Abstract
This chapter addresses the development of a new semi-analytical Lagrangian-Hamiltonian method for the three-dimensional solution of piezoelectric smart composite plates. It is based on the analytic state space symplectic Hamiltonian approach to fulfil the electromechanical multilayer interface continuity constraints and two-dimensional in-plane finite element (FE) numerical discretization to deal with arbitrary boundary conditions (BC) on the composite lateral edges. The originality of the proposed semi-analytical solution is that the latter feature (arbitrary BC handling) is reached through a mechanical displacements-electric potential primary variables based Lagrangian formalism, while the solution accuracy feature is reached through a primary and dual (transverse stresses and electric displacement) variables-based partial mixed Hamiltonian formalism. The transformation of the Lagrangian FE discretized formulation to a state space Hamiltonian one is made through the Legendre transformation. The proposed methodology is applied to the static actuation and sensing of piezoelectric hybrid laminated composite plates subjected to various BC. The obtained results comparison to reference ones of various benchmarks solutions, for non classical BC (cantilever), multilayer composite layups (angle-ply) and electromechanical loadings (uniform), from the open literature shows good computational convergence (coarse mesh), low cost (few FE degrees of freedom) and high accuracy (exact through-the-thickness) of the present new Hamiltonian semianalytical solutions. Thus, the provided tabulated numerical results can be used safely for benchmarking other closed-form, numerical or semi-analytical solutions.
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Andrianarison O, BenjeddouA(2012) Hamiltonian partial mixed finite element-state space symplectic semi-analytical approach for the piezoelectric smart composites and FGM analysis. Acta Mechanica 223:1597–1610
Benedetti, I., Aliabadi, M.H., Milazzo, A.: A fast BEM for the analysis of damaged structures with bonded piezoelectric sensors. Computer Methods in Applied Mechanics and Engineering 199(9), 490–501 (2010)
Benjeddou, A.: Advances in piezoelectric finite element modeling of adaptive structural elements: a survey. Computers & Structures 76, 347–363 (2000)
Benjeddou, A., Andrianarison, O.: A piezoelectric mixed variational theorem for smart multilayered composites. Mechanics of Advanced Materials and Structures 12(1), 1–11 (2005)
Benjeddou, A., Deu, J.F.: Piezoelectric transverse shear actuation and sensing of plates, part 1: A three-dimensional mixed state space formulation. Journal of Intelligent Material Systems and Structures 12(7), 435–449 (2001)
Benjeddou, A., Deu, J.F., Letombe, S.: Free vibrations of simply-supported piezoelectric adaptive plates: an exact sandwich formulation. Thin-Walled Structures 40(7), 573–593 (2002)
Boffi D, Brezzi F, Fortin M (2013) Mixed Finite Element Approach and Applications. Springer, Berlin, Heidelberg
Carrera, E., Buttner, A., Nali, P.: Mixed elements for the analysis of anisotropic multilayered piezoelectric plates. Journal of Intelligent Material Systems and Structures 21(7), 701–717 (2010)
Kapuria, S., Kumari, P., Nath, J.K.: Efficient modeling of smart piezoelectric composite laminates: a review. Acta Mechanica 214, 31–48 (2010)
Khandelwal, R.P., Chakrabarti, A., Bhargava, P.: An efficient hybrid plate model for accurate analysis of smart composite laminates. Journal of Intelligent Material Systems and Structures 24(16), 1927–1950 (2013)
Leung, A.Y.T., Zheng, J.J., Lim, C.W., Zhang, X., Xu, X.S., Gu, Q.: A new symplectic approach for piezoelectric cantilever composite plates. Computers & Structures 86(19), 1865–1874 (2008)
Li, D.: Layerwise theories of laminated composite structures and their applications: A review. Archives of Computational Methods in Engineering online first 1, 24 (2020). https://doi.org/10.1007/s11831-019-09392-2
Li, S., Huang, L., Jiang, L., Qin, R.: A bidirectional B-spline finite point method for the analysis of piezoelectric laminated composite plates and its application in material parameter identification. Composite Structures 107, 346–362 (2014)
Lim, C.W., Xu, X.S.: Symplectic Elasticity: Theory and Applications. Applied Mechanics Reviews 63(5), 1–10 (2011)
Liu, G.R., Dai, K.Y., Lim, K.M.: Static and vibration control of composite laminates integrated with piezoelectric sensors and actuators using the radial point interpolation method. Smart Materials and Structures 13(6), 1438–1447 (2004)
Moleiro, F., Mota Soares, C., Mota Soares, C., Reddy, J.: Layerwise mixed models for analysis of multilayered piezoelectric composite plates using least-squares formulation. Composite Structures 119, 134–149 (2015)
Pablo F, Bruant I, Polit O (2009) Use of classical plate finite elements for the analysis of electroactive composite plates. Numerical validations. Journal of Intelligent Material Systems and Structures 20(15):1861–1873
Phung-Van, P., Lorenzis, L.D., Thai, C.H., Abdel-Wahab, M., Nguyen-Xuan, H.: Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higher-order shear deformation theory and isogeometric finite elements. Computational Materials Science 96, 495–505 (2015)
Shan, L., Jun, L., Gao, L., Zhang, Z., Zhang, P.: The static solution for the layered piezoelectric bounded domain with side face load by the modified SBFEM. Advances in Applied Mathematics and Mechanics 10, 209–241 (2018)
Sze, K.Y., Pan, Y.S.: Hybrid finite element models for piezoelectric materials. Journal of Sound and Vibration 226(3), 519–547 (1999)
Tahani M, Naserian-NikAM(2013) Bending analysis of piezolaminated rectangular plates under electromechanical loadings using multi-term extended Kantorovich method. Mechanics of Advanced Materials and Structures 20(6):415–433
Tzou HS (1989) Development of a light-weight robot end-effector using polymeric piezoelectric bimorph. In: Proceedings, 1989 International Conference on Robotics and Automation, IEEE, Scottsdale, AZ, vol 3, pp 1704–1709
Tzou HS (1993) Piezoelectric Shells: Distributed Sensing and Control of Continua. Kluwer Academic Publishers
Vidal, P., D’Ottavio, M., Thaier, M.B., Polit, O.: An efficient finite shell element for the static response of piezoelectric laminates. Journal of Intelligent Material Systems and Structures 22(7), 671–690 (2011)
Wu, C.P., Liu, Y.C.: A review of semi-analytical numerical methods for laminated composite and multilayered functionally graded elastic/piezoelectric plates and shells. Composite Structures 147, 1–15 (2016)
Zhang, W.X., Wang, H.: Axisymmetric boundary condition problems for transversely isotropic piezoelectric materials. Mechanics Research Communications 87, 7–12 (2018)
Zhou, Y., Li, S., Zhou, H.: State space finite element analysis for piezoelectric precision positioning considering secondary converse piezoelectric effect. Finite Elements in Analysis and Design 102–103, 85–94 (2015)
Zhou, Y., Nyberg, T., Xiong, G., Li, S.: State space finite element analysis for piezoelectric laminated curved beam with variable curvature. Mechanics of Advanced Materials and Structures 27(4), 265–273 (2020)
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Andrianarison, O., Benjeddou, A. (2020). New Hamiltonian Semi-analytical Approach for 3D Solution of Piezoelectric Smart Composites. In: Altenbach, H., Chinchaladze, N., Kienzler, R., Müller, W. (eds) Analysis of Shells, Plates, and Beams. Advanced Structured Materials, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-030-47491-1_2
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