The displacement of relatively rigid beads within a relatively compliant, elastic matrix can be used to measure the mechanical properties of the matrix. For example, in mechanobiological studies, magnetic or reflective beads can be displaced with a known external force to estimate the matrix modulus. Although such beads are generally rigid compared to the matrix, the material surrounding the beads typically differs from the matrix in one or two ways. The first case, as is common in mechanobiological experimentation, is the situation in which the bead must be coated with materials such as protein ligands that enable adhesion to the matrix. These layers typically differ in stiffness relative to the matrix material. The second case, common for uncoated beads, is the situation in which the beads disrupt the structure of the hydrogel or polymer, leading to a region of enhanced or reduced stiffness in the neighborhood of the bead. To address both cases, we developed the first analytical solution of the problem of translation of a coated, rigid spherical inclusion displaced within an isotropic elastic matrix by a remotely applied force. The solution is applicable to cases of arbitrary coating stiffness and size of the coating. We conclude by discussing applications of the solution to mechanobiology.

References

1.
Freundlich
,
H.
, and
Seifriz
,
W.
,
1923
, “
Ueber die elastizität von solen und gelen
,”
Z. Phys. Chem.
,
104
(
1
), pp.
233
261
.
2.
Kupradze
,
V. D.
(
1965
).
Potential Methods in the Theory of Elasticity
,
Israel Program for Scientific Translations
,
Jerusalem
.
3.
Lurie
,
A.
, and
Belyaev
,
A.
,
2005
,
Theory of Elasticity (Foundations of Engineering Mechanics)
,
Springer
,
Berlin
, pp.
978
973
.
4.
De
,
V. I.
, and
Dekker
,
C.
,
2012
, “
Recent Advances in Magnetic Tweezers
,”
Annu. Rev. Biophys.
,
41
(
1
), pp.
453
472
.
5.
Lefèvre
,
V.
,
Danas
,
K.
, and
Lopezpamies
,
O.
,
2017
, “
A General Result for the Magnetoelastic Response of Isotropic Suspensions of Iron and Ferrofluid Particles in Rubber, With Applications to Spherical and Cylindrical Specimens
,”
J. Mech. Phys. Solids
,
107
, pp.
343
364
.
6.
Zhong
,
Q. Y.
, and
Selvadurai
,
A. P. S.
,
1995
, “
On the Mechanics of a Rigid Disc Inclusion Embedded in a Fluid Saturated Poroelastic Medium
,”
Int. J. Eng. Sci.
,
33
(
11
), pp.
1633
1662
.
7.
Giordano
,
S.
,
Palla
,
P. L.
,
Cadelano
,
E.
, and
Brun
,
M.
,
2012
, “
Elastic Behavior of Inhomogeneities With Size and Shape Different From Their Hosting Cavities
,”
Mech. Mater.
,
44
, pp.
4
22
.
8.
Huang
,
G.
,
Li
,
F.
,
Zhao
,
X.
,
Ma
,
Y.
,
Li
,
Y.
,
Lin
,
M.
,
Jin
,
G.
,
Lu
,
T. J.
,
Genin
,
G. M.
, and
Xu
,
F.
,
2017
, “
Functional and Biomimetic Materials for Engineering of the Three-Dimensional Cell Microenvironment
,”
Chem. Rev.
,
117
(
20
), pp.
12764
12850
.
9.
Hooper
,
J. B.
, and
Schweizer
,
K. S.
,
2006
, “
Theory of Phase Separation in Polymer Nanocomposites
,”
Macromolecules
,
39
(
15
), pp.
5133
5142
.
10.
Robin
,
G.
,
1886
, “
Sur la distribution de l’électricité à la surface des conducteurs fermés et des conducteurs ouverts
,”
Ann. Sci. Ecole Norm. Sup
,
3
(
2
), pp.
31
358
.
11.
Lurie
,
A. I.
,
1967
, “
Elastostatic Robin Problem for a Triaxial Ellipsoid
,”
Mech. Solids
,
1
(
2
), pp.
80
83
.
12.
Selvadurai
,
A.
,
1975
, “
The Distribution of Stress in a Rubber-Like Elastic Material Bounded Internally by a Rigid Spherical Inclusion Subjected to a Central Force
,”
Mech. Res. Commun.
,
2
(
3
), pp.
99
106
.
13.
Zureick
,
A.
,
1988
, “
Transversely Isotropic Medium With a Rigid Spheroidal Inclusion Under an Axial Pull
,”
J. Appl. Mech.
,
55
, pp.
495
497
.
14.
Selvadurai
,
A. P. S.
,
2016
, “
Indentation of a Spherical Cavity in an Elastic Body by a Rigid Spherical Inclusion: Influence of Non-Classical Interface Conditions
,”
Continuum Mech. Thermodyn.
,
28
(
1-2
), pp.
617
632
.
15.
Lin
,
D. C.
,
Yurke
,
B.
, and
Langrana
,
N. A.
,
2005
, “
Use of Rigid Spherical Inclusions in Young’s Moduli Determination: Application to DNA-Crosslinked Gels
,”
J. Biomech. Eng.
,
127
(
4
), pp.
571
579
.
16.
Selvadurai
,
A.
,
1976
, “
The Load–Deflexion Charateristics of a Deep Rigid Anchor in an Elastic Medium
,”
Geotechnique
,
26
(
4
), pp.
603
612
.
17.
Kanwal
,
R.
, and
Sharma
,
D.
,
1976
, “
Singularity Methods for Elastostatics
,”
J. Elast.
,
6
(
4
), pp.
405
418
.
18.
Hashin
,
Z.
,
1991
, “
The Spherical Inclusion With Imperfect Interface
,”
J. Appl. Mech.
,
58
(
2
), pp.
444
449
.
19.
Mura
,
T.
, and
Furuhashi
,
R.
,
1984
, “
The Elastic Inclusion With a Sliding Interface
,”
J. Appl. Mech.
,
51
(
2
), pp.
308
310
.
20.
Kamgoué
,
A.
,
Ohayon
,
J.
, and
Tracqui
,
P.
,
2007
, “
Estimation of Cell Young’s Modulus of Adherent Cells Probed by Optical and Magnetic Tweezers: Influence of Cell Thickness and Bead Immersion
,”
J. Biomech. Eng.
,
129
(
4
), pp.
523
530
.
21.
Walpole
,
L. J.
,
1991
, “
A Translated Rigid Ellipsoidal Inclusion in an Elastic Medium
,”
Proc. Math. Phys. Sci.
,
434
(
1892
), pp.
571
585
.
22.
Kachanov
,
M.
,
Shafiro
,
B.
, and
Tsukrov
,
I.
,
2003
,
Handbook of Elasticity Solutions
,
Kluwer Academic
,
Boston
.
23.
Eshelby
,
J. D.
,
1957
, “
The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems
,”
Proc. R. Soc. A
,
241
(
1226
), pp.
376
396
.
24.
Lur’Ea
,
I.
,
1964
,
Three-Dimensional Problems of the Theory of Elasticity
,
Interscience Publishers
,
New York
.
25.
Rabinowitsch
,
B.
,
1929
, “
Über die viskosität und elastizität von solen
,”
Z. Phys. Chem.
,
145
(
1
), pp.
1
26
.
26.
Harada
,
T.
,
Swift
,
J.
,
Irianto
,
J.
,
Shin
,
J.-W.
,
Spinler
,
K. R.
,
Athirasala
,
A.
,
Diegmiller
,
R.
,
Dingal
,
P. D. P.
,
Ivanovska
,
I. L.
, and
Discher
,
D. E.
,
2014
, “
Nuclear Lamin Stiffness Is a Barrier to 3D Migration, But Softness Can Limit Survival
,”
J. Cell Biol.
,
204
(
5
), pp.
669
682
.
27.
Li
,
Y.
,
Hong
,
Y.
,
Xu
,
G.-K.
,
Liu
,
S.
,
Shi
,
Q.
,
Tang
,
D.
,
Yang
,
H.
,
Genin
,
G. M.
,
Lu
,
T. J.
, and
Xu
,
F.
,
2018
, “
Non-Contact Tensile Viscoelastic Characterization of Microscale Biological Materials
,”
Acta Mech. Sin.
,
34
(
3
), pp.
589
599
.
28.
Legant
,
W. R.
,
Miller
,
J. S.
,
Blakely
,
B. L.
,
Cohen
,
D. M.
,
Genin
,
G. M.
, and
Chen
,
C. S.
,
2010
, “
Measurement of Mechanical Tractions Exerted by Cells in Three-Dimensional Matrices
,”
Nat. Methods
,
7
(
12
), pp.
969
971
.
29.
Yamada
,
S.
,
Wirtz
,
D.
, and
Kuo
,
S. C.
,
2000
, “
Mechanics of Living Cells Measured by Laser Tracking Microrheology
,”
Biophys. J.
,
78
(
4
), pp.
1736
1747
.
30.
Marquez
,
J. P.
,
Genin
,
G. M.
,
Zahalak
,
G. I.
, and
Elson
,
E. L.
,
2005
, “
The Relationship Between Cell and Tissue Strain in Three-Dimensional Bio-Artificial Tissues
,”
Biophys. J.
,
88
(
2
), pp.
778
789
.
31.
Crisp
,
M.
,
Liu
,
Q.
,
Roux
,
K.
,
Rattner
,
J.
,
Shanahan
,
C.
,
Burke
,
B.
,
Stahl
,
P. D.
, and
Hodzic
,
D.
,
2006
, “
Coupling of the Nucleus and Cytoplasm: Role of the LINC Complex
,”
J. Cell Biol.
,
172
(
1
), pp.
41
53
.
32.
Swift
,
J.
,
Ivanovska
,
I. L.
,
Buxboim
,
A.
,
Harada
,
T.
,
Dingal
,
P. D. P.
,
Pinter
,
J.
,
Pajerowski
,
J. D.
,
Spinler
,
K. R.
,
Shin
,
J.-W.
, and
Tewari
,
M.
,
2013
, “
Nuclear Lamin-A Scales with Tissue Stiffness and Enhances Matrix-Directed Differentiation
,”
Science
,
341
(
6149
), p.
1240104
.
33.
Babaei
,
B.
,
Davarian
,
A.
,
Lee
,
S.-L.
,
Pryse
,
K. M.
,
McConnaughey
,
W. B.
,
Elson
,
E. L.
, and
Genin
,
G. M.
,
2016
, “
Remodeling by Fibroblasts Alters the Rate-Dependent Mechanical Properties of Collagen
,”
Acta Biomater.
,
37
, pp.
28
37
.
34.
Trepat
,
X.
,
Deng
,
L.
,
An
,
S. S.
,
Navajas
,
D.
,
Tschumperlin
,
D. J.
,
Gerthoffer
,
W. T.
,
Butler
,
J. P.
, and
Fredberg
,
J. J.
,
2007
, “
Universal Physical Responses to Stretch in the Living Cell
,”
Nature
,
447
(
7144
), pp.
592
595
.
35.
Tirkkonen
,
L.
,
Halonen
,
H.
,
Hyttinen
,
J.
,
Kuokkanen
,
H.
,
Sievanen
,
H.
,
Koivisto
,
A. M.
,
Mannerstrom
,
B.
,
Sandor
,
G. K.
,
Suuronen
,
R.
,
Miettinen
,
S.
, and
Haimi
,
S.
,
2011
, “
The Effects of Vibration Loading on Adipose Stem Cell Number, Viability and Differentiation Towards Bone-Forming Cells
,”
J. R. Soc. Interface
,
8
(
65
), pp.
1736
1747
.
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