Issue 71

M. Abdulla et alii, Fracture and Structural Integrity, 71 (2025) 124-150; DOI: 10.3221/IGF-ESIS.71.10

Figure 4: Meshed view of plate.

Stress and Displacement Fields Near the Crack Tip The stress field near the crack tip in a linear elastic material is characterized by the Mode I (opening mode) stress intensity factor K I . The stress components in polar coordinates (r, θ ) around the crack tip are given in Eqn. (1) [4]:

K

    r ,

  

I



(1)

f

j

ij

i



r

2

where σ ij is the stress components, r is the distance from the crack tip, and θ is the angle with respect to the crack plane. The functions f ij ( θ ) for mode I are provided in Eqns. (2), (3) and (4) as follows:

I K

θ

θ

3 θ

 

  

σ xx ( r , θ ) =

(2)

  cos 1 sin sin 2

2 2

2 π r

I K

θ

θ

3 θ

 

  

(3)

σ yy ( r , θ ) =

  cos 1 sin sin 2

2 2

2 π r

I K

θ θ cos sin cos

3 θ

τ xy ( r , θ ) =

(4)

2 2 2

2 π r

The displacement components near the crack tip can be expressed as in Eqn. (5) and (6) [23]:

     

K r

  

   ,

2

I



    1 2sin

x u r

k

cos

(5)

    2

 





E

2

  2

                    1 2cos sin 2 2 k 2 

K r

   ,

I



u r

(6)

y





E

2

where u x and u y are the displacement components in the x and y directions respectively.  E is the effective Young’s modulus, given by  E = E for plane stress conditions, and k is a parameter that depends on the material properties and is given by k = 3- v for plane stress, where v is Poison’s ratio.

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