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

459

5

in which

(

)

1

1

−

−

1 =  + +  =   + +  1 u r u r u   − − 1 r r r r u r u r u − ,

s

1

r

(12)

(

)

,

s r

1

r r

r





 

(

)

(

) 2 , 2  ) r

1

2

−

−

s

u r u r

u u

( r r r =  + r r =  + 

r   +   −

2

r

r

(13)

)

(

1

2

−

−

s

u r u r

u u

   +   +

2





According to (Gourgiotis & Georgiadis, 2009), it is also useful to introduce the quantities s r and s θ representing the contribution of the classical elasticity to the equilibrium of the flexoelectric body as follows ( ) ( ) 1 2 1 2 1 2 , 1 2 . r r r s s s s s s      = + − = + − (14)

Equations (11)-(14) allow one to rewrite the equilibrium equation (8) in the form ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 2 2 0, 2 1 2 2 0. r r r r r r r s l s r s r s l s r s r s s l s r s r s l s r s r s              − − − − − − − − −  − −  − −  − −  = −  − +  − −  − +  =

(15)

It should be noted that neglecting the flexoelectricity parameters f 1 and f 2 in (9) simplifies the equilibrium equations (15) in the case of the strain gradient elasticity and neglecting l simplifies the equilibrium equations (15) in the case of the classical elasticity. 3. Some comments on matched asymptotic expansion method The matched asymptotic expansion method is an important mathematical tool in the asymptotic analysis mainly applied in the fluid mechanics. The matched asymptotic expansion method with the boundary layer approach is also increasingly adopted to get the change in potential energy of the notch or crack array if the very small and finite crack increment is considered at its tips (Leguillon et al, 2001; Martin et al, 2001; Leguillon et al, 2000; Kotoul et al, 2010; Vu-Quoc & Tran, 2006). The principle consists in construction of the so-called inner and outer asymptotic solutions which are then matched along a certain boundary. First of all it should be noted that an analytical solution governed by the direct flexoelectricity laws and that describes this effect in the nearby domain of the defect as well as its weakening in the remote parts of the material is not available. Instead of that, it is possible to find the so-called outer solution, which suppresses the influence of the strain gradients, and the so-called inner solution, which includes their dominancy. The outer solution for the crack tip ( , out out  u ) can be derived if the polar coordinate system ( ) , r    is introduced at the point x 0 inside or near the process zone whose dimension is proportional to the material length scale parameter l as shown in Fig. 1. The outer solution ( , out out  u ) is valid outside the zone 0 r x   , where the strain gradients ∇ are negligible small with respect to the magnitudes of the strains ∇ and the maximum error in the governing equilibrium equations (2) is of the order O ( l 4 )for 0 x l  . Omitting the double stress τ due to the negligible small values of the strain gradients ∇ outside the crack process zone, the traction boundary conditions (5) 3 prevailing at the tip of the crack can be supposed as