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Engineering Fracture Mechanics 75 (2008) 2087–2114
www.elsevier.com/locate/engfracmech
Numerical simulation of plasticity-induced fatigue crack
closure with emphasis on the crack growth
scheme: 2D and 3D analyses
P.F.P. de Matos, D. Nowell
*
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, United Kingdom
Received 28 February 2007; received in revised form 17 October 2007; accepted 18 October 2007
Available online 7 November 2007
Abstract
In this paper a numerical simulation of plasticity-induced fatigue crack closure is performed using the finite element
method. Emphasis is placed on the crack growth scheme usually adopted for modelling fatigue crack growth in crack
closure problems. The number of load cycles between node releases usually reported in the literature has been, in general,
one or two. The present work shows that increasing the number of load cycles between node releases has a strong effect on
the opening stresses, particularly, under plane strain conditions and 3D fatigue cracks, in contrast plane stress shows little
variation with increasing number of load cycles. This investigation also suggests that ratchetting may take place close to
the crack tip in both plane strain and 3D crack problems. The problem of discontinuous crack closure under plane strain
conditions, often reported in the literature, is also addressed.
2007 Elsevier Ltd. All rights reserved.
Keywords: Finite element method; Fatigue crack closure; Opening stresses; Plane strain; Plane stress; Discontinuous closure; Ratchetting
1. Introduction
The phenomenon of fatigue crack closure has been known over 30 years, Elber [1] first detected the effect in
1970. Since then it has been extensively investigated by many researchers. It has been found that the fatigue
crack closure phenomenon is an intrinsic aspect of the mechanics of growing cracks. In many engineering
problems the main source of fatigue crack closure is thought to be plasticity-induced. In a very simplified
way this phenomenon arises due to residual plastically deformed material which is left along the crack faces.
As a consequence, the crack faces of a growing crack contact each other and the resulting contact stresses
reduce the effective stress intensity factor at the crack tip and therefore the rate of crack propagation. Given
the relevance of this topic, a wide variety of techniques have been used to investigate the phenomenon. These
include experimental methods, numerical methods and also analytical models. Among the analytical models,
*
Corresponding author. Tel.: +44 01865 273184; fax: +44 01865 273906.
E-mail address: [email protected] (D. Nowell).
0013-7944/$ - see front matter 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engfracmech.2007.10.017
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strip yield closure models have provided useful insight into the behaviour of cracks under plane stress, however there is no consensual model for the plane strain situation. More general numerical methods, such as the
finite element method are often used to simulate plasticity-induced fatigue crack closure in 2D geometries
under plane strain [2–6], plane stress [3,8–10,7] and in three-dimensional analyses [11–14]. In this paper we
wish to give particular emphasis to the crack growth scheme usually adopted to simulate plasticity-induced
fatigue crack closure using finite element analysis. The focus of this work is the study of plane strain conditions, although the work will be complemented with some plane stress and 3D FEM analyses. Given the vast
amount of work carried out over the last few years using the FEM to study crack closure, it is appropriate to
start with a brief literature review giving particular attention to plasticity-induced crack closure under plane
strain conditions, crack growth schemes usually adopted.
2. Literature review
The phenomenon of plasticity-induced fatigue crack closure under plane strain conditions is one of the
most controversial topics concerning the mechanics of crack propagation. No general consensus exists among
the scientific community concerning the physical mechanism for crack closure under plane strain conditions.
Fleck [2], used finite elements to simulate plasticity-induced crack closure under plane strain conditions and
predicted that the nature of the closure process changes from continuous to discontinuous after a sufficient
increment of crack growth. Discontinuous closure is the phenomenon whereby the crack faces first contact
at a location remote from the crack tip. According to his work the source of discontinuous closure appears
to be a residual wedge of material on the crack flanks, located just ahead of the initial position of the crack
tip. He suggested that closure involves only a few elements relatively distant from the current crack tip and the
closure levels decay steadily as the crack grows beyond its initial length. In the limit closure would not occur at
all. A further attempt to understand the problem was undertaken by McClung et al. [3] who stated that steady
state closure does occur under plane strain. They found that crack opening levels are significantly lower inplane strain than in-plane stress and crack opening and closing is a continuous unzipping process for both
regimes. This observed behaviour is in strong contrast to the analysis of Fleck [2]. In addition McClung
et al. [3] reported that the residual plastic stretch in the wake of a growing plane strain fatigue crack is associated with the transfer of material from the in-plane transverse direction to the axial direction. This in-plane
contraction also leads to the generation of complex residual stress fields. Sehitoglu and Sun [15] addressed the
plane strain problem by introducing the crack tip tensile load parameter, which characterises the stress level at
which the stresses at the crack tip node change from compressive to tensile. They stated that crack advance is a
strong function of tensile stresses in front of the crack tip and crack growth into a wholly compressive zone is
highly unlikely. They further observed that a crack blunting mechanism in-plane strain competes with the closure mechanism. More recently Wei and James [16] reported that after growing a plane strain fatigue crack for
a few cycles, there is no contact in the region immediately behind the crack tip and the contact pressure along
the crack faces is discontinuous. These findings are in contrast with those of McClung et al. [3]. Zao et al. [17]
modelled a CT specimens under plane stress and plane strain. They did not observe plasticity-induced crack
closure under plane strain during steady state crack growth under cyclic tension, although they found significant levels of closure under plane stress.
Crack growth schemes adopted for simulating crack closure usually consist of releasing nodes ahead of the
initial crack tip. The stage at which the node release should be performed is unclear and crack tip advance can
be performed at the minimum load, maximum load, during the loading/unloading cycle or during the second
cycle. Recent work has shown that good agreement can be obtained independently of the node release scheme,
provided a suitable mesh refinement is used [10]. Table 1 presents details of some of the previous work
reported in the literature, where it can clearly be seen that usually one or two load cycles are performed
between node releases. However in 3D analyses usually only one load cycle is applied between node releases
[18,13,11,19,20]. One logical question is whether application of further cycles between node releases has any
effect on the result. The authors have not been able to find any investigation of this aspect of the problem in
the published literature. In current work this issue will be addressed using plane stress, plane strain and in 3D
simulations of crack closure. The influence of the number of load cycles on the opening and closing stresses
will be quantified.
Table 1
Previous work reported in the literature
Author
Material and material
model
Node release
scheme
Number
of load
cycles
Specimen type
Element type and state of stress
Some parameters assessed and
methods
1986
Fleck [2]
Al-alloy. EPP
25
Triangular three nodes Pr and Pe
1989
McClung and
Sehitoglu [8,9]
Al-alloy. Kin. hardening
H/E = 0.01 and 0.07
At max. load every
load cycle
At max. load every
load cycle
Opening load, node displacement
method
Opening load, node displacement
method
1990
McClung et al.
[3]
Sehitoglu and
Sun [15]
Sehitoglu and
Sun [21]
Wu and Ellyin
[22]
Al-alloy. Kin. hardening
H/E = 0.01 and 0.07
Steel. Kinematic
hardening H/E = 0.07
Steel. Kinematic
hardening H/E = 0.01
Steel. Isotropic hardening
Quadrilateral four nodes Pr and Pe
2002
Ellyin and Wu
[23]
Wei and James
[16]
Pommier [5]
2003
2004
Solanki et al.
[24,25]
Steel. Isotropic and
kinematic hardening
Polycarbonate. Kinematic
hardening
Steel. Cyclic strain
hardening and EPP
Steel. EPP
2004
Zhao et al. [17]
2005
Lee and Song
[6]
GonzálezHerrera and
Zapatero [26]
Alizadeh et al.
[27]
1991
1992
1996
1999
2000
2005
2006
2006
de Matos and
Nowell [7]
Nikel superlalloy. Kin.
and Iso. hardening
Al 2024-T351. Non-linear
kinematic hardening
Al-2024-T351. EPP,
Isotropic and kin.
hardening H/E = 0.03
Al 2024. EPP, Isotropic
and kinematic
Ti–6Al–4V. EPP
At min. load every
load cycle
At max. load every
load cycle
At max. load every
load cycle
At max. and min.
load every load
cycle
At max. load every
load cycle
At max. load every
load cycle
At min. load every
two load cycles
At max. load every
load cycle
20–50
CCT and bend
specimen
Rectangular
plate with
central hole
CCT
20
CT
Quadrilateral four nodes Pr and Pe
–
CCT and CT
Quadrilateral four nodes Pr and Pe
20
CCT
Quadrilateral four nodes Pr
Opening and closing loads, node
displacement method and CSCT
20
CCT
Quadrilateral four nodes Pr
20
CT
Triangular three nodes Pr and Pe
80
CCT
Quadrilateral four nodes Pe
Opening and
displacement
Opening and
displacement
Opening and
30
CT and CCT
Quadrilateral four nodes Pr and Pe
At max. load every
two load cycles
At min. load every
load cycle
At max. load every
load cycle
50
CT
Quadrilateral four nodes Pr and Pe
25
CCT
Quadrilateral four nodes Pe
–
CT
Quadrilateral four nodes Pr and Pe
50
CCT
120
CCT
Triangular six nodes and quadrilateral
four nodes Pr and Pe. Hexahedral
eight nodes 3D
Quadrilateral four nodes Pr
At min. load every
two load cycles.
Hardening
At min. load every
two load cycles
24
Quadrilateral four nodes Pr and Pe
Opening load. Node displacement
method
Tensile tip load, CSCT Opening
load, node displacement method
Tensile tip load, CSCT
closing loads, node
method and CSCT
closing loads, node
method and CSCT
closing loads
Opening and closing loads, node
displacement and contact stress
method
Opening and closing loads.
Compliance offset method
Opening and closing loads. Node
displacement method
Opening and closing loads, node
displacement method and CSCT
Opening and closing loads, node
displacement method
P.F.P. de Matos, D. Nowell / Engineering Fracture Mechanics 75 (2008) 2087–2114
Year
Opening and closing loads. Node
displacement, contact stress
method and CSCT
Pr – plane stress; Pe – plane strain; CCT – centre crack tension; CT – compact tension; CSCT – change in stress at crack tip.
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3. Finite element modelling
3.1. Geometry and material model
The geometry of the problem modeled is shown in Fig. 1. It consists of a square plate with a finite central
crack. For convenience only one quarter of the plate was analysed in 2D and one-eighth in 3D analyses as
shown in Fig. 1. The dimensions of the plate are W = 45 mm with an initial crack a0 = 1 mm, in addition
the 3D model has a thickness t of 1.0 mm. The material model used was elastic perfectly plastic with a von
Mises yield criterion. The material properties used are representative of the titanium alloy Ti–6Al–4V: the
yield stress is 1000 MPa, Young’s Modulus 110 GPa; and Poisson’s ratio 0.34.
3.2. Mesh and boundary conditions
Fig. 2 presents the mesh used for both plane stress and plane strain simulations. This particular mesh was
used previously by de Matos and Nowell [7] for modelling plane stress crack closure. The mesh was designed
with an increasing level of refinement towards the crack tip region. Two different mesh densities were used; the
difference between them concerns only the element size (5 and 10 lm) in the region where the crack is allowed
to grow (see Fig. 2b). In the case where the smallest element size is 5 lm the mesh has 15446 four noded elements and 15,753 nodes, in the second case 5410 elements and 5600 nodes. Fig. 3 shows mesh details of the 3D
model. As with the 2D model the mesh was designed with an increasing level of refinement towards the region
where the crack is allowed to grow, eight elements were used through the thickness of the model. Elements at
the free surface are one half the size of elements at the centre of the plate. Eight-noded brick elements were
used (8976 elements and 10,899 nodes). The possibility of contact between the two faces of the crack was taken
into account by modelling a rigid line and a rigid surface, on 2D and 3D models, respectively, and ascribing
contact to the elements along the crack plane. The ‘augmented Lagrange’ contact algorithm available in
ABAQUS [28] was employed. This algorithm uses a penalty function method for each iteration, allowing
an interpenetration of 0.001% of the characteristic contact length.
3.3. Crack growth modelling
The modelling of crack growth in plasticity-induced fatigue crack closure problems is, in general, performed by node release, i.e. nodes ahead of the initial crack tip (2D) or crack front (3D) are released sequentially by modifying the appropriate boundary conditions ascribed to the nodes. In most of the 2D FEM work
Fig. 1. Geometry of the 3D model. W = 45 mm and a0 = 1 mm and t = 1 mm (note that the geometry of the 2D model corresponds only
to the xy plane).
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Fig. 2. Mesh details: (a) mesh of a square plate with 45 · 45 mm with an initial crack (a0) of 1 mm; (b) mesh detail close to the crack tip,
smallest element size 10 lm.
performed over recent years, node release is performed every load cycle or every two load cycles at minimum
or maximum load, as shown in Table 1. In 3D analyses usually only one load cycle has been applied between
node releases [18,13,11,19,20], probably due to the computational cost of these simulations. One of the aims of
the present work is to clarify whether the number of load cycles before node release has any effect on the closure behaviour under plane stress, plane strain and 3D analyses. Fig. 4 shows different node release schemes
adopted in the present work. The maximum number of load cycles applied between node releases was 16 cycles
for a plane strain fatigue crack. The load cycle at which node release should be made is not entirely clear, but
recent work has shown that good results can be obtained independently of the node release, provided that a
suitable mesh refinement is used [10]. In the present study node release was specified at minimum load. This is
computationally easier, since there is no sudden change in crack displacement. The increment of crack growth
was equal to the element size. The computational cost of these simulations is high, given that multiple load
cycles may be required between node releases. In addition small elements are needed, with a large number
of load increments per cycle. Since the initial crack has no plastic wake, it is necessary to simulate a significant
number of load cycles in order to reach steady state closure behaviour. It should also be noted that there are
significant non-linearities in the problem due to the material model (elasto-plastic material behaviour) and due
to the contact along the crack faces.
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Fig. 3. Mesh details, 3D model: (a) mesh of a square plate with 45 · 45 mm with an initial crack (a0) of 1 mm and thickness (t) of 1 mm;
(b) mesh detail of the crack front, smallest element size on xy plane is 10 lm.
3.4. Crack opening and closing stresses
Two different techniques were used to calculate the opening stresses, the node displacement method and the
weight function technique. For convenience a brief description of each of these techniques will be given. The
node displacement method consists of monitoring the displacement of a node (typically the first or second
node behind the crack tip) as the load is applied. The opening stresses are found when the displacement of
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Fig. 4. Load cycles and node release scheme.
the node monitored became positive during the loading stage of a load cycle and the closing stresses are found
when the displacement of this node falls to zero during the unloading stage. Here results from the first node
behind the crack tip are presented.
The weight function or contact stress method can be used to calculate a residual stress intensity factor due
to the compressive residual stresses which exist along the crack faces, in the presence of closure, at minimum
load. A negative residual stress intensity factor does not, of course, have any physical meaning on its own, but
by employing a superposition argument it may be equated to the (nominal) opening stress intensity Kop needed
to overcome the residual stress field along the crack faces and open the crack to the tip. The contribution to
the residual stress intensity factor from a 2D linear displacement element is [7],
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiixiþ1
2a h
K ires ¼ pffiffiffiffiffiffi A arcsinðx=aÞ B a2 x2
;
ð1Þ
xi
pa
riþ1 xi
ri
and B ¼ rxiþ1
. ri and ri+1 are the contact stresses at the nodes.
where A ¼ ri xiþ1
xiþ1 xi
iþ1 xi
The total residual stress intensity factor may then be found by summing the contributions from the individual elements in contact, so that
N
X
K res ¼
K ires :
ð2Þ
i¼1
The opening stress is calculated by the following equation,
pffiffiffiffiffiffi
K res þ Crmin pa
pffiffiffiffiffiffi
rop ¼
;
C pa
ð3Þ
where rop is the applied stress corresponding to crack opening, rmin is the minimum applied stress and C is the
normal geometry factor in the stress intensity factor expression. For the current geometry and loading C 1
and rmin = 0. The use of this technique is straightforward, it is only necessary to calculate Kres at minimum
load. In the present work the calculation was performed in the load cycle before node release. The weight function technique was used here only in 2D models since weight functions for 3D cracks are less readily available,
although the method applies equally to the 3D situation.
4. Analysis of a 2D plane stress growing crack
In this section a 2D plane stress crack is studied. The phenomenon of plasticity-induced fatigue crack closure under plane stress is relatively well understood, although the effect of the number of load cycles before
node release has not been systematically studied. In the present work only a growing crack under a maximum
cyclic loading of rmax/ryield = 0.5 and R = 0 is examined. Four different node release schemes were used,
node release was performed at minimum load after one, two, three and four load cycles. Diagrams of the
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displacement of the first node behind the crack tip are plotted for each of the node release schemes studied.
Finally the influence of the number of load cycles on opening and closing stresses was quantified.
4.1. Load displacement diagrams
Figs. 5 and 6 present the displacement of the first node behind the crack tip as the remote applied stress is
applied. The number of load increments in the last load cycle was higher than in previous load cycles. This is
the reason why in both Figs. 5 and 6 the line describing the last load cycle is smoother. The results presented
for a/a0 = 1.0 concern a static crack, a/a0 = 2.0 concern the case of a growing crack where some plastic wake
has developed. Reverse plasticity and closure effects contribute for a decrease in uy/a, in the latter case of
growing cracks. Fig. 5b shows that the first load cycle is not stabilized, this is confirmed in Fig. 6 where three
load cycles were applied before node release. Fig. 6 shows that third cycle collapses onto the same curve as that
for the second load cycle, therefore load load cycles between node releases seems to be the best choice to simulate crack closure under plane stress. Similar results to those shown in Fig. 6 were obtained for four load
cycles between node releases.
4.2. Opening and closing stresses
Fig. 7 compares the opening and closing stresses obtained as a function of the number of load cycles
between node releases. The results presented in Fig. 7b are those for two, three and four load cycles between
Fig. 5. Load displacement diagrams for the first node behind the crack tip, two load cycles between node releases: (a) a/a0 = 1.0;
(b) a/a0 = 2.0. Plane stress, rmax =ryield ¼ 0:5 and R = 0.
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Fig. 6. Load displacement diagrams for the first node behind the crack tip, three load cycle between node releases: (a) a/a0 = 1.0;
(b) a/a0 = 2.0. Plane stress, rmax =ryield ¼ 0:5 and R = 0.
node releases, only one curve is plotted for each opening/closing stresses since the results for two, three and
four load cycles are indistinguishable. These results show that under plane stress (at least for this particular
load conditions) the opening and closing stresses obtained with each of these different techniques is not
affected by the number of load cycles between node releases, n, provided n P 2. The node displacement technique slightly underestimates the opening stresses since it is calculated a small distance behind the crack tip
whereas the weight function technique accounts for all the contact stress along the crack faces. As shown
in [7] these two approaches would converge to the same value, essentially that given by the weight function
method, if a much more refined mesh were used. However, it would be time consuming to simulate crack
growth with such a mesh, given the computational cost.
5. Analysis of a 2D plane strain growing crack
We now turn our attention to a 2D plane strain crack. As previously mentioned the phenomenon of plasticity-induced fatigue crack closure under plane strain is a controversial matter, and no consensus exists
among the scientific community concerning the existence of plane strain crack closure [3,17]. There is also little
consensus regarding whether any closure is continuous [3] or discontinuous [2]. In addition to these uncertainties the effect of the number of load cycles between node releases does not appear to have been studied. In the
present work a 2D plane strain growing crack is studied under a maximum cyclic loading of rmax =ryield ¼ 0:4,
0.5 and 0.7 with R = 0. For the loading conditions of rmax/ryield = 0.5 and R = 0, different numbers of load
cycles (1, 2, 3, 4, 8, 12 and 16) were applied between node releases. For rmax =ryield ¼ 0:4 and 0.7 with R = 0, a
more limited range of load cycles between node releases was studied (1, 2, 4 and 8). Due to lack of space, only
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Fig. 7. Opening and closing stresses as a function of the number of load cycles between node releases. Plane stress, rmax/ryield = 0.5 and
R = 0. (a) one load cycle; (b) two, three and four load cycles. Weight function technique and uy node displacement.
the results for the case of rmax =ryield ¼ 0:5 and R = 0, are presented in detail, for the other two load cases only
the opening and closing stresses will be presented. The opening stresses for these three different load conditions
were studied and compared with predictions from Newman’s model [29].
5.1. Load displacement diagrams
Figs. 8–10 present load displacement diagrams, for the first node behind the crack tip, for 2, 8 and 16 load
cycles, respectively at different stages of the crack growth. Fig. 8 shows the results obtained for the case
where two load cycles between node releases are applied. This figure shows that the first load cycle is not
stabilized. Fig. 9 presents the load displacement diagrams obtained for eight load cycles between node
releases. The analysis of this figure allows us to conclude two important things: firstly, for the case of a growing crack (a/a0) it is possible to see that at the end of each load cycle there is an increment of the displacement, i.e. the material is ratchetting due to the accumulation of plastic strain in the region close to the crack
tip. Secondly, it is possible to see that the first node behind the crack tip is not in contact with the crack plane
at the end of the eighth load cycle. Fig. 10 shows similar results for the case of 16 load cycles before node
release. For this particular case only results up to a/a0 = 1.5 were available due to the high computational
cost of the simulation. Results similar to those presented in Figs. 8 and 9 were also obtained for three, four
and 12 load cycles between node releases. The ratchetting effect seems to play an important role in these simulations since the load displacement diagrams show an increment on displacement for each new applied load
cycle. Fig. 11 shows the displacement of the first node behind the crack tip at maximum and minimum load
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Fig. 8. Displacement of the first node behind the crack tip as a function of the remote applied stress for different crack lengths:
(a) a/a0 = 1.0; (b) a/a0 = 2.0. Two load cycles between node releases. Plane strain, rmax/ryield = 0.5 and R = 0.
for the case where 12 load cycles are applied between node releases. It can be seen that for the case of a static
crack a/a0 = 1.0 the displacement at maximum and minimum load tends to stabilise, for the case of a growing crack a/a0 = 1.5 and 2.0 the trend consists of an increasing in displacement with increasing number of
load cycles. A more sensitive analysis can be performed by calculating the increment in displacement between
i ucyclei1 Þ at maximum
two consecutive load cycles. Fig. 12 shows the increment in displacement ðduy ¼ ucycle
y
y
and minimum loads for the cases when 8 and 16 load cycles are applied between node releases. Both cases
show similar results for a static crack a/a0 = 1.0. The increment in displacement decreases with increasing
number of load cycles. The increment in displacement duy tends asymptotically to zero with increasing number of load cycles. For the case of growing crack, a/a0 > 1.0, it is possible to conclude that the slope of the
increment in displacement shown in Fig. 12(a1) and (a2) happens because eight load cycles are insufficient to
reach a constant increment in displacement. As shown in Fig. 12b the increment in displacement does stabilise after a larger number of load cycles. An interesting feature of these three node release schemes is that
duy measured at maximum load decreases or vanishes at minimum load, up to a given number of cycles
(depending on the stage of crack growth), then stabilizes and has the same value at maximum and minimum
loads. This transition corresponds to the onset of crack tip opening at zero load; the accumulated plastic
strains become large enough to avoid the contact along a part of the crack length along the crack faces.
In order to quantify the increase of the plastic strains with increasing number of cycles, the increment in
equivalent plastic strains was plotted for three different points (P1, P2 and P3, shown in Fig. 12b2) for 16
load cycles between node releases. The difference in the equivalent plastic strain is defined in as
depeq ¼ epeq ðcyclei Þ epeq ðcyclei1 Þ at minimum load.
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Fig. 9. Displacement of the first node behind the crack tip as a function of the remote applied stress for different crack lengths, eight load
cycle between node releases: (a) a/a0 = 1.0; (b) a/a0 = 2.0. Plane strain, rmax/ryield = 0.5 and R = 0.
The results presented in Fig. 13 show the increment in the equivalent plastic strain for P1, P2 and P3. Little
difference is found between P2 and P3, and indeed for greater than 10 load cycles all curves collapse on the
curve presented for P3, for the case of a/a0 = 1.5.
The results presented here show that for more than one load cycle between node releases, ratchetting takes
place in the crack tip region. The increment in plastic strain tends to a constant value after a given number of
load cycles. In real problems the plastic strain cannot, of course, increase indefinitely. However, this feature is
difficult to investigate using the current approach due to the difficulty in simulating sufficient load cycles. From
a physical point of view, the accumulation of plastic strains in the crack tip region may lead to some crack tip
blunting. If this were to occur, the rachetting would decrease and eventually cease. On the other hand, the
accumulation of plastic strains in the region close to the crack tip may well result in localized damage and
propagation of the crack. Rachetting effects have been previously reported by other authors [5,32–34],
although this effect has not been systematically quantified. In addition, some of these analyses were more limited in terms of the number of crack increments simulated (e.g. Pommier [5] 40 crack increments, Toribio and
Kharin [33] 10 crack increments).
An additional consideration is the validity of the material model used in the present work. Here we have
used a simple elastic perfectly plastic description in order to separate the effects of the underlying mechanics
from those of more complex material behaviour. Nevertheless, it should be noted that the model does give
reasonably good accuracy for some materials (e.g. Ti–6Al–4V) so that it cannot necessarily be dismissed as
physically unrealistic. The material description is clearly not appropriate where there is significant strain hardening or softening. The influence of the number of load cycles on the closure behaviour for other elasto-plastic
material models (e.g. including isotropic/kinematic hardening), is an interesting area for future investigation.
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Fig. 10. Displacement of the first node behind the crack tip as a function of the remote applied stress for different crack lengths. Node
release every 16 load cycles. Plane strain, rmax =ryield ¼ 0:5 and R = 0. (a) a/a0 = 1.0; (b) a/a0 = 1.5.
Fig. 11. Displacement uy at maximum and minimum loads, first node behind the crack tip. Twelve load cycles between node releases.
Plane strain, rmax/ryield = 0.5 and R = 0.
Fig. 14 shows typical pictures for the forward and reverse plastic zones estimated by selecting elements
where the von Mises stress equals the yield stress at maximum and minimum loads respectively for the case
of two and eight load cycles between node releases at a/a0 = 2.0. It can be seen that increasing the number
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Fig. 12. Difference in displacement duy at maximum and minimum loads. First node behind the crack tip. Plane strain, rmax/ryield = 0.5
and R = 0. (a1) eight load cycles, maximum load; (a2) eight load cycles, minimum load; (b1) 16 load cycles, at maximum load; (b2) 16 load
cycles, minimum load.
Fig. 13. Difference in the equivalent plastic strains at three different points. Crack tip lies at a/a0 = 1.5, 16 load cycles between node
releases. Plane strain, rmax/ryield = 0.5 and R = 0.
of load cycles has a little influence on the size and shape of the forward plastic zone, the reverse plastic zone
becomes slightly more elongated along a plane inclined about 70 to the crack plane. The number of elements
inside the reverse plastic zone is roughly 12, 27 and 44 for a/a0 = 1.5, 2 and 2.5, respectively for the case of two
load cycles between node releases.
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Fig. 14. Forward and reverse plastic zones at a = 2.04 mm. Plane strain, rmax =ryield ¼ 0:5 and R = 0. (a) Forward plastic zones; (b) reverse
plastic zones.
5.2. Opening and closing stresses
Fig. 15 compares opening and closing stresses obtained with different techniques for the same loading conditions but different crack node release schemes. The results presented in Fig. 15 show that up from two load
cycles between node releases the weight function technique gives the upper limit of the opening stresses followed by the first node behind the crack tip. In contrast, the results presented in Fig. 15a, one load cycle
between node releases, the node displacement method gives the upper limit of the opening stresses. For these
loading conditions, rmax/ryield = 0.5 and R = 0, the comparison of results between the weight function technique and node displacement method show that they are in close agreement, for up to four load cycles between
node releases. However, for eight, 12 and 16 load cycles between node releases, the nodes behind the crack tip
are not in contact at minimum load for a/a0 > 1.25. Regions more remote from the crack tip are still in contact
at minimum load, and the opening stresses calculated using the weight function technique take these into
account. Fig. 16 shows the contact stress profile at different stages of growth at minimum load for the last load
cycle before node release. In Fig. 16a the highest contact stresses are in the region adjacent to the crack tip, so
that the crack tip is the last point to open as the remote applied stresses increases from minimum to maximum
load. For more than two load cycles between node releases the nature of closure changes from ‘continuous’ to
‘discontinuous’. It can be clearly seen that after some crack growth contact takes place only in the region close
to initial crack. It is worth mentioning that discontinuous closure under plane strain conditions has been
reported by other authors, e.g. Fleck [2] and Wei and James [16].
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Fig. 15. Opening and closing stresses for different number of load cycles between node releases: (a) one load cycle; (b) two load cycles; (c)
16 load cycles. Plane strain, rmax =ryield ¼ 0:5 and R = 0.
It is clear that independently of the number of load cycles between node releases, no steady state was found
for the opening and closing stresses of a crack growing under constant amplitude loading, and plane strain
conditions. In all cases opening stresses tend to zero as the crack grows, Fleck [2] and more recently Zhao
et al. [17] have also reported that on the limit closure does not occur under pure plane strain conditions.
The question of how to simulate correctly crack growth in crack propagation problems is difficult to
answer. The ideal approach would be to model the same crack increment per load cycle as that measured
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Fig. 16. Contact stresses along the crack faces at different crack lengths and different number of load cycles between node releases: (a) one
load cycle; (b) two load cycles; (c) eight load cycles. Plane strain, rmax/ryield = 0.5 and R = 0.
experimentally, although this is impracticable from the numerical point of view since very small elements
would be necessary in the early stages of the crack growth process. The present work highlights the fact
plane strain closure regime appears to depend on the rate of crack propagation (described here by the
number of load cycles between node releases) the closure behaviour under plane strain changes. Fig. 17
presents the development of opening stresses obtained with the weight function technique and it can be
clearly seen that the opening stresses decrease with increasing number of load cycles between node
releases. The results obtained in the present work are in strong contrast with the prediction from Newman’s model [29] for plane strain (a = 3). This model is based on Dugdale strip yield model [30] and
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makes use of the plastic constraint factor a to characterize the plastic constraint effect (a = 1, for plane
stress and a = 3 for plane strain). It is known that strip yield models are not suitable to accurately
describe the plane strain closure behaviour and indeed Fig. 17 quantifies the difference for this particular
problem. Based on the FEM results presented in Fig. 17 it can be speculated that, close to threshold crack
growth region, the opening stresses under plane strain will be very small or not existent, although as the
rate of crack propagation increases the level of closure may increase. It should be noted Newman et al.
[31] used a small-crack approach to predict fatigue crack lives in metallic structures. They modified the
strip yield model published in [29] by allowing the plastic constraint factor to change depending on the
rate of crack propagation. This plastic constraint factor was modified in such a way that lower opening
stresses were used at lower rates of crack propagation, and increasing the opening stress up to limit predicted for plane stress at higher rates of crack propagation.
5.3. Opening and closing stresses for other levels of remote applied stress
In order to confirm the trends presented in the previous section we have simulated fatigue crack closure for
two additional levels of remote applied stress rmax =ryield ¼ 0:4 and 0.7 with R = 0. Elements with 5 and 10 lm
were used in these two simulations, respectively. Due to the high computational cost inherent to these simulations only one, two, four and eight load cycles between node releases were modelled. Fig. 17 shows that small
differences are found in the opening stresses from 8 to 16 load cycles between node releases. Fig. 18 presents
the opening and closing stresses obtained for the remote applied stress (rmax =ryield ¼ 0:7) applying 1, 4 and 8
load cycles before node release. The results shown in Fig. 18 have the same trend as those reported for
rmax =ryield ¼ 0:5. The weight function technique accounts for the contact stresses along the crack faces and
therefore seems to be the most reliable method to calculate the opening stresses under plane strain conditions.
Fig. 19 compares the opening stresses obtained with this technique for different number of load cycles between
node releases, rmax =ryield ¼ 0:4 and 0.7 with R = 0. As previously reported the closure level decreases with
increasing number of load cycles before node release. The highest values of the opening stresses seem to be
related to the plastic deformation just ahead of the initial crack tip. In all cases the opening stresses decrease
asymptotically to zero. As shown in Fig. 19 the predictions from the yield strip model for plane strain conditions, are again in strong contrast to the FE results.
5.4. Nature of the closure phenomenon under plane strain conditions
As previously shown (see Fig. 16), up from two load cycles between node releases, the nature of the closure
phenomenon changes under plane strain conditions. At the beginning of the crack growth process the crack
Fig. 17. Opening stresses calculated using weight function technique as a function of the number of cycles between node releases. Plane
strain, rmax/ryield = 0.5 and R = 0.
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Fig. 18. Opening stresses and closing stresses for different number of load cycles between node releases: (a) one load cycle; (b) four load
cycles; (c) eight load cycles. Plane strain, rmax =ryield ¼ 0:7 with R = 0.
closes continuously at minimum load between the current and initial crack tip positions. As the crack grows
the crack closes first in a region remote from the crack tip and the contact stresses are, in general, discontinuous along the crack faces. At a later stage of crack growth only the region close to the initial crack is in contact. In order to understand the effect of both types of closure the artificial problems shown in Fig. 20a and b
were studied. These two problems consist of an infinite plate with a central crack under remote applied stress.
The first case, in Fig. 20a, is supposed to represent a crack which has grown from a0 to a having only some
residual closure stresses in a region remote from the crack tip, the second case Fig. 20b represents a crack
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Fig. 19. Opening stresses calculated using the weight function technique, for different number of load cycles between node releases.
(a) rmax =ryield ¼ 0:4 and R = 0; (b) rmax =ryield ¼ 0:7 and R = 0.
which has residual stresses only in a small region close to the crack tip. For simplicity the residual stress distribution considered has a constant value rres ¼ r0 , the final crack length a = 2a0, and the length over which
the residual stress acts w = 0.25 a0. Both problems were simulated using ABAQUS [28], using a linear elastic
analysis. Contact boundary conditions were prescribed along the crack plane, and the stress intensity factors
at the crack tip were assessed using the J-integral, in this particular case using 15 contours. The continuous line
in Fig. 20c shows the applied stress intensity factor in the absence of residual stresses acting along the crack
plane, the dashed-dot and dashed lines show the results obtained for the case where the residual stresses exist
in a remote region from the crack tip and at the crack tip, respectively. The effective stress intensity factor was
calculated as
K eff ¼ K app þ K res :
ð4Þ
The results obtained show that in the case of remote contact from the crack tip (Fig. 20a) the crack tip is open
and experiences non zero K as soon as the load is applied. For the loading conditions shown in Fig. 20b the
crack tip does not experience any stress intensity until Kapp > Kres, the superposition principle is directly applicable, in contrast with the previous case where the superposition principle is valid only when the crack is fully
open. In both cases, it should be noted that the effective stress intensity factor range DKeff is given by
DKeff = Kmax + Kres provided the crack is fully open at maximum load. The results presented in Fig. 20c also
highlight the fact that it might not be appropriate to assess crack closure predominantly under plane strain
conditions by looking at the crack tip. Closure would not be detected, although the stress intensity range is
still reduced at maximum load.
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Fig. 20. (a) Residual stresses at the initial crack (remote closure mechanism); (b) residual stresses close to the crack tip (continuous closure
mechanism); (c) stress intensity factor as a function of the remote applied stress during a load cycle.
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6. Analysis of a 3D growing crack
In this section we put our attention to the closure behaviour of a 3D growing crack. The main objective is to
show the effect of the number of load cycles between node releases on the 3D closure behaviour. The computational cost of the 3D FEM simulations of plasticity-induced crack closure is high, in the present work this is
aggravated by the fact that several (1, 2, 4 and 8) load cycles are applied between node releases. In the published literature only one load cycle between node releases appears to have been applied in previous 3D FEM
analyses [18,13,11,19,20]. Real fatigue cracks are inherently 3D having, for example thumbnail shape with a
natural aspect ratio. In a recent work Heyder et al. [35] have experimentally shown that in PMMA material
and for mode I loading, the crack front in their specimens was shaped so as to ensure square root singularity
along all crack front, one of the consequences of this is that the crack front breaks the free surface at a specific
angle, which is a function of the Poisson’s ratio of the material [36]. In the present work, this aspect of 3D
fracture mechanics was not taken into account since the crack front was straight, making an angle of 90 with
the free surface of the specimen. This geometry was chosen in order to facilitate comparison with 2D work,
however, it would be interesting to compare the difference in closure behaviour between a curved 3D crack
with a square root singularity everywhere and the straight crack front case where the singularity is only square
root singular in the centre of the plate.
6.1. Displacement diagrams
Fig. 21 shows the displacement uy of the first node behind the crack tip at five different nodes along the
crack front. The results presented are for the case where four load cycles are applied between node releases
at a/a0 = 1.5. This figure shows that in both cases crack closure is more pronounced at the surface of the specimen and as a consequence the magnitude of the displacement is lower at z/t = 0 (free surface) than at z/t = 0.5
(central pane of the plate). Fig. 21b shows that there is an increase in the uy displacement, along whole crack
front, with each new load cycle. This behaviour is similar to that reported in the earlier sections for plane
strain conditions.
6.2. Opening and closing stresses
Figs 22 and 23 present opening, and closing stresses for the cases where one and four load cycles are
applied between node releases. Similar results were obtained for the cases where two and eight load cycles
but space considerations prevent their presentation here. Fig. 22 shows the opening and closing stresses
for the case where one load cycle is applied between node releases. The opening and closing stresses
Fig. 21. Displacement uy for five nodes along the crack front, using the first node behind the crack front. Four load cycles between node
releases, rmax =ryield ¼ 0:5 and R = 0 at a/a0 = 1.5.
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Fig. 22. Opening and closing stresses (using first node uy) for one load cycle between node releases. 3D model rmax =ryield ¼ 0:5 and R = 0.
(a) Opening stresses; (b) closing stresses; (c) comparison of 2D and 3D opening stresses.
obtained using the first node behind the crack front are considerably higher than those calculated using
the second node, similar to the case of a plane strain crack. This difference is lower when more than
one load cycle is applied before releasing a node as shown in Fig. 23. The trend observed consists of
a decrease in the opening, closing and tensile tip stresses from the surface towards interior of the plate.
Figs. 22 and 23c compare the results obtained using the uy displacement of the first node behind the crack
tip from the 2D analyses with the results obtained for the 3D model at z/t = 0 and z/t = 0.5. As expected,
at the surface the results obtained with the 3D model are closer to those estimated with the plane stress,
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Fig. 23. Opening and closing stresses (using 1st node uy) for four load cycles between node releases. 3D model rmax =ryield ¼ 0:5 and R = 0.
(a) Opening stresses; (b) closing stresses; (c) comparison of 2D and 3D opening stresses.
whereas the results obtained at the interior of the plate (z/t = 0.5) are closer to those calculated using a
plane strain model. Fig. 24 compares the opening and closing stresses, calculated using the first behind the
crack front, for the different crack node release schemes and different positions along the crack front z/
t = 0, 0.202 and 0.5. As with the 2D plane strain case the opening stresses decrease with increasing number of load cycles between node releases. This effect is more pronounced at the centre of the plate (where
the stress state is very close to plane strain), but is also present at the free surface; which is often thought
to be equivalent to plane stress.
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Fig. 24. Comparison of the opening stresses (using 1st node uy) for different crack growth schemes. 3D model rmax/ryield = 0.5 and R = 0.
(a) opening stresses, z/t = 0; (b) opening stresses, z/t = 0.202; (c) opening stresses, z/t = 0.5.
7. Discussion of the results
The analyses presented show that the crack growth scheme adopted to model plasticity-induced fatigue
crack closure plays an important role. The number of load cycles between node releases adopted by other
researchers has normally been one or two load cycles. The current investigation suggests that two load cycles
is the best choice for plane stress conditions, since the plastic zone stabilizes in the first load cycle. It was
observed that the plastic region close to the crack tip is stabilized after one load cycle. The load displacement
diagrams for the first node behind the crack tip, all collapse onto the same curve for two or more cycles
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between node releases. After the first load cycle there is no net accumulation of plastic strain, and the material
experiences cyclic plasticity rather than ratchetting. A contrasting behaviour was found for the case of plane
strain growing crack, where plastic strains continue to accumulate during each load cycle (ratchetting).
Increasing the number of load cycles between node release decreases the opening stresses. This is due to
the accumulation of plastic strain close to the crack tip which decreases the magnitude of the contact stresses
along the crack faces. The results obtained, for an elastic perfectly plastic material, show that the opening
stresses decrease asymptotically to zero as the crack grows, as speculated in the early work of Fleck [2] and
recently confirmed by other authors (e.g. [17]). One of the reasons why some authors have reported some closure may be to do with the fact that a limited number of load cycles has been modelled (see Table 1). The
current work supports Fleck [2] in that the source of discontinuous closure seems to be the residual plastic
deformation left on the crack flanks just ahead of the initial crack together with the phenomenon of ratchetting when more than one load cycle are applied between node releases. It should be mentioned that the material
model adopted has some influence of the closure behaviour as reported by [5,37] for plane strain crack closure
analyses.
It has also been shown that under plane stress, the node displacement method and the weight function technique give similar opening stress values, in the limit these techniques lead to the same results provided a very
refined mesh is used has shown in [7].
It is clear from this investigation that the opening stresses are a function of the number of load cycles
applied between node releases. It can be speculated that during the earlier stages of fatigue crack propagation
closure under plane strain is less pronounced, although as the crack grows the rate of crack propagation
increases an therefore it is reasonable to expect an increasing of the opening stresses. It is known that at
the final stage of the cracking process, at higher rates of crack propagation and larger plastic zones close
to the crack tip the predominant mechanical behaviour of fatigue cracks is closer to ‘‘plane stress’’. One of
the reasons for this may be the fact that the opening stresses under plane strain conditions show some dependence on the rate of crack propagation. More experimental work is needed to quantify the three-dimensional
change in closure behaviour during fatigue crack propagation, for example measuring fatigue striations at different stages of crack growth [38,39], although this procedure can be time consuming and is not straight forward. On the numerical side it would be interesting to develop a model where the crack is allowed to grow
according to a physical criteria (e.g. a failure model including a critical value for the total accumulated tensile
strains, as reported by [40]) including more realistic plasticity laws (e.g., strain hardening or softening).
8. Conclusions
The main conclusions of the present work are as follows:
• The number of load cycles between node releases is an important aspect to take into account in 2D plane
strain and 3D FEM analyses of plasticity-induced fatigue crack closure, due to rachetting in the region
close to the crack tip;
• In 2D plane strain and 3D FEM analyses increasing the number of load cycles between node releases has a
significant decreasing effect on the opening stresses.
• This investigation suggests that the ‘‘optimum’’ number of load cycles between node releases is two for 2D
plane stress problems. For 2D plane strain and 3D FEM analyses and exact number of load cycles between
node releases was not established, although for more than eight load cycles between node releases the influence of the crack growth scheme on the opening stresses is very little.
• The number of load cycles between node releases needs to be chosen carefully to give a compromise
between accuracy and computational cost of the analysis.
• For 2D plane strain fatigue cracks, whose behaviour can be described using an elastic perfectly plastic
material model, crack closure under constant amplitude loading exists only at the beginning of the crack
growth process.
• The initial residual plastic deformation left on the crack flanks just ahead of the initial crack together with
the rachetting effect are responsible for the initial transient closure behaviour of 2D plane strain fatigue
cracks.
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• This investigation shows that in the case of 3D fatigue cracks closure is mainly a surface phenomenon so
that closure is likely to disappear altogether remote from the surfaces as the crack becomes fully developed.
Acknowledgements
Acknowledgements are due to the Portuguese Fundação para a Ciência e a Tecnologia (FCT) for providing
a D.Phil. scholarship for P.F.P. de Matos (reference SFRH/BD/12989/2003, financed by POSI). A number of
the finite element calculations reported here were undertaken at the Oxford Supercomputing Centre (OSC).
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