Fatigue Crack Paths 2003

form. Let us consider a model for evaluating the local stiffness matrix ks, as shown in

Fig.1, which indicates dimensions and sign conventions for forces and corresponding

displacements. Then the following relation is obtained:

F=kll

(2)

S

S S

whereFSI[ P1, V1, M1, P2, V2, M2]T and uSI[ a1 wI e‘1 u; w e2]T. The stiffness matrix of a

line-spring kS is given by Tharp [2] as follows:

"IA 0 ‘1% ‘1% 0 “A

0 % 0

0 —1, 0

k 4% 0 1% 1% 0 4%) F a m 4,,/ rm -/1,,/

(3)

A 0

PD %) 0

PD

—1

1

0

AW 0

0

0

251% 0 — k , %2 5 % 0 2 5 %

whereDIe'ppe'mm—e'2mp. In Eq. 3, e”, , e'mm , e‘W are complianceexpressions for extension,

bending and shear, respectively and em,=0, e'pvI0 are compliances for the coupling of

bending and shearing, extension and shearing respectively. The compliance matrix for

this cracked membermaybe derived according to the theory presented by Okamuraet

al. [3] and Carpinteri et al. [4] as

I :Mj[%j2dr1, i m f fi j a j a

P P

E 0

O M

(4)

AW I M A I [ @ J Z C J A” ,I M j [ & ] [ @ ) d r 1

E O V P E P

()

In Eqs 4, KIP and K Mare m o d eI stress intensity factors caused by axial load P and

bending momentM respectively, and KHV is mode 11 stress intensity factor caused by

shear force V. E and i are the Young’s modulus and the Poisson’s ratio, respectively.

Cracked area is denoted by A and its infinitesimal increment d A is equal to Bda, where

B is thickness and a is crack length. The following equations for KIP, K Mand KHVare

utilized in the present paper:

P KIP 2 5 % } ?

(5)