Page 484 - Biaxial Multiaxial Fatigue and Fracture

468 1L.Z SANTOS ET AL.'

amplitude on the critical plane and the normal stress acting on that plane is defined as the
fatigue damage correlation parameter.
For complex loading histories, the principal directions may rotate during a loading cycle
(e.g. see Ref. [9]). Therefore, Bannantine and Socie [5] suggested that the critical plane should
be identified as the plane experiencing the maximum damage, and the fatigue life of the
component is estimated from the damage calculations on this plane. The approach proposed by
Bannantine and Socie [5] defines the critical plane as the plane of maximum damage rather
than the plane of maximum shear stress (strain) amplitude, as defined by previous authors. This
approach evaluates the damage parameter on each material plane. The plane with the greatest
fatigue damage is the critical plane, by definition. For general random loading conditions, with
six independent stress components, the critical plane approaches have to be carried out for
plane angles 8 and Q varying from 0 to E. These procedures demand a great deal of calculations,
especially when small angle increments are used.
In the last decades, the critical plane approaches have found wide applications and also
received some criticism. The critical plane approach assumes that only the stress (strain) acting
on a fixed plane is effective to induce damage, and then, no interaction of the damages on the
different planes occurs. These assumptions are not always valid, and may considerably
underestimate fatigue damage. Zenner et al. [lo] also indicated by a typical example that the
hypotheses of the critical plane approach are not suitable for describing the effect of the phase
difference. The example considers the stress waves of Eq. (3), with phase shift angle Sxy= 90'
and stress amplitude q= 0.5ab. Under this load case, the shear stress amplitude has the same
magnitude in all planes.

Integral Approaches

Integral approaches are based on the Novoshilov's integration formulation, as a mean square
value of the shear stresses for all planes [lo]:

Equivalent-stress amplitude is yielded by an integration of the square of the shear stress
amplitude over all planes y+ for fully reversed stresses.
Further developments of the integral approaches led to various hypotheses such as the
effective shear stress hypothesis, the shear stress intensity hypothesis (SEI), etc. Generally, the
integral approach [lo] uses the average measure of the fatigue damage by integrating the
damage over all the planes. The integral approach considers all damaged planes of a specific
critical volume. The averaged stress amplitude of the shear stress intensity hypothesis (SEI) is
formulated as:

Papadopoulos' mesoscopic approach 11 11 is also formulated as an average measure, by
integration of the plastic strains accumulated in all the crystals, within the elementary volume: