Keywords: smart laminates, piezoelectric material, refined hybrid plate theory, finite element, static and dynamic response, structural control.

The paper presents a refined hybrid plate theory to model the coupled electro-mechanical field problem found in the analysis of smart laminates. Actually, a proper prediction of structural response of composite laminates having embedded/surface-bonded piezoelectric layers with an affordable computational involvement is a challenging problem. In this context, the most common approach is to take the piezoelectric layers as surface bonded and the interfaces of these layers with the core are grounded to make the electric potential zero over the entire core. In these studies, the structural deformation is modelled by a single layer theory (SLT). A major limitation of these approaches is that the piezoelectric layers cannot be embedded inside the laminate. Moreover, these plate theories (SLT) are not capable enough of predicting the stresses accurately. In case of a laminate having sandwich construction, even the global parameters like displacement cannot be predicted properly with the SLT.

The other extreme of the modelling of this problem is the use of three-dimensional finite element analysis but it requires a huge computational effort. A compromise between these two approaches is due to Heyliger and his associates [1] who have used layer-wise theory (LWT) to model the mechanical as well as the electric field. As the number of unknowns is dependent on the number of layers in LWT, its use may not be practicable for the analysis of multi-layered plates. In order to have more computational efficiency, Mitchell and Reddy [2] have proposed the concept of hybrid plate theory (HPT) where structural deformation is modelled by SLT and LWT is used for the electrical field. With the use of SLT having unknowns at the reference plane only, HPT becomes computationally efficient while generality of the problem is not affected due to the use of LWT for the electrical field. However, these hybrid models inherit the limitations of SLT in modelling layered plates as mentioned earlier.

In this context, it should be mentioned that transverse shear strains are found to be discontinuous at the layer interfaces in laminated plates due to difference in the rigidities of the adjacent layers. Unfortunately, the variation of in-plane displacements across the plate thickness is expressed by continuous functions in SLT; consequently, the transverse shear strains become continuous at the layer interfaces. Thus SLT cannot properly represent the transverse shear strains in layered plates and this drawback has inspired the researchers to develop LWT.

Keeping all these aspects in view, a new hybrid plate model that consists of refined higher order shear deformation theory (RHSDT) for the modelling of structural deformation and LWT for the electric field is proposed. This new model may be named as refined hybrid plate theory (RHPT). Interestingly, a RHSDT [3,4] possesses unknowns at the reference plane like SLT while it gives piece-wise parabolic variation of transverse shear strain across the plate thickness with discontinuity at the layer interfaces as desired in a layered plate. Moreover, the transverse shear stresses become zero at the plate top and bottom surfaces. Again the unknowns obtained at all the interfaces for electric potential are eliminated through static condensation before the system of governing equations is solved.

The finite element technique is used to implement the refined hybrid plate theory (RHPT). In this context, it should be noted that the resulting plate model requires C1 continuity of transverse displacement at the element interfaces and it is not easy to be satisfied in every situation. Actually, it is a well-known problem of plate finite element. Considering this aspect in view, a new rectangular element is developed where the continuity requirement is fully ensured.

The performance of the finite element model is first tested with a numerical example of a cross-ply laminate where the results obtained are found to be closer to the elasticity solution compared to those obtained from SLT. The model is then applied to the problem of a smart laminate under static load where the response is controlled by the supply voltage at the actuator layer taking sensor voltage as the feedback. Finally the dynamic response and its control of a smart laminate are studied in the last example.

1

P.R. Heyliger, G. Ramirez, D.A. Saravanos, "Coupled discrete-layer finite-element models for laminated piezoelectric plates", Communications in Numerical Methods in Engineering, 10(12), 971-981, 1994. doi:10.1002/cnm.1640101203

2

J.A. Mitchell, J.N. Reddy, "A refined hybrid plate-theory for composite laminates with piezoelectric laminate", Journal of Solids and Structures, 32(16), 2345-2367, 1995. doi:10.1016/0020-7683(94)00229-P

3

M. Di Sciuva, "Multilayered anisotropic plate models with continuous interlaminar stress", Computers and Structures, 22(3), 149-167, 1992. doi:10.1016/0263-8223(92)90003-U

4

M. Cho, R.R. Parmerter, "Efficient higher order plate theory for general lamination configurations", AIAA Journal, 31(7), 1299-1308, 1993. doi:10.2514/3.11767

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