A recent study demonstrated that three-dimensional (3D) continuous displacement fields in transparent soft gels can be constructed from discrete displacement data obtained by optically tracking fluorescent particles embedded in the gels. Strain and stress fields were subsequently determined from gradients of the displacement field. This process was achieved through the moving least-square (MLS) interpolation method. The goal of this study is to evaluate the numerical accuracy of MLS in determining the displacement, strain, and stress fields in soft materials subjected to large deformation. Using an indentation model as the benchmark, we extract displacement at a set of randomly distributed data points from the results of a finite-element model, utilize these data points as the input for MLS, and compare resulting displacement, strain, and stress fields with the corresponding finite-element results. The calculation of strain and stress is based on finite strain kinematics and hyperelasticity theory. We also perform a parametric study in order to understand how parameters of the MLS method affect the accuracy of the interpolated displacement, strain, and stress fields. We further apply the MLS method to two additional cases with highly nonuniform deformation: a plate with a circular cavity subjected to large uniaxial stretch and a plane stress crack under large mode I loading. The results demonstrate the feasibility of using optical particle tracking together with MLS interpolation to map local strain and stress field in highly deformed soft materials.

References

References
1.
Rogers
,
J. A.
,
Someva
,
T.
, and
Huang
,
Y.
,
2010
, “
Materials and Mechanics for Stretchable Electronics
,”
Science
,
327
(
5973
), pp.
1603
1607
.
2.
Yasuda
,
K.
,
Gong
,
J. P.
,
Katsuyama
,
Y.
,
Nakayama
,
A.
,
Tanabe
,
Y.
,
Kondo
,
E.
,
Ueno
,
M.
, and
Osada
,
Y.
,
2005
, “
Biomechanical Properties of High-Toughness Double Network Hydrogels
,”
Biomaterials
,
26
(
21
), pp.
4468
4475
.
3.
Beebe
,
D. J.
,
Moore
,
J. S.
,
Bauer
,
J. M.
,
Qing
,
Y.
,
Liu
,
R. H.
,
Devadoss
,
C.
, and
Jo
,
B. H.
,
2000
, “
Functional Hydrogel Structures for Autonomous Flow Control Inside Microfluidic Channels
,”
Nature
,
404
(
6778
), pp.
588
590
.
4.
Huang
,
J.
,
Lu
,
T.
,
Zhu
,
J.
,
Clarke
,
D. R.
, and
Suo
,
Z.
,
2012
, “
Large, Uni-Directional Actuation in Dielectric Elastomers Achieved by Fiber Stiffening
,”
Appl. Phys. Lett.
,
100
(
21
), p.
211901
.
5.
Haraguchi
,
K.
, and
Takehisa
,
T.
,
2002
, “
Nanocomposite Hydrogels: A Unique Organic-Inorganic Network Structure With Extraordinary Mechanical, Optical, and Swelling/De-Swelling Properties
,”
Adv. Mater.
,
14
(
16
), pp.
1120
1124
.
6.
Gong
,
J. P.
,
Katsuyama
,
Y.
,
Kurokawa
,
T.
, and
Osada
,
Y.
,
2003
, “
Double-Network Hydrogels With Extremely High Mechanical Strength
,”
Adv. Mater.
,
15
(
14
), pp.
1155
1158
.
7.
Sun
,
J. Y.
,
Zhao
,
X.
,
Illeperuma
,
W. R.
,
Chaudhuri
,
O.
,
Oh
,
K. H.
,
Mooney
,
D. J.
,
Vlassak
,
J. J.
, and
Suo
,
Z.
,
2012
, “
Highly Stretchable and Tough Hydrogels
,”
Nature
,
489
(
7414
), pp.
133
136
.
8.
Sun
,
T. L.
,
Kurokawa
,
T.
,
Kuroda
,
S.
,
Ihsan
,
A. B.
,
Akasaki
,
I.
,
Sato
,
K.
,
Haque
,
M. A.
,
Nakajima
,
T.
, and
Gong
,
J. P.
,
2013
, “
Physical Hydrogels Composed of Polyampholytes Demonstrates High Toughness and Viscoelasticity
,”
Nat. Mater.
,
12
(
10
), pp.
932
937
.
9.
Ducrot
,
E.
,
Chen
,
Y.
,
Bulters
,
M.
,
Sijbesma
,
R. P.
, and
Creton
,
C.
,
2014
, “
Toughening Elastomers With Sacrificial Bonds and Watching Them Break
,”
Science
,
344
(
6180
), pp.
186
189
.
10.
Zhao
,
X.
,
2014
, “
Multi-Scale Multi-Mechanism Design of Tough Hydrogels: Building Dissipation Into Stretch Network
,”
Soft Matter
,
10
(
5
), pp.
672
687
.
11.
Shull
,
K. R.
,
2006
, “
Fracture and Adhesion of Elastomers and Gels: Large Strains at Small Length Scales
,”
J. Polym. Sci., Part B
,
44
(
24
), pp.
3436
3439
.
12.
Long
,
R.
, and
Hui
,
C. Y.
,
2015
, “
Crack Tip Fields in Soft Elastic Solids Subjected to Large Quasi-Static Deformation—A Review
,”
Extreme Mech. Lett.
,
4
, pp.
131
155
.
13.
Creton
,
C.
, and
Ciccotti
,
M.
, “
Fracture and Adhesion of Soft Materials
,”
Rep. Prog. Phys.
(online).
14.
Hui
,
C. Y.
,
Jagota
,
A.
,
Bennison
,
S. J.
, and
London
,
J. D.
,
2003
, “
Crack Blunting and the Strength of Soft Elastic Solids
,”
Proc. R. Soc. London A
,
459
(
2034
), pp.
1489
1516
.
15.
Livne
,
A.
,
Bouchbinder
,
E.
,
Svetlizky
,
I.
, and
Fineberg
,
J.
,
2010
, “
The Near-Tip Fields of Fast Cracks
,”
Science
,
327
(
5971
), pp.
1359
1363
.
16.
Bouchbinder
,
E.
,
Livne
,
A.
, and
Fineberg
,
J.
,
2010
, “
Weakly Nonlinear Fracture Mechanics: Experiments and Theory
,”
Int. J. Fract.
,
162
(
1
), pp.
3
20
.
17.
Zhang
,
J.
,
An
,
Y.
,
Yazzie
,
K.
,
Chawla
,
N.
, and
Jiang
,
H.
,
2012
, “
Finite Element Simulation of Swelling-Induced Crack Healing in Gels
,”
Soft Matter
,
8
(
31
), pp.
8107
8112
.
18.
Zhang
,
T.
,
Lin
,
S.
,
Yuk
,
H.
, and
Zhao
,
X.
,
2015
, “
Predicting Fracture Energies and Crack-Tip Fields of Soft Tough Materials
,”
Extreme Mech. Lett.
,
4
, pp.
1
8
.
19.
Xiao
,
X.
,
Song
,
H.-P.
,
Kang
,
Y.-L.
,
Li
,
X.-L.
,
Tan
,
X.-H.
, and
Tan
,
H.-Y.
,
2012
, “
Experimental Analysis of Crack Tip Fields in Rubber Materials Under Large Deformation
,”
Acta. Mech. Sin.
,
28
(
2
), pp.
432
437
.
20.
Sutton
,
M.
,
Orteu
,
J. J.
, and
Schreier
,
H. W.
,
2009
,
Image Correlation for Shape, Motion and Deformation Measurements
,
Springer
,
New York
.
21.
Bay
,
B. K.
,
Smith
,
T. S.
,
Fyhrie
,
D. P.
, and
Saad
,
M.
,
1999
, “
Digital Volume Correlation: Three-Dimensional Strain Mapping Using X-Ray Tomography
,”
Exp. Mech.
,
39
(
3
), pp.
217
236
.
22.
Gates
,
M.
,
Lambros
,
J.
, and
Heath
,
M. T.
,
2011
, “
Towards High Performance Digital Volume Correlation
,”
Exp. Mech.
,
51
(
4
), pp.
491
507
.
23.
Pan
,
B.
,
Wu
,
D.
, and
Wang
,
Z.
,
2012
, “
Internal Displacement and Strain Measurement Using Digital Volume Correlation: A Least-Squares Framework
,”
Meas. Sci. Technol.
,
23
(
4
), p.
045002
.
24.
Franck
,
C.
,
Hong
,
S.
,
Maskarinec
,
S. A.
,
Tirrell
,
D. A.
, and
Ravichandran
,
G.
,
2007
, “
Three-Dimensional Full-Field Measurements of Large Deformations in Soft Materials Using Confocal Microscopy and Digital Volume Correlation
,”
Exp. Mech.
,
47
(
3
), pp.
427
438
.
25.
Huang
,
J.
,
Pan
,
X.
,
Li
,
S.
,
Peng
,
X.
,
Xiong
,
C.
, and
Fang
,
J.
,
2011
, “
A Digital Volume Correlation Technique for 3D Deformation Measurements of Soft Gels
,”
ASME Int. J. Appl. Mech.
,
3
(
2
), pp.
335
354
.
26.
Bar-Kochba
,
E.
,
Toyjanova
,
J.
,
Andrews
,
E.
,
Kim
,
K.-S.
, and
Franck
,
C.
,
2015
, “
A Fast Iterative Digital Volume Correlation Algorithm for Large Deformations
,”
Exp. Mech.
,
55
(
1
), pp.
261
274
.
27.
Style
,
R. W.
,
Boltyanskiy
,
R.
,
German
,
G. K.
,
Hyland
,
C.
,
MacMinn
,
C. W.
,
Mertz
,
A. F.
,
Wilen
,
L. A.
,
Xu
,
Y.
, and
Dufresne
,
E. R.
,
2014
, “
Traction Force Microscopy in Physics and Biology
,”
Soft Matter
,
10
(
23
), pp.
4047
4055
.
28.
Dembo
,
M.
, and
Wang
,
Y. L.
,
1999
, “
Stresses at Cell-to-Substrate Interface During Locomotion of Fibroblasts
,”
Biophys. J.
,
76
(
4
), pp.
2307
2316
.
29.
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
.
30.
Xu
,
Y.
,
Engl
,
W. C.
,
Jerison
,
E. R.
,
Wallenstein
,
K. J.
,
Hyland
,
C.
,
Wilen
,
L. A.
, and
Dufresne
,
E. R.
,
2010
, “
Imaging In-Plane and Normal Stresses Near an Interface Crack Using Traction Force Microscopy
,”
Proc. Nat. Acad. Sci.
,
107
(
34
), pp.
14964
14967
.
31.
Xu
,
Y.
,
German
,
G. K.
,
Mertz
,
A. F.
, and
Dufresne
,
E. R.
,
2013
, “
Imaging Stress and Strain in the Fracture of Drying Colloidal Films
,”
Soft Matter
,
9
(
14
), pp.
3735
3740
.
32.
Hall
,
M. S.
,
Long
,
R.
,
Hui
,
C. Y.
, and
Wu
,
M.
,
2012
, “
Mapping Three-Dimensional Stress and Strain Fields Within a Soft Hydrogel Using a Fluorescence Microscope
,”
Biophys. J.
,
102
(
10
), pp.
2241
2250
.
33.
Gao
,
Y.
, and
Kilfoil
,
M. L.
,
2009
, “
Accurate Detection and Complete Tracking of Large Population of Features in Three Dimensions
,”
Opt. Express
,
17
(
6
), pp.
4685
4704
.
34.
Feng
,
X.
,
Hall
,
M. S.
,
Wu
,
M.
, and
Hui
,
C.-Y.
,
2014
, “
An Adaptive Algorithm for Tracking 3D Bead Displacements: Application in Biological Experiments
,”
Meas. Sci. Technol.
,
25
(
5
), p.
055701
.
35.
Lancaster
,
P.
, and
Salkauskas
,
K.
,
1981
, “
Surfaces Generated by Moving Least-Squares Method
,”
Math. Comput.
,
37
(
155
), pp.
141
158
.
36.
Nayroles
,
B.
,
Touzot
,
G.
, and
Villon
,
P.
,
1992
, “
Generalizing the Finite Element Method: Diffusion Approximation and Diffuse Elements
,”
Comput. Mech.
,
10
(
5
), pp.
307
318
.
37.
Belytschko
,
T.
,
Lu
,
Y. Y.
, and
Gu
,
L.
,
1994
, “
Element-Free Galerkin Methods
,”
Int. J. Numer. Methods Eng.
,
37
(
2
), pp.
229
256
.
38.
Liu
,
G. R.
,
2009
,
Meshfree Methods: Moving Beyond the Finite Element Method
,
2nd ed.
,
CRC Press
,
Boca Raton, FL
.
39.
Holzapfel
,
G. A.
,
2001
,
Nonlinear Solid Mechanics: A Continuum Approach for Engineering
,
Wiley
,
West Sussex, UK
.
40.
Lengyel
,
T. H.
,
Long
,
R.
, and
Schiavone
,
P.
,
2014
, “
Effect of Interfacial Slippage on the Near-Tip Fields of an Interface Crack Between a Soft Elastomer and a Rigid Substrate
,”
Proc. R. Soc. A
,
470
(
2170
), p.
20140497
.
41.
Yeoh
,
O. H.
,
1990
, “
Characterization of Elastic Properties of Carbon-Black-Filled Rubber Vulcanizates
,”
Rubber Chem. Technol.
,
63
(
5
), pp.
792
805
.
42.
Arruda
,
E. M.
, and
Boyce
,
M. C.
,
1993
, “
A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials
,”
J. Mech. Phys. Solids
,
41
(
2
), pp.
389
412
.
43.
Gent
,
A. N.
,
1996
, “
A New Constitutive Relation for Rubber
,”
Rubber Chem. Technol.
,
69
(
1
), pp.
59
61
.
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