Issue 71

Ch. F. Markides et alii, Fracture and Structural Integrity, 71 (2025) 302-316; DOI: 10.3221/IGF-ESIS.71.22

Vnotch Rounded

2b 30cm 

5

-0.5 0.0 0.5 1.0

y

[cm]

y 17.07 

9.96



0

2 2 

x

c 2  x

32.07

y

-5

 



xx 

xx 

 

-1.0-0.5 0.0 0.5 1.0

 

2h  20cm

10.09



10.03 MPa



18.5

-10

32.07 MPa

Parabolic upper cavity

x

 

-0.5 0.0 0.5 1.0

-15

(

0)

    

1.73cm Span

32.07

3.17 

Parabolic upper cavity

-20

 

9.96

0.75



Lower cavity

10MPa

xx 

Length

-25

-1.0-0.5 0.0 0.5 1.0

-25 -20 -15 -10 -5 0 5 10 15 20 Doubly notched strip (c=0.5 cm; α =0.5 cm 1/2 ) [cm]

(a)

(b)

2b 30cm 

5

5

2b 30cm 

[cm]

y

[cm]

4.22 

y

2.74 

yy 

xy 

0

0

0.00

x

0.02

4.15

2.74 

0.00

x





(y 0.48) 

yy 

xy 

-5

-5



5.18MPa (y 0.13) 

yy 

yy 

0.00 xy 

xy 

2h  20cm

2h  20cm

0.16

0.04 

8.89MPa (y 0.20) 

-10

0.00

-10

4.22 

-15

-15



4.15

0.00

0.00

0.02 

-20

-20

yy 

 

xy 

 

 

 

2.74

-25

-25

-20 -15 -10 -5 0 5 10 15 20 Doubly notched strip(c=0.5 cm; α =0.5 cm 1/2 ) [cm]

-20 -15 -10 -5 0 5 10 15 20 Doubly notched strip (c=0.5 cm; α =0.5 cm 1/2 ) [cm]

(d)

(c)

Figure 7: (a) The variation of σ xx along the bisector of the notches and along the sides of the half strip, and the variation of σ ξξ along half the upper notch; (b) The geometric features of the parabolic cavities (vs the rounded V-notch); (c) The variation of σ yy along the bisector of the notches, along half the strip sides, and along half the upper notch; (d) The variation of σ xy along the bisector of the notches, along half the strip sides, and along half the upper notch. In Fig. 7a, the variation of the σ xx stress, as obtained from Eqns.(10), is plotted along the bisector of the notches (namely, along y-axis), as well as along the sides of the half (due to the symmetry) upper notch. As it is seen, at the midpoint of the bisector of the notches it holds that σ xx =10.09 MPa, a value almost identical to σ o holding in the case of the intact strip (“Problem 1”), thus approximating quite satisfactory the demand of non-interacting edge notches, set as a prerequisite while obtaining the present solution. At the bases (tips) of the notches σ xx attains (as it is expected) its maximum value equal to σ xx =32.07 MPa, a reasonable stress amplification (equal to about 3 times σ o ), taking into account the particular geometry of the notch (i.e., that of an almost semi-circular cavity). In addition, σ xx is around 10 MPa all along the sides of the strip. These results show that the short notches assumption suffices both insignificant stress disturbance along the sides of the strip, and, also, effective predictions about the stress concentration at the tips of the notches. The value σ xx =–3.17 MPa at the end points of the notches is to suffice stress equilibrium at the particular points. The only non-zero curvilinear stress component σ ξξ is, also, plotted in Fig. 7a, along the upper half notch (green color). Its maximum value σ ξξ =32.07 MPa, attained at the notch base, obviously coincides with the maximum value of σ xx at the same point (where σ xx = σ ξξ ). Similarly, the variation of the other two Cartesian stress components σ yy and σ xy are plotted in Figs. 7c and 7d, respectively. The very small value of σ yy (Fig. 7c) at the midpoint of the notches’ bisector (0.16 MPa) and the almost zero value attained along the sides of the strip guarantee further the validity of the solution, from the point of view of satisfying the boundary conditions of the problem.

309