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
206
5
z Z z Z − +
z Z z Z − +
z Z z Z − −
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
Z z
Z z
1
Z
Z
n
2 ( ) j j
i
i
j
−
j
j
j
iP
F e
e
+
−
+
+
m
m
c
2 z Z −
2
2 z Z −
2
2 (1 ) 1 2 (1 ) + +
z Z z Z − −
1
j
=
m
m
j
j
2 R Z z R Z z − + 2 j
2 R Z z R Z z − + 2 j
2 R Z z −
n
i
i
j
−
j
( ) F e
log
log
log
iP
e
+
+
+
m
j
j
c
2
R
m Z z
−
1
j
=
j
j
2
2
R Z Z R Z Z − −
2 2 R Z Z R Z Z R iP − − 2
n
i
j
j
j
j
j
( ) F e
+
−
+
−
m m
m m
(
) ( 2
)
(
) ( 2
)
c
j
j
2
2
2 (1 ) +
2 R Z z −
2 R Z z −
2 R Z z +
2 R Z z −
1
j
=
m
m
j
j
+
(
)
2
( 1) −
iP Z Z −
m m Z Z R +
2 R Z Z R Z z F − − 2 ( ) j j
2 R Z Z −
i
−
j
c
m m
j
j
j
z
e
+
−
(
)
(
) ( 2
)
2
4
2
2
2 (1 ) +
m m R Z Z R z Z Z z − + + m m
2
Z z
R
+
j
i
2
R
n
(
)
j
j
2 2 − − j e R Z Z R Z Z z 3 2
+
(
)
j
j
j
j
(1 ) 1 + −
8 4 2 R R Z Z z Z Z z − + + 2 2 2 2 4
1
j
=
j
j
j
j
1
−
(
)
i
−
j
3 2 Z Z Z Z z e + + 2 3 j j j j
2 2 − − R Z Z R Z Z z 3 2
3 2 Z Z Z Z z + 2 3
+
+
j
j
j
j
j
j
j
j
2
2
R
R
(6)
3. Analytical results 3.1 The stress- and displacement-fields The components of the stress- and displacement-fields are obtained in closed form by substituting Eqs.(5) and (6) into the well-known formulae (Muskhelishvili 1963): (7) Taking advantage of the as above obtained expressions, the variation of stresses is here plotted along two charac teristic loci, namely along the y -axis (in other words along the loading line) (Fig.5a), and, also, along the OL radius (Fig.5b), where L is the end- point of the cavity’s rim . The numerical values used to calculate the stresses were: R = 5 cm, R 1 =1 cm, h =5 R /6, E 1 =50 GPa, ν 1 =0.3, κ 1 =3 – 4 ν 1 , E 2 =210 GPa, ν 2 =0.3, κ 2 =3 – 4 ν 2 , and P frame =15 kN, resulting to R 2 =0.954 cm, ω 2 =74.88 ο and P c =122.11 MPa. Among the most interesting features of the stress distribution along the above loci is that that all three stress com ponents remain bounded (see Figs.5(a,b)) all over the loaded arc (i.e., the rim of the cavity), even at its end-points. 1 1 4 ( ), z + = 2 ( ) − = + ( ) z z z + ( ), 2 ( z ) + = ( ) − − ( ) ( ) z x y y xy i u i v z z z
-0.06 -0.04 -0.02 0 0.02 0.04 0.06 y [m]
10
-80 -60 -40 -20 0 20 0.00 0.01 0.02 0.03 0.04 0.05 x y
x
0
Deformed
0.00 0.01 0.02 0.03 0.04 0.05
-30 Stresses along OL [MPa] -20 -10
Undeformed
-120 -100 Stresses along y -axis [MPa]
xy
-0.06 -0.04 -0.02 0 0.02 0.04 0.06
Fixed po int •
y
-40
y -axis [m]
OL -radius [m]
x [m]
(a) (c) Fig. 5. Stresses along the loaded radius (a), and along the radius OL (b). The deformed versus the undeformed configuration (c). (b)
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