PSI - Issue 28

Per Stahle et al. / Procedia Structural Integrity 28 (2020) 2065–2071 P. Ståhle et al. / Structural Integrity Procedia 00 (2020) 000–000

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Fig. 3. Near tip mesh and the presence of plastic strain.

Fig. 4. Length of the cohesive zone.

The potential energy density, Ψ in the x - y plane caused by small anti-plane ( z -direction) displacements, w , is given by

Eh 2 12(1 − ν 2 )

2 − 2(1 − ν ) w

, xy + σ x w 2

, xx w , yy − w 2

, x + 2 τ xy w , x w , y + σ y w 2 , y .

( w , xx + w , yy )

(7)

2 Ψ =

Here indices preceded by a comma denote partial derivatives with respect to the specified coordinates, cf. e.g. Shaw and Huang (1990). At instability the variation of Ψ becomes independent of the magnitude of a distribution of w = w ( x , y ), i.e.,

δ Ψ δ w

= 0 .

(8)

Applying the Euler-Lagrange formalism for variations on Eqs. (7) and (8), gives the following eigenvalue equation, Eh 3 6(1 − ν 2 ) ( w , xxxx + 2 w , xxyy + w , yyyy + 2(1 − ν ) w , xxyy ) = σ x w , xx + 2 τ xy w , xy + σ y w , yy . (9) The stress field, σ x , σ y and τ xy is supposedly in equilibrium. It is given a priori and eigenvalues are calculated using Eq. (9) recognising the given boundary conditions, which here are traction free crack surfaces, and a remote uniaxial tensile stress perpendicular to the crack, i.e. σ yy = σ xy = 0 at | x | < a and y = 0 σ yy = σ ∞ and σ xy = 0 as x 2 + y 2 → ∞ (10) 3. Numerical considerations The finite element method is employed to do both the elastic-plastic pre-buckling calculation and for finding the eigenvalues. The mesh is generated using Delaunay triangulation with 20628 nodes and 6819 elements in a mix of six and eight node isoparametric shell elements. Due to the vertical and horizontal symmetries of the problem only a quarter of the full quadratic geometry, i.e., 0 ≤ x ≤ L , 0 ≤ y ≤ L , where L = 20 a , is meshed and computed with symmetry boundary conditions along vertical and horizontal cuts that pass through the centre of the crack. The crack length is believed to be su ffi ciently small to eliminate the influences of the finite external dimensions of the mesh. The smallest elements covers a thin layer extending straight ahead of the crack tip to a distance similar to the crack length from the crack tip. The linear extent of these elements is close to 2% of the crack length and 1 of the mesh size (see Fig. 3) The remote boundary conditions are prescribed displacement υ and vanishing in-plane shear tractions along the upper horizontal edge and absent in-plane tractions along the edges at y = 0 and L . The anti-plane displacements w are