Issue 68

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 68 (2024) 1-18; DOI: 10.3221/IGF-ESIS.68.01

The variation of the polar stress in the vicinity of the base (tip) of the notch In Fig.6, the variation of the polar stress components σ rr , σ θθ , σ r θ , is plotted along a semi-circular locus (red color) with its center at the origin of the reference system, enclosing a part of the strip in the vicinity of the base (tip) of the notch. Three different stretched, notched strip configurations are studied in accordance with the parameters set in previous section. The radii r of the circular loci around each one of the three notch bases were arbitrarily chosen as: r=10 α 2 for α =0.25 cm 1/2 , resulting to an angle range 53.13 ο ≤θ≤ –90 ο (Fig.6a), r=4 α 2 for α =0.5 cm 1/2 (resulting to 30 ο ≤θ≤ –90 ο ) (Fig. 6b), and r=3 α 2 for α =1.0 cm 1/2 (resulting to 19.47 ο ≤θ≤ –90 ο ) (Fig.6c). For the calculations of stresses, the general expressions given in the Appendix were used, in conjunction with the transformation formulae for the rotation of the reference system at an angle θ . The distributions of σ rr , σ θθ , σ r θ , resemble qualitatively the respective well established ones given in ref.[19], based on completely different analytical and numerical approaches, thus, justifying further the validity of the present solution.

1/2 0.25cm    

10

10 15 20 25

y

5

rr 

0

x

-5

r  

-10 -5 0 5 10

-5 0 5

y

 

o 53.13



2 

Polar stress components [MPa]

x

r

-90

-60

-30

0

30

60



 2 r 10  

o 90 

o 53.13

Angle θ [deg]

(a)

1/2 0.5cm    

10

24

y

5

18

rr 

0

x

12

-5

r  

-10 -5 0 5 10

6

O y

 

o 30

0

2 

Polar stress components [MPa]

x

r

-90

-60

-30

0

30

(b)

2 r 4    

Angle θ [deg]

o 90 

1/2 1.0cm    

18

10

y

5

12

0

rr 

x

-5

r  

-10 -5 0 5 10

6

O y

 

o 19.47

0

Polar stress components [MPa]

2 

x

r

-90

-60

-30

0

30

o 19.47

(c)

2 r 3    

Angle θ [deg]

o 90 

Figure 6: The variation of the σ rr , σ θθ and σ r θ , stress components along a circular locus around the base (tip) of the notch, for notches with (a) α =0.25, (b) α =0.5 and (c) α =1.0 cm 1/2 .

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