Thermal Stresses in Plates and Shells (Solid Mechanics and Its Applications, 277) 303149914X, 9783031499142

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Solid Mechanics and Its Applications

Mohammad Reza Eslami

Thermal Stresses in Plates and Shells

Solid Mechanics and Its Applications Volume 277

Series Editors J. R. Barber, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA Anders Klarbring, Department of Mechanical Engineering, Linköping University, Linköping, Sweden

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. Springer and Professors Barber and Klarbring welcome book ideas from authors. Potential authors who wish to submit a book proposal should contact Dr. Mayra Castro, Senior Editor, Springer Heidelberg, Germany, email: [email protected] Indexed by SCOPUS, Ei Compendex, EBSCO Discovery Service, OCLC, ProQuest Summon, Google Scholar and SpringerLink.

Mohammad Reza Eslami

Thermal Stresses in Plates and Shells

Mohammad Reza Eslami Department of Mechanical Engineering Amirkabir University of Technology Tehran, Iran

ISSN 0925-0042 ISSN 2214-7764 (electronic) Solid Mechanics and Its Applications ISBN 978-3-031-49914-2 ISBN 978-3-031-49915-9 (eBook) https://doi.org/10.1007/978-3-031-49915-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

The author dedicates this book to the memory of his parents: Mohammad Sadegh Eslami (1900–1980) Zinat Shahrestani (1925–2006)

Preface

In the solid mechanic problems, the principle law to be observed and considered in the process of solution is the simultaneous considerations of the constitutive law of material, equations of motion, and the compatibility condition. The plate and shell structures, however, are in general thin in thickness. In early stages of the development, theory of elasticity was properly modified for these types of structures by lumping the stresses across the thickness direction. Force and bending moment resultants were introduced and the equations of motion were replaced by the equilibrium of force and bending moment resultants, employing Newton’s law. The reason for this approach is that to be able to establish a connection between forces, moments, and deformation components of the middle plane of shell it is necessary to know how the stress vary across the thickness. Any attempt to describe the stresses and strains across the thickness of a general shell by the laws of three-dimensional theory of elasticity provides an extremely difficult mathematical problem mainly due to the geometrical reasons. This fact has been the main reason to initiate the thought for the need of a different approach for the plate and shell formulation. This is why the shell and plate theories deviate from the three-dimensional theory of elasticity. The essential problem in development of shell theory is an appropriate formulation of stresses and strains through the shell thickness. Once an assumption and its related formulation is adopted, the shell governing equations are obtained through either the force summation method based on a shell element in equilibrium or variational methods. The resulting governing equations are an approximation to the three-dimensional theory of elasticity to describe the shell behavior. The selection of a proper form of approximation has been the subject of many investigations in shell theory. This situation has led to the development of a large number of different general and specialized thin shell theories. Accordingly, the general shell theories are branched into five different classical theories. The mathematical formulations of plate and shell equations of motion, keeping in mind the different theories, require special considerations. The formulations do not directly obey the established basis of the theory of elasticity due to the lumped formulation across the thickness. In addition, how about the important law of compatibility, where any elastic component should obey to insure the uniqueness and continuity vii

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of the deformation components? And, most importantly, five different theories of plate and shell assumptions require its own types of formulations. The combination of these theories is also quite legitimate. These considerations and questions should be properly defined and reasonably explained. Now, why Chap. 2 is devoted to the tensor analysis? Tensors are one of the tough subjects in the mathematical science. So, why they are used and what is the advantage to use them in the description of physical laws? The most important advantage to use the tensors is when a physical law is established in one coordinate system, such as the Rectangular Cartesian Coordinate system (RCC), it holds its general form in any other general curvilinear coordinate systems. It is always the most convenient to formulate and write the physical laws in the RCC system. Once the formulations are established in the RCC system, the form of the equations are kept identical in the general curvilinear coordinate system and their physical components are then derived in this latter system by the tensor laws. Some physical laws written in terms of components in the general curvilinear coordinates system do not have any physical or geometrical interpretation for their components, while their RCC interpretation and derivations are clear and simple. As an example, the linear strain–displacement relations in the theory of elasticity are simply derived and represented in the RCC system by i j =

1 (u i, j + u j,i ) 2

Now, we may try to derive the same relation for the spherical shells, as an example. There is no direct method that the terms appeared in the strain–displacement relations for the spherical shells, for example, be geometrically interpreted. The form of strain– displacement relation in RCC is kept and transformed into the general curvilinear coordinates system to be i j =

1 (u i| j + u j|i ) 2

where u i| j is the covariant derivative of u i with respect to j. Now, it remains to derive the physical components in the curvilinear coordinate systems. The results are sometimes very complicated mathematical relations that many terms of them do not have any physical or geometrical justifications. This is why tensor analysis is important in the presentation of the geometries and physical laws of the shell structures. In conclusion, tensor analysis is a powerful method to derive the strain– displacement relations for any shell geometry. Another important reason that the tensor analysis is important in the shell analysis is the justifications why the compatibility equations are not used in the shell and plate theory. The Codazzi and Gauss conditions are replaced for the compatibility equations. These conditions enforce the continuity of the surface in three-dimensional space. The method is described in this chapter.

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The classical theory of plate is an approximation to the two-dimensional theory of elasticity, where the Love–Kirchhoff hypothesis are adopted. The so-called flexural theory of plates, which was first introduced by Love, is an attempt to formulate the equilibrium of a plate element based on simplified two-dimensional theory of elasticity. The plate element is lumped across the thickness, where the equilibrium of the plate element under the applied force and moment resultants is derived. It is further assumed that the normal stress and strain are zero. Apparently, the latter assumption violates the rules of the classical two-dimensional elasticity, where either plane-stress or plane-strain conditions are allowable. In either case, the other lateral normal strains or stresses are nonzero and are obtained from Hooke’s law. The general development of the classical plate theory is discussed in Chap. 3. The first-order plate theory, which is the basis of classical plate theory, is specially attractive in the mathematical treatment of plate problems. The governing equation for lateral deflection of plates, either circular or rectangular, is a biharmonic equation. Since there is always an analytical solution to the biharmonic partial differential equations, therefore, the plate problems of practical importance have been discussed and analyzed in literature since early stages of development of the theory. Many problems of circular and rectangular plates with different boundary conditions, mechanical and thermal loadings, and material properties such as isotropic, anisotropic types, and composites are developed and solved effectively by the analytical methods. For this reason, in Chap. 3, the classical theory of plates is discussed, and analytical solutions are presented. Chapter 4 of this book deals with the derivations of the shell equations. The geometry of shell structures is, in general, complicated as far as the classical theory of elasticity is concerned. It is not simple to model a shell segment by the three-dimensional theory of elasticity. The shell element is lumped in the thickness direction and consequently one of the three dimensions, the thickness variable, is removed from three variables remaining only two more variables to model the shell and write down the equilibrium equations or the equations of motion. This is the basic initiative assumption of Love, which led to the development of the broad science of plate and shell theory. A shell element is considered and the force and moment resultants are placed on the edges of the element and the equilibrium of the shell element is satisfied considering the balance of the forces and moments on the element. The resulting governing equations appear in terms of the force and moment resultants. The force and moment resultants are obtained employing Hooke’s law and the shell kinematic relations relating the strain–displacement relations are derived. Now, how about the strains and deformations? The main problem to obtain the strains and deformations in the shell structures is to derive the strain–displacement relations. While derivation of the strain–displacement relations in the rectangular Cartesian coordinates have clear geometrical meaning, it is not the same for the shell elements. There is no general method to derive the strain–displacement relations for shells using the simple geometrical relations and concepts. Now, either the analyst should forcefully accept these relations for different types of shell geometries, or he or she should have a mathematical tool to derive the relations regardless of the type of the shell geometry (cylindrical, spherical,

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conical, shell of revolutions, etc.). This mathematical tool is tensor analysis. This is the justification of why Chap. 2 is devoted to the treatment of tensors and details description of tensor mathematics. The strain–displacement relations in any general curvilinear coordinates system in tensor language were just described. Now, it is the responsibility of the analyst to derive the components of the above tensor equation in any desirable curvilinear coordinates system with no mathematical limitation. Another important justification of devoting a chapter to tensor analysis is the problem of compatibility of strain and displacement components in shell structures. In the classical theory of elasticity, the displacement components are eliminated among the strain components to come up with the compatibility equations. Then, it is proved that to have a continuous and single valued displacement components, for a simply connected region, is to satisfy the compatibility equations. This is not the same for the plates and shell theories. One may solve a simple shell problem and substitute the solved displacement components into the compatibility equations to be surprised to see that the compatibility equations derived for the theory of elasticity are not satisfied. The reason? plate and shell structures are lumped across the thickness direction. The question then arises about the continuity and single valuedness of the plate and shell displacement and strain components? Plates and shells are the extension of a two-dimensional surface into the three-dimensional space. A thickness is then added to the surface. Now, if the surface is continuous and single valued, the compatibility conditions are satisfied. This condition is enforced by the Coddazi and Gauss conditions, the discussion of which is given in Chap. 2 of tensor analysis. That is, the Codazzi and Gauss conditions for surface are replaced for the compatibility conditions in the theory of elasticity. The reader is now convinced why a whole chapter is devoted to tensor analysis. The justifications are the strain–displacement relations and the compatibility conditions of shell structures. Chapter 5 presents some basic analytical solutions for different shell geometries. Thin shells are one of the most frequently used members in the structural design problems. The real configurations of such members in the industrial equipments are far from simple geometrical shapes and for the precise deformation and stress analysis, numerical models such as finite element method are essential. It is in this stage that the designer needs to make sure that his or her numerical model, incorporating the actual model for the boundary conditions, is correct. The most reliable source for such validation is to have analytical solutions. The validation procedure is to make a simple model of the real shell structure under the actual loads, with proper boundary conditions, and compare the numerical results with some known analytical solution. In this chapter, it is intended to provide some examples of the classical shell problems, including the cylindrical, spherical, and conical shells under simple thermal loads that are frequently used in the engineering problems and come up with analytical solutions. The analytical solutions are intended to be in terms of the displacements and stresses. In physical applications of the shell and plate structures, mechanical or thermal loads may be applied in form of shocks. If the characteristic times of the structural and thermal disturbances are of comparable magnitudes, the equations of motion of

Preface

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the structure under such condition are coupled with the energy equation. The subjects fall into the theory of coupled or generalized theory of thermoelasticity. Chapter 6 considers this condition and presents a comprehensive mathematical discussion on the subject. Recently developed functionally graded materials (FGMs) show promise for their adaptability to high-temperature environments and have thus attracted attention. Therefore, it is desirable to analyze FGM structures subjected to thermal loadings, such as thermal shock, which have a wide range of applications in engineering problems. The same loading conditions have frequently occurred for the shell structures. Discussion is presented in Chap. 7. This chapter presents the coupled and generalized thermoelasticity of the cylindrical shells, spherical shells, conical shells, and shells of revolution. The shell material is assumed to be made of functionally graded, where by proper substitution for the power law index, response of shells of homogeneous material is obtained. Consider a shell structure under applied thermal load. The applied thermal load may be of different natures: steady-state thermoelasticity, thermal-induced vibrations, and the coupled or generalized thermoelasticity. The steady-state thermoelasticity of shells occurs when the shell is under steady-state thermal load, that is, under steady-state temperature distribution. The heat conduction equation is independently solved to derive the temperature distribution. The resulting temperature distribution is employed to substitute into the steady-state governing thermoelastic equations and solve for the displacement and stress components. In this type of analysis, the thermal energy equation and the thermoelastic shell equations are uncoupled and are solved independently. It is important to emphasize that sometimes the transient temperature distribution in shells is treated similar to the steady-state conditions, provided that the shell thickness is relatively thick, otherwise thermal-induced vibrations may appear. How much thick? This is not simply answered and requires detail analysis. The second type of shell behavior under applied thermal load is thermal-induced vibrations. In this condition, temperature distribution in the shell is of transient form, but its application to the shell results in shell vibrations. The reason? geometrical properties of the shell. In such conditions, the transient temperature distribution in the shell is derived through the heat conduction equation and substituted into the equations of motion of the shell considering the inertia terms. Simultaneous solution of the equations of motion reveals if the shell behavior falls into thermal-induced vibrations or not. A notable and critical parameter is the magnitude of the shell thickness. For thin enough shell, we may expect thermal-induced vibrations. The coupled thermoelasticity of shells occurs when the applied thermal load is in form of thermal shock. If the time period of application of the applied thermal load is much smaller compared to the time period of the lowest natural frequency of the shell, then the governing thermoelasticity equations are coupled with the thermal energy equations and the response of the shell should be derived employing the coupled thermoelastic equations. For extremely small thermal shock load periods, the generalized thermoelasticity should be considered. Under such thermal shock load, the shell material’s relaxation time is excited and should be included in the thermoelastic shell governing equations.

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Question of detection of the wave front in plates and shells under the coupled thermoelastic analysis should be now addressed. The stress, displacement, and temperature (second sound) wave fronts are expected to appear in structures under the coupled thermoelastic assumption. None of these wave fronts are detected in the analysis of plates and shells exposed to the thermal shock loads. The reason is that the flexural structures, such as beams, plates, and shells, are lumped across the thickness direction. Thermal shock loads applied to the surface of these structures should create the wave fronts across the thickness direction, where the structure is lumped in that direction. This is why no such wave fronts are detected across the thickness direction. That is, while the stress wave front is generated across the plates or shells under the lateral thermal shock loads, but the flexural theory is unable to detect it. The example is pouring hot water in a cold glass, where the large created wave front of stress results in the glass breakage. The wave fronts appear when the analysis is based on the three-dimensional theory of thermoelasticity. Under thermal shock loads, shells may lose their stability. The subject is well described in this chapter. In Chap. 8, thermal-induced vibrations of plates and shells are discussed. The subject is very interesting, as the classical heat conduction equation is of parabolic nature and the temperature change by time is of transient type. Under circumstances, the transient-type temperature distribution results in structural vibrations. In 1956 Boley was the first who introduced the thermally induced vibration phenomenon in a beam including simply supported ends and exposed to a sudden constant thermal shock at the top surface. Thermally induced vibration phenomenon was not only investigated for beams, other researchers extended Boley’s work to several structures or different thermal shock problems. Boley and Barber obtained the response of a thin rectangular plate with simply supported edges. The results showed that vibrations caused by rapid heating occur when inertia parameter is small enough. On the other hands, quasi-static response occurs when inertia parameter becomes very large. Thermal-induced vibrations of rectangular plates, cylindrical, and conical shells are discussed in Chap. 8. At this point, the author is obligated to express his deep appreciations to his many graduate students throughout these years who made this piece of work available. Valuable help of Mr. M. Javani, my Ph.D. student in particular, is appreciated. These young fellows were the most intelligent, dedicated, and hard workers. This piece of work, presented in this volume, is the result of over 50 years of teaching and collaborations with my fabulous students. There is no word that I can express for their appreciations. The author expresses his deep appreciations to Prof. M. Sabbaghian of Louisiana State University who taught him thermoelasticity. The constant support and encouragement of Prof. Richard B. Hetnarski is invaluable and appreciated. Tehran, Iran

Mohammad Reza Eslami

Contents

1 Introduction to Plate and Shell Structures . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5

2 Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Coordinate Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Euclidean Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Different Types of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Some Special Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Tensors in Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Christoffel Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Some Important Tensor Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Base Vectors in Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . 2.14 Physical Components of Vectors and Tensors . . . . . . . . . . . . . . . . . . 2.15 Theory of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Fundamental Quadratic Forms of Surfaces, Codazzi Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 11 12 13 15 17 18 18 20 22 24 25 33 35

3 Thermal Stresses in Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Classical First-Order Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Circular Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Navier Solution for Rectangular Plates . . . . . . . . . . . . . . . . . . . . . . . 3.6 Levy’s Solution for Rectangular Plates . . . . . . . . . . . . . . . . . . . . . . .

43 43 44 52 54 56 58

37 42 42

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3.7 3.8 3.9 3.10 3.11 3.12

General Solution for Rectangular Plates . . . . . . . . . . . . . . . . . . . . . . Circular Plate with Radial Temperature . . . . . . . . . . . . . . . . . . . . . . . Composite Rectangular Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Shear Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Analysis of FGM Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rectangular FGM Plates Under Thermal Shock . . . . . . . . . . . . . . . . 3.12.1 Temperature Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Active Piezothermoelastic Analysis of Composite Plates . . . . . . . . 3.13.1 First-Order Shear Deformation Theory . . . . . . . . . . . . . . . . 3.13.2 Sensor and Actuator Equations . . . . . . . . . . . . . . . . . . . . . . 3.13.3 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60 66 67 75 81 101 110 122 124 129 131 139

4 Theory of Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Shell Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Shells of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Analysis of Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Stress–Strain Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Stress Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Equations of Motion of Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Comparison of Different Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 147 157 162 173 175 182 192 195 196

5 Thermal Stresses in Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Symmetrically Loaded Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . 5.3 Axially Loaded and Heated Cylindrical Shells . . . . . . . . . . . . . . . . . 5.4 Thermal Stresses in Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Derivation of the Equilibrium Equations . . . . . . . . . . . . . . 5.4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Cylindrical Shells of Gas Turbine Combustor . . . . . . . . . . . . . . . . . . 5.5.1 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Simply Supported Cylindrical Panel . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Symmetrically Loaded Spherical Shells . . . . . . . . . . . . . . . . . . . . . . . 5.8 Axisymmetric Thermal Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Thermal Stresses in Spherical Domes . . . . . . . . . . . . . . . . . . . . . . . . 5.10 General Temperature Distribution Along Meridian . . . . . . . . . . . . . 5.11 Membrane Analysis of Spherical Shells . . . . . . . . . . . . . . . . . . . . . . . 5.12 Spherical Shells with Circular Holes . . . . . . . . . . . . . . . . . . . . . . . . . 5.12.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Thermal Stresses in Conical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13.1 Conical Shell Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13.2 Equilibrium Equations of Conical Shell . . . . . . . . . . . . . . .

197 197 197 202 205 206 212 215 216 216 222 228 234 237 239 242 247 251 253 254 258

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xv

5.14 Rotating Conical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 6 Coupled Thermoelasticity of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Rectangular FGM Plates, TSDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Coupled Thermoelasticity of FGM Annular Plate . . . . . . . . . . . . . . 6.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271 271 273 274 278 281 284 296 298 299 303 313 327

7 Couple Thermoelasticity of Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Coupled Thermoelasticity of Cylindrical Shells . . . . . . . . . . . . . . . . 7.2.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Strain–Displacement Relations . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.7 Parametric Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Cylindrical Shell; Effect of Normal Stress . . . . . . . . . . . . . . . . . . . . . 7.4 Composite Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Discussion and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Coupled Thermoelasticity of Spherical Shells . . . . . . . . . . . . . . . . . 7.5.1 Energy Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Composite Spherical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Coupled Thermoelasticity of Conical Shells . . . . . . . . . . . . . . . . . . . 7.7.1 Strain–Displacement Relations . . . . . . . . . . . . . . . . . . . . . . 7.7.2 Stress–Strain Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.4 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.5 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

329 329 330 332 334 335 337 338 341 343 354 361 364 366 367 378 380 382 384 389 392 398 399 401 401 403 404 406

xvi

Contents

7.8

Thermoelasticity of Shells of Revolution . . . . . . . . . . . . . . . . . . . . . . 7.8.1 Coupled Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.2 Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.3 Spherical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Stability of Cylindrical Shells, Coupled Thermoelastic Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.2 Equations of Motion and Boundary Conditions . . . . . . . . . 7.9.3 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.4 Finite Element Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.5 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

411 416 419 427

8 Thermal Induced Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 FGM Rectangular Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Temperature Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 FGM Conical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Fundamental Equations of the FG Conical Shell . . . . . . . . 8.3.2 Kinematic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Stress Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.5 Temperature Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.7 Comparison Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.8 Parametric Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Spherical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 HDQ Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Temperature Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.6 Comparison Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.7 Parametric Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

453 453 455 456 462 463 473 474 475 476 478 481 482 483 484 494 496 500 503 504 506 506 508 516

431 433 435 437 439 442 444 449

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

Chapter 1

Introduction to Plate and Shell Structures

Abstract Plates and shell structures, as a division of solid mechanics, are frequently used in the structural design problems. The variety of different plate and shell theories are developed and this chapter is intended to give the milestones of these historic development.

1.1 Introduction In the solid mechanic problems, one of the principal laws to be observed and considered in the process of solution is the constitutive law of material relating the state of stress to strain. The physical hypothesis expressed by these relations in connection with the equations of motion and compatibility is sufficient for the description of the state of deformation or stress in the shell. However, shell structures are in general thin in thickness, such that its dimension across the thickness is much smaller than any other dimensions. To be able to establish a connection between forces, moments, and deformation components of the middle plane of shell, it is necessary to know how the stress vary across the thickness. Any attempt to describe the stresses and strains through the thickness of a general shell by the laws of three-dimensional theory of elasticity provides an extremely difficult mathematical problem mainly due to the geometrical reasons. This fact has been the main reason to initiate the thought for a need of a different approach for the plate and shell formulation. This is why the shell and plate theories deviate from the three-dimensional elasticity theory. The essential problem in development of shell theory is an appropriate formulation of stress and strain through the shell thickness. Once an assumption and its related formulation is adopted, the shell governing equations are obtained through either the force summation method based on a shell element in equilibrium or variational methods. The resulting governing equations are an approximation to the three-dimensional theory of elasticity to describe the shell behavior.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. R. Eslami, Thermal Stresses in Plates and Shells, Solid Mechanics and Its Applications 277, https://doi.org/10.1007/978-3-031-49915-9_1

1

2

1 Introduction to Plate and Shell Structures

The selection of a proper form of approximation has been the subject of many investigations in shell theory. This situation has led to development of a large number of different general and specialized thin shell theories. Accordingly, the general shell theories are branched into five different classical theories. Specialized assumptions are allowed in each of the five classical theories. The classical shell theories are divided into the following five different categories: 1 - Membrane shell theory. 2 - First-order approximation shell theory. 3 - Second-order approximation shell theory. 4 - Shear deformation and special shell theories. 5 - Nonlinear shell theories. Each category is defined by the order of approximation of terms retained in the expression of strains and stresses through the thickness. The membrane shell theory admits only in plane tension and compression forces and ignores the flexural rigidity of shell, as given by Love [1, 2]. This assumption yields a system of equilibrium equations for shell where all expressions of moments are ignored. In general, shell structures are not basically designed to stand the large moment loads. A small amount of bending moment produce large amount of bending stress in shells. Therefore, if a loading condition is applied such that the resulting bending moments are small, the membrane shell theory may be quite adequate to provide acceptable analysis. The first-order shell theory was first introduced by Love as an approximation of the classical elasticity. Since the shell thickness is in general small compared to the other dimensions, the strain–displacement relations and the constitutive law can be simplified. On this basis, Love introduced the following assumptions known as Love–Kirchhoff hypothesis [1, 2]: 1 - Since the shell thickness .h is negligibly small compared to the least radius of curvature, . Rmin , of the middle surface, that is, .h/Rmin 10) the constants of integration, .C3 and .C4 , are taken equal to zero, as the edge effects decay very rapidly as the coordinate .x increases. Hence, only two boundary conditions are necessary at the “origin” end of the cylinder. If loads are applied at both edges of a long cylinder, two “origin” ends may be established and treated as if there are two long cylinders with loads at one edge of each. The constant of integration .C5 appears only in the formula for the axial displacement .u. Its value is obtained by specifying the displacement at some .x-coordinate value (usually at 0 or . L), which generally stipulates merely an axial displacement of the cylinder. Hence, .C 5 may usually be taken equal to zero. .

216

5 Thermal Stresses in Shells

5.5 Cylindrical Shells of Gas Turbine Combustor The firing temperatures of industrial gas turbines are increased since their initial designs. From a base-load firing temperature of approximately 816 .o C in 1961, the design working temperatures are increased in the advanced turbine models. This section addresses the problem of the structural integrity of the combustor due to the increase in firing temperature [8]. Combustors have been designed to release increased energy in a given volume to achieve higher firing temperatures. This higher volumetric heat release rate, together with metal cooling techniques, produces large temperature variations over the surface of the combustor. The combustor, usually made of high temperature-resistant alloys, is film air cooled to tolerate the hightemperature gases near the interior wall of the combustor. Several cooling systems have been used in combustor designs to reduce the average metal surface temperature.

5.5.1 Theoretical Formulation Consider a slot-cooled combustor in form of a cylindrical shell with periodically spaced circumferential bands of cooling holes [8]. Series of small holes are made along the circumference of the thin cylindrical shell. The longitudinal space between these circumferentially placed small holes is assumed to be . S. The cooling air which enters the combustor through the holes is assumed to produce a circumferentially cooled band at the hole locations. The shell sections between the bands of cooling holes are subjected to a higher temperature. We may assume that the cylindrical shell is subjected to a periodic temperature variation in the longitudinal direction as shown in Fig. 5.8. The lateral deflections and the thermal stresses in the shell can be determined using the shell theory. The approach used in this section is to determine an appropriate influence function which can be used to determine closedform solutions, valid for any longitudinally varying temperature field, expressible in terms of a Fourier series. The assumed cylindrical shell of this section is under axisymmetric temperature distribution. Figure 5.9 shows the geometry and coordinates used in the derivation of the solution. A thin cylindrical shell of radius . R, thickness .h, and length . L is considered. The (.x)-axis is in the longitudinal direction, while the (.z)-axis is drawn positive inward to the centerline of the cylinder. The governing differential equation for the radial displacement .(w) due to an external pressure .( p) may be obtained from Eq. (5.2.17) and is given by [8] .

d 4w p(x) + 4β 4 w = dx4 D

where .(D) is the flexural rigidity and the shell parameter .β is defined as

(5.5.1)

5.5 Cylindrical Shells of Gas Turbine Combustor

217

Fig. 5.8 Longitudinal cross section of combustor model of the maximum stress is shown to be: Showing periodic temperature distribution Fig. 5.9 Geometry and coordinate system for model

β4 =

.

3(1 − ν 2 ) R2h2

(5.5.2)

Note that for the assumed condition of temperature distribution of .T (x), the thermal bending moment is zero. A singular solution of differential equation (5.5.1) may be sought which can be used as an influence function to determine the displacement and stresses for any arbitrary axisymmetric shell load varying in the longitudinal direction. The pressure distribution in Eq. (5.5.1) is simply replaced by a concentrated load applied along the circumference of the circular shell. Reference [9] discusses the analogy between the solutions for a circular shell and a beam on an elastic foundation since the governing differential equations and boundary conditions are similar. Accordingly, one may simplify the geometric representation to that shown in Fig. 5.10. This analogy can furnish physical intuition to the problem under consideration. In addition, it will provide a simple method of obtaining influence functions from concentrated load solutions tabulated in Reference [9]. It should be noted that the method of solution to be obtained for an infinite cylinder can also be used to determine solution for finite length cylinders using the elastic foundation solutions in [9]. Using the elastic foundation analogy, consider the beam shown in Fig. 5.10, subjected to the concentrated load . Pδ(x − ξ), with the load applied at .x = ξ . The

218

5 Thermal Stresses in Shells

Fig. 5.10 Beam on an elastic foundation model—analogy to radially loaded circular shell

solution to the differential equation (5.5.1) can be obtained from the solution in reference [9] for a concentrated load applied to a beam on an elastic foundation. Therefore, the influence function solution for the radial deflection of an infinite circular shell subjected to a concentrated hoop load applied at .x = ξ is given by [8] as Pδ(x − ξ)e−β|x−ξ| (5.5.3) .w(x, ξ) = [sin β|x − ξ| + cos β|x − ξ|] 8β 3 D where the concentrated load is in form of the Dirac delta function .

P = Pδ(x − ξ) = 0 for x /= ξ and

= P for x = ξ

(5.5.4)

Solution (5.5.3) is the influence function and has the property of reciprocity as w(x, ξ) = w(ξ, x)

.

(5.5.5)

This solution reduces to the solution in reference [1] when the deflection at .x = 0 is determined for values of .ξ > 0. w(0, ξ) =

.

Pe−βξ [sin βξ + cos βξ] 8β 3 D

(5.5.6)

Solution (5.5.6) gives the deflection at origin .x = 0 when a concentrated load . P is applied at .x = ξ. The references [1] and [9] discuss the equivalency of an external pressure and temperature load. It can be shown that the external pressure can be expressed in terms of an equivalent thermal load given by .

P = pd x =

Eh αT d x R

(5.5.7)

where.α is the coefficient of thermal expansion and.(T ) is the temperature differential. The deflection .w at the origin due to a temperature differential applied at .x = ξ is determined from (5.5.6) w(0, ξ) =

.

EhαT (ξ)dξe−βξ [sin βξ + cos βξ] 8Rβ 3 D

(5.5.8)

5.5 Cylindrical Shells of Gas Turbine Combustor

219

The bending stress in shell is simply calculated using the bending moment .(Mξ ) as σ (ξ) =

. x

6Mξ 6 d 2w = h2 h 2 dξ 2

(5.5.9)

Substituting (5.5.8) into (5.5.9), the bending stress is 3 EαT (ξ)dξe−βξ [sin βξ − cos βξ] 2 Rβh

σ (0, ξ) =

. x

(5.5.10)

The circumferential stress is determined from the equation σ =

. θ

E (w + αRT ) R

(5.5.11)

Substituting (5.5.8) into (5.5.11), the circumferential stress is determined as σ (0, ξ) =

. θ

E R

{

} EhαT (ξ)dξe−βξ + EαT (0) βξ + cos βξ] [sin 8Rβ 3 D

(5.5.12)

where.T (0) is the temperature at the origin and.T (ξ) is the arbitrary axial temperature applied at .x = ξ. Solutions given in Eqs. (5.5.8), (5.5.10), and (5.5.12) are the influence functions for a temperature rise at .x = ξ. The deflection and stresses at the origin for an arbitrary temperature distribution along the longitudinal direction are determined using the principle of superposition. Let us assume that the temperature distribution on the outside surface of the shell extends in the period of .−x1 ≤ ξ ≤ +x1 . When the material properties are assumed to be temperature independent, the solutions for the lateral displacement, axial stress, and hoop stress at origin for the temperature distribution given in the above period can be expressed in terms of definite integrals w(0) =

.

Ehα 8Rβ 3 D

3Eα .σ x (0) = 2Rβh σ (0) =

. θ

E 2 hα 8R 2 β 3 D

{

{

−x1

{

+x1 −x1

+x1

+x1 −x1

T (ξ)e−β|ξ| [sin β|ξ| + cos β|ξ|] dξ

(5.5.13)

T (ξ)e−β|ξ| [sin β|ξ| − cos β|ξ|] dξ

(5.5.14)

T (ξ)e−β|ξ| [sin β|ξ| + cos β|ξ|] dξ + EαT (0) (5.5.15)

A periodic temperature field of the form shown in Fig. 5.8 generates the highest bending stresses in the vicinity of the cooling holes where the temperature is lowest and the stress concentration is the highest. Expanding the temperature distribution by the Fourier series

220

5 Thermal Stresses in Shells

) ∞ ( a0 E nπξ nπξ . T (ξ) = + + bn sin an cos 2 L L n=1

(5.5.16)

the axial stress may be obtained from (5.5.14) by substitution of Eq. (5.5.16). Thus, the axial stress at origin is

.

{ 3Eα a0 σx (0) = − sin βx1 e−βx1 2β Rh 2β | −βx1 ∞ 1 E e + an [(β − γn ) cos γn x1 − (β + γn ) sin γn x1 ] 2 n=1 β 2 + γn2 e−βx1 [(β − λn ) cos λn x1 − (β + λn ) sin λn x1 ] β 2 + λ2n | β − γn β − λn − 2 − 2 β + γn2 β + λ2n | ∞ 1 E e−βx1 + bn [(β + γn ) cos γn x1 + (β − γn ) sin γn x1 ] 2 n=1 β 2 + γn2

+

e−βx1 [(β + λn ) cos λn x1 + (β − λn ) sin λn x1 ] β 2 + λ2n |} β + γn β + λn − 2 + 2 β + γn2 β + λ2n

−

(5.5.17)

where the parameters .γn and .λn are .

nπ L nπ λn = β − L γn = β +

(5.5.18) (5.5.19)

Equation (5.5.17) is the solution for the axial thermal stress at the origin when the periodic temperature field varies from .−xl to .+x1 . Note that in the limiting case when .xl −→ ∞, which is the condition for a periodic temperature field applied over the entire infinite cylinder, the exponential terms will vanish and the stress is given by a simple series [8]. Let us assume that the temperature field on the slot-cooled combustor be expressed in terms of a cosine function .

T (x) = T0 (1 − cos

πx ) L

(5.5.20)

The Fourier coefficients for this type of temperature distribution are given by a = 2T0

. 0

a1 = −T0

(5.5.21)

5.5 Cylindrical Shells of Gas Turbine Combustor

221

Fig. 5.11 Normalized thermal stress as a function of non-dimensional cooling hole pitch

If the length of the combustor is large compared to the periodic spacing of the cooling holes, it is reasonable to assume that the cylinder is infinite and the temperature acts over the entire length. The bending stress given by Eq. (5.5.17) is σ (0) =

. x

3 EαT0 2 β Rh

|

−π/L π/L + 2 β 2 + (β + π/L)2 β + (β − π/L)2

| (5.5.22)

It may be shown that this solution possesses an extremum [8]. The maximum bending stress will occur when the spacing .(S) between the holes is given by .

2π S = 2L = √ 2β

(5.5.23)

The dimensionless thermal stress generated in an infinite circular shell subjected to a periodic axial temperature variation is determined from (5.5.22) and is expressed as | | 1 σx (0) 3 6(π/L)2 = (5.5.24) . EαT0 2 Rh [β 2 + (β + π/L)2 ][β + (β − π/L)2 ] The plot of this thermal stress as a function of the dimensionless length .(β L) is given in Fig. 5.11. This solution is equivalent to a similar solution obtained by Den Hartog (reference [10]), who solved the fourth-order shell equation directly for sinusoidal and triangular temperature variations in the axial direction. The axial stress given by Eq. (5.5.17) is more general and may be applied to any temperature distribution provided that its Fourier expansion exists. The extremum value obtained from this analysis may provide a good guide to reduce thermal stresses in a combustor. The critical half wave length is calculated from (5.5.23)

222

5 Thermal Stresses in Shells .

π L=√ 2β

(5.5.25)

√ This extremum value, shown in Fig. 5.11 for .β L = π/ 2, will generate the maximum thermal stress. If the cooling hole spacing .(S) is less than or greater than the critical value, the thermal stress produced will be less than the maximum. This criterion can be mathematically expressed as 2π 0 < F˜mn j >=< U˜ mn V˜mn W˜ mn X˜ mn Y˜mn o

(6.2.26)

the solution for the unknown variables in Eq. (6.2.24) in the Laplace domain can be given as Q ∗ (s) ˜mn j (s) = mn j .F (6.2.27) ∗ (s) s Pmn j ∗ where . Q ∗mn j (s) and . Pmn (s) are the polynomial functions of .s. The degree of polyj ∗ ∗ (s). Carrying out the analytical nomial . Q mn j (s) is less than that of polynomial . Pmn j Laplace inverse transform on Eq. (6.2.27) [30], solutions for the unknown variables in the real physical time domain are obtained as

.

Fmn j (t) =

Q ∗mn j (0) ∗ (0) Pmn j

+

nr E Q ∗mn j (srγ ) '

γ=1

∗ (s ) srγ Pmn rγ j

esrγ t + 2Re

ni E Q ∗mn j (siγ ) '

γ=1

∗ (s ) srγ Pmn iγ j

esiγ t

(6.2.28) ∗ where .srγ and .siγ are the real and complex roots of . Pmn (s), and .n r and .n i are the j number of real and imaginary roots, respectively. A prime indicates a derivative with respect to .s. Using Eqs. (6.2.21) and (6.2.28), the unknowns are obtained in the space and time domains. As an example, considering .m = n = 1, the lateral deflection and temperature functions at the midpoint of the middle plane of an FGM plate are obtained for the coupled assumption as .

w0c (a/2, b/2, 0, t) = A˜ 0 + A˜ 1 e−τc1 t + A˜ 2 e−τc2 t + A˜ 3 e−τc3 t + A˜ 4 e−τc4 t + e−τc5 t ( A˜ 5 cos(ωc1 t) + A˜ 6 sin(ωc1 t)) + e−τc6 t ( A˜ 7 cos(ωc2 t) + A˜ 8 sin(ωc2 t)) + e−τc7 t ( A˜ 9 cos(ωc3 t) + A˜ 10 sin(ωc3 t)) + e−τc8 t ( A˜ 11 cos(ωc4 t) + A˜ 12 sin(ωc4 t))

.

+e−τc9 t ( A˜ 13 cos(ωc5 t) + A˜ 14 sin(ωc5 t)) θ1c (a/2, b/2, 0, t) = B˜ 0 + B˜ 1 e−τc1 t + B˜ 2 e−τc2 t + B˜ 3 e−τc3 t + B˜ 4 e−τc4 t + e−τc5 t ( B˜ 5 cos(ωc1 t) + B˜ 6 sin(ωc1 t)) + e−τc6 t ( B˜ 7 cos(ωc2 t) + B˜ 8 sin(ωc2 t)) + e−τc7 t ( B˜ 9 cos(ωc3 t) + B˜ 10 sin(ωc3 t)) + e−τc8 t ( B˜ 11 cos(ωc4 t) + B˜ 12 sin(ωc4 t)) +e−τc9 t ( B˜ 13 cos(ωc5 t) + B˜ 14 sin(ωc5 t))

(6.2.29)

˜ and . Bs ˜ are where .τc is the time constant, .ωc is the frequency of oscillations, and . As the constants of functions. Similarly, for the uncoupled assumption, we have

284

6 Coupled Thermoelasticity of Plates .

w0uc (a/2, b/2, 0, t) = A´ 0 + A´ 1 e−τuc1 t + A´ 2 e−τuc2 t + A´ 3 e−τuc3 t + A´ 4 e−τuc4 t + A´ 5 cos(ωuc1 t) + A´ 6 sin(ωuc1 t) + A´ 7 cos(ωuc2 t) + A´ 8 sin(ωuc2 t) + A´ 9 cos(ωuc3 t) + A´ 10 sin(ωuc3 t) + A´ 11 cos(ωuc4 t) + A´ 12 sin(ωuc4 t) + A´ 13 cos(ωuc5 t) + A´ 14 sin(ωuc5 t) θ1uc (a/2, b/2, 0, t) = B´ 0 + B´ 1 e−τuc1 t + B´ 2 e−τuc2 t + B´ 3 e−τuc3 t + B´ 4 e−τuc4 t

(6.2.30) where the symbols have similar meaning as in the coupled case. Therefore, it can be deduced from Eqs. (6.2.29) and (6.2.30) that the coupling between the strain and temperature fields cause the damping effect on the lateral deflection .w.

6.2.4 Results and Discussion For the problem under consideration, there are no suitable comparison results of coupled thermoelasticity of FGM plates in the literature. To validate the results of the present method for FGM plates, three test examples are examined. First, static bending of a simply supported FGM square plate (.a = .b = 0.8m, .n p =4) under uniform lateral load .107 .N/m2 is considered. The material properties of metal and ceramic constituents are given in Table 6.1. To check the accuracy of the present method, the midpoint lateral deflection of the loaded FGM plate for different thickness ratios (.a/ h) based on the TSDT and FSDT is compared with the analytical solution of an FGM plate based on the classical plate theory [31] and is given in Table 6.2. By increasing .a/ h, results for the classical plate theory (CPT) become close to the results for the TSDT and FSDT models. For instance, for .a/ h = 40, the error of the CPT model is less than 0.5 percent. The first natural frequency of simply supported. Al/Al2 O3 FG square plates for different plate thickness ratios (.a/ h = 5, 10) and power law indices (.n p = 0, 0.5, 1, 10) [14] in Table 6.3. is compared with those reported by Zhao et al. [32] and Matsunaga √ The natural frequencies are given in dimensionless form as.ω¯ = ωh (ρc /E c ), where .ω is the natural frequency, . E c is Young’s modulus, and .ρc is the mass density. The subscript .c stands for ceramic properties. Close agreements are observed between the three studies. Finally, the static bending of a simply supported FGM square plate (.a =.b = 0.6m, 2 2 6 .h = 0.03, and.n p = 2) under thermal load.qh = 10 .W/m with.h c = 10000 W/m K is considered. The material properties of metal and ceramic constituents are the same as in the first example. To validate the accuracy of the present thermal solution method, the midpoint lateral deflection of the heated FGM plate based on the TSDT is compared with the analytical solution of an FGM plate based on the classical plate theory [12] in Fig. 6.2. As seen in the figure, the maximum deviation is less than 4 percent, which is occurred at the center of the plate. Furthermore, the obtained

6.2 Rectangular FGM Plates, TSDT

285

Table 6.1 Material properties of metal and ceramic constituents Metal: .T i − 6Al − 4V Ceramic: . Zr O2 = 66.2 (Gpa) = 0.322 −6 .(1/o K) .αm = 10.3.×10 3 3 .ρm = 4.41.×10 .(kg/m ) .km = 18.1 (W/mK) .cm = 808.3 (J/kg K)

= 117.0 (Gpa) = 0.322 −6 .(1/o K) .αc = 7.11.×10 3 3 .ρc = 5.6.×10 .(kg/m ) .kc = 2.036 (W/mK) .cc = 615.6 (J/kg K)

. Em

. Ec

.ν

.ν

Table 6.2 Deflection of FGM plates under the static mechanical loading, .a = .b =0.8m, uniform mechanical loading .107 .N/m2 , .n p = 4 a/h

TSDT

FSDT

CPT [31]

5 10 20 40

0.000670 0.004584 0.035122 0.27787

0.000644 0.004532 0.035019 0.27767

0.000541 0.004326 0.034605 0.27684

Table 6.3 Comparisons of dimensionless first natural frequency for the simply supported FG square plates .a/ h = 5 .a/ h = 10 .n p = 0 .n p = 0.5 .n p = 1 .n p = 10 .n p = 0 .n p = 0.5 .n p = 1 .n p = 10 Present [32] [14]

0.2142 0.2055 0.2121

0.1828 0.1757 0.1819

0.1650 0.1587 0.1640

0.1344 0.1284 0.1306

0.05794 0.05673 0.05777

0.04917 0.04818 0.04917

0.04438 0.04346 0.04426

0.03673 0.03591 0.03642

analytical solution for the couple thermoelasticity analysis is reduced to those results given by [24] for the thermal-induced vibration of an FGM plate. Consider an FGM plate with ceramic upper surface and metal lower surface. The material properties of metal and ceramic are given in Table 6.1. The mechanical boundary conditions at the edges of the plate are assumed to be simply supported. The thermal boundary conditions at the edges of the plate are assumed to be at ambient temperature .T0 = 293 K. The lower surface of the plate is subjected to a step function thermal shock, while the upper side is subjected to convection with ambient with coefficient .h c = 10000 W/m2 K . Figures 6.3 through 6.5 are presented to show the effect of the power law index of the functionally graded plate (.a = .b = 0.6m, and .a/ h = 20) subjected to a step function thermal shock with the magnitude of .qh = 105 W/m2 . Figures 6.6 and 6.7 are plotted to show the effect of the thickness ratio (.a/ h) of the functionally graded plate (.a = .b = 0.8m, and .n p = 0.5), where .qh = 108 W/m2 . The results in these figures are shown for the coupled thermoelasticity case. In Figs. 6.10 and 6.12, the

286

6 Coupled Thermoelasticity of Plates

0.025 TSDT [Present] CPT [21]

Deflection (m)

0.02

0.015

0.01

0.005

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Distance along x−direction (m)

Fig. 6.2 Comparison of the mid-plate deflection between TSDT and CPT under thermal load 900

800

n =0.5 p

n =2 p

700

n =4

Temperature Change

p

600

500

400

300

200

100

0 −0.015

−0.01

−0.005

0

0.005

0.01

0.015

Height (m)

Fig. 6.3 Temperature change distribution at the midpoint of the plate (.a = .b = 0.6m, = 20) across the thickness direction at .t = 1000s for different power law indices

.a/ h

coupled and uncoupled solutions of the functionally graded plate (.a = .b = 0.6m, and .n p = 2), where .qh = 105 W/m2 , are compared. Figure 6.3 shows the through-thickness distribution of the temperature at the midpoint of the FGM square plate for different power law indices at time .t = 1000s. The temperature distributions under the thermal shock changes nonlinearly through the thickness and the maximum temperature occurs at the bottom surface of the plate where the high heat flux is applied. It is concluded that for higher values of .n p the temperature distribution changes slightly across the thickness of the FGM plate. By increasing the ceramic share of the plate, the gradient of temperature increases in value due to the lower thermal conductivity of ceramic.

6.2 Rectangular FGM Plates, TSDT

287

−3

5

x 10

4.5 n =0.5 p

4

n =2

3.5

np=4

Deflection (m)

p

3 2.5 2 1.5 1 0.5 0 0

100

200

300

400

500 Time (s)

600

700

800

900

1000

Fig. 6.4 Lateral deflection history at the midpoint of the plate (.a = .b = 0.6m, .a/ h = 20) for different power law indices

250 n =0.5 p

n =2 p

Temperature Change

200

n =4 p

150

100

50

0 0

200

400

600

800

1000

1200

1400

1600

1800

2000

Time (s)

Fig. 6.5 Temperature change history at the midpoint of the plate (.a = .b = 0.6m, = 20) at the middle plane for different power law indices

.a/ h

Figure 6.4 plots the lateral deflection at the midpoint versus time for different values of the power law index .n p . It is seen in Fig. 6.4 that with the increase of the power law index .n p , the midpoint lateral deflection of the FGM plate is generally decreased because of a decline in the temperature gradient in the plate. The midpoint temperature history at the mid-plane of the plate is shown in Fig. 6.5. Due to the applied step function thermal shock, the beam temperature peaks to a maximum value, and then is diffused with the time. The figure shows that for more metal-rich FGM (larger values of .n p ) the temperature distribution decreases in value due to higher conductivity of metal. Figure 6.6 shows the midpoint lateral deflection of the loaded FGM plate with respect to time for different thickness ratios (.a/ h) based on the TSDT. As shown in

288

6 Coupled Thermoelasticity of Plates −3

x 10

8 7 a/h=5 a/h=10

6 Deflection (m)

a/h=20 a/h=50

5 4 3 2 1 0 0

0.2

0.4

0.6

0.8

1 Time (s)

1.2

1.4

1.6

2

1.8

4

x 10

Fig. 6.6 Lateral deflection history at the midpoint of the plate (.a = .b = 0.8m, .n p = 0.5) for different thickness ratios 1200

Temperature Change

1000 a/h=5 a/h=10 a/h=20 a/h=50

800

600

400

200

0 0

0.2

0.4

0.6

0.8

1 Time (s)

1.2

1.4

1.6

1.8

2 4

x 10

Fig. 6.7 Temperature change history at the midpoint of the plate (.a = .b = 0.8m, = 0.5) at the middle plane for different thickness ratios

.n p

Figure 6.6, the higher the magnitude of the thickness ratio, the larger the maximum of the lateral deflection. We see in Fig. 6.7 that as the thickness ratio (.a/ h) decreases, the temperature distribution versus time at the upper side of the plate also increases. The next few figures show the coupling effect for the functionally graded plate under thermal shock. Figure 6.8 shows the lateral deflection history with a proper scale at the midpoint of the FGM plate (.a = .b = 0.6m, .a/ h = 20, and .n p = 2) for the coupled solution, where the coupled solution vibrate with a very small amplitude. Investigating Eqs. (6.2.29) and (6.2.30), one sees that the coupling between the strain and temperature fields has a damping effect on the lateral deflection .w shown in Fig. 6.8. The midpoint temperature history at the middle plane of the FGM plate

6.2 Rectangular FGM Plates, TSDT

289

−3

x 10

2.5391

Deflection (m)

2.5391

2.5391

2.5391

2.5391

2.5391 0

1000

2000

3000

4000

5000

6000

7000

8000

Time (s)

Fig. 6.8 Lateral deflection history at the midpoint of the plate (.a = .b = 0.6m, .a/ h = 20, .n p = 2), showing the effects of coupling 151.1484

Temperature Change

151.1483

151.1482

151.1481

151.148

151.1479

151.1478

0

1000

2000

3000

4000

5000

6000

7000

8000

Time (s)

Fig. 6.9 Temperature change history at the midpoint of the plate (.a = .b = 0.6m, = 20, .n p = 2) at the middle plane, showing the effects of coupling

.a/ h

(.a = .b = 0.6m, .a/ h = 20, and .n p = 2) for coupled solution is shown in Fig. 6.9 with proper scale. Although the temperature distribution vibrates around the steady state temperature, the amplitude of vibration is negligible. Figure 6.10 shows the lateral deflection history at the midpoint of the FGM plate (.a = .b = 0.8m, .a/ h = 20, .n p = 0.5) for the coupled and uncoupled solutions. As shown in this figure, the difference between the coupled and uncoupled solutions for the lateral deflection is negligible. However, the difference of lateral deflection history between the coupled and uncoupled solutions, .wc − wuc , is shown in Fig. 6.11 for the FGM plate (.a = .b = 0.8m, .a/ h = 20, and .n p = 0.5). Figure 6.12 shows the midpoint temperature history at the middle plane of the FGM plate for coupled and uncoupled solutions where no considerable difference is observed.

290

6 Coupled Thermoelasticity of Plates 0.09 0.08

Coupled Uncoupled

0.07

Deflection (m)

0.06 0.05 0.04 0.03 0.02 0.01 0 0

500

1000

1500

2000

2500 Time (s)

3000

3500

4000

4500

5000

Fig. 6.10 Lateral deflection history at the midpoint of the plate (.a = .b = 0.8m, .a/ h = 20, .n p = 0.5), showing the effects of coupling −5

x 10 0 −0.5 −1

−2 −2.5

c

w −wuc (m)

−1.5

−3 −3.5 −4 −4.5 −5 0

500

1000

1500

2000

2500 Time (s)

3000

3500

4000

4500

5000

Fig. 6.11 Difference between the coupled and uncoupled solutions for the lateral deflection at the midpoint of the plate (.a = .b = 0.8m, .a/ h = 20, .n p = 0.5), showing the effects of coupling

In general, the coupled and uncoupled thermoelasticity solutions of the structural problems do not have significant differences in the distribution of displacement and stresses. However, for the coupled problems, the stress and temperature wave fronts appear in generalized thermoelasticity, which may cause structural damage. In addition, the coupled solution for the major part of the stress distribution (except around the wave front) is lower than that of the uncoupled solution. The magnitudes of the stress and temperature wave fronts depend on how short the thermal shock is applied over the structure. The shorter duration of the applied thermal shock, the higher the magnitude of the stress wave front becomes. In Figs. 6.8–6.12, wave fronts for the displacement and temperature do not appear. The reason is that the stress and temperature wave fronts appear in any structure when the solution is based on the gen-

6.2 Rectangular FGM Plates, TSDT

291

3500

Coupled Uncoupled

3000

Temperature Change

2500

2000

1500

1000

500

0 0

500

1000

1500

2000

2500 Time (s)

3000

3500

4000

4500

5000

Fig. 6.12 Temperature change history at the midpoint of the plate (.a =.b = 0.8m,.a/ h = 20,.n p = 0.5) at the middle plane, showing the effects of coupling

eralized thermoelasticity equations. Furthermore, when the flexural model is used, such as in beam, plate, and shell theories, the wave fronts do not appear in the solution. The reason is simple: we have lumped the stress through the thickness. Figure 6.12 for the temperature distribution does not show a wave front due to the classical coupled thermoelasticity assumptions used in this work. According to the classical coupled thermoelasticity theory, the temperature equation is of parabolic type and the speed of propagation for thermal wave is infinity. In the present section, the coupled thermoelasticity of a plate based on the thirdorder shear deformation theory with functionally graded materials is investigated. The plate is subjected to a thermal shock of step function type on the lower side. The upper side of the plate is assumed to have convection with the ambient. Boundary conditions of the plate are taken to be simply supported with ambient temperature at the ends of the plate. To solve the problem, the double Fourier series expansion is used. Moreover, to treat the time dependency, the Laplace transform technique is applied. The inverse Laplace transform is carried out analytically. Results show that for larger values of power law indices, which provide the most metal-rich FGM, the lateral deflection of an FGM plate due to an applied thermal shock decreases proportionally. By increasing the metal share of the FGM plate, the distribution of temperature changes slightly across the thickness of the plate. Generally, it can be said that there is no significant difference between the coupled and uncoupled solutions. However, the effect of coupling is in the form of damping; it decreases the amplitude of vibration and increases the frequency of the vibrations with the increase of time. Another important conclusion is related to the appearance of stress or temperature wave fronts. The wave fronts for displacement and temperature do not appear. The stress and temperature wave fronts appear in any structure under thermal shock load, but they are only detected when the solution is based on the generalized thermoelasticity equations. When the flexural model is used, the wave

292

6 Coupled Thermoelasticity of Plates

fronts do not appear in the solution. The reason is being that the stresses are lumped through the thickness.

Appendix

.

A¯ 1 =

{

E(z) dz, 1 − ν2

A¯ 2 =

{

E(z) dz 2(1 + ν)

{ { E(z) ν E(z) ¯ 4 = − cs E(z) z 3 dz dz + dz, A 1 − ν2 2(1 + ν) 1 − ν2 { { ν E(z) 3 E(z) 3 A¯ 5 = − cs z dz − cs z dz 1 − ν2 1+ν { { E(z) E(z) 3 A¯ 6 = z dz − cs z dz 1 − ν2 1 − ν2 { { E(z) 3 E(z) A¯ 7 = zdz − cs z dz 2(1 + ν) 2(1 + ν) { { { { ν E(z) 3 E(z) 3 E(z) ν E(z) A¯ 8 = zdz − c z dz + zdz − cs z dz s 1 − ν2 1 − ν2 2(1 + ν) 2(1 + ν) { { ν E(z)α(z) E(z)α(z) A¯ 9 = − dz − dz 1 − ν2 1 − ν2 { { E(z)α(z) z ν E(z)α(z) z A¯ 10 = − dz − dz 1 − ν2 h 1 − ν2 h { { E(z)α(z) z 2 ν E(z)α(z) z 2 A¯ 11 = − ( dz − ( ) dz ) 1 − ν2 h 1 − ν2 h { { ν E(z)α(z) z 3 E(z)α(z) z 3 A¯ 12 = − ( ) dz − ( ) dz 1 − ν2 h 1 − ν2 h

A¯ 3 =

{

A¯ 13 = −I0 , B¯ 1 =

.

B¯ 6 B¯ 7

E(z) dz + 2(1 + ν)

{

A¯ 15 = −J1

ν E(z) dz, 1 − ν2

B¯ 2 =

{

E(z) dz 2(1 + ν)

{ { E(z) ¯ 4 = − cs E(z) z 3 dz − cs ν E(z) z 3 dz dz, B 1 − ν2 1+ν 1 − ν2 { E(z) 3 z dz = − cs 1 − ν2 { { { { ν E(z) 3 E(z) 3 ν E(z) E(z) = zdz − c z dz + zdz − cs z dz s 1 − ν2 1 − ν2 2(1 + ν) 2(1 + ν) { { E(z) ν E(z) 3 = zdz − cs z dz 2(1 + ν) 2(1 + ν)

B¯ 3 = B¯ 5

{

A¯ 14 = cs I3 ,

{

6.2 Rectangular FGM Plates, TSDT

293

{ E(z) 3 E(z) zdz − cs z dz 1 − ν2 1 − ν2 { { E(z)α(z) ν E(z)α(z) B¯ 9 = − dz − dz 1 − ν2 1 − ν2 { { ν E(z)α(z) z E(z)α(z) z B¯ 10 = − dz − dz 1 − ν2 h 1 − ν2 h { { E(z)α(z) z 2 ν E(z)α(z) z 2 B¯ 11 = − ( ) dz − ( ) dz 1 − ν2 h 1 − ν2 h { { E(z)α(z) z 3 ν E(z)α(z) z 3 B¯ 12 = − ( ) dz − ( ) dz 1 − ν2 h 1 − ν2 h B¯ 13 = −I0 , B¯ 14 = cs I3 , B¯ 15 = −J1 B¯ 8 =

C¯ 1 = .

{

{

C¯ 3 =

cs {

E(z) 3 z dz, 1 − ν2

cs

{ cs

{ cs

E(z) 3 z dz + 1+ν

ν E(z) 3 z dz, 1 − ν2

{

C¯ 4 =

cs {

ν E(z) 3 z dz 1 − ν2

cs

E(z) 3 z dz 1 − ν2

{ E(z) 6 E(z) dz, C¯ 6 = − cs2 z dz 2(1 + ν) 1 − ν2 { { 2ν E(z) 6 ¯ 8 = (1 − cs' z 2 + cs' 2 z 4 ) E(z) dz C¯ 7 = − cs2 z dz, C 1 − ν2 2(1 + ν) { { E(z) 6 E(z) ' ' C¯ 9 = − cs2 dz z dz, C¯ 10 = (1 − cs z 2 + cs2 z 4 ) 1 − ν2 2(1 + ν) { { E(z) E(z) C¯ 11 = (cs z 4 − cs2 z 6 ) dz, C¯ 12 = (cs z 4 − cs2 z 6 ) dz 2 1−ν 1 − ν2 { { E(z) ¯ 14 = (1 − cs' z 2 + cs' 2 z 4 ) E(z) dz C¯ 13 = (cs z 4 − cs2 z 6 ) dz, C 1 − ν2 2(1 + ν) { { E(z)α(z) 3 E(z) C¯ 15 = (cs z 4 − cs2 z 6 ) z dz dz, C¯ 16 = − cs 1 − ν2 1−ν { { E(z)α(z) 3 E(z)α(z) z 4 z dz, dz C¯ 18 = − cs C¯ 17 = − cs 1−ν 1−ν h { { E(z)α(z) z 4 E(z)α(z) z 5 C¯ 19 = − cs dz dz, C¯ 20 = − cs 1−ν h 1 − ν h2 { { 6 E(z)α(z) z 5 ¯ 22 = − cs E(z)α(z) z dz dz, C C¯ 21 = − cs 1 − ν h2 1 − ν h3 { E(z)α(z) z 6 dz, C¯ 24 = −cs I3 , C¯ 25 = −cs I3 C¯ 23 = − cs 1 − ν h3 C¯ 5 =

{

E(z) 3 z dz + 1+ν

C¯ 2 =

'

C¯ 26 = −I0 , D¯ 1 = D¯ 3 =

'

(1 − cs z 2 + cs2 z 4 )

C¯ 27 = cs2 I6 ,

{ (z − cs z 3 )

E(z) dz, 2(1 − ν)

(z − cs z 3 )

E(z) dz, 1 − ν2

{

C¯ 28 = cs2 I6 , D¯ 2 = D¯ 4 =

{

C¯ 29 = −J4 ,

{ (z − cs z 3 )

C¯ 30 = −J4

E(z) dz 2(1 + ν)

(−cs z 4 + cs2 z 6 )

E(z) dz 1 − ν2

294

6 Coupled Thermoelasticity of Plates { E(z) ¯ 6 = (−1 + 2cs' z 2 − cs' 2 z 4 ) E(z) dz dz, D 1 − ν2 2(1 + ν) { E(z) 2 4 2 6 ¯ D7 = (z − 2cs z + cs z ) dz 2(1 − ν) { E(z) D¯ 8 = (z 2 − 2cs z 4 + cs2 z 6 ) dz 2(1 + ν) { E(z) D¯ 9 = (z 2 − 2cs z 4 + cs2 z 6 ) dz 1 − ν2 { { E(z) E(z)α(z) ' ' D¯ 10 = (−1 + 2cs z 2 − cs2 z 4 ) dz, D¯ 11 = (−z + cs z 3 ) dz 2(1 + ν) 1−ν { { E(z)α(z) z E(z)α(z) z 2 D¯ 12 = (−z + cs z 3 ) dz, D¯ 13 = (−z + cs z 3 ) ( ) dz 1−ν h 1−ν h { E(z)α(z) z 3 D¯ 14 = (−z + cs z 3 ) ( ) dz, D¯ 15 = −J1 1−ν h

D¯ 5 = .

{

(−cs z 4 + cs2 z 6 )

D¯ 16 = cs J4 , E¯ 1 = E¯ 3 = E¯ 5 = E¯ 7 = E¯ 9 =

{ (z − cs z 3 )

E(z) dz, 2(1 − ν)

(z − cs z 3 )

E(z) dz, 1 − ν2

{ {

(−cs z 4 + cs2 z 6 ) {

E¯ 2 = E¯ 4 =

E(z) dz, 1 − ν2

(z − cs z 3 )

(z 2 − 2cs z 4 + cs2 z 6 )

E(z) dz, 1 − ν2

E(z) dz 2(1 + ν)

(−cs z 4 + cs2 z 6 )

E¯ 6 =

E(z) dz, 2(1 − ν)

{

{

{

(z 2 − 2cs z 4 + cs2 z 6 ) {

{

E(z) dz 1 − ν2

'

'

(−1 + 2cs z 2 − cs2 z 4 ) E¯ 8 =

E¯ 10 =

{

{

E(z) dz 2(1 + ν)

(z 2 − 2cs z 4 + cs2 z 6 ) '

'

E(z) dz 2(1 + ν)

(−1 + 2cs z 2 − cs2 z 4 )

E(z) dz 2(1 + ν)

{ E(z)α(z) E(z)α(z) z dz, E¯ 12 = (−z + cs z 3 ) dz 1−ν 1−ν h { { E(z)α(z) z 2 E(z)α(z) z 3 ( ) dz, ( ) dz = (−z + cs z 3 ) E¯ 14 = (−z + cs z 3 ) 1−ν h 1−ν h

E¯ 11 = E¯ 13

D¯ 17 = −K 2

(−z + cs z 3 )

E¯ 15 = −J1 , E¯ 16 = cs J4 , E¯ 17 = −K 2 { { F¯1 = K (z)dz, F¯2 = K (z)dz { z z 1 K (z)dz, F¯5 = K (z)dz, F¯6 = h c h h 2 { { z z 1 F¯7 = ( )2 K (z)dz, F¯8 = ( )2 K (z)dz, F¯9 = − h c h h 4 { { z z 1 F¯10 = ( )3 K (z)dz, F¯11 = ( )3 K (z)dz, F¯12 = h c h h 8 { { { z z F¯13 = −ρ(z)cv (z)dz, F¯15 = −ρ(z)cv (z)( )2 dz F¯14 = −ρ(z)cv (z) dz, h h F¯3 = −h c ,

F¯4 =

{

6.2 Rectangular FGM Plates, TSDT F¯16 = F¯18 = F¯20 = F¯22 =

.

G¯ 2 =

{ {

z −ρ(z)cv (z)( )3 dz, h

F¯17 =

E(z) α(z)T0 dz, 1 − 2ν

F¯19 =

− {

cs z 3 {

295

E(z) α(z)T0 dz, 1 − 2ν

K (z)zdz,

h G¯ 3 = h c , 2

− {

F¯21 =

E(z) (−z + cs z 3 )α(z)T0 dz, 1 − 2ν

{

{

E(z) α(z)T0 dz 1 − 2ν

cs z 3 {

E(z) α(z)T0 dz 1 − 2ν

E(z) (−z + cs z 3 )α(z)T0 dz 1 − 2ν { F¯23 = 1, G¯ 1 = K (z)zdz

G¯ 4 =

{

z2 K (z)dz h

{ { 3 1 h hc z G¯ 6 = − − K (z)dz, G¯ 7 = )K (z)dz 2 2 h h2 { { 3 z ¯ 9 = h h c − K (z) 2z dz )K (z)dz, G G¯ 8 = h2 2 4 h2 { 4 { 4 z z K (z)dz, G¯ 11 = K (z)dz G¯ 10 = h3 h3 { { 3z 2 h hc − K (z) 3 dz, G¯ 13 = −ρ(z)cv (z)zdz G¯ 12 = − 2 8 h { { 2 z z3 G¯ 15 = −ρ(z)cv (z) 2 dz G¯ 14 = −ρ(z)cv (z) dz, h h { { 4 z E(z) α(z)zT0 dz G¯ 17 = − G¯ 16 = −ρ(z)cv (z) 3 dz, h 1 − 2ν { { E(z) E(z) G¯ 18 = − α(z)zT0 dz, α(z)T0 dz G¯ 19 = cs z 4 1 − 2ν 1 − 2ν { { E(z) E(z) G¯ 20 = cs z 4 α(z)T0 dz, α(z)T0 dz G¯ 21 = (−z 2 + cs z 4 ) 1 − 2ν 1 − 2ν { E(z) h G¯ 22 = (−z 2 + cs z 4 ) α(z)T0 dz, G¯ 23 = 1 − 2ν 2 { { h2 H¯ 2 = K (z)z 2 dz, H¯ 3 = − h c H¯ 1 = K (z)z 2 dz, 4 { 3 { 3 { z z h2 2z H¯ 4 = K (z)dz, H¯ 5 = K (z)dz, H¯ 6 = h c − K (z) dz h h 8 h { 4 { 4 z z K (z)dz, H¯ 8 = K (z)dz H¯ 7 = h2 h2 { { 5 h2 4z 2 z H¯ 9 = − h c − K (z) 2 dz, H¯ 10 = K (z)dz 16 h h3 { 5 { 2 3 z ¯ 12 = h h c − K (z) 6z dz H¯ 11 = K (z)dz, H h3 32 h3 G¯ 5 =

{

z2 K (z)dz, h

296

6 Coupled Thermoelasticity of Plates H¯ 13 = H¯ 15 = H¯ 17 = H¯ 19 = H¯ 21 =

{ −ρ(z)cv (z)z 2 dz, { −ρ(z)cv (z) { − {

z4 dz, h2

{

H¯ 16 =

−ρ(z)cv (z)

−ρ(z)cv (z)

H¯ 18 =

E(z) α(z)T0 dz, 1 − 2ν

H¯ 20 =

(−z 3 + cs z 5 )

z3 dz h

{

E(z) α(z)T0 z 2 dz, 1 − 2ν

cs z 5 {

H¯ 14 =

E(z) α(z)T0 dz, 1 − 2ν

{ − {

z5 dz h3

E(z) α(z)T0 z 2 dz 1 − 2ν E(z) α(z)T0 dz 1 − 2ν { E(z) α(z)T0 dz = (−z 3 + cs z 5 ) 1 − 2ν

cs z 5 H¯ 22

{ { h2 h3 , I¯1 = K (z)z 3 dz, hc I¯2 = K (z)z 3 dz, I¯3 = H¯ 23 = 4 8 { 4 { 4 { z z h3 3z 2 I¯4 = K (z)dz, I¯5 = K (z)dz, I¯6 = − h c − K (z) dz h h 16 h { 5 { 5 { 3 z z ¯8 = ¯9 = h h c − K (z) f rac6z 3 h 2 dz K (z)dz, I K (z)dz, I I¯7 = h2 h2 32 { 6 { 6 { 3 4 z z ¯11 = ¯12 = − h h c − K (z) 9z K (z)dz, I K (z)dz, I I¯10 = h3 h3 64 h3 { { { 4 z z5 I¯13 = −ρ(z)cv (z)z 3 dz, I¯15 = −ρ(z)cv (z) 2 dz I¯14 = −ρ(z)cv (z) dz, h h { { 6 z E(z) α(z)T0 z 3 dz I¯17 = − I¯16 = −ρ(z)cv (z) 3 dz, h 1 − 2ν { { E(z) E(z) I¯18 = − α(z)T0 z 3 dz, α(z)T0 dz, I¯19 = cs z 6 1 − 2ν 1 − 2ν { { E(z) E(z) I¯20 = cs z 6 α(z)T0 dz, α(z)T0 dz I¯21 = (−z 4 + cs z 6 ) 1 − 2ν 1 − 2ν { E(z) h3 α(z)T0 dz, I¯23 = I¯22 = (−z 4 + cs z 6 ) 1 − 2ν 8

6.3 Coupled Thermoelasticity of FGM Annular Plate The basic theory and applications of the coupled thermoelasticity problems in mechanics are well discussed and presented in [28] and the methods of solutions are given in [33]. Eslami et al. [34] formulated the coupled thermoelasticity of shell of revolution based on the Flugee second-order shell theory with the linear temperature distribution across the shell thickness. The effects of normal stress and coupling term were studied. Bahtui and Eslami [35] studied the coupled thermoelasticity

6.3 Coupled Thermoelasticity of FGM Annular Plate

297

problem of a titanium–zirconia functionally graded cylindrical shell under impulsive thermal shock load. The equations were obtained based on the second-order shear deformation shell theory and classic linear theory of thermoelasticity. Temperature distribution across the thickness was assumed to be linear and the results of .C 0 -continuous and .C 1 -continuous element types were compared to each other. Babaei et al. [36] presented the finite element solution of FGM beams subjected to lateral thermal shock loads. The results were obtained under coupled thermoelastic and Euler–Bernoulli beam assumption, where the .C 1 -continuous shape function was employed in the finite element model. Coupled thermoelasticity of constant cross section beam under thermal and mechanical shock loads is considered by Eslami and Vahedi [37]. The beam is modeled as a one-dimensional elasticity element, where the thermal shock is assumed to be a sudden temperature change at the end of the beam. Large amplitude thermoelastic vibrations of the Timoshenko beam, which is subjected to short heat flux and harmonic mechanical loading, are studied by Manoacha and Ribeirob [38]. Excitation frequencies of mechanical load are close to the natural frequencies of the beams. Their results show a significant influence of thermal loads on the dynamic behavior of structures. Golmakani and Kadkhodayan [39] have studied axisymmetric large deflection analysis of circular and annular plates under thermomechanical loading. The firstorder shear deformation theory is taken into account and temperature-dependent material properties are assumed. Also, they have compared the results obtained using the third-order and first-order shear deformation theories [40]. The temperature, displacement, and stress fields in a multilayered composite plate, which is suddenly heated, are discussed by Atarashi and Minagawa while using the coupled theory of thermoelasticity [41]. Nakajo and Hayashi [42] assumed a simply supported circular plate, where a uniform heat flux is applied to one side of it. They have given the analytical solution of the problem by considering the uncoupled theory of thermoelasticity. Irie et al. [43] have presented an analytical solution for the natural frequencies of an annular plate based on the Mindlin theory. The results are obtained under nine combinations of boundary conditions for the six modes. Davies and Martin [44] carried out a comprehensive review and comparison of the numerical inverse of Laplace transform methods in 1979. In 1999, Brancik [45] wrote a numeric code in MATLAB software environment based on the application of fast Fourier transforms where the Epsilon algorithm is used to speed up the convergence. The present section is focused on the study of an annular plate made of functionally graded materials under the assumption of coupled theory of thermoelasticity [46]. A uniform lateral impulsive heat flux is applied to the top surface of the plate, while the other surfaces are thermally insulated. The equations of motion are obtained in terms of the displacement components using the first-order shear deformation theory. A quadratic function of variable .z is assumed to show the temperature distribution in thickness direction and then the energy equation is accordingly obtained. The Laplace transformation and the Galerkin finite element method are applied to the dimensionless equations. A numerical code, which is written in MATLAB environment, is

298

6 Coupled Thermoelasticity of Plates

Fig. 6.13 Geometry of annular plate

employed to solve the coupled system of equations in the Laplace domain. Finally, to calculate the results in real-time domain, numerical inversion of the Laplace transform (NILT) is used. The effect of power law index, the coupling coefficient, and different geometric proportions are studied.

6.3.1 Governing Equations A circular FGM plate with a central hole and an inner radius . Rin , outer radius . Rout , and thickness .h, as shown in Fig. 6.13, is considered. The annular FGM plate is made of metal and ceramic, where the material properties continuously change in the thickness direction as a function of location. In this study, we assume that the power law represents the variation of material properties in the thickness direction. Thus, the effective material properties based on the linear rule of mixture are defined as (

h − 2z . Fz = Fc + (Fm − Fc ) 2h

)ξ (6.3.1)

where . Fz may be any of the FGM properties and .ξ is the power law index. Here, . Fm and . Fc are related properties of metal and ceramic constituents, respectively. Figure 6.14 shows changes in the volume fraction of metal across thickness of the plate.

6.3 Coupled Thermoelasticity of FGM Annular Plate

299

1.2 ξ=0 ξ=0.1 ξ=0.5 ξ=1 ξ=2 ξ=10

1

ξ=1010

Metal volume fraction

0.8

0.6

0.4

0.2

0

−0.2 −0.1

−0.05

0.05 0 Dimensionless thickness (h=0.2)

0.1

0.15

Fig. 6.14 Changes in the volume fraction of metal for different values of .ξ across the thickness

Figure 6.14 shows changes in the volume fraction of metal across thickness of the plate.

6.3.2 Equations of Motion In order to take into consideration the shear deformations (.γr z = γθz /= 0), components of displacement are assumed on the basis of first-order shear deformation plate approximation as follow u r (r, θ, z, t) = u 0 (r, θ, t) + zφr (r, θ, t) .

u θ (r, θ, z, t) = v0 (r, θ, t) + zφθ (r, θ, t)

(6.3.2)

u z (r, θ, z, t) = w0 (r, θ, t) where .u 0 , .v0 , and .w0 denote the displacement components in the plane .z = 0 and in directions of the .r , .θ, and .z axes, respectively. Also, .φr and .φθ represent the rotations of a transverse normal about the .θ and .r axes, respectively [47]. Since the rotations are not simply calculated in terms of the displacement components, the rotations are considered separately as unknown functions.

300

6 Coupled Thermoelasticity of Plates

The linear strain–displacement relations in polar coordinates are ∂φr ∂u 0 +z ∂r( ∂r ) ( ) 1 1 ∂v0 ∂φθ u0 + +z φr + εθθ = r ∂θ r ∂θ ) ) ( | ( | ∂v0 ∂φθ 1 ∂φr 1 ∂u 0 − v0 + +z − φθ + γr θ = . r ∂θ ∂r r ∂θ ∂r ∂w0 γr z = φr + ∂r 1 ∂w0 γθz = φθ + r ∂θ εzz = 0 εrr =

(6.3.3)

According to Hooke’s law in the state of plane stress, stress–strain relations for the linear elastic FGM are [28] ⎫ ⎧ ⎫ ⎤⎧ ⎡ 1ν 0 0 0 εrr − αz /T ⎪ σrr ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ν 1 0 0 0 ⎥⎪ εθθ − αz /T ⎪ ⎨ ⎬ ⎨ σθθ ⎪ ⎬ ⎥ ⎢ Ez ⎢ 1−ν ⎥ 0 0 0 0 σr θ = γ (6.3.4) . rθ 2 ⎥ 2 ⎢ ⎪ ⎪ 2 ⎪ ⎪ ⎪ σr z ⎪ ⎪ 1 − ν ⎣ 0 0 0 1−ν 0 ⎦ ⎪ ζ γ ⎪ ⎪ ⎪ ⎪ rz ⎪ ⎪ ⎪ ⎪ 2 ⎩ ⎭ ⎩ ⎭ ζ 2 γθz σθz 0 0 0 0 1−ν 2 where .αz is the coefficient of thermal expansion, . E z is Young’s modulus, .ν is Poisson’s ratio, and .ζ 2 is a shear correction coefficient and it is assumed to be .5/6. The variable ./T is the temperature change across the thickness. Due to the fact that the thickness of plate is thin, ./T is approximated as a quadratic polynomial function of . z as 2 ./T (r, θ, z, t) = T0 (r, θ, t) + zT1 (r, θ, t) + z T2 (r, θ, t) (6.3.5) Here, .T0 , T1 , and .T2 are unknown functions which should be calculated with the aid of energy equation. This is explained in Sect. 6.2.2. Since thickness of the plate is small, the stress distributions are integrated through the thickness of the plate and the force and moment resultants are defined. The force and moment resultants for the circular plate in terms of the stress components are { = { .

=

h/2

−h/2

h/2 −h/2

dz

dz

=

{

h/2 −h/2

zdz

(6.3.6)

6.3 Coupled Thermoelasticity of FGM Annular Plate

301

Substituting Hooke’s law in Eq. (6.3.6) and then substituting strain–displacement relations into the obtained results, components of the force and moment resultants in terms of the displacement components are obtained as | ( )| ( )| ∂φr ∂u 0 ν ν ∂v0 ∂φθ + u0 + + E2 + φr + Nrr = E 1 ∂r r ∂θ ∂r r ∂θ − β1 (T0 ) − β2 (T1 ) − β3 (T2 ) | | ( )| ( )| ∂φr ∂u 0 1 1 ∂v0 ∂φθ + u0 + + E2 ν + φr + Nθθ = E 1 ν ∂r r ∂θ ∂r r ∂θ − β1 (T0 ) − β2 (T1 ) − β3 (T2 ) ( ) ∂w0 Q r = ζ 2 E` 1 φr + ∂r ( ) (6.3.7) . 1 ∂w0 Q θ = ζ 2 E` 1 φθ + r ∂θ | | ( )| ( )| ∂u 0 ∂φr ν ν ∂v0 ∂φθ + u0 + + E3 + φr + Mrr = E 2 ∂r r ∂θ ∂r r ∂θ − β2 (T0 ) − β3 (T1 ) − β4 (T2 ) | | ( )| ( )| ∂φr ∂u 0 1 1 ∂v0 ∂φθ + u0 + + E3 ν + φr + Mθθ = E 2 ν ∂r r ∂θ ∂r r ∂θ − β2 (T0 ) − β3 (T1 ) − β4 (T2 ) |

where

{

) Ez ( 1, z, z 2 dz 2 −h/2 1 − v { h/2 ( ) = βz 1, z, z 2 , z 3 dz;βz = αz E z h/2

E 1,2,3 = .

β1,2,3,4 E` 1 =

{

(6.3.8)

−h/2

h/2 −h/2

Ez dz 2 (1 + ν)

Assuming that there are no edge loads and moments, the Hamilton principle is used to obtain the equations of motion for circular plate as { .

T 0

{ {

h/2

y

−h/2

{

T

− 0

{

y

|

| (σrr δεrr + σθθ δεθθ + σr θ δγr θ + σr z δγr z + σzθ δγzθ ) dzr dr dθdt − ρ (u˙ r δ u˙ r + u˙ θ δ u˙ θ + u˙ z δ u˙ z )

(P) δw0 r dr dθdt = 0

(6.3.9) Proper substitution for the stress components into the above equation and setting the coefficients of .δu 0 , δv0 , δw0 , δφr , and δφθ equal to zero, the governing equations of motion are obtained as

302

6 Coupled Thermoelasticity of Plates

| | ∂2u0 ∂ 2 φr 1 ∂ ∂ Nr θ + I0 2 + I1 2 = 0 δu : − (r Nrr ) − Nθθ + r ∂r ∂θ ∂t ∂t | | 2 ∂ v0 ∂ 2 φθ 1 ∂ Nθθ ∂ δv : − + (r Nr θ ) + Nr θ + I0 2 + I1 2 = 0 r ∂θ ∂r ∂t ∂t | | 2 1 ∂ ∂ w0 ∂ Qθ . δw : − + I0 −P=0 (r Q r ) + r ∂r ∂θ ∂t 2 | | ∂2u0 ∂ 2 φr 1 ∂ ∂ Mr θ δφr : − + Q r + I1 2 + I2 2 = 0 (r Mrr ) − Mθθ + r ∂r ∂θ ∂t ∂t | | ∂ ∂ 2 v0 ∂ 2 φθ 1 ∂ Mθθ + δφθ : − (r Mr θ ) + Mr θ + Q θ + I1 2 + I2 2 = 0 r ∂θ ∂r ∂t ∂t (6.3.10) where { h/2 ) ( . I0,1,2 = ρ 1, z, z 2 dz (6.3.11) −h/2

The natural boundary conditions at the edge of the plate are obtained as δu 0 = 0 or Nrr n r + Nr θ n θ = 0 δv0 = 0 or Nr θ n r + Nθθ n θ = 0 .

δφr = 0 or Mrr n r + Mr θ n θ = 0 δφθ = 0 or Mr θ n r + Mθθ n θ = 0

(6.3.12)

δw0 = 0 or Q r n r + Q θ n θ = 0 ) = 0 ; v0 = Considering the symmetry of geometry and loading condition, (. ∂(··· ∂θ 0 ; φθ = 0). Assuming symmetry condition and substituting Eq. (6.3.7) into Eq. (6.3.10), the governing equations of motion in terms of the displacement components are obtained as ( ) ( ) 1 1 ∂2 ∂2 1 ∂ 1 ∂ + 2 u 0 + E2 − 2 + + 2 φr δu 0 : E 1 − 2 + r r ∂r ∂r r r ∂r ∂r 2 2 ∂ u0 ∂ φr ∂ − (β1 T0 + β2 T1 + β3 T2 ) − I0 2 − I1 2 = 0 ∂r ∂t ∂t ( ( ) ) 1 1 ∂ ∂2 ∂ ∂ 2 w0 2 ` . δw0 : ζ E + φr + ζ 2 E` 1 + 2 w0 − I 0 +P=0 1 r ∂r r ∂r ∂r ∂t 2 ( ) ( ) 1 1 ∂2 ∂2 1 ∂ 1 ∂ δφr : E 2 − 2 + + 2 u 0 + E3 − 2 + + 2 φr − ζ 2 E` 1 φr r r ∂r ∂r r r ∂r ∂r 2 ∂ ∂w0 ∂ u0 ∂ 2 φr − − ζ 2 E` 1 (β2 T0 + β3 T1 + β4 T2 ) − I1 2 − I2 2 = 0 ∂r ∂r ∂t ∂t (6.3.13)

6.3 Coupled Thermoelasticity of FGM Annular Plate

303

The boundary conditions from Eq. (6.3.12) reduce to δu 0 = 0 or Nrr = 0 .

δφr = 0 or Mrr = 0 δw0 = 0 or Q r = 0

(6.3.14)

6.3.3 Energy Equation The energy equation based on the classical coupled theory of thermoelasticity for the isotropic heterogeneous medium (the FGM) appears in the form [28] ∇. (K z ∇T ) − ρz cz T˙ − Ta βz ∇.u˙ + R = 0

.

(6.3.15)

where . K z is the thermal conductivity, .ρz is the mass density, .cz is the specific heat, β is the stress-temperature moduli, .u is the displacement vector, .Ta is the reference temperature, .T is the absolute temperature, and . R is the internal heat generation per unit volume per unit time. The parameters with subscript .z for the FGM plate are assumed to be functions of the .z coordinate following Eq. (6.3.1). Thus, substituting Eq. (6.3.2) into Eq. (6.3.15), the energy equation for the symmetry condition, loading and geometrical, is obtained in terms of variables .r , .z, and .t as ) ( )| | ( 2 ∂ ∂ T ∂T 1 ∂T + Kz Kz + ∂r 2 r ∂r ∂z ∂z ( ) . (6.3.16) u˙ 0 ∂ u˙ 0 φ˙ r ∂ φ˙ r + +z +z − Ta βz − ρz cz T˙ + R = 0 r ∂r r ∂r . z

Now, the temperature distribution across the thickness may be approximated as /T (r, z, t) = T0 (r, t) + zT1 (r, t) + z 2 T2 (r, t)

.

(6.3.17)

The approximate equation (6.3.17) is substituted into Eq. (6.3.16) and the resulting residue . Res is made orthogonal with respect to .1, .z, and .z 2 to reduce the error. This provides three independent equations to calculate the unknown functions .T0 , T1 , T2 as {

h

/2

.

−h /2

{

h

/2

−h /2

{

h

/2

−h /2

Res.dz = 0 Res (z) .dz = 0 ( ) Res z 2 .dz = 0

(6.3.18)

304

6 Coupled Thermoelasticity of Plates

Now, we may assume that the internal heat generation is zero, the lower surface of the plate is thermally insulated and a heat flux .q '' is applied to the upper surface. Therefore )| )| ( ( ∂T || ∂T ||h/2 '' = q ; Kz =0 (6.3.19) . R = 0; K z ∂z | ∂z |−h/2 Substituting Eq. (6.3.19) into Eq. (6.3.18) and using integration by part, three differential equations are obtained as First energy equation: {

h/2 −h/2

.

Res.dz = 0

( ) ) u˙ 0 ∂ u˙ 0 ∂ φ˙ r φ˙ r + − Ta β2 + − Ta β1 r ∂r r ∂r ) ( 2 ∂ 1 ∂ + + (K 1 T0 + K 2 T1 + K 3 T2 ) ∂r 2 r ∂r − |1 T˙0 − |2 T˙1 − |3 T˙2 + q '' = 0 (

(6.3.20)

Second energy equation: {

h/2 −h/2

.

Res (z) .dz = 0

( ) ) u˙ 0 ∂ u˙ 0 ∂ φ˙ r φ˙ r + − Ta β3 + − Ta β2 r ∂r r ∂r ( 2 ) ∂ 1 ∂ + + (K 2 T0 + K 3 T1 + K 4 T2 ) ∂r 2 r ∂r (

− |2 T˙0 − |3 T˙1 − |4 T˙2 − K 1 T1 − 2K 2 T2 +

(6.3.21)

h ( '' ) q =0 2

Third energy equation: {

h/2 −h/2

.

( ) Res z 2 .dz = 0

( ) ) u˙ 0 ∂ u˙ 0 ∂ φ˙ r φ˙ r + − Ta β4 + − Ta β3 r ∂r r ∂r ) ( 2 ∂ 1 ∂ + + (K 3 T0 + K 4 T1 + K 5 T2 ) ∂r 2 r ∂r (

− |3 T˙0 − |4 T˙1 − |5 T˙2 − 2K 2 T1 − 4K 3 T2 +

(6.3.22) ( )2 h ( '' ) q =0 2

6.3 Coupled Thermoelasticity of FGM Annular Plate

where

{ (K 1 , K 2 , K 3 , K 4 , K 5 ) = {

.

(|1 , |2 , |3 , |4 , |5 ) =

h2

−h/2 h/2

−h/2

305

( ) K z 1, z, z 2 , z 3 , z 4 dz

( ) ρz cz 1, z, z 2 , z 3 , z 4 dz

(6.3.23)

To numerically solve the problem, the non-dimensional parameters are used as V 1 Ta rˆ = r ; tˆ = t ; Tˆa = l l Tb 1−ν 1−ν 1−ν uˆ = u 0 ; wˆ = w0 ; φˆ r = φr l (1 + ν) αm Tb l (1 + ν) αm Tb (1 + ν) αm Tb l 1 1 q '' ; Rˆ = R ; Pˆ = P qˆ '' = ρm cm V Tb ρm cm V Tb ρm αm V 2 Tb ( ) ) 1 ( 1 I0 I1 I2 T0 , lT1 , l 2 T2 ; Iˆ0,1,2 = Tˆ0,1,2 = , , Tb ρm l l 2 l 3 ( ) ( ) E1 E2 E3 β1 β2 β3 β4 1 1 ˆ ˆ ; β1,2,3,4 = , , , , , (6.3.24) . E 1,2,3 = Em l l2 l3 αm E m l l 2 l 3 l 4 1 ( α1 α2 α3 ) ˆ` E` 1 , 2 , 3 ; E1 = αˆ 1,2,3 = αm l l l l (E m ) ( ) |1 |2 |3 |4 |5 1 |ˆ 1,2,3,4,5 = , 2, 3, 4, 5 ; ρm cm l l l l l ) ( K K K K 1 1 2 3 4 K5 Kˆ 1,2,3,4,5 = , , , , ρm cm V l 2 l 3 l 4 l 5 l 6 1 1 1 Mˆ rr = Mrr ; Nˆ rr = Nrr ; Qˆ r = Qr 2 E m αm Tb l E m αm Tb l E m αm Tb l where .l, .V , and .Tb are defined as / E m (1 − ν) km ; Tb = 298 .V = ;l = ρm cm V (1 + ν) (1 − 2ν) ρm

(6.3.25)

The non-dimensional forms of the equations of motion and energy equations are obtained as ( ) ( ) 1 1 ∂2 ∂2 ˆ 1 ∂ 1 ∂ ˆ ˆ + 2 uˆ + E 2 − 2 + + 2 φr E1 − 2 + rˆ rˆ ∂ rˆ ∂ rˆ rˆ rˆ ∂ rˆ ∂ rˆ ) ( . ( ) 2 2ˆ 1 ∂ u ˆ ∂ φ ∂ ˆ ˆ r − β1 T0 + βˆ2 Tˆ1 + βˆ3 Tˆ2 + − Iˆ0 2 − Iˆ1 2 = 0 ∂ rˆ E ∂ tˆ ∂ tˆ

(6.3.26)

306

6 Coupled Thermoelasticity of Plates

ˆ` .ζ E 1 2

(

1 ∂ + rˆ ∂ rˆ

)

φˆ r + ζ 2 Eˆ` 1

(

1 ∂ ∂2 + 2 rˆ ∂ rˆ ∂ rˆ

)

( ) 1 ∂ 2 wˆ ˆ ˆ wˆ + − I0 2 + P = 0 E ∂ tˆ (6.3.27)

( ) ( ) 1 1 1 ∂ 1 ∂ ∂2 ∂2 ˆ ˆ ˆ E2 − 2 + + 2 uˆ + E 3 − 2 + + 2 φr − ζ 2 Eˆ` 1 φˆ r rˆ rˆ ∂ rˆ ∂ rˆ rˆ rˆ ∂ rˆ ∂ rˆ ) ( . ( ) 2 2ˆ ∂ ˆ ˆ ∂ u ˆ ∂ ˆ 1 φ r 2 ˆ` ∂ w − β2 T0 + βˆ3 Tˆ1 + βˆ4 Tˆ2 + − ζ E1 − Iˆ1 2 − Iˆ2 2 = 0 ∂ rˆ ∂ rˆ E ∂ tˆ ∂ tˆ (6.3.28) ) ) ( ( 1 ∂ 1 ∂ ∂2 ∂2 uˆ − /Tˆa βˆ2 − /Tˆa βˆ1 + + φˆ r rˆ ∂ tˆ ∂ rˆ ∂ tˆ rˆ ∂ tˆ ∂ rˆ ∂ tˆ ) ( 2 ) ∂ 1 ∂ (ˆ ˆ ˆ 2 Tˆ1 + Kˆ 3 Tˆ2 (6.3.29) . + K + + K T 1 0 ∂ rˆ 2 rˆ ∂ rˆ ) ∂ ( ˆ |ˆ 1 T0 + |ˆ 2 Tˆ1 + |ˆ 3 Tˆ2 + qˆ '' = 0 − ∂ tˆ ) ) ( ( 1 ∂ 1 ∂ ∂2 ∂2 ˆ ˆ ˆ ˆ uˆ − /Ta β3 − /Ta β2 + + φˆ r rˆ ∂ tˆ ∂ rˆ ∂ tˆ rˆ ∂ tˆ ∂ rˆ ∂ tˆ ) ( 2 ) ∂ 1 ∂ (ˆ ˆ ˆ 3 Tˆ1 + Kˆ 4 Tˆ2 K + + + K T 2 0 ∂ rˆ 2 rˆ ∂ rˆ ( ) hˆ ∂ |ˆ 2 Tˆ0 + |ˆ 3 Tˆ1 + |ˆ 4 Tˆ2 − Kˆ 1 Tˆ1 − 2 Kˆ 2 Tˆ2 + qˆ '' = 0 − 2 ∂ tˆ

(6.3.30)

) ) ( ( 1 ∂ 1 ∂ ∂2 ∂2 ˆ ˆ ˆ ˆ uˆ − /Ta β4 − /Ta β3 + + φˆ r rˆ ∂ tˆ ∂ rˆ ∂ tˆ rˆ ∂ tˆ ∂ rˆ ∂ tˆ ) ( 2 ) ∂ 1 ∂ (ˆ ˆ ˆ 4 Tˆ1 + Kˆ 5 Tˆ2 K + + + K T 3 0 ∂ rˆ 2 rˆ ∂ rˆ ( )2 ( ) ∂ hˆ |ˆ 3 Tˆ0 + |ˆ 4 Tˆ1 + |ˆ 5 Tˆ2 − 2 Kˆ 1 Tˆ1 − 4 Kˆ 3 Tˆ2 + − qˆ '' = 0 2 ∂ tˆ

(6.3.31)

.

.

where ./ and .E are the dimensionless parameters that are defined as /=

.

Tb αm 2 E m Em ;E = ρm cm ρm V 2

(6.3.32)

Also, the non-dimensional forms of the force and moment resultants in terms of the displacement components are

6.3 Coupled Thermoelasticity of FGM Annular Plate

307

| | | ( ) ( ) ( ) ˆr ∂ φ ∂ u ˆ ν ν ˆ + uˆ + Eˆ 2 + φr − βˆ1 Tˆ0 − βˆ2 Tˆ1 − βˆ3 Tˆ2 Nˆ rr = Eˆ 1 ∂ rˆ rˆ ∂ rˆ rˆ ( ) ˆ r = ζ 2 Eˆ` 1 φˆ r + ∂ wˆ .Q ∂ rˆ | | | | ( ) ( ) ( ) ˆr ∂ φ ∂ u ˆ ν ν + uˆ + Eˆ 3 + φˆ r − βˆ2 Tˆ0 − βˆ3 Tˆ1 − βˆ4 Tˆ2 Mˆ rr = Eˆ 2 ∂ rˆ rˆ ∂ rˆ rˆ (6.3.33) The superscript hat sign ( ˆ. ) indicates that the following parameters are in dimensionless form. Applying the Laplace transform to the derived governing equations, while assuming zero initial conditions, the following equations are obtained |

( ) ( ) 1 1 ∂2 ∂2 ¯ 1 ∂ 1 ∂ ˆ ˆ + 2 u¯ + E 2 − 2 + + 2 φr E1 − 2 + rˆ rˆ ∂ rˆ ∂ rˆ rˆ rˆ ∂ rˆ ∂ rˆ . ( ( ) ) 1 ∂ ˆ ¯ − Iˆ0 s 2 u¯ − Iˆ1 s 2 φ¯ r = 0 − β1 T0 + βˆ2 T¯1 + βˆ3 T¯2 + ∂ rˆ E ζ 2 Eˆ` 1

.

(

1 ∂ + rˆ ∂ rˆ

)

φ¯ r + ζ 2 Eˆ` 1

(

1 ∂ ∂2 + 2 rˆ ∂ rˆ ∂ rˆ

) w¯ +

(6.3.34)

) 1 ( ˆ 2 − I0 s w¯ + P¯ = 0 E (6.3.35)

( ) ( ) 1 1 ∂2 ∂2 ¯ 1 ∂ 1 ∂ ˆ ˆ + 2 u¯ + E 3 − 2 + + 2 φr − ζ 2 Eˆ` 1 φ¯ r E2 − 2 + rˆ rˆ ∂ rˆ ∂ rˆ rˆ rˆ ∂ rˆ ∂ rˆ . ( ) 1 ( ) ∂ ∂ w ¯ ˆ − − Iˆ1 s 2 u¯ − Iˆ2 s 2 φ¯ r = 0 − ζ 2 E` 1 βˆ2 T¯0 + βˆ3 T¯1 + βˆ4 T¯2 + ∂ rˆ ∂ rˆ E (6.3.36) ( ( ) ) s s ∂ ∂ ¯ +s u¯ − /Tˆa βˆ2 +s − /Tˆa βˆ1 φr rˆ ∂ rˆ rˆ ∂ rˆ ) ( 2 ) ∂ 1 ∂ (ˆ ¯ . (6.3.37) K 1 T0 + Kˆ 2 T¯1 + Kˆ 3 T¯2 + + 2 ∂ rˆ rˆ ∂ rˆ ) ( − s |ˆ 1 T¯0 + |ˆ 2 T¯1 + |ˆ 3 T¯2 + q¯ '' = 0

.

( ( ) ) s s ∂ ∂ ¯ +s u¯ − /Tˆa βˆ3 +s − /Tˆa βˆ2 φr rˆ ∂ rˆ rˆ ∂ rˆ ) ( 2 ) 1 ∂ (ˆ ¯ ∂ K 2 T0 + Kˆ 3 T¯1 + Kˆ 4 T¯2 + + 2 ∂ rˆ rˆ ∂ rˆ ) ( hˆ − s |ˆ 2 T¯0 + |ˆ 3 T¯1 + |ˆ 4 T¯2 − Kˆ 1 T¯1 − 2 Kˆ 2 T¯2 + q¯ '' = 0 2

(6.3.38)

308

6 Coupled Thermoelasticity of Plates

.

( ( ) ) s s ∂ ∂ ¯ +s u¯ − /Tˆa βˆ4 +s − /Tˆa βˆ3 φr rˆ ∂ rˆ rˆ ∂ rˆ ) ( 2 ) ∂ 1 ∂ (ˆ ¯ ˆ 4 T¯1 + Kˆ 5 T¯2 K + + + K T 3 0 ∂ rˆ 2 rˆ ∂ rˆ ( )2 ( ) hˆ − s |ˆ 3 T¯0 + |ˆ 4 T¯1 + |ˆ 5 T¯2 − 2 Kˆ 2 T¯1 − 4 Kˆ 3 T¯2 + q¯ '' = 0 2

(6.3.39)

Also, we have | | | ¯r ( ) ( ) ( ) ∂ φ ∂ u ¯ ν ν + u¯ + Eˆ 2 + φ¯ r − βˆ1 T¯0 − βˆ2 T¯1 − βˆ3 T¯2 N¯ rr = Eˆ 1 ∂ rˆ rˆ ∂ rˆ rˆ ( ) ¯ r = ζ 2 Eˆ` 1 φ¯ r + ∂ w¯ .Q ∂ rˆ | | | | ¯r ( ) ( ) ( ) ∂ φ ∂ u ¯ ν ν + u¯ + Eˆ 3 + φ¯ r − βˆ2 T¯0 − βˆ3 T¯1 − βˆ4 T¯2 M¯ rr = Eˆ 2 ∂ rˆ rˆ ∂ rˆ rˆ (6.3.40) The superscript bar sign ( ¯. ) denotes the Laplace transforms of the function. In the following equations, for convenience in writing, the (ˆ.) symbol is removed. The Galerkin finite element method is used to solve the problem. Since the highest order of partial derivative in the governing equations is second order, and also due to the geometric symmetry of the plate and the loading symmetry, straight linear elements may be employed to model the finite element formulations. Thus, the finite element approximations for the assumed field problem are [48] |

{ }(e) ¯ (e) ; w¯ (e) = (e) {w} ¯ (e) ; φ¯ r(e) = (e) φ¯ r u¯ (e) = (e) {u} . { }(e) { }(e) { }(e) ; T¯1(e) = (e) T¯1 ; T¯2(e) = (e) T¯2 T¯0(e) = (e) T¯0

(6.3.41)

where .{}(e) denotes the values of function on nodal point and .(e) is the shape function vector. Assuming linear element and defining local coordinate . R as .ri < R < r j by .r = r f + R with .r f being the coordinate of the first node of each element, the shape functions may be written in the form .

Ni = 1 −

R R ; Nj = L L

(6.3.42)

where . L = r j − ri . The procedure of applying the Galerking method is described for Eq. (6.3.34) and, likewise, others may be obtained. Therefore

6.3 Coupled Thermoelasticity of FGM Annular Plate

.

309

⎧ ⎫ ⎡ ⎤ ⎪ ⎪ 2 ⎪ ⎪ ν ∂ ∂ ∂ 1 − ν 1 ⎪ ⎪ ⎪ ⎪ ⎦ u¯ E 1 ⎣− ( + + + ⎪ ⎪ ) ⎪ ⎪ 2 2 ⎪ ⎪ R + r ∂ R R + r ∂ R ∂ R f f ⎪ ⎪ R +rf ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎤ ⎡ ⎪ { L⎪ ⎨ ⎬ 2 ∂ ∂ 1 − ν 1 ν ∂ Nm d R = 0 ¯ ⎦ ⎣ + φ − + E + + r⎪ ( )2 2 2 0 ⎪ ⎪ ⎪ R + r ∂ R R + r ∂ R ∂ R ⎪ ⎪ f f R +rf ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ( ) ⎪ ⎪ ¯ ¯ ¯ ∂ T ∂ T ∂ T 1 ⎪ ⎪ 0 1 2 2 2 ⎪ −β ⎪ ¯ ⎪ ⎪ −I s u ¯ − I s + φ − β − β r 1 2 3 0 1 ⎩ ⎭ ∂R ∂R ∂R E m = i, j

(6.3.43) The terms with the double underline are subjected to weak formulation. From the rule of differentiation, we have ( ) ∂ Nm ∂ u¯ ∂ u¯ ∂ 2 u¯ ∂ Nm − Nm = ∂ R2 ∂R ∂R ∂R ∂R ( ) ∂ φ¯ r ∂ 2 φ¯ r ∂ ∂ Nm ∂ φ¯ r Nm = Nm − 2 ∂R ∂R ∂R ∂R ∂R ) ( ( ) ∂ ν Nm ν Nm ν Nm ∂ u¯ ∂ (6.3.44) . u¯ = u¯ − R + rf ∂R ∂R R + rf ∂R R + rf ( ( ) ) ν Nm ∂ φ¯ r ν Nm ¯ ν Nm ∂ ∂ φr − = φ¯ r R + rf ∂R ∂R R + rf ∂R R + rf ) ∂ Nm ∂ ( ∂ T¯i = Nm T¯i − Nm T¯i ; i = 0, 1, 2 ∂R ∂R ∂R Using Eqs. (6.3.44), (6.3.43) may be written as

.

⎫ ⎧ ⎡ ⎤ Nm Nd Nm (1 − ν) ∂ Nd ⎪ ⎪ ⎪ ⎪ − + ( )2 ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ R + r ∂ R ⎪ ⎪ f R + r ⎪ ⎪ ⎢ ⎥ f ⎪ ⎪ E u ¯ ⎪ ⎪ 1⎢ ) ⎥ d ( ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ ∂ N ∂ ν N ∂ N ⎪ ⎪ m d m ⎪ ⎪ N − − ⎪ ⎪ d ⎪ ⎪ ⎪ ⎪ ∂ R ∂ R ∂ R R + r f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ N N ∂ N N − ν) (1 ⎪ ⎪ m d m d ⎪ ⎪ ⎪ − + { L⎪ ( )2 ⎬ ⎨ ⎢ ⎥ R + r ∂ R f R +rf ⎢ ⎥¯ dR φ + E ⎢ ⎥ ) ( 2 rd ⎪ 0 ⎪ ⎪ ⎪ ⎣ ⎦ ∂ ν Nm ∂ Nm ∂ Nd ⎪ ⎪ ⎪ ⎪ Nd − − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂R ∂R ∂R R + rf ⎪ ⎪ ⎪ ⎪ | | | | | | ⎪ ⎪ ⎪ ⎪ ∂ N ∂ N ∂ N ⎪ ⎪ m m m ⎪ ⎪ ¯ ¯ ¯ ⎪ + β + β + β T T T ) ) ) (N (N (N 1 d 0d 2 d 1d 3 d 2d ⎪ ⎪ ⎪ ⎪ ⎪ ∂ R ∂ R ∂ R ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ s s ⎪ ⎪ ⎭ ⎩ + (−I0 Nm Nd ) u¯ d + (−I1 Nm Nd ) φ¯ r d E E |L = −Nm N¯ rr |0 ; m = i, j (6.3.45)

310

6 Coupled Thermoelasticity of Plates

Similarly, other governing equations are obtained as follows: ⎤ ∂ Nd Nm Nm Nd ( ¯ ) ) φr d + ζ 2 E` 1 ( ) ζ 2 E` 1 ( (w¯ d ) ⎥ ⎢ R +rf R + rf ∂R ⎥ { L⎢ ⎥ ⎢ ⎢ − ζ 2 E` ∂ Nm (N ) φ¯ − ζ 2 E` ∂ Nm ∂ Nd (w¯ ) ⎥ d R ⎥ ⎢ d rd 1 d 1 0 ⎢ ∂R ∂R ∂R ⎥ ⎦ ⎣ s2 − (I0 Nm Nd ) w¯ d E { L ¯ |L P = −Nm Q¯ r |0 − (Nm ) d R ; m = i, j E 0

(6.3.46)

⎡ ⎧ ⎫ Nm Nd Nm (1 − ν) ∂ Nd ⎤ ⎪ ⎪ −( ⎪ ⎪ )2 + ⎪ ⎪ ⎥ ⎢ R + r ∂ R ⎪ ⎪ f R + r ⎪ ⎪ ⎢ ⎥ f ⎪ ⎪ ⎪ ⎪ ⎥ u¯ d ⎪ ⎪ ( ) E2 ⎢ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ν Nm ∂ Nm ∂ Nd ∂ ⎣ ⎦ ⎪ ⎪ ⎪ ⎪ − − Nd ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ R ∂ R ∂ R R + r ⎪ ⎪ f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎤ ⎡ ⎪ ⎪ ⎪ ⎪ ∂ N N N − ν) N (1 m m ⎪ ⎪ d d ⎪ ⎪ + − ⎪ ⎪ ( )2 ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ R +rf ∂R ⎥ ⎪ ⎪ R +rf ⎢ ⎪ ⎪ ⎪ { L⎪ ⎥ ¯ ⎨ + E3 ⎢ ⎬ ( ) ⎥ φr d ⎢ dR ν Nm ∂ ∂ Nm ∂ Nd ⎦ ⎣ ⎪ − − Nd ⎪ ⎪ . 0 ⎪ ⎪ ⎪ ∂ R ∂ R ∂ R R + r ⎪ ⎪ f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ | | | | | | ⎪ ⎪ ⎪ ∂ Nm ( ) ¯ ∂ Nm ( ) ¯ ⎪ ∂ Nm ( ) ¯ ⎪ ⎪ ⎪ ⎪ ⎪ N N N T T T + β + β + β 2 3 4 d 0d d 1d d 2d ⎪ ⎪ ⎪ ⎪ ⎪ ∂ R ∂ R ∂ R ⎪ ⎪ ⎪ ⎪ ⎪ ( ⎪ ) ( ⎪ ⎪ ) ⎪ ⎪ ∂ N ⎪ ⎪ d 2 E` N N φ 2 E` N ⎪ ⎪ ¯ ⎪ ⎪ w ¯ + −ζ + −ζ 1 m d 1 m d rd ⎪ ⎪ ⎪ ⎪ ∂ R ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ( 2 ( ⎪ ⎪ ) ) s s ⎩ ⎭ ¯ + −I1 Nm Nd u¯ d + −I2 Nm Nd φr d E E |L = −Nm M¯ rr | ; m = i, j

(6.3.47)

⎡

.

0

) ( ⎧ ⎫ s Nm Nd ∂ Nd ⎪ ⎪ ⎪ ⎪ − /T u ¯ + s N β ⎪ ⎪ a m 1 d ⎪ ⎪ ⎪ ⎪ R +rf ∂R ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ) ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s N ∂ N N m ⎪ ⎪ d d ¯rd ⎪ ⎪ φ − /T β + s N ⎪ ⎪ a m 2 ⎪ ⎪ ⎪ ⎪ R + r ∂ R f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ) ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Nm ∂ Nd ∂ Nm ∂ Nd ¯ ⎪ ⎪ { L⎪ ⎨ + (K 1 ) ( ⎬ ) T0d − ∂R ∂R R + r f ∂R dR ⎪ 0 ⎪ ) ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ Nm ∂ Nd ∂ Nm ∂ Nd ¯ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (K 2 ) ( R + r ) ∂ R − ∂ R ∂ R T1d ⎪ ⎪ ⎪ f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ( ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Nm ∂ Nd ∂ Nm ∂ Nd ¯ ⎪ ⎪ ⎪ ⎪ ( ) T + − (K ) ⎪ ⎪ 3 2d ⎪ ⎪ ∂ R ∂ R ∂ R ⎪ ⎪ R + r f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ − |1 (s) Nm Nd T¯0d − |2 (s) Nm Nd T¯1d − |3 (s) Nm Nd T¯2d { L =− q¯ '' Nm d R ; m = i, j 0

(6.3.48)

6.3 Coupled Thermoelasticity of FGM Annular Plate

311

) ( ⎫ ⎧ ∂ Nd s Nm Nd ⎪ ⎪ ⎪ ⎪ − /Ta β2 u¯ d + s Nm ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R +rf ∂R ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ) ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s Nm Nd ∂ Nd ¯ ⎪ ⎪ ⎪ ⎪ φr d + s Nm − /Ta β3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R +rf ∂R ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ) ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Nm ∂ Nd ∂ Nm ∂ Nd ¯ ⎪ ⎪ ⎪ ⎪ ( ) T + − (K ) ⎪ ⎪ 2 0d ⎪ ⎪ { L⎨ ⎬ ∂R ∂R R + r f ∂R dR ) ( ⎪ 0 ⎪ ⎪ ⎪ Nm ∂ Nd ∂ Nm ∂ Nd ¯ ⎪ ⎪ ⎪ ⎪ . ( ) T + − (K ) ⎪ ⎪ 3 1d ⎪ ⎪ ⎪ ⎪ ∂R ∂R R + r f ∂R ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ( ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Nm ∂ Nd ∂ Nm ∂ Nd ¯ ⎪ ⎪ ⎪ ⎪ + (K 4 ) ( ) T − ⎪ ⎪ 2d ⎪ ⎪ ⎪ ⎪ ∂ R ∂ R ∂ R R + r f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¯ ¯ ¯ ⎪ ⎪ − |2 (s) Nm Nd T0d − |3 (s) Nm Nd T1d − |4 (s) Nm Nd T2d ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ¯ ¯ − K 1 Nm Nd T1d − 2K 2 Nm Nd T2d { L h '' =− q¯ Nm d R ; m = i, j 0 2

(6.3.49)

) ( ⎫ ⎧ s Nm Nd ∂ Nd ⎪ ⎪ ⎪ ⎪ − /T + s N β u¯ d ⎪ ⎪ m a 3 ⎪ ⎪ ⎪ ⎪ R + r ∂ R f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ) ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s N ∂ N N m ⎪ ⎪ d d ¯rd ⎪ ⎪ φ + s N − /T β ⎪ ⎪ a m 4 ⎪ ⎪ ⎪ ⎪ R + r ∂ R f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ) ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ N N ∂ N ∂ N m m ⎪ ⎪ d d ⎪ ⎪ ¯ ( ) + − T (K ) ⎪ ⎪ 3 0d ⎪ { L⎪ ⎬ ⎨ ∂R ∂R R + r f ∂R dR ) ( ⎪ ⎪ 0 ⎪ ⎪ Nm ∂ Nd ∂ Nm ∂ Nd ¯ ⎪ ⎪ ⎪ ⎪ . ( ) + − T (K ) ⎪ ⎪ 4 1d ⎪ ⎪ ⎪ ⎪ ∂R ∂R R + r f ∂R ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ) ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ) ( ⎪ ⎪ ∂ N N ∂ N ∂ N m m ⎪ ⎪ d d ¯ ⎪ ⎪ ( ) + K − T ⎪ ⎪ 2d 5 ⎪ ⎪ ∂R ∂R ⎪ ⎪ R + r f ∂R ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¯0d − |4 (s) Nm Nd T¯1d − |5 (s) Nm Nd T¯2d ⎪ T − | N N (s) ⎪ ⎪ m 3 d ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ − 2K 2 Nm Nd T¯1d − 4K 3 Nm Nd T¯2d { L ( )2 h =− q¯ '' Nm d R ; m = i, j 2 0

(6.3.50)

Finally, the finite element equation in matrix form is obtained for the base element (e)

.

[K ](e) {X}(e) = {F}(e)

.

where the sub-matrices are

(6.3.51)

312

6 Coupled Thermoelasticity of Plates

⎡ | md || md || md || md || md || md |⎤ k11 k12 k13 k14 k15 k16 ⎢ |k md ||k md ||k md ||k md ||k md ||k md |⎥ ⎢ 21 22 23 24 25 26 ⎥ ⎥ ⎢| ⎢ k md ||k md ||k md ||k md ||k md ||k md |⎥ ⎢ 31 32 33 34 35 36 ⎥ (e) [K ] = ⎢ | md || md || md || md || md || md |⎥ ; ⎢ k41 k42 k43 k44 k45 k46 ⎥ ⎥ ⎢| ⎢ md || md || md || md || md || md |⎥ ⎣ k51 k52 k53 k54 k55 k56 ⎦ | md || md || md || md || md || md | k61 k62 k63 k64 k65 k66 . ⎧ | d |⎫ ⎧ | m |⎫ x11 ⎪ f 11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ | d |⎪ ⎪ ⎪ ⎪ | m |⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ |x21 |⎪ ⎪ | f 21 |⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ xd ⎬ ⎨ fm ⎪ ⎬ m = i, j 31 31 {X }(e) = | d | ;{F}(e) = | m | ; d = i, j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ |x41 |⎪ ⎪ | f 41 |⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m ⎪ d ⎪ ⎪ ⎪ ⎪ x51 ⎪ f 51 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ | d |⎪ ⎩ | m |⎪ ⎭ ⎭ f 61 x61

(6.3.52)

The matrix arrays of Eq. (6.3.52) is presented in Appendix A. Once all elements in the solution domain are assembled, the final equilibrium equation in the finite element form is obtained. Note that time does not appear in the resulting finite element equation, due to the Laplace transformation of the governing equations. The finite element equilibrium equation is solved in the Laplace domain and the inversion of the Laplace transformation into the physical time domain is carried out numerically. Let the original function . f (t) satisfy the condition .| f (t)| < K eγt , where . K and .γ are real positive constants. The inversion of integral expression for . f (t) is defined as follows [45]: 1 . f (t) = 2π j

w+ { j∞

F(s)est ds, where w > γ

(6.3.53)

w− j∞

series approximation formula for the discrete case in the interval .t ∈ | A Fourier ) 0; N T˘ may be written as [45] { |∞ | } E k k k ˜ .f = C Fn E n − F0 k = 0, 1, . . . , N − 1 (6.3.54) 2 Re n=0

where

.

( ) o wk T˘ ˘ e ;Fn = F (w − jno) ; E nk = e− jk T no f˜k = f˜ k T˘ ;C k = 2π

(6.3.55)

6.3 Coupled Thermoelasticity of FGM Annular Plate Table 6.4 Table of materials [35] Ti–6Al–4V

ZrO.2 = 117 GPa = 7.11 μm/m◦ C 3 .ρc = 5600 Kg/m . K c = 2.036 W/mK ◦ .C c = 615.6 J/Kg C ∗ .νc = 0.333

= 66.2 GPa = 10.3 μm/m◦ C 3 .ρm = 4410 Kg/m . K m = 18.1 W/mK ◦ .C m = 808.3 J/Kg C ∗ .νm = 0.321 . Em

. Ec

.αm

.αc

∗

.

313

A constant mean value of 0.327 is assumed for Poisson’s ratio (.ν).

Here, .T˘ and .o =

2π N T˘

are periodicity of original and transformed function, respec-

tively. Consider the solution time as .tm = (M − 1) T˘ with . M = N /2. Thus, .o and .w may be calculated using the following equation: w≈γ−

.

o π (1 − 1/M) ln Er ; o = 2π tm

(6.3.56)

where . Er is optional relative error.

6.3.4 Results and Discussion The FGM plate may be assumed to consist of a mixture of titanium and zirconium oxides. Thermal and mechanical properties of titanium and zirconium oxides are expressed in Table 6.4 [35]. Simply supported boundary conditions are considered for the inner and outer radii of the annular plate. The dimensionless outer radius is assumed as. Rout = 2. Since the difference between .200 and .400 elements along the radial direction were negligible, the analyses were performed by considering .200 elements. The dimensionless time increment for each step is also chosen less than .0.2. The following equation is used to describe the applied thermal shock: q '' = ate−bt

.

(6.3.57)

where.a and.b are constant coefficients and are assumed to be.a = 0.003 and.b = 0.8. Thus, Eq. (6.3.57) may be written as q '' = 0.003te−0.8t

.

(6.3.58)

The plot of applied thermal shock is shown in Fig. 6.15. To check the accuracy of the calculations, the first two natural frequencies of the isotropic annular plate are calculated by considering different geometric aspect ratios for various boundary

314

6 Coupled Thermoelasticity of Plates −3

x 10

1.4

1.2

1

q"

0.8

0.6

0.4

0.2

0

0

2

6

4

8

10

Time

Fig. 6.15 Heat flux function Table 6.5 Comparison of numerical (Prs.) and analytical (Ref.) solutions of natural frequencies of an isotropic annular plate [43] S-S* C-C ** F-C *** + ++ Ref. Prs. Ref. Prs. Ref. Prs. . Rin /Rout .h/Rout 0.1

0.1

0.1

0.3

0.1

0.5

0.1

0.6

0.2

0.5

0.3

0.4

* +

Simply-Simply Reference

1 2 1 2 1 2 1 2 1 2 1 2

13.87 13.88 46.95 47.00 20.22 20.23 71.71 71.82 37.33 37.36 127.17 127.43 56.08 56.14 182.10 182.55 31.87 31.93 90.64 90.98 20.26 20.32 54.06 54.29 ** Clamp-Clamp ++ Present

24.63 62.14 39.40 95.59 70.28 159.78 100.07 217.65 48.31 97.39 28.27 55.77

24.66 9.90 62.28 36.33 39.47 11.12 95.85 46.25 70.45 17.02 160.36 77.24 100.38 24.26 218.56 108.92 48.53 15.40 97.87 55.27 28.42 10.91 56.05 33.88 *** Free-Clamp

9.99 36.67 11.22 46.70 17.18 78.01 24.49 110.01 15.54 55.82 11.01 34.21

conditions. The results are compared with the analytical solution and are shown in Table 6.5 [43]. Close agreement is observed between two results. The deflection of simply supported FGM circular plate at the center (.W0 ) under uniform mechanical load for different geometries and power law indices are calculated and compared with the analytical solution and given in Table 6.6 [49]. The results are well compared. Finally, the quasi-static response of a simply supported circular plate under lateral thermal shock load, considering the uncoupled theory of thermoelasticity, is

6.3 Coupled Thermoelasticity of FGM Annular Plate

315

Table 6.6 Comparison of deflection at the center of a simply supported FGM circular plate (.W0 ) with analytical solution [49] 0.05 0.1 0.2 .h/Rout .ξ . W0

0 2 10

Ref. 10.396 5.497 4.667

Prs. 10.365 5.480 4.662

Fig. 6.16 Comparison of deflection at the center of an isotropic circular plate (.W0 ) for quasi-static analysis between numerical and analytical solutions

Ref. 10.481 5.539 4.712

Prs. 10.473 5.535 4.709

Ref. 10.822 5.708 4.885

Prs. 10.819 5.707 4.854

0.14

0.12 Present Analytical Solution

0.1

W0

0.08

0.06

0.04

0.02

0

0

0.1

0.2

0.3

0.4

τ

0.5

0.6

0.7

0.8

0.9

compared with the analytical solution [42] and is shown in Fig. 6.16. In this figure, τ is the dimensionless time (Fig. 6.17). Considering the above discussion, it may be concluded that the results are fairly well compared with the known data in literature. It is to be noted, however, that, in thermoelastic analysis of the FGM annular plates, many parameters are involved which do not necessarily have a similar impact. Thus, sometimes, prediction of the results without solving the problem seems to be not possible. To explain further results, natural frequencies of the FGM plate for different geometric aspect ratios and power law indices are calculated and are shown in Table 6.7. From Table 6.7, it is seen that increasing the values of . Rin /Rout and .h/Rout lead to increase the natural frequencies of the plate. Also, it may be noticed that by increasing the power law index, natural frequency of the plate becomes higher. These results are mentioned and shown in the next figures. The power law index is gradually increased from full metal .ξ = 0 until reaching full ceramic .ξ = 1010 . The values for the power law index of FGM may be selected as .ξ = 0, 0.1, 0.5, 1, 2, 10, and .1010 . The ratio of the inner radius to the outer radius is . Rin /Rout = 0.5 and the ratio of thickness to outer radius of .h/Rout = 0.1 is chosen. At the midpoint of the radius, graphs of the temperature changes and lateral displacement versus time in the midplane .z = 0 are drawn in Figs. 6.18 and 6.20, respectively.

.

316

6 Coupled Thermoelasticity of Plates

0.0243

0.015

0.0242

Temperature at middle radius for z=0

Temperature at middle radius for z=0

0.024

0.02

0.023 0.022 0.021 0.02 0.019 0.018 0.017

0.0241 0.024 0.0239 0.0238 0.0237 0.0236 0.0235 0.0234

0.016 3

4

5

6

7

8

9

10

42.2

11

42.4

42.6

42.8

43

43.2

43.4

Time

Time −4

x 10 2

0.01

Temperature at middle radius for z=0

Temperature at middle radius for z=0

0.025

0.005

ξ =0 ξ =0.1 ξ =0.5 ξ =1 ξ =2 ξ =10

1

10

0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

ξ =10

0.16

Time

0 0

5

10

15

20

25

30

35

40

Time

Fig. 6.17 5a Table 6.7 Natural frequencies of the FGM plate by considering different geometric aspect ratios and power law indices .ω ˆ + . Rin /Rout

= 0.5

.h/Rout

= 0.1

.ξ

0 .0.1 .0.5 .1 .2 .10 .10

.ξ

=1

. Rin /Rout

.h/Rout

+

.

= 0.5

= 0.1

.h/Rout . Rin /Rout

10

0.05 0.1 0.1 0.5

0.4668 0.4807 0.5054 0.5143 0.5194 0.5333 0.5508 0.2706 0.5143 0.1907 0.5143

Dimensionless natural frequency (.ωˆ = ω (l/V ))

Comparing the results for .ξ = 0.1 up to .ξ = 1010 in Fig. 6.18, it is seen that by increasing the power law index the value of temperature increases. It should be noticed that in the transient analysis, in addition to the thermal conductivity (. K z ), the value of .ρz cz plays a key role to determine the temperature changes of the plate, as the heat flux is applied only for a limited time. Here, the heat flux is approximately applied in the interval (.0 ≤ t ≤ 10). From Eq. (6.3.16), it is obtained that in the transient analysis the larger value of .ρz cz results into lower rate of temperature rise, while larger thermal conductivity causes lower temperature gradient and leads to higher midplane temperature. Both thermal conductivity (. K z ) and the value of .ρz cz

6.3 Coupled Thermoelasticity of FGM Annular Plate

317

Temperature at middle radius for z=0

0.024 0.0235 0.023 0.0225 0.022 0.0215

ξ=0 ξ=0.1 ξ=0.5 ξ=1 ξ=2 ξ=10

0.021 0.0205 0.02 0.0195

ξ=1010

0.019 40

35

30

25 Time

20

15

10

5

Fig. 6.18 Temperature changes in midplane for different FGM power law indices 0.025 0.019 0.018

0.02

0.017

Tz at center

0.016

Tz at center

0.015

0.015

Metal Midplane Temperature

0.014

Metal Lower Surface Temperature

0.013

Metal Upper Surface Temperature

0.012

Ceramic Midplane Temperature

0.011

Ceramic Lower Surface Temperature Ceramic Upper Surface Temperature

0.01 0.009

0.01

1.5

2

2.5

3

3.5

4

4.5

t −4

−3 −4

x 10

10 xx 10

8

12 8

7

7

10

6

6

8

5

Tz at center

Tz at center

0.005

46 34 2

5 4 3

2

2

1

0

0

0

00

0.2 0.05 0.4 0.10.6

0.8 0.15

tt

1 0.2 1.2

1.4 0.25

1

1.60.3 1.8

0 0

0.05

0.1

0.2

0.15

0.25

0.3

t

−0.005

0

5

10

15

20

25

30

35

40

45

t

Fig. 6.19 Upper surface, midplane, and lower surface temperature changes of the pure metal and pure ceramic plate

for the purely metal plate are larger than the ceramic one. Thus, these two parameters have an opposite influence on the midplane temperature. As a result, without solving the problem, it is not possible to determine the final temperature value. To clarify the topic, the upper surface, midplane, and lower surface temperatures of the plate are plotted all together in Fig. 6.19, while both pure metal and pure ceramic are considered. From Fig. 6.19, it is seen that at the first steps, due to the higher thermal conductivity of the metal, the midplane temperature of the metal is higher than ceramic, while the upper surface temperature of ceramic is always higher than metal. It is worth to mention that, as seen in Fig. 6.19, at the beginning of the time due to the lower thermal conductivity of ceramic, the temperature changes of metal is placed

318

6 Coupled Thermoelasticity of Plates

Fig. 6.20 Deflection at midpoint in midplane for different FGM power law indices

−3

6

x 10

ξ=0 ξ=0.1 ξ=0.5 ξ=1 ξ=2 ξ=10

5

W at middle radius

4

3

10

ξ=10

2

1

0

−1

5

0

10

15

25

20

30

35

40

45

Time

Fig. 6.21 Radial force resultant changes in outer radius (. Nrr | Rout ) for different FGM power law indices

−3

1

x 10

ξ=0 ξ=0.1 ξ=0.5 ξ=1 ξ=2 ξ=10

0 −1

Nrr at outer radius

−2 −3

ξ=1010

−4 −5 −6 −7 −8 −9

0

5

10

15

25

20

30

35

40

45

Time

between the graphs of ceramic. But, since the slope of temperature changes is higher for the ceramic plate, as time increases temperature rise increases. As a result, as time increases the graphs start to separate and finally the temperature of ceramic plate becomes higher than the metal plate. This is shown in Figs. 6.18 and 6.19. Parameters such as Young’s modulus . E z , thermal expansion coefficient .αz , power law index .ξ, temperature distribution, and boundary conditions can affect the respond of the plate to primary thermal excitation. The combined effect of all these parameters, which some have opposite influence on deflection of the plate, are shown in Fig. 6.20. As seen in Table 6.7, by increasing the power law index (which means much ceramic portion) the natural frequency of the plate is increased. This result is also shown in Fig. 6.20. Considering the results for .ξ = 0 up to .ξ = 0.5 in Fig. 6.20, it is seen that the amplitude and frequency of vibration increase. But a comparison between the results for .ξ = 0.5 up to .ξ = 1010 shows that, while the frequency of vibration increases, the amplitude is reduced. The graph of radial force resultant variation in outer radius (. Nrr | Rout ) versus time is plotted in Fig. 6.21. It is seen from Fig. 6.21 that

6.3 Coupled Thermoelasticity of FGM Annular Plate 0.025

0.02

0.015

W at middle radius

Fig. 6.22 Deflection at midpoint in midplane for different ratios of thickness to outer radius while . Rin /Rout = 0.5

319

0.01

0.005

0

h/Rout=0.05 h/Rout=0.1

−0.005

0

5

10

15

25

20

30

35

40

45

Time

0.05

0.04

Temperature for z=0

Fig. 6.23 Temperature changes in midplane for different ratios of thickness to outer radius, while . Rin /Rout = 0.5

0.03

0.02

0.01

0

h/Rout=0.05 h/Rout=0.1

−0.01

0

5

10

15

25

20

30

35

40

45

Time

by changing the material properties from pure metal to pure ceramic, the absolute value of the radial force resultant increases gradually. The problem is solved for different geometric aspect ratios when .ξ = 1, and the results are compared with each other. First, it is assumed that . Rin /Rout = 0.5. The midpoint deflection of annular plate for .h/Rout = 0.05 and .h/Rout = 0.1 is plotted in Fig. 6.22. When thickness of the plate is taken into account in the calculation of moment of inertia, it is obtained that the thicker plate has larger moments of inertia and, as seen in Fig. 6.22, by increasing the thickness of the maximum deflection of the plate is decreased. The midpoint temperature changes in the midplane .z = 0 and the radial force resultant variation at outer radius (. Nrr | Rout ) in terms of time are plotted in Figs. 6.23 and 6.24, respectively. From Figs. 6.22 and 6.24, it is seen that by increasing the ratio of thickness to outer radius, the frequency of vibration increases. The same results can be found in Table 6.7. Since the heat flux is the same for both cases, when the thickness is increased, the temperature is dropped, as shown in Fig. 6.23. Now, it is assumed that.h/Rout = 0.1. The midpoint deflection of annular plate for . Rin /Rout = 0.1 and . Rin /Rout = 0.5 is plotted in Fig. 6.25. In Fig. 6.25, it is shown

320 −3

0

x 10

−3

x 10 −7.4

−1

−7.5

−2 Nrr at outer radius

−3 −4 −5

h/R

=0.05

h/R

=0.1

out

−7.6

Nrr at outer radius

Fig. 6.24 Radial force resultant changes in outer radius (. Nrr | Rout ) for different ratios of thickness to outer radius, while . Rin /Rout = 0.5

6 Coupled Thermoelasticity of Plates

out

−7.7

−7.8

h/Rout=0.05

−7.9

h/R

=0.1

out

−8

−6 10

5

15

20

−7

25 Time

40

35

30

−8 −9

10

5

0

15

35

30

25

20

40

45

Time

Fig. 6.25 Deflection at midpoint in midplane for different ratios of inner radius to outer radius when .h/Rout = 0.1

−3

14

x 10

12

W at center

10

Rin/Rout=0.1 Rin/Rout=0.5

8 6 4 2 0 −2

0

5

10

15

25

20

30

35

40

45

t

that maximum deflection of the plate is decreased for a larger aspect ratio . Rin /Rout , since the radii difference of the plate is decreased. Also, increase in the ratio of inner radius to outer radius leads to increase in the frequency of vibration, as it can be obtained from Table 6.7. The midpoint temperature changes in the midplane .z = 0 and force resultant variation at the outer radius (. Nrr | Rout ) in terms of time are plotted in Figs. 6.26 and 6.27, respectively. As seen in Fig. 6.26, since the thickness and heat flux are assumed to be identical, changes in temperature versus time will be the same. In Fig. 6.27, it is seen that the increase of the ratio of inner radius to outer radius causes the frequency of vibration to increase and the amplitude to decrease. In this section, results for the coupled and uncoupled solutions are compared. To show the effect of coupling in equations, the coupling coefficient ./ is multiplied by .r, where .r is an arbitrary coefficient. Assuming this new coupling coefficient while . Rin /Rout = 0.5, .h/Rout = 0.05, and .ξ = 1, the problem is solved by considering the

6.3 Coupled Thermoelasticity of FGM Annular Plate

321

0.025

0.02

Tz=0 at center

0.015 Rin/Rout=0.1

0.01

Rin/Rout=0.5

0.005

0

−0.005

0

5

10

15

20

25

30

35

40

45

t

Fig. 6.26 Temperature changes in midplane for different ratios of inner radius to outer radius when = 0.1

.h/Rout

−3

1

x 10

−3

x 10 −7.4

−1

−7.5

−2

−7.6 Nrr at outer radius

Nrr at outer radius

0

−3 −4

−7.7

R /R in

R /R in

−7.8

=0.1

out

=0.5

out

−7.9

−5

−8

−6

−8.1 10

5

15

20

25 Time

30

35

40

−7 −8 −9 0

5

10

15

20

Time

25

30

35

40

45

Fig. 6.27 Radial force resultant changes in outer radius (. Nrr | Rout ) for different ratios of inner radius to outer radius when .h/Rout = 0.1

coupled theory of thermoelasticity. While.r = 1, as shown in Fig. 6.28, the difference between the coupled and uncoupled thermoelasticity theories for real values of the coupling coefficient is negligible. To better illustrate the differences between these solutions, the numerical value of .r is considered to be .105 . The results for time distribution of temperature, deflection, and radial force resultant for the coupled and uncoupled theories are shown in Figs. 6.29, 6.30 and 6.32, respectively. As seen in Figs. 6.29, 6.30, 6.31 and 6.32, by increasing the coupling coefficient the absolute value of deflection at the midpoint, radial force resultant in outer radius,

322

6 Coupled Thermoelasticity of Plates

Fig. 6.28 Comparison of the coupled and uncoupled solutions

0.05 Temperature

0.04

Results at midplane

Coupled solution Uncoupled solution 0.03

0.02

Deflection

0.01

0 Stress Resultant −0.01

0

5

10

25

20

15

35

40

45

700

800

900

800

900

30

Time

0.05

Temperature at middle radius for z=0

Fig. 6.29 The effect of coupling term on temperature at midpoint in midplane when assuming the new coupling coefficient

0.04

0.03

0.02 Coupled solution while ϒ=105 Uncoupled solution 0.01

0

−0.01

100

200

300

400 500 Time

600

0.025

0.02

W at middle radius

Fig. 6.30 The effect of coupling term on deflection at midpoint in midplane when assuming the new coupling coefficient

0

0.015

0.01

0.005

0 Coupled solution while ϒ=105 Uncoupled solution −0.005

0

100

200

300

400 500 Time

600

700

6.3 Coupled Thermoelasticity of FGM Annular Plate

323

Fig. 6.31 Magnified part of (Fig. 6.30)

W at middle radius

0.02

0.015

0.01

0.005

0

Coupled solution while ϒ=105 Uncoupled solution 0

Fig. 6.32 The effect of coupling term on radial force resultant in outer radius (. Nrr | Rout ) when assuming the new coupling coefficient

20

40

60 Time

80

100

−3

1

x 10

0 −1

Nrr at outer radius

−2 Coupled solution while ϒ=105 Uncoupled solution

−3 −4 −5 −6 −7 −8 −9

0

100

200

300

400 500 Time

600

700

800

900

and temperature at midpoint in midplane decrease. It is seen in Figs. 6.31 and 6.30 that vibration frequency of the plate becomes higher and is damped as time increases. The coupled theory of thermoelasticity of the FGM annular plate subjected to sudden heat flux is studied by considering the first order shear deformation plate theory. Laplace transforms and the Galerkin finite element method is used to obtain the results. The effect of coupling coefficient, power law index, and geometrical aspect ratios are discussed in the main body of the section. Some of the results are as follow: It is seen that a large coupling coefficient can influence the dynamic behavior of the plate, and thus significant differences between solutions of the coupled and uncoupled theories are clear. Also, the values of deflection and radial force resultant are decreased as the coupling coefficient becomes larger. It is shown that increase of vibration frequency and amplitude reduction may occur simultaneously when considering different geometries. Figure 6.21 shows that for the assumed materials, the radial force resultant increases gradually as the material properties change from pure metal to pure ceramic.

324

6 Coupled Thermoelasticity of Plates

Appendix A A-1: Nonzero arrays of the stiffness matrix: ⎡ { | md | =⎣ . k 11

L

⎡ { | md | =⎣ . k 13

L

0

0

⎧ ⎨

⎡

−

Nm Nd 2 R+r ( E 1 ⎣ ∂ N ∂f N) m ⎩ − ∂ R ∂ Rd

⎧ ⎨

⎡

−

Nm Nd 2 R+r ( E 2 ⎣ ∂ N ∂f N) ⎩ − ∂ Rm ∂ Rd

.

| md | k14 =

| md | = . k 15

.

| md | = . k 22

|{

L

| md | k16 =

{

|

ζ E` 1 2

0

.

| md | k23 =

⎧ { L⎨

⎡

| md | . k 31 =⎣

0

−

∂ ∂R

+

Nm (1−ν) ∂ Nd R+r f ∂R

−

∂ ∂R

L 0

|{

L 0

|{

L 0

L

(

(

ν Nm R+r f

ν Nm R+r f

⎤

⎤ 2

) Nd

0

−

| md | k32 =

⎤

⎦ − s (I1 Nm Nd ) d R ⎦ ⎭ E

{ ( )} | ∂ Nm β1 dR Nd ∂R { ( )} | ∂ Nm β2 dR Nd ∂R )} | { ( ∂ Nm dR β3 Nd ∂R |

} | s2 − (I0 Nm Nd ) d R E

|{

+

Nm (1−ν) ∂ Nd R+r f ∂R

−

∂ ∂R

L 0

(

ν Nm R+r f

⎤ 2

) Nd

( ) | ∂ Nd −ζ 2 E` 1 Nm dR ∂R

⎤⎫ ⎤ ⎧ ⎡ (1−ν) ∂ Nd − Nm Nd 2 + NmR+r ⎪ ⎪ ⎪ ⎪ ∂ R f ( ) ⎦⎬ ⎥ | md | ⎢ L ⎨ E 3 ⎣ ( R+r f ) ⎢ ∂ N ∂ N ν N ∂ d m m =⎣ . k 33 − ∂ R ∂ R − ∂ R R+r f Nd ⎪ d R ⎥ ⎦ 0 ⎪ ⎪ ⎪ ⎩ s2 ⎭ − E (I2 Nm Nd ) − ζ 2 E` E` 1 Nm Nd {

| md | k34 =

|{

L 0

⎫ ⎬

⎤

⎦ − s (I1 Nm Nd ) d R ⎦ ⎭ E

⎡

.

⎫ ⎬

{ ( )} | Nm Nd ∂ Nm ζ 2 E` 1 dR Nd − R +rf ∂R

Nm Nd 2 R+r ( E 2 ⎣ ∂ N ∂f N) ⎩ − ∂ Rm ∂ Rd

.

Nm (1−ν) ∂ Nd R+r f ∂R

Nm ∂ Nd ∂ Nm ∂ Nd ( ) − ∂R ∂R R + rf ∂R

|{

⎡

|{

⎫ ⎤ ⎬ 2 s ) ⎦ − (I0 Nm Nd ) d R ⎦ ⎭ E Nd

+

{ ( )} | ∂ Nm β2 dR Nd ∂R

6.3 Coupled Thermoelasticity of FGM Annular Plate

| md | = . k 35 | .

| md | . k 41 =

.

| md | k43 =

|{

|{

| md | = . k 46

| md | = . k 53

|{

L

{

( −/Ta β2 L

{

0

|{

L

{

0

|{

L

{

0

0 L

| md | = . k 56

L

|{

{

L

∂ Nm ∂ Nd ∂R ∂R

( Nm ∂ Nd − (K 3 ) ( R+r f ) ∂R −|3 (s) Nm Nd

∂ Nm ∂ Nd ∂R ∂R

)

} | φ¯ r d d R

)}

| dR

)}

| dR

)}

| dR

|{

L

0

s Nm Nd ∂ Nd + s Nm R +rf ∂R

( ∂ Nd Nm − (K 2 ) ( R+r f ) ∂R −|2 (s) Nm Nd

∂ Nm ∂ Nd ∂R ∂R

)

} | ¯ φr d d R

)}

| dR

{

)} | ( Nm ∂ Nd ∂ Nm ∂ Nd − (K 3 ) ( R+r ∂R ∂R f ) ∂R dR −|3 (s) Nm Nd − K 1 Nm Nd

{

)} | ( ∂ Nd ∂ Nm ∂ Nd Nm − (K 4 ) ( R+r ∂R ∂R f ) ∂R dR −|4 (s) Nm Nd − 2K 2 Nm Nd

0

0

s Nm Nd ∂ Nd + s Nm R +rf ∂R

( Nm ∂ Nd − (K 2 ) ( R+r f ) ∂R −|2 (s) Nm Nd

(

0

L

| dR

∂ Nm ∂ Nd ∂R ∂R

−/Ta β3

|{

|{

)}

( Nm ∂ Nd − (K 1 ) ( R+r f ) ∂R −|1 (s) Nm Nd

{

0

| md | = . k 55

∂ Nm Nd ∂R

{ ( ) } | s Nm Nd ∂ Nd −/Ta β2 u¯ d d R + s Nm R +rf ∂R

L

|{

( β4

0

{

|{

|{

| md | = . k 54

| md | = . k 61

0

)} | { ( ∂ Nm dR β3 Nd ∂R

0

| md | = . k 45

| md | k51 =

L

L

{ ( ) } | s Nm Nd ∂ Nd −/Ta β1 u¯ d d R + s Nm R +rf ∂R

L

0

| md | . k 44 =

.

| md k36 =

|{

325

) } | { ( s Nm Nd ∂ Nd u¯ d d R −/Ta β3 + s Nm R +rf ∂R

326

6 Coupled Thermoelasticity of Plates

|

md . k 63

|

|

|{ =

| md

|

md . k 65

|

{

( −/Ta β4

0

k64

.

L

md . k 66

|{

L

=

{

0

|

|{

L

= |{

L

=

( Nm ∂ Nd − (K 3 ) ( R+r f ) ∂R −|3 (s) Nm Nd

∂ Nm ∂ Nd ∂R ∂R

)

} | ¯ φr d d R

)}

{

)} | ( Nm ∂ Nd ∂ Nm ∂ Nd − (K 5 ) ( R+r ∂R ∂R f ) ∂R dR −|5 (s) Nm Nd − 4K 3 Nm Nd

A-2: Arrays of unknown matrix: |

| d x11 = [u d ] | d| x21 = [wd ] | d| | | x31 = φr d | d| x41 = [T0d ] | d| x51 = [T1d ] | d| x61 = [T2d ] A-3: Arrays of force matrix: |L | | | m = − Nm N¯ rr |0 f 11 | | { L ¯ |L | m| P | ¯ f 21 = − Nm Q r 0 − (Nm ) d R 0 E | | |L | m| f 31 = − Nm M¯ rr |0 | | { L | m| Nm d R f 41 = −q ''

|

|

0

| { h '' L Nm d R f 51 = − q 2 0 | ( ) | { | m| h 2 '' L f 61 = − q Nm d R 2 0 | m

dR

)} | ( Nm ∂ Nd ∂ Nm ∂ Nd − (K 4 ) ( R+r ∂R ∂R f ) ∂R dR −|4 (s) Nm Nd − 2K 2 Nm Nd

0

|

|

{

0

|

∂ Nd s Nm Nd + s Nm R +rf ∂R

References

327

References 1. Massalas CV, Kalpakidis VK (1983) Coupled thermoelastic vibration of a simply supported beam. J Sound Vibr 88:425–429 2. Massalas CV, Kalpakidis VK (1984) Coupled thermoelastic vibration of a timoshenko beam. Lett Appl Eng Sci 22:459–465 3. Atarasha T, Minagawa VK (1992) Transient coupled thermoelastic problem of heat conduction in a multilayered composite plate. Int J Eng Sci 30(10):1443–1550 4. Mukherjee N, Sinha PK (1996) Thermal shocks in composite plates: a coupled thermoelastic finite element analysis. Compos Struct 34:1–12 5. Reddy JN, Chin CD (1998) Thermomechanical analysis of functionally graded cylinders and plates. J Therm Stresses 21(6):593–626 6. Praveen GN, Reddy JN (1998) Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Int J Solids Struct 35(33):4457–4476 7. Reddy JN (2000) Analysis of functionally graded plates. Int J Numer Meth Eng 47:663–684 8. Han X, Liu GR, Xi ZC, Lam KY (2001) Transient waves in a functionally graded cylinder. Int J Solids Struct 38:3021–3037 9. Vel SS, Batra RC (2002) Exact solution for thermoelastic deformation of functionally graded thick rectangular plates. AIAA J 40:1421–1433 10. Vel SS, Batra RC (2002) Three-dimensional analysis of transient thermal stresses in functionally graded plates. Int J Solids Struct 40:1421–1433 11. Bahtui A, Eslami MR (2007) Coupled thermoelasticity of functionally graded cylindrical shells. Mech Res Commun 34(1):1–18 12. Chung YL, Chang HX (2008) Mechanical behavior of rectangular plates with functionally graded coefficient of thermal expansion subjected to thermal loading. J Therm Stress 31:368– 388 13. Babaei MH, Abbasi M, Eslami MR (2008) Coupled thermoelasticity of functionally graded beams. J Therm Stress 31:1–18 14. Matsunaga H (2008) Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory. Compos Struct 82:499–512 15. Brischetto S, Leetsch R, Carreral E, Wallmersperger T, Kroplin B (2008) Thermo-mechanical bending of functionally graded plates. J Thermal Stress 31:286–308 16. Li Q, Iu VP, Kou KP (2009) Three-dimensional vibration analysis of functionally graded material plates in thermal environment. J Sound Vibr 324:733–750 17. Abbasi M, Sabbaghian M, Eslami MR (2010) Exact closed-form solution of the dynamic coupled thermoelastic response of a functionally graded timoshenko beam. J Mech Mater Struct 5(1):79–94 18. Bodaghi M, Saidi AR (2011) Thermoelastic buckling behavior of thick functionally graded rectangular plates. Arch Appl Mech 81:1555–1572 19. Bouazza M, Tounsi A, Adda-Bedia EA, Megueni A (2010) Thermoelastic stability analysis of functionally graded plates: An analytical approach. Comput Mater Sci 49(4):865–870 20. Rong LS, Xi ZF, Ming LY, Yan Y (2010) Generalized thermoelastic responses of functionally graded materials. Appl Mech Mater 29–32:1954–1959 21. Zhou FX, Li SR, Lai YM (2011) Three-dimensional analysis for transient coupled thermoelastic response of a functionally graded rectangular plate. J Sound Vib 330(16):3990–4001 22. Akbarzadeh AH, Hosseini ZSK, Sadighi M, Eslami MR (2009) Analytical solution of functionally graded rectangular plates under mechanical loading. Proc 15th Int Conf composite structures (ICCS15). Faculdade de Engenharia da Universidade de Porto, Porto, Portugal, pp 15–17 23. Akbarzadeh AH, Komeili A, Kiani Y, Sadough A, Eslami MR (2009) Vibration analysis of functionally graded plates under dynamic mechanical loading. In: Proceeding of 1st international conference on composite materials and structures, Algeria

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24. Akbarzadeh AH, Komeili A, Abbasi M, Eslami MR (2010) Coupled thermoelasticity of functionally graded plates based on the first-order shear deformation theory. In: Proc. ISME annual conf, Sharif Univ., Tehran 25. Akbarzadeh AH, Abbasi M, Hosseini zad SK, Eslami MR (2011) Dynamic analysis of functionally graded plates using the hybrid fourier-laplace transform under thermomechanical loading. Meccanica 46(6):1373–1392. https://doi.org/10.1007/s11012-010-9397-6 26. Akbarzadeh AH, Abbasi M, Eslami MR (2012) Coupled thermoelasticity of functionally graded plates based on the third-order shear deformation theory. Thin-Wall Struct PIT: s0263B231(12)00012-2, 53:141–155. https://doi.org/10.1016/j.tws.2012.01.009 27. Reddy JN (2004) Mechanics of laminated composite plates and shells, 2nd edn.. CRC Press LLC 28. Hetnarski RB, Eslami MR (2019) Thermal stresses: advanced theory and applications. Springer, Switzerland 29. McQuillen EJ, Brull MA (1970) Dynamic thermoelastic response of cylindrical shell. J Appl Mech ASME 37(3):661–670 30. Krylov VI, Skoblya NS (1977) A handbook of methods of approximate fourier transformation and inversion of the laplace transformation. Mir Publishers, Moscow 31. Chi SH, Chung YL (2006) Mechanical behavior of functionally graded material plates under transverse load- Part I: analysis. Int J Solids Struct 43:3657–3674 32. Zhao X, Lee YY, Liew KM (2009) Free vibration analysis of functionally graded plates using the element-free Kp-Ritz method. J Sound Vib 319:918–939 33. Eslami MR, Hetnarski RB, Ignaczak J, Noda N, Sumi N, Tanigawa Y (2013) Theory of elasticity and thermal stresses. Springer, Dordrecht 34. Eslami MR, Shakeri M, Ohadi AR, Shiari B (1999) Coupled thermoelasticity of shells of revolution: effect of normal stress and coupling. AIAA J 37(4):496–504 35. Bahtui A, Eslami MR (2007) Coupled thermoelasticity of functionally graded cylindrical shells. Mech Res Commun 34(1):1–18 36. Babaei MH, Abbasi M, Eslami MR (2008) Coupled thermoelasticity of functionally graded beams. J Thermal Stress 31(8):680–697 37. Eslami MR, Vahedi H (1989) Coupled thermoelasticity beam problems. AIAA J 27(5):662–665 38. Manoach E, Ribeiro P (2004) Coupled, thermoelastic, large amplitude vibrations of timoshenko beams. Int J Mech Sci 46(11):1589–1606 39. Golmakani ME, Kadkhodayan M (2011) Large deflection analysis of circular and annular FGM plates under thermo-mechanical loadings with temperature-dependent properties. Compos Part B: Eng 30 42(4):614–625 40. Golmakani ME, Kadkhodayan M (2011) Nonlinear bending analysis of annular FGM plates using higher-order shear deformation plate theories. Compos Struct 31 93(2):973–982 41. Atarashi T, Minagawa S (1992) Transient coupled-thermoelastic problem of heat conduction in a multilayered composite plate. Int J Eng Sci 31 30(10):1543–IN10 42. Nakajo Y, Hayashi K (1984) Response of circular plates to thermal impact. J Sound Vibr 95(2):213–222 43. Irie T, Yamada G, Takagi K (1982) Natural frequencies of thick annular plates. J Appl Mech 49(3):633–638 44. Davies B, Martin B (1979) Numerical inversion of the laplace transform: a survey and comparison of methods. J Comput Phys 33(1):1–32 45. Brancik L (1999) Programs for fast numerical inversion of laplace transforms in MATLAB language environment. In: Conference MATLAB 99-Praha. MATLAB 99, Praha: Konference MATLAB 99, Praha, pp 27–39 46. Jafarinezhad MR, Eslami MR (2017) Coupled thermoelasticity of FGM annular plate under lateral thermal shock. Compos Struct 168:758–771 47. Nosier A, Fallah F (2008) Reformulation of mindlin-reissner governing equations of functionally graded circular plates. Acta Mech 198(3–4):209–233 48. Eslami MR (2014) Finite elements methods in mechanics. Springer, Switzerland 49. Reddy JN, Wang CM, Kitipornchai S (1999) Axisymmetric bending of functionally graded circular and annular plates. Eur J Mech-A/Solids 18(2):185–199

Chapter 7

Couple Thermoelasticity of Shells

Abstract Shell structures under thermal shock loads are frequently encountered in the structural design problems. This chapter presents the coupled and generalized thermoelasticity of the cylindrical shells, spherical shells, conical shells, and shells of revolution. The shell material is assumed to be made of functionally graded, where by proper substitution for the power law index, the response of shells of homogeneous material is obtained.

7.1 Introduction Consider a shell structure under applied thermal load. The applied thermal load may be of different natures; steady-state thermoelasticity, thermal induced vibrations, and the coupled or generalized thermoelasticity. The steady-state thermoelasticity of shells occurs when the shell is under steadystate thermal load, that is under steady-state temperature distribution. The heat conduction equation is independently solved to derive the temperature distribution. The resulting temperature distribution is employed to substitute into the steady-state governing thermoelastic equations and solve for the displacement and stress components. In this type of analysis, the thermal energy equation and the thermoelastic shell equations are uncoupled and are solved independently. It is important to emphasize that sometimes the transient temperature distribution in shells is treated similarly to the steady-state conditions, provided that the shell thickness is relatively thick, otherwise thermally induced vibrations may appear. How much thick? This is not simply answered and requires to detain analysis. The second type of shell behavior under applied thermal load is thermal induced vibrations. In this condition, the temperature distribution in the shell is of transient form, but its application to the shell results in shell vibrations. The reason? geometrical properties of the shell. In such conditions, the transient temperature distribution in the shell is derived through the heat conduction equation and substituted into the equations of motion of the shell considering the inertia terms. Simultaneous solution of the equations of motion reveals if the shell behavior falls into thermal induced © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. R. Eslami, Thermal Stresses in Plates and Shells, Solid Mechanics and Its Applications 277, https://doi.org/10.1007/978-3-031-49915-9_7

329

330

7 Couple Thermoelasticity of Shells

vibrations or not. A notable and critical parameter is the magnitude of the shell thickness. For thin enough shell, we may expect thermal induced vibrations. The coupled thermoelasticity of shells occurs when the applied thermal load is in the form of thermal shock. If the time period of application of the applied thermal load is much smaller compared to the time period of the lowest natural frequency of the shell, then the governing thermoelasticity equations are coupled with the thermal energy equations and the response of the shell should be derived employing the coupled thermoelastic equations. For extremely small thermal shock load periods, the generalized thermoelasticity should be considered [1]. Under such thermal shock load, the shell material’s relaxation time is excited and should be included in the thermoelastic shell governing equations. Question of detection of the wave front in plates and shells under the coupled thermoelastic analysis should be now addressed. The stress, displacement, and temperature (second sound) wave fronts are expected to appear in structures under the coupled thermoelastic assumption. None of these wave fronts are detected in the analysis of plates and shells exposed to the thermal shock loads. The reason is that the flexural structures, such as plates and shells, are lumped in the thickness direction. Thermal shock loads applied to the surface of these structures should create the wave fronts across the thickness direction, where the structure is lumped in that direction. This is why no such wave fronts are detected across the thickness direction. The wave fronts appear when the analysis is based on the three-dimensional theory of thermoelasticity.

7.2 Coupled Thermoelasticity of Cylindrical Shells Rapid heating of engineering structures frequently occurs in industrial applications. Physically, rapid heating of structures can lead to various critical conditions. One of the most important critical conditions that may occur due to rapid heating is the propagation of stress front waves or heat-induced vibrations in these structures. When the time of application of the thermal shock load is much less than the period of the lowest natural frequency of the structure, the stress analysis is recommended to be made through the coupled thermoelasticity [1]. Researches are conducted on the coupled thermoelastic analysis of structural members. McQuillen and Brull [2] proposed an analytical solution for the dynamic thermoelastic response of homogeneous cylindrical shells of finite and infinite lengths. They concluded that in thermal shock, dynamic analysis is more important than static analysis, and this becomes even more important as the shell becomes thinner. They also mentioned the use of semi-coupled analysis and complete coupled thermoelasticity analysis. Eslami et al. [3] considered a thin homogeneous cylindrical shell under thermal and mechanical shocks and solved the problem in space and time domains using the Galerkin and Newmark methods, respectively. They considered the equations of motion and energy as couples. The coupled thermoelasticity of thick plate is discussed by Sherief and Hamza [4]. They applied symmetrical axial thermal

7.2 Coupled Thermoelasticity of Cylindrical Shells

331

and mechanical shock loads and solved the equations of motion and energy simultaneously using the Laplace method and the Fourier expansion. Reddy and Chin [5] applied thermal shock to the two geometries of cylinder and plate made of functionally graded material and investigated the dynamic thermoelasticity response using the finite element method, and compared the results of the coupled and non-coupled equations. They considered the properties to be temperature-dependent. Eslami et al. [6] applied the Galerkin finite element method to the shell of revolution using the Lord–Shulman coupled thermoelasticity model and studied the effect of normal stress and coupling. Shiari and Eslami [7] considered a composite cylindrical shell and applied thermal and mechanical shock loads on it. They used the Flugee second-order shell theory, which results in the normal stress considered, and used the finite element method based on the Galerkin technique to solve the problem in the space domain and the Newmark method in the time domain and compared the results for coupled and uncoupled modes. A one-dimensional generalized thermoelasticity model of a disk based on the Lord–Shulman theory is presented by Bagri and Eslami [8]. They solved the problem in the space and time domain using Laplace and Galerkin techniques, respectively, and studied the dynamic thermoelastic response of the disk under axisymmetric thermal shock load and the effects of the relaxation time and coupling coefficient. Bakhshi et al. [9] studied the coupled thermoelastic response of functionally graded annular disk based on the classical theory of thermoelasticity. The response of a circular cylindrical thin shell made of the functionally graded material based on the generalized theory of thermoelasticity is obtained by Bahtui and Eslami [10]. They examined the effects of the temperature field for linear and nonlinear distributions across the shell thickness. Babaei et al. [11] presented the finite element solution of an Euler–Bernoulli beam with functionally graded material (FGM) subjected to lateral thermal shock loads. They used the Laplace transform and Galerkin finite element techniques and studied the effect of the coupling coefficient. The generalized coupled thermoelasticity based on the Lord–Shulman model that admits the second sound effect was considered to study the dynamic thermoelastic response of functionally graded annular disk by Bagri and Eslami [12]. They first removed time variation from the equations using the Laplace method, then solved the equations in the space domain using the Galerkin method and obtained the answers in the time domain by the numerical inverse Laplace transformation. They studied the effect of the relaxation time parameter and coupling coefficient. Hosseini [13] presented a coupled thermoelasticity analysis of finite FGM hollow cylinder using the Green–Naghdi (GN) theory of the type of without energy dissipation. They obtained the time history of radial displacement, temperature, radial, and hoop stress distribution across the thickness of the cylinder for various values of the power law exponents in three sections across the length. Heydarpour and Aghdam [14] employed the generalized coupled thermoelasticity based on the Lord–Shulman (LS) theory to study the transient thermoelastic behavior of rotating functionally graded (FG) truncated conical shell subjected to thermal shock with different boundary conditions. They analyzed the governing equations using the mapping differential quadrature method and the Newmark technique. Jafarinezhad and Eslami [15] focused on the study of an annular plate made of functionally graded materials under

332

7 Couple Thermoelasticity of Shells

the assumption of the coupled theory of thermoelasticity. They studied the effects of the power law index, the coupling coefficient, and different geometric proportions. Large amplitude thermally induced vibrations of cylindrical shells made of functionally graded material (FGM) investigated by Esmaeili et al. [16]. They assumed that the thermo-mechanical properties of the FGM shell be functions of temperature and thickness coordinates. Zeverdejani and Kiani [17] investigated the coupled and nonlinear thermo-mechanical response of a functionally graded material (FGM) hollow sphere under thermal shock. They used the GDQ method, the Newmark time marching scheme, and the Picard successive algorithm and solved the governing equations to obtain the temperature, displacement, and stresses within the body as a function of time. The generalized thermoelastic response of a beam subjected to a partial lateral thermal shock was analyzed by Sakha and Eslami [18]. They employed the Galerkin finite element method along with the time marching algorithm to solve the one-dimensional dynamic problem. Also, they used the quadratic shape functions for the temperature field and .C 1 -continuous shape function for the displacement field. In this section, the coupled thermoelastic response of a cylindrical shell under local axisymmetrical heat shock along the shell length is investigated [19]. The shell is made of functionally graded material and follows Hooke’s law. The shell displacements under the applied loads are assumed to be small, and as a result, linear strain–displacement relationships are used to analyze the behavior of the shell. Hamilton’s principle is used to derive the complete set of equations of motion and boundary conditions governing the shell. The equations of motion, along with the coupled heat conduction equation, are considered. The equations are solved using the Galerkin finite element method and the Newmark technique in space and time domains, respectively, and the results are presented for different values of the coupling coefficient and the volume ratios of the materials in FGM. The necessity of using coupling thermoelasticity in thermal loading speed is also investigated.

7.2.1 Analysis The material properties of the functionally graded shell, such as Young’s modules E(z), thermal expansion coefficient .α(z), thermal conduction coefficient . K (z), specific heat conduction .c(z), and density .ρ(z) must be described across the shell thickness, where .z is the thickness coordinate .(−h/2 ≤ z ≤ h/2), and .h is the thickness of shell. We assume that the functionally graded shell is comprised of metal phase and ceramic phase. Let us assume that .Vm and .Vc represent the volumes of metal and ceramic phases, respectively. Let the volume fraction of each constituent material be denoted by

.

f =

. m

Vm Vm + Vc

fc =

Vc Vm + Vc

7.2 Coupled Thermoelasticity of Cylindrical Shells

333

Here, . f m and . f c are the volume fractions of metal and ceramic of FGM, respectively, and satisfy the following equation f + f c = 1,

. m

The volume fraction is a spatial function. Using the combination of these functions, the effective material properties of functionally graded materials may be expressed as . Fe f (z) = Fm f m + Fc f c , where . Fe f is the effective material property of functionally graded material, and . Fm and . Fc are the persistent material properties of each phase. The volume fraction is assumed to follow a power law function as .

2z + h k ) 2h fc = 1 − fm fm = (

(7.2.1)

where volume fraction index .k represents the material variation profile through the shell thickness, which is always greater than or equal to zero, and may be varied to obtain the optimum distribution of the constituent materials. The value of .k equal to zero represents a fully metal and infinity represents a fully ceramic shell. From Eq. (7.2.1), the effective material properties of a functionally graded cylindrical shell may be written as .

E(z) = E c + E mc f m α(z) = αc + αmc f m K (z) = K c + K mc f m

(7.2.2)

ρ(z) = ρc + ρmc f m c(z) = cc + cmc f m where .

E mc = E m − E c αmc = αm − αc K mc = K m − K c ρmc = ρm − ρc cmc = cm − cc

It is assumed that Poisson’s ratio .ν is constant across the shell thickness.

(7.2.3)

334

7 Couple Thermoelasticity of Shells

7.2.2 Strain–Displacement Relations Consider a functionally graded circular cylindrical shell of finite length. L, wall thickness .h, and mean radius . R. The cylindrical coordinates .(x, θ, z) are considered along the axial, circumferential, and radial directions, respectively. In order to include the shear deformations.(γx z , γθz /= 0), we may assume that the displacement components based on the first-order approximation are represented as .

u(x, θ, z) = u 0 (x, θ) + z ψx (x, θ, 0) v(x, θ, z) = v0 (x, θ) + z ψθ (x, θ, 0)

(7.2.4)

w(x, θ, z) = w0 (x, θ) where .u 0 , .v0 , and .w0 represent the components of the displacement vector of a point on the middle plane, and the quantities .ψx and .ψθ represent the rotations of tangents to the middle plane along the.x and.θ axes, respectively. The strain distributions across the shell thickness are given as (Kraus [20]) .

εx x = ε0x x + zk x 1 εθθ = (ε0 + zkθ ) 1 + z/R θθ 0 + zk xθ γxθ = γxθ 0 γx z = γx z 1 γθz = γ0 1 + z/R θz

(7.2.5)

where the strains and curvatures are defined as .

ε0x x = u 0,x 1 w0 ε0θθ = v0,θ + R R 0 γxθ = v0,x +

1 u 0,θ R(1 + z/R)

γx0z = w0,x + ψx 1 v0 0 γθz = w0,θ − + ψθ R R k x = ψx,x 1 kθ = ψ R θ,θ 1 k xθ = ψθ,x + ψ 1 + z/R x,θ

A comma denotes partial differentiation with respect to the space variables.

(7.2.6)

7.2 Coupled Thermoelasticity of Cylindrical Shells

335

The stress–strain relations for a functionally graded shell based on the assumed displacement model, including the shear deformations, are ⎧ σx x ⎪ ⎪ ⎪ σθθ ⎨ τxθ . ⎪ ⎪ ⎪ ⎩ τx z τθz

⎡

⎫ ⎪ ⎪ ⎪ ⎬

1 ⎢ν E(z) ⎢ ⎢0 = ⎪ 1 − ν2 ⎢ ⎪ ⎣0 ⎪ ⎭ 0

ν 1 0 0 0

0 0 1−ν 2

0 0

0 0 0 1−ν 2

0

0 0 0 0 1−ν 2

⎤⎧ ⎫ ⎡ ⎤ ΔT εx x ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎨ εθθ ⎪ ⎢ΔT ⎥ ⎬ ⎥ ⎥ ⎥ γxθ − E(z)α(z) ⎢ ⎢ 0 ⎥ ⎥⎪ 2 ⎪ ⎣ 1 − ν ⎪ γx z ⎪ ⎦⎪ 0 ⎦ ⎪ ⎩ ⎭ γθz 0

(7.2.7)

where .ΔT (x, θ, z, t) is the temperature change. The force and moment resultants from the second-order shell theory are

.

⎧ ⎫ ⎧ ⎫ ⎨ N x x ⎬ { ⎨ σx x ⎬ z N xθ = τxθ (1 + )dz ⎩ ⎭ ⎩ ⎭ R z Qxx τx z ⎫ ⎧ ⎫ ⎧ ⎨ Nθθ ⎬ { ⎨ σθθ ⎬ Nθx = τθx dz ⎭ ⎩ z⎩τ ⎭ Q θθ θz } { { } { z σx x Mx x = (1 + )zdz Mxθ τ R xθ z { } { { } Mθθ σθθ = zdz Mθx τθx z

(7.2.8)

7.2.3 Equations of Motion Love’s equations of motion in cylindrical coordinates, including the shear deformations (see Sect. 4) [21], are .

1 R 1 N xθ,x + R 1 Q x x,x + R 1 Mx x,x + R 1 Mxθ,x + R N x x,x +

Nθx,θ = I0 u¨ 0 + I1 ψ¨ x 1 Q θθ = I0 v¨0 R 1 Q θθ,θ − Nθθ = pz (t) + I0 w¨ 0 R

Nθθ,θ +

Mθx,θ − Q x = I1 u¨ 0 + I2 ψ¨ x Mθ,θ − Q θ = I2 ψ¨θ

(7.2.9)

336

7 Couple Thermoelasticity of Shells

A dot denotes partial differentiation with respect to time, where { .

I0 =

ρ(z) dz {

z

I1 =

ρ(z)z dz {z

I2 =

ρ(z)z 2 dz

(7.2.10)

z

and . pz (t) is the applied lateral force. In terms of the applied forces to the inside and outside surfaces of the cylindrical shell, . pz (t) is .

pz (t) = pz+ (t)(1 +

h h ) + pz− (t)(1 − ) 2R 2R

(7.2.11)

where . pz+ and . pz− are the external and internal pressures, respectively. The sign of . pz (t) is assumed positive toward the center. Since the thermal and mechanical applied loads are assumed to be axisymmetric, then ∂ =0 . ∂θ leaving the following three equations of motion to be simultaneously solved with the energy equation .

∂ Nx x = I0 u¨ 0 + I1 ψ¨ x ∂x Nθθ ∂ Qx − = I0 w¨ 0 ∂x R ∂ Mx x − Q x = I1 u¨ 0 + I2 ψ¨ x ∂x

(7.2.12)

For the thin cylindrical shells, temperature distribution across the shell thickness may be assumed to be linear as .

ΔT (x, z, t) = T0 (x, t) + zT1 (x, t)

(7.2.13)

where .T0 , and .T1 are some unknown functions, and must be obtained through the coupled system of equations. Substituting Eq. (7.2.4) into Eqs. (7.2.5), (7.2.6), and (7.2.7) and using Eq. (7.2.8) and finally substituting the resulting equations into the equations of motion (7.2.12), the equations of motion are obtained in terms of the displacement components as

7.2 Coupled Thermoelasticity of Cylindrical Shells

.

337

∂2u0 ∂w0 ∂ 2 ψx ∂T0 ∂T1 + A − B4 = I0 u¨ 0 + I1 ψ¨ x + A − B3 1 4 ∂x 2 ∂x ∂x 2 ∂x ∂x ) ( ∂u 0 ∂ψx ∂ 2 w0 , ∂ψx − + A − B A A, 1 + A w + A T − B T 1 1 6 0 2 1 0 2 1 ∂x 2 ∂x ∂x ∂x = I0 w¨ 0 ) ( ∂2u0 ∂w0 ∂ 2 ψx ∂T0 ∂T1 , ∂w0 , + A − B − A + A A4 + A − B ψ 1 1 x 2 5 4 5 ∂x 2 ∂x ∂x 2 ∂x ∂x ∂x = I1 u¨ 0 + I2 ψ¨ x A3

(7.2.14) where . A,1 , A j ( j = 1..6), and . Bi (i = 1..5) are constants given in the Appendix. Three equations of motion contain five unknown dependent functions .u 0 , .w0 , .ψx , .T0 , and .T1 . This means that two more equations are needed to complete the necessary equations and calculate the dependent functions. These two equations are derived by employing the energy equation.

7.2.4 Energy Equation The first law of thermodynamics for heat conduction equation in the assumed cylindrical functionally graded shell, when the temperature distribution is assumed to be a function of time and thickness direction [1], is ¯ a ε˙ii = (kT,i ),i ρcε T˙ + βT

.

(7.2.15)

Eα , .Ta where .ρ is the mass density, .cε is the specific heat at constant strain, .β¯ = 1−2ν is the reference temperature, and .k is the heat conduction coefficient. This equation may be written in expanded form for the assumed conditions. We move all parts of the equation to the left side of the equation and call it the residue . Re. The resulting residue . Re is made orthogonal with respect to 1 and .z. This yields two independent equations for .T0 and .T1 , as it is made orthogonal with respect to 1 and .z for .T0 and . T1 [2]

{

h/2

.

−h/2 { h/2 −h/2

Re × dz = 0 Re × zdz = 0

(7.2.16)

Five governing equations, including the equations of motion and the energy equations, must be simultaneously solved to obtain the displacements and temperature functions.

338

.

7 Couple Thermoelasticity of Shells

As the thermal boundary conditions, it is considered that the heat flux . Q in and Q out are applied on the inside and outside surfaces of the shell | ( ) / ∂T | | = h in T − T in , z = − h .k(−h 2) ∞ ∂z |− h 2

(7.2.17)

| / ∂T | ( ) | = h out T − T out , z = h .k(h 2) ∞ | h ∂z 2

(7.2.18)

2

2

Using Eqs. (7.2.13) and (7.2.16) for the linear approximation of temperature distribution across the thickness direction, two energy equations for the cylindrical shell are obtained as ∂ u˙ 0 ∂ ψ˙ x ∂ 2 T0 ∂ 2 T1 + B , 4 w˙ + B , 2 + F1 T˙0 + F2 T˙1 − D1 − D − D4 T1 2 ∂x ∂x( ∂x 2 ∂x 2 ( ) ) h h +h in T0 − T1 + h out T0 + T1 = h in (Tin (t) − Ta ) 2 2 ˙ ∂ u˙ 0 ∂ ψx ∂ 2 T0 ∂ 2 T1 + B , 5 w˙ + B , 3 + F2 T˙0 + F3 T˙1 − D2 B,2 − D + D5 T1 3 ∂x ( ∂x 2 ∂x 2 ) ∂x ( ) h h h h h − h in T0 − T1 − h out T0 + T1 = − h in (Tin (t) − Ta ) 2 2 2 2 2 (7.2.19) B,1

.

where the constants . F j ( j = 1..3), Bi, , Di (i = 1..5) are given in the Appendix.

7.2.5 Numerical Solution Consider a cylindrical shell under an axisymmetric partial thermal shock load along the length of the shell. Temperature distribution across the shell thickness is assumed to be linear. Under such assumed conditions five unknown functions .u 0 , w0 , ψ, T0 , T1 , as given by Eqs. (7.2.14) and (7.2.19), appear in the governing equations. These equations are transformed into the dimensionless form by the following dimensionless parameters .

x 1 = x xc l T T0 0 T¯0 = = T0c Ta 1 u0 = u0 u¯ 0 = uc lαm Ta

x¯ =

t c1 = t tc l T l 1 = T1 T¯1 = T1c Ta w0 1 w¯ 0 = = w0 wc lαm Ta t¯ =

ψx 1 = ψx ψ¯ x = ψc αm Ta (7.2.20)

7.2 Coupled Thermoelasticity of Cylindrical Shells

/

where c =

. 1

l=

.

339

Em ( ) 1 − ν 2 ρm km ρm cm c1

Here, the subscript .m denotes the material properties of metal constituent. With the assumed dimensionless parameters, consider the quadratic shape function for the base element .(e) of the dependent functions as ⎧ ⎫(e) ⎨u¯ 01 ⎬ } { (e) N N u¯ 02 N u¯ (e) = (x) 1 2 3 0 ⎩ ⎭ u¯ ⎧ 03 ⎫(e) }(e) ⎨w¯ 01 ⎬ { w¯ 02 w¯ 0(e) (x) = N1 N2 N3 ⎩ ⎭ w¯ 03 ⎧ ⎫(e) ¯ ψ } ⎨ x1 ⎬ { ¯ (e) (x) = N1 N2 N3 (e) ψ¯ x2 .ψ (7.2.21) x ⎩¯ ⎭ ψ ⎧ x3⎫(e) ¯ ⎨ }(e) T01 ⎬ { T¯0(e) (x) = N1 N2 N3 T¯02 ⎩¯ ⎭ T ⎧ 03 ⎫(e) ¯ }(e) ⎨T11 ⎬ { (e) ¯ ¯ T1 (x) = N1 N2 N3 T ⎩ ¯12 ⎭ T13 )( ) ( x¯ x¯ 1− N1 (x) ¯ = 1−2 L¯ (e) L¯ (e) ( ) x¯ x¯ 1− . N2 ( x) ¯ =4 L¯ (e) L¯ (e) ( ) x¯ x¯ 1−2 N3 (x) ¯ =− L¯ (e) L¯ (e)

where

(7.2.22)

and, . L¯ (e) being the length of base element .(e). Applying the Galerkin method to the system of five equations and employing the weak formulations, yield [19] ⎤ wc ∂ N m ψc ∂ N m ∂ N i ¯ u c ∂ N m ∂ Ni A + A w ¯ + A u ¯ N ψ 3 0i 1 i 0i 4 xi 2 2 ⎥ ⎢ xc ∂ x¯ ∂ x¯ xc ∂ x¯ xc ∂ x¯ ∂ x¯ ⎥ { ⎢ ⎥ ⎢ T0c ∂ Nm T ∂ N ⎥ ⎢ 1c m ¯ ¯ . Ni T0i − B4 Ni T1i ⎥d x¯ = N x x Nm |0L ⎢ − B3 ⎥ ⎢ xc ∂ x¯ xc ∂ x¯ ⎥ ⎢ L¯ (e) ⎣ ⎦ ψ uc c ¨ + I0 2 Nm Ni u¨¯ 0i + I1 2 Nm Ni ψ¯ xi tc tc ⎡

340

7 Couple Thermoelasticity of Shells ) ( ⎤ uc ∂ Ni , wc ∂ N m ∂ N i w ¯ N + A w N N u ¯ + A A m 0i 6 c m i 0i 1 1 ⎥ ⎢ xc ∂ x¯ xc2 ∂ x¯ ∂ x¯ ⎥ { ⎢ ) ⎥ ⎢ ( ⎥ ⎢ ∂ N ψ ∂ N ψ m c i c , ¯ . ⎥d x¯ = Q x x Nm |0L ⎢ + A1 ψ + A N N i 2 m xi ⎥ ⎢ x ∂ x ¯ x ∂ x ¯ c c ⎥ ⎢ L¯ (e) ⎣ ⎦ w c − B1 T0c Nm Ni T¯0i − B2 T1c Nm Ni T¯1i + I0 2 Nm Ni w¨¯ 0i tc ⎡

) ( ⎤ u c ∂ N m ∂ Ni wc ∂ N m ∂ Ni , wc w ¯ + A + A N A u ¯ N 4 0i 2 i m 0i 1 ⎢ ⎥ xc2 ∂ x¯ ∂ x¯ xc ∂ x¯ xc ∂ x¯ ⎢ ( ⎥ ) ⎢ ⎥ ∂ N ψ ∂ N ⎢ ⎥ m i c , ¯ ⎥ + A 1 ψc Nm Ni ψxi { ⎢ + A5 2 ⎢ ⎥ xc ∂ x¯ ∂ x¯ ⎢ ⎥d x¯ = Mx x Nm | L . 0 ⎢ ⎥ ∂ N T ∂ N T 0c m 1c m ⎢ ⎥ Ni T¯0i − B5 Ni T¯1i ⎥ L¯ (e) ⎢ − B4 ⎢ ⎥ x ∂ x¯ xc ∂ x¯ ⎢ ( c ⎥ ) ⎣ ⎦ ψ uc c + I1 2 Nm Ni u¨¯ 0i + I2 2 Nm Ni ψ¨¯ xi tc tc ⎡

⎤ ∂ Ni ˙ wc uc u¯ + B , 4 N Nm Ni w˙¯ 0i B, ⎥ ⎢ 1 xc tc m ∂ x¯ 0i tc ⎥ ⎢ ⎥ ⎢ ∂ N T T ψ c i ˙¯ 0c ¯˙ + F 1c N N T˙¯ ⎥ ⎢ + B, T ψ N + F N N m 1 m i 0i 2 m i 1i 2 ⎥ ⎢ xi { x t ∂ x ¯ t t ⎥ ⎢ c c c c ⎥d x¯ ⎢ ( ) 2 ⎥ ⎢ ⎥ ⎢ − D1 T0c Nm ∂ Ni − T0c (h in + h out ) Nm Ni T¯0i ⎥ 2 2 . L¯ (e) ⎢ xc ∂ x¯ ⎥ ⎢ ⎢ ( ) ⎥ 2 ⎦ ⎣ T1c ∂ Ni h − (−h in + h out ) T1c Nm Ni T¯1i − D4 T1c Nm Ni + D2 2 Nm xc ∂ x¯ 2 2 { Nm h in (Tin (t) − Ta )d x¯ = ⎡

L¯ (e)

⎡ ⎢ ⎢ ⎢ ⎢ { ⎢ ⎢ ⎢ ⎢ ⎢ . L¯ (e) ⎢ ⎢ ⎢ ⎣ { = L¯ (e)

⎤ uc ∂ Ni ¯ wc u˙ 0i + B , 5 Nm Nm Ni w¯˙ 0i ⎥ x c tc ∂ x¯ tc ⎥ ⎥ ∂ Ni ¯˙ T0c T 1c ¯ ¯ , ψc ⎥ ψ xi + F2 +B3 Nm Nm Ni T˙ 0i + F3 Nm Ni T˙ 1i ⎥ x c tc ∂ x¯ tc tc ⎥ ⎥d x¯ ) ( 2 ⎥ h T0c ∂ Ni ⎥ ¯ T T + + h N N − D2 2 N m (h ) in out 0c m i 0i ⎥ xc ∂ x¯ 2 2 ⎥ ⎥ ( ) ⎦ ∂ 2 Ni T1c h2 ¯1i T T − D3 2 N m − D T N N − − h N N (h ) 1c m i in out 1c m i 5 xc ∂ x¯ 2 4 h − h in (Tin (t) − Ta )Nm d x¯ 2

B,2

(7.2.23)

7.2 Coupled Thermoelasticity of Cylindrical Shells

341

where .Tin (t) is the applied thermal shock. Set of Eq. (7.2.23) are assembled for all the finite elements of the solution domain and the final equation is given as .

{ } { } [M] X¨ + [C] X˙ + [K ] {X } = {F (t)}

(7.2.24)

where .[M], .[C], and .[K ] are the global mass, damping, and stiffness matrix. The matrix .{F(t)} is the global force matrix and .{X } is the global unknown matrix in form of the non-dimensional displacement components defined by Eq. (7.2.21). The finite element equation (7.2.24) may be solved in the time domain by many techniques. In this paper, the Newmark method is used. In this technique, the velocity and acceleration matrices at time .t + Δt are approximated in terms of their values at time .t as [22] .

{ } X˙

t+Δt

{ } X¨

t+Δt

( ( ) ) ) { } α { ˙} α α ( {X }t+Δt − {X }t + 1 − X t + 1− Δt X¨ t βΔt β 2β ) ( { } ( ) 1 1 { ¨} 1 {X }t+Δt − {X }t − X˙ t + 1 − X t = 2 βΔt βΔt 2β (7.2.25)

=

where the coefficients .α and .β are parameters which determine the accuracy and stability of the numerical technique. For time .t + Δt, we get .

[ ] { } Kˆ {X }t+Δt = Fˆ

(7.2.26)

where .

] [ ] [ 1 α Kˆ = + + [M] [C] [K ] βΔt 2 βΔt ( ) ) ( { } { } 1 1 { ˙} 1 ¨ {X } X − 1 X Fˆ = {F}t+Δt + [M] + + t t t βΔt 2 βΔt 2β ( ) ) ) ( ( { } { } α α α {X }t + − 1 X˙ t + − 1 Δt X¨ t + [C] βΔt β 2β (7.2.27)

7.2.6 Results and Discussion Verification The present analysis is validated with those given by Eslami et al. [6] and Bahtui et al. [21]. In both references, thermal shock is applied to the inside surface of cylindrical shell but dimensions, material, and heat flux rate are different. Two different types of thermal shock loads are considered in these papers, slow rate shock

342

7 Couple Thermoelasticity of Shells

2500

Temperatur(K)

2000

1500

1000

500

Present work

Eslami et al.

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time(sec)

Fig. 7.1 Temperature versus time for slow rate load 10-3

4

3.5

Radial Displacement(m)

3

2.5

2

1.5

1

0.5

0 Eslami et al.

Present work

-0.5 0

1

2

3

4

5

Time(sec)

Fig. 7.2 Radial displacement versus time for slow rate load

ΔT = 2207(1 − exp (−1.31t)) and fast rate shock .ΔT =2207(1− exp (−13100t)). Applications of these two thermal shock loads on the shell are shown in Figs. 7.1, 7.2, 7.3 and 7.4 for the temperature and radial displacement. As seen in Figs. 7.1, 7.2, 7.3 and 7.4, errors of the temperature and radial displacement are less than 2 percent. This is due to the different shell theories considered. Bahtui et al. [21] applied impact heat flux with very fast rate causing lateral vibration. As shown in Fig. 7.5, difference between frequencies obtained in [21] is less than 4 percent but it is about 100 percent for amplitude. These errors exist due to 36 percent difference between coefficients in the stress–strain relations and 100 percent due to the assumed coupling coefficients in the present and the above reference.

.

7.2 Coupled Thermoelasticity of Cylindrical Shells

343

600

550

Temperatur(K)

500

450

400

350

300

Eslami et al.

Present work

250 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.03

0.035

0.04

Time(sec)

Fig. 7.3 Temperature versus time for fast rate load 10-4

1.8

1.6

Radial Displacement(m)

1.4

1.2

1

0.8

0.6

0.4

0.2 Eslami et al.

Present work

0 0

0.005

0.01

0.015

0.02

0.025

Time(sec)

Fig. 7.4 Radial displacement versus time for fast rate load

7.2.7 Parametric Studies Consider a simply supported functionally graded cylindrical shell under the inside axisymmetric axially local thermal shock load. The length, diameter, and thickness of the shell are 500, 125, and 6.25 mm, respectively. The functionally graded shell is assumed to be made of a combination of metal (Ti-6Al-4V) and ceramic (ZrO2), with the material properties shown in Table 7.1. The initial temperature of the shell is assumed to be 298.15.◦ K.

344

7 Couple Thermoelasticity of Shells 10-9

2.5

Radial Displacement(m)

2

1.5

1

0.5

0

Present work

Bahtui et al.

-0.5 0

0.5

1

1.5

2

2.5

3

Time(sec)

3.5

4 10-4

Fig. 7.5 Radial displacement versus time Table 7.1 Material properties of functionally graded constituent materials Metal Ceramic . Em .αm .ρm .km .cm .νm

= 66.2 Gpa = 10.3 .×10−6 (1/K ) = 4.41 .×103 (kg/m3 ) = 18.1 (W/mK) = 808.3 (J/kg.K) = 0.321

= 117 Gpa = 7.11 × 10−6 (1/K ) 3 3 .ρc = 5.6 × 10 (kg/m. ) .kc = 2.036 (W/mK) .cc = 615.6 (J/kg.K) .νc = 0.333 . Ec .αc

The shell is ceramic rich at the inside and metal rich at the outside surfaces, respectively. Temperature field across the shell thickness is assumed to be of linear type. Thermal shock is applied to the inside surface in the interval of .150mm ≤ x ≤ 300 mm for two different types of thermal shock loads. The lowest natural frequency of the mentioned cylindrical shell for the first mode in axial and circumferential directions (for the axisymmetric load) is 5200 cycle/s [23], therefore, it’s time period is 0.192 ms. Two different thermal shock loads are considered to be applied to the inside surface, slow rate shock .ΔT = 2207(1 − exp (−1.31t)) and fast rate shock .ΔT = 2207(1 − exp (−13100t)). The slow rate shock, case (1), temperature increases to its maximum value in 0.5 ms which is larger than the lowest time period of the cylindrical shell 0.192. The fast rate shock, case (2), temperature increases to its maximum value in 0.05 ms which is much smaller than the lowest time period of the cylindrical shell 0.192. The response of the assumed cylindrical shell to the low rate thermal shock is expected to be transient and smooth. On the other hand, The response of the assumed cylindrical shell to the fast rate thermal shock is expected to be of a vibratory nature [19]. It is recalled that the wave fronts for displacement and

7.2 Coupled Thermoelasticity of Cylindrical Shells

345

3000

2500

Ti(K)

2000

1500

1000

500

0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

time(s)

Fig. 7.6 Applied thermal shock, case 1 299

Inside Temperature(K)

298

297

296

295

294

k=0

k=4

k=1

k=infinity

293 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Time(sec)

Fig. 7.7 Inside temperature of the shell at.x = 225 mm versus time for different power law indices, case (1)

stress are not expected to be detected to the assumption of the lumped formulation. For both cases, the boundary conditions at the ends of the shell are assumed to be thermally insulated and also the number of the elements is 150. Results for the slow rate thermal shock load are shown in Figs. 7.6, 7.7, 7.8, 7.9, 7.10, 7.11, 7.12, 7.13, 7.14 and 7.15. Figure 7.7 shows the temperature of the shell middle length versus time. This figure shows that for pure ceramic shell (.k = ∞), temperature distribution becomes higher, as the ceramic conductivity is lower compared to metal. The reason is that lower thermal conductivity of ceramics increases the temperature of the structure under applied thermal shock load.

346

7 Couple Thermoelasticity of Shells 10-6

2 1.8 1.6

Radial Displacement(m)

1.4 1.2 1 0.8 0.6 0.4 0.2 k=0

k=1

k=4

k=infinity

0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Time(sec)

Fig. 7.8 Radial displacement of the shell at .x = 225 mm versus time for different power law indices, case (1)

The centre of the shocked lenght Nxx(N/m)

0

-500

-1000

-1500

-2000

k=0

k=1

k=4

k=infinity

-2500 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Time(sec)

Fig. 7.9 Axial force resultant of the shell at .x = 225 mm versus time, case (1)

The lateral deflection of the shell at.x = 225 mm versus time for different values of the power law index is shown in Fig. 7.8 [19]. Since the thermal expansion coefficient of metal is larger than ceramic, as .k increases the lateral deflection increases. Figure 7.9 shows the axial force of the shell at point .x = 225 mm versus time. The axial moment of the shell, which is shown in Fig. 7.10, increases as the metal volume fraction increases. The variations of inside temperature, radial displacement, force, and moment along the shell length are shown in Figs. 7.11, 7.12, 7.13 and 7.14. In these diagrams, the index .k is considered to be constant and equal to one. These

7.2 Coupled Thermoelasticity of Cylindrical Shells

347

The centre of the shocked lenght Mxx(N.m/m)

35

30

25

20

15

10

5

k=0

k=1

k=4

k=infinity

0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.45

0.5

Time(sec)

Fig. 7.10 Axial moment resultant of the shell at .x = 225 mm versus time, case (1) 299

298

Inside Temperature(K)

297

296

295

294

293

t= 20 ms

t= 5 ms

t= 1 ms

t= 0.5 ms

292 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Lenght(m)

Fig. 7.11 Variation of inside temperature versus shell length at different times, .k = 1, case (1)

diagrams are plotted at four different times. The errors in .x = 150 mm and .x = 300 mm areas are due to the discontinuity of applied thermal shock. The patterns of the field variables shown for the case (1) are all smooth and non-vibratory. The reason is the time duration of the assumed applied thermal shock load, being larger than the time period of the first natural frequency of the assumed cylindrical shell. In the second case, the applied thermal shock load, as shown in Fig. 7.15, is ten times faster than the case (1) and its time period is much smaller than the time period of the first natural frequency of the assumed cylindrical shell. The inside temperature versus time is displayed in Fig. 7.16 [19]. In general, the temperature

348

7 Couple Thermoelasticity of Shells 10-7

20 18 16

Radial Displacement(m)

14 12 10 8 6 4 2 t= 20 ms

t= 5 ms

t= 1 ms

t= 0.5 ms

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Lenght(m)

Fig. 7.12 Variation of radial displacement versus shell length at different times, .k = 1, case (1) 0

-500

Force Nxx(N/m)

-1000

-1500

-2000

-2500

-3000

t= 20 ms

t= 5 ms

t= 1 ms

t= 0.5 ms

-3500 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Length(m)

Fig. 7.13 Variation of force resultant versus shell length at different times, .k = 1, case (1)

distribution versus time for both cases are similar. Due to the basic assumptions of the shell theories and the classical coupled thermoelasticity theory, temperature distribution versus time remains smooth and non-vibratory for the type of assumed applied thermal shock load. The lateral displacement appears in the form of vibrations due to the assumed applied thermal shock, as seen in Fig. 7.17. Since the heat conduction coefficient of ceramic is higher than metal, as the power law index .k increases, the inside temperature increases too. The frequency of lateral vibration of the shell due to the increase of young modulus, as shown in Fig. 7.17, is increased. Also, the displacement amplitude decreases by the increase of index .k. The axial

7.2 Coupled Thermoelasticity of Cylindrical Shells

349

35

30

Moment Mxx(N.m/m)

25

20

15

10

5

0 t= 20 ms

t= 5 ms

t= 1 ms

t= 0.5 ms

-5 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Lenght(m)

Fig. 7.14 Variation of moment resultant versus shell length at different times, .k = 1, case (1) 3000

2500

Ti(K)

2000

1500

1000

500

0 0

0.2

0.4

0.6

0.8

1

time(s)

1.2

1.4

1.6

1.8

2 10-3

Fig. 7.15 Applied thermal shock, case (2)

force for pure metal is less than pure ceramic and for the FGM is higher than both, as illustrated in Fig. 7.18. This behavior is also reported by Bahtui et al. [10]. The axial moment versus time for different power law indexes .k is indicated in Fig. 7.19. The results are compared for different coupling coefficients and demonstrated in Figs. 7.20, 7.21, 7.22 and 7.23. For .β = 0, the mechanical coupling term from the energy equation equals zero, and the problem is decoupled. By increasing the coupling coefficient, the effects of damping increase, as demonstrated in Fig. 7.21. The axial force decreases by increasing the coupling coefficient, as illustrated in Fig. 7.22, but the axial moment does not change much, as shown in Fig. 7.23.

350

7 Couple Thermoelasticity of Shells

293.6

Inside Temperature(K)

293.5

293.4

293.3

293.2

293.1

k=0

k=1

k=4

k=infinity

293 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 10-3

Time(sec)

Fig. 7.16 Inside temperature of the shell at.x = 225 mm versus time for different power law indices, case (2) 10-7

2.5

Radial Displacement(m)

2

1.5

1

0.5

k=0

k=1

k=4

k=infinity

0 0

0.2

0.4

0.6

0.8

1

Time(sec)

1.2

1.4

1.6

1.8

2 10-3

Fig. 7.17 Radial displacement of the shell at .x = 225 mm versus time for different power law indices, case (2)

7.2 Coupled Thermoelasticity of Cylindrical Shells

351

0

The centre of the shocked lenght Nxx(N/m)

-50

-100

-150

-200

-250

-300

k=0

k=1

k=4

k=infinity

-350 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 10-3

Time(sec)

Fig. 7.18 Axial force resultant of the shell at .x = 225 mm versus time for different power law indices, case (2)

The centre of the shocked lenght Mxx(N.m/m)

3.5

3

2.5

2

1.5

1

0.5

k=0

k=4

k=1

k=infinity

0 0

0.2

0.4

0.6

0.8

1

Time(sec)

1.2

1.4

1.6

1.8

2 10-3

Fig. 7.19 Axial moment resultant of the shell at .x = 225 mm versus time for different power law indices, case (2)

352

7 Couple Thermoelasticity of Shells

293.6

Inside Temperature(K)

293.5

293.4

293.3

293.2

293.1

=2.3046*10 6

=0

=107

=5*107

293 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 10-3

Time(sec)

Fig. 7.20 Inside temperature of the shell at .x = 225 mm versus time for different mechanical coupling coefficient, .k = 1, case (2) 10-7

2.5

Radial Displacement(m)

2

1.5

1

0.5

=2.3046*10 6

=0

=107

=5*107

0 0

0.2

0.4

0.6

0.8

1

Time(sec)

1.2

1.4

1.6

1.8

2 10-3

Fig. 7.21 Radial displacement of the shell at .x = 225 mm versus time for different mechanical coupling coefficient, .k = 1, case (2)

7.2 Coupled Thermoelasticity of Cylindrical Shells

353

0

The centre of the shocked lenght Nxx(N/m)

-50

-100

-150

-200

-250

-300 =2.3046*10 6

=0

=107

=5*107

-350 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 10-3

Time(sec)

Fig. 7.22 Axial force resultant of the shell at .x = 225 mm versus time for different mechanical coupling coefficient, .k = 1, case (2)

The centre of the shocked lenght Mxx(N.m/m)

3

2.5

2

1.5

1

0.5

=2.3046*10 6

=0

=107

=5*107

0 0

0.2

0.4

0.6

0.8

1

Time(sec)

1.2

1.4

1.6

1.8

2 10-3

Fig. 7.23 Axial moment resultant of the shell at .x = 225 mm versus time for different mechanical coupling coefficient, .k = 1, case (2)

Appendix { A1,2,3,4,5,6 = A, 1 =

{ z

z

E (z) 1 − ν2

(

) ν ν 1 z ( z ) 2( z) , z ,. ,1 + ,z 1 + ,z 1+ dz R R z+R R R R

z) E (z) ( 1+ dz 2 (1 + ν) R

354

7 Couple Thermoelasticity of Shells

) z) ( 1 z 2 ( z ) 2( z) , ,z , 1+ ,z 1+ ,z 1+ dz R R R R R z ( ) { E (z) α(z) 1 z , 2 1, z, z , , dz B 1,2,3,4,5 = 1 − 2ν z+R z+R z ) ( { 1 R 2 , dz D1,2,3,4,5 = K (z) 1, z, z , z+R z+R z { ( ) F1,2,3 = ρ(z) c(z) 1, z, z 2 dz {

B1,2,3,4,5 =

E (z) α(z) 1−ν

(

z

(7.2.28)

7.3 Cylindrical Shell; Effect of Normal Stress Coupled thermoelasticity of cylindrical shells under thermal shock loads is addressed by Eslami et al. in references [24–27]. A general discussion of the cylindrical shells under thermal shock loads is presented in these references. Mathematical formulations of the coupled thermoelasticity of thick cylinders and their solution in the Laplace domain are discussed by Bagri and Eslami [28, 29]. The analytical solution of the coupled thermoelasticity of thick cylinders is given by Jabbari et al. [30], where the solution is obtained in the real time domain employing the general series solution. When lateral shock loads or highly concentrated loads are applied to shells, the effect of normal stress becomes significant [31]. The equations of motion of axially symmetric cylindrical shells under axisymmetric loads with the inclusion of normal stress and transverse shear effect and rotary inertia are [32] .

1 Nθx,θ + qx = I1 u ,tt + I2 ψ1,tt R 1 1 , ,, + 21 I3 w,tt Q x,x + Q θ,θ − Nθ − qx = I1 w,tt + I2 w,tt R R 1 Mx,x + Mθx,θ − Q x + m x = I2 u ,tt + I3 ψ1,tt R 1 1 , ,, Sx,x + + 21 I4 w,tt Sx,x − M − A − m z = I2 w,tt + I3 w,tt R R 1 1 h2 Tx,x + Tθ,θ − Pθ − B − qz = 21 I3 w,tt R R 8 1 , ,, + 21 I4 w,tt + I5 w,tt 4 N x,x +

(7.3.1)

7.3 Cylindrical Shell; Effect of Normal Stress

355

where Ii = ρ

.

{ h/2

(1 + z/R)(z i−1 )dz

i = 1, 2, ..5

−h/2 + qi = qi (1 + h/2R) + qi− (1 − h/2R) i = x, z m i = h/2 [qi+ (1 + h/2R) − qi− (1 − h/2R)] i = x, z

(7.3.2) The signs .(+) and .(−) refer to the application of load to outside and inside surfaces, respectively. The definition of force and moment resultants form Eqs. (7.3.1) and (7.3.2) are obtained as ⎡ .

⎤ ⎡ ⎤ { h/2 σx Nx ⎣ N xθ ⎦ = ⎣τxθ ⎦ (1 + z )dz R −h/2 τ Qx 1z ⎤ ⎡ ⎤ ⎡ { h/2 σθ Nθ ⎣τθx ⎦ dz ⎣ Nθx ⎦ = −h/2 τ Qθ θz ⎤ ⎡ ⎤ ⎡ { h/2 σx Mx ⎣τxθ ⎦ z (1 + z )dz ⎣ Mxθ ⎦ = R −h/2 τ Sx xz ⎡ ⎤ ⎡ ⎤ { h/2 σθ Mθ ⎣ Mθx ⎦ = ⎣τθx ⎦ z dz −h/2 τ Sθ θz [ ] { h/2 [ ] σx 1 Px = (1 + z/R)z 2 dz 2 Tx τ xz −h/2 [ ] { h/2 [ ] σθ 1 2 Pθ = z dz 2 Tθ τ θz −h/2 { h/2 { h/2 A= σz (1 + z/R)dz B = σz (1 + z/R)zdz −h/2

−h/2

(7.3.3)

In addition, the general displacement components are described in terms of the displacement components of shell middle plane as .

U (x, θ, z) = u(x, θ) + zψ1 (x, θ) V (x, θ, z) = v(x, θ) + zψ2 (x, θ) W (x, θ, z) = w(x, θ) + zw, (x, θ) +

z 2 ,, w (x, θ) 2

(7.3.4)

356

7 Couple Thermoelasticity of Shells

Using the general strain–displacement relations and the constitutive law, the equations of motion (7.3.1) in terms of the displacement components reduce to the following [31] 1 1 , a12 w,x + b11 ψ1,x x + a12 w,x R 1 1 ,, c12 )w,x +(b12 + + qx = I1 u ,tt + I2 ψ1,tt 2R 1 1 2 − a12 u ,x + a44 w,x x − 2 a11 w + a44 ψ1,x R R 1 1 2 ,, +b44 w,x x − (a12 + b11 )w , + 21 c44 w,x R R 1 2 ,, , ,, − + 21 I3 w,tt c w − qz = I1 w,tt + I2 w,tt 2R 11 1 1 1 b11 u ,x x − a44 w,x + c11 ψ1,x x − a44 ψ1 − (b44 − b12 1 , ,, − c12 )w,x − 21 (c44 − 2c12 )w,x + mx R = I2 u ,tt + I3 ψ1,tt 1 1 2 1 −a12 u ,x + b44 w,x x − (a12 + b11 )w R R 1 1 1 , +(b44 − b12 − c12 )ψ1,x + c44 w,x x R 1 2 1 ,, −(a11 + 2 c11 )w + 21 d44 w,x x R 1 2 1 1 (3c12 + d11 )]w ,, − m z = I2 w,tt −[b11 + 2R R , ,, +I3 w,tt + 21 I4 w,tt

a11 u ,x x +

.

.

1 2 1 c12 )u ,x + 21 c44 w,x x − c w 2R 2R 2 11 1 , + 21 (c44 − 2c12 )ψ1,x + 21 d44 w,x x 1 2 1 1 1 , ,, (3c12 + d11 )]w + e44 w,x −[b11 + x 2R R 4 2 1 2 ,, h qz = 21 I3 w,tt e ]w − −[c11 + 4R 2 11 8 1 , ,, + I5 w,tt + 21 I4 w,tt 4

1 −(b12 +

where the constant coefficients are defined as { h/2 . (ai j , ci j , ei j ) = C i j (1, z 2 , z 4 )dz −h/2

(7.3.5)

7.3 Cylindrical Shell; Effect of Normal Stress

{ (bi1j ,

di1j )

= Ci j

h/2

−h/2

357

(1 + z/R)(z, z 3 )dz {

(ai2j , bi2j , ci2j , di2j , ei2j ) = Ci j

h/2 −h/2

1 (1, z, z 2 , z 3 , z 4 )dz (1 + z/R) (7.3.6)

and .

E(1 − ν) (1 + ν)(1 − 2ν) νE = (1 − ν)(1 − 2ν) E = 2(1 + ν)

C11 = C22 = C33 = C12 = C13 = C23 C44 = C55 = C66

(7.3.7)

All other .Ci j = 0. It is noticed that the normal stress .σz is directly included in the equations of motion through the force and moment per unit length. A and. B. The effect of normal stress was also carried on through the strain–displacement and stress–strain relations and therefore the equations of motion (7.3.5) are consistent with the kinematical relations and the constitutive law. The effect of transverse shear and rotary inertia is also included in Eq. (7.3.5). The five dependent functions .u, w, ψ1 , w , , and .w ,, are functions of the variables .x and .t. The Galerkin method is used to obtain the finite element model of shell. A Kantrovich type of approximation for space and time domain is employed and the space domain is approximated by linear shape functions [33]. The dependent functions in equations of motion (7.3.1) are three, namely, .u, .w, and .ψ1 , and in the equations of motion (7.3.5) are five as .u, .w, .ψ1 , .w , , and .w ,, . Considering a base element .(e), the linear approximation yields .

u =< N1 > {U } w =< N1 > {W } ψ1 =< N1 > {ψ1 } w, =< N1 > {w , } w,, =< N1 > {w ,, }

(7.3.8)

where .< N1 > stands for linear approximation. Applying the formal Galerkin method to the equations of motion (7.3.5) and (7.3.1) results into the following the dynamic finite element equation of motion [M]{ X¨ } + [K ]{X } = {F}

.

(7.3.9)

The elements of mass matrix .[M], stiffness matrix .[K ], and the force matrix .{F} are obtained by employing Eq. (7.3.5) and the Galerkin method [24, 25]. The members

358

7 Couple Thermoelasticity of Shells

of force matrix are composed of two parts. The first part is directly related to the integration of external applied force over the element .(e). The second part is obtained through the weak formulation of higher order derivative terms in the equations of motion and their evaluation on the boundary. Therefore, the first part of the force matrix has force basis, while the second part of force matrix has kinematical basis. The kinematical force member is rather important in constructing the Galerkin finite element model, as the weak formulation should provide expressions which have kinematical meaning on the shell boundary [22]. Once the finite element equation of motion is established, different numerical methods can be employed to solve them in space and time domains. The Newmark direct integration method with a suitable time step is used and the equation of motion is solved. Consider a thin cylindrical shell of mean radius . R = 15 cm, thickness .h = 0.5 cm and length . L = 100 cm. The material properties are .ν = 0.3, . E = 196 Gpa, .ρ = 7904 kg/cm. The dynamic lateral inside pressure shock is assumed to be [31] .

P = 8 × 105 (1 − e−13100 t ) Pa

(7.3.10)

This pressure is uniformly distributed along the length of shell. Both ends of the cylindrical shell is assumed clamped. The Newmark method is used to integrate the dynamic equations of motion. The accuracy of the Newmark method strongly depends upon the parameters .α and .β (the accuracy and stability coefficients) as well as the time increment .Δt. For this problem with the assumed numerical values, the time increment .Δt = 5E − 6 was found to be suitable and the results converged and coincided with that of the Houbolt method. The behavior of radial displacement versus time and length from equations of motion (7.3.9) is plotted in Figs. 7.24 and 7.25, as given by Eslami et al. [3]. All four classical theories predict identical values for lateral displacement. The variation 2.5

× 10 -4

2

W(m)

1.5

1

(ALL THEORIES)

0.5 SOLID : AT THE MIDDLE OF THE LENGTH(X=0.5 m) DASHED(-) : AT THE QUARTER OF THE LRNGTH (X=0.25 m)

0 0

0.1

0.2

0.3

0.4

0.5

t(sec)

Fig. 7.24 Radial displacement versus time

0.6

0.7

0.8

0.9

1 × 10 -3

7.3 Cylindrical Shell; Effect of Normal Stress 1

359

× 10 -5

(ALL THEORIES)

0.8

0.6

0.4

U(m)

0.2

0

-0.2

-0.4

-0.6

SOLID : AT TIME 0.8 msec DASHED(-) : AT TIME 0.2 msec

-0.8

-1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t(sec)

Fig. 7.25 Axial displacement versus length 400

(ALL THEORIES) 200

0

MX(Nm/m)

-200

-400

-600

-800

SOLID : AT TIME 0.8 msec

-1000

DASHED(-) : AT TIME 0.2 msec -1200 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

L(m)

Fig. 7.26 Axial moment versus length

of axial displacement versus length based on the classical theories is plotted in Fig. 7.26. The results show identical values for different theories. The plots of radial and axial displacement show symmetry with respect to the middle section of shell, as expected. The axial force and moment also coincide for different classical theories, as shown in Figs. 7.27 and 7.28. However, for Sanders and Reissner the circumferential moment differs from Vlasov and Flugge theories as shown in Fig. 7.29. The second-order shell theories are relatively the most complete classical theory describing the shell behavior. These theories predict more accurate results compared to the first-order theory, especially when the shell thickness is considerably thick. However, while they predict about the same numerical values of displacement for

360

7 Couple Thermoelasticity of Shells AT THE QUARTER OF LENGTH

30

20

MT(N.m/m)

10

0

-10

-20

-30 SOLID : FLUGGE & VLASOV THEORY DASHED(-) : RETSSNER & SANDERS THEORY -40 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 × 10 -3

t(sec)

Fig. 7.27 Circumferential moment versus time 2.5

AT THE MIDDLE OF LENGTH

× 10 -4

2

W(m)

1.5

1

SOLID : WITH NORMAL STRESS DASHED(-) : WITHOUT NORMAL STRESS

0.5

0 0

0.1

0.2

0.3

0.4

0.5

t(sec)

0.6

0.7

0.8

0.9

1 × 10 -3

Fig. 7.28 Lateral displacement versus time

a loaded shell, the other shell parameters such as forces and moments may differ slightly for different theories, which in turn cause different values for stresses in shell. For thicker shells, or when the material shear modulus is relatively low, the effect of transverse shear deformation and rotary inertia becomes significant. In addition, type of applied load has influence on the response of shell and the shell theories are sensitive in this respect. For applied lateral shock and blast loads, or for highly concentrated loads, the effect of normal stress, in addition to the other secondorder assumptions, becomes important such that it effects even the displacement components.

7.4 Composite Cylindrical Shells 2

361 AT THE QUARTER OF LENGTH

× 10 -5

1.5

U(m)

1

0.5

0

SOLID : WITH NORMAL STRESS

-0.5

DASHED(-) : WITHOUT NORMAL STRESS

-1 0

0.1

0.2

0.3

0.4

0.5

t(sec)

0.6

0.7

0.8

0.9

1 × 10 -3

Fig. 7.29 Axial displacement versus time

7.4 Composite Cylindrical Shells In this section thermoelasticity of composite laminated cylindrical shell is investigated. The dynamic response of shell under thermal and mechanical shock loads is discussed. Thermo-mechanical coupling is also considered. Dynamic thermoelasticity of orthotropic cylindrical shells and multilayer cylindrical shells are discussed by Wu et al. [34], and Wang et al. [35]. The conventional classic coupled thermoelasticity assumptions are considered in these papers. Elasticity solution for thick laminated circular cylindrical shallow and non-shallow panels under dynamic loads, thick laminated anisotropic cylindrical panels, and multilayered shallow panels using the theory of elasticity are discussed by Alibiglu et al. [36–38] and Shakeri et al. [39–43]. These references assume the uncoupled thermoelasticity theory to derive their analysis. In this section, a multilayer orthotropic composite cylindrical shell under the thermal and mechanical shocks resulting in coupled thermoelastic field is considered. The full stress–strain relations of the three-dimensional theory of elasticity are considered and the Flugge second-order shell theory, which is the most complete compared to the other classical second-order theories, is employed to formulate the coupled thermoelasticity of shells of revolution. The results are obtained with the formulation where the normal stress is considered and the effect of normal stress in composite shell response is evaluated. Furthermore, the effect of coupling and layer stacking is studied and the results are compared for coupled and uncoupled models. Consider a thin cylindrical shell of thickness .h and mean radius . R with the principle coordinate system .(x, θ, z) as shown in Fig. 7.30, where .z−coordinate measures from the shell middle plane [7]. The shell is assumed to be under the thermo-mechanical loadings, where the load is assumed to be axisymmetric and

362

7 Couple Thermoelasticity of Shells

Fig. 7.30 Shell coordinates

uniformly distributed along the .x−axis. To write the shell governing equations, we start from the assumption of displacement distribution across the shell thickness. The basic assumption to consider the normal stress and strain in the shell equations requires to relate the displacement components along the principle orthogonal curvilinear coordinates of the shell, namely,.U ,.V , and.W , to the displacement components on the middle plane as given by the following relations .

U (x, θ, z) = u 0 (x, θ) + zψx (x, θ) V (x, θ, z) = v0 (x, θ) + zψθ (x, θ) W (x, θ, z) = w0 (x, θ) + zψz (x, θ) +

(7.4.1) 2

z φz (x, θ) 2

where .(x, θ, z) are the principle orthogonal curvilinear coordinates of the shell and u , v , w0 are the middle plane displacements, .ψx and .ψθ are rotations of the tangent line to the middle plane along .x and .θ axes, respectively, and .ψz and .φz are related to the nonzero transverse normal strains. The strain–displacement relations for the second-order shell theory follow to be

. 0 . 0 .

.

εx = u 0,x + zψx,x 1 1 1 2 εθ = z [ (v0,θ + w0 + zψθ,θ + zψz + z φz )] 1+ R R 2 εz = ψz + zφz 1 1 γxθ = v0,x + zψθ,x + [ (u 0,θ + zψx,θ )] 1 + Rz R 1 γx z = w0,x + ψx + zψz,x + z 2 φz,x 2 1 1 1 γθz = [ (w0,θ − v0 + Rψθ + zψz,θ + z 2 φz,θ )] 1 + Rz R 2

(7.4.2)

where .ε’s are the normal strains and .γi j are the shear strains. The sign .(, ) in the subscript indicates partial derivative. These relations are obtained based on Flugge

7.4 Composite Cylindrical Shells

363

second-order shell theory where the term .z/R is retained in the equations compared to the unity. The forces and moments resultants based on the second-order shell theory are defined as { +h z . < N x , A, N xθ , Q x >= (σx , σz , τxθ , τx z )(1 + )dz R −h { +h < Nθ , Nθx , Q θ >= (σθ , τxθ , τθz )dz −h

{

< Mx , B, Mxθ , Sx >= { < Mθ , Mθx , Sθ >= Pθ =

1 2 {

Tx = { Tθ =

{

+h

−h +h

−h +h

−h

+h

−h +h

−h

(σx , σz , τxθ , τx z )(1 +

z )zdz R

(σθ , τxθ , τθz )zdz

(7.4.3)

σθ z 2 dz

τx z (1 +

z 2 )z dz R

τθz (1 +

z 2 )z dz R

where . Ni j and . Mi j are the forces and moments per unit length of shell, . Q i is the transverse shear force, . A, . B, . Si , . Pθ , and .Ti are just some shell generalized forces as defined in Eq. (7.4.3). The equations of motions may be obtained using Hamilton’s variational principle. For this general case, where the normal stress and strain are included in the governing equations, Hamilton’s principle yields the following equilibrium equations .

Nθx,θ = I1 u¨ 0 + I2 ψ¨ x R Nθ,θ Qθ + N xθ,x + = I1 v¨0 + I2 ψ¨θ R R Nθ Q θ,θ I3 − + Q x,x + − Rqz = I1 w¨ 0 + I2 ψ¨ z + φ¨ z R R 2 Mθx,θ Mx,x + − Q x = I2 u¨ 0 + I3 ψ¨ x R Mθ,θ + Mxθ,x − Q θ = I2 v¨0 + I3 ψ¨θ R Mθ Sθ,θ I4 − + Sx,x − A + = I2 w¨ 0 + 2I3 ψ¨ z + φ¨ z R R 2 Pθ Tθ,θ I3 I4 ¨ I4 h2 − − B + Tx,x − − qz = w¨ 0 + ψz + φ¨ z R R 8 2 2 4 N x,x +

(7.4.4)

364

7 Couple Thermoelasticity of Shells

where the moments of inertia . In are defined as { .

In =

h −h

ρz (n−1) (1 +

z )dz R

n = 1, 2, .., 5

(7.4.5)

Here, .qz and .m z are the components of external forces and moments acting on the middle plane of the shell. These forces and moments are related to the external applied forces .qz+ and .qz− as .

.

z z ) + qz− (1 − ) R R z z z + − m z = [qz (1 + ) − qz (1 − )] 2 R R

qz = qz+ (1 +

(7.4.6)

The stress–strain relation for the k-th orthotropic layer bounded by surfaces at z = h k−1 and .z = h k is given by ⎡

⎤ ⎡ Q¯ 11 Q¯ 12 σx ⎢ σθ ⎥ ⎢ Q¯ 21 Q¯ 22 ⎢ ⎥ ⎢ ⎢ σz ⎥ ⎢ ¯ ¯ ⎥ = ⎢ Q 31 Q 32 .⎢ ⎢ τθz ⎥ ⎢ 0 0 ⎢ ⎥ ⎢ ⎣ τx z ⎦ ⎣ 0 0 τxθ k Q¯ 61 Q¯ 62

Q¯ 13 Q¯ 23 Q¯ 33 0 0 Q¯ 63

0 0 0 Q¯ 44 Q¯ 54 0

0 0 0 Q¯ 45 Q¯ 55 0

⎤ ⎡ ⎤ ⎡ ¯ ⎤ Q¯ 16 βx εx ⎢ εθ ⎥ ⎢ β¯θ ⎥ Q¯ 26 ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ¯ ⎥ [ ] Q¯ 36 ⎥ ⎥ ⎢ εz ⎥ − ⎢ βz ⎥ T (7.4.7) ⎢ ⎥ ⎢ 0 ⎥ k 0 ⎥ ⎥ ⎢ γθz ⎥ ⎥ ⎢ ⎣ 0 ⎦ 0 ⎦ ⎣ γx z ⎦ ¯ Q 16 k γxθ k β¯xθ k

where .[ Q¯ i j ]k and .[β¯i ]k are the stiffness and thermoelastic matrices, respectively [44]. The lamination constitutive relations are given by matrix .(a − 1) in Appendix A. Substituting the element of matrix .(a − 1) into the motion equations (7.4.4) results into seven Navier’s type equations for cylindrical shells in terms of the displacement components .u 0 , .v0 , .w0 , and rotations .ψθ , .ψx and the transverse strains .ψz , .φz .

7.4.1 Energy Equation The thermoelastic coupled heat conduction equation for the anisotropic media may be derived from the energy conservation relation based on the first law of thermodynamics and the definition of specific entropy in the form .

ki j T,i j −[cν ρT˙ + Ta βi j ε˙i j ] = 0

(7.4.8)

where the constants in Eq. (7.4.8) are defined in the Nomenclature. We may assume a linear temperature distribution across the shell thickness as [6, 44]

7.4 Composite Cylindrical Shells .

365

T (x, θ, z, t) = T0 (x, θ, t) + zT1 (x, θ, t)

(7.4.9)

where .T0 and .T1 are unknown functions to be obtained through the coupled system of equations. Following McQuillen and Brull [2], the traditional Galerkin method is used to derive two independent heat conduction equations of the shell of revolution from Eq. (7.4.8) by averaging it across the shell thickness .z. Due to the assumption of linear temperature variation across the shell thickness, as given by Eq. (7.4.9), two unknowns .T0 and .T1 appear in the energy equation. For multilayer composite cylindrical shells under axisymmetric thermal shock, uniformly distributed along the .x−axis, the energy equation (7.4.8) in terms of the displacement components reduces to .

βθθ ˙ Residual = ρc T˙ + Ta [βx x U˙ ,x + + W R+z ∂2 T ∂2 T 1 ∂T ) +βzz W˙ ,z + βxθ V˙,x ] − k x x 2 − k zz ( 2 + ∂x ∂z R + z ∂z

(7.4.10)

The following two integrals of the energy equation provide two independent energy equations for two independent functions .T0 and .T1 as follow [2, 6] { (Residual).(1).dz = 0

.

z Rx(1) u˙0,x

(1) ˙ (1) Rc(1) T˙0 + Rc(2) T˙1 + + Rx(2) ψ˙ x,x + Rθx v˙0,x + Rz(2) ψ˙ z + Rx(2) φ˙ z ψθ,x + Rθx 1 1 (3) ˙ 1 (1) 1 (1) (2) R φz + Rkx T0,x x + Rkx T1,x x − Rkz T1 + Rθ(1) w˙0 + Rθ(2) ψ˙ z + R R 2R θ R −(h i − h o )To + h(h i − h o )T1 + [h i Ti (t) − h o T∞ ] = 0 (7.4.11)

{ (Residual).(z).dz = 0

.

z (2) Rx u˙0,x

(2) ˙ (2) Rc(2) T˙0 + Rc(3) T˙1 + + Rx(3) ψ˙ x,x + Rθx v˙0,x + Rz(2) ψ˙ z + Rx(3) φ˙ z ψθ,x + Rθx 1 1 (4) ˙ 1 (2) 1 (2) (3) Rθ φz + Rkx T0,x x + Rkx T1,x x − Rkz T1 + Rθ(2) w˙0 + Rθ(3) ψ˙ z + R R 2R R −h[(h i − h o )To − (h i − h o )T1 − (h i Ti (t) − h o T∞ )] = 0 (7.4.12)

where .

(1) Rkz =

N Σ

i k zz (h i − h i−1 )

i=1

(2) . Rkz

=

N { Σ i=1

hj h j−1

k zzj zdz

366

7 Couple Thermoelasticity of Shells

.

.

(i) Rkx =

Rc(i) =

N { Σ i=1

h j−1

N { Σ

hj

=

N { Σ j=1

.

Rθ(i) =

hj

=

N { Σ

(i) Rxθ =

hj

N { Σ j=1

< ρcν > j z (i−1) dz

i = 1, 2, 3

< Ta βx x > j z (i−1) dz

i = 1, 2, 3

< Ta βθθ > j z (i−1) dz

i = 1, 2, 3

< Ta βzz > j z (i−1) dz

i = 1, 2, 3

< Ta βxθ > j z (i−1) dz

i = 1, 2, 3

h j−1

N { Σ j=1

.

i = 1, 2, 3

h j−1

j=1

(i) . Rz

k xj x z (i−1) dz

h j−1

j=1

(i) . Rx

hj

hj h j−1

hj h j−1

The seven equations of motion (7.4.4), written in terms of the displacement and rotation components, and the two energy equations (7.4.11) and (7.4.12) constitute the governing equations for the nine dependent unknown functions .u 0 , .v0 , .w0 , .ψθ , .ψ x , .ψz , .φz , . T0 and . T1 . The governing equations are solved by means of the Galerkin finite element method using the initial and boundary conditions.

7.4.2 Finite Element Solution The analytical solution of cylindrical shell with the thermoelastic equilibrium equations (7.4.4) and the coupled energy equations (7.4.11) and (7.4.12) is practically not possible. The finite element technique based on the Galerkin method is very efficient to solve the coupled thermoelastic problems, including the coupled thermoelasticity of shells [3, 6]. This method is used to obtain the solution of the coupled equations (7.4.4), (7.4.11), and (7.4.12). The nodal degrees of a shell under axisymmetric load and coupled thermoelastic assumption are seven shell variables .u 0 , .v0 , .w0 , .ψθ , .ψx , .ψz , .φz and two temperature variables . T0 and . T1 . A linear set of test functions are adopted to model the dependent functions. Considering identical shape functions for all nine degrees of freedom and applying the formal Galerkin method to the system of seven shell equations and two energy equations (7.4.11) and (7.4.12) results into the following finite element equation.

7.4 Composite Cylindrical Shells .

¨ + [C]{d} ˙ + [K ]{d} = {F} [M]{d}

367

(7.4.13)

The force matrix is divided into two terms, one term composed of the terms obtained through the weak formulation of the governing equations and the resulting natural boundary conditions, and the second term which include the components of external applied forces and thermal shocks. The process of the weak formulation and the terms which are selected for weak formulation in the governing equations are very important in regard to the resulting natural boundary conditions. The natural boundary conditions which are obtained as the result of the weak formulation should have either a kinematical meaning on the boundary or add up to make a traction boundary condition. Therefore, it is essential to set up a possible kinematic and forced shell boundary conditions in advance and try to obtain them by the weak formulation. The finite element equilibrium equation (7.4.13) may be solved in the time domain by many techniques such as Newmark, Houbolt, Wilson , and other time integration techniques. In this section, the .α-method is used. According to this method the equilibrium equations are transformed into the following form in the time domain [22]. .

[M]{a}n+1 + (1 + α)[C]{v}n+1 − α[C]{v}n +(1 + α)[K ]{d}n+1 − α[K ]{d}n = {F(tn+α) }

(7.4.14)

˙ and .a = .d. ¨ The .α−method where .tn+α = (1 + α)tn+1 − αtn = tn+1 + α Δ t, .v = .d, become Newmark method when .α = 0. The displacement and velocity matrices at time step .(n + 1) is written in terms of their values at time step n as Δt 2 [(1 − 2β){a}n + 2β{a}n+1 ] 2 = {v}n + Δt[(1 − γ){a}n + γ{a}n+1

.

{d}n+1 = {d}n + Δt{v}n +

(7.4.15)

.

{v}n+1

(7.4.16)

where .γ and .β are the accuracy and stability parameters, respectively. Using Eqs. (7.4.14), (7.4.15) and (7.4.16) we can obtain the unknown matrices .{d}n+1 , .{v}n+1 and .{a}n+1 in terms of their values at .tn . The stability condition of the.α-method, similar to other methods such as the Newmark, is based on positive definite matrices. Application of the Galerkin method to this problem results into non-axisymmetric stiffness and damping matrices and therefore the resulting solution must be checked for its convergency. The selection of the time increment is important and has an absolute effect on the solution convergence.

7.4.3 Discussion and Results Consider a multilayer thin cylindrical shell of thickness .h, mean radius . R, and length L. The material constants of each individual layer are given in Table 7.2, where the definition of terms are given in the Nomenclature. The indices 1,2, and .n define the

.

368

7 Couple Thermoelasticity of Shells

Table 7.2 Material properties of three orthotropic layers of 90/0/90 . E 11 . E 22 . E 1n 196 .GN/m.2 4.83 .GN/m.2 4.83 .GN/m.2 .G 2n

.ν12

.ν2n

3.44 .GN/m.2 .k22 67 W/mk .α33 −6 .15 ∗ 10 1/k

0.05 .k33 67 W/mk .h i 2 .10000 W/m. k

0.3 .α11 −6 .1.3 ∗ 10 1/k .h o 2 .200 W/m. k

.G 12

3.44 .GN/m.2 .k11

180 W/mk .α22 .15

∗ 10−6 1/k

directions along the cylindrical shell axis, circumferential direction, and thickness direction, respectively, see Fig. 7.30. We further assume that the boundary conditions along the edges .x = 0 and .x = L are clamped and zero initial conditions for displacements and velocities are considered. The thermal boundary conditions at the ends .x = 0 and .x = L are assumed isolated, where the inside and outside surfaces transfer the heat to the ambient by convection (with convective coefficients of .h i and .h o ). As the first example, we may consider a thin cylindrical shell of three orthotropic layers with ply angle stacking of [90/0/90] under pressure shock. The shell length, radius, and thickness are assumed to be . L = 1 m, . R = 0.15 m, and .h = 0.015 m, respectively. We assume that the inside pressure shock is uniformly applied by the following equation 6 . Pi (t) = 8 × 10 [1 − exp (−1300t)] (7.4.17) The inside pressure from Eq. (7.4.17) reaches to a sustained maximum value of .8 Mpa within .0.4 ms. Shell is divided into 200 elements along the .x−axis, and time increment is selected .Δt = 5 × 10−6 s. Shell behavior is studied up to the final time of .1 ms. The convergence of the finite element model used in this analysis depends on the total number of elements, time increment, and the numerical values of the stability coefficients .α and .β in the .α−method algorithm. The selected values of .Δt and the total number of elements along with the values of .α and .β are all within the proper range of convergency. Figure 7.31 shows the variation of the radial deflection of the shell middle plane at shell mid-length versus time for composite [90/0/90] and isotropic .(E = E 11 = 196 Mpa) cylindrical shells. Due to the higher stiffness of the composite shell compared to the assumed isotropic shell, its lateral deflection is considerably smaller. The result is well compared with the results of Noor and Burton [45]. Figure 7.32 is the plot of radial deflection versus the length of the shell for two different layers stacking [90/0/90] and [0/90/0]. The shell with layer stacking [90/0/90] has higher stiffness and thus its deflection is lower. Although the energy equation is coupled with the shell thermoelastic equation, the temperature rise due to the applied inside pressure was obtained to be negligible and is not shown for this example. In addition to this result,

7.4 Composite Cylindrical Shells

Fig. 7.31 Radial deflection of cylindrical shell versus time under inside pressure shock

Fig. 7.32 Effect of layer stacking of cylindrical shell under inside pressure shock

369

370

7 Couple Thermoelasticity of Shells

Fig. 7.33 Effect of thermo-mechanical coupling on the temperature time history

it was found that for the assumed non-symmetric stacking sequence (three layers composite) circumferential displacement is nonzero under axisymmetric loading. The second example is a composite cylindrical shell with the material properties given in Table 7.2. The mechanical and thermal boundary conditions are the same as the first example. Shell is assumed to have three layers with stacking sequence [90/0/90] and is considered to be exposed to an inside temperature shock given by T (t) = 2207(1 − e−13100t ) + 293◦ k

. i

(7.4.18)

The temperature of the inside surface raise rapidly from .293 to 2500 .◦ K in .1 ms, and the shell behavior is studied up to .4 s. This time period of study is about 90 times larger than the time required for the thermal shock to reach its steady-state condition. Shell is divided into 100 elements along its length. The time increment is selected .Δt = 10−5 s. The .α−method stability coefficients are selected as .α = 0.5 and .β = 0.25. The inside temperature versus time is shown in Fig. 7.33. The effect of thermomechanical coupling is shown in this figure. The curves in the time period .3 ≤ t ≤ 3.09 is zoomed to show more clearly the difference between the coupled and uncoupled theories. For the uncoupled theory, the parameter .βi j in Eq. (7.4.8) is set to zero, and the energy equation is independently solved for .T0 and .T1 . The results are then used in Eq. (7.4.4) to obtain the thermoelastic response of the shell. It is observed from this figure that the coupled theory predicts slightly lower temperature distribution by time. This conclusion is validated by McQuillen and Brull [2] and is

7.4 Composite Cylindrical Shells

371

Fig. 7.34 Effect of thermo-mechanical coupling on the radial displacement time history

also in agreement with the results of Eslami et al. [3, 6]. It is generally accepted that the coupled thermoelasticity theory predicts slightly lower temperature distribution in the structures [46]. In Fig. 7.34 this comparison is shown for the middle plane radial deflection. It is noticed that while at .t = 0.45 ms the thermal shock is reached to its steady-state condition, the lateral deflection is still increasing. The reason is that the characteristic time of heat transfer is much larger than the mechanical characteristic time for stress wave. This behavior is different when the shell is under pressure shock [6]. In Figs. 7.33 and 7.34, the values of temperature and displacement for coupled condition are less than the values for uncoupled condition. This means that the coupling effect acts like a damper and thus it could be regarded as thermoelastic damping. At the beginning of the shock, due to the lower values of strains, the difference between the coupled and uncoupled theories is negligible but as time increases the difference increases. This difference between the coupled and uncoupled solutions eventually vanishes for a large enough time, as expected. When the temperature reaches its steady-state condition, the strains reach their maximum values while their time rate is decreased and the effect of mechanical coupling is increased. In Figs. 7.35 and 7.36 time history of axial force and axial moment at inner surface is shown. Although both temperature and radial displacement are related to the axial force, due to their signs and magnitudes the effect of temperature is dominant and the axial force is negative. The variation of axial force and axial moment versus the shell length is shown in Figs. 7.37 and 7.38. As expected, the ply angle will have an important effect on the response of heated shell. The results are presented in Figs. 7.39 and 7.40 for axial displacement versus length and axial stress versus thickness.

372

7 Couple Thermoelasticity of Shells

Fig. 7.35 Effect of thermo-mechanical coupling on the axial force time history

Fig. 7.36 Effect of thermo-mechanical coupling on the axial moment time history

7.4 Composite Cylindrical Shells

Fig. 7.37 Effect of thermo-mechanical coupling on the axial force versus shell length

Fig. 7.38 Effect of thermo-mechanical coupling on the axial moment versus the shell length

373

374

7 Couple Thermoelasticity of Shells

Fig. 7.39 Effect of layer stacking on axial displacement in coupled theory

Fig. 7.40 Effect of layer stacking on axial stress distribution through the thickness

7.4 Composite Cylindrical Shells

375

Fig. 7.41 Time history of radial displacement at middle length of shell for two theories . R/ h = 30

The effect of normal stress is studied in the next example. A simply supported cylindrical shell under inside uniform axisymmetric pressure and thermal shocks of .

P(t) = 8 ∗ 106 (1 − e−13100t ) Ti (t) = 2207(1 − e−13100t ) + 293◦ k

(7.4.19)

is considered. Pressure reaches its maximum value at .1 ms. Figures 7.41 and 7.42 show the time history of the radial deflection in middle length of shell for two theories; when normal stress is considered (.w is quadratic function of .z), and when normal stress is not considered (.w is constant across the thickness) for. R/ h = 30 and . R/ h = 10, respectively. Table 7.3 gives the radial displacement for middle length at .4.5 ms. The difference between two cases are about 1.4 percent for . R/ h = 30 and 7.35 percent for . R/ h = 10. In Fig. 7.43 time history of circumferential moment at middle length of shell for . R/ h = 30 is plotted. It is observed that the effect of normal stress is to increase circumferential moment.

376

7 Couple Thermoelasticity of Shells

Fig. 7.42 Time history of radial displacement at middle length of shell for two theories . R/ h = 10

Appendix A 1 ⎤ ⎡ a11 Nx a12 ⎢ Nθ ⎥ ⎢ ⎢ ⎥ ⎢ 2 ⎢ ⎢ A ⎥ ⎢ a13 ⎢ ⎥ ⎢ 1 ⎢ N xθ ⎥ ⎢ a16 ⎢ ⎥ ⎢ Nθx ⎥ ⎢ a ⎢ ⎥ ⎢ 16 1 ⎢ Mx ⎥ ⎢ ⎢ ⎥ ⎢ b11 ⎢ Mθ ⎥ ⎢ ⎢ ⎥ ⎢ b12 ⎢ B ⎥ ⎢ 2 ⎢ ⎥ ⎢ ⎢ b13 . ⎢ M x,θ ⎥ = ⎢ b1 ⎢ ⎥ 16 ⎢ Mθx ⎥ ⎢ ⎢ ⎥ ⎢ b16 ⎢ ⎢ 2Px ⎥ ⎢ 1 ⎢ ⎥ c11 ⎢ 2Pθ ⎥ ⎢ ⎢ ⎥ ⎢ c12 ⎢ ⎢ Qθ ⎥ ⎢ ⎢ ⎥ 0 ⎢ Sθ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎢ ⎢ Sx ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎣ 2Tθ ⎦ ⎣ 0 0 2Tx 0

⎡

2 a12 a13 3 a4 a22 23 4 a5 a23 33 2 a26 a36 4 a26 a36 2 b12 b13 3 b4 b22 23 4 b5 b23 33 2 b26 b36 3 b4 b26 36 2 c12 c13 3 c4 c22 23

1 a 1 a16 16 b11 b12 3 b 3 a26 a26 12 b22 2 4 2 4 a36 a36 b13 b23 1 1 a66 a66 b16 b26 3 b 3 a66 a66 16 b26 1 b 1 c b16 c 16 11 12 3 c b26 b26 11 2 4 2 b36 b36 c13 1 b 1 b66 66 c16 3 c b66 b66 16

3 c22 4 c23 c26 3 c26

1 c 1 c16 16 d11 d12 3 3 c26 c26 d12 d22

2 b13 4 b23 5 b33 2 b36 4 b36 2 c13 4 c23 5 c33 2 c36 4 c36 2 d13 4 d23

1 b b16 16 3 b26 b26 2 4 b36 b36 1 b b66 66 3 b66 b66 1 c c16 16

1 c c11 12

0

0

0

0

0

3 c12 c22 2 4 c13 c23 1 c16 c26 3 c16 c26

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

d11 d12

3 d c26 c26 12 2 4 2 c36 c36 d13 1 c 1 c66 66 d16 3 d c66 c66 16

3 d22 4 d23

0

0

0

0

0

0

0

0

0

0

d26

0

0

0

0

0

3 d26

0

0

0

0

0

1 d 1 d16 16 e11 e12 3 3 d26 d26 e12 e22

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

3 a a44 45 1 a45 a55 3 b44 b45 1 b45 b55 3 c c44 45 1 c45 c55

3 b b44 45 1 b45 b55 3 c44 c45 1 c45 c55 3 d d44 45 1 d45 d55

3 c44

0

⎤

⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ c45 ⎥ 1 ⎥ c55 ⎥ d45 ⎥ ⎥ 1 d55 ⎦ 0

c45 3 d44 d45 3 e e44 45 1 e45 e55

7.4 Composite Cylindrical Shells

377

Table 7.3 Effect of normal stress (cylindrical shells) w(m) (R/h .= 30) Theory including .σn Theory excluding .σn Percent of difference

.3.1868E .3.1417E

−4 −4

1.4

w(m) (R/h .= 10) .5.0391E .4.6686E

−5 −5

7.35

Fig. 7.43 Time history of axial moment at middle length of shell for two theories . R/ h = 10

⎡

ε01 ε02 w1 β10 β20 ε,1 ε,2 w2 β1, β2, 1 ,, ε 2 ,,1 1 ε 2 2 μ02 μ01 μ,2 μ,1

⎤

⎡

at12 ⎥ ⎢ at 4 ⎢ ⎥ ⎢ 25 ⎢ ⎥ ⎢ at ⎢ ⎥ ⎢ 2n ⎢ ⎥ ⎢ at ⎢ ⎥ ⎢ 12 ⎢ ⎥ ⎢ at 4 ⎢ ⎥ ⎢ 122 ⎢ ⎥ ⎢ bt ⎢ ⎥ ⎢ 14 ⎢ ⎥ ⎢ bt ⎢ ⎥ ⎢ 25 ⎢ ⎥ ⎢ bt ⎢ ⎥ ⎢ 2n ⎢ ⎥ ⎢ bt ⎢ ⎥ − ⎢ 12 .⎢ ⎥ ⎢ bt 4 ⎢ ⎥ ⎢ 12 ⎢ ⎥ ⎢ ct 2 ⎢ ⎥ ⎢ 14 ⎢ ⎥ ⎢ ct ⎢ ⎥ ⎢ 2 ⎢ ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎢ ,, ⎥ ⎢ 0 ⎣ 1μ ⎦ ⎣ 0 2 2,, 1 0 μ 2

1

⎤ bt12 bt24 ⎥ ⎥ btn5 ⎥ ⎥ 2 ⎥ bt12 ⎥ 4 ⎥ bt12 ⎥ ct12 ⎥ ⎥ ct24 ⎥ ⎥ ctn5 ⎥ ⎥[ ] 2 ⎥ ct12 ⎥ T0 4 ⎥ ct12 ⎥ T1 dt12 ⎥ ⎥ dt24 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ 0

378

7 Couple Thermoelasticity of Shells

where .aimj , bimj , cimj , dimj , eimj are

Σ N { hk 2 3 4 (ai j , bi j , ci j , di j , ei j )=(ai4j , bi4j , ci4j , di4j , ei4j ) = ( Q¯ i j )k k=1 h k−1 (1, z, z , z , z )dz 1 1 1 1 1 2 2 2 2 2 .(ai j , bi j , ci j , di j , ei j ) = (ai j , bi j , ci j , di j , ei j ) ΣN { hk 5 5 5 2 3 4 ¯ i j )k k=1 .= (ai j , bi j , ci j ) = ( Q h k−1 (1, z, z , z , z )dz { Σ hk N 1 3 3 3 3 3 2 3 4 ¯ i j )k k=1 .(ai j , bi j , ci j , di j , ei j ) = ( Q h k−1 (1+z/R) (1, z, z , z , z )dz { Σ hk 1 2 2 2 2 2 3 ¯ i )k N .(ati , bti , cti , dti ) = (ati , bti , cti , dti ) = (β k=1 h k−1 1+z/R (1, z, z , z )dz { Σ hk 1 1 1 1 2 3 ¯ i )k N .(ati , bti , ci , di ) = (β k=1 h k−1 (1, z, z , z )dz .

ε0 = u o,x , .ε0θ = vRo,θ + wRo , .ε,x = ψx,x , .ε,θ = ψRθ,θ + ψRz , ,, φz , .ε x = 0, .ε = , .βx0 = vo,x , .βθ0 = u o,θ R, θ R wo,θ −vo , 0 , 0 .β x = ψθ,x , .β = ψ x,θ , .μ x = wo,x + ψ x , .μ = + ψθ , θ θ R ,, ,, ψ ψ z,θ z,θ , , , .μx = φz,x , .μθ = R .μ x = ψz,x , .μ = θ R . x

7.5 Coupled Thermoelasticity of Spherical Shells Many researches are conducted on the coupled thermoelastic analysis of structural members and reported in literature. The coupled thermoelasticity of spherical shells, however, is limited in literature. Using the Ritz finite element method, Chang and Shyong [47] presented an investigation on the vibrational behavior of laminated cylindrical shell panels subjected to a surface thermal shock. Through numerical results of this work, effects of themo-mechanical coupling, boundary conditions, ply angles, and radius of curvature on dynamic thermoelastic response of laminated cylindrical shell panels were studied. Coupled thermoelasticity of thick spherical shells made of isotropic and homogeneous material is given by Eslami and Vahedi [48]. The Galerkin finite element method, employing the displacement formulation, is used to derive the response of the thick sphere. Hata [49] has presented the coupled thermoelasticity of hollow spheres under rapid uniform heating. For thin spherical shells under uncoupled thermoelastic loads, we may refer to the work of Eslami and Alizadeh [50, 51]. In these papers, the Galerkin finite element method is employed to analyze the spherical shells under non-axisymmetric loads. The coupled thermoelasticity of thin spherical shells is discussed by Amiri et al. [52]. In this paper the shell material is assumed to be isotropic and homogeneous and the classical coupled thermoelasticity theory is used for the derivation of the shell response. This section presents the dynamic coupled thermoelastic response of spherical shells to suddenly applied thermal shocks. The work is formulated based on a firstorder shear-deformable shell model. The shell is considered to obey Hooke’s law for their mechanical material properties and to accept infinitesimal strains under the applied load conditions. The classical dynamic coupled energy equation is used to obtain the temperature distribution through the shells. Also, caused by thinness assumption for shells, the temperature field in the shells thickness direction

7.5 Coupled Thermoelasticity of Spherical Shells

379

is approximated to vary linearly. Based on the governing equations, a flexural coupled thermoelastic element is derived and its applicability is proved in comparison with the ABAQUS software. Also, the effects of shells slenderness and thermal edge conditions on the dynamic coupled thermoelastic response of thin spherical shells to lateral thermal shocks are investigated. The distribution of displacement field is approximated as (7.5.1)

.

U (φ, θ, z, t) = u 0 (φ, θ, t) + z ψφ (φ, θ, t) V (φ, θ, z, t) = v0 (φ, θ, t) + z ψθ (φ, θ, t)

(7.5.1)

W (φ, θ, z, t) = w0 (φ, θ, t) where .z expresses the thickness coordinate of shells such that the radial position for material points of the shell is expressed by . R + z. Shells under applied loads are considered to have infinitesimal deformations [53]. Accordingly, in spherical coordinates, strain–displacement relations are obtained as (7.5.2) εφφ εθθ .

εφz

1 = R(1 + Rz )

(

∂u 0 ∂ψφ + w0 + z ∂φ ∂φ

)

( ) 1 ∂v0 1 z + cot φ u 0 + w0 + (cos φ ψφ ) = R(1 + Rz ) sin φ ∂θ sin φ ) ( 1 ∂w0 1 u0 + ψ = − φ (1 + Rz ) R ∂φ R

(7.5.2)

εθz = 0 εφθ = 0 The material properties of shell are assumed to be linearly thermoelastic so as the constitutive equations become σφφ = σθθ = .

σθz = σφz = σφθ =

E (εφφ + ν εθθ ) − 1 − ν2 E (εθθ + ν εφφ ) − 1 − ν2 E εθz 2(1 + ν) E εφz 2(1 + ν) E εφθ 2(1 + ν)

Eα T 1−ν Eα T 1−ν (7.5.3)

Using Hamilton’s principle and variational calculus the equations of motion for spherical shells become [53]

380

7 Couple Thermoelasticity of Shells

) ( ∂ (R sin φ Nφφ ) − R cos φ Nθθ + R sin φ Q φ = I0 u¨ 0 + I1 ψ¨φ R 2 sin φ ∂φ ) ( ∂ (R sin φ Q φ ) − R sin φ Nφφ + Nθθ = I0 w¨ 0 R 2 sin φ . ∂φ ) ( ∂ (R sin φ Mφφ ) − R cos φ Mθθ − R 2 sin φ Q φ = I1 u¨ 0 + I2 ψ¨φ R 2 sin φ ∂φ (7.5.4) where . Ii s are integral constants and are { I =

. i

h 2

− h2

ρ z i dz

(7.5.5)

Also, stress resultants appeared in (7.5.4) are defined as (7.5.6) { Ni j = { .

Mi j = { Qi =

h 2

− h2 h 2

− h2 h 2

− h2

σi j dz, σi j z dz, σi z dz,

i j = φφ, θθ, φθ i j = φφ, θθ

(7.5.6)

i = φ, θ

Along with the motion equations, the associated boundary conditions for spherical shells are obtained. These boundary conditions could be presented as (7.5.7).

.

δu 0 = 0

or

δψφ = 0

or

Nφφ = N¯ φφ Mφφ = M¯ φφ

or

Q φ = Q¯ φ

δw0 = 0

(7.5.7)

7.5.1 Energy Equations In the Cartesian coordinate system, the energy equation associated with linear dynamic coupled thermoelasticity using indicial notation and summation convention is as given by (7.5.8) [1]. For more information on the linear dynamic coupled thermoelasticity see [54, 55]. ρ Cε T˙ + T0 βi j ε˙i j = (ki j T, j ),i

.

(7.5.8)

7.5 Coupled Thermoelasticity of Spherical Shells

381

where .i, j = 1, 2, 3. This equation in spherical coordinates and for axisymmetric analysis becomes ( ) ( )) 1 1 ∂ ∂T 1 ∂ 2 ∂T R + 2 sin φ .k R 2 ∂z ∂z R sin φ ∂φ ∂φ −ρ Cε T˙ − β T0 (ε˙zz + ε˙φφ + ε˙θθ ) = 0 (

(7.5.9)

where .β = (3λ + 2μ) α. In this work, it is assumed that the temperature varies linearly through the shell thickness. So, the mathematical statement of temperature distribution through the shell is as given by (7.5.10) .

T (φ, θ, z, t) = T0 (φ, θ, t) + z T1 (φ, θ, t)

(7.5.10)

For imposing the prescribed temperature distribution on the energy equation, the Galerkin method is utilized. In this way, the coefficients of Eq. (7.5.10) are made orthogonal to Eq. (7.5.9) in thickness direction of shells. Implementing the mentioned process leads to two reduced energy equations, i.e., Eq. (7.5.11), that determine distribution of .T0 and .T1 through the spherical shells. h ρ Cε h T˙0 + β T0 (u˙ 0 + 2 w˙ 0 + u˙ 0 cot φ) − qz R ( ) ∂ ∂T0 kh sin φ =0 − 2 R sin φ ∂φ ∂φ h3 ˙ h3 ˙ h .ρ C ε T1 + β T0 ψφ (1 + cot φ) − qz 12 12R 2 ( ) 3 ∂ ∂T1 kh sin φ =0 + k h T1 − 12R 2 sin φ ∂φ ∂φ .

(7.5.11a)

(7.5.11b)

Derivation of weak form for governing equations is rooted in the concept of weighted residual integrals. The process begins with the orthogonalization of motion and energy equations to the test functions, i.e., .δu 0 , .δw0 , .δψφ , .δT0 , and .δT1 . In this section, .S indicates the curved-path along meridional direction of shells, and located at their mid-surface. For spherical shells .S = R φ, and .dS = R dφ. The weighted integral-form associated with governing equations of spherical shells may be written as ( { ∂ (R sin φ Nφφ ) − R cos φ Nθθ + R sin φ Q φ . δu 0 ∂φ S ) ( ) − I0 u¨ 0 + I1 ψ¨φ R 2 sin φ dS = 0 (7.5.12a) ( { ∂ (R sin φ Mφφ ) − R cos φ Mθθ − R 2 sin φ Q φ . δw0 ∂φ S ) ( ) − I1 u¨ 0 + I2 ψ¨φ R 2 sin φ dS = 0 (7.5.12b)

382

7 Couple Thermoelasticity of Shells

(

{ .

S

δψφ

) ( ) ∂ (R sin φ Q φ ) − R sin φ Nφφ + Nθθ − I0 w¨ 0 R 2 sin φ dS = 0 ∂φ (7.5.12c)

( h . δT0 ρ Cε h T˙0 + β T0 (u˙ 0 + 2 w˙ 0 + u˙ 0 cot φ) − qz R S ( )) ∂T0 ∂ kh sin φ dS = 0 − 2 R sin φ ∂φ ∂φ ( { h3 ˙ h3 ˙ h . δT1 ρ Cε T1 + β T0 ψφ (1 + cot φ) − qz 12 12R 2 S ( )) 3 ∂ ∂T1 kh sin φ dS = 0 +k h T1 − 2 12R sin φ ∂φ ∂φ {

(7.5.12d)

(7.5.12e)

Integrating Eq. (7.5.12) by part and using Gauss’s theorem in conjunction with some variational principles, the weak form of governing equations are obtained [22].

7.5.2 Solution Procedure In this part, the weak form of governing equations is spatially discretized using a standard displacement-based finite element procedure. For this purpose, filed variables, namely .u 0 , .w0 , .ψφ , .T0 , and .T1 , are discretized as given by Eq. (7.5.13) u 0 (S, t) ≈ Nu 0 (S) · Xu 0 (t) w0 (S, t) ≈ Nw0 (S) · Xw0 (t) .

ψφ (S, t) ≈ Nψφ (S) · Xψφ (t)

(7.5.13)

T0 (S, t) ≈ N (S) · X (t) T0

T0

T1 (S, t) ≈ NT1 (S) · XT1 (t) Substituting Eq. (7.5.13) into weak form of governing equations along with some mathematical operations leads to associated finite element equations and may be arranged in the following matrix form ⎡

Miuj0 u 0 ⎢ w0 u 0 ⎢ Mi j ⎢ ψφ u 0 . ⎢M ⎢ ij ⎢ T0 u 0 ⎣ Mi j MiTj1 u 0

Miuj0 w0 Miwj 0 w0 ψ w Mi j φ 0 MiTj0 w0 MiTj1 w0

u ψ

Mi j0 φ w ψ Mi j 0 φ ψ ψ Mi j φ φ Tψ Mi j0 φ Tψ Mi j1 φ

Miuj0 T0 Miwj 0 T0 ψ T Mi j φ 0 MiTj0 T0 MiTj1 T0

⎤⎡ ⎤ Miuj0 T1 X¨ uj 0 (t) ⎥ ¨ w0 ⎥ Miwj 0 T1 ⎥ ⎢ ⎢ X j (t)⎥ ⎢ ψφ ⎥ ψφ T1 ⎥ ⎥ Mi j ⎥ ⎢ X¨ j (t)⎥ + ⎥ ⎢ T0 ⎥ MiTj0 T1 ⎦ ⎣ X¨ j (t) ⎦ X¨ Tj 1 (t) M T1 T1 ij

7.5 Coupled Thermoelasticity of Spherical Shells

⎤⎡ ⎤ u ψ Ciuj0 u 0 Ciuj0 w0 Ci j0 φ Ciuj0 T0 Ciuj0 T1 X˙ u 0 (t) ⎢ w0 u 0 w0 w0 w0 ψφ w0 T0 w0 T1 ⎥ ⎢ ˙ wj 0 ⎥ ⎢Ci j Ci j ⎥ X j (t)⎥ C C C ⎢ ψφ u 0 ψφ w0 iψjφ ψφ iψjφ T0 iψjφ T1 ⎥ ⎢ ⎢ ˙ ψφ ⎥ ⎢C X j (t)⎥ + Ci j Ci j Ci j ⎥ ⎢ i j Ci j ⎥⎢ ⎥ ⎢ T0 u 0 T0 w0 T0 ψφ T0 T0 T0 T1 ⎥ ⎢ ⎣ X˙ Tj 0 (t) ⎦ C C C C C ⎣ ij ij ij ij ij ⎦ Tψ X˙ Tj 1 (t) CiTj1 u 0 CiTj1 w0 Ci j1 φ CiTj1 T0 CiTj1 T1 ⎡ uu ⎤⎡ ⎤ ⎡ u0 ⎤ u ψ K i j0 0 K iuj0 w0 K i j0 φ K iuj0 T0 K iuj0 T1 X uj 0 (t) F (t) ⎢ w0 u 0 w0 w0 w0 ψφ w0 T0 w0 T1 ⎥ ⎢ w0 ⎥ ⎢ wj 0 ⎥ ⎢Ki j Ki j Ki j K i j K i j ⎥ ⎢ X j (t)⎥ ⎢ F j (t)⎥ ⎢ ψφ u 0 ψφ w0 ψφ ψφ ψφ T0 ψφ T1 ⎥ ⎢ ψφ ⎥ ⎢ ψφ ⎥ ⎢K = ⎢ F (t)⎥ Ki j Ki j Ki j Ki j ⎥ ⎢ ij ⎥ ⎢ X j (t)⎥ ⎥ ⎢ jT0 ⎥ T0 ⎢ T0 u 0 T0 w0 T0 ψφ T0 T0 T0 T1 ⎥ ⎢ ⎣ ⎣ F j (t) ⎦ ⎦ X (t) Ki j Ki j Ki j Ki j ⎦ ⎣ Ki j j T1 T1 ψφ T1 u 0 T1 w0 T1 T0 T1 T1 X j (t) F jT1 (t) Ki j Ki j Ki j Ki j Ki j

383

⎡

(7.5.14)

The Newmark method is adjusted for temporal discretization of Eq. (7.5.14). This is a dynamic updating procedure which converts the differential-algebraic equations to the algebraic system of equations. Assume that the global finite element equations obtained in the previous section could be shown by Eq. (7.5.15). ¨ +CX ˙ +KX+f = 0 MX

.

(7.5.15)

Based on Newmark’s method, the unknown vector and its time derivatives at the time step .n + 1 are approximated according to Eq. (7.5.16) ¨ n+γ ˙ n + 1 (Δt)2 X Xn+1 = Xn + Δt X 2 . ˙ ˙ n + Δt X ¨ n+ϕ Xn+1 = X

(7.5.16)

¨ n+ϕ = (1 − ϕ)X ¨n +ϕ X ¨ n+1 X where .γ and .ϕ are the parameters that determine the explicity/implicity form of the solution procedure. Substituting Eq. (7.5.16) into Eq. (7.5.15) leads to the following system of algebraic equation K˘ n+1 Xn+1 + f˘n,n+1 = 0

.

(7.5.17)

where K˘ n+1 = K n+1 + a3 M n+1 + a6 C n+1 M C .f˘n,n+1 = fn+1 + M n+1 yn + C n+1 yn M ˙ n + a5 X ¨n .yn = a3 Xn + a4 X .

˙ n + a8 X ¨n = a6 Xn + a7 X 2 1 .a3 = , a4 = Δt a3 , a5 = − 1 γ(Δt)2 γ

C .yn

(7.5.18a) (7.5.18b) (7.5.18c) (7.5.18d)

384

7 Couple Thermoelasticity of Shells

a6 =

2ϕ 2ϕ , a7 = − 1, a8 = Δt γΔt γ

(

) ϕ −1 γ

(7.5.18e)

˙ 0 , and .X ¨ 0 are required. The values of .X0 For the first time step, knowledge of .X0 , .X ˙ 0 are provided by physical initial conditions and the value of .X ¨ 0 is obtained and .X using Eq. (7.5.15). Also, at the end of each time step the new first and second time derivatives of unknowns are computed as

.

¨ n+1 = a3 (Xn+1 − Xn ) − a4 X ˙ n − a5 X ¨n X ˙ n + a2 X ˙ n+1 = X ¨ n + a1 X ¨ n+1 X

(7.5.19)

where .a1 = ϕ Δt and .a2 = (1 − ϕ) Δt.

7.5.3 Results and Discussions The finite element model presented in this work can be used for numerical investigation on the response of thin spherical shells under axisymmetric loading conditions. In this section, a spherical shell of revolution witch occupies the region between ◦ ◦ .φ = 30 and .φ = 60 is considered. The thermo-mechanical material properties of the shell are presented in Table 7.4. Shell is initially considered to be at.293 K temperature. Then, the outer surface of the shell is considered to be subjected by a step-type heat flux loading that its magnitude is .104 W/m2 . It should be noted that the inner surface of the shell is considered to be thermally insulated, and the boundary edges of the shell are mechanically assumed to be clamped. For the purpose of validation, the performance of the developed flexural element is compared with that of ABAQUS .C3D8T element. The shells under investigation are considered to have the geometrical slenderness of . R/ h = 50. Also, the boundary edges of the shells during the analysis are assumed to be kept at the initial temperature. Under the applied thermal shock, response of the shell is illustrated in Fig. 7.44. In this figure, the response of considered spherical shell has been obtained both with flexural element developed in this work, and the .C3D8T element of ABAQUS software. As seen, Figs. 7.44 and 7.45 show that for both lateral deflection and temperature rise the results of two elements are in good agreement with each other. This agreement Table 7.4 Thermomechanical properties of spherical shells

Parameter

Value

.ρ

.7904

.E .ν .k .α .C ε

kg/m3 .200 gPa .0.3 .50 W/mK −6 1/K .11.8 × 10 .500 J/kgK

7.5 Coupled Thermoelasticity of Spherical Shells

385

−4

x 10

w0 [m]

2

1

0 0

P resent W ork ABAQU S C3D8T Element 100

200

300

t[s]

400

500

600

700

Fig. 7.44 Comparison between the results of flexural element developed in this work and .C3D8T element of ABAQUS software; Lateral deflection of point .φ = 45◦ with respect to time 90 80 70

T − T0

60 50 40 30 20 10 0 0

P resent W ork ABU QU S C3D8T Element 100

200

300

t[s]

400

500

600

700

Fig. 7.45 Comparison between the results of flexural element developed in this work and .C3D8T element of ABAQUS software; Temperature of point .z = 0, φ = 45◦ with respect to time

is seen for non-steady as well as steady parts of solutions. Also, the validation study has been performed for distribution of temperature rise through the thickness of shells, Fig. 7.46. The maximum difference seen between results of two elements is about .2 percent. This means that the linear temperature approximation in the thickness direction is a reasonable choice for dynamic coupled thermoelastic analysis of thin spherical shells. That is, the performance of the developed flexural element for coupled dynamic thermoelastic analysis of thin spherical shells is confirmed (Fig. 7.45).

386

7 Couple Thermoelasticity of Shells

Fig. 7.46 Comparison between the results of flexural element developed in this work and .C3D8T element of ABAQUS software; Distribution of temperature through the thickness of shell at point ◦ .φ = 45

Fig. 7.47 Effect of slenderness on the response of thin spherical shells with isothermal boundary edges; Lateral deflection through the meridian direction of shells

In this section, the effects of slenderness parameter and thermal conditions at boundary edges on the response of thin spherical shells are studied. Consider Figs. 7.48 and 7.49. Figures reveal that the larger slenderness ratio for a shell result in larger temperature rise in a certain time period. Because of thermally insulated inner surface of the shell, the larger temperature rise is followed by larger lateral deflection for both types of thermal boundary edges. Also, comparing of Figs. 7.50, 7.51, and 7.52 reveals that the insulating the boundary edges of the shell results in larger temperature elevation through the shell at any arbitrary frame of time. In addition, caused by heat transfer at isothermal boundary edges, both lateral deflection and temperature rise of the shells reach a steady-state condition. However, these steady states do not appear in the results belong to thermally insulated edges. This section addresses the dynamic coupled thermoelastic response of thin spherical shells to suddenly applied lateral thermal shocks. The formulations are based on the first-order shear-deformable shell model. The shell is considered to obey Hooke’s

7.5 Coupled Thermoelasticity of Spherical Shells

387

Fig. 7.48 Effect of slenderness on the response of thin spherical shells with isothermal boundary edges; Lateral deflection of point ◦ .φ = 45 with respect to time

Fig. 7.49 Effect of slenderness on the response of thin spherical shells with isothermal boundary edges; Temperature of point ◦ .z = 0, φ = 45 with respect to time

law and to accept infinitesimal strains under the applied load conditions. The classical dynamic coupled energy equation is used to obtain the temperature distribution in the shell. Also, the linear approximation function is considered for temperature distribution in the thickness direction of the shell. Based on the governing equations, a flexural coupled thermoelastic element is derived and its applicability is confirmed in comparison with the ABAQUS .C3D8T element. Results reveal the effects of shell slenderness and thermal edge conditions on the dynamic coupled thermoelastic response of thin spherical shells to lateral thermal shocks. Accordingly, the following points are highlighted: • Developed flexural element could be a reliable alternative for .3D coupled thermoelastic elements. • The linear approximation seems to be good enough for temperature evaluation in thickness direction of shells.

388 Fig. 7.50 Effect of slenderness on the response of thin spherical shells with thermally insulated boundary edges: Lateral deflection through the meridian direction of shells

Fig. 7.51 Effect of slenderness on the response of thin spherical shells with thermally insulated boundary edges: Lateral deflection of point .φ = 45◦ with respect to time

Fig. 7.52 Effect of slenderness on the response of thin spherical shells with thermally insulated boundary edges: Temperature of point ◦ .z = 0, φ = 45 with respect to time

7 Couple Thermoelasticity of Shells

7.6 Composite Spherical Shells

389

• The more slender shells experience more temperature rise and consequently more lateral deflection under the applied thermal shock and boundary conditions. • Although the isothermal edge condition results in the steady-state temperature rise and lateral deflection, these steady states do not occur for shells with thermally insulated edge conditions.

7.6 Composite Spherical Shells The discussion of the previous section was limited to shells of isotropic and homogeneous material. In this section the thermoelasticity of composite spherical shells is discussed. There are limited literature on the coupled or dynamic thermoelasticity of the composite spherical shells. Coupled thermoelasticity of thick spherical shells are given by Eslami and Vahedi [56], Jabbari et al. [57], and Eghbalian and Eslami [58]. While the first and third references use the Galerkin finite element method for the analysis, the paper by Jabbari et al. [57] presents fully analytical method based on the series expansion to solve the coupled problem. All these mentioned references assume isotropic/homogeneous materials. Shakeri et al. [59] present the analysis of thin spherical shells under dynamic load considering the transverse shear deformation. Asadzadeh et al. [60] and Javani et al. [61] have considered thermal induced vibrations of thin spherical shells. The later references consider uncoupled thermoelasticity theory. The coupled thermoelasticity of composite spherical shells is given by Eslami et al. [62]. The discussion in this section is limited to dynamic response of laminated orthotropic spherical shells under mechanical and thermal shock loads. The full stress–strain relations of the three-dimensional theory of elasticity are considered and the Flugge second-order shell theory is employed to formulate the thermoelasticity of spherical shells. The equations of motion for the spherical shell are derived through the general relations presented in Chap. 4 for the shells of revolution. Results of parametric studies are presented to bring out the effects of the material orthotropy and normal stress on the transient response. The basic assumption to consider the normal stress and strain in the shell equations requires to relate the displacement components along the principle orthogonal curvilinear coordinates of the shell to the displacement components on the middle plane, as given by [62] .

U (φ, θ, z) = u 0 (φ, θ) + zψx (φ, θ) V (φ, θ, z) = v0 (φ, θ) + zψθ (φ, θ) W (φ, θ, z) = w0 (φ, θ) + zψz (φ, θ) +

(7.6.1) 2

z φz (φ, θ) 2

390

7 Couple Thermoelasticity of Shells

where .(φ, θ, z) are the principle orthogonal curvilinear coordinates of the shell and u, .v, .w are the middle plane displacements, .ψx and .ψθ are the rotations of the tangent line to the middle plane along .φ and .θ axes, respectively, and .ψz and .φz represents the none-zero transverse normal strains. From the general strain–displacement relations in the curvilinear coordinates

.

.

εφ =

1 1+

z R

1 z z2 [ (u 0,φ + w0 ) + (ψφ,φ + ψz ) + φz ] R R 2R

1 1 z z2 (ψφ cotgφ + ψz ) + φz ] z [ (u 0 cotgφ + w0 ) + 1+ R R R 2R εz = ψz + zφz (7.6.2) 2 1 1 z z [ (w0,φ − u 0 ) + ψφ + ψz,φ + φz,φ ] γφz = 1 + Rz R R 2R 1 v0 [− + ψθ ] γθz = 1 + Rz R 1 1 z [ (v0,φ − v0 cotgφ) − (ψθ cotgφ − ψθ,φ )] γφθ = 1 + Rz R R εθ =

These relations are obtained based on the Flugge second-order shell theory, where the term .z/R is retained in the equations compared to the unity. The forces and moments resultants based on the second-order shell theory are defined as { +h z . < Nφ , Nφθ , Q φ >= (σφ , τφθ , τφz )(1 + )dz R −h { +h z < Nθ , Nθφ , Q θ >= (σθ , τθφ , τθz )(1 + )zdz R −h { +h z < Mφ , Mφθ >= (σφ , τφθ )(1 + )zdz R −h { +h z < Mθ , Mθφ >= (σθ , τθφ )(1 + )zdz (7.6.3) R −h { 1 +h < Pφ , Pθ >= (σφ , σθ )zdz 2 −h { +h z < Sφ , Sθ >= (τφz , τθz )(1 + )zdz R −h { +h z A= σz (1 + )2 dz R −h { +h z B= σz (1 + )2 zdz R −h

7.6 Composite Spherical Shells

391

The equations of motion can be obtained using Hamilton’s variational principle. For this general case, where the normal stress and strain are included in the governing equations, Hamilton’s principle yields the following equations of motion .

Nφ cotgφ + Nφ,φ − Nθ cotgφ + Q φ + Rqφ = R(I1 u¨ 0 + I2 ψ¨ φ ) 2Nφθ cotgφ − Nφθ,φ + Q θ + Rqθ = R(I1 v¨0 + I2 ψ¨ θ ) I3 ¨ φz ) 2 Mφ,φ − Mφ cotgφ − R Q φ − Mθ cotgφ + Rm φ = R(I2 u¨ 0 + I3 ψ¨ φ ) Mφθ,φ − 2Mφθ cotgφ − R Q θ + Rm θ = R(I2 v¨0 + 2I3 ψ¨ θ ) Q φ,φ + Q φ cotgφ − Nθ − Nφ + Rqz = R(I1 w¨ 0 + I2 ψ¨ z +

I4 −Mθ − Mφ − R A − Rm z + Sφ,φ + Sφ cotgφ = R(I2 w¨ 0 + I3 ψ¨ z + φ¨ z ) 2 h2 I3 I4 ¨ I5 ¨ Tφ cotgφ + Tφ,φ − Pθ − Pφ − R B − qz = R( w¨ 0 + ψz + φz ) 8 2 2 4

(7.6.4) where { I =

. n

h

N Σ

−h k=1

(ρ)k z (i−1) (1 +

z 2 ) dz R

(i = 1, 2, ..., 5)

(7.6.5)

where .qφ , .qθ , .qz , .m φ , .m θ , and .m z are the components of external forces and moments acting on the middle plane of the shell along the directions .φ, .θ and z, respectively, and are given as .

.

z z ) + qz− (1 − ) R R z z m z = qz+ (1 + ) − qz− (1 − ) R R

qz = qz+ (1 +

(7.6.6)

The stress–strain relation for the k-th orthotropic layer bounded by surfaces at z = h k and .z = h k−1 are given by [63] ⎡ ⎤ Q¯ 11 σφ ⎢ Q¯ 21 ⎢ σθ ⎥ ⎢ ⎢ ⎥ ⎢ Q¯ 31 ⎢ σz ⎥ .⎢ ⎥ =⎢ ⎢ 0 ⎢ τθz ⎥ ⎢ ⎣τ ⎦ ⎣ 0 φz τφθ k Q¯ 61 ⎡

Q¯ 12 Q¯ 22 Q¯ 32 0 0 Q¯ 62

Q¯ 13 Q¯ 23 Q¯ 33 0 0 Q¯ 63

0 0 0 Q¯ 44 Q¯ 54 0

0 0 0 Q¯ 45 Q¯ 55 0

⎤ ⎡ ⎤ ⎡ ⎤ Q¯ 16 β¯ φ εφ ⎢ β¯ θ ⎥ Q¯ 26 ⎥ ε ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ θ ⎥ ⎢ β¯ z ⎥ [ ] Q¯ 36 ⎥ ε ⎢ ⎥ z ⎥ ⎢ ⎥ ⎢ − ⎥ ⎢ 0 ⎥ T k γθz ⎥ 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎣γ ⎦ ⎦ ⎣ 0 ⎦ 0 φz β¯ φθ k Q¯ 16 k γφθ k

(7.6.7) where .[ Q¯ i j ]k and .[β¯i ]k are the stiffness and thermoelastic matrices, respectively. The temperature distribution across the shell thickness is assumed linear as .

T (φ, θ, z, t) = T0 (φ, θ, t) + zT1 (φ, θ, t)

(7.6.8)

392

7 Couple Thermoelasticity of Shells

The lamination constitutive relation is given by the matrix at the end of the Section. Substituting the element of this matrix into the equations of motion (7.6.4) results into seven Navier’s type equations for spherical shells in terms of the displacement components .u 0 , .v0 , .w0 , and rotations .ψθ , .ψφ , and the transverse strains .ψz , .φz . The unknown functions .T0 and .T1 also appear in these equations, requiring two more independent equations to be included into the governing equations. The latter two independent equations are obtained by the energy equation.

7.6.1 Energy Equation The thermoelastic coupled heat conduction equation for the anisotropic media may be derived from the energy conservation relation based on the first law of thermodynamics and the definition of specific entropy in the form [1] .

ki j T,i j −[cν ρT˙ + Ta βi j ε˙i j ] = 0

(7.6.9)

Following McQuillen and Brull [2], the Galerkin method is used to derive two independent heat conduction equations of the shell of revolution from Eq. (7.6.8) by averaging it across the shell thickness .z. Due to the assumption of linear temperature variation across the shell thickness, as given by Eq. (7.6.8), two unknowns .T0 and . T1 appear in the energy equation. For a multilayer composite spherical shells under axisymmetric thermal shock the energy equation (7.6.9) in terms of the displacement components reduce to .

βφθ ¨ βθθ V,φ + βzz W¨ ,z + cotgφU¨ R+z R+z βφθ βθθ − βφφ ¨ −2 cotgφV¨ + W R+z R+z βφφ ˙ βφθ ˙ βθθ ρc T˙ + Ta [ U,φ + V,φ + βzz W˙ z + cotgφU˙ R+z R+z R+z βφθ βθθ − βφφ ˙ −2 (7.6.10) cotgφV˙ + W] R+z R+z 2 1 cotgφ −k zz T,zz − kφφ T,φφ − kφφ T,φφ = 0 k zz T,z − 2 R+z (R + z) (R + z)2

Substituting Eq. (7.6.8) into Eq. (7.6.10) and averaging the residuals across the thickness of the shell results into the following two sets of equations for .T0 and .T1 and the other shell dependent functions [6]

7.6 Composite Spherical Shells

393

{ (Residual) × (1) × dz = 0

.

z

1 1 (1) ˙ 1 (1) 1 (1) v˙0,φ ψθ,φ + Rφθ Rφ u˙0,φ + Rφ(2) ψ˙ φ,φ + Rφθ R R R R Cotgφ ˙ Cotgφ (1) 2Cotgφ u˙0 + Rθ(2) v˙0 +Rz(1) ψ˙ z + Rz(2) φ˙ z + Rθ(1) ψφ − Rφθ R R R w˙0 ψ˙ z φ˙ z (2) 2Cotgφ ˙ + (Rθ(2) − Rφ(2) ) + (Rθ(3) − Rφ(3) ) −Rφθ ψθ + (Rθ(1) − Rφ(1) ) R R R 2R 2 (1) 1 (1) 1 (2) Cotgφ (1) − Rkz T1 − 2 Rkφ T0,φφ − 2 Rkφ T1,φφ − Rkφ T0,φ R R R R2 Cotgφ (2) Rkφ T1,φ − (h i − h o )To + h(h i − h o )T1 − [h i Ti (t) − h o T∞ ] = 0 − R2 (7.6.11) Rc(1) T˙0 + Rc(2) T˙ +

{ (Residual) × (z) × dz = 0

.

z

1 1 (2) ˙ 1 (2) 1 (2) R u˙,φ + Rφ(3) ψ˙ φ,φ + Rφθ v˙0,φ + Rz(2) ψ˙ z ψθ,φ + Rφθ R Rφ R R R Cotgφ Cotgφ ˙ (2) 2Cotgφ (3) 2Cotgφ ˙ u˙0 + Rθ(2) v˙0 − Rφθ +Rz(3) φ˙ z + Rθ(2) ψφ − Rφθ ψθ R R R R ψ˙ z φ˙ z w˙0 2 (2) T1 +(Rθ(2) − Rφ(2) ) + (Rθ(3) − Rφ(3) ) + (Rθ(4) − Rφ(4) ) − Rkz R R 2R R 1 (3) Cotgφ (2) 1 (2) T0,φφ − 2 Rkφ T1,φφ − Rkφ T0,φ − 2 Rkφ R R R2 Cotgφ (3) Rkφ T1,φ − h[(h i − h o )To + h(h i − h o )T1 ] − h[h i Ti (t) − h o T∞ ] = 0 − R2 (7.6.12) Rc(2) T˙0 + Rc(3) T˙ +

where .

(1) Rkz =

N Σ

i k zz (h i − h i−1 )

i=1

.

(2) Rkz =

N { Σ i=1

.

(i) Rkφ =

N { Σ i=1

.

Rc(i) =

N { Σ j=1

hj h j−1

hj h j−1

hj h j−1

k zzj zdz

kφφ z (i−1) dz j

< ρcν > j z (i−1) dz

i = 1, 2, 3

394

7 Couple Thermoelasticity of Shells

.

Rφ(i) =

N { Σ j=1

(i) .R θ

=

hj

Rz(i) =

N { Σ

=

hj

N { Σ

hj

i = 1, 2, 3

< Ta βzz > j z (i−1) dz

i = 1, 2, 3

< Ta βφθ > j z (i−1) dz

i = 1, 2, 3

h j−1

N { Σ j=1

< Ta βθθ > j z (i−1) dz

h j−1

j=1

(i) .R φθ

i = 1, 2, 3

h j−1

j=1

.

< Ta βφφ > j z (i−1) dz

hj h j−1

The seven equations of motion (7.6.4), written in terms of the displacement and rotation components, and the two energy equations (7.6.11) and (7.6.12) constitute the governing equations for the nine unknown dependent functions .u 0 , .v0 , .w0 , .ψθ , .ψφ , .ψz , .φz , . T0 and . T1 . The governing equations are solved by means of the Galerkin finite element method using the initial and boundary conditions. The Galerkin finite element method is used to analyze the coupled thermoelastic shell equations. The nodal degrees of freedom for an axisymmetric coupled field are seven shell variables .u 0 , .v0 , .w0 , .ψθ , .ψφ , .ψz , .φz and two temperature variables .T0 and . T1 . It is verified that a linear test function for the shell variables provide accurate enough approximation compared to higher polynomials [44]. Considering identical shape functions for all nine degrees of freedom and applying the formal Galerkin method to the system of seven shell equations (7.6.4) and two energy equations (7.6.11) and (7.6.12) results into the following finite element equation .

¨ + [C]{d} ˙ + [K ]{d} = {F} [M]{d}

(7.6.13)

where for the base element .(e) .

< d >e =< u 0

w0

ψφ

ψθ

ψz

φz

T0

T1 >

(7.6.14)

The force matrix is divided into two terms, one term composed of the terms obtained through the weak formulation of the governing equations and the resulting natural boundary conditions, and the second term which include the components of external applied forces and thermal shocks. The process of weak formulation and terms which are selected for weak formulation in the governing equations is very important in regard to the resulting natural boundary conditions. The natural boundary conditions which are obtained as the result of weak formulation should either have a kinematical meaning on the boundary or add up to make a traction boundary condition. Therefore, it is essential to set up a possible kinematic and forced shell boundary conditions in advance and try to obtain them by weak formulation.

7.6 Composite Spherical Shells

395

The finite element equilibrium equation (7.6.13) may be solved in time domain by many techniques such as Newmark, Houbolt, Wilson-.θ and other methods [22]. In this section the .α−method is used. According to this method the equilibrium equation (7.6.13) is transformed into the following form in time domain [M]{a}n+1 + (1 + α)[C]{v}n+1 − α[C]{v}n +(1 + α)[K ]{d}n+1 − α[K ]{d}n = {F(t(n+α) }

.

(7.6.15)

˙ and .a=.d. ¨ The .α−method where .tn+α = (1 + α)tn+1 − αtn = tn+1 + α Δ t, .v=.d, become Newmark method when .α = 0. The displacement and velocity matrices at time step .(n + 1) is written in terms of their values at time step .n as Δt 2 [(1 − 2β){a}n + 2β{a}n+1 ] 2 = {v}n + Δt[(1 − γ){a}n + γ{a}n+1

.

{d}n+1 = {d}n + Δt{v}n +

(7.6.16)

.

{v}n+1

(7.6.17)

where .γ and .β are the accuracy and stability parameters, respectively. Using Eqs. (7.6.16) and (7.6.17), we can obtain the unknown matrices .{d}n+1 , .{v}n+1 and .{a}n+1 in terms of their values at .tn . Let us define the following matrices Δt 2 (1 − 2β){a}n 2 = {v}n + Δt (1 − γ){a}n

.

¯ n+1 = {d}n + Δt{v}n + {d}

(7.6.18)

.

{v} ¯ n+1

(7.6.19)

From Eqs. (7.6.15), (7.6.16) and (7.6.18) .

{a}n+1 = −

1 ¯ n+1 ] [{d}n+1 − {d} β(Δt)2

(7.6.20)

and from Eqs. (7.6.17) and (7.6.19) .

{v}n+1 = {v} ¯ n+1 +

γ ¯ n+1 ] [{d}n+1 − {d} β(Δt)

(7.6.21)

Substituting Eq. (7.6.18) through (7.6.20) into Eq. (7.6.15) we obtain the main equation to solve for .{d}n+1 : .

1 γ [C] + (1 + α)[K ]{d}n+1 [M] + (1 + α) 2 β(Δt) β(Δt) 1 γ ¯ n+1 [C]]{d} = {F(tn+α } − [ [M] + (1 + α) 2 β(Δt) β(Δt) −(1 + α)[C]{v} ¯ n+1 + α[C]{v}n + α[K ]{d}n [

(7.6.22)

The condition of stable solution is not only based on the values of .γ and .β, it is also very sensitive to the choice of .Δt. For Newmark method the values .2β > γ > 1/2

396

7 Couple Thermoelasticity of Shells

result into unconditionally stable solution provided that the choice of .Δt is correct. It is suggested that the following values be selected for .α-method [22] .

αε[−1/3, 0],

γ=

1 − 2α , 2

β=(

1 − 2α 2 ) 2

(7.6.23)

The numerical method will be unconditionally stable, will have second-order accuracy, and is self initiate. It is further important to note that sometimes in the solution of Eq. (7.6.14) higher frequency modes are artificially appeared which do not belong to the equation. It is therefore required to delete these disturbing frequencies by increasing the artificial damping. This is possible by .α−method as by decreasing the value of .α, the artificial numerical damping is increased without effecting the problem accuracy. The stability condition of the .α−method, similar to the other method, such as the Newmark, is based on the positive definite matrices. The application of Galerkin method to this problem results to non-axisymmetric stiffness and damping matrices and therefore the resulting solution must be checked for its convergency. The selection of time increment is important and has absolute effect on the solution convergency. A computer program is written based on the solution presented herein for the analysis of thermoelastic coupled response of laminated spherical shells subjected to time-dependent thermal and pressure shocks. In all numerical results presented herein, zero initial conditions for displacements and velocities are assumed. Material properties of the shell are given in Table 7.5. Consider a hemispherical shell with a clamped edge under a thermal shock load applied to its inside surface. The shell is made of three-layer laminates (0/90/0) and (90/0/90). The thermal condition at the end of shell is assumed isolated and the shell is considered to be exposed to an inside thermal shock load given by the following equation .

Ti (t) = 2207(1 − e−13100t ) + 293◦ k

(7.6.24)

The temperature of the inside surface raise from .293 to 2500.◦ K in 0.45 ms. The effect of normal stress is studied in this example. A simply supported hemispherical shell made of three-layer laminates under inside pressure and thermal shock Table 7.5 Material properties . E 11 . E 22 4.83 .gPa 196 .gPa .G 1n = G 2n .ν12 = ν1n 0.05 3.44 .gPa .K3 .K2 67 W/mk 67 W/mk .h i .α3 −6 2 .10000 W/m. k .15 ∗ 10 1/k

. E nn

4.83 .gPa .ν2n 0.2 .α1 −6 .1.3 ∗ 10 1/k .h o 2 .200 W/m. k

.G 12

3.44 .gPa .K1

180 W/mk .α2

∗ 10−6 1/k R/h 10 m

.15

7.6 Composite Spherical Shells

397

Table 7.6 Effect of normal stress (spherical shells) . w(m) (R/h .= 30) Theory including .σn Theory excluding .σn Percent of difference

.4.5583E .4.5315E

0.58

−4 −4

w(m) (R/h .= 10) .5.0397E .4.7336E

−5 −5

6.1

load is considered. The profile of the shocks is given by the following equations .

P(t) = 8 ∗ 106 (1 − e−1.3100t ) Ti (t) = 2207(1 − e−13100t ) + 293◦ k

(7.6.25)

Pressure reaches its maximum value at 0.45 ms. Table 7.6 gives the radial displacement for the crown of the shell at .t = 1. E − 3 s. The difference between two cases are about 0.58 percent for . R/ h = 30 and 6.1 percent for . R/ h = 10. The effect of layer stacking and orthotropic sequence is given by Eslami et al. [62]. A hemispherical shell made of three cross-ply (0/90/0) and (90/0/90) layer laminates is considered. Curves presented in this article show the effect of thermoelastic coupling and stacking sequence on radial displacement of the shell. The lamination constitutive relations with the definition of terms are given in the following: ⎡ ⎤ ⎤ 1 a 2 1 1 2 1 1 N1 ⎡ a11 0 0 0 0 0 12 a13 a16 a16 b11 b12 b13 b16 b16 c11 c12 0 ⎢ N2 ⎥ ⎢ a12 a3 a4 a26 a3 b12 b3 b4 b26 b3 c12 c3 0 0 0 0 0 0 ⎥ 22 23 26 22 23 26 22 ⎢ ⎥ ⎥ ⎢ A ⎥⎢ 2 a 4 a 5 a 2 a 4 b2 b4 b5 b2 b4 c2 c4 a13 0 0 0 0 0 ⎥ 23 33 36 36 13 23 33 36 36 13 23 0 ⎢ ⎥⎢ ⎢ ⎢ N12 ⎥ ⎢ a1 a26 a2 a1 a66 b1 b26 b2 b1 b66 c1 c26 0 0 0 0 0 0 ⎥ ⎥ 36 66 16 36 66 16 ⎢ ⎥ 16 ⎥ ⎢ N21 ⎥ ⎢ 4 a 3 3 4 3 3 a16 a26 a36 0 0 0 0 0 ⎥ 66 a66 b16 b26 b36 b66 b66 c16 c26 0 ⎢ ⎥⎢ ⎢ ⎢ M1 ⎥ ⎢ b1 b12 b2 b1 b16 c1 c12 c2 c1 c16 d11 d12 0 0 0 0 0 0 ⎥ ⎥ 13 16 11 13 16 ⎢ ⎥ 11 ⎥ ⎢ M2 ⎥ ⎢ 3 3 4 3 3 3 b4 b 0 0 0 0 0 ⎥ b12 b22 23 26 b26 c11 c22 c23 c26 c26 d12 d22 0 ⎢ ⎥⎢ ⎢ ⎢ B ⎥ ⎢ b2 b4 b5 b2 b4 c2 c4 c5 c2 c4 d 2 d 4 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ 13 23 33 36 36 13 23 33 36 36 13 23 ⎥ .⎢ M12 ⎥ ⎢ b1 b26 b2 b1 b66 c1 c26 c2 c1 c66 d 1 d26 0 0 0 0 0 0 ⎥ 36 66 16 36 66 16 ⎢ ⎥ ⎢ 16 ⎢ M21 ⎥ ⎢ b16 b3 b4 b66 b3 c16 c3 c4 c66 c3 d16 d 3 0 0 0 0 0 0 ⎥ ⎥ 66 26 36 66 26 26 36 ⎢ ⎥ ⎥ ⎢ 2P1 ⎥ ⎢ 1 c 2 1 1 2 1 1 c11 0 0 0 0 0 ⎥ 12 c13 c16 c16 d11 d12 d13 d16 d16 e11 e12 0 ⎢ ⎥⎢ ⎢ ⎢ 2P2 ⎥ ⎢ c12 c3 c4 c26 c3 d12 d 3 d 4 d26 d 3 e12 e3 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎥ 22 23 26 22 23 26 22 ⎥ ⎢ Q2 ⎥ ⎢ 3 a 3 3 0 0 0 0 0 0 0 0 0 0 0 0 a44 45 b44 b45 c44 c45 ⎥ ⎢ ⎥⎢ ⎢ 1 1 1 ⎢ S2 ⎥ ⎢ 0 0 0 0 0 0 0 0 0 0 0 0 a45 a55 b45 b55 c45 c55 ⎥ ⎢ ⎥ 0 0 0 0 0 0 0 0 0 0 0 0 b3 b c3 c d 3 d ⎥ ⎥ ⎢ S1 ⎥ ⎢ 44 45 44 45 44 45 ⎥ 1 1 1 c ⎢ ⎥⎢ 0 0 0 0 0 0 0 0 0 0 0 0 b45 b55 45 c55 d45 d55 ⎦ ⎣ ⎣ 2T2 ⎦ 0 0 0 0 0 0 0 0 0 0 0 0 c3 c d 3 d e3 e 44 45 44 45 44 45 1 1 1 d 0 0 0 0 0 0 0 0 0 0 0 0 c45 c55 2T1 45 d55 e45 e55

398

7 Couple Thermoelasticity of Shells

⎡

⎤⎡ ⎤ ε01 at 2 bt 2 ⎢ ε0 ⎥ ⎢ 14 14 ⎥ ⎢ 2 ⎥ ⎢ at2 bt2 ⎥ ⎢ w1 ⎥ ⎢ 5 ⎢ 0 ⎥ ⎢ atn btn5 ⎥ ⎥ ⎢ β ⎥⎢ 2 2 ⎥ ⎢ 10 ⎥ ⎢ at12 bt12 ⎥ ⎢ β ⎥⎢ 4 4 ⎥ ⎢ ,2 ⎥ ⎢ at12 bt12 ⎢ ε ⎥⎢ 2 2 ⎥ ⎢ ,1 ⎥ ⎢ bt1 ct1 ⎥ ⎢ ε ⎥⎢ 4 4 ⎥ ⎢ 2 ⎥ ⎢ bt2 ct2 ⎥ ⎢ w2 ⎥ ⎢ 5 5 ⎥ ⎢ , ⎥ ⎢ btn ctn ⎥ [ ] ⎢ β ⎥⎢ 2 2 ⎥ T0 ⎢ 1, ⎥ ⎢ bt12 ct12 ⎥ ⎥ .⎢ ⎥⎢ 4 4 ⎥ β 2 bt ct ⎢ 1 ⎥ ⎢ 12 12 ⎥ T1 ⎢ ε,, ⎥ ⎢ 2 ⎢ 2 1 ⎥ ⎢ ct1 dt12 ⎥ ⎥ ⎢ 1 ,, ⎥ ⎢ 4 ⎢ 2 ε2 ⎥ ⎢ ct2 dt24 ⎥ ⎥ ⎢ 0 ⎥⎢ ⎢ μ2 ⎥ ⎢ 0 0 ⎥ ⎥ ⎢ 0 ⎥⎢ ⎢ μ1 ⎥ ⎢ 0 0 ⎥ ⎥ ⎢ 0, ⎥ ⎢ ⎢ μ2 ⎥ ⎢ 0 0 ⎥ ⎥ ⎢ , ⎥⎢ ⎢ μ1 ⎥ ⎢ 0 0 ⎥ ⎥ ⎢ 1 ,, ⎥ ⎣ 0 0 ⎦ ⎣ 2 μ2 ⎦ 1 ,, 0 0 μ 2 1 where .aimj ,.bimj ,.cimj , .dimj , .eimj are

.

(ai j , bi j , ci j , di j , ei j ) = (ai1j , bi1j , ci1j , di1j , ei1j ) = (ai3j , bi3j , ci3j , di3j , ei3j ) N { hk Σ = ( Q¯ i j )k (1, z, z 2 , z 3 , z 4 )dz k=1

h k−1

(ai2j , bi2j , ci2j , di2j , ei2j ) = ( Q¯ i j )k N { Σ

N { Σ k=1

hk

hk h k−1

(1 +

z )(1, z, z 2 , z 3 , z 4 )dz R

z 2 ) (1, z, z 2 )dz R h k−1 k=1 N { Σ (ati , bti , cti , dti ) = (ati2 , bti2 , cti2 , dti2 ) = (β¯i )k

(ai5j , bi5j , ci5j ) = ( Q¯ i j )k

(ati1 , bti1 , ci1 , di1 ) = (β¯i )k

N { Σ k=1

(1 +

k=1 hk

hk h k−1

1 (1, z, z 2 , z 3 )dz 1 + z/R

(1, z, z 2 , z 3 )dz

h k−1

7.7 Coupled Thermoelasticity of Conical Shells This section deals with the dynamic coupled thermoelastic response of truncated conical shells to suddenly applied thermal shocks. Coupled thermoelasticity of thin

7.7 Coupled Thermoelasticity of Conical Shells

399

conical shells under thermal shock load is considered by Soltani et al. [64]. The shell material is considered to be made of homogeneous/isotropic material, temperature distribution across the shell thickness is assumed to be a quadratic function of the thickness variable and the Galerkin finite element method is employed to solve the coupled equations. The coupled equations are directly solved in the space and time domain using the Newmark method. Coupled thermoelasticity of rotating truncated conical shells based on the Lord–Shulman model is presented by Heydarpour and Aghdam [14]. While the title of the paper is conical shells, the authors consider a thick shell segment and use the theory of elasticity and the equations of motion along the axial and radial directions. The differential quadrature method is used to solve the governing equations. Since the theory of elasticity is used to model the problem, the temperature and stress wave fronts should be detected in the corresponding figures. No clear discussion for the wave fronts is given in the numerical result section of the article. This conclusion is the critical issue and the authors should have provided a brief discussion on the interaction of the speed of rotation and the magnitudes and locations of the temperature and stress wave fronts. The formulation in this section is based on the first-order shear-deformable shell theory [65]. The shell is considered to obey Hooke’s law for their mechanical material properties and to accept infinitesimal strains under the applied loading conditions. The classical dynamic coupled energy equation is used to obtain the temperature, displacement, and axial force and moment resultant distributions along the shells. The temperature field across the shell thickness is approximated to vary linearly. The effects of shell slenderness and thermal edge conditions on the dynamic coupled thermoelastic response of thin conical shells to lateral thermal shock loads are investigated. A truncated conical FGM shell with radius. Ri and. Ro , and thickness.h, as shown in Fig. 7.53, is considered. The functionally graded shell is made of metal and ceramic, where the material properties continuously change in the thickness direction as a function of location. The material properties such as Young’s modules . E(z), coefficient of thermal expansion .α(z), coefficient of heat conduction . K (z), specific heat .C(z), and the mass density .ρ(z) are described across the shell thickness by the power law function, where .z is the shell thickness coordinate between .−h/2 and .h/2. The volume fraction of constituent material is denoted by Eq. (7.2.1). Note that the power law index .k represents the material variation through the shell thickness. When value of .i equals to zero, it represents a fully metal and infinity represents a fully ceramic shell. It is assumed that Poisson’s ratio is constant across the shell thickness.

7.7.1 Strain–Displacement Relations Structure in this problem is a functionally graded truncated conical shell. The conical coordinates .(x, θ, z) are considered along the axial, circumferential, and normal to shell surface directions. The displacement components based on the first-order approximation are represented as

400

7 Couple Thermoelasticity of Shells

Fig. 7.53 Truncated conical shell with displacement fields

.

u(x, θ, z) = u 0 (x, θ) + zψx (x, θ, 0) v(x, θ, z) = v0 (x, θ) + zψθ (x, θ, 0) w(x, θ, z) = w0 (x, θ)

(7.7.1)

where .u 0 , .v0 , and .w0 represent the components of displacement vector in the middle plane of the shell at a point along the .x, .θ, and .z-directions, respectively. The strain–displacement relations for the conical shell based on first-order shear deformation theory are given as .

∂u ◦ ∂ψx +z ∂x )] ( [ ∂x 1 u◦ w◦ z ∂ψθ 1 ∂v◦ sin β + sin β + cos β + + ψ εθ = x 1 + Rzv R ∂θ R R R ∂θ ) ( v◦ 1 1 ∂w◦ γθz = + ψθ − cos β z 1 + Rv R ∂θ R εx =

∂w◦ + ψx ∂x ] [ 1 ∂u ◦ ∂v◦ ∂ψθ 1 v◦ z ∂ψx ) = +z +( − sin β + ( − ψθ sin β) ∂x ∂x 1 + Rzv R ∂θ R R ∂θ

γx z = γxθ

(7.7.2) where in these equations . R = R(x) and . Rv = R cos β.

7.7 Coupled Thermoelasticity of Conical Shells

401

7.7.2 Stress–Strain Relations The stress–strain relations for a functionally graded shell based on the assumed displacement model, including the shear deformations, are ⎫ ⎧ ⎫ ⎤⎧ ⎡ 1υ 0 0 0 εx − α(z)(ΔT ) ⎪ σx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢υ 1 0 0 0 ⎥⎪ εθ − α(z)(ΔT ) ⎪ ⎨ ⎬ ⎨ σθ ⎪ ⎬ ⎥ ⎢ E(z) ⎢ 1−υ ⎥ 0 0 0 0 τxθ = γ . (7.7.3) xθ 2 ⎥ ⎢ ⎪ ⎪ ⎪ ⎪ 1 − υ2 ⎣ ⎪ ⎪ τx z ⎪ γx z 0 ⎦⎪ 0 0 0 1−υ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎩ ⎭ ⎩ ⎭ γθz τθz 0 0 0 0 1−υ 2 The force and moment resultants from the first-order shell theory are { .

< N x x , Nθθ , Nθx , N xθ , Q x x , Q θθ >= { < Mx x , Mθθ , Mθx , Mxθ >=

h/2 −h/2

h/2

−h/2

< σx x , σθθ , τθx , τxθ , τx z , τθz >dz

< σx x , σθθ , τθx , τxθ >zdz

(7.7.4)

7.7.3 Equations of Motion The following equations are obtained by the Hamilton principle, in conical coordinates, including the shear deformations [20] .

N x x,x .R(x) + Nθx,θ + (N x x − Nθθ ) sin α = R(x)(I◦ u¨ ◦ + I1 ψ¨ x )

N xθ,x .R(x) + Nθθ,θ + (N xθ + Nθx ) sin α + Q θ . sin α = R(x)(I◦ v¨◦ + I1 ψ¨θ ) Q x,x .R(x) + Q θ,θ + Q x . sin α − Nθθ . cos α = R(x)(q + I◦ w¨ ◦ )

Mx x .sinα + Mx x,x .R(x) + Mθx,θ − Mθθ . sin α − R(x)Q x = R(x)(I1 u¨ ◦ + I2 ψ¨ x ) Mxθ,x .R(x) + Mθθ,θ + (Mxθ + Mθx ) sin α − R(x)Q θ = R(x)(I1 v¨◦ + I2 ψ¨θ ) (7.7.5) where {

h/2

.

−h/2 { h/2 −h/2 { h/2 −h/2

ρ(z)dz = I0 ρ(z)zdz = I1 ρ(z)z 2 dz = I2

For the axisymmetric loading conditions, Eq. (8.2.9) reduce to

(7.7.6)

402

7 Couple Thermoelasticity of Shells

.

sin α = I◦ u¨ ◦ + I1 ψ¨ x R cos α sin α − Nθθ = q + I◦ w¨ ◦ Q x,x + Q x . R R sin α − Q x = I1 u¨ ◦ + I2 ψ¨ x Mx x,x + (Mx x − Mθθ ) R N x x,x + (N x x − Nθθ )

(7.7.7)

Temperature distribution across the shell thickness may be assumed to be linear, as .

T (x, θ, z, t) − Ta = T0 (x, θ, t) + zT1 (x, θ, t)

(7.7.8)

Substituting Eq. (8.2.5) into Eq. (8.2.7) and using Eqs. (8.2.10) and (8.2.12) and finally substituting the resulting equations into the equations of motion (8.2.11), the equations of motion are obtained in terms of the displacement components as .

sin α ∂ sin2 α ∂2 cos α ∂ ∂2 − A + A − I )u 0 + (−A, 111 111 111 0 ∂x 2 R ∂x R2 ∂t 2 R ∂x 2 2 ∂ sin α ∂ sin α sin α cos α − B111 2 )w0 + (B111 2 + B111 −A111 R2 ∂x R ∂x R ∂ ∂ ∂2 −I1 2 )ψx − (G 11 )T0 − (G 22 )T1 = 0 ∂t ∂x ∂x ∂ cos α sin α ∂2 sin α ∂ cos α ,, − A111 (−A, 111 )u + (A + A111 111 0 R ∂x R2 ∂x 2 R ∂x ∂ cos2 α ∂ ∂ cos α sin α − B , 111 + A,, 111 − I0 2 )w0 + (A,, 111 −A111 2 R ∂t ∂x R ∂x R cos α cos α cos α sin α −B111 )T0 + (G 22 )T1 = 0 )ψx + (G 11 R2 R R ∂2 sin α ∂ sin2 α ∂2 cos α ∂ − B111 2 − I1 2 )u 0 + (−B , 111 (B111 2 + B111 ∂x R ∂x R ∂t R ∂x 2 ∂ ∂ ∂ sin α sin α cos α − A,, 111 )w0 + (C111 2 + C111 −B111 R2 ∂x ∂x R ∂x 2 2 sin α ∂ ∂ ∂ −C111 2 − I2 2 − A,, 111 )ψx − (G 22 )T0 − (G 33 )T1 = 0 R ∂t ∂x ∂x (7.7.9)

(A111

, where. A111 , A,111 , A,,111 , B111 , B111 , C111 , G ii (i = 1..3) are constants given at the end of this section. Three equations of motion contain five unknown dependent functions .u 0 , .w0 , .ψ x , . T0 , and . T1 . This means that two more equations are needed to complete the necessary equations and calculate the dependent functions. These two equations are derived by employing the energy equation.

7.7 Coupled Thermoelasticity of Conical Shells

403

7.7.4 Energy Equation The first law of thermodynamics for heat conduction equation is assumed for the truncated conical functionally graded shell. The classical theory of coupled thermoelasticity for the FG truncated conical shell is considered as [1] ¯ a ε˙ii = (kT,i ),i ρcε T˙ + βT

.

(7.7.10)

where.ρ is the mass density,.cε is the specific heat at constant strain,.β¯ = α(3λ + 2μ), T is the reference temperature, and.k is the heat conduction coefficient. This equation may be written in expanded form for the assumed conditions. We move all parts of the equation to the left side of the equation and calling it . Re. The resulting residue . Re is made orthogonal with respect to 1 and . z. This yields two independent equations for .T0 and .T1 , as it is made orthogonal with respect to 1 and .z for .T0 and .T1 . . a

{

h/2

.

−h/2 { h/2 −h/2

Re × dz = 0 Re × zdz = 0

(7.7.11)

Five governing equations, including the equations of motion and the energy equations, must be simultaneously solved to obtain the displacements and temperature functions. As the thermal boundary conditions, it is considered that the heat flux . Q in and . Q out are applied on the inside and outside surfaces of the cone | ( ) / ∂T | | = h in T − T in , z = − h .k(−h 2) ∞ | ∂z − h 2

(7.7.12)

| ( ) / ∂T | | = h out T − T out , z = h k(h 2) ∞ | ∂z h 2

(7.7.13)

2

.

2

Using Eq. (8.2.12) for the linear approximation of temperature distribution across the thickness direction, two energy equations for the conical shell are obtained as [ ( 2 )] [ ( )] ∂ Ta H1 ∂x∂t + sinRα ( ∂t∂ ) u 0 + Ta H1 cosR α ∂t∂ w0 ( 2 [ )] ∂ + Ta H2 ∂x∂t + sinRα ( ∂t∂ ) ψx ] [ ( ) sin α ∂ ∂2 ∂ .+ K − J − J 1 R ∂x 1 ∂x 2 T0 1 ] [ ( ∂t ) ∂2 ∂ T1 − J2 ∂x + K 2 ∂t∂ − J2 sinRα ∂x 2 = Q out − Q in

404

7 Couple Thermoelasticity of Shells

[

( 2 )] ( )] [ ∂ Ta H2 ∂x∂t + sinRα ( ∂t∂ ) u 0 + Ta H2 cosR α ∂t∂ w0 ( 2 )] [ ∂ + sinRα ( ∂t∂ ) ψx + Ta H3 ∂x∂t ] [ ( ) sin α ∂ ∂2 ∂ .+ K 2 ∂t − J2 R ∂x − J2 ∂x 2 T0 ] [ ( ) ∂ ∂2 + K 3 ∂t∂ − J3 sinRα ∂x − J3 ∂x 2 + J1 T1 = h2 (Q out + Q in ) (7.7.14) where the constants . H j , K j , J j ( j = 1...3) are given at the end of this section.

7.7.5 Numerical Solution Consider a truncated conical shell under axisymmetric thermal shock load. Temperature distribution across the shell thickness is assumed to be linear. Under such assumed conditions five unknown functions .u 0 , w0 , ψ, T0 , T1 , as given by Eqs. (8.2.13) and (8.2.17), appear in the governing equations. The governing equations are transformed into the Laplace domain. In order to solve the governing equations in the Laplace domain, the Galerkin finite element technique is used. Linear shape functions and linear elements with two nodes are used. Applying the Galerkin technique to the system of five Laplace transferred equations and employing the weak formulations, yields [K (s)]{X } = {F(s)}

(7.7.15)

.

The linear shape function is .

Ni = 1 − Nj =

x¯ L

x¯ L (7.7.16)

where .[K (s)] is the global stiffness matrix, .{F(s)} is the force matrix, and .{X } is the unknown matrix in terms of the non-dimensional displacement components in the Laplace transformed domain. The governing equations are changed into the dimensionless form through the following formulas x¯ = xδ t¯ = tCδ 1 T −Ta ¯ 1 ¯ T0 = TTa0 T¯1 = αT . ΔT = Ta Ta (λm +2μm )w0 (λm +2μm )u 0 w¯ 0 = αγm Ta ψ¯ x = u¯ 0 = αγm Ta σi j σ¯ i j = γm Ta

(λm +2μm )ψx γm Ta

(7.7.17)

7.7 Coupled Thermoelasticity of Conical Shells

405

where .γm = E m αm indicating that .γm is evaluated for the metal constituent and / C1 =

.

δ=

.

λm + 2μm ρm

km ρm cm C1

Since the time domain is transformed into the .s-domain (Laplace), the mass, capacitance, and stiffness element matrices are compressed into one matrix, called the global stiffness matrix .[K (s)](e). The governing equations are changed into dimensionless form through the following formulas {l 0

∂ Nl α ((−A111 (λm +2μ m ) ∂x

∂N ∂x

α − A,,, 111 sinR 2α (λm +2μ Nl N m) 2

∂ Nl α α N + A,,, 111 sinRα (λm +2μ Nl ∂∂xN −A, 111 sinRα (λm +2μ m ) ∂x m) α Nl N −I0 ρm1α s 2 Nl N )u¯ 0 + (−A,,, 111 cos αR 2sin α (λm +2μ m) . ∂ Nl ∂ Nl ∂ N cos α α α , −A 111 R (λm +2μm ) ∂x N )w¯ 0 + (−B111 (λm +2μ m ) ∂x ∂x 2 ∂ Nl α α Nl N − B , 111 sinRα (λm +2μ N −B ,,, 111 sinR 2α (λm +2μ m) m ) ∂x sin α ∂N 1 α ,,, 2 ¯ +B 111 R (λm +2μm ) Nl ∂x − I1 ρm α2 s Nl N )ψx ¯ l +(G 11 γ1m ∂∂xNl N )T¯0 + (G 22 γm1α ∂∂xNl N )T¯1 )d x = −N x¯ x.N

{l 0

α α ((−A111 (λm +2μ Nl ∂∂xN − A, 111 cos αR 2sin α (λm +2μ Nl N )u¯ 0 m) m)

∂ Nl ∂ N α α Nl ∂∂xN − A,, 111 (λm +2μ +(A,, 111 sinRα (λm +2μ m) m ) ∂x ∂x cos2 α α 1 2 Nl N − I0 ρm α s Nl N )w¯ 0 . −A 111 (λm +2μm ) R 2 α α α cos α +(−B111 (λm +2μm ) R Nl ∂∂xN − B , 111 sin αRcos NN 2 (λm +2μm ) l ∂ Nl sin α α α ,, ,, −A 111 (λm +2μm ) ∂x N + A 111 R (λm +2μm ) Nl N )ψ¯ x ¯ l +(G 11 cosR α γ1m Nl N )T¯0 + (G 22 cosR α γm1α Nl N )T¯1 )d x = −Q x¯ x.N

{l 0

∂ Nl 1 ((−B111 (λm +2μ m ) ∂x

∂N ∂x

1 − B ,,, 111 sinR 2α (λm +2μ Nl N m) 2

∂ Nl 1 1 N + B ,,, 111 sinRα (λm +2μ Nl ∂∂xN −B , 111 sinRα (λm +2μ m ) ∂x m) 1 Nl N −I1 ρm1α s 2 Nl N )u¯ 0 + (−B ,,, 111 cos αR 2sin α (λm +2μ m) ∂ Nl 1 cos α 1 ∂N ,, , N − A N ¯0 −B 111 (λm +2μm ) l ∂x )w 111 R (λm +2μm ) ∂x . ∂ Nl ∂ N sin2 α 1 1 ,,, +(−C111 (λm +2μm ) ∂x ∂ x¯ − C 111 R 2 (λm +2μm ) Nl N ∂ Nl 1 1 N + C ,,, 111 sinRα (λm +2μ Nl ∂∂xN −C , 111 sinRα (λm +2μ m ) ∂x m) 1 −I2 ρm1α2 s 2 Nl N − A,, 111 (λm +2μ Nl N )ψ¯ x + (G 22 γm1α ∂∂xNl N )T¯0 m) 1 ∂ Nl ¯ l +(G 33 γm α2 ∂x N )T¯1 )d x = −M x¯ x.N

406

7 Couple Thermoelasticity of Shells

)] {l [ ( 1 ( H1 γm Ta (s Nl ∂∂xN ) + sinRα γm1Ta (s Nl N ) u¯ 0 0 [ ] + H1 cosR α γm1Ta (s Nl N ) w¯ 0 [ ( )] + H2 αγ1m Ta (s Nl ∂∂xN ) + sinRα αγ1m Ta (s Nl N ) ψ¯ x

.

m )Ta sin α m +2μm )Ta +(K 1 (λα(γ Nl ∂∂xN (s Nl N ) − J1 (λ(γm +2μ 2 2 R m Ta ) m Ta ) C 1 −J1 (λm +2μ2m )Ta ∂ Nl ∂ N )T¯0 + (K 2 (λm2 +2μm )T2 a (s Nl N )

(γm Ta ) C1

∂x ∂x

α (γm Ta )

m +2μm )Ta m +2μm )Ta sin α −J2 (λ Nl ∂∂xN − J2 (λ α(γm Ta )2 C1 R α(γm Ta )2 C1 {l = (− Q¯ in + Q¯ out )Nl d x

∂ Nl ∂ N ¯ )T1 )d x ∂x ∂x

0

)] {l [ ( 1 ( H2 αγm Ta (s Nl ∂∂xN ) + sinRα αγ1m Ta (s Nl N ) u¯ 0 0 [ ] + H2 cosR α αγ1m Ta (s Nl N ) w¯ 0 )] [ ( + H3 α2 γ1m Ta (s Nl ∂∂xN ) + sinRα α2 γ1m Ta (s Nl N ) ψ¯ x

.

m )Ta m +2μm )Ta sin α +(K 2 (λαm2 +2μ Nl ∂∂xN (s Nl N ) − J2 (λ (γm Ta )2 α(γm Ta )2 C1 R −J2 (λm +2μm2)Ta ∂ Nl ∂ N )T¯0 + (K 3 (λm3 +2μm )T2 a (s Nl N )

α(γm Ta ) C1 ∂x ∂x sin α Nl ∂∂xN R m a 1

m )Ta −J3 α(λ2m(γ+2μ T )2 C

m )Ta +J2 α(λ3m(γ+2μ )T¯1 )d x = T )2 C m a

1

α (γm Ta )

m )Ta − J3 α(λ2m(γ+2μ T )2 C

{l 0

m a

h ( Q¯ in 2α2

1

∂ Nl ∂ N ∂x ∂x

+ Q¯ out )Nl d x (7.7.18)

Here, . Q in and . Q out are the inner and outer non-dimension heat flux. To solve the problem we used a method for a fast numerical inversion of Laplace transforms developed to run in Matlab language environment that was presented by Brancik [66]. The methods are based on the application of fast Fourier transformation followed by so-called e-algorithm to speed up the convergence of infinite complex Fourier series.

7.7.6 Results and Discussion Consider a simply supported functionally graded conical shell under the inside impulsive thermal shock. The ratio of thickness to little radius is assumed to be 0.1 and ratio of little radius to length of conical shell is assumed to be 0.1. The functionally graded shell is assumed to be made of combination of metal (Ti-6Al-4V) and ceramic (ZrO2), at the initial temperature 298.15.K, with the material properties shown in Table 7.7. The shell is ceramic rich at the inside and metal rich at the outside surfaces, respectively. Temperature field across the shell thickness is assumed to be of linear

7.7 Coupled Thermoelasticity of Conical Shells

407

Table 7.7 Material properties of functionally graded constituent materials Metal Ceramic E .= 66.2 Gpa = 10.3 .×10−6 (1/K ) 3 3 .ρ = 4.41 .×10 (kg/m. ) .k = 18.1 (W/mK) c .= 808.3 (J/kg.K) .ν = 0.321 .α

E .= 117 Gpa = 7.11 × 10−6 (1/K ) 3 3 .ρ = 5.6 × 10 (kg/m. ) .k = 2.036 (W/mK) c .= 615.6 (J/kg.K) .ν = 0.333 .α

type. Thermal shock is of impulsive type and is applied to the inside surface. The equation of thermal shock is ¯ 1 = 50t¯e−10t¯ .Q (7.7.19) The boundary conditions at the ends of the shell are assumed to be thermally insulated. For the FG shell, the power law index is assumed to be .i = 0, 5, and .i = ∞. Figure 7.54 shows the lateral deflection of the shell middle length versus time for different values of the power law index. Since the modulus of elasticity of ceramic is larger than metal, as the power law index .i increases, the normal frequency which is directly proportional to the modulus of elasticity increases. When the power law index .i increases, the displacement amplitude decreases too. Figure 7.55 shows the temperature of the shell middle length versus time. This figure shows that for pure ceramic shell (.i = ∞), temperature distribution becomes higher, as the ceramic conductivity is lower compared to metal. The reason is that lower thermal conductivity of ceramics increases the temperature of the structure under applied thermal shock load. Figure 7.56 shows the lateral deflection of the shell middle length versus time for different cone angles. This figure shows that the lateral displacement increases and its frequency decreases as the cone angle increases. When the cone angle is increased, the shell approaches to an annular plate and when the cone angle is decreased it approaches to cylindrical shell. The conical shells with larger half-apex cone angle tend to have larger lateral deflection under internally applied thermal shock loads. Now, consider a conical shell with similar material and thermal shock load of the previous example. The ratio of thickness to small radius is assumed to be 0.1 and ratio of small radius to length of the conical shell is assumed to be 0.1 and 1. Figure 7.57 shows the lateral deflection of the shell middle length versus time. This figure shows that, for the same ratio of .h/R, when the ratio of small radius to length of conical shell is increased, the lateral deflection is also increased. Figure 7.58 shows the axial moment of the shell middle length versus time. Figure 7.59 shows the axial force of the shell middle length versus time. Now consider the same type of applied thermal shock load, as given by Eq. (8.2.23), but with different amplitudes. The shock amplitude is assumed to be 50, 100, and 200. The plots of thermal shock loads are shown in Fig. 7.60.

408

7 Couple Thermoelasticity of Shells −3

6

x 10

k=0 k=5 k=inf

Non−Dim radial displacement

5

4

3

2

1

0

−1

4

3

2

1

0

5

10

9

8

7

6

Non−Dim Time

4

x 10

Fig. 7.54 Radial displacement of middle length of the shell versus time for different power law indices 304

k=0 k=5 k=inf

303

Temperature

302

301

300

299

298 0

10

20

30

40

50

60

70

80

90

100

Non−Dim Time

Fig. 7.55 Temperature of middle length and midplane of the shell versus time for different power law indices

Figures 7.61 and 7.62 show the lateral deflection and temperature of the shell middle length versus time. When the shock amplitude is increased, the radial middle length deflections of shell versus time and the induced temperatures are increased. {

E(z) dz = A111 1 − υ2 { E(z) υzdz = B , 111 1 − υ2 { E(z) dz = A,,, 111 1+υ

{

E(z) . zdz = B111 1 − υ2 { E(z) dz = A,, 111 2(1 + υ) { E(z) zdz = B ,,, 111 1+υ

{

E(z) υdz = A, 111 1 − υ2 { E(z) zdz = B ,, 111 2(1 + υ) { E(z) 2 υ z dz = C111 1 − υ2

7.7 Coupled Thermoelasticity of Conical Shells

409

−3

10

x 10

pi/4 pi/5 pi/3

Non−Dim radial displacement

8

6

4

2

0

−2 0

10

9

8

7

6

5

4

3

2

1

Non−Dim Time

4

x 10

Fig. 7.56 Radial displacement of middle length of the shell versus time for different cone angles 0.025

R/L=0.1 , h/R=0.1 R/L=1 , h/R=0.1

Non−Dim radial displacement

0.02

0.015

0.01

0.005

0

−0.005 0

3

2

1

6

5

4

7

10

9

8

Non−Dim Time

4

x 10

Fig. 7.57 Radial displacement of middle length of the shell versus time −0.008

non−dim Nxx

−0.009 −0.01

−0.011 −0.012 −0.013 −0.014 0

0.5

1

1.5

2

non−dim Time

Fig. 7.58 Axial moment of middle length of the shell versus time

2.5

3.5

3

−8

x 10

410

7 Couple Thermoelasticity of Shells −13

2.5

x 10

Non−Dim Mxx

2 1.5 1 0.5 0 0

3.5

3

2.5

2

1.5

1

0.5

Non−Dim Time

−8

x 10

Fig. 7.59 Axial force of middle length of the shell versus time 8

Non−dimensional Shocks

7

50t*exp(−10t) 100t*exp(−10t) 200t*exp(−10t)

6

5

4

3

2

1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Non−dimensional Time

Fig. 7.60 Shocks with different amplitudes

Non−dimensional Radial displacement

0.025

Q=50t*exp(−10t) Q=100t*exp(−10t) Q=200t*exp(−10t)

0.02

0.015

0.01

0.005

0

−0.005 0

1

2

3

4

5

6

7

8

Non−dimensional Time

Fig. 7.61 Radial displacement of middle length of shell versus time with shocks

9

10 4

x 10

7.8 Thermoelasticity of Shells of Revolution

411

350

Non−dimensional Temperature

345

Q=50t*exp(−10t) Q=100t*exp(−10t) Q=200t*exp(−10t)

340 335 330 325 320 315 310 305 300 0

10

20

30

40

50

60

70

80

90

100

Non−dimensional Time

Fig. 7.62 Temperature of middle length and midplane of the shell versus time with shocks

{

{ E(z) 2 E(z).α(z) , z dz =C dz = G 11 = H1 111 1 − υ2 1−υ { { E(z) 2 E(z).α(z) z dz =C ,,, 111 zdz = G 22 = H2 1+υ 1−υ { { { E(z).α(z) 2 ρ(z).c(z)dz = K 1 k(z)dz = J1 z dz = G 33 = H3 1−υ { { { ρ(z).c(z)zdz = K 2 k(z)zdz = J2 ρ(z).c(z)z 2 dz = K 3 { k(z)z 2 dz = J3 (7.7.20) υ

7.8 Thermoelasticity of Shells of Revolution Thin shell response to the applied thermal or mechanical impulsive loads to its inside or outside surfaces may differ significantly when normal stress is considered. The effect is significant compared to the consideration of the transverse shear and rotary inertia, which are occasionally considered to improve the accuracy. It is shown that even second-order shell theory is sensitive to the inclusion of normal stress for such types of loading [32, 33]. In this section, the full stress–strain relations of the three-dimensional theory of elasticity are considered and the Flugge second-order shell theory, which is the most complete compared to the other classical second-order theories, is employed to formulate the thermoelasticity of shells of revolution [6]. The results are obtained with the formulation where the normal stress is not considered and are compared with those where the normal stress is considered. For the classical shell geometries

412

7 Couple Thermoelasticity of Shells

Fig. 7.63 Shell coordinate directions

and material properties, significant difference in shell response between two cases is observed. Furthermore, the effect of the thermo-mechanical coupling term is studied and the results are compared for coupled and uncoupled equations. The basic assumption to consider the normal stress and strain in the shell equations requires to relate the displacement components along the principle orthogonal curvilinear coordinates of shell to the displacement components on middle plane as given by the following relations .

u(α1 , α2 , z) = u 0 (α1 , α2 ) + zβ1 (α1 , α2 ) v(α1 , α2 , z) = v0 (α1 , α2 ) + zβ2 (α1 , α2 ) w(α1 , α2 , z) = w(α1 , α2 ) + zw , (α1 , α2 ) +

z 2 ,, w (α1 , α2 ) 2 (7.8.1)

where .α1 , α2 , and .z are the principle orthogonal curvilinear coordinates of shell (as shown in Fig. 7.63) and .u 0 , v0 , and .w0 are the middle plane displacements, .β1 and .β2 are the rotations of the tangent line to the middle plane along .α1 and .α2 axes, respectively, and .w , and .w ,, are related to the nonzero transverse normal strain. Consideration of these two forms violate the third Love’s assumption (which states .γn = 0) and part of Love’s fourth assumption (which states .εn = 0). Furthermore, if the transverse shear strains .γ1n and .γ2n are not made zero, the rotations .β1 and .β2 are no longer simply described in terms of the middle plane displacements, and the restrictions imposed by the other part of Love’s fourth assumption (which state .γ1n = .γ2n = 0) is removed. From the general strain–displacement relations in the curvilinear coordinates, for the second-order shell theory, we have be

7.8 Thermoelasticity of Shells of Revolution

.

ε1 =

1 z 2 ,, 0 , ε 1) z (ε 1 + zε 1 + 1 + R1 2

ε2 =

1 z 2 ,, 0 , ε 2) z (ε 2 + zε 2 + 1 + R2 2

413

εn = w , + zw ,, 1 z 2 ,, 0 , μ ) γ1n = z (μ 1 + zμ 1 + 1 + R1 2 1 γ2n =

1 z 2 ,, 0 , μ ) (μ + zμ + 2 2 1 + Rz2 2 2

γ12 =

1 1 (β 0 1 + zβ , 1 ) + (β 0 2 + zβ , 1 ) 1 + Rz1 1 + Rz2 (7.8.2)

where the definition of terms given in Eq. (7.8.2) are .

1 ∂u v ∂ A1 w + + A1 ∂α1 A1 A2 ∂α2 R1 ∂v ∂ A 1 u w 2 ε0 2 = + + A2 ∂α2 A1 A2 ∂α1 R2 β2 ∂ A1 1 ∂β1 + k1 = A1 ∂α1 A1 A2 ∂α2 β2 ∂ A2 1 ∂β2 + k2 = A2 ∂α2 A1 A2 ∂α1 ∂v 1 u ∂ A1 β01 = − A1 ∂α1 A1 A2 ∂α2 v ∂ A2 1 ∂u − β02 = A2 ∂α2 A1 A2 ∂α1 ∂β β1 ∂ A1 1 2 − β,1 = A1 ∂α1 A1 A2 ∂α2 1 ∂β1 β2 ∂ A2 , β2= − A2 ∂α2 A1 A2 ∂α1 u 1 ∂w − + β1 μ0 1 = A1 ∂α1 R1 1 ∂w , , μ,, 1 = μ, 1 = A1 ∂α1 v 1 ∂w − + +β1 μ0 2 = A2 ∂α2 R2 1 ∂w , , μ,, 1 = μ, 2 = A2 ∂α2 ε0 1 =

1 ∂w ,, A1 ∂α1

1 ∂w ,, A2 ∂α2

414

7 Couple Thermoelasticity of Shells

ε, 1 = k1 + ε,, 1 =

w ,, R1

w, R1

ε, 2 = k 2 +

, ,

ε, 2 = k 2 +

w ,, R2

w, R2 (7.8.3)

These relations are obtained based on the Flugge second-order shell theory, where the term .z/R is retained in the equations compared to the unity. The forces and moments resultants based on the second-order shell theory are defined as { .

< N1 , N12 , Q 1 >=

+h −h +h

(σ1 , τ12 , τ1n )(1 +

z )dz R2

(σ2 , τ21 , τ2n )(1 +

z )dz R1

{ < N2 , N21 , Q 2 >= { < M1 , M12 >= { < M2 , M21 >=

+h −h +h −h

−h

(σ1 , τ12 )(1 +

z )zdz R2

(σ2 , τ21 )(1 +

z )zdz R1

{ 1 +h z < Si , Pi , Ti >= (τin z, 21 σn z 2 , 21 τin z 2 )(1 + )dz 2 −h Ri { +h z z < A, B >= (σn , σn z)(1 + )(1 + )dz R R 1 2 −h

(7.8.4)

i, j = 1, 2

The temperature distribution across the shell thickness is assumed linear as .

T (α1 , α2 , z, t) − Ta = T0 (α1 , α2 , t) + zT1 (α1 , α2 , t) (7.8.5)

The equations of motions can be obtained using Hamilton’s variational principle. For this general case, where the normal stress and strain are included in the governing equations, Hamilton’s principle yields the following equations of motion .

∂ A1 ∂(A2 N1 ) ∂(A1 N21 ) + + N12 ∂α1 ∂α2 ∂α2 ∂ A2 Q 1 A1 A2 + + A1 A2 {q1 −N2 ∂α1 R1 2 h2 1 1 ¨ h ( )u¨ + + )β1 ]} = 0 −ρh[(1 + 12R1 R2 12 R1 R2 ∂ A2 ∂ A1 ∂(A1 N2 ) ∂(A2 N12 ) Q 1 A1 A2 + + N21 − N1 + ∂α2 ∂α1 ∂α1 ∂α2 R2

7.8 Thermoelasticity of Shells of Revolution

415

h2 h2 1 1 ¨ )v¨ + + ( + )β2 ]} = 0 12R1 R2 12 R1 R2 ∂( A2 Q 1 ) ∂(A1 Q 2 ) N1 N2 + − A1 A2 ( + ) + A1 A2 {qn − ρh[(1 ∂α1 ∂α2 R1 R2 h2 3h 2 h2 1 1 h2 + ( (1 + )w¨ + + )w¨ , + w¨ ,, ]} = 0 12R1 R2 12 R1 R2 24 20R1 R2 ∂ A1 ∂ A2 ∂(A2 M1 ) ∂(A1 M21 ) + + M12 − M2 ∂α1 ∂α2 ∂α2 ∂α1 ρh 3 1 1 3h 2 ¨ [( −Q 1 A1 A2 + A1 A2 {m 1 − + )u¨ + (1 + β1 ]} = 0 12 R1 R2 20R1 R2 ∂ A2 ∂ A1 ∂( A1 M2 ) ∂(A2 M12 ) + + M21 − M1 ∂α2 ∂α1 ∂α1 ∂α2 ρh 3 1 1 3h 2 ¨ [( −Q 2 A1 A2 + A1 A2 {m 2 − + )v¨ + (1 + β2 ]} = 0 12 R1 R2 20R1 R2 ∂(A2 S1 ) ∂(A1 S2 ) M1 M2 + − A1 A2 ( + ) − A A1 A2 ∂α1 ∂α2 R1 R2 3 1 ρh 1 3h 2 [( +A1 A2 {m n − + )w¨ + (1 + w¨ , 12 R1 R2 20R1 R2 3h 2 1 1 ( + + )w¨ ,, ]} = 0 40 R1 R2 h2 ∂( A2 T1 ) P1 P2 − A1 A2 ( + ) − B A1 A2 + A1 A2 {−qn ∂α1 R1 R2 8 3 2 2 2 ρh 3h 5h 2 h 1 1 3h − [(1 + ( (1 + )w¨ + + )w¨ , + w¨ ,, ]} = 0 24 20R1 R2 20 R1 R2 40 28R1 R2 +A1 A2 {q2 − ρh[(1 +

(7.8.6) In these equations of motion .q1 , q2 , and .qn are the components of the external force and .m 1 , m 2 , and .m n are the components of the external moment acting on the middle plane of shell. These forces and moments are related to the external applied forces + − .q i and .q i as .

z z ) + qz− (1 − ) R R z z m z = qz+ (1 + ) − qz− (1 − ) R R

qz = qz+ (1 +

(7.8.7)

where .q + i acts on the outer surface and .q − i acts on the inner surface of shell. The force.qn is selected positive in the opposite direction of the outer normal to the middle plane. The sixth and seventh of Eq. (7.8.6) are obtained due to the consideration of ,, the normal stress and the introduction of parameters .w , and .w . For axisymmetric loading condition, the following simplifications are made .

v = β2 =

∂ ,, ,, = β 0 1 = β 0 2 = β , 1 = β 1 = μ2 = μ, 2 = μ 2 = 0 ∂α2 (7.8.8)

416

7 Couple Thermoelasticity of Shells

Substitution of these conditions into the strain–displacement equations (7.8.2) and (7.8.3) and finally, with the help of Eqs. (7.8.4) and (7.8.5), into the equations of motion (7.8.6), yield the second and fifth equations to be identically satisfied and the equations of motion reduce to a set of five equations. For the spherical, cylindrical, and conical shells these equations are given in Chap. 4, Eqs. (4.7.21), (4.7.26), and (4.7.27).

7.8.1 Coupled Problems When the time period of excitation or application of a thermal external agency .tT applied into a structure is comparable to the time period of structural disturbance .t S , the thermal stress waves are produced and the problem solution must be obtained through the coupled field equations. In this case, the strain tensor is introduced into the expression of entropy of continuum. This results into an expression for the conduction equation based on the first law of thermodynamics which include the first time rate of strain tensor and temperature. The results are the classical coupled thermoelasticity theory. As explained in Chap. 2 of [1], according to the classical theory the speed of propagation of thermal wave is infinity, which is physically questionable. Lord and Shulman (L.S) theory resolved this discrepancy by suggesting the relaxation time .τ0 . The coupled thermoelasticity of shells is studied by McQuillen and Brull [2]. They applied the traditional Galerkin method to the equations of classical coupled thermoelasticity of thin shells and obtained an approximate solution. They considered the first-order shell theory based on the Love assumptions and essentially ignored the normal stress, transverse shear, and rotary inertia, but assumed a nonlinear temperature distribution across the shell thickness. They concluded that the difference between the coupled and uncoupled solutions is about one percent. Later on, Sabbaghian [68], Li et al. [69], Eslami and Vahedi [24], and Hata [49] used the analytical and finite element methods and solved the coupled problems of thick cylinders and spheres. Thin cylindrical shells under thermal shock are studied by Takazono et al. [67], where uncoupled equations are considered. The coupled thermoelasticity of thin cylindrical shells based on the first-order shell theory and the Love assumptions using proper Galerkin finite element method is studied by Eslami et al. [3]. The coupled thermoelasticity of cylindrical and spherical shells based on the second-order shell theory is given by Eslami et al. [51]. A comprehensive discussion and the general formulations of the coupled thermoelasticity of single layer shells of revolution with isotropic and homogeneous material are presented in the latter paper. In this paper the shell equations are derived on the Flugge assumption of second-order shell theory, where, in addition, the transverse shear stress, rotary inertia, and the normal stress and strains are considered. The classical coupled energy equation as well as the energy equation based on the Lord and Shulman theory are considered and the results are discussed. The following presentation is based on the latter paper. Based on the Lord and Shulman theory, the classical Fouries’s law is improved by consideration of the relaxation time and, thus, the resulting energy equation is

7.8 Thermoelasticity of Shells of Revolution

k T,i j − (1 + τ0

. ij

417

∂ )[cv ρT˙ + Ta βi j ε˙i j ] = 0 ∂t

(7.8.9)

Considering a linear distribution for temperature across the shell thickness, as given by Eq. (7.8.5), two unknowns .T0 and .T1 appear in the energy equation. The following two integrals of the energy equation provides two independent energy equations as follows {

∂ )[cv ρT˙ + Ta βi j ε˙i j ]dz = 0 ∂t ∂ )[cv ρT˙ + Ta βi j ε˙i j ]zdz = 0 − (1 + τ0 ∂t

[ki j T,i j − (1 + τ0

.

z

{

[ki j T,i j

.

z

(7.8.10) (7.8.11)

The Galerkin finite element method is used to analyze the uncoupled and coupled thermoelastic shell equations. The nodal degrees of freedom for an axisymmetric ,, field are five shell variables .u, .w, .ψ, .w , , and .w whereas for the coupled problems two temperature variables .T0 and .T1 are added to the five shell variables. A linear shape function for the shell variables may be considered for this problem. Considering identical shape functions for all seven degrees of freedom and applying the formal Galerkin method to the system of five shell equations and two energy equations (7.8.10) and (7.8.11), results into the finite element equation as ¨ + [C]{d} ˙ + [K ]{d} = {F} [M]{d}

(7.8.12)

.

where for the base element (e) .

< d >e =< u

w

ψ

w,

w

,,

T0

T1 > (7.8.13)

The force matrix is divided into two terms, one term composed of the terms obtained through the weak formulation of the governing equations and the resulting natural boundary conditions, and the second term which includes the components of external applied forces and thermal shocks. The process of weak formulation and terms which are selected for weak formulation in the governing equations is very important in regard to the resulting natural boundary conditions. The natural boundary conditions, which are obtained as the result of weak formulation, should either have a kinematical meaning on the boundary or add up to make a traction boundary condition. Therefore, it is essential to set up a possible kinematic and forced boundary conditions in the shell in advance, and try to obtain them by weak formulation. The finite element equations of motion (7.8.12) may be solved in time domain by many techniques such as Newmark, Houbolt, Wilson-.θ and other methods. In this section, the .α−method is used. According to this method, the equation of motion (7.8.12) is transformed into the following form in time domain [22]

418

7 Couple Thermoelasticity of Shells

[M]{a}n+1 + (1 + α)[C]{v}n+1 − α[C]{v}n

.

+(1 + α)[K ]{d}n+1 − α[K ]{d}n = {F(t(n+α) }

(7.8.14)

where.tn+α = (1 + α)tn+1 − αtn = tn+1 + αΔt. The.α−method becomes Newmark method when .α = 0. The displacement and velocity matrices at time step .(n + 1) are written in terms of their values at time step .n as Δt 2 [(1 − 2β){a}n + 2β{a}n+1 ] 2 = {v}n + Δt[(1 − γ){a}n + γ{a}n+1

{d}n+1 = {d}n + Δt{v}n +

.

{v}n+1

(7.8.15)

where .γ and .β are the accuracy and stability parameters, respectively. Using Eqs. (7.8.14) and (7.8.15), we obtain the unknown matrices .{d}n+1 , .{v}n+1 , and .{a}n+1 in terms of their values at .tn . Let us define the following matrices Δt 2 (1 − 2β){a}n 2 = {v}n + Δt (1 − γ){a}n

¯ n+1 = {d}n + Δt{v}n + {d}

.

{v} ¯ n+1

.

(7.8.16) (7.8.17)

From Eqs. (7.8.15) and (7.8.16) {a}n+1 = −

.

1 ¯ n+1 ] [{d}n+1 − {d} β(Δt)2

(7.8.18)

and from Eqs. (7.8.15) and (7.8.17) {v}n+1 = {v} ¯ n+1 +

.

γ ¯ n+1 ] [{d}n+1 − {d} β(Δt)

(7.8.19)

Substituting Eq. (7.8.16) through (7.8.19) in Eq. (7.8.14), we obtain the main equation to solve for .{d}n+1 as .

1 γ [C] + (1 + α)[K ]]{d}n+1 [M] + (1 + α) 2 β(Δt) β(Δt) 1 γ ¯ n+1 [C]]{d} = {F(tn+α )} − [ [M] + (1 + α) 2 β(Δt) β(Δt) −(1 + α)[C]{v} ¯ n+1 + α[C]{v}n + α[K ]{d}n

[

(7.8.20)

The condition of stable solution is not only based on the values of .γ and .β, it is also very sensitive to the choice of .Δt. For the Newmark method the values .2βγ1/2 result into unconditionally stable solution, provided that the choice of .Δt is correct. It is suggested that if the following values are selected for .α−method

7.8 Thermoelasticity of Shells of Revolution

419

Table 7.8 Geometrical and material properties .E .k .α3 ◦ −6 1/k 200 .GN/m.2 .50 W/m. k .15 ∗ 10

.L .0.40

.R

.ν

.ρ

.h

0.1085 .m .cv ◦ .500 J/kg. k

0.3 .h i 2 .10000 W/m. k

7904 .kg/m.3 .h o 2 .200 W/m. k

.0.002

.

αε[−1/3, 0]

,

γ=

1 − 2α 2

m

. Ta .293

,

◦k

β=(

1 − 2α 2 ) 2 (7.8.21)

the numerical method is unconditionally stable [22]. It is further important to note that sometimes in the solution of Eq. (7.8.20) higher frequency modes are artificially appeared which do not belong to the equation. It is, therefore, required to delete these disturbing frequencies by increasing the artificial damping. This is possible by.α−method, as by decreasing the value of.α, the artificial numerical damping is increased without effecting the problem accuracy. The stability condition of the .α−method, similar to the other methods such as Newmark, is based for the positive definite matrices. The application of Galerkin method to this problem results in non-axisymmetric stiffness and damping matrices and, therefore, the resulting solution must be checked for its convergency. The selection of time increment is important and has absolute effect on the solution convergency.

7.8.2 Cylindrical Shells Consider a thin cylindrical shell of clamped edges and following geometrical and material properties given in Table 7.8. The thermal conditions at the ends of shell are assumed isolated, and the shell is considered to be exposed to the inside thermal shock given by the following equation .

Ti (t) = 2207(1 − e−13100t ) + 293◦ k

(7.8.22)

The temperature of the inside surface raise from .293 to 2500 .◦ K in 0.45 ms, and the shell behavior is studied up to 0.04 s, which is about 90 times the time period required for the thermal shock to reach its steady-state condition. Shell is divided into 50 elements along its length and the time increment is .1E − 6 s. In Fig. 7.63 the inside temperature versus time is shown. The effect of mechanical coupling (1−ν)α2 Ta E is shown in this figure. For .δ = 0, the mechanical coupling term is .δ = (1−ν)(1−2ν)ρcv

420

7 Couple Thermoelasticity of Shells

Fig. 7.64 The inside temperature versus time, example 1

ignored from the energy equation and the problem is decoupled. For the given shell δ = 0.0095 and it is noticed that the effect of damping is negligible. For larger .α, or smaller .cv , the value of .δ is larger. For .δ = 0.095 the mechanical coupling has noticeable effect. In Fig. 7.64 this comparison is shown for the middle plane lateral deflection. It is noticed that while at .t = 0.04 s the thermal shock is reached to its steady-state condition, the lateral deflection is still increasing. The reason is that the characteristic time of heat transfer is much larger than the mechanical characteristic time for stress wave. This behavior is different when the shell is under pressure shock [33, 70]. Now consider the same shell under thermal shock, as a second example. The equation of temperature shock uniformly applied to the inside surface of the cylindrical shell is −1.3t . Ti (t) = 2207 × (1 − e ) + 293◦ k (7.8.23)

.

The rate of temperature variation with respect to time is slower compared to Eq. (7.8.22). Temperature reaches to its maximum value within 35 s. Time increment is selected .Δt = 0.01 s and the shell behavior is studied up to 5 s. In Figs. 7.65 and 7.66 time history of inside surface temperature and radial displacement is shown. Similar to Figs. 7.63 and 7.64, the values of temperature and displacement for coupled condition (.δ = 0.095) are less than the values for semi-coupled condition (.δ = 0). This means that the coupled effect acts like a damper and thus it could be regarded as thermoelastic damping. At the beginning of the shock, due to the lower values of strains, the difference between coupled (.δ = 0.095) and semi-coupled (.δ = 0) is negligible and as time increases this difference also increases. When temperature reaches its steady-state condition, the strains reach their maximum values while their time rate is decreased and the effect of mechanical coupling increases. In Figs. 7.67, 7.68, and 7.69 the time history of axial force, axial moment, and axial stress at inner

7.8 Thermoelasticity of Shells of Revolution

421

Fig. 7.65 The middle plane lateral deflection, example 1

Fig. 7.66 Time history of inside surface temperature

surface is shown. Although both temperature and radial displacement are related to the axial force, due to their signs and coefficients, the effect of temperature is dominant and the axial force is negative. Since the axial moment is small, the axial stress follows the pattern of the axial force. The variation of radial displacement, axial force, axial moment, and axial stress versus the shell length are shown in Figs. 7.70, 7.71, 7.72 and 7.73. The value of normal stress around the shell edge is seen from Fig. 7.73 to be large. In all figures describing the time history, the effect of mechanical coupling decreases as time increases, as expected.

422

7 Couple Thermoelasticity of Shells

Fig. 7.67 Time history of radial displacement, example 2

Fig. 7.68 Time history of axial force, example 2

The effect of normal stress is studied in the third example. A simply supported cylindrical shell of . L = 1.0 .m, . R = 0.15 .m, .h = 0.005 .m, . E = 196 .Gpa, .ρ = 8000 kg/m.3 , and .ν = 0.3 under inside uniform pressure shock of .

P(t) = 8 × 106 (1 − e−1.3100 t )

(7.8.24)

is considered. Pressure reaches its maximum value at 0.45 ms. Figure 7.74 shows the time history of the middle length of shell for two theories; when normal stress is considered (.w is quadratic function of .z), and when normal stress is not considered

7.8 Thermoelasticity of Shells of Revolution

423

Fig. 7.69 Time history of axial moment, example 2

Fig. 7.70 Time history of axial stress at inner surface, example 2

and .w is constant across the thickness. Table 7.9 gives the radial displacement for middle length at .t = 5E − 4 s. The difference between two cases are about .%0.7 for . R/ h = 30 and .%2 for . R/ h = 10. In Fig. 7.75 time history of axial moment at middle length of shell for . R/ h = 30 is plotted. It is observed that the effect of normal stress causes considerable increase of axial moment. Figure 7.76 shows the time history of normal stress for middle length. At inside surface, the normal stress is at equilibrium with inside pressure and at outside surface is zero. Figures 7.77 and 7.78 show the time history of the axial force and moment of inside shell surface at middle length for . R/ h = 10. Figure 7.79 is the plot of axial stress versus time at the same location for . R/ h = 10. The axial stress is the sum of . N x / h + 12z Mx / h 3 . Since the axial force is dominant, the

424

7 Couple Thermoelasticity of Shells

Fig. 7.71 The variation of radial displacement versus shell length, example 2

Fig. 7.72 The variation of axial force versus shell length, example 2

axial stress follows its pattern. Figure 7.80 illustrate the difference between the values calculated under two theories; the classic coupled thermoelasticity (.τ0 = 0) and the Lord–Shulman theory (.τ0 = 1.5 × 10−6 ). The axial stress curve for L-S theory becomes smaller compared to the classical theory. Maximum value of difference is about 38 percent, which occurs at .t = 0.4 × 10−3 (s) at outer surface of the shell. It is to be noticed that the above large difference between the two theories is due to the artificially large value of .τ0 , which for metallic materials range about .10−12 s. The results can be compared with the work of McQuillen and Brull who applied the traditional Galerkin method to thin cylindrical shells to obtain the approximate solution of coupled thermoelastic response of cylindrical shells [2]. They considered the first-order shell theory based on the Love assumptions and essentially ignored

7.8 Thermoelasticity of Shells of Revolution

425

Fig. 7.73 The variation of axial moment versus shell length, example 2

Fig. 7.74 The variation of axial stress versus shell length, example 2

the normal stress, transverse shear stress, and rotary inertia, but assumed a nonlinear temperature distribution across the shell thickness. They concluded that the difference between the coupled and uncoupled solutions is about .%1. Thin cylindrical shells under thermal shock are studied by Eslami et al. [3]. This work is based on the first-order shell theory and the classical coupled energy equation. The paper disregards the axial displacement on the judgment that for the long circular cylindrical shells it may be ignored.

426

7 Couple Thermoelasticity of Shells

Table 7.9 Radial displacement for middle length at .t = 5E − 4 s w(m) (R/h .= 30) w(m) (R/h .= 10) Theory including .σn Theory excluding .σn .% of difference

.0.20976E .0.20825E .%0.7

−6 −3

.0.67291E .0.65881e

−4 −4

.%2

Fig. 7.75 Time history of radial displacement at middle length of shell for two theories, example 3

Fig. 7.76 Time history of axial moment at middle length of shell for . R/ h = 30, example 3

7.8 Thermoelasticity of Shells of Revolution

427

Fig. 7.77 Time history of normal stress at middle length, example 3

Fig. 7.78 Time history of axial force of inside shell surface at middle length for . R/ h = 10, example 3

7.8.3 Spherical Shells The fourth example is a hemispherical shell with the following data . R = 0.3 .m, h = 0.01 .m, . E = 200 .Gpa, .ρ = 8000 kg/m.3 , .ν = 0.3. Sphere is clamped at edges. The pressure pulse of Eq. (7.8.24) is applied and the lateral deflection at the crown of the shell is given in Table 7.10. It is noted that the consideration of normal stress improves the results up to .%1.4 for . R/ h = 30 and .%4.6 for . R/ h = 10. The fifth example is a hemispherical shell with clamped edges under thermal shock given by Eq. (7.8.23) with the material properties given in Table 7.11. In Figs. 7.81 and 7.82 the time history of the radial deflection and inside temperature

.

428

7 Couple Thermoelasticity of Shells

Fig. 7.79 Time history of axial moment of inside shell surface at middle length for . R/ h = 10, example 3

Fig. 7.80 Axial stress versus time at middle length for . R/ h = 10, example 3 Table 7.10 Lateral deflection at the crown of the shell w(m) (R/h .= 30) Theory including .σn Theory excluding .σn .% of difference

−4 .1.736E − 4 .%1.4 .1.731E

w(m) (R/h .= 10) .5.3871E .5.6347E .%4.6

−5 −5

7.8 Thermoelasticity of Shells of Revolution Table 7.11 Material properties .E .k ◦ 200 .GN/m.2 .50 W/m. k .ν .R 0.3 0.15 .m .h i .cv ◦ 2 .10000 W/m. k .500 J/kg. k

429

.α3 .11.8

.G

∗ 10−6 1/k

.76.9

.ρ

.h

7904 .kg/m.3 .h o 2 .200 W/m. k

.0.15

GN/m.2 m

. Ta .293

◦k

Fig. 7.81 Variation of shear stress versus shell thickness for . R/ h = 10, example 3

Fig. 7.82 Variation of normal stress versus shell thickness for . R/ h = 10, example 3

430

7 Couple Thermoelasticity of Shells

Fig. 7.83 The time history of radial deflection at the crown of shell, example 5

Fig. 7.84 Time history of inside temperature at the crown of shell, example 5

at the crown of shell are given. It is noted that the mechanical coupling .δ = 0.095 has considerable effect on the shell response and causes the reduction of .w and . T . In Figs. 7.83, 7.84, 7.85, and 7.86 the time history radial displacement, inside temperature, meridional force, and meridional moment of the same point is shown. Since the moment is dominant, the stress distribution follows its pattern. The positive moment of Fig. 7.84 causes compressive stress at the inside surface.

7.9 Stability of Cylindrical Shells, Coupled Thermoelastic Assumption

431

Fig. 7.85 Time history of meridional force at the crown of shell, example 5

Fig. 7.86 Time history of meridional moment at the crown of shell, example 5

7.9 Stability of Cylindrical Shells, Coupled Thermoelastic Assumption Given discussions in the previous sections consider linear governing equations, where the kinematical and energy equations are both considered to be linear. We may consider nonlinear kinematical relations for the strain–displacement relations and solve a set of governing equations, with the assumption of uncoupled thermoelasticity theory, and come up with thermal induced vibrations. Under such assumption, the possibility of structural stability should be examined and checked. Chang and Shyong [47] considered the laminated circular cylindrical shell panels and derived the

432

7 Couple Thermoelasticity of Shells

thermally induced vibrations of the panels under thermal shock load employing the hyperbolic thermoelastic plate equation. Asadzadeh and Eslami [60] and Javani et al. [61] considered functionally graded spherical shells and, using the nonlinear kinematical relations for the strain–displacement relations, derived the performance of the shell in the form of thermal induced vibrations. The heat conduction equation in this set of problems is uncoupled. In the later article, the material properties of the shell were assumed to be temperature dependent. Another type of nonlinearity in the analysis of structural element is the assumption of thermally nonlinear coupled heat conduction equation. Under such assumption, we may assume that the temperature difference with the reference temperature is so high that the term .(T − Ta )/Ta may not be ignored compared to 1, .Ta being the reference temperature [1]. Thermally nonlinear behavior of spheres [71], layers [54, 72], and disks [73] are reported in the given references and the influence of thermal boundary conditions for thermally nonlinear behavior of structures under the Lord–Shulman generalized thermoelasticity is discussed by Bateni and Eslami [55]. The classical or generalized coupled thermoelasticity of structural flexural elements discussed in the above references are all based on the linear strain–displacement relations, where the structural behavior is stable in nature, as mathematically inherent in the assumed basic equations. Extensive discussion for the stability of flexural element under thermal loads is already presented in [74], but all discussions are limited to the uncoupled heat conduction equation. The question may arise regarding the stability of the flexural elements under thermal shock load. In many structural design problems, thermal shock loads may have sufficiently small time period of application with large thermal shock load magnitude, where the coupled thermoelasticity assumption is justified. Will the flexural element stands the load and does not lose its stability and buckle. Discussion in this section is aimed to answer this question. No such a discussion is given in literature. The present section deals with the coupled thermoelasticity problem of a Titanium– Zirconia functionally graded thin cylindrical shell under impulsive thermal shock load, where the kinematical relations are assumed to be nonlinear. Under this assumption, the possibility of instability may be examined [75, 76]. The material properties are graded across the thickness direction according to a volume fraction power law distribution. The governing equations are based on Sanders theory for nonlinear strain–displacement relations. The Galerkin finite element method is used to solve the nonlinear governing equations. The Newmark method for the nonlinear equations is used to obtain the solution in the time domain. The finite element analysis is carried out with the .C 0 -continuous element. The solution is verified with the known data in the literature for the linear coupled thermoelasticity of thin cylindrical shells. The numerical results are obtained for the lateral displacement component under several types of thermal shocks loads and the occurrence of thermal buckling is examined.

7.9 Stability of Cylindrical Shells, Coupled Thermoelastic Assumption

433

7.9.1 Analysis Consider a functionally graded thin cylindrical shell, as shown in Fig. 7.87. The material properties of the functionally graded shell, such as Young’s modules . E(z), thermal expansion coefficient.α(z), thermal conduction coefficient.k(z), specific heat conduction .c(z), and density .ρ(z) are described across the shell thickness, where h ≤ z ≤ h2 ), and .h is thickness of the shell. We . z is the thickness coordinate .(− 2 assume that the functionally graded shell is made of metal phase and ceramic phase. It is assumed that .Vm and .Vc represent the volumes of metal and ceramic phases, respectively, and the volume fraction of each constituent material is denoted by f =

. m

Vm Vm + Vc

fc =

Vc Vm + Vc

(7.9.1)

Here, . f m and . f c are the volume fractions of metal and ceramic of the FGM, respectively, and satisfy the following equation f + fc = 1

(7.9.2)

. m

The volume fraction is a spatial function. Using the combination of these functions, the effective material properties of functionally graded materials are expressed as .

Fc f (z) = Fm f m + Fc f c

(7.9.3)

where . Fc f is the effective material property of functionally graded material and . Fm and . Fc are the material properties of each phase. The volume fraction is assumed to follow the power law distribution as ( f =

. m

2z + h 2h

)i ,

fc = 1 − f m

i ≥ 0,

(7.9.4)

where the volume fraction index .i represents the material variation profile through the shell thickness and is always larger than or equal to zero. It may be varied to obtain the optimum distribution of the constituent materials. The value of .i equal to zero represents a fully metal and infinity represents a fully ceramic shell. E(z) = E c + E mc f m α(z) = αc + αmc f m . K (z) = k c + k mc f m ρ(z) = ρc + ρmc f m c(z) = cc + cmc f m where

(7.9.5)

434

7 Couple Thermoelasticity of Shells

Fig. 7.87 Cylindrical shell with displacement fields

E mc = E m − E c αmc = αm − αc . K mc = K m − K c ρmc = ρm − ρc cmc = cm − cc

(7.9.6)

It is assumed that Poisson’s ratio is constant in terms of shell thickness due to its negligible variations for the constituent materials. Consider a thin FGM cylindrical shell with length . L, thickness .h, and midthickness radius . R. The cylindrical coordinates .(x, θ, z) are considered along the axial, circumferential, and radial directions, respectively, as shown in Fig. 7.87. It is assumed that the thermal shock load is axisymmetric. Considering the first-order shear deformation theory, the displacement components in terms of the midplane displacements are u(x, z, t) = u 0 (x, t) + zψx (x, t) . (7.9.7) w(x, z, t) = w0 (x, t) where .u 0 and .w0 represent the displacement components of the middle surface of the shell and .ψx is the rotation of tangent line to the middle plane about the .x-axis. Normal and shear strains at any point across the thickness of the cylindrical shell at a distance .z away from the middle plane according to the nonlinear Sanders assumption are x 0 2 ◦ + z ∂ψ + 21 ( ∂w ) εx = ∂u ∂x ∂x ∂x w0 . εθ = (7.9.8) r 0 γx z = ψx + ∂w ∂x According to Hooke’s law, the stress–strain relationship is ⎧ ⎫ ⎡ ⎫ ⎤⎧ Q 11 Q 12 0 ⎨ εx − αΔT ⎬ ⎨ σx ⎬ σθ = ⎣ Q 12 Q 22 0 ⎦ εθ − αΔT . ⎩ ⎩ ⎭ ⎭ τx z 0 0 Q 55 γx z where

(7.9.9)

7.9 Stability of Cylindrical Shells, Coupled Thermoelastic Assumption

.

Q 11 = Q 22 =

E(z) , 1 − ν2

Q 12 =

ν E(z) , 1 − ν2

Q 55 =

435

E(z) 2(1 + ν)

(7.9.10)

Using Eq. (8.2.7) through (7.9.10), the force and bending moment resultants are obtained in terms of the displacement components as ⎧ ⎫ ⎡ A11 Nx x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎨ Nθθ ⎪ ⎬ ⎢ A12 Mx x = ⎢ . ⎢ B11 ⎪ ⎪ ⎪ ⎪ ⎣ B12 M ⎪ ⎪ θθ ⎪ ⎪ ⎩ ⎭ Qxz 0

A12 A22 B12 B22 0

B11 B12 D11 D12 0

⎧ T ⎤ ⎧ ∂u 2⎫ 0 0 N 0 + 21 ( ∂w ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂x ∂x ⎪ ⎪ ⎪ w0 ⎪ ⎪ ⎪ NT 0 ⎥ ⎨ ⎨ ⎬ ⎥ R ∂ψ ⎥ x 0 ⎥ − MT ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎦⎪ ⎪ MT ⎪ 0 ⎪ ⎪ ⎩ ∂w0 ⎩ ⎭ ⎪ 0 A55 + ψx

B12 B22 D12 D22 0

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

(7.9.11)

∂x

where . Ai j , . Bi j , and . Di j are defined in the Appendix. The thermal force and bending moment resultants are defined as {h/2 .

N = T

−h/2

{h/2 M = T

−h/2

E(z)α(z) ΔT dz 1 − ν2 E(z)α(z) ΔT zdz 1 − ν2

(7.9.12)

7.9.2 Equations of Motion and Boundary Conditions The Hamilton principle is used to obtain the equations of motion of a thin cylindrical shell. According to Hamilton’s principle, the equations of motion are derived through the following variational equation { δ

.

t1

(π − e)dt = 0

(7.9.13)

t0

where the first variation of the strain energy is { .

{t1 { δπdt =

(σx δε0 x + σθ δε0 θ + zσx δκ0 x + zσθ δκ0 θ + σx z δγ 0 x z )r dzdθd xdt t0

v

(7.9.14) In terms of the force and bending moment resultants

436

7 Couple Thermoelasticity of Shells

{

{t1 { (N x δε0 x + Nθ δε0 θ + Mx δκ0 x + Mθ δκ0 θ + Q x δγ 0 x z )r dθd xdt

δπdt =

.

v

t0

(7.9.15) The first variation of the kinetic energy is { = .

=

δe.dt =

t0

=−

v

t0

{t1 { t0 {t1 {

{t1 {

v

ρ(z)(uδ ˙ u˙ + wδ ˙ w)r ˙ dzdθd xdt

ρ(z)(u˙ 0 δ u˙ 0 + u˙ 0 zδ ψ˙ x + ψ˙ x zδ u˙ 0 + ψ˙ x z 2 δ ψ˙ x + w˙ 0 δ w˙ 0 )r dzdθd xdt {

(I1 u˙ 0 δ u˙ 0 + I2 u˙ 0 δ ψ˙ x + I2 ψ˙ x δ u˙ 0 + I3 ψ˙ x δ ψ˙ x + I1 w˙ 0 δ w˙ 0 )r dθd xdt

{t1 { {

(I1 u¨ 0 δu 0 + I2 u¨ 0 δψx + I2 ψ¨ x δu 0 + I3 ψ¨ x δψx + I1 w¨ 0 δw0 )r dθd xdt

t0

(7.9.16) where { .

I1 = I2 = I3 =

{ {

ρ(z)dz ρ(z)zdz ρ(z)z 2 dz

(7.9.17)

Substituting Eqs. (8.2.14) and (8.2.15) into Eq. (8.2.12) and applying the Euler equations to the functional of Eq. (8.2.12), in the absence of external mechanical load, results into the nonlinear equations of motion of the assumed thin cylindrical shell as .

N x x,x + I1 u¨ ◦ + I2 ψ¨ x = 0 N x x,x (w0,x ) − N x x (w0,x x ) + Nθθ /R − Q x z,x + I1 w¨ 0 = 0 Mx x,x + Q x z + I2 u¨ ◦ + I3 ψ¨ x = 0

(7.9.18)

Using Hamilton’s principle, the force and kinematical natural boundary conditions are derived as .

N x x (δu 0 ) = N¯ x x Vx (δw0 ) = Q¯ x + N¯ x x w¯ 0,x Mx x (δψx ) = M¯ x x

or

δu 0 = δ u¯ 0

or or

δw0 = δ w¯ 0 δψx = δ ψ¯ x

(7.9.19)

The barred symbols here represent the known values of the parameters. Using Eqs. (8.2.11) and (7.9.12) for the force and bending moment resultants and substituting into the equations of motion, the equations are obtained in terms of the displacement components as

7.9 Stability of Cylindrical Shells, Coupled Thermoelastic Assumption

437

First equation of motion 0 −A11 . ∂∂xu20 − A11 . ∂w ∂x ∂ψx +I2 ∂t 2 = 0 2

.

∂ 2 w0 ∂x 2

− A12 r1

∂w0 ∂x

∂T1 ∂u 0 0 − B11 .( ∂∂xψ2x ) + (R1 + R12 ) ∂T ∂x + (R2 + R22 ) ∂x + I1 ∂t 2 2

(7.9.20)

Second equation of motion ∂ ψx ∂w0 2 ∂ w0 ∂w0 ∂T0 ∂w0 1 ∂w0 2 0 −A11 . ∂∂xu20 ( ∂w ∂x ) − A11 .( ∂x ) ( ∂x 2 ) − A12 r ( ∂x ) − B11 .( ∂x 2 )( ∂x ) + (R1 + R12 ) ∂x ( ∂x ) 2

2

2

∂u 0 ∂ 2 w0 1 ∂w0 2 ∂ 2 w0 1 ∂w0 +(R2 + R22 ) ∂T ∂x ( ∂x ) − A11 ∂x ( ∂x 2 )_A11 . 2 ( ∂x ) ( ∂x 2 ) 2w 2w 2 2 ∂ψ ∂ ∂ 1 x 0 0 . −A12 w0 ( ) − B11 .( ∂x 2 )( ∂x ) + (R1 + R12 )T0 ( ∂∂xw20 ) + (R2 + R22 )T1 ( ∂∂xw20 ) r ∂x 2 1 ∂u 0 1 1 ∂w0 2 1 ∂ψx 1 1 +A12 . r ∂x + A12 . r 2 ( ∂x ) + A22 r 2 w0 + B12 . r ( ∂x ) − (R1 + R12 ) r T0 − (R2 + 2 ∂ 2 w0 x −A55 .( ∂∂xw20 + ∂ψ ∂x ) + I1 ∂t 2 = 0

R22 ) r1 T1

(7.9.21)

Third equation of motion 0 0 0 −B11 . ∂∂xu20 − B11 ( ∂∂xw20 )( ∂w ) − Br12 ( ∂w ) − D11 ( ∂∂xψ2x ) + (R2 + R22 )( ∂T ) ∂x ∂x ∂x ∂ 2 ψx ∂2 u0 ∂w0 ∂T1 +(R3 + R33 )( ∂x ) + A55 ( ∂x + ψx ) + I2 ∂t 2 + I3 ∂t 2 = 0 (7.9.22) where . R1 , . R2 , . R3 , . R12 , . R22 and . R33 are given in the Appendix. 2

2

2

.

7.9.3 Energy Equation The energy equation based on the generalized theory of thermoelasticity is [1] (

∂ ∂2 + (t0 + t1 ) 2 .ρc ∂t ∂t

)

( T + βi j Ta

∂ ∂2 + t0 2 ∂t ∂t

)

( ) u i, j = kT, j ,i

(7.9.23)

where .Ta represents the reference temperature. According to the formulation, three generalized theories are defined as 1- Classic .t0 = t1 = 0 2- Green-Lindsey .t0 = 0 t1 /= 0 3- Lord–Shulman .t0 /= 0 t1 = 0 The classical coupled thermoelasticity theory is used in this section. Due to the assumed thin shell thickness, temperature distribution may be considered linear across the shell thickness as .

T (x, θ, z, t) − Ta = T0 (x, θ, t) + zT1 (x, θ, t)

(7.9.24)

The assumed temperature distribution is substituted into the energy equation (8.2.22) and the resulting residue . Re(T, z) is made orthogonal with respect to 1 and .z. This

438

7 Couple Thermoelasticity of Shells

yields two independent equations for .T0 and .T1 as { .

{

Re.dz = 0 Re.z.dz = 0

(7.9.25)

The boundary conditions for energy equation are considered as | / ∂T | | = Qi , z = − h k(−h 2) ∂z |− h 2

(7.9.26)

|h / ∂T | 2 | = Qo , z = h .k(h 2) ∂z | 2

(7.9.27)

.

2

k

| ∂T || = 0, x = 0 ∂x |0

(7.9.28)

k

| ∂T || L = 0, x = L ∂x |

(7.9.29)

.

.

where . L is the cylinder length and . Q i and . Q o are the heat flux applied to the inside and outside shell surfaces, respectively. The two energy equations in terms of the displacement components are written as follow [76]. The coefficients are given in the Appendix. First: [ ( 2 )] [ ( )] ∂ Ta (R1 + R12 ) ∂x∂t u 0 + 21 Ta (R1 + R12 ) ∂t∂ (w0,x )2 )] [ ( [ ( )] ∂2 ψx + Ta (R1 + R12 ) r1 ∂t∂ w0 + Ta (R2 + R22 ) ∂x∂t (7.9.30) . ] ] [ [ ) ( (∂) 2 2 ∂ ∂ T0 + C22 . ∂t∂ − K 2 . ∂x T1 + C11 . ∂t − K 1 . ∂x 2 2 = Qo − Qi Second: [ ( 2 )] ( )] [ ∂ Ta (R2 + R22 ) ∂x∂t u 0 + 21 Ta (R2 + R22 ) ∂t∂ (w0,x )2 [ ( 2 )] ( )] [ ∂ ψx + Ta (R2 + R22 ) r1 ∂t∂ w0 + Ta (R3 + R32 ) ∂x∂t . ] ] [ [ ( ) (∂) ∂2 ∂ ∂2 + C22 . ∂t − K 2 . ∂x 2 T0 + C33 ∂t − K 3 . ∂x 2 + K 1 T1

(7.9.31)

= h2 (Q o + Q i ) The resulting energy equations, as given by Eqs. (8.2.29) and (8.2.30), are nonlinear due to the assumed nonlinear kinematical relation.

7.9 Stability of Cylindrical Shells, Coupled Thermoelastic Assumption

439

The simultaneous solution of the nonlinear equations of motion and nonlinear energy equations provides a system of equations to be used to determine the stability of the shell under the applied thermal shock load. This system of equations involves five unknown functions .u 0 , .w0 , .ψx , .T0 , and .T1 as functions of time for the coupled thermoelasticity problem of functionally graded cylindrical shell. A numerical technique is used to solve the governing equations in the time domain and the Galerkin finite element method is employed to obtain the solution in the space domain. This technique allows to analyze and study the behavior of the system in very short time intervals. Before solving the problem, the governing equations are made dimensionless. For this purpose, the following dimensionless parameters are defined [1] dimensionless length: x .x ¯= (7.9.32) L dimensionless time: t¯ =

.

tC1 L

(7.9.33)

dimensionless temperature: ΔT¯ =

.

T¯0 =

T −Ta Ta

T0 Ta

T¯1 =

L T1 Ta

(7.9.34)

dimensionless displacement field: u¯ =

. 0

(λm + 2μm )u 0 w¯ 0 = Lγm Ta

(λm +2μm )w0 Lγm Ta

dimensionless stress: σ¯ =

. ij

where

L=

ψ¯ x =

(λm +2μm )ψx γm Ta

σi j γm Ta

(7.9.35)

(7.9.36)

km ρm cm C1

.

C1 =

/

(7.9.37) λm +2μm ρm

and .γm , λm , μm , ρm , cm , km , and .υ are the metal physical constants.

7.9.4 Finite Element Modeling The dimensionless equations of motion and energy are solved with the Galerkin method in space domain. The linear shape function is employed to model the finite element equations. Proper explanations for the accuracy and convergence of this model are provided by Hetnarski and Eslami in reference [1]. Due to the assumed symmetric thermal loading condition, only the longitudinal element is required. Call-

440

7 Couple Thermoelasticity of Shells

ing the element function .φ(e) along the longitudinal direction, the variation in the base element .(e) is (e) .φ = Ni φi + N j φ j (7.9.38) and the shape functions in terms of the local coordinate .ξ are { {N } =

.

Ni Nj

{1

} =

(1 − ξ)

2 1 (1 2

}

+ ξ)

The variations of the unknown functions involved in the governing equations in the base element .(e) are } { U0i (t) u¯ 0 (x, t) =< Nu0i Nu0 j > {U0 j (t) } W0i (t) w¯ 0 (x, t) =< Nw0i Nw0 j > W { 0 j (t)} ¯ x (x, t) =< Nψx i Nψx j > φxi (t) .ψ { φx j (t) } TT 0i (t) T¯0 (x, t) =< N T0 i N T0 j > T { T 0 j (t) } TT 1i (t) T¯1 (x, t) =< N T1 i N T1 j > TT 1 j (t) The dimensionless nonlinear equations of motion and energy, employing the assumed shape functions given by Eq. (8.4.37) and proper weak formulations for the coupled problem [22], are written as [76] {1 {

2γm Ta α A11 ∂ Nu0l ∂ Nw0 Nw0 . ∂ξ W0 (t). ∂ ∂ξ W0 (t) (λm +2μm )2 L 2 ∂ξ ∂ Nu0l ∂ Nψx 2B11 1 ∂ Nu0l 12 + (λm A+2μ .Nw0 .W0 (t) + (λm +2μ . ∂ξ .φx (t) ∂ξ ∂ξ m) r m )αL (R2+R22) (R1+R12) ∂ N ∂ N u0l u0l .− .N T 0 .TT 0 (t) − γm α2 .N T 1 .TT 1 (t) γm α ∂ξ ∂ξ } |1 2 2 + 2ρI1mLα2 Nu0l .Nu0 ∂ U02(t) + 2ρI2mLα2 Nu0l .Nψx ∂ φx2(t) dξ = −Nu0l .N x x |−1 ∂t ∂t −1

∂ Nu0l ∂ Nu0 2 A11 . ∂ξ U0 (t) (λm +2μm )L ∂ξ

+

l = i, j

(7.9.39)

7.9 Stability of Cylindrical Shells, Coupled Thermoelastic Assumption {

441

4γm Ta α.A11 ∂ Nw0l ∂ Nw0 Nu0 . ∂ξ W0 (t). ∂ ∂ξ U0 (t) (λm +2μm )2 L 2 ∂ξ + 2γm Ta α.A212 ∂ N∂ξw0l .Nw0 .W0 (t). ∂ N∂ξw0 W0 (t) + γm Ta α.A122 . r1 .Nw0l . ∂ N∂ξw0 .W0 (t). ∂ N∂ξw0 .W0 (t) (λm +2μm ) r L (λm +2μm ) L ∂ Nw0l ∂ Nw0 L .A22 55 + (λm2.A .W (t) + N .Nw0 .W0 (t) . 0 ∂ξ +2μm )L ∂ξ 2(λm +2μm )r 2 w0l 2(R2 +R22 )Ta ∂ Nw0l ∂ Nw0 1 +R12 )Ta ∂ Nw0l ∂ Nw0 − 2(R .W (t)N .T (t) − 0 T0 T0 ∂ξ ∂ξ .W0 (t)N T 0 .TT 1 (t) (λm +2μm )L ∂ξ (λm +2μm )αL ∂ξ

.

∂ Nw0l A12 1 ( −1 (λm +2μm )r ∂ξ

.Nu0 .U0 (t) +

+ 4(γm Ta )

2 α2 .A 11 ∂ Nw0l ∂ Nw0 . ∂ξ . ∂ξ W0 (t). ∂ N∂ξw0 .W0 (t). ∂ N∂ξw0 .W0 (t) (λm +2μm )3 L 3 2Ta .(R2 +R22 ) ∂ Nw0l ∂ Nw0 a .(R1 +R12 ) ∂ Nw0l ∂ Nw0 + 2T (λm +2μm )L ∂ξ . ∂ξ W0 (t).N T 0 .TT 0 (t) + (λm +2μm )αL ∂ξ . ∂ξ W0 (t).N T 1 .TT 1 (t) ∂ Nψ x ∂ Nψ x 4γm Ta .B11 ∂ Nw0l ∂ Nw0 ∂ Nw0l A55 B12 . ∂ξ W0 (t) ∂ξ φx (t) + (λm +2μ + ∂ξ .Nψx .φx (t) + (λm +2μm )αr Nw0l ∂ξ m )α (λm +2μm )2 L 2 ∂ξ (R2 +R22 )L 1 +R12 )L 1 − (R12γ r Nw0l .N T 0 .TT 0 (t) − r Nw0l .N T 1 .TT 1 (t) 2γ α2 mα

.φx (t)

m

− 2ρI1 Lα2 m

2 Nw0l .Nw0 ∂ W02(t) )dξ ∂t

|1 |1 | m Ta )Nw0l . ∂ N∂ξw0 W0 (t).N x x | = − Nw0l .Q x |−1 −( (λ2αγ m +2μm )L

−1

l = i, j

(7.9.40) {

∂ Nψxl ∂ Nu0 2γm Ta B11 ∂ Nψxl ∂ Nw0 2B11 ∂ Nw0 1 −1 ( (λm +2μm )αL ∂ξ . ∂ξ U0 (t) + (λm +2μm )2 L 2 ∂ξ . ∂ξ W0 (t). ∂ξ W0 (t) B12 A55 Nw0 1 ∂ Nψxl .Nw0 .W0 (t) + (λm +2μ Nψxl ∂ ∂ξ .W0 (t) + (λm +2μ ∂ξ m )α r m )α ∂ Nψxl ∂ Nψx L .A55 2D11 + (λm +2μm )α2 L ∂ξ . ∂ξ .φx (t) + 2(λm +2μm )α2 Nψxl .Nψx .φx (t) . 22 ) ∂ Nψxl 32 ) ∂ Nψxl − (Rγ2 m+R .N T 0 .TT 0 (t) − (Rγ3 m+R .N T 1 .TT 1 (t) α2 ∂ξ α3 ∂ξ |1 ∂ 2 φx (t) ∂ 2 U0 (t) I3 L I2 L + 2ρm α3 Nψxl .Nu0 + 2ρm α3 Nψxl .Nψx )dξ = −Nψxl .M x x |−1 2 2 ∂t ∂t

l = i, j

(7.9.41)

{

2 (R1 +R12 )Ta ∂ Nu0 ∂U0 (t) Nw0 Nw0 ∂W0 (t) 1 a (R1 +R12 )γm + 2T(λ N T 0l . ∂ ∂ξ W0 (t). ∂ ∂ξ 2 −1 ( (λm +2μm )α N T 0l . ∂ξ ∂t ∂t m +2μm ) L ∂ Nψx ∂φx (t) ∂W0 (t) Ta (R2 +R22 ) a (R1 +R12 ) L 1 N .N . N . + + T(λ T 0l w0 T 0l 2 (λm +2μm )α ∂ξ ∂t ∂t m +2μm ) 2α r ∂ N T 0l ∂ N T 0 T 0 (t) 1 . ∂ξ .TT 0(t) + γm2K + Cγm11 2αL 2 N T 0l .N T 0 . ∂T∂t C1 αL ∂ξ . NT 1 T 1 (t) + Cγm22 2αL 3 N T 0l .N T 1 . ∂T∂t + γm C2K1 α2 2 L ∂ N∂ξT 0l . ∂ ∂ξ .TT 1 (t) |1 { { | 1 L dz)| ) = γm Ta αC1 ( 1−1 N T 0l .(Q o − Q i ) 2 dξ + N T 0l ( K (z) ∂T ∂ξ −1

l = i, j

(7.9.42)

{

2 )γm ∂ Nu0 ∂U0 (t) 1 ( (R2 +R22 )Ta N N T 1l . ∂ N∂ξw0 W0 (t). ∂ N∂ξw0 ∂W∂t0 (t) + 2Ta (R2 +R22 −1 (λm +2μm )α2 T 1l . ∂ξ ∂t (λm +2μm )2 Lα ∂ Nψx ∂φx (t) Ta (R3 +R32 ) a (R2 +R22 ) L 1 + T(λ N T 1l .Nw0 . ∂W∂t0 (t) + (λ 2 N T 1l . ∂ξ ∂t m +2μm ) 2α2 r m +2μm )α ∂ N T 1l ∂ N T 0 ∂TT 0 (t) C22 L 2K 2 + γm 2α3 N T 1l .N T 0 . ∂t + γ C α2 L ∂ξ . ∂ξ .TT 0 (t) m 1 . ∂ N T 1l ∂ N T 1 2K 3 + Cγm33 2αL 4 N T 1l .N T 1 . ∂TT∂t1 (t) + γ C 3 ∂ξ . ∂ξ .TT 1 (t) m 1α L { { K 1 .L 1 + 2α3 C γ N T 1l .N T 1 .TT 1 (t) = γ T α2 C ( 1−1 N T 1l . h4L .(Q 0 + Q i )dξ + N T 1l ( K (z)z ∂T ∂ξ m a 1 1 m

|1 | dz)| )

l = i, j

−1

(7.9.43) Note that in the above equations the following matrix notations are considered

442

7 Couple Thermoelasticity of Shells

} U0i (t) {U0 j (t) } W0i (t) W0 (t) = W { 0 j (t)} φxi (t) . φ x (t) = φ { x j (t) } TT 0i (t) TT 0 (t) = T { T 0 j (t) } TT 1i (t) TT 1 (t) = TT 1 j (t) {

U0 (t) =

and .

Nuo =< Nu0i

Nu0 j >

Nwo =< Nw0i Nw0 j > Nψx =< Nψx i Nψx j > N T0 =< N T0 i N T1 =< N T1 i

N T0 j > N T1 j >

(7.9.44)

The set of equations given by Eqs. (8.4.39) to (8.4.43) for the base element .(e) with nodes .i and . j are assembled for all element in the solution domain to receive at the following general finite element equation .

[M] { X¨ } + [C( X˙ )]{ X˙ } + [K (X )] {X } = {F(t)}

(7.9.45)

It is noted that due to the nonlinear nature of the governing finite element equations, the capacitance and stiffness matrices .[C] and .[K ] are variable dependent.

7.9.5 Numerical Method Due to the nonlinearity of the governing equations and coupling of the displacement terms and their derivatives in these equations, an incremental solution method is adopted to check the thermal stability of the shell. By assuming an incremental solution procedure, the real time system is approximated by a step-by-step time marching, assuming time-invariance within each time step. The nonlinear finite element equation (7.9.45) is solved by time marching method assuming that the initial conditions . X˙ 0 and . X 0 are given [77, 78]. The force function .{F} is the assumed thermal shock load known in terms of time and is given at any step time. The numerical solution is accomplished through the following steps [76, 79]. Step 1. The cylindrical shell is discretized into a number of elements along the axial direction with axisymmetric assumption.

7.9 Stability of Cylindrical Shells, Coupled Thermoelastic Assumption

443

Step 2. The initial values of the middle surface displacement components, rotation, and temperatures .T0 and .T1 in the unknown matrix .{X 0 } are set to zero. The initial first time rate of these functions, .{ X˙ 0 }, are also set equal to zero. Step 3. The mass, stiffness, and capacitance matrices are calculated using the initial values of midplane displacement components given in Step 2. Step 4. The matrix .{ X¨ 0 } is calculated from Eq. (7.9.45) using the mass, stiffness, and capacitance matrices with the given . X˙ 0 and . X 0 . .

( ) X¨ 0 = [M]−1 {F(0)} − [K (X 0 )]{X 0 } − [C(X 0 )]{ X˙ }0

(7.9.46)

Step 5. Time is incremented. Choosing .Δt and calculating .tn+1 = Δt + tn , where n is the step time counter. Step 6. Calculating the Newmark integration constants according to the following equations. Two parameters .γ and .β are selected. The parameter .γ = 1/2 ensures second-order accuracy and .β = 1/4 makes the algorithm implicit and equivalent to the trapezoidal rule [22].

.

[M] [C] a1 = βΔt 2 + γ βΔt γ [M] a2 = βΔt + ( β − 1)[C] . γ 1 − 1)[M] + Δt ( 2β − 1)[C] a3 = ( 2β [K˜ ] = [K ] + a1

(7.9.47)

˜ and Step 7. Calculating the equivalent stiffness and force matrices .[ K˜ ] and .[ F] predicting with assumption .{X }n+1 = {X }n . 2 ˜ n+1 = {F}n+1 + a1 × {X }n + a2 × ∂{X }n + a3 × ∂ {X }n {F} ∂t ∂t 2

.

[K˜ ]n+1 = K n+1 + a1

.

(7.9.48) (7.9.49)

Step 8. Calculate the residual vector .

˜ n+1 − [K (X n+1 )]{X }n+1 − a1 × {X }n+1 ˜ n+1 = {F} {R}

(7.9.50)

Step 9. Solve the term .Δ{X } from the equation Δ{X } =

.

˜ n+1 {R} [K˜ ]n+1

(7.9.51)

Step 10. If .|Δ{X }| ≤ {η}, .{η} being the convergence parameter vector, calculate the velocity and acceleration vectors from the next step. Step 11. Calculate the nodes velocity .{X˙ } and acceleration .{X¨ } from the following equations and enter them into the next time step.

444

7 Couple Thermoelasticity of Shells

.

d{X }n+1 −{X }n γ d 2 {X }n }n = γ {X }n+1 + (1 − βγ ) d{X + Δt (1 − 2β ) dt 2 dt βΔt dt 2 2 {X }n+1 −{X }n d {X }n+1 γ d {X }n 1 d{X }n = + βΔt dt + (1 − 2β ) dt 2 dt 2 βΔt 2

(7.9.52)

Step 12. If .|Δ{X }| > {η}, we consider .Δ{X }m instead of .Δ{X }, where .m is the iteration counter and m Δ{X }m = {X }m+1 n+1 − {X }n+1

.

(7.9.53)

m Step 13. Calculate .{X }m+1 n+1 in above equation, where .{X }n+1 at the first iteration 1 is equal to .{X }n+1 or .{X }n+1 predicted in Step 7.

˜ n+1 and .[K˜ ]n+1 with .{X }m+1 Step 14. Recalculate .{R} n+1 . ˜ n+1 and .[K˜ ]n+1 and check the convergence. Step 15. Calculate .Δ{X } with new .{R} If converged, go to step 11 otherwise go to step 12 for the next iteration .m.

7.9.6 Results and Discussion The formulations and method of solution derived in this section are validated with those reported by Bahtui and Eslami [10]. For the comparison with this reference, the displacement fields are calculated based on the linear coupled thermoelasticity of the FG cylindrical shell by eliminating the nonlinear terms in the strain–displacement relations. The dimensions of the cylindrical shell, material, boundary conditions, thermal shock, and dimensionless methods are selected similar to those used in Behtui and Eslami [10] article. The radial displacement versus time at the middle length of the functionally graded cylindrical shell is presented in Fig. 7.88. The material is Ti6Al4V/. Zr O2 and the power law index is selected equal to zero. Similar to reference [10], the clamped cylindrical shell under inside impulsive thermal shock load is considered. The ratio of length of shell to radius is equal to 5 and the ratio . R/ h is 10, where .h and . R are the thickness and radius of the shell, respectively. The equation of thermal shock is .

T (t) = 2201.85 × aT e−13100bt + 298.15

(7.9.54)

where .a and .b are selected such that the impulsive shock period becomes .10−6 s and . T (t) is the time-dependent temperature. This equation generates a convective heat flux equal to .2201.85 × h in × W/m2 in 0.5 .µs. The thermal boundary conditions at the ends of the shell are assumed to be insulated. The coefficient of thermal convection of the ceramic rich inside surface is (.h in = 10,000 .W/m.2 K) and that of the metal rich outside surface is (.h out = 200 .W/m.2 K). The comparison shown in Fig. 7.88 is well justified. Now, consider a simply supported functionally graded cylindrical shell under the outside impulsive thermal shock [76]. The ratio of the length to cylinder diameter is

7.9 Stability of Cylindrical Shells, Coupled Thermoelastic Assumption

445

Radial displacement (Nano meter)

1.2 Bahtui

1

Present 0.8 0.6 0.4 0.2 0 −0.1 0

37

74

111

148

222 185 Time (Micro second)

259

296

333

370

400

Fig. 7.88 Comparison between the radial displacement of middle length of the shell versus time with that reported in [10]

assumed to be 2, and the ratio . R/ h is considered to be 30. The functionally graded shell is assumed to be made of combination of metal (Ti-6Al-4V) and ceramic (ZrO2), at the initial temperature 298.15 .K, with the material properties given in Table 7.12 [10]. The shell is ceramic rich in inside and metal rich at outside surfaces, respectively. The same boundary conditions at both ends of the cylindrical shell are considered. The aim of this research is to check the stability of the shells under an applied thermal shock load. This is the main reason that nonlinear kinematical relations is considered into the coupled thermoelasticity shell formulations. Another important question to be considered is that the reader should not expect wave fronts of any nature in the response of the flexural elements under coupled or generalized thermoelasticity assumption as these structures such as beams, plates, and shells are lumped across the thickness direction. The details may be studies in reference [1] given by Hetnarski and Eslami. The functionally graded shell is studied for the power law indices of .i = 0 and .i = 10. The plots of radial displacement versus time for different values of the heat fluxes are presented in the figures. The response of the shell for three different types of thermal shock loads is studied. The thermal shock load is considered in form of . Q = At¯e−B t¯ heat flux applied to the outside surface of the shell, where . A and . B are some constant parameters. These constant parameters are selected for three different types of shocks. The first thermal shock load, calling . Q 1 , is selected with values . A = 27.0608 and . B = 0.3383. The time period of . Q 1 is less than the lowest natural frequency of the shell. Figure 7.89 shows the radial displacement of middle length of the shell versus time (100 times of non-dimension time) under the first shock load. Q 1 , when the power law index.(i) is equal to zero and 10 with the assumption of nonlinear strain–displacement relations. Radial response of the shell is stable, vibratory, with small time periods, and follows a long range vibrations as time advances. The amplitudes for the power law index .i = 0 are larger compared to .i = 10, as the power law index .i = 0 represents pure metal with lower modulus of elasticity compared to the ceramic for larger power law indices of the FGM shell. As the power law index increases, the material changes from a metal state to a combination of ceramic and metal. In this condition and under

446

7 Couple Thermoelasticity of Shells

Table 7.12 Material properties of FGM Metal 66.2 0.327 −6 .7.11 ∗ 10 3 .8.9 ∗ 10 6.08 625

. E(Gpa) .υ .α(1/K ) .ρ(kg/m

Ceramics

3)

. K (W/m K ) .c(J/kgK )

.E

117 0.327 −6 .18.48 ∗ 10 3 .2.37 ∗ 10 1.775 529

.υ .α .ρ .K .c

50 45 i=0 i=10

Non−dim Radial displacement

40 35 30 25 20 15 10 5 0 0

10

20

30

40

50 Non−dim time

60

70

80

90

100

Fig. 7.89 Radial displacement of middle length of the shell versus time under the first shock . Q 1 and power law indices 0 and 10, considering the nonlinear strain–displacement relations

the same thermal shock load, the amplitude of vibrations decreases and the frequency of vibrations increases with the increase of the power law index. Figure 7.90 shows radial displacement of middle length of the shell versus time (100 times of non-dimension time) with the first shock load when the power law indices .(i) are equal to zero and 10 with linear strain–displacement relations. This figure clearly shows how the linear and nonlinear kinematical assumption affects the period of vibrations. Under the linear kinematical assumption, the shell response is stable regardless of the period and magnitude of the applied thermal shock load. To better distinguish the behavior of shell under linear and nonlinear kinematical relations, Fig. 7.91 is plotted. This figure presents the plots of dimensionless radial displacement versus time under the first thermal shock load applied to the outside surface of the cylindrical shell. Comparison between the nonlinear and linear strain– displacement relations is shown in this figure. For the linear relations, the amplitude of radial vibrations is around an almost straight line. Now consider the second and third thermal shock loads with equations . Q 2 = (0.1)At¯e−(0.1)B t¯ and . Q 3 = 10 At¯e−B t¯. The period of application of . Q 1 is higher

7.9 Stability of Cylindrical Shells, Coupled Thermoelastic Assumption

447

50 i=0 i=10

45

Non−dim Radial displacement

40 35 30 25 20 15 10 5 0 −5 0

10

20

30

40

50 Non−dim time

60

70

80

90

100

Fig. 7.90 Radial displacement of middle length of the shell versus time under the first shock . Q 1 and power law indices 0 and 10 with linear strain–displacement relations 45 40

Linear Non−Linear

Non−dim Radial displacement

35 30 25 20 15 10 5 0 −5 0

10

20

30

40

50 Non−dim time

60

70

80

90

100

Fig. 7.91 Radial displacement of middle length of the shell versus time under. Q 1 shock with linear and nonlinear strain–displacement relations for .i = 0

than the period of lowest natural frequency of the shell. Under this type of thermal shock load, we should expect a smooth radial displacement although the coupled thermoelasticity formulation is considered. The nature of the third type of thermal shock load . Q 3 is the same as . Q 2 , except that the magnitude of shock is much higher and the possibility of shell buckling exist. Figure 7.92 shows the radial displacement of middle length of the shell versus time with the first shock . Q 1 , second shock (. Q 2 = (0.1)At¯e−(0.1)B t¯), and third shock (. Q 3 = 10 At¯e−B t¯). The figure shows the dimensionless radial displacement versus time for the first, second, and third shock loads applied to the outside surface of the cylindrical shell. The first shock period is smaller than the natural frequency of the shell. Period of the second shock is much larger than the natural frequency of shell

448

7 Couple Thermoelasticity of Shells

80 Q2 Q3 Q1

Non−dim Radial displacement

70 60 50 40 30 20 10 0 0

10

20

30

40

50 Non−dim time

60

70

80

90

100

Fig. 7.92 Radial displacement of middle length of the shell versus time under three different shocks, =0

.i

40 R/h=30 R/h=20 R/h=10

Non−dim Radial displacement

35 30 25 20 15 10 5 0 0

5

10

15 Non−dim time

20

25

30

Fig. 7.93 Radial displacement of middle length of the shell versus time with . R/ h ratio. linear or nonlinear, .i = 0

and the period of third shock is smaller than the lowest natural frequency of shell, similar to the first shock, but the amplitude of the third shock is much higher than the first. The response of the shell under . Q 2 has no vibration. The response of the shell under . Q 1 is similar to Figs. 7.89 and 7.91. The response of the shell under . Q 3 is interesting. The third shock is similar to the first shock but with higher amplitude. The buckling phenomenon is observed in this curve due to the high thermal shock amplitude applied to the shell. Figure 7.93 shows the radial displacement of middle length of the shell versus time under the first thermal shock load. This figure shows that for higher ratio of radius to shell thickness, the amplitude of vibrations increases.

References

449

The aim of this research is to examine the possibility of thermal buckling of a structure under thermal shock loads with very small short time period, where the coupled thermoelasticity assumption is essential [76]. This condition is possible, when the flexural elements are under thermal shock loads producing compressive stresses. The classical coupled thermoelasticity theory is formulated employing the linear strain–displacement relations. With this assumption, the possibility of losing stability is lost in the analysis. A comprehensive presentation of the thermal stability of the flexural elements of all types is given by Eslami [74], where uncoupled thermoelasticity theory is basically used throughout. When the coupled assumption is considered, all the governing equations, including the energy equation, become nonlinear where the solution of which is extremely complicated. The analysis presented in this section proves the mathematical complicity incurred. Appendix

{ A11 = {A22 = E(z)dz A12 = { E(z)υdz E(z) dz A55 = 2(1+υ) { B11 = {B22 = E(z)zdz B12 = E(z)zυdz { D11 = {D22 = E(z)z 2 dz D12 ={ E(z)z 2 υdz R1 = { E(z)α(z)dz R12 ={ E(z)α(z)υdz . R2 = { E(z)α(z)zdz R22 ={ E(z)α(z)zυdz R3 = { E(z)α(z)z 2 dz R32 = { E(z)α(z)z 2 υdz C11 = { ρ(z)c(z)dz C22 = { ρ(z)c(z)zdz C33 ={ ρ(z)c(z)z 2 dz K 1 = { K (z)dz K 2 = { K (z)zdz K 3 = K (z)z 2 dz

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Chapter 8

Thermal Induced Vibrations

Abstract Thermal induced vibrations occur in flexural elements such as beams, plates, and shells. Whenever structures are exposed to transient thermal conditions, thermal induced vibrations may occur. This phenomenon was first observed in the satellites behavior in orbit. The occurrence of thermal induced vibrations depends upon the geometrical properties of the element in consideration. This chapter presents thermal induced vibrations of rectangular plates and the spherical and conical shells. The analysis is based on the solution of transient heat conduction equation along with the equations of motion of plates and shells. Limitations for the occurrence of induced vibrations are discussed in detail.

8.1 Introduction In 1956 Boley [1] was the first who introduced the thermally induced vibration phenomenon in a beam including simply supported ends and exposed to a sudden constant thermal shock at the top surface. Thermally induced vibration phenomenon was not only investigated for beams, other researchers extended Boley’s work to several structures or different thermal shock problems. Boley and Barber [2] obtained the response of a thin rectangular plate with simply supported edges. The results showed that vibrations caused by rapid heating occur when inertia parameter is small enough. On the other hands, quasi-static response occurs when inertia parameter becomes very large. Kraus [3] analyzed the axisymmetric response of a deep spherical shell, whose outer side was under thermal shock. Results of his work confirmed the qualitative results of previous researches. As a general conclusion, he presented that the inertia terms in beams are more important than plates and plates over shells, which is due to the existence of extensional-bending coupling in the kinematics of the shells. By using the superposition method, Venkataramana and Jana [4] established an analytical solution for beams subjected to a harmonically varying temperature. Bruch et al. [5] presented the deflection and velocity of a beam subjected to the thermal shock. A control strategy is also provided in this work. The solution of the control problem is obtained by means of the eigenfunction expansions. Manolis and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. R. Eslami, Thermal Stresses in Plates and Shells, Solid Mechanics and Its Applications 277, https://doi.org/10.1007/978-3-031-49915-9_8

453

454

8 Thermal Induced Vibrations

Beskos [6] presented three examples for beam-like structures under rapid heating loads considering the in-plane loading and damping effects. Malik and Kadoli [7, 8] presented the thermoelastic response of a functionally graded beam subjected to various thermal loads and surface heating. Vibrations of an orthotropic plate under rapid heating are analyzed by Tauchert [9]. He utilized the superposition method to investigate the response of the rectangular plate with various types of edge supports. Das [10] studied the thermally induced vibration of a plate with an arbitrary polygonal shape based on the complex variable theory. Using 2D finite element method, Chang [11] investigated the vibrations of a composite plate under thermal shock. Stroud and Mayers [12] proposed an alternative method based on the Reissner variational energy principle to analyze the thermally induced response of rapidly shocked rectangular plates. He assumed the elasticity modulus and thermal expansion coefficient as temperature-dependent variables. The obtained results showed that the removal of temperature dependence causes considerable error. Alipour [13] utilized the 2D differential quadrature method to analyze the thermally induced vibrations of an FGM rectangular plate. All thermomechanical properties of the plates are assumed to be temperature dependent. Nakajo and Hayashi [14] studied the axisymmetric response of circular plates by different methods. An analytical solution based on the separation of variables, a numerical method based on the finite elements, and an experimental study are presented. As shown in their results, for circular plate with clamped edge, which is affected greatly by the membrane force, the geometrically linear theory is inefficient. Also, Kiani and Eslami [15] studied the axisymmetric dynamics and quasi-static response of a functionally graded solid circular plate. Ritz method is applied to analyze the geometrically nonlinear problem. By means of a hybrid GDQ-Newmark method, Javani et al. [16] obtained the quasi-static and vibrational response of an FGM annular plate. In this work it is shown that for the clamped plates with immovable edges there is the possibility of dynamic instability under thermal shock. Ghiasian et al. [17] investigated the thermally induced vibration response of the FGM beams using the Ritz method. They showed that flat beams with clamped edges remain flat up to a critical time. Clamped beams remain in the flat shape until a prescribed time, in which dynamic instability occurs and structure bends. Hill and Mazumdar [18] presented a method for thermal dynamic response of plates and shallow shells made of viscoelastic materials based on geometrically nonlinear deflection. The method of analysis basically is originated from the other works of the researchers for the case of linear strain–displacement relations [19–21]. However, solution methodology may be valid even when deflections are of the order of the thickness of the structure. The Navier method suitable for movable simply supported edges is used for thermally induced vibrations of doubly curved shells in symmetric or anti-symmetric cross-ply lamination schemes by Huang and Tauchert [22]. Also Huang and Tauchert [23] modified their previous work and used the nonlinear strain–displacement relations (von Kármán type) to analyze the laminated shells under thermal shock. Results showed that a comprehensive dynamic response indeed captures the dynamic snap-through phenomenon, whereas classical quasistatic deflection results in numerical divergence.

8.2 FGM Rectangular Plates

455

Khdier [24] reported an analytical solution for the vibrational response of crossply laminated shallow shells subjected to rapid heating. Its solution is applicable to shells whose parallel edges are simply supported and other edges are clamped. Khdier [25] also employed an exact formulation to survey the dynamic deflection of layered orthotropic shallow arches under thermal shock. Keibolahi et al. [26] analyze the thermally induced vibrations in a shallow arch subject to rapid surface heating. Finite element method in two-dimensional space within the coupled thermoelasticity formulation is investigated by Chang and Shyong [27] to analyze the thermally induced vibrations of composite circular cylindrical shells. Thermally induced vibrations of FGM sandwich plate and shell panels are investigated by Pandey and Pradyumna [28]. They proposed a finite element method based on a higher order layerwise theory to obtain the response of shell panels which are made of temperaturedependent material properties and are also under thermal shock. Present investigation aims to analyze the large amplitude forced vibrations in an annular plate induced by rapid surface heating. To this end, the axisymmetric firstorder shear deformation plate theory and von Kármán type of kinematic relations are used. The temperature profile within the plate domain is extracted by means of the solution of the heat conduction equation. The temperature profile is inserted into the nonlinear equations of motion and the mentioned equations are solved employing the GDQ method and the Newton–Raphson technique. Numerical results are provided to explore the effects of different parameters.

8.2 FGM Rectangular Plates Consider a rectangular plate made of FGMs with thickness .h, length .a, and width .b [13]. Conventional Cartesian coordinates system .(x, y, z) with its origin located at the corner midplane of the plate is considered. In this system, .0 ≤ x ≤ a, .0 ≤ y ≤ b, and .−h/2 ≤ z ≤ +h/2 represent, respectively, the through the length, through the width, and through the thickness directions. Due to its simplicity and effectiveness, the Voigt rule of mixture is commonly used to obtain the equivalent thermo-mechanical properties of the FGM plates in terms of the thermo-mechanical properties of its constituents. Therefore, the thermomechanical properties such as Young’s modulus . E, Poisson’s ratio .ν, thermal expansion coefficient .α, mass density .ρ, specific heat .Cv , and thermal conductivity . K are assumed as the linear function of the volume fractions of the ceramic .Vc and metal . Vm . Thus, as a function of thickness direction, the non-homogeneous property of the plate . P may be expressed by [15, 17, 29] .

P(z, T ) = Pm (T )Vm (z) + Pc (T )Vc (z)

(8.2.1)

In the above equation and in the rest of this work, subscripts .m and .c are associated with the properties of metal and ceramic constituents, respectively. Temperature dependence of the FGM constituents is frequently expressed based on the Touloukian

456

8 Thermal Induced Vibrations

formula [15, 17, 29] where higher order dependence to temperature is included. According to this formula, each property of the metal or ceramic may be written in the form [29] .

P(T ) = P0 (P−1 T −1 + 1 + P1 T + P2 T 2 + P3 T 3 )

(8.2.2)

In this equation .T is temperature which is measured in Kelvin and . Pi ’s are temperature dependence coefficients, unique to each property of the constituent. To completely define the properties in an FGM media, the distribution of volume fractions should be known. The power law type of ceramic volume fraction dispersion is frequently used [29]. Accordingly, dispersion of ceramic volume fraction .Vc and metal volume fraction .Vm may be considered as ) z ζ 1 + . Vc = 2 h ) ( z ζ 1 + Vm = 1 − 2 h (

(8.2.3)

In this equation .ζ ≥ 0 is the power law index and dictates the property dispersion profile.

8.2.1 Governing Equations To capture through the thickness shear deformations and rotary inertia effects, the first-order shear deformation theory (FSDT) of plates is used to formulate the governing equations of the plate. Based on the FSDT, components of the displacement on a generic point of the plate may be represented according to the mid-surface characteristics such that u (x, y, z, t) = u 0 (x, y, t) + zϕx (x, y, t) v (x, y, z, t) = v0 (x, y, t) + zϕ y (x, y, t)

.

w (x, y, z, t) = w0 (x, y, t)

(8.2.4)

In the above equations .u, v, and .w are through the length, through the width, and through the thickness displacements, respectively. A subscript .0 indicates the characteristics of the mid-surface. Besides, .ϕ y and .ϕx are, respectively, the transverse normal rotations about the .x and . y axes. According to the FSDT, the components of strain field on an arbitrary point of the plate may be obtained in terms of those which belong to the mid-surface of the plate and change of curvatures. Consequently, one may write

8.2 FGM Rectangular Plates

457

⎧ ⎧ ⎫ ⎧ ⎫ ⎫ εx x0 ⎪ κx x ⎪ εx x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ε yy0 ⎪ ⎨ κ yy ⎪ ⎨ ε yy ⎪ ⎬ ⎪ ⎬ ⎬ γx y = γx y0 + z κx y . ⎪ ⎪ ⎪ ⎪ ⎪ γx z0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ κx z ⎪ ⎪ γx z ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎩ ⎭ ⎭ ⎭ γ yz γ yz0 κ yz

(8.2.5)

where the components of the strain associated with the mid-surface of plate are ⎧ ⎫ ⎧ ⎫ u 0,x εx x0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v0,y ⎨ ⎨ ε yy0 ⎪ ⎬ ⎪ ⎬ γx y0 = u 0,y + v0,x . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕx + w0,x ⎪ ⎪ γx z0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎪ ⎭ γ yz0 ϕ y + w0,y

(8.2.6)

and the components of change in curvature compatible with the FSDT are ⎧ ⎫ ⎧ ⎫ ϕx,x κx x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ y,y ⎨ ⎨ κ yy ⎪ ⎬ ⎪ ⎬ κx y = ϕ y,x + ϕx,y . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ κx z ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎪ ⎭ κ yz 0

(8.2.7)

where in Eqs. (8.2.6) and (8.2.7), .(),x and .(),y denote, respectively, the derivatives with respect to the .x and . y directions of the plate. For the case when material properties of the plate are isotropic and linearly thermoelastic, components of stress in terms of strains and temperature change are evaluated as ⎧ ⎧ ⎫ ⎡ ⎫ ⎫⎞ ⎤ ⎛⎧ εx x ⎪ Q 11 Q 12 0 0 0 T − T0 ⎪ σx x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪⎟ ⎪ ⎜⎪ ⎪ ⎪ ⎨ T − T0 ⎪ ⎨ σ yy ⎪ ⎬ ⎢ Q 12 Q 22 0 0 0 ⎥ ⎬⎟ ⎥ ⎜⎨ ε yy ⎬ ⎢ ⎟ ⎜ ⎥ σx z = ⎢ 0 0 Q 44 0 0 ⎥ ⎜ γ yz − α 0 . (8.2.8) ⎟ ⎪ ⎪ ⎪ ⎪ ⎣ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎠ ⎦ ⎝⎪ γx z ⎪ σ 0 0 0 0 Q 0 ⎪ ⎪ ⎪ ⎪ x z 55 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎩ ⎭ ⎭ ⎭ σx y γx y 0 0 0 0 Q 66 0 where . Q i j ’s .(i, j = 1, 2, 6) are the reduced material stiffness coefficients and are obtained as follow

. Q 11

= Q 22 =

E(z, T ) , 1 − ν 2 (z, T )

Q 12 =

ν(z, T )E(z, T ) , 1 − ν 2 (z, T )

E(z, T ) Q 44 = Q 55 = Q 66 = 2(1 + ν(z, T ))

(8.2.9)

The components of stress resultants are obtained using the components of stress field as

458

8 Thermal Induced Vibrations

⎧ ⎫ ⎧ ⎫ ⎨ N x x ⎬ { +0.5h ⎨ σx x ⎬ N yy = σ yy dz, . ⎩ ⎭ −0.5h ⎩ σ ⎭ Nx y xy ⎧ ⎫ ⎫ ⎧ { ⎨ Mx x ⎬ +0.5h ⎨ σx x ⎬ M yy = z σ yy dz, ⎩ ⎭ ⎭ ⎩ −0.5h Mx y σx y } { +0.5h { } { Qxz σx z = dz κ Q yz σ yz −0.5h

(8.2.10)

In the above equation,.κ is the correction shear factor of FSDT. As known, adoption of a shear correction factor may result in more accurate results. Since shear correction factor depends on the boundary conditions, material properties, and loading type, determination of its exact value is not straight forward. However, the approximate values of .κ = 5/6 or .κ = π 2 /12 are extensively used. In this research, the shear correction factor is set equal to .κ = 5/6. Substitution of Eq. (8.2.8) into Eq. (8.2.10) with the simultaneous aid of Eqs. (8.2.5)–(8.2.7) generates the stress resultants in terms of the mid-surface characteristics of the plate as ⎫ ⎧ T⎫ ⎤⎧ εx x0 ⎪ N ⎪ 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ε yy0 ⎪ ⎪ ⎪ 0 0 0 ⎥ NT ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γx y0 ⎪ ⎪ B66 0 0 ⎥ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎨ ⎨ ⎬ T ⎬ 0 0 0 ⎥ ⎥ κx x − M T 0 0 0 ⎥ κ yy ⎪ M ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ κx y ⎪ ⎪ ⎪ D66 0 0 ⎥ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γ yz0 ⎪ 0 κA44 0 ⎦ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ γx z0 0 0 κA55 0 (8.2.11) In the above equations, the constant coefficients. Ai j ,. Bi j , and. Di j indicate the stretching, bending-stretching, and bending stiffnesses, respectively, which are calculated by ⎧ ⎫ ⎡ A11 ⎪ Nx x ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ N yy ⎪ A12 ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ Nx y ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ ⎢ B11 Mx x ⎢ =⎢ . ⎪ M yy ⎪ ⎢ B12 ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ Mx y ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ Q yz ⎪ ⎪ ⎣ 0 ⎪ ⎪ ⎩ ⎭ Qxz 0

A12 A22 0 B12 B22 0 0 0

0 0 A66 0 0 B66 0 0

B11 B12 0 D11 D12 0 0 0

B12 B22 0 D12 D22 0 0 0

{ (Ai j , Bi j , Di j ) =

+0.5h

.

−0.5h

(Q i j , z Q i j , z 2 Q i j )dz

(8.2.12)

Besides . N T and . M T are the thermally induced force and moment resultants which are defined as { (N , M ) =

.

T

T

+0.5h

−0.5h

(1, z)

E(z, T )α(z, T ) (T − T0 )dz 1 − ν(z, T )

(8.2.13)

The complete set of the equations of motion and boundary conditions of an FGM rectangular plate may be obtained based on the generalized Hamilton principle. Statement of Hamilton’s principle in the absence of external loads reads

8.2 FGM Rectangular Plates

{ δ

t2

459

(K − U )dt = 0

.

t1

at t = t1 , t2 : δu 0 = δv0 = δw0 = δϕx = δϕ y = 0

(8.2.14)

where in the above equation, .δK is the virtual kinetic energy of the plate which is equal to {

a

δK =

{

b

{

+0.5h

.

0

−0.5h

0

ρ(z, T ) (uδ ˙ u˙ + vδ ˙ v˙ + wδ ˙ w) ˙ dzdyd x

(8.2.15)

Here, a .(˙) indicates the derivative with respect to time .t. Besides, .δU is the virtual strain energy of the plate which may be calculated as { .δU

a

= 0

{

b 0

{

+0.5h

( ) σx x δεx x + σ yy δε yy + σx y δγx y + κσx z δγx z + κσ yz δγ yz dzdyd x

−0.5h

(8.2.16) Integrating the expressions (8.2.15) and (8.2.16) with respect to .z coordinate and performing the Green–Guass theorem to relieve the virtual displacement gradients results in the expressions for linear equations of motion of the FGM plate as

.

N x x,x + N x y,y = I1 u¨ 0 + I2 ϕ¨ x N x y,x + N yy,y = I1 v¨0 + I2 ϕ¨ y Q x z,x + Q yz,y = I1 w¨ 0 Mx x,x + Mx y,y − Q x z = I2 u¨ 0 + I3 ϕ¨ x Mx y,x + M yy,y − Q yz = I2 v¨0 + I3 ϕ¨ y

(8.2.17)

where the following definitions apply { (I1 , I2 , I3 ) =

+0.5h

.

−0.5h

ρ(z, T )(1, z, z 2 )dz

(8.2.18)

The complete set of boundary conditions is revealed through the process of virtual displacement relieving. For . y = 0 and . y = b, the boundary conditions are extracted as .

N x y δu 0 = N yy δv0 = Q yz δw0 = Mx y δϕx = M yy δϕ y = 0

(8.2.19)

and similarly for the two other edges, i.e., .x = 0 and .x = a the boundary conditions are .

N x x δu 0 = N x y δv0 = Q x z δw0 = Mx x δϕx = Mx y δϕ y = 0

(8.2.20)

460

8 Thermal Induced Vibrations

Equations of motion may be obtained in terms of displacements, when Eqs. (8.2.11) are substituted into Eqs. (8.2.17). Upon substitution, the following equations are established

.

A11 u 0,x x + A12 v0,x y + B11 ϕx,x x + B12 ϕ y,x y + A66 (u 0,yy + v0,x y ) + B66 (ϕx,yy + ϕ y,x y ) = I1 u¨ 0 + I2 ϕ¨ x A22 v0,yy + A12 u 0,x y + B22 ϕ y,yy + B12 ϕx,x y + A66 (v0,x x + u 0,x y ) + B66 (ϕ y,x x + ϕx,x y ) = I1 v¨0 + I2 ϕ¨ y

κA44 (ϕ y,y + w0,yy ) + κA55 (ϕx,x + w0,x x ) = I1 w¨ 0 B11 u 0,x x + B12 v0,x y + D11 ϕx,x x + D12 ϕ y,x y + B66 (u 0,yy + v0,x y ) + D66 (ϕx,yy + ϕ y,x y ) − κA55 (ϕx + w0,x ) = I2 u¨ 0 + I3 ϕ¨ x B22 v0,yy + B12 u 0,x y + D22 ϕ y,yy + D12 ϕx,x y + B66 (v0,x x + u 0,x y ) + D66 (ϕ y,x x + ϕx,x y ) − κA44 (ϕ y + w0,y ) = I2 v¨0 + I3 ϕ¨ y (8.2.21) To discrete the above equations along the length and width, the GDQ method is adopted. Distribution of nodal points along the length and width is obtained based on the Chebyshev–Gauss–Lobatto distribution, which reads ( ) a i −1 a − cos π , .xi = 2 2 Nx − 1 ( ) i −1 b b π , yi = − cos 2 2 Ny − 1

i = 1, 2, ..., N x j = 1, 2, ..., N y

(8.2.22)

where . N x and . N y are the number of grid points along, respectively, the .x and . y directions. The discreted equations based on the GDQ method are not given here for the sake of brevity, meanwhile one may refer to the other available works, e.g., [31]. Considering the boundary conditions given in Eqs. (8.2.19) and (8.2.20), two commonly used types of boundary conditions may be defined which are .

• Simply Supported (S): x = 0, a : u 0 = v0 = Mx x = ϕ y = w0 = 0 y = 0, b : u 0 = v0 = M yy = ϕx = w0 = 0 • Clamped (C): x = 0, a : u 0 = v0 = ϕx = ϕ y = w0 = 0 y = 0, b : u 0 = v0 = ϕx = ϕ y = w0 = 0

(8.2.23)

Similar to the equations of motion, the boundary conditions should be discreted by means of the GDQ method, which again are not repeated herein for the sake of

8.2 FGM Rectangular Plates

461

brevity. In a compact form, the equations of motion, after imposing the boundary conditions, may be expressed as ¨ + KX = F MX

(8.2.24)

.

In the above equations, .M is the mass matrix, .K is the stiffness matrix, .F is the force vector, and .X is the unknown displacement vector. It is of worth-noting that, since temperature dependence is taken into consideration in this work, the elements of mass and stiffness matrices and force vector have to be calculated at current temperature. The force vector appearing in Eq. (8.2.24) is induced by the thermal loads. As seen from Eq. (8.2.21), thermal force and moments are absent in the governing equations of motion, whereas thermally induced force and moments are present in the boundary conditions. Equation (8.2.24) is traced in time by means of the well-known Newmark time marching scheme. According to the constant average acceleration method .(α N = 0.5, β N = 0.25) from the Newmark family, Eq. (8.2.24) alternates to ^ j+1 = ^ .KX F j, j+1 (8.2.25) where in the above equation, the following definitions are assumed ^ j+1 = K j+1 + a3 M j+1 K ( ) ˙ j + a5 X ¨j ^ F j, j+1 = F j+1 + M j+1 a3 X j + a4 X

.

(8.2.26)

and the constants appearing in Eq. (8.2.26) are a =

. 3

1 , β N Δt 2

a4 =

1 , β N Δt

a5 =

1 − 2β N 2β N

(8.2.27)

Once the solution .X is known at .t j+1 = ( j + 1)Δt, the first and second derivatives of .X at .t j+1 (or equivalently the velocity and acceleration vectors) can be computed from ( ) ¨ j+1 = a3 X j+1 − X j − a4 X ˙ j − a5 X ¨j X ˙ j+1 = X ¨ j + a1 X ¨ j+1 ˙ j + a2 X X

.

(8.2.28)

where a = α N Δt,

. 1

a2 = (1 − α N )Δt

(8.2.29)

The resulting equations are solved at each time step using the information known ˙ and from the preceding time step solution. At time .t = 0, the initial values of .X, .X, ¨ are known or obtained by solving Eq. (8.2.24) at time .t = 0 and are used to initiate .X the time marching procedure. Since the plate is initially at rest, the initial values of ˙ are assumed to be zero. In other words, the initial conditions to begin the .X and .X time marching are

462

8 Thermal Induced Vibrations

t = 0 : u 0 = u˙ 0 = v0 = v˙0 = w0 = w˙ 0 = ϕx = ϕ˙ x = ϕ y = ϕ˙ y = 0

.

(8.2.30)

8.2.2 Temperature Profile Temporal evolution of temperature through the plate domain is obtained in this section. In an FGM plate, generally the one-dimensional heat conduction equation across the plate thickness is considered, compatible with the desired applications [29]. The transient one-dimensional Fourier type of heat conduction equation in the absence of heat generation takes the form [30] ( .

K (z, T )T,z

) ,z

= ρ(z, T )Cv (z, T )T˙

(8.2.31)

The solution of the above equation requires initial and boundary conditions. Since before heating plate is initially at reference temperature, the initial condition may be written as . T (z, 0) = T0 (8.2.32) Various types of boundary conditions may be assumed on the top and bottom surfaces of the plate. Here, it is assumed that the top surface which is ceramic rich is subjected to sudden temperature or heat flux (rapid heating), whereas the bottom surface which is metal rich is at a lower temperature or thermally insulated. Three different cases of thermal boundary conditions may be defined which are Case 1 : T (+0.5h, t) = Tc (t), Case 2 : T (+0.5h, t) = Tc (t),

.

Case 3 : K (+0.5h, t)T,z (+0.5h, t) = Q c (T ),

T (−0.5h, t) = Tm (t) T,z (−0.5h, t) = 0 T,z (−0.5h, t) = 0 (8.2.33)

Upon solution of Eq. (8.2.31) with considerations of Eqs. (8.2.32) and (8.2.33) temporal evolution of temperature along the plate thickness is achieved. When material properties are temperature independent, the heat conduction equation is linear and may be solved by means of the classical separation of variables method. However, under the temperature-dependent material properties assumption, thermal conductivity is a function of temperature, and heat conduction equation becomes nonlinear. In this case, analytical solution of Eq. (8.2.31) is complicated to find. Therefore, a numerical technique should be implemented to solve this equation. Similar to the equations of motion, the heat conduction equation is solved by means of the GDQ method, where distribution of nodal points across the plate thickness is dictated by h cos .z i = − 2

(

) i −1 π , Nz − 1

i = 1, 2, ..., Nz

(8.2.34)

8.2 FGM Rectangular Plates

463

Upon applying the GDQ method to the heat conduction equation (8.2.31) and imposing the boundary conditions (8.2.33) to the resulting system of equations, the matrix representation of the heat conduction equation may be written as CT (T)T˙ + KT (T)T = F(T)

.

(8.2.35)

The above equation, as expected and mentioned earlier, is nonlinear since the elements of the damping matrix .CT and stiffness matrix .KT are function of nodal temperatures. To solve the above equation at each time step, an iterative scheme should be employed. At each time step, first the thermo-mechanical properties are evaluated at reference temperature and Eq. (8.2.35) is solved to obtain the temperature vector .T. This solution provides the nodal temperatures of the grid points. Thermomechanical properties then have to be evaluated at the obtained nodal temperatures and stiffness and damping matrices are again constructed. Equation (8.2.35) should be solved to obtain the nodal temperatures. This procedure should be repeated until the desired accuracy for the convergence of the temperature at the nodal points is achieved. The profile of temperature is obtained in time domain by means of the wellknown fourth-order Runge–Kutta method. After evaluation of temperature profile, the thermo-mechanical property is obtained for each of the constituents at each time step and each point of the plate. Thermal force and moment resultants and inertia terms are obtained according to Eqs. (8.2.12), (8.2.13), and (8.2.18).

8.2.3 Results and Discussion The procedure outlined in the previous sections is implemented herein to analyze the vibrations induced by rapid surface heating in the FGM rectangular plates. In the numerical results, thermal loadings are assumed to be of uniform magnitude with infinite duration. Furthermore, the midpoint deflection of the plate .w0 (a/2, b/2, t) is denoted by.W . Furthermore, the following convention is used for boundary conditions of the plate. For example, a . ABC D indicates a plate which has . A type of boundary condition at .x = 0, . B type at . y = 0, .C type at .x = a and . D type at . y = b.

Comparison study The problem of thermally induced vibration of rectangular FGM plates is carried out in this research. To assure the validity and accuracy of the present formulation and solution method, two comparison studies are conducted. In the first comparison study, fundamental frequency parameter of a rectangular FGM plate with various boundary conditions is evaluated and compared with the available data in the open literature. Comparison is done in Table 8.1. In this study an FGM plate whose properties are linearly graded across the thickness is considered. Metal and ceramic constituents are Al and .Al2 O3 . Properties of the constituents are

464

8 Thermal Induced Vibrations

Table 8.1 A comparison of fundamental frequency parameter of linearly graded .Al/Al2 O3 square plates with .a/ √h = 10 subjected to various boundary conditions. Frequency parameter is defined as 2 .Ω = ωa / h ρc /E c SSSS SCSC SCSS CCCC . Nx = N y 5 7 9 11 13 15 17 –

.4.3633

.6.1695

.5.1141

.7.5517

.4.4085

.6.2264

.5.1949

.7.6158

.4.4077

.6.2267

.5.1980

.7.6164

.4.4104

.6.2277

.5.2009

.7.6164

.4.4163

.6.2300

.5.2009

.7.6164

.4.4163

.6.2300

.5.2059

.7.6164

.5.2075

.7.6164

.4.4185 .4.4220

.6.2308

a

.6.2222

a

a

.5.2039

.7.6120

b

a . Results b Result

.

of Hosseini-Hashemi et al. [32] of Pradyumna and J.N. Banyopadhyay [33]

E m = 70 GPa, . E c = 380 GPa, .ρm = 2702 kg/m3 , .ρc = 3800 kg/m3 and .νm = νc = 0.3. Geometric characteristics of the plate are .a/ h = b/ h√= 10. Non-dimensional frequency parameter of the plate is defined as .Ω = ωa 2 / h ρc /E c . It is seen that for number of nodal points . N x = N y = 17 the results are in reasonable agreement with the available data in the open literature. Therefore, in the rest of this work 17 nodal points are chosen along both .x and . y directions. In the second comparison study which deals with thermally induced vibrations, comparison is confined to the case of an isotropic homogeneous plate. A square plate with .a = b = 0.254 m and thickness .h = 0.006a is considered. The plate is made from Aluminium with material properties . E = 72.4 GPa, .ν = 0.33, 3 .ρ = 2790 kg/m , . K = 132 W/mK and .C v = 964 J/kgK. Case 3 of thermal loading is considered where the bottom surface is thermally insulated and the top one is subjected to rapid surface heating with magnitude of . Q c = 16.4 × 106 W/m2 . Numerical results of this study are compared with those of Tauchert [9] for both dynamic and quasi-static cases. In quasi-static response, inertia terms are excluded from the equations of motion, and time-dependent temperature profile is inserted into the equilibrium equations. Illustrations are provided in Fig. 8.1. It is seen that results of this study match well with the analytical results of Tauchert [9] which guarantees the validity and accuracy of the solution method. It is of worth-noting that the number of nodal points along the thickness direction is also considered to be . N z = 17 which seems to be accurate enough. Therefore, in the rest of this research number of nodal points through the thickness to solve the heat conduction equation is chosen as . Nz = 17. .

8.2 FGM Rectangular Plates

465

14 12

1 : Quasi − static 2 : Dynamic 2

10

1

W (mm)

8 6

a = b = 0.254 m, h/a = 0.006 E = 72.4 GP a, ν = 0.33 ρ = 2790 kg/m 3 , K = 132 W/mK C v = 964 J/kgK C ase 3 : Qc = 16.4 × 10 6 W/m 2

4 2 0 −2 −4 0

T auchert[16] P resent 0.01

0.02

t(s)

0.03

0.04

0.05

Fig. 8.1 A comparison of quasi-static and dynamic responses of a square Aluminium plate subjected to rapid surface heating. For the sake of comparison, results of Tauchert [9] are extracted from graph

Parametric studies A ceramic–metal functionally graded material plate is considered. The top surface of the plate which is ceramic rich is exposed to thermal shock, whereas the bottom one is kept at the reference temperature or thermally insulated. Silicon Nitride (.Si3 N4 ) is considered as the ceramic constituent and Stainless Steel (SUS304) as the metal constituent. As previously noted, the thermo-mechanical properties of the aforementioned materials are highly temperature dependent where the dependence is described in terms of higher order Touloukian functions (8.2.2). The coefficients . Pi ’s for each of the properties and constituents are given in Table 8.2. In all of the parametric studies, the abbreviate TD considers the temperature-dependent material properties whereas TID case is associated with the conditions where the material properties are evaluated at reference temperature .T0 = 300 K. Unless otherwise stated, a plate with geometrical characteristics .a = b = 100 mm is considered.

Influence of temperature dependence In this section, influence of temperature dependence assumption is examined for a moderately thick FGM plate. A square plate with .h/a = 0.07 is considered which is subjected to rapid surface heating .Tc = 700 K where the bottom surface is kept at reference temperature. Two power law indices are considered. The case .ζ = 0 whose results are provided in Fig. 8.2 and the case of .ζ = 2 whose results are provided in

466

8 Thermal Induced Vibrations

Table 8.2 Temperature-dependent coefficients for SUS304 and .Si3 N4 [15] Material

Property

. P−1

. P0

. P1

. P2

. P3

. SU S304

.α[1/K ]

.0

−6 .201.04e + 9 .15.379 .8166 .0.3262 .496.56

−4 .3.079e − 4 .−1.264e − 3 .0 .−2.002e − 4 .−1.151e − 3

.0

.0

−6 .348.43e + 9 .13.723 .2370 .0.24 .555.11

−4 .−3.07e − 4 .−1.032e − 3 .0 .0 .1.016e − 3

. Si 3 N4

. E[Pa]

.0

. K [W/m K ]

.0

.ρ[kg/m 3 ]

.0

.ν

.0

.C v [J/kg K ]

.0

.α[1/K ]

.0

. E[Pa]

.0

. K [W/m K ]

.0

.ρ[kg/m 3 ]

.0

.ν

.0

.C v [J/kg K ]

.0

.12.33e

.5.8723e

.8.086e

.9.095e

.−6.534e

.0

.2.092e

−7 −6

.−7.223e

.0 .3.797e .1.636e

−7 −6

.0 .2.16e

.0 .−5.863e

− 10

.0

−7 −7

.5.466e

.−8.946e .−7.876e

.0

.0

.0

.0

.2.92e

− 10

.0

−7

.−1.67e

− 11 − 11

− 10

Fig. 8.3. In each case, temporal evolutions of temperature at .z = 0, center point of the plate deflection, thermally induced axial force resultant, and thermally induced bending moment resultant are provided. For each case both TD and TID results are given. It is seen that at a specific time and under TD assumption, temperature is underestimated whereas thermally induced bending moment and axial force and also lateral deflection are overestimated. It is observed that temperature dependence is an important factor to accurate estimation of deflection and temperature. The influence of temperature dependence is more revealed at higher temperature levels and higher times. It is of worth-noting that for the moderately thick studied plates, thermally induced vibrations are hard to recognize. Consequently, one may conclude that for moderately thick and thick plates the quasi-static analysis suffices and inertia terms may be dropped out of the governing equations. In the rest of this work, only TD case of material properties is considered since consideration of temperature dependence, which is the real state of material properties, provides more accurate results.

Influence of power law index As shown in the previous parametric study, for moderately thick plates no specific thermally induced vibrations are observed, as formerly reported by other investigations, see, e.g., [1, 2]. In this example, the previous problem is repeated only for the case of a thin plate. Four various values of power law indices are considered for the plate. In each case, dynamic response, quasi-static response, and the associated static response are provided. Plate thickness is chosen as .h = 1 mm. Plate is subjected to the case 1 of thermal loading where the top surface is subjected to . Tc = 310 K and bottom one is kept at reference temperature. Numerical results are provided in Fig. 8.4. Furthermore, the influence of power law index on the thermally induced vibrations may be discussed in detail since the only variable parameter in this figure is the power law index. It is evident that, unlike the results for the case of a

8.2 FGM Rectangular Plates

467

500 1

475

2

450

T [K]

425 400

1 : T ID 2 : TD

375 350 325

(a)

300 0

0.07

6

2

0.06

N T [N/m] × 10 6

0.05

W/h

2

5

1

0.04 1 : T ID 2 : TD

0.03

3

2.5

2

1.5 t(s)

1

0.5

0.02

1

4 3

1 : T ID 2 : TD

2 1

0.01

(b) 0 0

0.5

1

1.5 t(s)

(c) 0 0

3

2.5

2

0.5

1

7

1.5 t(s)

2

2.5

3

2

6

1

M T [N ] × 10 3

5 4 1 : T ID 2 : TD

3 2 1

(d) 0 0

0.5

1

1.5 t(s)

2

2.5

3

Fig. 8.2 Influences of temperature dependence on characteristics of a square FGM plate with = 7 mm and .ζ = 0 subjected to case 1 of thermal loading with .Tc = 700 K and .Tm = 300 K. a Temporal evolution of temperature at .z = 0. b Temporal evolution of midpoint deflection. c Temporal evolution of thermally induced axial force resultant. d Temporal evolution of thermally induced bending moment resultant

.h

468

8 Thermal Induced Vibrations 500 475 1 2

1 : T ID 2 : TD

450

T [K]

425 400 375 350 325

(a)

300 0

0.10

6

2

0.09 0.08

2 5

1 N T [N/m] × 10 6

0.07

W/h

0.06 0.05

1 : T ID 2 : TD

0.04 0.03 0.02

1

4 3

1 : T ID 2 : TD

2 1

0.01 0 0

3

2.5

2

1.5 t(s)

1

0.5

(b) 0.5

1

1.5 t(s)

(c) 0 0

3

2.5

2

0.5

1

1.5 t(s)

2

2.5

3

8 2

7 1

M T [N ] × 10 3

6 5 4

1 : T ID 2 : TD

3 2 1 0 0

(d) 0.5

1

1.5 t(s)

2

2.5

3

Fig. 8.3 Influences of temperature dependence on characteristics of a square SSSS FGM plate with.h = 7 mm and.ζ = 2 subjected to case 1 of thermal loading with.Tc = 700 K and.Tm = 300 K. a Temporal evolution of temperature at .z = 0. b Temporal evolution of midpoint deflection. c Temporal evolution of thermally induced axial force resultant. d Temporal evolution of thermally induced bending moment resultant

8.2 FGM Rectangular Plates

469

thick plate in previous example, for thin plates thermally induced vibrations indeed exist. In each figure, as seen, the dynamic response and quasi-static response are thoroughly different. Dynamic response fluctuates around the quasi-static response. It is evident that as the time passes and temperature becomes approximately steady state, the quasi-static response and static response become united. A comparison of the numerical results for various values of the power index reveals that as the power law index decreases (the volume fraction of Si3N4 increases), the magnitude of thermally induced vibration diminishes. Furthermore, as the power law index increases, natural period of the SUS304/Si3N4 FGM plate increases.

Influence of in-plane boundary conditions In this example, the influence of in-plane boundary conditions is depicted. It should be point-out that, depending on the type of in-plane edge supports, four different types of a boundary condition (for example four different types of simply supported edge) may be defined. In boundary conditions (8.2.23) for simply supported edge, the in-plane motion is restrained at the edges. This type of boundary conditions may be called immovable (IM). On the other hand, when .u 0 = 0 is replaced by . N x x = 0 on .x = 0, a and similarly .v0 = 0 is replaced by . N yy = 0 on . y = 0, b, free to move (FM) type of simply supported edge is established. In this example, the influences of in-plane edge supports are examined. Geometrical characteristics of the plate are the same as those used in the previous example and similar thermal loading is used. Power law index is chosen as .ζ = 2. Numerical results are provided in Fig. 8.5. As seen, the magnitude of thermally induced vibration in an IM plate is more than an FG plate which is expected since in IM plate the induced compressive axial forces at the boundaries enhance the thermally induces vibrations. However, the influence of inplane boundary conditions on natural period of the plate is approximately negligible.

Influence of type of thermal loading In this example, a moderately thick square plate with.a/ h = 20 is considered. Plate is subjected to the case 2 of thermal loading where the intensity of thermal shock on its upper surface is considered as .Tc = 600 K. Responses of the plate for various values of power law index are provided in Fig. 8.6. A comparison on the results of this figure and those provided in Fig. 8.2 reveals that types of thermal boundary conditions are important on dynamic behavior of the plate. In this case of thermal loading (Case 2) where one surface is experiencing thermal shock and the other one is thermally insulated, plates reach a maximum deflection during heating. When thermal shock is applied to the top surface, plates deflect upward since upper surfaces inherent more thermal elongations. As the time passes, temperature of bottom surfaces increases and deflection decreases. For the case of an isotropic homogeneous plate .(ζ = 0), when temperature becomes steady, deflection through the plate is identically zero since the thermally induced bending moment vanishes after the establishment of a

470 0.08 0.07 0.06

W/h

0.05 0.04

Dynamic Quasi − static Static

0.03 0.02 0.01 0 0

(a) 0.005

0.01

t(s)

0.015

0.02

0.025

0.11 0.10 0.09 0.08

W/h

0.07 0.06 Dynamic Quasi − static Static

0.05 0.04 0.03 0.02 0.01 0 0

(b) 0.005

0.01

t(s)

0.015

0.02

0.025

0.12 0.11 0.10 0.09 0.08 W/h

0.07 0.06

Dynamic Quasi − static Static

0.05 0.04 0.03 0.02

(c)

0.01 0 0

0.005

0.01

t(s)

0.015

0.02

0.025

0.13 0.12 0.11 0.10 0.09 0.08 W/h

Fig. 8.4 Influences of the power law index on thermally induced vibrations of a square SSSS FGM plate with .h = 1 mm subjected to case 1 of thermal loading with .Tc = 310 K and . Tm = 300 K. a .ζ = 0. b .ζ = 1. c .ζ = 2. d .ζ = 5

8 Thermal Induced Vibrations

0.07 Dynamic Quasi − static Static

0.06 0.05 0.04 0.03 0.02

(d)

0.01 0 0

0.005

0.01

t(s)

0.015

0.02

0.025

8.2 FGM Rectangular Plates 0.12 0.11 0.10 0.09 0.08 0.07 W/h

Fig. 8.5 influence of in-plane boundary conditions on the temporal evolution of plate deflection. A square SSSS FGM plate with .h = 1 mm subjected to the case 1 of thermal loading with .Tc = 310 K, . Tm = 300 K and .ζ = 2 is considered

471

0.06 0.05 0.04 0.03

FM IM

0.02 0.01 0 0

0.01

0.02 t(s)

0.03

0.04

steady-state temperature profile. On the other hand, for heterogeneous plates.(ζ > 0), even after establishment of the steady-state temperature profile, the FGM plates have a nonzero dynamic deflection. In the FGM plates, various surfaces of the plate have different coefficient of thermal expansion, elasticity modulus, and Poisson’s ratio. Therefore, at the steady-state condition when the temperature profile is unified at different surfaces, nonzero thermal moments are generated. The induced thermal moments result in nonzero deflection which is also observed in Fig. 8.6. Note that, when temperature profile is steady, the FGM plate bends downward since the thermal expansion of metal constituent is more than that of ceramic and bottom surfaces experience more thermal elongations.

Influence of out-of-plane boundary conditions A comparison is made in this section to analyze the temporal evolution of FGM plates with SSSS, SCSS, SCSC, and CCCC edge supports. A thin plate .(a/ h = 100) subjected to the case 1 of thermal loading with thermal shock intensity .Tc = 320 K is considered. Midpoint lateral deflection for each case of boundary conditions is given in Fig. 8.7. As seen from this figure, a plate which is clamped all around experiences no lateral deflection when is subjected to rapid surface heating. In a clamped plate, the induced thermal moments are suppressed by the clamping condition of the boundaries, and lateral deflection does not take place. It is of worth-noting that such behavior is valid only under geometrically linear assumption which is formerly reported by Kiani and Eslami [15]. Under geometrically nonlinear formulation, plate may become unstable as shown by Kiani and Eslami [15] and Ghiasian et al. [17]. Among the other cases of boundary conditions, it is seen that as the boundaries become stiffer (a simply supported edge changes to a clamped edge) lateral dynamic deflection and natural period both decrease. This is expected since by clamping an edge, the local stiffness of the plate near the boundary increases.

472

8 Thermal Induced Vibrations 0.14

600

0.12

1

550

4

0.08

2

3

0.06

W/h

500 T [K]

4 3 2

0.1

450

1

0.04 0.02

400

1 2 3 4

350 300 0

1

2

:ζ :ζ :ζ :ζ

3

= = = = 4

0

0 1 2 5

−0.04

(a) 5 t(s)

6

7

8

9

−0.06 0

10

4

7 6

2

= = = =

3

0 1 2 5

(b)

4

5 t(s)

6

7

8

9

10

3

2.5

2

2

1

1.5

5

M T [N ] × 10 3

N T [N/m] × 10 6

1

:ζ :ζ :ζ :ζ

3

8

4

1

0.5

3 1 2 3 4

2 1 0 0

1 2 3 4

−0.02

1

2

:ζ :ζ :ζ :ζ 3

= = = =

0 1 2 5 4

1 2 3 4

0 −0.5

(c) 5 t(s)

6

7

8

9

−1 0

10

1

2

:ζ :ζ :ζ :ζ

= = = =

3

0 1 2 5

2 5 t(s)

4

6

7

3

1 4

(d)

8

9

10

Fig. 8.6 Influences of power law index and type of thermal loading on thermally induced vibrations of a square SSSS FGM plate with.h = 5 mm subjected to case 2 of thermal loading with.Tc = 600 K. a Temporal evolution of temperature at .z = 0. b Temporal evolution of midpoint deflection. c Temporal evolution of thermally induced axial force resultant. d Temporal evolution of thermally induced bending moment resultant 0.25

SSSS

0.2

0.15 W/h

Fig. 8.7 Influence of out-of-plane boundary conditions on the temporal evolution of plate deflection. A square FGM plate with .h = 1 mm and .ζ = 0.5 subjected to case 1 of thermal loading with . Tc = 320 K, . Tm = 300 K is considered

SC SS

0.1

SC SC

0.05

0 0

CCCC 0.01

0.02

t(s)

0.03

0.04

0.05

8.3 FGM Conical Shells

473

It may be concluded that: .• Thermally induced vibrations exist for thin plates. As the plates become thicker, thermally induced vibrations disappear and the dynamic and quasi-static responses become similar. .• Temperature dependence is an important factor in accurate estimation of plate response, especially for severe thermal shocks. Neglecting the temperature dependence assumption results in an underestimation of the plate deflection. .• In-plane type of boundary conditions is influential on thermally induced vibration characteristics of the plate. Plates with free to move edge support experience less lateral deflection in comparison with the plates with immovable edge supports. .• Power law index changes the graded profile of the plate and affects the characteristics of the plate. For a SUS304/Si3N4 plate, as the power law index increases (or the metal volume fraction increases), the amplitude of static, dynamic, and quasi-static responses increases, whereas natural period of the plate decreases. .• Under geometrically linear assumption, a fully clamped plate does not experience any lateral deflection. For other combinations of edge supports, as an edge becomes stiffer, thermally induced dynamic lateral deflection and natural period of the plate decrease.

8.3 FGM Conical Shells An analytical solution based on the Green function method is proposed to investigate the thermally induced vibration of beams under concentrated heat source by KidawaKukla [34]. The influences of damping and axial compressive forces on the dynamic behavior of beams are also investigated in this research. Adam et al. [35] examined the thermally excited vibrations of composite viscoelastic beams with consideration of interfacial slip. The solution of the associated equations of motion is acquired as the sum of quasi-static and complementary dynamic responses. Stability of thermally induced vibration of a beam subjected to solar heating is examined by Zhang et al. [36]. They proposed a criterion model which indicates that the thermal flutter may occur if the incidence angle of solar heat flux is less than the bending angle of the beam free end at its quasi-static equilibrium position. Displacement control of rectangular and circular plates under rapid heating using piezo-thermoelastic layers is demonstrated by Tauchert et al. [37]. Hong et al. [38] investigated the thermally induced response of rapidly heated laminated tubes. The computational GDQ method adopted in this research provides the natural frequencies, displacements, and thermal stresses. Piezo-hygro-thermoelastic composite laminated plates and shells are modeled by Raja et al. [39]. Thermally induced vibration control is attempted using piezoelectrically developed active damping. Kumar et al. [40] proposed a finite element model of piezolaminated structure under electrical, mechanical, and thermal loads. Modeling is based on the shear deformation and linear thermo-piezo-electric theory. Based on a higher order layerwise

474

8 Thermal Induced Vibrations

theory, thermally induced vibration of the FGM sandwich plates and shell panels is investigated by Pandey and Pradyumna [28]. In this research, thermo-mechanical properties are assumed to be temperature dependent. For the first time, axisymmetric thermally induced vibrations are investigated for the FGM conical shells. Based on the first-order shear deformation theory which is efficient for moderately thick shells and the von Kármán type of geometrical nonlinearity which is applicable for small strain, moderately rotation, and large deflection response, rapidly heated conical shells are analyzed. Because of temperature dependence and geometrically nonlinear theory, heat conduction and equations of motion are obtained as nonlinear governing equations which should be solved using numerical methods. A hybrid GDQ-Crank–Nicolson–Picard method is applied to obtain the temperature profile across the thickness. The GDQ-Newmark–Newton–Raphson method is used to capture the dynamic response of the shell. It is illustrated that temperature dependence, in-plane boundary conditions, type of thermal boundary conditions, cone semi-vertex angle, shell length, and thickness of the shell affect the temporal evolution characteristics of the conical shell.

8.3.1 Fundamental Equations of the FG Conical Shell Consider a conical shell of finite length. L, thickness.h, and semi-vertex angle.β0 made of functionally graded materials where properties are distributed in . Z direction [41]. The curvilinear coordinates system is defined as .(X, θ, Z ), where . X and .θ indicate the generator and circumferential direction, respectively, . Z is measured across the thickness, positive outward, and variable . X is measured from the apex of the cone, as shown in Fig. 8.8. Due to the axisymmetric thermal loads and boundary conditions, displacement components in .θ direction are vanished. The Voigt rule of mixtures is generally postulated to capture the equivalent thermo-mechanical properties of the FGM shells because of its simplicity and productivity [42–44]. According to this rule, the mechanical and thermal properties of the conical shells, such as Young’s modulus . E, Poisson’s ratio .ν, thermal expansion coefficient .α, mass density .ρ, specific heat .C v , and thermal conductivity . K are expressed as the linear functions of the volume fractions of the ceramic.Vc and metal.Vm constituents. Therefore, a non-homogeneous property of the conical shell . P is expressed as a function of thickness direction and may be written in the form

.

P(Z , T ) = Pm (T ) + Vc (Z )Pcm (T ), Pcm (T ) = Pc (T ) − Pm (T )

(8.3.1)

where the subscripts .m and .c represent the virtues of metal and ceramic constituents, respectively. The Touloukian model for temperature dependence of the material properties is used to estimate the properties of the cone as [45] .

P(T ) = P0 (P−1 T −1 + 1 + P1 T + P2 T 2 + P3 T 3 )

(8.3.2)

8.3 FGM Conical Shells

475

Fig. 8.8 Schematic and geometric characteristic of a conical shell

β0

R

r x R

where .T is the temperature measured in Kelvin and . Pi ’s are constants which are unique to each of the constituents. Following Reddy and Chin [45], a power law distribution of the constituents through the shell thickness may be used to represent the ceramic volume fraction .Vc and metal volume fraction .Vm such that ( .

Vc =

Z 1 + 2 h

)ζ

, Vm = 1 − Vc

(8.3.3)

Here, .ζ is a positive constant called the power law index and illustrates the property distribution profile. Obviously, ceramic rich surface is the outer surface of the shell (. Z = +h/2) and metal rich surface is the inner surface of the shell (. Z = −h/2).

8.3.2 Kinematic Assumptions Thermally induced vibrations of the shells are assumed only in the . X − Z plane due to axisymmetric loads and boundary conditions. Also, displacement field is postulated based on the first-order shear deformation theory (FSDT) consistent with the Timoshenko assumptions. This theory is suitable for moderately thick shells and its components may be written in terms of the characteristics of midplane of the shell and cross-sectional rotations as u¯ ∗ (X, Z , t) = u ∗ (X, t) + Z ϕ∗ (X, t) w¯ ∗ (X, Z , t) = w ∗ (X, t)

.

(8.3.4)

476

8 Thermal Induced Vibrations

In the above equation .u ∗ and .w ∗ represent the displacements at the mid-surface along the . X − and . Z −directions, respectively. Besides, transverse normal rotation about the .θ axis is denoted by .ϕ∗ . The von Kármán type of geometrical nonlinearity, compatible with the small strains, moderate rotations, and large displacements in the curvilinear coordinates take the form 1 2 = u ,X + w,X 2 w u + εθθ = X X tan(β0 ) γ X Z = u ,Z + w,X ε

. XX

(8.3.5)

In order to easily formulate and validate with other investigations, the displacement field and strain–displacement relations are obtained using the definition of the changing variable .r (x) = R0 + x sin(β0 ) and based on the transfer coordinates R0 as follow system . X = x + sin(β 0) u(x, z, t) = u 0 (x, t) + zϕ(x, t)

.

w(x, z, t) = w0 (x, t)

(8.3.6)

and 1 2 = u ,x + w,x 2 cos(β0 ) sin(β0 ) u+ w εθθ = r (x) r (x) γx z = u ,z + w,x ε

. xx

(8.3.7)

where .εx x and .εθθ represent the longitudinal normal and circumferential normal strains and .γx z denotes the shear strain component. Here and in the rest, a comma indicates the partial derivative with respect to its afterward.

8.3.3 Stress Resultants The stress–strain relations are considered in the context of linear thermoelasticity. Thus the constitutive law for the FGM conical shell subjected to thermal loadings is

8.3 FGM Conical Shells

477

⎧ ⎫ ⎡ ⎫ ⎧ ⎫⎞ ⎤ ⎛⎧ Q 11 Q 12 0 ⎨ εx x ⎬ ⎨ σx x ⎬ ⎨α⎬ σθθ = ⎣ Q 12 Q 22 0 ⎦ ⎝ εθθ − ΔT α ⎠ . ⎩ ⎩ ⎭ ⎭ ⎩ ⎭ 0 τx z γx z 0 0 Q 55

(8.3.8)

where .ΔT demonstrates the temperature difference and the other constants are .

Q 11 = Q 22 =

E(z, T ) , 1 − ν 2 (z, T )

Q 12 =

ν(z, T )E(z, T ) , 1 − ν 2 (z, T )

Q 55 =

E(z, T ) 2(1 + ν(z, T )) (8.3.9)

Based on the FSDT, the components of stress resultants are acquired using the components of the stress field through the thickness as { (N x x , Nθθ , Mx x , Mθθ , Q x z ) =

+0.5h

.

−0.5h

(σx x , σθθ , zσx x , zσθθ , τx z )dz

(8.3.10)

Substituting Eq. (8.3.8) into Eq. (8.3.10) with the aid of Eqs. (8.3.6) and (8.3.7), the stress resultants in terms of the midplane displacements are obtained as ⎧ ⎫ 1 2 ⎪ ⎪ ⎪ ⎪ u w + ,x ⎧ ⎪ ⎫ ⎧ ⎫ ⎡ ⎪ ⎤⎪ ,x ⎪ 2 ⎪ ⎪ ⎪ NT ⎪ A N A B B 0 ⎪ ⎪ ⎪ x x 11 12 11 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ cos(β0 ) ⎪ sin(β0 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T⎪ ⎪ ⎪ ⎥⎪ ⎨ Nθθ ⎪ ⎬ ⎢ ⎢ A12 A22 B12 B22 0 ⎥ ⎨ r (x) u + r (x) w ⎬ ⎨ N T ⎬ ⎢ ⎥ Mx x = ⎢ B11 B12 D11 D12 0 ⎥ − M . ϕ,x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎪ ⎪ ⎪ Mθθ ⎪ MT ⎪ B12 B22 D12 D22 0 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sin(β ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎩ ⎪ ⎪ ⎭ ⎩ ⎭ ⎪ ϕ Qxz 0 0 0 0 0 A55 ⎪ ⎪ ⎪ ⎪ ⎪ r (x) ⎪ ⎪ ⎩ ⎭ w,x + ϕ (8.3.11) In the above equations, the constant coefficients . Ai j , . Bi j , and . Di j display the stretching, bending-stretching, and bending stiffness, respectively, which are calculated by { (Ai j , Bi j , Di j ) =

+0.5h

.

−0.5h

(Q i j , z Q i j , z 2 Q i j )dz,

i, j = 1, 2, 5

(8.3.12)

Besides, . N T and . M T are the in-plane thermal force and thermal moment resultants which are given by { (N , M ) =

.

T

T

+0.5h

−0.5h

(1, z)

1 E(z, T )α(z, T )(T − T0 )dz 1 − ν(z, T )

(8.3.13)

478

8 Thermal Induced Vibrations

8.3.4 Equations of Motion Axisymmetric equations of motion of FGM conical shells based on the uncoupled thermoelasticity assumptions may be obtained by applying the principle of virtual displacements as { t2 . (δT − δV − δU )dt = 0 (8.3.14) t1

Using definition for the cone length as . L = the conical shell .δU can be written as { δU =

L

{

+0.5h

.

0

−0.5h

R1 −R0 , sin(β0 )

the total virtual strain energy of

(σx x δεx x + σθθ δεθθ + τx z δγx z ) 2πr (x)dzd x

(8.3.15)

Here, .δV is the virtual potential energy of the external applied loads which is absent in this research. Also, the kinetic energy .δT is obtained by {

L

δT =

{

+0.5h

ρ(z, T ) (uδ ˙ u˙ + wδ ˙ w) ˙ 2πr (x)dzd x =

.

0

{

L

−

−0.5h

{(I1 u¨0 + I2 ϕ) ¨ δu 0 + I1 w¨0 δw0 + (I2 u¨0 + I3 ϕ) ¨ δϕ} 2πr (x)d x

(8.3.16)

0

where a .( ˙ ) illustrates a derivative with respect to time and the inertia terms . I1 , I2 , and . I3 are defined by { (I1 , I2 , I3 ) =

+0.5h

.

−0.5h

ρ(z, T )(1, z, z 2 )dz

(8.3.17)

Substituting Eqs. (8.3.15), (8.3.16) into Eq. (8.3.14) and by means of the suitable mathematical simplifications, the expressions for the equations of motion for the FGM conical shell take the form

δu

:

δw

:

δϕ

:

.

sin(β0 ) (N x x − Nθθ ) = I1 u¨0 + I2 ϕ¨ r (x) ) 1 ( sin(β0 ) cos(β0 ) r (x) N x x w0,x ,x + Q x z,x + Qxz − Nθθ = I1 w¨0 r (x) r (x) r (x) sin(β0 ) Mx x,x + (8.3.18) (Mx x − Mθθ ) − Q x z = I2 u¨0 + I3 ϕ¨ r (x)

N x x,x +

8.3 FGM Conical Shells

479

The equations of motion in terms of the displacement components for an axisymmetric FGM conical shell may be obtained using Eqs. (8.3.11) and (8.3.18). The resulting equations can be written as

. A11 (u 0,x x

+

B11 (ϕ,x x +

sin(β0 ) sin2 (β0 ) sin(β0 ) cos(β0 ) sin(β0 ) 2 u 0,x − 2 u0 − w0 + w0,x w0,x x + w ) r (x) r (x) r 2 (x) 2r (x) 0,x sin(β0 ) sin2 (β0 ) cos(β0 ) sin(β0 ) 2 ¨ ϕ,x − 2 ϕ) + A12 ( w0,x − w ) = I1 u¨ 0 + I2 ϕ r (x) r (x) r (x) 2r (x) 0,x

cos(β0 ) sin(β0 ) cos2 (β0 ) sin(β0 ) 3 2 u0 − w0 + u 0,x x w0,x + u 0,x w0,x x + u 0,x w0,x + w0,x x w0,x r 2 (x) r 2 (x) r (x) 2 sin(β0 ) 3 cos(β0 ) sin(β0 ) sin(β0 ) w ) + B11 (− ϕ + ϕ,x x w0,x + ϕ,x w0,x x + ϕ,x w0,x ) 2r (x) 0,x r 2 (x) r (x) A12 cos(β0 ) 2 (− cos(β0 )u 0,x + sin(β0 )u 0,x w0,x + sin(β0 )u 0 w0,x x + cos(β0 )w0,x x w0 + w0,x ) r (x) 2 sin(β0 ) B12 (− cos(β0 )ϕ,x + sin(β0 )ϕ,x w0,x + sin(β0 )ϕw0,x x ) + A55 (ϕ,x + ϕ + w0,x x r (x) r (x) sin(β0 ) sin(β0 ) cos(β0 ) T w0,x ) − N T (w0,x x + w0,x ) + N = I1 w¨ 0 r (x) r (x) r (x)

A11 (− + + + +

B11 (u 0,x x +

sin(β0 ) sin2 (β0 ) sin(β0 ) cos(β0 ) sin(β0 ) 2 u 0,x − 2 u0 − w0 + w0,x w0,x x + w ) r (x) r (x) r 2 (x) 2r (x) 0,x

+ D11 (ϕ,x x +

sin(β0 ) sin2 (β0 ) cos(β0 ) sin(β0 ) 2 ϕ,x − 2 ϕ) + B12 ( w0,x − w ) r (x) r (x) r (x) 2r (x) 0,x

− A55 (ϕ + w0,x ) = I2 u¨ 0 + I3 ϕ ¨

(8.3.19)

The complete set of the boundary conditions is extracted using the procedure of virtual displacement relieving. For .x = 0 and .x = L, the boundary conditions are given by

.

N x x δu = Mx x δϕ = (Q x z + N x x w,x )δw = 0

(8.3.20)

To discrete the above equations along the cone domain, the GDQ method is used. Distribution of nodal points is proposed by means of the Chebyshev–Gauss–Lobatto distribution which reads ( x =L

. i

( )) (i − 1)π 1 1 − cos , 2 2 Nx − 1

i = 1, 2, . . . , N x

(8.3.21)

where . N x is the number of grid points along the cone length. Utilizing the boundary conditions given in Eq. (8.3.20), several set of boundary conditions suitable for ther-

480

8 Thermal Induced Vibrations

mally induced deflection analysis are obtained. Consequently, edges of the conical shell may take one of the following boundary conditions IM-S : u = w = Mx x = 0 M-S : N x x = w = Mx x = 0 IM-C : u = w = ϕ = 0

.

M-C

:

Nx x = w = ϕ = 0

(8.3.22)

For a solid conical shell, the point .x = 0 is in the shell domain. Hence, the boundary condition at . R0 = 0 is described according to the following equation u = Q x z + N x x w,x = ϕ = 0

.

(8.3.23)

In Eq. (8.3.22) IM-S, M-S, IM-C, and M-C are associated with the immovable simply support, movable simply support, immovable clamped, and movable clamped boundary conditions, respectively. Similar to the equations of motion, discretization of the boundary conditions should be performed using the GDQ method. discretized equations of motion and boundary conditions into nodal points in conical shell domain and edges support are given in the Appendix. Finally, the discretized equations (8.3.19) based on the generalized differential quadrature method, the equations of motion take the following matrix form .

{ } ¨ + [K(T,X)] {X} = {F(T)} [M(T)] X

(8.3.24)

It is worth to note that due to the consideration of the von Kármán type of geometrical nonlinearity, the generalized stiffness matrix is a function of unknown timedependent nodal vector .{X}. Here, the Newmark direct integration scheme based on the constant average acceleration method .(α = 0.5, β = 0.25) is applied to approximate the system of Eq. (8.3.24) in terms of the modified stiffness matrix and force vector. Applying the Newmark method to Eq. (8.3.24) yields .

[ ] { } ^ {X} j+1 = ^ K(T,X) F(T) j, j+1

(8.3.25)

where the modified stiffness matrix and force vector take the form

.

and

[ ] ^ K(T,X) = [K(T,X)] + a0 [M(T)] ( { } { } { } ) ˙ + a2 X ¨ ^ F(T) = {F(T)} j+1 + [M(T)] a0 {X} j + a1 X j j

(8.3.26)

8.3 FGM Conical Shells

a =

. 0

481

1 , βΔt 2

a1 =

1 , βΔt

a2 =

1 − 2β 2β

(8.3.27)

Once the solution .{X} is known at .t j+1 = ( j + 1)Δt, the velocity and acceleration vectors at .t j+1 can be obtain from

.

{ } ( ) { } { } ¨ ˙ − a2 X ¨ X = a0 {X} j+1 − {X} j − a1 X j+1 j j { } { } { } { } ˙ ˙ ¨ ¨ X j+1 = X j + a3 X j + a4 X j+1

(8.3.28)

and a = (1 − α)Δt,

. 3

a4 = αΔt

(8.3.29)

The resulting equations are solved at each time step using the information known { } ˙ , from the preceding time step solution. At time .t = 0, the initial values of .{X}, . X { } ¨ are known or obtained by solving Eq. (8.3.24) at time .t = 0 and are utilized and . X to initiate the time { }marching procedure. Since the shell is initially at rest, the initial ˙ are considered to be zero. An iterative scheme should be applied values .{X} and . X to Eq. (8.3.21) to solve the resulting highly nonlinear algebraic equations. In this study, the well-known Newton–Raphson iterative method is used in which the tangent stiffness matrix is evaluated based on the developed method in [46].

8.3.5 Temperature Profile This section deals with the temporal evolution of temperature profile for conical shells exposed to rapid heating in the thickness direction. It is assumed that the temperature profile varies only through the thickness due to the assumption that . z/R ≪ 1. Consequently, in the absence of heat generation, one-dimensional transient heat conduction equation takes the form [30] ( .

K (z, T )T,z

) ,z

= ρ(z, T )Cv (z, T )T˙

(8.3.30)

Because the shell is at the reference temperature before the loading, the initial condition is obtained as . T (z, 0) = T0 (8.3.31) To solve the heat conduction Eq. (8.3.30), several types of thermal boundary conditions may be considered on the top and bottom surfaces of the FGM shell. Here, it is assumed that the top surface of the shell, which is ceramic rich, is exposed to a time-dependent sudden temperature (rapid heating), whereas the opposite surface which is metal rich may undergo thermally insulated boundary condition or the tem-

482

8 Thermal Induced Vibrations

perature specified time-dependent boundary condition (rapid heating). Two different types of thermal boundary conditions can be described which are Case 1 : T (+0.5h, t) = Tc (t),

T (−0.5h, t) = Tm (t)

Case 2 : T (+0.5h, t) = Tc (t),

T,z (−0.5h, t) = 0

.

(8.3.32)

Upon solution of Eq. (8.3.30) with consideration of Eqs. (8.3.31) and (8.3.32), temperature profile along the shell thickness is attained. Thermal conductivity is a function of temperature due to temperature-dependent material properties assumption. Therefore, the heat conduction equation becomes nonlinear. The heat conduction equation is solved by means of the GDQ method, analogous to the equations of motion. According to the GDQ method, distribution of nodal points across the shell thickness can be written as ( ) (i − 1)π h cos , i = 1, 2, . . . , Nz .z i = − (8.3.33) 2 Nz − 1 Utilizing the GDQ method for the heat conduction (8.3.30) in space domain (thickness) and imposing the boundary conditions Eq. (8.3.32) to the resulting system of ordinary differential equations, the matrix form of the heat conduction equation becomes { } ˙ + [KT (T)] {T} = {FT (T)} . [CT (T)] T (8.3.34) Since the material properties are temperature dependent, in Eq. (8.3.34) damping matrix .[CT (T)], stiffness matrix .[KT (T)], are functions of the nodal temperatures. Hence, based on the Crank–Nicolson time marching scheme, at each time step, an iterative procedure should be applied to obtain the temperature profile of the conical shell under the assumption of temperature-dependent thermo-mechanical properties. In this section, the Picard method is utilized. At each time step, thermal properties are evaluated at reference temperature .T0 . Material properties are then evaluated at captured nodal temperatures.{T} and Eq. (8.3.34) is solved repeatedly. This procedure is repeated until the temperature profile converges at the current time step. The Crank– Nicolson procedure is adopted to solve Eq. (8.3.34) along with the initial conditions (8.3.31). Details on the procedure of the Crank–Nicolson method are available in [46].

8.3.6 Results and Discussion The developed solution method and formulation in the previous sections may be used to analyze the thermally induced vibration phenomenon in conical shells made of FGMs subjected to rapid surface heating. In this section, the FGM shell is assumed to be made of SUS304 as metal phase and Si.3 N.4 as ceramic phase. All the thermomechanical properties of the assumed constituents are highly temperature dependent,

8.3 FGM Conical Shells

483

Table 8.3 Temperature-dependent coefficients for . SU S304 and . Si 3 N4 [45] Material

Property

. P−1

. P0

. SU S304

.α[1/K ]

.0

.12.33e

. Si 3 N4

. P1

−6 +9

.8.086e

−4

.3.079e

−4

. E[Pa]

.0

.201.04e

. K [W/m K ]

.0

.15.379

.−1.264e

.ρ[kg/m 3 ]

.0

.8166

.0

−3

. P2

. P3

.0

.0

.−6.534e .2.092e

−7

−6

.0

.0 .−7.223e

.ν

.0

.0.3262

.−2.002e

−4

.3.797e

−7

.0

.C v [J/kg K ]

.0

.496.56

.−1.151e

−3

.1.636e

−6

.−5.863e

.α[1/K ]

.0

.5.8723e

−6 +9

. E[Pa]

.0

.348.43e

. K [W/m K ]

.0

.13.723

.9.095e

−4

.−3.07e

−4

.−1.032e

−3

.0 .2.16e

−7

.5.466e

−7

.−8.946e

− 11

.−7.876e

− 11

.ρ[kg/m 3 ]

.0

.2370

.0

.0

.0

.0

.0.24

.0

.0

.0

.C v [J/kg K ]

.0

.555.11

.1.016e

.2.92e

− 10

.0

.ν

−3

− 10

.0

−7

.−1.67e

− 10

where the dependence is described by the Touloukian expression (8.3.2). The coefficients . Pi for the constituents are provided in Table 8.3. In this section, at first a comparison study is presented to show the validity and accuracy of the developed formulation. Afterward, novel numerical results are developed for the conical shells made of FGMs. In this section, reference temperature is set as .T0 = 300 K. As a convention, CS indicates a shell which is clamped at .x = 0 and simply supported at . x = L.

8.3.7 Comparison Study In this section, a comparison study is performed for the case of solid circular plates made of FGMs subjected to rapid surface heating. The presented formulation may be reduced easily to a circular plate when the inner radius of the cone is set equal to ◦ . R1 = 0 and the semi-vertex angle of the cone is set equal to .β0 = 90 . The boundary conditions for the inner of the circular plate are chosen according to Eq. (8.3.23) while the boundary condition on the outer edge of the circular plate are simply supported. The comparison study is performed in Fig. 8.9. Comparison is performed between the results of this study and those given by Kiani and Eslami [15], which are obtained using the Ritz method with simple polynomial functions. The ceramic rich surface is subjected to sudden 10K temperature rise where the other surface is kept at reference temperature. The power law index is set equal to .ζ = 5 and geometric characteristics are .h = 1 mm, . R0 = 0 mm, and . R1 = 80 mm. It is seen that our results match well with those of Kiani and Eslami [15], which shows the accuracy and correctness of the developed formulations and solution method.

484

8 Thermal Induced Vibrations

0.8 0.6 0.4 0.2 0

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Fig. 8.9 Comparison of thermally induced vibrations in a solid circular plate with immovable simply supported boundary conditions. Center point deflection of the circular plate is denoted by.W

8.3.8 Parametric Studies After validating the proposed formulation and solution method, novel numerical results are given in this section. In this section, .W indicates the deflection at the point .x = L/2.

Influence of temperature dependence For the first study in this section, an analysis is performed to show the importance of temperature dependence. Results of this study are depicted in Figs. 8.10 and 8.11. In these figures, the FGM shells with power law indices .ζ = 0 and .ζ = 2 are analyzed, respectively. The conical shells with immovable simply supported supports are considered. Geometric characteristics of the shell in these figures are .β0 = 45◦ , .h = 5 mm, . R1 = 350 mm, and . R0 = 250 mm. The metal rich surface is kept at reference temperature while the ceramic rich surface is subjected to .Tc = 900 K. Both temperature-dependent material properties (denoted by TD) and temperatureindependent material properties (denoted by TID) are considered. In the TID case, material properties are evaluated at reference temperature .T0 = 300 K. In both of these figures it is demonstrated that under temperature-dependent material properties, which is the real state of materials, temperatures within the shell are underestimated. Also, deflection induced by heating are overestimated. This is expected, since under temperature-dependent material properties assumption the elasticity modulus decreases as the temperature elevates resulting in a reduction in material stiffness and higher lateral deflection. Both thermally induced force and moment increase when the temperature-dependent material properties are considered. As seen from the results of Figs. 8.10 and 8.11, thermally induced bending moment becomes steady much faster than the thermally induced force. As the time evolutes, deflections within the shell tend to the static response of the shell which is expected since temperature becomes steady state within the shell body. It is seen that for both of the cases in Figs. 8.10 and 8.11, no specific vibrations are detected. This is to the fact that shell

8.3 FGM Conical Shells

485

650 600 550 500 450 400 350 300

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

106

6 5 4 3 2 1 0

6000 5000 4000 3000 2000 1000 0

Fig. 8.10 Effect of temperature dependence on the thermally induced vibration of conical shells with.ζ = 0. Other characteristic of the shell are. R0 .= 250 mm,. R1 .= 350 mm,.β0 = 45◦ ,.h = 5 mm, . Tc = 900 K, and . Tm .= 300 K. Boundary conditions are IM-SS

486

8 Thermal Induced Vibrations 600 550 500 450 400 350 300

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

106

7 6 5 4 3 2 1 0

6000 5000 4000 3000 2000 1000 0

Fig. 8.11 Effect of temperature dependence on the thermally induced vibration of conical shells with.ζ = 2. Other characteristic of the shell are. R0 .= 250 mm,. R1 .= 350 mm,.β0 = 45◦ ,.h = 5 mm, . Tc .= 900 K, and . Tm .= 300 K. Boundary conditions are IM-SS

8.3 FGM Conical Shells

487

geometry is thick, as previously discussed by many other authors. Under such condition, the quasi-static and dynamic responses are the same and the inertia terms may be dropped out from the equations of motion. As shown in this study, the role of temperature dependence is influential on the accurate estimation of dynamic deflection of conical shell under thermal shock. As a result, in the following numerical examples, only the temperature-dependent material properties are considered.

Influence of inertia In the next example, the importance of inertia effect on the response of FGM conical shells subjected to rapid heating is depicted. A thin conical shell with geometrical properties . R0 .= 300 mm, . R1 = 350 mm, .β0 = 60◦ , and .h =1 mm is considered. In this study, conical shells with different power law indices are considered which are .ζ = 0, 0.5, 1, 5. The metal rich surface is kept at reference temperature while the ceramic rich surface is subjected to sudden temperature rise of .Tc = 450 K. Shell is simply supported and immovable on both ends. Numerical results are provided in Fig. 8.12. It is seen that the dynamic response of the shell is different from the quasi-static response of the shell, which shows the importance of the inertia effects. In dynamic analysis, the time-dependent temperature profile is inserted into the equations of motion while in quasi-static analysis the temperature profile is inserted into the equilibrium equations. As the time evolutes, the quasi-static response tends to the steady-state response. Also, dynamic response of the shell oscillates around the quasi-static response. Numerical results of this figure also show that the power law index is an important factor in deflections induced by surface heating in the FGM shells. For the constituents of this study, as the power law index increases deflections of the shell also increases.

Influence of length In this example, the effect of shell length on the response of the conical shell subjected to rapid surface heating is analyzed. In this example, conical shell with geometrical parameters . R0 = 200 mm, .β0 = 45◦ , and .h = 1 mm are considered. Different shell lengths are assumed, which are . L = 50, 75, 275, 575 mm. A linearly graded conical shell is considered where both ends of the shell are axially immovable and simply supported. Case 1 of thermal loading is considered, where the metal rich surface is kept at reference temperature and the ceramic rich surface is subjected to.Tc = 450 K. The deflected shape of the conical shell for different shell lengths are computed at .t = 0.1s. Results are provided in Fig. 8.13. If is seen that for longer shells, the maximum deflection takes place near the edge supports due to the effect of boundary layer near the edges. On the other hand, for shorter shells the maximum deflection occurs near the mid-span of the conical shell. It is seen that for long shells, the effect

488

8 Thermal Induced Vibrations

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.01

0.02

0.03

0.04

0.05

0

0.01

0.02

0.03

0.04

0.05

0

0.01

0.02

0.03

0.04

0.05

Fig. 8.12 A study on the effect of power law index and inertia on the dynamic response of FGM conical shells subjected to thermal shock

8.3 FGM Conical Shells

489

10-3

1.2 1 0.8 0.6 0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.16

0.18

0.2

Fig. 8.13 Deflected shape of the conical shell for different shell lengths 1.3 1.1 0.9 0.7 0.5 0.3 0.1 -0.1 -0.3 -0.5 -0.7

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Fig. 8.14 The thermally induced vibrations on linearly graded FGM conical shell with various semi-vertex angles of the conical shell

of edge supports vanishes far away from the boundaries where the membrane smooth stresses are revealed.

Influence of semi-vertex angle The aim beyond this parametric study is to discuss the influence of the semi-vertex angle of the cone of the thermally induced vibrations. In this example, cones with geometrical parameters . R0 = 300 mm, . L = 70 mm, and .h = 1 mm are considered. A linearly graded (.ζ = 1) shell is considered where both ends of the shell are simply supported and axially movable. Different semi-vertex angles for the cone are assumed which are .β0 = 0◦ , 30◦ , 60◦ , 90◦ . The case 2 of thermal loading is considered, where the metal rich surface is thermally insulated and ceramic rich surface is subjected to temperature elevation of .Tc = 450 K. Results of this study are provided in Fig. 8.14. It is seen that as the semi-vertex angle of the cone increases, the induced deflection in the cone increases. Also, the amplitude of vibrations enhances as the semi-vertex angle of the cone increases.

490

8 Thermal Induced Vibrations

0.8 0.6 0.4 0.2 0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1 0.8 0.6 0.4 0.2 0

Fig. 8.15 Importance of in-plane edge supports on the thermally induced vibrations of FGM conical shells

Influence of in-plane edge supports In this study, the influence of in-plane edge supports is investigated. As mentioned earlier, two types of boundary conditions are considered which are movable and immovable. In these example, the response of conical shells with geometrical characteristics . R0 = 350 mm, . R1 = 400 mm, .β0 = 45◦ , and .h = 1 mm are considered. Power law index is set equal to .ζ = 2. Two different types of thermal boundary conditions are analyzed. For the first case, the ceramic rich surface is subjected to . Tc = 450 K and the metal rich surface is kept at reference temperature. For the second case, the metal rich surface is thermally insulated and the ceramic rich surface is subjected to .Tc = 450 K. Conical shells are assumed to be simply supported on both ends. However, they are either movable or immovable. Numerical results are illustrated in Fig. 8.15. It is seen that for the case of immovable edge supports, the induced deflections in the cone are higher, as expected, due to the axial compressive forces acting on the cone. Furthermore, the amplitude of vibrations is higher for the case of movable shells due to the less constraints acting on the shell. The required time to reach the steady-state response of the structure is much higher for the immovable shells.

8.3 FGM Conical Shells

491

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.01

0.02

0.03

0.04

0.05

0

0.01

0.02

0.03

0.04

0.05

0.18 0.15 0.12 0.09 0.06 0.03 0

Fig. 8.16 Importance of geometrical nonlinearity on the response of FGM conical shells under thermal shock

Influence of geometrical nonlinearity The aim of the final example is to show the importance of geometrical nonlinearity on the thermally induced deflections of the FGM conical shells. In this example, conical shells with clamped at.x = 0 and simply supported at.x = L are analyzed. Both of the immovable and movable types of boundary conditions are examined. Results of this study are demonstrated in Fig. 8.16. Geometrical characteristics of the shell are . R0 = 350 mm, . R1 = 400 mm, .β0 = 75◦ , and .h = 1 mm. Linearly graded conical shells are considered and for both cases the responses under linear and nonlinear theories are given. As expected, for immovable shells the effect of nonlinearity is more significant due to the induced compressive forces at the supports. For immovable shells, the divergence of responses based on linear and nonlinear theories is not ignorable. However, when edges are movable, the responses under two theories are the same as expected since the induced forces on the boundaries are ignorable. As the conclusion, we may consider that: .• Under the assumption of temperature-dependent material properties, which is the real state of material properties, the temperature is underestimated and deflections are overestimated in comparison with the analysis under constant material properties.

492

8 Thermal Induced Vibrations

.• The power law index is an important factor which changes the response of FGM cones under thermal shock. For the constituents of this study, as the power law index increases, the dynamic deflections of the cone increase. .• For sufficiently thick shells, thermally induced vibration almost disappears. Under such condition, dynamic and quasi-static responses are the same. Therefore, the response of the structure may be obtained when temperature profile is inserted in the equilibrium equations. However, for thin class of conical shells, thermally induced vibrations take place and dynamic and quasi-static responses are different. .• For longer shells, except for a narrow zone near the boundaries, the membrane stresses are produced. Under such conditions, the maximum deflection occurs near the edge supports. For short shells, however, the maximum deflection in the shell is observed approximately in the mid-span of the shell. .• Semi-vertex angle of the cone is an important factor on the temporal evolution of the cone displacement. Results show that as the semi-vertex angle of the cone increases, deflections in the cone increase. Also, the amplitude of vibrations in the cone increases when the semi-vertex angle becomes larger. .• Type of the in-plane boundary conditions has an important role on the deflections of the cone induced by rapid surface heating. The magnitude of deflections for immovable shells is much higher than the movable shells. The amplitude of oscillations is higher in movable shells. The required time to reach to the steady motion of the shell is higher in immovable shells. .• Geometrical nonlinearity is an important factor for accurate estimation of cone deflections. In general, this factor is more observed for immovable shells due to the induced compressive forces at the boundaries of immovable shells.

Appendix Discretized equations of motion based on GDQ method into the nodal points of shell length domain may be written as

. A11 (

Nx Σ j=1

+

Nx Nx Nx Σ Σ sin(β0 ) Σ sin2 (β0 ) sin(β0 ) cos(β0 ) ¯ i j w j )( ui − w B¯ i j u j + A¯ i j u j − 2 B A¯ ik wk ) + ( i r (xi ) r (xi ) r 2 (xi )

sin(β0 ) ( 2r (xi )

+ A12 (

j=1

Nx Σ

A¯ i j w j )(

j=1

cos(β0 ) r (xi )

Nx Σ

j=1

A¯ ik wk )) + B11 (

k=1 Nx Σ j=1

sin(β0 ) A¯ i j w j − ( 2r (xi )

Nx Σ j=1

Nx Σ j=1

sin(β0 ) B¯ i j ϕ j + r (xi )

A¯ i j w j )(

Nx Σ

k=1

Nx Σ j=1

k=1

sin2 (β0 ) A¯ i j ϕ j − 2 ϕi ) r (xi )

A¯ ik wk )) = I1 u¨ i + I2 ϕ ¨i

(8.3.35)

8.3 FGM Conical Shells

493

x x x x Σ Σ Σ Σ cos(β0 ) sin(β0 ) cos2 (β0 ) ¯ i j u j )( ¯ ik wk ) + ( ¯ i j u j )( B A A B¯ ik wk ) − + ( u w i i r 2 (xi ) r 2 (xi )

N

. A11 (−

N

j=1

+

+

sin(β0 ) ( r (xi )

Nx Σ

A¯ i j u j )(

j=1

k=1

+(

Nx Σ

k=1

B¯ i j ϕ j )(

j=1

Nx Σ

k=1

A¯ i j w j )(

j=1

Nx Σ

j=1

Nx Σ

A¯ ik wk )(

k=1

k=1

B¯ il wl )

l=1

A12 (− cos(β0 ) r (xi )

sin(β0 )(

j=1

A¯ ik wk ) + (

Nx Σ

Nx Σ

A¯ i j ϕ j )(

j=1 Nx Σ

Nx Σ

k=1

A¯ i j u j + sin(β0 )(

j=1

Nx Σ

x x Σ sin(β0 ) Σ B¯ ik wk ) + A¯ i j ϕ j )( A¯ ik wk )) ( r (xi )

N

N

j=1

A¯ i j u j )(

j=1

Nx Σ

Nx Σ

k=1

A¯ ik wk ) + sin(β0 )u i

k=1

Nx Σ

B¯ i j w j

j=1

x x x Σ Σ cos(β0 ) Σ B12 B¯ i j w j + A¯ i j w j )( A¯ ik wk )) + A¯ i j ϕ j + ( (− cos(β0 ) 2 r (xi ) j=1 j=1 k=1 j=1

N

A¯ i j ϕ j )(

j=1 Nx Σ

l=1

k=1

+ cos(β0 )wi

+

3 A¯ ik wk ) + ( 2

Nx Σ

N

Nx Nx Nx Σ Σ sin(β0 ) Σ cos(β0 ) sin(β0 ) ( A¯ i j w j )( A¯ ik wk )( A¯ il wl )) + B11 (− ϕi 2r (xi ) r 2 (xi ) j=1

+

Nx Σ

N

Nx Σ

N

A¯ ik wk ) + sin(β0 )ϕi

k=1

Nx Σ

N

B¯ i j w j + A55 (

j=1

Nx Σ j=1

sin(β0 ) A¯ i j ϕ j + ϕi r (xi )

Nx Nx Nx Σ sin(β0 ) Σ sin(β0 ) Σ cos(β0 ) T N = I1 w¨ i B¯ i j w j + A¯ i j w j ) − N T ( B¯ i j w j + A¯ i j w j ) + r (xi ) r (xi ) r (xi ) j=1

j=1

j=1

(8.3.36)

. B11 (

Nx Σ j=1

Nx Nx Nx Σ Σ sin(β0 ) Σ sin2 (β0 ) sin(β0 ) cos(β0 ) ui − wi + ( B¯ i j u j + A¯ i j u j − 2 B¯ i j w j )( A¯ ik wk ) 2 r(xi ) r (xi ) r (xi ) j=1

j=1

k=1

Nx Nx Nx Nx Σ Σ sin(β0 ) Σ sin(β0 ) Σ sin2 (β0 ) ( ϕi ) + A¯ i j w j )( A¯ ik wk )) + D11 ( B¯ i j ϕ j + A¯ i j ϕ j − 2 2r(xi ) r(xi ) r (xi ) j=1

+ B12 (

cos(β0 ) r(xi )

k=1

Nx Σ j=1

j=1

sin(β0 ) ( A¯ i j w j − 2r(xi )

Nx Σ

A¯ i j w j )(

j=1

j=1

Nx Σ

A¯ ik wk )) − A55 (ϕi +

k=1

Nx Σ

A¯ i j w j ) = I2 u¨ i + I3 ϕ¨ i

j=1

(8.3.37) where .i = 2, 3, . . . , N x − 1. Also, boundary condition should be discretized using GDQ method as follow

.u s

=0

or

A11 (

Nx Σ j=1

+ B11 (

x x Σ 1 Σ sin(β0 ) cos(β0 ) us + ws ) A¯ s j w j + ( A¯ s j w j )( A¯ sk wk )) + A12 ( 2 r (xs ) r (xs )

Nx Σ j=1

N

j=1

N

k=1

sin(β0 ) A¯ s j ϕ j ) + B12 ( ϕs ) − N T = 0, s = 1, N x r(xs )

(8.3.38)

494

8 Thermal Induced Vibrations

ws = 0

Nx Σ

or

.

A¯ s j w j = 0, s = 1, N x

(8.3.39)

j=1

.ϕs

=0

or

B11 (

Nx Σ

N

j=1

+ D11 (

x x Σ 1 Σ sin(β0 ) cos(β0 ) A¯ s j w j + ( A¯ s j w j )( A¯ sk wk )) + B12 ( us + ws ) 2 r (xs ) r (xs )

N

j=1

Nx Σ j=1

k=1

sin(β0 ) A¯ s j ϕ j ) + D12 ( ϕs ) − M T = 0, s = 1, N x r (xs )

(8.3.40)

where . A¯ i j and . B¯ i j are weighting coefficients of the first- and second-order derivative, respectively, and are defined as follow ⎧ ϒ(ξi ) ⎪ when i /= j ⎨ (ξi −ξ j )ϒ(ξ j ) Nx ¯i j = Σ .A ⎪ A¯ ik when i = j ⎩− k=1,k/=i

i, j = 1, 2, ..., N x

(8.3.41)

in which Nx | |

ϒ(ξi ) =

.

(ξi − ξk )

(8.3.42)

k=1,k/=i

and [ .

( B¯ i j = 2 A¯ ii A¯ i j −

i, j = 1, 2, ..., N x ⎧ N ⎪ ⎨ B¯ = − Σx B¯ ii ik k=1,k/=i ⎪ ⎩i = 1, 2, ..., N x

A¯ i j (ξi −ξ j )

) when i /= j

when i = j

(8.3.43)

8.4 Spherical Shells In this section, the nonlinear axisymmetric thermally induced vibration analysis of the FGM shallow spherical shells is discussed. The thermo-mechanical properties of the shallow spherical shell are assumed to be temperature and position depen-

8.4 Spherical Shells

495

Fig. 8.17 Schematic and geometric characteristic of a shallow spherical shell

dent. The heat conduction equation is established for the shell through the thickness direction. This equation is nonlinear due to the assumption of temperature dependence material properties. Hybrid GDQ-Crank–Nicolson method will be applied to obtain the temperature distribution through the thickness direction of the shell. Using Hamilton’s principle, the equations of motion based on the first-order shear deformation shell theory and geometrically nonlinear theory are established. Axisymmetric differential equations based on the HDQ method are discretized into a number of grid points in shell domain and then utilizing the Newmark method, system of ordinary differential equations is converted into a system of nonlinear algebraic equations which is solved by implementation of the well-known Newton–Raphson method. Influence of different parameters is investigated on thermally induced deflection of rapidly heated shallow spherical shell [47]. A shallow spherical shell made of functionally graded material with thickness .h, opening angle .2ϕ0 , and radius of curvature . R is considered. The curvilinear coordinates are designated by .ϕ and .θ, which are located in the shell middle surface. The coordinate .z is measured through the shallow shell thickness from the reference middle plane and is considered to be positive outward, as shown in Fig. 8.17. Due to the axisymmetric rapid heating and boundary conditions, displacement components in .θ direction are vanished. Thermo-mechanical properties of an FGM thin structure should be defined according to a convenient homogenization procedure. The Voigt rule is commonly used for this reason [15, 17, 48]. Conforming to this method, the mechanical and thermal properties of the FGM shallow spherical shell, such as Young’s modulus . E, mass density .ρ, Poisson’s ratio .ν, specific heat .Cv , thermal conductivity . K , and thermal

496

8 Thermal Induced Vibrations

expansion coefficient .α are assumed to be linear function of the volume fractions of the ceramic .Vc and metal .Vm constituents. Therefore, as a function of thickness direction, a non-homogeneous property of the shell, . P, may be expressed in the form .

P(z, T ) = Pm (T ) + Vc (z)Pcm (T ), Pcm (T ) = Pc (T ) − Pm (T )

(8.4.1)

where the subscripts .m and .c illustrate the attributes of metal and ceramic constituents, respectively. Touloukian model for temperature-dependent material properties is considered, which reads [45] .

P(T ) = P0 (P−1 T −1 + 1 + P1 T + P2 T 2 + P3 T 3 )

(8.4.2)

where .T is the temperature measured in Kelvin and . Pi ’s are constant and unique to each of the constituents. The power law profile of the constituents may be utilized to demonstrate the ceramic volume fraction .Vc and metal volume fraction .Vm such as ( .

Vc =

z 1 + 2 h

)ζ

, Vm = 1 − Vc

(8.4.3)

Here, .ζ is a positive constant called the power law index and assigns the property dispersion profile. Obviously, ceramic rich surface is at the top of the shell (.z = h/2) and metal rich surface is at the bottom of the shell (.z = −h/2).

8.4.1 Governing Equations Thermally induced deflection of the shells is considered only along the .ϕ − z direction due to axisymmetric thermal loads and boundary conditions. Also, displacement components are described based on the first-order shear deformation theory (FSDT) consistent with the Timoshenko assumptions. This theory is suitable for thin and moderately thick shells and its components may be written in terms of the midplane displacement and rotations as u(ϕ, ¯ z, t) = u 0 (ϕ, t) + zψ(ϕ, t) w(ϕ, ¯ z, t) = w0 (ϕ, t)

.

(8.4.4)

In the above equations,.u 0 and.w0 indicate the displacements at the mid-surface along the .ϕ− and .z−directions, respectively. Besides, transverse normal rotation about the .θ axis is denoted by .ψ. The nonlinear strain–displacement relations based on the von Kármán theory, consistent with the small strains, moderate rotations, and large displacements in the curvilinear coordinates are

8.4 Spherical Shells

497

1 2 1 1 u¯ ,ϕ + w¯ + w¯ R R 2R 2 ,ϕ 1 cot(ϕ) u¯ + w¯ εθθ = R R 1 γϕz = u¯ ,z + w¯ ,ϕ R ε

. ϕϕ

=

(8.4.5)

where .εϕϕ and .εθθ demonstrate the longitudinal normal and circumferential normal strains and .γϕz denotes the shear strain component. Here and in the rest, a comma indicates the partial derivative with respect to its afterward. Linear thermoelastic stress–strain relations are assumed in this research. The constitutive law for the FGM shallow spherical shells subjected to thermal load takes the form ⎧ ⎫ ⎡ ⎫ ⎧ ⎫⎞ ⎤ ⎛⎧ Q 11 Q 12 0 ⎨ εϕϕ ⎬ ⎨ σϕϕ ⎬ ⎨α⎬ σθθ = ⎣ Q 12 Q 22 0 ⎦ ⎝ εθθ − ΔT α ⎠ . (8.4.6) ⎩ ⎩ ⎭ ⎭ ⎩ ⎭ 0 τϕz γϕz 0 0 Q 55 where .ΔT is the temperature difference between the temperature profile within the shell and the reference temperature and . Q i j ’s (i, j = 1, 2, 6) are the reduced material stiffness coefficients and are obtained as .

Q 11 = Q 22 =

E(z, T ) , 1 − ν 2 (z, T )

Q 12 =

ν(z, T )E(z, T ) , 1 − ν 2 (z, T )

Q 55 =

E(z, T ) 2(1 + ν(z, T )) (8.4.7)

Based on the FSDT, the membrane stress resultants . Nϕϕ and . Nθθ , the membrane out-of-plane shear stress resultants . Q ϕz , and the bending stress resultants . Mϕϕ and . Mθθ may be expressed upon integration of the stress components as { (Nϕϕ , Nθθ , Mϕϕ , Mθθ , Q ϕz ) =

+0.5h

.

−0.5h

(σϕϕ , σθθ , zσϕϕ , zσθθ , τϕz )dz

(8.4.8)

Substituting Eq. (8.4.6) into Eq. (8.4.8) with the aid of Eqs. (8.4.4) and (8.4.5), the stress resultants in terms of the displacement components are obtained as ⎧ ⎫ 1 1 2 ⎪ ⎪ ⎪ ⎪ (u + w ) + w 0 ⎪ R 0,ϕ ⎪ 2R 2 0,ϕ ⎪ ⎧ T⎫ ⎪ ⎪ ⎧ ⎫ ⎡ ⎤⎪ ⎪ ⎪ cot(ϕ) 1 ⎪ N ⎪ A11 A12 B11 B12 0 ⎪ Nϕϕ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u 0 + w0 ⎪ T⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ R R ⎨N ⎪ ⎬ ⎪ ⎬ ⎨ Nθθ ⎪ ⎬ ⎢ A12 A22 B12 B22 0 ⎥ ⎥⎨ 1 Mϕϕ = ⎢ B11 B12 D11 D12 0 ⎥ − MT . ψ,ϕ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ R ⎪ ⎪ ⎪ B12 B22 D12 D22 0 ⎦ ⎪ Mθθ ⎪ MT ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ cot(ϕ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎭ ⎩ ⎭ ⎪ ψ Q ϕz 0 0 0 0 0 A55 ⎪ ⎪ ⎪ ⎪ ⎪ R ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ ⎭ w0,ϕ + ψ R (8.4.9)

498

8 Thermal Induced Vibrations

In the above equations, the constant coefficients . Ai j , . Bi j , and . Di j represent the stretching, bending-stretching, and bending stiffness in the FG media, respectively, which are calculated by { (Ai j , Bi j , Di j ) =

+0.5h

.

−0.5h

(Q i j , z Q i j , z 2 Q i j )dz,

i, j = 1, 2, 5

(8.4.10)

Besides, . N T and . M T are the in-plane thermal force and thermal moment resultants which are given by { (N T , M T ) =

+0.5h

.

−0.5h

(1, z)

1 E(z, T )α(z, T )(T − T0 )dz 1 − ν(z, T )

(8.4.11)

The temperature profile within the shell domain should be defined to obtain the thermal force and thermal moment resultants. Calculation of the temperature profile within the shell domain will be discussed in the next section. The equations of axisymmetric motion of FGM shallow spherical shells based on the uncoupled thermoelasticity may be obtained using the principle of virtual displacements as [49] { t2

.

(δT − δV − δU )dt = 0

(8.4.12)

t1

The total virtual strain energy of the shallow spherical shells .δU can be computed as { δU =

ϕ

{

2π

{

+0.5h

.

0

0

−0.5h

( ) σϕϕ δεϕϕ + σθθ δεθθ + τϕz δγϕz R 2 sin(ϕ)dz dθ dϕ

(8.4.13) Similarly, .δV is the virtual potential energy of the external applied loads which is absent in this paper. Also, the kinetic energy .δT may be obtained by {

ϕ

δT =

{

2π

{

+0.5h

.

0

{

ϕ

− 0

{

0 2π

{(

−0.5h

( ) ˙¯ u˙¯ + wδ ˙¯ w˙¯ R 2 sin(ϕ)dz dθ dϕ = ρ(z, T ) uδ

( ) ) } I1 u¨ 0 + I2 ψ¨ δu 0 + I1 w¨ 0 δw0 + I2 u¨ 0 + I3 ψ¨ δψ R 2 sin(ϕ) dθ dϕ

0

(8.4.14) where the .( ˙ ) indicates a derivative with respect to time and the inertia terms . I1 , I2 , and . I3 are defined by { (I1 , I2 , I3 ) =

+0.5h

.

−0.5h

ρ(z, T )(1, z, z 2 )dz

(8.4.15)

8.4 Spherical Shells

499

Substitution of Eqs. (8.4.13) and (8.4.14) into Eq. (8.4.12) and with the aid of the suitable mathematical simplifications, the expressions for the equations of motion of the shallow spherical shell take the form ) 1 cot(ϕ) ( Nϕϕ,ϕ + Nϕϕ − Nθθ = I1 u¨ 0 + I2 ψ¨ R R ) ) ( 1 1 cot(ϕ) 1 ( Q ϕz − Nϕϕ + Nθθ = I1 w¨ 0 δw0 : sin(ϕ) Nϕϕ w0,ϕ ,ϕ + Q ϕz,ϕ + R R R R 2 sin(ϕ) ) 1 cot(ϕ) ( Mϕϕ,ϕ + Mϕϕ − Mθθ − Q ϕz = I2 u¨ 0 + I3 ψ¨ δψ : (8.4.16) R R

.δu 0

:

The equations of motion in terms of the displacement components for an axisymmetric FGM shallow spherical shell may be evaluated by utilizing Eqs. (8.4.9) and (8.4.16). The resulting equations may be written as

.

1 R2

{ A11 u 0,ϕϕ + A11 cot(ϕ)u 0,ϕ −

+ B11 ψ,ϕϕ + B11 cot(ϕ)ψ,ϕ − +(A11 + A12 )w0,ϕ +

A12 sin2 (ϕ)

B12 sin2 (ϕ)

u 0 + (A12 − A11 ) cot 2 (ϕ)u 0

ψ + (B12 − B11 ) cot 2 (ϕ)ψ

} A11 (A11 − A12 ) 2 w0,ϕ w0,ϕϕ + cot(ϕ)w0,ϕ = I1 u¨ 0 + I2 ψ¨ R 2R

1 { −R(A11 + A12 )u 0,ϕ − R(A11 + A12 ) cot(ϕ)u 0 + (A11 + A12 ) cot(ϕ)u 0,ϕ w0,ϕ R3 ( ) ( ) 1 + A11 u 0,ϕ w0,ϕϕ + u 0,ϕϕ w0,ϕ + A12 cot 2 (ϕ)u 0 w0,ϕ + cot(ϕ)u 0 w0,ϕϕ − u 0 w0,ϕ 2 sin (ϕ) + R(−B11 + B12 )ψ,ϕ − R(B11 + B12 ) cot(ϕ)ψ + (B11 + B12 ) cot(ϕ)ψ,ϕ w0,ϕ ( ) ( ) 1 + B11 ψ,ϕ w0,ϕϕ + ψ,ϕϕ w0,ϕ + A12 cot2 (ϕ)ψw0,ϕ + cot(ϕ)ψw0,ϕϕ − ψw 0,ϕ sin2 (ϕ) ( ) ( ) 2 + R A55 cot(ϕ)ψ + ψ,ϕ + R A55 cot(ϕ)w0,ϕ + w0,ϕϕ − 2R(A11 + A12 )w0 ( ) ) ( A11 3 2 1 2 1 3 w0,ϕ w0,ϕϕ + cot(ϕ)w0,ϕ + + (A11 + A12 ) cot(ϕ)w0,ϕ w0 + w0,ϕϕ w0 + w0,ϕ 2 R 2 2 } ( ) T 2 T −R N cot(ϕ)w0,ϕ + w0,ϕϕ + 2R N = I1 w¨ 0

1 R2

{ B11 u 0,ϕϕ + B11 cot(ϕ)u 0,ϕ −

+ D11 ψ,ϕϕ + D11 cot(ϕ)ψ,ϕ − +(B11 + B12 )w0,ϕ +

B12 sin2 (ϕ)

D12 sin2 (ϕ)

u 0 + (B12 − B11 ) cot2 (ϕ)u 0

ψ + (D12 − D11 ) cot 2 (ϕ)ψ − R 2 A55 ψ

} B11 (B11 − B12 ) 2 − RA w ¨ w0,ϕ w0,ϕϕ + cot(ϕ)w0,ϕ 55 0,ϕ = I2 u¨ 0 + I3 ψ R 2R

(8.4.17)

500

8 Thermal Induced Vibrations

The boundary conditions are evaluated by means of the procedure of virtual displacement relieving. For .ϕ = 0 and .ϕ = ϕ0 , the boundary conditions are given by ( .

Nϕϕ δu 0 =

Q ϕz

) 1 + Nϕϕ w0,ϕ δw0 = Mϕϕ δψ = 0 R

(8.4.18)

Utilizing the boundary conditions given in Eq. (8.4.18), several sets of boundary conditions appropriate for thermally induced vibration behavior may be defined. Consequently, the edge of the shallow spherical shell may take one of the following boundary conditions .

IM − S

:

u = w = Mϕϕ = 0

M − S : Nϕϕ = w = Mϕϕ = 0 IM −C : u = w = ψ = 0 M − C : Nϕϕ = w = ψ = 0

(8.4.19)

and at the point.ϕ = 0 the boundary condition is described according to the following equation u = Q ϕz +

.

1 Nϕϕ w,ϕ = ψ = 0 R

(8.4.20)

In the above equations IM-S, M-S, IM-C, and M-C indicate immovable simply support, movable simply support, immovable clamped, and movable clamped, respectively.

8.4.2 HDQ Discretization The first and second derivatives of a function . f at a sample grid point may be expressed in terms of the values of the function at the grid points as [50, 51] Σ d f (ϕ) |ϕ=ϕi = . A¯ i j f j , dϕ j=1 Nϕ

Σ d 2 f (ϕ) | = B¯ i j f j , ϕ=ϕ i dϕ2 j=1 Nϕ

i = 1, 2, ..., Nϕ (8.4.21)

According to the above definition, the equations of motion for .i-th nodal point may be written as

8.4 Spherical Shells

501

⎧ Nϕ Nϕ Σ Σ 1 ⎨ A12 . B¯ i j u j + A11 cot(ϕi ) A¯i j u j − u + (A12 − A11 ) cot 2 (ϕi )u i A11 2 (ϕ ) i 2 R ⎩ sin i j=1 j=1 Nϕ Σ

Nϕ Σ

B12 ψ + (B12 − B11 ) cot2 (ϕi )ψi 2 (ϕ ) i sin i j=1 j=1 ⎞⎛ ⎞ ⎞ ⎛ ⎛ Nϕ Nϕ Nϕ Nϕ Σ Σ Σ A11 − A12 A11 ⎝Σ ¯ ¯ ¯ ¯ ⎠ ⎝ ⎠ ⎝ cot(ϕi ) + (A11 + A12 ) Ai j w j + Ai j w j Bik wk + Ai j w j ⎠ R 2R j=1 j=1 k=1 j=1 ⎛ ⎞⎫ Nϕ ⎬ Σ ¯ ⎝ (8.4.22) Aik wk ⎠ = I1 u¨ i + I2 ψ¨ i ⎭ + B11

B¯ i j ψ j + B11 cot(ϕi )

A¯i j ψ j −

k=1

⎧ ⎛ ⎞ Nϕ Nϕ Σ Σ 1 ⎨ . A¯i j u j − R(A11 + A12 ) cot(ϕi )u i + (A11 + A12 ) cot(ϕi ) ⎝ A¯i j u j ⎠ −R(A11 + A12 ) R3 ⎩ j=1 j=1 ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ Nϕ Nϕ Nϕ Nϕ Nϕ Σ Σ Σ Σ Σ ¯ ¯ ¯ ¯ ¯ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ Aik wk + A11 Ai j u j Bik wk + A11 Bi j u j Aik wk ⎠ k=1

⎛

j=1

k=1

⎛

Σ

⎞

A¯i j w j ⎠ + A12 cot(ϕi )u i ⎝

j=1

Nϕ

Σ

⎞

k=1

B¯i j w j ⎠ −

j=1

j=1

⎛

Nϕ

k=1 Nϕ

Σ

⎞

k=1

⎛ ⎞⎛ ⎞ ⎛ ⎞ Nϕ Nϕ Nϕ Σ Σ Σ Σ Σ 2 ¯ ¯ ¯ ¯ ¯ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ + B11 Ai j ψ j Bik wk + B11 Bi j ψ j Aik wk + B12 cot (ϕi )ψi Ai j w j ⎠ Nϕ

⎞⎛

⎛

j=1

⎛ ⎞ Nϕ Σ A12 ¯ ⎝ w u A i ij j⎠ sin2 (ϕi ) j=1 j=1 j=1 ⎛ ⎞⎛ ⎞ Nϕ Nϕ Nϕ Σ Σ Σ − R(B11 + B12 ) A¯i j ψ j − R(B11 + B12 ) cot(ϕi )ψi + (B11 + B12 ) cot(ϕi ) ⎝ A¯i j ψ j ⎠ ⎝ A¯ik wk ⎠

+ A12 cot2 (ϕi )u i ⎝

Nϕ

j=1

⎞

k=1

j=1

⎛ ⎞ ⎛ ⎞ Nϕ Nϕ Σ Σ 2 ¯ ¯ ⎝ ⎠ ⎝ Ai j w j + R A55 cot(ϕi )ψi + Ai j ψ j ⎠ ψi

B12 2 (ϕ ) sin i j=1 j=1 j=1 ⎛ ⎛ ⎞ Nϕ Nϕ Nϕ Σ Σ Σ + R A55 ⎝cot(ϕi ) A¯i j w j + B¯i j w j ⎠ − 2R(A11 + A12 )wi + (A11 + A12 ) ⎝cot(ϕi )wi A¯i j w j

+ B12 cot(ϕi )ψi ⎝

⎛

B¯i j w j ⎠ −

j=1

j=1

j=1

j=1

⎞

j=1

⎞⎛ ⎞⎞ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎛ Nϕ Nϕ Nϕ Nϕ Nϕ Σ Σ Σ Σ Σ 1 3A 11 ⎝ +wi ⎝ B¯i j w j ⎠ + ⎝ A¯i j w j ⎠ ⎝ A¯ik wk ⎠⎠ + A¯i j w j ⎠ ⎝ A¯ik wk ⎠ ⎝ B¯il wl ⎠ 2 2R j=1 j=1 k=1 j=1 k=1 l=1 ⎞⎛ ⎞⎛ ⎞ ⎛ Nϕ Nϕ Nϕ Σ Σ A11 ⎝Σ ¯ ¯ ¯ ⎠ ⎝ ⎠ ⎝ + Ai j w j Aik wk Ail wl ⎠ 2R j=1 k=1 l=1 ⎫ ⎛ ⎞ Nϕ Nϕ ⎬ Σ Σ −R N T ⎝cot(ϕi ) (8.4.23) A¯i j w j + B¯i j w j ⎠ + 2R 2 N T = I1 w¨ i ⎭ Nϕ Σ

502

8 Thermal Induced Vibrations ⎧ Nϕ Nϕ Σ Σ 1 ⎨ B12 . B¯ i j u j + B11 cot(ϕi ) A¯i j u j − u + (B12 − B11 ) cot2 (ϕi )u i B11 2 (ϕ ) i 2 R ⎩ sin i j=1 j=1 Nϕ Σ

Nϕ Σ

D12 ψ + (D12 − D11 ) cot 2 (ϕi )ψi 2 (ϕ ) i sin i j=1 j=1 ⎞⎛ ⎞ ⎞ ⎛ ⎛ Nϕ Nϕ Nϕ Nϕ Σ Σ Σ B11 − B12 B11 ⎝Σ ¯ ¯ ¯ ¯ ⎠ ⎝ ⎠ ⎝ cot(ϕi ) + (B11 + B12 ) Ai j w j + Ai j w j Bik wk + Ai j w j ⎠ R 2R j=1 j=1 k=1 j=1 ⎛ ⎞ ⎛ ⎞⎫ Nϕ Nϕ ⎬ Σ Σ 2 ¯ ¯ ⎝ ⎠ ⎝ (8.4.24) Aik wk − A55 R ψi + R Ai j w j ⎠ = I1 u¨ i + I2 ψ¨ i ⎭ + D11

B¯ i j ψ j + D11 cot(ϕi )

k=1

A¯i j ψ j −

j=1

where .i = 2, 3, . . . , Nϕ − 1. Also, boundary conditions should be discretized using the HDQ method as follow ⎛

.u s

=0

or

⎛ ⎞⎛ ⎞⎞ ( Nϕ Nϕ Nϕ Σ Σ Σ 1 1 1 ¯sj w j ⎠ ⎝ ¯ sk wk ⎠⎠ + A12 cot(ϕs ) u s ⎝ A11 ⎝ A¯ s j u j + ws + A A R R 2R 2 R j=1

+

1 ws R

j=1

)

k=1

Nϕ

+ B11

1 Σ ¯ cot(ϕs ) As j ψ j + B12 ψs − N T = 0, s = 1, Nϕ R R

(8.4.25)

j=1

⎧ ⎞ ⎛ ⎛ ⎞ Nϕ Nϕ Nϕ ⎨ Σ Σ Σ 1 1 1 1 1 ⎝ A55 ⎝ A¯ s j w j + ψs ⎠ + A¯ s j u j + ws + A¯ s j w j ⎠ A11 ⎝ R R⎩ R R 2R 2 ⎛

.ws

=0

or

j=1

Nϕ

)

j=1

Nϕ

cot(ϕs ) 1 Σ ¯ 1 As j ψ j u s + ws + B11 R R R k=1 j=1 ⎞ ⎛ } Σ Nϕ cot(ϕs ) +B12 A¯ sl wl ⎠ = 0, s = 1, Nϕ ψs − N T ⎝ R

(

Σ

j=1

(

A¯ sk wk )) + A12

(8.4.26)

l=1

⎛ .ψs

=0

or

1 B11 ⎝ R +

1 ws R

Nϕ Σ j=1

⎛ ⎞⎛ ⎞⎞ ( Nϕ Nϕ Σ cot(ϕs ) 1 1 ⎝Σ ¯ ¯ ¯ ⎠ ⎝ us As j u j + ws + As j w j Ask wk ⎠⎠ + B12 2 R 2R R

)

j=1

k=1

Nϕ

+ D11

1 Σ ¯ cot(ϕs ) As j ψ j + D12 ψs − M T = 0, s = 1, Nϕ R R

(8.4.27)

j=1

where. A¯ i j and. B¯ i j are weighting coefficients of the first and second-order derivatives, respectively, and are defined as follow

8.4 Spherical Shells

503

.

A¯ i j =

⎧ ⎪ ⎪ ⎨

( π2 )ϒ(ϕi ) π(ϕi −ϕ j ) sin( )ϒ(ϕ j ) 2 Nϕ

Σ ⎪ ⎪ ⎩−

A¯ ik

k=1,k/=i

when i /= j when i = j

i, j = 1, 2, ..., Nϕ

(8.4.28)

in which ϒ(ϕi ) =

Nϕ | |

sin(

.

k=1,k/=i

π(ϕi − ϕk ) ) 2

(8.4.29)

and [ .

( ) π(ϕ −ϕ ) B¯ i j = A¯ i j 2 A¯ ii − π cot( i2 j )

i, j = 1, 2, ..., Nϕ ⎧ Nϕ ⎪ ⎨ B¯ = − Σ B¯ ii ik k=1,k/=i ⎪ ⎩i = 1, 2, ..., N ϕ

when i /= j

when i = j

(8.4.30)

Dispensation of nodal points in the shell domain is proposed by means of the Chebyshev–Gauss–Lobatto distribution which reads ( ϕi = ϕ0

.

1 1 − cos 2 2

(

i −1 π Nϕ − 1

)) ,

i = 1, 2, . . . , Nϕ

(8.4.31)

where . Nϕ is the number of grid points along the shell length.

8.4.3 Solution Procedure Three governing equations (8.4.22), (8.4.23), and (8.4.24) along with a proper choice of the boundary conditions (8.4.25), (8.4.26), and (8.4.27) may be written in a compact form as { } ¨ + [K(T,X)] {X} = {F(T)} . [M(T)] X (8.4.32) In this equation .[M] is the mass matrix, .[K] is the stiffness matrix, and .{F} is the force vector. Also, .{X} is the displacement vector including the unknowns .u i , .wi , and .ψi where .i = 0, 1, ..., Nϕ . It is evident that the generalized stiffness matrix depends upon the unknown time-dependent nodal vector .{X} due to the consideration of the von Kármán type of geometrical nonlinearity. Here, the Newmark direct integration scheme based on the constant average acceleration method .(α = 0.5, β = 0.25) is

504

8 Thermal Induced Vibrations

applied to approximate the system of Eq. (8.4.32) in terms of the modified stiffness matrix and force vector. Implementation of the time marching Newmark method to Eq. (8.4.32) yields [ ] { } ^ {X} j+1 = ^ . K(T,X) F(T) j, j+1 (8.4.33) where modified stiffness matrix and force vector take the following form .

[ ] ^ K(T,X) = [K(T,X)] + a0 [M(T)] ( { } { } { } ) ˙ + a2 X ¨ ^ F(T) = {F(T)} j+1 + [M(T)] a0 {X} j + a1 X j j

(8.4.34)

and a =

. 0

1 , βΔt 2

a1 =

1 , βΔt

a2 =

1 − 2β 2β

(8.4.35)

Once the solution .{X} is known at .t j+1 = ( j + 1)Δt, the velocity and acceleration vectors at .t j+1 can be obtained from

.

{ } ( ) { } { } ¨ ˙ − a2 X ¨ X = a0 {X} j+1 − {X} j − a1 X j+1 j j { } { } { } { } ˙ ¨ ¨ ˙ X j+1 = X j + a3 X j + a4 X j+1

(8.4.36)

and a = (1 − α)Δt,

. 3

a4 = αΔt

(8.4.37)

The displacement vector can be obtained at each time step by means of the known information { } from { }the precedent time step solution. At time .t = 0, the initial values of ˙ , and . X ¨ are known or extracted by solving Eq. (8.4.32) at time .t = 0 and .{X}, . X are applied to initiate the time { }marching scheme. Since the shell is initially at rest, ˙ are assumed to be zero. A suitable iterative method the initial values of .{X} and . X should be applied to Eq. (8.4.32) to solve the resulting highly nonlinear algebraic equations. In this work, the well-known Newton–Raphson iterative method is implemented in which the tangent stiffness matrix is evaluated based on the developed method in [46, 52].

8.4.4 Temperature Profile Due to the axisymmetric thermal loads, the heat conduction differential equation for radial temperature distribution in spherical coordinates reduces to [30]

8.4 Spherical Shells

505

.

) 1 ( 2 r K (z, T )T,r ,r = ρ(z, T )Cv (z, T )T˙ 2 r

(8.4.38)

where.r = R + z and.( ),r = ( ),z . For .( Rz ≪ 1), which is valid for moderately thick shells, one can assume .r ≈ R. Consequently, in the absence of heat generation the one-dimensional transient heat conduction equation for temperature-dependent FG media takes the form ( .

K (z, T )T,z

) ,z

= ρ(z, T )Cv (z, T )T˙

(8.4.39)

Since, prior to loading the shell is at reference temperature, the initial condition can be written as . T (z, 0) = T0 (8.4.40) To obtain the temperature profile from the heat conduction Eq. (8.4.39), various types of thermal boundary conditions may be applied on the top and bottom surfaces of the shell. Hence, it is assumed that the top surface of the shell, which is ceramic rich, is subjected to a time-dependent sudden temperature (rapid heating), whereas the opposite surface which is metal rich may undergo thermally insulated boundary condition or the temperature specified time-dependent boundary condition (rapid heating). Three different types of thermal boundary conditions can be defined as Case 1 : T (+0.5h, t) = Tc (t),

.

Case 2 : T (+0.5h, t) = Tc (t), Case 3 : K (+0.5h, T )T,z (+0.5h, t) = Q c ,

T (−0.5h, t) = Tm (t) T,z (−0.5h, t) = 0 T,z (−0.5h, t) = 0 (8.4.41)

Upon solution of Eq. (8.4.39) with consideration of Eqs. (8.4.40) and (8.4.41), temperature profile across the shell thickness is obtained. Thermal conductivity is a function of temperature due to temperature-dependent material properties assumption. Therefore, the heat conduction differential equation becomes nonlinear. The heat conduction equation is solved by means of the generalized differential quadrature method. According to the GDQ method, distribution of nodal points across the shell thickness can be written as ( ) (i − 1)π h cos , i = 1, 2, . . . , Nz .z i = − (8.4.42) 2 Nz − 1 applying the GDQ method to the heat conduction Eq. (8.4.39) through the thickness direction and imposing the boundary conditions, the matrix form of the heat conduction equation takes the form .

{ } ˙ + [KT (T)] {T} = {FT (T)} [CT (T)] T

(8.4.43)

Details on applying the GDQ method are not given here, however one may refer to [53–56]. Due to the temperature dependence of the material properties, in Eq. (8.4.43)

506

8 Thermal Induced Vibrations

damping matrix .[CT (T)], stiffness matrix .[KT (T)], and even force vector .{FT (T)} are functions of the nodal temperatures. Accordingly, based on the Crank–Nicolson time marching scheme, at each time step an iterative procedure should be followed to obtain the temperature profile of the shell under the assumption of temperaturedependent thermo-mechanical properties. In this work, the Picard iterative method is utilized. At each time step, thermal properties are evaluated at reference temperature . T0 . Material properties are then evaluated at given nodal temperatures .{T} and Eq. (8.4.43) is solved repeatedly. This method is repeated until the temperature profile converges at the current time step. The Crank–Nicolson procedure is adopted to solve Eq. (8.4.43) along with the initial conditions (8.4.40). Details on the Crank–Nicolson and Picard methods are available in [46] and are not given in this research for the sake of brevity.

8.4.5 Results and Discussion The procedure outline in the previous sections may be used here to analyze the large amplitude rapid surface heating of spherical shells made of functionally graded materials. In this section, first two comparison studies are performed to show the effectiveness and correctness of the developed formulations and the solution method. Afterward, novel numerical results are provided for the case of thermally induced vibrations in FGM shallow spherical shells. Unlike the comparison study of Table 8.5, the constituents are assumed as SUS304 for the metal phase and Si.3 N.4 for the ceramic phase. The properties of these constituents are highly temperature dependent according to the Touloukian representation (8.4.1). The coefficients of the Touloukian model for each property of the constituents are provided in Table 8.4. In the numerical results, deflection at the apex of the spherical shell is denoted by .W . The reference temperature is set equal to .T0 = 300 K. Two types of analysis are done. The temperature-dependent (TD) analysis is where the material properties are evaluated at current temperature and the temperature-independent (TID) analysis is where properties are obtained at reference temperature.

8.4.6 Comparison Study In this section, two comparison studies are provided. For the first comparison study, fundamental natural frequency parameter of a spherical shell is evaluated for different number of nodal diameters in the HDQ method. Results of this study are also compared with those of Xu and Chia [57] using the Galerkin method for the case of IM-C shells. Comparison is shown in Table 8.5. It is shown that the results are in excellent agreement with those of Xu and Chia [57]. This comparison validates the mass and stiffness matrices of the recent formulation. Especially, since the fre-

8.4 Spherical Shells

507

Table 8.4 Temperature-dependent coefficients for . SU S304 and . Si 3 N4 [45] Material

Property

. P−1

. P0

. SU S304

.α[1/K ]

.0

.12.33e

. Si 3 N4

. P1

−6 +9

.8.086e

−4

.3.079e

−4

. E[Pa]

.0

.201.04e

. K [W/m K ]

.0

.15.379

.−1.264e

.ρ[kg/m 3 ]

.0

.8166

.0

−3

. P2

. P3

.0

.0

.−6.534e .2.092e

−7

−6

.0

.0 .−7.223e

.ν

.0

.0.3262

.−2.002e

−4

.3.797e

−7

.0

.C v [J/kg K ]

.0

.496.56

.−1.151e

−3

.1.636e

−6

.−5.863e

.α[1/K ]

.0

.5.8723e

−6 +9

. E[Pa]

.0

.348.43e

. K [W/m K ]

.0

.13.723

.9.095e

−4

.−3.07e

−4

.−1.032e

−3

.0 .2.16e

−7

.5.466e

−7

.−8.946e

− 11

.−7.876e

− 11

.ρ[kg/m 3 ]

.0

.2370

.0

.0

.0

.0

.0.24

.0

.0

.0

.C v [J/kg K ]

.0

.555.11

.1.016e

.2.92e

− 10

.0

.ν

−3

− 10

.0

−7

.−1.67e

− 10

/ Table 8.5 Comparison of the first linear frequency parameters .^ ω = ω R 2 sin2 (ϕ0 ) ρ/Eh 2 for spherical shells with .ν = 0.3, .ϕ0 = 0.001◦ and IM-C type of boundary condition R . (1 − Source . Nϕ = 5 . Nϕ = 7 . Nϕ = 9 . Nϕ = 11 . Nϕ = 13 . Nϕ = 15 h cos(ϕ0 )) 0

2

5

Present Xu and Chia [57] Present Xu and Chia [57] Present Xu and Chia [57]

3.161

3.090

3.091

3.091

3.091

3.091 3.091

6.513

6.375

6.375

6.375

6.375

6.375 6.367

14.031

13.276

13.267

13.266

13.266

13.266 13.291

quency remains unchanged for . Nϕ > 12, for the subsequent results, the number of nodal diameters in the HDQ method is set equal to . Nϕ = 13. As mentioned in the literature review section, there is no reported work on the thermally induced vibration of shallow spherical FGM shells. However, for the case of solid circular plates, results of Kiani and Eslami [15] are available in the open literature. Since the shallow spherical shell reduces to a circular plate when the rise in the shell is set equal to zero, a comparison may be done between the results of the present formulation and those of Kiani and Eslami [15]. For the sake of comparison, the radius of curvature is set equal to a large number, i.e., . R = 1000 m. The thickness is set equal to.h = 1 mm and the half opening angle is.ϕ0 = 0.08 × 10−3 . Under such conditions, the radius of circular plate will be . Rϕ0 = 80 mm which is the same with the one used by Kiani and Eslami [15]. A shell with case 1 of thermal loading is considered where the metal rich surface is kept at reference temperature and the

508

8 Thermal Induced Vibrations

0.8 0.6 0.4 0.2 0

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Fig. 8.18 Comparison of thermally induced vibrations in a solid circular plate with the IM-S boundary conditions with those of Kiani and Eslami [15]

ceramic rich surface is experiencing 10K temperature elevation. The power law index for this example is set equal to .ζ = 5. Comparison is done in Fig. 8.18. It is seen that results are in excellent agreement with those of Kiani and Eslami [15], which are obtained by means of the polynomial Ritz method.

8.4.7 Parametric Studies After validating the present formulation and solution method, novel numerical results are given in this section. For the first study in this section, the influence of temperature dependence of the constituents is examined. Two different studies are done and the results are illustrated in Figs. 8.19 and 8.20. These figures are associated with spherical shells of .ζ = 0 and 2, respectively. The other properties of the shells are . R = 800 mm, ◦ .φ0 = 15 , . Tm = 300 K, . Tc = 1000 K, .h = 5 mm, with the IM-S type of boundary conditions. For each figure, the temporal evolution of temperature at the mid-surface, the deflection at the apex, the thermally induced force resultant, and the thermally induced bending moment resultants are provided. Two types of analysis, namely TD and TID are performed. It is seen that under the TD analysis, which is the real state of material properties, the temperature profile is lower compared to the TID curve. Also, the magnitude of thermal bending moment increases when the temperature-dependent analysis is performed. The accurate values of thermal force and lateral deflection also depend on the temperature dependence of the constituents. It is further observed that the nature of temperature profile, thermal bending moment, and thermal force are all of diffusive type since the heat conduction equation is a parabolic equation. For the studied cases in Figs. 8.19 and 8.20, deflections in the shell are also of diffusive type, and no specific vibrations are detected. This is due to the fact that the geometries of the shell belong to a thick shell. For such cases, only the quasi-static response is observed and the inertia terms are not exited and may be dropped out from the equations of motion.

8.4 Spherical Shells

509

700 650 600 550 500 450 400 350 300

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0.5 0.4 0.3 0.2 0.1 0

106

7 6 5 4 3 2 1 0

7000 6000 5000 4000 3000 2000 1000 0

Fig. 8.19 Characteristics of an FGM spherical shell with IM-S edge under rapid surface heating with = 0, . R = 800 mm, .ϕ0 = 15◦ , .Tm = 300 K, .Tc = 1000 K, and .h = 5 mm. From top to bottom, temperature at the mid-surface, the apex deflection, thermally induced force resultant, and thermally induced bending resultant are depicted

.ζ

510

8 Thermal Induced Vibrations 650 600 550 500 450 400 350 300

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

106

8 7 6 5 4 3 2 1 0

7000 6000 5000 4000 3000 2000 1000 0

Fig. 8.20 Characteristics of an FGM spherical shell with IM-S edge under rapid surface heating with = 2, . R = 800 mm, .ϕ0 = 15◦ , .Tm = 300 K, .Tc = 1000 K, and .h = 5 mm. From top to bottom, temperature at the mid-surface, the apex deflection, thermally induced force resultant, and thermally induced bending resultant are depicted

.ζ

8.4 Spherical Shells

511

Since temperature dependence is an important factor for accurate evaluation of deflection, thermal force and moment, and temperature profiles, therefore the TD case of material properties is considered in the subsequent sections.

Influence of inertia In this example, the case of a thin FGM spherical shell subjected to rapid surface heating is considered. For the numerical results of this study, which are shown in Fig. 8.21, the following properties are used. Shells with various power law indices, ◦ . R = 800 mm, .φ0 = 5 , . Tm = 300 K, . Tc = 480 K, and .h = 1 mm are considered. For this example various power law indices are taken into consideration, which are .ζ = 0, 0.5, 1, 5. It is seen that, unlike the observations from the previous example, in this example the dynamic and quasi-static responses are totally different. This is also verified previously by many other authors, see, e.g., Kiani and Eslami [15]. In thin class of shells under rapid surface heating the complete dynamic analysis should be considered to reach the accurate deflections in the shell. Also, examination of power law index reveals that for the constituents of this study as the power law index increases, the deflection of the shell also increases. This is expected since as the power law index increases, the volume fraction of ceramic phase decreases. Since for the constituents of this study the elasticity modulus of metal is less than that of ceramic, an increase in the power law index results in the lower stiffness of the shell and therefore higher lateral deflection. As expected, when time evolutes the quasistatic response tends to the steady-state response. During deformation, the dynamic response oscillates around the quasi-static response. Influence of opening angle (.ϕ0 ) The next study in this research aims to analyze the importance of opening angle on the characteristics of the FGM spherical shells subjected to rapid surface heating. Results from this study are depicted in Figs. 8.22 and 8.23. The characteristics of the shell in this example are as follows: .ζ = 1, . R = 1000 mm, .Tm = 300 K, .Tc = 480 K, and .h = 1 mm. The boundary conditions are assigned as the IM-S type. In Fig. 8.22 the deflected shapes of the spherical shells at .t = 0.1 s are given for different opening angles of the shell. It is seen that for lower values of the opening angle of the shell, the maximum deflection is located at the apex. However, when the opening angle increases the maximum deflection moves toward the edge of the shell. The variation of maximum deflection with respect to opening angle is not monotonic. Figure 8.23 also performs a study on the effect of opening angle of the spherical shell on the dynamic characteristics of the shell under rapid heating. In this example shells with IM-S boundary conditions are assumed where the properties are .ζ = 1, . R = 1000 mm, . Tm = 300 K, . Tc = 480 K, and .h = 1 mm. The magnification factor in the shell is defined by the ratio of maximum dynamic to static deflections. This ratio is plotted versus the opening angle. It is concluded that, in general, as the opening angle

512

8 Thermal Induced Vibrations

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.02

0.04

0.06

0.08

0.1

0

0.02

0.04

0.06

0.08

0.1

0

0.02

0.04

0.06

0.08

0.1

Fig. 8.21 Thermally induced vibrations in thin shallow spherical shells with different power law indices under case 1 of rapid surface heating

8.4 Spherical Shells

513

1.2 1 0.8 0.6 0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 8.22 The deflected shape of the IM-S shallow spherical shell for different opening angles at .t = 0.1 s. Characteristics of the shell are .ζ = 1, . R = 1000 mm, .Tm = 300 K, .Tc = 480 K, and .h = 1 mm 1.25 1.2 1.15 1.1 1.05 1 0.3

3.3

6.3

9.3

12.3

15

Fig. 8.23 The value of dynamic magnification factor for the FGM IM-S spherical shells subjected to rapid surface heating. Characteristics of the shell are .ζ = 1, . R = 1000 mm, .Tm = 300 K, .Tc = 480 K and .h = 1 mm

increases the magnification factor enhances. Higher magnification factor indicates that the role of inertia parameter is prominent in the accurate evaluation of the shell response under rapid surface heating.

Influence of in-plane edge supports The aim beyond this example is to show the importance of the in-plane edge supports on the dynamic deflections of the spherical shells subjected to rapid surface heating. A shell with characteristics .ζ = 2, .ϕ0 = 3◦ , . R = 1000 mm, and .h = 1 mm is considered. Two different types of loads are assumed. The case 1 of loading, where the inner surface of the shell is kept at reference temperature and the other surface is subjected to rapid heating .Tc = 450 K. The case 3 of thermal loading, where the inner surface is thermally insulated and the ceramic rich surface is subjected to heat flux. Results are provided in Fig. 8.24 for both M-S and IM-S types of boundary conditions. It is seen that for both cases of boundary conditions the induced vibrations in IM-S case is less than the M-S case due to the higher constrains on the support. Also, for

514

8 Thermal Induced Vibrations

1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.02

0.04

0.06

0.08

0.1

0

0.02

0.04

0.06

0.08

0.1

1 0.8 0.6 0.4 0.2 0

Fig. 8.24 Importance of in-plane boundary conditions on the temporal evolution of shallow spherical shells subjected to rapid surface heating. Shells with.ζ = 2,.ϕ0 = 3◦ ,. R = 1000 mm,.h = 1 mm, and different types of heating are considered

case 1 of loading higher thermally induced vibrations occur. Comparison of IM-S and M-S boundary conditions reveals that, deflection in M-S shells becomes steady faster than the IM-S shells.

Influence of geometrical nonlinearity The importance of geometrical nonlinearity is depicted in Fig. 8.25 for shallow spherical shells subjected to case 2 of thermal loading. Here, the ceramic rich surface is subjected to .Tc = 400 K while the opposite surface is thermally insulated. Results are obtained for both IM-S and M-S shells where the characteristics are as .ζ = 1, ◦ .ϕ0 = 3 , . R = 1000 mm, and .h = 1 mm. It is seen that geometrical nonlinearity is an important factor in accurate estimation of the shell deflection. For the studied case in this example, linear analysis results in overestimation of the response of the shell. An analysis is performed in this work to explore the thermally induced vibration phenomenon in shallow spherical shells made of FGMs. All the thermo-mechanical properties of the shell are assumed to be position and temperature dependent. The governing equations of the shell are established using the Donnell shallow shell theory, the von Kármán nonlinearity and the first-order shear deformation shell theory. The three coupled partial differential equations are obtained and solved in the space

8.4 Spherical Shells 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

515

0

0.02

0.04

0.06

0.08

0.1

0

0.02

0.04

0.06

0.08

0.1

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1

Fig. 8.25 Importance of geometrical nonlinearity on the deflection of the spherical shell subjected to rapid surface heating. Shells with .ζ = 1, .ϕ0 = 3◦ , . R = 1000 mm, .h = 1 mm, and case 2 of thermal loading with .Tc = 400 K are assumed

domain using the HDQ method. These equations are highly coupled and nonlinear and should be solved using a linearization technique such as the Newton–Raphson. Temporal evolution of displacement components is achieved using the Newmark time marching scheme. The heat conduction equation is established through the thickness direction. This equation is also nonlinear since the properties are assumed to be temperature dependent. The temporal evolution of temperature profile is also obtained using the GDQ method and the Crank–Nicolson method. The adopted numerical methods may be used for arbitrary choices of thermal and mechanical boundary conditions. Numerical results indicate that in thin spherical shells thermally induced vibrations indeed exist. However, as the shell becomes thicker, vibrations fade. Therefore, for sufficiently thick shells only the quasi-static response suffices. Numerical results indicate that the type of thermal and mechanical boundary conditions are both important on the response of the shell. Power law index is a controlling parameter for the shell response. For the constituents of this study, as the power law index increases the deflection in the shell increases.

516

8 Thermal Induced Vibrations

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53. Tornabene F, Viola E (2009) Free vibration analysis of functionally graded panels and shells of revolution. Meccanica 44(3):255–281 54. Tornabene F, Viola E (2009) Free vibrations of four-parameter functionally graded parabolic panels and shells of revolution. Eur J Mechanics-A/Solids 28(5):991–1013 55. Tornabene F, Viola E (2013) Static analysis of functionally graded doubly-curved shells and panels of revolution. Meccanica 48(4):901–930 56. Malekzadeh P, Rahideh H, Karami G (2006) A differential quadrature element method for nonlinear transient heat transfer analysis of extended surfaces. Int J Comput Methodol 49(5):511– 523 57. Xu CS, Chia CY (1992) Nonlinear vibration of symmetrically laminated spherical caps with flexible supports. Acta Mechanica 94(1–2):59–72

Index

A Annular plates, coupled equations, 301 Annular plates, coupled thermoelasticity, 297 Annular plates, energy equation, 303 B Base vectors, cross product, 28 Bending moment–deflection relations, 49, 55 Built-in edge, 52 C Chebyshev–Gauss–Lobatto distribution, 503 Christoffel symbol, 20 Christoffel symbol of the first kind, 20 Christoffel symbol of the second kind, 20 Circular plates, 54 Circular plates, equilibrium equations, 55 Circular plates, radial temperature, 66 Classical shell theories , 2 Codazzi condition, 164, 146 Comparison of different theories, 192 Compatibility equations, curvilinear coordinates, 25 Compatibility in surface theory, 153 Compatibility of shells, 154 Conical shells, axisymmetric loading, 401 Conical shells, rotating, 263 Conical shells, strain–displacement relations, 172 Conical shells, thermal stresses, 253 Contravariant base vectors, 26 Coordinate transformation, 11 Coupled, cylindrical, 332 Coupled cylindrical shell, 419 Coupled energy equation, 277, 278, 364 Coupled energy equation, conical shell, 403 Coupled energy equation, cylindrical shell, 337

Coupled energy equation, spherical shell, 380, 392 Coupled equations of motion, conical shell, 401 Coupled equations of motion, cylindrical shell, 363 Coupled equations of motion, spherical shell, 379, 391 Coupled spherical shell, 427 Coupled thermoelasticity, composite cylinder, 361 Coupled thermoelasticity, composite spherical shell, 389 Coupled thermoelasticity, conical shells, 399 Coupled thermoelasticity, effect of normal stress, 354 Coupled thermoelasticity, shell of revolution, 411 Coupled thermoelasticity, spherical shells, 378 Covariant base vectors, 26 Covariant derivative, 22 Covariant derivative, contravariant tensor, 23 Covariant derivative, covariant tensor, 23 Covariant derivative, higher order tensors, 23 Covariant derivatives of vectors, 30 Cylindrical panel, 223 Cylindrical shell, equilibrium equations, 199 Cylindrical shells, 159 Cylindrical shells, axially loaded, 202 Cylindrical shells, force resultants, 180 Cylindrical shells, strain–displacement relations, 172 Cylindrical shells, symmetrically loaded, 198 Cylindrical shells, thermal stresses, 206 Cylindrical shells, turbine combustor, 216

D Double Fourier series expansion, 279

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. R. Eslami, Thermal Stresses in Plates and Shells, Solid Mechanics and Its Applications 277, https://doi.org/10.1007/978-3-031-49915-9

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520 Dummy index, 8

E Energy equation, generalized, 437 Equations of motion, conical shells, 189, 478 Equations of motion, curvilinear coordinates, 24 Equations of motion, cylindrical shell with normal stress, 354, 362 Equations of motion, cylindrical shells, 191, 335 Equations of motion, Flugge second-order theory, 414 Equations of motion of shells, 182 Equations of motion, rectangular plates, 459 Equations of motion, spherical shells, 188, 499 Equilibrium equation, composite plates, 68 Equilibrium equations, in-plane forces, 51 Euclidean metric tensor, 12

F Finite element, coupled spherical shell, 382 Finite element equation, 311 Finite element formulation, 131 Finite element solution, 366 Finite element solution, coupled conical shell, 404 Finite element solution, spherical shell, 394 First fundamental quadratic form, 39 Flugge second-order shell theory, 412 Fourier transformations, 89 Free edge, 53 Free index, 9

G Galerkin finite element method, 308 Gauss condition, 146 Gaussian coordinates, 35 General differential quadrature, 462, 479, 492, 500, 505

H Heat conduction equation, 110, 462, 481 Heat conduction equation, FGM, 84 Heat conduction equation, spherical, 504

I Induced vibrations, conical shells, 473, 491

Index Induced vibrations, equations of motion, conical shells, 479 Induced vibrations, equations of motion, rectangular plates, 460 Induced vibrations, equations of motion, spherical shells, 499 Induced vibrations, rectangular plates, 456, 473 Induced vibrations, spherical shells, 495, 508

J Jacobian determinant, 11

K Kronecker delta, 10

L Lamè parameters, 147 Laplace transform, 281 Laplace transform, inverse, 283 Levy solution, 58 Lord and Shulman theory, 416 Love–Kirchhoff hypothesis, 2

N Navier solution, 56 Newmark method, 341, 367, 383, 395, 417, 480, 504 Newmark time marching method, 461 Nonlinear equations of motion, 440 Nonlinear heat conduction equation, 462 Nonlinear strain–displacement relations, 476, 496

P Permutation symbol, 10 Permutation tensor, 15 Piezoelectric plate, 125 Piezoelectric plate, equations of motion, 128 Piezoelectric stress resultants, 125 Plate, boundary conditions, 52 Plate, equilibrium equation, 48, 49 Plates, displacement equations of motion, 87 Plates, dynamic solution, 88 Plates, equations of motion, 86, 108 Plates, FGM, 83 Plates, first-order shear deformation, 105 Plates, GDQ method, 109 Plates, general solution, 60

Index Plates, higher order theory, 84 Plates, piezoelectric, 124 Plates, third-order shear deformation, 274 Plate theory, classic, 44 Plate theory, force–moment resultants , 45 Plate theory, strain–displacement, 45 R Rectangular FGM plates, thermal shock, 104 Rectangular plates, composite, 67 Rectangular plates, coupled equations, 276 Rectangular plates, shear deformation, 75 S Second fundamental quadratic form, 40 Second-order shells theory, 175 Sensor and actuator equations, 129 Shell base vectors, 156 Shell, Euclidean metric tensors, 156 Shell geometry, 147 Shell of revolution, Lamè parameters, 160 Shell, position vector, 154 Shells, effect of normal stress, 170 Shells, first-order shear deformation, 166 Shells, Flugge theory, 166 Shells, general strain–displacement relations, 163 Shells, Hooke’s law, 173 Shells, Love–Kirchhoff hypothesis, 166 Shells, Love–Timoshenko theory, 167 Shells of revolution, 157 Shells of revolution, Christoffel symbols, 159 Shells, physical components, 163 Shells, Reissner–Naghdi theory, 167 Shells, strain–displacement relations, 164 Shells, strain–displacement relations with normal strain, 170 Shells, strain tensor, 162 Shells, Vlasov theory, 168 Simply supported edge, 52 Spherical shells, 160 Spherical shells, force resultants, 179 Spherical shells, membrane analysis, 242 Spherical shells, strain–displacement relations, 171 Spherical shells, symmetrically loaded, 228 Spherical shells, with circular holes, 247

521 Stability, coupled cylindrical shell, 432 Stability, energy equations, 438 Stability equations of motion, 436 Stability, numerical method, 442 Stress–displacement, cylindrical shells, 175 Stress–displacement, spherical shells, 174 Surface, base vectors, 148 Surface, Codazzi and Gauss conditions, 153 Surface, Codazzi equations, 40 Surface, fundamental quadratic forms, 37 Surface, metric tensors, 36 Surface, partial derivative of unit vectors, 152 Surface, permutation symbol, 37 Surface theory, 35 Surface, unit vectors relations, 148

T Tensor, associated metric, 15 Tensor, contravariant, 14 Tensor, covariant, 14 Tensor, mixed, 15 Tensor, mixed metric, 15 Tensor, rank zero, 13 Tensors, area element, 29 Tensors, base vectors, 25 Tensors, contraction, 18 Tensors, covariant and contravariant vector components, 29 Tensors, cylindrical coordinates, 31 Tensors in Cartesian coordinates, 17 Tensors, partial derivatives, 18 Tensors, physical components, 33 Tensors, spherical coordinates, 32 Tensors, volume element, 29 Touloukian coefficients, 464, 483, 506 Touloukian formula, 104, 456 Transformation, improper, 12 Transformation, proper, 12

V Von Kármán geometrical nonlinearity, 476, 496

W Weak formulation, 309, 440