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: Kwong-Lai.Tsang@manchester.ac.uk Email: S.O.Oyadiji@manchester.ac.uk

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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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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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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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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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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( )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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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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( )( )

( )( )

( ) ( )( )( )

( ) ( ) ( ) ( )

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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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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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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( ) ( ) ( )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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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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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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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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