Abstract

This article presents the displacement field produced by a point force acting on an athermal random fiber network (the Green function for the network). The problem is defined within the limits of linear elasticity, and the field is obtained numerically for nonaffine networks characterized by various parameter sets. The classical Green function solution applies at distances from the point force larger than a threshold which is independent of the network parameters in the range studied. At smaller distances, the nonlocal nature of fiber interactions modifies the solution.

References

1.
Mauri
,
A.
,
Hopf
,
R.
,
Ehret
,
A. E.
,
Picu
,
C. R.
, and
Mazza
,
E.
,
2015
, “
A Discrete Network Model to Represent the Deformation Behavior of Human Amnion
,”
J. Mech. Behav. Biomed. Mater.
,
58
, pp.
45
56
.
2.
Kang
,
H.
,
Wen
,
Q.
,
Janmey
,
P. A.
,
Tang
,
J. X.
,
Conti
,
E.
, and
MacKintosh
,
F. C.
,
2009
, “
Nonlinear Elasticity of Stiff Filament Networks: Strain Stiffening, Negative Normal Stress, and Filament Alignment in Fibrin Gels
,”
J. Phys. Chem. B
,
113
(
12
), pp.
3799
3805
.
3.
Vader
,
D.
,
Kabla
,
W.
,
Weitz
,
D.
, and
Mahadevan
,
L.
,
2009
, “
Strain-Induced Alignment in Collagen Gels
,”
PLoS One
,
4
(
6
), p.
e5902
.
4.
Head
,
D. A.
,
Levine
,
A. J.
, and
MacKintosh
,
F. C.
,
2003
, “
Distinct Regimes of Elastic Response and Deformation Modes of Cross-Linked Cytoskeletal and Semiflexible Polymer Networks
,”
Phys. Rev. E
,
68
(
6
), p.
061907
.
5.
Love
,
A. E. H.
,
1990
,
A Treatise on the Mathematical Theory of Elasticity
,
Dover Publication
,
New York
.
6.
Broedersz
,
C. P.
, and
MacKintosh
,
F. C.
,
2014
, “
Modeling Semiflexible Polymer Networks
,”
Rev. Mod. Phys.
,
86
(
3
), pp.
995
1036
.
7.
Picu
,
C. R.
,
2020
, “Mechanics of Random Fiber Networks: Structure–Properties Relation,”
Mechanics of Fibrous Materials and Applications: Physical and Modeling Aspects
,
C.
Picu
, and
J.-F.
Ganghoffer
, eds.,
Springer International Publishing
,
Cham
, pp.
1
61
.
8.
Picu
,
R. C.
,
2011
, “
Mechanics of Random Fiber Networks—A Review
,”
Soft Matter
,
7
(
15
), pp.
6768
6785
.
9.
Kabla
,
A.
, and
Mahadevan
,
L.
,
2006
, “
Nonlinear Mechanics of Soft Fibrous Networks
,”
J. R. Soc., Interface.
10.
Picu
,
R. C.
,
Deogekar
,
S.
, and
Islam
,
M. R.
,
2018
, “
Poisson’s Contraction and Fiber Kinematics in Tissue: Insight From Collagen Network Simulations
,”
ASME J. Biomech. Eng.
,
140
(
2
), p.
021002
.
11.
Shivers
,
J. L.
,
Arzash
,
S.
, and
MacKintosh
,
F. C.
,
2019
, “
Nonlinear Poisson Effect Governed by Mechanical Critical Transition
,” arXiv:1905.09844 [cond-mat, physics:physics].
12.
Berkache
,
K.
,
Deogekar
,
S.
,
Goda
,
I.
,
Picu
,
R. C.
, and
Ganghoffer
,
J.-F.
,
2017
, “
Construction of Second Gradient Continuum Models for Random Fibrous Networks and Analysis of Size Effects
,”
Compos. Struct.
,
181
, pp.
347
357
.
13.
Berkache
,
K.
,
Deogekar
,
S.
,
Goda
,
I.
,
Picu
,
R. C.
, and
Ganghoffer
,
J.-F.
,
2019
, “
Homogenized Elastic Response of Random Fiber Networks Based on Strain Gradient Continuum Models
,”
Math. Mech. Solids
,
24
(
12
), pp.
3880
3896
.
14.
Berkache
,
K.
,
Deogekar
,
S.
,
Goda
,
I.
,
Picu
,
R. C.
, and
Ganghoffer
,
J.-F.
,
2019
, “
Identification of Equivalent Couple-Stress Continuum Models for Planar Random Fibrous Media
,”
Continuum Mech. Thermodyn.
,
31
(
4
), pp.
1035
1050
.
15.
Tyznik
,
S.
, and
Notbohm
,
J.
,
2019
, “
Length Scale Dependent Elasticity in Random Three-Dimensional Fiber Networks
,”
Mech. Mater.
,
138
, p.
103155
.
16.
Head
,
D. A.
,
Levine
,
A. J.
, and
MacKintosh
,
F. C.
,
2005
, “
Mechanical Response of Semiflexible Networks to Localized Perturbations
,”
Phys. Rev. E
,
72
(
6
), p.
061914
.
17.
Notbohm
,
J.
,
Lesman
,
A.
,
Rosakis
,
P.
,
Tirrell
,
D. A.
, and
Ravichandran
,
G.
,
2015
, “
Microbuckling of Fibrin Provides a Mechanism for Cell Mechanosensing
,”
J. R. Soc., Interface
,
12
(
108
), p.
20150320
.
18.
Grimmer
,
P.
, and
Notbohm
,
J.
,
2018
, “
Displacement Propagation in Fibrous Networks Due to Local Contraction
,”
ASME J. Biomech. Eng.
,
140
(
4
), p.
041011
.
19.
Burkel
,
B.
,
Proestaki
,
M.
,
Tyznik
,
S.
, and
Notbohm
,
J.
,
2018
, “
Heterogeneity and Nonaffinity of Cell-Induced Matrix Displacements
,”
Phys. Rev. E
,
98
(
5
), p.
052410
.
20.
Humphries
,
D. L.
,
Grogan
,
J. A.
, and
Gaffney
,
E. A.
,
2017
, “
Mechanical Cell–Cell Communication in Fibrous Networks: The Importance of Network Geometry
,”
Bull. Math. Biol.
,
79
(
3
), pp.
498
524
.
21.
Ronceray
,
P.
,
Broedersz
,
C. P.
, and
Lenz
,
M.
,
2016
, “
Fiber Networks Amplify Active Stress
,”
Proc. Natl. Acad. Sci.
,
113
(
11
), pp.
2827
2832
.
22.
Deogekar
,
S.
, and
Picu
,
R. C.
,
2017
, “
Structure-Properties Relation for Random Networks of Fibers With Noncircular Cross Section
,”
Phys. Rev. E
,
95
(
3
), p.
033001
.
23.
Merson
,
J.
, and
Picu
,
R. C.
,
2020
, “
Size Effects in Random Fiber Networks Controlled by the Use of Generalized Boundary Conditions
,”
Int. J. Solids Struct.
,
206
, pp.
314
321
.
24.
Glüge
,
R.
,
Weber
,
M.
, and
Bertram
,
A.
,
2012
, “
Comparison of Spherical and Cubical Statistical Volume Elements With Respect to Convergence, Anisotropy, and Localization Behavior
,”
Comput. Mater. Sci.
,
63
, pp.
91
104
.
25.
Zienkiewicz
,
O. C.
,
Emson
,
C.
, and
Bettess
,
P.
,
1983
, “
A Novel Boundary Infinite Element
,”
Int. J. Numer. Methods Eng.
,
19
(
3
), pp.
393
404
.
26.
Fish
,
J.
,
2010
,
Multiscale Methods: Bridging the Scales in Science and Engineering
,
Oxford University Press on Demand
,
Oxford
.
27.
Fish
,
J.
,
Nuggehally
,
M. A.
,
Shephard
,
M. S.
,
Picu
,
C. R.
,
Badia
,
S.
,
Parks
,
M. L.
, and
Gunzburger
,
M.
,
2007
, “
Concurrent AtC Coupling Based on a Blend of the Continuum Stress and the Atomistic Force
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
45–48
), pp.
4548
4560
.
28.
Mathew
,
N.
,
Picu
,
C. R.
, and
Bloomfield
,
M.
,
2011
, “
Concurrent Coupling of Atomistic and Continuum Models at Finite Temperature
,”
Comput. Methods Appl. Mech. Eng.
,
200
(
5–8
), pp.
765
773
.
29.
Miller
,
R. E.
, and
Tadmor
,
E. B.
,
2002
, “
The Quasicontinuum Method: Overview, Applications and Current Directions
,”
J. Comput.-Aided Mater. Des.
,
9
(
3
), pp.
203
239
.
30.
Rudd
,
R. E.
, and
Broughton
,
J. Q.
,
2000
, “
Concurrent Coupling of Length Scales in Solid State Systems
,”
Phys. Status Solidi B
,
217
(
1
), pp.
251
291
.
31.
Tadmor
,
E. B.
,
Ortiz
,
M.
, and
Phillips
,
R.
,
1996
, “
Quasicontinuum Analysis of Defects in Solids
,”
Philos. Mag. A
,
73
(
6
), pp.
1529
1563
.
32.
Wagner
,
G. J.
, and
Liu
,
W. K.
,
2003
, “
Coupling of Atomistic and Continuum Simulations Using a Bridging Scale Decomposition
,”
J. Comput. Phys.
,
190
(
1
), pp.
249
274
.
33.
Xu
,
M.
,
2010
,
Multiscale Methods: Bridging the Scales in Science and Engineering
,
J.
Fish
, ed.,
Oxford University Press
,
Oxford, UK
, pp.
93
133
.
34.
Deogekar
,
S.
, and
Picu
,
R. C.
,
2018
, “
On the Strength of Random Fiber Networks
,”
J. Mech. Phys. Solids
,
116
, pp.
1
16
.
35.
Onck
,
P. R.
,
Koeman
,
T.
,
van Dillen
,
T.
, and
van der Giessen
,
E.
,
2005
, “
Alternative Explanation of Stiffening in Cross-linked Semiflexible Networks
,”
Phys. Rev. Lett.
,
95
(
17
).
36.
Shahsavari
,
A. S.
, and
Picu
,
R. C.
,
2013
, “
Size Effect on Mechanical Behavior of Random Fiber Networks
,”
Int. J. Solids Struct.
,
50
(
20
), pp.
3332
3338
.
You do not currently have access to this content.