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

2509

6

3 Π 8

5 Π 8

3 Π 4

7 Π 8

3 Π 8

5 Π 8

3 Π 4

7 Π 8

Π 8

Π 4

Π 2

Π 8

Π 4

Π 2

0

0

Π

Π

1.0

1.0

M A

M A

0.8

0.8

Q A

Q A

M S

M S

0.6

0.6

Re Λ

Re Λ

x

x

0.4

0.4

γ

γ

γ

γ

y

y

0.2

0.2

0.0

0.0

3 Π 8

3 Π 4

7 Π 8

3 Π 8

3 Π 4

7 Π 8

Π 8

Π 4

Π 2

Π 8

Π 4

Π 2

5 Π 8

5 Π 8

0

0

Π

Π

Γ

Γ

(a) 0-degree fibre direction.

(b) 90-degree fibre direction.

Fig. 3: Singularity exponent λ in dependence of the angle γ between the notch faces and the x -axis for di ff erent fibre directions and stress-free notch faces.

2 Π

4 Π

5 Π

7 Π

8 Π

Π

2 Π

Π

2 Π

4 Π

5 Π

7 Π

8 Π

Π

2 Π

Π

0

0

9

9

9

9

9

9

3

3

9

9

9

9

9

9

3

3

10

1.0

y

y

R

8

0.8

R

Asymptotic FEM

φ

φ

2 π 9

2 π 9

M i M x Φ� 0

M i M x Φ� 0

x

x

6

0.6

M y

M y

4

0.4

Asymptotic FEM

0.2

2

M x

M x

0.0

0

Π

Π

Π

Π

2 Π

4 Π

2 Π

7 Π

8 Π

5 Π

2 Π

4 Π

2 Π

7 Π

8 Π

5 Π

0

0

9

3

9

3

9

9

3

9

9

9

9

9

3

9

9

9

Φ

Φ

(a) 0-degree fibre direction.

(b) 90-degree fibre direction.

Fig. 4: Comparison between finite element results and the asymptotic solution for the angular distribution of the bending moments M x and M y .

Using Eq. (12) for the deflection w and the inclination angles ψ x and ψ y the relations in Eq. (22) imply A 1 = B 1 , A 2 = B 2 , A 3 = B 3 , symmetric , − A 1 = B 1 , − A 2 = B 2 , − A 3 = B 3 , antisymmetric . Consequently, the deformation modes can also be characterised by the structure of the eigenvector a and Eq. (23) allows for linking the singularity exponent with its associated deformation mode. In the following, the considered plates are single unidirectional fibre-reinforced composite layers with the material properties: E 11 = 135 GPa , E 22 = 9 . 4 GPa , G 12 = 4 . 85 GPa , G 23 = 3 . 24 GPa and ν 12 = 0 . 35. Let us first consider the case of a symmetric notch with a plane of symmetry along the x -axis and stress free notch edges. The boundary conditions can be expressed as ( M φ + i M r φ ) Γ = 0 , Q φ Γ = 0 , (24) where Γ denotes the notch faces. In Fig. 3a the influence of the angle γ between the x -axis and the edges of the notch on the singularity exponent λ for a composite layer with 0-degree fibre direction is depicted, in which the fibres are oriented along the x -axis. Calculating the eigenvector related to each singularity exponent and using Eq. (21) together with Eq. (23), it can be shown that the dashed line is associated to singular transverse shear forces induced by an antisymmetric loading ( Q A ). The solid lines represent singularities of the bending moments due to a (23)