- Email: [email protected]
AFM-based indentation stiffness tomography—An asymptotic model
Journal of the Mechanics and Physics of Solids 70 (2014) 190–199
Contents lists available at ScienceDirect
Journal of the Mechanics and Physics of Solids journal homepage: www.elsevier.com/locate/jmps
AFM-based indentation stiffness tomography—An asymptotic model I.I. Argatov Engineering Mechanics Laboratory, University of Oulu, 90014 Oulu, Finland
a r t i c l e in f o
abstract
Article history: Received 21 November 2013 Received in revised form 28 May 2014 Accepted 4 June 2014 Available online 12 June 2014
The so-called indentation stiffness tomography technique for detecting the interior mechanical properties of an elastic sample with an inhomogeneity is analyzed in the framework of the asymptotic modeling approach under the assumption of small size of the inhomogeneity. In particular, it is assumed that the inhomogeneity size and the size of contact area under the indenter are small compared with the distance between them. By the method of matched asymptotic expansions, the first-order asymptotic solution to the corresponding frictionless unilateral contact problem is obtained. The case of an elastic half-space containing a small spherical inhomogeneity has been studied in detail. Based on the grid indentation technique, a procedure for solving the inverse problem of extracting the inhomogeneity parameters is proposed. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Depth-sensing indentation Indentation stiffness Indentation tomography Elastic inhomogeneity Asymptotic model
1. Introduction In recent years, the atomic force microscope (AFM) has become an indispensable tool for the indentation based characterization (Fischer-Cripps, 2004) of mechanical properties of living samples at the nanometric scale (Kasas and Dietler, 2008; Loparic et al., 2010; Plodinec et al., 2012; Kasas et al., 2013). Based on the best-fitting analysis of the forceindentation (FI) curve, the so-called indentation stiffness tomography technique was proposed by Roduit et al. (2009) for distinguishing structures of different stiffness buried into the bulk of the sample. The developed methodology assumes that the presence of an inhomogeneity (inclusion) changes the FI curve in a deterministic way reflecting the relative hardness/ softness property of the inclusion and the depth of the inclusion. The validity of this concept was verified by finite element models and was proven useful in AFM-based indentation experiments on living cells (Rheinlaender et al., 2011). Three-dimensional contact problems for the special case of an elastic half-space with inhomogeneities were studied by numerical techniques in a number of publications (Leroux et al., 2010; Zhou et al., 2011). An extensive review of works on the elastic problems for inclusions in an elastic half-space was very recently presented by Zhou et al. (2013). In the present paper, we consider the frictionless unilateral contact problem for a homogeneous linearly elastic body with a small homogeneous inclusion (with no eigenstrains). Based on the obtained first-order asymptotic solution, we address the identification problem resulting in the identification of some parameters of the small inhomogeneity. The dynamic problem of stress-wave identification of material defects (e.g., cavities, cracks, inclusions) has been the subject of extensive research (Alves and Ammari, 2001; Guzina et al., 2003; Bonnet and Constantinescu, 2005; Ammari and Kang, 2006; Guzina and Chikichev, 2007). At the same time, the quasi-static problem of indentation stiffness tomography has its own peculiarities, and the asymptotic model proposed herein for extracting the inhomogeneity parameters based on the
E-mail address: [email protected] http://dx.doi.org/10.1016/j.jmps.2014.06.001 0022-5096/& 2014 Elsevier Ltd. All rights reserved.
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
191
grid indentation data has not been developed elsewhere. The objective of this study was to develop a simple mathematical model for the indentation stiffness tomography. The rest of the paper is organized as follows. In Section 2, we formulate the mathematical model of frictionless indentation. By means of the method of matched asymptotic expansions, in Section 3 an approximate analytical solution is obtained. The first-order asymptotic model for the indentation test is developed in Section 4. The case of a small spherical inhomogeneity in an elastic half-space is studied in detail in Section 5, where a procedure for extracting the inhomogeneity parameters is proposed. Finally, in Sections 6 and 7, respectively, we outline a discussion of the results obtained and formulate our conclusions. 2. Mathematical model Suppose a homogeneous linearly elastic body without inhomogeneity occupies a three-dimensional domain Ω. Let ωε be a small inhomogeneity with the center at a point x0 and the diameter proportional to a small positive parameter ε such that ωε lies within the domain Ω. Thus, the domain Ωε ¼ Ω\ω ε will represent the reference configuration of the homogeneous deformable body with the inhomogeneity ωε . In the absence of volume forces, the displacement vectors u ¼ ðu1 ; u2 ; u3 Þ in Ωε and u0 ¼ ðu01 ; u02 ; u03 Þ in ωε satisfy the Lamé differential systems
μ∇x ∇x uðxÞ þ ðλ þ μÞ∇x ∇x uðxÞ ¼ 0;
x A Ωε ;
μ0 ∇x ∇x u0 ðxÞ þðλ0 þ μ0 Þ∇x ∇x u0 ðxÞ ¼ 0;
ð1Þ
x A ωε :
ð2Þ
We assume that the inhomogeneity ωε is perfectly bonded to the surrounding medium, and the continuity and equilibrium conditions along the interface ∂ωε are formulated as follows: uðxÞ ¼ u0 ðxÞ;
σ ðnÞ ðu; xÞ ¼ σ 0ðnÞ ðu0 ; xÞ; x A ∂ωε :
ð3Þ
The outer surface of the body Ωε is assumed to be decomposed into three mutually disjoint parts: ∂Ω ¼ Γ u [ Γ σ [ Γ c . Over Γu, the body is held fixed, while the surface Γ σ is assumed to be traction free, i.e., uðxÞ ¼ 0; x A Γ u ;
σ ðnÞ ðu; xÞ ¼ 0; x A Γ σ :
ð4Þ
Finally, Γc is the potential contact boundary, over which contact of the deformable body Ωε with a rigid indenter may take place. Let us introduce a Cartesian coordinate system with the center at a point O on Γc, which is supposed to be the initial point of contact in the case of a convex indenter (in the unloading state, the indenter touches the surface Γc at the point O). To fix our ideas, we will assume that the x3-axis is directed inside the body Ωε with the plane Ox1 x2 being tangent to the surface Γc (see Fig. 1). At the initial moment, the surface of the indenter is specified by the equation x3 ¼ Φðx1 ; x2 Þ;
ð5Þ
and during the normal indentation of the indenter into the deformable body (see Fig. 2), the indenter surface will be given by the equation x3 ¼ w Φðx1 ; x2 Þ;
ð6Þ
where w is a small displacement of the indenter. To formulate on Γc the boundary conditions of unilateral contact (Kikuchi and Oden, 1988; Shillor et al., 2004), we denote by gN the variable gap between the indenter surface and Γc. Note that gN is measured in a direction normal to the body surface and g N ðx1 ; x2 Þ ¼ g 0N ðx1 ; x2 Þ þ
w ; cos ðnðxÞ; x3 Þ
ð7Þ 0 gN
where n is the outer unit normal vector to the boundary of the domain Ω, is the initial gap between the body and the indenter. It is assumed that the gap function g 0N ðx1 ; x2 Þ takes on a zero value at the origin, so that w represents the current depth of the indenter's penetration into the body. Thus, the Signorini boundary conditions of frictionless contact may be stated as
σ TðnÞ ðu; xÞ ¼ 0;
x A Γc ;
ð8Þ
Fig. 1. Deformable body and reference coordinate system.
192
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
Fig. 2. Indentation test schematics.
Fig. 3. Schematics for the “outer” displacement field.
σ ðnÞ uN ðxÞ g N ðx1 ; x2 Þ r 0; N ðu; xÞ r0; σ ðnÞ ðu; xÞðu ðxÞ g N ðx1 ; x2 ÞÞ ¼ 0; x A Γ c : N N
ð9Þ
ðnÞ ðnÞ Here, σ ðnÞ N and σ T are the normal and tangential components of the stress vector σ . Observe that the second constraint in (9) means that the normal displacement uN ¼ u n cannot exceed the current gap gN, which is governed by the vertical position of the indenter (see Eq. (7)). Under the assumption that the size of the contact area is small compared to the curvature radii of the surface Γc at the point O, we will have
g N ðx1 ; x2 Þ C Φðx1 ; x2 Þ w;
ð10Þ
where Φðx1 ; x2 Þ is the indenter shape function introduced by Eq. (5). Note that the approximate equation (10) becomes exact when a neighborhood on the surface Γc around the point O is flat. Local equilibrium at the contact interface in the x3-direction requires that the contact pressure should be balanced by the contact force, P, that is ðnÞ ðu; xÞ cos ðnðxÞ; x3 Þ dSx : P ¼ ∬Γ c σ N
ð11Þ
Now, combining (1)–(11), we formulate a mathematical model of unilateral contact for the linearly elastic body Ωε with the small inhomogeneity ωε . 3. Asymptotic approximation for the displacement field away from the indenter At a distance from the contact zone as well as from the inhomogeneity, the stress–strain state of the body Ωε is approximated by the following so-called “outer” displacement vector-field (see Fig. 3): 6
vðxÞ ¼ PGðxÞ þ ∑ Q k MðkÞ ðx0 ; xÞ:
ð12Þ
k¼1
Here, GðxÞ is the solution of the elastic problem in the domain Ω of the action of a unit point force applied at the point O in the x3-direction, MðkÞ ðx0 ; xÞ are the solutions of the elastic problem in the domain Ω of the action of force dipoles at the point x0 . For the simplicity sake let us assume that the principal curvatures of the surface Γc at the point O are zero. Then, the vector-function G satisfies the following equations:
μ∇x ∇x GðxÞ þ ðλ þ μÞ∇x ∇x GðxÞ ¼ 0; σ ðnÞ ðG; xÞ ¼ 0; x A Γ σ [ ðΓ c \OÞ; GðxÞ ¼ TðxÞ þOð1Þ; x-O:
x A Ω;
GðxÞ ¼ 0; x A Γ u ;
ð13Þ
Here, T is the solution of the Boussinesq problem (see, e.g., Johnson, 1985) of the action on the boundary of an elastic halfspace x3 40 of a unit force directed along the x3-axis, i.e., 1 xi x3 μ xi ; i ¼ 1; 2; T i ðx Þ ¼ 3 4πμ jxj λ þ μ jxjðjxj þ x3 Þ 2 x3 λ þ 2μ 1 1 : ð14Þ þ T 3 ðx Þ ¼ 4πμ jxj3 λ þ μ jxj
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
193
To describe the vector-functions MðkÞ , following Zorin et al. (1990), we introduce vector polynomials Vð1Þ ðxÞ ¼ ðx1 ; 0; 0Þ;
Vð2Þ ðxÞ ¼ ð0; x2 ; 0Þ;
Vð4Þ ðxÞ ¼ p1ffiffi2 ðx2 ; x1 ; 0Þ;
Vð3Þ ðxÞ ¼ ð0; 0; x3 Þ;
Vð5Þ ðxÞ ¼ p1ffiffi2 ð0; x3 ; x2 Þ;
Vð6Þ ðxÞ ¼ p1ffiffi2 ðx3 ; 0; x1 Þ:
ð15Þ
Let also SðxÞ be the Kelvin–Somigliana fundamental matrix, i.e., xi xj λ þ3μ δij λþμ ; i; j ¼ 1; 2; 3: þ Sij ðxÞ ¼ 3 8πμðλ þ2μÞ jxj λ þ μ jxj Then, the vector-functions MðkÞ ðx0 ; xÞ with singularities of order Oðjx x0 j 2 Þ at the point x0 are defined as follows:
μ∇x ∇x MðkÞ ðx0 ; xÞ þ ðλ þ μÞ∇x ∇x MðkÞ ðx0 ; xÞ ¼ 0; σ ðnÞ ðMðkÞ ; xÞ ¼ 0; x A Γ σ [ Γ c ;
x A Ω\x0 ;
MðkÞ ðx0 ; xÞ ¼ 0; x A Γ u ;
MðkÞ ðx0 ; xÞ ¼ VðkÞ ð∇x ÞSðx x0 Þ þ Oð1Þ;
x-x0 :
ð16Þ
Further, to construct the boundary layer around the inhomogeneity ωε let us assume that ωε ¼ fxjε ðx x Þ A ωg, where ω is a fixed three-dimensional domain, from which the small inhomogeneity ωε with the center at the point x0 is rescaled by means of the so-called stretched coordinates ξ ¼ ε 1 ðx x0 Þ, so that the two domains ω (in the ξspace) and ωε (in the xspace) are related through a linear transform x ¼ x0 þ εξ. In what follows, we make use of the Taylor expansion 1
0
6
GðxÞ ¼ Gðx0 Þ þ ðx x0 Þ ωðx0 Þ þ ∑ ϵ0j VðjÞ ðx x0 Þ þ Oðjx x0 j2 Þ;
ð17Þ
j¼1
where ωðxÞ ¼ ð1=2Þ∇x GðxÞ, VðjÞ are the vector polynomials (15), and the strain components
ϵ0i ¼ εii ðG; x0 Þ; i ¼ 1; 2; 3; pffiffiffi pffiffiffi ϵ04 ¼ 2ε12 ðG; x0 Þ; ϵ05 ¼ 2ε23 ðG; x0 Þ;
ϵj are defined as 0
pffiffiffi
ϵ06 ¼ 2ε13 ðG; x0 Þ:
ð18Þ
Thus, in view of (12), (16)3 and (17), the following asymptotic expansion takes place: ( ) 6
6
j¼1
k¼1
vðx0 þ εξÞ ¼ P Gðx0 Þ þ εξ ωðx0 Þ þ ε ∑ ϵ0j VðjÞ ðξÞ þOðε2 jξj2 Þ þ ∑ Q k fε 2 VðkÞ ð∇ξ ÞSðξÞ þOð1Þg:
ð19Þ
By the method of matched asymptotic expansions (Van Dyke, 1964; Il'in, 1992), the stress–strain state of the body Ωε in the vicinity of the inhomogeneity is approximated by the vector-function ( ) 6 x x0 x x0 ¼ P G x0 þ x x0 ω x0 þ ∑ ϵ0j WðjÞ : ð20Þ w
ε
ε
j¼1
Here, WðjÞ ðξÞ are the unique solutions of the following elastic problem (Zorin et al., 1990):
μ∇ξ ∇ξ WðjÞ ðξÞ þ ðλ þ μÞ∇ξ ∇ξ WðjÞ ðξÞ ¼ 0;
ξ A R3 \ ω ;
μ0 ∇ξ ∇ξ W0ðjÞ ðξÞ þ ðλ0 þ μ0 Þ∇ξ ∇ξ W0ðjÞ ðξÞ ¼ 0; WðjÞ ðξÞ ¼ W0ðjÞ ðξÞ;
ξ A ω;
σ ðnÞ ðWðjÞ ; ξÞ ¼ σ 0ðnÞ ðW0ðjÞ ; ξÞ; ξ A ∂ω;
WðjÞ ðξÞ ¼ VðjÞ ðξÞ þ oð1Þ;
jξj-1:
The solutions of the above problem have the following expansions at the infinity (Zorin et al., 1990): 6
WðjÞ ðξÞ VðjÞ ðξÞ ¼ ∑ P jk VðkÞ ð∇ξ ÞSðξÞ þOðjξj 3 Þ:
ð21Þ
k¼1
Here, P jk are the components of the so-called elasticity polarization matrix of the inhomogeneity ω (Zorin et al., 1990; Lewiński and Sokołowski, 2003; Ammari and Kang, 2007). Thus, in view of (19) and (21), the asymptotic matching of the outer asymptotic representation (12) and the inner asymptotic representation (20) implies the equations 6
Q k ¼ ∑ P ϵ0j P εjk ;
ð22Þ
j¼1
where P εjk ¼ ε3 P jk are the polarization matrix components for the inhomogeneity ωε . Note that the polarization matrix possesses a number of important properties (Nazarov and Sokołowski, 2003; Ammari and Kang, 2007) and has been encountered in the method of topological sensitivity (Bonnet and Guzina, 2004; Guzina and
194
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
Chikichev, 2007; Novotny and Sokołowski, 2013). Note also that in regard to the case of detecting a small inclusion, complete asymptotic expansions for the elastic displacement fields vðxÞ and wðξÞ were constructed by Ammari et al. (2002). 4. Asymptotic model for the indentation test To construct the boundary layer type solution around the indenter, we consider the following asymptotic expansion (see (12) and (13)3): 6
vðxÞ ¼ PðTðxÞ þgðOÞÞ þ ∑ Q k MðkÞ ðx0 ; OÞ þ OðjxjÞ:
ð23Þ
k¼1
Here, gðxÞ ¼ GðxÞ TðxÞ is the regular part of Green's function GðxÞ. In asymptotic analysis of frictionless contact problems, an important role is played by the asymptotic constant g 3 ðOÞ, which can be normalized as follows (Argatov, 2010): g 3 ð OÞ ¼
1 ν a0 ; 2πμ h
ð24Þ
where a0 is a dimensionless quantity, which depends on Poisson's ratio ν, h is a characteristic size of the domain Ω (see Fig. 1). Following Argatov (1999), the first order asymptotic model of unilateral contact can be formulated in the form pðx1 ; x2 Þ 4 0; ðx1 ; x2 Þ A Σ c ;
pðx1 ; x2 Þ ¼ 0; ðx1 ; x2 Þ A ∂Σ c ;
ð25Þ
6
0 ∬Σ c T 3 ðx1 y1 ; x2 y2 ; 0ÞpðyÞ dy ¼ w Pg3 ðOÞ ∑ Q k M ðkÞ 3 ðx ; OÞ k¼1
Φðx1 ; x2 Þ;
ðx1 ; x2 Þ A Σ c :
ð26Þ
Here, pðx1 ; x2 Þ is the contact pressure, Σc is the contact domain, which should be determined as a part of solution of the integral equation (26) under the conditions (25). At that, according to (14), the kernel function of the integral equation (26) is given by 1ν 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: T 3 y1 ; y2 ; 0 ¼ 2πμ y2 þy2 1 2 Let us consider the case of blunt indenter with the shape function
Φðx1 ; x2 Þ ¼ Aðx21 þ x22 Þβ=2 :
ð27Þ
Note that the cases β ¼ 1; 2; and 1 correspond to a conical, spherical, and cylindrical indenter, respectively. In the axisymmetric case (27), using the known analytical solution (Galin, 2008; Sneddon, 1965), we can get a closedform solution to Eq. (26). In particular, the indenter displacement w, the contact force P, and the contact radius a are related by the following two equations, which are exact consequences of Eq. (26) (Argatov, 2011): w m03 P ¼ AN1 ðβÞaβ ;
ð28Þ
π P ¼ θ1 A N 2 β aβ þ 1 : 2
ð29Þ
Here we introduced the following notation (with Γ ðxÞ being the gamma function): 2 2 2E θ1 ¼ ; 1 ν2
N1 β ¼ 2 6
β2
Γ
β
2 ; β Γ ðβ Þ
0 m03 ¼ g 3 ðOÞ þ ∑ ϵ0j P εjk M ðkÞ 3 ðx ; OÞ:
Γ
β
2β 1 β 2 N2 β ¼ π ðβ þ1Þ Γ ðβÞ
2
ð30Þ
j;k ¼ 1
Eqs. (28) and (29) imply that during the depth-sensing indentation, the incremental indentation stiffness can be evaluated according to the relation dP θ1 a ¼ ; dw 1 þm03 θ1 a
ð31Þ
which represents the first-order asymptotic model for the indentation stiffness in the axisymmetric case. We emphasize that the coefficients of the asymptotic representation (31) do not depend on the dimensionless parameter β, which describes the indenter shape (Argatov and Sabina, 2013). It should be noted that Eq. (31) is not convenient from the point of view of the indentation stiffness tomography method, because the contact radius a is not an easily controlled parameter. Based on the asymptotic analysis given (Argatov and Sabina,
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
2013), the following first-order asymptotic model for the indentation stiffness can be derived from Eqs. (28) and (29): 1=β ( 1=β ) dP w w 0 ðβ þ 2Þ C θ1 1 θ 1 m3 : dw AN1 ðβÞ ðβ þ 1Þ AN 1 ðβÞ
195
ð32Þ
Thus, if the indentation testing is performed with the same indentation depth, denoting by Sε the left-hand side of Eq. (32), we can rewrite it as follows: ðβ þ2Þ Sε C S0 1 m03 S0 : ð33Þ ðβ þ1Þ Here, S0 is the indentation stiffness of the bulk material under the specified level of indentation, i.e., 1=β w : S0 ¼ θ1 AN 1 ðβÞ
ð34Þ
From (33), it immediately follows that Sε S0 ðβ þ 2Þ S0 : C m03 S0 ðβ þ 1Þ
ð35Þ
Eq. (34) assumes that near the point of indentation, the elastic body Ω is approximated by an elastic half-space. This assumption is valid only for relatively small values of the contact radius. Finally, using the second Betti's formula ∭Ω ðGðxÞ Lð∇x ÞMðkÞ ðx0 ; xÞ MðkÞ ðx0 ; xÞ Lð∇x ÞGðxÞÞ dx ¼ ∬∂Ω ðσ ðnÞ ðG; xÞ MðkÞ ðx0 ; xÞ Gðx0 ; xÞ σ ðnÞ ðMðkÞ ; xÞÞ dSx and taking into account the relations σ ðnÞ ðG; xÞ ¼ δðxÞe3 (for x on Γc near the point O) and Lð∇x ÞMðkÞ ðx0 ; xÞ ¼ VðkÞ ð∇x Þðδðx x0 Þe1 þ δðx x0 Þe2 þ δðx x0 Þe3 Þ as well as the boundary conditions in (13) and (16), we obtain after integration by parts that MðkÞ ðx0 ; OÞ ¼ ϵ0k ;
k ¼ 1; 2; …; 6:
ð36Þ
Thus, formulas (30) and (36) imply that 6
m03 ¼ g 3 ðOÞ ∑ ϵ0j ϵ0k P εjk ;
ð37Þ
j;k ¼ 1
where the strain components
ϵj are defined in (18). 0
5. Example. Spherical inhomogeneity in an elastic half-space According to Argatov and Sevostianov (2011), the polarization matrix P ε for a spherical inhomogeneity ωε of radius r ε is related to the stiffness contribution tensor Nε of the inhomogeneity by the formula 0 ε 1 N1111 N ε1122 Nε1122 0 0 0 B ε C ε ε 0 0 0 C B N1122 N 1111 N1122 B ε C ε ε BN 0 0 0 C B 1122 N 1122 N1111 C ε P ¼ VB C; 0 0 C 0 0 2Nε1212 B 0 B C ε B 0 0 C 0 0 0 2N 1212 A @ 0
0
0
0
0
2Nε1212
where V is a certain reference volume. By using a known solution (Kachanov et al., 2003), we will have
Nεijkl ¼ K εs δij δkl þGεs J ijkl 13 δij δkl where coefficients Vε K 1 ð1 κ Þ 1 ; K εs ¼ K 1 1 þ G V
Gεs ¼
Vε 2G1 ð5 2κ Þ 1 G1 1 þ ; 15G V
and where J ijkl ¼ ðδik δlj þ δil δkj Þ=2 are components of the fourth rank unit tensor, K 0 ¼ λ0 þ 2G0 =3 and K ¼ λ þ2G=3 are the bulk moduli of the sphere and the matrix, respectively, G0 ¼ μ0 and G ¼ μ are the corresponding shear moduli, κ ¼ ðλ þ GÞ=ðλ þ 2GÞ, and V ε ¼ ð4π =3Þr3ε is the sphere volume. Thus, in the case of spherical inhomogeneity, we obtain P εij ¼ EV ε pij ðα; ν; ν0 Þ;
ð38Þ
196
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
where α ¼ E0 =E is the inclusion–matrix elastic moduli ratio, 2 g pii ¼ ks þ g s ði ¼ 1; 2; 3Þ; pij ¼ ks s ði a j; i; j ¼ 1; 2; 3Þ; 3 3 pii ¼ g s ði ¼ 4; 5; 6Þ; pij ¼ 0 ði a j; i; j ¼ 4; 5; 6Þ; ks ¼
gs ¼
ð1 νÞðαð1 2νÞ þ 2ν0 1Þ
ð1 2νÞ2 ðαð1 þ νÞ þ 2ð1 2ν0 ÞÞ
;
15ð1 νÞð1 þ ν0 αð1 þ νÞÞ : 2ð1 þ νÞð2αð1 þ νÞð5ν 4Þ þ ð1 þ ν0 Þð5ν 7ÞÞ
ð39Þ ð40Þ
ð41Þ
Note that in the cases of spherical cavity and rigid sphere we, correspondingly, have cavity
ks
rigid
ks
¼
¼
ð1 νÞ
2ð1 2νÞ2
;
1ν ; ð1 þ νÞð1 2νÞ
g cavity ¼ s
g rigid ¼ s
15ð1 νÞ ; 2ð1 þ νÞð7 5νÞ
15ð1 νÞ : 4ð1 þ νÞð4 5νÞ
ð42Þ
ð43Þ
Now, let us introduce a coordinate system in such a way that the center of the sphere ωε has the coordinates ð0; 0; dÞ. To fix our ideas, we consider the indentation testing of the elastic half-space with the inclusion by moving the indentation point along the x1-axis (see Fig. 4). Thus, the strain components in the Boussinesq problem can be represented as
ϵ0j ¼
1 þ ν ϵj 2π E d2
0
ðj ¼ 1; 2; …; 6Þ;
ð44Þ
where
! 2 1 3ξ ξ2 1 ξ2 þ ϵ ¼ 3 5 þ ð1 2νÞ 2 ; r r r ðr þ 1Þ2 rðr þ 1Þ r 3 ðr þ1Þ 0 1
ϵ 02 ¼
1 ð1 2νÞ ; r 3 rðr þ 1Þ
ϵ 04 ¼ ϵ 05 ¼ 0;
ϵ 06 ¼
ϵ 03 ¼
2ν 3 ; r3 r5
pffiffiffi 3 2ξ ; r5
r¼
qffiffiffiffiffiffiffiffiffiffiffiffiffi ξ2 þ 1:
ð45Þ
Because the elastic body Ω coincides with an elastic half-space, formula (37) reduces to the following one: 6
m03 ¼ ∑ ϵ0j ϵ0k P εjk :
ð46Þ
j;k ¼ 1
Now, taking into account Eqs. (38), (39), and (44), we obtain o ð1 þ νÞ2 V ε 2 m03 ¼ ks ðϵ 01 þ ϵ 02 þ ϵ 03 Þ2 þg s ðϵ 06 Þ2 þ g s ðϵ 01 Þ2 þ ðϵ 02 Þ2 þðϵ 03 Þ2 ϵ 01 ϵ 02 ϵ 02 ϵ 03 ϵ 03 ϵ 01 ; 4 3 4π 2 Ed
ð47Þ
where the expression in the braces in (47) depends on the relative distance from the indentation point to the epicenter of the spherical inhomogeneity, ξ, the relative stiffness of the inhomogeneity, α, and Poisson's ratios, ν and ν0.
Fig. 4. Schematics of the indentation test for an elastic half-space with a spherical inhomogeneity.
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
197
Because a value of 0:5 is usually assumed for Poisson's ratio of cells, let us consider a special case of incompressible bulk material when ν ¼ 0:5. Then, according to Eq. (45), we will have pffiffiffi 2 1 3ξ 1 1 3 3 2ξ ϵ 01 ¼ 3 5 ; ϵ 02 ¼ 3 ; ϵ 03 ¼ 3 5 ; ϵ 06 ¼ 5 : ð48Þ r r r r r r Now, substituting the expressions (40), (41), and (48) into Eq. (47) and passing to the limit as ν-0:5 in the obtained result, we arrive at the formula m03 ¼
15 V ε ð3α 2ð1 þ ν0 ÞÞ 1 : ðα þ 1 þ ν0 Þ ðξ2 þ 1Þ3
ð49Þ
ð4π Þ2 Ed2
On the other hand, formula (35) can be written as ðβ þ 1Þ ðSε S0 Þ C m03 ; ðβ þ 2Þ S20
ð50Þ
where Sε and S0 are the incremental indentation stiffnesses of the elastic half-space with the inclusion and without the inclusion under the same level of indentation. Because the epicenter of the inclusion ωε is not known a priori, the so-called grid indentation technique (Constantinides et al., 2006) should be employed to collect the indentation stiffness data Siε at points ðxi1 ; xi2 Þ, i ¼ 1; 2; …; N. According to the relations (49) and (50), the experimental data should be fitted with the following five-parameter approximation: S ðx1 ; x2 Þ ¼ S0 þ
C0 2
ðd þðx1 x01 Þ2 þðx2 x02 Þ2 Þ3
:
ð51Þ
Here, d is the (unknown) depth of the inclusion, ðx01 ; x02 Þ are the coordinates of its epicenter, S0 is the indentation stiffness of the bulk material, and C0 is some coefficient, which reflects the presence of the inclusion. The inclusion parameter identification can be performed as follows. First, by the least square method, the five unknowns on the right-hand side of (51) are determined from the solution of the optimization problem N
min
∑ ðSðxi1 ; xi2 Þ Siε Þ2 :
d;x01 ;x02 ;S0 ;C 0 i ¼ 1
Second, in view of (49)–(51), the following relationship must be satisfied: 3α 2ð1 þ ν0 Þ ð4π Þ2 ðβ þ1Þ EC 0 Vε ¼ : α þ1 þ ν0 15 ðβ þ2Þ S2 d2
ð52Þ
0
Since the right-hand side of Eq. (52) is supposed to be already known, we have only one equation for determining the three parameters α, V ε , and ν0. As usual in the indentation–identification testing, we can assume some value for Poisson's ratio of the inclusion, ν0. After that, Eq. (52) allows us to determine either of the two parameters α and V ε provided the other is known. Let the dimensionless factor appearing on the left-hand side of Eq. (52) be denoted as f ðα; ν0 Þ ¼
3α 2ð1 þ ν0 Þ : α þ 1 þ ν0
ð53Þ
Fig. 5. Variation of the dimensionless factor (53) (a) as a function of ν0 for different values of α indicated on the graph and (b) as a function of α for some values of ν0.
198
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
It is clear that C0, and correspondingly f ðα; ν0 Þ, have the same sign as Sε S0 (see (50) and (51)). Fig. 5a shows that in the case of an incompressible bulk material (ν ¼ 0:5) when the elastic moduli coincide (α ¼1), any difference in Poisson's ratios produces a reinforcing effect. It is interesting that, even when the inclusion is more compliant than the matrix (α 5 1), the reinforcing effect can be observed for the inclusion made from an auxetic material with negative Poisson's ratio ν0 o 1:5α 1. 6. Discussion First of all, observe that the elastic polarization matrix P ε is positive (negative) definite for relatively hard (soft) inclusions (Zorin et al., 1990). Correspondingly, the quadratic form ϵ0j ϵ0k P εjk will be sign definite. This implies that a hard inclusion produces a steeper force–displacement curve than a softer one. It should also be emphasized that though this conclusion is drawn from an asymptotic model, it has a general character (Roduit et al., 2009). Second, the proposed identification method assumes the knowledge of both Poisson's ratios ν0 and ν. Note that even in the extreme cases (cavity (42) or rigid inclusion (43)), Poisson's ratio ν is required to extract the bulk elastic modulus E from the bulk indentation stiffness S0. Third, Fig. 5b illustrates the fact that the error of determining the elastic moduli ratio α ¼ E0 =E from Eq. (52) increases when the inclusion becomes stiffer (E0 increases relative to E). In this case, the rigid inclusion model based on Eqs. (43) and (47) should be employed to verify the obtained results. On the other hand, if α is known (along with ν0), then Eq. (52) immediately yields the inclusion volume V ε , and the corresponding result will be more sensitive to the error in the elastic moduli ratio for relatively compliant inclusions. We emphasize that formula (50) represents the leading-order asymptotic model, and its accuracy becomes more precise as the value of the parameter ε decreases. The question of error estimates for asymptotic expansions like those in Section 3 was considered in Ammari et al. (2002) and Nazarov and Sokołowski (2003). It should be observed that the asymptotic analysis employs only one small parameter, though the problem under consideration contains several dimensionless ratios including diamðωε Þ=h, diamðΣ c Þ=h and d=h, where h is a characteristic size of the deformable domain and d ¼ distfO; x0 g. The first two ratios are assumed to be small, or more exactly, the inclusion diameter diamðωε Þ and the diameter of the contact zone diamðΣ c Þ should be small compared to both lengths d and h. The interrelation between d and h depends on the location of the inhomogeneity center x0 with respect to the indentation point O, and generally speaking, d ¼ OðhÞ, whereas the condition d ¼ oðhÞ corresponds to the case studied in Section 5. Note that following Ammari et al. (2008), the presented analysis can be extended to the time-harmonic regime where the indenter tip vibrates with a given frequency (Foschia et al., 2009). Finally, it is to note that the same identification procedure can be used in the case of a small inhomogeneity buried in an elastic layer provided Eq. (45) are replaced with the corresponding equations for the strains in the elastic layer without any inhomogeneity at the point coinciding with the small inhomogeneity center. 7. Conclusions In this study, the problem of indentation stiffness tomography for sensing of a small inhomogeneity in a homogeneous reference elastic body is investigated via an asymptotic modeling approach. With the reference body modeled as a homogeneous elastic half-space, the inverse problem of the spherical inhomogeneity parameter identification is reduced to the least-square based minimization of a misfit between the asymptotic predictions for the indentation stiffness and the massive array of experimental data for the indentation stiffness under a specified level of indentation depth collected on the body surface by means of grid indentation technique.
Acknowledgment The author would like to thank Dr. M. Loparic (Biozentrum and the Swiss Nanoscience Institute, University of Basel) for bringing his attention to the paper of Roduit et al. (2009). References Alves, C., Ammari, H., 2001. Boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium. SIAM J. Appl. Math. 62, 94–106. Ammari, H., Calmon, P., Iakovleva, E., 2008. Direct elastic imaging of a small inclusion. SIAM J. Imaging Sci. 1, 169–187. Ammari, H., Kang, H., 2006. Reconstruction of elastic inclusions of small volume via dynamic measurements. Appl. Math. Optim. 54, 223–235. Ammari, H., Kang, H., 2007. Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory. Applied Mathematical Sciences, vol. 162. Springer-Verlag, New York. Ammari, H., Kang, H., Nakamura, G., Tanuma, K., 2002. Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion. J. Elast. 67, 97–129. Argatov, I.I., 1999. The indentation of a punch in the form of an elliptic paraboloid into the plane boundary of an elastic body. J. Appl. Math. Mech. 63, 641–649. Argatov, I.I., 2010. Frictionless and adhesive nanoindentation: asymptotic modeling of size effects. Mech. Mater. 42, 807–815.
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
199
Argatov, I.I., 2011. Depth-sensing indentation of a transversely isotropic elastic layer: second-order asymptotic models for canonical indenters. Int. J. Solids Struct. 48, 3444–3452. Argatov, I.I., Sabina, F.J., 2013. Asymptotic analysis of the substrate effect for an arbitrary indenter. Quart. J. Mech. Appl. Math. 66, 75–95. Argatov, I., Sevostianov, I., 2011. Rigid toroidal inhomogeneity in an elastic medium. Int. J. Eng. Sci. 49, 61–74. Bonnet, M., Constantinescu, A., 2005. Inverse problems in elasticity. Inverse Probl. 21, R1–R50. Bonnet, M., Guzina, B.B., 2004. Sounding of finite solid bodies by way of topological derivative. Int. J. Numer. Methods Eng. 61, 2344–2373. Constantinides, G., Ravi Chandran, K.S., Ulm, F.-J., Van Vliet, K.J., 2006. Grid indentation analysis of composite microstructure and mechanics: principles and validation. Mater. Sci. Eng. A 430, 189–202. Fischer-Cripps, A.C., 2004. Nanoindentation. Springer-Verlag, New York. Foschia, R., Jobin, M., Hengsberger, S., 2009. Local dynamic mechanical analysis. Micron 40, 51–55. Galin, L.A., 2008. In: Gladwell G.M.L. (Ed.), Contact Problems: The Legacy of L.A. Galin. Springer, Dordrecht. Guzina, B.B., Chikichev, I., 2007. From imaging to material identification: a generalized concept of topological sensitivity. J. Mech. Phys. Solids 55, 245–279. Guzina, B.B., Nintcheu, S.F., Bonnet, M., 2003. On the stress-wave imaging of cavities in a semi-infinite solid. Int. J. Solids Struct. 40, 1505–1552. Il'in, A.M., 1992. Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Translations of Mathematical Monographs, vol. 102. AMS, Providence, RI. Johnson, K.L., 1985. Contact Mechanics. Cambridge University Press, Cambridge, UK. Kachanov, M., Shafiro, B., Tsukrov, I., 2003. Handbook of Elasticity Solutions. Kluwer Academic Publishers, Dordrecht. Kasas, S., Dietler, G., 2008. Probing nanomechanical properties from biomolecules to living cells. Pflugers Arch.—Eur. J. Physiol. 456, 13–27. Kasas, S., Longo, G., Dietler, G., 2013. Mechanical properties of biological specimens explored by atomic force microscopy. J. Phys. D: Appl. Phys. 46, 133001. 12 pp. Kikuchi, N., Oden, J.T., 1988. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia. Leroux, J., Fulleringer, B., Nélias, D., 2010. Contact analysis in presence of spherical inhomogeneities within a half-space. Int. J. Solids Struct. 47, 3034–3049. Lewiński, T., Sokołowski, J., 2003. Energy change due to the appearance of cavities in elastic solids. Int. J. Solids Struct. 40, 1765–1803. Loparic, M., Wirz, D., Daniels, A.U., Raiteri, R., VanLandingham, M.R., Guex, G., Martin, I., Aebi, U., Stolz, M., 2010. Micro- and nanomechanical analysis of articular cartilage by indentation-type atomic force microscopy: validation with a gel-microfiber composite. Biophys. J. 98, 2731–2740. Nazarov, S.A., Sokołowski, J., 2003. Asymptotic analysis of shape functionals. J. Math. Pure Appl. 82, 125–196. Novotny, A.A., Sokołowski, J., 2013. Topological Derivatives in Shape Optimization. Springer, Heidelberg. Plodinec, M., Loparic, M., Monnier, C.A., Obermann, E.C., Zanetti-Dallenbach, R., Oertle, P., Hyotyla, J.T., Aebi, U., Bentires-Alj, M., Lim, R.Y.H., Schoenenberger, C.-A., 2012. The nanomechanical signature of breast cancer. Nat. Nanotechnol. 7, 757–765. Rheinlaender, J., Geisse, N.A., Proksch, R., Schäffer, T.E., 2011. Comparison of scanning ion conductance microscopy with atomic force microscopy for cell imaging. Langmuir 27, 697–704. Roduit, C., Sekatski, S., Dietler, G., Catsicas, S., Lafont, F., Kasas, S., 2009. Stiffness tomography by atomic force microscopy. Biophys. J. 97, 674–677. Shillor, M., Sofonea, M., Telega, J.J., 2004. Models and Analysis of Quasistatic Contact: Variational Methods. Springer, Berlin. Sneddon, I.N., 1965. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3, 47–57. Van Dyke, M.D., 1964. Perturbation Methods in Fluid Mechanics. Academic Press, New York. Zhou, K., Chen, W.W., Keer, L.M., Ai, X., Sawamiphakdi, K., Glaws, P., Wang, Q.J., 2011. Multiple 3D inhomogeneous inclusions in a half space under contact loading. Mech. Mater. 43, 444–457. Zhou, K., Hoh, H.J., Wang, X., Keer, L.M., Pang, J.H.L., Song, B., Wang, Q.J., 2013. A review of recent works on inclusions. Mech. Mater. 60, 144–158. Zorin, I.S., Movchan, A.B., Nazarov, S.A., 1990. Application of tensors of elastic capacity, polarization and associated deformation. Studies on Elasticity and Plasticity, Leningrad University, vol. 16, pp. 75–91 (in Russian).
Contents lists available at ScienceDirect
Journal of the Mechanics and Physics of Solids journal homepage: www.elsevier.com/locate/jmps
AFM-based indentation stiffness tomography—An asymptotic model I.I. Argatov Engineering Mechanics Laboratory, University of Oulu, 90014 Oulu, Finland
a r t i c l e in f o
abstract
Article history: Received 21 November 2013 Received in revised form 28 May 2014 Accepted 4 June 2014 Available online 12 June 2014
The so-called indentation stiffness tomography technique for detecting the interior mechanical properties of an elastic sample with an inhomogeneity is analyzed in the framework of the asymptotic modeling approach under the assumption of small size of the inhomogeneity. In particular, it is assumed that the inhomogeneity size and the size of contact area under the indenter are small compared with the distance between them. By the method of matched asymptotic expansions, the first-order asymptotic solution to the corresponding frictionless unilateral contact problem is obtained. The case of an elastic half-space containing a small spherical inhomogeneity has been studied in detail. Based on the grid indentation technique, a procedure for solving the inverse problem of extracting the inhomogeneity parameters is proposed. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Depth-sensing indentation Indentation stiffness Indentation tomography Elastic inhomogeneity Asymptotic model
1. Introduction In recent years, the atomic force microscope (AFM) has become an indispensable tool for the indentation based characterization (Fischer-Cripps, 2004) of mechanical properties of living samples at the nanometric scale (Kasas and Dietler, 2008; Loparic et al., 2010; Plodinec et al., 2012; Kasas et al., 2013). Based on the best-fitting analysis of the forceindentation (FI) curve, the so-called indentation stiffness tomography technique was proposed by Roduit et al. (2009) for distinguishing structures of different stiffness buried into the bulk of the sample. The developed methodology assumes that the presence of an inhomogeneity (inclusion) changes the FI curve in a deterministic way reflecting the relative hardness/ softness property of the inclusion and the depth of the inclusion. The validity of this concept was verified by finite element models and was proven useful in AFM-based indentation experiments on living cells (Rheinlaender et al., 2011). Three-dimensional contact problems for the special case of an elastic half-space with inhomogeneities were studied by numerical techniques in a number of publications (Leroux et al., 2010; Zhou et al., 2011). An extensive review of works on the elastic problems for inclusions in an elastic half-space was very recently presented by Zhou et al. (2013). In the present paper, we consider the frictionless unilateral contact problem for a homogeneous linearly elastic body with a small homogeneous inclusion (with no eigenstrains). Based on the obtained first-order asymptotic solution, we address the identification problem resulting in the identification of some parameters of the small inhomogeneity. The dynamic problem of stress-wave identification of material defects (e.g., cavities, cracks, inclusions) has been the subject of extensive research (Alves and Ammari, 2001; Guzina et al., 2003; Bonnet and Constantinescu, 2005; Ammari and Kang, 2006; Guzina and Chikichev, 2007). At the same time, the quasi-static problem of indentation stiffness tomography has its own peculiarities, and the asymptotic model proposed herein for extracting the inhomogeneity parameters based on the
E-mail address: [email protected] http://dx.doi.org/10.1016/j.jmps.2014.06.001 0022-5096/& 2014 Elsevier Ltd. All rights reserved.
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
191
grid indentation data has not been developed elsewhere. The objective of this study was to develop a simple mathematical model for the indentation stiffness tomography. The rest of the paper is organized as follows. In Section 2, we formulate the mathematical model of frictionless indentation. By means of the method of matched asymptotic expansions, in Section 3 an approximate analytical solution is obtained. The first-order asymptotic model for the indentation test is developed in Section 4. The case of a small spherical inhomogeneity in an elastic half-space is studied in detail in Section 5, where a procedure for extracting the inhomogeneity parameters is proposed. Finally, in Sections 6 and 7, respectively, we outline a discussion of the results obtained and formulate our conclusions. 2. Mathematical model Suppose a homogeneous linearly elastic body without inhomogeneity occupies a three-dimensional domain Ω. Let ωε be a small inhomogeneity with the center at a point x0 and the diameter proportional to a small positive parameter ε such that ωε lies within the domain Ω. Thus, the domain Ωε ¼ Ω\ω ε will represent the reference configuration of the homogeneous deformable body with the inhomogeneity ωε . In the absence of volume forces, the displacement vectors u ¼ ðu1 ; u2 ; u3 Þ in Ωε and u0 ¼ ðu01 ; u02 ; u03 Þ in ωε satisfy the Lamé differential systems
μ∇x ∇x uðxÞ þ ðλ þ μÞ∇x ∇x uðxÞ ¼ 0;
x A Ωε ;
μ0 ∇x ∇x u0 ðxÞ þðλ0 þ μ0 Þ∇x ∇x u0 ðxÞ ¼ 0;
ð1Þ
x A ωε :
ð2Þ
We assume that the inhomogeneity ωε is perfectly bonded to the surrounding medium, and the continuity and equilibrium conditions along the interface ∂ωε are formulated as follows: uðxÞ ¼ u0 ðxÞ;
σ ðnÞ ðu; xÞ ¼ σ 0ðnÞ ðu0 ; xÞ; x A ∂ωε :
ð3Þ
The outer surface of the body Ωε is assumed to be decomposed into three mutually disjoint parts: ∂Ω ¼ Γ u [ Γ σ [ Γ c . Over Γu, the body is held fixed, while the surface Γ σ is assumed to be traction free, i.e., uðxÞ ¼ 0; x A Γ u ;
σ ðnÞ ðu; xÞ ¼ 0; x A Γ σ :
ð4Þ
Finally, Γc is the potential contact boundary, over which contact of the deformable body Ωε with a rigid indenter may take place. Let us introduce a Cartesian coordinate system with the center at a point O on Γc, which is supposed to be the initial point of contact in the case of a convex indenter (in the unloading state, the indenter touches the surface Γc at the point O). To fix our ideas, we will assume that the x3-axis is directed inside the body Ωε with the plane Ox1 x2 being tangent to the surface Γc (see Fig. 1). At the initial moment, the surface of the indenter is specified by the equation x3 ¼ Φðx1 ; x2 Þ;
ð5Þ
and during the normal indentation of the indenter into the deformable body (see Fig. 2), the indenter surface will be given by the equation x3 ¼ w Φðx1 ; x2 Þ;
ð6Þ
where w is a small displacement of the indenter. To formulate on Γc the boundary conditions of unilateral contact (Kikuchi and Oden, 1988; Shillor et al., 2004), we denote by gN the variable gap between the indenter surface and Γc. Note that gN is measured in a direction normal to the body surface and g N ðx1 ; x2 Þ ¼ g 0N ðx1 ; x2 Þ þ
w ; cos ðnðxÞ; x3 Þ
ð7Þ 0 gN
where n is the outer unit normal vector to the boundary of the domain Ω, is the initial gap between the body and the indenter. It is assumed that the gap function g 0N ðx1 ; x2 Þ takes on a zero value at the origin, so that w represents the current depth of the indenter's penetration into the body. Thus, the Signorini boundary conditions of frictionless contact may be stated as
σ TðnÞ ðu; xÞ ¼ 0;
x A Γc ;
ð8Þ
Fig. 1. Deformable body and reference coordinate system.
192
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
Fig. 2. Indentation test schematics.
Fig. 3. Schematics for the “outer” displacement field.
σ ðnÞ uN ðxÞ g N ðx1 ; x2 Þ r 0; N ðu; xÞ r0; σ ðnÞ ðu; xÞðu ðxÞ g N ðx1 ; x2 ÞÞ ¼ 0; x A Γ c : N N
ð9Þ
ðnÞ ðnÞ Here, σ ðnÞ N and σ T are the normal and tangential components of the stress vector σ . Observe that the second constraint in (9) means that the normal displacement uN ¼ u n cannot exceed the current gap gN, which is governed by the vertical position of the indenter (see Eq. (7)). Under the assumption that the size of the contact area is small compared to the curvature radii of the surface Γc at the point O, we will have
g N ðx1 ; x2 Þ C Φðx1 ; x2 Þ w;
ð10Þ
where Φðx1 ; x2 Þ is the indenter shape function introduced by Eq. (5). Note that the approximate equation (10) becomes exact when a neighborhood on the surface Γc around the point O is flat. Local equilibrium at the contact interface in the x3-direction requires that the contact pressure should be balanced by the contact force, P, that is ðnÞ ðu; xÞ cos ðnðxÞ; x3 Þ dSx : P ¼ ∬Γ c σ N
ð11Þ
Now, combining (1)–(11), we formulate a mathematical model of unilateral contact for the linearly elastic body Ωε with the small inhomogeneity ωε . 3. Asymptotic approximation for the displacement field away from the indenter At a distance from the contact zone as well as from the inhomogeneity, the stress–strain state of the body Ωε is approximated by the following so-called “outer” displacement vector-field (see Fig. 3): 6
vðxÞ ¼ PGðxÞ þ ∑ Q k MðkÞ ðx0 ; xÞ:
ð12Þ
k¼1
Here, GðxÞ is the solution of the elastic problem in the domain Ω of the action of a unit point force applied at the point O in the x3-direction, MðkÞ ðx0 ; xÞ are the solutions of the elastic problem in the domain Ω of the action of force dipoles at the point x0 . For the simplicity sake let us assume that the principal curvatures of the surface Γc at the point O are zero. Then, the vector-function G satisfies the following equations:
μ∇x ∇x GðxÞ þ ðλ þ μÞ∇x ∇x GðxÞ ¼ 0; σ ðnÞ ðG; xÞ ¼ 0; x A Γ σ [ ðΓ c \OÞ; GðxÞ ¼ TðxÞ þOð1Þ; x-O:
x A Ω;
GðxÞ ¼ 0; x A Γ u ;
ð13Þ
Here, T is the solution of the Boussinesq problem (see, e.g., Johnson, 1985) of the action on the boundary of an elastic halfspace x3 40 of a unit force directed along the x3-axis, i.e., 1 xi x3 μ xi ; i ¼ 1; 2; T i ðx Þ ¼ 3 4πμ jxj λ þ μ jxjðjxj þ x3 Þ 2 x3 λ þ 2μ 1 1 : ð14Þ þ T 3 ðx Þ ¼ 4πμ jxj3 λ þ μ jxj
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
193
To describe the vector-functions MðkÞ , following Zorin et al. (1990), we introduce vector polynomials Vð1Þ ðxÞ ¼ ðx1 ; 0; 0Þ;
Vð2Þ ðxÞ ¼ ð0; x2 ; 0Þ;
Vð4Þ ðxÞ ¼ p1ffiffi2 ðx2 ; x1 ; 0Þ;
Vð3Þ ðxÞ ¼ ð0; 0; x3 Þ;
Vð5Þ ðxÞ ¼ p1ffiffi2 ð0; x3 ; x2 Þ;
Vð6Þ ðxÞ ¼ p1ffiffi2 ðx3 ; 0; x1 Þ:
ð15Þ
Let also SðxÞ be the Kelvin–Somigliana fundamental matrix, i.e., xi xj λ þ3μ δij λþμ ; i; j ¼ 1; 2; 3: þ Sij ðxÞ ¼ 3 8πμðλ þ2μÞ jxj λ þ μ jxj Then, the vector-functions MðkÞ ðx0 ; xÞ with singularities of order Oðjx x0 j 2 Þ at the point x0 are defined as follows:
μ∇x ∇x MðkÞ ðx0 ; xÞ þ ðλ þ μÞ∇x ∇x MðkÞ ðx0 ; xÞ ¼ 0; σ ðnÞ ðMðkÞ ; xÞ ¼ 0; x A Γ σ [ Γ c ;
x A Ω\x0 ;
MðkÞ ðx0 ; xÞ ¼ 0; x A Γ u ;
MðkÞ ðx0 ; xÞ ¼ VðkÞ ð∇x ÞSðx x0 Þ þ Oð1Þ;
x-x0 :
ð16Þ
Further, to construct the boundary layer around the inhomogeneity ωε let us assume that ωε ¼ fxjε ðx x Þ A ωg, where ω is a fixed three-dimensional domain, from which the small inhomogeneity ωε with the center at the point x0 is rescaled by means of the so-called stretched coordinates ξ ¼ ε 1 ðx x0 Þ, so that the two domains ω (in the ξspace) and ωε (in the xspace) are related through a linear transform x ¼ x0 þ εξ. In what follows, we make use of the Taylor expansion 1
0
6
GðxÞ ¼ Gðx0 Þ þ ðx x0 Þ ωðx0 Þ þ ∑ ϵ0j VðjÞ ðx x0 Þ þ Oðjx x0 j2 Þ;
ð17Þ
j¼1
where ωðxÞ ¼ ð1=2Þ∇x GðxÞ, VðjÞ are the vector polynomials (15), and the strain components
ϵ0i ¼ εii ðG; x0 Þ; i ¼ 1; 2; 3; pffiffiffi pffiffiffi ϵ04 ¼ 2ε12 ðG; x0 Þ; ϵ05 ¼ 2ε23 ðG; x0 Þ;
ϵj are defined as 0
pffiffiffi
ϵ06 ¼ 2ε13 ðG; x0 Þ:
ð18Þ
Thus, in view of (12), (16)3 and (17), the following asymptotic expansion takes place: ( ) 6
6
j¼1
k¼1
vðx0 þ εξÞ ¼ P Gðx0 Þ þ εξ ωðx0 Þ þ ε ∑ ϵ0j VðjÞ ðξÞ þOðε2 jξj2 Þ þ ∑ Q k fε 2 VðkÞ ð∇ξ ÞSðξÞ þOð1Þg:
ð19Þ
By the method of matched asymptotic expansions (Van Dyke, 1964; Il'in, 1992), the stress–strain state of the body Ωε in the vicinity of the inhomogeneity is approximated by the vector-function ( ) 6 x x0 x x0 ¼ P G x0 þ x x0 ω x0 þ ∑ ϵ0j WðjÞ : ð20Þ w
ε
ε
j¼1
Here, WðjÞ ðξÞ are the unique solutions of the following elastic problem (Zorin et al., 1990):
μ∇ξ ∇ξ WðjÞ ðξÞ þ ðλ þ μÞ∇ξ ∇ξ WðjÞ ðξÞ ¼ 0;
ξ A R3 \ ω ;
μ0 ∇ξ ∇ξ W0ðjÞ ðξÞ þ ðλ0 þ μ0 Þ∇ξ ∇ξ W0ðjÞ ðξÞ ¼ 0; WðjÞ ðξÞ ¼ W0ðjÞ ðξÞ;
ξ A ω;
σ ðnÞ ðWðjÞ ; ξÞ ¼ σ 0ðnÞ ðW0ðjÞ ; ξÞ; ξ A ∂ω;
WðjÞ ðξÞ ¼ VðjÞ ðξÞ þ oð1Þ;
jξj-1:
The solutions of the above problem have the following expansions at the infinity (Zorin et al., 1990): 6
WðjÞ ðξÞ VðjÞ ðξÞ ¼ ∑ P jk VðkÞ ð∇ξ ÞSðξÞ þOðjξj 3 Þ:
ð21Þ
k¼1
Here, P jk are the components of the so-called elasticity polarization matrix of the inhomogeneity ω (Zorin et al., 1990; Lewiński and Sokołowski, 2003; Ammari and Kang, 2007). Thus, in view of (19) and (21), the asymptotic matching of the outer asymptotic representation (12) and the inner asymptotic representation (20) implies the equations 6
Q k ¼ ∑ P ϵ0j P εjk ;
ð22Þ
j¼1
where P εjk ¼ ε3 P jk are the polarization matrix components for the inhomogeneity ωε . Note that the polarization matrix possesses a number of important properties (Nazarov and Sokołowski, 2003; Ammari and Kang, 2007) and has been encountered in the method of topological sensitivity (Bonnet and Guzina, 2004; Guzina and
194
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
Chikichev, 2007; Novotny and Sokołowski, 2013). Note also that in regard to the case of detecting a small inclusion, complete asymptotic expansions for the elastic displacement fields vðxÞ and wðξÞ were constructed by Ammari et al. (2002). 4. Asymptotic model for the indentation test To construct the boundary layer type solution around the indenter, we consider the following asymptotic expansion (see (12) and (13)3): 6
vðxÞ ¼ PðTðxÞ þgðOÞÞ þ ∑ Q k MðkÞ ðx0 ; OÞ þ OðjxjÞ:
ð23Þ
k¼1
Here, gðxÞ ¼ GðxÞ TðxÞ is the regular part of Green's function GðxÞ. In asymptotic analysis of frictionless contact problems, an important role is played by the asymptotic constant g 3 ðOÞ, which can be normalized as follows (Argatov, 2010): g 3 ð OÞ ¼
1 ν a0 ; 2πμ h
ð24Þ
where a0 is a dimensionless quantity, which depends on Poisson's ratio ν, h is a characteristic size of the domain Ω (see Fig. 1). Following Argatov (1999), the first order asymptotic model of unilateral contact can be formulated in the form pðx1 ; x2 Þ 4 0; ðx1 ; x2 Þ A Σ c ;
pðx1 ; x2 Þ ¼ 0; ðx1 ; x2 Þ A ∂Σ c ;
ð25Þ
6
0 ∬Σ c T 3 ðx1 y1 ; x2 y2 ; 0ÞpðyÞ dy ¼ w Pg3 ðOÞ ∑ Q k M ðkÞ 3 ðx ; OÞ k¼1
Φðx1 ; x2 Þ;
ðx1 ; x2 Þ A Σ c :
ð26Þ
Here, pðx1 ; x2 Þ is the contact pressure, Σc is the contact domain, which should be determined as a part of solution of the integral equation (26) under the conditions (25). At that, according to (14), the kernel function of the integral equation (26) is given by 1ν 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: T 3 y1 ; y2 ; 0 ¼ 2πμ y2 þy2 1 2 Let us consider the case of blunt indenter with the shape function
Φðx1 ; x2 Þ ¼ Aðx21 þ x22 Þβ=2 :
ð27Þ
Note that the cases β ¼ 1; 2; and 1 correspond to a conical, spherical, and cylindrical indenter, respectively. In the axisymmetric case (27), using the known analytical solution (Galin, 2008; Sneddon, 1965), we can get a closedform solution to Eq. (26). In particular, the indenter displacement w, the contact force P, and the contact radius a are related by the following two equations, which are exact consequences of Eq. (26) (Argatov, 2011): w m03 P ¼ AN1 ðβÞaβ ;
ð28Þ
π P ¼ θ1 A N 2 β aβ þ 1 : 2
ð29Þ
Here we introduced the following notation (with Γ ðxÞ being the gamma function): 2 2 2E θ1 ¼ ; 1 ν2
N1 β ¼ 2 6
β2
Γ
β
2 ; β Γ ðβ Þ
0 m03 ¼ g 3 ðOÞ þ ∑ ϵ0j P εjk M ðkÞ 3 ðx ; OÞ:
Γ
β
2β 1 β 2 N2 β ¼ π ðβ þ1Þ Γ ðβÞ
2
ð30Þ
j;k ¼ 1
Eqs. (28) and (29) imply that during the depth-sensing indentation, the incremental indentation stiffness can be evaluated according to the relation dP θ1 a ¼ ; dw 1 þm03 θ1 a
ð31Þ
which represents the first-order asymptotic model for the indentation stiffness in the axisymmetric case. We emphasize that the coefficients of the asymptotic representation (31) do not depend on the dimensionless parameter β, which describes the indenter shape (Argatov and Sabina, 2013). It should be noted that Eq. (31) is not convenient from the point of view of the indentation stiffness tomography method, because the contact radius a is not an easily controlled parameter. Based on the asymptotic analysis given (Argatov and Sabina,
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
2013), the following first-order asymptotic model for the indentation stiffness can be derived from Eqs. (28) and (29): 1=β ( 1=β ) dP w w 0 ðβ þ 2Þ C θ1 1 θ 1 m3 : dw AN1 ðβÞ ðβ þ 1Þ AN 1 ðβÞ
195
ð32Þ
Thus, if the indentation testing is performed with the same indentation depth, denoting by Sε the left-hand side of Eq. (32), we can rewrite it as follows: ðβ þ2Þ Sε C S0 1 m03 S0 : ð33Þ ðβ þ1Þ Here, S0 is the indentation stiffness of the bulk material under the specified level of indentation, i.e., 1=β w : S0 ¼ θ1 AN 1 ðβÞ
ð34Þ
From (33), it immediately follows that Sε S0 ðβ þ 2Þ S0 : C m03 S0 ðβ þ 1Þ
ð35Þ
Eq. (34) assumes that near the point of indentation, the elastic body Ω is approximated by an elastic half-space. This assumption is valid only for relatively small values of the contact radius. Finally, using the second Betti's formula ∭Ω ðGðxÞ Lð∇x ÞMðkÞ ðx0 ; xÞ MðkÞ ðx0 ; xÞ Lð∇x ÞGðxÞÞ dx ¼ ∬∂Ω ðσ ðnÞ ðG; xÞ MðkÞ ðx0 ; xÞ Gðx0 ; xÞ σ ðnÞ ðMðkÞ ; xÞÞ dSx and taking into account the relations σ ðnÞ ðG; xÞ ¼ δðxÞe3 (for x on Γc near the point O) and Lð∇x ÞMðkÞ ðx0 ; xÞ ¼ VðkÞ ð∇x Þðδðx x0 Þe1 þ δðx x0 Þe2 þ δðx x0 Þe3 Þ as well as the boundary conditions in (13) and (16), we obtain after integration by parts that MðkÞ ðx0 ; OÞ ¼ ϵ0k ;
k ¼ 1; 2; …; 6:
ð36Þ
Thus, formulas (30) and (36) imply that 6
m03 ¼ g 3 ðOÞ ∑ ϵ0j ϵ0k P εjk ;
ð37Þ
j;k ¼ 1
where the strain components
ϵj are defined in (18). 0
5. Example. Spherical inhomogeneity in an elastic half-space According to Argatov and Sevostianov (2011), the polarization matrix P ε for a spherical inhomogeneity ωε of radius r ε is related to the stiffness contribution tensor Nε of the inhomogeneity by the formula 0 ε 1 N1111 N ε1122 Nε1122 0 0 0 B ε C ε ε 0 0 0 C B N1122 N 1111 N1122 B ε C ε ε BN 0 0 0 C B 1122 N 1122 N1111 C ε P ¼ VB C; 0 0 C 0 0 2Nε1212 B 0 B C ε B 0 0 C 0 0 0 2N 1212 A @ 0
0
0
0
0
2Nε1212
where V is a certain reference volume. By using a known solution (Kachanov et al., 2003), we will have
Nεijkl ¼ K εs δij δkl þGεs J ijkl 13 δij δkl where coefficients Vε K 1 ð1 κ Þ 1 ; K εs ¼ K 1 1 þ G V
Gεs ¼
Vε 2G1 ð5 2κ Þ 1 G1 1 þ ; 15G V
and where J ijkl ¼ ðδik δlj þ δil δkj Þ=2 are components of the fourth rank unit tensor, K 0 ¼ λ0 þ 2G0 =3 and K ¼ λ þ2G=3 are the bulk moduli of the sphere and the matrix, respectively, G0 ¼ μ0 and G ¼ μ are the corresponding shear moduli, κ ¼ ðλ þ GÞ=ðλ þ 2GÞ, and V ε ¼ ð4π =3Þr3ε is the sphere volume. Thus, in the case of spherical inhomogeneity, we obtain P εij ¼ EV ε pij ðα; ν; ν0 Þ;
ð38Þ
196
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
where α ¼ E0 =E is the inclusion–matrix elastic moduli ratio, 2 g pii ¼ ks þ g s ði ¼ 1; 2; 3Þ; pij ¼ ks s ði a j; i; j ¼ 1; 2; 3Þ; 3 3 pii ¼ g s ði ¼ 4; 5; 6Þ; pij ¼ 0 ði a j; i; j ¼ 4; 5; 6Þ; ks ¼
gs ¼
ð1 νÞðαð1 2νÞ þ 2ν0 1Þ
ð1 2νÞ2 ðαð1 þ νÞ þ 2ð1 2ν0 ÞÞ
;
15ð1 νÞð1 þ ν0 αð1 þ νÞÞ : 2ð1 þ νÞð2αð1 þ νÞð5ν 4Þ þ ð1 þ ν0 Þð5ν 7ÞÞ
ð39Þ ð40Þ
ð41Þ
Note that in the cases of spherical cavity and rigid sphere we, correspondingly, have cavity
ks
rigid
ks
¼
¼
ð1 νÞ
2ð1 2νÞ2
;
1ν ; ð1 þ νÞð1 2νÞ
g cavity ¼ s
g rigid ¼ s
15ð1 νÞ ; 2ð1 þ νÞð7 5νÞ
15ð1 νÞ : 4ð1 þ νÞð4 5νÞ
ð42Þ
ð43Þ
Now, let us introduce a coordinate system in such a way that the center of the sphere ωε has the coordinates ð0; 0; dÞ. To fix our ideas, we consider the indentation testing of the elastic half-space with the inclusion by moving the indentation point along the x1-axis (see Fig. 4). Thus, the strain components in the Boussinesq problem can be represented as
ϵ0j ¼
1 þ ν ϵj 2π E d2
0
ðj ¼ 1; 2; …; 6Þ;
ð44Þ
where
! 2 1 3ξ ξ2 1 ξ2 þ ϵ ¼ 3 5 þ ð1 2νÞ 2 ; r r r ðr þ 1Þ2 rðr þ 1Þ r 3 ðr þ1Þ 0 1
ϵ 02 ¼
1 ð1 2νÞ ; r 3 rðr þ 1Þ
ϵ 04 ¼ ϵ 05 ¼ 0;
ϵ 06 ¼
ϵ 03 ¼
2ν 3 ; r3 r5
pffiffiffi 3 2ξ ; r5
r¼
qffiffiffiffiffiffiffiffiffiffiffiffiffi ξ2 þ 1:
ð45Þ
Because the elastic body Ω coincides with an elastic half-space, formula (37) reduces to the following one: 6
m03 ¼ ∑ ϵ0j ϵ0k P εjk :
ð46Þ
j;k ¼ 1
Now, taking into account Eqs. (38), (39), and (44), we obtain o ð1 þ νÞ2 V ε 2 m03 ¼ ks ðϵ 01 þ ϵ 02 þ ϵ 03 Þ2 þg s ðϵ 06 Þ2 þ g s ðϵ 01 Þ2 þ ðϵ 02 Þ2 þðϵ 03 Þ2 ϵ 01 ϵ 02 ϵ 02 ϵ 03 ϵ 03 ϵ 01 ; 4 3 4π 2 Ed
ð47Þ
where the expression in the braces in (47) depends on the relative distance from the indentation point to the epicenter of the spherical inhomogeneity, ξ, the relative stiffness of the inhomogeneity, α, and Poisson's ratios, ν and ν0.
Fig. 4. Schematics of the indentation test for an elastic half-space with a spherical inhomogeneity.
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
197
Because a value of 0:5 is usually assumed for Poisson's ratio of cells, let us consider a special case of incompressible bulk material when ν ¼ 0:5. Then, according to Eq. (45), we will have pffiffiffi 2 1 3ξ 1 1 3 3 2ξ ϵ 01 ¼ 3 5 ; ϵ 02 ¼ 3 ; ϵ 03 ¼ 3 5 ; ϵ 06 ¼ 5 : ð48Þ r r r r r r Now, substituting the expressions (40), (41), and (48) into Eq. (47) and passing to the limit as ν-0:5 in the obtained result, we arrive at the formula m03 ¼
15 V ε ð3α 2ð1 þ ν0 ÞÞ 1 : ðα þ 1 þ ν0 Þ ðξ2 þ 1Þ3
ð49Þ
ð4π Þ2 Ed2
On the other hand, formula (35) can be written as ðβ þ 1Þ ðSε S0 Þ C m03 ; ðβ þ 2Þ S20
ð50Þ
where Sε and S0 are the incremental indentation stiffnesses of the elastic half-space with the inclusion and without the inclusion under the same level of indentation. Because the epicenter of the inclusion ωε is not known a priori, the so-called grid indentation technique (Constantinides et al., 2006) should be employed to collect the indentation stiffness data Siε at points ðxi1 ; xi2 Þ, i ¼ 1; 2; …; N. According to the relations (49) and (50), the experimental data should be fitted with the following five-parameter approximation: S ðx1 ; x2 Þ ¼ S0 þ
C0 2
ðd þðx1 x01 Þ2 þðx2 x02 Þ2 Þ3
:
ð51Þ
Here, d is the (unknown) depth of the inclusion, ðx01 ; x02 Þ are the coordinates of its epicenter, S0 is the indentation stiffness of the bulk material, and C0 is some coefficient, which reflects the presence of the inclusion. The inclusion parameter identification can be performed as follows. First, by the least square method, the five unknowns on the right-hand side of (51) are determined from the solution of the optimization problem N
min
∑ ðSðxi1 ; xi2 Þ Siε Þ2 :
d;x01 ;x02 ;S0 ;C 0 i ¼ 1
Second, in view of (49)–(51), the following relationship must be satisfied: 3α 2ð1 þ ν0 Þ ð4π Þ2 ðβ þ1Þ EC 0 Vε ¼ : α þ1 þ ν0 15 ðβ þ2Þ S2 d2
ð52Þ
0
Since the right-hand side of Eq. (52) is supposed to be already known, we have only one equation for determining the three parameters α, V ε , and ν0. As usual in the indentation–identification testing, we can assume some value for Poisson's ratio of the inclusion, ν0. After that, Eq. (52) allows us to determine either of the two parameters α and V ε provided the other is known. Let the dimensionless factor appearing on the left-hand side of Eq. (52) be denoted as f ðα; ν0 Þ ¼
3α 2ð1 þ ν0 Þ : α þ 1 þ ν0
ð53Þ
Fig. 5. Variation of the dimensionless factor (53) (a) as a function of ν0 for different values of α indicated on the graph and (b) as a function of α for some values of ν0.
198
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
It is clear that C0, and correspondingly f ðα; ν0 Þ, have the same sign as Sε S0 (see (50) and (51)). Fig. 5a shows that in the case of an incompressible bulk material (ν ¼ 0:5) when the elastic moduli coincide (α ¼1), any difference in Poisson's ratios produces a reinforcing effect. It is interesting that, even when the inclusion is more compliant than the matrix (α 5 1), the reinforcing effect can be observed for the inclusion made from an auxetic material with negative Poisson's ratio ν0 o 1:5α 1. 6. Discussion First of all, observe that the elastic polarization matrix P ε is positive (negative) definite for relatively hard (soft) inclusions (Zorin et al., 1990). Correspondingly, the quadratic form ϵ0j ϵ0k P εjk will be sign definite. This implies that a hard inclusion produces a steeper force–displacement curve than a softer one. It should also be emphasized that though this conclusion is drawn from an asymptotic model, it has a general character (Roduit et al., 2009). Second, the proposed identification method assumes the knowledge of both Poisson's ratios ν0 and ν. Note that even in the extreme cases (cavity (42) or rigid inclusion (43)), Poisson's ratio ν is required to extract the bulk elastic modulus E from the bulk indentation stiffness S0. Third, Fig. 5b illustrates the fact that the error of determining the elastic moduli ratio α ¼ E0 =E from Eq. (52) increases when the inclusion becomes stiffer (E0 increases relative to E). In this case, the rigid inclusion model based on Eqs. (43) and (47) should be employed to verify the obtained results. On the other hand, if α is known (along with ν0), then Eq. (52) immediately yields the inclusion volume V ε , and the corresponding result will be more sensitive to the error in the elastic moduli ratio for relatively compliant inclusions. We emphasize that formula (50) represents the leading-order asymptotic model, and its accuracy becomes more precise as the value of the parameter ε decreases. The question of error estimates for asymptotic expansions like those in Section 3 was considered in Ammari et al. (2002) and Nazarov and Sokołowski (2003). It should be observed that the asymptotic analysis employs only one small parameter, though the problem under consideration contains several dimensionless ratios including diamðωε Þ=h, diamðΣ c Þ=h and d=h, where h is a characteristic size of the deformable domain and d ¼ distfO; x0 g. The first two ratios are assumed to be small, or more exactly, the inclusion diameter diamðωε Þ and the diameter of the contact zone diamðΣ c Þ should be small compared to both lengths d and h. The interrelation between d and h depends on the location of the inhomogeneity center x0 with respect to the indentation point O, and generally speaking, d ¼ OðhÞ, whereas the condition d ¼ oðhÞ corresponds to the case studied in Section 5. Note that following Ammari et al. (2008), the presented analysis can be extended to the time-harmonic regime where the indenter tip vibrates with a given frequency (Foschia et al., 2009). Finally, it is to note that the same identification procedure can be used in the case of a small inhomogeneity buried in an elastic layer provided Eq. (45) are replaced with the corresponding equations for the strains in the elastic layer without any inhomogeneity at the point coinciding with the small inhomogeneity center. 7. Conclusions In this study, the problem of indentation stiffness tomography for sensing of a small inhomogeneity in a homogeneous reference elastic body is investigated via an asymptotic modeling approach. With the reference body modeled as a homogeneous elastic half-space, the inverse problem of the spherical inhomogeneity parameter identification is reduced to the least-square based minimization of a misfit between the asymptotic predictions for the indentation stiffness and the massive array of experimental data for the indentation stiffness under a specified level of indentation depth collected on the body surface by means of grid indentation technique.
Acknowledgment The author would like to thank Dr. M. Loparic (Biozentrum and the Swiss Nanoscience Institute, University of Basel) for bringing his attention to the paper of Roduit et al. (2009). References Alves, C., Ammari, H., 2001. Boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium. SIAM J. Appl. Math. 62, 94–106. Ammari, H., Calmon, P., Iakovleva, E., 2008. Direct elastic imaging of a small inclusion. SIAM J. Imaging Sci. 1, 169–187. Ammari, H., Kang, H., 2006. Reconstruction of elastic inclusions of small volume via dynamic measurements. Appl. Math. Optim. 54, 223–235. Ammari, H., Kang, H., 2007. Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory. Applied Mathematical Sciences, vol. 162. Springer-Verlag, New York. Ammari, H., Kang, H., Nakamura, G., Tanuma, K., 2002. Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion. J. Elast. 67, 97–129. Argatov, I.I., 1999. The indentation of a punch in the form of an elliptic paraboloid into the plane boundary of an elastic body. J. Appl. Math. Mech. 63, 641–649. Argatov, I.I., 2010. Frictionless and adhesive nanoindentation: asymptotic modeling of size effects. Mech. Mater. 42, 807–815.
I.I. Argatov / J. Mech. Phys. Solids 70 (2014) 190–199
199
Argatov, I.I., 2011. Depth-sensing indentation of a transversely isotropic elastic layer: second-order asymptotic models for canonical indenters. Int. J. Solids Struct. 48, 3444–3452. Argatov, I.I., Sabina, F.J., 2013. Asymptotic analysis of the substrate effect for an arbitrary indenter. Quart. J. Mech. Appl. Math. 66, 75–95. Argatov, I., Sevostianov, I., 2011. Rigid toroidal inhomogeneity in an elastic medium. Int. J. Eng. Sci. 49, 61–74. Bonnet, M., Constantinescu, A., 2005. Inverse problems in elasticity. Inverse Probl. 21, R1–R50. Bonnet, M., Guzina, B.B., 2004. Sounding of finite solid bodies by way of topological derivative. Int. J. Numer. Methods Eng. 61, 2344–2373. Constantinides, G., Ravi Chandran, K.S., Ulm, F.-J., Van Vliet, K.J., 2006. Grid indentation analysis of composite microstructure and mechanics: principles and validation. Mater. Sci. Eng. A 430, 189–202. Fischer-Cripps, A.C., 2004. Nanoindentation. Springer-Verlag, New York. Foschia, R., Jobin, M., Hengsberger, S., 2009. Local dynamic mechanical analysis. Micron 40, 51–55. Galin, L.A., 2008. In: Gladwell G.M.L. (Ed.), Contact Problems: The Legacy of L.A. Galin. Springer, Dordrecht. Guzina, B.B., Chikichev, I., 2007. From imaging to material identification: a generalized concept of topological sensitivity. J. Mech. Phys. Solids 55, 245–279. Guzina, B.B., Nintcheu, S.F., Bonnet, M., 2003. On the stress-wave imaging of cavities in a semi-infinite solid. Int. J. Solids Struct. 40, 1505–1552. Il'in, A.M., 1992. Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Translations of Mathematical Monographs, vol. 102. AMS, Providence, RI. Johnson, K.L., 1985. Contact Mechanics. Cambridge University Press, Cambridge, UK. Kachanov, M., Shafiro, B., Tsukrov, I., 2003. Handbook of Elasticity Solutions. Kluwer Academic Publishers, Dordrecht. Kasas, S., Dietler, G., 2008. Probing nanomechanical properties from biomolecules to living cells. Pflugers Arch.—Eur. J. Physiol. 456, 13–27. Kasas, S., Longo, G., Dietler, G., 2013. Mechanical properties of biological specimens explored by atomic force microscopy. J. Phys. D: Appl. Phys. 46, 133001. 12 pp. Kikuchi, N., Oden, J.T., 1988. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia. Leroux, J., Fulleringer, B., Nélias, D., 2010. Contact analysis in presence of spherical inhomogeneities within a half-space. Int. J. Solids Struct. 47, 3034–3049. Lewiński, T., Sokołowski, J., 2003. Energy change due to the appearance of cavities in elastic solids. Int. J. Solids Struct. 40, 1765–1803. Loparic, M., Wirz, D., Daniels, A.U., Raiteri, R., VanLandingham, M.R., Guex, G., Martin, I., Aebi, U., Stolz, M., 2010. Micro- and nanomechanical analysis of articular cartilage by indentation-type atomic force microscopy: validation with a gel-microfiber composite. Biophys. J. 98, 2731–2740. Nazarov, S.A., Sokołowski, J., 2003. Asymptotic analysis of shape functionals. J. Math. Pure Appl. 82, 125–196. Novotny, A.A., Sokołowski, J., 2013. Topological Derivatives in Shape Optimization. Springer, Heidelberg. Plodinec, M., Loparic, M., Monnier, C.A., Obermann, E.C., Zanetti-Dallenbach, R., Oertle, P., Hyotyla, J.T., Aebi, U., Bentires-Alj, M., Lim, R.Y.H., Schoenenberger, C.-A., 2012. The nanomechanical signature of breast cancer. Nat. Nanotechnol. 7, 757–765. Rheinlaender, J., Geisse, N.A., Proksch, R., Schäffer, T.E., 2011. Comparison of scanning ion conductance microscopy with atomic force microscopy for cell imaging. Langmuir 27, 697–704. Roduit, C., Sekatski, S., Dietler, G., Catsicas, S., Lafont, F., Kasas, S., 2009. Stiffness tomography by atomic force microscopy. Biophys. J. 97, 674–677. Shillor, M., Sofonea, M., Telega, J.J., 2004. Models and Analysis of Quasistatic Contact: Variational Methods. Springer, Berlin. Sneddon, I.N., 1965. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3, 47–57. Van Dyke, M.D., 1964. Perturbation Methods in Fluid Mechanics. Academic Press, New York. Zhou, K., Chen, W.W., Keer, L.M., Ai, X., Sawamiphakdi, K., Glaws, P., Wang, Q.J., 2011. Multiple 3D inhomogeneous inclusions in a half space under contact loading. Mech. Mater. 43, 444–457. Zhou, K., Hoh, H.J., Wang, X., Keer, L.M., Pang, J.H.L., Song, B., Wang, Q.J., 2013. A review of recent works on inclusions. Mech. Mater. 60, 144–158. Zorin, I.S., Movchan, A.B., Nazarov, S.A., 1990. Application of tensors of elastic capacity, polarization and associated deformation. Studies on Elasticity and Plasticity, Leningrad University, vol. 16, pp. 75–91 (in Russian).