Solving Problems with the Functionally Graded Anisotropic Piezoelectric Plates Via a BFEM
Main Article Content
Abstract
A boundary-based finite element method (BFEM) is developed for solving stress and contact problems in two-dimensional multilayered and functionally graded piezoelectric plates. In the proposed formulation, the plate is discretized into multiple material sublayers, each modeled as an individual finite element constructed solely from boundary nodes. By establishing direct relations between boundary tractions and electric displacements and nodal forces, the governing electromechanical coupling equations are transformed into a boundary-based finite element framework, thereby avoiding volumetric discretization. When contact problems are considered, the contact constraints are incorporated into the BFEM formulation through appropriate contact conditions, allowing the unknown contact regions and contact tractions to be determined as part of the solution. The proposed approach effectively accounts for multilayered configurations, functionally graded material properties, anisotropy, and electromechanical coupling effects. Numerical examples are presented to validate the method's accuracy and convergence. In addition, parametric studies are conducted to investigate the influences of material gradation and anisotropic properties on the electromechanical responses of the plates.
Keywords:
BFEM, functionally graded, multilayered, anisotropic, piezoelectricity, contact analysis.
References
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[6] X. F. Li, X. L. Peng, K. Y. Lee, Radially Polarized Functionally Graded Piezoelectric Hollow Cylinders as Sensors and Actuators, European Journal of Mechanics - A/Solids, Vol. 29, No. 4, 2010, pp. 704-713, https://doi.org/10.1016/j.euromechsol.2010.02.003.
[7] H. L. Dai, L. Hong, Y. M. Fu, X. Xiao, Analytical Solution for Electromagnetothermoelastic Behaviors of a Functionally Graded Piezoelectric Hollow Cylinder, Applied Mathematical Modelling, Vol. 34, No. 2, 2010,
pp. 343-357, https://doi.org/10.1016/j.apm.2009.04.008.
[8] A. L. Shuvalov, E. L. Clezio, G. Feuillard, The State-vector Formalism and the Peano-series Solution for Modelling Guided Waves in Functionally Graded Anisotropic Piezoelectric Plates, International Journal of Engineering Science, Vol. 46, No. 9, 2008, pp. 929-947, https://doi.org/10.1016/j.ijengsci.2008.03.007.
[9] M. K. Pal, A. K. Singh, Analysis of Reflection and Transmission Phenomenon at Distinct Bonding Interfaces in a Rotating Pre-stressed Functionally Graded Piezoelectric-Orthotropic Structure, Analysis of Reflection and Transmission Phenomenon at Distinct Bonding Interfaces in a Rotating Pre-stressed Functionally Graded Piezoelectric-Orthotropic Structure, Vol. 409, 2021, pp. 126398, https://doi.org/10.1016/j.amc.2021.126398.
[10] F. Li, J. Xie, O. Shi, Multilayer Homogeneous Model for Functionally Graded Piezoelectric Structure with Arbitrary Property: From Mechanical Analysis to Optimization, Composite Structures, Vol. 384, 2026, pp. 120173, https://doi.org/10.1016/j.compstruct.2026.120173.
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pp. 114130, https://doi.org/10.1016/j.tws.2025.114130.
[12] T. K. H. Nguyen, V. T. Nguyen, V. L. Nguyen, X. T. Nguyen, Pin-Loaded Hole Contact in Anisotropic Multi-Layered/Functionally Graded Composite Plate, Composite Structures, Vol. 382, 2026, pp. 120103, https://doi.org/10.1016/j.compstruct.2026.120103.
[13] S. Jie, L. L. Ke, Y. S. Wang, Axisymmetric Frictionless Contact of a Functionally Graded Piezoelectric Layered Half-Space Under a Conducting Punch, International Journal of Solids and Structures, Vol. 90, 2016, pp. 45-59, https://doi.org/10.1016/j.ijsolstr.2016.04.011.
[14] X. T. Nguyen, V. T. Nguyen, N. D. Duc. Indentation on Multilayered and Functionally Graded Anisotropic Elastic Plates, Acta Mechanica, 2026, https://doi.org/10.1007/s00707-026-04658-w.
[15] B. Yang, E. Pan, V. K. Tewary, Static Responses Of A Multilayered Anisotropic Piezoelectric Structure To Point Force And Point Charge, Smart Materials and Structures, Vol. 13, 2003, pp. 175, https://doi.org/10.1088/0964-1726/13/1/020.
[16] Y. T. Zhou, K. Y. Lee, Frictional Contact of Anisotropic Piezoelectric Materials Indented by Flat and Semi-Parabolic Stamps, Archive of Applied Mechanics, Vol. 83, 2013, pp. 73-95, https://doi.org/10.1007/s00419-012-0633-5.
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[18] V. T. Nguyen, V. S. Pham, N. D. Duc, T. Q. Bui, Semi-analytical Method for Frictional Sliding Contact of a Piezoelectric Layer-Substrate, Mechanics of Materials, Vol. 213, 2026, pp. 105572, https://doi.org/10.1016/j.mechmat.2025.105572.
[19] V. T. Nguyen, T. K. H. Nguyen, C. M. Nguyen, Semi-Analytical Method for Pin-Loaded Joint Contact in Anisotropic Piezoelectric Plates, International Journal of Mechanical Sciences, Vol. 288, 2025, pp. 110001, https://doi.org/10.1016/j.ijmecsci.2025.110001.
[20] J. Sladek, V. Sladek, H. H. -H. Lu, D. L. Young, The FEM Analysis of FGM Piezoelectric Semiconductor Problems, Composite Structures, Vol. 163, 2017, pp. 13-20, https://doi.org/10.1016/j.compstruct.2016.12.019.
[21] X. Q. He, K. M. Liew, T. Y. Ng, S. Sivashanker, A FEM Model for the Active Control of Curved FGM Shells Using Piezoelectric Sensor/Actuator Layers, International Journal for Numerical Methods in Engineering, Vol. 54, 2002, pp. 853-870, https://doi.org/10.1002/nme.451.
[22] M. N. Balci, S. Dag, B. Yildirim, Subsurface Stresses in Graded Coatings Subjected to Frictional Contact with Heat Generation, Journal of Thermal Stresses, Vol. 40, 2016, pp. 517-534, https://doi.org/10.1080/01495739.2016.1261261.
[23] M. A. Güler, A. Kucuksucu, K. B. Yilmaz, B. Yildirim, On the Analytical and Finite Element Solution of Plane Contact Problem of a Rigid Cylindrical Punch Sliding Over a Functionally Graded Orthotropic Medium, International Journal of Mechanical Sciences, Vol. 120, pp. 2017, pp. 12-29, https://doi.org/10.1016/j.ijmecsci.2016.11.004.
[24] X. Tang, W. Yang, J. Liu, K. Qi. A High-Efficiency Versatile Adhesive Contact Model for Layered Media. Tribology International, Vol. 212, 2025, pp. 110926, https://doi.org/10.1016/j.triboint.2025.110926.
[25] Y. Liu, H. Fan, Analysis of Thin Piezoelectric Solids by the Boundary Element Method, Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 21-22, 2002, pp. 2297-2315, https://doi.org/10.1016/S0045-7825(01)00410-8.
[26] V. L. Nguyen, V. T. Nguyen, N. D. Duc, Pin-loaded Hole Contact in an Anisotropic Magnetoelectroelastic Plate, International Journal of Mechanical Sciences, Vol. 302, 2025, pp. 110551, https://doi.org/10.1016/j.ijmecsci.2025.110551.
[27] V. T. Nguyen, C. Hwu, Boundary Element Method for Two-Dimensional Frictional Contact Problems of Anisotropic Elastic Solids, Engineering Analysis with Boundary Elements, Vol. 108, 2019, pp. 49-59, https://doi.org/10.1016/j.enganabound.2019.08.010.
[28] X. T. Nguyen, V. T. Nguyen, N. D. Duc, Boundary-Based Finite Element Method for Anisotropic Functionally Graded Materials, Engineering Analysis with Boundary Elements, Vol. 181, 2025, pp. 106526, https://doi.org/10.1016/j.enganabound.2025.106526.
[29] C. Hwu, S. T. Huang, C. C. Li, Boundary-Based Finite Element Method for Two-Dimensional Anisotropic Elastic Solids with Multiple Holes and Cracks, Engineering Analysis with Boundary Elements, Vol. 79, 2017,
pp. 13-22, https://doi.org/10.1016/j.enganabound.2017.03.003.
[30] C. Hwu, Anisotropic Elasticity with Matlab, Springer Cham, 2021.