PSI - Issue 52

T. Profant et al. / Procedia Structural Integrity 52 (2024) 455–471 T. Profant et al / Structural Integrity Procedia 00 (2023) 000 – 000

461

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The inner solution for the fracture problem in the isotropic dielectric material obeying the direct flexoelectric equilibrium equations (2) is based on the asymptotic plane strain analyses provided in (Mao & Purohit, 2015)or (Gourgiotis & Georgiadis, 2009) for the case of the strain gradient elasticity. A semi-infinite plane crack in the isotropic flexoelectric material with the implemented polar coordinate system ( , ) at its tip is considered and depicted in Fig. 2. The traction-free crack faces given by the external normal = (0, ±1) and associated with the boundary conditions 0, for 0 and r   = =  = t r (21) at the tip of the crack are supposed. Moreover, the insulating or conductive crack faces are considered for mode I or mode II, respectively

0 or

0 for

0 and

r E

r



=

=



  =

(22)

Fig. 2. The traction free and isolated/conducted faces of the semi-infinite crack of the fracture problem in the material exhibits the flexoelectric properties as the inner solution , or , . The solution of this fracture problem is given in (Profant et al, 2023). It shows, that the fulfilling of the boundary conditions (21) and (22) leads to the asymptotic solution whose exponent of the dominant terms for the displacement field is =3⁄2 associated with generally five independent amplitude factors. The inner solution requires the scaling-up of the coordinate system, because of the vanishingly small dimension of the region on which is defined. Hence, the gauge = ⁄ is introduced as well as the scaled-up radius vector = ⁄ . Then the asymptotic form of the inner solutions for the mode I or II loading with insulated or conducting crack faces, respectively, can be written as

3 3 2 2

   

    

1 2

5 2

3 2



   +

, in ID r

cos2

cos

cos

cos

,

u

1 2 =  + R 

R A 

A

A







+

+

1

2

3

1 1

  

f





(

)

, in ID

0 2 2

1  − + +  12 3 4 5 3 3 5 A l A A

C R 



= +

2

a

(23)

  

1

1 2

f

  

 

 

     

  

(

)

(

)

1

, in ID

  −

5 3 5 − + + − A A f

5 3 3 5 A l

cos

u

2 1 f A l 

A A

+

− + +

− 

(

)

12

3

4

1

12

3 4

 

2 1 a +

a



0

3 3 2 2

   

   

1 2

5 2

3 2

, in ID u 

sin2

sin

sin

sin ,  

R =−  

R A 

A

l A

  +





−

−

2

4

2

12 3