Issue 20

R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01 9 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  . ( ) ( ) ( ) 0 ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) ( , ) ( ) ( , ) ( ) ( , ) x i x i P i P i y y i y i P i P y i xy i xy i P i P i P r P r P r                        (4) 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 fluctuations evaluated in P in an elastic infinite plane containing a single inclusion i ( 1, 2, 3, 4,..... i  ), see Eq. 2, under the 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. 0E+000 4E-004 8E-004 Position (m) -0.004 -0.002 0 0.002 0.004 dimensionless stresses,  x /  0y ,  xy /  0y 0.998 0.999 1 1.001 1.002 dimensionless stresses,  y /  0y  y /  0y  x /  0y  xy /  0y (a) 0E+000 4E-004 8E-004 Position (m) -0.008 -0.004 0 0.004 0.008 dimensionless stresses,  x /  0y ,  xy /  0y 0.996 1 1.004 dimensionless stresses,  y /  0y  y /  0y  x /  0y  xy /  0y (b) 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.