Thermo Dynamics of Plates and Shells (Springer 2007)

Thermo Dynamics of Plates and Shells (Springer 2007) This monograph is devoted to the investigation of nonlinear dynamics of plates and shells embedded in a temperature field. Numerical approaches and rigorous mathematical proofs of solution existence in certain classes of differential equations with various dimensions are applied. Both closed shell-type constructions and sectorial shells are studied.

Foundations of Engineering Mechanics

Series Editors: V.I Babitsky, J Wittenburg

Jan Awrejcewicz Vadim A Krysko

Anton V Krysko

Thermo-Dynamics of Plates and Shells

f ¨ur

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ISBN-13: 978-3-540-34261-8

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Jan Awrejcewicz

Department of Automatics and Biomechanics

Faculty of Mechanical Engineering

Technical University of Lodz

410054 Saratov, Russia

A E

LT X

© Springer-Verlag Berlin Heidelberg 2007

The present monograph is devoted to nonlinear dynamics of thin plates and shellswith termosensitive excitation Since the investigated mathematical models are ofdifferent sizes (two- and three-dimensional differential equation) and different types(differential equations of hyperbolic and parabolic types with respect to spatial coor-dinates), there is no hope to solve them analytically On the other hand, the proposedmathematical models and the proposed methods of their solutions allow to achievemore accurate approximation to the real processes exhibited by dynamics of shell(plate) - type structures with thermosensitive excitation Furthermore, in this mono-graph an emphasis is put into a rigorous mathematical treatment of the obtaineddifferential equations, since it helps efficiently in further developing of various suit-able numerical algorithms to solve the stated problems

It is well known that designing and constructing high technology electronic vices, industrial facilities, flying objects, embedded into a temperature field is ofparticular importance Engineers working in various industrial branches, and partic-ularly in civil, electronic and electrotechnic engineering are focused on a study ofstress-strain states of plates and shells with various (sometimes hybrid types) damp-ing along their contour, with both mechanical and temperature excitations, with

de-a simultde-aneous de-account of hede-at sources influence de-and vde-arious temperde-ature tions Very often heat processes decide on stability and durability of the mentionedobjects Since numerous empirical measurement of heat processes are rather ex-pensive, therefore the advanced precise and economical numerical approaches arehighly required

condi-A brief monograph description follows Chapter 1 of this monograph is devoted

to a study of three-dimensional problems of theory of plates in a temperature field.First, a brief historical outline as well as a state-of-art of the mentioned problems isdescribed in introductional section In Section 1.2, the system of differential equa-tions governing a broad class of problems in the coupled dynamic theory of ther-moelasticity in three-dimensional formulation is derived A difference variationalapproximation is given and the difference scheme error is derived Also stability of

an explicit difference scheme is rigorously studied

Section 1.3 includes a comparison of solving systems governed by either bolic or elliptic equations through various iterative methods

hyper-In section 1.4 numerous results of solutions of broad class of elasticity and moelasticity problems including coupling of temperature and deformations, are il-lustrated and discussed.

ther-In Chapter 2, after a brief historical research review, the variational equationsfor shallow anisotropic shells embedded into a temperature are derived Couplingconditions and stress-strain state of shallow shells are formulated In section 2.2universality and efficiency of finite difference method devoted to boundary valueproblems for elliptic equations if outlined Difference schemes for series of multi-dimensional stationary heat transfer equations are proposed in both sections 2.2 and2.3 In the last section 2.4, influence of heat sources on a shell stress-strain and itsstability is studied

Chapter 3 is devoted to analysis of dynamical behaviour and stability of closedcylindrical shells subject to continuous thermal load A brief historical background

is followed by variational formulation of the coupled dynamical problem of moelasticity Hybrid-type variational equations of thin conical composite orthotropicthermosensitive shells are derived, and a problem of their solution is rigorously dis-cussed Furthermore, a solution to the biharmonic equation in relation to forcingfunction, as well as reliability of the obtained results, are addressed Dynamical sta-bility loss and non-uniform thermal loading are also studied

ther-Dynamical behaviour and stability of rectangular shells is addressed in Chapter

4 In section 4.1, the computational algorithm to analyse differential equations withthe associated boundary conditions is derived The associated finite difference equa-tions are given, and reliability of the results are verified Stationary state method toanalyse statical and dynamical problems is illustrated in section 4.1.4 Various vibra-tional phenomena and stability loss are studied Stability of thin shallow shells withboth transversal and heat loads are examined in section 4.2 Section 4.3 is devoted

to stability of thin conical shells subject to both longitudinal load and heat flow.Finally, dynamical stability of flexurable conical shells with convection is studied insection 4.4

In Chapter 5 dynamics and stability of flexurable sectorial shells with thermalloads are addressed First, theory of flexurable sectorial shells is introduced Thefundamental relations are assumed, differential equations are derived and initial con-ditions are given After introduction of a thermal field the numerical “set-up” tech-nique is illustrated and discussed, and numerical results reliability is outlined Thenvarious examples of stability of sectorial shells with finite deflections are studied Inaddition, chaotic dynamics of sectorial shells and its control is addressed

Chapter 6 is devoted to a study of coupled problems of thin shallow shells in perature field within the Kirchhoff-Love kinematic model Fundamental assump-tions and relations are introduced, and the differential equations are derived Thefinite difference model of a solution to three dimensional heat conductivity equation

tem-is formulated Numerical algorithm to solve the obtained equations tem-is proposed, andthen numerous examples of investigation of stability loss of shallow rectangularshells follow Additional original method to solve a coupled thermoelastic problem

is also proposed

Preface VII

In chapter 7 a novel optimal and exact method of solving large systems of linearalgebraic equations In the approach under consideration the solution of a system ofalgebraic linear equations is found as a point of intersection of hyperplanes, whichneeds a minimal amount of computer operating storage The proposed methodmakes it possible to benefit from the essential advantages of both the direct method(universality, finitness of a computational process, exactness) and the iterational one(minimal amount of operational storage) Two examples are given In the first ex-ample, the boundary value problem for a three-dimensional stationary heat transferequation in a parallelepiped in R3is considered, where boundary value problems ofthe 1st, 2nd or 3rd order, or their combinations are taken into account The govern-ing differential equations are reduced to algebraic ones with the help of the finiteelement and the boundary element methods for different meshes applied The ob-tained results are compared with known analytical solutions The second exampleconcerns computation of a non-homogeneous shallow physically and geometricallynon-linear shell subject to transversal uniformly distributed load The partial differ-ential equations are reduced to a system of non-linear algebraic equations with the

ther-a differentither-al equther-ations governing vibrther-ations of ther-a plther-ate, i.e the modified Germther-ain-Lagrange equation of thermal conductivity (a parabolic equation)

Germain-Second, a coupled thermo-mechanical of non-homogeneous shells with variablethickness and variable Young modulus (a so-called Timoshenko type model) is stud-ied The investigated problem is reduced to uniformly correct problem in the firstform of a first order difference equation

Third, boundary conditions for a non-homogeneous first order operator – ential equation possessing a unique solution are derived Two important theoremsare formulated

A.V Krysko

Preface V

1 Three–Dimensional Problems of Theory of Plates in Temperature

Field 1

1.1 Introduction 2

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid 11

1.2.1 Variational equations 11

1.2.2 Differential equations 13

1.2.3 Difference approximation 21

1.2.4 Difference approximation Error 26

1.2.5 Difference approximation Stability 29

1.3 Methods of Solving Difference Equations 39

1.3.1 Dimensionless Equations 39

1.3.2 Systems of Elliptic Difference Equations 41

1.3.3 Systems of Parabolic and Hyperbolic Difference Equations 47 1.3.4 Algorithm 48

1.3.5 Reliability 54

1.3.6 Numerical Experiments 57

1.4 Linear Problems in the Theory of Plates in 3D Space 59

1.4.1 Static Problems 59

1.4.2 Dynamic problems 79

1.4.3 Non-stationary temperature field 82

1.4.4 Comparison of Solutions – non-isothermal Processes in Static Problems 87

1.4.5 Inner heat sources 109

1.4.6 Deformation and Temperature 114

1.5 3D Physically Non-Linear Problems 129

1.5.1 Differential equations and difference approximation 130

1.5.2 Algorithm 133

1.5.3 Estimation of Convergence 134

1.5.4 Temperature and Deformation Coupling 136

2 Stability of Rectangular Shells within Temperature Field 149

2.1 Introduction 150

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields 152

X Table of Contents

2.2.1 Problem formulation and assumptions 152

2.2.2 Fundamental relations 153

2.2.3 Variational and differential equations 159

2.2.4 Boundary and compatibility conditions 167

2.2.5 Compatibility conditions for shallow shells equations 177

2.2.6 Temperature field 183

2.3 Solution of 3D Stationary Heat Transfer Equation 186

2.3.1 The method 186

2.3.2 Construction of difference schemes 194

2.3.3 A priori convergence estimation 206

2.3.4 Algorithm of computation and compatibility conditions 209

2.3.5 Problems 215

2.4 Algorithm for Difference Equations 227

2.4.1 Construction of difference equations 227

2.4.2 Stability problems 232

2.4.3 Reliability of obtained results 234

2.4.4 Transversal load 236

2.4.5 Different boundary conditions 238

2.5 Computations of Plates and Shells in a Temperature Field 252

2.5.1 Stress-strain state 252

2.5.2 Stress-strain state and shells stability 264

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells 267

3.1 Introduction 267

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells 276

3.2.1 General Introduction 276

3.2.2 Variational Formulation 278

3.2.3 Hybrid-Type Variational Equations 283

3.2.4 Solution Existence 297

3.2.5 Classification 304

3.3 Computational Algorithms 310

3.3.1 Finite Difference Equations 310

3.3.2 Solution to Biharmonic Equation 315

3.3.3 Reliability of the Obtained Results 320

3.3.4 Modified Relaxation Method 328

3.4 Dynamical Stability Loss with Ununiform Force Excitation 334

3.4.1 Criteria of Dynamical Stability Loss (A Review) 334

3.4.2 Nonuniform Impulse External Pressure 342

3.5 Dynamical Stability Loss and Non-uniform Thermal Load 366

3.5.1 Thermal Field Computation 366

3.5.2 Influence of Time, Shell Geometry and Load 373

3.5.3 Combined Static and Thermal Loads 382

4 Dynamical Behaviour and Stability of Rectangular Shells with

Thermal Load 395

4.1 Introduction 395

4.2 Algorithm 402

4.2.1 Differential Equations, Boundary and Initial Thermoelastic Conditions 402

4.2.2 Finite Difference Equations 411

4.2.3 Reliability of the Results 412

4.2.4 Stationary State Method 420

4.3 Stability of Thin Shallow Shells 432

4.3.1 Influence of Heat Stream Intensity 432

4.3.2 Shells with Transversal Load and Heat Flow 436

4.3.3 Influence of Thermal and Mechanical Characteristics 446

4.4 Stability of Thin Conical Shells 458

4.4.1 Boundary Conditions and Surrounding Medium 458

4.4.2 Constant Compressing Load and Heat Flow 464

4.4.3 Harmonic Longitudinal Load and Heat Flow 466

4.5 Stability of Flexurable Conical Shells with Convection 479

4.5.1 Problem Formulation 479

4.5.2 Boundary and Thermal Fields Conditions 480

4.5.3 Critical Temperature Versus Heat Transfer Coefficient 483

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells 493

5.1 Introduction 493

5.2 Flexurable Conical Sectorial Shells Computations 498

5.2.1 Fundamental Relations, Differential Equations, Boundary and Initial Conditions 499

5.2.2 Thermal Field and Set-Up Method 509

5.2.3 Results 515

5.3 Stability of Sectorial Shells with Finite Deflections 520

5.3.1 Influence of the Sector’s Angle 522

5.3.2 Set-Up Method and Determination of Critical Loads 560

5.3.3 Heat Impact 595

5.3.4 Local Surface Load With Infinite Duration 603

5.4 Chaotic Dynamics of Sectorial Shells 614

5.4.1 Statement of the problem and computational algorithm 614

5.4.2 Static problems and reliability of results 617

5.4.3 Convergence of a finite difference method along spatial coordinates for non-stationary problems 618

5.4.4 Investigation of chaotic vibrations of spherical sector-type shells 625

5.4.5 Transitions from harmonic to chaotic vibrations 627

5.4.6 Control of chaotic vibrations of flexible spherical sector-type shells 630

XII Table of Contents

6 Coupled Problems of Thin Shallow Shells in a Temperature Field 633

6.1 Introduction 633

6.2 Fundamental Assumptions and Relations 634

6.3 Differential Equations 636

6.4 Boundary and Initial Conditions 639

6.5 Solution to 3D heat conductivity equation 640

6.6 Algorithm 642

6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields 645

6.8 Additional Method 666

7 Novel Solution Method for a System of Linear Algebraic Equations 671

7.1 Introduction 671

7.2 Elimination method for equations 673

7.3 Numerical solution of a three-dimensional equation of elliptic type 684 7.4 Computation of geometrically non-linear non-homogenous shallow shells 696

8 Mathematical Approaches to Coupled Termomechanical Problems 705

8.1 Existence and Uniqueness of Solution of One Coupled Plate Thermomechanics Problem 705

8.1.1 Introduction 705

8.1.2 Basic assumption 707

8.1.3 Main results 708

8.2 On the Solution of a Coupled Thermo-mechanical Problem 713

8.2.1 Introduction and Statement of the Problem 713

8.2.2 Method 715

8.3 On the Solvable Operators Generated by Uniformly Correct Problems 719

8.3.1 Introduction 719

8.3.2 Method 719

References 723

Index 767

Temperature Field

In section 1.1 historical outline putting emphasis on not solved problems in dimensional formulation of plates thermoelastic theory is given

three-Section 1.2 presents a system of differential equations describing a broad class

of problems of the coupled dynamic theory of thermoelasticity in a complete, dimensional formulation including material’s non-homogeneity The investigatedsystem of equations has been supplemented with an equation at singular points ofthe examined space (a cubicoid), such as ribs, corners and simple points where vari-ous boundary conditions meet A difference approximation of the initial differentialsystem has been formulated with the use of the variational-difference method (themethod of integral identity) The margin of the difference scheme error has beenestimated A theorem concerning stability of an explicit difference scheme has beenproven and the condition of stability that guarantees weak convergence of the dif-ference scheme’s solution towards the solution of a differential system has beenobtained

three-Section 1.3 contains a comparison of solving systems of hyperbolic equations(using an explicit difference scheme based on applying Runge-Kutta’s method withautomatic choice of an integration step and Runge-Kutta’s method with a constantstep) Additionally, the section presents a comparison of applied iterative methods

of solving systems of elliptic equations (Seidel’s method, the upper relaxation, theexplicit and implicit methods of variable directions, and the explicit method of vari-able directions with the so-called Chebyshev’s acceleration) Several model prob-lems have been used to draw the comparisons and the most economical methodshave been applied as far as accuracy of solutions and computation time are con-cerned Algorithms of the described methods have been formulated and a package

of programs for solving problems of statics, quasistatics and elasticity and moelasticity dynamics has been created An optimum choice of a spatial mesh stepand an integration step within a time interval has been made and legitimacy of thetheoretically obtained (in the first section) stability condition has been numericallyconfirmed Feasibility of the obtained results has also been proven by means ofcomparison with real processes

ther-Section 1.4 presents numerous results of solutions to a broad class of elasticityand thermoelasticity problems within the range of static, quasistatic and dynamicproblems There is also an analysis of the influence of the temperature and deforma-tions’ coupling’s effect using some examples of thermal and mechanical impacts

2 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

Finally, section 1.5 contains formulation of the equations of coupled dynamicthree-dimensional problems with physical non-linearities Moreover, the finite dif-ference methods, Runge-Kutta’s method and the method of additional loads havebeen combined to form a numerical algorithm of solutions Convergence of an ap-proximate solution to the real one (the one searched for) has been analysed Theresults of problems concerning thermal and mechanical impacts beyond the elas-ticity fields have been presented and the effects of the influence of reciprocal tem-perature and deformation fields’ coupling on the analysed processes have also beeninvestigated in this chapter

1.1 Introduction

While designing and constructing electronic devices, industrial facilities, flying jects or technological instrumentation, the problems related to heat processes areparticularly important They appear due to the use of new materials, more complexloads affecting every single element of analysed objects, and also due to an increase

ob-of permissible heat loads in devices ob-of smaller and smaller dimensions As it isgenerally known, heat processes determine stability of functioning and durability

of analysed objects On the other hand though, numerous empirical measurements

of heat processes are extremely complex and expensive Therefore, exact tional analyses (numerical, as well as analytical) ought to be conducted in order toobtain constructions of optimum characteristics

computa-In fact, non-stationary temperature reactions in surrounding environment requiremore accurate calculations than classic modelling of thermomechanical phenomena

In 1845, Duhamel [188] was the first to formulate the theory of elasticity regardingthermal stresses However it was not until 1956, that Biot [107] introduced a dissi-pation function into a thermal conduction equation to account for the heat caused

by the material’s deformation Thus, the problem of thermoelasticity and the ational principle of coupled theory of thermoplasticity were first formulated Sincethen there has been a great interest in that sort of problems

vari-Earlier works on the theory of thermoelasticity [188] presented a dominatingview that a change of temperature within a time interval is small, and therefore itwas possible to apply a simplified (quasistatic) method, that is to neglect inertialterms in equations of motion, without the risk of major errors The next step, in-troduced by means of the theory of thermoelasticity to simplify the problem, wasneglecting dilatation terms in heat conduction equations Sometimes, when both ofthe above mentioned terms are neglected in differential equations [598], the solution

of a static problem is found It turns out though, that due to the significance of theproblems such simplifications ought not to be made Among such problems are: theproblem of investigating stress waves in deformable bodies; the problems related todetermining thermoelastic vibrations; the problems related to investigating stabil-ity of conservative elastic systems [119, 164, 267, 316, 356, 466] In their works,Danilovskoya [160, 161, 162, 163, 164], Kartashova and Shefter [316] analysed theinfluence of inertial terms on bodies’ behaviour considering the inertia forces They

also proved that neglecting a dilatation term does not ensure qualitatively tory results due to inefficient examination of the coupling coefficient’s influence onthe phenomenon.

satisfac-All the factors mentioned above caused a growth of interest in complete (i.e notsimplified) problems which fruited in numerous analytical works

Works of Karlsoy and Eger [315], Lykov [451], Kovalenko [355] and Nowacki[512] contain analyses and generalisation of two, so far independent disciplines, i.e.the theory of elasticity and the theory of heat conduction, and also a definition of socalled coupled problem A full formulation of the principles of variational theories

of thermoelasticity is to be found in works [107, 265] Betti’s theorem on reciprocity

of virtual works is discussed in monograph [516], and a generalisation of Maizel’smethod may be found in work [453] Formulation of flat and space problems of cou-pled quasistatic theory of thermoelasticity is described in the works of Podstrigach,Schvetz, and Nowacki [512, 516, 545, 546, 547, 548] Nowacki’s monograph [513]introduces equations of the coupled theory of thermoelasticity into wave equationsand a method of solving linear and non-linear variants of the problems listed above.Many popular methods of solving the equations of Galerkin’s [215] or Papkovich’s[528] classic theories of elasticity are generalized in Podstrigach’s or Nowacki’sworks and applied into the theory of coupled thermoelasticity The method of solv-ing problems of the coupled theory of thermoelasticity in case of a boundless spacewas proposed by Zorski [727], who used Green’s function to solve a heat conduc-tion equation and considered dilatation to be a heat source Chadwick’s work [145]takes up generalized problems of solving boundary problems of the coupled theory

of thermoelasticity with the use of integral methods, whereas Souler and Brul usethe small parameter method [632]

The problems related to accuracy of formulated boundary problems of the pled theory of thermoelasticity were described first in book [119], which investi-gates an initial boundary problem for an isotropic body, later extended also onto ananisotropic body in Ionescu work [277]

cou-Numerous dynamical problems of mathematical physics apply various integraltransformations, including Laplace’s transformation [294], the solution of which

is related to the use of Fourier’s series In their work, Kupradze and others [398]propose their theory of multidimensional singular integral equations that makes itpossible to investigate the static and dynamic problems of stabilised continuous sys-tems’ vibrations Hybrid problems, investigated by Magnaradze [452], Kupradzeand Burchuadze [397] may be solved with generalized integrals that correspond todifferential equations with the use of harmonic and analytical functions

Defermos’ work [175] contains many theorems concerning basic problems ofthe theory of thermoelasticity, including their proofs Work [101] investigates theso-called second and third boundary and initial boundary problems of the coupledtheory of thermoelasticity with the use of the method of potential and Laplace’stransformation Work [397] analyses four basic three-dimensional boundary prob-lems of the theory of thermoelasticity in case of harmonic vibrations of a ho-mogeneous isotropic medium with the following conditions set in its boundaries:

4 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

1) displacement and distribution of temperature; 2) thermal stress and thermalflux; 3) displacement and thermal flux; 4) thermal stress and distribution of tem-perature In addition, the authors formulate and prove many theorems concerningthe existence and uniqueness of the above mentioned problems The solutions toall of the four types of boundary conditions, presented in the form of generalizedFourier’s series, are to be found in Burchuladze’s work [135] Fundamental resultsreferring to the initial boundary problems of the theory of thermoelasticity have beenobtained in the work of Kachnashviliev [294] Nevertheless, fundamental solutionsare still being perceived as classic The conditions of smoothness appear to be too

difficult to achieve for solutions of a wave equation describing impact processes.Due to the fact that such solutions do not have derivatives of the first order, theyneed to be examined from a generalized perspective Integral relations contain in-formation about solutions and emphasise physical phenomena because information

on solution’s smoothness is partially lost in differential equations

The generalized mathematical theory on differential equations of the coupledtheory of thermoelasticity described by means of both hyperbolic and parabolicequation has been formulated relatively recently The works of Ladyzhenskaya[405] and Ilyisn [276] that were published in early fifties, contain numerous vitalresults referring to the theory of boundary problems for one hyperbolic or parabolicequation of a general type In order to prove the existence and uniqueness of a gen-eralized equation, it is necessary to make an entirely new a priori estimation thatwould take into account the right parts of equations in the form of the weakest normand thus would accurately emphasise the physical aspect of the problem

Qualitatively most adequate examinations of general solutions seem to be theones that apply the finite difference method The method definitely stands out amongmany other approximate methods Owing to continuing research of Samarskiy,Gulin, Nikolaev [591, 593, 594, 595], a large number of problems concerning sta-bility of difference schemes for all types of one-dimensional equations in mathemat-ical physics have been solved This also started the research on difference schemes

in the theory of elasticity Let us list only a few examples of important results tained with the use of the theory of difference schemes Work [419] describes an

ob-a priori estimob-ation of ob-a solution in spob-aces W22,2, W2,1

2 made by means of energy equalities for dynamic problems of the theory of thermoelasticity using Dirichlet’shomogeneous boundary conditions The authors have also constructed and exam-ined a non-overt difference scheme and proved its convergence In his work [483],Moskalkov presents a method of constructing difference schemes for the coupledtheory of thermoelasticity boundary problems that is also useful for the equations ofvariable or discontinuous coefficients Work [541] proposes a variational-differenceformulation of the difference scheme of the coupled theory of thermoelasticity prob-lems Work [341] proves convergence of the difference solution towards the solution

in-of a general hybrid problem for a hyperbolic equation with variable coefficients Italso shows how to improve the accuracy of presently applied difference schemes Inworks [419, 694], the relation between the smoothness of a solution to the coupledtheory of thermoelasticity one-dimensional dynamic problems and the smoothness

of input data is examined Smoothness is examined with the use of terminology plied for Hilbert and Sobolev’s spaces Two difference scheme families have beenconstructed and their stability and convergence have been studied Works [419, 693]extend the investigated problems by taking into account two-dimensionality or manyso-called layer problems It is worth noticing that at present, many finite differen-tial problems modelling the flat problem of the dynamic theory of elasticity and thetheory of thermoelasticity have already been solved A large number of schemesdescribed by displacements of high accuracy, stability and short computation timehave also been presented [79, 96, 97, 345, 484, 591, 592, 664] Among the lessthoroughly examined problems are the ones that refer to the differential method ofsolving initial-boundary problems of the three-dimensional theory of elasticity andthe theory of thermoelasticity A review work by Suslova [643] contains a broadbibliography of works on research focused on solving boundary problems of thethree-dimensional theory of elasticity It also lists several works concerning the the-ory of thermoelasticity [142, 293, 643] In works [198, 199] Ermolenko describesconstructing the solution of a hybrid problem for a cubicoid by cutting the finitespace out and he proves stability and convergence of the cubic difference process byapplying the transformation of Lam´e’s equations He compares the result obtained

ap-in this way to the accurate one In works [339, 340] Konovalov describes stabilityconditions for difference schemes for two-dimensional dynamic and static hybridproblems

The development of computational methods using computers and special rithms has led to a sudden progress in the discussed field of science A major contri-bution in the development of computational methods in the research on the dynamics

algo-of continuous media has been brought by the works algo-of Godunov [224], Kukudzanov[393], Neuman [500], Rachmatulin [561, 562], Richtmyer [572], Wilkins [703] andJanenko [287] Numerous examples of computations regarding the mechanics of acontinuous medium are included in monographs [225, 287, 394, 573] The problem

of the coupled theory of thermoelasticity still remains a live issue due to its potentialapplication and the numerical methods allow drawing a great deal of conclusions of

a general nature The examples of these may be the research and solutions of pled thermoelasticity problems with the use of numerical methods for a number ofparticular issues: in work [546], Galerkin’s method is applied for solving a coupledproblem in a finitely dimensional space with the use of a three-dimensional model;

cou-in work [616], the same method is applied to solve a two-dimensional problem; cou-inwork [430], a half-space finite difference method is applied for a one-dimensionalproblem, and in works [220, 721] – for a three-dimensional problem

In work [266], Huang and Shich compare solutions of free vibration problemsregarding thermal processes in plates and spherical shells by applying dynamicand quasistatic theories Non-stationary thermoelasticity problems for an infinitetwo-layered and initially heated plate consisting of various materials and thermallyprocessed through interaction with fluids within Newton’s laws, have been examined

in work [646] Work [649] analyses stress-strain states of thick two-layered sphereswith regard to axially symmetrical heat sources (the problem has been solved with

6 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

the use of the quasistatic theory) Work [648] investigates a system of coupled moelasticity differential equations with the use of a cylindrical coordinate system.Fourier’s method has been used to examine stress-strain states in a long circularcylinder with inserted rigid rings in work [504] The finite difference method hasbeen used to solve the problem of thermoelasticity for a rectangular orthotropic platewith regard to the dependence of its certain characteristics on temperature in work[641] Work [663] investigates a non-stationary coupled thermoelasticity problemfor an infinitely long, thick plate The plate’s surfaces have been subjected to in-tensive heating and the coupling between the temperature field and the deformationhas been analysed The distribution of the temperature field in time has also beenexamined, as well as concentration of the stresses depending on the size of the stressfield and the material’s thermodynamic properties Dynamic loss of stability of thinplates has been analysed with the use of finite difference method in work [191],taking into account the effect of reciprocal coupling of the temperature field andthe deformation field Work [324] presents a solution to the coupled thermoelastic-ity problem for a thin rectangular shell affected by a three-dimensional temperaturefield It also mathematically proves the convergence of the obtained approximatesolution

ther-All of the above mentioned works point out the differences which appear in tions if the coupling of the deformation (strain) fields and the temperature fields arenot taken into account An increase of the coupling coefficient leads to an increase

solu-of interactions, which consequently leads to damping solu-of the produced thermoelasticwaves Works by Karnauchov [312] and Pobedria [541] are focused on the problem

of coupling in the theory of thermoelasticity The influence of coupling on the strain state of elastic and elastoplastic constructions has been investigated in work[359] Several works of Day [169, 170, 171, 172, 173, 174] are also worth attentionsince the author investigates the conditions of legitimacy of applying approxima-tions of unbounded theory of thermoelasticity and also the conditions of applyingthe properties of the solutions of heat conductivity equations to the solutions of aonedimensional dynamic coupled thermoelasticity problem’s equations

stress-Research on thermal processes with regard to finite velocity of heat transfer

is another direction in the development of the theory of thermoelasticity, since anentire class of physical processes (highly intensive thermal processes, laser rays)should be presented from the perspective of generalized Fourier’s law [451] Works[323, 429, 495, 496, 558, 627] have been dedicated to the research on dynamicprocesses in solid bodies with regard to the heat transfer finite speed In the works

of Engelbrecht and Ivanov [285], an analysis of one- and two-dimensional models ofwave processes have been made In Kolyano and Shter’s work [337], a variationalprinciple of reciprocal coupling of thermoelasticity for non-homogeneous mediahas been investigated using a cantilever beam as an example Coupling of the defor-mation field and the temperature field significantly affects the solution’s character,especially in the problems of spreading impact fields in thermoelastic bodies There-fore, the research on the dynamic coupling effects occurring in thermoelastic bodiessubjected to simultaneous thermal, impulse, impact and mechanic treatment is one

of the most important issues these days Danilovskaya [163, 164] was the first to amine the dynamic effect in the “impact” problem along a half-space The researchwas consequently carried on by Mura [489] If the temperature on the surface of abody changes at a limited speed instead of sudden leaps, then the problem may besolved with a small parameter method [494] In Pobrushin’s work [544], an analysis

ex-of some one-dimensional initial-boundary problems with thermal and mechanicalimpacts along the symmetry axis of an infinite rod has been made The dynamiccoupled thermoelasticity problem for a half-infinite plate at a simultaneous increase

of temperature on its edge and with the use of Laplace’s integral transformationincluding the small parameter method has been solved in Sidlar’s work [617] Dy-namic behaviour of thin cylindrical shells subjected to impetuous thermal treatmenthas been investigated in work [632] A coupled system of differential equations

is derived with the use of Bubnov-Galerkin method and variational theorems, andalso a simple-supported infinite cylindrical shell is investigated Work [359] investi-gates dynamic thermoelastic processes during heat impacts in such construction ele-ments as plates or spherical and cylindrical shells The research has been conductedwith the use of dynamic coupled thermoelastic equations and dynamic non-coupledequations of thermoelastoplasticity, and with the method of reduction to a series

of non-coupled quasistatic problems, which in turn have been solved with Bogolubov method In Kuvyrkin’s work [402], a heat impact in the surface layer

Krylov-of a body limited by a curvilinear surface has been investigated Shatalov’s work[608] shows that a decrease of equations’ couplings leads to a decrease of strain

in the front of a thermoelastic wave A method of expansion into power series inregard to a small parameter being the thermomechanical coupling has been applied

in that case Gayvas’ work [221] presents an analytical solution to a ity problem for a plate with discontinuity caused by heat impact The behaviours ofplates subjected to steady mechanical load and rapid thermal transients on their bothsurfaces have been investigated in work [231] Few of the solved problems that arerelated to impacts belong to the class of problems with aperiodic excitations In thisrespect the theory of thermoelasticity seems to be a little underdeveloped and it facessome significant mathematical problems Due to simultaneous mechanical and ther-mal impacts in constructions some small plastic deformations are ignored The firstwork focused on investigation of elastoplastic stress states was published by Iliushin[272], and later by Rogoshinskov, who took non-uniformity of heating into account.Many works analyse also particular problems Ionov’s works [278, 498] based on thetheory of small elastoplastic deformations are among them Work [148] describes astress-strain state of an infinitely long cylindrical shell subjected to heating

thermoelastic-In a series of works by Piskun [538, 539], cylindrical shells subjected tonon-uniform heating and internal pressure have been examined Work [307] con-tains some computations of thermoplastic deformations based on the variational-difference method, and work [109] describes a stress-strain state of rotational shells

in conditions of axially symmetrical heating Monographs [609, 610] present a ory and computational methods concerning many problems of thermoplasticity atvariable loads including also the history of loading (the objects of study included

the-8 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

cylinders, disks and low lift rotational shells) Work [126] applies Iliushin’s theory

of plasticity to deal with heating an isotropic sphere with heat impacts of variousshape and length (the problem was solved as a non-coupled one) Analytical descrip-tion of thermoelastoplastic deformations is published in work [583] In work [242],Birger’s method is applied to solve non-linear elasticity problems Many interestingconclusions concerning dependence of physical and material parameters on temper-ature and work regime related to cooling shells and plates have been drawn in work[417] Work [399] formulates a functional in order to find a variational solution

to a plasticity theory problem at changing temperature for an elastoplastic ial Work [261] investigates the influence of the temperature load history, and work[150] analyses unique and continuous dependence on initial conditions in dynamicproblems of non-linear thermoelasticity A theory and a method of solving problems

mater-of thin-walled constructions heated by stationary and non-stationary heat sources aredescribed in work [336], in which the dependence of physical and mechanical char-acteristics on temperature has been taken into account A combination of the methodapplied for the theory of thermoelasticity with Vlasov’s variational method has beenused to solve a three-dimensional problem of non-linear thermoelasticity in work[357] It needs to be emphasised that coupling of the temperature and deformationfields (also in a quasistatic case) for problems of non-elastic material characteristics

is taken into account only in selected works [180, 217, 259, 350, 584]

A recent Polish publication edited by Wo´zniak [708] contains a synthetic andabundant presentation of the level of modern knowledge of the theory of elasticplates and shells with specific reference to the contribution of Polish scientists inits development In contrast to that approach this monograph puts more light tothe contribution of scientists from the former eastern bloc into the development

of the theory of plates in the temperature field It is worth emphasising that names

of the two first authors of this book are connected with a series of monographs on thetheory of plates and shells published in Polish [37, 38, 39, 48, 50, 51, 53] The latesttheoretical achievements in non-classic analyses of the thermoelastic shell theoryproblems are described in monograph [39]

Numerous aspects of non-linear dynamics of shells and plates, including furcations, chaos and solitons, have been analysed in other works of the two firstauthors of this monograph [41, 45, 46, 47, 49, 52, 55, 56, 57, 58, 60, 61, 62, 63, 64,

bi-65, 66, 67, 68, 69, 70, 389, 390], which also seem to be worth recommendation forreaders who wish to broaden their knowledge in the field of shells and plates

At this point, several conclusions need to be drawn (i) All of the above tioned works investigate classic initially-boundary problems, while a typical (com-bined) boundary conditions are the most important in the theory of elasticity andthermoelasticity There is a noticeable lack of solutions of that type in both linearand non-linear problems (ii) There is no evidence for stability of difference schemes

men-of the coupled theory men-of thermoelasticity in three-dimensional formulation for a bicoid (iii) Complexity of a physically non-linear system of differential equationslimits the number of examples of solutions to thermoelastoplastic problems to only

cu-a few

The authors of this chapter focused their attention on solving the following lems: 1) construct a system of differential equations of the coupled dynamic theory

prob-of thermoelasticity taking into account a three-dimensional model and singularities

of all kinds; 2) apply the variational-difference method for solving the coupled moelasticity theory problems; 3) prove stability of the difference approximation forthe examined class of problems; 4) solve a typical problems of the theory of elastic-ity and the theory of thermoelasticity; 5) formulate a method and solve physicallynon-linear, initially-boundary problems for a three-dimensional plate in the dynamiccoupled approach, and examine the influence of temperature and deformation fields’coupling

ther-The following notation is used:

x i , i = 1, , 3 - coordinate of a point in space;

n - normal unit vector directed outside the field: ni , j+m = cos(n i , j+m , x i);

U(u1, u2, u3) - displacement vector;

T = T0+ θ - absolute temperature;

T0 - absolute temperature in a stress-free state;

θ - temperature increase;

ατ - linear coefficient of thermal expansion;

λq - heat conduction coefficient;

λ - heat emission coefficient;

c - thermal capacity;

ei j - strain tensor coefficient;

σi j - stress tensor coefficient;

10 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

lα - plate’s dimension along xαaxis;

Nα - set of points of division towards xαaxis;

W21(Ω) - space of elements L2(Ω) with generalized derivatives of the first

order due to Ω and of the following properties:

(u, υ)(1)2,Ω=

Ω

∼ (u, u)(1)2, ¯ω1

;

W21,0(Ω) - Hilbert’s space composed of elements u(x, τ) belonging to space

L2(Qτ), which have generalized derivatives of the first order due to

Qτof the following properties:

(u, υ)(1,0)2,Qτ =



Qτ(uυ + u xυx )dxdτ,

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

This chapter presents a variational method-based derivation of a system of coupledthermoelasticity differential equations for a three-dimensional plate, taking into ac-count material’s non-homogeneity The system includes equations within the plate’sfield, at its edges, ribs in its corners and at simple contact points of numerous bound-ary conditions, which allows solving a substantial number of problems A differencesystem is derived with the use of the variational-difference method by approximat-ing the initial differential system with accuracy of such small values as O(h2) Theobtained difference scheme’s stability theorem has been proven

1.2.1 Variational equations

We shall consider interaction between an elastic non-homogeneous body Ω and amedium that surrounds it in conditions in which thermal and mechanical processesare taken into account Let us assume that at time instant τ = τ0 the body doesnot remain in the state of stress, i.e the thermodynamic quantities that characterise

the body such as absolute temperature T = T0, strain and stress tensor nents and displacement vector components are equal to zero Mechanical interactionmakes displacement fields appear in the body In every general case they accompanythe change of the temperature field Heating the body also causes perturbations in theinvestigated fields Heat conductivity involves producing entropy, and strains cause

compo-a decrecompo-ase of it, which in result lecompo-ads to producing hecompo-at Although thermoelcompo-asticdamping is usually weak and for a short time interval it may be neglected (the non-coupled thermoelasticity theory), the relatively long-lasting processes require takingenergy dissipation into account (the combined theory of thermoelasticity)

Dissipation energy can be described by the following relation [63]:

2

Ω

T0

λq

∂S2

12 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

S = cθ

where: S (S1, S2, S3) is an entropy vector

Body Ω remains in motion, therefore according to Hamilton-Ostrogradski’sprinciple, integral

3

Ω

c

T0θ2dΩ +

Ω

3

denotes the external forces’ work

In spite of the fact that principle (1.3) does not take dissipation energy intoaccount, it is essential to do it in energetic conditions of the coupled theory of ther-moelasticity That is why equation (1.4) takes the following form:

δ

τ1

τ

Using Cauchy’s dependences

and Duhamel-Neuman’s dependences

σii = 2µe i j + λe − βθ, σ i j = µe i j, (1.12)and expression (1.2), which is equivalent to the following dependence:

14 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

can be called singular points An analytical solution of the coupled thermoelasticitytheory problems in the described field requires taking the field’s singularities intoaccount For instance, when applying the method of mesh it is necessary to thickenthe mesh during the approach to the singular points In this way the computationtime needed to solve the problem will suddenly prolong In order to avoid undesir-able effects it is necessary to create additional equations in the singular points, fromnow on called the consistency conditions, that will constitute a part of the differen-tial equations which function as Euler’s system for functional (1.10), used by theauthors of work [429] In addition, surface integrals will be included to describe thewhole of the additional conditions imposed on the plate at its edges, in its cornersand places where boundary conditions meet

2C i +mn

λqλ

∂S i2

∂τ n i ,i+m+1

respectively at the limit of the function of the heat flux and the medium’ temperature;

D i j +m , A i +m , B i +mΛ , C i +mΠ– are constants that assume values 0 or 1 depending on the

type of the boundary conditions; indeces Λ and Π define parts of the plate’s wall(left or right), where the function is set

l k

0

l k/2

0

β∂θ

∂x i δu i dΩ +

Ω

i +m δu i d∂Ωi +m, (1.18)

16 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

δB =

Ω

δD =

Ω

∗∗C i +mn

λqλ

∂

∂x j

∂S i

Considering the fact that some integrals are equal to zero, expression δR can be

reduced The integrals marked with one star equal zero because integrands are stressderivatives, which in turn occur to be constant in relation to the variable, for which

a derivative is calculated The integrals marked with two stars are also equal to zerobecause integrands are constant derivatives in relation to the variable, according towhich differentiation is made

Substituting expressions (1.16)–(1.22) into (1.10) and assuming u i , S i, θ as dependent variables (their variations are arbitrary), we obtain the following system

in-of differential equations:

3 α,β=1

3

3

3

3

18 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

3

3

3

3

j +m , f j +mΛ, are set functions corresponding to the

sur-face forces, whereas q(x), q s

i +m , A s k +m , E s j +m , A i +m , B i +m , C i +mare equal

to 0 or 1, we shall obtain boundary conditions well-known in the theory of elasticityand thermoelasticity [6, 198]:

1 Rigid fixing (the first boundary problem)

a) D i j +m = 0, E j

i +m = 1, j = 1, , 3, or b) D i j +m = D i

– wall ∂Ω1 is free and insulated from heat sources, thus (D11 = D3

20 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

(λ + 2µ)∂u ∂x1

1

+ λ∂u ∂x22

+ λ∂u ∂x33

∂x1 + θ = T0

3(0, x2, 0, τ) Compatibility conditions (1.29), (1.30) for corner (0,0,0) in the investigated caseare as follows:

The problems described by the system of equations (1.23)–(1.33) are going to be

solved with the method of mesh The method makes it possible to bring a system of

partial differential equations to a system of algebraic equations There are a lot ofapproximations of the same problem Among them there is one that provides a re-quired approximation order and is stable (the so-called convergent approximation)

If it is possible, a difference scheme should model the primary differential problem.Variational-difference methods are the most appropriate for analysing the problemsdiscussed in this chapter and they retain the properties of a differential system Inorder to build a difference scheme we are going to use the method of integral iden-tity [429], which is based on an assumption that the energy functional is expressed

3

22 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

s=1

3 α,β=1

3

0, , Nα, j = 0, , M} For every continuous function f (x, τ) set in field Qτ,

func-tions f i jk (x, x4)= f (x i , x j , x k , x4) are going to be constructed and defined within ωτ.Identity (1.34) consists not only of equations (1.23), (1.24), but also of conditions(1.25)–(1.33) We are going to find out about it when making a transformation ofdependence (1.34) The integral identity is approximated with a summing identityreplacing the integrals with quadratic and derivative forms - difference quotients.Integrals (1.34) will be replaced with quadratic forms describing trapezoids, and in-tegrals (1.35) will be replaced with linear combinations of various relations of leftand right rectangles Strictly speaking:

An approximate solution of problem (1.23)–(1.33) will be such two mesh

func-tions y i , i= 1, , 4, set on ¯ωτ, which for arbitrary net functions ηi , i= 1, , 4, set inthe same net field ¯ωτ, fulfil the following identity:

Eventually we obtain the following system of equations:

α=1

24 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

D k k +m nk ,k+m Hk¯

$(λ+ 2µ) y k

of the node’s location, in which the direction of the coordinate axis coincides withpossibility of notating the derivative within the field in the same direction (the rightderivative) The lower notation corresponds to such a location, in which both thedirection of the axis and the direction of the derivative notation are opposite each

other (the left derivative) The note that belongs to edge (l1, 0, 0), (l1, 0, l3) will serve

as an example of our investigation Equations (1.42), (1.43) have the following form:

3

4+ D3 2

26 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

To start an investigation of the a priori characteristics in the form of a differencesystem it is necessary to begin with determining the order of the difference approx-imation error, since the difference scheme’s accuracy depends on it Moreover, the

function’s decomposition into Taylor’s series is used in this case Let u i

hbe a

pro-jection of solution u i, θ onto the mesh field ωτ, step h – a vector with norm |h| > 0, and let u ihave a sufficient number of generalized derivatives

We shall investigate deviation Ψ = L h u h − P h − L u − P, where L his a difference

operator, and L is a differential operator We say that L h approximates L with order

n on mesh ωτ, if|Ψ| < Mh n , where M − const > 0 does not depend on h Let us

make a separate estimation of the deviation of each of equations (1.36)–(1.38) Weshall assume that index n, the smallest for all the equations, will be the system’sapproximation order’s error Let us analyse the error of equations (1.38) in detail:

θ¯x i x i + P4− αθˆx4− β

3

i=1

u i ˆx

i x4=3

1

+h2

∂

∂x i(λ+ 2µ)∂u ∂x1

1

+ λ∂u ∂x22+

We are going to prove that the edge equations contain approximation errors of

the O(h2) order Let us consider a deviation expression of one in 48 similar equations(1.42), (1.43):

Ψ4(0, 0, x3)= h2

$(λ + 2µ) u1

28 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

1

+h12

+h2312

1

+h12

h1h2

h1h2+ h2h3+ h3h1

µ

∂u1

∂x2 +h22

∂u1

∂x +h32

h1h2+ h2 h3+ h3 h1

µ

Taking equations (1.23), (1.25), (1.51) into account, we obtain Ψ5(0, 0, 0) =

O(h2) Every equation of system (1.38)–(1.45) belongs to one of the five investigated

forms of equations This being so, the approximation error does not exceed O(h2),and the general error’s order in each node of field ωτis not smaller than the other

1.2.5 Difference approximation Stability

It may turn out that the knowledge of a difference scheme’s approximation error’sorder is insufficient to estimate the scheme’s quality That is why, having determinedthe approximation order, it is necessary to analyse the scheme’s stability The a

priori estimation for y iis an essential part of the analysis of a difference scheme Ifthe system approximates the problem and it is stable, then its solution leads to thesolution of a differential problem

The difference problem obtained with variational-difference methods are stable[384] However, derivation of estimations imposes bounds upon the right parts andthe coefficients of equations and also upon the steps of the mesh field hi, which isparticularly important while making calculations

Our reasoning will be based on mesh space W21(ω), W21,0(ωτ) and the followingenergy estimation:

– differential transformation (a uni-dimensional case):

(ω, v) x ,i= ωx ,i vi+ ωi+1vx ,i= ωx ,i vi+ ωi+1v ¯x,i+1,

(ω, v) ¯x,i= ω¯x,i v i+ ωi−1v ¯x,i= ω¯x,i v i+ ωi−1v x ,i−1; (1.53)

– summation [429] (p.225) (a one-dimensional case):

30 1 Three–Dimensional Problems of Theory of Plates in Temperature Field

– multi-dimensional summation with zero boundary values of mesh functions

¯ ω

hωx ,i v i= −

¯ ω

The one-dimensional relations have been shown due to the fact that the initialtransformations are derived only for one variable Additionally we apply the follow-ing equalities:

a two-dimensional case Ladyzhenskaya’s work contains derivations of the firstinitially-boundary problem for a parabolic and a hyperbolic equation in a generalform Works [231, 241, 492] address extended research into hybrid types of prob-lems Treating those references as basis we are going to prove a theorem that refers

to stability of approximate solutions to the coupled thermoelasticity problems forthree-dimensional plates