Theory of Thermal Stresses

by Bruno A. Boley, Jerome H. Weiner

Overview

Elevated temperatures and extreme temperature gradients arise in a large variety of engineering problems, and often produced thermal stresses and thermal deformations that crucially affect the life of the materials and the systems involved. Early examples arose with the advent of high-speed rocket-powered flight and the development of nuclear energy sources. More recent applications can be found in fields ranging from reentry heating and ablation in space flight to the localized heat generation in computer chips, produced by high temperature during fabrication and by high current density during service.
This highly regarded text, aimed both at the researcher and the practicing engineer, as well as the student, presents a detailed discussion of fundamental aspects of the theory, accompanied by detailed solutions of typical and illustrative problems. The book is divided into four parts: Part I develops the fundamentals of thermoelasticity, starting with a presentation of the thermodynamic foundations of the subject and leading to various alternate formulations and methods of solutions of thermoelastic problems. Part II discusses the physical basis of heat transfer theory and methods of solution of heat conduction boundary-value problems. Part III covers more practical aspects of thermal stress analysis, mainly from the strength-of-materials viewpoint. Finally, Part IV presents the manner in which temperature effects can be included in inelasticity theory.
The result is an extremely useful resource which presents the salient features of the subject in a single volume from a unified and basic theoretical point of view.

Product Details

ISBN-13:	9780486143866
Publisher:	Dover Publications
Publication date:	04/25/2012
Series:	Dover Civil and Mechanical Engineering
Sold by:	Barnes & Noble
Format:	eBook
Pages:	608
File size:	46 MB
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CHAPTER 1

MECHANICAL AND THERMODYNAMICAL FOUNDATIONS

1.1 Introduction

The mathematical formulations describing the behavior of solid media under the combined action of heat and external loads are examined in this chapter. This examination makes use of the basic concepts which underlie all investigations regarding continuous media, concepts which arise from two different disciplines, namely mechanics and thermodynamics. Since the mechanical concepts enter, as well, into the consideration of constant- temperature problems and are discussed thoroughly in many works on elasticity theory, they are treated here only briefly with frequent references to the existing literature. It is, however, essential to consider the thermodynamic aspects of the problem only when nonconstant temperature problems are under consideration; for this reason, this subject is treated in only a cursory fashion in most elasticity textbooks. It is therefore necessary to devote a major portion of this chapter to an exposition of the principles of thermodynamics, starting first with the classical theory as relating to uniform systems and using it as a basis for the subsequent choice of postulates suitable for the nonuniform systems with which this book is concerned.

Thus the theory presented in this chapter forms the basis of all the subsequent developments in linear thermoelasticity in this book, and leads to a general formulation (summarized in Article 1.14) of the coupled thermoelastic boundary-value problem; simplifications and alternative formulations of this problem, which are usually necessary for the derivation of workable solutions, are discussed in the chapters that follow. In addition, a brief treatment is given in Article 1.13 of a thermodynamic approach to the theory for a linear viscoelastic material; that theory is introduced here only parenthetically, a discussion of various types of inelastic problems from another viewpoint being postponed until Chapter 14.

1.2 Notation

It will be convenient in this and the following three chapters to use the compact notation often referred to as indicial notation. A short outline of this notation is given here for purposes of reference. The discussions in which this notation will be used are of a general theoretical character and do not refer to any specific geometry or loading pattern; it is therefore sufficient at this point always to refer all quantities to a rectangular cartesian coordinate system. The coordinate axes of this system will be denoted by x1, x2, and x3 or, more briefly, by xi,i = 1, 2, 3. Consider now a vector f with components f1, f2, and f3 in the x1, x2, and x directions respectively; more simply, we may then say that f is a vector with components fi, i = 1, 2, 3; or, more simply yet, a vector fi. Clearly, it is immaterial what letter is used as subscript, provided of course that the values to be taken by it are prescribed. In what follows, Latin subscripts, unless otherwise specified, assume the values 1,2, 3, and this will not, in general, be stated explicitly; on the other hand, Greek subscripts or superscripts will not denote vector components and will extend over a range to be specified in each case.

The inner, or dot, product, f · g, of two vectors (f with components fi and g with components gi) is, by definition,

[MATHEMATICAL EXPRESSION OMITTED] (1.2.1)

Since summations of this type occur frequently, a summation convention is adopted whereby a repeated subscript automatically implies a summation so that the summation sign itself may be omitted. Thus, 1.2.1 may be written simply as

[MATHEMATICAL EXPRESSION OMITTED] (1.2.1a)

It is again immaterial what letter is used for the repeated subscript or index in eq. 1.2.1a Such an index is referred to as a bound index since it must assume all three values, as opposed to a free index which may assume any one of the three values 1, 2, 3.

The cross product of two vectors, f · g, may be expressed in indicial notation by use of the alternating tensor whose components γijk are defined as follows:

[MATHEMATICAL EXPRESSION OMITTED] (1.2.2)

Then f · g is a vector h with components hi where

[MATHEMATICAL EXPRESSION OMITTED] (1.2.3)

or, in extended notation, h1 = (f2g3 – f3g2), h2 = (f3g1 - f1g3), and h3 = (f1g2 - f2g1).

If a different vector is associated with each point of some region of space, we then speak of a vector field; it may be denoted alternatively by the symbols f(P) or fi(x1, x2, x3), where f(P) is the vector with components fi associated with the point P whose coordinates are x1, x2, x3. If these functions are differentiable, it may then be verified that the nine partial derivatives, [MATHEMATICAL EXPRESSION OMITTED], i = 1, 2, 3, and j = 1, 2, 3, are the components of a second-rank cartesian tensor. In indicial notation, it is customary to employ a comma to denote partial differentiation with respect to a spatial coordinate, as follows:

[MATHEMATICAL EXPRESSION OMITTED] (1.2.4)

As in this equation, the tensorial rank is always indicated by the number of free indices appearing in its generic component. Thus, the inner product of two vectors, eq. 1.2.1a, is a tensor of rank zero or a scalar, whereas the cross product of two vectors, eq. 1.2.3, is a tensor of rank one, or a vector. This fact is greatly useful in verifying the tensorial validity of an equation, that is, in assuring that only terms of the same tensorial rank are added. Note that in such an equation the same subscripts must be used for the free indices in each term to indicate that corresponding, components of each tensor are used in the addition.

1.3 Deformation; Small-Strain Tensor

When the particles of a continuous substance move so that distances between particles are changed, the substance is said to be deformed. The concept of deformation is a purely geometrical or kinematical one and is independent of the nature of the medium and of the causes of the deformations. In this chapter, attention is restricted to deformations which obey the restrictions assumed in the linear theory of elasticity.

Let the displacement of a particle at the point P with coordinates xi and at time t be denoted by the vector ui (x1, x2, x3, t). Then under the above assumptions, a measure of the deformation is given by the so-called small-strain tensor with components εeij defined as follows:

[MATHEMATICAL EXPRESSION OMITTED] (1.3.1)

The velocity vector vi of the particle is

[MATHEMATICAL EXPRESSION OMITTED] (1.3.2)

where the superposed dot denotes partial differentiation with respect to time.

The components of the rotation tensor, ωij, are defined as follows:

[MATHEMATICAL EXPRESSION OMITTED] (1.3.3)

so that

[MATHEMATICAL EXPRESSION OMITTED] (1.3.4)

that is, the tensor is antisymmetric. In three dimensions, therefore, this tensor has (except for sign) only three, nonzero, distinct components and we may represent it by a three-component vector without any loss of information. This may be done by use of the alternating tensor as follows; let

[MATHEMATICAL EXPRESSION OMITTED] (1.3.5)

so that

[MATHEMATICAL EXPRESSION OMITTED]

The vector ω with components ωi is termed the infinitesimal rotation vector. It may be shown (cf. Novozhilov, footnote 3), under the geometrical assumptions of linear elasticity, that for a given deformation a volume element undergoes an average rotation given by the value of the vector ω at the point in question, that is, the element rotates through an angle |ω| with the vector ω as axis, in the direction of a right-handed screw for right-handed coordinate systems. If we multiply eq. 1.3.5 by σrsk and use the identity (with δij defined in eq. 1.4.9):

[MATHEMATICAL EXPRESSION OMITTED] (1.3.6)

we may invert eq. 1.3.5 to obtain

[MATHEMATICAL EXPRESSION OMITTED] (1.3.7)

1.4 Equations of Motion

The equations of motion are an expression of the laws of Newtonian mechanics and again apply, therefore, equally well to any type of continuum and process. Consider therefore a substance of unspecified nature which occupies a closed region D + B with B as its boundary.

Each point P of B is subjected to traction [MATHEMATICAL EXPRESSION OMITTED] (force per unit area) where the superscript (n) refers to the outward normal, ni, to B at P, and the subscript refers again to the coordinate directions. Each mass element of the body is subjected to a body force per unit mass fi, which includes any inertia forces present. The laws of mechanics require that the resultant force and the resultant moment of all the forces acting on the body be both equal to zero. These then lead to the following two sets of equations:

[MATHEMATICAL EXPRESSION OMITTED] (1.4.1)

where ρ = ρ (x1x2x3) is the mass density of the material at the point in space with coordinates xk. In order to proceed further, it is necessary to introduce the postulate known as the stress-principle, which states that tractions exist across every internal surface element of the body and that, in terms of these, the laws of motion, eqs. 1.4.1 and 1.4.2, apply for any region interior to the body as well as to the body as a whole. The same notation is used for the interior tractions as for the actual boundary tractions; thus if an interior element has a surface normal ni then [MATHEMATICAL EXPRESSION OMITTED] denotes the force per unit area, conceived as exerted by the material on the positive side of the normal upon the material on the negative side of the normal. With this postulate, eq. 1.4.1 may be applied to a vanishingly small tetrahedron, and if the body forces are bounded, this process leads to the existence of a stress tensor, σij such that

[MATHEMATICAL EXPRESSION OMITTED] (1.4.3)

for either an internal surface or a bounding surface. By substitution of this result into 1.4.1 we obtain

[MATHEMATICAL EXPRESSION OMITTED] (1.4.4)

By use of the divergence theorem, the surface integral may be transformed to a volume integral over D, leading to the equation

[MATHEMATICAL EXPRESSION OMITTED] (1.4.5)

Since D is any arbitrary portion of the body, it follows that, wherever the integrand is continuous,

[MATHEMATICAL EXPRESSION OMITTED] (1.4.4)

Similarly, substitution of eq. 1.4.3 into eq. 1.4.2 leads to

[MATHEMATICAL EXPRESSION OMITTED] (1.4.7)

The surface integral may be transformed by use of the divergence theorem as follows:

[MATHEMATICAL EXPRESSION OMITTED] (1.4.8)

Here the relation xk,m = δkm has been used, where δkm is the Kronecker delta defined as follows:

[MATHEMATICAL EXPRESSION OMITTED] (1.4.9)

Substitution of 1.4.8 into 1.4.7 and use of 1.4.6 then leads to the equation

[MATHEMATICAL EXPRESSION OMITTED] (1.4.10)

Since this applies for any interior region and for each value of i, it follows that

[MATHEMATICAL EXPRESSION OMITTED] (1.4.11)

Equations 1.4.6 and 1.4.11 are the equations of motion which apply under the assumptions stated to any continuous medium. If, in addition to applied body forces Fi per unit mass exerted by some external agency, inertia forces are present, then we may write

[MATHEMATICAL EXPRESSION OMITTED] (1.4.12)

and eq. 1.4.6 assumes the form,

[MATHEMATICAL EXPRESSION OMITTED] (1.4.13)

1.5 Thermodynamics; Basic Definitions

The next step in the development of the theory requires the introduction of a connection between the variables describing deformations and the stress. This connection depends on the existence of quantities such as energy and entropy and therefore is a part of the subject of thermodynamics. The outline of thermodynamics which follows will emphasize the concepts most useful in the study of continuous media, including the effects of nonuniformity and dissipation, and is presented in the following four main parts: (1) the basic definitions are given in the remainder of this article, (2) uniform systems are studied in Article 1.6, (3) nonuniform systems in Articles 1.7 to LI 1, and (4) specific applications to elastic and viscoelastic solids are given in Articles 1.12 to 1.14.

The literature on thermodynamics is, of course, quite extensive and it is difficult to give a complete listing of works which may usefully be consulted on the subject. As general references the reader may take the texts of Zemansky, Keenan or Pippard for classical thermodynamics, and the works of Prigogine, DeGroot, or Biot for the thermodynamics of irreversible processes.

The particular collection of matter which is being studied will be referred to as a system. Only closed systems will be considered here, that is, systems which do not exchange mass with their surroundings. Occasionally, the further restriction is made that no interaction between the system and its surroundings occur; the system is then said to be isolated. It is assumed that the system occupies a closed region of space. It will be furthermore assumed that the material is chemically inert.

Consider now a system all of whose particles are at rest. When all the information required for the complete characterization of the system for the purposes at hand is available, it will be said that its state is known. This information takes the form of the numerical values of several quantities, called state variables, each of which describes a different property of the system. Some of these properties will be extensive or additive, for example, volume, whereas others will be intensive, for example, pressure. If for such a system the values of the state variables are independent of time, the system is said to be in thermodynamic equilibrium. A system is said to be uniform if the values of the intensive state variables do not vary in space.

The state variables may be measured for systems not in thermodynamic equilibrium and will then generally vary with time; the system is then said to undergo a process. During a process it may be that the system is not completely characterized by the state variables alone, but may require the specification of some additional quantities (see Article 1.9c).

In the next article attention will be restricted to uniform systems and to processes for the entire duration of which the system is uniform, for the purpose of introducing certain basic concepts; these will later be generalized for nonuniform systems.

1.6 Thermodynamics of Uniform Systems

(a) Equations of state

The question of whether all the state variables are independent has, thus far, been left open. It is found from experimental observation, however, that at thermodynamic equilibrium, once the values of a certain number of state variables have been determined, the values of the others are fixed; in other words, a number of functional relationships exist between the state variables. If for a system with (m + n) state variables X α (α = 1, ..., m + n) there exist m independent functional relations, then any n of the state variables (say X1 ..., Xn) may be taken as the independent ones, their selection being a matter of convenience; the others may be expressed in terms of them as follows:

[MATHEMATICAL EXPRESSION OMITTED] (1.6.1)

and will be referred to as state functions.

Equations of this type are known as equations of state. The number and form of these equations depend on the particular system under consideration. Only in the simplest systems (characterized, that is, by few state variables, as for example a gas) is it possible to establish them directly by an analysis of experimental dat#. It is often possible, however, to derive a considerable amount of information regarding their form from thermodynamic principles, provided only that the choice of independent variables is made.

(Continues…)

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Table of Contents

Note on Bibliographical References
Part 1 BASIC THEORY
Chapter 1. Mechanical and Thermodynamical Foundations
1.1 Introduction
1.2 Notation
1.3 Deformation; Small-Strain Tensor
1.4 Equations of Motion
1.5 Thermodynamics; Basic Definitions
1.6 Thermodynamics of Uniform Systems
1.7 Transition to Nonuniform Systems
1.8 Conservation of Energy in Nonuniform Systems
1.9 Preliminaries to the Second Law of Thermodynamics for Continua
1.10 Carathéodory's Statement of the Second Law of Thermodynamics and Its Consequences
1.11 Irreversible Theromodynamics; Entropy Production
1.12 Stress-Strain Relations and Energy Equation
1.13 Stress-Strain Relations and Energy Equation for an Isotropic Elastic Solid
1.14 Summary of Linear Coupled Thermoelastic Theory; Uniqueness Theorem
Chapter 2. Uncoupled Quasi-Static Thermoelastic Theory
2.1 Introduction
2.2 General Remarks on the Effects of Coupling and Inertia
2.3 Solution of a Coupled Thermoelastic Problem
2.4 Discussion of Article
2.5 Effect of Inertia
2.6 Uncoupled Quasi-Static Foundation
2.7 "Uniqueness Theorems for the Uncoupled, Quasi-Static Thermoelastic Theory"
Appendix Thermoelastic Damping
Chapter 3. Alternate Formulations of Thermoelastic Problems
3.1 Introduction
3.2 Displacement Formulation
3.3 Body-Force Analogy
3.4 Reduction of the Thermoelastic Problem to One at Constant Temperature with No Body Forces; Goodier's Method
3.5 Use of Boussinesq-Papkovich Functions
3.6 Stress-Formulation
3.7 Necessity of Compatibility Equations
3.8 Stress-Formulation for Multiply Connected Bodies
3.9 Temperature Distributions Which Result in Zero Stress
3.10 Dislocations
Chapter 4. Two-Dimensional Thermoelastic Formulations
4.1 Introduction
4.2 Plane-Strain Thermoelastic Problems
4.3 Boundary Conditions on the End Faces for the Case of Plane Strain
4.4 Stress Formulation of the Plane-Strain Problems
4.5 Stress Formulation in Terms of a Stress Function
4.6 Plane-Stress Thermoelastic Problems
4.7 Discussion of the Plane-Stress Solutions
4.8 Plane Stress as a Limiting Case of a Three-Dimensional State of Stress for Thin Slices
4.9 Steady-State Temperature Distributions
4.10 Dislocation Analogy
Part 2 HEAT CONDUCTION
Chapter 5. The Formulation of Heat Transfer Problems
5.1 Introduction
5.2 Modes of Heat Transfer
5.3 The Fourier Heat Conduction Equation
5.4 Initial and Boundary Conditions
5.5 Dimensionless Parameters
5.6 Discussion of the Boundary Conditions
5.7 Uniqueness Theorem
5.8 One-Dimensional Formulations for Thin Sections
Chapter 6. Some Basic Problems in Heat Conduction
6.1 Introduction
6.2 Sources and Sinks in an Infinite Solid
6.3 A More General Solution of Eq. 6.2.3b
6.4 The Semi-Infinite Solid under Time-Dependent Boundary Conditions
6.5 Solutions Obtained by Superposition and Imaging of Sources Conditions
6.6 Alternative Forms of Series Solutions; Poisson's Formula
6.7 Temperatures Due to Sources Regarded as Fundamental Solutions (Green's Functions)
6.8 Saint-Venant's Principle in Heat Conduction Problems
6.9 Upper and Lower Bounds on the Temperature
6.10 Over-All Heat Balance; the Melting Slab
Chapter 7. Methods of Solution of Heat Conduction Problems
7.1 Introduction
7.2 Separation of Variables (Method of Characteristic Functions)
7.3 Laplace Transforms
7.4 Conformal Mapping
7.5 Numerical Methods
7.6 Electrical Analogy
7.7 Approximate Analytical Procedures
7.8 Some Techniques for Extending Previous Solutions
Part 3 THERMAL STRESS ANALYSIS FOR ELASTIC SYSTEMS
Chapter 8. Summary of the Formulation of Thermoelastic Problems
8.1 Introduction
8.2 Thermoelastic Stress-Strain Relations
8.3 Equations of Equilibrium
8.4 Strain-Displacement Relations
8.5 Boundary Conditions
8.6 Mathematical Formulation of the Problem of Thermoelasticity
8.7 Principle Stresses and Strains
8.8 Separation of Stresses Due to Temperature and to External Loads
8.9 Alternative Formulations of the Problem of Thermoelasticity
8.10 Two-Dimensional Formulations
8.11 Energy Methods
8.12 Methods of Solution of Thermoelastic Problems
Chapter 9. Some Basic Problems in Thermoelasticity
9.1 Introduction
9.2 Three-Dimensional Problems in Which the Stresses Are Zero
9.3 Three-Dimensional Problems in Which the Displacements Are Zero
9.4 Two-Dimensional Problems in which the Stresses in the Plane Are Zero
9.5 Free Plate with Temperature Variation through the Thickness Only
9.6 Rectangular Beam with Temperature Variation through the Depth Only
9.7 Discussion of Articles 9.5 and 9.6
9.8 Example for Articles 9.5 and 9.6
9.9 Slowly Heated Beam or Plate
9.10 Circular Disc or Cylinder with Radial Temperature Variation
9.11 Circular Disc or Cylinder with Plane-Harmonic Temperature Distribution
9.12 Additional References on Thermal Stresses in Cylinders
9.13 Circular Rectangular Beam with Radial Temperature Variation
9.14 Solid or Hollow Sphere under Radial Temperature Variation
9.15 Over-All Thermoelastic Deformation
Chapter 10. Thermal Stresses in Beams
10.1 Introduction
10.2 Elementary Formulas for Normal Thermal Stresses in Free Beams
10.3 Thermal Deflections of Beams
10.4 Beam End-Conditions; Statically Indeterminate Beams
10.5 Thermal Shear Stresses in Thin-Walled Beams
10.6 Exact Two-Dimensional Thermoelastic Solution for Rectangular Beams under Arbitrary Temperature Distributions
10.7 Discussion of Article 10.6; Relation to Strength-of-Materials Theory
10.8 Exact Theory for Free Beams of Arbitrary Simply-Connected Cross Section with Linear Spanwise Temperature Distributions
10.9 Discussion of Article 10.8; Relation to Strength-of-Materials Theory
10.10 Use of Dummy Loads for the Calculation of Beam Deflections
10.11 Thermally Induced Vibrations of Beams
Appendix The End-Problem in Beams; Saint-Venant's Principle
"Chapter 11. Thermal Stresses in Curved Beams, Rings, Trusses, Frames, and Built-up Structures"
11.1 Introduction
11.2 Strength-of-Materials Theory for Thermal Stresses in Curved Beams
11.3 Discussion of Article 11.2; Relation to Exact and to Straight-Beam Analyses
11.4 Thermal Stresses in Rings
11.5 Thermal Stresses in Statically Determinate Trusses
11.6 Thermal Stresses in Statically Indeterminate Trusses
11.7 Thermal Stresses in Rigid Frames
11.8 Use of Influence Coefficients
11.9 References on the Analysis of Reinforced Sheet Structures
Chapter 12. Thermal Stresses in Plates
12.1 Introduction
12.2 Basic Plate Equations
12.3 Plate Boundary Conditions
12.4 Solutions of Thermoelastic Plate Problems
12.5 Plates with Temperature Distributions Varying Through the Thickness Only
12.6 Relation of Thin-Plate Theory to Exact Thermoelastic Solutions
12.7 Thermally Induced Vibrations of Plates
Chapter 13. Thermoelastic Stability and Related Problems
13.1 Introduction
13.2 Heated Beam-Columns with Ends Axially Unrestrained
13.3 Heated Beam-Columns with Ends Axially Restrained
13.4 Heated Beams under Axial Loads: General Theory
13.5 Discussion of Article
13.6 Bending and Buckling of Bimetallic Beams
13.7 Thermal Buckling of Plates
13.8 Buckling of Plates Subjected to Heat and No Transverse Loads with Edges Unrestrained in the Plane
13.9 Buckling of Plates Subjected to Heat and Loads in the Plane; Edges Unrestrained in the Plane
13.10 Plates with Their Edges Restrained in the Plane
13.11 Large Deflections and Post-Buckling Behavior of Plates
Part 4 THERMAL STRESS ANALYSIS FOR INELASTIC SYSTEMS
Chapter 14. The Formulation of Inelastic Thermal Stress Problems
14.1 Introduction
14.2 Stress Relaxation and Creep
14.3 Plastic Flow and Work-Hardening
14.4 Idealized Theories and Materials
14.5 Viscoelastic Stress-Strain Relations
14.6 Idealized Plasticity Theory: Work-Hardening Solid
14.7 Idealized Plasticity Theory: Perfectly Plastic Solid
14.8 Uniqueness Theorem for Perfectly Plastic Solid
14.9 The Mises Yield Condition
14.10 The Tresca Yield Condition
14.11 Combined Viscoelastic and Plastic Effects
Chapter 15. Viscoelastic Stress Analysis
15.1 Introduction
15.2 Viscoelastic-Elastic Analogy
15.3 Discussion of the Viscoelastic-Elastic Analogy
15.4 Example for the Viscoelastic-Elastic Analogy
15.5 Linear Viscoelastic Strength-of-Materials Theory
15.6 Initial Conditions for a Linear Viscoelastic Solid
15.7 Nonlinear Viscoelastic Analyses
15.8 Nonlinear Viscoelastic Strength-of-Materials Theory; Creep Rupture in Tension
15.9 Creep Buckling
15.10 Further Investigations of Creep Buckling
Chapter 16. Plastic Stress Analysis
16.1 Introduction
16.2 Elastoplastic Free Plate Analysis
16.3 Two Examples of Elastoplastic Plate Analysis
16.4 "Free Plate Analysis, Including a Temperature-Dependent Yield Condition and Viscoelastic Effects"
16.5 Elastoplastic Cylinder Analysis-Tresca Condition
16.6 Elastoplastic Cylinder Analysis-Mises Condition
AUTHOR INDEX
SUBJECT INDEX