Crack initiation and propagation in ductile specimens with notches: experimental and
numerical study
Alessandro Schiavone*, Gayan Abeygunawardana-Arachchige, Vadim V. Silberschmidt
Wolfson School of Mechanical and Manufacturing Engineering
Loughborough University, LE11 3TU, Loughborough, UK
*Corresponding author: Alessandro Schiavone; Email: A.Schiavone@Lboro.ac.uk
Abstract
Failures of components and structures are often related to the presence of notches of different
shapes. Damage modelling techniques have been proven capable of modelling the crack initiation
and propagation in ductile materials (such as Al alloys). The Gurson-Tvergaard-Needleman (GTN)
method and Extended Finite Element Method (XFEM) are compared against original experiments to
study the crack initiation and propagation processes in aluminium specimens with different notch
shapes (V-shape, U-shape and square). Two regimes are considered in this study: quasi-static and
impact uniaxial tensile loading. Results show that the load-bearing capability predicted with the two
methods is somewhat lower compared to experiments; still, the crack shapes were predicted
correctly, with the exception of the square-notch case, for which XFEM was unable to predict the
correct shape due to limitations in the model formulation. This study provides information useful for
design of components with stress raisers that are exposed to different loading regimes and shows
limitations in both the GTN- and XFEM-based approaches that in many cases underestimate the load-
bearing capacity
Keywords: Ductile damage; GTN model; XFEM; Notched specimens; Finite element
1 Introduction
Failure of many components and structures is related to cracks initiated at locations of stress raisers,
often related to notches of various shapes and dimensions. One of the typical examples is a screen
used in shakers for separation of particulate matter. Screens have patterns of perforation (holes) with
different geometry and, though made of ductile materials, fail as a result of crack emanating from
these holes. A dynamic regime of loading as a result of repetitive impact of particles, defines specific
features of this process.
Damage modelling techniques have been widely used to model crack initiation and propagation in
ductile materials, such as aluminium alloys and stainless steel. Ruzicka et al. (2012) showed that by
calibration of damage parameters against experimental uniaxial tensile test data, damage modelling
can predict force-displacement curves in good accordance with the experiments for different notched
specimens, i.e. tensile bar, notched bar and butterfly specimens. Another work by Zuo et al. (2003)
studied a failure process in ductile materials and implications for its prediction, based on a relationship
between the failure initiation and the stress constraint. Their work demonstrated that a two-parameter
model is capable to predict the failure of ductile materials by modelling the growth and nucleation of
voids. These mechanisms can be described using several approaches, two of which are used in this
paper
The first - a Gurson-Tvergaard-Needleman (GTN) method - is based on simulation of softening and
failure of ductile material based on the introduced approximations for void nucleation, growth and
coalescence in their microstructure, into the models of the plastic flow, hardening behaviour and
failure (Gurson, 1977; Tvergaard, 1980; Needleman et al., 1984). The GTN method was employed,
for instance, to simulate the fracture in aluminium round bars (Anvari et al., 2007) and compared with
experimental results for both quasi-static and dynamic loading conditions. These results showed that
the simulations provided reliable account for the rate dependency and that the diameter reduction of
the specimens was also predicted properly, even though the final diameter differed from that in
experiments. Additionally, Xue et al. (2010) conducted a calibration study of an extended Gurson
constitutive model to predict the processes of ductile fracture in steel, which demonstrated that after
proper calibration this model could predict the damage initiation and propagation in a ductile structural
alloy under a wide range of stress states. This was achieved by calibrating the model with three types
of tests: uniaxial tension, mode-I and mode-II cracking. The GTN method was also employed to
simulate ductile fracture in steel specimens with notches under a high stress-triaxiality regime (Kiran
et al., 2013), successfully predicting the ductility of structural components in the case of bars with
holes, a plate with reduced section and plates with holes.
Ductile fracture can also be modelled with a cohesive-crack model, in which the propagation of the
crack is governed by a traction-separation law across the crack faces at the tip. When the crack path
is not known, this method can be used within the framework of Extended Finite Element Model
(XFEM), which simulates the discontinuity in the displacement field along the crack path (Moës,
2002). Compared with other schemes, for instance cohesive-zone elements, this method allows the
crack to grow within the elements and along their interfaces, a feature that considerably reduces the
mesh dependency of results (Moës, 1999). Although this method has recently become very popular
for the modelling of brittle fracture, the validity of XFEM for the modelling of crack initiation and
propagation in ductile materials is an open subject.
Seabra et al. (2012) compared the XFEM method for B-bar and F-bar formulations with other
competing methods, i.e. the enhanced strain formulation. The study showed that XFEM can handle
efficiently problems with ductile materials that would require a very high computational effort if solved
with more traditional techniques. For example, XFEM combined with Lemaitre ductile damage
modelling has been used to solve the problem of fracture in ductile metals (Seabra et al., 2013) in the
case of double-notched specimens. Results showed that XFEM can deal with ductile fracture without
additional material parameters, without knowing the location of the crack and a considerably reduced
mesh dependency, compared to traditional damage models. A further study of application of XFEM
for the crack growth in ductile materials was conducted by Kumar et al. (2013), who applied this
method to solve of two problems: compact tension and three point bending. The obtained load-
displacement curves showed a very good agreement with the literature data for both type of
specimens, demonstrating that XFEM is an effective tool for the modelling of stable crack growth in
ductile materials.
The response of ductile materials to impact loading is important for a large number of applications,
especially manufacturing, e.g. high speed blanking and hole flanging processes. This regime covers
also a wide range of loading cases - hypervelocity impact, blast loading, jet impact, projectile
penetration, dropped-object loading, structural crushing and so on. As a result, multiple studies have
been conducted for a wide range of materials under these conditions. An impact-test approach was
used to study the impact behaviour of monolithic materials (Ali et al., 2011). Recently conducted
studies employed Charpy impact tests for the measure of toughness under different temperature
conditions for rectangular hollow sections (Sun et al., 2014). Influence of notch severity on the impact
behaviour of an Al-alloy was studied by Champlin et al. (1999) with notch angles of 45°, 75° and 90°.
The studied samples were notched only on one side and a standard Charpy impact test was
conducted. Still, a dynamic response to impact loading, including fracture initiation, crack propagation
and crack shapes, was studied insufficiently for standard notched specimens made of ductile metals
(i.e. aluminium).
The aim of this study is to compare different strategies for modelling of crack initiation and
propagation in ductile materials by validation against experimental results. In the first part of this work,
the GTN and XFEM methods employed in numerical simulations of aluminium specimens with three
different types of notches: V-shape, U-shape and square - are compared against the original results
from quasi-static uniaxial tension experiments. The force-displacement curves obtained in the
experiments and with numerical simulations are compared as well as the crack shapes. In the second
part, analyses of the same specimens were conducted under dynamic loading conditions. For these
analyses, GTN and XFEM were adopted, monitoring force-time curves and crack shapes.
2 Methodology
2.1 Experimental setup
Experiments were performed to study the crack initiation and propagation in aluminium specimens in
the presence of macroscopic notches. The specimens were prepared according to standard ASTM
E8-13a for tension testing of metallic materials. The effect of three different notches was studied - V-
shape, U-shape and square; the notches were designed according to standard ASTM E399-12 for the
measurement of the strain fracture toughness of metallic materials. The used specimen and notches
with respective dimensions are shown in Figure 1. The aluminium alloy employed to manufacture the
specimens was Al 1050A (EN AW-1050A-H14), which is 99.5% aluminium, strain hardened to half-
hard temper.
All the specimens were machined from raw aluminium sheets using a wire cutting technique; the
obtained dimensions of the specimens were then analysed a using Smart-scope 200 automatic
measuring machine. It uses a video measurement system, which - along with 3 axis of movement -
allows very accurate measurements to be taken of parts placed onto the machine bed. The
measurements were taken under 32.5% magnification and 44% lighting which allowed any deviations
from the actual dimensions to be detected. Analysis of the images was performed by means of
equally spaced points (0.02 mm of distance) automatically detected by the software on the border of
the specimens. This test ensured that the dimensions of the specimens were in good agreement with
those specified for machining with a tolerance of 0.05 mm.
Uniaxial tensile tests with the manufactured specimens were performed according to standard ASTM
E8-13a, an Instron 3369 universal testing machine, with a maximum load of 5 kN. Furthermore, the
tensile tests were performed at three different extension rates: i.e. 1 mm/min, 0.5 mm/min and 5
mm/min, to ensure that there was no rate dependency on our results. Each test was repeated 5 times
for all the three extension rates, with a total number of 15 tests for each type of specimen.
2.2 GTN technique
GTN is a method capable to describe the damage-induced softening of the material. It is based on
theory of plasticity of materials with voids and simulates micro-void nucleation, growth and failure in
terms of the effective porosity factor
, assuming isotropic behaviour of the material (Gurson et al.,
1977; Tvergaard, 1982; Needleman et al., 1984)
The constitutive equation (Abaqus Documentation, 2013) is given by
,
=
+ 2
cosh
3
2
(
1 +
)
where =
3
2
is von Mises stress, is the deviatoric stress tensor, is the hydrostatic
pressure,
is the yield stress,
,
and
are arbitrary GTN model parameters.
The effective porosity
is given by the following equation:
+
+
(
)
< <
where
is the magnitude of void volume fraction that triggers their coalescence in the material and
= (
+
)
is the void volume fraction at which the material losses its load-bearing
capacity.
At first, the GTN parameters were found in the literature (He, 2011), where used for aluminium
specimens then they were adjusted by varying the total void volume fraction at failure (
) and the
critical void volume fraction (
) so that the displacement at failure matched our experimental results.
The parameters for the GTN that showed the best fit of the force-displacement curve for the studied
material are shown in Table 1.
2.3 XFEM technique
Displacement approximation in XFEM is based on enrichment of the finite element (FE)
approximation with discontinuous functions. The displacement of the enriched nodes is given by the
following equation (Abaqus Documentation, 2013):
=
(
)
+
(
)
+
()
,
where is the total number of nodes,
(
)
is the shape function of the node ,
(
)
is the jump
enrichment function,
() is the tip enrichment function,
,
are additional degrees of freedom and
and are the sets of all nodes and the tip enriched nodes respectively.
A traction-separation law was used to model damage in XFEM, based on the Maximum Principal
Stress criterion for the crack initiation, while the damage evolution was modelled based on the
definition of the facture energy. Similarly to the case of GTN the parameters were calibrated based on
the obtained experimental results, in order to predict the correct displacement at failure in the case of
the uniaxial tension of the specimen without notch. The XFEM parameters that produced the best
results are given in Table 2.
2.4 Finite-element modelling
The geometry of all studied specimens was reproduced in Abaqus finite-element software, according
to the dimension and shapes defined in our experimental work. The specimen without notch was
modelled as a shell part, while those with notches were modelled as square shell parts of a gauge
length, with dimensions of 1 mm in thickness and 20 mm in length and width. A symmetry boundary
condition was used for the line of the specimen’s main axis, reducing thus computational efforts. For
the GTN, the specimen with a square notch was also modelled as a square shell part, with
dimensions of 1 mm and 40 mm to cover the entire width of the modelled specimens; this was due to
the asymmetric crack path expected for this notch shape. All the notches were modelled based on the
shape and dimensions of the experimental specimens - shown in Figure 1. The notched parts with 20
mm in length and width were modelled with a single notch on the right edge of the part, whilst the
square notch specimen with dimensions of 40 mm had two notches on both edges of the part.
For the GTN-based modelling, the Abaqus/Explicit solver was used to perform the analyses. The
parts were meshed with around 10000 elements for the notched specimens, with hexahedral 4-node
shell elements with reduced integration and plain-stress formulation. The Abaqus/Standard solver
was employed for the XFEM analyses; the parts were meshed using some 5000 elements of the
same type. The reason behind different mesh densities is due to the fact that GTN modelling is based
on element deletion, which makes it strongly mesh dependent compared to XFEM. The used mesh
and geometry of the models are shown in Figure 2.
Material properties for Al 1050a were extracted directly from the data obtained with the uniaxial tensile
test performed on specimens without notch. Its Young’s modulus was calculated to be 70 GPa, the
Poisson’s ratio 0.33, yield stress 85 MPa and the plastic region was fitted with a multi-linear curve
based on the points of our experimental tests. These data were introduced into the FE models.
For the quasi-static uniaxial tension simulation, boundary conditions (BCs) were used on the bottom
and top of each specimen, the bottom edge was fixed in the Y direction, whilst the top specimen had
a displacement BC in the Y direction, which magnitude depended on the simulation, i.e. 20 mm for the
specimen without notch and 2 mm for the specimens with notches. A symmetry BC was applied to the
left edge of the 20 mm notched specimens. Since in the cases of the specimen without notch and the
40 mm square-notch part the symmetry boundary condition cannot be applied, the nodes on their
bottom and top edges were also constrained in the X direction to avoid a rigid-body motion. Figure 3
illustrates the BCs used for the case of 20 mm and 40 mm specimens.
In the impact loading analysis, a hammer of Resil impactor used in the tests was introduced into
transient simulations employing a 3D arrangement as shown in Figure 4a. Still, a 2D plane-stress
formulation is adequate for the modelling needs. The surface energy γ of the aluminium was
considered 1500 J/m
2
, equal to half of the fracture energy used for the XFEM model (Table 2). The
nominal cross section was also calculated for the notched specimens. The total energy required to
fracture the sample should be given in the form of kinetic energy of the impactor. Based on the
energy conservation, the velocity of the impactor was calculated as 361 mm/s to completely fracture
the specimen. The kinetic energy carried by the impactor can be transferred to the specimen
efficiently with the rigid body impactor.
Displacements at the fixed edge of the specimen were
constrained, with regard to rotation and displacement along the X and Y directions (Figure 4b), and
the calculated velocity was imposed to the assigned reference point on the impactor.
A typical stress pattern obtained with XFEM analysis of specimen with V-notch in both quasi-static
and impact loading conditions shows that in dynamic loading simulation the von Mises stress is more
localised (Figure 5b) compared to quasi-static simulations (Figure 5a). Examples of distribution of
voids - in terms of void volume fraction - in GTN-based simulations of the specimen with the U-shape
notch for quasi-static and impact loading (Figures 5c and 5d, respectively) demonstrate similar
patterns, with a concentration of voids in the vicinity of the crack tip and around the point of crack
initiation.
3 Results and Discussion
3.1 Specimen without notch
The simulation involving the dog bone specimen without notch was used to calibrate our material
model, especially the GTN and XFEM parameters against the experimental results. Figure 6 shows
the force-displacement curve curves for the specimen without notch, obtained in the experimental
study as well as with XFEM and GTN simulations.
