© 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Damage and Fracture Mechanics VII, CA Brebbia, & SI Nishida (Editors). ISBN 1-85312-926-7

A novel method for the solution of the threedimensional dynamic crack problems E. 1, Shifrinl &A,

Staroselsk/

‘Moscow Aviation Technology University, Moscow, Russia ‘United Technologies Research Center, Hartjord, Connecticut,

USA

Abstract We present a novel,

efficient

and robust

numerical

method

to handle

spatial problems for practically arbitrarily shaped plane cracks or crack systems under static or dynamic loading. A two-parametric method previously

used for static analysis is generalized

for dynamic

shear problems.

A system of integral-differential equations for a displacement jump in the crack plane was derived for generalized loading conditions and solved by this method.

In this paper, values of Stress Intensity

tained for a penny-shaped loads.

crack,

subjected

The results are shown to be in good

results for static loading

and numerical

Factors

(SIF)are

to either harmonic accord

ob-

or impact

with known analytical

results of other investigators.

Introduction Dynamic response of elastic media subject to transient excitations is extremely import ant for two reasons. First, interaction of a planar crack in three-dimensional for developing the scattering

a point of practical ing activities considered,

solids with time-harmonic

and theoretical

where the response is damage

used in high technology prediction

elastic waves is a foundation

non-destructive methods of crack detection. In particular, of non-stationary time-dependent loads/waves by cracks is

tolerant

The second

and life cycle design,

industrial

of crack behavior

interest.

of cracks to dynamic

subject

applications. to dynamic

area of engineerloading

must be

which is now widely

This practice loading.

requires the

© 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Damage and Fracture Mechanics VII, CA Brebbia, & SI Nishida (Editors). ISBN 1-85312-926-7

48

Damage and FractureMechanics VII Both wave reflection

the Navier equations presents

a real challenge

ited number Difficulties inertia

and crack-tip

field analysis require the solution

[1] for three-dimensional

of complete

even for static solutions

arise when dynamic

problems

problems. (governing

and for problems

integral equation

method

mechanics.

It reduces

(BIE)

contain

the

method

has

problems

[2]. However

(a fine mesh is needed near

in an infinite

domain.

The boundary

[3] has also been widely used in continuum which is well suited for

[4], however, the system matrix is full and un-

symmetric

so BIE is also computationally

analytical

methods

has been analyzed

cracks.

equations

the order of the problem,

with singularities

a lim-

shaped

The finite element

with singularities

the crack front)

problems

Consequently,

used for solving fracture mechanics

it is not effective for problems

of

Spatial crack analysis

is known for arbitrarily

‘2U ) are to be considered. term pm

been extensively

space.

expensive.

the case of crack interaction by Mal [5], Martin

Using BIE and semi-

with longitudinal

[6], and Budreck

waves

and Achenbach

[7].

Recently, there have been a number of works (for example, Annigeri and Cleary [4]) aimed at developing a hybrid method combining advantages of both numerical schemes, however, there is a need in an efficient and robust numerical

method

to handle spatial problems

for nearly arbitrarily

plane cracks or crack systems under static or dynamic We present a novel two-parametric to use a few elements solving

problems

method

of basic functions.

with arbitrarily

that gives us an opportunity

This approach

shaped

shaped

loading. can be applied

planar cracks.

for

In this paper we

apply this method to obtain a complete solution for the scattering of shear harmonic loads by a planar crack in three-dimensional space. We consider the case of wave scattering

by a penny-shaped

crack as a model

to verify our results against results of other investigators, Shippy

and Rizzo

[8] who gave the BIE solution

for both a longitudinal

and a shear harmonic

The plan of this paper is as follows. governing

system

for a penny-shaped

Jia, crack

wave.

In the next section

of integral-differential

example

particularly

equations

we deduce the

and state the physical

problem. We will also show that the solution for an arbitrary incident wave can be represented as the superposition of two independent problems: shear and longitudinal. Next we briefly describe the major idea of the numerical method and derive the principal relationships. In the following section we evaluate the applicability wave interaction

of this method

with a penny-shaped

that the two-parametric a class of problems.

method

for the analysis of harmonic

crack.

is extremely

Our calculation

efficient and accurate

We close in the last section

shear

results show for such

with some final remarks.

It is important to note that due to the paper size limitation, we have omitted most of technical details and derivations. We just attempt to explain the main idea and the spirit of the numerical applicability.

method

and demonstrate

its

© 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Damage and Fracture Mechanics VII, CA Brebbia, & SI Nishida (Editors). ISBN 1-85312-926-7

Damage and FractureMechanics VII

Fundamental

equations

The three-dimensional normally

49

steady-state

harmonic displacement

to the crack G in plane Z3 = O. This problem

wave projects

can be reduced

to

that of a crack whose upper and lower surfaces are forced with equal effort in opposite directions, such that they relieve the external stresses defined by the incident

wave.

It is assumed

with each other and Sommerfeld’s

that crack surfaces

radiation

conditions

do not interact

are assumed

at in-

finity. For the initial problem, the stresses applied to the upper side of where t = (tl, tz,ts),tl=@~ul/c., the crack are equal to t ezp(–iw~),

t2=piwUz/c., transverse

t3 = (A + 2p)itiU3/c~. Here c. = (~)*

wave, cd = (~)*

p are Lame’s

is the velocity

elastic constants;

is the velocity

of a

of a dilat ational wave, ~ and

p is the density.

If applied

loads have the

the displacements are also harmonically dependent on term t ezp(–iw~), time as UexP(–iw~), where u = (UI, U2, U3) is a vector of displacement amplitudes. The Navier equations

for the amplitudes

(A+ P)uj,ij

P“i,jj

+

have the form: =

(1)

‘fxJ2ui

Application of the Fourier transform reduces the problem a system of integral equations in the frequency domain. Denote by ~(t)

, c= (&l, Ez) the Fourier transformation

After taking the second derivative transformations,

(~ d2

the equations

—n~—

We use the following

and &=

d2

dx:

notations:

for nl and nz. Suppose

–i~,

of the function

and making elementary

~(z):

but cumbersome

reduce to the form Itou [9]:

n: = &’2- P2,

~-w, branches

)(

to the solution of Let z = (zl, Z2 ).

— n;

‘iii = O.

i=l–s

(2)

) n? = ( 2 –clZ;

p=@w,

cl=

&2 = c? + f;. Further, let us choose the that 6

is a positive

number

if s >0,

if s

and d = j+.

at points

an,

< 1;

1;

1

The circles have radii equal to aa

belonging

in G. In other words,

we cover the

crack surface by overlapping circles, define finite “bell” type basis functions on each of them and then calculate A~q5~. After that, we integrate the product (ej, A* ~k) over the crack area G and find the solution of the linear system (9). As noted above, the functions @k are a linear combination of the functions $bk and by linearity of the operator A, we calculate A~k easily. Thus, the numerical

procedure

of the solution to this problem

is nearly

complete.

Results

of computations

In this paper, values of Stress Intensity Factors are obtained for a pennyshaped crack of a unit radius and treated with harmonic shear loading. Here, the system of basis functions ~gi(~).

Without

has the form of symmetry functions

~(z)

=

ei (z) has been chosen in form e~(z) =

10SS of generality, (Const,

in this case.

O). The

Taking

it was assumed

solution

them

possesses

into account,

that the load some

we choose

features vector

gi in the form:

o)

gi(z) = (cos(j7rT)cos(2rr@, gi(z)

= (O, cos(km)sin(2nO)),

where 8 is the angle in the polar frame and

j, m, k, n are whole numbers.

The solution of the system (9) gives us values of the coefficients we determine

the jumps

of displacement

in the crack region

Ci, whereby [ul] and [U2].

After that, we have the opportunity to compute all of the important characteristics of the solution. The vector of displacement jumps is approximated by the sum ~i=l ciei . Let us denote ~i=l cigi = [u”] = ([u?], [u;]). It immediately definition

follows that [uj] =