Application of running-crack eigenfunctions to finite-element simulation of crack propagation

MECH. RES. COMM.

APPLICATION SIMULATION

OF OF

Vol.3, 197-202, 1976.

RUNNING-CRACK CRACK

Pergamon Press.

EIGENFUNCTIONS

Printed in USA.

TO FINITE-ELEMENT

PROPAGATION

W.W. K i n g , J.F. M a l l u c k , J.A. A b e r s o n a n d J.M. A n d e r s o n G e o r g i a I n s t i t u t e of T e c h n o l o g y , A t l a n t a , G e o r g i a (Received 28 January

1976; accepted as ready for print 23 February

1976)

Introduction

In the last several years, the motions of elastic bodies containing rapidly propagating cracks have received a great deal of attention from experimentalists and analysts. Understandably, the vast majority of analytical investigations have been concerned with rather fundamental problems of semi-infinite cracks in infinite bodies or of determining the nature of the stresses and deformations in a near neighborhood of the tip of a running crack. While such analyses are invaluable, it is reasonable to anticipate that approximate methods will play increasingly significant roles in the solutions of problems where the finite dimensions of bodies cannot be ignored. The finite-element method has been particularly prominent in studies of structures containing stationary cracks [i]; consequently, it is natural to consider utilization of this technique for problems of running cracks. In this paper, we suggest a procedure for extracting stress-intensity factors from nodal displacements near the tip of a rapidly propagating crack. The analysis utilizes eigenfunctions associated with a restricted class of movingcrack problems, and feasibility of the proposed approach is demonstrated by the results of a straightforward finite-element analysis of a problem to which Broberg [2] has obtained an exact solution.

Eigenfunctions Associated with a Propagating Crack

Rice [3] has developed eigensolutions to the problem of steady motion of a homogeneous, linearly elastic, isotropic body containing a crack in rectilinear propagation.

Consider the plane-strain opening-mode propagation of a

crack at constant speed c, the crack tip being located at x = ct, y = O.

The

equations of motion are satisfied if the usual displacement potentials [4], and ~, are taken to be = Ar~ cos me I

Scientific Communication

- abbreviated

197

198

W.W. KING, J.F. MALLUCK,

and

J.A. ABERSON and J.M. ANDERSON

~ = Br~ sin m92

where A, B and m are arbitrary constants,

Vol.3, No.3

, and

r I, r 2, 81 and e2 are defined by

i9 r e where

= (x-ct) + is y s

~ = 1,2

= (i - c2/c2) ½ ,

in which c I and c 2 are the propagation The motions under consideration

speeds of longitudinal

are elastodynamic

and shear waves.

states in which fixed

patterns of stress and deformation are convected through the body at constant speed c.

A characteristic

free crack ~aces has roots

equation arising from the requirement (eigenvalues),

of traction-

yielding bounded displacements

at

the crack tip, given by m = i +½n The eigenfunctions

,

n = 0,1,2 ....

(displacements)

may be written compactly as

½n Un = A n(½n + l){r I cos(½n0 I) - ½g(n)r~ n cos(½nO2)} and

v

n

= A

where

n

(1)

, . ~2n sin(½nO2)} ' (½n + l){-slr~n sin(½n9 I) + ½(g(n)/s2)r 2 4SlS2/(l + s2), for n

odd,

2 1 + s2 ,

even.

and

g(n) =

It is of interest

to note that n = 0 corresponds

along the crack-tip trajectory. the familiar r functions

for n

to rigid-body

The case n = 1 yields singular stresses

(of

form) at the crack tip; this is the only one of the eigen-

given explicitly by Rice

[3].

factor, KI, is of course proportional lira K I = x÷ct +

[/2~x

The opening-mode

where ~ is the shear modulus.

stress-intensity

to AI; defining K I by o

yy

(x-ct,O)]

22 3/2T ~4SlS2(i + s2) KI = 4 ~ 2 (i + s 2)

then

translation

The eigenfunctions

'

] AI '

developed by Williams

[5,6]

for the equilibrium problem can be developed formally by dividing each of the 22 eigenfunctions (i) by 4SlS 2 - (i + s2) and taking the limit, c + O. Since Williams'

eigenfunctions

deformations cracks

are widely used to approximate near-tip stresses

in finite-element

analyses of problems

involving stationary

[7], it is natural to consider a similar implementation

of the

and

VOl. 3, No. 3

eigenfunctions

RUNNING-CRACK EIGENFUNCTIONS

199

(i) for problems of propagating cracks.

be of value even for the case of unsteady propagation

This approach may well (6 ~ 0) since several

investigators, e.g. [8], recently have shown that the eigenfunction corresponding to n = i expresses the asymptotic form of near-tip stresses;

i.e.,

the angle-dependence of stress is a function only of the instantaneous speed of propagation.

Finite-Element Analysis of Broberg's Problem

Broberg's problem [2} consists of an infinite body in equilibrium with a uniform uniaxial tension, o Y begins to grow symmetrically,

= o, prior to crack extension.

At t = 0 a crack

from an initial length of zero, at constant rate

2c (each tip moves at speed c).

Broberg's analytical solution shows that the

ratio of dynamic stress-intensity factor to static stress-intensity factor is a function only of the crack speed. Shown in Fig. I is a finite-element model of one quadrant of a rectangular region of sufficient size to simulate, for the time interval of interest, the infinite space of Broberg's problem.

Crack propagation is simulated by the

Y

-=

3.2 L

~--

\/\/\

CVVVVVV ~ - FIG. 1

6 L

--J --I

k- -q

Finite-element model of Broberg's problem.

200

W.W. KING, J.F. MALLUCK,

sequential

removal of restraints

(crack-tip

trajectory).

triangle, crack.

Each element

impression

distance

small.

a Rayleigh wave speed,

effects would not be manifested time, T = c2t/L,

of adjacent nodes

interval

the equations

interval

of incremental

were also employed

is not likely to have influenced

in attempts

analyses

of statics

cannot provide displacements

nearest

problems

that converge,

the crack tip are especially

described

inadequately

been overcome

to positions

the results

significantly,

from dis-

from early finite elements

with mesh refinement,

suspect.

wave across

factors

Conventional

to the ~-r-

displacements

of nodes

On the other hand, remote nodes

at which the continuum displacements (~-) expressions.

the nodal displacements

Thus a similar utilization

are

This problem has

in a not-too-near

hood of the crack tip to a finite series of eigenfunctions problem E7J.

integration

e.g. at c/c 2 = 0.8

are well-known

[7i.

an

time steps in the numer-

Consequently,

by the asymptotic

by matching

Within

time of a longitudinal

elements

associated with the crack tip.

are likely to correspond

cases;

to extract stress-intensity

obtained with conventional

dimensionless

using the

ten time steps of numerical

The use of different

arising

boundary

Within this

AT = 0.05, were used.

for each of the other crack-speed

Difficulties

dependence

Introducing

integrated numerically

ten time steps,

of the finite elements.

finite-element

to a continuum analysis,

is T = 5, and the interval between

since the largest step used was the transit

placements

Crack speeds c/c 2 = 0.2, 0.4,

during propagation.

the smallest

to the

were chosen so that Cl/C 2 = 2,

according

propagation,

of the

near the crack are

is AT = 0.5 for the case c/c 2 = 0.2.

the time step was AT = 0.0125. ical integrations

thus contrary

i, the elements

c R = 0.933 c 2.

of motion were

(8 = ~) method;

is L/10;

properties

the time of propagation

releases

Newmark-~

by Fig.

thus,

is a constant-strain

L, is the largest half-length

between nodes

Material

0.6 and 0.8 were considered;

of the model

length,

that might be conveyed

not particularly

Vol.3, No.3

on the nodes at the base of the model

and the characteristic

The smallest

yielding

J.A. ABERSON and J.M. ANDERSON

neighbor-

of the equilibrium

of the running-crack

eigenfunctions

in the problem at hand is suggested. Stress-intensity

factors were computed using the nodal displacements

at the times at which

the crack tip would coincide with the apparent

of the finite-element

model.

functions

A series of the first four

(i) was used to describe

coefficients

representation

of the displacement

crack tip

(n = 0,1,2,3)

the near-tip displacement

in the series were determined

occurring

by a least-squares

pattern. matching

field to the thirteen displacement

eigenThe of this components

Vol.3, No.3

RUNNING-CRACK EIGENFUNCTIONS

201

computed for the nodes shown by filled circles in Fig. 2. Stress-intensity factors resulting from these calculations are shown in Fig. 2, where it appears that the more accurate stress-intensity factors are those corresponding to the higher propagation speeds. suggest that this ought to be the case:

Two features of the analysis

(a) the lower the crack speed the

larger was the time interval between releases of successive nodes, and (b) crack propagation at the lower speeds allowed the remote and relatively coarse portions of the finite-element model to participate significantly in the motion.

In fact, however, the apparent differences in accuracies suggested by

Fig. 2 are due primarily to the scale of the figure.

For c/c 2 = 0.2 the maxi-

mum error is about 15%, which is the same maximum error for c/c 2 = 0.8.

At

c/c 2 = 0.4 and 0.6 the maximum errors are less than 10%. Calculations based upon a series of thirteen eigenfunctions produced stressintensity factors of accuracies comparable to those in Fig. 2 at the higher propagation speeds; at the lower speeds, the scatter was noticeably greater than that shown in Fig. 2.

i°0

0.8

0 "~'/c2 = 0.4

~

'

c/c 2 = 0.2

~ [] c/c 2 = 0.4

0.6

/~ c/c 2 = 0.6

m

0

c/c 2 = 0.8

0.4 c/c 2 = 0.8

0.2

l__lJ

o 0

0.2

[ 0.4

0.6

L -- 0.8

i°0

CRACK HALF - LENGTH, a/L FIG. 2.

Least-squares four-term fit of finite-element results.

202

W.W. KING, J.F. MALLUCK, J.A. ABERSON and J.M. ANDERSON

Vol.3, No.3

Conclusion

A finite-element analysis of Broberg's problem has produced stress-intensity factors in substantial agreement with the exact solution.

These results are

particularly encouraging in light of an obvious criticism which can be made of the method of extracting stress-intensity factors from nodal displacements; that is, there is no reason to believe that the running-crack eigenfunctions are complete except for "convecting-deformation-pattern" Broberg's problem is not one of these.

However, given that a finite number of

approximating functions are to be employed, appeal in that:

problems, and

the eigenfunctions have practical

(a) they satisfy the field equations and the boundary condi-

tions of traction-free crack faces, and (b) at zero propagation speed they reduce to Williams'

functions for the equilibrium problem.

sive numerical experiments yield positive results, cal applications

Should more exten-

there are immediate practi-

for this analysis in the interpretation of experiments

designed for the determination of dynamic fracture toughnesses.

References

i. 2. 3. 4. 5. 6. 7. 8.

E . F . Rybicki and S. E. Benzley (ed.), Computational Fracture Mechanics, ASME (1975). K . B . Broberg, Arkiv for Fysik, 18, p. 159 (1960). J . R . Rice, in Fracture II, H. Liebowitz (ed.), p. 191, Acad. Press. (1968). Y . C . Fung, Foundations of Solid Mechanics, p. 184, Prentice-Hall (1965). M . L . Williams, J. Appl. Mech., 24, p. 109 (1957). M . L . Williams, J. Appl. Mech., 28, p. 78 (1961). r P . D . Hilton and G. C. Sih, in Mechanics of Fracture I, G. C. Sih (ed.), p. 426, Noordhoff, Leyden (1973). F. Nilsson, J. Elas., i, p. 73 (1974).

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