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

new formulation inspired by reduced integration finite elements was proposed, which projects the shear strain onto a lower order space to avoid locking [7]. This has the advantage of naturally having only 3 n v DoFs for an element with n v vertices, which has not achieved without static condensation of additional DoFs in previous VEM formulations. This element was found to perform well in clamped and simply supported bending compared to reduced integration finite elements, as well as against previous VEM formulations.This paper explores further numerical tests of the reduced order shear projection VEM, and for the first time examines the suitability of VEM for the free vibration and buckling problems and calculation of the Mode 1 bending stress intensity factor, K 1 B of Mindlin plates. Section 2 introduces Mindlin plate theory, which is used in Section 3 to briefly explain the formulation of the element used in the numerical tests. Section 4 details the construction of the VEM mass and geometric sti ff ness matrices, and the J-integral method for calculation of the bending stress intensity factor. In Section 5 a number of numerical tests are performed for each problem and the results are compared to FEM and analytical methods, and the conclusions and plans for future work are presented in Section 6.

2. Mindlin Plate Theory

This section presents the Mindlin plate theory used in the development of the VEM formulation discussed in Section 3, and is used in the further developments of the method in in Section 4. The undeformed position of the plate is given by Ω × ( − t / 2 , t / 2) where Ω is a domain on R 2 , which represents the midsurface of the plate, and t is the thickness of the plate. A Cartesian coordinate system is defined such that x and y lie on Ω ,with z orthogonal. Mindlin plate theory gives the displacements, ( u , v , w ), from the undeformed position of the plate as

u ( x , y , z ) = z θ x ( x , y ) , v ( x , y , z ) = z θ y ( x , y ) , w ( x , y , z ) = w ( x , y ) ,

(1) (2) (3)

where θ x and θ y are the rotations of the transverse normal around the y and x axes respectively [8]. The collection of θ x , θ y and w is referred to as the generalised displacements. The rotations are written in vector form as θ =  θ x ,θ y  ⊤ . The generalised strains are the curvature, χ =  χ xx ,χ yy ,χ xy  ⊤ and the shear strain, γ =  γ xz ,γ yz  ⊤ . The strains can be written χ =    ∂/∂ x 0 0 ∂/∂ y ∂/∂ y ∂/∂ x     θ x θ y  = L χ θ , (4) γ =  ∂/∂ x ∂/∂ y  w +  θ x θ y  = L γ w + θ , (5) where L χ and L γ are the compatibility operators for bending and shear respectively. Assuming the plate to have a linear isotropic response, with Young’s Modulus, E , and Poisson’s ratio, ν , the moment, M R , and shear force, Q R resultants are given by

   = D χ ,

  

   χ xx χ yy χ xy

= D   

1 ν 0 ν 1 0 0 0 (1 − ν ) 2

M R

(6)

= C 

1 0 0 1 

γ xz γ yz 

Q R

= C γ ,

(7)