8/12/2019 Geometrically Nonlinear Finite Element Analysis of Sandwich Plates

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Geometrically nonlinear finite element analysis of sandwich plates using

normal deformation theory

Madhukar S., M.K. Singha ⇑

Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi 110 016, India

a r t i c l e i n f o

 Article history:

Available online xxxx

Keywords:

Sandwich plate

Normal deformable theory

Bending

Vibration

Nonlinear finite element

a b s t r a c t

The geometrically nonlinear bending and vibration behavior of soft core sandwich plates is investigated

here using higher order finite element model incorporating transverse shear and normal deformation.

The geometric nonlinearity, based on von Kármán’s assumption is introduced and the nonlinear govern-

ing equations of motion are derived considering in-plane and rotary inertia. The nonlinear governing

equation is solved by Newton–Raphson iteration technique for the nonlinear bending problem, whereas,

the harmonic balance method is employed to obtain the frequency versus amplitude relationships for the

large amplitude free and forced vibration of sandwich plates. The results obtained from the normal defor-

mation theory are compared with the results of first-order and third-order shear deformation theories.

Limited parametric study is conducted to examine the influences of  span-to-thickness  ratio and  core-to-

 face sheet thickness ratio of soft core sandwich plates.

  2012 Elsevier Ltd. All rights reserved.

1. Introduction

Composite materials are increasingly used in structural compo-

nents of aircrafts, automobile, submarines, and other high perfor-

mance application areas because of their high  stiffness-to-weight 

and  strength-to-weight   ratios. Further, sandwich panels with the

addition of a thick soft core provide higher bending rigidity, ther-

mal and noise isolations and overall damping characteristics of 

the structure. These sandwich panels are sometimes subjected to

dynamic loads and oscillate with large amplitude. Hence, the geo-

metrically nonlinear bending and vibration characteristics of sand-

wich plates are important problems to be investigated for the

effective design of such structures.

Several analytical and finite element investigations are reported

in the literature on the bending, buckling and free vibration behav-

ior of sandwich plates. Bending and vibration characteristics of soft

core sandwich plates has been investigated using three-dimen-

sional elasticity solutions [1–3] and two dimensional plate bending

theories [4–17]. The commonly employed two-dimensional theo-

ries are the first-order shear deformation theory (FSDT), third order

shear deformation theory (TSDT), trigonometric shear deformation

theory, zig-zag theory and several versions of higher order shear

and normal deformation theory (HSDT). The efficiencies of the

above theories for the static and dynamic analysis of sandwich

plates have been critically reviewed by Mallikarjuna and Kant

[18], Hu et al.  [19] and Carrera and Brischetto [20].

It is observed from the literature that most of works have dealt

with the linear bending and vibration behavior of sandwich plates

and geometrically nonlinear analysis of sandwich plates is limited.

Rajagopal et al. [21] employed a rectangular finite element having

five degrees of freedom at each node for the non-linear free vibra-

tion analysis of a three-layered sandwich plates. Cheng et al.  [22]

reported an analytical study on the nonlinear vibration behavior

of rectangular Reissner sandwich plates. Ganapathi et al. [23] em-

ployed higher-order theories for the nonlinear dynamic analysis of 

thick composite and sandwich plates. Du and Ma [24] studied the

nonlinear vibration and buckling behavior of circular sandwich

plates, whereas, Chakrabarti and Bera [25] studied the large ampli-

tude flexural vibration of elliptical sandwich plates. Yongqiang et

al. [26,27] reported analytical studies on the nonlinear free vibra-

tion characteristics of simply supported and clamped symmetric

rectangular honeycomb sandwich panels based on third order

shear deformation theory, whereas, Chandrashekhar and Ganguli

[28] employed Reddy’s third order plate theory for the nonlinear

vibration analysis of sandwich plates with random material

properties.

In the present paper, a four node shear flexible rectangular plate

bending element is developed (similar to Refs. [29,30]) to study the

geometrically nonlinear static and dynamic behavior of sandwich

plates made with soft core. Here, nonlinear finite element model

is developed based on higher order displacement model incorpo-

rating shear and normal deformation (ND). The developed finite ele-

ments are free from shear locking and any spurious zero energy

modes. Two additional models similar to the  first order shear defor-

mation theory   (FSDT) and   third-order shear deformation theory

0263-8223/$ - see front matter    2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.compstruct.2012.10.034

⇑

Corresponding author. Tel.: +91 11 2659 6445; fax: +91 11 2658 1119.E-mail address: maloy@am.iitd.ac.in (M.K. Singha).

Composite Structures xxx (2012) xxx–xxx

Contents lists available at  SciVerse ScienceDirect

Composite Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p s t r u c t

Please cite this article in press as: Madhukar S, Singha MK. Geometrically nonlinear finite element analysis of sandwich plates using normal deformation

theory. Compos Struct (2012), http://dx.doi.org/10.1016/j.compstruct.2012.10.034