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Buckling and vibrations of FGM circular plates in thermal environment
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Procedia Structural Structural IntegrityIntegrity Procedia1400(2019) (2016)362–374 000–000
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2nd International Conference on Structural Integrity and Exhibition 2018 2nd International Conference on Structural Integrity and Exhibition 2018
Buckling and vibrations of FGM circular plates in thermal Buckling and vibrations of FGM circular plates in thermal environment XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal environment � Rahul Saini�,� , Shivam Saini�� , Roshan Lal , Indra Vir Singh�� blade of an Thermo-mechanical a high pressure turbine �,� modeling of � Rahul Saini , Shivam Saini , Roshan Lal , Indra Vir Singh 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 airplane gas turbine engine 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖, 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 � �
� �
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖, 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼
P. Brandãoa, V. Infanteb, A.M. Deusc*
AbstractaDepartment of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Abstract Portugal b IDMEC, is Department Mechanical Engineering, Superioron Técnico, Universidade devibrations Lisboa, Av.ofRovisco Pais, 1, 1049-001 Lisboa, An analysis presentedoffor the study of thermalInstituto environment free axisymmetric functionally graded circular Portugal An analysis is presented for the study of thermal environment on free axisymmetric vibrations of functionally graded circular plates subjected to uniform in-plane peripheral loading and non-linear temperature distribution along the thickness direction. It is c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, plates subjected uniform in-plane peripheral loadingdirection and non-linear temperature distribution along the thickness direction. It is assumed that thetoplate material is graded in thickness and mechanical properties are temperature-dependent. Hamilton’s Portugal assumed the plate is graded in thickness directionfor and mechanical properties areand temperature-dependent. Hamilton’s principle that has been usedmaterial in obtaining the governing equations thermo-elastic equilibrium vibration for such a plate model principle hasof been used inplate obtaining governing differential equations for thermo-elastic equilibrium a plate values model on the basis classical theory.the Generalized quadrature rule has been usedand in vibration evaluatingfor thesuch numerical on the basis displacements of classical plate Generalized rule has beenplates used in the for numerical forAbstract thermal andtheory. frequencies in casedifferential of clampedquadrature and simply supported at evaluating the periphery the firstvalues three for thermal displacements and frequencies in caseforofwhich clamped the reported peripheryasfor the first three modes of vibration. Compressive in-plane loads the and platesimply ceasessupported to vibrateplates have at been critical buckling modes of vibration. Compressive loads forcomponents which thevibration plate ceases to to vibrate have reported criticaland buckling loads. Effects various parameters have been analyzed on the characteristics for all been thedemanding modes. Foras uniform linear During theirofoperation, modernin-plane aircraft engine are subjected increasingly operating conditions, loads. Effectsdistribution, of high various parameters been analyzed onconditions the vibration the different modes. uniform andhaving linear temperature the benchmark results haveSuch been computed. Ascharacteristics athese special case, a all study with theFor plate material especially the pressure turbinehave (HPT) blades. cause parts tofor undergo types of time-dependent temperature distribution, the isbenchmark results havethe been computed. As acompared special case, study withinthe platetomaterial having degradation, one of which creep. A been model using finite element method (FEM) wasathe developed, order be able to predict temperature-independent properties has performed. Results have been with published work. the creep behaviour ofproperties HPT blades. Flight data records (FDR) for a compared specific aircraft, a commercial aviation temperature-independent has been performed. Results have been with the provided published by work. were used to obtainbythermal mechanical data for three different flight cycles. In order to create the 3D model © company, 2018 The Authors. Published Elsevierand B.V. © 2019 The Authors. Published by Elsevier B.V. B.V. scrap was scanned, and its chemical composition and material properties were needed for the FEM analysis, HPT © 2018 The Authors. Published byaElsevier This is an open access article under the CCblade BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access undergathered the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) obtained. The dataarticle that was wasBY-NC-ND fedofinto thelicense FEM model and different of simulations were organizers. run, first with a simplified 3D This is an and open access article under the CC (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection peer-review under responsibility Peer-review under responsibility the SICE 2018 Selection and peer-review under responsibility of establish Peer-review under responsibility of thethe SICE 2018 organizers. rectangular block shape, in order to better the model, and responsibility then with realthe 3DSICE mesh2018 obtained from the blade scrap. The Selection and peer-review under responsibility of Peer-review under of organizers. overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore Keywords: Circular plates; functionay graded; in-plane force; temperature-dependent material; non-linear temperature; generalized differentialsuch a model can be useful the goal of predicting turbine blade life, given a setmaterial; of FDRnon-linear data. quadrature rule. Keywords: Circular plates;infunctionay graded; in-plane force; temperature-dependent temperature; generalized differential quadrature rule.
© 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation.
* Corresponding author. Tel.: +91-812-651-5226 . address:author. [email protected] *E-mail Corresponding Tel.: +91-812-651-5226 . E-mail address: [email protected] 2452-3216 © 2018 The Authors. Published by Elsevier B.V.
This is an © open article under theby CCElsevier BY-NC-ND 2452-3216 2018access The Authors. Published B.V. license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection under responsibility of Peer-review responsibility of the SICE 2018 organizers. This is an and openpeer-review access article under the CC BY-NC-ND licenseunder (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under * Corresponding author. Tel.: +351responsibility 218419991. of Peer-review under responsibility of the SICE 2018 organizers. E-mail address: [email protected]
2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016.
2452-3216 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of Peer-review under responsibility of the SICE 2018 organizers. 10.1016/j.prostr.2019.05.045
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
363 2
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 �𝑅𝑅, 𝜃𝜃, 𝑧𝑧�, consider an FGM circular plate of radius 𝑎𝑎, thickness ℎ, density 𝜌𝜌, subjected to uniform, tensile in-plane force 𝑁𝑁� and 𝑧𝑧 � � being the middle surface of the plate. The line 𝑅𝑅 � � is the axis of the plate. The top surface 𝑧𝑧 � �ℎ/2 is taken as ceramic rich while the bottom 𝑧𝑧 � �ℎ/2 as metal rich. The plate is subjected to a temperature distribution 𝑇𝑇 varying along the thickness i.e. 𝑇𝑇�� 𝑇𝑇�𝑧𝑧�� such that 𝑇𝑇 � 𝑇𝑇� and 𝑇𝑇 � 𝑇𝑇� at the ceramic and metallic rich surfaces, respectively (Fig. 1). The typical effective material properties 𝑃𝑃�𝑧𝑧, 𝑇𝑇� of fabricated FGM plate are related with the material properties of metal 𝑃𝑃� �𝑇𝑇� and ceramic 𝑃𝑃� �𝑇𝑇� in the following manner (Malekzadeh et al., 2011), 𝑃𝑃�𝑧𝑧, 𝑇𝑇� � 𝑃𝑃� �𝑇𝑇� � �𝑃𝑃� �𝑇𝑇� � 𝑃𝑃� �𝑇𝑇�� 𝑉𝑉� �𝑧𝑧�,
where, 𝑉𝑉� describe the volume fraction of ceramic at any point 𝑧𝑧 and defined as, 𝑉𝑉� �𝑧𝑧� � �
���� � ��
� ,
�1�
�2�
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
364
3
Fig. 1. FGM circular plate in thermal environment (a) cross-section; (b) top view.
𝑝𝑝�� �� being a parameter and represents the shape of the volume fraction. Following (Swaminathan and Sangeetha, 2017), the material properties 𝑃𝑃� �𝑇𝑇� and 𝑃𝑃� �𝑇𝑇� are given by 𝑃𝑃� �𝑇𝑇� � 𝑃𝑃� �𝑃𝑃�� 𝑇𝑇 �� � � � 𝑃𝑃� 𝑇𝑇 � 𝑃𝑃� 𝑇𝑇 � � 𝑃𝑃� 𝑇𝑇 � �,
𝑏𝑏 � � , 𝑐𝑐 .
���
The constituent materials of the plate are taken as titanium (𝑇𝑇𝑖𝑖 � ��� � 4�) for metal and zirconia (𝑍𝑍𝑍𝑍𝑍𝑍� ) for ceramic. The experimental values of the coefficients 𝑃𝑃� �𝑖𝑖 � ��, � , � , 2 , �� are taken from reference (Malekzadeh et al., 2011). The symbol ‘𝑃𝑃’ has been used for the material properties such as Young’s modulus 𝐸𝐸, mass density 𝜌𝜌, expansion coefficient 𝛼𝛼 and thermal conductivity 𝑘𝑘. 2.1. Thermal Stress Analysis
Assuming that the plate material is unidirectionally non-homogenous in thickness direction and due to that any kind of temperature variation applied to such plates will produce variations only in thickness direction. Though, numerous studies with constant/linear variation of temperature across the thickness of the circular plate are available in the literature (Khorshidvand et al., 2012; Swaminathan and Sangeetha, 2017), but authors have not come across any study dealing with non-linear temperature variation. In the present work, the non-linear variation of temperature is assumed to arise from the solution of one-dimensional steady state static heat conduction equation without heat flux (Swaminathan and Sangeetha, 2017), 𝑑𝑑 𝑑𝑑𝑑𝑑 �𝑘𝑘�𝑧𝑧� � � �, 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
subject to the boundary conditions: 𝑇𝑇 � 𝑇𝑇� at 𝑧𝑧 � �ℎ/2 and 𝑇𝑇 � 𝑇𝑇� at 𝑧𝑧 � �ℎ/2 . Using equation (1) for the power law distribution of 𝑘𝑘�𝑧𝑧�, the solution of equation (4) gives: 𝑇𝑇�𝑧𝑧� � 𝑇𝑇� �
�
� 𝑘𝑘�� 2𝑧𝑧 � ℎ ���� ∆𝑇𝑇 � ����� � � , � �𝑖𝑖𝑖𝑖 � ��𝑘𝑘� 2ℎ 𝐶𝐶 ∗ ���
∗
�
where 𝑘𝑘�� � 𝑘𝑘� � 𝑘𝑘� , ∆𝑇𝑇 � 𝑇𝑇� � 𝑇𝑇� , 𝐶𝐶 � ������ ���
�4�
�5� � 𝑘𝑘�� � �𝑖𝑖𝑖𝑖 � ��𝑘𝑘�
and 𝑁𝑁 represents the number of terms in the binomial expansion. Due to axisymmetric temperature distribution, the displacement components 𝑢𝑢 � , 𝑣𝑣� , 𝑤𝑤� and thermal stresses at an arbitrary point �𝑅𝑅, 𝜃𝜃, 𝑧𝑧� on the mid plane of the plate are (Reddy, 2008, p. 153) 𝑢𝑢 � �𝑅𝑅, 𝑧𝑧� � �𝑑𝑑
𝑑𝑑𝑤𝑤� , 𝑑𝑑𝑑𝑑
𝑣𝑣� �𝑅𝑅, 𝑧𝑧� � �,
𝑤𝑤� �𝑅𝑅, 𝑧𝑧� � 𝑤𝑤� �𝑅𝑅�,
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
𝑑𝑑 � 𝑤𝑤� 𝜈𝜈 𝑑𝑑𝑤𝑤� � � � 𝑇𝑇� � , 𝑑𝑑𝑅𝑅� 𝑅𝑅 𝑑𝑑𝑑𝑑 𝑑𝑑 � 𝑤𝑤� 1 𝑑𝑑𝑤𝑤� � �𝐸𝐸� �𝑧𝑧, 𝑇𝑇� �𝑧𝑧 �𝜈𝜈 � � � 𝑇𝑇� � , 𝑑𝑑𝑅𝑅� 𝑅𝑅 𝑑𝑑𝑑𝑑 � 𝜎𝜎��� � 𝜎𝜎��� � � ,
𝜎𝜎��� � �𝐸𝐸� �𝑧𝑧, 𝑇𝑇� �𝑧𝑧 �
𝜎𝜎���
𝜎𝜎���
𝐸𝐸�𝑧𝑧, 𝑇𝑇� �
𝐸𝐸�𝑧𝑧, 𝑇𝑇� , �1 � 𝜈𝜈 � �
⎫ ⎪
365 4
���
⎬ ⎪ ⎭
𝑇𝑇� � ��𝑧𝑧, 𝑇𝑇� ∆𝑇𝑇�𝑧𝑧� �1 � 𝜈𝜈� , ∆𝑇𝑇�𝑧𝑧� � 𝑇𝑇�𝑧𝑧� � 𝑇𝑇� ,
where, 𝑇𝑇� being the initial temperature and subscript 𝑇𝑇 is used for the deformations due to thermal environment. Introducing the non-dimensional variables 𝑊𝑊� � 𝑤𝑤� /𝑎𝑎, 𝑍𝑍 � 𝑧𝑧/𝑎𝑎 and 𝑟𝑟 � 𝑅𝑅/𝑎𝑎, the thermo-elastic equilibrium equation of the plate has been obtained using Hamilton’s principle, given by �
� 𝑄𝑄� ���
𝑑𝑑 � 𝑊𝑊� � �, 𝑑𝑑𝑟𝑟 �
���
together with the conditions at the periphery: �𝑖𝑖� Clamped �C� ∶
�𝑖𝑖𝑖𝑖� Simply supported �S� ∶
where , 𝑄𝑄� � 𝑁𝑁 ∗ �
𝑑𝑑𝑊𝑊� � ��, 𝑑𝑑𝑑𝑑 ��� 𝑑𝑑 � 𝑊𝑊� 𝐷𝐷�∗ 𝑑𝑑𝑊𝑊� �𝑊𝑊� ���� � �𝑄𝑄� � � 𝑚𝑚∗� � � �, 𝑑𝑑𝑟𝑟 � 𝑟𝑟 𝑑𝑑𝑑𝑑 ���
�𝑊𝑊� ���� � �
⎫ ⎪
⎬ ⎪ ⎭
���, ���
1 1 2 𝐷𝐷� 𝐷𝐷� �𝑄𝑄� � 𝑟𝑟 � 𝑁𝑁 ∗ �, 𝑄𝑄� � � � �𝑄𝑄� � 𝑟𝑟 � 𝑁𝑁 ∗ �, 𝑄𝑄� � 𝑄𝑄� , 𝑄𝑄� � 𝐷𝐷�∗ � ∗ , 𝐷𝐷�∗ � ∗ � 𝐷𝐷 𝐷𝐷 𝑟𝑟 𝑟𝑟 𝑟𝑟
𝑎𝑎𝑚𝑚 � 𝑎𝑎� 𝑁𝑁� , 𝑚𝑚∗� � ∗ , 𝐷𝐷 ∗ 𝐷𝐷
and �𝐷𝐷� , 𝐷𝐷� � � �
�/�
𝑚𝑚 � � �
�/�
�𝑧𝑧 . 𝑇𝑇� . 𝐸𝐸�𝑧𝑧, 𝑇𝑇��𝑑𝑑𝑧𝑧 , 𝐷𝐷 ∗ �
��/�
𝐸𝐸� ℎ� value 𝐸𝐸� is with respect to , � � 12�1 � 𝜈𝜈 � � the ceramic constituent
𝑧𝑧 � 𝐸𝐸�𝑧𝑧, 𝑇𝑇��1, 𝜈𝜈�𝑑𝑑𝑑𝑑 are the flexural rigidities of the plate.
��/�
2.2. Vibration analysis
For axisymmetric motion of the plate, the displacement components are given by 𝑢𝑢�𝑅𝑅, 𝑧𝑧, 𝑡𝑡� � �𝑧𝑧
𝑑𝑑𝑑𝑑 , 𝑑𝑑𝑑𝑑
𝑣𝑣�𝑅𝑅, 𝑧𝑧, 𝑡𝑡� � �,
𝑤𝑤�𝑅𝑅, 𝑧𝑧, 𝑡𝑡� � 𝑤𝑤�𝑅𝑅, 𝑡𝑡�.
To analyse the effect of thermal environment on the vibration characteristics of the plate, the total displacement components of point �𝑅𝑅, 𝜃𝜃, 𝑧𝑧� becomes 𝑢𝑢�𝑅𝑅, 𝑧𝑧, 𝑡𝑡� � 𝑢𝑢 � �𝑅𝑅, 𝑧𝑧� , 𝑣𝑣�𝑅𝑅, 𝑧𝑧, 𝑡𝑡� � 𝑣𝑣� �𝑅𝑅, 𝑧𝑧� and 𝑤𝑤�𝑅𝑅, 𝑧𝑧, 𝑡𝑡� � 𝑤𝑤� �𝑅𝑅, 𝑧𝑧� , respectively. Using the strain-displacement relations (Malekzadeh et al., 2011) 𝑒𝑒�� �
𝑑𝑑𝑑𝑑 1 𝑑𝑑𝑑𝑑 � 𝑑𝑑𝑑𝑑 � � � � �� � , 𝑑𝑑𝑑𝑑 2 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
𝑒𝑒�� �
𝑢𝑢 𝑢𝑢� � �, 𝑅𝑅 2𝑅𝑅
𝑒𝑒�� � 𝑒𝑒�� � 𝑒𝑒�� � � ,
the stress-displacement relations for small amplitude vibrations around the thermal equilibrium position are given by
Rahul Saini Procedia Structural Integrity 14 (2019) 362–374 R. Saini, S. Saini, R. Lal, andetI.al. V. /Singh / Structural Integrity Procedia 00 (2018) 000–000
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𝜎𝜎�� � 𝐸𝐸�𝑧𝑧, 𝑇𝑇� �
𝑑𝑑𝑑𝑑 𝑢𝑢 𝑑𝑑𝑑𝑑 𝑢𝑢 � 𝜈𝜈 � , 𝜎𝜎�� � 𝐸𝐸�𝑧𝑧, 𝑇𝑇� �𝜈𝜈 � �, 𝑑𝑑𝑑𝑑 𝑅𝑅 𝑑𝑑𝑑𝑑 𝑅𝑅
5
𝜎𝜎�� � 𝜎𝜎�� � 𝜎𝜎�� � 0 .
Here, the consideration of non-linear terms in strain-displacement relations doesn’t mean that non-linear vibrations are being considered. The equation of motion has been obtained using Hamilton’s principle, given by �𝐷𝐷� � 𝑆𝑆��� �
𝑑𝑑 � 𝑤𝑤 2 𝑑𝑑 � 𝑤𝑤 𝑑𝑑𝑆𝑆��� 1 𝑑𝑑 � 𝑤𝑤 � �𝐷𝐷� � 𝑆𝑆��� � � �𝐷𝐷� � 𝑆𝑆��� � 𝑑𝑑 � 𝑅𝑅� �𝑁𝑁��� � 𝑁𝑁� �� � � � � 𝑑𝑑𝑅𝑅 𝑅𝑅 𝑑𝑑𝑅𝑅 𝑑𝑑𝑑𝑑 𝑅𝑅 𝑑𝑑𝑅𝑅� �𝐷𝐷� � 𝑆𝑆��� � 𝑑𝑑
together with the conditions at the periphery: �𝑖𝑖� Clamped �C� ∶ �𝑤𝑤���� � 0,
�
𝑑𝑑𝑑𝑑 � �0, 𝑑𝑑𝑑𝑑 ���
𝑑𝑑𝑆𝑆��� 1 𝑑𝑑𝑑𝑑 𝑑𝑑 � 𝑤𝑤 � 𝑅𝑅� �𝑁𝑁��� � 𝑁𝑁� �� � � 𝐴𝐴� ℎ � � 0, �10� 𝑑𝑑𝑑𝑑 𝑅𝑅 𝑑𝑑𝑑𝑑 𝑑𝑑𝑡𝑡
�𝑖𝑖𝑖𝑖� Simply supported �S� ∶ �𝑤𝑤���� � 0 , ��𝐷𝐷� � 𝑆𝑆��� � where,
� 𝑆𝑆��� , 𝑆𝑆��� � � �
���
𝑧𝑧 � �𝜎𝜎��� , 𝜎𝜎��� �𝑑𝑑𝑧𝑧 ,
����
𝑑𝑑 � 𝑤𝑤 𝐷𝐷� 𝑑𝑑𝑑𝑑 � � � 0, 𝑑𝑑𝑅𝑅� 𝑅𝑅 𝑑𝑑𝑑𝑑 ���
�𝑁𝑁��� , 𝑁𝑁��� � � �
���
����
�𝜎𝜎��� , 𝜎𝜎��� �𝑑𝑑𝑧𝑧 .
Assuming 𝑤𝑤�𝑎𝑎 � 𝑊𝑊�𝑟𝑟�𝑒𝑒 ��� (for harmonic solution), the non-dimensional form of equation (10) and periphery conditions lead to, �
� 𝑆𝑆� ���
𝑑𝑑 � 𝑊𝑊 � 0, 𝑑𝑑𝑟𝑟 �
�𝑊𝑊���� � � where, 𝑆𝑆� �
𝑑𝑑𝑑𝑑 𝑑𝑑 � 𝑊𝑊 𝐷𝐷�∗ 𝑑𝑑𝑑𝑑 � � 0 ; �𝑊𝑊���� � �𝑆𝑆� � � �0, 𝑑𝑑𝑑𝑑 ��� 𝑑𝑑𝑟𝑟 � 𝑟𝑟 𝑑𝑑𝑑𝑑 ���
𝑆𝑆� � �𝑆𝑆� Ω� ,
𝑆𝑆� �
1 1 𝑑𝑑𝑆𝑆��� � ∗ �𝑆𝑆��� � 𝑟𝑟 � 𝑎𝑎� 𝑟𝑟 � 𝑁𝑁��� � � 𝑄𝑄� � 𝑟𝑟 � 𝑁𝑁 ∗ � , � 𝑟𝑟 𝐷𝐷 𝑑𝑑𝑑𝑑
1 1 𝑑𝑑𝑆𝑆��� � �𝑟𝑟 � 𝑆𝑆��� � 𝑎𝑎� 𝑟𝑟 � 𝑁𝑁��� � � 𝑄𝑄� � 𝑟𝑟 � 𝑁𝑁 ∗ �, 𝑟𝑟 � 𝐷𝐷 ∗ 𝑑𝑑𝑑𝑑
𝑆𝑆� � 𝑄𝑄� �
�11� �12��1��
2 𝑆𝑆��� 𝑆𝑆� � �𝑄𝑄� � ∗ �, 𝑟𝑟 𝐷𝐷
𝐴𝐴� 1 ��� 𝑆𝑆��� 𝜌𝜌� 𝜔𝜔� ℎ � , 𝑆𝑆 � , Ω � , 𝐴𝐴 � � 𝜌𝜌�𝑧𝑧, 𝑇𝑇�𝑑𝑑𝑑𝑑, � � 𝐷𝐷 ∗ 𝐷𝐷 ∗ 𝜌𝜌� ℎ ����
value 𝜌𝜌0 is with respect to the ceramic constituent and ω is the radian frequency.
Equations (7) and (11) are the fourth-order differential equations with variable coefficients whose exact solutions are not possible. Then approximate solutions with appropriate boundary conditions have been obtained by using GDQR.
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3. Method of solution Let 𝑟𝑟� , 𝑟𝑟� , … , 𝑟𝑟� be the 𝑛𝑛 grid points in range �0, 1� of the plate. According to GDQR (Wu et al. 2002), the 𝑚𝑚�� order derivative of transverse displacement 𝑊𝑊�𝑟𝑟� with respect to 𝑟𝑟 at the 𝑖𝑖 �� point 𝑟𝑟� is given by �
���
���
���
𝑑𝑑 � 𝑊𝑊�𝑟𝑟� � � � �𝑟𝑟 �𝑊𝑊 � � � �𝑟𝑟� �𝑊𝑊� � ℎ�� � � � ℎ�� � � � � ℎ�� �𝑟𝑟� �𝑊𝑊� � � 𝐸𝐸�� 𝑈𝑈� , �𝑖𝑖 � 1,2, … , 𝑛𝑛�, 𝑑𝑑𝑑𝑑 � ��� �
�14�
where, 𝐸𝐸��� are the weighting coefficients of 𝑚𝑚�� order derivative at point 𝑟𝑟� and ℎ�� �𝑟𝑟� are the Hermite interpolation functions. Here, for convenience the unknowns 𝑊𝑊’s have been replaced by 𝑈𝑈’s in equation (14) as follows: � 𝑊𝑊� , 𝑊𝑊�� , 𝑊𝑊� , 𝑊𝑊� , … , 𝑊𝑊��� , 𝑊𝑊��� , 𝑊𝑊� , 𝑊𝑊�� � � � 𝑈𝑈� , 𝑈𝑈� , … , 𝑈𝑈��� , 𝑈𝑈��� �.
The co-ordinates of 𝑛𝑛 grid points have been generated as per the Chebyshev Gauss-Lobatto grid distribution given by 1 𝑖𝑖 � 1 ��� , 𝑟𝑟� � �1 � ��� � 2 𝑛𝑛 � 1
𝑖𝑖 � 1, 2, … , 𝑛𝑛�
3.1. Thermal analysis
Discretizing equation (7) at the internal grid points 𝑟𝑟� , 𝑖𝑖 � 2, �, … , �𝑛𝑛 � 1�, using equation (14) by putting 𝑊𝑊� for 𝑊𝑊, one gets ���
���
���
���
���
����𝑄𝑄� ��� 𝐸𝐸�� � �𝑄𝑄� ��� 𝐸𝐸�� � �𝑄𝑄� ��� 𝐸𝐸�� � �𝑄𝑄� ��� 𝐸𝐸�� �𝑈𝑈�� � � �𝑄𝑄� ��� 𝑈𝑈�� � 0, ���
�15�
where the superscript 𝑇𝑇 over 𝑈𝑈� i.e. 𝑈𝑈�� represents the unknowns 𝑈𝑈� , 𝑈𝑈� , … , 𝑈𝑈��� , 𝑈𝑈��� for thermal displacement as � � 𝑈𝑈�� , 𝑈𝑈�� , … , 𝑈𝑈��� , 𝑈𝑈��� . The satisfaction of equation (15) at �𝑛𝑛 � 2� internal grid points together with the regularity conditions �𝑑𝑑𝑊𝑊� /𝑑𝑑𝑑𝑑���� � �𝑑𝑑 � 𝑊𝑊� /𝑑𝑑𝑟𝑟 � ���� � 0, provides a set of 𝑛𝑛 equations in 𝑛𝑛 � 2 unknowns 𝑈𝑈�� , � � 1, 2, �, … , �𝑛𝑛 � 2�. The resulting system of equations can be written in matrix form as, �𝑀𝑀� �𝑋𝑋� � 0,
� �� where 𝑀𝑀 is a matrix of order 𝑛𝑛 � �𝑛𝑛 � 2� and 𝑋𝑋 is a column vector given by �𝑈𝑈�� , 𝑈𝑈�� , … , 𝑈𝑈��� .
�16�
By satisfying the periphery conditions for a C-plate (8), a set of two homogenous equations is obtained. These equations together with the field equation (16) give a complete set of �𝑛𝑛 � 2� equations in �𝑛𝑛 � 2� unknowns, which can be written as 𝑀𝑀 �𝑋𝑋� �0� � � , 𝑀𝑀�
�
where 𝑀𝑀� is the matrix of order 2 � �𝑛𝑛 � 2�.
�17�
Similarly, for S-plate (9), a set of two non-homogenous equations have been obtained which with the field equation (16) give a complete set of �𝑛𝑛 � 2� equations in �𝑛𝑛 � 2� unknowns and can be written as
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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𝑀𝑀 �𝑋𝑋� 0 � � � � �, 𝑀𝑀 � 𝐹𝐹
7
�18�
�
respectively. � � , 𝑈𝑈��� using the computer software The equations �17� and �18� have been solved for unknowns 𝑈𝑈�� , 𝑈𝑈�� , … , 𝑈𝑈��� MATLAB. The thermal stresses raised in the plate due to thermal environment have been obtained as follows: Step 1. The equation ��� have been discretized by using equation (14), ���
���
𝜎𝜎��� � �𝐸𝐸� �𝑧𝑧, 𝑇𝑇� �𝑧𝑧 �� �𝐸𝐸�� � ���
���
���
𝜈𝜈 ��� � ⎫ 𝐸𝐸�� � 𝑈𝑈� � � 𝑇𝑇� � ,⎪ 𝑟𝑟� ⎪
𝜎𝜎��� � �𝐸𝐸� �𝑧𝑧, 𝑇𝑇� �𝑧𝑧 �� �𝜈𝜈𝜈𝜈�� � ���
𝜎𝜎��� � 𝜎𝜎��� � 𝜎𝜎��� � 0,
1 ��� � ⎬ 𝐸𝐸 � 𝑈𝑈� � � 𝑇𝑇� � , ⎪ 𝑟𝑟� �� ⎪ ⎭
�19�
� � Step 2. Substitution of the numerical values for 𝑈𝑈�� , 𝑈𝑈�� , … , 𝑈𝑈��� , 𝑈𝑈��� (obtained from solving equations �17,18�) in �19� provide the thermal stresses for C and S-plates, respectively.
3.2. Vibration analysis
For vibration analysis, equation (11) has been discretized by using equation (14) for various order derivatives of 𝑊𝑊 and at 𝑖𝑖 �� grid point 𝑟𝑟� , it leads to ���
���
���
���
���
����𝑆𝑆� ��� 𝐸𝐸�� � �𝑆𝑆� ��� 𝐸𝐸�� � �𝑆𝑆� ��� 𝐸𝐸�� � �𝑆𝑆� ��� 𝐸𝐸�� �𝑈𝑈� � � �𝑆𝑆� ��� 𝑈𝑈� � 0. ���
�20�
Now, the satisfaction of equation (20) at �𝑛𝑛 � 2� internal grid points together with the regularity conditions provides a set of 𝑛𝑛 equations in terms of �𝑛𝑛 � 2� unknowns 𝑈𝑈� , � � 1, 2, 3, … , �𝑛𝑛 � 2�. These resulting 𝑛𝑛 equations together with the two more equations obtained by satisfying the periphery conditions, give a complete set of �𝑛𝑛 � 2� equations in �𝑛𝑛 � 2� unknowns. For C-plate, the set of these �𝑛𝑛 � 2� homogenous equations can be written as 𝑉𝑉 �𝑌𝑌� �0�, � � 𝑉𝑉 �
�
�21�
𝑉𝑉 � � � � 0. 𝑉𝑉
�22�
𝑉𝑉 � � � � 0, 𝑉𝑉
�23�
where 𝑉𝑉, 𝑉𝑉 � are the matrices of order 𝑛𝑛 � �𝑛𝑛 � 2�, 2 � �𝑛𝑛 � 2�, respectively and 𝑌𝑌 � �𝑈𝑈� , 𝑈𝑈� , … , 𝑈𝑈��� �� . For a non-trivial solution of equation (21), the frequency determinant must vanish and hence
Similarly, for S-plate, the frequency determinant can be written as
respectively.
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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Table 1. Comparison of Ω for first three modes, (material properties (Lal and Ahlawat, 2015)). 𝑁𝑁 ∗ ↓
‐10
𝐶𝐶
𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝
𝐼𝐼
0
5
𝐼𝐼𝐼𝐼
5.8274
�
35.3161
0
0
10.2158
39.7711
5
20
0
5
10.216�
7.5738
7.5738
7.576 �
39.7711�
39.773�
89.1041�
-
4.9351�
29.4854
29.496
66.0598
4.935�
‐
3.6588
15.4897
Periphery ↓ \ 𝑝𝑝 𝑝
35.8936
1
2
3
4
7.4955
6.8967
6.4689
4.1978
�
8.6933
14.6820
S
70.6780
�
38.6369�
0
4.1978�
Table 3. Values of frequency parameter 𝛀𝛀𝛀
8.6933
2.4855
2.4855�
7.4955 �
2.1431
2.1431�
54.9777
35.8936
12.8819
�
‐
12.8819
6.8967 �
1.9719
1.9719�
�
22.052 �
14.6820 �
74.1561
74.1561�
83.6475
80.3700
∗ Table 2. Comparison of critical buckling load N�� in compression.
22.0338
48.1241
48.1241�
38.6369
41.7750
�
29.736�
68.9221�
11.8425
11.8425�
80.3700
29.7200
29.72�
68.9221
‐
3.659 �
97.6309
9.1895�
𝐼𝐼𝐼𝐼𝐼𝐼
22.0338
�
97.6309�
�
15.2490
3.6588
�
47.4235�
41.7750
�
4.9351
66.0598
�
47.4235
15.2490
C
89.1041
�
9.1895
*
�
15.4897�
57.5734
29.4854
�
*
24.0538
24.0538
57.5734�
20.7194�
10.2158�
𝐼𝐼𝐼𝐼
*
�
84.5161
20.7194
*
𝐼𝐼
*
84.5161
�
5.8274 *
𝐼𝐼𝐼𝐼𝐼𝐼
35.3161
𝑆𝑆
54.9777� 83.6475� 70.6780�
5
6.4689 �
1.8496
1.8496�
6.1267
6.1267�
1.7517
1.7517�
0
1
2
3
4
5
C
𝑁𝑁 ∗ ↓ \ 𝑝𝑝 𝑝
‐20
*
*
*
*
*
*
‐10
0.0551
*
*
*
*
*
0
8.4466
7.9447
7.7508
7.6372
7.5582
7.4988
10
11.8288
11.1873
10.9647
10.8436
10.7639
10.7064
20
14.3824
13.6266
13.3752
13.2428
13.1580
13.0980
Periphery
30
16.5133
15.6593
15.3816
15.2380
15.1475
15.0842
S
‐20
*
*
*
*
*
*
‐10
*
*
*
*
*
*
0
3.0960
3.0333
2.9024
2.7983
2.7149
2.6470
10
8.2215
7.8559
7.7029
7.6143
7.5547
7.5111
20
11.2072
10.6866
10.4985
10.3972
10.3321
10.2863
30
13.5478
12.9093
12.6910
12.5770
12.5056
12.4560
1
Values from reference Lal and Ahlawat, 2015,
Values from reference Pradhan and Chakraverty, 2015, * Values does not exist due to buckling, 2
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4. Numerical results The lowest three roots of frequency equations ���� ��� have been computed using MATLAB and retained as the ∗ has been calculated using Bisection method. values of the frequency parameter Ω. The critical buckling load 𝑁𝑁�� The numerical values for TI material properties are taken from reference (Malekzadeh et al., 2011). From the literature, the values of various parameters are taken as follows: material graded index 𝑝𝑝 � �� 1� �� �� �� �� in-plane force parameter 𝑁𝑁 ∗ � ���� �1�� �� 1�� ��� ��, temperatures 𝑇𝑇� � ����, 𝑇𝑇� � �����, ����� � 𝑇𝑇� � �����, thickness � � ��1 and Poisson’s ratio � � ���. 4.1. Convergence and comparison study
During the computations, the number of grid points 𝑛𝑛 has been fixed as 16 since there was no difference up to the fourth place of decimal between two successive values of frequency parameter Ω for varying values of 𝑝𝑝, 𝑁𝑁 ∗ and 𝑇𝑇� keeping 𝑇𝑇� � �����. In this regard, the computer program developed to evaluate Ω was run for these values taking 𝑛𝑛 � 6����� � �1� and observe that �Ω���� � Ω�� � � ������� for all the three modes � � 1� �� �. In graphical form, the percentage error �1 � Ω� ⁄Ω�� � � �1�� for a specified plate 𝑝𝑝 � � , 𝑁𝑁 ∗ � ��� 𝑇𝑇� � 𝑇𝑇� � ����� and 𝑇𝑇� � ����� has been shown in Fig. 2. The deviations obtained for this set of values of the parameters for all the three modes of vibration were maximum. Further, it is worth to mention that for evaluation of the frequencies one has to consider a finite number of terms (𝑁𝑁) in the binomial expansion (5). Being an infinite series, its accuracy will depend upon the value of 𝑁𝑁. In the present analysis, 𝑁𝑁 has been taken as 20, as further increase in this value does not improve the result, except at the fifth place of decimal. A comparison of results in the absence of thermal environment �� � �� 𝑇𝑇 � �� for different values of in-plane force �𝑁𝑁 ∗ � �1�� �� ��) and graded index (𝑝𝑝 � �� �) with those obtained by DTM (Lal and Ahlawat, 2015) and Rayleigh-Ritz method (Pradhan and Chakraverty, 2015) has been presented in Table 1. Further, by allowing the ∗ in compression, for varying values of 𝑝𝑝 has been compared with frequency to approaches zero, the values of 𝑁𝑁�� DTM (Lal and Ahlawat, 2015) and reported in Table 2. A close agreement has been observed which describes the versatility of the present technique.
Fig. 2. Percentage error in Ω: ‘○’ – I mode; ‘△’ – II mode; ‘□’ – III mode (a) C-Plate (b) S-Plate.
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4.2. Parametric discussion To analyze the effect of various parameters on frequency parameter Ω, the results have been presented in Figs. 3-6, however for selected data the numerical values are also given in Table 3. From the results, it has been noticed that the values of Ω for S-plate are smaller than that for the corresponding C-plate. The value of Ω is found to decrease with the increasing value of 𝑝𝑝 i.e. as the nature of plate changes from ceramic to metallic for both the boundary conditions for the same set of the values of other parameters. The effect of temperature difference ∆𝑇𝑇 on Ω for an FGM plate with 𝑝𝑝 � � for two type of plate materials: (i) when the material properties are taken as TI; (ii) when the material properties are TD and varying nonlinearly in thickness direction for 𝑁𝑁 ∗ � ���� ��� has been shown in Fig. 3. Ω is found to decrease with the increasing values of ∆𝑇𝑇 for both the C and S-plates, whatever be the values of other parameters. It can be seen that the values of Ω for TD material are less than those for TI material for the same set of the values of other parameters. This effect is more pronounced for C-plate as compared to S-plate and increases with the increase in the number of modes. It has been noticed that the rate of decay in the value of Ω increases as the nature of in-plane force changes from tensile to compressive for both the plates. This rate of decay decreases with the increase in the number of modes. As a special case, a study for uniform temperature distribution (UTD): 𝑇𝑇�𝑧𝑧� � � � 𝑇𝑇� � ∆𝑇𝑇 as well as linear temperature distribution (LTD): 𝑇𝑇�𝑧𝑧� � 𝑇𝑇� � ∆𝑇𝑇��𝑧𝑧 � ������ ∗ in equation (5) have been made. These numerical results together with non-linear temperature distribution (NTD) have been plotted in Fig. 4 for varying value of ∆𝑇𝑇, for 𝑁𝑁 ∗ � ���� �� and fixed value of 𝑝𝑝 � �. It has been noticed that Ω decreases with the increase in the values of ∆𝑇𝑇 for all the three types of temperature distributions in the order UTD > LTD > NTD for both C and S-plates. The rate of decay for UTD is much higher as compared to LTD as well as NTD for the same set of the values of other parameters. This decay is higher for compressive force as compared to the tensile for both the plates and more pronounced in case of S-plate as compared to C-plate, keeping other parameters fixed. Fig. 5 presents the behaviour of Ω for varying values of 𝑁𝑁 ∗ for two values of 𝑝𝑝 � �� ��and ∆𝑇𝑇 � ��� for C and S-plates. It can be seen that the value of Ω increases with the increase in the value of 𝑁𝑁 ∗ for both the values of 𝑝𝑝, keeping other parameters fixed. The rate of increase for 𝑝𝑝 � � is higher than that for 𝑝𝑝 � �.
Fig. 3. Ω vs ∆T, 𝑁𝑁 ∗ � ���: ‘○’- TI, ‘△’- TD;
𝑁𝑁 ∗ � ���: ‘●’- TI, ‘△’- TD;
C - plate - ‘────’, S - plate - ‘- - - - - -’.
∗ The numerical results for critical buckling load 𝑁𝑁�� have been plotted in Fig. 6a for 𝑝𝑝 � �� � and ∆𝑇𝑇 � ��� ∗ when the plate is vibrating in first mode of vibration. It has been noticed that the values of 𝑁𝑁�� for C-plate are higher ∗ than that for the corresponding S-plate. The values of 𝑁𝑁�� decrease with the increase in the value of 𝑝𝑝 from 0 to 5 i.e. as the nature of plate changes from fully ceramic (isotropic) to metallic. Two-dimensional mode shapes for a specified plate with 𝑝𝑝 � �� 𝑁𝑁 ∗ � �������∆𝑇𝑇 � ��� have been presented in
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Fig. 6b. It is found that the nodal circles for C-plate are smaller than that for S-plate.
Fig. 4. Ω vs ∆T: ‘●’- 𝑁𝑁 ∗ � ��, ‘○’ - 𝑁𝑁 ∗ � ���; C - plate - ‘────’, S - plate - ‘- - - - - -’; UTD - ‘ ’, LTD - ‘△’, NTD -‘○’.
Fig. 5. Ω vs 𝑁𝑁 ∗ : ‘○’ - � � �, ‘△’-�� � �; C - plate - ‘────’, S - plate - ‘- - - - - -’; I Mode - ‘────’, II Mode - ‘────’, III Mode - ‘────’.
5. Conclusions
The combined effect of in-plane force on the axisymmetric vibration of FGM circular plate vibrating in thermal environment has been presented using GDQR. Being TD material properties, the temperature distribution along the thickness of the plate is assumed to be non-linear. Numerical results show that:
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∗ Fig. 6. C - plate - ‘────’, S - plate - ‘- - - - - -’; (a) Ω vs 𝑁𝑁�� : ‘○’- 𝑝𝑝 � 𝑇, ‘△’- 𝑝𝑝 � �, ‘□’- 𝑝𝑝 � �, ‘⁎’-𝑇𝑝𝑝 � �; (b) First three mode shapes.
Ω� � Ω� , for same set of the values of parameters. For both C and S-plates, the frequency parameter Ω for fully ceramic plate (Ω��� � is always higher than that for the corresponding FGM plate (Ω��� ). The value of Ω decreases with the increase in the value of temperature difference and it is found that Ω∆��� � Ω∆��� , whatever be the values of other parameters. The percentage decrease in the value of Ω for 𝑝𝑝 � �� 𝑁𝑁 ∗ � �𝑇𝑇with the change in temperature difference ∆𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 to 400 are 6.58, 7.86 for C, S-plates, respectively, with TD material properties, when the plate is vibrating in the first mode. These variations are differing only at the first place of decimal for fully ceramic plate. In view of the TI material properties, the percentage variation in Ω with varying values of ∆𝑇𝑇 from 0 to 400, for 𝑝𝑝 � �𝑇� 𝑁𝑁 ∗ � �𝑇 are -3.81 and -5.69 for C and S-plates, respectively. However, for 𝑁𝑁 ∗ � ��𝑇, values do not exist due to buckling. The behaviour of Ω with the change in the nature of temperature distribution from NTD to LTD and then to UTD is found in the order of Ω��� � Ω��� � Ω��� . The percentage variation in the values of Ω with varying values of ∆𝑇𝑇 from 0 to 400 for 𝑝𝑝 � ��𝑇for UTD, LTD and NTD are (-15.39, -7.50, -6.58) and (-18.35, -9.06, -7.86) for 𝑁𝑁 ∗ � �𝑇 while values do not exist due to buckling for 𝑁𝑁 ∗ � ��𝑇 for both the plates vibrating in the first mode of vibration. ∗ decreases as the nature of the plate changes from ceramic to metallic i.e. 𝑝𝑝 The value of critical buckling load 𝑁𝑁�� changes from 0 to 5 for both the plates. The above analysis will be of great practical use to a design engineer dealing with FGM plates in obtaining the desired frequency by controlling one or more parameters involved in the present study.
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Acknowledgments The financial support provided by MHRD, India: Grant No. MHRD-02-23-200-44 in carrying this research work is gratefully acknowledged by Rahul Saini. References Behravan Rad, A., 2015. Thermo-elastic analysis of functionally graded circular plates resting on a gradient hybrid foundation. Appl. Math. Comput. 256, 276–298. doi:10.1016/j.amc.2015.01.026 Fazzolari, F.A., 2016. Modal characteristics of P- and S-FGM plates with temperature-dependent materials in thermal environment. J. Therm. Stress. 39, 854–873. doi:10.1080/01495739.2016.1189772 Khorshidvand, A.R., Jabbari, M., Eslami, M.R., 2012. Thermoelastic buckling analysis of functionally graded circular plates integrated with piezoelectric layers. J. Therm. Stress. 35, 695–717. doi:10.1080/01495739.2012.688666 Kiani, Y., Eslami, M.R., 2014. Thermal postbuckling of imperfect circular functionally graded material plates: examination of Voigt, Mori– Tanaka, and Self-Consistent schemes. J. Press. Vessel Technol. 137, 21201. doi:10.1115/1.4026993 Koizumi, M., 1997. FGM activities in Japan. Compos. Part B Eng. 28, 1–4. doi:10.1016/S1359-8368(96)00016-9 Koizumi, M., 1993. The concept of FGM. Ceram. Trans. Func. Grad. Mater. 34, 3–10. Lal, R., Ahlawat, N., 2015. Axisymmetric vibrations and buckling analysis of functionally graded circular plates via differential transform method. Eur. J. Mech. A/Solids 52, 85–94. doi:10.1016/j.euromechsol.2015.02.004 Malekzadeh, P., Haghighi, M.R.G., Atashi, M.M., 2011. Free vibration analysis of elastically supported functionally raded annular plates subjected to thermal environment. Meccanica 46, 893–913. doi:10.1007/s11012-010-9345-5 Pradhan, K.K., Chakraverty, S., 2015. Static analysis of functionally graded thin rectangular plates with various boundary supports. Arch. Civ. Mech. Eng. 15, 721–734. doi:10.1016/j.acme.2014.09.008 Pradhan, K.K., Chakraverty, S., 2015. Free vibration of functionally graded thin elliptic plates with various edge supports. Struct. Eng. Mech. 53, 337–354. Prakash, T., Ganapathi, M., 2006. Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method. Compos. Part B Eng. 37, 642–649. doi:10.1016/j.compositesb.2006.03.005 Reddy J N, 2008. Theory and Analysis of Elastic Plates and Shells, CRC Press, London, New York. doi:10.1002/zamm.200890020 Suresh, S., Mortensen, A., 1998. Fundamentals of Functionally Graded Materials. London, U.K. Swaminathan, K., Sangeetha, D.M., 2017. Thermal analysis of FGM plates – A critical review of various modeling techniques and solution methods. Compos. Struct. 160, 43–60. doi:10.1016/j.compstruct.2016.10.047 Wu TY, Wang YY, Liu GR, 2002. Free vibration analysis of circular plates using generalized differential quadrature rule. Comput. Methods Appl. Mech. Engrg. 191, 5365–5380.
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Buckling and vibrations of FGM circular plates in thermal Buckling and vibrations of FGM circular plates in thermal environment XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal environment � Rahul Saini�,� , Shivam Saini�� , Roshan Lal , Indra Vir Singh�� blade of an Thermo-mechanical a high pressure turbine �,� modeling of � Rahul Saini , Shivam Saini , Roshan Lal , Indra Vir Singh 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 airplane gas turbine engine 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖, 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 � �
� �
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖, 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼
P. Brandãoa, V. Infanteb, A.M. Deusc*
AbstractaDepartment of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Abstract Portugal b IDMEC, is Department Mechanical Engineering, Superioron Técnico, Universidade devibrations Lisboa, Av.ofRovisco Pais, 1, 1049-001 Lisboa, An analysis presentedoffor the study of thermalInstituto environment free axisymmetric functionally graded circular Portugal An analysis is presented for the study of thermal environment on free axisymmetric vibrations of functionally graded circular plates subjected to uniform in-plane peripheral loading and non-linear temperature distribution along the thickness direction. It is c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, plates subjected uniform in-plane peripheral loadingdirection and non-linear temperature distribution along the thickness direction. It is assumed that thetoplate material is graded in thickness and mechanical properties are temperature-dependent. Hamilton’s Portugal assumed the plate is graded in thickness directionfor and mechanical properties areand temperature-dependent. Hamilton’s principle that has been usedmaterial in obtaining the governing equations thermo-elastic equilibrium vibration for such a plate model principle hasof been used inplate obtaining governing differential equations for thermo-elastic equilibrium a plate values model on the basis classical theory.the Generalized quadrature rule has been usedand in vibration evaluatingfor thesuch numerical on the basis displacements of classical plate Generalized rule has beenplates used in the for numerical forAbstract thermal andtheory. frequencies in casedifferential of clampedquadrature and simply supported at evaluating the periphery the firstvalues three for thermal displacements and frequencies in caseforofwhich clamped the reported peripheryasfor the first three modes of vibration. Compressive in-plane loads the and platesimply ceasessupported to vibrateplates have at been critical buckling modes of vibration. Compressive loads forcomponents which thevibration plate ceases to to vibrate have reported criticaland buckling loads. Effects various parameters have been analyzed on the characteristics for all been thedemanding modes. Foras uniform linear During theirofoperation, modernin-plane aircraft engine are subjected increasingly operating conditions, loads. Effectsdistribution, of high various parameters been analyzed onconditions the vibration the different modes. uniform andhaving linear temperature the benchmark results haveSuch been computed. Ascharacteristics athese special case, a all study with theFor plate material especially the pressure turbinehave (HPT) blades. cause parts tofor undergo types of time-dependent temperature distribution, the isbenchmark results havethe been computed. As acompared special case, study withinthe platetomaterial having degradation, one of which creep. A been model using finite element method (FEM) wasathe developed, order be able to predict temperature-independent properties has performed. Results have been with published work. the creep behaviour ofproperties HPT blades. Flight data records (FDR) for a compared specific aircraft, a commercial aviation temperature-independent has been performed. Results have been with the provided published by work. were used to obtainbythermal mechanical data for three different flight cycles. In order to create the 3D model © company, 2018 The Authors. Published Elsevierand B.V. © 2019 The Authors. Published by Elsevier B.V. B.V. scrap was scanned, and its chemical composition and material properties were needed for the FEM analysis, HPT © 2018 The Authors. Published byaElsevier This is an open access article under the CCblade BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access undergathered the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) obtained. The dataarticle that was wasBY-NC-ND fedofinto thelicense FEM model and different of simulations were organizers. run, first with a simplified 3D This is an and open access article under the CC (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection peer-review under responsibility Peer-review under responsibility the SICE 2018 Selection and peer-review under responsibility of establish Peer-review under responsibility of thethe SICE 2018 organizers. rectangular block shape, in order to better the model, and responsibility then with realthe 3DSICE mesh2018 obtained from the blade scrap. The Selection and peer-review under responsibility of Peer-review under of organizers. overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore Keywords: Circular plates; functionay graded; in-plane force; temperature-dependent material; non-linear temperature; generalized differentialsuch a model can be useful the goal of predicting turbine blade life, given a setmaterial; of FDRnon-linear data. quadrature rule. Keywords: Circular plates;infunctionay graded; in-plane force; temperature-dependent temperature; generalized differential quadrature rule.
© 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation.
* Corresponding author. Tel.: +91-812-651-5226 . address:author. [email protected] *E-mail Corresponding Tel.: +91-812-651-5226 . E-mail address: [email protected] 2452-3216 © 2018 The Authors. Published by Elsevier B.V.
This is an © open article under theby CCElsevier BY-NC-ND 2452-3216 2018access The Authors. Published B.V. license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection under responsibility of Peer-review responsibility of the SICE 2018 organizers. This is an and openpeer-review access article under the CC BY-NC-ND licenseunder (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under * Corresponding author. Tel.: +351responsibility 218419991. of Peer-review under responsibility of the SICE 2018 organizers. E-mail address: [email protected]
2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016.
2452-3216 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of Peer-review under responsibility of the SICE 2018 organizers. 10.1016/j.prostr.2019.05.045
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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 �𝑅𝑅, 𝜃𝜃, 𝑧𝑧�, consider an FGM circular plate of radius 𝑎𝑎, thickness ℎ, density 𝜌𝜌, subjected to uniform, tensile in-plane force 𝑁𝑁� and 𝑧𝑧 � � being the middle surface of the plate. The line 𝑅𝑅 � � is the axis of the plate. The top surface 𝑧𝑧 � �ℎ/2 is taken as ceramic rich while the bottom 𝑧𝑧 � �ℎ/2 as metal rich. The plate is subjected to a temperature distribution 𝑇𝑇 varying along the thickness i.e. 𝑇𝑇�� 𝑇𝑇�𝑧𝑧�� such that 𝑇𝑇 � 𝑇𝑇� and 𝑇𝑇 � 𝑇𝑇� at the ceramic and metallic rich surfaces, respectively (Fig. 1). The typical effective material properties 𝑃𝑃�𝑧𝑧, 𝑇𝑇� of fabricated FGM plate are related with the material properties of metal 𝑃𝑃� �𝑇𝑇� and ceramic 𝑃𝑃� �𝑇𝑇� in the following manner (Malekzadeh et al., 2011), 𝑃𝑃�𝑧𝑧, 𝑇𝑇� � 𝑃𝑃� �𝑇𝑇� � �𝑃𝑃� �𝑇𝑇� � 𝑃𝑃� �𝑇𝑇�� 𝑉𝑉� �𝑧𝑧�,
where, 𝑉𝑉� describe the volume fraction of ceramic at any point 𝑧𝑧 and defined as, 𝑉𝑉� �𝑧𝑧� � �
���� � ��
� ,
�1�
�2�
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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3
Fig. 1. FGM circular plate in thermal environment (a) cross-section; (b) top view.
𝑝𝑝�� �� being a parameter and represents the shape of the volume fraction. Following (Swaminathan and Sangeetha, 2017), the material properties 𝑃𝑃� �𝑇𝑇� and 𝑃𝑃� �𝑇𝑇� are given by 𝑃𝑃� �𝑇𝑇� � 𝑃𝑃� �𝑃𝑃�� 𝑇𝑇 �� � � � 𝑃𝑃� 𝑇𝑇 � 𝑃𝑃� 𝑇𝑇 � � 𝑃𝑃� 𝑇𝑇 � �,
𝑏𝑏 � � , 𝑐𝑐 .
���
The constituent materials of the plate are taken as titanium (𝑇𝑇𝑖𝑖 � ��� � 4�) for metal and zirconia (𝑍𝑍𝑍𝑍𝑍𝑍� ) for ceramic. The experimental values of the coefficients 𝑃𝑃� �𝑖𝑖 � ��, � , � , 2 , �� are taken from reference (Malekzadeh et al., 2011). The symbol ‘𝑃𝑃’ has been used for the material properties such as Young’s modulus 𝐸𝐸, mass density 𝜌𝜌, expansion coefficient 𝛼𝛼 and thermal conductivity 𝑘𝑘. 2.1. Thermal Stress Analysis
Assuming that the plate material is unidirectionally non-homogenous in thickness direction and due to that any kind of temperature variation applied to such plates will produce variations only in thickness direction. Though, numerous studies with constant/linear variation of temperature across the thickness of the circular plate are available in the literature (Khorshidvand et al., 2012; Swaminathan and Sangeetha, 2017), but authors have not come across any study dealing with non-linear temperature variation. In the present work, the non-linear variation of temperature is assumed to arise from the solution of one-dimensional steady state static heat conduction equation without heat flux (Swaminathan and Sangeetha, 2017), 𝑑𝑑 𝑑𝑑𝑑𝑑 �𝑘𝑘�𝑧𝑧� � � �, 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
subject to the boundary conditions: 𝑇𝑇 � 𝑇𝑇� at 𝑧𝑧 � �ℎ/2 and 𝑇𝑇 � 𝑇𝑇� at 𝑧𝑧 � �ℎ/2 . Using equation (1) for the power law distribution of 𝑘𝑘�𝑧𝑧�, the solution of equation (4) gives: 𝑇𝑇�𝑧𝑧� � 𝑇𝑇� �
�
� 𝑘𝑘�� 2𝑧𝑧 � ℎ ���� ∆𝑇𝑇 � ����� � � , � �𝑖𝑖𝑖𝑖 � ��𝑘𝑘� 2ℎ 𝐶𝐶 ∗ ���
∗
�
where 𝑘𝑘�� � 𝑘𝑘� � 𝑘𝑘� , ∆𝑇𝑇 � 𝑇𝑇� � 𝑇𝑇� , 𝐶𝐶 � ������ ���
�4�
�5� � 𝑘𝑘�� � �𝑖𝑖𝑖𝑖 � ��𝑘𝑘�
and 𝑁𝑁 represents the number of terms in the binomial expansion. Due to axisymmetric temperature distribution, the displacement components 𝑢𝑢 � , 𝑣𝑣� , 𝑤𝑤� and thermal stresses at an arbitrary point �𝑅𝑅, 𝜃𝜃, 𝑧𝑧� on the mid plane of the plate are (Reddy, 2008, p. 153) 𝑢𝑢 � �𝑅𝑅, 𝑧𝑧� � �𝑑𝑑
𝑑𝑑𝑤𝑤� , 𝑑𝑑𝑑𝑑
𝑣𝑣� �𝑅𝑅, 𝑧𝑧� � �,
𝑤𝑤� �𝑅𝑅, 𝑧𝑧� � 𝑤𝑤� �𝑅𝑅�,
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𝑑𝑑 � 𝑤𝑤� 𝜈𝜈 𝑑𝑑𝑤𝑤� � � � 𝑇𝑇� � , 𝑑𝑑𝑅𝑅� 𝑅𝑅 𝑑𝑑𝑑𝑑 𝑑𝑑 � 𝑤𝑤� 1 𝑑𝑑𝑤𝑤� � �𝐸𝐸� �𝑧𝑧, 𝑇𝑇� �𝑧𝑧 �𝜈𝜈 � � � 𝑇𝑇� � , 𝑑𝑑𝑅𝑅� 𝑅𝑅 𝑑𝑑𝑑𝑑 � 𝜎𝜎��� � 𝜎𝜎��� � � ,
𝜎𝜎��� � �𝐸𝐸� �𝑧𝑧, 𝑇𝑇� �𝑧𝑧 �
𝜎𝜎���
𝜎𝜎���
𝐸𝐸�𝑧𝑧, 𝑇𝑇� �
𝐸𝐸�𝑧𝑧, 𝑇𝑇� , �1 � 𝜈𝜈 � �
⎫ ⎪
365 4
���
⎬ ⎪ ⎭
𝑇𝑇� � ��𝑧𝑧, 𝑇𝑇� ∆𝑇𝑇�𝑧𝑧� �1 � 𝜈𝜈� , ∆𝑇𝑇�𝑧𝑧� � 𝑇𝑇�𝑧𝑧� � 𝑇𝑇� ,
where, 𝑇𝑇� being the initial temperature and subscript 𝑇𝑇 is used for the deformations due to thermal environment. Introducing the non-dimensional variables 𝑊𝑊� � 𝑤𝑤� /𝑎𝑎, 𝑍𝑍 � 𝑧𝑧/𝑎𝑎 and 𝑟𝑟 � 𝑅𝑅/𝑎𝑎, the thermo-elastic equilibrium equation of the plate has been obtained using Hamilton’s principle, given by �
� 𝑄𝑄� ���
𝑑𝑑 � 𝑊𝑊� � �, 𝑑𝑑𝑟𝑟 �
���
together with the conditions at the periphery: �𝑖𝑖� Clamped �C� ∶
�𝑖𝑖𝑖𝑖� Simply supported �S� ∶
where , 𝑄𝑄� � 𝑁𝑁 ∗ �
𝑑𝑑𝑊𝑊� � ��, 𝑑𝑑𝑑𝑑 ��� 𝑑𝑑 � 𝑊𝑊� 𝐷𝐷�∗ 𝑑𝑑𝑊𝑊� �𝑊𝑊� ���� � �𝑄𝑄� � � 𝑚𝑚∗� � � �, 𝑑𝑑𝑟𝑟 � 𝑟𝑟 𝑑𝑑𝑑𝑑 ���
�𝑊𝑊� ���� � �
⎫ ⎪
⎬ ⎪ ⎭
���, ���
1 1 2 𝐷𝐷� 𝐷𝐷� �𝑄𝑄� � 𝑟𝑟 � 𝑁𝑁 ∗ �, 𝑄𝑄� � � � �𝑄𝑄� � 𝑟𝑟 � 𝑁𝑁 ∗ �, 𝑄𝑄� � 𝑄𝑄� , 𝑄𝑄� � 𝐷𝐷�∗ � ∗ , 𝐷𝐷�∗ � ∗ � 𝐷𝐷 𝐷𝐷 𝑟𝑟 𝑟𝑟 𝑟𝑟
𝑎𝑎𝑚𝑚 � 𝑎𝑎� 𝑁𝑁� , 𝑚𝑚∗� � ∗ , 𝐷𝐷 ∗ 𝐷𝐷
and �𝐷𝐷� , 𝐷𝐷� � � �
�/�
𝑚𝑚 � � �
�/�
�𝑧𝑧 . 𝑇𝑇� . 𝐸𝐸�𝑧𝑧, 𝑇𝑇��𝑑𝑑𝑧𝑧 , 𝐷𝐷 ∗ �
��/�
𝐸𝐸� ℎ� value 𝐸𝐸� is with respect to , � � 12�1 � 𝜈𝜈 � � the ceramic constituent
𝑧𝑧 � 𝐸𝐸�𝑧𝑧, 𝑇𝑇��1, 𝜈𝜈�𝑑𝑑𝑑𝑑 are the flexural rigidities of the plate.
��/�
2.2. Vibration analysis
For axisymmetric motion of the plate, the displacement components are given by 𝑢𝑢�𝑅𝑅, 𝑧𝑧, 𝑡𝑡� � �𝑧𝑧
𝑑𝑑𝑑𝑑 , 𝑑𝑑𝑑𝑑
𝑣𝑣�𝑅𝑅, 𝑧𝑧, 𝑡𝑡� � �,
𝑤𝑤�𝑅𝑅, 𝑧𝑧, 𝑡𝑡� � 𝑤𝑤�𝑅𝑅, 𝑡𝑡�.
To analyse the effect of thermal environment on the vibration characteristics of the plate, the total displacement components of point �𝑅𝑅, 𝜃𝜃, 𝑧𝑧� becomes 𝑢𝑢�𝑅𝑅, 𝑧𝑧, 𝑡𝑡� � 𝑢𝑢 � �𝑅𝑅, 𝑧𝑧� , 𝑣𝑣�𝑅𝑅, 𝑧𝑧, 𝑡𝑡� � 𝑣𝑣� �𝑅𝑅, 𝑧𝑧� and 𝑤𝑤�𝑅𝑅, 𝑧𝑧, 𝑡𝑡� � 𝑤𝑤� �𝑅𝑅, 𝑧𝑧� , respectively. Using the strain-displacement relations (Malekzadeh et al., 2011) 𝑒𝑒�� �
𝑑𝑑𝑑𝑑 1 𝑑𝑑𝑑𝑑 � 𝑑𝑑𝑑𝑑 � � � � �� � , 𝑑𝑑𝑑𝑑 2 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
𝑒𝑒�� �
𝑢𝑢 𝑢𝑢� � �, 𝑅𝑅 2𝑅𝑅
𝑒𝑒�� � 𝑒𝑒�� � 𝑒𝑒�� � � ,
the stress-displacement relations for small amplitude vibrations around the thermal equilibrium position are given by
Rahul Saini Procedia Structural Integrity 14 (2019) 362–374 R. Saini, S. Saini, R. Lal, andetI.al. V. /Singh / Structural Integrity Procedia 00 (2018) 000–000
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𝜎𝜎�� � 𝐸𝐸�𝑧𝑧, 𝑇𝑇� �
𝑑𝑑𝑑𝑑 𝑢𝑢 𝑑𝑑𝑑𝑑 𝑢𝑢 � 𝜈𝜈 � , 𝜎𝜎�� � 𝐸𝐸�𝑧𝑧, 𝑇𝑇� �𝜈𝜈 � �, 𝑑𝑑𝑑𝑑 𝑅𝑅 𝑑𝑑𝑑𝑑 𝑅𝑅
5
𝜎𝜎�� � 𝜎𝜎�� � 𝜎𝜎�� � 0 .
Here, the consideration of non-linear terms in strain-displacement relations doesn’t mean that non-linear vibrations are being considered. The equation of motion has been obtained using Hamilton’s principle, given by �𝐷𝐷� � 𝑆𝑆��� �
𝑑𝑑 � 𝑤𝑤 2 𝑑𝑑 � 𝑤𝑤 𝑑𝑑𝑆𝑆��� 1 𝑑𝑑 � 𝑤𝑤 � �𝐷𝐷� � 𝑆𝑆��� � � �𝐷𝐷� � 𝑆𝑆��� � 𝑑𝑑 � 𝑅𝑅� �𝑁𝑁��� � 𝑁𝑁� �� � � � � 𝑑𝑑𝑅𝑅 𝑅𝑅 𝑑𝑑𝑅𝑅 𝑑𝑑𝑑𝑑 𝑅𝑅 𝑑𝑑𝑅𝑅� �𝐷𝐷� � 𝑆𝑆��� � 𝑑𝑑
together with the conditions at the periphery: �𝑖𝑖� Clamped �C� ∶ �𝑤𝑤���� � 0,
�
𝑑𝑑𝑑𝑑 � �0, 𝑑𝑑𝑑𝑑 ���
𝑑𝑑𝑆𝑆��� 1 𝑑𝑑𝑑𝑑 𝑑𝑑 � 𝑤𝑤 � 𝑅𝑅� �𝑁𝑁��� � 𝑁𝑁� �� � � 𝐴𝐴� ℎ � � 0, �10� 𝑑𝑑𝑑𝑑 𝑅𝑅 𝑑𝑑𝑑𝑑 𝑑𝑑𝑡𝑡
�𝑖𝑖𝑖𝑖� Simply supported �S� ∶ �𝑤𝑤���� � 0 , ��𝐷𝐷� � 𝑆𝑆��� � where,
� 𝑆𝑆��� , 𝑆𝑆��� � � �
���
𝑧𝑧 � �𝜎𝜎��� , 𝜎𝜎��� �𝑑𝑑𝑧𝑧 ,
����
𝑑𝑑 � 𝑤𝑤 𝐷𝐷� 𝑑𝑑𝑑𝑑 � � � 0, 𝑑𝑑𝑅𝑅� 𝑅𝑅 𝑑𝑑𝑑𝑑 ���
�𝑁𝑁��� , 𝑁𝑁��� � � �
���
����
�𝜎𝜎��� , 𝜎𝜎��� �𝑑𝑑𝑧𝑧 .
Assuming 𝑤𝑤�𝑎𝑎 � 𝑊𝑊�𝑟𝑟�𝑒𝑒 ��� (for harmonic solution), the non-dimensional form of equation (10) and periphery conditions lead to, �
� 𝑆𝑆� ���
𝑑𝑑 � 𝑊𝑊 � 0, 𝑑𝑑𝑟𝑟 �
�𝑊𝑊���� � � where, 𝑆𝑆� �
𝑑𝑑𝑑𝑑 𝑑𝑑 � 𝑊𝑊 𝐷𝐷�∗ 𝑑𝑑𝑑𝑑 � � 0 ; �𝑊𝑊���� � �𝑆𝑆� � � �0, 𝑑𝑑𝑑𝑑 ��� 𝑑𝑑𝑟𝑟 � 𝑟𝑟 𝑑𝑑𝑑𝑑 ���
𝑆𝑆� � �𝑆𝑆� Ω� ,
𝑆𝑆� �
1 1 𝑑𝑑𝑆𝑆��� � ∗ �𝑆𝑆��� � 𝑟𝑟 � 𝑎𝑎� 𝑟𝑟 � 𝑁𝑁��� � � 𝑄𝑄� � 𝑟𝑟 � 𝑁𝑁 ∗ � , � 𝑟𝑟 𝐷𝐷 𝑑𝑑𝑑𝑑
1 1 𝑑𝑑𝑆𝑆��� � �𝑟𝑟 � 𝑆𝑆��� � 𝑎𝑎� 𝑟𝑟 � 𝑁𝑁��� � � 𝑄𝑄� � 𝑟𝑟 � 𝑁𝑁 ∗ �, 𝑟𝑟 � 𝐷𝐷 ∗ 𝑑𝑑𝑑𝑑
𝑆𝑆� � 𝑄𝑄� �
�11� �12��1��
2 𝑆𝑆��� 𝑆𝑆� � �𝑄𝑄� � ∗ �, 𝑟𝑟 𝐷𝐷
𝐴𝐴� 1 ��� 𝑆𝑆��� 𝜌𝜌� 𝜔𝜔� ℎ � , 𝑆𝑆 � , Ω � , 𝐴𝐴 � � 𝜌𝜌�𝑧𝑧, 𝑇𝑇�𝑑𝑑𝑑𝑑, � � 𝐷𝐷 ∗ 𝐷𝐷 ∗ 𝜌𝜌� ℎ ����
value 𝜌𝜌0 is with respect to the ceramic constituent and ω is the radian frequency.
Equations (7) and (11) are the fourth-order differential equations with variable coefficients whose exact solutions are not possible. Then approximate solutions with appropriate boundary conditions have been obtained by using GDQR.
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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3. Method of solution Let 𝑟𝑟� , 𝑟𝑟� , … , 𝑟𝑟� be the 𝑛𝑛 grid points in range �0, 1� of the plate. According to GDQR (Wu et al. 2002), the 𝑚𝑚�� order derivative of transverse displacement 𝑊𝑊�𝑟𝑟� with respect to 𝑟𝑟 at the 𝑖𝑖 �� point 𝑟𝑟� is given by �
���
���
���
𝑑𝑑 � 𝑊𝑊�𝑟𝑟� � � � �𝑟𝑟 �𝑊𝑊 � � � �𝑟𝑟� �𝑊𝑊� � ℎ�� � � � ℎ�� � � � � ℎ�� �𝑟𝑟� �𝑊𝑊� � � 𝐸𝐸�� 𝑈𝑈� , �𝑖𝑖 � 1,2, … , 𝑛𝑛�, 𝑑𝑑𝑑𝑑 � ��� �
�14�
where, 𝐸𝐸��� are the weighting coefficients of 𝑚𝑚�� order derivative at point 𝑟𝑟� and ℎ�� �𝑟𝑟� are the Hermite interpolation functions. Here, for convenience the unknowns 𝑊𝑊’s have been replaced by 𝑈𝑈’s in equation (14) as follows: � 𝑊𝑊� , 𝑊𝑊�� , 𝑊𝑊� , 𝑊𝑊� , … , 𝑊𝑊��� , 𝑊𝑊��� , 𝑊𝑊� , 𝑊𝑊�� � � � 𝑈𝑈� , 𝑈𝑈� , … , 𝑈𝑈��� , 𝑈𝑈��� �.
The co-ordinates of 𝑛𝑛 grid points have been generated as per the Chebyshev Gauss-Lobatto grid distribution given by 1 𝑖𝑖 � 1 ��� , 𝑟𝑟� � �1 � ��� � 2 𝑛𝑛 � 1
𝑖𝑖 � 1, 2, … , 𝑛𝑛�
3.1. Thermal analysis
Discretizing equation (7) at the internal grid points 𝑟𝑟� , 𝑖𝑖 � 2, �, … , �𝑛𝑛 � 1�, using equation (14) by putting 𝑊𝑊� for 𝑊𝑊, one gets ���
���
���
���
���
����𝑄𝑄� ��� 𝐸𝐸�� � �𝑄𝑄� ��� 𝐸𝐸�� � �𝑄𝑄� ��� 𝐸𝐸�� � �𝑄𝑄� ��� 𝐸𝐸�� �𝑈𝑈�� � � �𝑄𝑄� ��� 𝑈𝑈�� � 0, ���
�15�
where the superscript 𝑇𝑇 over 𝑈𝑈� i.e. 𝑈𝑈�� represents the unknowns 𝑈𝑈� , 𝑈𝑈� , … , 𝑈𝑈��� , 𝑈𝑈��� for thermal displacement as � � 𝑈𝑈�� , 𝑈𝑈�� , … , 𝑈𝑈��� , 𝑈𝑈��� . The satisfaction of equation (15) at �𝑛𝑛 � 2� internal grid points together with the regularity conditions �𝑑𝑑𝑊𝑊� /𝑑𝑑𝑑𝑑���� � �𝑑𝑑 � 𝑊𝑊� /𝑑𝑑𝑟𝑟 � ���� � 0, provides a set of 𝑛𝑛 equations in 𝑛𝑛 � 2 unknowns 𝑈𝑈�� , � � 1, 2, �, … , �𝑛𝑛 � 2�. The resulting system of equations can be written in matrix form as, �𝑀𝑀� �𝑋𝑋� � 0,
� �� where 𝑀𝑀 is a matrix of order 𝑛𝑛 � �𝑛𝑛 � 2� and 𝑋𝑋 is a column vector given by �𝑈𝑈�� , 𝑈𝑈�� , … , 𝑈𝑈��� .
�16�
By satisfying the periphery conditions for a C-plate (8), a set of two homogenous equations is obtained. These equations together with the field equation (16) give a complete set of �𝑛𝑛 � 2� equations in �𝑛𝑛 � 2� unknowns, which can be written as 𝑀𝑀 �𝑋𝑋� �0� � � , 𝑀𝑀�
�
where 𝑀𝑀� is the matrix of order 2 � �𝑛𝑛 � 2�.
�17�
Similarly, for S-plate (9), a set of two non-homogenous equations have been obtained which with the field equation (16) give a complete set of �𝑛𝑛 � 2� equations in �𝑛𝑛 � 2� unknowns and can be written as
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𝑀𝑀 �𝑋𝑋� 0 � � � � �, 𝑀𝑀 � 𝐹𝐹
7
�18�
�
respectively. � � , 𝑈𝑈��� using the computer software The equations �17� and �18� have been solved for unknowns 𝑈𝑈�� , 𝑈𝑈�� , … , 𝑈𝑈��� MATLAB. The thermal stresses raised in the plate due to thermal environment have been obtained as follows: Step 1. The equation ��� have been discretized by using equation (14), ���
���
𝜎𝜎��� � �𝐸𝐸� �𝑧𝑧, 𝑇𝑇� �𝑧𝑧 �� �𝐸𝐸�� � ���
���
���
𝜈𝜈 ��� � ⎫ 𝐸𝐸�� � 𝑈𝑈� � � 𝑇𝑇� � ,⎪ 𝑟𝑟� ⎪
𝜎𝜎��� � �𝐸𝐸� �𝑧𝑧, 𝑇𝑇� �𝑧𝑧 �� �𝜈𝜈𝜈𝜈�� � ���
𝜎𝜎��� � 𝜎𝜎��� � 𝜎𝜎��� � 0,
1 ��� � ⎬ 𝐸𝐸 � 𝑈𝑈� � � 𝑇𝑇� � , ⎪ 𝑟𝑟� �� ⎪ ⎭
�19�
� � Step 2. Substitution of the numerical values for 𝑈𝑈�� , 𝑈𝑈�� , … , 𝑈𝑈��� , 𝑈𝑈��� (obtained from solving equations �17,18�) in �19� provide the thermal stresses for C and S-plates, respectively.
3.2. Vibration analysis
For vibration analysis, equation (11) has been discretized by using equation (14) for various order derivatives of 𝑊𝑊 and at 𝑖𝑖 �� grid point 𝑟𝑟� , it leads to ���
���
���
���
���
����𝑆𝑆� ��� 𝐸𝐸�� � �𝑆𝑆� ��� 𝐸𝐸�� � �𝑆𝑆� ��� 𝐸𝐸�� � �𝑆𝑆� ��� 𝐸𝐸�� �𝑈𝑈� � � �𝑆𝑆� ��� 𝑈𝑈� � 0. ���
�20�
Now, the satisfaction of equation (20) at �𝑛𝑛 � 2� internal grid points together with the regularity conditions provides a set of 𝑛𝑛 equations in terms of �𝑛𝑛 � 2� unknowns 𝑈𝑈� , � � 1, 2, 3, … , �𝑛𝑛 � 2�. These resulting 𝑛𝑛 equations together with the two more equations obtained by satisfying the periphery conditions, give a complete set of �𝑛𝑛 � 2� equations in �𝑛𝑛 � 2� unknowns. For C-plate, the set of these �𝑛𝑛 � 2� homogenous equations can be written as 𝑉𝑉 �𝑌𝑌� �0�, � � 𝑉𝑉 �
�
�21�
𝑉𝑉 � � � � 0. 𝑉𝑉
�22�
𝑉𝑉 � � � � 0, 𝑉𝑉
�23�
where 𝑉𝑉, 𝑉𝑉 � are the matrices of order 𝑛𝑛 � �𝑛𝑛 � 2�, 2 � �𝑛𝑛 � 2�, respectively and 𝑌𝑌 � �𝑈𝑈� , 𝑈𝑈� , … , 𝑈𝑈��� �� . For a non-trivial solution of equation (21), the frequency determinant must vanish and hence
Similarly, for S-plate, the frequency determinant can be written as
respectively.
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Table 1. Comparison of Ω for first three modes, (material properties (Lal and Ahlawat, 2015)). 𝑁𝑁 ∗ ↓
‐10
𝐶𝐶
𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝
𝐼𝐼
0
5
𝐼𝐼𝐼𝐼
5.8274
�
35.3161
0
0
10.2158
39.7711
5
20
0
5
10.216�
7.5738
7.5738
7.576 �
39.7711�
39.773�
89.1041�
-
4.9351�
29.4854
29.496
66.0598
4.935�
‐
3.6588
15.4897
Periphery ↓ \ 𝑝𝑝 𝑝
35.8936
1
2
3
4
7.4955
6.8967
6.4689
4.1978
�
8.6933
14.6820
S
70.6780
�
38.6369�
0
4.1978�
Table 3. Values of frequency parameter 𝛀𝛀𝛀
8.6933
2.4855
2.4855�
7.4955 �
2.1431
2.1431�
54.9777
35.8936
12.8819
�
‐
12.8819
6.8967 �
1.9719
1.9719�
�
22.052 �
14.6820 �
74.1561
74.1561�
83.6475
80.3700
∗ Table 2. Comparison of critical buckling load N�� in compression.
22.0338
48.1241
48.1241�
38.6369
41.7750
�
29.736�
68.9221�
11.8425
11.8425�
80.3700
29.7200
29.72�
68.9221
‐
3.659 �
97.6309
9.1895�
𝐼𝐼𝐼𝐼𝐼𝐼
22.0338
�
97.6309�
�
15.2490
3.6588
�
47.4235�
41.7750
�
4.9351
66.0598
�
47.4235
15.2490
C
89.1041
�
9.1895
*
�
15.4897�
57.5734
29.4854
�
*
24.0538
24.0538
57.5734�
20.7194�
10.2158�
𝐼𝐼𝐼𝐼
*
�
84.5161
20.7194
*
𝐼𝐼
*
84.5161
�
5.8274 *
𝐼𝐼𝐼𝐼𝐼𝐼
35.3161
𝑆𝑆
54.9777� 83.6475� 70.6780�
5
6.4689 �
1.8496
1.8496�
6.1267
6.1267�
1.7517
1.7517�
0
1
2
3
4
5
C
𝑁𝑁 ∗ ↓ \ 𝑝𝑝 𝑝
‐20
*
*
*
*
*
*
‐10
0.0551
*
*
*
*
*
0
8.4466
7.9447
7.7508
7.6372
7.5582
7.4988
10
11.8288
11.1873
10.9647
10.8436
10.7639
10.7064
20
14.3824
13.6266
13.3752
13.2428
13.1580
13.0980
Periphery
30
16.5133
15.6593
15.3816
15.2380
15.1475
15.0842
S
‐20
*
*
*
*
*
*
‐10
*
*
*
*
*
*
0
3.0960
3.0333
2.9024
2.7983
2.7149
2.6470
10
8.2215
7.8559
7.7029
7.6143
7.5547
7.5111
20
11.2072
10.6866
10.4985
10.3972
10.3321
10.2863
30
13.5478
12.9093
12.6910
12.5770
12.5056
12.4560
1
Values from reference Lal and Ahlawat, 2015,
Values from reference Pradhan and Chakraverty, 2015, * Values does not exist due to buckling, 2
8
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9
4. Numerical results The lowest three roots of frequency equations ���� ��� have been computed using MATLAB and retained as the ∗ has been calculated using Bisection method. values of the frequency parameter Ω. The critical buckling load 𝑁𝑁�� The numerical values for TI material properties are taken from reference (Malekzadeh et al., 2011). From the literature, the values of various parameters are taken as follows: material graded index 𝑝𝑝 � �� 1� �� �� �� �� in-plane force parameter 𝑁𝑁 ∗ � ���� �1�� �� 1�� ��� ��, temperatures 𝑇𝑇� � ����, 𝑇𝑇� � �����, ����� � 𝑇𝑇� � �����, thickness � � ��1 and Poisson’s ratio � � ���. 4.1. Convergence and comparison study
During the computations, the number of grid points 𝑛𝑛 has been fixed as 16 since there was no difference up to the fourth place of decimal between two successive values of frequency parameter Ω for varying values of 𝑝𝑝, 𝑁𝑁 ∗ and 𝑇𝑇� keeping 𝑇𝑇� � �����. In this regard, the computer program developed to evaluate Ω was run for these values taking 𝑛𝑛 � 6����� � �1� and observe that �Ω���� � Ω�� � � ������� for all the three modes � � 1� �� �. In graphical form, the percentage error �1 � Ω� ⁄Ω�� � � �1�� for a specified plate 𝑝𝑝 � � , 𝑁𝑁 ∗ � ��� 𝑇𝑇� � 𝑇𝑇� � ����� and 𝑇𝑇� � ����� has been shown in Fig. 2. The deviations obtained for this set of values of the parameters for all the three modes of vibration were maximum. Further, it is worth to mention that for evaluation of the frequencies one has to consider a finite number of terms (𝑁𝑁) in the binomial expansion (5). Being an infinite series, its accuracy will depend upon the value of 𝑁𝑁. In the present analysis, 𝑁𝑁 has been taken as 20, as further increase in this value does not improve the result, except at the fifth place of decimal. A comparison of results in the absence of thermal environment �� � �� 𝑇𝑇 � �� for different values of in-plane force �𝑁𝑁 ∗ � �1�� �� ��) and graded index (𝑝𝑝 � �� �) with those obtained by DTM (Lal and Ahlawat, 2015) and Rayleigh-Ritz method (Pradhan and Chakraverty, 2015) has been presented in Table 1. Further, by allowing the ∗ in compression, for varying values of 𝑝𝑝 has been compared with frequency to approaches zero, the values of 𝑁𝑁�� DTM (Lal and Ahlawat, 2015) and reported in Table 2. A close agreement has been observed which describes the versatility of the present technique.
Fig. 2. Percentage error in Ω: ‘○’ – I mode; ‘△’ – II mode; ‘□’ – III mode (a) C-Plate (b) S-Plate.
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4.2. Parametric discussion To analyze the effect of various parameters on frequency parameter Ω, the results have been presented in Figs. 3-6, however for selected data the numerical values are also given in Table 3. From the results, it has been noticed that the values of Ω for S-plate are smaller than that for the corresponding C-plate. The value of Ω is found to decrease with the increasing value of 𝑝𝑝 i.e. as the nature of plate changes from ceramic to metallic for both the boundary conditions for the same set of the values of other parameters. The effect of temperature difference ∆𝑇𝑇 on Ω for an FGM plate with 𝑝𝑝 � � for two type of plate materials: (i) when the material properties are taken as TI; (ii) when the material properties are TD and varying nonlinearly in thickness direction for 𝑁𝑁 ∗ � ���� ��� has been shown in Fig. 3. Ω is found to decrease with the increasing values of ∆𝑇𝑇 for both the C and S-plates, whatever be the values of other parameters. It can be seen that the values of Ω for TD material are less than those for TI material for the same set of the values of other parameters. This effect is more pronounced for C-plate as compared to S-plate and increases with the increase in the number of modes. It has been noticed that the rate of decay in the value of Ω increases as the nature of in-plane force changes from tensile to compressive for both the plates. This rate of decay decreases with the increase in the number of modes. As a special case, a study for uniform temperature distribution (UTD): 𝑇𝑇�𝑧𝑧� � � � 𝑇𝑇� � ∆𝑇𝑇 as well as linear temperature distribution (LTD): 𝑇𝑇�𝑧𝑧� � 𝑇𝑇� � ∆𝑇𝑇��𝑧𝑧 � ������ ∗ in equation (5) have been made. These numerical results together with non-linear temperature distribution (NTD) have been plotted in Fig. 4 for varying value of ∆𝑇𝑇, for 𝑁𝑁 ∗ � ���� �� and fixed value of 𝑝𝑝 � �. It has been noticed that Ω decreases with the increase in the values of ∆𝑇𝑇 for all the three types of temperature distributions in the order UTD > LTD > NTD for both C and S-plates. The rate of decay for UTD is much higher as compared to LTD as well as NTD for the same set of the values of other parameters. This decay is higher for compressive force as compared to the tensile for both the plates and more pronounced in case of S-plate as compared to C-plate, keeping other parameters fixed. Fig. 5 presents the behaviour of Ω for varying values of 𝑁𝑁 ∗ for two values of 𝑝𝑝 � �� ��and ∆𝑇𝑇 � ��� for C and S-plates. It can be seen that the value of Ω increases with the increase in the value of 𝑁𝑁 ∗ for both the values of 𝑝𝑝, keeping other parameters fixed. The rate of increase for 𝑝𝑝 � � is higher than that for 𝑝𝑝 � �.
Fig. 3. Ω vs ∆T, 𝑁𝑁 ∗ � ���: ‘○’- TI, ‘△’- TD;
𝑁𝑁 ∗ � ���: ‘●’- TI, ‘△’- TD;
C - plate - ‘────’, S - plate - ‘- - - - - -’.
∗ The numerical results for critical buckling load 𝑁𝑁�� have been plotted in Fig. 6a for 𝑝𝑝 � �� � and ∆𝑇𝑇 � ��� ∗ when the plate is vibrating in first mode of vibration. It has been noticed that the values of 𝑁𝑁�� for C-plate are higher ∗ than that for the corresponding S-plate. The values of 𝑁𝑁�� decrease with the increase in the value of 𝑝𝑝 from 0 to 5 i.e. as the nature of plate changes from fully ceramic (isotropic) to metallic. Two-dimensional mode shapes for a specified plate with 𝑝𝑝 � �� 𝑁𝑁 ∗ � �������∆𝑇𝑇 � ��� have been presented in
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Fig. 6b. It is found that the nodal circles for C-plate are smaller than that for S-plate.
Fig. 4. Ω vs ∆T: ‘●’- 𝑁𝑁 ∗ � ��, ‘○’ - 𝑁𝑁 ∗ � ���; C - plate - ‘────’, S - plate - ‘- - - - - -’; UTD - ‘ ’, LTD - ‘△’, NTD -‘○’.
Fig. 5. Ω vs 𝑁𝑁 ∗ : ‘○’ - � � �, ‘△’-�� � �; C - plate - ‘────’, S - plate - ‘- - - - - -’; I Mode - ‘────’, II Mode - ‘────’, III Mode - ‘────’.
5. Conclusions
The combined effect of in-plane force on the axisymmetric vibration of FGM circular plate vibrating in thermal environment has been presented using GDQR. Being TD material properties, the temperature distribution along the thickness of the plate is assumed to be non-linear. Numerical results show that:
12
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∗ Fig. 6. C - plate - ‘────’, S - plate - ‘- - - - - -’; (a) Ω vs 𝑁𝑁�� : ‘○’- 𝑝𝑝 � 𝑇, ‘△’- 𝑝𝑝 � �, ‘□’- 𝑝𝑝 � �, ‘⁎’-𝑇𝑝𝑝 � �; (b) First three mode shapes.
Ω� � Ω� , for same set of the values of parameters. For both C and S-plates, the frequency parameter Ω for fully ceramic plate (Ω��� � is always higher than that for the corresponding FGM plate (Ω��� ). The value of Ω decreases with the increase in the value of temperature difference and it is found that Ω∆��� � Ω∆��� , whatever be the values of other parameters. The percentage decrease in the value of Ω for 𝑝𝑝 � �� 𝑁𝑁 ∗ � �𝑇𝑇with the change in temperature difference ∆𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 to 400 are 6.58, 7.86 for C, S-plates, respectively, with TD material properties, when the plate is vibrating in the first mode. These variations are differing only at the first place of decimal for fully ceramic plate. In view of the TI material properties, the percentage variation in Ω with varying values of ∆𝑇𝑇 from 0 to 400, for 𝑝𝑝 � �𝑇� 𝑁𝑁 ∗ � �𝑇 are -3.81 and -5.69 for C and S-plates, respectively. However, for 𝑁𝑁 ∗ � ��𝑇, values do not exist due to buckling. The behaviour of Ω with the change in the nature of temperature distribution from NTD to LTD and then to UTD is found in the order of Ω��� � Ω��� � Ω��� . The percentage variation in the values of Ω with varying values of ∆𝑇𝑇 from 0 to 400 for 𝑝𝑝 � ��𝑇for UTD, LTD and NTD are (-15.39, -7.50, -6.58) and (-18.35, -9.06, -7.86) for 𝑁𝑁 ∗ � �𝑇 while values do not exist due to buckling for 𝑁𝑁 ∗ � ��𝑇 for both the plates vibrating in the first mode of vibration. ∗ decreases as the nature of the plate changes from ceramic to metallic i.e. 𝑝𝑝 The value of critical buckling load 𝑁𝑁�� changes from 0 to 5 for both the plates. The above analysis will be of great practical use to a design engineer dealing with FGM plates in obtaining the desired frequency by controlling one or more parameters involved in the present study.
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