PSI - Issue 80

Luke Wyatt et al. / Procedia Structural Integrity 80 (2026) 31–42

39

L. Wyatt et. al. / Structural Integrity Procedia 00 (2023) 000–000

9

Table 3. Comparison of the buckling loads of the thick clamped / simply supported square plate Buckling Mode VEMQuad VEMVor FEMQuad

Analytical [17]

a 2 λ a 2 λ a 2 λ

2 D ) 2 D ) 2 D )

1 / ( π 2 / ( π 4 / ( π

6.186 7.320

6.202 7.294

6.216 7.329

6.370

- -

11.726

11.733

11.796

Table 4. Comparison of the buckling loads of the thick simply supported square plate Buckling Mode VEMQuad VEMVor

FEMQuad

Analytical [17]

a 2 λ a 2 λ a 2 λ

2 D ) 2 D ) 2 D )

1 / ( π 2 / ( π 4 / ( π

3.743 5.328

3.740 5.338

3.739 5.333

3.786

- -

11.617

11.619

11.631

5.3. Bending Stress Intensity Factor Results

The crack problems were performed on a rectangular domain, of dimensions (2 b × 2 c ), with a moment M 0 applied in the x direction on the outer edges. For a crack perpendicular to the applied moment, this produces a purely Mode 1 crack. Two di ff erent crack configurations were tested, a central crack, and two identical edge cracks, which are shown in Figure 6.

a

M 0

M 0

2 a

2 c

2 c

a

2 b

2 b

(a) Crack problem 1- rectangular plate subjected to a constant moment with a central crack

(b) Crack problem 2- rectangular plate subjected to a constant moment with two identical edge cracks

Fig. 6. Diagrams of the two problems that K 1 B was computed

The dimensions and material properties chosen for the numerical tests were b / c = 2, Ec 2 / M 0 = 210 , 000, ν = 0 . 3 and k = 5 / 6. The method was tested for moderately thick and thick plates, t / c = 0 . 1 and t / c = 0 . 5, to compare with available analytical results [18]. Both the Quad and Vor meshes tested were refined uniformly, without special consideration to the crack, apart from ensuring that there is a vertex of the mesh at the crack tip to ensure the geometry is exact.. While this does not lead to particularly accurate estimates of the stresses at the crack tip, as the J-integral is an energy measure accurate results can be obtained with coarse meshes for linear analyses. To simplify analysis, only the top left quarter of the domain was simulated, and the appropriate symmetry constraints were imposed. Figure 7 shows the convergence of K 1 B normalised by the analytical solution, ¯ K 1 B [18] for crack problem 1 with decreasing mesh size for both thicknesses of plate. In this test a = c / 2. The convergence of the Quad mesh is consid erably smoother than that of the Vor mesh, but for finer meshes the accuracy is comparable for both plate thicknesses. Figures 8 and 9 shows the change in K 1 B with varying crack length a / c , with the VEM results compared to those previously published [18]. For these tests the finest meshes in Figure 7 were used, and the radius of the path for the J-integral was chosen to be r = ( c − a ) / 2, with 50 integration points. The results of both meshes show good agreement with the reference values for both plate thicknesses. The results diverge from the reference furthest where either a or c − a is very small, where the size of the geometric features are closer in size to the mesh, so further or local mesh refinement would provide improved results.