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2013 | Buch | 1. Auflage

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Theory of Elasticity and Thermal Stresses

Explanations, Problems and Solutions

verfasst von: M. Reza Eslami, Richard B. Hetnarski, Józef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Verlag: Springer Netherlands

Buchreihe : Solid Mechanics and Its Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book contains the elements of the theory and the problems of Elasticity and Thermal Stresses with full solutions. The emphasis is placed on problems and solutions and the book consists of four parts: one part is on The Mathematical Theory of Elasticity, two parts are on Thermal Stresses and one part is on Numerical Methods.

The book is addressed to higher level undergraduate students, graduate students and engineers and it is an indispensable companion to all who study any of the books published earlier by the authors. This book links the three previously published books by the authors into one comprehensive entity.

Inhaltsverzeichnis

Frontmatter

Erratum to: Thick Cylinders and Spheres

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

The Mathematical Theory of Elasticity

Frontmatter

Chapter 1. Mathematical Preliminaries

Abstract

In this chapter the basic definitions of vector and tensor algebra, elements of tensor differential and integral calculus, and concept of a convolutional product for two time-dependent tensor fields are recalled. These concepts are then used to solve particular problems related to the Mathematical Preliminaries.

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 2. Fundamentals of Linear Elasticity

Abstract

In this chapter a number of concepts are introduced to describe a linear elastic body. In particular, the displacement vector, strain tensor, and stress tensor fields are introduced to define a linear elastic body which satisfies the strain-displacement relations, the equations of motion, and the constitutive relations. Also, the compatibility relations, the general solutions of elastostatics, and an alternative definition of the displacement field of elastodynamics are discussed. The stored energy of an elastic body, the positive definiteness and strong ellipticity of the elasticity fourth-order tensor, and the stress-strain-temperature relations for a thermoelastic body are also discussed.

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 3. Formulation of Problems of Elasticity

Abstract

In this chapter both the basic boundary value problems of elastostatics and initial-boundary value problems of elastodynamics are recalled; in particular, the mixed boundary value problems of isothermal and nonisothermal elastostatics, as well as the pure displacement and the pure stress problems of classical elastodynamics are discussed. The Betti reciprocal theorem of elastostatics and Graffi’s reciprocal theorem of elastodynamics together with the uniqueness theorems are also presented. An emphasis is made on a pure stress initial-boundary value problem of incompatible elastodynamics in which a body possesses initially distributed defects.

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 4. Variational Formulation of Elastostatics

Abstract

In this chapter the variational characterizations of a solution to a boundary value problem of elastostatics are recalled. They include the principle of minimum potential energy, the principle of minimum complementary energy, the Hu-Washizu principle, and the compatibility related principle for a traction problem. The variational principles are then used to solve typical problems of elastostatics.

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 5. Variational Principles of Elastodynamics

Abstract

In this chapter both the classical Hamilton-Kirchhoff Principle and a convolutional variational principle of Gurtin’s type that describes completely a solution to an initial-boundary value problem of elastodynamics are used to solve a number of typical problems of elastodynamics.

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 6. Complete Solutions of Elasticity

Abstract

In this chapter general solutions of the homogeneous isotropic elastostatics and elastodynamics are discussed. The general solutions are related to both the displacement and stress governing equations, and emphasis is made on completeness of the solutions [See also Chap. 16.]

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 7. Formulation of Two-Dimensional Problems

Abstract

In this chapter a class of problems is discussed in which an elastic state depends on two space variables only, or an elastic process depends on two space variables and time only. In particular, problems related to a plane strain state and a generalized plane stress state of homogeneous isotropic elastostatics, a plane strain process, and a generalized plane stress process of homogeneous isotropic elastodynamics are discussed. The problems related to a two-dimensional homogeneous isotropic elastodynamics described in terms of stresses only are also considered. [See also Chaps. 16 and 17].

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 8. Solutions to Particular Three-Dimensional Boundary Value Problems of Elastostatics

Abstract

In this chapter the boundary value problems related to torsion of a prismatic bar bounded by a cylindrical lateral surface and by a pair of planes normal to the lateral surface, are discussed. It is assumed that a resultant torsion moment is applied at one of the bases while the other is subject to a warping and the lateral surface is stress free. In each of the problems an approximate three-dimensional formulation is reduced to a two-dimensional one for Laplace’s or Poisson’s equation on the cross section of the bar.

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 9. Solutions to Particular Two-Dimensional Boundary Value Problems of Elastostatics

Abstract

In this chapter a number of two-dimensional boundary value problems for a body under plane strain conditions or under generalized plane stress conditions are solved. The problems include: (i) a semispace subject to an internal concentrated body force, (ii) an elastic wedge subject to a concentrated load at its tip, and (iii) an infinite elastic strip subject to a discontinuous temperature field. To solve the problems a two-dimensional version of the Boussinesq-Papkovitch-Neuber solution as well as an Airy stress function method, are used.

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 10. Solutions to Particular Three-Dimensional Initial-Boundary Value Problems of Elastodynamics

Abstract

In this chapter a number of spherically symmetric initial-boundary value problems of the dynamic theory of thermal stresses for a homogeneous isotropic infinite elastic body are solved. The problems include: (i) the dynamic thermal stresses due to an instantaneous temperature distributed on a spherical surface in \({E}^3\), (ii) the dynamic thermal stresses due to a time-dependent spherically symmetric temperature field that satisfies a parabolic heat conduction equation in \({E}^3\), and (iii) the dynamic thermal stresses propagating in an infinite body with a stress free spherical cavity, corresponding to an instantaneous temperature distributed on a spherical surface lying inside the body. To solve the problems a method of the dynamic thermoelastic displacement potential in spherical coordinates is used.

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 11. Solutions to Particular Two-Dimensional Initial-Boundary Value Problems of Elastodynamics

Abstract

The particular solutions discussed in this chapter include: (i) dynamic thermal stresses in an infinite elastic sheet subject to a discontinuous temperature field, and (ii) dynamic thermal stresses produced by a concentrated heat source in an infinite elastic body subject to plane strain conditions.

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 12. One-Dimensional Solutions of Elastodynamics

Abstract

In this chapter a number of typical one-dimensional initial-boundary value problems of homogeneous isotropic isothermal and nonisothermal elastodynamics are solved in a closed-form using the Laplace transform technique. The isothermal solutions include: (a) one-dimensional displacement waves in a semispace subject to a uniform dynamic boundary pressure, (b) one-dimensional displacement waves in a semispace subject to the initial disturbances, and (c) one-dimensional stress waves in an infinite space composed of two homogeneous isotropic elastic semispaces of different material properties. The nonisothermal solutions include: (i) one-dimensional dynamic thermal stresses produced by a plane source of heat that varies harmonically with time in an infinite elastic solid, (ii) one-dimensional dynamic thermal stresses produced by a plane nucleus of thermoelastic strain in an infinite elastic solid, and (iii) one-dimensional dynamic thermal stresses in a semispace due to the action of a plane internal nucleus of thermoelastic strain.

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Thermal Stresses

Frontmatter

Chapter 13. Thermal Stresses in Bars

Abstract

In this chapter the concept of thermal stresses in bars is introduced for the simple case of a perfectly clamped bar subjected to arbitrary temperature change. The problems and solutions related to thermal stresses in bars are: a perfectly clamped bar, a clamped bar with a small gap, a clamped circular frustum, a bar with variable cross-sectional area, two bars attached to each other, three bars fastened to each other, truss of three bars, and three bars hanging from a rigid plate.

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 14. Thermal Stresses in Beams

Abstract

In this chapter, based on the Bernoulli-Euler hypothesis, thermal stresses in beams subjected to thermal and mechanical loads are recalled. Thermal stresses in composite and curved beams, and thermal deflections in beams subjected to a symmetrical thermal load are treated. Furthermore, solutions for stresses in curved beams are included. Problems and solutions for beams subjected to various temperature field or various boundary conditions are presented. [see also Chap. 23.]

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 15. Heat Conduction

Abstract

In this chapter the Fourier heat conduction equation along with the boundary conditions and the initial conditions for various coordinate systems are recalled. One-dimensional heat conduction problems in Cartesian coordinates, cylindrical coordinates and spherical coordinates are treated for both the steady and the transient temperature fields. The particular problems and solutions for heat conduction in a strip, a solid cylinder, a hollow circular cylinder and a hollow sphere are presented for various boundary conditions. [See also Chap. 22.]

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 16. Basic Equations of Thermoelasticity

Abstract

In this chapter the basic governing equations of thermoelasticity for three-dimensional bodies are recalled. The equilibrium equations of stresses, Cauchy’s relations between the tractions and stresses, and the compatibility equations of strains in Cartesian coordinates are presented. The formulae for coordinate transformation of stress, strain and displacement components are included. A solution of Navier’s equations is carried out wherein Goodier’s thermoelastic potential is used in conjunction with harmonic functions of various types. The equilibrium equations, stress, strain, the compatibility equations, Navier’s equations in cylindrical and spherical coordinates are also presented. [See also Chaps. 2, 3, 6, and 7.]

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 17. Plane Thermoelastic Problems

Abstract

In this chapter the basic treatment of plane thermoelastic problems in a state of plane strain and a plane stress are recalled. Typical three methods for the solution of plane problems are presented: the thermal stress function method for both simply connected and multiply connected bodies, the complex variable method with use of the conformal mapping technique, and potential method for Navier’s equations [See also Chap. 7].

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 18. Thermal Stresses in Circular Cylinders

Abstract

In this chapter various techniques are presented to determine the thermal stresses in solid and hollow cylinders. The one-dimensional problems of cylindrical bodies are treated by the displacement method. Plane problems for infinitely long cylinders and for circular plates are treated by the thermal stress function method. Two-dimensional axisymmetric problems and three-dimensional problems are treated with Goodier’s thermoelastic potential and the Boussinesq harmonic functions or Michell’s biharmonic function. The derivation and the general solution of the basic equations related to thermal stresses in circular cylinders are treated in a number of problems. [See also Chapter 24.]

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 19. Thermal Stresses in Spherical Bodies

Abstract

In this chapter the thermal stresses in spherical bodies are presented. First, one-dimensional problems for a solid and a hollow sphere are discussed. Next, two-dimensional axisymmetric problems are treated by Goodier’s thermoelastic potential and the Boussinesq harmonic functions. Problems and solutions for thermal stresses in a solid and a hollow cylinder subjected to the steady and the transient temperature field are presented.

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 20. Thermal Stresses in Plates

Abstract

In this chapter the thermal stresses and the deflection in thin rectangular plates subjected to the temperature change in the thickness direction only are recalled. The basic equations are developed with the Kirchhoff-Love hypothesis. Next, the basic equations for the thermal bending of circular plates with various boundary conditions are summarized. A number of problems for rectangular and circular plates are presented

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 21. Thermally Induced Instability

Abstract

In this chapter the thermoelastic buckling of beam-columns subjected to both in-plane and lateral loads is recalled for a built-in edge, a simply supported edge and a free edge. Furthermore, the thermoelastic buckling of rectangular and circular plates is also recalled. The problems and solutions for the buckling behavior of beam-columns with various boundary conditions are given. The stress-displacement relations, the relations between the resultant forces and the stress function are treated in illustrative problems.

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Thermal Stresses—Advanced Theory and Applications

Frontmatter

Chapter 22. Heat Conduction

Abstract

In this chapter heat conduction problems are presented. Employing the first law of thermodynamics, the problems in the rectangular Cartesian coordinates, cylindrical coordinates, and the spherical coordinates are solved. The method of treatments of the nonhomogeneous boundary and differential equations are given and the lumped formulation of the heat conduction problems are discussed.

Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 23. Thermal Stresses in Beams

Abstract

Beam are one of the basic elements of structural design problems. Thermal stresses in beams are discussed in this chapter.

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 24. Thick Cylinders and Spheres

Abstract

Thick cylinders and spheres are components of many structural systems. Due to their capacity to withstand high pressures, radial loads, and radial temperature gradients, the problem of thermal stress calculations is an important design issue in these types of problems. This chapter presents the method to calculate thermal stresses in such structural members which are made from the homogeneous/isotropic materials. The application of Michell conditions to derive thermal stresses in a multiply-connected region, such as thick walled cylinders, is shown through the solution of problems.

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 25. Piping Systems

Abstract

Piping systems are essential components in many industries such as refineries, power plants, and chemical plants, where their prime purpose is the transport of fluid from one piece of equipment to another. Normally, the content fluid of the pipe is hot, and since the piping system is initially designed at reference temperature, the temperature change causes thermal expansion.

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 26. Coupled Thermoelasticity

Abstract

When a structure is under the thermal shock load, the governing equation of thermoelasticity and the first law of thermodynamics are coupled. This thermal shock may be applied to the surface of a body or may be caused through the body heating.

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Numerical Methods

Frontmatter

Chapter 27. The Method of Characteristics

Abstract

The purpose of this chapter is to develop the method of characteristics for the solution of dynamic problems in thermoelasticity. The theoretical analysis of dynamic stresses due to impact loadings has generally been performed by the Laplace transform method. Due to inversion difficulties, the Laplace transform method is usually limited to simple wave problems. The need for the numerical methods to the solution of dynamic problems is dictated by the well-known difficulty of obtaining the exact solutions. Among the various numerical methods, the method of characteristics has the advantages of giving a simple description of the wave fronts and it can give numerical solutions readily to problems with any types of input functions. In mathematics, the method of characteristics is a method of numerical integration of a system of partial differential equations of hyperbolic type. The method is to reduce the hyperbolic partial differential equations to a family of ordinary differential equations, each of which is valid along a different family of characteristic lines (called the characteristics). These equations (called the characteristic equations) are more suitable for numerical analysis because the use of these equations makes it possible to obtain the solutions by a step-by-step integration procedure.

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 28. Finite Element of Coupled Thermoelasticity

Abstract

Due to the mathematical complexities encountered in analytical treatment of the coupled thermoelasticity problems, the finite element method is often preferred. The finite element method itself is based on two entirely different approaches, the variational approach based on the Ritz method, and the weighted residual methods. The variational approach, which for elastic continuum is based on the extremum of the total potential and kinetic energies has deficiencies in handling the coupled thermoelasticity problems due to the controversial functional relation of the first law of thermodynamics. On the other hand, the weighted residual method based on the Galerkin technique, which is directly applied to the governing equations, is quite efficient and has a very high rate of convergence.

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Chapter 29. Boundary Element, Coupled Thermoelasticity

Abstract

In this chapter, considering the Lord and Shulman’s theory, a Laplace-transform boundary element method is developed for the dynamic problems in coupled thermoelasticity with relaxation time involving a finite two dimensional domain. The boundary element formulation is presented and a single heat excitation is used to drive the boundary element formulations. Aspect of numerical implementation are discussed. It is shown that the distributions of temperature, displacement, and stress show jumps at their wave fronts. The thermo-mechanical waves propagation in a finite domain and the influence of relaxation time on them are presented. The results of this section are compared with the classical coupled theory (CCT) and the Green-Lindsay theory (GL). It is verified that the LS theory of the generalized thermoelasticity results into significant differences in the patterns and wave fronts of temperature, displacement, and stress compared to the CCT and GL theories, although the material and geometrical properties of the solution domain are identical. The details of these differences are given in the result section.

M. Reza Eslami, Richard B. Hetnarski, Jozef Ignaczak, Naotake Noda, Naobumi Sumi, Yoshinobu Tanigawa

Backmatter

Titel: Theory of Elasticity and Thermal Stresses
verfasst von: M. Reza Eslami
Richard B. Hetnarski
Józef Ignaczak
Naotake Noda
Naobumi Sumi
Yoshinobu Tanigawa
Verlag: Springer Netherlands
Electronic ISBN: 978-94-007-6356-2
Print ISBN: 978-94-007-6355-5
DOI: https://doi.org/10.1007/978-94-007-6356-2

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