Li et al [8], proposed that the elastic stress concentration K_{t}
introduced by multiple micro-notches in SP, is somehow lower than the one
determined in the case of a single notch of similar depth and width. The above
finding reflects the uniformity of the micro-notches on the surface. According
to Li, the resulting K_{t }from SP is given by,

(3)

where the parameters R_{t} and S are respectively the mean
of peak-to-valley heights and the mean spacing of adjacent peaks in the surface
roughness profile. In the case of a semi-elliptical notch and a high degree of
uniformity (SP coverage percentage of more than 100%), Eq.(3) can be written
as,

(4)

At the beginning of this section it was pointed out that the
bluntness of the notch can significantly affect the strain generated at the
root of the notch and consequently the propagation rate of the crack. In light
of that, Smith and Miller [4] proposed that K_{t }should be determined by,

(5)

where r is the notch root radius.
In the case of a semi-elliptical notch, the notch root radius can be
approximated by and thus Eq.(5) can
be rewritten as,

(6)

where g is the notch half width that considers the bluntness of the notch.
By equating Eq.(6) with Eq.(4), the stress concentration due to multiple
micro-notches can be expressed in terms of a single notch by,

(7)

**Modelling the Fatigue Life in CSP components**

In [3] it was proposed that the fatigue life of polycrystalline
materials can be determined by,

(8)

where A_{2},
m_{2} are parameters from the Paris law of crack propagation, CTOD is
the crack tip opening displacement and , are limit values of n_{1} as defined in Fig. 1.

**Figure 1.** Schematic
showing the position of n_{1} prior (n_{c}) and after (n_{s})
the unblocking of the crack tip plasticity.

In Eq.(8) the parameter and represent
respectively, the position of the crack tip at the beginning and end of each
interval i of crack growth. These two parameters are calculated by,

(9)

where s_{2} is the flow resistance of the material and is the Kitagawa-Takahashi formula for a plain specimen,

(10)

with s_{FL} denoting the fatigue limit of the
plain material. In Equation 10 the parameter m_{i}/m_{1}
represents the effect of the grain orientation factor. More details can be
found in [3, 9].

From Eq.(8) it is clear that the number of cycles required by the
crack to propagate an i number of half grains, depends solely on the parameter . For SP components, the parameter has to be modified in
a way that it would take into account the roughening of the surface and the
crack closure stresses generated by the residual stresses,

(11)

where the parameter Z_{i} is
taken by Equation 2. The Kitagawa-Takahashi formula for the case of SP,, is given by [3],

(12)

where . Hence, Equation 12 is rewritten as,

(13)

**Introducing the Improvement Life Factor (ILF)**

In order to increase the life consumed at each grain and consequently
the overall life of the SP component, we make use of a predetermined ILF,

(14)

where the values of ILF are in percentage. Solution of Equation 14
in terms of CTOD yields,

(15)

In the case of a plain/unpeened material, Equation 15 is written
as,

(16)

The fact that the value of CTOD at the position n_{c}, where
the crack tip plasticity is able to overcome the microstructural barrier, is identical
for both the peened and the unpeened material (for the same loading conditions)
allows Equations 15, 16 to be equated,

(17)

Simplification of Equation 17 gives,

(18)

From Equation 18, the
closure stress can be determined. It should be noted that due to the
complexity of Equation 18, a computational solution is advised. Fig. 2 shows
the calculated crack closure, , for several conditions of loading and treatment.

**Discussion and Conclusions**

In this work the effects of SP are analysed and modelled according
to their effects on fatigue damage. Surface roughness is modelled in a way that
would increase the far-field stress. Hence, the component appears having an
amplified capacity of initiating and propagating short fatigue cracks.
Compressive residual stresses translated as closure stresses are regarded as one
of the beneficial effect of SP. Residual stresses tend to reduce the
application of the far-field stress by introducing a closure stress on the
crack flanks. Thus, the propagation of the crack is expected to be somewhat
reduced compared to the unpeened condition. Finally, strain hardening is
expected to reduce the propagation of short fatigue cracks by increasing the
ability of the material to localised straining (crack tip plastic zone).

The introduction of a predetermine improvement in terms of fatigue
life (ILF) has been achieved by introducing the ILF into the number of fatigue
cycles consumed in every grain. The above approach allows the mathematical
modelling of the conflict between the beneficial and detrimental SP effects. At
first, the analysis reveals that the magnitude of the closure stresses should
always attain a maximum at the surface. Such distribution minimises the
premature initiation of a ¡°visible¡± fatigue crack. Secondly, the depth
distribution should be able to follow the stress gradient generated by the
surface roughness. Further analysis allows to scrutinise against parameters
like the far-field stress level, the ILF and the surface roughness. From Figure
2, the effect of the above parameter is classified in the following order,
starting from the most decisive: a) *Surface Roughness*. The analysis
reveals that a 12% increase, measured in terms of K_{t}, in the surface
roughness requires a 47% increase in the closure stress magnitude to allow a 5%
increase in the per grain life. Additionally, a higher K_{t} would
require deeper closure stresses; b) *Far-Field Stress Level*. In principle
high far-field stress levels require high magnitude and deeper closure
stresses. This comes as a verification to the fact, published extensively in the
literature, that SP will have a minimum effect, or in some cases a detrimental
effect, on the low cycle fatigue region; and c) *ILF*. The analysis
reveals that SP components are not so sensitive to different ILF values. The
above finding is in accordance to many experimental data where short cracks
were found to propagate almost irrespective of the crack closure stress levels.

It should be noted that due to the fact that the methodology is
expressed in terms of crack length, it can be easily adjusted to incorporate
relaxation profiles of residual stress and strain hardening.

**References**

[1] S. Curtis, E. R. de los Rios, C. A. Rodopoulos^{ }and A.
Levers, Inter. J. of Fatigue, 25 (2003), 59-66.

[2] C. Vallellano, A. Navarro and J. Dom¨ªnguez, Fatigue Fracture
Engineering Materials Structures, 2000, 23, 113-121.

[3] E. R. de los Rios, M. Trull and A. Levers, Fatigue Fracture
Engineering Materials Structures, 2000, 23, 709-716.

[4] R. A. Smith and K. J. Miller, Inter. J. Mech. Sci., 1978, 20,
201-206.

[5] K. Tanaka, Inter. J. Fract., 1983, 22, R39-R45.

[6] S. Suresh, Fatigue of Materials, Cambridge University Press,
1991.

[7] C. Vallellano, A. Navarro and J. Dom¨ªnguez, Fatigue Fracture Engineering Materials
Structures, 2000, 23, 123-128.

[8] J. K. Li, M. Yao, D. Wang., R. Wang, Fatigue Fracture
Engineering Materials Structures, 1999, 15(12), 1271 ¨C 1279.

[9] E. R. de los Rios
and A. Navarro, (1990) Philosophical Magazine 1990, 61, 435-449.