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9
th
International Congress of Croatian Society of Mechanics
18-22 September 2018
Split, Croatia
On characterising fracture resistance in mode-I
delamination
Leo ŠKEC
*
, Giulio ALFANO
+
, Gordan JELENIĆ
*
*
University of Rijeka, Faculty of Civil Engineering, Radmile Matejčić 3, 51000 Rijeka,
Croatia
E-mails: {leo.skec, gordan.jelenic}@uniri.hr
+
Brunel University London, Kingston Lane, UB8 3PH, Uxbridge, UK
E-mail: giulio.alfano@brunel.ac.uk
Abstract. In this work we focus on the mode-I quasi-static crack propagation in
adhesive joints or composite laminates. For this problems a number of di
ff
erent
standards have been approved. The most widely used are based on the double
cantilever beam (DCB) test and on linear elastic fracture mechanics (LEFM) but
di
ff
er in some aspects of the testing procedure and the recommended data-reduction
schemes. The applicability of these methods is still a matter of debate in the
scientific community, particularly in the case of ductile interfaces. We revisit the
accuracy of the most used standards and compare it with other methods based on
either LEFM or J-integral theory. All the methods analysed in our work are based
on either Euler-Bernoulli or Timoshenko beam theories. We present a number of
numerical examples where we compare di
ff
erent expressions for fracture resistance
obtained with di
ff
erent methods. The input for the analysis, which includes applied
load, cross-head displacement and rotation, crack length and cohesive zone length,
is obtained from the numerical model which simulates real experiments. In these
simulations, we use a Timoshenko beam model with a bi-linear CZM, which allows
us accurate comparison with analytical formulas for fracture resistance based on
Euler-Bernoulli and Timoshenko beam theory.
1 Introduction
In recent years, the use of linear elastic fracture mechanics (LEFM) for the
experimental determination of the fracture resistance during adhesive joint debonding or
composite delamination in presence of `large-scale' fracture processes, has been
seriously questioned. In general, it is widely accepted that “LEFM is not applicable to
those specimens containing large fracture process zone around the delamination front”
[1]. Instead, there is a general consensus that, in presence of large process zones, J-
integral theory [2] provides a more accurate framework to determine the fracture
resistance [3].
Because for the aforementioned types of problems, a very accurate characterisation
of the fracture cannot be obtained using a one-parameter fracture-mechanics theory, the
richer modelling framework of cohesive-zone models (CZMs) is an alternative that is
usually considered [4]. In this work we will consider the critical energy release rate, ܩ
,
the critical value of the J integral, ܬ
, and the area under the traction separation law Ω as
candidates to characterise fracture resistance by a single energy value. Because, in the
general case, Ω is the only parameter that can be considered as an interface property, we
2
will assess the accuracy of methods based on either ܩ
or ܬ
by evaluating how closely
they predict Ω.
When it comes to mode-I delamination or debonding, today we have many different
standard procedures for determining the fracture resistance of adhesive joints and
composite laminates [5], [6]. Due to its simple geometry and a rather simple testing
procedure used, the double cantilever beam (DCB) is the most commonly used
specimen in all the standards. All standards use analytical formulae to compute ܩ
based
on LEFM and simple beam theories (Euler-Bernoulli or Timoshenko), where it is
assumed that the DCB arms are clamped at the crack tip. Furthermore, in order to
compute ܩ
, the measurement of the crack length, which is usually done optically by
means of a travelling microscope or a high-resolution camera, is required. However,
determining the exact position of the crack tip is extremely difficult and time-
consuming, and it can introduce significant uncertainty in the determination of ܩ
.
Although formulae for ܩ
that do not require measurement of the crack length have been
already proposed in the literature [7], [8], they do not take into account the difference
between the actual crack length, ܽ, and the equivalent crack length, ܽ
, which is the
length that makes simple beam deflection formulae valid for a measured pair of force
and displacement.
In this work we first discuss the difference between ܩ
, ܬ
and Ω in Section 2. Then,
for the case of a DCB with prescribed displacement, in Section 3 we derive the correct
formula for ܩ
. Using the numerical data produced from ‘virtual’ experiments, the
accuracy of different formulae for ܩ
and ܬ
is compared in Section 4. Finally, the most
important conclusions are presented in Section 5. This work is a brief summary of [9],
where a more detailed discussion about the J integral, the case of DCB with prescribed
rotations and additional numerical comparisons can be found.
2 Relationship between ࡳ
ࢉ
, ࡶ
ࢉ
and ષ
Let us consider the case of a DCB specimen that is modelled as a 2D solid pinned in
two points, subject to a monotonically increasing prescribed (opening) displacement, ݒ,
on one of its pinned ends as shown in Figure 1. An initial crack of length ܽ
is assumed
to be present in a plane of geometric, material and loading symmetry of the body so that
crack propagates in pure mode I.
Figure 1: Geometry of a DCB with prescribed displacement
The interface of the DCB considered is modelled with the CZM shown in Figure 2(a)
where the behaviour is linear elastic with progressive damage. As clarified in Figure
2(a), ߜ
and ߜ
are the limit values of the relative displacement, ߜ, at the interface,
whereas ߪ
௫
is the limit value of the contact traction ߪ. It is very important to notice
3
that in a DCB new damage dissipation occurs only ahead of the crack tip on the part of
the interface where ߜ
< ߜ ≤ߜ
. Once ߜ
is reached all the energy,
Ω, is dissipated
(see Figure 2(b)).
Figure 2: DCB with a bi-linear CZM with progressive failure: traction separation law for (a) a
partially damaged point and (b) a fully damaged point, and (c) the associated structural response
When the body with an initial crack ܽ
reaches the force ܨ
and the displacement ݒ
,
which is the point at which ߜ reaches ߜ
at the initial crack tip, the triangle OAA'O
represents the part of the external work stored as elastic energy, which in this case is the
total potential energy Π
, whereas the part OAO (where OA is a curve and AO a straight
line) is energy dissipated due to damage at the interface ahead of the crack tip, and will
be here denoted by Π
.
. As soon as at the crack tip ߜ =ߜ
, the crack will start
propagating if the prescribed displacement is further increased. In other words, when the
prescribed displacement is ݒ
>ݒ
, the crack length will become ܽ
+Δܽ, while the
force decreases from ܨ
to ܨ
. The potential energy stored at point B, Π
, is the area of
the triangle OBB'O.
The unloaded body from point B would not be the same as one with an initial crack
equal to ܽ
+Δܽ, because the latter would have no damage developed ahead of the
crack tip. The area OBO (where OB is a curve and BO is a straight line), shaded in
yellow in Figure 2(c), represents the energy dissipated ahead of the crack tip at point B
when the linear elastic CZM with progressive failure is used, and is therefore denoted by
Π
.
. Grey area in Figure 2(c) represents the energy dissipated on a newly created crack
surface. In general, at all times during crack propagation the total (nonlinear) potential
energy, Π
ே
, can be written as
Π
ே
=Π+Π
(1)
where, as discussed above, Π is the energy stored, while Π
ே
is the energy dissipated
ahead of the crack tip. As shown in [9], we can derive the following relation:
4
Ω= −
ଵ
பஈ
ವ
డ
−
ଵ
ୢஈ
ವ
ୢ
=ܩ
−
ଵ
ୢஈ
ವ
ୢ
, (2)
or, for the case of an homogeneous material, where
Ω = ܬ
,
ܬ
=ܩ
−
ଵ
ୢஈ
ವ
ୢ
. (3)
The above results show that the difference between ܩ
and Ω, and between ܩ
and ܬ
for a homogeneous interface, is not to be attributed to the size of the cohesive zone, but
to the variation of the amount of energy already dissipated ahead of the crack tip during
crack propagation. In other words, if the profile of the specific energy dissipated ahead
of the crack tip remains unaltered during crack propagation, and therefore translates in a
steady-state fashion together with the crack tip, then ܩ
=ܬ
=Ω. It is worth
mentioning, that in [9] it has been shown that for a case of a non-homogeneous interface
(which can be modelled as variable Ω over the interface), ܬ
≠Ω.
3 Determination of ࡳ
ࢉ
for a DCB with prescribed displacement
We will assume that the DCB (as shown in Figure 1) is a 2D body undergoing an
isothermal, quasi-static, rate-independent deformation process, where the length, width
and depth of each arm are denoted by ܮ, ܾ and ℎ, respectively, with ℎ ≪ ܮ. We will also
assume that strains, displacements and rotations are sufficiently small so that a
geometrically linear beam model is sufficiently effective. Because the behaviour is
linear elastic with damage, using geometrically linear beam theories, the total potential
energy is given by
Πሺݒ,ܽሻ=
ி ௩
ଶ
, (4)
where ܨ = ܨሺݒ,ܽሻ is the reaction force.
ܨ, ݒ and ܽ can be related using formulae from simple beam theories, but we have to
be aware that the crack length in these formulae is not the actual crack length ܽ, but an
equivalent crack length, which will be denoted by ܽ
.ா
or ܽ
.்
to specify that Euler-
Bernoulli or Timoshenko beam theory is considered, respectively. Beam deflection
formulae are based on the assumption that the arms of the DCB are clamped at the crack
tip, but cross sections at the crack tip are normally characterised by both a displacement
and a rotation as a result of the deformation of the interface and of the beam in front of
the crack tip. Therefore, the equivalent crack length is defined as the length that the
crack should have to make the formulae correct if the arms were really clamped at the
crack tip, for given values of ܨ and ݒ.
For Euler-Bernoulli beam theory we have
ܽ
.ா
=ට
ଷ ௩ ாூ
ଶ ி
య
, (5)
where ܧܫ is the bending stiffness of a single DCB arm. Because the total potential
energy can be now expressed as
Πሺݒ,ܽሻ=Π
ா
ሺݒ,ܽ
.ா
ሺܽሻሻ, (8)
5
we can derive
ܩ
=−
ଵ
డஈ
డ
=−
ଵ
డஈ
ಶ
డ
.ಶ
డ
.ಶ
డ
, (9)
which can be written as
ܩ
=ܩ
ாడ
.ಶ
డ
, (10)
where
ܩ
ா
=
ி
మ
.ಶ
మ
ாூ
. (11)
The derivative
ୢ
.ಶ
ୢ
in Equation (10) also defines how close to being steady state the
crack propagation is. Therefore, for
ୢ
.ಶ
ୢ
=1, we have ܩ
=ܬ
=Ω, which, as shown
in [9], is the case for a DCB with prescribed rotations. Analogous procedure can be
applied also for Timoshenko beam theory, as shown in [9].
4 Numerical Examples
By using a DCB numerical model [10] consisting of Timoshenko beam finite
elements with an interface with a bi-linear CZM (see Figure 2(a)), we created ‘virtual’
experimental data (ܨ, ݒ and ܽ), which are then used in various formulae for ܩ
and ܬ
. In
our numerical model Ω is a known input value. By keeping Ω constant and changing
ߪ
௫
(and ߜ
), we can obtain a range of behaviours at the interface, which vary from an
extremely brittle one to a extremely ductile one (with a relatively large damage process
zone). All the data used in the numerical examples are given in Tables 1 and 2, where ܧ
represents the Young’s modulus, ߥ is Poisson’s ratio (shear stiffness is computed as ߤ =
ா
ଶሺଵାఔሻ
) and ݇
௦
is the shear correction coefficient.
Table 1: Geometric data used in the virtual experiments
ܮ
ℎ
ܾ
ܽ
[mm]
[mm]
[mm]
[mm]
200 6 25 25
Table 2: Material data used in the virtual experiments
ܧ
ߥ
݇
௦
Ω
ߪ
݉ܽݔ
ߜ
ܿ
ߜ
0
[GPa]
[-] [-] [N/mm]
[MPa] [mm] [mm]
70 1/3
5/6
1 {7.5, 15, 30, 60, 120}
2
Ω
/
ߪ
݉ܽݔ
0
.
01
ߜ
In Figure 3, values of ܩ
ா
(as defined in Equation (11)) and dܽ
.ா
/dܽ are given for
different values of ߪ
௫
. It can be noted that as ߪ
௫
increases the behaviour at the
interface becomes more brittle and the crack propagation becomes extremely close to
being steady state. In the limit case (ߪ
௫
→∞),
ୢ
.ಶ
ୢ
=1 and ܩ
ா
=ܩ
=Ω.
However, even for the most ductile case (ߪ
௫
=7.5 MPa), we can see that the crack
6
propagation is very close to being steady state and ܩ
ா
is indeed a very good
approximation of Ω. Moreover, formula (11) for ܩ
ா
does not require the measurement
of the crack length.
Figure 3: Values of (a) ܩ
ா
/Ω and (b) dܽ
.ா
/dܽ for different values of ߪ
݉ܽݔ
By multiplying ܩ
ா
and dܽ
.ா
/dܽ, i.e. Figures 3(a) and 3(b), we get the actual value
of ܩ
. In Figure 4(a) we can see that the same values of ܩ
are obtained regardless the
beam theory used (EBT stands for Euler-Bernoulli, while TBT for Timoshenko beam
theory). We can also see that as ߪ
௫
→∞, ܩ
→Ω. In Figure 4(b), a comparison is
made between different formulae for ܩ
and ܬ
, where, according to [9],
ܬ
்
=
ி
మ
ቀ
ଵ
ఓ
ೞ
−
ଶ ఏ
ೡ
ி
ቁ, (12)
is the critical value of the J integral with shear deformability of the arms taken into
account. ߠ
௩
represents the rotation of the DCB arm at the point where the displacement
ݒ is prescribed. For Euler-Bernoulli beam theory, ܬ
ா
is obtained by letting ߤܣ
௦
→∞ in
Equation (12). It can be noted that, for ߠ
௩
obtained from a virtual experiment, Equation
(12) for ܬ
்
is the only formula capable of giving the input value of Ω. ܩ
and Ω
are
values of ܩ
and Ω computed using the areas under the ܨ −ݒ diagram, as discussed in
Section 2. We can appreciate that the accuracy of all formulas presented in Figure 4(b)
is very high, considering that the behaviour of the interface is extremely ductile.
5 Conclusions
In this work, for a case of a mode-I delamination in a DCB we derived the
relationship between the critical energy released rate, ܩ
, the critical value of the J
integral, ܬ
, and the area under the traction-separation law of a CZM, Ω. We showed
that, their difference is not to be attributed to the size of the damage process zone, but to
how close to being steady state crack propagation is. As shown in Section 4, even for
relatively large damage process zones, the difference between ܩ
(which is derived from
LEFM) and Ω is extremely small. Moreover, simple analytical formulae derived from
LEFM and geometrically linear beam theories, namely ܩ
ா
and ܩ
்
, are very accurate
7
approximations of Ω that, unlike the formulae used in standards, do not require the
measurement of the crack length.
In [9], a detailed derivation of the relationship between ܩ
, ܬ
and Ω, derivation of the
expressions for ܩ
்
, ܬ
்
and ܬ
ா
can be found. Furthermore, the case of a DCB with
prescribed rotations is investigated. Various numerical examples are presented where
the accuracy of different expressions for ܩ
and ܬ
is assessed and compared with data-
reduction schemes from standards.
Figure 4: Values of: (a) ܩ
/Ω for different values of ߪ
݉ܽݔ
and (b) different fracture
resistance parameters normalised with respect to Ω obtained for ߪ
݉ܽݔ
=7.5
MPa
Supplementary data
Supplementary material related to this article can be found on-line at
http://dx.doi.org/10.17633/rd.brunel.6194483.
Acknowledgements
This project has received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Sklodowska-Curie grant agreement No.
701032. The third author wishes to acknowledge the financial support of the Croatian
Science Foundation (Research Project IP-2016-06-4775).
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8
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