Fatigue Crack Paths 2003

Fundamental system of equations

EJmQs.

EA fl — l— G i A fl + l + r+p = 0

ds r

r ds r

G A fl + l + ¢ E@ — +Xq = 0

ds r

r ds r

[ E l d ¢ ) — G A [ d v + u + o ) + m = 0

ds

a's

r

Displacements vector

\ Exte,rnal forces vector

6 l P

on

i

Indefinite equilibrium equations

d 1 _ i L 0

i — l 0

p

is rd

N

8

ds r

u

q I —- —I 0 T

y I I i 1

v

r

ds

r ds

m

d M

Z

d (p

0

1

— i

0

i

ds

ds

T Internal forces vector

Constitutive relations

Strain vector

l

/N S: T

N EA 0 0 e T = 0 GA y X

Y

i;

‘

8 X

I < i ‘ s II y

M 0

El

/

\

Figure 2. Arch problem in the form of Tonti Diagram.

XIIIIIXI/IZ

=0

(1)

together with two differential equations: the first involving tangential displacement and

curvature, the second involving radial displacement and curvature. Then, the general

solution of such system can be expressed as follows [5]:

u: —C2+C4r & — Lcos£+ C1—C3r & — Lsin£+C4Assin£+CA

s 3 s cos——C5rs— C6r

2 A r

2 A r 2

r

I’

(2)

C31 ,

s C41

s

.

s

s

v=C1cos—+C2s1n——

s cos——C5r2

s 81H — +

r

r

r

2

r 2

. s (p : C3r2s1n;— C4r2cos;+ C5s+ C6 s