PSI - Issue 2_A

J. Felger et al. / Procedia Structural Integrity 2 (2016) 2504–2511 J. Felger, W. Becker / Structural Integrity Procedia 00 (2016) 000–000

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which describes the dominating part of the local solution in the vicinity of the notch tip. The obtained reduced PDE system in Eq. (9) can be solved by means of a complex potential approach based on the Lekhnitskii-formalism. The PDE-system (9) can be rewritten symbolically in the form D 1 D 2 w = 0

  with D k =

∂ ∂ y −

∂ ∂ x

D 3 D 4 D 5 D 6 ψ x = 0 D 3 D 4 D 5 D 6 ψ y = 0

(10)

µ k

.

The complex parameters µ k are the roots of the algebraic characteristic equations corresponding to the PDE-system (9) which are purely imaginary in the considered case of orthotropic material behaviour and appear in complex conjugated pairs. They can be given explicitly as

   i

A 55 A 44

for k = 1

i

±   

(11)

µ k =

  2 −

D 11 D 22 − 2 D 12 D 66 − D 2 12 2 D 22 D 66

D 11 D 22 − 2 D 12 D 66 − D 12 2 2 D 22 D 66

D 11 D 22

for k = 2 , 3 ,

where µ 4 = µ 1 , µ 5 = µ 2 and µ 6 = µ 3 . Integrating the decoupled system in Eq. (10) finally yields the general solution

ψ x = 2Re    3 k = 2

Φ k ( z k )]  

   3 k = 2

γ k Φ k ( z k )]    .

w = 2Re[ Φ 1 ( z 1 )] ,

 , ψ y = 2Re

(12)

Here, Φ 1 ( z 1 ) , Φ 2 ( z 2 ) and Φ 3 ( z 3 ) are arbitrary holomorphic potentials and

D 11 + D 66 µ 2 k ( D 12 + D 66 ) µ k ,

z k = x + µ k y

(13)

and γ k =

where z k represent generalised complex coordinates. In order to express the stress resultants through the complex potentials, the general solution representation in Eq. (12) is substituted into the constitutive relations. Considering only leading order terms finally yields

Re    3 k = 2 Re    3 k = 2 Re    3 k = 2

( D 11 + D 12 γ k µ k ) Φ k ( z k )    ( D 12 + D 22 γ k µ k ) Φ k ( z k )    D 66 ( γ k + µ k ) Φ k ( z k )        .

   

  = 2

,   M x M y M xy

Q y

Re[ A 55 Φ 1 ( z 1 )] Re[ A 44 µ 1 Φ 1 ( z 1 )]

Q x

= 2

(14)

The stress resultants corresponding to a cylindrical coordinate system { r , φ, z } can be obtained from Eq. (14) by con ventional tensor transformation.

3. Singularity analysis

The underlying PDE-system is fulfilled for an arbitrary choice of the holomorphic potentials. Once the three com plex potentials are known, all field variables can be deduced from Eq. (12) and Eq. (14). In order to solve a boundary value problem, the three complex potentials have to be chosen such that the prescribed boundary conditions are ful filled. As typical for asymptotic solutions only Dirichlet or Neumann boundary condition along the edges of the notch are taken into account. Within an asymptotic near-field analysis, a suitable ansatz for the complex potentials can be written as Φ i ( z i ) = A i z λ i + B i z λ i , with λ ∈ C , i ∈ { 1 , 2 , 3 } , (15)