CAVITATION, INDENTATION AND PENETRATION

364 D. Durban, R. Masri
with
η T =
Pc =
Pc = 1
√ Σy 1 +
3
2
Σy 1 +
4 − ηT 1 − η T ln
n 1
2Σy
1 π2 2Σy 6
F(n) − 1
n
− 1 − ln
1 − ln
1
2Σy
2Σy
n F(n) − 1
,
n
(10)
n 4 1
+ F(n) −
3 2Σy
1
,
n
(11)
1
√ 3Σy
and F(n) = ζ(1 + n)Γ(1 + n), (12)
where n is the hardening exponent and ζ and Γ denote the Zeta and Gamma
functions, respectively.
Durban and Masri[4] have examined the relation between the average indentation
pressure of standard conical indenter, identified with material hardness
H, and quasi-static cavitation pressures. It turns out that for power law response,
in plastic range, there is a good agreement between Pc of (8) and the
hardness (H/E).
3. Penetration
Following Goodier[5] it is assumed that the pressure on the penetrator is given
by the local dynamic cavitation pressure. Considering steady-state dynamic
expansion of a spherical cavity, Masri and Durban [10], we introduce the nondimensional
coordinate ξ = R/A, where R is the radial coordinate and A the
instantaneous cavity radius. For self similar expansion fields we use the transformation
( ˙)
= ˙ d( )
ξ
dξ = ˙ A )
(V − ξ)d(
A dξ , where V = ˙ R/ ˙ A. (13)
With the J2 theory of plasticity it is possible to reduce the governing equations
to the non-linear couple
βΣ ′ β
r +
2 Σ′ + 1
2 ǫ′ = 1
β
−
1 − e 2
ξ
Σ+3 2ǫ , (14)
Σ ′ r − 2
ξ Σ = m2 ξ 2 βΣ ′ r + βΣ ′ − ǫ ′ e −3βΣr−βΣ−3ǫ , (15)
where radial stress Σr and effective stress Σ are nondimensionalized with respect
to E and a superposed prime denotes differentiation with respect to ξ.
Numerical solutions of (14)-(15) provide the dependence of Pc on m. However,
for elastic/perfectly-plastic solids it is instructive to peruse a power expansion