Classical and Computational Solid Mechanics

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CLASSICAL AND

COMPUTATIONAL

SOLID

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CLASSICAL AND

COMPUTATIONAL

SOLID

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CLASSICAL AND

COMPUTATIONAL

SOLID

MECHANICS

Y. C. FUNG

University of California,

San

Diego

PIN TONG

Hong Kong University of Science

& Technology

World Scientific

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World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, Farrer Road, Singapore 912805

USA oftice: Suite lB, 1060 Main Street, River Edge, NJ 07661 UK oftice: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library

CLASSICAL AND COMPUTATIONAL SOLID MECHANICS Advanced Series in Engineering Science -Volume 1

Copyright 0 2001 by World Scientific Publishing Co. Pte. Ltd.

Allrights resewed. Thisbook, orpartsthereof; may notbereproducedinanyformorbyanymeans, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-3912-2 ISBN 981-02-4124-0 (pbk)

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DEDICATION

To my teacher Professor Fung and to Mrs. Fung who have been the constant

source of caring, inspiration and guidance; to my parents who were the source of love and discipline; to my wife who has been the source of love, companionship and encouragement; to my children and grandchildren who have been the source of joy and love; to Denny who will be forever in my heart.

Pin Tong To Luna, Conrad, and Brenda

Y. c.

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PREFACE

The objective of this book is to offer students of science and engineering a concise, general, and easy-to-understand account of some of the most important concepts and methods of classical and computational solid mechanics. The classical part is mainly a re-issue of F‘ung’s Foundations of

Solid Mechanics, with a major addition to the modern theories of plasticity, and a major revision of the theory of large elastic deformation with finite strains. The computational part consists of five new chapters, which focus on numerical methods to solve many major linear and nonlinear boundary- value problems of solid mechanics.

We hold the principle of easy-to-understand for the readers as an objective of our presentation. We believe that to be easily understood, the presentation must be precise, the definitions and hypotheses must be clear, the arguments must be concise and with sufficient details, and the con- clusion has to be drawn very carefully. We strive to pay strict attention to these requirements. We believe that the method must be general and the notations should be unified. Hence, we presented the tensor analysis in general coordinates, but kept the indicia1 notations for tensors in the first fifteen chapters. In Chapters 16-21, however, the dyadic notations of tensors and the notations for matrix operations are used to shorten the formulas.

This book was written for engineers who invent and design things for human kind and want to use solid mechanics to help implement their designs and applications. It was written also for engineering scientists who enjoy solid mechanics as a discipline and would like to help develop and advance the subject further. It was further designed to serve those physical and natural scientists and biologists and bioengineers whose activities might be helped by the classical and computational solid mechanics. For example, biologists are discovering that the functional behavior of cells depends on the stresses acting on the cell. It is widely recognized that the molecular mechanics of the cell must be developed as soon as possible.

Solid mechanics deals with deformation and motion of “solids.” The displacement that connects the instantaneous position of a particle to its

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position in an “original” state is of general interest. The preoccupation about particle displacements distinguishes solid mechanics from that of fluids.

This book begins with an introductory chapter containing a brief sketch of history, an outline of some prototypes of theories, and a description of some more complex features of solid mechanics. In Chapter 2, an introduction to tensor analysis is given. The bulk of the text from Chapters 3 to 16 is concerned with the classical theory of elasticity, but the discussion also in- cludes thermodynamics of solid, thermoelasticity, viscoelasticity, plasticity, and finite deformation theory. Fluid mechanics is excluded, but methods that are common to both fluid and solid mechanics are emphasized. Both dynamics and statics are treated; the concepts of wave propagation are introduced at an early stage. Variational calculus is emphasized, since it provides a unified point of view and is useful in formulating approximate theories and computational methods. The large deflection theory of plates presented in the concluding section of Chapter 16 illustrates the elegance of the general approach to the large deformation theory.

Chapters 17 to 21 are devoted to computational solid mechanics to deal with linear, nonlinear, and nonhomogeneous problems. It is recognized that the incremental approach is the most practical approach, hence the updated Lagrangian description. Chapter 17 develops the incremental t h e ory in considerable detail. Chapter 18 is devoted to numerical methods, with the finite element methods singled out for a detailed discussion of the theory of elasticity. Chapter 19 presents methods of calculations based on the mixed and hybrid variational principles, illustrating the broadening of the computational power with a less restrictive (or weaker) hypothesis informulating the variational principle. Chapter 20 deals with finite element methods for plates and shells, making the computational methods accessible to the analysis of the structures of aircraft, marine architecture, land vehicles, and shell-like structures in human being, animals, plants, earth, globe, and space. Finally, the book concludes with Chapter 21 dealing with finite element modeling of nonlinear elasticity, viscoelasticity, plasticity, viscoplasticity, and creep. Thus, a broad sweep of modern, advanced topics are covered.

Overall, this book lays emphasis on general methodology. It prepares the students to tackle new problems. However, as it was said in the original preface of the Foundations of Solid Mechanics, no single path can embrace the broad field of mechanics. As in mountain climbing, some routes are safe to travel, others more perilous; some may lead to the summit, others to different vistas of interest; some have popular claims, others are less

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PREFACE ix

traveled. In choosing a particular path for a tour through the field, one is influenced by the curricular, the trends in literature, and the interest in engineering and science. Here, a particular way has been chosen to view some of the most beautiful vistas in classical and computational mechanics. In making this choice, we have aimed at straightforwardness and interest, and practical usefulness in the long run.

Holding the book to a reasonable length did not permit inclusion of many numerical examples, which have to be supplemented through problems and references. Fortunately, there are many excellent references to meet this demand. We have presented an extensive bibliography in this book, but we suggest that the reader consults the review journal Applied Mechanics Reviews (AMR) published by the American Society of Mechan- ical Engineers International since 1947, for the current information. The reader is referred to the periodic in-depth reviews of the literature in specific issues of AMR.

We are indebted to many authors and colleagues a s acknowledged in the preface of the Foundations of Solid Mechanics. In the preparation of the present edition, we are especially indebted to Professors Satya Atluri and Theodore Pian. We would like to record our gratitude to many colleagues who wrote us to discuss various points and sent us errata in the Foundations of Solid Mechanics, especially to Drs. Pao-Show

D.

Cheng, Shun Cheng, Ellis

H.

Dill, Clive L. Dym, 3. B. Haddow, Manohar P.

Ka-

mat, Hans Krumhaar, T.

D.

Leko, Howard A, Magrath, Sumio Murakami, Theodore Pian, R. S. Rivlin, William P. Rodden, Bertil Storakers, Howard J. White, Jr. and John C. Yao. We would also like to thank professors

Y.

Ohashi, S. Murakami and

N.

Kamiya for translating the Foundations

book into Japanese, Professors Oyuang Zhang, Ma Wen-Hua and Wang

Kai-Fu for translating the

Foundations book into Chinese.

On the cover of this book, portraits of some pioneers of mechanics are presented. They are arranged, from left to right and top to bottom, according to their birthdays. They are: Galileo Galilei, 1564-1642, Isaac Newton, 1642-1727, Daniel Bernoulli, 1700-1782, Leonhard Euler, 1707-1783, Joseph Louis Lagrange, 1736-1813, Claude Louis Marie Henri Navier, 1785-1836, Augustin Louis Cauchy, 1789-1857, Lord Kelvin, William Thompson, 1824-1907, Gustave Robert Kirchhoff, 1824-1889, Lord Rayleigh, John William Strutt, 1842-1919, Ludwig Edward Boltsmann, 18441906, August Edward Hugh Love, 1863-1940, Stephen P. Timoshenko, 1878-1972. These portraits were supplied by Dr Stephen Juhasz, who was the editor of the Applied Mechanics Reviews for 30 years, from 1953- 1983. In 1973, Dr Juhasz published an article entitled “Famous Mechanics

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Scientists” in A p p . Mech. Rev. Vol. 26, No. 2. These portraits are from that article, in which acknowledgement to original sources is recorded. We thank you, Dr Juhasz.

We would like to take this opportunity to mention a few of editorial notes:

(1) The bibliography is given at the end of the book.

(2) Equations in each Section are numbered sequentially. When referring to equations in other Sections, we use the format (Sec. no: Eq. no), e.g. (5.6:4).

(3) Formulas are concise way of saying lots of things. The most important formulas are marked with a triangular star, A. They are worthy of being committed to memory.

Y. C. F’ung P. Tong

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127

127 128 131 133 134

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CONTENTS xiii

5.6. Moment of Momentum

5.7. Other Field Equations

6 ELASTIC AND PLASTIC BEHAVIOR OF MATERIALS 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10. 6.11. 6.12. 6.13. 6.14. 6.15. 6.16.

Generalized Hooke’s Law

Stress-Strain Relationship for Isotropic Elastic Materials

Ideal Plastic Solids

Some Experimental Information

A Basic Assumption of the Mathematical Theory of Plasticity

Loading and Unloading Criteria Isotropic Stress Theories of Yield Function

Further Examples of Yield Functions Work Hardening - Drucker’s Hypothesis and Definition

Ideal Plasticity

Flow Rule for Work-Hardening Materials Subsequent Loading Surfaces

-

Isotropic and Kinematic Hardening Rules

Mroz’s, Dafalias and Popov’s, and Valanis’ Plasticity Theories Strain Space Formulations Finite Deformation

Plastic Deformation of Crystals

7 LINEARIZED THEORY OF ELASTICITY

7.1. Basic Equations of Elasticity for Homogeneous Isotropic Bodies 7.2. Equilibrium of an Elastic Body

Under Zero Body Force

7.3. Boundary Value Problems

7.4. Equilibrium and Uniqueness of 7.5.

Solutions

Saint Venant’s Theory of Torsion

135 136 138 138 140 143 146 150 156 157 159 166 167 171 177 189 195 199 200 203 203 206 207 210 213

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7.6. Soap Film Analogy 7.7. Bending of Beams 7.8. Plane EIastic Waves 7.9. Rayleigh Surface Wave 7.10. Love Wave

8 SOLUTIONS OF PROBLEMS IN LINEARIZED

THEORY OF ELASTICITY BY POTENTIALS

8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9. 8.10. 8.11. 8.12. 8.13. 8.14. 8.15.

Scalar and Vector Potentials for Displacement Vector Fields

Equations of Motion in Terms of Displacement Potentials

Strain Potential Galerkin Vector

Equivalent Galerkin Vectors

Example - Vertical Load on the Horizontal Surface of a Semi-Infinite Solid

Love’s Strain Function

Kelvin’s Problem -

A

SingIe Force Acting in the Interior of an Infinite Solid

Perturbation of Elasticity Solutions by a Change of Poisson’s Ratio

Boussinesq’s Problem On Biharmonic Functions

Neuber-Papkovich Representation

Other Methods of Solution of Elastostatic Problems Reflection and Refraction of PIane P and S Waves Lamb’s Problem - Line Load Suddenly Applied on Elastic Half-Space

9 TWO-DIMENSIONAL PROBLEMS IN

LINEARIZED THEORY OF ELASTICITY

9.1. 9.2. 9.3.

9.4. General Case

Plane State of Stress or Strain

Airy Stress Functions for Two-Dimensional Problems

Airy Stress Function in Polar Coordinates

222 224 229 231 235 238 238 241 243 246 249 250 252 254 259 262 263 268 270 270 273 280 280 282 288 295

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CONTENTS XV 9.5. Representation of Two-Dimensional Biharmonic

Functions by Analytic Functions of

a Complex Variable 299

9.6. Kolosoff-Muskhelishvili Method 301

10 VARIATIONAL CALCULUS, ENERGY THEOREMS,

SAINT-VENANT'S PRINCIPLE 313 10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7. 10.8. 10.9. 10.10. 10.11. 10.12. 10.13. 10.14. 10.15. Minimization of F'unctionals

hnctional Involving Higher Derivatives of the Dependent Variable

Several Unknown Functions Several Independent Variables

Subsidiary Conditions - Lagrangian Multipliers Natural Boundary Conditions

Theorem of Minimum Potential Energy Under Small Variations of Displacements

Example of Application: Static Loading on a Beam - Natural and Rigid End Conditions The Complementary Energy Theorem Under Small Variations of Stresses

Variational Functionals Fkequently Used in Computational Mechanics

Saint-Venant's Principle

Saint-Venant 's Principle-Boussinesq-Von

Mises-Sternberg Formulation

Practical Applications of Saint-Venant 's Principle Extremum Principles for Plasticity

Limit Analysis 313 319 320 323 325 328 330 335 339 346 355 359 362 365 369

11 HAMILTON'S PRINCIPLE, WAVE PROPAGATION, APPLICATIONS OF GENERALIZED

COORDINATES 379

11.1. Hamilton's Principle

11.2. Example of Application - Equation of Vibration of a Beam

11.3. Group Velocity

379 383 393

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11.4. Hopkinson’s Experiment 11.5. Generalized Coordinates

11.6. Approximate Representation of Functions 11.7.

11.8.

Approximate Solution of Differential Equations Direct Methods of Variational Calculus

12 ELASTICITY AND THERMODYNAMICS

12.1. 12.2. 12.3. 12.4. 12.5. 12.6. 12.7. 12.8. 12.9.

The Laws of Thermodynamics The Energy Equation

The Strain Energy Function

The Conditions of Thermodynamic Equilibrium The Positive Definiteness of the

Strain Energy Function

Thermodynamic Restrictions on the Stress-Strain Law of an Isotropic Elastic Material

Generalized Hooke’s Law, Including the Etfect of Thermal Expansion

Thermodynamic h n c t i o n s for Isotropic Hookean Materials

Equations Connecting Thermal and Mechanical Properties of a Solid

13 IRREVERSIBLE THERMODYNAMICS AND

VISCOELASTICITY 13.1. 13.2. 13.3. 13.4. 13.5. 13.6. 13.7. 13.8. 13.9. Basic Assumptions

One-Dimensional Heat Conduction

Phenomenological Relations-Onsager Principle Basic Equations of Thermomechanics

Equations of Evolution for a Linear Hereditary MateriaI

Relaxation Modes Normal Coordinates

Hidden Variables and the Force-Displacement Relationship

Anisotropic Linear Viscoelastic Materials

396 398 399 402 402 407 407 412 414 416 418 419 421 423 425 428 428 431 432 436 440 444 447 450 454

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CONTENTS 14 THERMOELASTICITY 14.1. 14.2. 14.3. 14.4. 14.5. 14.6. 14.7. 14.8. 14.9. 14.10. 14.11. 14.12. 14.13. Basic Equations

Thermal Effects Due to a Change of Strain; Kelvin’s Formula

Ratio of Adiabatic to Isothermal Elastic Moduli Uncoupled, Quasi-Static Thermoelastic Theory Temperature Distribution

Thermal Stresses

Particular Integral: Goodier’s Method Plane Strain

An Example - Stresses in a Turbine Disk

Variational Principle for Uncoupled Thermoelasticity Variational Principle for Heat Conduction

Coupled Thermoelasticity

Lagrangian Equations for Heat Conduction and Thermoelasticity

15 VISCOELASTICITY

15.1. Viscoelastic Material

15.2. Stress-Strain Relations in Differential Equation Form 15.3. Boundary-Value Problems and Integral

Transformations

15.4. Waves in an Infinite Medium 15.5. Quasi-Static Problems 15.6. Reciprocity blations 16 LARGE DEFORMATION 16.1. 16.2. Deformation Gradient 16.3. Strains 16.4. 16.5. Strain Rates 16.6. 16.7. Stresses

Coordinate Systems and Tensor Notation

Right and Left Stretch Strain and Rotation Tensors Material Derivatives of Line, Area, and

Volume Elements xvii 456 456 459 459 461 462 464 466 467 470 473 474 478 48 1 487 487 491 497 500 503 507 514 514 521 525 526 528 529 532

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16.8. 16.9. 16.10. 16.11. 16.12. 16.13. 16.14. 16.15. 16.16.

Example: Combined Tension and Torsion Loads Objectivity

Equations of Motion

Constitutive Equations of Thermoelastic Bodies More Examples

Variational Principles for Finite Elasticity: Compressible Materials

Variational Principles for Finite Elasticity: Nearly Incompressible or Incompressible Materials Small Deflection of Thin Plates

Large Deflection of PIates

17 INCREMENTAL APPROACH TO SOLVING

SOME NONLINEAR PROBLEMS

17.1. 17.2. 17.3. 17.4. 17.5. 17.6. 17.7. 17.8. 17.9. 17.10. 17.11.

Updated Lagrangian Description Linearized Rates of Deformation Linearized Rates of Stress Measures Incremental Equations of Motion Constitutive Laws

Incremental Variational Principles in Terms of T Incremental Variational Principles in Terms of :*

Incompressible and Nearly Incompressible Materials

Updated Solution Incremental Loads

Infinitesimal Strain Theory

0 539 543 548 550 557 562 568 573 581 587 587 590 593 597 598 604 610 612 617 620 622

18 FINITE ELEMENT METHODS 624

18.1. Basic Approach 626

18.2. One Dimensional Problems Governed by a Second

Order Differential Equation 629

18.3. Shape hnctions and Element Matrices for Higher

Order Ordinary Differential Equations 638

18.4. Assembling and Constraining Global Matrices 643

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CONTENTS xix 18.6. 18.7. 18.8. 18.9. 18.10. 18.11. 18.12. 18.13. 18.14. 18.15. 18.16. 18.17. 18.18. 18.19. 18.20. 18.21. 18.22.

Two Dimensional Problems by One-Dimensional Elements

General Finite Element Formulation Convergence

Two-Dimensional Shape Functions Element Matrices for a Second-Order Elliptical Equation

Coordinate Transformation

Triangular Elements with Curved Sides Quadrilateral Elements

Plane Elasticity

Three-Dimensional Shape Functions Three Dimensional Elasticity Dynamic Problems of Elastic Solids Numerical Integration

Patch Tests

Locking-F'ree Elements

Spurious Modes in Reduced Integration Perspective

19 MIXED AND HYBRID FORMULATIONS 19.1. Mixed Formulations

19.2. Hybrid Formulations

19.3. Hybrid Singular Elements (Super-Elements)

19.4. Elements for Heterogeneous Materials

19.5. Elements for Infinite Domain

19.6. Incompressible or Nearly Incompressible Elasticity

20 FINITE ELEMENT METHODS FOR PLATES AND SHELLS

20.1.

20.2. Reissner-Mindlin Plates

20.3.

20.4. Hybrid Formulations for Plates

20.5.

20.6. General Shell Elements

20.7.

Linearized Bending Theory of Thin Plates Mixed Functionals for Reissner Plate Theory Shell as an Assembly of Plate Elements Locking and Stabilization in Shell Applications

655 657 664 665 672 676 679 682 690 702 708 714 726 73 1 735 750 754 756 756 760 767 782 782 788 795 795 805 813 819 822 832 843

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21 FINITE ELEMENT MODELING OF NONLINEAR ELASTICITY, VISCOELASTICITY, PLASTICITY, VISCOPLASTICITY AND CREEP

21.1. 21.2. 21.3. 21.4. 21.5. 21.6. 21.7. 21.8.

Updated Lagrangian Solution for Large Deformation Incremental Solution

Dynamic Solution

Newton-Raphson Iteration Method Viscoelasticity Plasticity Viscoplasticity Creep BIBLIOGRAPHY AUTHOR INDEX 849 852 854 855 857 859 869 870 873 909 SUBJECT INDEX 919

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1 INTRODUCTION

Mechanics is the science of force and motion of matter. Solid mechanics is the science of force and motion of matter in the solid state. Physicists are of course interested in mechanics. The greatest advances in physics in the twentieth century are identified with mechanics: the theory of relativity, quantum mechanics, and statistical mechanics. Chemists are interested in the mechanics of chemical reaction, the formation of molecular aggre- gates, the formation of crystals, or the creation of new materials with de- sirable properties, or polymerization of larger molecules, etc. Biologists are interested in biomechanics that relates structure to function at all hierar- chical levels: from biomolecules to cells, tissues, organs, and individuals. Although a living cell is not a homogeneous continuum, it is a protein ma- chine, a protein factory, with internal machinery that moves and functions in an orderly way according to the laws of mechanics. Therefore, all scientists are interested in mechanics and mechanics is developed by scientists continuously.

Engineers, especially aeronautical, mechanical, civil, chemical, materials, biomedical, biotechnological, space, and structural engineers, are real developers and users of fluid and solid mechanics because of their profes- sional needs. They design. They invent. They are concerned about the safety and economy of their products. They want to know the function of their products its precisely as possible. They want results fast. They experiment. They theorize. They test, compute, and validate. To them mechanics is a toy, a bread and butter, a feast or delicacy.

Engineering is quite different from science. Scientists try to understand nature. Engineers try to make things that do not exist in nature. Engineers stress invention. To embody an invention the engineer must put his idea in concrete terms, and design something that people can use. That something

c a n be a device, a gadget, a material, a method, a computing program, an

innovative experiment, a new solution to a problem, or an improvement on what is existing. Since a design has to be concrete, it must have its geome- try, dimensions, and characteristic numbers. Almost all engineers working on new designs find that they do not have all the needed information. Most

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often, they are limited by insufficient scientific knowledge. Thus they study mathematics, physics, chemistry, biology and mechanics. Often they have to add to the sciences relevant to their profession. Thus engineering sciences are born.

This book is written by engineering scientists, for engineering scientists, and this determines its style. The qualities we want are:

0 Easy to read,

0 Precise, concise, and practical,

0 First priority on the formulation of problems, 0 Presenting the classical results as gold standard, and 0 Numerical approach as everyday tool to obtain solutions.

If the book is a banquet, we offer some hors d’oeuvres in this introductory chapter.

1.1. HOOKE’S LAW

Historically, the notion of elasticity was first announced in 1676 by Robert Hooke (1635-1703) in the form of an anagram, ceiiinosssttuw. He explained it in 1678 as

U t tensio sic v i s

,

or “the power of any springy body is in the same proportion with the extension.” t

As stated in the original form, Hooke’s law is not very clear. Our first task is to give it a precise expression. Historically, this was done in two different ways. The first way is to make use of the common notion of “springs,” and consider the load-deflection relationship. The second way is to state it as a tensor equation connecting the stress and strain. Although the second way is the proper way to start a general theory, the first, simpler and more restrictive, is not without interest. In this section, we develop the first alternative as a prototype of the theory of elasticity.

Let us consider the static equilibrium state of a solid body under the action of external forces (Fig. 1.1:l). Let the body be supported in some manner so that at least three points are fixed in a space which is de- scribed with respect to a rectangular Cartesian frame of reference. We shall make three basic hypothesis regarding the properties of the body under considerat ion.

(Hl) The body is continuous and remains continuous under the action of external forces.

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Sec. 1.1 HOOKE’SLAW 3

Under this hypothesis the atomistic structure of the body is ignored and the body is idealized into a geometrical copy in Euclidean space whose

Fig. 1.1:l. Static equilibrium of a body under external forces.

points are identified with the material particles of the body. Continuity is defined in mathematical sense as an iso- morphism of the real number system. Neighboring points remain as neigh- bors under any loading condition. No cracks or holes may open up in the interior of the body under the action of external load.

A material satisfying this hypothesis is said to be a continuum. The study of the deformation or motion of a continuum under the action of forces is called the continuum mechanics. To introduce the second hypothesis, let us consider the action of a set of forces on the body. Let every force be fixed in direction and in point of application, and let the magnitude of all the forces be increased or decreased together: always bearing the same ratio to each other. Let the forces be denoted by PI, Pa,.

. .

,

P n and their magnitude by PI, P2,

.

. .

,

Pn. Then

the ratios PI : P2 :

.

..

: P,, remain fixed. When such a set of forces is applied on the body, the body deforms. Let the displacement at an arbitrary point in an arbitrary direction be measured with respect to a rectangular Cartesian frame of reference fixed with the supports. Let this displacement be denoted by u. Then our second hypothesis is

(H2)

Hooke’s law;

where al, a2,.

. .

,a, are constants independent of the magnitude of PI,

Pz,.

. . ,

P,. The constants a l l a2,.

. .

,a, depend, of course, on the loca- tion of the point at which the displacement component is measured and on the directions and points of application of the individual forces of the loading.

Hooke’s law in the form

(H2)

is one that can be subjected readily to direct experimental examination.

To complete the formulation of the theory of elasticity, we need a third hypothesis:

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(H3) There exists a unique unstressed state of the body, to which the

body returns whenever all the external forces are removed.

A body satisfying these three hypothesis is called a linear elastic solid. A number of deductions can be drawn from these assumptions. We shall list a few important ones.

(A) Principle of superposition

By a combination of (H2) and (H3), we can show that Eq. (1) is valid not only for systems of loads for which the ratios PI : P2 :

. . .

: P, remain fixed as originally assumed, but also for an arbitrary set of loads P I , P2,

.

,

P,. In other words, Eq. (1) holds regardless of the order in which the loads are applied.

Proof. If a proof of the statement above can be established for an arbitrary pair of loads, then the general theorem can be proved by mathematical induction.

Let P1 and P2 (with magnitudes PI and P2) be a pair of arbitrary loads acting at points 1 and 2, respectively. Let the deflection in a specific direction be measured at a point 3 (see Fig. 1.1:l). According to (H2), if P1 is applied alone, then a t the point 3 a deflection u3 = ~ 3 1 P1 is produced. If Pz is applied alone, a deflection u3 = c32P2 is produced. If P1 and Pa are applied together, with the ratio PI : Pz fixed, then according to (H2) the deflection can be written as

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u3 = C k 1 Pl

+

CL2 P2

.

The question arises whether

cil

= ~ 3 1 , cb2 = ~ 3 2 . The answer is af- firmative, as can be shown as follows. After P1 and P2 are applied, we take away P I . This produces a change in deflection, -c&P1, and the total deflection becomes

(b) u3 = C k 1 Pl

+

Ck2P2 - c;1 Pl

.

Now only P2 acts on the body. Hence, upon unloading Pz we shall have

Now all the loads are removed, and u3 must vanish according to (H3).

Rearranging terms, we have

Since the only possible difference of cbl and c& must be caused by the action of P z , the difference

cil

- cgl can only be a function of P2 (and not

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Sec. 1.1 HOOKE’SLAW 6 of

PI).

Similarly, ~ 3 2

-

cL2 can only be a function of

4 .

If we write Eq. (d) as

then the left-hand side is a function of P2 alone, and the right-hand side is a function of PI alone. Since PI and Pz are arbitrary numbers, the only possibility for Eq. (e) to be valid is for both sides to be a constant k which is independent of both PI and P2. Hence,

But a substitution of (f) into (a) yields

The last term is nonlinear in P I , P2, and Eq. (g) will contradict (H2) unless

k

vanishes. Hence, k = 0 and ci2 = ~ 3 2 . An analogous procedure shows

Thus the principle of superposition is established for one and two forces.

An

entirely similar procedure will show that if it is valid for m forces, it is also valid for m

+

1 forces. Thus, the general theorem follows by mathematical induction. Q.E.D.

The constants ~ 3 1 , ~ 3 2 , etc., are seen to be of significance in defining the elastic property of the solid body. They are called influence coeficients or, more specifically, flexibility influence coefficients.

C i l = cp1 = c31.

(B) Corresponding forces and displacements and the unique meaning of the total work done by the forces.

Let us now consider a set of external forces P I ,

. .

.

,

P, acting on the body and define the set of displacements at the points of application and in the direction of the loads as the displacements “corresponding” to the forces at these points. The reactions at the points of support are considered as external forces exerted on the body and included in the set of forces.

Under the loads

P I , .

.

,

P,, the corresponding displacements may be written as u1 = C11Pl

+

c12P2

+ .

.

*

+

ClnPn, 212 = C21Pl

+

c22P2

+

‘ * .

+

% P n , 21n=cn1P1 + c n 2 P 2 + * * * + c n n P n .

...

(2)

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If we multiply the first equation by P I , the second by P2, etc., and add, we obtain

(3) 9.1

+

P2u2

+

. . .

+

Pnu, = CllP?

+

C12PlP2

+

. .

*

+

ClnPlP,

+

C21PlP2

+

c22Pz”

+

. . .

+

c,,P,2.

The quantity above is independent of the order in which the loads are applied. It is the total work done by the set of forces.

( C ) Maxwell’s reciprocal relation

The influence coeficients for corresponding forces and displacements are symmetric.

c . . %J

-

- c , , 3 2 *

In other words, the displacement at a point i due to a unit load at another point j is equal to the displacement at j due to a unit load at a, provided that the displacements and forces ‘%orrespond,” i.e., that they are measured in the same direction at each point.

The proof is simple. Consider two forces P1 and P2 (Fig. 1.1:l). When the forces are applied in the order P I , P2, the work done by the forces is easily seen to be

(4)

1 2

W = -(C11P,”

+

c22P3

+

Cl2PlP2.

When the order of application of the forces is interchanged, the work done is

1

2

W’ = - ( C 2 2 r ) 2 2

+

CllP,”)

+

C21PlPZ.

But according to (B) above, W =

M”

for arbitrary P I , P2. Hence, c12 = c21, and the theorem is proved.

(D) Betti-Rayleigh reciprocal theorem

Let a set of loads P I , P2,

. . .

,

P , produce a set of corresponding displace- ments u1, u2,.

. .

,u,. Let a second set of loads P i , P h , . .

.

,PL, acting in the same directions and having the same points of application as those of the first, produce the corresponding displacements ui, ui,

.

. .

,

uk. Then (5) Plu:

+

P 2 U k

+

. . .

+

P,.:, = P[u,

+

* .

.

+

P{u,,

In other words, in a linear elastic solid, the work done by a set of forces acting through the corresponding displacements produced b y a second set of

(28)

Sec. 1.1 HOOKE'SLAW 7 forces is equal t o the work done by the second set of forces acting through

the corresponding displacements produced b y the first set of forces.

A straightforward proof is furnished by writing out the u( and u: in terms of Pi and Pi, (i = 1,2,.

.

, n ) , with appropriate influence coefficients, comparing the results on both sides of the equation, and utilizing the symmetry of the influence coefficients.

In the form of Eq. ( 5 ) , the reciprocal theorem can be generalized to include moments and rotations as the corresponding generalized forces and generalized displacements. An illustration is given in Fig. 1.1:2. These theorems are very useful in practical applications.

For the same beom, cPf = ct2. ( a ) Forces and corresponding displacements.

(b) Generolized force (moment) and the corresponding generolized displocement (moment -rotation of angle).

Fig. 1.1:2. Illustration of the reciprocal theorem.

(E)

Strain energy

Further insight can be gained from the first law of thermodynamics. When a body is thermally isolated and thermal expansions are neglected the first law states that the work done on the body by the external forces in a certain time interval is equal to the increase in the kinetic energy and internal energy in the same interval. If the process is so slow that the kinetic energy can be ignored, the work done is seen to be equal to the change in internal energy.

If the internal energy is reckoned as zero in the unstressed state, the stored internal energy shall be called strain energy. Writing

U

for the

(29)

strain energy, we have, from (3) and (4),

If we differentiate Eq. (6) with respect to

Pi,

we obtain

But, the right-hand side is precisely ui; hence, we obtain (F) Castigliano’s theorem

(7) - dU = u a ,

dPa

i = 1 , 2

, . * . ,

72.

i = l ,

...,

n .

In other words, if a set of loads P I , .

. . ,

P, is applied on a perfectly elastic body as described above and the strain energy is expressed as a function of the set P I , .

.

,

P,, then the partial derivative of the strain energy, with

respect to a particular load, gives the corresponding displacement at the point of application of that particular load in the direction of that load.

(E) T h e principle of virtual work

On the other hand, for a body in equilibrium under a set of external forces, the principle of virtual work can be applied to show that, i f the strain energy is expressed as a function of the corresponding displacements, then

- dU =Pa,

d U i i = 1,

...,

n

The proof consists in allowing a virtual displacement 6u to take place in the body in such a manner that Su is continuous everywhere but vanishes at all points of loading except under P , . Due to 6u, the strain energy changes by an amount 6 U , while the virtual work done by the external forces is the product of Pi times the virtual displacement, i.e., P i h i . According to the principle of virtual work, these two expressions are equal,

SU

=

Pidui.

On rewriting it in the differential form, the theorem is established.

The important result (8) is established on the principle of virtual work as applied to a state of equilibrium under the additional assumption that a strain energy function that is a function of displacement exists. It is a p plicable also to elastic bodies that follow the nonlinear load-displacement relationship.

(30)

Sec. 1.2 LINEAR SOLIDS WITH MEMORY: MODELS OF

. . .

9 1.2. LINEAR SOLIDS WITH MEMORY: MODELS OF

VISCOELASTICITY

Most structural metals are nearly linear elastic under small strain, as measurements of load-displacement relationship reveal. The existence of normal modes of free vibrations which are simple harmonic in time, is of- ten quoted as an indication (although not as a proof) of the linear elastic character of the material. However, when one realizes that the vibration of metal instruments does not last forever, even in a vacuum, it becomes clear that metals deviate somewhat from Hooke’s law. Thus, other constitutive laws must be considered. The need for such an extension becomes particularly evident when organic polymers are considered.

In this section we shall consider a simple class of materials which retains linearity between load and deflection, but the linear relationship depends on a third parameter, time. For this class of material, the present state of deformation cannot be determined completely unless the entire history of loading is known.

A

linear elastic solid may be said to have a simple memory: it remembers only one configuration; namely, the unstrained natural state of the body. Many materials do not behave this way: they remember the past. Among such materials with memory there is one class that is relatively simple in behavior. This is the class of materials named above, for which the cause and effect are linearly related.

Fig. 1.2:l. Models of linear viscoelasticity: (a) Maxwell, (b) Voigt, (c) Kelvin.

Let us consider some simple examples. In Fig. 1.2:l are shown three mechanical models of material behavior, namely, the Maxwell model, the Voigt model, and the Kelvin model, all of which are composed of combina- tions of linear springs with spring constant p and dashpots (pistons moving in a viscous fluid) with coefficient of viscosity q.

A

linear spring is s u p posed to produce instantaneously a deformation proportional to the load.

A

dashpot is supposed to produce a velocity proportional to the load at any instant. The load-deflection relationship for these models are

(31)

(1) Maxwell model: u = - + - F F

P r l

(2) Voigt model: F = PU

+

qU, u(0) = 0 ,

(3) Kelvin model: F

+

T ~ F = ER(U

+

T , I ~ ) , T,F(O) =

ERT,u(O),

where T ~ , T~ are two constants. When these equations are t o be integrated,

the initial conditions at t = 0 must be prescribed as indicated above. The creep functions c ( t ) , which are the displacement u ( t ) in response to a unit-step force

F ( t )

= l ( t ) defined in Eq. (7) below, are the solution of Eqs. (1)-(3). They are:

(4) Maxwell solid:

c ( t ) =

(1

+

' t ) l ( t )

,

P r l

(5) Voigt solid:

(6) Kelvin solid:

where the unit-step function l ( t ) is defined as 1 when t > O ,

l ( t ) = - when t = 0 ,

i:

0 when t < O . (7)

A

body which obeys a load-deflection relation like that given by Maxwell's model is said to be a Maxwell solid. Similarly, Voigt and Kelvin solids are defined. These models are called models of wiscoelasticity.

Interchanging the roles of F and u , we obtain the relaxation function as a response F ( t ) = k ( t ) corresponding to an elongation u ( t ) = l ( t ) , Eq.

(7).

( 8 ) Maxwell solid:

k ( t ) = p e - ( p ' q ) ' l ( t ) , (9) Voigt solid:

(32)

Sec. 1.2 LINEAR SOLIDS WITH MEMORY: MODELS OF

. . .

11 (10) Kelvin solid:

Here we have used the symbol s ( t ) to indicate the unit-impulse function,

or Dimc-delta function, which is defined as a function with a singularity at the origin:

q t )

= 0 for t < 0, and t > 0 ,

(‘

f ( t ) b ( t ) d t = f(0)

,

E > O ,

J - - E

where f ( t ) is an arbitrary function continuous at t = 0. These functions,

c(t) and k ( t ) , are illustrated in Figs. 1.2:2 and 1.2:3, respectively, for which we add the following comments.

Fig. 1.2:2. Creep function of (a) Maxwell, (b) Voigt, (c) Kelvin solid. A negative phase is superposed at the time of unloading.

For the Maxwell solid, a sudden application of a load induces an immediate deflection by the elastic spring, which is followed by “creep” of the dashpot. On the other hand, a sudden deformation produces an immediate reaction by the spring, which is followed by stress relaxation according to an exponential law Eq. (8). The factor ~ / p , with dimengions of time, may be called a relaxation time: it characterizes the rate of decay of the force.

For the Voigt solid, a sudden application of force will produce no immediate deflection because the dashpot, arranged in parallel with the spring, will not move instantaneously. Instead, as shown by Eq. (5) and Fig. 1.2:2(b), a deformation will be gradually built up, while the spring takes a greater and greater share of the load. The dashpot displacement relaxes exponentially. Here the ratio q / p is again a relaxation time: it characterizes the rate of decay of the deflection.

(33)

al

e

l.2

E L .c 0 0 ) Q ‘0 Time

I

Time

Fig. 1.2:3. Relaxation function, of (a) Maxwell, (b) Voigt, (c) Kelvin solid. For the Kelvin solid, a similar interpretation is applicable. The con- stant rE is the time of relaxation of load under the condition of constant deflection [see Eq. (lo)], whereas the constant ro is the time of relaxation of deflection under the condition of constant load [see Eq. (4)]. As t -+ 00,

the dashpot is completely relaxed, and the load-deflection relation becomes that of the springs, as is characterized by the constant ER in Eqs. (4) and (10). Therefore,

ER

is called the relaxed elastic modulus.

Load-deflection relations such as (1)-(3) were proposed t o extend the classical theory of elasticity to include anelastic phenomena. Lord Kelvin (Sir William Thomson, 1824-1907), on measuring the variation of the rate of dissipation of energy with frequency of oscillation in various materials, showed the inadequacy of the Maxwell and Voigt equations. A more successful generalization using mechanical models was first made by John H. Poynting (1852-1914) and Joseph John Thomson (“J.J.,” 1856-1940) in their book Properties of Matter (London: C . Griffin and Co., 1902).

1.3. SINUSOIDAL OSCILLATIONS IN A VISCOELASTIC

MATERIAL

It is interesting to examine the relationship between the load and the deflection in a body when it is forced t o perform simple harmonic oscillations. A simple harmonic oscillation can be described by the real or imaginary part of a complex variable. Thus, if Fo is a complex variable

FO = Aeib = A(cos

4 +

i sin

d),

then a simple harmonic oscillatory force can be written as

(1) F ( t ) =

FoeiWt

= A(cos4

+

isin4)(coswt

+

isinwt)

(34)

Sec. 1.3 SINUSOIDAL OSCILLATIONS IN A VISCOELASTIC

. . .

13 Similarly, if uo = Bei$, then a simple harmonic oscillatory displacement

u ( t )

= uOeiWt is either the real part or the imaginary part of (2) u ( t ) = uOeiWt = B cos(wt

+

$1

+

i~ sin(wt

+

$)

.

On substituting (1) and (2) into Eqs. (1)-(3) of Sec. 1.2, we can obtain the ratio uo/Fo, which is a complex number. The inverse,

FO/UO,

is called the

complex modulus of a viscoelastic material, and is often designated by

M :

(3)

M

= FO/UO =

lMleis

where

[MI

is the magnitude and 6 is the phase angle by which the strain lags behind the stress. The tangent of 6 is often used as a measure of the internal friction of a linear viscoelastic material:

imaginary part of

M

tan6 =

real part of

M

*

(4)

For the Kelvin model, we have

(5)

M =

1

+

iwr, E R 1

IMI=

( 1 + w 2 r ~ ) 1 ' 2 E R ,

1 f w".," 1

+

iwr,

When

[MI

and t a n 6 in (5) and ( 6 ) are plotted against the logarithm of

w , curves as shown in Fig. 1.3:l are obtained. Experiments with torsional

I I I

I

-

,n -.

I

c .- c Internal friction - 0 Y .- _- ₂ 0.01 0. I 10 100 0-

(35)

Frequency

Fig. 1.3:2. A typical relaxation spectrum. (After C. M. Zener, Elasticity and Anelas- ticity of Metals, The University of Chicago Press, 1948.)

oscillations of metal wires a t various temperatures, reduced to room temperature according t o certain thermodynamic formula, yield a typical “relaxation spectrum” as shown in Fig. 1.3.2. Many peaks are seen in the internal-friction-versus-frequency curve. It has been suggested that each peak should be regarded as representing an elementary process as described above, with a particular set of relaxation times T,, r E . Each set of

relaxation times T,, T~ can be attributed t o some process in the atomic or

microscopic level. A detailed study of such a relaxation spectrum tells a

great deal about the structures of metals; and the study of internal friction has provided a very effective key to metal physics.

1.4. PLASTICITY

Take a small steel rod. Bend it. When the deflection is small, the rod will spring back to its original shape when you release the load. This is elasticity. When the deflection is sufficiently large, a permanent deformation will remain when the load is released. That is plasticity. The load-deflection relationship of an ideal plastic material is shown in Fig. 1.4:1(a). For an ideal plastic material, the deflection is zero when the load is smaller than a critical value. Then, when the load reaches a critical value, the deflection continues to increase (the material flows) as long as the same load remains. If, a t a given deflection, the load becomes smaller than the critical value, then the flow stops. There is no way the load can be increased beyond the critical value.

(36)

Sec. 1.5 VIBRATIONS 15

Deflection

(a) Ideal plasticity

..il.

Deflection

(b) Linear elastic - ideal plastic material

Fig. 1.4:l.

Structural steel behaves pretty much like an ideal plastic material, except that when the load (measured in terms of the maximum shear stress) is smaller than the critical load (called the yield shear stress), the load- deflection curve is an inclined straight line (Hooke’s Law). Upon unloading, there is a small rebound. There are some details at the yield point that were ignored in the statement above. Other metals, such as copper, aluminum, lead, stainless steel, etc., behave in a somewhat similar, but more complex manner. Metals at a sufficiently high temperature may behave more like

a fluid. Theories that deal with these features of materials are called the theories of plasticity, which are presented in Chapter 6 .

1.5. VIBRATIONS

We know vibrations by experience while driving a car, flying an airplane, playing a musical instrument. The trees sway in the wind. A building shakes in an earthquake. Sometimes we want to know if a structure is safe in vibration. Sometimes we want to design a cushion that isolates an instrument from vibrations. A prototype of this kind of problem is shown in Fig. 1.5:1(a). A body with mass M is attached to an initially vertical massless spring, which has a spring constant

k,

and a damping constant c, and is “built-in” to a “ground” which moves horizontally with a displacement history s ( t ) . Let x ( t ) denote the horizontal displacement of the mass and a dot over x or s denote a differentiation with respect to time. Then 2 is the acceleration of the body, k ( x

-

s ) is the spring force acting on the body, and c(k - S) is the viscous damping force acting on the body. Newton’s second law requires that

(1)

MX

+

~ ( 2

-

S)

+

k ( ~

-

S) = 0 .

If we let

(37)

Fig. 1.5:l.

represent the displacement of the body relative to the ground, then Eq. (1) may be written as

(3) M y + c y + k y = - M s .

The solution of Eq. (3) represents a forced vibration of a damped system. If the forcing function MS(t) were zero, the equation

(4) My

+

qj

+

Icy = 0 ,

describes the free vibration of the damped system. If the damping constant

c vanishes, then the free vibration of the undamped system is described by the equation

(5) M y + k y = 0 .

Equation (5) is satisfied by the solution

y = Acoswt

+

B s i n w t ,

(6)

in which A and B are arbitrary constants, and

(7) w =

m,

as can be verified by direct substitution of ( 6 ) into (5). When

k

and M are real values, w is real. The motion y(t) given by (6) is an oscillation of the

(38)

Sec. 1.5 VIBRATIONS 17 body at a circular frequency of w rad/sec. It can exist without external load. Hence w is called the natural frequency of a free vibration.

If

the damping constant c were zero, and the forcing function i is periodic with the same frequency as the natural one, then the amplitude of the oscillation is unbounded, and we have the phenomenon of resonance.

The solution of Eq. (4) must be an exponential function of time, because the derivative of an exponential function is another exponential function. Thus

(8)

On substituting (8) into Eq. (4), we have

y(t) = Ae" implies

y ( t )

= AXe"

,

y(t) = A X 2 e x t .

(9)

MA2

+

C X

+

k

= 0 .

If we write

(10) k / M = w 2 , & = c / ( 2 & 7 ) , then the two roots of Eq. (9) may be written as XI and X2:

(11) X I = -&W + i W J S , Xz = -&W -

i w J 1 - E 2 .

When

k

and M are real positive numbers, w is real, and the solution of Eq. (9) under the initial conditions

(12) y(0) = yo

,

y(0) =

yo

when

t

= 0 . is

--EW t

(13) y ( t ) = yoe c o s w & 2 t

Now, we can return to the solution of Eq. (3). A particular solution of Eq. (3) can be obtained by Laplace or Fourier transformation or other methods. By direct substitution, it can be verified that a particular solution satisfying the initial conditions y(t) = y ( t ) = 0 at

t

= 0 is:

The general solution of Eq. (3) is the sum of the functions given in (13) and (14). From this solution we can examine the nature of the forced oscillation

(39)

and its dependence on the parameters w , E , and the frequency spectrum of

the forcing function s ( t ) .

T h e simple solution given by Eqs. (13) and (14) has important applications to the problems of isolation of delicate instruments in shipping, response of buildings to earthquake, impact of an airplane on landing, landing

a robot on the moon. A frequently asked quention is: what is the maximum absolute value of the displacement y, or the acceleration jj as functions of the peak values of the ground displacement s, ground acceleration 6 , the time course of the ground motion, the system characteristis MI c ,

k,

and the initial conditions yo, I j , ie.,

(15) max I y ( t ) l / max Is(t)l = f u n c t i o n of . s ( t ) , M , c, Ic, yo, Ij,

max Iy(t)l/ max Is(t)l = f u n c t i o n s of s ( t ) , MI c , C, yo, yo

.

These functions are called shock spectra. Studies of shock spectra are important not only for technological applications, but also for mathematics. Note that although Eq. (3) is linear in y ( t ) , the shock spectra are nonlinear functions of the parameters of the system. Qualitatively, we notice that the most important characteristic time of the system. Qualitatively, we notice that the most important characteristic time of the free oscillation is

l/w. The most important characteristic time of the excitation s ( t ) is the length of time it takes t o rise from 0 t o the peak value, t,. We call t , the rise tame of the signal. Alluring to resonance, we see that a parameter of importance is the ratio of the two charateristic times l / w and t,. Hence the shock spectra are principally functions of wt,. See some references in the bibliography at the end of the book.

The picture shown in Fig. 1.5:1(b) may represent, in a very crude way, a concrete floor of a steel frame building during an earthquake. A system shown in Fig. 1.5:1(c) may represent an astronaut in flight. A dumbbell shown in Fig. 1.5:1(d) has been used to model a molecule. It is clear that fairly comprehensive models of dynamic systems can be devised by this approach, see, for example, Fig. 1.5:1(e). When there are too many particles of mass and springs of elasticity, however, simplicity may be lost, and one turns naturally to the continuum approach outlined in the next section.

1.6. PROTOTYPE OF WAVE DYNAMICS

A wire with an infinite number of particles of mass connected together elastically is an obvious candidate for the continuum approach. As an example, consider a wire as shown in Fig. 1.6:l. Let an axis z run along

(40)

Sec. 1.6 PROTOTYPE OF WAVE DYNAMICS 19

Fig. 1.6:l.

Fig. 1.6:2.

the length of the wire, with an origin 0 chosen at the lower end. When the wire is loaded, each particle in the wire will be displaced longitudinally from its original position by an amount u. We shall consider only axial loading and assume that the plane cross sections remain plane, so that u is parallel to the x-axis and is a function of z. We shall assume further that u is infinitesimal, so that the strain in the wire is

The change in cross-sectional dimensions of the wire due to the axial load will be ignored. The wire is assumed to be elastic, and is so thin that all stress components other than the axial may be neglected. Then the axial stress is given by Hooke’s law

dU

dX

0 = Ee = E - ,

in which E is a constant called the Young’s modulus of elasticity. Now consider an element of the wire of a small length dx (see Fig. 1.6:2). The

(41)

force acting at the lower end is u A , where A is the cross-sectional area of the wire. The force acting a t the upper end is A ( a

+

da). For a continuous function a(.) the differential da is equal to ( da /d x)d x. The acceleration of the element, d 2 u / d t 2, must be caused by the difference in axial forces. Now, the mass of the element is pAdx, with p denoting the density of the material. Hence, equating the mass times acceleration with the axial load

Ada, and canceling the nonvanishing factor Adz, we obtain the equation of motion

(3) p - = - a2u aa

at2 ax

A substitution of Eq. (2) yields the wave equation (4)

in which the constant c is the wave speed:

(5)

The wave speed c is a characteristic constant of the material. The general solution of Eq. (4) is

( 6 ) u = f (- ~Ct)

+

F ( x

+

ct)

,

where f and F are two arbitrary functions. This can be verified by substi- tuting Eq. (6) directly into Eq. (4). The function u = f ( x - ct) represents a wave propagating in the positive x direction. The function u = F ( z

+

c t )

represents a wave propagating in the negative x direction. In either case we have

with u = &/at denoting the particle velocity. A substitution of (7) and (5) into (2) yields the formula

E

a = f - w = f p c v ,

where the - sign applies to a wave propagating in the positive x direction, and the

+

sign applies to a wave in the other direction. This result is remarkable. It says that the stress is equal to the product of the mass density of the material, the velocity of the sound wave, and the longitudinal velocity of the particles of the wire. This result has a very important