l2k 1h ðnÞ ðnÞ ð2k þ 1Þð2k þ 4 4nÞwk1 þ a2k1 wk ð4k þ 1Þ G k¼0 " # ∞ X l 4n þ 1 4k þ 3 ðnÞ ðnÞ ð1 2nÞ ð3 4nÞw0 þ ð1 2nÞ wk ¼ 2G 2nð2n þ 1Þ ð2k þ 1Þð2k þ 2Þ
∞ X Bk
ðn ¼ 1; 2; …Þ:
k¼0
(21)
From the first boundary condition (5), we get
i lF0 h An ð2n þ 1Þð2n þ 2Þ An1 2nð2n 1Þ ðnÞ ðnÞ a2n þ 2nþ1 b2n2 ð1 2nÞd0 þ 2ð2 nÞw0 þ 2nþ3 4n þ 3 4n 1 2 l l ( ∞ i X B ð2k þ 2Þð2k þ 3Þ h ð2n þ 1Þð2n þ 2Þ 2nð2n 1Þ ðnÞ ðn Þ k a2kþ2 wk þ b2kþ1 wkþ1 An l2n þ b2n4 þ An1 l2n2 a2n1 2kþ4 4k þ 5 4n þ 3 4n 1 l k¼0 " " #) # ∞ X 2kð2k þ 1Þ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðn ¼ 0; 1; 2; …Þ; þ Bk l2k1 b2k5 wk1 þ a2k1 wk ¼ ð1 2nÞd0 þ 2ð2 nÞw0 4k þ 1 2
(22)
k¼1
where
bn ¼ ðn þ 2Þðn þ 5Þ 2n;
3. Equilibrium of a two-layer hemispherical solid
bn ¼ ðn þ 2Þðn þ 5Þ 2n:
Finally, the second boundary condition (5) yields
In addition to the coefficients An ; Bn ; An , and Bn , the system (20)e(23) contains the unknown F0. It was suggested (Kouris and
" # ∞ X F0 4n þ 1 4k þ 3 An ð2n þ 2Þ A ð2n 1Þ ðnÞ ðnÞ w0 þ wk a þ n1 a ð1 2nÞ þ 2nþ3 2nð2n þ 1Þ ð2k þ 1Þð2k þ 2Þ ð4n þ 3Þ 2n l2nþ1 ð4n 1Þ 2n1 2 l k¼0 ∞ i X Bk a2kþ2 h ð2n þ 2Þ ð2n 1Þ ðnÞ ðnÞ ð2k þ 3Þwk þ ð2k þ 2Þwkþ1 þ An l2n þ a þ An1 l2n2 a 2kþ4 ð4n þ 3Þ 2n ð4n 1Þ 2n1 ð4k þ 5Þ l k¼0 " # ∞ ∞ i h X X a 1 4n þ 1 4k þ 3 ðnÞ ðnÞ ðnÞ ðnÞ w0 þ wk þ Bk l2k1 2k1 ð2k þ 1Þwk1 þ 2kwk ¼ ð1 2nÞ 2 2nð2n þ 1Þ ð2k þ 1Þð2k þ 2Þ 4k þ 1
ðn ¼ 1; 2; …Þ:
k¼0
k¼1
Now, provided F0 is known, the solution for the surface Green function can be obtained by solving the linear algebraic system (20)e(23) for the unknown coefficients An ; Bn ; An , and Bn . In view of the relation ðu3 Þx3 ¼0 ¼ ðuf Þf¼p=2 , the asymptotic constants an entering (7) can be expressed in terms of the coefficients Bn as follows:
(23)
Tsuchida, 1992) that the unknown constant F0 should be determined by the interfacial boundary conditions, that is from the system (20)e(23). Here we show that
F0 ¼ 1:
(25)
Indeed, let us consider a two-layer hemispherical solid R < L,
I.I. Argatov, F.J. Sabina / European Journal of Mechanics A/Solids 53 (2015) 151e162
155
where L > l. The vertical equilibrium equation for the two-layer solid can be written as
Zp=2 2
P ¼ 2pL
0
sR cosf tRf sinf R¼L sinf df
Z1 ¼ 2pL
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi sR m tRf 1 m2
(26) dm:
R¼L
0
In view of (10) and (19), we will have
1 m2 vF0 1 m2 v2 F0 v2 F0 þ vm vRvm R vR2 R2 vF 2 v F 3 3 þ m2 R m2 þ 1 2n vR vR2 1 vF v2 F 3 3 þ m 1 m2 : 2m 1 m2 R vm vRvm (27)
sR cosf tRf sinf ¼ m
Fig. 3. Asymptotic constants a0 > 0 and a1 < 0 for a clamped elastic hemisphere.
a2nþ1 ð2n þ 4Þ ð2n þ 7 4nÞ þ Anþ1 l2nþ3 4n þ 7 4n þ 3 ∞ i X l2k h ðnþ1Þ ðnþ1Þ ð2k þ 1Þð2k þ 4 4nÞwk1 þ a2k1 wk Bk þ 4k þ 1 k¼0 " l 4n þ 5 ðnþ1Þ ð1 2nÞ ð3 4nÞw0 ¼ þ ð1 2nÞ 2 ð2n þ 2Þð2n þ 3Þ # ∞ X 4k þ 3 ðnþ1Þ w ðn ¼ 0; 1; 2; …Þ: ð2k þ 1Þð2k þ 2Þ k
An l2nþ1 The subsequent substitution of (27) and (14)e(16) into Eq. (26) yields
P ¼ pl2 p0 F0 þ Oðl=LÞ:
(28)
Now, due to Eq. (12), the asymptotic relation (28), as L/∞, implies Eq. (25). Eq. (25) should be employed to simplify the system (20)e(23).
k¼0
(30)
4. Small-scale indentation of a clamped elastic hemisphere In this section we consider the first limit case when G/0 (see Fig. 2a) and Eqs. (20) and (21) reduce to the system
2n ð2n þ 1Þð2n þ 2Þ ð2n 2 þ 4nÞ a þ An l2nþ1 ð4n 1Þ 2n1 4n þ 3 ∞ i X ð2k þ 1Þ h ðnÞ ðnÞ 2kð2k 3 þ 4nÞwk1 þ a2k1 wk þ Bk l2k ð4k þ 1Þ k¼0 i lh ðnÞ ðnÞ ðn ¼ 0; 1; 2; …Þ; ¼ ð1 2nÞd0 þ 4ð1 nÞw0 2 (29)
An1 l2n1
Fig. 3 shows the behavior of the main asymptotic constants a0 and a1 with respect to Poisson's ratio n. Observe that the functions a0 ðnÞ and a1 ðnÞ have minima 1.45 and 0.39, respectively. The problem under consideration was solved in (Argatov, 2002) by another method based on the Weber harmonic potentials. For the considered specific set of n, the numerical results for a0 ðnÞ and a1 ðnÞ obtained by the two methods coincide. Let us compare the first asymptotic constant a0 with the corresponding quantity for an elastic layer of thickness l firmly attached (bonded) to a rigid base. According to Vorovich et al. (1974), we will have
aLn ¼
ð1Þn ½ð2nÞ!!2
Z∞ ½1 LðuÞu2n du; 0
Fig. 2. Two limit cases: (a) A clamped elastic hemisphere; (b) A rigid hemisphere embedded at the free surface of an elastic half-space.
(31)
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I.I. Argatov, F.J. Sabina / European Journal of Mechanics A/Solids 53 (2015) 151e162
5. Indentation of a rigid hemisphere embedded at the free surface of an elastic half-space In this section, we consider the case when the absolutely rigid inhomogeneity u (see Fig. 2b) receives a unit rigid body displacement we3, which can be described by the following boundary conditions:
ðu3 ÞR¼l ¼ w;
ður ÞR¼l ¼ 0;
ðuq ÞR¼l ¼ 0:
(32)
By converting Eq. (32) from cylindrical to spherical coordinates, we arrive at the boundary conditions
ðuR ÞR¼l ¼ wm;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi uf R¼l ¼ w 1 m2 :
(33)
According to (14)e(16) with F0 ¼ 1, Eq. (33) yield the following system of linear algebraic equations: Fig. 4. Asymptotic constants a0 and aL0 for a clamped elastic hemisphere and a bonded elastic layer, respectively.
where
LðuÞ ¼
2ksh2u 4u ; 2kch2u þ 1 þ k2 þ 4u2
k ¼ 3 4n:
Fig. 4 presents the behavior of the constants a0 and aL0 for different values of Poisson's ratio n. From the figure, we see that a0 > aL0 for all possible values of n. We emphasize that the Poisson's ratio effect is more pronounced for a clamped elastic hemisphere than for a bonded elastic layer. Recall that for an isotropic elastic layer resting on a smooth rigid base the asymptotic constants do not depend on its Poisson's ratio (Vorovich et al., 1974; Alexandrov and Pozharskii, 2001; Argatov et al., 2012). Note also that the numerical calculations show that the function a1 ðnÞ aL1 ðnÞ is not positive definite. Finally, note that the axisymmetric frictionless contact problem of the theory of elasticity for a clamped truncated sphere (a clamped elastic hemisphere is a special case of this) was studied by Alexandrov and Pozharskii (1997).
An1 2nð2n 1Þ An ð2n þ 1Þ ð2n þ 3 4nÞ þ 2nþ2 a ð4n þ 3Þ 2n l2n ð4n 1Þ l ∞ i X B ð2k þ 2Þ h ðnÞ ðnÞ k a2kþ2 wk þ ð2k þ 3Þð2k þ 6 4nÞwkþ1 þ 2kþ3 ð4k þ 5Þ l k¼0 i lh ðnÞ ðnÞ ð1 2nÞd0 þ 4ð1 nÞw0 ¼ 2 2G ðnÞ w w ðn ¼ 0; 1; 2; …Þ; p0 0 (34) An1 ð2n 1Þ An a2n ð2n 4 þ 4nÞ þ 2nþ2 ð4n þ 3Þ l2n ð4n 1Þ l ∞ h i X Bk 1 ðnÞ ðnÞ a2kþ2 wk þ ð2k þ 2Þð2k 1 þ 4nÞwkþ1 þ 2kþ3 ð4k þ 5Þ l k¼0 ( " # ∞ X l 4n þ 1 4k þ 3 ðnÞ ð1 2nÞ þ wk ¼ 2 2nð2n þ 1Þ ð2k þ 1Þð2k þ 2Þ k¼0 ) 2G ðnÞ ðnÞ w w ðn ¼ 1; 2; …Þ: þ ð3 4nÞw0 p0 0 (35) In addition to the unknowns An and Bn, the system (34), (35) contains the unknown parameter p0, which is related to the total force P by Eq. (12). It is to emphasize that we cannot derive the additional equation from the equilibrium Eq. (28) with L ¼ l, because the value of P is not known in advance, if w is given a priori. Further, observe that Eqs. (21) and (23) were derived on the basis of the expansion
! ∞ ∞ X X 1 4k þ 3 4n þ 1 ðnÞ ¼ w P 0 ðmÞ; 1 þ m n¼1 ð2k þ 1Þð2k þ 2Þ k 2nð2n þ 1Þ 2n k¼0
which does not converge to 1 as m / 0. On the other hand, we have ðu3 Þx3 ¼0 ¼ ðuf Þf¼p=2 . Therefore, in view of (33), we obtain ∞ X k¼0
Fig. 5. Stiffness coefficients C3(n) and c3(n) for a rigid hemisphere and a bonded flatended cylindrical punch, respectively.
ð1Þk
Bk ð2k þ 1Þ! l Gw ¼ þ : 2 ð1 nÞp0 l2kþ3 22k ðk!Þ2
(36)
Thus, following Argatov (2002), the force-displacement relationship can be written as
I.I. Argatov, F.J. Sabina / European Journal of Mechanics A/Solids 53 (2015) 151e162
P¼
2pG lC ðnÞw; 1n 3
(37)
where lC3(n) represents the translational capacity of the rigid hemispherical inhomogeneity, and C3(n) is a dimensionless quantity given by
C3 ðnÞ ¼
ð1 nÞp0 l : 2Gw
(38)
C1;0 ðnÞ ¼
157
ð2n þ 1Þ ð2n 2 þ 4nÞ; ð4n þ 3Þ
D1 ðn; kÞ ¼
C1;1 ðnÞ ¼
1 a ; ð4n þ 7Þ 2nþ2
h i 1 ðnþ1Þ ðnþ1Þ þ ð2k þ 2Þð2k 1 þ 4nÞwkþ1 ; a2kþ2 wk ð4k þ 5Þ
c1;0 ðnÞ ¼
1 a2nþ1 ; G ð4n þ 3Þ
1 ð2n þ 4Þ ð2n þ 7 4nÞ; c1;1 ðnÞ ¼ G ð4n þ 7Þ
Fig. 5 shows the variation of C3(n) with the change of Poisson's ratio n. The numerical data were obtained based on Formula (38) by solving the system (34)e(36) for w ¼ l (note that the problem (32) is linear). Also, for comparison, Fig. 5 presents a dashed line for the translational capacity lc3(n) of a flat-ended cylindrical punch of radius l attached to an elastic half-space. According to Mossakovskii (1963); Borodich and Keer (2004), in the last case, we will have
h 1 1 ðnþ1Þ ð2k þ 1Þð2k þ 4 4nÞwk1 d1 ðn; kÞ ¼ G ð4k þ 1Þ i ðnþ1Þ ; þ a2k1 wk
2pG lc ðnÞw; P¼ 1n 3
1 F1 ðnÞ ¼ 2
(39)
ð4n þ 5Þ ðnþ1Þ ð3 4nÞw0 ð2n þ 2Þð2n þ 3Þ # N2 X 4k þ 3 ðnþ1Þ wk þ ð1 2nÞ ð2k þ 1Þð2k þ 2Þ k¼0 " 1 ð4n þ 5Þ ðnþ1Þ ð3 4nÞw0 ð1 2nÞ 2G ð2n þ 2Þð2n þ 3Þ # N2 X 4k þ 3 ðnþ1Þ wk þ ð1 2nÞ ; ð2k þ 1Þð2k þ 2Þ
where w is the indenter displacement, P is the contact force, and
c3 ðnÞ ¼
2ð1 nÞ lnð3 4nÞ: pð1 2nÞ
"
(40)
ð1 2nÞ
Note that the normalization in (40) was chosen such that Formulas (37) and (39) have the same structure. Observe also that C3(0.5) z 0.728 and c3(0.5) ¼ 2/p z 0.637. Finally, note that the basis of homogeneous polynomial elastic solutions produced by the III Boussinesq potentials FII0 , FII3 , FIII 0 , and F3 coincides, up to normalization, with the basis constructed by Bogovoi and Nuller (1971).
C2;1 ðnÞ ¼
2nð2n 1Þ b ; ð4n 1Þ 2n2
6. Small-scale indentation of an elastic hemisphere embedded at the free surface of an elastic half-space
D2 ðn; kÞ ¼
i ð2k þ 2Þð2k þ 3Þ h ðnÞ ðnÞ a2kþ2 wk þ b2kþ1 wkþ1 ; ð4k þ 5Þ
For this general case, according to the approach described in Sections 2 and 3, the unknown coefficients of the Boussinesq potentials can be obtained by solving the system of linear algebraic Eqs. (20)e(23) with the relation F0 ¼ 1 taken into account. Correspondingly, the asymptotic constant a0 can be evaluated using Eq. (24). First, let us introduce the notation
C0;1 ðnÞ ¼
2nð2n 1Þ ð2n þ 3 4nÞ; 4n 1
C0;0 ðnÞ ¼
ð2n þ 1Þ a ; ð4n þ 3Þ 2n
i ð2k þ 2Þ h ðnÞ ðnÞ a2kþ2 wk þ ð2k þ 3Þð2k þ 6 4nÞwkþ1 ; D0 ðn; kÞ ¼ ð4k þ 5Þ 1 2n a ; G ð4n 1Þ 2n1 1 ð2n þ 1Þð2n þ 2Þ ð2n 2 þ 4nÞ; c0;0 ðnÞ ¼ G ð4n þ 3Þ
k¼0
C2;0 ðnÞ ¼
ð2n þ 1Þð2n þ 2Þ a2n ; ð4n þ 3Þ
2nð2n 1Þ c2;1 ðnÞ ¼ a ; ð4n 1Þ 2n1 ð2n þ 1Þð2n þ 2Þ c2;0 ðnÞ ¼ b2n4 ; ð4n þ 3Þ i 2mð2m þ 1Þ h ðnÞ ðnÞ b2m5 wm1 þ a2m1 wm ; d2 ðn; mÞ ¼ ð4m þ 1Þ F2 ðnÞ ¼
i 1h ðnÞ ðnÞ ð1 2nÞd0 þ 2ð2 nÞw0 2 i 1h ðnÞ ðnÞ ð1 2nÞd0 þ 2ð2 nÞw0 ; 2
c0;1 ðnÞ ¼
C3;0 ðnÞ ¼
ð2n þ 1Þ a ; ð4n þ 3Þ 2nþ1
i 1 ð2k þ 1Þ h ðnÞ ðnÞ 2kð2k 3 þ 4nÞwk1 þ a2k1 wk ; d0 ðn; kÞ ¼ G ð4k þ 1Þ
D3 ðn; kÞ ¼
i 1h 1 h ðnÞ ðnÞ ðnÞ ð1 2nÞd0 þ 4ð1 nÞw0 ð1 2nÞd0 F0 ðnÞ ¼ 2 2G i ðnÞ þ 4ð1 nÞw0 ;
c3;0 ðnÞ ¼
ð2n þ 4Þ a ; ð4n þ 7Þ 2nþ2
i a2kþ2 h ðnþ1Þ ðnþ1Þ ð2k þ 3Þwk þ ð2k þ 2Þwkþ1 ; ð4k þ 5Þ
ð2n þ 1Þ a ; ð4n þ 3Þ 2nþ1
d3 ðn; mÞ ¼
C3;1 ðnÞ ¼
c3;1 ðnÞ ¼
ð2n þ 4Þ a ; ð4n þ 7Þ 2nþ2
i a2m1 h ðnþ1Þ ðnþ1Þ ; ð2m þ 1Þwm1 þ 2mwm 4m þ 1
158
F3 ðnÞ ¼
I.I. Argatov, F.J. Sabina / European Journal of Mechanics A/Solids 53 (2015) 151e162
" 1 ð4n þ 5Þ ðnþ1Þ ð1 2nÞ w0 2 ð2n þ 2Þð2n þ 3Þ þ "
N2 X k¼0
4k þ 3 ðnþ1Þ w ð2k þ 1Þð2k þ 2Þ k
#
1 ð4n þ 5Þ ðnþ1Þ ð1 2nÞ w0 2 ð2n þ 2Þð2n þ 3Þ þ
N2 X k¼0
# 4k þ 3 ðnþ1Þ w : ð2k þ 1Þð2k þ 2Þ k
Second, for a given natural number N1, we define an (4N1 þ 4) (4N1 þ 4)-matrix M and an (4N1 þ 4)-vector f as follows. Let the indices n and m run from 1 to N1, and the index k run from 0 to N1. Then, we set
M0;0 ¼ C0;0 ð0Þ; M0;1 ¼ c0;0 ð0Þ; M0;4kþ2 ¼ D0 ð0; kÞ; M0;4kþ3 ¼ d0 ð0; kÞ; f0 ¼ F0 ð0Þ; M1;0 ¼ C1;0 ð0Þ; M1;1 ¼ c1;0 ð0Þ; M1;4 ¼ C1;1 ð0Þ; M1;5 ¼ c1;1 ð0Þ; M1;4kþ2 ¼ D1 ð0; kÞ; M1;4kþ3 ¼ d1 ð0; kÞ; f1 ¼ F1 ð0Þ; M2;0 ¼ C2;0 ð0Þ; M2;1 ¼ c2;0 ð0Þ; M2;4kþ2 ¼ D2 ð0; kÞ; M2;4mþ3 ¼ d2 ð0; mÞ; f2 ¼ F2 ð0Þ; M3;0 ¼ C3;0 ð0Þ; M3;1 ¼ c3;0 ð0Þ; M3;4 ¼ C3;1 ð0Þ; M3;5 ¼ c3;1 ð0Þ; M3;4kþ2 ¼ D3 ð0; kÞ; M3;4mþ3 ¼ d3 ð0; mÞ; f3 ¼ F3 ð0Þ; M4n;4n4 ¼ C0;1 ðnÞ; M4n;4n3 ¼ c0;1 ðnÞ; M4n;4n ¼ C0;0 ðnÞ; M4n;4nþ1 ¼ c0;0 ðnÞ; M4n;4kþ2 ¼ D0 ðn; kÞ; M4n;4kþ3 ¼ d0 ðn; kÞ; f4n ¼ F0 ðnÞ;
Fig. 6. Asymptotic constant a0 for a hemispherical inhomogeneity as a function of its Poisson's ratio: (a) Relatively soft inhomogeneity (G 1); (b) Relatively stiff inhomogeneity (G 1).
M4nþ1;4n ¼ C1;0 ðnÞ; M4nþ1;4nþ1 ¼ c1;0 ðnÞ; M4nþ1;4nþ4 ¼ C1;1 ðnÞ; M4nþ1;4nþ5 ¼ c1;1 ðnÞ; M4nþ1;4kþ2 ¼ D1 ðn; kÞ; M4nþ1;4kþ3 ¼ d1 ðn; kÞ; f4nþ1 ¼ F1 ðnÞ; M4nþ2;4n4 ¼ C2;1 ðnÞ; M4nþ2;4n3 ¼ c2;1 ðnÞ; M4nþ2;4n ¼ C2;0 ðnÞ; M4nþ2;4nþ1 ¼ c2;0 ðnÞ; M4nþ2;4kþ2 ¼ D2 ðn; kÞ; M4nþ2;4mþ3 ¼ d2 ðn; mÞ; f4nþ2 ¼ F2 ðnÞ; M4nþ3;4n ¼ C3;0 ðnÞ; M4nþ3;4nþ1 ¼ c3;0 ðnÞ; M4nþ3;4nþ4 ¼ C3;1 ðnÞ; M4nþ3;4nþ5 ¼ c3;1 ðnÞ; M4nþ3;4kþ2 ¼ D3 ðn; kÞ; M4nþ3;4mþ3 ¼ d3 ðn; mÞ; f4nþ3 ¼ F3 ðnÞ: Third, taking l as unit of length, we introduce an (4N1 þ 4)-vector X of the unknown coefficients An, Bn, An , Bn by formulas
X4k ¼ Ak ;
X4kþ1 ¼ Ak ;
X4kþ2 ¼ Bk ;
X4kþ3 ¼ Bk :
Thus, the problem of approximate solution of the system (20)e(23) is reduced to the system of 4(N1 þ 1) linear algebraic equations MX ¼ f, from which it immediately follows that X ¼ M1f.
The results of numerical calculations are presented on Figs. 6 and 7. All calculations were performed using Mathcad software (MathSoft Inc., Cambridge, MA). The accuracy of numerical approximations depends on the numbers N1 and N2 of series members in the Legendre series expansions (15)e(18) and in the series on the right-hand sides of Eqs. (21), (23), (30) and (35), the maxima of which were taken to be N1 ¼ 40 and N2 ¼ 100, respectively. Convergence was checked by varying the numbers N1 and N2. In particular, Fig. 6 shows the behavior of the coefficient of local compliance a0 as a function of the inhomogeneity Poisson's ratio n for different values of the inhomogeneity-matrix shear moduli ratio G ¼ G=G, whereas the variation of a0 as a function of the matrix Poisson's ratio n is shown in Fig. 7. In order to illustrate the behavior of the asymptotic contact a0, we compare the case of an elastic hemisphere embedded at the free surface of an elastic half-space with the case of an elastic layer bonded to an elastic substrate (see Figs. 8 and 9). This latter case was studied in detail in (Argatov, 2010; Argatov and Sabina, 2014), and the corresponding asymptotic contact is denoted by aLS 0 . An extensive asymptotic analysis of the indentation stiffness has been recently given by Argatov and Sabina (2013), whereas the asymptotics of the contact pressure was obtained earlier by Argatov (2001). Following Argatov and Sabina (2013, 2014), we rewrite Eq. (1) in the form
I.I. Argatov, F.J. Sabina / European Journal of Mechanics A/Solids 53 (2015) 151e162
159
Fig. 7. Asymptotic constant a0 for a hemispherical inhomogeneity in an elastic halfspace as a function of the half-space Poisson's ratio: (a) Relatively soft inhomogeneity (G 1); (b) Relatively stiff inhomogeneity (G 1).
Fig. 8. Asymptotic constants a0 and aLS 0 for a hemispherical inhomogeneity in an elastic half-space and an inhomogeneity in the form of elastic layer bonded to an elastic semi-infinite substrate as functions of their Poisson's ratios: (a) Relatively soft inhomogeneities (G 1); (b) Relatively stiff inhomogeneities (G 1).
dP ¼ 2aEeff K ; dw
Poisson's ratio n ¼ 0.3. Note that the curve corresponding to n ¼ 0:3 (dashed line) crosses the abscissa axis at E=E ¼ 1. Fig. 11 shows that the effect of the matrix Poisson's ratio is less pronounced than the effect of the inhomogeneity Poisson's ratio (cf. Fig. 10). Now, let us outline the method of solving the inverse identification problem based on Eqs. (41) and (42) under the assumption that the matrix modulus of elasticity E as well as both Poisson's ratios n and n are known. Moreover, for the sake of simplicity, we neglect the deformation of the indenter assuming that it is absolutely rigid. In this case, in view of (2) and (42), Eq. (41) takes the form
(41)
where we introduced the corresponding indentation scaling factor, which accounts for the effect of localized inhomogeneity in the frictionless small-scale indentation of an elastic hemispherical inhomogeneity embedded at the free surface of an elastic semiinfinite matrix. At that, according to (1), the following first-order asymptotic formula takes place:
a1 K x 1 a0 : l
(42)
Here we introduced the notation
a0 ¼
2a0 : p
(43)
Recall that a0 is a dimensionless asymptotic constant (normalized coefficient of local compliance), which possesses information about the relative stiffness of the inhomogeneity and also depends on the boundary conditions imposed at the interface between the inhomogeneity and the matrix. Fig. 10 presents the behavior of a0 as a function of the elastic moduli ratio E=E ¼ Gð1 þ nÞ=ð1 þ nÞ for a special case of the matrix
dP 2aE a1 ¼ 1 a : 0 dw 1 n2 l
(44)
Further, denoting the right-hand side of Eq. (41) by S ðaÞ and introducing a dimensionless unknown
x¼
E ; E
(45)
we can rewrite Eq. (44) as follows:
x¼
1 n2 a S ðaÞ 1 a0 ðxÞ : l 2aE
(46)
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Fig. 9. Asymptotic constants a0 and aLS 0 for a hemispherical inhomogeneity in an elastic half-space and an inhomogeneity in the form of elastic layer bonded to an elastic semi-infinite substrate as functions of the half-space Poisson's ratios: (a) Relatively soft inhomogeneities (G 1); (b) Relatively stiff inhomogeneities (G 1).
It can be shown that if the incremental indentation stiffness S ðaÞ is measured in the small-scale range (i.e., when a=l≪1), Eq. (46) can be solved by an iterative method as
xn ¼
1 n2 a S ðaÞ 1 a0 ðxn1 Þ ; l 2aE
n ¼ 1; 2; …;
starting with the zeroth approximation
x0 ¼
1 n2 S ðaÞ: 2aE
Finally, note that this method can be straightforwardly generalized to take into account the indenter deformation through Formula (2).
7. Discussion Let us compare the developed first-order asymptotic model (44) with the analytical approximation of Batog et al. (2008) obtained by fitting finite-element numerical results, which in our notation is represented in the form
Fig. 10. Asymptotic constant a0 (see Eg. (43)) for a hemispherical inhomogeneity in an elastic half-space as a function of the ratio E=E. Effect of the inhomogeneity Poisson's ratio: (a) Relatively soft inhomogeneity (E=E 1); (b) Relatively stiff inhomogeneity (E=E 1).
dP 2aE a1 ¼ 1 þ bðxÞ ; 2 dw 1 n l
(47)
where
bðxÞ ¼
0:85ðx 1Þ for x > 1; x1 for x < 1:
(48)
Apparently, the relation n ¼ n was assumed by Batog et al. (2008), so that we have x ¼ E=E ¼ G=G. Thus, comparing Formulas (44) and (47), we arrive at the approximation a0 ðxÞz bðxÞ. For instance, in the cases n ¼ n ¼ 0:3 and n ¼ n ¼ 0:5, the maximum relative errors of this approximation are 6% (for 0.1 < x < 1), 15% (for 1 < x < 5) and 40% (for 0.1 < x < 1), 30% (for 1 < x < 5), respectively, and decrease as x increases. Note that Formula (48) was based on the FEM simulations of the contact problem for a cylindrical indenter, while based on the numerical results presented on Fig. 10, one can obtain more accurate linear approximations for the asymptotic constant a0(x) for different values of Poisson's ratio n ¼ n. It should be emphasized that we consider a perfect inhomogeneity/matrix bond and suppose that the continuity conditions for stresses and displacements are satisfied along the inhomogeneity/matrix interface. Following Sabina et al. (2012), the
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References
Fig. 11. Asymptotic constant a0 (see Eg. (43)) for a hemispherical inhomogeneity in an elastic half-space as a function of the ratio E=E. Effect of the matrix Poisson's ratio: (a) Relatively soft inhomogeneity (E=E 1); (b) Relatively stiff inhomogeneity (E=E 1).
mechanical imperfect conditions can be employed to model a transition zone (interphase) between the inhomogeneity and the matrix. Finally, we note that using the analytical results obtained by Kouris and Tsuchida (1992), one can generalize the developed firstorder asymptotic model for the case of a semi-ellipsoidal elastic inhomogeneity, whose geometry is described in the prolate spheroidal coordinates. This case is related to the problem of fiber indentation (Muki and Sternberg, 1969; Andrianov et al., 2009). On the other hand, applying finite element analysis (Kral et al., 1993; Kogut and Komvopoulos, 2004), the case of frictionless indentation of an elasticeplastic inhomogeneity can be studied. Numerical techniques (Zhou et al., 2011) will be also indispensable to extend the analysis to account for inhomogeneities of irregular shapes (see also a recent review by Zhou et al. (2013)) as well as to study the effect of neighboring inhomogeneities, in particular, with the aim of determining the critical distance between two inhomogeneities when Eq. (44) can be recommended. At that, it should be noted that interpretation of the experimental data depends on a number of factors (Fischer-Cripps, 2000; Gouldstone et al., 2007). Acknowledgment The support of Conacyt project number 129658 and Fenomec, rez UNAM is gratefully acknowledged. The authors thank Ana Pe vez Tovar for computational support. Arteaga and Ramiro Cha
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Small-scale indentation of a hemispherical inhomogeneity in an elastic half-space Ivan I. Argatov, Federico J. Sabina* ticas Aplicadas y en Sistemas, Universidad Nacional Auto noma de M Instituto de Investigaciones en Matema exico, Apartado Postal 20-726, n Alvaro Obregon, 01000, M Delegacio exico, D. F., Mexico
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 December 2014 Accepted 8 April 2015 Available online 18 April 2015
The axisymmetric problem of small-scale frictionless indentation of an elastic hemispherical inhomogeneity embedded at the free surface of a semi-infinite elastic matrix is considered. It is assumed that the radius of contact area is relatively small compared with the radius of the inhomogeneity. The first-order asymptotic model for the incremental indentation stiffness is presented in terms of the coefficient of local compliance, which is evaluated based on the analytical solution for the surface Green's function. The influence of both Poisson's ratios on the corresponding indentation scaling factor, which reflects the effect of localized inhomogeneity, is studied in detail. © 2015 Elsevier Masson SAS. All rights reserved.
Keywords: Indentation testing Hemispherical inhomogeneity Asymptotic model
1. Introduction Indentation technique (Hardy et al., 1971; Follansbee and Sinclair, 1984; Hill et al., 1989) has been proved very useful in testing mechanical properties of small material samples and, in particular, thin films (Antunes et al., 2007; Hemmouche et al., 2013). By modeling the sample configuration as an elastic layer bonded to a rigid base or to an elastic substrate, one can study the corresponding thickness effect (Hayes et al., 1972; Argatov et al., 2013) or the substrate effect (Yu et al., 1990; Gao et al., 1992; Chen and Vlassak, 2001; Perriot and Barthel, 2004; Argatov and Sabina, 2014). In recent years, the AFM indentation tests have been applied for characterizing composite materials (Gregory and Spearing, 2005; Constantinides et al., 2006), for which new identification methods should be developed in order to take into account the effect of material inhomogeneity (Kabele et al., 2008). In particular, an important sample geometry is represented by a hemispherical inhomogeneity embedded at the free surface of an elastic half-space made of another material (Fig. 1a). Axiallysymmetric finite-element solutions of the indentation problems for an elastic hemispherical inhomogeneity were obtained by Batog et al. (2008) and Kabele et al. (2008) under simplifying assumptions
* Corresponding author. E-mail addresses: [email protected] (I.I. Argatov), [email protected]. mx (F.J. Sabina). http://dx.doi.org/10.1016/j.euromechsol.2015.04.003 0997-7538/© 2015 Elsevier Masson SAS. All rights reserved.
that the values of Poisson's ratio is assumed to be the same in the hemispherical inhomogeneity and the semi-infinite matrix. In the present paper, we develop the first-order asymptotic model for the incremental indentation stiffness, which in the axisymmetric case can be written as follows (Argatov, 2010; Argatov and Sabina, 2014):
dP 2aEeff x : dw 1 ε 2a0
(1)
p
Here, P is the contact force, w is the indenter displacement, ε ¼ a/l is a small parameter, a is the radius of contact area, l is the radius of inhomogeneity, Eeff is the effective elastic modulus defined through the formula
1 1 n 1 n20 þ ¼ : Eeff E0 2G
(2)
Recall that the effective modulus Eeff is used to account for the effect of elastic deformation of the indenter, whose Young's modulus and Poisson's ratio are denoted by E0 and n0, respectively. Formula (1) contains the so-called (Argatov, 2002) coefficient of local compliance a0, which bears information about the inhomogeneity geometry, the interface conditions between the inhomogeneity and matrix, and depends on Poisson's ratios n and n as well as on the inhomogeneity-matrix shear moduli ratio G ¼ G=G.
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2. The hemispherical inhomogeneity subjected to a concentrated force In this section, we briefly outline the analytical solution obtained by Tsuchida et al. (1990) for the three-dimensional mixed boundary value problem of a hemispherical inhomogeneity, u, embedded at the free surface of an elastic half-space, ℝ3þ ¼ fx ¼ ðx1 ; x2 ; x3 Þ : x3 > 0g, and subjected to a concentrated force, P. By the way, we correct misprints and shortcomings in the mentioned study. We assume that the inhomogeneity u is perfectly bonded to the semi-infinite medium, and the continuity and equilibrium conditions along the interface g ¼ vu∩ℝ3þ are formulated as follows:
uðxÞ ¼ uðxÞ;
s
ðnÞ
ðxÞ ¼ s
ðnÞ
ðxÞ;
x2g:
(3)
Here, u and sðnÞ , u and sðnÞ are the displacement and stress vectors in the inhomogeneity u and in the matrix ℝ3þ nu, respectively. Let us denote the systems of cylindrical and spherical coordinates by (r,q,z) and (R,q,f) respectively (see Fig. 1). Then, in the spherical coordinates, the boundary conditions (3) can be rewritten as
ðuR ÞR¼l ¼ ðuR ÞR¼l ; ðsR ÞR¼l ¼ ðsR ÞR¼l ; Note that tRq ¼ tRq ¼ 0.
in
uf
R¼l
¼ uf R¼l ;
axisymmetric
G¼
G G
(8)
as well as on Poisson's ratios n and n, where G and n are the shear modulus and Poisson's ratio of the semi-infinite matrix. 2.2. Boussinesq potentials In the case of axisymmetry about the x3-axis, the general solution to the elasticity equations in the matrix is given by the Boussinesq potentials
vF0 vF þ m R 3 ð3 4nÞF3 ; uq ¼ 0; vR vR sinf vF0 vF þ sinf m 3 þ ð3 4nÞF3 ; 2Guf ¼ R vm vm 2GuR ¼
where F0 and F3 are harmonic functions, i. e., V2 F0 ¼ V2 F3 ¼ 0,
(5) case,
uq ¼ uq ¼ 0
V2 ¼
and
v 1 v 1 v 2 v 2 R þ 1 m ; vR vm R2 vR R2 vm
m ¼ cosf:
The corresponding stress field is given by1
1 m2 vF3 vF3 2n ; sR ¼ þ mR 2ð1 nÞm vR vm R vR2 vR2 i 1 vF h 1 vF0 m vF0 vF 3 2 þ ð1 2nÞm 3 2n þ ð1 2nÞm2 ; sq ¼ R vR R vm vm vR R i 1 vF h 1 m2 v2 F0 m vF0 1 vF0 m 1 m2 v2 F3 vF 3 sf ¼ þ þ ; þ ð1 2nÞm 3 1 þ ð1 2nÞ 1 m2 2 2 2 vm 2 R R vm R vR vR R R vm vm ! 1 vF0 1 v2 F0 vF v2 F 3 m vF3 þ ð1 2nÞ 3 m þ 2ð1 nÞ tRf ¼ sinf 2 ; R vRvm R vm vR vRvm R vm v2 F0
(9)
(4)
tRf R¼l ¼ tRf R¼l : the
Here, G and n are the shear modulus and Poisson's ratio of the inhomogeneity, l is the radius of the inhomogeneity. The asymptotic constants a0,a1,… are called the coefficients of local compliance (Argatov, 2002). We emphasize that the local-compliance coefficients are dimensionless and depend on the inhomogeneitymatrix shear moduli ratio
v2 F 3
(10)
tfq ¼ tRq ¼ 0:
2.1. Green's function and asymptotic coefficients of local compliance At the coordinate center, O, the vector-function u should satisfy the following asymptotic condition:
The solution T of the Boussinesq problem is expressed in terms of the Boussinesq potentials as
uðxÞ ¼ PTðxÞ þ Oð1Þ;
p l2 0 F0 ¼ 0 ð1 2nÞlnðR þ x3 Þ; 2
x/O:
(6)
Here, T is the solution of the Boussinesq problem for an elastic half-space comprised of the same material properties as the inhomogeneity. In the case of a unit force P, the solution to the elastic problem (3), (6) represents Green's vector-function with a pole at the point O. The structure of the next terms in the asymptotic expansion (6) was studied in detail by Argatov (2002). It was shown that in asymptotic analysis of frictionless contact problems, the following asymptotic expansion plays an important role: ∞ n 2pG 1 1X an 2 2 x u3 ðx1 ; x2 ; 0Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ x : 1 2 2n ð1 nÞP x2 þ x2 l n¼0 l 1
2
p l2 1 0 ; F3 ¼ 0 2 R
(11)
where p0 is the equivalent pressure given by
p0 ¼
P ; pl2
(12)
and the quantities denoted by a bar refer to the inhomogeneity.
(7) 1 Formulas (10) coincide with the corresponding relations used by Tsuchida et al. (1990) except for the formula for sf.
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153
Fig. 1. Hemispherical inhomogeneity under a concentrated force in the spherical (a) and cylindrical (b) coordinates.
The potentials (11) yield the following displacements2:
1 uR ¼ ½4ð1 nÞcosf ð1 2nÞ; 2 p 0 l2
2RG 1 1 2n sinf ð3 4nÞ u ¼ : f 2 1 þ cosf p 0 l2
V
F0 ¼ ð1 2nÞp0
2RG
p l2 1 FI3 ¼ 0 F0 ; R 2
(13)
P ðmÞ ¼ 2ð1 nÞp0 An 2n ; R2nþ1 n¼0
∞ X ðmÞ P ð2n þ 1ÞAn 2nþ1 ; FII3 ¼ p0 2nþ2 R n¼0
FIII 0 ¼ ð1 2nÞp0
∞ X
Bn
n¼0
FIII 3
¼ p0
∞ X
ð2n þ 2ÞBn
n¼0
P2nþ1 ðmÞ ; R2nþ2
P2nþ2 ðmÞ R2nþ3
(14)
IV
∞ X
¼ p0
∞ X
for the inhomogeneity (R < l). Here, Pn(m) are the Legendre functions of the first kind, F0 is an unknown quantity, and An, Bn, An , Bn are series coefficients. 2.3. Resulting linear algebraic system
0
(15)
(16)
IV
V
Fi ¼ Fi þ Fi þ Fi ;
Fi ¼ FIi þ FIIi þ FIII i
ði ¼ 0; 3Þ
(19)
for the inhomogeneity and the matrix, respectively, one can express stresses and displacements in terms of the unknown coefficients An, Bn, An , Bn . In order to apply the boundary conditions along the interface (4) and (5), the Legendre polynomials of odd order are expanded in terms of the Legendre polynomials of even order using the following “half-range expansion” (Tsuchida et al., 1990):
P2kþ1 ðmÞ ¼
∞ X
ðnÞ
wk P2n ðmÞ;
0 P2kþ1 ðmÞ ¼
n¼0
∞ X
ðnÞ
0 wk P2n ðmÞ:
n¼0
0 ðmÞ is the derivative of the Legendre polynomial P (m) Here, P2n 2n of degree 2n, and we used the notation
An R2nþ2 P2nþ2 ðmÞ;
n¼0 IV F3
(18)
∞ X ¼ p0 ð2n þ 1ÞBn R2n P2n ðmÞ
Using the Boussinesq potentials
for the matrix (R > l), and
F0 ¼ 2ð1 nÞp0
V F3
n¼0
∞ X
FII0
Bn R2nþ1 P2nþ1 ðmÞ;
n¼0
Following Tsuchida et al. (1990), the required auxiliary Boussinesq potentials are taken in the form
p l2 FI0 ¼ 0 F0 ð1 2nÞlnðR þ x3 Þ; 2
∞ X
(17)
ð2n þ 2ÞAn R2nþ1 P2nþ1 ðmÞ;
n¼0
ðnÞ
wk ¼
ð4n þ 1Þð2k þ 1Þ P ð0ÞP2k ð0Þ: ð2k þ 1 2nÞð2k þ 2 þ 2nÞ 2n
Thus, the first boundary condition (4) implies3
i lF0 h An ð2n þ 1Þ A 2nð2n 1Þ ðnÞ ðnÞ a þ n1 ð2n þ 3 4nÞ ð1 2nÞd0 þ 4ð1 nÞw0 þ 2nþ2 4n þ 3 2n 2 l l2n ( 4n 1 ∞ i 1 X Bk ð2k þ 2Þ h ð2n þ 1Þð2n þ 2Þ ðnÞ ðnÞ a ð2n 2 þ 4nÞ þ þ w þ ð2k þ 3Þð2k þ 6 4nÞw An l2nþ1 2kþ2 k kþ1 2kþ3 ð4k þ 5Þ G 4n þ 3 l k¼0 ) ∞ i X 2n ð2k þ 1Þ h ðnÞ ðnÞ 2kð2k 3 þ 4nÞwk1 þ a2k1 wk a2n1 þ Bk l2k þ An1 l2n1 ð4n 1Þ ð4k þ 1Þ k¼0 i l h ðnÞ ðnÞ ð1 2nÞd0 þ 4ð1 nÞw0 ðn ¼ 0; 1; 2; …Þ; ¼ 2G
2 Formulas (13) differ from the corresponding relations used by Tsuchida et al. (1990) by the factor 1/2 on the right-hand sides instead of the factor 2 that appeared in Tsuchida et al. (1990).
(20)
3 Note that the term with An1 in Eq. (20) differs from the corresponding term in Eq. (2.15) derived by Tsuchida et al. (1990) by an additional factor of n.
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ðnÞ
where di
denotes the Kronecker delta, and
an ¼ ðn þ 1Þ2 2 þ 2n;
an ¼ ð1Þn
an ¼ ðn þ 1Þ2 2 þ 2n:
ð2n þ 1Þ! 22n1 ðn!Þ2
l2n1 Bn :
(24)
Note that according to (22), the coefficients An ; Bn ; An , and Bn have dimensions of L2nþ3,L2nþ4,L2n, and L12n, respectively.
From the second boundary condition (4) it follows that
"
# ∞ X 4n þ 1 4k þ 3 An 1 A ð2n 1Þ ðnÞ ðnÞ ð3 4nÞw0 þ ð1 2nÞ w a þ n1 ð2n 4 þ 4nÞ ð1 2nÞ þ 2nþ2 2n ð4n 1Þ 2nð2n þ 1Þ ð2k þ 1Þð2k þ 2Þ k 4n þ 3 2n l l k¼0 ∞ h i X Bk 1 ð2n þ 2Þ 1 a 1 ðnÞ ðnÞ a2kþ2 wk þ ð2k þ 2Þð2k 1 þ 4nÞwkþ1 An l2nþ1 ð2n þ 5 4nÞ An1 l2n1 2n1 þ 2kþ3 ð4k þ 5Þ ð4n þ 3Þ G G ð4n 1Þ l lF 0 2
k¼0
i l2k 1h ðnÞ ðnÞ ð2k þ 1Þð2k þ 4 4nÞwk1 þ a2k1 wk ð4k þ 1Þ G k¼0 " # ∞ X l 4n þ 1 4k þ 3 ðnÞ ðnÞ ð1 2nÞ ð3 4nÞw0 þ ð1 2nÞ wk ¼ 2G 2nð2n þ 1Þ ð2k þ 1Þð2k þ 2Þ
∞ X Bk
ðn ¼ 1; 2; …Þ:
k¼0
(21)
From the first boundary condition (5), we get
i lF0 h An ð2n þ 1Þð2n þ 2Þ An1 2nð2n 1Þ ðnÞ ðnÞ a2n þ 2nþ1 b2n2 ð1 2nÞd0 þ 2ð2 nÞw0 þ 2nþ3 4n þ 3 4n 1 2 l l ( ∞ i X B ð2k þ 2Þð2k þ 3Þ h ð2n þ 1Þð2n þ 2Þ 2nð2n 1Þ ðnÞ ðn Þ k a2kþ2 wk þ b2kþ1 wkþ1 An l2n þ b2n4 þ An1 l2n2 a2n1 2kþ4 4k þ 5 4n þ 3 4n 1 l k¼0 " " #) # ∞ X 2kð2k þ 1Þ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðn ¼ 0; 1; 2; …Þ; þ Bk l2k1 b2k5 wk1 þ a2k1 wk ¼ ð1 2nÞd0 þ 2ð2 nÞw0 4k þ 1 2
(22)
k¼1
where
bn ¼ ðn þ 2Þðn þ 5Þ 2n;
3. Equilibrium of a two-layer hemispherical solid
bn ¼ ðn þ 2Þðn þ 5Þ 2n:
Finally, the second boundary condition (5) yields
In addition to the coefficients An ; Bn ; An , and Bn , the system (20)e(23) contains the unknown F0. It was suggested (Kouris and
" # ∞ X F0 4n þ 1 4k þ 3 An ð2n þ 2Þ A ð2n 1Þ ðnÞ ðnÞ w0 þ wk a þ n1 a ð1 2nÞ þ 2nþ3 2nð2n þ 1Þ ð2k þ 1Þð2k þ 2Þ ð4n þ 3Þ 2n l2nþ1 ð4n 1Þ 2n1 2 l k¼0 ∞ i X Bk a2kþ2 h ð2n þ 2Þ ð2n 1Þ ðnÞ ðnÞ ð2k þ 3Þwk þ ð2k þ 2Þwkþ1 þ An l2n þ a þ An1 l2n2 a 2kþ4 ð4n þ 3Þ 2n ð4n 1Þ 2n1 ð4k þ 5Þ l k¼0 " # ∞ ∞ i h X X a 1 4n þ 1 4k þ 3 ðnÞ ðnÞ ðnÞ ðnÞ w0 þ wk þ Bk l2k1 2k1 ð2k þ 1Þwk1 þ 2kwk ¼ ð1 2nÞ 2 2nð2n þ 1Þ ð2k þ 1Þð2k þ 2Þ 4k þ 1
ðn ¼ 1; 2; …Þ:
k¼0
k¼1
Now, provided F0 is known, the solution for the surface Green function can be obtained by solving the linear algebraic system (20)e(23) for the unknown coefficients An ; Bn ; An , and Bn . In view of the relation ðu3 Þx3 ¼0 ¼ ðuf Þf¼p=2 , the asymptotic constants an entering (7) can be expressed in terms of the coefficients Bn as follows:
(23)
Tsuchida, 1992) that the unknown constant F0 should be determined by the interfacial boundary conditions, that is from the system (20)e(23). Here we show that
F0 ¼ 1:
(25)
Indeed, let us consider a two-layer hemispherical solid R < L,
I.I. Argatov, F.J. Sabina / European Journal of Mechanics A/Solids 53 (2015) 151e162
155
where L > l. The vertical equilibrium equation for the two-layer solid can be written as
Zp=2 2
P ¼ 2pL
0
sR cosf tRf sinf R¼L sinf df
Z1 ¼ 2pL
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi sR m tRf 1 m2
(26) dm:
R¼L
0
In view of (10) and (19), we will have
1 m2 vF0 1 m2 v2 F0 v2 F0 þ vm vRvm R vR2 R2 vF 2 v F 3 3 þ m2 R m2 þ 1 2n vR vR2 1 vF v2 F 3 3 þ m 1 m2 : 2m 1 m2 R vm vRvm (27)
sR cosf tRf sinf ¼ m
Fig. 3. Asymptotic constants a0 > 0 and a1 < 0 for a clamped elastic hemisphere.
a2nþ1 ð2n þ 4Þ ð2n þ 7 4nÞ þ Anþ1 l2nþ3 4n þ 7 4n þ 3 ∞ i X l2k h ðnþ1Þ ðnþ1Þ ð2k þ 1Þð2k þ 4 4nÞwk1 þ a2k1 wk Bk þ 4k þ 1 k¼0 " l 4n þ 5 ðnþ1Þ ð1 2nÞ ð3 4nÞw0 ¼ þ ð1 2nÞ 2 ð2n þ 2Þð2n þ 3Þ # ∞ X 4k þ 3 ðnþ1Þ w ðn ¼ 0; 1; 2; …Þ: ð2k þ 1Þð2k þ 2Þ k
An l2nþ1 The subsequent substitution of (27) and (14)e(16) into Eq. (26) yields
P ¼ pl2 p0 F0 þ Oðl=LÞ:
(28)
Now, due to Eq. (12), the asymptotic relation (28), as L/∞, implies Eq. (25). Eq. (25) should be employed to simplify the system (20)e(23).
k¼0
(30)
4. Small-scale indentation of a clamped elastic hemisphere In this section we consider the first limit case when G/0 (see Fig. 2a) and Eqs. (20) and (21) reduce to the system
2n ð2n þ 1Þð2n þ 2Þ ð2n 2 þ 4nÞ a þ An l2nþ1 ð4n 1Þ 2n1 4n þ 3 ∞ i X ð2k þ 1Þ h ðnÞ ðnÞ 2kð2k 3 þ 4nÞwk1 þ a2k1 wk þ Bk l2k ð4k þ 1Þ k¼0 i lh ðnÞ ðnÞ ðn ¼ 0; 1; 2; …Þ; ¼ ð1 2nÞd0 þ 4ð1 nÞw0 2 (29)
An1 l2n1
Fig. 3 shows the behavior of the main asymptotic constants a0 and a1 with respect to Poisson's ratio n. Observe that the functions a0 ðnÞ and a1 ðnÞ have minima 1.45 and 0.39, respectively. The problem under consideration was solved in (Argatov, 2002) by another method based on the Weber harmonic potentials. For the considered specific set of n, the numerical results for a0 ðnÞ and a1 ðnÞ obtained by the two methods coincide. Let us compare the first asymptotic constant a0 with the corresponding quantity for an elastic layer of thickness l firmly attached (bonded) to a rigid base. According to Vorovich et al. (1974), we will have
aLn ¼
ð1Þn ½ð2nÞ!!2
Z∞ ½1 LðuÞu2n du; 0
Fig. 2. Two limit cases: (a) A clamped elastic hemisphere; (b) A rigid hemisphere embedded at the free surface of an elastic half-space.
(31)
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I.I. Argatov, F.J. Sabina / European Journal of Mechanics A/Solids 53 (2015) 151e162
5. Indentation of a rigid hemisphere embedded at the free surface of an elastic half-space In this section, we consider the case when the absolutely rigid inhomogeneity u (see Fig. 2b) receives a unit rigid body displacement we3, which can be described by the following boundary conditions:
ðu3 ÞR¼l ¼ w;
ður ÞR¼l ¼ 0;
ðuq ÞR¼l ¼ 0:
(32)
By converting Eq. (32) from cylindrical to spherical coordinates, we arrive at the boundary conditions
ðuR ÞR¼l ¼ wm;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi uf R¼l ¼ w 1 m2 :
(33)
According to (14)e(16) with F0 ¼ 1, Eq. (33) yield the following system of linear algebraic equations: Fig. 4. Asymptotic constants a0 and aL0 for a clamped elastic hemisphere and a bonded elastic layer, respectively.
where
LðuÞ ¼
2ksh2u 4u ; 2kch2u þ 1 þ k2 þ 4u2
k ¼ 3 4n:
Fig. 4 presents the behavior of the constants a0 and aL0 for different values of Poisson's ratio n. From the figure, we see that a0 > aL0 for all possible values of n. We emphasize that the Poisson's ratio effect is more pronounced for a clamped elastic hemisphere than for a bonded elastic layer. Recall that for an isotropic elastic layer resting on a smooth rigid base the asymptotic constants do not depend on its Poisson's ratio (Vorovich et al., 1974; Alexandrov and Pozharskii, 2001; Argatov et al., 2012). Note also that the numerical calculations show that the function a1 ðnÞ aL1 ðnÞ is not positive definite. Finally, note that the axisymmetric frictionless contact problem of the theory of elasticity for a clamped truncated sphere (a clamped elastic hemisphere is a special case of this) was studied by Alexandrov and Pozharskii (1997).
An1 2nð2n 1Þ An ð2n þ 1Þ ð2n þ 3 4nÞ þ 2nþ2 a ð4n þ 3Þ 2n l2n ð4n 1Þ l ∞ i X B ð2k þ 2Þ h ðnÞ ðnÞ k a2kþ2 wk þ ð2k þ 3Þð2k þ 6 4nÞwkþ1 þ 2kþ3 ð4k þ 5Þ l k¼0 i lh ðnÞ ðnÞ ð1 2nÞd0 þ 4ð1 nÞw0 ¼ 2 2G ðnÞ w w ðn ¼ 0; 1; 2; …Þ; p0 0 (34) An1 ð2n 1Þ An a2n ð2n 4 þ 4nÞ þ 2nþ2 ð4n þ 3Þ l2n ð4n 1Þ l ∞ h i X Bk 1 ðnÞ ðnÞ a2kþ2 wk þ ð2k þ 2Þð2k 1 þ 4nÞwkþ1 þ 2kþ3 ð4k þ 5Þ l k¼0 ( " # ∞ X l 4n þ 1 4k þ 3 ðnÞ ð1 2nÞ þ wk ¼ 2 2nð2n þ 1Þ ð2k þ 1Þð2k þ 2Þ k¼0 ) 2G ðnÞ ðnÞ w w ðn ¼ 1; 2; …Þ: þ ð3 4nÞw0 p0 0 (35) In addition to the unknowns An and Bn, the system (34), (35) contains the unknown parameter p0, which is related to the total force P by Eq. (12). It is to emphasize that we cannot derive the additional equation from the equilibrium Eq. (28) with L ¼ l, because the value of P is not known in advance, if w is given a priori. Further, observe that Eqs. (21) and (23) were derived on the basis of the expansion
! ∞ ∞ X X 1 4k þ 3 4n þ 1 ðnÞ ¼ w P 0 ðmÞ; 1 þ m n¼1 ð2k þ 1Þð2k þ 2Þ k 2nð2n þ 1Þ 2n k¼0
which does not converge to 1 as m / 0. On the other hand, we have ðu3 Þx3 ¼0 ¼ ðuf Þf¼p=2 . Therefore, in view of (33), we obtain ∞ X k¼0
Fig. 5. Stiffness coefficients C3(n) and c3(n) for a rigid hemisphere and a bonded flatended cylindrical punch, respectively.
ð1Þk
Bk ð2k þ 1Þ! l Gw ¼ þ : 2 ð1 nÞp0 l2kþ3 22k ðk!Þ2
(36)
Thus, following Argatov (2002), the force-displacement relationship can be written as
I.I. Argatov, F.J. Sabina / European Journal of Mechanics A/Solids 53 (2015) 151e162
P¼
2pG lC ðnÞw; 1n 3
(37)
where lC3(n) represents the translational capacity of the rigid hemispherical inhomogeneity, and C3(n) is a dimensionless quantity given by
C3 ðnÞ ¼
ð1 nÞp0 l : 2Gw
(38)
C1;0 ðnÞ ¼
157
ð2n þ 1Þ ð2n 2 þ 4nÞ; ð4n þ 3Þ
D1 ðn; kÞ ¼
C1;1 ðnÞ ¼
1 a ; ð4n þ 7Þ 2nþ2
h i 1 ðnþ1Þ ðnþ1Þ þ ð2k þ 2Þð2k 1 þ 4nÞwkþ1 ; a2kþ2 wk ð4k þ 5Þ
c1;0 ðnÞ ¼
1 a2nþ1 ; G ð4n þ 3Þ
1 ð2n þ 4Þ ð2n þ 7 4nÞ; c1;1 ðnÞ ¼ G ð4n þ 7Þ
Fig. 5 shows the variation of C3(n) with the change of Poisson's ratio n. The numerical data were obtained based on Formula (38) by solving the system (34)e(36) for w ¼ l (note that the problem (32) is linear). Also, for comparison, Fig. 5 presents a dashed line for the translational capacity lc3(n) of a flat-ended cylindrical punch of radius l attached to an elastic half-space. According to Mossakovskii (1963); Borodich and Keer (2004), in the last case, we will have
h 1 1 ðnþ1Þ ð2k þ 1Þð2k þ 4 4nÞwk1 d1 ðn; kÞ ¼ G ð4k þ 1Þ i ðnþ1Þ ; þ a2k1 wk
2pG lc ðnÞw; P¼ 1n 3
1 F1 ðnÞ ¼ 2
(39)
ð4n þ 5Þ ðnþ1Þ ð3 4nÞw0 ð2n þ 2Þð2n þ 3Þ # N2 X 4k þ 3 ðnþ1Þ wk þ ð1 2nÞ ð2k þ 1Þð2k þ 2Þ k¼0 " 1 ð4n þ 5Þ ðnþ1Þ ð3 4nÞw0 ð1 2nÞ 2G ð2n þ 2Þð2n þ 3Þ # N2 X 4k þ 3 ðnþ1Þ wk þ ð1 2nÞ ; ð2k þ 1Þð2k þ 2Þ
where w is the indenter displacement, P is the contact force, and
c3 ðnÞ ¼
2ð1 nÞ lnð3 4nÞ: pð1 2nÞ
"
(40)
ð1 2nÞ
Note that the normalization in (40) was chosen such that Formulas (37) and (39) have the same structure. Observe also that C3(0.5) z 0.728 and c3(0.5) ¼ 2/p z 0.637. Finally, note that the basis of homogeneous polynomial elastic solutions produced by the III Boussinesq potentials FII0 , FII3 , FIII 0 , and F3 coincides, up to normalization, with the basis constructed by Bogovoi and Nuller (1971).
C2;1 ðnÞ ¼
2nð2n 1Þ b ; ð4n 1Þ 2n2
6. Small-scale indentation of an elastic hemisphere embedded at the free surface of an elastic half-space
D2 ðn; kÞ ¼
i ð2k þ 2Þð2k þ 3Þ h ðnÞ ðnÞ a2kþ2 wk þ b2kþ1 wkþ1 ; ð4k þ 5Þ
For this general case, according to the approach described in Sections 2 and 3, the unknown coefficients of the Boussinesq potentials can be obtained by solving the system of linear algebraic Eqs. (20)e(23) with the relation F0 ¼ 1 taken into account. Correspondingly, the asymptotic constant a0 can be evaluated using Eq. (24). First, let us introduce the notation
C0;1 ðnÞ ¼
2nð2n 1Þ ð2n þ 3 4nÞ; 4n 1
C0;0 ðnÞ ¼
ð2n þ 1Þ a ; ð4n þ 3Þ 2n
i ð2k þ 2Þ h ðnÞ ðnÞ a2kþ2 wk þ ð2k þ 3Þð2k þ 6 4nÞwkþ1 ; D0 ðn; kÞ ¼ ð4k þ 5Þ 1 2n a ; G ð4n 1Þ 2n1 1 ð2n þ 1Þð2n þ 2Þ ð2n 2 þ 4nÞ; c0;0 ðnÞ ¼ G ð4n þ 3Þ
k¼0
C2;0 ðnÞ ¼
ð2n þ 1Þð2n þ 2Þ a2n ; ð4n þ 3Þ
2nð2n 1Þ c2;1 ðnÞ ¼ a ; ð4n 1Þ 2n1 ð2n þ 1Þð2n þ 2Þ c2;0 ðnÞ ¼ b2n4 ; ð4n þ 3Þ i 2mð2m þ 1Þ h ðnÞ ðnÞ b2m5 wm1 þ a2m1 wm ; d2 ðn; mÞ ¼ ð4m þ 1Þ F2 ðnÞ ¼
i 1h ðnÞ ðnÞ ð1 2nÞd0 þ 2ð2 nÞw0 2 i 1h ðnÞ ðnÞ ð1 2nÞd0 þ 2ð2 nÞw0 ; 2
c0;1 ðnÞ ¼
C3;0 ðnÞ ¼
ð2n þ 1Þ a ; ð4n þ 3Þ 2nþ1
i 1 ð2k þ 1Þ h ðnÞ ðnÞ 2kð2k 3 þ 4nÞwk1 þ a2k1 wk ; d0 ðn; kÞ ¼ G ð4k þ 1Þ
D3 ðn; kÞ ¼
i 1h 1 h ðnÞ ðnÞ ðnÞ ð1 2nÞd0 þ 4ð1 nÞw0 ð1 2nÞd0 F0 ðnÞ ¼ 2 2G i ðnÞ þ 4ð1 nÞw0 ;
c3;0 ðnÞ ¼
ð2n þ 4Þ a ; ð4n þ 7Þ 2nþ2
i a2kþ2 h ðnþ1Þ ðnþ1Þ ð2k þ 3Þwk þ ð2k þ 2Þwkþ1 ; ð4k þ 5Þ
ð2n þ 1Þ a ; ð4n þ 3Þ 2nþ1
d3 ðn; mÞ ¼
C3;1 ðnÞ ¼
c3;1 ðnÞ ¼
ð2n þ 4Þ a ; ð4n þ 7Þ 2nþ2
i a2m1 h ðnþ1Þ ðnþ1Þ ; ð2m þ 1Þwm1 þ 2mwm 4m þ 1
158
F3 ðnÞ ¼
I.I. Argatov, F.J. Sabina / European Journal of Mechanics A/Solids 53 (2015) 151e162
" 1 ð4n þ 5Þ ðnþ1Þ ð1 2nÞ w0 2 ð2n þ 2Þð2n þ 3Þ þ "
N2 X k¼0
4k þ 3 ðnþ1Þ w ð2k þ 1Þð2k þ 2Þ k
#
1 ð4n þ 5Þ ðnþ1Þ ð1 2nÞ w0 2 ð2n þ 2Þð2n þ 3Þ þ
N2 X k¼0
# 4k þ 3 ðnþ1Þ w : ð2k þ 1Þð2k þ 2Þ k
Second, for a given natural number N1, we define an (4N1 þ 4) (4N1 þ 4)-matrix M and an (4N1 þ 4)-vector f as follows. Let the indices n and m run from 1 to N1, and the index k run from 0 to N1. Then, we set
M0;0 ¼ C0;0 ð0Þ; M0;1 ¼ c0;0 ð0Þ; M0;4kþ2 ¼ D0 ð0; kÞ; M0;4kþ3 ¼ d0 ð0; kÞ; f0 ¼ F0 ð0Þ; M1;0 ¼ C1;0 ð0Þ; M1;1 ¼ c1;0 ð0Þ; M1;4 ¼ C1;1 ð0Þ; M1;5 ¼ c1;1 ð0Þ; M1;4kþ2 ¼ D1 ð0; kÞ; M1;4kþ3 ¼ d1 ð0; kÞ; f1 ¼ F1 ð0Þ; M2;0 ¼ C2;0 ð0Þ; M2;1 ¼ c2;0 ð0Þ; M2;4kþ2 ¼ D2 ð0; kÞ; M2;4mþ3 ¼ d2 ð0; mÞ; f2 ¼ F2 ð0Þ; M3;0 ¼ C3;0 ð0Þ; M3;1 ¼ c3;0 ð0Þ; M3;4 ¼ C3;1 ð0Þ; M3;5 ¼ c3;1 ð0Þ; M3;4kþ2 ¼ D3 ð0; kÞ; M3;4mþ3 ¼ d3 ð0; mÞ; f3 ¼ F3 ð0Þ; M4n;4n4 ¼ C0;1 ðnÞ; M4n;4n3 ¼ c0;1 ðnÞ; M4n;4n ¼ C0;0 ðnÞ; M4n;4nþ1 ¼ c0;0 ðnÞ; M4n;4kþ2 ¼ D0 ðn; kÞ; M4n;4kþ3 ¼ d0 ðn; kÞ; f4n ¼ F0 ðnÞ;
Fig. 6. Asymptotic constant a0 for a hemispherical inhomogeneity as a function of its Poisson's ratio: (a) Relatively soft inhomogeneity (G 1); (b) Relatively stiff inhomogeneity (G 1).
M4nþ1;4n ¼ C1;0 ðnÞ; M4nþ1;4nþ1 ¼ c1;0 ðnÞ; M4nþ1;4nþ4 ¼ C1;1 ðnÞ; M4nþ1;4nþ5 ¼ c1;1 ðnÞ; M4nþ1;4kþ2 ¼ D1 ðn; kÞ; M4nþ1;4kþ3 ¼ d1 ðn; kÞ; f4nþ1 ¼ F1 ðnÞ; M4nþ2;4n4 ¼ C2;1 ðnÞ; M4nþ2;4n3 ¼ c2;1 ðnÞ; M4nþ2;4n ¼ C2;0 ðnÞ; M4nþ2;4nþ1 ¼ c2;0 ðnÞ; M4nþ2;4kþ2 ¼ D2 ðn; kÞ; M4nþ2;4mþ3 ¼ d2 ðn; mÞ; f4nþ2 ¼ F2 ðnÞ; M4nþ3;4n ¼ C3;0 ðnÞ; M4nþ3;4nþ1 ¼ c3;0 ðnÞ; M4nþ3;4nþ4 ¼ C3;1 ðnÞ; M4nþ3;4nþ5 ¼ c3;1 ðnÞ; M4nþ3;4kþ2 ¼ D3 ðn; kÞ; M4nþ3;4mþ3 ¼ d3 ðn; mÞ; f4nþ3 ¼ F3 ðnÞ: Third, taking l as unit of length, we introduce an (4N1 þ 4)-vector X of the unknown coefficients An, Bn, An , Bn by formulas
X4k ¼ Ak ;
X4kþ1 ¼ Ak ;
X4kþ2 ¼ Bk ;
X4kþ3 ¼ Bk :
Thus, the problem of approximate solution of the system (20)e(23) is reduced to the system of 4(N1 þ 1) linear algebraic equations MX ¼ f, from which it immediately follows that X ¼ M1f.
The results of numerical calculations are presented on Figs. 6 and 7. All calculations were performed using Mathcad software (MathSoft Inc., Cambridge, MA). The accuracy of numerical approximations depends on the numbers N1 and N2 of series members in the Legendre series expansions (15)e(18) and in the series on the right-hand sides of Eqs. (21), (23), (30) and (35), the maxima of which were taken to be N1 ¼ 40 and N2 ¼ 100, respectively. Convergence was checked by varying the numbers N1 and N2. In particular, Fig. 6 shows the behavior of the coefficient of local compliance a0 as a function of the inhomogeneity Poisson's ratio n for different values of the inhomogeneity-matrix shear moduli ratio G ¼ G=G, whereas the variation of a0 as a function of the matrix Poisson's ratio n is shown in Fig. 7. In order to illustrate the behavior of the asymptotic contact a0, we compare the case of an elastic hemisphere embedded at the free surface of an elastic half-space with the case of an elastic layer bonded to an elastic substrate (see Figs. 8 and 9). This latter case was studied in detail in (Argatov, 2010; Argatov and Sabina, 2014), and the corresponding asymptotic contact is denoted by aLS 0 . An extensive asymptotic analysis of the indentation stiffness has been recently given by Argatov and Sabina (2013), whereas the asymptotics of the contact pressure was obtained earlier by Argatov (2001). Following Argatov and Sabina (2013, 2014), we rewrite Eq. (1) in the form
I.I. Argatov, F.J. Sabina / European Journal of Mechanics A/Solids 53 (2015) 151e162
159
Fig. 7. Asymptotic constant a0 for a hemispherical inhomogeneity in an elastic halfspace as a function of the half-space Poisson's ratio: (a) Relatively soft inhomogeneity (G 1); (b) Relatively stiff inhomogeneity (G 1).
Fig. 8. Asymptotic constants a0 and aLS 0 for a hemispherical inhomogeneity in an elastic half-space and an inhomogeneity in the form of elastic layer bonded to an elastic semi-infinite substrate as functions of their Poisson's ratios: (a) Relatively soft inhomogeneities (G 1); (b) Relatively stiff inhomogeneities (G 1).
dP ¼ 2aEeff K ; dw
Poisson's ratio n ¼ 0.3. Note that the curve corresponding to n ¼ 0:3 (dashed line) crosses the abscissa axis at E=E ¼ 1. Fig. 11 shows that the effect of the matrix Poisson's ratio is less pronounced than the effect of the inhomogeneity Poisson's ratio (cf. Fig. 10). Now, let us outline the method of solving the inverse identification problem based on Eqs. (41) and (42) under the assumption that the matrix modulus of elasticity E as well as both Poisson's ratios n and n are known. Moreover, for the sake of simplicity, we neglect the deformation of the indenter assuming that it is absolutely rigid. In this case, in view of (2) and (42), Eq. (41) takes the form
(41)
where we introduced the corresponding indentation scaling factor, which accounts for the effect of localized inhomogeneity in the frictionless small-scale indentation of an elastic hemispherical inhomogeneity embedded at the free surface of an elastic semiinfinite matrix. At that, according to (1), the following first-order asymptotic formula takes place:
a1 K x 1 a0 : l
(42)
Here we introduced the notation
a0 ¼
2a0 : p
(43)
Recall that a0 is a dimensionless asymptotic constant (normalized coefficient of local compliance), which possesses information about the relative stiffness of the inhomogeneity and also depends on the boundary conditions imposed at the interface between the inhomogeneity and the matrix. Fig. 10 presents the behavior of a0 as a function of the elastic moduli ratio E=E ¼ Gð1 þ nÞ=ð1 þ nÞ for a special case of the matrix
dP 2aE a1 ¼ 1 a : 0 dw 1 n2 l
(44)
Further, denoting the right-hand side of Eq. (41) by S ðaÞ and introducing a dimensionless unknown
x¼
E ; E
(45)
we can rewrite Eq. (44) as follows:
x¼
1 n2 a S ðaÞ 1 a0 ðxÞ : l 2aE
(46)
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Fig. 9. Asymptotic constants a0 and aLS 0 for a hemispherical inhomogeneity in an elastic half-space and an inhomogeneity in the form of elastic layer bonded to an elastic semi-infinite substrate as functions of the half-space Poisson's ratios: (a) Relatively soft inhomogeneities (G 1); (b) Relatively stiff inhomogeneities (G 1).
It can be shown that if the incremental indentation stiffness S ðaÞ is measured in the small-scale range (i.e., when a=l≪1), Eq. (46) can be solved by an iterative method as
xn ¼
1 n2 a S ðaÞ 1 a0 ðxn1 Þ ; l 2aE
n ¼ 1; 2; …;
starting with the zeroth approximation
x0 ¼
1 n2 S ðaÞ: 2aE
Finally, note that this method can be straightforwardly generalized to take into account the indenter deformation through Formula (2).
7. Discussion Let us compare the developed first-order asymptotic model (44) with the analytical approximation of Batog et al. (2008) obtained by fitting finite-element numerical results, which in our notation is represented in the form
Fig. 10. Asymptotic constant a0 (see Eg. (43)) for a hemispherical inhomogeneity in an elastic half-space as a function of the ratio E=E. Effect of the inhomogeneity Poisson's ratio: (a) Relatively soft inhomogeneity (E=E 1); (b) Relatively stiff inhomogeneity (E=E 1).
dP 2aE a1 ¼ 1 þ bðxÞ ; 2 dw 1 n l
(47)
where
bðxÞ ¼
0:85ðx 1Þ for x > 1; x1 for x < 1:
(48)
Apparently, the relation n ¼ n was assumed by Batog et al. (2008), so that we have x ¼ E=E ¼ G=G. Thus, comparing Formulas (44) and (47), we arrive at the approximation a0 ðxÞz bðxÞ. For instance, in the cases n ¼ n ¼ 0:3 and n ¼ n ¼ 0:5, the maximum relative errors of this approximation are 6% (for 0.1 < x < 1), 15% (for 1 < x < 5) and 40% (for 0.1 < x < 1), 30% (for 1 < x < 5), respectively, and decrease as x increases. Note that Formula (48) was based on the FEM simulations of the contact problem for a cylindrical indenter, while based on the numerical results presented on Fig. 10, one can obtain more accurate linear approximations for the asymptotic constant a0(x) for different values of Poisson's ratio n ¼ n. It should be emphasized that we consider a perfect inhomogeneity/matrix bond and suppose that the continuity conditions for stresses and displacements are satisfied along the inhomogeneity/matrix interface. Following Sabina et al. (2012), the
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161
References
Fig. 11. Asymptotic constant a0 (see Eg. (43)) for a hemispherical inhomogeneity in an elastic half-space as a function of the ratio E=E. Effect of the matrix Poisson's ratio: (a) Relatively soft inhomogeneity (E=E 1); (b) Relatively stiff inhomogeneity (E=E 1).
mechanical imperfect conditions can be employed to model a transition zone (interphase) between the inhomogeneity and the matrix. Finally, we note that using the analytical results obtained by Kouris and Tsuchida (1992), one can generalize the developed firstorder asymptotic model for the case of a semi-ellipsoidal elastic inhomogeneity, whose geometry is described in the prolate spheroidal coordinates. This case is related to the problem of fiber indentation (Muki and Sternberg, 1969; Andrianov et al., 2009). On the other hand, applying finite element analysis (Kral et al., 1993; Kogut and Komvopoulos, 2004), the case of frictionless indentation of an elasticeplastic inhomogeneity can be studied. Numerical techniques (Zhou et al., 2011) will be also indispensable to extend the analysis to account for inhomogeneities of irregular shapes (see also a recent review by Zhou et al. (2013)) as well as to study the effect of neighboring inhomogeneities, in particular, with the aim of determining the critical distance between two inhomogeneities when Eq. (44) can be recommended. At that, it should be noted that interpretation of the experimental data depends on a number of factors (Fischer-Cripps, 2000; Gouldstone et al., 2007). Acknowledgment The support of Conacyt project number 129658 and Fenomec, rez UNAM is gratefully acknowledged. The authors thank Ana Pe vez Tovar for computational support. Arteaga and Ramiro Cha
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