Non-linear response of laminated composite plates under thermomechanical loading

doi:10.1016/j.ijnonlinmec.2004.11.003

International Journal of Non-Linear Mechanics

Volume 40, Issue 7, September 2005, Pages 971-985

https://doi.org/10.1016/j.ijnonlinmec.2004.11.003 Get rights and content

Abstract

The non-linear response of laminated composite plates under thermomechanical loading is studied using the third-order shear deformation theory (TSDT) that includes classical and first-order shear deformation theories (CLPT and FSDT) as special cases. Geometric non-linearity in the von Kármán sense is considered. The temperature field is assumed to be uniform in the plate. Layers of magnetostrictive material, Terfenol-D, are used to actively control the center deflection. The negative velocity feedback control is used with the constant gain value. The effects of lamination scheme, magnitude of loading, layer material properties, and boundary conditions are studied under thermomechanical loading.

Introduction

Active vibration/deflection control of small to large structural components in aerospace and civil engineering have been studied extensively. A smart structure is the one that has surface mounted or embedded sensors and actuators so that it has the capability to sense the input and take appropriate correcting actions. Piezoelectric, magnetostrictive, and electrostrictive materials, and shape memory alloys are the common examples of smart materials [1], [2], [3], [4], [5], [6], [7], [8].

Terfenol-D is a commercially available magnetostrictive material and an alloy of terbium, iron, and dysprosium. It has a unique advantage over the other smart materials in that it has the ability to produce large actuation forces. Terfenol-D material can also be easily embedded into the host material of the structures such as a CFRP composite. There are a number of studies that investigate applications of magnetostrictive materials as active actuators in composite structures. They presented finite element formulations and analytical solutions for simply supported boundary conditions of laminated composite beams and plates with embedded active layers by the classical and the first-order shear deformation theories [9], [10], [11], [12], [13], [14], [15], [16]. The previous works addressed neither the effect of the geometric non-linearity nor the effect of thermal loads on deflection control. Recently, Lee et al. [17] and Lee and Reddy [18] presented finite element analyses of vibration/deflection control of laminated composite plates. They studied the effect of simply supported, clamped, free, and their combined boundary conditions with other parameters such as lamination scheme and loading type on deflection control times by the third-order shear deformations theory. They also considered von Kármán-type geometric non-linearity in laminated composite plates under mechanical loads.

In the present paper, the complete derivation of the governing equations of the third-order shear deformation theory including thermal effects and von Kármán non-linear strains is presented. The temperature field is assumed to be uniform and the material properties are assumed to be independent of temperature. The Newmark scheme is used to approximate the time derivatives and the resulting non-linear finite element equations are solved using the Newton–Raphson iteration scheme.

Section snippets

Displacements and strains

The third-order shear deformation theory (TSDT) is based on the following displacement field (see Reddy [19]): $u (x, y, z, t) = u_{0} (x, y, t) + z φ_{x} (x, y, t) - c_{1} z^{3} (φ_{x} + \frac{\partial w_{0}}{\partial x}),$ $v (x, y, z, t) = v_{0} (x, y, t) + z φ_{y} (x, y, t) - c_{1} z^{3} (φ_{y} + \frac{\partial w_{0}}{\partial y}),$ $w (x, y, z, t) = w_{0} (x, y, t),$ where t denotes time, $(u_{0}, v_{0}, w_{0})$ are the displacements of a point on the plane $z = 0$ , and $φ_{x} (x, y, t)$ and $φ_{y} (x, y, t)$ denote the rotations of a transverse normal about the y- and x-axes, respectively. The constant $c_{1}$ is given by $4 / 3 h^{2}$ , h being the total thickness of the laminate.

The

Non-linear finite element analysis

We assume finite element approximation of the generalized displacements in the form $u_{0} (x, y, t) = \sum_{i = 1}^{m} u_{i}^{e} (t) ψ_{i}^{e} (x, y),$ $v_{0} (x, y, t) = \sum_{i = 1}^{m} v_{i}^{e} (t) ψ_{i}^{e} (x, y),$ $w_{0} (x, y, t) = \sum_{i = 1}^{m} {\bar{Δ}}_{i}^{e} (t) ϕ_{i}^{e} (x, y),$ $φ_{x} (x, y, t) = \sum_{i = 1}^{m} X_{i}^{e} (t) ψ_{i}^{e} (x, y),$ $φ_{y} (x, y, t) = \sum_{i = 1}^{m} Y_{i}^{e} (t) ψ_{i}^{e} (x, y),$ where $ψ_{i}^{e}$ denotes the Lagrange and $ϕ_{i}^{e}$ are the Hermite interpolation functions. The conforming element, that has eight degrees of freedom $(u_{0} v_{0}, w_{0}, w_{0, x}, w_{0, y}, w_{0, xy}, φ_{x}, φ_{y})$ is used in this study. Substitution of Eq. (14) into the weak form of the governing equation (6

Numerical results

The temperature field is assumed to be uniform over the plate surface and linear through the thickness [i.e., $T_{0}$ is constant and $T_{1} = 0$ in Eq. (13b)]. The mechanical load is applied downward on the top surface so that it bends the plate concave up.

A laminated composite square plate with both the upper and lower surfaces mounted with magnetostrictive materials layers is considered. The laminate is composed of a total of 10 layers and all layers are assumed to be of the same thickness. Three

Conclusions

In this paper, structural response of laminated composite plates with surface-mounted smart material layers is studied under thermomechanical loading using the third-order laminate plate theory. Finite element formulation that accounts for the geometric non-linearity in the von Kármán sense and temperature effects is presented. The material properties are assumed to be independent of temperature. The Newmark time integration scheme is used to reduce the semidiscretized finite element equations

Acknowledgements

The second author gratefully acknowledges the support of this work by ILIR Contract DAAE07-03-C-L103 from the US Army Tank-Automotive Tank and Armaments Command, Warren, Michigan.

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View full text

Non-linear response of laminated composite plates under thermomechanical loading

Abstract

Introduction

Section snippets

Displacements and strains

Non-linear finite element analysis

Numerical results

Conclusions

Acknowledgements

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Finite Elements Anal. Des.

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Smart Structures

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J. Intel. Mater. Systems Struct.

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J. Intel. Mater. Systems Struct.