A semi-analytical solution for static and dynamic analysis of plates with piezoelectric patches

Abstract

A modified mixed variational principle for piezoelectric materials is established and the state-vector equation of piezoelectric plates is deduced directly from the principle. Then the exact solution of the state-vector equation is simply given, and based on the semi-analytical solution of the state-vector equation, a realistic mathematical model is proposed for static analysis of a hybrid laminate and dynamic analysis of a clamped aluminum plate with piezoelectric patches. Both the plate and patches are considered as two three-dimensional piezoelectric bodies, but the same linear quadrilateral element is used to discretize the plate and patches. This method accounts for the compatibility of generalized displacements and generalized stresses on the interface between the plate and patches, and the transverse shear deformation and the rotary inertia of the plate and patches are also considered in the global algebraic equation system. Meanwhile, there is no restriction on the thickness of plate and patches. The model can be also modified to achieve a semi-analytical solution for the transient responses to dynamic loadings and the vibration control of laminated plate with piezoelectric patches or piezoelectric stiffeners.

A fast BEM for the analysis of damaged structures with bonded piezoelectric sensors

“…The state-space formulation can be deduced starting from a generalized hybrid functional [18] whose variation with respect to the generalized variables U and R z gives the generalized statespace equation…”

“…The dual boundary element method allows to model the presence of cracks with accuracy, and its performance in terms of memory storage and solution time is improved using the hierarchical solver. The attached sensors, as well as the adhesive layer, are modeled using a 3D state-space finite element approach [18], taking into account the full electro-mechanical coupling in the piezoelectric layer. The suitable boundary conditions are embodied in the sensor model which is eventually expressed in terms of interface variables, allowing straightforward coupling with the underlying host structure.…”

Hamiltonian partial mixed finite element-state space symplectic semi-analytical approach for the piezoelectric smart composites and FGM analysis

“…They include the so-called finite layer method (FLM), which combines a thickness polynomial approximation with in-plane trigonometric series that should satisfy a priori the layer edge BC [13]; it is based on the classical displacement-potential piezoelectric VF. Extension of the semi-analytical solution for elastic bodies of [7] to 3D plates with piezoelectric patches can be seen in [14]; it used an in-plane FE bilinear approximation of the partial mixed variables and the precise integration method [12] for the state matrix exponential evaluation. Use of the differential quadrature method in the semi-analytical solution was presented recently in [15] for the cylindrical bending of laminates.…”

“…The reformulation of the equations of 3D [7,11] or 2D [18,19] elasticity and 3D [14] or 2D [16] piezoelectricity in the Hamiltonian framework allows natural introduction of, respectively, the transverse stresses and electric displacement as primary unknowns, thanks to Legendre transformations. This makes the final partial differential equations (PDE) lower in order, compared to the classical displacement or displacement and potential-based formulations.…”

“…This feature is quite interesting in the perspective of developing numerical tools for multilayer plates since the propagator matrix approach, used for analytical methods [6], can then be used. However, the first-order state space equation, resulting from the Hamiltonian system, leads automatically to a state matrix exponential that is computed using either the power series expansion of the exponential, as in [11] for elastic laminated plates and in [14] for plates with piezoelectric patches, or using the symplectic approach, as in [18,19] for 2D elastic plates and in [16] for FGM plane problems.…”