dynamic crack growth in tdcb specimens

Pergamon

0020-7403(95)00114-X

Int. d. Mech. Sci. Voh 38, No. 10. pp. 1073 1088, 1996 Copyright F 1996 Elsevier Science Lid

Printed in Great Britain. All rights reserved 0020-7403/96 $15.00 + 0.00

D Y N A M I C C R A C K G R O W T H IN T D C B S P E C I M E N S

Y. WANG and J. G. WILLIAMS Room D310, Wilton Centre, ICI Engineering Technology, Middlesborough, Cleveland, TS90 8JE, U.K.

(Received 20 March 1995, and in revised form 20 September 1995)

AbstraeI--Dynamic crack propagation in tapered double cantilever beam (TDCB) specimens is analysed via beam theory and the finite element method. Steady state and transient solutions of the energy release rate G are given for various load conditions. Finite element analysis is performed to obtain the dynamic G at given crack speed or the crack history for a given fracture toughness. The stress wave effects on the dynamic G are discussed. The beam solutions are compared with the finite element results and some experimental phenomena are explained. Copyright © 1996 Elsevier Science Ltd.

Keywords: energy release rate, tapered double cantilever beam, steady state crack growth, fracture toughness, dynamic effects.

N O T A T I O N

G~ static energy release rate G dynamic energy release rate

Gc fracture toughness a crack length

a0 crack length at initiation d crack speed I second moment of area E Young's modulus p mass density C longitudinal wave speed of materials, C = v/(E/p) fl Rayleigh damping factor

[k] stiffness matrix U~ strain energy Uk kinetic energy

Uox, external work done by the applied load c compliance

Cr flexural wave speed A correction factor on crack length t time

P applied load v o applied displacement

1. I N T R O D U C T I O N

The double cantilever beam (DCB) specimen is one of the most commonly used test configurations for measuring fracture toughness of composites, polymers and adhesives. Some theoretical analyses of the DCB specimen, via either simple beam or shear beam theories, can be found in previous literature [1-4], and solutions based on the energy balance have been given for rapid crack propagation and arrest. The application of beam theory to dynamic crack propagation is particularly attractive because it is a one-dimensional analysis and the inclusion of the time variable does not introduce further mathematical difficulty. Although the beam analysis cannot predict the details of crack tip stresses or strains, it does provide an accurate account of energy quantities which form the basis of the Griffith fracture criterion.

Parallel to the analytical research, the finite element method provides a powerful alternative of analysing most real crack propagation specimen configurations. Finite element analyses have been carried out to simulate dynamic fractures using the node release technique [5-9]. Detailed singular stress or strain fields and energy quantities may be obtained to a great extent of accuracy provided sufficiently fine mesh is used. Efforts have also been made to validate the numerical results by

1073

1074 Y. Wang and J. G. Williams.

examining some simple geometries such as finite strip or parallel strip where the exact solutions are available.

Through these researches, it was found that the energy release rate G in the DCB specimen is intrinsically prone to the dynamic effects due to its low flexural stiffness of the arm, as demonstrated by Wang and Williams [9]. This problem becomes more pronounced in composite tests where specimens are usually very slender. As a result, the relationship between the fracture toughness and crack speeds cannot be found easily. It is therefore suggested that the tapered double cantilever beam (TDCB) should be used for its large stiffness of the arm, hence less the stress wave effect. Also because of its large stiffness, very high crack speeds can be achieved by applying relatively slow loading rates. In fact, this test configuration has been adopted by the ASTM standard [10] for testing adhesive strength under static conditions (Fig. 1). By making a contour profile on the arm, i.e. h 3 = 3xZ/m where m is a constant, the compliance becomes linear to crack length and the energy release rate G can be expressed as a function of the measured load only. For a material of a constant fracture toughness, the measured load will be constant.

Some analyses of TDCB tests can be found in previous literature [11, 12]. For example, Yaniv and Daniel [12] used TDCB specimens to investigate the rate sensitivity of delamination in composites at crack speeds up to 26 m s - l , and assumed a logarithmic polynomial expression to fit the experimental results to correlate fracture toughness and crack velocity. However the crack velocities they tested were very low compared to the wave speed of the material ( ~ 10,000 m s- 1) where the dynamic effects were negligible.

In this paper, attention is focused on the analysis of dynamic crack growth in the TDCB specimen. The intention is to improve the understanding of the specimen behaviour at crack speeds of a fraction of the characteristic wave speed of the material. Simple beam solutions for steady-state and transient crack propagations are given for fixed load, fixed displacement and fixed displacement rate loading. The solution for fixed displacement rate loading is particularly useful because experiments are usually carried out under this condition. The effects of transverse shear and rotation of the beam are considered by using the beam-on-elastic foundation model [-3, 13] which is applicable for both isotropic materials and composites. A finite element analysis is also performed for the purpose of comparison. The facts revealed in the paper provide a useful explanation to some experimental observations [15].

2. T H E O R E T I C A L

Owing to symmetry, only half of the specimen is analysed. At the crack tip, a clamped boundary condition is assumed which moves at a certain crack speed. At the loading point, a concentrated load P is applied (Fig. 1). The equation of motion for the beam under bending is

Ox 2 E1 8xZj + p A ~ 2 -- 0 (1)

where I = Bh3/12, A = Bh, E is the Young's modulus, p the density of the material, B the uniform width of the arm and h the thickness of the arm. The close form solution to Eqn (1) with a moving boundary is somewhat intractable. Instead Berry's method [1] is used to obtain an approximate solution where the dynamic energy terms are calculated on the basis of equilibrium field. The dynamic energy release rate G is then determined through the total energy balance

1 (,~Uext c~Us OUk~ G = ~ \ 8a 8a ~a / (2)

where Us is the strain energy, Uk the kinetic energy and Uoxt the work done by the applied load.

2.1. Steady-state crack propagation First, dynamic crack growth in the TDCB specimen under a fixed displacement Vo is examined. If

the crack begins to grow from its initial length ao = 0 at a constant speed & a steady state crack growth can be assumed on the basis that the crack has all the dynamic information available during propagation. This is proved later in this paper by a two-dimensional finite element analysis. From

Dynamic crack growth 1075

P

P V

lx

h 3 3x z = , m = 2

m

h

Fig. 1. Tapered double cantilever beam specimen.

B

simple beam theory with the effect of shear deformation ignored, the displacement profile of the arm can be expressed as

v = v o l X l n ( X ) + (1 - x ) l (3)

where Vo is the end deflection under the concentrated load P

4raP Vo = ~ a (4)

and the compliance of the arm c = vo/P is linearly proportional to the crack length a. If only the motion of crack advance is considered, the velocity distribution along the arm is

~ t - 7 \ 5 / (5)

and the bending curvature of the arm is

The kinetic energy Uk in the arm is

( ? 2 v Vo (6) c~x 2 ax"

U k = o 2 \~t;] dx

( 3 ) 3 / d '~2 /3~113 = H EBv2[-C)[maa) (7)

where C = x/(E/p) the longitudinal wave speed of the material. The strain energy U~ in the arm is

( aEBh3(~2y~2 c~ = jo - S U \ ~ / d x

EBv~ - (8)

8ma

In the case of the fixed displacement loading, Ue~t = O. From Eqn (2), the dynamic G is determined

G--~ = 1 + mh(a) (9)


where h(a) is the beam thickness at the crack tip, G~ is the static energy release rate

E~V0~ 2 (10) Gs = 8 m \ a ) "

The effects of shear deformation and root rotation at the crack tip may be included by applying a correction factor A on crack length [3, 13]. Equation (10) can be rewritten as

E ( Vo ~2 (11)

where A = 0.67 h(a) for isotropic materials. The contribution of the kinetic energy is characterized by the second term in Eqn (9) which can be rewritten into the usual form used in dynamic fracture as

G = 1 + (12) G~

where Cv is the flexural wave speed

11312C

Cv - ,,/(72 mh(a))"

For the beam under a constant displacement rate V, the solution for a steady-state crack growth can be obtained in a similar manner. Here the end deflection Vo = Vt, and the displacement profile along the arm is

v = v t l X l n ( X ) + ( l _ X ) ] .

If d is constant, then the velocity distribution is

(13)

and the curvature of the arm is

Ot V 1 (14)

02V V - ( 1 5 )

OX 2 xa"

The kinetic energy Uk and the strain energy Us in the arm are, respectively

27

U s = 8m\d/"

(16)

There is the work done by the applied load, which can be determined from static equilibrium

(17)

Tm

and at x = 0

(18)

~V V ~a d (19)


From Eqn (2), the dynamic G is

G = l _ 9 ( d ) 2 G-~ -~ mh(a} (20)

where Gs is the static energy release rate

G~ = ~ {21)

and the characteristic fiexural wave speed is

~/(11)c c v - 3~/ (mh{a)) {22)

This velocity of flexural waves is crack length dependent. The limiting case is d --+ CF, giving a bound to the behaviour.

Equation (20) indicates that the dynamic G decreases at a constant crack speed as the beam thickness h(a) increases with crack length. To achieve a steady-state crack growth with a constant G, crack speed must slow down gradually. The relation between crack length and time can be estimated via Eqn (20) which can be rewritten as

9 a 2 G = ~ m m

For G = Gc where Gc is constant, we have

/{8mG¢ 9mh(a}~ t = a 4 t - E - ~ - + l l C : ]" (24)

It shows that given a fixed loading rate V, the crack speed depends solely on the fracture toughness Gc of the material at that crack speed. For example, for a 300 mm long aluminium bonded joint under a fixed loading rate of V = 2 m s- 1, given a constant Gc = 1 kJ m - z, the crack speed will initiate at 132ms -1 and gradually reduce to 126ms -1. In reality, it is difficult to detect and measure such a small change and a constant crack velocity can be reasonably assumed. Therefore, in this circumstance, the second term in the square root in Eqn (24) can be ignored and a constant crack velocity is a good approximation provided that the material's rate dependency on such small crack speed variation is negligible. Hence we have

= "x/\8~Oc)

which is the static solution Eqn (21). However, for tests of materials of low Gc-values and high crack speeds, the dynamic effects cannot be ignored and Eqn (24) must be used to correlate crack speed and fracture toughness.

From Eqn (18), it is noticed that given a constant crack speed and a constant displacement rate, the applied load is constant. Therefore the solution also applies to the beam under a fixed load, in which case the static energy release rate can be written as

2raP 2 G s - EB 2 . (26)

2.2. Transient crack propagations The steady-state solutions given above are valid only where the crack initiates from zero length, in

which case the crack tip is constantly informed about the change of the boundary condition by the continuous waves reflected from the loading end. If the crack initiates at a finite length, the dynamic G becomes transient as a result of the delay of the stress wave reflection from the loading end. The dynamic G will drop from its initial value due to the increase of the kinetic energy in the beam, and

1078 Y. W a n g and J. G. Wil l iams.

then remain more or less constant until the first reflected wave arrives. Another drop of the G-value will take place to accommodate the new boundary condition brought back by the stress wave. And this behaviour is repeated for further crack extension, giving a step-wise decrease of the dynamic G. Following the method used in Ref. [-7], the transient energy release rate at each step can be estimated by averaging the steady-state solution during the wave reflection period. The solution can be expressed in terms of a series.

Given a fixed displacement Vo, when ao < a < al , a l = a0 ((1 + ~)/(1 - ~))

1 -~) G = G o ( 1 + a z ) (27)

where Go = E/8m(vo/ao) 2, ~ = a/Cv and Cv is determined from Eqn (12) at a = ao. For the next step, ao takes the value of al and a new G-value can be determined. Repeating this procedure gives G-values at various crack lengths. When ~ is small, ~2 in Eqn (27) is negligible. Equation (27) becomes analogous to the transient solution of a parallel strip under axial loading [7], as expected since both the geometries have a linear compliance with respect to crack length.

Under fixed displacement rate control, transient behaviour is again observed when the crack initiates at a finite length. The interval for the first wave arrival at the crack tip is given by

a, = ao ~ _ ~) (28)

where ~ = d/Cv, and Cv is determined from Eqn (22) at a = ao. However the transient solution cannot be obtained in this case apart from that the dynamic G will oscillate about the steady-state solution as a result of wave reflections.

3. N U M E R I C A L

A finite element analysis of high speed TDCB tests was carried out using a F O RTRA N programme written by Keegstra [5]. Some modifications were made for the present calculations. The dynamic G-values were computed for both steady-state and transient crack propagations discussed in the early section. The static G-values were also calculated via the virtual crack closure method [14] and the results were compared with that of the beam analysis.

A two-dimensional plane stress FE model for the TDCB specimen (Fig. 2) was set-up with a total of 3208 d.f. Linear triangular elements with equally distributed lump masses were used with in a nominal element size of 2 mm. The crack growth is simulated by releasing the nodal constraints along the pre-defined crack path at a given crack speed. In this node release technique, a "holding back force" F is postulated at the node immediately adjacent to the crack tip node, and decays linearly from the initial value to zero as the crack tip propagates from the released node to the crack tip node. In the finite element analysis, the static solution is always obtained prior to the crack initiation. Once the crack starts to propagate, the equation of motion in the two-dimensional domain is solved by an iterative time integration scheme. The dynamic G is then computed through the total energy balance and the work done by the "holding back force". Both the schemes should give the same G-values. Small viscous damping was applied in the region of the beam ahead of the

e j v 0

A ~ x.I'MM'M'M ",,IX,J'M'M

l . . ~ N,[/]x,]",J'Q",]M"OM'MM'M'M "xl M'NI'M "xl N N N NI'M NI ' x lNfM_~NNNN ~ . , ~ < 1 r ' ,J ' , ,N",s ' ,a~",s '4 ~',4"4",4 ",,r',a ",d "a "4~_ _M_"4 " , ! ~ _ " 4 _ " 4 _ " 4 ~ ~

L = 3 0 0 m m

Fig. 2. A schemat ic finite e lement mesh of T D C B specimen.

Dynamic crack growth

Table 1. Material properties and dimensions of the specimen

E(GPa) v p(kgm z) h(a)(mm) B(mm) Cims 1) L(mm)

70 0.3 2720 (1.5a2) l/a 12.5 5073 300

Note: the constant rn = 2, a is the crack length.

1079

crack tip to reduce the effect of stress waves reflected from the free end [9]. The damping takes the simple form of/3 [k], where [k] is the stiffness matrix and/3 the Rayleigh damping factor. It was found that /3 ~ 0.1 was appropriate. The dissipated work done by the damping force was also included in the total energy balance for G calculations.

The material of the TDCB specimen in the present analysis is an aluminium alloy, for which is often used in adhesive joint tests. The total length of the specimen, L, is 300 mm, and the thickness h varies along with crack length. The boundary conditions of the beam are shown in Fig. 2. Dynamic crack propagations at speeds ranging from 100 to 1000 ms -1 are examined in order to see the dynamic effect. They cover the range of crack speeds usually seen in practical tests. The details of the material properties and dimensions are given in Table 1.

4, R E S U L T S

4.1. Static analysis First a static stress analysis was carried out via the finite element method. The compliance of the

beam was calculated at various crack lengths with a fixed displacement vo = 0.5 mm applied at the loading end. From beam theory, the compliance of the beam is

4 m c = g k a. (a9)

Compared with Eqn (29), the finite element predicted good compliance values even at short crack lengths (Fig. 3). The longer the crack length, the better the agreement. However as the crack approached to the free end (a > 250 mm), a large discrepancy was seen. This is clearly due to the change of the boundary condition. When the remaining ligament on the crack path gets very short, the assumption of the clamped end condition is no longer valid and large shear deformation and rotation will occur, resulting in a sharp increase in the compliance and static G-values (Fig. 4). On the other hand, the discrepancy at very short crack length (a < 30 mm) can be attributed to: (1) beam theory is not applicable for short beam as the effect of shear deformation becomes more pronounced; (2) triangular finite elements tend to be over-stiff and (3) only a small number of elements employed in the region will usually underestimate the beam compliance.

4.2. Steady-state analysis The dynamic G-values of steady state crack growth under fixed displacement control were

computed at three different crack speeds (200, 500 and 1000 m s 1). The beam solution shows that for crack speeds less than 200 m s-1 the behaviour is quasi-static. In the computation, the initial crack length ao was set to a very small value (one element length) to approximate the steady-state condition (ao = 0). The results are presented in Figs 5-7 in terms of G/Gs vs a. It can be seen that the steady-state solution is indeed a good approximation to the behaviour. The discrepancy immediately after the onset of crack growth is due to possible spurious stress waves generated by the irregular finite element mesh in the region as well as the reasons mentioned in the previous section. In general, the solution will stabilize at a > 50 mm.

Three sets of the finite element results for dynamic crack growth under fixed displacement rate control are also presented (Figs 8-10). Again good agreement between the finite element and beam theory was found. According to Eqn (20), at fi = 1000 m s- 1, the dynamic G vanishes at a ~ 50 mm, which is also clearly predicted by the finite element (Fig. 10). As mentioned earlier, under fixed displacement rate control, the applied load is constant. The force at the loading point computed by the finite element compares very well with that of the beam solution, as shown in Fig. I1 for


6-

5-

4 .

3-

2

1

f 0 ~ L

0 100 200 a (ram)

Fig. 3. Comparison of compliance of TDCB specimen.

)

12

I

300

10

~=

6-

4 j

:1 0 100 200

a (mm)

Fig. 4. Comparison of static G in TDCB specimen.

Fixed displacement, v o = 0.5 mm, a o = O, ~ = 200 rn/s

- - - - o - - Finite element Beam theory

a ~

300

Fig. 5. Comparison of dynamic G-values.

I , J , i

100 200 300 a (rnm)


r3

t5

5 -

4 .

Fixed displacement, v o = 0.5 mm, a 0 = O, t l = 500 m/s

- - ~ o - - Finite element

Beam theory

4.

3

5-

5 -

Fixed displacement, v o = 0.5 mm, a o = O, h = 1000 m/s

- - - - o - - Finite element

Beam theory

O

100 200 300 a (ram)


Fixed displacement rate, V = 5 m/s, a o = O, ,i = 200 m/s


Beam theory

3-

2-

O

0 ! , , • , . .

0 1 O0 200 300 a (ram)


3 .

2.

0 I 1 t

0 100 200 300 a (mm)

Fig, 6. Comparison of dynamic G-values.


kb

5-

4-

Fixed displacement rate, V = 5 m/s, ao ~ O, /= = 500 m/s


Beam theory

0 100 200 300 a (mm)


L3

3-

Fixed displacement rate, V = 5 m/s, a o .-= O, ~ = 1000 m/s

- ~ . o - - Finite element

Beam theory

o

~ * '!'"L' ' " i . . . . . . . . . . . . . . . . . . . ,3331331

1 O0 200 300 a (mm)


fi = 500 m s-1. This argument is further confirmed by the finite element result with fixed load control. Figure 12 shows the G-values from beam theory and the finite element for ci = 500 m s - 1 and a constant P = 1093 (N). Figure 13 shows that the displacement rate at the loading point calculated by the finite element is identical to that of the fixed displacement rate case of V = 5 m s - ~.

In order to achieve a constant G crack growth, the crack speed has to decrease according to Eqn (24). If the fracture toughness is assumed to be a constant Gc = 437.5 J m -2 and a constant displacement rate V = 5 m s-1 is applied, then the crack speed by Eqn (24) will decrease from 500 m s - 1 to 320 m s - 1. The dynamic G computed by using this varying crack speed, as shown in Fig. 14, is indeed approximately equal to the constant value of 437.5 J m 2. The oscillation at the beginning and near the end of the beam is due to the boundary effects. This case can be compared with the previous one with a fixed loading rate V = 5 m s - 1 and fi = 500 m s - 1, as both have the same initial G-value but one is a constant G crack propagat ion and the other has a constant crack velocity.

A different finite element approach was also carried out to compute crack speed variat ion at a given G-value. An example is shown in Fig. 15 with a fixed Gc = 437.5 J m -2 and loading rate ~


3000

2000

v

" 1 o

q

1000

5 -

Lb

E v

Fixed displacement rate, V = 5 m/s, ao = 0, & = 500 m/s

- - - o - - Finite element

- - Beam theory

0 100 200 a (mm)

Fig. 11. Comparison of applied load.

3OO

2-

0

0

5-

Fixed load, P = 1093 N, ao = 0, & = 500 m/s

- - - e - - Finite element

Beam theory

100 20O 300 a (ram)


Fixed load, P = 1093 N, a o = 0, ~ = 500 mls 4

3

2

'1 0 • - , i

0 100 200 300 a (ram)

Fig. 13. End displacement vs crack length.

1084 Y. Wang and J. G. Williams,

400

5-

Fixed d~splacement rate, V = 5 m/s, ao = 0, ~i varies (equation 24)

----o--- Finite element Beam theory

0 ~ + • t

0 100 200 300 a (mm)

Fig. 14. Comparison of G-values.

300

2oo

100

0 n • T , , ,

0.0 0.2 0.4 0.6 0.B

Equation (24) - - - -o - - Finile element

t (ms)

Fig. 15. Comparison of crack length vs time.

L~

400 -

300

200

100-

0

0

Fixed displacement, vo = 0.5 ram, ao = 60 ram, & = 100 m/s


i Beam theory

o gb

100 200 300 a (mm)



MS 38-I0-D

N--.

E

&...

N"

E

400 -

300.

2 0 0

100

0

0

6000

5000

4000.

3000.

2000

1000

6000

5000

4000

3000

2000

1000

Fixed displacement, % = 0.5 mm, ao = 60 ram, & = 200 m/s


o Beam theory

100 200 300 a(mm)


Fixed displacement rate, V = 2 m/s, a o = 60 mm, ~i = 100 m/s


Beam theory

o

% o

100 200 300 a (mm)


Fixed displacement rate, V = 5 m/s, ao = 60 ram, f = 200 m/s

Beam theory


• Wave intervals o

1 ~ T

100 200 300 a (mm)

Fig, 19. Comparison of dynamic G-values,


L~

0 _J

1000

900 1 800J

300~

20O

100

0

Fixed displacement rate, V = 5 m/s, ao = 60 ram, f = 500 m/s

Beam theory

~ - - Finite element

100 200 300 a (mm)


6000

5000

4000

3000

2000

1000

Fixed displacement rate, V = 5 m/s, a o = 60 ram, & = 200 m/s

~ - - Finite element

Beam theory

o

i i i

100 200 300 a (mm)

Fig. 21. Comparison of the end load.

2O00

1800

1600

14 O0

1200

~1000

800

6oo~ 40O

2O0

Fixed displacement rate, V = 5 m/s, ao = 60 mm

Equation (24)


0

0.0

' ° ° o

o l ooo o ooo

i l i ~ i i

0.1 0.2 0,3 0.4 0.5 0.6 l (ms)

i

0.7

Fig. 22. Comparison of crack speed.


E g

500 -

400

300

200

100

Fixed displacement rate, V = 5 m/s, ao = 60 mm

- . - - o - - Finite e lement

Equat ion (24) ~ ,, ~ /

. . . . . Equat ion (25) ,- / " /

/ i / / f / / / ~ ~ 1 0"0

0

0.0 0.1 0,2 o.a 0.4 0 .5 0.6 0.7

t (ms)

Fig. 23. Comparison of crack length vs time.

V = 5 m s- t. The result, given in terms of crack length vs time, is compared with Eqn (24), showing a good agreement.

4.3. Transient analysis In practical tests, we usually initiate the crack at a finite length. Transient crack propagation will

occur and its effect becomes more pronounced at high loading rates. This can be easily identified by the severe oscillation of the measured load. To examine this transient effect, a finite initial crack length of 60 mm was used with both fixed displacement and fixed displacement rate loading applied. Two finite element results are given in Figs 16 and 17 at crack speeds of 100 and 200 m s-1 with a fixed displacement vo -- 0.5 mm. It can be seen that the dynamic G oscillates as a result of the wave effects but the trend follows that of the transient beam solution. Similar bebaviour was seen for the cases of fixed displacement rate control (Figs 18-20) at fi = 100, 200 and 500 ms -~. Analytical solution is not available in this case. Nevertheless the steady state solution appears to give the mean value of the transient G. The wave reflection intervals can be determined by Eqn (22), which have an obvious effect on the dynamic G-values (Fig. 19). As expected, the end load exhibits a large oscillation, as shown in Fig. 21 for V = 5 m s- t and ci = 200 m s- t. This is typically seen in high rate experiments.

Now if a fixed fracture toughness is given, the crack speed will accelerate and decelerate as the driving force G changes with time. Figure 22 shows the finite element prediction of the crack speed at V = 5 m s - t with a fixed Gc = 437.5 J m-2. The steady state crack speed predicted by Eqn (24) is also plotted in the same figure, which appears to be slightly lower than the average crack speed computed by the finite element. A comparison of crack length vs time relation is given in Fig. 23 which shows that in high rate tests, Eqn (24) gives a better prediction of the crack history than Eqn (25), the quasi-static solution.

5. C O N C L U S I O N S

(1) The analysis provides a simple solution of dynamic crack propagation in TDCB specimens which seems to give a good correlation compared to the finite element results.

(2) The dynamic effects are negligible for ~i < 100 m s- 1 in aluminium specimens, i.e. ti < 0.02C. (3) Both the static and dynamic G-values exhibit a sharp increase when the crack approaches to

the end of the specimen, resulting in unstable crack growth (a > 250 mm) as seen in experiments. (4) Under fixed displacement rate control, G decreases at a constant ~i. If Gc is constant, then ~i will

decreases. The relation of crack length vs time is given for a constant G crack propagation by beam theory.

(5) For ci < 200 m s- t, the dynamic effects are small. The fracture toughness can be approximately estimated by measuring the average crack speed [Eqn (25)]. Or if the fracture toughness is known, crack velocity can be predicted at a given loading rate.


(6) The transient behaviour of crack growth initiating from a finite length is governed by the velocity of flexural wave propagation. The higher the crack speed, the larger the transient effect on the G-values. The wave speed can be calculated and the wave effect intervals predicted.

Acknowledgements~he authors wish to thank EPSRC for providing the financial support for the project. Useful discussions with colleagues are acknowledged, particularly those with Dr B. Blackman and Mr A. Taylor.

R E F E R E N C E S

1. J. P. Berry, Some kinetic considerations of the Griffith criterion for fracture--l. Equations of motion at constant force. J. Mech. Phys. Solids, 8, 194~206 (1960).

2. J. P. Berry, Some kinetic considerations of the Griffith criterion for fracture--II. Equations of motion at constant deformation. J. Mech. Phys. Solids, 8, 207-216 (1960).

3. M.F. Kanninen, A dynamic analysis of unstable crack propagation and arrest in the DCB test specimen. Int. J. Fract., 10, 415~431 (1974).

4. L. B. Freund, A simple model of the double cantilever beam crack propagation specimen. J. Mech. Phys. Solids, 25, 69-79 (1977).

5. 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).

6. G. Rydholm, B. Fredriksson and F. Nilsson, Numerical investigations of rapid crack propagation. Numerical Method in Fracture Mechanics (edited by A. R. Luxmore and D. J. R. Owen), pp. 660-672. University of Swansea, Swansea, U.K. (1978).

7. Y. Wang and J. G. Williams, Dynamic G calculations for an axially loaded parallel strip. Engng Fract. Mech., 46, 617 631 (1993).

8. S. N. Atluri and T. Nishioka, Numerical studies in dynamic fracture mechanics. Int. J. Fract., 27, 245 261 (1985). 9. Y. Wang and J. G. Williams, A numerical study of dynamic crack growth in isotropic DCB specimens. Composites, 25,

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dynamic crack growth in tdcb specimens

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