PSI - Issue 80

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Procedia Structural Integrity 80 (2026) 31–42 Structural Integrity Procedia 00 (2023) 000–000 Structural Integrity Procedia 00 (2023) 000–000

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© 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Ferri Aliabadi © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of Professor Ferri Aliabadi. Keywords: Virtual Element Method; Mindlin Plates; Free Vibration; Plate Cracks Abstract This paper introduces extensions and further verification of the newly developed reduced order shear projection virtual element for Mindlin plates. The construction of the mass matrix and the buckling geometric sti ff ness matrix are developed and the J-integral for Mindlin plates is implemented. For the first time the Virtual Element Method (VEM) is used to solve the free vibration and buckling problems for Mindlin plates, and to calculate the bending stress intensity factor, K 1 B for cracked plates. Numerical tests are performed for all of the problems, and are compared to Finite Element Method (FEM) results, and previously published analytical results. The VEM results agree closely with the FEM and analytical results, with the first natural frequency found with the VEM converging to within 0.4% of the FEM result, and the VEM result of the first buckling load is over 1% more accurate than that from the FEM for thin plates. The value of K 1 B calculated by the VEM agrees closely with published results, even on coarse meshes, and those within special consideration of the crack tip. © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of Professor Ferri Aliabadi. Keywords: Virtual Element Method; Mindlin Plates; Free Vibration; Plate Cracks Fracture, Damage and Structural Health Monitoring Development and Verification of Shear Projection Virtual Element LukeWyatt a, ∗ , Zahra Sharif Khodaei a , M.H. Ferri Aliabadi a a Structural Integrity & Health Monitoring Group, Department of Aeronautics, Imperial College London, SW7 2AZ, London, UK Abstract This paper introduces extensions and further verification of the newly developed reduced order shear projection virtual element for Mindlin plates. The construction of the mass matrix and the buckling geometric sti ff ness matrix are developed and the J-integral for Mindlin plates is implemented. For the first time the Virtual Element Method (VEM) is used to solve the free vibration and buckling problems for Mindlin plates, and to calculate the bending stress intensity factor, K 1 B for cracked plates. Numerical tests are performed for all of the problems, and are compared to Finite Element Method (FEM) results, and previously published analytical results. The VEM results agree closely with the FEM and analytical results, with the first natural frequency found with the VEM converging to within 0.4% of the FEM result, and the VEM result of the first buckling load is over 1% more accurate than that from the FEM for thin plates. The value of K 1 B calculated by the VEM agrees closely with published results, even on coarse meshes, and those within special consideration of the crack tip. Fracture, Damage and Structural Health Monitoring Development and Verification of Shear Projection Virtual Element LukeWyatt a, ∗ , Zahra Sharif Khodaei a , M.H. Ferri Aliabadi a a Structural Integrity & Health Monitoring Group, Department of Aeronautics, Imperial College London, SW7 2AZ, London, UK The Mindlin plate theory is an e ff ective tool for analysis of both thick and thin plates, and can be used e ffi ciently for numerical methods such as the Finite Element Method (FEM). However, basic low order isoparametric finite element can su ff er from shear locking, where the Kirchho ff constraint cannot be satisfied in thin plates, causing large spurious strain energies which reduce the rate of convergence. The simplest method of alleviating locking in finite elements is selective reduced integration, where the shear part of the sti ff ness matrix is under integrated, which improves the element performance considerably [1]. The Virtual Element Method (VEM) was developed to solve problems on very arbitrary meshes, without the mesh requirements of FEM [2]. By using appropriate projections over the element, the approximation functions do not have to be explicitly defined, and the sti ff ness matrix can be found using only the degrees of freedom (DoFs) of the element. A number of VEM formulations for Mindlin plates have been developed, using mixed interpolation [3] [4], higher order approximation of the displacements [5], and a selective stabilisation scheme [6] to alleviate locking. Recently a The Mindlin plate theory is an e ff ective tool for analysis of both thick and thin plates, and can be used e ffi ciently for numerical methods such as the Finite Element Method (FEM). However, basic low order isoparametric finite element can su ff er from shear locking, where the Kirchho ff constraint cannot be satisfied in thin plates, causing large spurious strain energies which reduce the rate of convergence. The simplest method of alleviating locking in finite elements is selective reduced integration, where the shear part of the sti ff ness matrix is under integrated, which improves the element performance considerably [1]. The Virtual Element Method (VEM) was developed to solve problems on very arbitrary meshes, without the mesh requirements of FEM [2]. By using appropriate projections over the element, the approximation functions do not have to be explicitly defined, and the sti ff ness matrix can be found using only the degrees of freedom (DoFs) of the element. A number of VEM formulations for Mindlin plates have been developed, using mixed interpolation [3] [4], higher order approximation of the displacements [5], and a selective stabilisation scheme [6] to alleviate locking. Recently a 1. Introduction 1. Introduction

∗ Corresponding author. Tel.: + 447861018101 E-mail address: luke.wyatt19@imperial.ac.uk ∗ Corresponding author. Tel.: + 447861018101 E-mail address: luke.wyatt19@imperial.ac.uk

2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Ferri Aliabadi 10.1016/j.prostr.2026.02.004 2210-7843 © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of Professor Ferri Aliabadi. 2210-7843 © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of Professor Ferri Aliabadi.