PSI - Issue 42
Christos F. Markides et al. / Procedia Structural Integrity 42 (2022) 202–209 Stavros K. Kourkoulis and Christos F. Markides et al. / Structural Integrity Procedia 00 (2022) 000 – 000
205
4
f rame P
f rame P
1 R
Indenter
Parabolic distribution of concentrated forces
Part of the low impact on elastic response compete disc of its global
1
j Y
j
2 1
j j ( ) F
2 1 R R L
j Y
L
j ( ) P
j X
1 Z
Parabolic p r essure
j t
j Z
n Z m Z
CTBD
Complete disc
j X
Internal arc
Cavity
Fig. 3. Substitution of the parabolic contact pressure by a parabolic distribution of point forces.
The problem is solved in the mathematical ζ = ρ e i θ plane and for this reason the CTBD lying in the z = r e i θ plane, is mapped onto the unit CTBD by the function ω ( ζ )= R ζ , with s=e i θ being the point ζ on the unit circle γ , Fig.4.
m Y
Part of the compete disc of low impact on its global elastic response
j Y
j Y
L −
L
j Z −
j Z
j X
m Z j
j
j −
m
i e z r =
y
i e =
( ) R = =
z
r
x
O
O
j 1
R
i e s =
m
j −
m Z
j X
j
j Z
j Z −
L
L −
j Y
j Y
m Y
Fig. 4. The conformal mapping.
By demanding now that the periphery of the disc must be free from stresses (or, equivalently, that the condition ( ) ( ) ( ) 0 s s s s + + = must be fulfilled), Muskhelishvili’s complex potentials can be obtained in terms of the vari able ζ , and in turn, by inversing the transformation z = ω ( ζ )= R ζ , in terms of the variable z , as:
z Z z Z + −
z Z z Z + −
z Z z Z − −
1
n
i
i
j
−
j
j
j
( ) z
log
( ) F e
log
log
iP
e
=
+
+
m
c
j
j
2 (1 ) +
1
j
=
m
j
j
2 Z z R Z − 2
2 Z z R Z − 2
iP z Z −
z Z −
z
n
i
i
j
−
j
j
j
j
j
( ) F e
e
+
−
+
+
c
m
m
j
j
2 R Z z R Z z − − 2
4 R Z z −
2 2
4 R Z z −
2 2
(1 ) 2 +
1
j
=
m
m
j
j
(5)
2 R Z z R Z z + − 2 j
2 R Z z R Z z + − 2 j
2 R Z z R Z z − − 2 m
n
i
i
j
−
j
log
( ) F e
log
log
iP
e
+
+
+
c
j
j
2 (1 ) 1 4 (1 ) + −
1
j
=
m
j
j
(
)
z
n
(
)
2 ( ) j j
i
i
−
j
iP Z Z
j F e Z e Z +
+
− −
c
m m
j
j
2
R
+
1
j
=
Made with FlippingBook - Online catalogs