Issue 20
R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01
Figure 3 : Equally-spaced circular inclusions in an infinite domain arranged in a hexagonal cell pattern having characteristic size d , under remote uniform tensile stress 0 y .
x
( ) P
r
(
,
)
( ) i x i P i P ( ) ( )
i
0 y
0
( ) P
r
(
,
)
(4)
y
i y
( ) i P i P ( )
y
( )
i
( ) P
r
(
,
)
xy
( ) i xy i P i P ( ) ( )
i
where the cartesian stress tensor components ( ) ( ) ( ) ( , ) i x i P i P r , ( ) ( ) ( ) 0 ( ( , ) ) i y i P i P y r , ( ) ( ) ( ) ( , ) i xy i P i P r
indicate the stress
1, 2, 3, 4,..... i ), see Eq. 2, under the
fluctuations evaluated in P in an elastic infinite plane containing a single inclusion i (
remote stress 0 y . In the above expressions, the summation might be performed by taking into account all the inclusions that are within a significant influence region around the point P under consideration, since the inclusions located at a sufficiently large distance from P produce vanishing fluctuations of the stress components. In Fig. 4, sample spatial distributions of the fluctuating stress components along different lines normal to the remote loading axis are shown.
-0.008 0.008 dimensionless stresses, x / 0y , xy / 0y
1.002
-0.004 0.004 dimensionless stresses, x / 0y , xy / 0y (a) -0.002 0 0.002
dimensionless stresses, y / 0y
1.004
dimensionless stresses, y / 0y -0.004 0 0.004 (b)
1.001
1
1
0.999
y / 0y x / 0y xy / 0y
y / 0y x / 0y xy / 0y
0.996
0.998
0E+000 4E-004 8E-004 Position (m)
0E+000 4E-004 8E-004 Position (m)
Figure 4 : Stresses along a horizontal straight path (dashed line) located at (a) half distance and (b) one-third distance between two lines of inclusions, in an infinite plane under plane stress remote uniform tension stress 0 y . Dots indicate the positions of inclusions in the material.
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