Fatigue Crack Paths 2003

Figure 2: Typical fatigue crack growth rate diagram

cwk(x0)uk(x0) + Z

K Twk(x0,x)uk(x)dK(x)

Z K Uwk(x0,x)tk(x)dK(x) + h Z

l Uwk(x0,X)fq(X)dl(X)

=

(4)

and for plate bending

(x0)wj(x0) + Z

K Pij(x0,x)w

c ij

j ( x ) d K ( x )

Z K W i j ( x 0 , x ) p j ( x ) d K ( x ) + Z

=

l Wi3(x0,X)q3(X)dl(X)

(2)

where uk are in-plane displacements, wj are the rotations in x and y, tk in-plane

tractions, pj are the bending momentsand the shear tractions, q3 is the internal

pressure. Twk(x0,x) and Uwk(x0,x) represent the Kelvin fundamental solutions for

plane stress elasticity, while Pij(x0,x), Wij(x0,x) are the Reissner plate fundamental

solutions [4]. Greek indices vary from 4 to 2, Romanindices from 4 to 3.

T h eDualBoundaryM e t h o d

The Dual Boundary Method(DBM)for modelling cracks in plates was formulated

by Dirgantara and Aliabadi[3]. The main idea of D B Mis to model the crack as

two separate surfaces facing each other, with coincident discretisation points. Then

the displacement boundary integral equations are applied onto the upper surface

K+ meanwhile the traction boundary integral equations onto the lower surface K3.

The latter have been found via the application of the Hookelaws of elasticity to the

derivative of the displacement boundary integral equations. Theresulting equations

for uniform pressure and free-traction crack are the following:

pk(x0) + nq(x03) = Z

K Pkq (x03,x)w ( x ) d K ( x )