compressive stress,
σ
c
for two spheres in contact (Timoshenko and Goodier, 1970):
σ
c
=

0
.
62
P
Δ
2
1
R
+
1
R
s
2
1
/
3
(2)
where
P
is the contact force,
E
is the elastic modulus,
ν
is the Poisson’s ratio,
R
is the shot radius and
(1
/E
0
) =
1
E
s
(1

ν
s
)
2
+
1
E
(1

ν
)
2
represented by Δ. The subscript
s
denotes the shot and the variables without
any subscript point to the target material. Satraki
et al.
(2005) have found that the analytical solution proposed
by Hertz represents the maximum compressive stress for two spheres in contact.
Based on the Hertz contact
analysis approach, AlHassani (1981) proposed an estimate of the plasticized depth as a function of peening
parameters using a damage number represented by
ρV
2
/p
from the following equation of motion:
4
π
3
ρR
3
dV
dt
=

πa
2
p
(3)
where
p
is the average pressure resisting the motion and is evaluated as,
p
σ
y
= 0
.
6 +
2
3
Ea
σ
y
R
(4)
The plasticized depth due to single impact is calculated as:
h
p
R
= 2
.
57(
2
3
)
(1
/
4)
(
ρV
2
p
)
(1
/
4)
(5)
where
h
p
is the thickness of plastic layer. This value of
h
p
is used in further calculations of the residual stresses
in thin plates.
The approach for calculating the RCS is by using the concept of ’source’ stress introduced
by Flavenot and Nikulari (1977). The residual stresses are evaluated by superimposing the bending and axial
stresses onto the ’source’ stress which is a function of the plastic layer thickness. AlObaid (1995) has introduced
a different scaling factor of 3 in stead of 2.57 in the previous expression. Watanabe and Hasegawa (1996) has
modified the expression for thickness of plastic layer with more terms that contain the cubic expressions.
AlHassani (1982) further elaborated on the role played by shakedown, reverse yielding, Bauschinger effects
and strainrate in the accurate prediction of RCS. Li
et al.
(1990b) have introduced the concept of internal
fatigue strength as their theory relates the fatigue failures to subsurface residual tensile stresses in stead of the
residual compressive stresses at the surface layers. On the contrary, in their experimental studies on Ti3Al2V
alloy, Hanyuda
et al.
(1993) conclude that the fatigue strength improvement depends mainly on the magnitude
of surface RCS and not on the tensile stresses developed inside the material.
Based on experimental observations, Johnson (1987) proposed a different approach in which the spherical
indentation in an elastoplastic halfspace is considered. It is equivalent to a spherical cavity expanding in an
infinite medium with the same elastoplastic property, which is commonly cited as the expanding cavity model
(ECM). In a spherical coordinate system, for
p >
2
σ
y
/
3, the stress field in the plastic zone (
a < r < c
) is given
40
Baskaran et al. / International Journal of Structural Changes in Solids 2(2010) 3363
Figure 6:
Comparison of residual stresses obtained by analytical method with experimental results reproted by Cammett
et al.
You've reached the end of your free preview.
Want to read all 31 pages?
 Fall '15
 RichardAyala
 Materials Science, Strength of materials, Fracture mechanics, Residual stress, Shot peening