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Journal ArticleDOI

Toughness of natural rubber compounds under biaxial loading

01 Nov 2015-Engineering Fracture Mechanics (Pergamon)-Vol. 149, pp 250-261

Abstract: Strain induced molecular orientation effect on the fracture toughness of natural rubber based compounds was studied under biaxial loading conditions, using non-linear elastic fracture mechanics. The J -integral at fracture was evaluated using the finite element method. Fracture toughness can be severely influenced by strain induced molecular orientation up to a material dependent threshold, above which toughness becomes constant. The effect of the fracture phenomenology shown by two carbon black filled compounds, for which propagation is preceded by branching at the tip (sideways crack propagation), is shown to remove any significant stress concentration at the original crack tip, enhancing the apparent fracture resistance.
Topics: Fracture mechanics (69%), Crack growth resistance curve (65%), Fracture toughness (65%), Toughness (61%), Stress concentration (58%)

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Toughness of natural rubber
compounds under biaxial loading
Francesco Caimmi, Roberto Calabrò, Francesco Briatico-Vangosa, Claudia Marano, Marta Rink
Chemistry, Materials and Chemical Engineering Department Giulio Natta”, Politecnico di Milano, P.za Leonardo da
Vinci 32, I-20133 Milano, Italy
Engineering Fracture Mechanics 149 (2015) 250-261
ABSTRACT
Strain induced molecular orientation effect on the fracture toughness of natural rubber based
compounds was studied under biaxial loading conditions, using non-linear elastic fracture mechanics.
The J-integral at fracture was evaluated using the finite element method. Fracture toughness can be
severely influenced by strain induced molecular orientation up to a material dependent threshold, above
which toughness becomes constant.
The effect of the fracture phenomenology shown by two carbon black filled compounds, for which
propagation is preceded by branching at the tip (sideways crack propagation), is shown to remove any
significant stress concentration at the original crack tip, enhancing the apparent fracture resistance.
This is the post-print author version of the work.
Publisher's version of the paper is available at http://dx.doi.org/10.1016/j.engfracmech.2015.08.003
This work is licensed under Creative Commons Attribution Non-Commercial No Derivatives License
3.0

Toughness of natural rubber compounds under biaxial loading
Francesco Caimmi
a,
, Roberto Calabrò
a
, Francesco Briatico-Vangosa
a
, Claudia Marano
a
, Marta Rink
a
a
Chemistry, Materials and Chemical Engineering Department "Giulio Natta", Politecnico di Milano, P.za Leonardo da
Vinci 32, I-20133 Milano, Italy
Abstract
Strain induced molecular orientation effect on the fracture toughness of natural rubber based compounds
was studied under biaxial loading conditions, using non-linear elastic fracture mechanics. The J-integral at
fracture was evaluated using the finite element method. Fracture toughness can be severely influenced by
strain induced molecular orientation up to a material dependent threshold, above which toughness becomes
constant.
The effect of the fracture phenomenology shown by two carbon black filled compounds, for which propa-
gation is preceded by branching at the tip (sideways crack propagation), is shown to remove any significant
stress concentration at the original crack tip, enhancing the apparent fracture resistance.
Keywords: Natural Rubber Compounds, J-integral, biaxial tests
1. Introduction
It is well known that crystallising rubbers develop a strong strength anisotropy upon stretching: broadly
speaking, if stretched along some direction they can become easier to break upon subsequent stretching
perpendicularly to this direction [13]. This phenomenon is rather complicated and is linked with molecular
orientation and stress induced crystallisation which have a significant impact on fracture toughness [4].
Gent and Kim [2] tested notched rubber strips that were stretched, before testing, in a direction per-
pendicular to the testing one; their grip system allowed them to keep the pre-stretching level fixed during
testing. For crystallising rubbers, they found a very strong decrease of the tear energy of the material with
increasing elongation.
In a previous work by some of the authors of this paper, natural rubber compounds were studied by
applying a biaxial load using a cross-shaped specimen containing a central notch [3]. In those tests the
samples were first stretched in a direction parallel to the notch, in order to induce a certain level of orientation
in the material, and then the notch was opened by loading along the perpendicular direction. Different levels
of orientation were induced in the material by applying different stretch levels during the first part of the
test. The results were basically analysed in the framework of linear elastic fracture mechanics: an apparent
critical stress intensity factor was evaluated from the fracture load at fracture initiation. The authors
concluded that orientation produced a significant decrease in the resistance to crack propagation. Due to
the non-linear nature of the materials used, such an analysis was to some extent not correct; in principle
the trend observed could be apparent and caused by the use of a parameter which was not adequate to take
into account the material behaviour and large deformations.
In addition to the uncertainties introduced by the analysis, further complications were brought in by
the complex fracture phenomenology. Under certain conditions rubbers may exhibit a crack branching
phenomenon that has been variously termed in literature as “sideways crack propagation” [3] or “knotty
tearing” [2, 4, 5]. It takes place under load at the tip of a crack, where the front splits into two cracks that
Corresponding author:
Email address: francesco.caimmi@polimi.it (Francesco Caimmi)
Preprint submitted to Elsevier July 7, 2015

Nomenclature
a
0
initial crack length
B specimens thickness
C
i
constants for the description of branch shapes
J J-integral
J
c
fracture toughness
n number of terms in Ogden’s strain energy density function
p hydrostatic pressure
P
i
components of first Piola-Kirchhoff stress tensor
x, y coordinate system
U strain energy
W specimen width
δ
y
boundary displacement
γ
i
Ogden’s model material parameters (stretch exponents)
λ
1
, λ
2
, λ
3
principal stretches
λ
c
stretch level needed to suppress sideways propagation
λ
x
, λ
y
stretches
µ
i
Ogden’s model material parameter (modulus-like)
Ψ strain energy density
start to grow, more or less simultaneously, along an initial direction that is approximately perpendicular to
the initial crack plane; often these sideways cracks follow a curved path for a short distance and then stop,
due to the fact that, loosely speaking, the energy release rate associated with their propagation decreases
while they become deeper (see [6] for an approximated analysis). After sideways crack propagation, if the
load is further increased, propagation of a crack along the initial notch direction plane takes place; this
subsequent fracture event will be termed “forward propagation”, following the nomenclature proposed in [3];
often a very significant increment is needed.
Sideways cracks origin is not completely understood; it has been proposed that they may be linked with
molecular orientation in front of the notch due to opening stresses [4], which therefore creates “weak planes”
along the opening direction, but also that cavitation may play an important role in their formation because of
the peculiar stress field that develops near the tip of a crack in hyperelastic materials [7]. On a similar line of
reasoning, cavitation around carbon black particles may promote sideways crack formation [5]. Interestingly
the phenomenon is suppressed if enough pre-stretch in a direction parallel to the initial notch is applied; the
threshold level depends on the carbon black content [3]. Sideways cracks initiation and growth effectively
reduces the stress concentration at the initial notch tip, thus greatly influencing fracture toughness.
This work builds upon the previous results by Marano et al. [3] by re-analysing them, this time in the
framework of non-linear elastic fracture mechanics, hence using the critical J-integral, J
c
, as a parameter to
describe fracture resistance, as it has been widely done in literature (e.g. [6, 811]). The approach follows
closely the linear elastic fracture mechanics one: fracture is assumed to take place when the J-integral
reaches a critical value J
c
. The J-integral can be evaluated using its definition, i.e. as the flux of the
Eshelby’s tensor, if the proper stress and deformation measures are used [12]. It is worth recalling that, at
variance with the linear case, for flat and sharp crack problems involving non-linear material models and
large deformations, there is no universal solution for the asymptotic crack tip fields as in the linear case, but
the asymptotic behaviour depends on the constitutive law; a review of the available results can be found in
the work by Gao et al. [13]. Generally speaking, as in the linear case, the displacement field still follows a
power law as the distance from the crack tip tends to zero; the power law exponent however is not equal to ½
but it depends on the material model; for instance for Ogden model [14], which is the one which will be used
in this work, it depends on the largest stretch exponent in the strain energy function (see eq. 1)[15]. Usually,
considering mode I plane problems, there is only one singular Cauchy stress component, the opening one,
as, because of the finite deformations, the crack faces open deforming into some curve (the crack “blunts”).
Therefore, owing to equilibrium, at the tip of the crack the only non-vanishing stress component is the one
2

tangent to the crack faces
1
. On the other hand, the strain energy density is still singular and shows a 1/r
power-law behaviour, exactly as in the linear case, at least for all the hyperelastic models for which the
stress grows indefinitely with deformation. That is, crack blunting due to the large deformation does not
imply boundness of the stresses at the crack tip, and therefore the problem can be tackled, in principle,
by assuming an infinitesimal process zone and a one parameter description of fracture. Some limits of this
approach will be discussed in the rest of the paper.
In addition to the results on the cross shaped specimen, some tests were run using an ad-hoc developed
square shaped specimen that can be fitted in a biaxial dynamometer. Both the tests are briefly reviewed in
Sec.2.
In order to evaluate the J-integral in a non-linear setting, Ogden’s model for isotropic isocoric hyperelastic
materials [14] was chosen. The tests used to identify the material parameters are described in Section 2.2,
while Section 2.5 describes the finite element (FE) models that were developed to calculate the J-integral
in the square and cross shaped specimens. This was necessary as there are no available analytical solutions.
The results of this analysis are presented in Section 3.1.
A brief analysis of the effect of sideways crack propagation on the elastostatic fields near the crack tip,
again by using the FE method, is also presented in this work. Section 2.4 presents the methods used to
approximately derive sideways crack shape in the undeformed configuration; this shape was used in a FE
analysis to study their effect on J. The result are presented in Section 3.2.
2. Materials and methods
2.1. Materials
Natural rubber (NR0) and two carbon black filled natural rubber compounds (NR25 and NR50) were
considered in this study. The filled compounds contain 25 and 50 phr of N330 carbon black. The materials
were kindly supplied by Bridgestone T.C.E. (Roma, Italy). Sulphur content in the rubber was 1.3 phr, and
in addition stearic acid (2 phr), tert-Butyl-2-benzothiazole (0.8 phr) and Zinc oxide (3 phr) were added to
the rubbers for the vulcanisation process. Rubber sheets were compression moulded at 160
C and 8 MPa for
15 min so as to assure complete sulfur vulcanisation. Uniaxial tensile test-pieces were cut using a die from
flat sheets, while biaxial and pure shear specimens (see section 2.2) were compression moulded in specially
devised moulds. The nominal thickness of the sheets was 1 mm.
2.2. Material behaviour characterisation
It was assumed that the natural rubber compounds used in this study could be described by Ogden’s
hyperelastic law for incompressibile materials [14]. The corresponding strain energy density is given by
Ψ =
n
X
i=1
µ
i
γ
i
(λ
γ
i
1
+ λ
γ
i
2
+ λ
γ
i
3
3) , (1)
where λ
j
(j = 1, 2, 3) are the principal stretch ratios and µ
i
and γ
i
, with i = 1, . . . , n, are material parameters.
The number of terms in the sum, n, may be regarded as a material parameter as well, however, it is common
practice to fix it a priori; in this work n was arbitrarily fixed to 3. Hence a six parameter model was used.
The shear modulus in the undeformed configuration, µ, satisfies the following relationship [14]
2µ =
n
X
i=1
µ
i
γ
i
. (2)
The principal first Piola-Kirchhoff stresses are found from direct differentiation of Eq.1 as [16]:
1
Stress measures referred to the undeformed configuration usually show three singular stress components as in the linear
case, but generally different stress components present singularities with different strengths.
3

P
i
=
p
λ
i
+
U
λ
i
, (3)
(no summation on i). In the last equation, p is the hydrostatic pressure that can be determined by the stress
boundary conditions pertaining to a specific load case [16].
In order to identify the model constants, three types of tests were employed: uniaxial, pure shear (PS)
and equibiaxial tension.
Uniaxial tensile tests were run using a dumbbell test specimen as prescribed by ASTM D638 (type B-IV)
and are described elsewhere [17].
PS tests (see e.g. [18]) were run on strips 100 mm wide with a 10 mm gauge length. These tests were run
on an Instron 5800 dynamometer, at a prescribed crosshead speed of 30 mm min
1
(which provides a strain
rate similar to that of the tensile tests in [17]).
Equibiaxial tensile tests were perfomed on unnotched square specimens (2a
0
= 0 in Fig.1(a)). The
specimen edges are thicker than the rest, in order to connect the specimen to a biaxial dynamometer. Five
separate mounting positions were prepared for each side of the specimen, corresponding to the five clamps
on each side of the clamping system, by cutting the thick edge. At the end of each cut a circular hole was
cut in the specimen to reduce the stress concentration. Similar solutions have been adopted in the past for
biaxial tensile tests [19, 20]. Nevertheless it is to be noted these cuts acts as stress raiser, reducing the range
of strains for which the biaxial material behaviour can be measured, as they cause premature failure at the
edges.
The tests were run on a custom-built dynamometer; the experimental setup can be seen in Fig. 1(b).
The crosshead displacement rate was 60 mm min
1
.
Further details on the biaxial dynamometer and on the characterisation tests can be found in [21].
Here only the uniaxial nominal stress-stretch curves are shown to give an idea of the different mechanical
behaviour of the three compounds (Fig.2). As expected, the more carbon black is added, the stiffer the
resulting material.
To identify the model parameters, data from uniaxial, pure-shear and equibiaxial tests were simulta-
neously fitted; the fitting procedure and its results are discussed in [21] where it is also shown, through a
series of validation checks, that by using the identified material constants the mechanical behaviour of the
materials is adequately described. The reader is therefore referred to that work for additional details and
results. The identified parameters are given in Table 1.
As an example, the results of the fitting procedure are shown (solid lines) in Fig.2 for the uniaxial tensile
test.
2.3. Fracture Tests
To study the effect of orientation on toughness, two fracture tests were considered in this work. As
outlined in the Introduction, the fracture tests on cross shaped specimens performed by Marano et al. [3]
were analysed anew by FEM; further, new fracture tests were performed on notched square specimens
(Fig.1(a)). The reader is referred to the cited work for the details on the cross shaped specimen. Here it
will suffice to recall that the dimensions are those given in Fig. 3. The dark areas in Fig. 3 are regions were
a reinforced rubber was co-cured with the material to confer proper stiffness to the gripped region avoiding
problems with material flow from the grips or slippage; they do not contribute significantly to deformation.
It should also be recalled that the tests were run in two steps: first a pre-stretch is applied by prescribing a
fixed displacement rate to the arms parallel to the x direction, up to some value of the displacement δ
x
(Fig. 3)
while keeping the load applied along y as close as possible to zero; in the second step a fixed displacement
rate is applied to the arms parallel to the y direction while keeping the displacement of the transverse arms
fixed. Before the second step the load in the x direction was allowed to relax at a fixed displacement to an
equilibrium value, although the visco-elastic behaviour of these rubber was not pronounced (and was not
taken into account during modelling).
It is not easy to quantify into a single descriptor the molecular rubber orientation inside the material,
especially considering the complex strain distribution near the crack tip. To keep the description of the
4

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