If an elastomeric material is subjected to sufficiently large deformations, it eventually fractures. There are two typical micromechanisms of failure in such materials: chain scission and crosslink failure. The chain scission failure mode is mainly observed in polymers with strong covalent crosslinks, while the crosslink failure mode is observed in polymers with weak crosslinks. In two recent papers, we have proposed a theory for progressive damage and rupture of polymers with strong covalent crosslinks. In this paper, we extend our previous framework and formulate a theory for modeling failure of elastomeric materials with weak crosslinks. We first introduce a model for the deformation of a single chain with weak crosslinks at each of its two ends using statistical mechanics arguments, and then upscale the model from a single chain to the continuum level for a polymer network. Finally, we introduce a damage variable to describe the progressive damage and failure of polymer networks. A central feature of our theory is the recognition that the free energy of elastomers is not entirely entropic in nature; there is also an energetic contribution from the deformation of the backbone bonds in a chain and/or the crosslinks. For polymers with weak crosslinks, this energetic contribution is mainly from the deformation of the crosslinks. It is this energetic part of the free energy which is the driving force for progressive damage and fracture of elastomeric materials. Moreover, we show that for elastomeric materials in which fracture occurs by crosslink stretching and scission, the classical Lake–Thomas scaling—that the toughness Gc of an elastomeric material is proportional to 1/G0, with G0=NkBϑ the ground-state shear modulus of the material—does not hold. A new scaling is proposed, and some important consequences of this scaling are remarked upon.

References

References
1.
Lake
,
G. J.
, and
Thomas
,
A. G.
,
1967
, “
The Strength of Highly Elastic Materials
,”
Proc. R. Soc. London A
,
300
(
1460
), pp.
108
119
.
2.
Akagi
,
Y.
,
Sakurai
,
H.
,
Gong
,
J. P.
,
Chung
,
U.
, and
Sakai
,
T.
,
2013
, “
Fracture Energy of Polymer Gels With Controlled Network Structures
,”
J. Chem. Phys.
,
139
(
14
), p.
144905
.
3.
Sakai
,
T.
,
2013
, “
Gelation Mechanism and Mechanical Properties of Tetra-Peg Gel
,”
Reactive Funct. Polym.
,
73
(
7
), pp.
898
903
.
4.
Mao
,
Y.
,
Talamini
,
B.
, and
Anand
,
L.
,
2017
, “
Rupture of Polymers by Chain Scission
,”
Extreme Mech. Lett.
,
13
, pp.
17
24
.
5.
Talamini
,
B.
,
Mao
,
Y.
, and
Anand
,
L.
,
2018
, “
Progressive Damage and Rupture in Polymers
,”
J. Mech. Phys. Solids
,
111
, pp.
434
457
.
6.
Grindy
,
S. C.
,
Lenz
,
M.
, and
Holten-Andersen
,
N.
,
2016
, “
Engineering Elasticity and Relaxation Time in Metal-Coordinate Cross-Linked Hydrogels
,”
Macromolecules
,
49
(
21
), pp.
8306
8312
.
7.
Miehe
,
C.
, and
Schänzel
,
L.-M.
,
2014
, “
Phase Field Modeling of Fracture in Rubbery Polymers—Part I: Finite Elasticity Coupled With Brittle Failure
,”
J. Mech. Phys. Solids
,
65
, pp.
93
113
.
8.
Raina
,
A.
, and
Miehe
,
C.
,
2016
, “
A Phase-Field Model for Fracture in Biological Tissues
,”
Biomech. Model. Mechanobiol.
,
15
(
3
), pp.
479
496
.
9.
Miehe
,
C.
,
Welschinger
,
F.
, and
Hofacker
,
M.
,
2010
, “
Thermodynamically Consistent Phase-Field Models of Fracture: Variational Principles and Multi-Field Fe Implementations
,”
Int. J. Numer. Methods Eng.
,
83
(
10
), pp.
1273
1311
.
10.
Bourdin
,
B.
,
Francfort
,
G. A.
, and
Marigo
,
J.-J.
,
2000
, “
Numerical Experiments in Revisited Brittle Fracture
,”
J. Mech. Phys. Solids
,
48
(
4
), pp.
797
826
.
11.
Francfort
,
G. A.
, and
Marigo
,
J.-J.
,
1998
, “
Revisiting Brittle Fracture as an Energy Minimization Problem
,”
J. Mech. Phys. Solids
,
46
(
8
), pp.
1319
1342
.
12.
Creton
,
C.
,
2017
, “
50th Anniversary Perspective: Networks and Gels: Soft but Dynamic and Tough
,”
Macromolecules
,
50
(
21
), pp.
8297
8316
.
13.
Gurtin
,
M. E.
,
Fried
,
E.
, and
Anand
,
L.
,
2010
,
The Mechanics and Thermodynamics of Continua
,
Cambridge University Press
, New York.
14.
Mao
,
Y.
, and
Anand
,
L.
,
2018
, “
A Theory for Fracture of Polymeric Gels
,”
J. Phys. Mech. Solids
,
115
, pp.
30
53
.
15.
Kuhn
,
W.
, and
Grün
,
F.
,
1942
, “
Beziehungen Zwischen Elastischen Konstanten Und Dehnungs-Doppelbrechung Hochelastischer Stoffe
,”
Kolloid-Z.
,
101
(
3
), p.
248
.
16.
Doi
,
M.
,
1996
,
Introduction to Polymer Physics
,
Oxford University Press
, New York.
17.
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
.
18.
Anand
,
L.
,
1996
, “
A Constitutive Model for Compressible Elastomeric Solids
,”
Comput. Mech.
,
18
(
5
), pp.
339
355
.
19.
Schröder
,
J.
, and
Neff
,
P.
,
2003
, “
Invariant Formulation of Hyperelastic Transverse Isotropy Based on Polyconvex Free Energy Functions
,”
Int. J. Solids Struct.
,
40
(
2
), pp.
401
445
.
20.
Miehe
,
C.
,
Hofacker
,
M.
, and
Welschinger
,
F.
,
2010
, “
A Phase Field Model for Rate-Independent Crack Propagation: Robust Algorithmic Implementation Based on Operator Splits
,”
Comput. Methods Appl. Mech. Eng.
,
199
(
45–48
), pp.
2765
2778
.
21.
Gaston
,
D.
,
Newman
,
C.
,
Hansen
,
G.
, and
Lebrun
,
G. D.
,
2009
, “
Moose: A Parallel Computational Framework for Coupled Systems of Nonlinear Equations
,”
Nucl. Eng. Des.
,
239
(
10
), pp.
1768
1778
.
22.
Ayachit
,
U.
,
2015
, “
The Paraview Guide: A Parallel Visualization Application
,” Kitware, Inc., Clifton Park, NY.
23.
Zhao
,
X.
,
2014
, “
Multi-Scale Multi-Mechanism Design of Tough Hydrogels: Building Dissipation Into Stretchy Networks
,”
Soft Matter
,
10
(
5
), pp.
672
687
.
24.
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
.
25.
Mao
,
Y.
,
Lin
,
S.
,
Zhao
,
X.
, and
Anand
,
L.
,
2017
, “
A Large Deformation Viscoelastic Model for Double-Network Hydrogels
,”
J. Mech. Phys. Solids
,
100
, pp.
103
130
.
You do not currently have access to this content.