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Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
DYNAMIC FRACTAL FINITE ELEMENT METHOD FOR A PENNY-SHAPED CRACK
SUBJECT TO MODE I DYNAMIC LOADING
D. K. L. Tsang1 and S. O. Oyadiji1
1School of Mechanical, Aerospace and Civil Engineering, The
University of Manchester, Manchester M13 9PL, United Kingdom
Email: [email protected] Email:
[email protected]
Received 1 January 2005; accepted 24 February 2005
ABSTRACT The fractal finite element method (FFEM) is an accurate
method to determine the stress intensity factors around crack tips.
The method has been developed to study all kinds of static
two-dimensional crack problems. In this paper we demonstrate how
the method can be extended to include inertia effect. We present
the calculation of dynamic mode I stress intensity factors for
penny shaped cracks in a cylinder subjected to time dependent
axisymmetric loading. The effect of damping is also presented. The
precise time integration scheme is used to perform the time
integration. Our numerical results show that the fractal finite
element method together with the precise time step integration
method gives very accurate dynamic stress intensity factor.
Keywords: Finite element method, dynamic stress intensity factor,
fractal finite element method, penny-shaped crack. 1 INTRODUCTION
The stress intensity factor (SIF) technique in linear elastic
fracture mechanics is very successful in predicting unstable crack
propagation. The process involves the SIF calculation for a crack
in a structure under an applied load. Different modes of SIF are
related to the coefficient of square root singularity in different
Williams eigenfunction stress series. Unstable crack propagation is
assumed to follow when the SIF reaches its critical value cK . When
the load is applied rapidly to the structure, the same phenomenon
is expected to apply; at least as far as the initiation of the
crack motion is concerned. However, the problem is more difficult
than the static case since inertia effect must be considered.
Dynamic fracture mechanics is a branch of fracture mechanics where
the effect of material inertia becomes significant. Dynamic
fracture phenomenon can be further characterised by various dynamic
states of a crack tip. The dynamic states of the crack tip are
induced by impact or dynamic loads applied to a stationary cracked
solid or by fast motions of the crack tip itself. Thus, depending
upon the dynamic states of the crack tip, dynamic fracture
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Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
41
mechanics may be further classified into impact fracture
mechanics and fast fracture mechanics. In the case of impact
loading, the influence of the loads is transferred to the crack by
means of stress waves through the material. It is necessary to
calculate the transient driving force acting on the crack, in order
to check whether or not a crack will propagate due to the stress
wave loading. In the case of fast crack propagation, material
particles on opposite crack faces displace with respect to each
other once the crack edge has passed. The inertia resistance to
this motion can also influence the driving force, and it must be
taken into account in a complete description of the process. The
interaction of a stress wave due to rapid loading and the crack is
a complicated problem. When the stress wave passes through the
crack, an incident wave is reflected by the crack surface and
diffracted from the crack tip. The analytical study of the problem
is restricted to relatively simple cases. However, the study of the
crack interaction is also of great interest from the point of view
of quantitative non-destructive evaluation of structural materials.
The scattered field carries information on the location and the
size of the crack. There is considerable interest in scattering by
cracks, with a view towards solving the inverse problem to obtain
the crack geometry from the scattered fields. Use of the SIF in
examining crack stability requires an accurate knowledge of the
stress field in the vicinity of the crack tip for the given
structural geometry, loading and boundary conditions. The dynamic
response of a crack under the action of impact loading has been
treated analytically by many authors, for example: Sih and Ravera
1972; Thau and Lu 1971; Achenbach and Nuismer 1971; Chen and Sih
1977. These solutions are limited to structures of very large
dimensions compared with that of the crack such that boundary
effects may be neglected in the analysis. The inclusion of the
finite boundaries on the problem poses additional analytic
difficulties due to the interactions between the crack and the edge
of the structure. Consequently, analytical solutions only exist for
selected relatively simple cases due to the complicated boundary
conditions associated with the governing equations. The failure of
analytical investigation in general crack problems has caused many
experimental approaches to be devised for studying the process of
dynamic fracture. However the specimens used in these laboratory
tests have relatively small dimensions. Moreover, due to the very
short time spans of the dynamic fracture event, direct measurement
of the higher order physical quantities such as energy
distribution, instantaneous dynamic energy release rate and dynamic
stress field close to the crack tip is very difficult to achieve in
these test specimens. The usefulness of the stress intensity factor
in the analysis of crack problems has resulted in the determination
of stress intensity factors. Over the last decade or so, the finite
element method has become firmly established as a standard
procedure for the solution of crack problems. Stresses within
regular finite elements are generally represented by functions
containing at most quadratic terms. Since the stresses in the
neighbourhood of a crack tip are square root singular, these
regular finite elements are impotent to describe singular
behaviour. To overcome this difficulty, two basic directions are
taken. Either a refined mesh in the crack tip region is employed or
singular elements are used. In 2D crack problems, it is possible to
employ a refined mesh around the crack tip. However in 3D crack
problems, a large refined mesh needs considerable computational
power.
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D. K. L. Tsang and S. O. Oyadiji
Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
42
Over the years, there have been many singular elements
developed, for examples: Barsoum 1974; Atluri et al. 1975. After
the stress and displacement fields in a crack body are obtained,
the SIF can be determined by either direct or indirect methods.
With direct methods, the SIF is found from the stresses or
displacements obtained from the finite element analysis. Indirect
methods are energy-based methods, which include Griffiths energy,
the J-integral and the stiffness derivative technique. The fractal
finite element method by (Leung and Su 1994; Leung and Su 1995a;
Leung and Su 1995b; Leung and Su 1996a) is a very robust approach.
The method has been extended to cracked plate problems (Leung and
Su 1996b; Leung and Su 1996c), mode III crack problems (Leung and
Tsang 2000) and three dimensional crack problems (Su and Leung
2001). The fractal finite element method has also been used to
study multiple crack problems (Tsang et al. 2003). The method
separates the cracked elastic body into singular and regular
regions. The regular region is modelled by conventional finite
elements. Within the singular region, large number of conventional
finite elements is generated by a self-similarity process to model
the crack tip singular behaviour. The resulting large number of
nodal variables is reduced effectively to a small set of global
variables by global interpolating transformation with SIFs as
primary unknowns. As the global interpolating transformation can be
performed at element level, the order of matrices involved is very
small. Consequently, computer storage and solution time can be
reduced significantly. Also there is no need to generate any new
finite element; any existing standard finite element can be used to
perform the transformation. Furthermore, the stress intensity
factors can be determined directly from the global variables.
Previously, the global interpolating function was sought
analytically using the Williams eigenfunction expansion method.
However, the difficulties and limitations of this approach were
described recently by (Tsang et al. 2004). To overcome these
problems, a hybrid analytical cum numerical method based on the
finite difference approach was proposed and applied in the
determination of the global interpolating function. As a result of
this recent development, it is hoped that subsequently, the fractal
finite element method can be easily applied to anisotropic material
or three dimensional crack problems, where the singularity may not
be a square root function. In dynamic problems, the equations of
motion arising from a finite element analysis consist of a system
of linear ordinary differential equations with constant
coefficients. The most crucial step in time analysis is to choose
an efficient integration method. There are many time integration
methods in the literature. They can be classified into two
categories: explicit and implicit integration schemes. The explicit
integration scheme is very efficient for one time step computation.
However, to ensure stability of integration a very small time step
size must be selected. The implicit integration scheme can be made
unconditionally stable by proper selection of the integration
parameters, such that a large value of time step size can be
selected. However, because the time step size is large, the
vibration components with higher frequencies will be distorted
after several integration steps. Therefore the system invariants,
such as system energy or momentum in a conservative system, cannot
be maintained. The most commonly used time integration schemes in
finite element applications are the method of central difference,
the Newmark method by (Newmark 1959) and the Wilson method by
(Wilson et al. 1973). Zhong (Zhong and Williams 1994) proposed a
high-precision numerical time step integration method for a linear
time invariant structural dynamic system. The method was based
on
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43
Molers work (Moler and van Loan 1978) which was called matrix
scaling and squaring technique. Its numerical results are almost
identical to the precise solution and are almost independent of the
time step size for a wide range of step sizes. The errors are due
solely to computer round-off accuracy. Also the main computations
are all matrix multiplication and so can be efficiently executed on
parallel computers. This method, which is called the precise time
step integration method (PTSIM), is employed in the present work.
In this paper, we demonstrate that the fractal finite element
method can be easily extended to include inertia effect.
Axisymmetric crack problems with time dependent boundary loading
will be considered. The fractal finite element discretization is
used for space coordinates, while the precise time step integration
method is implemented for the direct time domain integration of the
resulting equations of motion. The effect of damping is also
included in the calculation. 2 THE FRACTAL FINITE ELEMENT METHOD In
the fractal finite element method, the overall crack problem is
divided into near field and far field regions as shown in Figure 1.
The curve that delineates the two regions is denoted as
0 . The far field is modelled by conventional finite element
method. In the near field, infinite numbers of layers of
conventional finite elements are generated layer by layer in a
self-similar manner with a scaling ratio . We take the crack tip as
the centre of similarity. By assuming that the value of lies
between 0 and 1, an infinite set of curves { }1 2, , L similar to
the shape of 0 with proportionality constants { }1 2, , L is
generated inside the singular region. The region in between any two
consecutive curves is called the nth layer. The nodes located on
the curve 0 are called the master nodes. A set of straight lines
that emanate from the similarity centre is connected to the master
nodes. Thus each layer is divided into a set of elements with a
similar pattern. All the nodes inside the curve 0 are called the
slave nodes. The grading of the mesh inside the singular region can
be controlled by the proportionality constant 0 1< < . Higher
values of will produce finer grade of mesh and vice versa. An
example of a fractal finite element mesh is depicted in Figure
2.
Figure 1: Near field and far field in a cracked structure
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D. K. L. Tsang and S. O. Oyadiji
Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
44
Figure 2: Fractal element mesh in the near field region Let r
and z denote the radial and axial coordinates of a point
respectively with respect to the origins of a polar coordinate
system. Without loss of generality, a 6-noded triangular element is
considered for axisymmetric stress analysis. Let ( ),u v=w be the
displacement field. The shape functions and the displacement
function may be expressed in an isoparametric form as
( ) ( )6 61 1
, , ,i i i ii i
r N r z N z = =
= = (1) and
( ) ( )6 61 1
, , ,i i i ii i
u N u v N v = =
= = (2) where ( ),i ir z and ( ),i iu v are the nodal
coordinates and the nodal displacements of an element respectively
and ( ),iN are the shape functions. The resulting local static
equilibrium equation is
=Kw f (3) The associated element stiffness matrix K is
calculated by
T
VdV= K B DB (4)
where V is the volume of the element,
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Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
45
( )1 6
0
0, , , , 1, 6
0
i
i
ii
i i
Nz
Nr i
NrN Nz r
= = =
B B B BL L (5)
and
( )( )( )
( )
1 01 1
1 01 1 1
1 1 2 1 01 1
1 20 0 02 1
E
= +
D (6)
where E and are Youngs modulus and Poissons ratio respectively.
Within the regular region, the static equilibrium equations can be
written as
0
0 00 0 0
,rr r r rr
= K K d fK K d f
(7)
where d and f are displacements and force vector respectively.
The subscript r and 0 represent the values in the regular region
and on the 0 curve respectively. Similarly the static equilibrium
equation of the layer n in the singular region is
11 12
1 121 22
n nn n
n nn n+ +
= d fK K
d fK K (8)
After we assemble all the elements, the global static stiffness
equation of the problem is
01 1
0 00 00 11 121 1 2 2
1 121 22 11 122 2 3 3
2 221 22 11 12
21 22
r rrr r
r
=
d fK Kd fK K + K Kd fK K + K Kd fK K + K K
d fK KM MO O O
(9)
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D. K. L. Tsang and S. O. Oyadiji
Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
46
Applying the transformation i i=d Tc , where iT is evaluated
from Williams eigenfunction and c is the vector of generalized
coordinates to be determined, one has
01
0 00 11 0 0 0
0
rr r r r
r s
s ss s
+ =
K K d fK K K K d f
K K c f (10)
where [ ] 10 0 12 0Ts s= =K K K T , [ ]1 Ts k kk==f T f and
11 22 1 11 12 21 22
2
T T k T k T k T kss k k k k k k k k
k
= = + + + + K T K T T K T T K T T K T T K T . (11)
It is well known that the stiffness of an isoparametric finite
element is independent of the actual dimension. Therefore, the
stiffness matrix of every layer in the singular region is the same.
As a result, the summation in equation (11) is a geometric
progression series with the geometric progression ratio being equal
to the scaling factor . Hence the infinite sum can be obtained
analytically without the need to perform transformations on all the
inner element layers. Although in Figure 2 several layers are
shown, only the first layer is enough to generate all the local
stiffness matrix. Hence, the FFEM becomes a meshless finite element
method. After the transformation, the unknowns become rd , 0d and c
instead of id , 1i = to . An additional advantage is that the SIFs
are included in c and no post-processing is required. Essentially,
the original infinite matrices are compressed to a finite one. 3
EIGENFUNCTION FOR AXISYMMETRIC CRACK In this section the
eigenfunction for axisymmetric crack needed in the fractal finite
element analysis of a penny-shaped crack will be described. More
detail descriptions can be found in (Tsang et al. 2004). The
non-dimensional displacement in radial and angular directions
within the singular region can be written as
( ) ( )2 20 0
,n n
r n nn n
u r f u r g
= == = (12)
where r and denote the non-dimensional radial and angular
coordinates of a point, respectively, with respect to the local
coordinate system around the crack tip. The eigenfunctions ( )nf
and ( )ng satisfy the following
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Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
47
( )( )
( )( )
( ) ( )( )( )
( ) ( ) ( ) ( )
1
1 1
2 2
2211 2 1 2
21 2 1 2
1 1 21 1
0 1 2 1 2
21 1
0 1 2 1 2
46 22 2
coscos sin 1 sin2
sin 2 2 coscos 1
n nn
k nn n
k
kn n
k
c nc c n cd f dg fd c c d c c
dfn c cn c f gc c c c d
c ck g fc c c c
=
=
+ ++ + = + +
+
(13)
( )( ) ( )
( ) ( )
( ) ( )( )1 1
1 1
2 2
221 2 1 2 1 2
21 1
1 2 1 21
0 1 1
2
0
4 12 4 28 8
cos coscos sin2 2
cos 1 sin cos sin
n nn
k n nn n
k
kn n
k
n c c c c n c cd g dfgd c c d
df dgc c c cf n gc d d c
k f g
=
=
+ + + + = + + + +
+
(14)
where 1 2 2n n k= and 2 4 2n n k= , respectively. The stress
free boundary conditions for ( )f and ( )g at = are
1
21 1 2
02n
n nk
dgnc c f c c fd
=
+ + = (15)
( )2 2 0n ndf n gd + = (16) Equations (13) and (14) are a system
of ordinary differential equations. The solutions of the ordinary
differential equations are composed of two parts, the complementary
functions and the particular integrals, i.e.
( ) ( ) ( )c pn n nf f f = + (17)
( ) ( ) ( )c pn n ng g g = + (18) where superscript c and p
represent complementary functions and particular integrals,
respectively. The complementary functions cnf and
cng can be evaluated by considering the
homogeneous parts of equations (13) and (14) and the boundary
conditions given by equations (15) and (16), which can be found
analytically. After some derivations, the solutions are obtained in
the form
( ) ( ) ( )1 2,1 ,2c c cn n n n nf a f a f = + (19)
( ) ( ) ( )1 2,1 ,2c c cn n n n ng a g a g = + (20)
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Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
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where
( ) ( ) ( )( ) ( )( ) ( )( ) ( )
1 2 1 2,1
1 2 1 2
2 1 6 22 2sin sin6 2 2 6 2 2
n
cn
c c n c n c nn nfc n c n c n c n
+ + ++ = + + + + + (21)
( ) ( ) ( )( ) ( )( ) ( )( ) ( )
1 2 1 2,2
1 2 1 2
2 1 6 22 2cos cos6 2 2 6 2 2
n
cn
c c n c n c nn nfc n c n c n c n
+ + + ++ = + + + + (22)
( ) ( ) ( )( ) ( )1 2
,11 2
2 12 2cos cos2 6 2 2
n
cn
c c nn ngc n c n
+ + = + + + (23)
( ) ( ) ( )( ) ( )1 2
,21 2
2 12 2sin sin2 6 2 2
n
cn
c c nn ngc n c n
+ + + = + + + (24)
For the particular integrals, a finite difference method by
(Tsang et al. 2004) has been employed to determine all the
eigenfunctions numerically. 4 PRECISE TIME STEP INTEGRATION METHOD
To perform dynamic analysis of a bounded medium in the time domain,
the mass matrix M is required in addition to the static stiffness
matrix K. The analytical expression for the mass matrix is
T
VdV= M N N (25)
where is the density. The mass matrix is transformed following
the procedure outline above for the stiffness matrix. After finite
element discretization, the usual dynamic equation of motion is
( )2 2d dd d tt t+ + =x xM G Kx r (26)
with the initial condition being the given vectors ( )0x and (
)0x& , where G is the damping matrix, x is the displacement
vector to be solved, ( )tr is the external force vector and the dot
denotes differentiation with respect to time t. In a time invariant
system, that is M, G and K are independent of time, the precise
time integration method gives highly precise results, even when G
is a general damping matrix. The scheme transforms the second order
differential equation (26) into a first order differential
equation, whose general solution can be easily found by using an
integrating factor. The general solution involves exponential
matrix
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Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
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computations. Zhong (Zhong and Williams 1994) gives a precise
computation algorithm for an exponential matrix in which he used
exponential matrix multiplication approach, which is similar to the
scaling and squaring method in (Moler and van Loan 1978). The
algorithm is highly precise. Numerical error is solely due to
computer round-off error. Since the integration method uses the
general solution of the dynamic equations, some of the problems of
other numerical integration methods such as stability and stiffness
do not exist. We define
1 1 11, , , ,
2 2 4 2
= = + = = =
Gx M G GM G GMD M P Mx A B K C
Then equation (26) can be rewritten as
1,= +v Hv r& (27)
where , = = x A D
v Hp B C
and 1 =
0r
r.
Equation (27) is inhomogeneous. From the theory of ordinary
differential equations, the general solution of its homogeneous
equation should be found first. The general solution can be given
as
( ) 0exp t=v H v (28) where ( )0 0=v v is the initial
conditions. Now let the time step be . Then ( ) 0 =v Sv , where (
)exp =S H . After finding the matrix S precisely, the time step
integration is
1k k=v Sv , 1, 2,k = L . A precise computation algorithm for an
exponential matrix is given in (Zhong and Williams1994). It uses
the theorem of exponential function
( )exp expm
m =
HH (29)
where m is an arbitrary integer. If 2Nm = is selected, t m =
will be an extremely small time interval for 20m . Therefore the
following truncated Taylor series expansion can be used
( ) ( ) ( ) ( )2 3 4
exp2 3! 4!
a
t t tt t
+ + + += +
H H HH I H
I S (30)
where I is an unit matrix and
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Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
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( ) ( ) ( )2
2 3 122a
t tt t
+ + = + I H HS H H (31) Also S can be factorized as
( )( ) ( ) ( ) ( )( ) ( )
1 1
1
2
2 2
2
N
N N
N
a
a a
a a a a
+= + += + + +
S I S
I S I S
I S S S S
(32)
and the factorization can be repeated recursively. Assuming that
the inhomogeneous term is linear within time step ( )1,k kt t + ,
then equation (27) can be written as
( )0 1 kt t= + + v Hv r r& (33) where 0r and 1r are given
vectors. Let ( )kt tY be the solution of the homogeneous equation Y
= HY& , ( )0 =Y I . Then the solution of equation (33) can be
derived as
( ) ( ) ( ) ( )1 1 1 1 10 1 0 1 1k k kt t t t = + + + v Y v H r
H r H r H r H r (34) Now substituting 1kt t += and ( ) ( )1k kt t +
= =Y Y S , then the one step integration equation is given by
( ) ( )1 1 1 11 0 1 0 1 1k k + = + + + + v S v H r H r H r H r r
(35) 5 NUMERICAL EXAMPLE In this paper, a 4h long cylinder with
radius h is considered. A penny-shaped crack is located at the
middle of the cylinder. The crack radius is ca . We define a
dimensionless parameter as the ratio of the crack radius to the
cylinder radius ca h = . We assume that a uniform, time-dependent
tensile force (mode I) loading is applied at both ends of the
cylinder. The loading increases linearly with time for the first lt
seconds and then remains constant. The Poissons ratio of the
cylinder material is assumed to be 0.25. In another paper (Tsang et
al. 2004), the fractal finite element method was used to find the
mode I SIF for a circular crack in a cylinder. In that work the
dimensionless SIF for 0.2 = , 0.4, 0.6 and 0.8 are 0.640, 0.660,
0.730 and 0.957 respectively.
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Figure 3 depicts the variation of the dimensionless SIF with
dimensionless time lt t for a ramp duration of 10lt = ms, values of
0.2 = , 0.4, 0.6 and 0.8, and for zero damping. It is seen that
there are harmonic oscillations in the SIF profile. These
oscillations are due to multiple reflections of stress waves from
the ends of the cylinder. We can see that the final mean values of
the dynamic SIF are approximately the same as the static SIF. These
SIF values increase very slightly as increases from 0.2 to 0.6. But
as increases further to 0.8, the SIF values increase rapidly.
Figure 3: The nondimensional mode I SIF time history without
damping for 0.2 = , To model the damping effect, we use the so
called Rayleigh damping, that is 1 2 = +G M K , where 1 and 2 are
two parameters. In this study we simplify the damping further by
assuming that 1 2 = = , i.e. ( )= +G M K . Figures 47 show the SIF
time histories for
61 10 = , 51 10 , 41 10 and 31 10 respectively. With 61 10 = ,
Figure 4 shows waves in the SIF profiles in the transient step
stage and then harmonic oscillations in the constant stage. However
the amplitude of these oscillatory waves decrease with time
gradually due to the effect of the damping. The SIF response in
this case is similar to the response of an undercritically damped
vibrating system. As increase to
51 10 Figure 5 shows that there are few oscillations in the SIF
response at the beginning of each stage. The SIF response at bigger
values of need a longer time to approach its corresponding static
value. The figure also shows that the waves are suppressed after
about 10 to 15 cycles of multiple reflections in the cylinder. For
the larger cracks, represented by the larger values of , the
amplitude of the waves as well as the time taken for them to die
out is larger. Also the SIF response in this case is similar to
that of an undercritically damped vibrating system. Figure 6 shows
SIF profiles with 41 10 = . It is seen that there is just a single
oscillation at the begin of the constant stage in each profile. For
this level of damping, the SIF response is seen to be similar to
that of a critically damped vibrating system. Finally with 31 10 =
,
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Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
52
Figure 7 shows that there are no oscillations in the SIF
profiles at all. In this case, the SIF response partly resembles
the response of an overcritically-damped vibrating system. However,
for 0.2 > , it is seen that the final value of the dynamic SIF
is greater than the corresponding static SIF value. This is likely
due to the damping coefficient being in excess of the critical
damping value.
Figure 4: The nondimensional mode I SIF time history with
damping ( )61 10 = for 0.2 = , 0.4, 0.6 and 0.8. Symbol represents
static SIF.
Figure 5: The nondimensional mode I SIF time history with
damping ( )51 10 = for 0.2 = , 0.4, 0.6 and 0.8. Symbol represents
static SIF.
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Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
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Figure 6: The nondimensional mode I SIF time history with
damping ( )41 10 = for 0.2 = , 0.4, 0.6 and 0.8. Symbol represents
static SIF.
Figure 7: The nondimensional mode I SIF time history with
damping ( )31 10 = for 0.2 = , 0.4, 0.6 and 0.8. Symbol represents
static SIF.
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D. K. L. Tsang and S. O. Oyadiji
Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
54
6 DISCUSSION AND CONCLUSIONS There has been considerable amount
of research directed towards the solution of crack problems in an
effort to improve an understanding of the behaviour of material
failure under dynamic loading. One of the main difficulties is the
stress singularity around a crack tip. Usually numerous degrees of
freedom or special singular elements have been used in order to
accurately represent the stress field singularities. Another
difficulty is the selection of a well-behaved time integration
method, since there are so many different integration schemes in
the literature. In this paper, we have presented a scheme to extend
our fractal finite element method to include time-dependent
boundary conditions. We used fractal finite element discretization
for space coordinates. In the singular region we transform the
nodal displacements to global variables, where SIFs are included in
the global variables. For the dynamic problem, we do a similar
transformation for the mass matrix. We then employ the precise time
step integration method for the resulting equation of motion. The
fractal finite element is an accurate method to determine stress
intensity factors. It uses analytical solutions, the Williams
eigenfunction series, in the singular region to model crack tip
singularity. In effect, the fractal finite element technique is a
semi-analytical method for crack problems. For the determination of
dynamic SIF, it is necessary to choose a time integration scheme
that can match the accuracy in space discretization. We employ
PTSIM because it is also a semi-analytical method. The method
transforms the second order ordinary differential equation of
motion into a first order differential equation. The general
solution can be easily found using an integration factor. The
integration factor is an exponential matrix, which can be computed
precisely. By using the PTSIM, the dynamic fractal finite element
method remains a semi-analytical method. Our numerical example
demonstrates the accuracy of the dynamic fractal finite element
method. Unlike the Newmark method or central difference, which
require matrix inversion at every time step, the precise time
integration method computes the exponential matrix at the beginning
of the time integration and performs only two matrix inversions for
the whole time duration. Once the exponential matrix has been
found, the time integration step is just a series of matrix
multiplications. Consequently, it is more efficient to use PTSIM
for long time integration. The calculations of dynamic mode I
stress intensity factors for a penny-shaped crack subjected to time
dependent axisymmetric loading have been presented. The numerical
results show that when there is no damping, the final mean value of
the dynamic SIF is identical to the static SIF values. When the
damping is below critical, the final value of the dynamic SIF is
equal to the static SIF. But when the damping is above critical,
the dynamic SIF is greater than the static SIF. These results show
that the dynamic fractal finite element method provides very
accurate computation of dynamic SIF values. ACKNOWLEDGEMENT We
gratefully acknowledge support of the research described in this
paper from the EPSRC of the United Kingdom.
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Dynamic fractal finite element method
Int. J. of Appl. Math. and Mech. Vol. 2 (2005), 40-56
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