Dynamic G calculations for an axially loaded parallel strip

Engineering

Fracture

Mechanics

Vol. 46, No. 4, pp. 617-631,

1993

0013-7944/93 $6.00 + 0.00 Q 1993 Pergamon Press Ltd.

Printed in Great Britain.

DYNAMIC

G CALCULATIONS FOR AN AXIALLY PARALLEL STRIP Y. WANG

and

LOADED

J. G. WILLIAMS

Mechanical Engineering Department, Imperial College of Science, Technology and Medicine, London SW7 2BX, U.K. Abstract-A dynamic finite element analysis of crack propagation is carried out for an axially loaded elastic parallel strip under various loading conditions. The analytical solution of the dynamic energy release rate G is used for the purposes of comparison to verify the numerical solution.

1. INTRODUCTION THE APPLICATION of numerical methods to dynamic fracture analysis has received considerable attention over many years, largely because the very complicated nature of dynamic fracture phenomena prevents easy access to dynamic solutions via analytical methods. A number of finite element (and finite difference) schemes have been reported [l-4]. Because of the lack of analytical solutions, the justification of the numerical solutions has largely relied on experimental data. However, both the experimental and computational results, when expressed in terms of the critical stress intensity factor, K,, or the critical energy release rate, G,, versus crack speed, ri, often show considerable scatter and the vital issue of the uniqueness in such a relationship remains unresolved. Nevertheless, analytical solutions of dynamic crack growth for some simple geometries (the parallel strip under an axial load, for example) can be obtained because of the simplicity of one-dimensional analysis. These can therefore be used as verification of numerical solutions. In the present paper a dynamic finite element programme [4], working in generation mode (i.e. to determine the energy release rate G for a prescribed crack history), is used to calculate the dynamic G-values of the parallel strip crack growth at constant crack speed. Plane stress conditions are assumed in all the calculations. 2. ANALYTICAL

SOLUTION

FOR ENERGY RELEASE

RATE G

For an elastic strip geometry under a constant force P or displacement loading u, (Fig. l), the solution for rapid crack growth can be obtained analytically by using a simplified model: due to symmetry, only half of the geometry is considered (Fig. 2a). A predominantly mode II fracture occurs when this geometry is subjected to an axial load. The presence of a small fraction of mode I energy dissipation is mainly due to the Poisson ratio effect. Hutchinson and Suo [5] provided the partitioning solution via a local stress field analysis, which shows about 9% G, out of the total energy release rate G. However, the partitioning of G is not a prime subject in the present discussion and in the analytical model the strip is assumed to be a simple beam clamped at one end of the initial crack length a,, and subject to an axial load at the opposite end (Fig. 2b). Crack propagation is simulated by removing the clamped boundary at corresponding crack speeds h. Stress waves

Fig. 1. The parallel strip under axial loading. 617

Y. WANG

618

and J. G. WILLIAMS

(a)

p,uo,

,-

4

I I

h t

I % Clamped

boundary

moving

at 6

ib) Fig. 2. Analytical

model

of the parallel

strip.

generated by the crack propagation will travel through the beam and reflect at the loading end. The reflected stress waves are allowed to pass the clamped boundary and disappear, as the beam is assumed to be infinitely long. The only variables that need be considered in the analysis are the displacement u and its derivatives. The mode I component of the energy release rate becomes negligible for these boundary conditions. Solving the one-dimensional stress wave equation with the proper boundary conditions enables the energy release rate G to be calculated via the balance of the strain energy, kinetic energy and external work (see details in ref. [6]). For the convenience of the following analysis, the static energy release rate GSTis first given by using the conventional compliance method, as follows: Under a constant displacement, uO,

and under a constant force loading, P,

where a is the crack length, E is Young’s modulus, B the width of the strip and h the thickness of the strip. In analysing dynamic crack propagation of the parallel strip, a generation mode was assumed, i.e. to compute the energy release rate G for a given crack speed. Various loading conditions (constant force, constant displacement or displacement rate) and different initial crack lengths a, were used to achieve either steady-state or transient behaviour. The analytical solutions of dynamic CD versus crack length are summarized as follows 161. (1) Under a constant displacement u,, with a, = 0, steady-state propagation will occur, and a constant G,/Gsr ratio, depending only on crack speed, is obtained:

where u = b/C, ri is the constant crack speed, C = J(E/p), G,, is given in eq. (1).

the longitudinal

wave speed, and

Dynamic G calculations for a parallel strip

619

(2) If a, takes a finite value, the behaviour of the crack propagation The G,/Gsr ratio can be expressed in terms of a series:

becomes transient.

whereN=1,2,3 ,..., and the crack length, a, varies for each increment from a,--~,,~, u,Pu,r2, u0&u0~3 and so on, where r = (1 + a/l -a). (3) When a constant force P is applied, the dynamic G will remain constant regardless of the initial crack length, a,,: $=(l

-c?),

ST

where GSTis given in eq. (2). The load point displacement increases linearly under these conditions. Thus, eq. (5) also applies if a constant displacement rate ri is used, and GSr in this case can be expressed as GSr= Eh/2(ti/h)‘. However, a,, must be zero to achieve a steady-state solution [6]. 3. FINITE ELEMENT MODELLING A finite element programme (BOLKNAK) originally written by Keegstra [4] was used in the following analysis, which is capable of calculating the dynamic energy release rate for mode I fracture (double cantilever beam and three-point bend) tests. Modifications were made to enable mode II calculations to be made. A two-dimensional mesh was generated to model the parallel strip with all the nodes behind the boundary at the initial a, constrained in both the x- and y-directions (Fig. 3). As it is generally accepted that dynamic behaviour prior to crack initiation is almost identical to that of static cases, a static stress-displacement analysis was always performed first at the given loading condition to establish the stress field in the strip before switching into the dynamic time marching process. In the dynamic part of the programme, crack propagation is simulated by releasing the nodal constraints (along the clamped boundary) at pre-defined time intervals according to different crack speeds and nodal spacing. When each nodal constraint is released, a stress impulse is produced. The equation of motion is then established, and an iterative time integration scheme is performed to obtain dynamic solutions:

(6) whereJ4] is the mass matrix, [k] the stiffness matrix, {F} the external force vector, and {6}, {g} and (6) the displacement, velocity and acceleration vectors respectively. A simple form of damping /3[k], proportional to velocity, is incorporated in the analysis and will be discussed later. According to the equation of motion, eq. (6), the energy in the system is conserved. In order to calculate the energy release rate G, an energy sink term (i.e. the energy dissipated at crack growth) in Keegstra’s model is obtained by postulating a set of viscous “holding back forces” F Clamped

boundary

moving

I--

L

IY

-I-

x

Fig. 3. The finite element model.

at crack

speed d

Y. WANG and J. G. WILLIAMS

620

at the released nodes, and decaying from their initial value -F, as the simulated crack tip approaches the next node (d = D): F=-F,

at that moment (d = 0) to zero,

1-g. (

)

where D is the nodal spacing along the crack path. The holding back force is assumed to act parallel to the crack surface, resulting in a certain amount of mode II fracture energy being removed from the system. The energy release rate G is then calculated via the global and local schemes. In the global scheme, G is calculated by the equilibrium of the total energy:

where Aa is the increment of the crack length, AU,,, the work done by external force during the crack growth Aa, AU,Ythe change of the strain energy and AU, the change of the kinetic energy in the system. In the local scheme, G is calculated from the holding back forces and the displacements of the nodes upon which the holding back forces are applied: (9) where {F}, and {F};_ l are the holding back forces of the released nodes at the current and previous crack positions respectively, with {c?}~and (~5}~_,being the corresponding nodal displacements. The global and local schemes should give the same G-values if the solution is stabilized. This can be used as a criterion for judging the performance of the programme. Because the boundary at the crack tip is clamped, stress waves generated from nodal releases will reflect between the two ends of the beam. The reflections of stress waves cause a great deal of displacement oscillation and generate a large amount of kinetic energy and strain energy en route. The effect of stress wave reflections is clearly illustrated in an extreme case where F, (the holding back force) was set to zero so that maximum displacement oscillations could be observed. An initial displacement u0 = 1.0 mm was applied to the strip with a, = 0.4 mm; the clamped boundary was released at a speed ri = 800 m/set. Since no holding back force was imposed (i.e. no energy was removed from the system), the kinetic energy and strain energy finally reached a balanced state, sharing the total energy input equally (Fig. 4). As a result, the displacement profile, u, along the beam axis exhibited tremendous oscillations, as expected (Fig. 5). Applying the holding back force reduces some of the stress wave reflections and displacement oscillations, but a substantial amount of them will remain. This behaviour prevents the holding back force from extracting appropriate amounts of energy from the system, since the kinetic energy and strain energy (generated by the reflections of stress waves) account for some of the energy input.

5

40 48

w

32

=

24 16 8 0

0

2

4

6

8

10

a

(mm)

12

14

16

18

20

Fig. 4. Energy variation (u,, = 1min, ci = 800 mjsec, F, = 0)

Dynamic G calculations for a parallel strip

B

2

18

4

a

12

14

621

16

(mm)

Fig. 5. Displacement profile at various crack lengths (ug = 1 mm, 6 = 800 m/set, F, = 0).

A viscous damper, parallel to the elastic stiffness, is therefore employed adjacent to the clamped boundary (1-2 columns of elements, the shaded elements in Fig. 3) to absorb the reflected stress waves so that an infinitely long beam can be simulated. The damping matrix takes the form of /?[k], where /3 is the damping factor (representing the fraction of the applied damping to the critical damping) and [k] the stiffness matrix. It was found that /I = 0.1-0.4 is generally appropriate, and higher /? is not necessary. As the damping is applied, a certain amount of work is done by the viscous damping force. However, because only a few elements are damped, the work done by the damping force is very small compared to the other forms of energy in the system. This is confirmed by the balance of the energy input with the sum of U,, U, and Uf (fracture energy) during crack propagation, and the convergence of the local and global G-values. Three types of loading, i.e. uniform displacement, uniform displacement rate and uniformly distributed force, can be applied to the strip edge (Fig. 3). Only distributed loading was used in order to eliminate any possible shear deformation, so that only the dilatational stress wave would propagate in the x-direction, as assumed in the analytical model. Triangular elements of constant strain were used with equally distributed lump masses on each nodal point. A uniform mesh was chosen to avoid the generation of spurious stress waves at element boundaries [3], with a total of 600 degrees of freedom. A finer mesh (1800 degrees of freedom) was also formulated, although the improvement to the G results was insignificant. The coarse mesh was therefore used throughout the present exercise. A symmetric arrangement of elements was used to obtain balanced nodal displacements and forces with respect to the central line. 4. RESULTS AND DISCUSSION The material and dimensional parameters used in the finite element analysis are given in Table 1. The initial crack length, a,, is taken as 0.4 mm for steady-state propagation and 5.2 mm Table 1 Young’s modulus E @Pa)

Poisson’s ratio

1.5

0.33

EFM W&F

V

P (kg/d

1500.0

L (mm)

20.0

h (mm)

1.6

B (mm)

20.0

Y. WANG and J. G. WILLIAMS

622

Cl

k-r--r-r-r-vi iJ

2

4

-7-

f.

8

10

i

----r-T-P-_;

1‘1

16

18

2n

a (mm:

Fig. 6. Static G comparison.

for transient cases. The crack propagation speeds are selected from 100 to 800 mjsec so that a complete range of dynamic behaviour can be examined within the range of the longitudinal stress wave speed C = J(E/p) of the material, which in this case is 1000 m/set, whilst the shear wave speed is 613 m/set and the Rayleigh wave speed is 570 m/set. 4.1. Static analysis A static finite element analysis was first carried out for the elastic strip under fixed displacement loading u,,. Extracting the nodal displacements and forces at the crack tip, the static energy release rate G,, can be calculated via the virtual crack closure method [7]. Given a constant displacement u0 = 1.0 mm at the strip edge, G,, at various crack lengths was computed and compared with the analytical solution [eq. (l)]. Good agreement was found, as shown in Fig. 6. A parameter A, defined in eq. (lo), can be used to measure the discrepancy:

Evaluated from the finite element G,, data, A showed a variation between -0.024 and - 0.035 mm for crack lengths of 0.4-20 mm. This implies that the finite element results are slightly higher than those predicted by eq. (l), and the maximum error which occurred at the shortest crack length, a = 0.4 mm, gave about 13% difference. However, this discrepancy decayed rapidly, and became negligible for longer crack lengths and, for example, when a > 3 mm, the error was less than 3%. Static G-values under constant force P = 100 N were also calculated for the crack length range from 4.8 to 20 mm. A constant G-value of 5.21 J/m2 was obtained in comparison with the analytical Gsr value of 5.21 J/m2 from eq. (2). 4.2. Steady-state dynamic analysis 4.2.1. Fixed displacement control. The analytical solution indicates that steady-state crack propagation will occur under fixed displacement loading u0 only when a, = 0. This is of interest because it represents a limiting case when the wave effects disappear from the system. The finite element formulation, however, cannot meet this requirement. An approximation was made by locating the initial boundary at the nodes immediately adjacent to the strip edge, leaving only the first column of elements subject to the loading of the initial displacement u0 = 1 mm. By varying the element size, a, can be changed. However, the transient effect will always exist no matter how small a, is. The nodes behind the initial boundary are all fixed at the instant of t = 0, and then released row by row according to various crack speeds. The dynamic energy release rate G, was calculated at five different crack speeds, 100, 200,400, 600 and 800 m/set. The results of GO, normalized by the static G,, of eq. (l), are shown in Figs 7-12, where the analytical solution G,/GsT is also plotted (note that the global and local GD/GsT of the finite element solutions overlapped in most cases). In Fig. 7, where damping was not used, pronounced oscillations in the numerical GD are observed at the crack speed of 100 mlsec,

623

Dynamic G calculations for a parallel strip

1

0 0

2

4

6

8

a

10 (mm)

12

14

16

18

20

Fig. 7. Dynamic G comparison with fixed displacement control (q, = I mm, d = 100 m/set, undamped).

0

2

4

6

1.4

8 l&12

16

18

20

Fig. 8. Dynamic G comparison with fixed displacement control (uO= 1 mm, 6 = 100 mjsec).

DISP LOWNC !_bl.O

o-1 0

2

4

6

8

(mm}, dn/dk r 200 (m/s)

10 a (mm)

12

14

16

18

20

Fig. 9. Dynamic G comparison with fixed dis@acetnent control (us = 1 mm, d = 2M)m&c.).

Y. WANG and J. G. WILLIAMS

624

>,iP

iQ&‘,,PiG

i,g=

1.0

jrnl,

al,‘dt

= ‘WC ,m

5! * -Q -

i!

INDE'X GLOBALG/Gs, (FE) LWALG/Gsr (FE) ANALYTICAL G/Gsr

Fig. 10. Dynamic G comparison with fixed displacement control (q, = 1 mm, ci = 400 m/see).

DISP

5 4

L@ADiNC ull=1.0

Immj

dO,dl

= 600

(m/S,

INDEX m*GLOBAL GIGS, (FE) -.a- LCCALG/Gs, (FE) ANALYTICAL GIGS, ~~~_ .___._~

Fig. Il. Dynamic G comparison with fixed displacement control (q, = 1 mm, ci = 600 mjsec).

Fig. 12. Dynamic G comparison with fixed displacement control (u,, = 1 mm, ri = 800 m/se@.

Dynamic G calculations for a parallel strip

5

60

6 c 2 w

50

625

40

0

2

4

6

810 a (mm)

12

14

16

18

2(Ii

Fig. 13. Energy variation (ug = 1 mm, ri = 800 m/set). Fixed displacement control.

and the oscillation became more severe as the crack propagated. For higher crack speeds, it was found that not only did the oscillation become more pronounced, but also G,/Gsr increased rapidly and became unstable (i.e the local G and global G diverged). Clearly this behaviour was mainly due to the effect of the stress wave reflections, establishing a large amount of kinetic energy in the system and resulting in higher GD/GST. Such dynamic effects were also observed by Kalthoff in his caustic experiments of double cantilever beam (DCB) specimens [S]. Secondly, the G,-values were normalized with GST, which decreased rapidly. As a result, the oscillation of GD/Gsr became more pronounced. Thirdly, stress waves also induce oscillations of the holding back force, from which G is directly calculated. Finally, in the finite element analysis, the absolute steady-state condition (i.e. a, = 0) cannot be achieved, and a transient mode will always exist which will also have an effect on the results. A damping scheme, as mentioned earlier, was therefore applied to absorb the reflected stress waves. The results with damping applied are shown in Figs 8-12, and good agreement between the theoretical and finite element results is achieved. The initial drop of the finite element results, particularly in the high crack speed cases, is due to the finite crack length a, used in the finite element model, which resulted in transient behaviour at initiation. From the above analysis, it can be concluded that the present finite element model is capable of simulating the crack propagation process with a viscous damper applied at the moving boundary. The agreement between the numerical and theoretical results is excellent. The work done by the damping force is negligible, as shown in Fig. 13, where U,, the sum of the kinetic energy, U,, strain energy, U,, and fracture energy, U,, is balanced by the initial energy input during crack propagation (note the external work U,,, = 0 under the fixed displacement control).

0

2

4

6

8

a

10 (mm)

12

14

16

18

20

Fig. 14. Dynamic G comparison with fixed load control (P = 100 N, 6 = 100 m/se@.

Y. WANG and J. G. WILLIAMS

626

INDEX GLOBAL G&I (FE) LOCALG/Gsr (FE) ANALYTICAL WGsr

.

1! -y

1

O/‘!‘!‘I’l’I’!‘i 0

2

4

I’I”

E

8

10 a

12

14

16

18

20

(mm)

Fig. 15. Dynamic G comparison with fixed load control (P = 100 N, ci = 200 mjsec).

FORCE LOADNC

P =

100 (N)

do,dt

i

4-

= 400 (In/S)

IhvW GLDBALG/Gsr (FE) LOCAL G/Qs (FE) ANALYTICAL GIGS,

-

O/‘I’,‘l’I’i’~‘I 0

2

4

/ 6

10

8

a

12

14

,‘!‘I

16

18

20

(mm)

Fig. 16. Dynamic G comparison with fixed load control (P = 100 N, ti = 400 m/set).

FORCE LaADING

P =

100 (N)

drl,dl

= 600 (ml/S)

5

INDEX GLOBALGIGs (FE) LOCALG/Gn (FE) ANALYTICAL GIGS,

~+ i ~-

4

3 LIFL GST 2

.0

en,‘!‘/‘~‘I’~‘~

0

2

4

6

8

10

a

12

14

16

’ 18

1 20

(mm)

Fig. 17. Dynamic G comparison with fixed load control (P = 100 N, a = 600 m/set)

Dynamic G calculations for a parallel strip FORCE

ImING

P =

100

(N).

da/d1

= 800

627

(m/s)

5

GST

2

1 R

old 0

,~,‘,~,‘/‘,~,‘,‘I’/

2

4

6

8

a

10 (mm)

12

14

16

18

20

Fig. 18. Dynamic G comparison with fixed load control (P = 100 N, ri = 800 m/set).

DlSP

PATE

LOADING

R=D.DOl,

da/di-

ID0

(m/r)

5-I /-I

4

i

01,,,,~I~,‘,,,~,~,~,~, 0

2

4

6

a

a

10 (mm)

12

14

16

18

20

Fig. 19. Dynamic G comparison with fixed displacement rate control (R = 0.001, ri= 100 m/set).

5

4

1

DISP

RATE

L04DINC

R=O.OCl.

j-1

do/dk

200

(m/s)

3

.!b %T

0% 0

2

4

6

8

a

10 (mm)

12

14

16

16

20

Fig. 20. Dynamic G comparison with fixed displacement rate control (R = 0.001, d = 200 m/set).

Y. WANG

628

and J. G. WILLIAMS

! I +

4

5

GLOBALGlGrr LOCALGIGs ANALYTICAL

1 -

(FE) (FE) G/Gs,

~~~~~_

~~~

3-1 1!

GST 21

oa 0

2

4

6

8 a

Fig. 21. Dynamic

G comparison

10 (mmi

12

with fixed displacement

14

16

rate control

18

2C‘

(R = 0.001,

ci = 400 mjsec).

1 -1 0. .0

e

‘I ii

2

#‘I’I’I’I’I’! 4 6

e

:o a

Fig. 22. Dynamic

G comparison

*TE

KADINC

:4

R=0.001

rate control

4

6

8

10

(R = 0.001, ci = 600 mjsec).

~.~.____._ r ~~-~ INDEX m*m

2

:‘ :‘

BOO (m/al

do/dl=

-

0

1” 1F

1e

(mm)

with fixed displacement

DisP

1:

12

14

OLOBALG/Ga LCCALG/Grr ANALYTICAL

‘6

(FE) (FE) G/Gsr

18

20

a (mm)

Fig. 23. Dynamic

G comparison

with fixed displacement

rate control

(R = 0.001, Li = 800 m/set.)

Dynamic G calculations for a parallel strip

629

4.2.2. Fixed load control. Under the loading of a constant force P, steady-state crack propagation will occur. The theory predicts GD= G&l - a2) with a0 taking any finite value. Unlike the fixed displacement loading case, the oscillation of G-values was much less pronounced when normalized by constant GST. Reasonably good agreement was obtained at low crack speeds (100 m/set) with no damping applied (Fig. 14). However, at high crack speeds (especially 600 and 800 m/set), the G-values were much lower than the predicted value of G,,(l - a2) when damping was not used, because of the reasons discussed in the previous section. Therefore, at high crack speeds (h > 200 m/set), damping (/3 = 0.1-0.2) was applied on the elements adjacent to the clamped boundary. It can be seen that this treatment was very successful and the G-values stabilized at values in good agreement with those of the theory (Figs 15-18). 4.2.3. Fixed displacement rate control. Another exercise carried out applied a fixed displacement rate at the loading end of the strip. In fact, this loading condition is equivalent to the constant force case. Under a fixed displacement rate ti, the theory gives:

However, to obtain a steady-state crack propagation, the initial crack length a, has to be zero, similar to those in the fixed displacement cases. In these calculations, the loading was applied such that the ratio R = (C/d) was kept constant at 0.001 in all cases and the external work, U,,,, was calculated via the reaction nodal forces at the loading edge. In the static run, an initial displacement u. was applied so that (h/a,) = (C/b).

ol~,‘,~,~,~‘~,~,~l~,~, 10 a (mm)

02468,

12

14

16

18

20

Fig. 24. Dynamic G comparison in transient mode (a0 = 5.2 mm, u, = 1 mm, ci = 100 m/xc).

INDEX --.- OLOBALGlQa (FE) -oLocALWOsr (FE) ANALYTICAL QlQsr

3 ZrL GST 2 !

O!.,‘,~,., 0

2

I.,,,.,.,.,

4

6

8

10

a

12

14

15

18.20

tmm)

Fig. 25. Dynamic G comparison in transient mode (a0 = 5.2 mm, u, = 1mm, 6 = 200 mkc).

630

Y. WANG

and J. G. WILLIAMS

I --

ANALYTICAL

18

16

14

G&r

2'2

a (mm)

1mm, b = 400 mjsec).

Fig. 26. Dynamic

G comparison

in transient

mode (a, = 5.2 mm, I+, =

Fig. 27. Dynamic

G comparison

in transient

mode (a0 = 5.2 mm, u,, = 1 mm, ci = 600 mjsec).

0

2

-1

c

a

10

I?

14

16

18

L^O

a (mm) Fig. 28. Dynamic

G comparison

in transient

mode (a0 = 5.2 mm, u, =

I mm, ri = 800 m/set).

Dynamic G calculations for a parallel strip

631

It can be seen that similar results of GD/GSTwere obtained compared with those of the constant force loading cases, as shown in Figs 19-23. 4.3. Transient dynamic analysis When the initial crack length a0 has a finite value, crack propagation exhibits a transient behaviour under fixed displacement control and the theoretical solution can be given in the form of a series. Dynamic Go values drop from the initial Gsr by the ratio of (1 - a/l + a) from step to step and remain constant within each step. If normalized by GST, given in eq. (l), the ratio G,/G,, will rise gradually from its lower bound to an upper bound within each step. Taking an arbitrary initial crack length, a, = 5.2 mm, the dynamic G-values were evaluated for various crack speeds under a fixed displacement u, = 1.Omm. The results are given in Figs 24-28. Again, good agreement was obtained with damping applied at the clamped boundary. The finite element results also showed that at the theo~tically predicted peak values, a smearing phenomenon occurred, which mainly depends on the mesh size. Finer meshes can be used, but only little improvement is obtained for a huge increase in computing time. 5. CONCLUSIONS From this analysis, it appears that dynamic crack propagation in a parallel strip, in either steady-state or transient modes, can be simulated by using Keegstra’s nodal release model. Using the damping mechanism enables an infinitely long beam to be modelled so that a direct comparison can be made between the finite element and quasi-static theory. Without damping (i.e. for a beam with a finite length), stress wave reffections within the beam lead to the recovery of kinetic energy and affect G-values. In light of this result, it can be concluded that dynamic effects (i.e. the stress wave re%ctions in the system) are very impo~ant in analysing high speed fracture problems, especially when interpreting experimental results, because test specimens are always made to finite dimensions. Taking account of such dynamic effects, measured by comparing the finite element results of infinite and finite size specimens, it is possible to determine dynamic G, values versus crack speed, d, via experimental data using quasi-static procedures. Acknowledgement-The

authors would like to thank the SERC for providing the financial support for this project.

REFERENCES [I] S. N. Atluri and T. Nishioka, Numerical studies in dynamic fracture mechanics. Znf. J. Fructure 27, 245-261 (1985). [2] L. Hodulak et al., A critical examination of a numerical fracture dynamic code. Fracture Mechzics: 12rh Conference, ASTM STY 780, 174-188 (1980). [3] 3. Brickstad, A FEM analysis of crack arrest experiments. 2~t. J. Fraciure 21, 177-194 (1983). [4] P. N. R. Keegstra, Computer studies of dynamic crack propagation in elastic continua. Ph.D. Thesis, Imperial College of Science, Technology and Medicine, London (1977). [5] J. W. Hutchinson and Z. Suo, Mixed-mode cracking in layered materials, in Advances in Applied Mechanics (Edited by J. W. Hutchinson and T. Y. Wu) Volume 28, pp. 63-191. Academic Press, London (1991). [6] J. G. Williams, A review of the determination of energy release rates for strips in tension and bending. fnr. J. Fracture (submitted for publication, 1992). [7] E. F. Rybicki and M. F. Kanninen, A finite element calculation of stress intensity factors by a modified crack closure integral. Engng Fracture Me&. 9, 931-938 (1977). [8] J. F. Kalthoff, On the measurement of dynamic fracture toughnesses-a review of recent work. Inr. J. Fracture 27, 277-298 (1985). (Received 5 October 1992)