PSI - Issue 80

ScienceDirect Available online at www.sciencedirect.com ScienceDirect Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2022) 000 – 000 Available online at www.sciencedirect.com Procedia Structural Integrity 80 (2026) 451–461 Structural Integrity Procedia 00 (2022) 000 – 000

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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.2026.02.044 2452-3216 © 2023 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 Professor Ferri Aliabadi 2452-3216 © 2023 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 Professor Ferri Aliabadi © 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 Abstract The collocation mixed FEM is developed for 3-d crack analyses in solids with the direct flexoelectricity. In the direct flexoelectricity the polarization vector is proportional to the strain gradients. In contrast to piezoelectric polarization, this phenomenon is occurring in all dielectric materials. Since the large strain gradients arise at the crack tip vicinity, the large flexoelectric effect is localized around the crack tip. A standard domain discretization method with the C 0 continuous approximations is insufficient for solution of boundary value problems, since strain gradients are occurring in constitutive equations. To deal with higher order derivatives in governing equations of the gradient theory, it is developed the collocation mixed FEM (CMFEM). In CMFEM the C 0 continuous approximations are applied independently to displacements and strains. The constraint between the independent mechanical strains and displacement is satisfied by a collocation method at considered collocation points on each finite element. The J-integral expression for 3D crack problems in strain-gradient piezoelectricity is derived in this paper. It is shown that 3D J-integral expression is supplied with some additional surface integrals in comparison with 2D case. Keywords: Direct flexoelectricity, gradient theory elasticity, mixed finite element method, crack problem, J integral 1. Introduction Smart materials with multi-field coupling properties play an important role in developing innovative, versatile, and efficient devices. Specifically, magnetically or electrically controllable materials with outstanding mechanical properties are well-suited for applications at the micro and nanoscale. Flexoelectric materials are a new type of smart Fracture, Damage and Structural Health Monitoring Collocation Mixed FEM for 3-d Crack Analyses in Flexoelectric Solids Xinpeng Tian a , Yuchen Hu b,c , Jan Sladek d *, Vladimir Sladek d , Qian Deng b,c a Department of Engineering Mechanics, Xi’an University of Technology, Xi’an 710048, China b Department of Engineering Mechanics, School of Aerospace Engineering, Huazhong, University of Science and Technology, Wuhan 430074, China c Hubei Key Laboratory of Engineering Structural Analysis and Safety Assesment, Luovu Road 1037, Wuhan 430074, China d Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia Abstract The collocation mixed FEM is developed for 3-d crack analyses in solids with the direct flexoelectricity. In the direct flexoelectricity the polarization vector is proportional to the strain gradients. In contrast to piezoelectric polarization, this phenomenon is occurring in all dielectric materials. Since the large strain gradients arise at the crack tip vicinity, the large flexoelectric effect is localized around the crack tip. A standard domain discretization method with the C 0 continuous approximations is insufficient for solution of boundary value problems, since strain gradients are occurring in constitutive equations. To deal with higher order derivatives in governing equations of the gradient theory, it is developed the collocation mixed FEM (CMFEM). In CMFEM the C 0 continuous approximations are applied independently to displacements and strains. The constraint between the independent mechanical strains and displacement is satisfied by a collocation method at considered collocation points on each finite element. The J-integral expression for 3D crack problems in strain-gradient piezoelectricity is derived in this paper. It is shown that 3D J-integral expression is supplied with some additional surface integrals in comparison with 2D case. Keywords: Direct flexoelectricity, gradient theory elasticity, mixed finite element method, crack problem, J integral 1. Introduction Smart materials with multi-field coupling properties play an important role in developing innovative, versatile, and efficient devices. Specifically, magnetically or electrically controllable materials with outstanding mechanical properties are well-suited for applications at the micro and nanoscale. Flexoelectric materials are a new type of smart Fracture, Damage and Structural Health Monitoring Collocation Mixed FEM for 3-d Crack Analyses in Flexoelectric Solids Xinpeng Tian a , Yuchen Hu b,c , Jan Sladek d *, Vladimir Sladek d , Qian Deng b,c a Department of Engineering Mechanics, Xi’an University of Technology, Xi’an 710048, China b Department of Engineering Mechanics, School of Aerospace Engineering, Huazhong, University of Science and Technology, Wuhan 430074, China c Hubei Key Laboratory of Engineering Structural Analysis and Safety Assesment, Luovu Road 1037, Wuhan 430074, China d Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia * Corresponding author. Tel.: +421904885687; fax: +421254773548. E-mail address: jan.sladek@savba.sk * Corresponding author. Tel.: +421904885687; fax: +421254773548. E-mail address: jan.sladek@savba.sk