An experimental method for estimating the tearing energy in rubber-like materials using the true stored energy

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A method for determining the critical tearing energy in rubber-like materials is proposed. In this method, the energy required for crack propagation in a rubber-like material is determined by the change of recovered elastic energy which is obtained by deducting the dissipated energy due to different inelastic processes from the total strain energy applied to the system. Hence, the classical method proposed by Rivlin and Thomas using the pure shear tear test is modified using the actual stored elastic energy. The total dissipated energy is evaluated using cyclic pure shear and simple shear dynamic experiments at the critical stretch level. To accurately estimate the total dissipated energy, the unloading rate is determined from the time the crack takes to grow an increment. A carbon-black-filled natural rubber is examined in this study. In cyclic pure shear experiment, the specimens were cyclically loaded under quasi-static loading rate of [Formula: see text] and for different unloading rates, i.e. [Formula: see text], [Formula: see text] and [Formula: see text]. The simple shear dynamic experiment is used to obtain the total dissipated energy at higher frequencies, i.e. [Formula: see text]-[Formula: see text] which corresponds to unloading rates [Formula: see text]-[Formula: see text], using the similarities between simple and pure shear deformation. The relationship between dissipated energy and unloading stretch rate is found to follow a power-law such that cyclic pure shear and simple shear dynamic experiments yield similar result. At lower unloading rates (i.e. [Formula: see text]), Mullins effect dominates and the viscous dissipation is minor, whereas at higher unloading rates, viscous dissipation becomes significant. At the crack propagation unloading rate [Formula: see text], the viscous dissipation is significant such that the amount of dissipated energy increases approximately by [Formula: see text] from the lowest unloading rate. The critical tearing energy is obtained to be [Formula: see text] using classical method and [Formula: see text] using the proposed method. Hence, the classical method overestimates the critical tearing energy by approximately [Formula: see text].

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# An experimental method for estimating the tearing energy in rubber‑like materials using the true stored energy

Elsiddig Elmukashfi

A method for determining the critical tearing energy in rubber-like materials is proposed. In this
method, the energy required for crack propagation in a rubber-like material is determined by the
change of recovered elastic energy which is obtained by deducting the dissipated energy due to
different inelastic processes from the total strain energy applied to the system. Hence, the classical
method proposed by Rivlin and Thomas using the pure shear tear test is modified using the actual
stored elastic energy. The total dissipated energy is evaluated using cyclic pure shear and simple shear
dynamic experiments at the critical stretch level. To accurately estimate the total dissipated energy,
the unloading rate is determined from the time the crack takes to grow an increment. A carbon-blackfilled natural rubber is examined in this study. In cyclic pure shear experiment, the specimens were
cyclically loaded under quasi-static loading rate of 0.01 s − 1 and for different unloading rates, i.e. 0.01,
0.1 and 1.0 s − 1 . The simple shear dynamic experiment is used to obtain the total dissipated energy
at higher frequencies, i.e. 0.5-18 Hz which corresponds to unloading rates 0.46-16.41 s − 1 , using the
similarities between simple and pure shear deformation. The relationship between dissipated energy
and unloading stretch rate is found to follow a power-law such that cyclic pure shear and simple shear
dynamic experiments yield similar result. At lower unloading rates (i.e. - ˙ U < 1.0 s − 1 ), Mullins effect
dominates and the viscous dissipation is minor, whereas at higher unloading rates, viscous dissipation
becomes significant. At the crack propagation unloading rate 125.2 s − 1 , the viscous dissipation is
significant such that the amount of dissipated energy increases approximately by 125.4% from the
lowest unloading rate. The critical tearing energy is obtained to be 7.04 kJ / m 2 using classical method
and 5.12 kJ / m 2 using the proposed method. Hence, the classical method overestimates the critical
tearing energy by approximately 37.5%.

The tearing energy, as a fracture mechanics concept, was proposed by Rivlin and Thomas 1 as an analogy to the
energy release rate 2 to study fracture in rubber and rubber-like materials. They assumed that Griffith’s approach
is valid for the case of large deformation, the irreversible changes in energy due to crack growth take place only
in the crack tip vicinity, and the change in energy is independent of the geometry. Therefore, the crack growth
is governed by the critical tearing energy criterion that is defined by

Tc = − ∂U, (1)

∂A

δc
����

where Tc is the critical tearing energy, A is the surface area of one face of the crack, U is the potential energy
stored in the system and the suffix (•)δc denotes that the differentiation is carried out at a constant displacement
δc, i.e. the external forces do not produce work. The approach was experimentally verified, concluding that the
tearing energy vs. rate of tearing relation is a fundamental material property 1,3–5 . Furthermore, the J-integral
approach 6 was later extended to rubber and rubber-like materials by Chang 7 as an alternative approach and the
critical J-value ( Jc ) was introduced as equivalence. Moreover, the critical tearing energy has been widely used
to study crack initiation and growth in other soft materials such as hydrogels 8 and fibrous biological tissues 9 .

1Department of Solid Mechanics, Royal Institute of Technology, Teknikringen 8D, 114 28 Stockholm,
Sweden. 2 Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK. email:
elsiddig.elmukashfi@eng.ox.ac.uk

Figure 1. The schematic of the pure shear tear specimen.

Several experimental techniques using different specimens were proposed for determining the critical tearing
energy. Rivlin and Thomas 1 introduced the trouser, pure shear, angled and split specimens and since then new
specimens have been continuously proposed in the literature, e.g. the single edge notch in tension (SENT) 10,
the double cantilever beam (DCB) 11,12, tensile strip test 5,13, the doubly cracked pure shear specimen (DCPS) 14
and the circumferentially-cracked cylindrical specimen (CCC) 15 . The evaluation of the critical tearing energy is
generally accomplished by the determination of the potential decrease due to crack growth, i.e. ∂U/∂A . Analytically, in specimens of simple geometries, the tearing energy is obtained from the energy balance assuming that
the decrease in potential energy is due to creation of new crack surfaces 1,11,12,14,15 . Other methods are based on
constructing a relation between the total energy stored in the system at the crack initiation and the crack length
experimentally using specimens with different initial crack length 10,16–18 . These methods assume a purely elastic
material and ignore inelastic deformation effects.

Many experimental investigations reveal that rubber and rubber-like materials experience remarkable microstructural changes during deformation such as the development of cavitation damage, breakage of filler-polymer
bonds and crystallisation 19–24 . These processes result in inelastic changes including stress softening (Mullins softening effect 25 ), hysteresis, permanent set and induced anisotropy. Thus, a significant amount of the strain energy
can be dissipated during deformation which may lead to inaccurate estimation of the critical tearing energy.
It is worth mentioning that some soft materials experience similar inelastic changes during deformation such
as hydrogels and fibrous biological tissues (e.g. see Long and Hui 8 and Humphrey 26 ). In the context of fracture
mechanics, Andrews 27 showed, in his theoretical study of an infinite inelastic lamina containing a crack, that
the inelastic deformation has a significant role in the total energy change due to crack propagation. Early studies
attempted to extend the analytical formula of the tearing energy to materials that dissipate energy by assuming
that the stored energy density available for crack extension is the recoverable energy density under the unloading
curve without exploring the unloading conditions and the self-similarity of the crack tip fields 28–30 . Recently, Qi
et al. 31 proposed theoretical and computational frameworks to study fracture toughness under steady-state crack
propagation assuming neo-Hookean solid with rate-independent hysteresis described by the Mullins effect. It is
worth mentioning that the determination of intrinsic fracture energy of rubber-like solids—the energy needed
to propagate a crack by a unit area without bulk dissipation—is still under investigation 32 . Thus, an accurate
estimation of the critical tearing energy might help in the determination of the intrinsic fracture energy.

In the present work, a new method for determining the critical tearing energy in rubber and rubber-like
materials is introduced. The critical tearing energy is determined from the actual stored elastic energy such that
the effect of dissipated energy due to the inelastic deformation is taken into account. The actual stored elastic
energy is estimated experimentally using cyclic pure shear and simple shear dynamic experiments. The unloading rate is experimentally obtained from the time the crack takes to grow an increment. This method can also
be used in other soft materials that experience energy dissipation during loading (i.e. manifested in the form of
hysteresis) such as hydrogels and fibrous biological tissues.

Analysis of crack propagation in pure shear tear test
Rivlin and Thomas 1 proposed the pure shear tear test for characterising the tearing energy in rubber-like materials which are assumed to exhibit purely elastic behaviour. In this setup, a pre-cracked pure shear specimen
with a low ratio between the height and width is used. A typical specimen is illustrated in Fig. 1 in which the
undeformed width, height, and thickness of the specimen are denoted by W0, H0 and B0, respectively, and the
crack length by a . The loading is defined by the displacement δL and its rate δ ˙ L on the boundary where the applied
force is F . Moreover, the critical tearing energy is generally estimated under quasi-static loading conditions (i.e.

δ˙L → 0 ) in which the loading rate effects are minimal.

The classical method. In order to study the crack propagation, the specimen is divided into four different
regions, based on the deformation state: ( _i_ ) region A behind the crack tip in which the material is unloaded, ( _ii_ )
region B is in a state of pure shear deformation, ( _iii_ ) region C, between regions A and B, is in a complicated state
of deformation, and ( _iv_ ) region D is between the pure shear region and the traction-free edge. The propagation of

_F_

Ψ _V_ 0

_δ_ c

_a_ = 0

_a >_ 0

Crack
initiation

_δ_

Figure 2. The schematic of the load-displacement curve: the shaded area is the strain energy stored in the
uncracked specimen at the critical displacement δc.

the crack is assumed to take place at a fixed separation between the clamps, i.e. δL = const., and is seen as a shift
of region C in the direction of the propagation. Consequently, region A will increase while region B will decrease
by the same amount. Hence, the propagation of the crack by da (measured in the undeformed configuration) is a
process of unloading a volume of H0 B0 da from the pure shear deformation to the undeformed state. The change
of the potential energy dU in the specimen is defined as

dU = U(a + da) − U(a) = −H0 B0 da �, (2)

where � is the elastically stored energy per unit referential volume of the material in a state of pure shear at the
critical displacement δL = δc (i.e. at which the crack propagation takes place). It should be noted that the change
in the total energy is equal to the change in the elastic stored energy in the case of purely elastic materials. Additionally, the stress-strain state in the vicinity of the crack tip in region C is taken to be self-similar during crack
initiation as da → 0 . Therefore, using Eq. (1), the critical tearing energy can be determined as

Tc = � H0. (3)

The load-displacement relation of a pure shear test under quasi-static loading conditions is assumed to
take the general form F = F�δL, a� . Therefore, the stored energy per unit referential volume � is obtained by

graphical integration under the load-displacement curve of an uncracked pure shear specimen of the material take the general form F = F�δL, a� . Therefore, the stored energy per unit referential volume � is obtained by
( a = 0 ), see Fig. 2, as

�δL, a

= 1

- δc = 1 F δL, 0 dδL, (4)

V0
 -  - 0  -

�δL, 0

- dδL,

δc

where V0 = W0H0B0 is the volume of the specimen in the reference configuration. Strictly speaking, this relation
is valid under the assumption of an ideal rubber-like solid in which the deformation is assumed to be purely
elastic and the load-displacement relation is reversible.

The modified method. The load-displacement relation for a non-ideal rubber-like material in the pure
shear test is irreversible and may depend on the unloading rate 33, see Fig. 3. The total change in the total internal
energy per unit reference volume of the uncracked pure shear specimen, E ˙, can be divided into the change in
the _elastic_ free energy per unit reference volume � ˙, _heat_ and _dissipation_ energy per unit reference volume Q ˙, and
_free_ energy in other forms per unit reference volume � ˙ ′ (e.g. free energy stored as surface energy between the
amorphous and crystalline phase during the strain-induced crystallisation):

E˙ = �˙ + Q˙ + �˙ ′ . (5)

In Fig. 4, the elastically stored energy is the area under the unloading curve and the area between the loading
and unloading curves is associated with Q and � ′ . Hence, the actual change in the stored energy due to crack
propagation in a pure shear tear specimen is associated with the true elastic energy � rather than the assumed
elastic energy.

and The irreversible load-displacement relation in a pure shear test can be expressed as δ ˙ are the displacement and its rate; and κ are some internal variables that describe the different inelastic pro-F = F�δ, δ ˙, a; κ�, where δ
cesses and determine the changes in the load-displacement behaviour. The internal variables, κ, are history and

The irreversible load-displacement relation in a pure shear test can be expressed as F = F

�δ, δ ˙, a; κ

_F_

_δ_ ˙U

_δ_

Figure 3. The schematic of the load-displacement curve of a non ideal rubber-like material. The inelastic effects
are demonstrated by the irreversibility of the loading-unloading behaviour for different unloading rates δ ˙ U.

_F_

_Q_ + Ψ _V_ 0

Ψ _V_ 0

_δ_ p _δ_ c

_a_ = 0

_a >_ 0

Crack
initiation

_δ_

Figure 4. The schematic of the load–displacement curve for a non ideal rubber-like material. The lighter
shaded area is the elastic stored energy, � V0, and the darker shaded area is the summation of the heat and
dissipation energy and the free energy in other forms, �Q + � ′� V0, in the uncracked specimen at the critical

displacement dissipation energy and the free energy in other forms, δc . δp is the permanent deformation after unloading.�Q + � ′� V0, in the uncracked specimen at the critical

�Q + � ′�

rate dependent; and can be determined by a set of evolution laws that may take the general form κ˙ = κ˙ (δ, δ ˙, �, κ),
where � is the absolute temperature. The actual elastic energy is then obtained by

= 1

- δc, δ ˙ U = 1 F δU, δ ˙ U, 0; κ dδU, (6)

V0
 -  -  -
δc

- dδU,

δp

δc

�δU, δ ˙ U, 0; κ

and the summation of the heat and dissipation energy becomes

Q + � ′ = 1 F δ, δ ˙, 0; κ dδ, (7)

- dδ,

�δ, δ ˙, 0; κ

δp

where δp is the permanent deformation after the pure shear specimen is completely unloaded and κ account
for the evolution of inelastic effects during loading and unloading. It should be noted that the forms of the
load-displacement and internal variables relations are out of the scope of the current study and we use direct

NR CB Plasticizer Additives Shore A

100 54 13 19 50

Table 1. The mix formulation in parts per hundred rubber by weight (phr) and Shore A hardness of carbonblack-filled natural rubber (NR3233).

experimental measurements. Further, it is not an easy task to obtain the exact unloading rate of region B during
crack growth. Therefore, the unloading rate effect on the recovered elastic energy will be investigated.

The unloading rate can directly be estimated from the experimentally measured crack growth rate. Thus,
denoting the time taken by the crack to grow an increment �a by tc, the average unloading rate due crack growth
is determined as

�˙ U = �c − �p, (8)

where � ˙ U = δ ˙ U/H0 is the unloading stretch rate, �c = 1 + δc/H0 is the critical stretch and �p = 1 + δp/H0 is
the permanent stretch.

The actual elastic and; heat and dissipation energies, can be estimated using different experiments. One
method is to use _cyclic pure-shear experiment_, see Fig. 4. Here, uncracked pure-shear specimens are cyclically
loaded from the undeforemed state (i.e. δ = 0 ) to the critical displacement (i.e. δ = δc ) and then are unloaded to
zero force (i.e. F = 0 and δ = δp ). The loading and unloading rates are set to be equal to the rates in the case of
cracked pure-shear specimen. It follows that the actual elastic and; heat and dissipation energies are determined
graphically from the load-displacement response as in Eqs. (6) and (7), respectively. Another method is to use the
_dynamic simple shear experiment_ in which the storage and loss shear moduli data, i.e. G ′ and G ′′, respectively, are
determined for a wide range of frequencies. Typically, a harmonic shear strain is applied, i.e. γ (t) = γa cos (ω t),
where γa is the amplitude and ω is the frequency in rad/s, which results in a harmonic shear stress with a phase
shift τ(t) = τa cos (ω t + φ), where τa is the shear stress amplitude and φ is the equivalent loss angle. Hence, the
storage and loss moduli are determined from G ′ = τa/γa and G ′′ = G ′ tan(φ), respectively. The dissipated energy
during half a cycle (i.e. includes one loading and unloading path) at a given frequency can then be expressed by

Q + � ′ = 1 τ dγ = 1 (9)

- 2 π τa γa tan(φ).

- τ dγ = 1 2

1

2 π τa γa tan(φ).

The simple shear is assumed to differ from pure shear only by a rotation 34, therefore, the test is equivalent to
pure shear test taking

γa = � - - 1 , and τa = - 2 + 1 σ , (10)

where � and σ are the stretch and Cauchy stress in pure shear. Additionally, the relationship between shear
strain and stretch rates can directly be obtained by differentiation of Eq. (10) with respect to time. It should be
mentioned that the response of simple and pure shear can be equivalent for small range of stretch in the large
deformation regime 35, i.e. � ≤ 1.4 . Additionally, the test loading frequency should be similar to the loading and
unloading rates of the cracked pure-shear specimen. Thus, the knowledge of the dissipated energy and the total
energy from the cracked pure-shear specimen test allows to obtain the actual elastic energy using Eq. (5).

Experimental studies of the deformation field of a stationary crack in rubber-like materials suggest that
the deformation field near the crack tip can be characterised by three regions: ( _i_ ) closest to the crack tip where
deformation is significantly large and damage takes place; ( _ii_ ) the interim region that contains inhomogeneous
moderate deformation; and ( _iii_ ) far from the crack tip which is characterised by homogenous moderate remote
deformation 32,36–38 . Therefore, for a very small crack growth increment, the dissipation in the bulk due to unloading is predominantly taking place in the far region. Additionally, self-similarity of the crack tip fields is valid if
the increment is very small in comparison with the specimen dimensions, i.e. da/H0 → 0.

Experimental work
A carbon-black-filled natural rubber material is investigated in this experimental study. The material is manufactured by TrelleborgVibracoustic under the designation NR3233 and its chemical properties are listed in
Table 1. Two types of specimens have been used in this investigation, i.e. the uncracked and cracked pure-shear
specimens. The specimens were of width W0 = 110 mm, height H0 = 30 mm, and thickness B0 = 2.5 mm .
In the cracked pure-shear specimens, initial cracks of length a = 30 mm were created using razor blades. A
standard servo-hydraulic test machine of load capacity 50 KN, was used and the different tests were performed
at ambient temperature between 22-25 ◦ C and relative humidity 60% . Additionally, a high speed camera at up to
7000 frames/s was used to detect the crack propagation onset and to follow the progress of the crack and later
a post-processor was used to obtain the crack trajectory and velocity. The experiment setup is shown in Fig. 5.

Four cracked pure-shear specimens were monotonically loaded at cross head speed of 0.3 mm/s until complete failure. It is worth noting that, at this loading rate, the material shows very limited rate sensitivity. The
load-displacement graphs were recorded and the crack growth points were marked during the test, and the
critical displacement was determined. The uncracked pure-shear specimens were subjected to a cycle of loading

Figure 5. The experimental setup: (a) the loading machine, the cracked pure shear specimen and the high
speed camera; (b) the cracked pure shear specimen, the loading direction and the experiment frame.

and unloading. They were monotonically loaded until the critical displacement, i.e. obtained from the cracked
pure-shear specimen, was reached and then they were unloaded completely. The loading cross head speed was
kept at 0.3 mm/s ( 0.01 s −1 ), as in the case of the cracked specimens. During unloading, the cross head speed was
varied to investigate the effect of the unloading rate. Therefore, three unloading cross head speeds were used,
i.e. 0.3, 3.0 and 30.0 mm/s ( 0.1, 0.1, and 1.0 s −1 ). Three specimens per unloading rate were tested such that nine
specimens were used in total. In addition to these experiments, dynamic simple shear experimental data are
available in Österlöf et al. 39 for the same material.

Results and discussion
Figure 6 shows the load-displacement records of four cracked pure-shear specimens in which the solid circles
are the crack initiation points. The average critical displacement is determined to be δ ¯ c = 13.131 mm with a
standard deviation of 1.897 mm which corresponds to the average stretch � ¯ c = 1.44.

Figure 7 illustrates the relationship between the crack extension and time at the onset of propagation. The
result shows that the crack propagates in a stick-slip fashion with average increment of �a ≈ 0.6 mm . The average crack velocity is determined to be a˙ = 52 mm/s with a standard deviation of 2.5 mm/s . Thus, the average
crack propagation time over the increment is tc ≈ �a/a˙ = 11.5 ms . The average unloading rate due crack growth
is determined as � ˙ U = ��¯ c −¯�p�/tc ≈ 125.2 s −1 where � ¯ p ≈ 1 is the average permanent stretch that will be

is determined as � ˙ U = ��¯ c −¯�p�/tc ≈ 125.2 s −1 where � ¯ p ≈ 1 is the average permanent stretch that will be

discussed in the next paragraph.

Typical load-displacement records for an uncracked pure-shear specimen are shown in Fig. 8, wherein the
specimens are loaded under controlled deformation until the average critical displacement δ ¯ c and then unloaded.
The material shows nonlinear large deformation behaviour and significant energy dissipation. The average permanent deformation is determined to be δ ¯ p = 0.4, 0.66 and 0.66 mm for the unloading rates 0.01, 0.01 and 1.0 s −1,
respectively. Therefore, the result suggests that the permanent stretch is insignificant, i.e. � ¯ p = 1 + δ ¯ p/H0 ≈ 1.

The total energy E and actual elastic energy � per unit volume were evaluated numerically using the load–displacement records of the uncracked pure-shear specimens at different unloading rates. The critical tearing energy
was then determined using the classical and proposed methods using Eqs. (4) and (6), respectively; together with
Eq (3). Using the classical method, the critical tearing energy is found to be T ¯ c = 7.04 kJ/m 2 with a standard
deviation of 0.43 kJ/m 2 . Table 2 shows the critical tearing energies using the true stored elastic energy at different

��¯ c −¯�p

Figure 6. The load-displacement curves of four cracked pure-shear specimens where the solid circles denote
the crack initiation points.

Figure 7. The crack extension vs time.

unloading rates � ˙ U . The result suggests that the recovered elastic energy weakly depends on the unloading rate for
the given experimental range ( � ˙ U ≤ 1.0 s −1 ) which indicates that Mullins effect is the dominant contributor. The
experimental unloading rate falls outside the experimental unloading rates which requires further investigates.

To determine the dissipation at the experimental unloading rate, we consider the storage and loss shear
moduli data available in Österlöf et al. 39 for the same material and obtained from simple shear experiment
with a harmonic excitation. Hence, the loss angle φ is obtained from the storage and loss shear moduli data 39,
i.e. φ = arctan �G ′′ /G ′� . Fig. 9 shows the loss angle as a function of the strain amplitude and for different fre

quencies. The result implies that the dissipation depends on the strain amplitude and frequency. At lower and i.e. φ = arctan �G ′′ /G ′� . Fig. 9 shows the loss angle as a function of the strain amplitude and for different frehigher strain amplitudes, the dissipation weakly depends on the frequency which suggests that the dissipation
is mainly due to Mullins effect and the viscous dissipation is a minor part. At intermediate strain amplitudes,
the viscous dissipation appears to dominate. The equivalent shear strain for the critical stretch �c is calculated
using Eq. (10) to be γa ≈ 0.75 . The shear stress is then calculated using the storage modulus, i.e. τa = G ′ γa, and

�G ′′ /G ′�

Figure 8. The load–displacement curves of uncracked pure-shear specimens at loading rate of 0.3 mm/s
( 0.01 s −1 ) and different unloading rates. The red, blue and green lines represent the unloading rates of 0.3, 3.0,
and 30.0 mm/s (i.e. 0.01, 0.1, and 1.0 s −1 ), respectively, and fill areas represent the experiment scatter (error
bars). The unloading point is taken at δ ¯ c = 13.131 mm.

�˙ U [s − 1 ] T ¯ c [kJ / m 2 ] σ˜ Tc [kJ / m 2 ]

0.01 6.17 0.49

0.1 6.00 0.54

1.0 5.82 0.48

Table 2. The average critical tearing energy T ¯ c and its standard deviation σ˜Tc using the true stored elastic
energy for different unloading rates � ˙ U from the cyclic pure shear experiment.

Figure 9. The relationship between the tangent of the loss angle and the shear strain amplitude for different
frequencies using simple shear dynamic experiment 39 .

Pure Shear
Simple Shear
Fit

20
10 _−_ 3 10 _−_ 2 10 _−_ 1 10 0 10 1 10 2 10 3

_λ_ ˙ U [1 _/_ s]

Figure 10. The relationship between the total dissipation and the unloading rate for pure shear and simple
shear dynamic experiments.

the tangent of the equivalent loss angle is obtained from Fig. 9. Thus, for γa ≈ 0.75, the dissipation can be estimated using Eq. (9) for the range of frequencies. The equivalent average unloading stretch rate can be obtained

periodic time. The unloading rates for the frequency range by � ˙ U = 4��c − �p�/T, where T = 2 π/ω is the periodic time such that the unloading path takes quarter the 0.5-18 Hz is obtained to be 0.46-16.41 s −1 . It is worth
noting that the shear strain rate varies during the simple shear dynamic test and we limit our considerations
to the average stretch rate. Additionally, direct differentiation of Eq. (10) with respect to time yields similar
stretch rates. Figure 10 illustrates the total dissipated energy as a function of the unloading stretch rate. The
result implies that cyclic pure shear and simple shear dynamic tests yield similar outcome where the amount of
dissipated energy increases with the unloading rate. The rate sensitivity of the dissipation is due to the increase
of viscous contribution. The critical tearing energy as a function of the unloading rate is then calculated using
Eq. (3). Figure 11 shows the relationship between critical tearing energy and unloading rate wherein the difference between the classical and proposed methods increases with the increase of unloading rate due to the
increase in viscous dissipation. For the crack propagation unloading rate 125.2 s −1, using the fitting in Fig. 10,
the dissipated energy is 64.23 kJ/m 3 which corresponds to 125.4% increase from the lowest unloading rate. Thus,
using the fitting in Fig. 11, the average tearing energy becomes 5.12 kJ/m 2 which is 37.5% less than the tearing
energy estimated using the classical method.

Conclusions
In conclusion, a modified method for estimation of the critical tearing energy in rubber-like solids has been
presented. The method is a modification of the classical method proposed by Rivlin and Thomas 1 using the pure
shear tear test. In this method, the total energy stored in a rubber-like material is divided into elastic and inelastic
contributions taking into account the different inelastic processes. Hence, the energy required for crack propagation is determined by the change of the elastically stored energy only rather that the total energy in the case
of the classical method. The true elastic stored energy is determined at unloading rate obtained experimentally
from the time the crack takes to grow an increment. A carbon-black-filled natural rubber material is experimentally investigated. The crack growth average increment is significantly smaller than the specimen’s dimensions,
i.e. �a/a = �a/H0 ≈ 0.02, which suggests that the self-similarity of the crack field can be a valid assumption
(experimentally observed damage zone in rubber-like materials is about 20% of the crack length 36,37 ). The total
dissipated energy is evaluated using cyclic pure shear experiment and simple shear dynamic data available in
Österlöf et al. 39 at the critical stretch level. A power-law relationship between dissipated energy and unloading
stretch rate is determined using the cyclic pure shear and simple shear dynamic experiments. The result implies
that the dissipated energy can be estimated using cyclic pure shear and simple shear dynamic tests. Strictly
speaking, to minimise the uncertainty in estimating the total dissipation, it should be obtained using the loading and unloading rates of the cracked pure shear specimen. The analysis of dissipation suggests that, at the
critical stretch level, Mullins effect and viscous dissipation contribute to the total dissipation with the viscous
contribution increases with the increase of unloading rate. For the crack propagation unloading rate 125.2 s −1,
the viscous dissipation is significant such that the total dissipation is ≈ 64.23 kJ/m 3 (i.e. 125.4% increase in the
total dissipation from the lowest unloading rate). The average critical tearing energy becomes 5.12 kJ/m 2 which

Pure Shear
Simple Shear
Proposed Method
Classical Method

2
10 _−_ 3 10 _−_ 2 10 _−_ 1 10 0 10 1 10 2 10 3

_λ_ ˙ U [1 _/_ s]

Figure 11. The relationship between the critical tearing energy and the unloading rate for pure shear and
simple shear dynamic experiments using the proposed method where the dashed black line is the fitting. The
red chain line represents the average tearing energy using the classical method and fill area represents the
experiment scatter (error bar).

is 37.5% less than the tearing energy estimated using the classical method. It is worth mentioning that more
accurate estimation of the critical tearing energy should be done by considering experiments at the unloading rate
125.2 s −1 . However, the results presented here give strong indication of the overestimation in the critical tearing
energy due to the excellent correlation between the unloading rate and dissipated energy in Fig. 10. Therefore,
to accurately measure the critical tearing energy, the recovered elastic energy should be experimentally measured using cyclic pure shear and simple shear dynamic tests which depends on the material, loading rate, crack
propagation unloading rate and the critical stretch �c . The unloading rate should experimentally be determined
for the time the crack takes to propagate an increment. Additionally, the simple shear dynamic experiment offers
excellent alternative for determination of the dissipation at different frequencies (unloading rates). The proposed
method should be used to study other soft materials that experience energy dissipation during loading such as
hydrogels and fibrous biological tissues. Additionally, similar approach can be adopted to estimate the actual
tearing in specimens of simple geometries such as the trouser and single edge notch in tension (SENT) specimens.

Received: 10 January 2021; Accepted: 6 July 2021

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Acknowledgements
The author is grateful for valuable discussions with Professor Martin Kroon. The author would like also to
thank Dr. Rickard Österlöf for providing the material, Mr. Kurt Lindqvist and Mr. Göran Rådberg for helping
in manufacturing the specimens, and M.Sc. Martin Öberg and Dr. Irene Arregui for their assistance during the
experimental realisation of this work.

Author contributions
The author has formulated the theory, performed the experiment, interpreted the results and written the
manuscript.

Funding
Open access funding provided by Royal Institute of Technology (KTH).

Competing interests
The author declares no competing interests.

Additional information
Correspondence and requests for materials should be addressed to E.E.

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