COMBINATION OF REGIONS 187
Figure 7.6 First and second regions
From these results the stress intensity factor was calculated,
according to the linear theory of fracture mechanics to be
K
Y
= lim σ Jlnr
r - 0
The variation of
stress
intensity factor along the crack tip is shown in
Figure 7.10.
7.3 APPROXIMATE BOUNDARY SOLUTIONS
We have already indicated that approximate solutions can be used to
form boundary elements. These solutions are of practical interest
188 COMBINATION OF REGIONS
Figure 7.7 Third and fourth regions
when the fundamental solution is difficult to obtain or cumbersome
to use.
To illustrate the technique consider the case of the Helmholtz
equation, i.e.
V
2
w + K
2
U = 0 (7.7)
K
is the wavenumber and u a potential. Let us consider that the
domain can be divided into two regions, Ω
γ
where a finite element
discretisation is applied and Ω
2
which extends to infinity. The two
regions are connected at the interface Γ
ι
(Figure 7.11).
"20
' 60 ' Too ' ιϊο ' m
r (mm)
Figure 7.8 Fifth region
Figure 7.9 Variation of
the
calculated direct stress σ
190 COMBINATION OF REGIONS
0 ~S/2 5
Figure 7.10 Variation of
the
stress intensity factor along the crack tip
Let us analyse the Ω
2
region considering that the elements on the
interface are boundary elements. The fundamental solution of the
Helmholtz equation for two dimensions is (assuming a time de-
pendence
e
icüi
)
U* = ^HV{KT) (7.8)
where r is the distance from the point under consideration to any
other point on Γ
ν
300
250
200
150
inn
COMBINATION OF REGIONS 191
Region Ω
Figure 7.11 Finite and boundary elements domain: (a) domain under consideration;
(b) interface
Weighting equation (7.7) by
M*
and integrating by parts we obtain,
(V
2
w* + K
2
u*)udß= -^-tidr- ιι*γάΓ (7.9)
As the function u* satisfies the Helmholtz equation with δ
{
on the
right-hand side we can write,
r
dn J dn
(7.10)
for x
i
inQ
1
. Once the form of the fundamental solution is substituted
into (7.10) we can write,
H
{
o
2)
(Kf,
r
on
ΟΗ
{
0
2)
(ΚΓ)
ü
dr
ί^
n J
r
dn
üdr
(7.11)
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