Indentation using the atomic force microscope (AFM) has potential to measure detailed micromechanical properties of soft biological samples. However, interpretation of the results is complicated by the tapered shape of the AFM probe tip, and its small size relative to the depth of indentation. Finite element models (FEMs) were used to examine effects of indentation depth, tip geometry, and material nonlinearity and heterogeneity on the finite indentation response. Widely applied infinitesimal strain models agreed with FEM results for linear elastic materials, but yielded substantial errors in the estimated properties for nonlinear elastic materials. By accounting for the indenter geometry to compute an apparent elastic modulus as a function of indentation depth, nonlinearity and heterogeneity of material properties may be identified. Furthermore, combined finite indentation and biaxial stretch may reveal the specific functional form of the constitutive law—a requirement for quantitative estimates of material constants to be extracted from AFM indentation data.

1.
Bottemley
L. A.
,
Coury
J. E.
, and
First
P. N.
, “
Scanning probe microscopy
,”
Anal. Chem.
, Vol.
68
,
1996
, pp.
185R–230R
185R–230R
.
2.
Radmacher
M.
, “
Measuring the elastic properties of biological samples with the AFM
,”
IEEE Eng. Med. Biol. Mag.
, Vol.
16
,
1997
, pp.
47
57
.
3.
Hertz
H.
, “
U¨ber die beru¨hrung fester clastischer ko¨rper (On the contact of elastic solids)
,”
J. Reine Angew. Mathematik
, Vol.
92
,
1881
, pp.
156
171
.
4.
Love
A. E. H.
, “
Boussinesq’s problem for a rigid cone
,”
Q. J. Math
., Vol.
10
,
1939
, pp.
161
175
.
5.
Johnson, K. L., Contact Mechanics, Cambridge University Press, New York, 1985.
6.
Weisenhorn
A. L.
,
Khorsandi
M.
,
Kasas
S.
,
Gotzos
V.
, and
Butt
H.-J.
, “
Deformation and height anomaly of soft surfaces studied with an AFM
,”
Nanotech.
, Vol.
4
,
1993
, pp.
106
113
.
7.
Radmacher
M.
,
Fritz
M.
, and
Hansma
P. K.
, “
Imaging soft samples with the atomic force microscope: Gelatin in water and propanol
,”
Biophys. J.
, Vol.
69
,
1995
, pp.
264
270
.
8.
Sneddon
I. N.
, “
The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile
,”
Int. J. Engng. Sci.
, Vol.
3
,
1965
, pp.
47
57
.
9.
Radmacher
M.
,
Fritz
M.
,
Kacher
C. M.
,
Cleveland
J. P.
, and
Hansma
P. K.
, “
Measuring the viscoelastic properties of human platelets with the atomic force microscope
,”
Biophys. J.
, Vol.
70
,
1996
, pp.
556
567
.
10.
Hofmann
U. G.
,
Rotsch
C.
,
Parak
W. J.
, and
Radmacher
M.
, “
Investigating the cytoskeleton of chicken cardiocytes with the atomic force microscope
,”
J. Struct. Biol.
, Vol.
119
,
1997
, pp.
84
91
.
11.
A-Hassan
E.
,
Heinz
W. F.
,
Antonik
M. D.
,
D’Costa
N. P.
,
Nagaswaran
S.
,
Schoenenberger
C.-A.
, and
Hoh
J. H.
, “
Relative microelastic mapping of living cells by atomic force microscopy
,”
Biophys. J.
, Vol.
74
,
1998
, pp.
1564
1578
.
12.
Fung, Y. C., Biomechanics: Mechanical Properties of Living Tissues, Springer-Verlag, New York, 1981.
13.
Albrecht
T. R.
,
Akamine
S.
,
Carver
T. E.
, and
Quate
C. F.
, “
Microfabrication of cantilever styli for the atomic force microscope
,”
J. Vac. Sci. Technol. A
, Vol.
8
,
1990
, pp.
3386
3396
.
14.
Tortonese
M.
, “
Cantilevers and tips for atomic force microscopy
,”
IEEE Eng. Med. Biol. Mag.
, Vol.
16
,
1997
, pp.
28
33
.
15.
Green, A. E., and Zerna, W., Theoretical Elasticity, Oxford University Press, London, 1968.
16.
Beatty
M. F.
, and
Usmani
S. A.
, “
On the indentation of a highly elastic half-space
,”
Quart. J. Mech. Appl. Math.
, Vol.
28
,
1975
, pp.
47
62
.
17.
Humphrey
J. D.
,
Halperin
H. R.
, and
Yin
F. C. P.
, “
Small indentation superimposed on a finite equibiaxial stretch: Implications for cardiac mechanics
,”
ASME Journal of Applied Mechanics
, Vol.
58
,
1991
, pp.
1108
1111
.
18.
Hayes
W. C.
,
Keer
L. M.
,
Herrmann
G.
, and
Mockros
L. F.
, “
A mathematical analysis for indentation tests of articular cartilage
,”
J. Biomech.
, Vol.
5
,
1972
, pp.
541
551
.
19.
Matthewson
M. J.
, “
Axi-symmetric contact on thin compliant coatings
,”
J. Mech. Phys. Solids
, Vol.
29
,
1981
, pp.
89
113
.
20.
Jaffar
M. J.
, “
A numerical solution for axisymmetric contact problems involving rigid indenters on elastic layers
,”
J. Mech. Phys. Solids
, Vol.
36
,
1988
, pp.
401
416
.
21.
Jaffar
M. J.
, “
Stresses and deformations in elastic layers indented by a conical punch
,”
Proc. Instn. Mech. Engrs.
, Vol.
209
,
1995
, pp.
201
205
.
22.
Batra, R. C., “Quasistatic indentation of a rubberlike layer by a rigid cylinder,” Proc. Int. Conf. Finite Elements in Computational Mechanics, Kant, T., ed., 1985, pp. 345–357.
23.
Duszyk
M.
,
Schwab
B.
,
Zahalak
G. I.
,
Qian
H.
, and
Elson
E. L.
, “
Cell poking: Quantitative analysis of indentation of thick viscoelastic layers
,”
Biophys. J.
, Vol.
55
,
1989
, pp.
683
690
.
24.
Zahalak
G. I.
,
McConnaughey
W. B.
, and
Elson
E. L.
, “
Determination of cellular mechanical properties by cell poking, with an application to leukocytes
,”
ASME JOURNAL OF BIOMECHANICAL ENGINEERING
, Vol.
112
,
1990
, pp.
283
294
.
25.
Chang
W. V.
, and
Sun
S. C.
, “
Nonlinear elastic analysis of the hardness test on rubber-like materials
,”
Rubber Chem. Technol.
, Vol.
64
,
1991
, pp.
202
210
.
26.
Laursen
T. A.
, and
Simo
J. C.
, “
A study of the mechanics of microindentation using finite elements
,”
J. Mater. Res.
, Vol.
7
,
1992
, pp.
618
626
.
27.
Suh
J.-K.
, and
Spilker
R. L.
, “
Indentation analysis of biphasic articular cartilage: Nonlinear phenomena under finite deformation
,”
ASME JOURNAL OF BIOMECHANICAL ENGINEERING
, Vol.
116
,
1994
, pp.
1
9
.
28.
Karduna
A. R.
,
Halperin
H. R.
, and
Yin
F. C. P.
, “
Experimental and numerical analyses of indentation in finite-sized isotropic and anisotropic rubber-like materials
,”
Ann. Biomed. Eng.
, Vol.
25
,
1997
, pp.
1009
1016
.
29.
Prost-Domasky
S. A.
,
Szabo
B. A.
, and
Zahalak
G. I.
, “
Large-deformation analysis of nonlinear elastic fluids
,”
Comput. Struct.
, Vol.
64
,
1997
, pp.
1281
1290
.
30.
Oden, J. T., Finite Elements of Nonlinear Continua, McGraw-Hill, New York, 1972.
31.
ABAQUS/Standard, v5.5. Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, RI, 1995.
32.
Speneer, A. J. M., Continuum Mechanics, Longman Press, London, 1980.
33.
Haydon
P. G.
,
Lartius
R.
,
Parpura
V.
, and
Marchese-Ragona
S. P.
, “
Membrane deformation of living glial cells using atomic force microscopy
,”
J. Microsc.
, Vol.
182
,
1996
, pp.
114
120
.
34.
Schaus
S. S.
, and
Henderson
E. R.
, “
Cell viability and probe-cell membrane interactions of XR1 glial cells imaged by atomic force microscopy
,”
Biophys. J.
, Vol.
73
,
1997
, pp.
1205
1214
.
35.
Doerner
M. F.
, and
Nix
W. D.
, “
A method for interpreting the data from depth-sensing indentation instruments
,”
J. Mater. Res.
, Vol.
1
,
1986
, pp.
601
609
.
36.
Briscoe
B. J.
,
Sebastian
K. S.
, and
Adams
M. J.
, “
The effect of indenter geometry on the elastic response to indentation
,”
J. Phys. D: Appl. Phys.
, Vol.
27
,
1994
, pp.
1156
1162
.
37.
Schwarz
U. D.
,
Haefke
H.
,
Reimann
P.
, and
Gu¨ntherodt
H. J.
, “
Tip artefacts in scanning force microscopy
,”
J. Microsc.
, Vol.
173
,
1994
, pp.
183
197
.
38.
Bilodeau
G. G.
, “
Regular pyramid punch problem
,”
J. Appl. Mech.
, Vol.
59
,
1992
, pp.
519
523
.
39.
Hoh
J. H.
, and
Schoenenberger
C.-A.
, “
Surface morphology and mechanical properties of MDCK monolayers by atomic force microscopy
,”
J. Cell Sci.
, Vol.
107
,
1994
, pp.
1105
1114
.
40.
Burridge
K.
,
Fath
K.
,
Kelly
T.
,
Nuckolls
G.
, and
Turner
C.
, “
Focal adhesions: transmembrane junctions between the extracellular matrix and the cytoskeleton
,”
Ann. Rev. Cell Biol.
, Vol.
4
,
1988
, pp.
487
525
.
41.
Mak
A. F.
,
Lai
W. M.
, and
Mow
V. C.
, “
Biphasic indentation of articular cartilage—1: Theoretical analysis
,”
J. Biomech.
, Vol.
20
,
1987
, pp.
703
714
.
42.
Hanson
M. T.
, “
The elastic field for conical indentation including sliding friction for transverse isotropy
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
59
,
1992
, pp.
S123–S130
S123–S130
.
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