Crack Paths 2012

—(X'E'q 2 w 4at 1?‘ Kl,em(t)=—-f — e e a t p 8 7 t h It r“ r 2 \ / r — r R ‘—’Z

“ 4 —riElf—rl‘ —rZE;i _—'”2 air

(21)

r12 r 2 \ / r — ] / ' R

After integration (with the help of the Mathematica software) it becomes:

2 _ r R _ol’E’q 2 — 20at(e4at—l) — _—r2

22

K1,,emp(r)-

80)» J; 407r,,+—r13?/+25\/rREz 4at’?

( )

1/4 _ 2 m+10(at) _32F2(-_1,l),(l,§, rR +2F2(1,§))(i,l) rR _ 2

F 1

4 4 4a2t

44 4 2 4 4at

4

2

2

_ grR

5 1 7 i —r;R

3 5 —rR

1,4

2 2 4 ’ 4 ’ 4 ’ 2 ’2 42 a 4 ’ 2 ’ 4 ’ 4 a t

(at) F —

with the hypergeometric function:

+°° (al)k'"(a )k Zk

F a,...,a ,'b

,‘zI

— p —23

,

p qll 1

Pii 1

‘1i

)

Zk:0(b1)k___(bq)k _/

( )

where (a)kIr ( a + k )

=a(a+1)(a+2)...(a+k—1) is the Pochhammer symbol and

F(a)

+00 _ — x . . F(x)=j ux 1e du the Euler Gamma function. For instance in our case .

2F,i[a1, a2i;lbl’b2i;Z):Z:)0

% '

%

The evolution of Kuemp (the thermal correction on K ) versus time is presented in Figure 5 for a unit line heat source (qIl Wm'l) and the following typical material

characteristics:

pI7800kg.m‘3, CI460JK'1kg'1, 7tI52 Wm'lK-l, otI12.10‘° and

EI210GPa. For instance, after times of 10s and 100 s the thermal correction on the

stress intensity factor is respectively -l.2><10‘3MPa\/m and -2.1><10'3MPa\/m for a reverse cyclic plastic zone radius of 4 p m and qIl Wm'l. These values are negative

because the temperature field generates compressive circumferential normal stresses

56