Issue 71

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

y

-0.3 0.0 0.3 0.6 0.9 -1.2 -0.9 -0.6 -0.3 0.0 0.3 0.6 0.9 1.2 y 2 1 c 2 1  7 V 5 V 6 V 4 V 3 V 2 V 1 V 1 x [cm]

V 1 V

3 V 2

4 V

0.9

5 V

6 V

[cm]

7 V

0.6

7 7 c 0.75cm    2

0.3

Vnotch Rounded

1 1 c 0.75   

0.0

Parabolic upper cavity

-0.3

-1.2 -0.9 -0.6 -0.3 0.0 0.3 0.6 0.9 1.2 [cm]

[cm]

(b)

(a)

60

7 (V )

5

y

56.13

[cm]

50

0

1 x

2 1 1 c 0.75cm    1 V notch 's depth 

40

-5

k

2h  20cm

30

18.5cm

-10

20

6 (V )

2b 30cm 

-15

0.75cm

5 (V )

10

4 (V )

Stress concentration k

3 (V )

2 (V )

1 (V )

-20

o 

o 

3.21

1 V notch 's span  1.73cm

0

10MPa

0.0

0.1

0.2

0.3

0.4

0.5

-25

[cm]

-20 -15 -10 -5 0 5 10 15 Doubly notched strip (for V 1 - notches)

0.18 0.26 0.34 0.42

0.02

α [cm 1/2 ]

(c)

(d)

Figure 9: (a) Seven geometries for parabolic notches of the same depth (length) equal to 0.75 cm; (b) The parabolic cavities versus the respective approximated rounded V j notches; (c) The doubly notched strip in case of the V 1 notches, as a typical example; (d) The variation of the Stress Concentration Factor k versus the parameter α . equal to k=3.21 (already mentioned previously, with regard to Fig. 7a). Obviously, as α is getting smaller and smaller, k increases monotonically up to the value equal to k=56.13 for the V 7 -edge notches, i.e., the case corresponding to a value of α equal to α =0.02 cm 1/2 . For α → 0, Eqn.(12) yields k → ∞ , which is the case when the edge notches become “mathematical” edge cracks, and the concept of stress concentration k should be replaced by the concept of the stress intensity factor K I . The stress intensity factor K I in the stretched doubly notched strip Assuming α = 0 the two parabolic notches become edge “mathematical” cracks (i.e., edge notches of zero distance between their lips). As it was mentioned before in that case, by means of Eqn.(12), the stress concentration k becomes infinite, and the concept of stress intensity factor K I is now the adequate quantity to describe the stress state at the tip of the cracks. In this context, a definition is provided next for K I in the case α = 0, as follows:

 α 0 K : lim[ σ α ]   I xx,cr

(13)

Substituting from Eqn.(12) in Eqn.(13), yields:

4 σ

4

(14)

 K : lim[ σ α ] xx,cr I

 



c

σ π c

o

3/2 o

π

π

 α 0

Noticing that for α =0 the crack length is d=c+ α 2 =c, K I provided by Eqn.(14) is actually the Stress Intensity Factor at the tip of an internal crack of length 2c under mode I loading in an infinite plate, i.e., o σ π c , multiplied by the factor

311