PSI - Issue 80

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

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

Mesh

n

Integration point

y

n

x

ΩΓ C

CrackTip

Γ

(a) Local coordinate system and path integral contour for the J-integral

(b) Example of the implementation of the J-integral on a square mesh, show ing the integration points, and the elements which require postprocessing shown in grey

Fig. 1. The continuous and discretised crack tip, showing the J-integral contour

5. Results

This section presents numerical evaluation of the VEM formulation for free vibrations and buckling problems, and the suitability of VEM for the calculation of K 1 B using the J-integral. The results are compared to those from a selective reduced integration FEM [14], and analytical solutions or previously published results where possible. The meshes used to test the VEM and FEM are structured quadrilateral (Quad) and Voronoi (Vor) meshes. The Voronoi meshes were generated using Polymesher [15].

5.1. Free Vibration Results

The free vibration tests are performed on the square domain, Ω ∈ [0 , a ] 2 , on thin ( t / a = 0 . 01) and thick ( t / a = 0 . 1) plates. The dimensions and material constants used in these tests were a = 1mm, ρ = 1 , 000kgmm -3 , E = 10 , 920MPa and ν = 0 . 3. The shear correction factor was taken as k = 0 . 8601 for the clamped test and k = 0 . 833 for the simply supported case to be consistent with the published analytical results. Figures 2 and 3 show the convergence of the first natural frequency, normalised by the analytical solution [16], ¯ ω 1 , with average element diameter, h mean , for the clamped and simply supported tests. The clamped test shows that for the coarsest mesh the VEM prediction on both meshes is poorer than the FEM result, however with decreased mesh size the results of the VEM are within 0.4% of the FEM. The coarse mesh accuracy of the VEM can be improved by modification of the weighting of the stability part of the sti ff ness matrix. For the simply supported plate the VEM and FEM on the Quad mesh converge at almost identical rates, with the VEM on the Vor mesh being initially more accurate for the coarser meshes. Tables 1 and 2 show the converged VEM and FEM results of the 1 st , 2 nd and4 th natural frequencies of the clamped and simply supported plates and the analytical solutions. The tables show that the VEM results are almost identical to the FEM results and converged well towards the analytical solution. The one exception is ω 4 of the clamped plate, which both VEM meshes underestimates. By increasing the scaling parameter of the stability part of the sti ff ness matrix, the VEM result for ω 4 can be greatly improved, without impacting the accuracy of the other frequencies.

Table 1. Comparison of the natural frequencies of the thick clamped square plate Vibrational Mode VEMQuad VEMVor

FEMQuad

Analytical [16]

ω 1 a ρ/ G ω 2 a ρ/ G ω 4 a ρ/ G

1.590 3.043 3.984

1.590 3.033 4.106

1.590 3.042 4.262

1.594 3.039 4.265