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 3 where D = Et 3 / 12(1 − ν 2 ) and C = kEt / 2(1 + ν ) are the sti ff nesses in bending and shear respectively and k is the shear correction factor.

3. Reduced Order Shear Projection VEM

This section briefly explains the formulation of the reduced order shear projection virtual element method for Mindlin plates, which has been previously published [7]. The VEM uses orthogonal projections of the required vari ables onto a polynomial basis, such that the projected variables are exact up to the order of the chosen polynomial basis. The use of the projection means that the approximation functions do not have to be exactly defined, and only their value at the degrees of freedom of the problem are required. The consistency sti ff ness matrix can then be com puted using the projection, which ensures that the solution is exact up to the chosen polynomial order, and stability part is added, which approximates missing higher order energies to ensure the correct rank of the system [2]. This allows for very general geometry of elements, e.g. concave elements and hanging nodes, and also means that the element generation procedure is the same for an element with any number of sides. The VEM formulation that will be discussed in this paper uses a lower order projection of the rotations in the shear part of the problem to avoid shear locking. This technique is inspired by selective reduced integration in FEM, which is in e ff ect a projection of the shear onto a lower order space than is suggested by the compatibility equations. This VEM formulation was used to make a first order element which, by choosing the correct projections gives an element with 3 degrees of freedom at each vertex. The trial functions used in the projection are

θχ = θγ = 1 0 − η η ξ 0 0 0 0 0 1 ξ ξ 0 η 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 N P w = 0 0 0 0 0 0 1 ξ η .

N P

(8)

N P

(9)

(10)

where N P θχ is used in the projection of the rotations in the bending part of the integration and N P θγ and N P w are used in the projection of the rotations and displacement in the shear part of the integration. This ensures that the element can correctly represent thin plates, as the Kirchho ff constraint can be satisfied without higher order terms causing spurious strain energies. A more complete discussion of the formulation of the element is available [7]. This element was proven to be e ff ective for both clamped and simply supported tests both with distributed and point loads, performing favourably when compared to the equivalent 4-noded FEM formulation, and VEM formulations with more DoFs per node. In particular, the VEM was found to be superior to the reduced integration FEM for thin plates with unstructured quadrilateral meshes.

4. Extensions to the VEM Formulation

This section presents new extensions to the VEM formulation in Section 3 to test the performance of the element in further detail. Section 4.1 details the calculation of the mass matrix for the element, and the eigenproblem for the free vibration of Mindlin plates. Section 4.2 presents the calculation of the geometric sti ff ness matrix and the stability problem for in-plane buckling. Section 4.3 introduces the J-integral for the Mindlin plate problem, and the calculation of K 1 B .

4.1. Free Vibrations

The equations of motion of Mindlin plates can be expressed as M ¨ V + KV = F ,

(11)