Travelling cracks in elastic materials under longitudinal shear

TRAVELLING

CRACKS

UNDER F.

I3y

llepartment

IX

ELASTIC

LONGITUDINAL

A.

SHEAR*

and S. P.

MCCLIPI‘TOCK

of Rleclnmical Engineering,

MATERIALS

SUKHATME

Massachusetts

Institute

of Technology

SUMMARY THE solution

for tbc stress field around the tip of a crack subjected to longitudinal shear (antiplane strain) and travelling with a constant velocity in an elastic medium, as given by BILBY and BULLOUGH (1954), has been extended to a configuration analogous to the tensile case studied by CRACGS (1960). As in the oaae of tensile cracks, the applied stress required for constant velocity is lower for higher crack velocity and there is a critical velocity approximately 0.0 times that of the shear wave velocity above which the crack will branch. Similar stress levels are found using two different fracture criteria : the Griffith energy criterion and the criterion of critical shear strain averaged over a critical area.

1.

the analysis

SIN(IE

shear

(anti-plane

tension,

of stress

strain)

and

INTR~DU~TI~I~

one may be tempt.ed

around

strain

is much

easier than

to extend

cracks

subjected

t,hat around

to longitudinal

cracks

the results of an analysis

subjected

to

of the shear

case to the tensile case by analogy. A comparison of the results when both modes of loa,ding can be analysed, is helpful in determining the validity or the limitations of such an analogy. have been studied

Two configurations by Y~FFB (1951)

results will be compared

of the travelling and

CRAWS

crack subjected

(1960),

with those for the corresponding

to tension

and in this paper these configurations

subjected

to shear. In addition, the analysis for longitudinal shear will be used to compare the energy criterion for fracture due to GRIFFITH (1920) with the criterion of critical average strain in a local area, as used by NEUUEK. (1937) and MCCLINTOCK (1958). While the two criteria turn out to be similar in the elastic case, the criterion of a limiting value of the local average shear strain seems more appropriate for use when plastic deformation is present and the strain history becomes important. Because OF mathematical difficulties associated with the growing crack, it is convenient region

to study

remains

implausible. taneously, of Figs.

configurations

constant

even

in which

though

the length

such configurations

of the crack or stressed are physically

rather

of Fig. 1, in which the crack reseals itself sponto one considered by YOFE% (19%). In the t!onfigurations

The ~onfiguratiorl is analogous

z and

s it is supposed

that

the

crack

length

I is nlaintained

constant

by

*This researchwas supported by t.heUnited States Air Force, through the XiF Office of’ ScientificResearch end Ueveloplnexrtcommnnd under Contract No. AF 18 (fiOO-957. Reproduction is permitted for any purpose of the United States government.

187

E’. A. MCCLINT~CKand S. 1’. SUK~ATME

188

removing

material

at the face a distance 1 behind the tip of the crack at the rate The stress fields of Figs. 1 and 2 are identical

at which the crack is progressing.

since there can be no shear stress on the plane X = difference

between

-

1 in Fig. 1, while the only

the fields of Figs. 2 and 3 is a constant

shear stress, q.

The

Y

FIG. 2. Crack from eroding face under external stress.

FIG. 1. Travelling crack under external stress.

X

FIG. 3. Crack from eroding face under internal stress.

stress distribution

FIG. 4. Internal stress following tip of infmite crack.

in these cases has been given by BILBY and BULLOUGH (1954).

configuration of Fig. 4 is analogous to that studied by CRAGGS (1960). The crack is semi-infinite, the shear stress q being applied over a length 1 behind the tip. In all cases the crack is supposed to be moving to the right with a constant The

velocity

V in an elastic medium 2.

For the conjigurations

(a)

The local distribution

of density

p and modulus

of rigidity

p.

STRESS DISTRIBUTIONS

of Figs.

1, 2 and 3

of stress around

the tip of the crack can be found

from

BILBY and BULLOUGII (1954) in terms of the applied shear stress q, the crack dimension I, a constant factor m which in this case turns out to be l/2/2 and a factor

fi defined by B = 2/l

-

( V/c,,”

(1)

Travelling

cracks

in elastic materials

under longitudinal

where c2 is the shear wave velocity given by c2 = d(p/p). is

(b)

shear

189

The resulting expression

For the configuration of Fig. 4

The solution to this is found by transforming the dynamic problem into a static problem, which is solved with the aid of a conformal mapping. In terms of coordinates moving with the crack at constant velocity, the equilibrium equation relates the shear stress gradients to the acceleration of the warping displacement, u : (3)

Expressing the stresses in terms of strains, and the strains in terms of displacements and making the substitution (4)

reduces (3) to Laplace’s equation

subject to the boundary condition &/3y = r/p along - l//l < x < 0. Thus the dynamic problem has been reduced to one of statics. It is most easily solved in terms of the stress function, 4, defined through

For equilibrium 4 must satisfy Laplace’s equation. 4 = -

- 1/p < x < 0, at

co < x <

The boundary conditions are 71(x + VP),

- E/B, $b= 0,

00,

4 =

0.

(7)

i

With the loaded length taken as unity for simplicity, the given region is mapped into a unit circle in the plane re,= u + iv by the transformation

.( - >.

x=$

w+1

2

(8)

w-l

In the notation of Fig. 5, the boundary conditions are 4 (19 #) = 0 and

d (1, #) = -

on the arc ADC, 71[l -

t cot2 M/2)]

on the arc ABC.

F. A.

190

The

tip of the crack

which is non-singular,

~CLINTOCK

and

has been mapped the first order

S. P. SUKHATME

into the point

approximation

B.

In the vicinity

to the sohltion

equation will be a plane whose equation can bc fonnd by letting and 7’ = 0. E:valuation of the Poisson integral formllla givrs

and C$is locally independent

w -

Vrc. 5. Notation for transformed

back

u = -

1 + E

of V.

D

Transforming

of B,

of Laplace’s

to the

z+/

plane

and

”

pione

pro&m.

using

polar

coordinates

r

and

19 gives 7=71*-.

2

l//3 -F1 7T JC

and

6 = 8 ‘2,

where 8 is the angle made by the total shear stress Q-with the y-axis, being

positive

in a counter-clockwise

(11) the sense

direction.

Since the stress distribution given by (11) is identical to that for the finite crack in the static case (BILLY and BUI~LOIJGH)with the exception that the value of the constant m is different, the dynamic state of stress around the tip of the infinite crack will also be given by (2) with a value of rn = 2/n. Equation (2) indicates that the shear stress on the plane Y = 0: directly ahead of the moving crack, is unaffected by the crack velocity but that the shear stresses elsewhere are increased as a result of increasing crack velocities. 3.

BR.\KCHTNGVELOCII’Y

Following YOFF~~ (1951) it will be assumed that a crack will fork whenever the maximum shear stress is attained on two radial planes on either side of the plane of the crack rather than on the plane of the crack itself. When (2) is transformed to give the shear stress on a radial plane, 7oz, (in terms of polar coordinates h?, 0) the relation plotted in Fig. 8 is obtained. For velocities greater than 0.57 c2 where

Travelling cracks in elastic materials under longitudinal shear c2 denotes the shear wave velocity,

the crack will tend to fork.

by YOFFB (1951)

value of O-6 obtained

and also to the values

This is similar to the of O-612 to 0.667

(as Poisson’s ratio varies from $ to 3) given by CRAMS (1960) rraxbk with internal pressure following its tip.

Angle

with

the

X - axis,

191

for the infinite

0

FIG.A. Distribution of radial shear stress.

4.

APPLIED

STRESS

The applied stress required to maintain estimated

from

either

the Griffith

average strain over a certsin Gri~th

energy

CRKK

FOR

PROPAGATION

a constant

energy

criterion

velocity

of the crack can be

or the criterion

of a critical

region.

criterion

In order to apply

the criterion

the energy flowing

into the region around

the

tip of the crack per unit, increase in crack length per unit height in the Z-direction will be equated

to twice the surface

energy,

2u =

s 5x

a.

3W

We have ds

(12)

where the integral is taken around the boundary of an area enclosing the tip of the crack. Choosing, for the sake of convenience, the shape of the region around the crack tip to be a square with the tip of the crack as its centre and using (2), we get

Criterion

of local average &ear strain

According

to this criterion

[NEUBER (1937) and MCCLINTOCK(1X%)],

fracture

will occur when the average shear strain over a distance reIated to the structure of the material along the line directly ahead of the tip of the crack reaches some

b’. A. MCCLINTOCK

192 critical

value yf.

In an elastic material

atomic or molecular

spacing,

a. Applying

and S. 1’. SUKHATMF

this structural

distance

is taken to be an

this criterion to (2) we get (14)

Local AveraqeSheor Strofn Criterion ________ ____

-*. . .

c

.

9

“2 0

I

/

Crack Velocity

FIG. 7. Variation

Comparison

..

\\

..

. I3

0.57

ShearWave

Velocity

of applied shear stress with crack velocity.

between, the two criteria

The two criteria can be compared if a relation between yf and u is obtained on physical grounds. Continuing with a linear stress-strain relation, the energy per unit area between Equating

two

crystallographic

planes

a distance

this to twice the surface energy since two surfaces

for y,, and substituting

in

(14)

a apart

is spy//%.

are formed,

solving

yields 2 _ 171 - P -’

ma

In the

static

case the Griffith criterion ET12=

(13) reduces to 0.60

c”“. m2

Thus the two criteria give similar results. Some of the difference is no doubt due to the non-linear stress-strain relation at high strain. In the dynamic case the energy criterion indicates a decrease in applied stress with increasing crack velocity,

while the local average shear-strain

criterion indicates

no effect of velocity provided the average is taken along the line directly in front of the crack (Fig. 7). If, however, the average is taken over a region around the tip, the increase in stress at the sides of the crack will again result in a decrease in required applied stress with increasing crack velocity. The physical explanation for this decrease from the energy point of view seems to be that, if the applied stress were constant, the displacements would be greater at higher crack velocity so that energy would be fed in at a higher rate per unit crack growth. From the strain point of view the decrease in required applied stress is due to the higher local stresses and strains at either side of the tip of the crack.

Travelling cracks in elastic materials under longitudinal shear

5.

193

CONCLUSION

For shear cracks the dependence of applied stress on crack velocity and the tendency of forking to occur above a given velocity are shown to be similar to what has previously been found for tensile cracks. Thus, in more difficult problems, such as the growing crack rather than the travelling crack, or the elastic-plastic case, there is reason to hope that a simpler shear analysis will provide some insight into the more difficult

but more practical

aid of the shear analysis two fracture

criteria,

problem

the Griffith

of tension. energy

With

criterion

the local average shear strain criterion, are shown to yield similar results.

REFERENCES BILBY, B. A. and BULLOUGH,R. CRAGGS,J. W. GRIFFITH,A. A. MCCLINTOCK,F. A. NEUBEB, H. YOFFI~,E. H.

1954 1960

1920 1958 1937 1951

Phil. Mag. 45, 631. J. Mech. Phys. Solids 8, 66. Phil. Trans. Roy. Soe. A 221, 163. J. Appl. Mech. 25, 582. Kerbspannungskhre (Springer, 2nd Ed. 1953). Phil. Msg. 42, 739.

the and