Page 366 - Biaxial Multiaxial Fatigue and Fracture

350 c. CAL~ R. CITARELLA AND M PERRELLA

Three dimensional simulation: theoretical aspects and results.
Surface crack solutions are widely used in applications of fracture mechanics to fatigue and
monotonic loading. A semi-elliptical surface crack lying perpendicular to the surface and
subject to applied stresses with no shear component parallel to the crack, experience mode I
conditions around its edge, as in the problem presented in the following. Generally this
orientation is the critical one, also in mixed mode conditions if the fatigue crack threshold is
governed by (AG I)W, where G is the energy release rate.

The dual boundary element method. BEASY uses dual elements for 3D crack growth analysis.
The dual boundary element method (DBEM) incorporates two independent boundary integral
equations: the displacement equation applied at the collocation point on one of the crack
surfaces and the traction equation on the other surface. Application of the DBEM to three-
dimensional crack growth analysis has been presented in [ 1 1-1 31.
In the absence of body force, the displacement boundary equation can be written as:

cIJ(xI).u,(xI)+ ~~,(x~,x).u,(x').(rr(x) lu,(.',x).t,(x').~(x)
=
r r
where iJ denote Cartesian components, r, (x 'J) and Uv(x 'J) represent the Kelvin traction and

displacement fundamental solutions at a boundary point x, respectively. The symbol stands
for the Cauchy principal value integral.
Assuming continuity of both strains and traction at x' on a smooth boundary, the stress
components G,, are given by:

where Tkv (x 'J) and uk,(x 'J) contain derivatives of z, (x f) and uv(x 'T), respectively.
The symbol stands for the Hadamard principal value integral. The traction components 6

are given by:

where n, (x ') denotes the component of outward unit normal to the boundary at x '.
Equations (2) and (4) constitute the basis of DBEM. A more complete description is given in
u11.