(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 .
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 .
PS tests (see e.g. ) 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
(which provides a strain
rate similar to that of the tensile tests in ).
Equibiaxial tensile tests were perfomed on unnotched square specimens (2a
= 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
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
Further details on the biaxial dynamometer and on the characterisation tests can be found in .
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
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  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
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. 
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
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