Keywords: layerwise theory, piezoelectricity, adaptive structures, plates, laminated composites, finite element method.

In this paper the layerwise theory of Reddy [1,2] is extended to 3D general shape laminated adaptive plate structures, with piezolaminated layers or patches. For the electric problem it is assumed an electroquasistatic approximation. It is considered that the electric field and polarization along with the mechanical variables are the only important interactions when describing the motion and deformation of the material. The behaviour of the structure is obtained solving the mechanical and electric coupled problem, described by the elasticity and electrostatics equations and the respective boundary conditions.

Considering the approach proposed by Reddy, for the finite element models, the layerwise theory primary variables, displacement field components , , and electric potential , are assumed to be approximated inside each element as products of functions of by continuous functions of :

where the thickness direction functions , and are constructed with Lagrange polynomials. The numbers , , and of interpolation functions , , and , depend of the interpolation scheme associated with each lamina thickness function , within the element. The finite element model is obtained from a variational formulation.

The predictions of the different developed models, for the static analysis of two rectangular piezolaminated plates, are compared among themselves and also with a closed form solution programmed for the purpose.

From the analysis of the distributions of the primary variables and secondary variables, for a defined surface discretization of the plate and for whichever interpolation is used on the surface, it can be observed that:

a)

For linear and quadratic approximations across the thickness, the primary and secondary variables distributions along the thickness do not agree well with the exact solution. Although for the primary variables the results are good in general, the secondary variables are highly influenced by chosen thickness approximation.

b)

A cubic approximation along the thickness of each layer gives very good predictions for all primary and secondary variables, with distributions almost coincident with the closed form solutions.

c)

Approximations along the thickness higher than third order, only improve slightly the predictions, but at the expense of a high computational effort.

The distributions of the in-plane stresses along the thickness compare always very well with the corresponding predictions of the closed form solution, even when linear or quadratic thickness approximations are used.

The values of the secondary variables are highly dependent of the discretization and interpolation used in the surface directions. The Lagrangean interpolation in the surface gives better predictions than the Hermite interpolation, even when the Lagrange interpolation is quadratic.

We can conclude that diverse agreements are found between the proposed alternative finite element models predictions, using different approximations across the thickness, and also with the exact solution. An excellent agreement is obtained for all the models using cubic approximation in the thickness direction, where all the primary and secondary variables are predicted with very high precision.

Due to their computational cost, these layerwise models should only be applied in certain areas of the laminated structure where the interlaminar stresses are of primary importance, namely in the neighbourhood of geometric discontinuities and delamination.

Based on the studies carried out, the best compromise is the third order polynomial approximation across the thickness and cubic Lagrangean interpolation functions on the surface.

1

J.N. Reddy, "A generalization of two-dimensional theories of laminated composite plates", Communications in Applied Numerical Methods, 3, 173-180, 1987. doi:10.1002/cnm.1630030303

2

J.N. Reddy, "Mechanics of laminated composite plates - Theory and analysis", CRC Press, Boca Raton, 1997.

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