COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
A Frequency-Domain Approach for Transient Dynamic Analysis Using
Scaled Boundary Finite Element Method (II): Application to Fracture
Problems
Z. J. Yang 1, 2, A. J. Deeks 2*, H. Hao 2
1
2
College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, 310027 China
School of Civil and Resource Engineering, the University of Western Australia, WA 6009 Australia
Email: [email protected]
Abstract The scaled boundary finite element method (SBFEM) allows mixed-mode stress intensity factors (SIFs) to
be accurately determined directly from the definition because the stress solutions in the radial direction are analytical.
This makes the SBFEM superior to the traditional finite element method and boundary element method when
calculating SIFs, since the other approaches generally need special crack-tip treatment, such as refining the crack-tip
mesh or using singular elements. In addition, anisotropic material behaviour can be handled with ease by the SBFEM.
These advantages are expoited in this study, in which the new frequency-domain approach combining the Frobenius
solution procedure [1] and the fast Fourier transform technique, as developed and validated in the accompanying paper
[2], is applied to model transient dynamic fracture problems. Two benchmark problems with isotropic and anisotropic
material behaviour are modelled with a small number of degrees of freedom. Excellent agreement is observed between
the results of this study and those in published literature. The new frequency-domain approach thus provides a
competitive alternative to model dynamic fracture problems with high accuracy and efficiency.
Keywords: linear elastic fracture mechanics, transient dynamic analysis, scaled boundary finite element method,
Frobenius solution procedure, frequency-domain approach, fast Fourier transform
INTRODUCTION
Dynamic stress intensity factors (DSIFs) are the most important parameters in the linear elastic fracture mechanics for
predicting when and in which direction a new crack initiates or an existing crack propagates in structures subjected to
dynamic loads. Numerous numerical methods have been developed and applied to evaluate the DSIFs. In terms of how
the modelled domain is discretised in space, these methods can generally be broken down into four groups: the finite
difference method (FDM), the finite element method (FEM), the boundary element method (BEM) and the meshless or
meshfree method. In terms of how the time histories of DSIFs are computed, these methods can be classified into
time-domain methods based on the direct time integration, and integral transform methods including the Laplace
transform and the Fourier transform (i.e., the frequency-domain method based on frequency analysis, and the discrete
or fast Fourier transform (FFT)). Excellent literature reviews of these methods have been reported recently by [3] and
[4], among others.
Because of the difficulty in handling irregular geometries, the traditional FDM has rarely been used to calculate DSIFs,
although the problem of a plate with a central crack, first investigated by Chen using the FDM [5], has since become a
benchmark for validating different numerical methods. The FEM soon became the method of choice for calculating
DSIFs, because of its capability of modelling complex geometries and material properties. A comprehensive
time-domain analysis carried out by Murti and Villiappan [6] using singular quarter-point elements found that an
optimal size of these elements exists at the crack tip, dependent upon various inherent FEM restrictions, such as the
mesh size and uniformity, the formulation of the mass matrix, the choice of time-steps, and the frequency content of
external loads. A FEM implemented in the frequency domain is rare for DSIFs computation. The application of the
BEM to compute DSIFs has been attracting much attention at present, probably because only domain boundaries are
discretised and representation of cracks is simpler than the FEM. Extensive studies have been published, notably by
⎯ 765 ⎯
Domingues and his co-workers in both the time domain [7, 8] and the frequency domain [3, 7], using the singular
quarter-point elements. The dual boundary element method (DBEM), which uses simultaneously the displacement and
traction integral equations, has also been applied to compute the DSIFs, notably by Fedelinski and Aliabadi and their
co-workers in the time-domain [9] and using the Laplace transform [10]. Although the BEM has gained considerable
success in computing the DSIFs, the need for a fundamental solution limits its applicability considerably. Anisotropic
material behaviour has also been sporadically investigated, e.g., by Zhu et al [11] and by Albuquerque et al [12].
The scaled boundary finite-element method (SBFEM), developed recently by Wolf and Song [13-14], is a
semi-analytical method combining the advantages of the FEM and the BEM, and also possessing its own attractions,
including: (1) it reduces the modelled spatial dimensions by one and discretises the boundaries only as the BEM, but
does not need fundamental solutions; (2) anisotropic constitutive behaviour can be easily modelled [14]; (3) no
discretization of crack surfaces, boundaries and bi-material interfaces that are connected to scaling centres is needed,
potentially reducing the computational cost significantly; and (4) the displacement and stress fields are analytical in
the radial direction. The analytical stress field explicitly represents stress singularities at crack tips, which allows
accurate SIFs to be computed directly from the definition. This has been well demonstrated by recent studies
calculating static SIFs for isotropic materials [15] and anisotropic materials [16]. The above features of the SBFEM
have also been exploited in developing a very simple remeshing procedure for automatic modelling of mixed-mode
crack propagation problems [17]. The only application of the SBFEM to the dynamic fracture problems appears to be
the work of Song in 2004 [4] using a standard time-integration scheme. The time-domain method requires a static
stiffness matrix and a mass matrix. The stiffness matrix can be calculated readily after an eigenproblem is solved. The
mass matrix derived by Song [4] corresponds to the low-frequency expansion of the dynamic stiffness matrix [14] and
thus can only accurately represent the inertial effects at low frequencies. This requires that the size of the subdomains
or super-elements be small enough to account for the highest frequency component of interest [4]. This may lead to
considerable computational cost in solving a significant number of eigenproblems in each time step.
A frequency-domain approach has certain advantages over the time-domain approach, such as no need for a mass
matrix, so that coarser meshes may be used, and once a complex frequency-response function is obtained, it can be
used in combination with FFT and inverse FFT (IFFT) to calculate transient responses for various forms of dynamic
loads. However, calculating transient DSIFs in the frequency-domain using the SBFEM has been hampered by the lack
of an effective analytical solution to the governing non-homogeneous second-order differential equations in the radial
direction. The only reported solution in the frequency domain for bounded media was developed by Song and Wolf
[18]. This approach has not been adopted widely, probably due to the following reasons: (1) the derivation procedure
is complex and hard to follow; (2) the analytical displacement solution is in the form of an infinite matrix series. The
number of terms in the solution must be selected by the analyst, and evaluating the stress field from the displacement
solution is not straightforward; and (3) the resultant analytical formulae for displacements are difficult to interpret
physically. The authors recently developed a new analytical series solution in the frequency domain using the
Frobenius procedure [1]. Compared with Song and Wolf’s solution [18], our solution does not suffer any of the
above-mentioned problems. The solution procedure is very easy to follow. Like the static solution, the
frequency-domain solution has clear physical meaning, and calculating the stress field is trivial. This makes
calculating DSIFs from the definition as simple as from the static stress field. In addition, the dynamic stiffness matrix
is explicitly formed and the number of series terms in the solution is determined by the desired accuracy.
This study applies the new frequency-domain approach using the SBFEM, which is presented in the accompanying
paper [2], to model transient dynamic fracture problems. The newly-developed Frobenius solution [1] in the frequency
domain is employed to calculate complex frequency-DSIFs functions, which are subsequently used with FFT and
IFFT to calculate time histories of DSIFs. The following sections first descirbe the technique used to extract DSIFs
from the solution. Two benchmark problems with isotropic and orthotropic material behaviour are then modelled in
detail using the new frequency-domain method and the results are discussed.
CALCULATION OF DYNAMIC STRESS INTENSITY FACTORS
The DSIFs can be directly extracted from the newly-developed Frobenius solution by definition. The interested
readers are referred to a brief description of the solution in the accompanying paper [2]. The detailed derivation of the
solution procedure can be found in [1]. The solution with (k+1) series is
k +1
n
n
n
u = ∑ ci ξ ( λi )φi + ∑ ξ ( λi )ci ( 2 gi ) + " ∑ ξ (
i =1
1
i =1
2
k +1
λi )
ci ( k +1 gi )
i =1
Explanations of the entities in Eq. (1) can be found in the accompanying paper [2].
⎯ 766 ⎯
(1)
Assuming the (k+1)th solution (Eq. (1)) meets the specified convergence criterion, the displacement field can be
recovered as
n
n
⎡ n
⎤
1
2
k +1
u(ξ, s ) = N(s ) ⎢ ∑ ci ξ ( λi )φi + ∑ ξ ( λi )ci ( 2 gi ) + " ∑ ξ ( λi )ci ( k +1 gi )⎥
⎢⎣ i =1
⎥⎦
i =1
i =1
(2)
For linear elastic constitutive behaviour and small deformation assumption, the stress field can be calculated from Eq. (2)
n
n
⎡ n
⎤
k +1
1
( 1λi −1)
2
( 2λi −1) 2
⎢
σ(ξ, s ) = DB (s ) ∑ ci ( λi )ξ
φi + ∑ ci ( λi )ξ
( gi ) + " ∑ ci ( k +1λi )ξ ( λi −1)( k +1 gi )⎥
⎢⎣ i =1
⎥⎦
i =1
i =1
1
n
n
⎡ n
⎤
1
2
k +1
+ DB2 (s ) ⎢ ∑ ci ξ ( λi −1)φi + ∑ ci ξ ( λi −1)( 2 gi ) + " ∑ ci ξ ( λi −1)( k +1 gi )⎥
⎢⎣ i =1
⎥⎦
i =1
i =1
(3)
where D is the elasticity matrix and B1(s) and B2(s) are relevant matrices [14].
Elastostatics can be regarded as a particular case of elastodynamics. Only the first term in Eq. (2) remains in the
solution to static problems, i.e.,
n
u = ∑ ci ξ λi φi
(4)
i =1
The displacement and stress fields for elastostatics are then simplified respectively as
⎡ n
⎤
1
u(ξ, s ) = N(s ) ⎢ ∑ ci ξ ( λi )φi ⎥
⎢⎣ i =1
⎥⎦
(5)
⎡ n
⎤
⎡ n
⎤
σ(ξ, s ) = DB1 (s ) ⎢ ∑ ciλi ξ (λi −1)φi ⎥ + DB2 (s ) ⎢ ∑ ci ξ (λi −1)φi ⎥
⎢⎣ i =1
⎥⎦
⎢⎣ i =1
⎥⎦
(6)
Eq. (6) can be re-expressed as
n
σ(ξ, s ) = ∑ ci ξ (λi −1)ψ i (s )
(7)
i =1
where each term in Eq. (7) can be interpreted as a stress mode where ψi(s) depends on the circumferential coordinate s
⎪⎧⎪ψxx (s )⎪⎫⎪
⎪⎪
⎪⎪
ψ i (s ) = ⎪⎨ψyy (s )⎪⎬ = D ⎡⎣λi B1 (s ) + B2 (s )⎤⎦ φi
⎪⎪
⎪
⎪⎪ψxy (s )⎪⎪⎪
⎩⎪
⎭⎪i
(8)
Fig. 1 shows a cracked domain modelled by the SBFEM. The scaling center is placed at the crack tip. The square-root
singularity which occurs at crack tip in a homogeneous plate is most widely studied. In this case, the mode-I and
mode-II SIFs are defined as
⎧⎪ 2πr σyy
⎧ K I ⎪⎫
⎪
⎪
⎪ ⎪=
⎨ ⎬ lim ⎨⎪
⎪
K ⎪
r →0 ⎪
2πr σxy
⎪
⎩ II ⎪⎭
⎩⎪⎪
⎫
⎪
⎪
⎬⎪
⎪
θ =0 ⎪
⎪
⎭
θ =0
(9)
where r and θ are the polar coordinates with the origin at the crack tip as shown in Fig. 1. Note the relationship between
r and ξ (ξ=0 at the crack tip and 1 at the boundary, see Fig. 1 in [2]) is
r = ξL(θ)
(10)
where L(θ) is the distance between the crack tip and the intersection point of the polar line r and the domain boundary.
Substituting Eq. (7) into Eq. (9) and considering Eq. (10) we have
⎯ 767 ⎯
n
⎧
⎪
⎪
2πL(θ) ∑ ci ξ (λi −0.5)ψyy (s )i
⎪
⎧ K I ⎪⎫
⎪
⎪
i =1
⎪ ⎪=
⎨ ⎬ lim ⎪⎨
n
⎪
K ⎪
r →0 ⎪
⎪
⎪ 2πL(θ) ∑ ci ξ (λi −0.5)ψxy (s )i
⎩ II ⎪⎭
⎪
⎪
i =1
⎪
⎩
⎫
⎪
⎪
⎪
⎪
⎬⎪
⎪
⎪
θ =0 ⎪
⎪
⎪
⎭
θ =0
y
B
C
O
(11)
r
L(θ)
θ x
A(ξ=1, s=sA)
L0
Nodes
Figure 1: A cracked domain modelled by the scaled boundary finite-element method
Eq. (6) indicates that all the stress modes with λi ≥ 1 vanish for ξ→0 or r→0. Only two modes with λi = 0.5 leads to
singular stresses for ξ→0 or r→0. This also applies to dynamic stress solution Eq. (3) in which all the added
summation terms have mλi ≥ 2 (2≤ m ≤ k+1) considering Eq. (4f) in [2]. Therefore, Eq. (11) is valid for both static and
dynamic problems. Denoting these two stress modes as mode I and mode II with λi = 0.5 and calculating the limit in Eq.
(11) analytically leads to the following SIFs [4]:
⎧
⎪
2πL(θ)ci ψyy (s )i
⎧⎪ K I ⎫⎪ ⎪⎪i∑
⎪ ⎪ = ⎪ =I,II
⎨ ⎬ ⎨
⎪
⎪K II ⎪
⎪ ⎪⎪
⎪ ∑ 2πL(θ)ci ψxy (s )i
⎩
⎭
⎪
⎩i =I,II
⎫⎪
⎛ ⎧⎪ψyy (s = sA )⎫⎪ ⎞⎟
⎪⎪⎪
⎜ ⎪
⎪ ⎟⎟
⎬ = 2πL0 ∑ ⎜⎜⎜ci ⎨
⎬⎟
⎪
⎜ ⎩⎪⎪⎪ψxy (s = sA )⎭⎪⎪⎪i ⎠⎟⎟
i = I,II ⎝
θ =0 ⎪
⎪
⎪
⎭
θ =0
(12)
where L0=L(θ=0) which is the distance between the crack tip and the point A(ξ =1, s = sA) at the crack surface direction
on the boundary (Fig. 1).
It should be noted that the above definition is based on the local coordinate system where the cracking direction is
assumed to coincide with the global x axis. For a domain with a crack surface at arbitrary direction as usually occurs in
crack propagation modelling [17], the stresses in Eq. (16) should first be transformed by the standard procedure to
normal (mode I) stress σn and shear (mode II) stress τn on the cracking surface plane at the point A so that Eq. (12)
becomes
⎛ ⎪σn (s = sA )⎫⎪ ⎞⎟
⎧K ⎫
⎜⎜ ⎪⎧
⎪⎟
⎪⎪ I ⎪⎪ = 2πL
⎨ ⎬
⎬ ⎟⎟
0 ∑ ⎜ci ⎨
⎜
⎪
τ (s = sA )⎪⎪ ⎠⎟⎟
i = I,II ⎝
⎜ ⎪⎪⎩
⎪K II ⎪⎭⎪
⎩
⎪ n
⎭⎪i
(13)
It may also be noted that the point A does not need to be an existing node. Its stress vectors can be conveniently
obtained from Eq. (6).
NUMERICAL EXAMPLES, RESULTS AND DISCUSSION
In this study, two transient dynamic fracture problems are modelled using the developed frequency-domain method
based on the SBFEM. For each problem, the complex frequency-response functions (CFRFs) for a wide range of
frequencies are first computed using the Frobenius solution procedure. The dynamic stress intensity factors KI and KII
are then extracted directly from the stress responses, as presented in the previous section. This is followed by a FFT of
the transient load and an inverse FFT of the CFRFs to calculate the time history of DSIFs. The readers are referred to
[19] for details of the FFT. The functions fft() and ifft() in MATLAB [20] are used to conduct FFT and IFFT
respectively.
As in [7], the damping effect is taken into account by modifying the elastic moduli to incorporate an internal damping
coefficient β, forming the complex Young’s modulus Ec = E(1+i2β) and the complex shear modulus Gc = G(1+i2β),
where E and G are the elastic Young’s modulus and the elastic shear modulus respectively. For each example, damping
⎯ 768 ⎯
coefficients of β = 0.0, 0.001, 0.01, 0.025 and 0.05 are modelled. A convergence tolerance of α = 1×10−3 in Eq. (5a) in
[2] is used for all the analyses. Two-node linear line elements are used to model both examples. Quantitative
comparisons of the proposed method with the FEM, the BEM and the time-domain SBFEM [4] are not conducted with
respect to the computational costs, as the results would be influenced greatly by the algorithms used in the many
different steps in the respective method.
1. Example 1: rectangular plate with a central crack The first example modelled here is the classical problem first
investigated by Chen [5] and adopted in most subsequent studies on DSIFs [6-12]. A rectangular plate with a central
crack is subjected to uniform tractions on its upper and lower surfaces. The geometry, boundary and loading conditions
are shown in Fig. 2(a). The crack length is 2a = 4.8mm. The isotropic material properties are: Young’s modulus
E = 200GPa, Poisson’s ratio ν = 0.3 and density ρ = 5000kg/m3. This is a mode-I fracture problem. A Heaviside step
function representing a suddenly applied load, as shown in Fig. 2(a) is modelled. All SIFs reported below are
normalised by a factor P0(πa)1/2 where P0 is the magnitude of the transient loading and a is the half crack length. A
plane strain condition is assumed.
Only one quarter of the plate is modelled due to the structural symmetry. The domain is divided into two subdomains.
The subdomain with the crack has its scaling centre at the crack tip and the other at its geometrical centre. Fig. 2(b) and
Fig. 2(c) show two meshes with 36 nodes and 72 nodes respectively. The two edges connected to the crack tip are not
discretised. The stresses at the point A in Fig. 2(a) are used to extract SIFs according to Eq. (13).
P
P0
P
t
Heaviside loading
40mm
2a=4.8mm
A
20mm
(a) Dimensions, material
properties and loading conditions
(b) Coarse mesh with 36 nodes
(c) Fine mesh with 72 nodes
Figure 2: Example 1: A rectangular plate with a central crack
21
2.8
14
2.4
Normalised Dynamic KI
Normalised KI
7
0
-7
-14
Real part
Imaginary part
-21
-28
2
1.6
1.2
0.8
0.4
Chirino et al (1994)
Chen (1975)
Coarse mesh (36 nodes), β=0.01
0
-35
-0.4
-42
0
50
100
150
200
250 300
f (103 Hz)
350
400
450
500
Figure 3: Frequency f-normalised KI curves
for Example 1 from β = 0.01 and Δf = 1000Hz
0
2
4
6
8
Time (10-6 s)
10
12
Figure 4: Normalised dynamic KI of Example 1:
comparison with other methods
⎯ 769 ⎯
14
2.8
Normalised Dynamic KI
Normalised Dynamic KI
2.4
2
1.6
1.2
0.8
0.4
Fine mesh (72 nodes), β=0.01
Coarse mesh (36 nodes), β=0.01
0
-0.4
0
2
4
6
8
Time (10-6 second)
10
12
14
Figure 5: Normalised dynamic KI of Example 1: mesh
convergence
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
Coarse Mesh (36 nodes), β=0.01
Song (2004)
Albuquerque et al (2002)
Zhu et al (1996)
0
2.5
5
7.5
-6
Time (10 s)
10
12.5
15
Figure 6: Normalised dynamic KI of Example 1:
anisotropic material
The normalised static mode-I stress intensity factors KI for f = 0Hz (static case) are 1.042, 1.042, 1.041, 1.039 and
1.034 for β = 0.0 (purely elastic material), β = 0.001, β = 0.01, β = 0.025 and β = 0.05 respectively for the coarse mesh
(Fig. 2(b)). They are 1.038, 1.037, 1.037, 1.036 and 1.033 respectively for the fine mesh (Fig. 2(c)). These values agree
well with KI =1.034 for a coarse mesh with 6 subdomains and 54 nodes, and KI =1.036 for a fine mesh with 12
subdomains and 141 nodes in Song [4].
Fig. 3 shows an example of f-normalised KI curves from β = 0.01, with the real part and imaginary part drawn
separately. These complex frequency response curves are used in IFFT to calculate time histories of DSIFs.
Fig. 4 presents the modelled time history of DSIFs using the developed frequency-domain SBFEM for β = 0.01 and the
coarse mesh, compared with those from the DFM by Chen [5] and those from the BEM by Chirino et al [7]. It can be
seen that the overall agreement is excellent, with about 2-4% differences between the peaks from these methods.
Fig. 5 shows the time histories of DSIFs for β = 0.01 from the coarse mesh and the fine mesh. They are in very good
agreement, which indicates that the coarse mesh with only 2 subdomains and 36 nodes leads to sufficiently accurate
results.
The orthotropic case of the same rectangular plate in Fig. 2(a) in plane stress is also modelled using the same meshes
(Fig. 2(b) and Fig. 2(c)). The material properties in the principal material axes are: Young’s modulus E1 = 18.3GPa,
E2 = 54.8GPa, shear modulus G12 = 8.79GPa, Poisson’s ratio ν12 = 0.083 and mass density ρ = 1900kg/m3. The same
problem was analysed by Song using the time-domain SBFEM [4], Zhu et al using FDM [11] and Albuquerque et al
using the BEM [12]. Fig. 6 compares the time histories of DSIFs calculated by these methods and the developed
frequency-domain method in this study. Again, excellent agreement is evident between these results. The current
results appear to be closer to those reported in [11] and [12] than the results of Song [4].
2. Example 2: rectangular plate with a slanted edge crack The second example is a rectangular plate with a slanted
edge crack subjected to uniform traction on its upper surface. The geometry, boundary and loading conditions are
shown in Fig. 7(a). The isotropic material properties are: the shear modulus G = 29.4GPa, Poisson’s ratio ν = 0.286 and
density ρ = 2450kg/m3. This is a mixed-mode fracture problem which has been modelled by various studies [7, 10]. The
Heaviside step loading as shown in Fig. 7(a) is modelled. All SIFs reported in this section are normalised by P0(πa)1/2
where P0 is the magnitude of the transient loading and a = 22.63mm is the crack length. A plane strain condition is
assumed.
The domain is represented by one subdomain with its scaling centre at the crack tip. Fig. 7(b) and Fig. 7(c) show two
meshes with 41 nodes and 83 nodes respectively. The two edges connected to the crack tip representing the crack
surfaces are not discretised. The stresses at the point A in Fig. 7(a) are used to extract SIFs according to Eq. (13).
Fig. 8 and Fig. 9 show the time histories of KI and KII respectively computed using the coarse mesh with various
material damping coefficients β, compared with the results obtained by Fedelinski et al in 1996 using the BEM [10].
There is very good agreement between the results of this study using β = 0.01 and those of [10] for both KI and KII. The
coarse mesh and the fine mesh lead to virtually identical results, as shown in Fig. 10 and Fig. 11, which indicate that
the coarse mesh with only 41 nodes is fine enough for this example. The same isotropic problem with slightly different
material properties was also modelled by Song [4] using the time-domain SBFEM. A much finer mesh with 23
subdomains (super-elements) and over 200 nodes was used in [4]. This reflects one of the disadvantages of
time-domain methods, i.e., a mass matrix must be used in time-integration. Because the mass matrix used in [4] is the
low-frequency expansion of the dynamic stiffness matrix [14], it can only represent the inertial effects at low
frequencies. The higher frequency components of interest must be represented by fine meshes with small sizes. The
⎯ 770 ⎯
frequency-domain method developed in this study, on the contrary, does not need mass matrices. This allows accurate
calculation of time responses using coarse meshes such as Fig. 2(b) and Fig. 7(b).
P
32mm
A
P0
P
44mm
t
Heaviside loading
a
a) Dimensions, boundary and
loading conditions
b) Coarse mesh with 41 nodes
c) Fine mesh with 72 nodes
1.75
1.2
1.5
1
Normalised Dynamic KII
Normalised Dynamic KI
Figure 7: Example 2: A rectangular plate with a slanted edge crack
1.25
1
0.75
0.5
β=0.01
0.25
β=0.025
0
Fedlinski's results (1996)
-0.25
0.6
0.4
β=0.01
0.2
β=0.25
0
Fedlinski's results (1996)
-0.2
0
2.5
5
7.5
10 12.5 15 17.5
Time (10-6 second)
20
22.5
25
Figure 8: Normalised dynamic KI of Example 2 using the
coarse mesh: effect of material damping coefficient β
0
1.75
1.2
1.5
1
1.25
1
0.75
0.5
0.25
Coarse mesh (41 nodes), β=0.01
Fine mesh (83 nodes), β=0.01
0
5
7.5
10 12.5 15 17.5
-6
Time (10 second)
20
7.5
10 12.5 15 17.5
-6
Time (10 second)
20
22.5
25
0.8
0.6
0.4
0.2
β=0.01, Coarse mesh (41 nodes)
β=0.01, Fine mesh (83 nodes)
0
-0.2
-0.25
2.5
5
Fedlinski's results (1996)
Fedlinski's results (1996)
0
2.5
Figure 9: Normalised dynamic KII of Example 2 using the
coarse mesh: effect of material damping coefficient β
Normalised Dynamic KII
Normalised Dynamic KI
0.8
22.5
25
0
2.5
5
7.5
10 12.5 15 17.5
-6
Time (10 second)
20
22.5
25
Figure: 10 Normalised dynamic KI of Example 2: effect of Figure: 11 Normalised dynamic KII of Example 2: effect
mesh density
of mesh density
The orthotropic case of the same rectangular plate in Fig. 7(a) in plane stress is also modelled using the same coarse
mesh (Fig. 7(b)). The material properties in the principal material axes are: Young’s modulus E1 = 82.4GPa, E2 = 2E1,
shear modulus G12 = 29.4GPa, Poisson’s ratio ν12 = 0.4006 and the mass density ρ = 2450kg/m3. The same problem was
analysed by Song using the time-domain SBFEM [4] and Albuquerque et al 2002 using the BEM [12]. Fig. 12 and
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Fig. 13 compare the time histories of KI and KII calculated by these methods and the present frequency-domain method,
respectively. Good agreement is observable between the results from different methods.
1.8
E2=2E1, Song (2004)
1.4
E2=2E1, Present result
E2=2E1, Song (2004)
1
E2=2E1, Albuquerque et al. (2002)
E2=2E1, Albuquerque et al. (2002)
1.2
Normalised KII
Normalised KI
1.2
E2=2E1, Present result
1.6
1
0.8
0.6
0.4
0.2
0.8
0.6
0.4
0.2
0
0
-0.2
-0.2
0
4
8
12
16
20
0
Time (10-6 s)
4
8
12
16
20
Time (10-6 s)
Figure: 12 Normalised dynamic KI of Example 2 with
orthotropic material properties
Figure: 13 Normalised dynamic KII of Example 2 with
orthotropic material properties
CONCLUSIONS
The newly-developed frequency-domain SBFEM is applied to calculate DSIFs of fracture problems subjected to
transient loading. Using a recently developed Frobenius solution technique in the frequency domain for solving the
governing equations of SBFEM, the mixed-mode DSIFs are directly extracted from the stress solutions for a wide
range of frequencies, leading to complex frequency-DSIFs response functions. These functions are then used in
conjunction with the FFT and the IFFT to generate time histories of the DSIFs. Two benchmark problems with both
isotropic and orthotropic material behaviour have been modelled using the new appraoch. Numerical results show that
this approach is capable of computing time histories of DSIFs accurately and effectively with a small number of
degrees of freedom.
Acknowledgements
This research is supported by the Australian Research Council (Discovery Project No. DP0452681) and partly by the
National Natural Science Foundation of China (No. 50579081).
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