propagation rate, and the effect of changes in variables on these factors. At present, prediction of fretting
fatigue is less developed than for plain (unnotched) fatigue. The main limitation is that continuum-mechanics
approaches do not consider microstructural inhomogeneities, and crack nucleation is controlled by such factors
as well as “short” fatigue-crack propagation.
Most of the early work in stress-field modeling for fretting fatigue uses, as a starting point, analysis similar to
that used by Mindlin (Ref 6) of a sphere pressed into a half plane and expands this to consider other geometries
and imposed shear loads. If stress fields are computed using FEM analysis, an assumed contact geometry and
coefficient of friction are used, and loads are imposed at various mesh locations in order to compute stresses
and subsequent strains. Alternatively, displacements can be imposed and strains and stresses computed. To
facilitate modeling, the stress singularity associated with an abrupt contact geometry change, such as at the edge
of a bolted flange or a hub/shaft interface, is accommodated by plastic deformation, and a limiting stress is
assumed. Current models are limited in that changes in contact geometry due to wear and variations in
coefficient of friction due to lubrication or debris accumulation are difficult to take into account.
Experiments have been undertaken and models have been proposed for both the full- and partial-slip regimes
and are based on empirical observations. Full-slip and partial-slip conditions can be achieved by varying the
test configurations.
In addition, while most fretting contacts are some combination of load- and displacement-controlled conditions,
laboratory experiments can be designed either to drive the fretting pads independently (displacement
controlled) or to allow them to move as a consequence of the clamping force and displacement of the “beam”
sample (load controlled).
For fretting under conditions of full slip, two early models predict the SRF due to fretting. Nishioka and
Hirakawa (Ref 13) derived the following equation to describe the fretting fatigue strength limit determined
using their displacement-controlled experiment setup with full-slip conditions under the fretting pads (Fig. 10).
σ
fw1
= σ
w1
- μp
o
{1 - e
(-d/K)
}
(Eq 2)
where σ
fw1
is the fretting fatigue strength, σ
w1
is the plain fatigue strength, p
o
is the clamping pressure, δ is the
slip amplitude (in mm), and K is a constant dependent on the material and surface condition (on the order of 3.4
× 10
-3
mm, or 1.34 × 10
-4
in., in Ref 13).
In later work of Wharton et al. (Ref 14), a similar form was developed in which notch sensitivity of the base
material was taken into account. The reduction in fatigue strength due to fretting was then proposed to also be
proportional to the shear stress resulting from the contact pressure of the cylindrical fretting pads, inversely
proportional to the contact width, and given by the equation:
σ
wf
= σ
wo
-q(8μP/πb)
(Eq 3)
where σ
wf
is the fatigue strength with fretting, σ
wo
is the fatigue strength without fretting, q is the notch
sensitivity factor, P is the load per unit length, and b is the contact width under the fretting pads (mm). Note
that both these predictions show that SRF is worse as μP or μP
o
is increased.
For probable location of fretting fatigue crack nucleation in the partial-slip regime, the approach of Ruiz and
Chen (Ref 9) can be used. In their analysis of a dovetail interface, a fretting parameter representing the energy
available for causing fretting damage, and given by the product of the slip amplitude, δ, and shear stress, τ, at
points under the interface, was computed. Next, the product στδ, called the fretting fatigue parameter, is
computed, where σ is the maximum surface tensile stress (resulting from the bulk cyclic loading). A fretting
fatigue crack is predicted to occur where the local value of στδ in the interface exceeds an empirically
determined critical value, or, fretting occurs when:
στδ ≥ στδ
crit
(Eq 4)
If στδ and στδ
crit
can be experimentally determined, then the designer can use this value as a design guide. Note
that both the slip amplitude and the shear stress depend on the coefficient of friction (with opposite responses)
and the imposed loading. Analysis by Nowell and Hills (Ref 15) of this work provided a theoretical justification
and a possible method for predicting “initiation” (or nucleation) time based on the total accumulated
incremental strain. With further effort, it appears that the composite parameter approach can be applied to
fretting fatigue in the full-slip regime and can be expanded to include plasticity.
Other models may be used to determine whether the conditions at the interface will be of full or partial slip and
to predict the location of the partial slip. These are generally FEM studies and make use of assumed
macroscopic contact conditions and bulk material properties. The text by Hills and Nowell covers this area; yet