PSI - Issue 14

Rahul Saini et al. / Procedia Structural Integrity 14 (2019) 362–374 R. Saini, S. Saini, R. Lal, and I. V. Singh / Structural Integrity Procedia 00 (2018) 000–000

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1. Introduction There has been a growing interest in the analysis of the behaviour of functionally graded materials (FGMs), since their invention in 1984 (Koizumi, 1997, 1993). Being a mixture of ceramic and metal, FGMs are microscopically inhomogeneous, in which, the mechanical properties vary continuously and smoothly from one surface to the other. This is achieved by gradually varying the volume fraction of the constituent material (Suresh and Mortensen, 1998). Plate-type structural elements of FGMs have their wide applications in energy conservation devices, gas turbines, chemical plants, plasma facings for fusion reactors, spacecraft heat shields, engine components, high-power electrical components etc. and particularly, in defence - as penetration resistance materials used for armour plates and bullet-proof vests. In recent years, numerous studies on the static/dynamic behaviour of FGM plates of different geometries have been made and important ones are reported in references (Fazzolari, 2016; Kiani and Eslami, 2014; Malekzadeh et al., 2011; Pradhan and Chakraverty, 2015; Prakash and Ganapathi, 2006; Swaminathan and Sangeetha, 2017). Out of these, reference (Swaminathan and Sangeetha, 2017) is a critical review of the work up to 2016 on the thermal analysis of FGM plates with various mathematical idealization of materials, temperature profiles, modelling techniques and solution methods. Malekzadeh et al. (2011) used differential quadrature method (DQM) in analysing the vibrational behaviour of FG annular plates on the basis of first order shear deformation theory (FSDT). Thermally induced vibrations of FGM circular plate using FSDT and Newton-Raphson Newmark Scheme, have been presented by Kiani and Eslami (2014). Very recently, Khorshidvand et al. (2012) presented the thermo-elastic analysis for the buckling of FGM circular plates with linear strain and integrated with piezoelectric layers using Bessel’s functions. In his study, Behravan Rad (2015) used a combination of state space method and DQM to analyse the thermo-elastic behaviour of FGM circular plates with non-uniform asymmetric mechanical and uniform thermal loads. In these studies, the mechanical properties of the plate material are assumed to be temperature dependent (TD) (Fazzolari, 2016; Kiani and Eslami, 2014; Malekzadeh et al., 2011) as well as temperature independent (TI) (Behravan Rad, 2015; Khorshidvand et al., 2012; Pradhan and Chakraverty, 2015; Prakash and Ganapathi, 2006). In this paper, an attempt has been made to extend free axisymmetric vibration results of FGM circular plate subjected to in-plane force and non-linear temperature distribution along the thickness direction. The mechanical properties of the plate material are assumed to be TD and vary according to a power law model. The governing differential equation for such a plate model with clamped and simply supported at the periphery have been solved by generalized differential quadrature rule (GDQR) (Wu et al., 2002). The effect of various parameters such as temperature difference, material graded index and in-plane force parameter on the natural frequencies and critical buckling load has been analyzed. Two-dimensional plate configurations have been shown for clamped and simply supported plates. 2. Geometrical description and formulation Referred to cylindrical co-ordinate system � , , � , consider an FGM circular plate of radius , thickness ℎ , density , subjected to uniform, tensile in-plane force � and � � being the middle surface of the plate. The line � � is the axis of the plate. The top surface � �ℎ/2 is taken as ceramic rich while the bottom � �ℎ/2 as metal rich. The plate is subjected to a temperature distribution varying along the thickness i.e. �� � �� such that � � and � � at the ceramic and metallic rich surfaces, respectively (Fig. 1). The typical effective material properties � , � of fabricated FGM plate are related with the material properties of metal � � � and ceramic � � � in the following manner (Malekzadeh et al., 2011), � , � � � � � � � � � � � � � �� � � �, �1� where, � describe the volume fraction of ceramic at any point and defined as, � � � � � �� � � � � � � , �2�