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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