Fiber networks are assemblies of one-dimensional elements representative of materials with fibrous microstructures such as collagen networks and synthetic nonwovens. The mechanics of random fiber networks has been the focus of numerous studies. However, fiber crimp has been explicitly represented only in few cases. In the present work, the mechanics of cross-linked networks with crimped athermal fibers, with and without an embedding elastic matrix, is studied. The dependence of the effective network stiffness on the fraction of nonstraight fibers and the relative crimp amplitude (or tortuosity) is studied using finite element simulations of networks with sinusoidally curved fibers. A semi-analytic model is developed to predict the dependence of network modulus on the crimp amplitude and the bounds of the stiffness reduction associated with the presence of crimp. The transition from the linear to the nonlinear elastic response of the network is rendered more gradual by the presence of crimp, and the effect of crimp on the network tangent stiffness decreases as strain increases. If the network is embedded in an elastic matrix, the effect of crimp becomes negligible even for very small, biologically relevant matrix stiffness values. However, the distribution of the maximum principal stress in the matrix becomes broader in the presence of crimp relative to the similar system with straight fibers, which indicates an increased probability of matrix failure.

References

References
1.
Picu
,
R. C.
,
2011
, “
Mechanics of Random Fiber Networks—A Review
,”
Soft Matter
,
7
(
15
), pp.
6768
6785
.
2.
Broedersz
,
C.
, and
MacKintosh
,
F.
,
2014
, “
Modeling Semiflexible Polymer Networks
,”
Rev. Mod. Phys.
,
86
(
3
), pp.
995
1036
.
3.
Licup
,
A.
,
Munster
,
S.
,
Sharma
,
A.
,
Sheinman
,
M.
,
Jawerth
,
L.
,
Fabry
,
B.
,
Weitz
,
D.
, and
MacKintosh
,
F.
,
2015
, “
Stress Controls the Mechanics of Collagen Networks
,”
Proc. Natl. Acad. Sci. U. S. A.
,
112
(
31
), pp.
9573
9578
.
4.
van Dillen
,
T.
,
Onck
,
P.
, and
Van der Giessen
,
E.
,
2008
, “
Models for Stiffening in Cross-Linked Biopolymer Networks: A Comparative Study
,”
J. Mech. Phys. Solids
,
56
(
6
), pp.
2240
2264
.
5.
Onck
,
P.
,
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
), pp.
178102
178105
.
6.
Shahsavari
,
A.
, and
Picu
,
R. C.
,
2013
, “
Elasticity of Sparsely Cross-Linked Random Fibre Networks
,”
Philos. Mag. Lett.
,
93
(
6
), pp.
356
361
.
7.
Broedersz
,
C. P.
,
Mao
,
X.
,
Lubensky
,
T. C.
, and
MacKintosh
,
F. C.
,
2011
, “
Criticality and Isostaticity in Fibre Networks
,”
Nat. Phys.
,
7
(
12
), pp.
983
988
.
8.
Shahsavari
,
A.
, and
Picu
,
R. C.
,
2012
, “
Model Selection for Athermal Cross-Linked Fiber Networks
,”
Phys. Rev. E
,
86
(
1
), p.
011923
.
9.
Ban
,
E.
,
Barocas
,
V. H.
,
Shephard
,
M.
, and
Picu
,
R. C.
,
2015
, “
Softening in Random Networks of Non-Identical Beams
,”
J. Mech. Phys. Solids
,
87
, pp.
38
50
.
10.
Zhang
,
L.
,
Lake
,
S.
,
Barocas
,
V. H.
,
Shephard
,
M. S.
, and
Picu
,
R. C.
,
2013
, “
Cross-Linked Fiber Network Embedded in an Elastic Matrix
,”
Soft Matter
,
9
(
28
), pp.
6398
6405
.
11.
Zhang
,
L.
,
Lake
,
S.
,
Lai
,
V.
,
Picu
,
R. C.
,
Barocas
,
V. H.
, and
Shephard
,
M. S.
,
2013
, “
A Coupled Fiber-Matrix Model Demonstrates Highly Inhomogeneous Microstructural Interactions in Soft Tissues Under Tensile Load
,”
ASME J. Biomech. Eng.
,
135
(
1
), p.
011008
.
12.
Cacho
,
F.
,
Elbischger
,
P.
,
Rodríguez
,
J.
,
Doblaré
,
M.
, and
Holzapfel
,
G.
,
2007
, “
A Constitutive Model for Fibrous Tissues Considering Collagen Fiber Crimp
,”
Int. J. Nonlinear Mech.
,
42
(
2
), pp.
391
402
.
13.
Lake
,
S.
,
Hadi
,
M.
,
Lai
,
V.
, and
Barocas
,
V. H.
,
2012
, “
Mechanics of a Fiber Network Within a Non-Fibrillar Matrix: Model and Comparison With Collagen-Agarose Co-Gels
,”
Ann. Biomed. Eng.
,
40
(
10
), pp.
2111
2121
.
14.
Legerlotza
,
K.
,
Dornb
,
J.
,
Richtera
,
J.
,
Rauschc
,
M.
, and
Leupin
,
O.
,
2014
, “
Age-Dependent Regulation of Tendon Crimp Structure, Cell Length and Gap Width With Strain
,”
Acta Biomater.
,
10
(
10
), pp.
4447
4455
.
15.
Franchi
,
M.
,
Raspanti
,
M.
,
Dell'Orbo
,
C.
,
Quaranta
,
M.
,
De Pasquale
,
V.
,
Ottani
,
V.
, and
Ruggeri
,
A.
,
2008
, “
Different Crimp Patterns in Collagen Fibrils Relate to the Subfibrillar Arrangement
,”
Connect. Tissue Res.
,
49
(
2
), pp.
85
91
.
16.
Raina
,
A.
, and
Linder
,
C.
,
2014
, “
A Homogenization Approach for Nonwoven Materials Based on Fiber Undulations and Reorientation
,”
J. Mech. Phys. Solids
,
65
, pp.
12
34
.
17.
Kabla
,
A.
, and
Mahadevan
,
L.
,
2007
, “
Nonlinear Mechanics of Soft Fibrous Networks
,”
J. R. Soc., Interface
,
4
(
12
), pp.
99
106
.
18.
Huisman
,
E.
,
van Dillen
,
T.
,
Onck
,
P.
, and
Van der Giessen
,
E.
,
2007
, “
Three-Dimensional Cross-Linked F-Actin Networks: Relation Between Network Architecture and Mechanical Behavior
,”
Phys. Rev. Lett.
,
99
(
20
), pp.
208103
208106
.
19.
Wen
,
Q.
,
Basu
,
A.
,
Janmey
,
P.
, and
Yodh
,
A.
,
2012
, “
Non-Affine Deformations in Polymer Hydrogels
,”
Soft Matter
,
8
(
31
), pp.
8039
8049
.
20.
Janmey
,
P.
,
Peetermans
,
J.
,
Zaner
,
K. S.
,
Stossel
,
T. P.
, and
Takana
,
T.
,
1986
, “
Structure and Mobility of Actin Filaments as Measured by Quasielastic Light Scattering, Viscometry, and Electron Microscopy
,”
J. Biol. Chem.
,
261
(
18
), pp.
8357
8362
21.
Hsiao
,
H.
, and
Daniel
,
I.
,
1996
, “
Effect of Fiber Waviness on Stiffness and Strength Reduction of Unidirectional Composites Under Compressive Loading
,”
Compos. Sci. Technol.
,
56
(
5
), pp.
581
593
.
22.
Lai
,
V.
,
Frey
,
C.
,
Kerandi
,
A.
,
Lake
,
S.
,
Tranquillo
,
R.
, and
Barocas
,
V. H.
,
2013
, “
Microstructural and Mechanical Differences Between Digested Collagen-Fibrin Co-Gels and Pure Collagen and Fibrin Gels
,”
Acta Biomater.
,
8
(
11
), pp.
4031
4042
.
23.
Love
,
A.
,
1944
,
A Treatise on the Mathematical Theory of Elasticity
,
Dover Publications
,
New York
.
24.
Zagar
,
G.
,
Onck
,
P.
, and
Van der Giessen
,
E.
,
2011
, “
Elasticity of Rigidly Cross-Linked Networks of Athermal Filaments
,”
Macromolecules
,
44
(
17
), pp.
7026
7033
.
25.
Zagar
,
G.
,
Onck
,
P.
, and
Van der Giessen
,
E.
,
2015
, “
Two Fundamental Mechanisms Govern the Stiffening of Cross-Linked Networks
,”
Biophys. J.
,
108
(
6
), pp.
1470
1479
.
26.
Discher
,
D.
,
Mooney
,
D.
, and
Zandstra
,
P.
,
2009
, “
Growth Factors, Matrices, and Forces Combine and Control Stem Cells
,”
Science
,
324
(
5935
), pp.
1673
1677
.
27.
Barthelat
,
F.
,
2007
, “
Biomimetics for Next Generation Materials
,”
Phil. Trans. R. Soc. A
,
365
(
1861
), pp.
2907
2919
.
You do not currently have access to this content.