ORIGINAL CONTRIBUTION
doi: 10.1111/ffe.12292
Non-planar mixed-mode growth of initially straight-fronted surface
cracks, in cylindrical bars under tension, torsion and bending, using the
symmetric Galerkin boundary element method-finite element method
alternating method
L. -G. TIAN1,2, L. -T. DONG2, N. PHAN3 and S. N. ATLURI2
1
Department of Geotechnical Engineering, College of Civil Engineering, Tong ji University, Shanghai 200092, China, 2Center for Aerospace Research and
Education, University of California, Irvine USA, 3Structures Division, Naval Air Systems Command, Patuxent River MD, USA
Received Date: 1 August 2014; Accepted Date: 19 January 2015; Published Online: 2015
A B S T R A C T In this paper, the stress intensity factor (SIF) variations along an arbitrarily developing
crack front, the non-planar fatigue-crack growth patterns, and the fatigue life of a round
bar with an initially straight-fronted surface crack, are studied by employing the 3D
symmetric Galerkin boundary element method-finite element method (SGBEM-FEM)
alternating method. Different loading cases, involving tension, bending and torsion of
the bar, with different initial crack depths and different stress ratios in fatigue, are
considered. By using the SGBEM-FEM alternating method, the SIF variations along
the evolving crack front are computed; the fatigue growth rates and directions of the
non-planar growths of the crack surface are predicted; the evolving fatigue-crack growth
patterns are simulated, and thus, the fatigue life estimations of the cracked round bar are
made. The accuracy and reliability of the SGBEM-FEM alternating method are verified
by comparing the presently computed results to the empirical solutions of SIFs, as well as
experimental data of fatigue crack growth, available in the open literature. It is shown
that the current approach gives very accurate solutions of SIFs and simulations of fatigue
crack growth during the entire crack propagation, with very little computational
burden and human–labour cost. The characteristics of fatigue growth patterns of initially
simple-shaped cracks in the cylindrical bar under different Modes I, III and mixed-mode
types of loads are also discussed in detail.
Keywords fatigue crack growth; round bar; SGBEM-FEM alternating method; stress
intensity factor.
NOMENCLATURE
a
a0
A
BIE
c
C
CT
CB
CTR
CTCA
da/dN
D
Da
E
FEM
FEAM
= crack depth
= initial crack depth
= area of the cross section of the round bar
= boundary integral equation
= crack surface arc length of the round bar
= parameter of Paris Law
= loading type, cyclic tension
= loading type, cyclic bending
= loading type, cyclic torsion
= loading type, cyclic torsion and cyclic axial loading
= fatigue crack growth rate
= diameter of the round bar
= surface tangential operator
= Young’s modulus
= finite element method
= finite element alternating method
Correspondence: L.-T. Dong. E-mail: dong.leiting@gmail.com
© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13
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2
L.-G. TIAN et al.
= the moment of initial of the cross section of the round bar
= mode I stress intensity factor
= mode II stress intensity factor
= mode III stress intensity factor
= fracture toughness of the material
= length of the specimen
= the bending moment applied to the specimen
= parameter of Paris Law
= number of load cycles
= the axial loading applied to the specimen
= stress ratio
= symmetric Galerkin boundary element method
= stress intensity factor
= range of stress intensity factor
= crack growth length in x-direction
= crack growth length in y-direction
= Poisson’s ratio
= the angle between the crack growth orientation and the plane of the cross
section of the round bar
σ = the applied nominal stress
σ 0 = monotonic yield strength of the material
σ b = the applied bending nominal stress
σ m = ultimate tensile strength of the material
σ t = the applied tension nominal stress
I
KI
KII
KIII
KIC
L
M
n
N
P
R
SGBEM
SIF
ΔK
Δx
Δy
ν
θ
INTRODUCTION
Round bars are widely used in various mechanical
systems to connect structural components. They may
be subjected to different types of loads, such as tension,
compression, bending and torsion. Fatigue failure often
occurs when such round bars are subjected to cyclic
loads, and such failures mostly originate from small
surface flaws. Several studies have been carried out to
compute the stress intensity factors (SIFs) along the
crack front of defective round bars and to estimate
the shape of the crack surface during fatigue
growth.1–5 The crack front is most usually treated in
prior literature as an equivalent elliptical arc during
fatigue growth, but the initial surface flaw is often
approximated as a straight-fronted crack. Also,
straight-edged surface flaws are often manually created
in experiments, to investigate the fatigue growth of
small surface flaws in round bars.
The SIFs, and fatigue growth of cracks in a round
bar, have been studied using various numerical
methods6–10 and by conducting experiments.11 Lin
and Smith12 used the 3D finite element analyses to
simulate Mode I fatigue crack propagation in a round
bar. It was shown that the crack shape after fatigue
growth is not sensitive to the aspect ratio of the initial
small surface flaw. Tschegg13 studied the fatigue
growth of Mode III cracks in a round bar and showed
that a significant reduction of crack growth rate was
observed. This is because of the ‘crack closure’ effect
due to the interaction between the fatigue crack surfaces. By employing the R-curve method, Yu et al.14 estimated the crack extension rate and discussed the
influence of crack surface contact on the crack growth
rate. Meanwhile, several experiments were conducted
to study the fatigue behaviour of cracked round bars.
Branco et al.15 determined the Paris Law constants in
a round bar made of S45 steel. They used a three-step
reverse engineering technique, from the analysis of
final fracture surfaces. Yang and Kuang16,17 conducted
experiments to study the fatigue crack growth of
surface cracks in round bars under various types of
loads. The fatigue life, surface crack extension direction
and the crack growth rate were investigated in their
experiments. It was shown that a compression load
superimposed on the cyclic Mode III (torsion) load
leads to a significant reduction on the crack growth
rates.
In spite of its widespread popularity, the traditional
finite element method (FEM), with simple polynomial
interpolations, is unsuitable for modelling cracks and
their propagations. This is partially because of the highinefficiency of approximating stress and strain singularities using polynomial FEM shape functions. In order to
overcome this difficulty, embedded-singularity elements
by Tong, Pian and Lasry,18 Atluri, Kobayashi and
© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13
MIXED-MODE FATIGUE ANALYSES OF CYLINDRICAL BARS USING THE SGBEM-FEM ALTERNATING METHOD
Nakagaki,19 and singular quarter-point elements by
Henshell and Shaw20 and Barsoum,21 were developed in
order to capture the crack tip/crack front singular field.
Many such related developments were summarised in
the monograph by Atluri,22 and they are now widely
available in many commercial FEM software. However,
the need for constant re-meshing makes the automatic
fatigue crack growth analyses, using traditional FEM, to
be very difficult.
In a fundamentally different mathematical way,
after the derivation of analytical solutions for embedded elliptical cracks in an infinite body whose faces
are subjected to arbitrary tractions,23 the first paper
on a highly accurate Finite Element (Schwartz–Neumann)
Alternating Method was published by Nishioka and
Atluri.24 The finite element (Schwartz–Neumann)
alternating method uses the Schwartz–Neumann alternation between a crude and simple finite element
solution for the uncracked structure and the analytical
solution for the crack embedded in an infinite body.
Subsequent 2D and 3D variants of the finite element
alternating methods were successfully developed and
applied to perform structural integrity and damage
tolerance analyses of many practical engineering
structures.25 Recently, the symmetric Galerkin boundary element method-finite element method (SGBEMFEM) alternating method, which involves the alternation between the very crude FEM solution of the
uncracked structure and an SGBEM solution for a small
region enveloping the arbitrary non-planar 3D crack,
was developed for arbitrary 3D non-planar growth of
embedded and surface cracks by Nikishkov, Park and
Atluri26 and Han and Atluri.27 An SGBEM ‘super
element’ was also developed in Dong and Aluri28–31 for
the direct coupling of SGBEM and FEM in fracture
and fatigue analysis of complex 2D and 3D solid
structures and materials. The motivation for this series
of works, by Atluri and many of his collaborators since
the 1980s, is to explore the advantageous features of
each computational method: model the complicated
uncracked structures with simple FEMs and model the
crack singularities by mathematical methods such as
complex variables, special functions, boundary integral
equations (BIEs) and by SGBEMs.
In the present paper, by employing the SGBEMFEM alternating method, the SIFs at the evolving front
of a non-planarly growing surface crack in round bars
are computed and compared with the available empirical solutions. The process of fatigue growth of the
initial surface crack is also simulated, considering
various types of load cases. The SIFs along the crack
front are computed during each step of crack increment. An additional layer of boundary elements is
added to the crack front, in the direction determined
© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13
3
by the Eshelby force vector,32 with the crack front
advance size determined locally by Paris’ Law. The
fatigue crack growth patterns and fatigue lives are
thereafter predicted for a damage tolerance evaluation.
The validity of the SGBEM-FEM alternating method
is illustrated, through the comparison of the present
SGBEM-FEM alternating results to the available
empirical and other numerical solutions and experimental observations in the open literature.
SGBEM-FEM ALTERNATING METHOD: THEORY
AND FORMULATION
The SGBEM has several advantages over traditional and
dual BEMs by Rizzo,33 Portela, Aliabadi and Rooke,34
such as resulting in a symmetrical coefficient matrix of
the system of equations and the avoidance of the need
to treat sharp corners specially. Early derivations of
SGBEMs involve regularisation of hypersingular
integrals; see the work by Frangi and Novati35; Bonnet,
Maier and Polizzotto36; Li, Mear and Xiao37; and Frangi,
Novati, Springhetti and Rovizzi.38 A systematic procedure to develop weakly singular symmetric Galerkin
boundary integral equations was presented by Han and
Atluri.39,40 The derivation of this simple formulation
involves only the non-hypersingular integral equations
for tractions, based on the original work reported in
Okada, Rajiyah and Atluri.41,42 It was used to analyse
cracked 3D solids with surface flaws by Nikishkov, Park
and Atluri26 and Han and Atluri.27
For a domain of interest as shown in Fig. 1, with
source point x and target point ξ, 3D weakly singular
symmetric Galerkin BIEs for displacements and tractions
were developed by Han and Atluri.39
Fig. 1 A solution domain with source point x and target point ξ.
L.-G. TIAN et al.
4
The displacement BIE is as follows:
1
2
¼
∫
∫
v ðxÞup ðxÞdSx
∂Ω p
∫
*p
v ðxÞdSx ∂Ω t j ðξÞuj
∂Ω p
∫
þ∫
þ
ðx; ξÞdSξ
∫
v ðxÞdS ∫
*p
v ðxÞdSx ∂Ω Di ðξÞuj ðξÞGij
∂Ω p
∂Ω p
*p
x ∂Ω ni ðξÞuj ðξÞφij
(1)
ðx; ξÞdSξ
ðx; ξÞdSξ :
And the corresponding traction BIE is as follows:
¼
1
2
∫
∫
Da ðxÞwb ðxÞdSx
∂Ω
∫
þ∫
∂Ω
wb ðxÞt b ðxÞdSx
∫
∫
*q
t ðξÞGab ðx; ξÞdSξ
∂Ω q
∂Ω
wb ðxÞdSx
∂Ω
Da ðxÞwb ðxÞdSx
*q
(2)
n ðxÞt q ðξÞφab ðx; ξÞdSξ
∂Ω a
∫
∂Ω
Dp ðξÞuq ðξÞH abpq ðx; ξÞdSξ :
In Eqs. (1) and (2), Da is a surface tangential operator
∂
∂ξ s
∂
Da ðxÞ ¼ nr ðxÞersa :
∂xs
Da ðξÞ ¼ nr ðξÞersa
(3)
*p *q *q
Kernels functions uj ; Gab ; φab ; H abpq can be found in
the paper by Han and Atluri,39 which are all weakly
singular, making the implementation of the current BIEs
very simple.
As shown in Fig. 2, by applying Eq. (1) to Su, where
displacements are prescribed, and applying Eq. (2) to St
where tractions are prescribed, a symmetric system of
equations can be obtained as follows:
2
38 9 8 9
A pp A pq A pr >
=
< fp >
= >
<p>
6A
7
A
A
(4)
¼
fq ;
q
4 qp
qq
qr 5
>
;
: >
; >
: >
A rp A rq A rr
fr
r
where p, q, r denotes the unknown tractions at Su,
unknown displacements at St ’, and unknown displacement discontinuities at Sc, respectively.
With the displacements and tractions being first
determined at the boundary and crack surface, the
displacements, strains and stress at any point in the
domain can be computed using the non-hyper singular
BIEs in the paper by Okada, Rajiyah and Atluri.41,42
Therefore, the path-independent or domain-independent
integrals can be used to compute the SIFs. Alternatively,
with the singular quarter-point boundary elements at the
crack face, SIFs can also be directly computed using the
displacement discontinuity at the crack front elements;
see the paper by Nikishkov, Park and Atluri26 for detailed
discussion. For fatigue growth of cracks, there is no need
to use any other special techniques to describe the crack
surface, such as the level sets. The crack surface is already
efficiently described by boundary elements. In each fatigue
step, a minimal effort is needed: One can simply extend the
crack by adding a layer of additional elements at the crack
tip/crack front. This greatly saves the computational time
for fatigue crack growth analyses.
However, SGBEM is unsuitable for carrying out
large-scale simulations of complex structures. This is
because the coefficient matrix for SGBEM is fully
populated. To further explore the advantages of both
FEM and SGBEM, Han and Atluri27 coupled FEM and
SGBEM indirectly, using the Schwartz–Neumann
alternating method. As shown in Fig. 3, simple FEM is
used to model the uncracked global structure, and
SGBEM is used to model the local cracked subdomain.
By imposing residual loads at the global and local
boundaries in an alternating procedure, the solution of
the original problem is obtained by superposing the
solution of each individual sub-problem. The detailed
procedures are described as follows.
(1) As shown in Fig. 3(a), firstly solve the problem of
the entire domain ΩFEM using FEM, with all the
externally prescribed displacements and tractions
but without considering the cracks. The resulting
can be obtained
tractions on crack surfaces SSGBEM
c
SGBEM
FEM
¼ pc .
as pc
(2) As shown in Fig. 3(b), model a local subdomain
ΩSGBEM with all the cracks by SGBEM, only consideron the crack surface SSGBEM
.
ing the tractions pSGBEM
c
c
SGBEM
SGBEM
on St
are set
The prescribed tractions t
to be 0. Then the resulting tractions on the
intersection surface SI are obtained, denoted as
pSGBEM
on SSGBEM
.
u
u
(3) Applying the tractions on the intersection surface
as residual loads to ΩFEM, denoted as pFEM ¼
pSGBEM
on SI in Fig. 3(c). Resolve the problem by
u
FEM, and obtain again the residual crack surface
on SSGBEM
.
tractions pSGBEM
c
c
(4) Repeat steps 2–3 until pFEM are very small. The
solution of the original problem is obtained by
the superposition of the FEM solution and the
SGBEM solution, as shown in Fig. 3(d).
The great flexibility of this SGBEM-FEM alternating
method is obvious. The SGBEM mesh of the cracked
sub-domain is totally independent of the crude FEM
mesh of the uncracked global structure. Because the
SGBEM is used to capture the stress singularity at crack
© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13
MIXED-MODE FATIGUE ANALYSES OF CYLINDRICAL BARS USING THE SGBEM-FEM ALTERNATING METHOD
5
tips/crack fronts, a very coarse mesh can be used for the
FEM model of the uncracked global structure. Because
FEM is used to model the uncracked global structure,
large-scale structures can be efficiently modelled.
In this study, the SGBEM-FEM alternating method is
used to simulate the fatigue growth of both Mode I and
mixed-mode cracks in round bars. The detailed discussions of numerical simulations and their comparisons to
the available empirical solutions and experimental results
are given in the next section.
GEOMETRY OF SPECIMENS AND EXPERIMENTAL
CONFIGURATIONS
Fig. 2 A defective solid with arbitrary cavities and cracks.
The SGBEM-FEM models in the present study are
created on the basis of the geometries given in the studies
by Yang and Kuang,16,17 in which many experiments were
carried out on the fatigue behaviour of round bars with an
initially straight-fronted surface crack. The geometrical
configuration of the specimen is shown in Fig. 4. The
diameter D is 12 mm, and the length L is 90 mm. Straightfronted surface cracks were cut in the mid-plane using
linear cutting machines, with different initial depths a0.
(a)
(b)
(c)
(d)
Fig. 3 Superposition principle for SGBEM-FEM alternating method: (a) the uncracked body modelled by simple FEM, (b) the local domain
containing cracks modelled by SGBEM, (c) FEM model subjected to residual loads and (d) alternating solution for the original problem.
© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13
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L.-G. TIAN et al.
Fig. 4 Geometry of the specimen, all dimensions are in millimetre.
The material is carbon steel S45, and the mechanical
properties of the test material at room temperature are
listed in Table 1. According to Yang and Kuang,17 fatigue
growth of cracks for this material under cyclic loading
can be characterised by Paris Law43
da
¼ C ðΔK Þn
dN
C ¼ 2:48661014 ; n ¼ 3:2560:
(5)
pffiffiffiffiffiffiffiffi
In Eq. (5), da/dN is in mm/cycle, and ΔK is in MPa mm.
Various far field sinusoidal cycling loads are applied to
the specimen. As listed in Table 2, four types of load
cases are considered in this study.
Type I Cyclic tension (CT) represents a simple CT load
with stress ratio 0.1, to study the Mode I SIFs and the
Mode I fatigue behaviour of the straight-fronted crack
in the round bar with pure tension load.
Type II Cyclic bending represents a simple cyclic
bending load with stress ratio 0 to study the Mode I SIFs
Table 1 Mechanical properties of S45 steel at room temperature
Ultimate tensile strength, σ m
775 MPa
Monotonic yield strength, σ 0
Young’s modulus, E
Poisson’s ratio, ν
Fracture toughness, KIC
635 MPa
206 GPa
0.3 pffiffiffiffi
104 MPa m
and the Mode I fatigue behaviour of the straight-fronted
crack in the round bar with pure bending load.
Type III Cyclic torsion represents a simple cyclic torsion
load with stress ratio 0 to study the mixed-mode fatigue
behaviour of the straight-fronted crack in the round bar
with pure torsion load.
Type IV Cyclic torsion and cyclic axial loading (CTCA)
represents a combination of cyclic torsion with CT/
compression to investigate the effect of cyclic axial stress.
The cyclic torsion and cyclic axial loads are synchronous
sinusoidal with the same frequency. The stress ratios for
tension/compression and torsion loads are both 0.
These load cases are modelled by the SGBEM-FEM alternating method. As shown in Fig. 5, the meshes for the
SGBEM-FEM alternating method consist of only 515 finite
elements and 14 boundary elements. On the other hand, for
pure finite element-based methods, it is typical to use more
than a million degrees of freedom to model a cracked 3D
complex structure; see Levén and Rickert44 for examples.
Thus, the current SGBEM-FEM alternating method
greatly reduces the burden of computation and the human
labour for preprocessing, compared with pure FEM-based
numerical methods, by several orders of magnitude.
NUMERICAL RESULTS
In this section, the computational results of the SIFs and
fatigue growths of cracks in round bars are presented.
Table 2 Loading conditions for the specimens
Cyclic axial
loading (N)
Model
ID
CT1
CT2
CB
CTR
CTCA1
CTCA2
Cyclic bending
moment (N m)
Cyclic torsion
moment (N m)
Initial crack
depth a0 (mm)
Pmax
Pmin
Mmax
Mmin
Tmax
Tmin
0.918
1.2
0.918
0.918
1.0
1.0
25 000
25 000
0
0
15 000
0
2500
2500
0
0
0
15 000
0
0
45
0
0
0
0
0
0
0
0
0
0
0
0
100
100
100
0
0
0
0
0
0
CT, cyclic tension; CB, cyclic bending; CTR, cyclic torsion; CTCA, cyclic torsion and cyclic axial loading.
© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13
MIXED-MODE FATIGUE ANALYSES OF CYLINDRICAL BARS USING THE SGBEM-FEM ALTERNATING METHOD
7
a
a 2
a 3
K
pffiffiffiffiffi ¼ 1:04 3:64
þ 16:86
32:59
(7)
D
D
D
σ πa
a 4
þ28:41
:
D
(a)
(b)
Fig. 5 Meshes for the problem of a round bar with a straight-fronted
surface crack: (a) the uncracked global structure is moulded with 515
finite elements (10-noded tetrahedron elements); (b) the initial crack
surface is moulded independently with only 14 boundary elements
(eight-noded quadrilateral elements).
Comparisons are made between the results obtained by
SGBEM-FEM alternating method and other empirical/
numerical solutions and experimental data reported in
the literature. All the numerical simulations using the
SGBEM-FEM alternating method are carried out on a
PC with an Intel Core i7 CPU.
Stress intensity analyses
In order to verify the accuracy of the SGBEM-FEM
alternating method, the computed SIFs are compared
with the available empirical solutions. James and Mills45
obtained an expression for the SIF at the midpoint of a
straight-fronted surface crack in a round bar subjected
to tension, by fitting a polynomial through several other
analytical and empirical results. The equation they
obtained is
a
a 2
K
pffiffiffiffiffi ¼ 0:926 1:771
þ 26:421
D
D
σ πa
a 3
a 4
78:481
þ 87:911
;
D
D
It should be noted that, in their derivation, the thickness
was changed across the round bar diameter to represent the
circular contour of the bar, thus, Eq. (7) only predicts an
‘average’ value of SIF along the straight crack front.
On the basis of Eq. (6), the empirical normalised SIF at
the midpoint of the crack front for the cracked round
bar subjected to axial tension load is 0.9130. The
corresponding computed SIF by SGBEM-FEM alternating method is 0.9555, with 515 finite elements and 14
boundary elements as shown in Fig. 5. On the basis of
Eq. (7), the empirical normalised SIF for the cracked round
bar subjected to a bending moment along the straight crack
front is 0.8466 (an ‘average’ value), and the corresponding
computed SIF by SGBEM-FEM alternating method is
0.8617. The computed SIFs along the initial crack front
for the round bar subjected to a bending moment are
compared with the empirical solution in Fig. 6. As
expected, the computed SIFs are higher in the midpoint
and lower at the free-boundary, in contrast to the constant
value solution of Daoud and Cartwright.46
It is also worth to mention that the computation time
for the presently reported SGBEM-FEM alternating
method for this problem is about 6 s in a regular PC with
Intel i7 CPU. In addition, the human–labour cost is also
very minimal, to generate the independent coarse meshes
of FEM and SGBEM.
Evolving non-planar crack surface under fatigue
The process of fatigue crack growth for each specimen is
divided into nine analysis steps. In each fatigue analysis
(6)
where K is the Mode I SIF, D is the diameter of the
round bar, a is the depth at the midpoint of the crack
front and σ is the applied nominal stress.
Daoud and Cartwright46 also derived an equation for
the SIF of a straight-fronted surface crack in a round
bar subjected to bending load, by a 2D finite element
analyses with variable thickness
© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13
Fig. 6 Normalised SIFs at the initial crack front for the round bar
subjected to a bending moment.
8
L.-G. TIAN et al.
step, the SIFs along the crack front are firstly computed
by the SGBEM-FEM alternating method. The size of
crack advance at each SGBEM node along the crack
front is then determined by the Paris’ Law, based on
the computed SIFs at each node. The direction of crack
propagation at each SGBEM node is further considered
to be the same as the Eshelby force vector (the vector
J-integral), which can be obtained by their relations with
the computed SIFs.32 And finally, the crack growth in an
arbitrary 3D surface direction is modelled by adding a
layer of additional elements at the crack front, with the
determined crack advance sizes and directions at each
SGBEM node.
First, pure cyclic tensile load is considered. The
maximum magnitude of the axial cyclic tensile load is
25 KN, and the stress ratio is 0.1. The numerical model
is denoted as ‘CT1’ in Table 2. The relevant material
constants are listed in Eq. (5). The non-planar evolution
of the crack surface of model ‘CT1’ is shown in Fig. 7.
The photograph of the final-cracked specimen and the
simulated crack surface by SGBEM-FEM alternating
method is shown in Fig. 8.
The load case presented earlier is relatively simple,
because pure Mode I SIFs result in an in-plane crack
growth. However, if cyclic torsion load is considered,
the initial crack will grow in a non-planar way. The value
of crack growth angle θ is a function of the load, material
properties, stress ratio and geometry of the specimen. In
the studies by Makabe and Socie47 and Liu and Wang,48
it was found that when the shear stress ratio R is negative,
the crack often branches into two directions on both sides
of the crack, as shown in Fig. 9(a). But when the shear
stress ratio R is non-negative, the crack propagates only
along one direction at each side of the crack, as shown
in Fig. 9(b). In this case, the non-planar evolution of
the crack surface can be depicted by Δx (the projection
Fig. 7 The propagation of a straight-fronted surface crack in a round bar subjected to cyclic tension (Model ID: CT1): 2D view.
(a)
(b)
Fig. 8 Final fatigue crack shape of model CT1 for the cracked round bar: (a) photograph of the cross section of the specimen, (taken from Yang
and Kuang17); (b) simulated by SGBEM-FEM alternating method.
© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13
MIXED-MODE FATIGUE ANALYSES OF CYLINDRICAL BARS USING THE SGBEM-FEM ALTERNATING METHOD
9
The computational results show that, when the round
bar is subjected to both cyclic torsion and tension, the
initially straight-fronted planar crack grows into a nonplanar surface. In fact, both the cyclic torsion and CT
loads contribute to the crack propagation. When the
crack depth is rather small, it is the torsion that predominates, and the crack growth angle is very large. As the
crack depth gradually increases and the uncracked area
of the round bar decreases, the bending effect at the crack
surface is magnified, and the crack extension angle θ is
gradually decreased.
Fatigue life estimation
In this section, the fatigue life of the cracked round bar is
estimated by using the SGBEM-FEM alternating
method. Paris Law is used to characterise the fatigue
crack growth in the round bar. According to Paris Law,
the fatigue life of the cracked round bar is as follows:
Fig. 9 Surface crack extension behaviour of the round bar under cyclic torsion loading.
of the actual crack growth length on the horizontal
x-direction) and θ (the angle between the crack growth
direction and the cross section of the round bar), which
are shown in Fig. 10.
In the following case, we consider that the cracked
round bar is subjected to a combination of cyclic torsion
and CT load. The cyclic torsion and tension loads are
synchronous sinusoidal functions with the same stress
ratio. The numerical model used here is labelled as
‘CTCA1’, as detailed in Table 2. The mixed-mode
fatigue growth of the crack is shown in Fig. 11 in a 2D
view. The photograph of the final crack shape by experiment and the simulated crack shape by SGBEM-FEM
alternating method is compared in Fig. 12. The final
fatigue crack shape is also plotted in different perspectives in Fig. 13.
Fig. 10 Surface crack extension paths for the cracked round bar.
© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13
N¼
∫
af
a0
1
da:
C ðΔK Þn
(8)
Thus, at each fatigue growth step, the maximum value
of ΔK along crack front is computed by the SGBEMFEM alternating method. And Eq. (8) is numerically
evaluated using the Trapezoidal rule. For the cracked
round bar under CT (numerical model labelled as
‘CT1’), the predicated fatigue life is plotted in Fig. 14,
and the final fatigue life of the specimen is listed and
compared with the experimental result in Table 3. Excellent agreements are found between the simulation by
SGBEM-FEM alternating method and experimental
results.
For the round bar under cyclic tensile loading with
initial crack depth of 1.2 mm (model labelled as ‘CT2’),
the computed fatigue life by SGBEM-FEM alternating
method is compared with the experimental results by
Yang and Kuang17 in Fig. 15, which also agree well with
the experimental results.
10
L.-G. TIAN et al.
Fig. 11 The propagation of a straight-fronted surface crack in a round bar subjected to cyclic torsion and cyclic tension (Model ID: CTCA1):
2D view.
(a)
(b)
Fig. 12 Final fatigue crack shape of model CTCA1 for the cracked round bar: (a) photograph of the crack path on the surface of the specimen,
(taken from Yang and Kuang16); (b) simulated by SGBEM-FEM alternating method.
Fig. 13 The final crack shape of model CTCA1 after fatigue growth under combined cyclic torsion and cyclic tension from different
perspectives.
© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13
MIXED-MODE FATIGUE ANALYSES OF CYLINDRICAL BARS USING THE SGBEM-FEM ALTERNATING METHOD
Fig. 14 Surface crack arc length versus the number of load cycles for
the round bar subjected to cyclic tension (model labelled as CT1).
Table 3 The predicted fatigue life of model CT1 by SGBEMFEM alternating method and the corresponding experimental
result by Yang and Kuang17
Method
Fatigue life
of specimen
Experiment
(cycles)
SGBEM-FEM alternating
method (cycles)
323 637
333 137
CT, cyclic tension; SGBEM-FEM, symmetric Galerkin boundary
element method-finite element method.
11
Fig. 16 Fatigue crack surface arc length versus the number of load
cycles for the round bar subjected to cyclic torsion and cyclic axial
loading (model labelled as CTCA1 and CTCA2).
the round bar is plotted against the number of load cycles
for models ‘CTCA1’ and ‘CTCA2’ in Fig. 16. It is clearly
shown that the axial CT superimposed on the cyclic
torsion will significantly reduce the fatigue life of the
cracked round bar. On the contrary, for the model of
‘CTCA2’, the axial compression reduces the SIFs during
the process of fatigue crack growth and significantly
increases the fatigue life of the round bar.
It should be noted that, for modes II and III crack
propagations, plasticity-induced or roughness-induced
crack closure may significantly affect lifetime estimation
of the specimen. In this paper, simple Paris Law is used
to predict the fatigue growth rate. However, other
models that account for the crack closure effects can also
be incorporated in the framework of the current
SGBEM-FEM alternating method, which will be
implemented in our future studies.
CONCLUSIONS
Fig. 15 Fatigue crack surface arc length versus the number of load
cycles for the round bar subjected to cyclic tensile loading (model
labelled as CT2).
The fatigue lives of the cracked round bar under cyclic
torsion and cyclic axial loadings are also studied using the
current method. The crack arc length on the surface of
© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13
In the present paper, an elastic round bar with an
initially straight-fronted surface crack is studied by
employing the 3D SGBEM-FEM alternating method.
Several numerical models of the cracked round bar
are built, considering the different loading cases, different initial crack depths and different stress ratios. SIF
variations, the evolving non-planar surface crack patterns and fatigue lives of the cracked round bars are
thoroughly investigated using the 3D SGBEM-FEM
alternating method. By carefully comparing the numerical results with empirical solutions and experimental
observations available in the open literature, the
following conclusions can be drawn:
12
L.-G. TIAN et al.
(a) The SGBEM-FEM alternating method requires very
simple independent and very coarse meshes for both
the uncracked global structure and the crack surface;
it only requires minimal computational burden and
very minimal human–labour efforts for modelling
the fatigue growth of cracks in 3D structures.
(b) The computed SIF variations for the cracks using the
SGBEM-FEM alternating method are in good
agreement with the empirical solutions.
(c) The predicted non-planar fatigue crack growth
patterns and fatigue lives of the cracked round bars
by using the SGBEM-FEM alternating method
agree well with the experimental observations,
showing the reliability and high accuracy of the
current method.
(d) For the cyclic pure tension of the round bar, the
crack propagates in the same plane, but the final
crack front does not strictly stay in a straight-line;
for combined cyclic torsion and CT loads, the crack
growth is along a curved path, and the crack
extension angle decreases during the propagation.
(e) The axial cyclic loading plays an important role in
the fatigue life of the cracked round bar. A significant
reduction of crack growth rate can be achieved, by
superimposing axial compression load to the torsion
load.
Acknowledgements
The first author gratefully acknowledges the financial
support of China Scholarship Council (Grant No.
201306260034); National Basic Research Program of
China (973 Program: 2011CB013800); and New Century
Excellent Talents Project in China (NCET-12-0415).
This work was supported at UCI by a collaborative
research agreement with ARL. The encouragement of
Messrs. Dy Le and Jarret Riddick is thankfully
acknowledged.
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