**Principles of Solid Mechanics** – Evolving from more than 30 years of research and teaching experience, Principles of Solid Mechanics offers an in-depth treatment of the application of the full-range theory of deformable solids for analysis and design. Unlike other texts, it is not either a civil or mechanical engineering text, but both. It treats not only analysis but incorporates design along with experimental observation. Principles of Solid Mechanics serves as a core course textbook for advanced seniors and first-year graduate students.

The author focuses on basic concepts and applications, simple yet unsolved problems, inverse strategies for optimum design, unanswered questions, and unresolved paradoxes to intrigue students and encourage further study. He includes plastic as well as elastic behavior in terms of a unified field theory and discusses the properties of field equations and requirements on boundary conditions crucial for understanding the limits of numerical modeling.

Designed to help guide students with little experimental experience and no exposure to drawing and graphic analysis, the text presents carefully selected worked examples. The author makes liberal use of footnotes and includes over 150 figures and 200 problems. This, along with his approach, allows students to see the full range, non-linear response of structures.

**Contents**show

## Principles of Solid Mechanics – Preface

There is no area of applied science more diverse and powerful than the mechanics of deformable solids nor one with a broader and richer history. From Galileo and Hooke through Coulomb, Maxwell, and Kelvin to von Neuman and Einstein, the question of how solids behave for structural applications has been a basic theme for physical research exciting the best minds for over 400 years. From fundamental questions of solid-state physics and material science to the mathematical modeling of instabilities and fracture, the mechanics of solids remains at the forefront of today’s research. At the same time, new innovative applications such as composites, prestressing, silicone chips, and materials with memory appear everywhere around us. To present to a student such a wonderful, multifaceted, mental jewel in a way that maintains the excitement while not compromising elegance and rigor, is a challenge no teacher can resist. It is not too difficult at the undergraduate level where, in a series of courses, the student sees that the simple solutions for bending, torsion, and axial load lead directly to analysis and design of all sorts of aircraft structures, machine parts, buildings, dams, and bridges. However, it is much more difficult to maintain this enthusiasm when, at the graduate level, the next layer of sophistication is necessary to handle all those situations, heretofore glassed over and postponed, where the strength-ofmaterials approach may be inaccurate or where a true field theory is required immediately.Principles of Solid Mechanics

This book has evolved from over 30 years of teaching advanced seniors and first-term graduate students a core course on the application of the full-range field theory of deformable solids for analysis and design. It is presented to help teachers meet the challenges of leading students in their exciting discovery of the unifying field theories of elasticity and plasticity in a new era of powerful machine computation for students with little experimental experience and no exposure to drawing and graphic analysis. The intention is to concentrate on fundamental concepts, basic applications, simple problems yet unsolved, inverse strategies for optimum design, unanswered questions, and unresolved paradoxes in the hope that the enthusiasm of the past can be recaptured and that our continued fascination with the subject is made contagious. Principles of Solid Mechanics

In its evolution this book has, therefore, become quite different from other texts covering essentially the same subject matter at this level.2 First, by including plastic as well as elastic behavior in terms of a unified theory, this text is wider in scope and more diverse in concepts. I have found that students like to see the full range, nonlinear response of structures and more fully appreciate the importance of their work when they realize that incompetence can lead to sudden death. Moreover, limit analysis by Galileo and Coulomb historically predates elastic solutions and is also becoming the preferred method of analysis for design not only in soil mechanics, where it has always dominated, but now in most codes for concrete and steel structures. Thus in the final chapters, the hyperbolic field equations of plasticity for a general Mohr-Coulomb material and their solution in closed form for special cases is first presented. The more general case requiring slip-line theory for a formal plasticity solution is then developed and applied to the punch problem and others for comparison with approximate upper-bound solutions.Principles of Solid Mechanics

Secondly, while the theory presented in the first three chapters covers familiar ground, the emphasis in its development is more on visualization of the tensor invariants as independent of coordinates and uncoupled in the stress–strain relations. The elastic rotations are included in anticipation of Chapter 4 where they are shown to be the harmonic conjugate function to the first invariant leading to flow nets to describe the isotropic field and closed-form integration of the relative deformation tensor to determine the vector field of displacements. Although the theory in three dimensions (3D) is presented, the examples and chapter problems concentrate on two-dimensional (2D) cases where the field can be plotted as contour maps and Mohr’s circle completely depicts tensors so that the invariants are immediately apparent. Students often find the graphic requirements difficult at first but quickly recognize the heuristic value of field plots and Mohr’s circle and eventually realize how important graphic visualization can be when they tackle inverse problems, plasticity, and limit analysis. In addition to the inclusion of elastic rotations as part of the basic field equations, the discussion in Chapter 4 of the properties of field equations and requirements on boundary conditions is normally not included in intermediate texts.Principles of Solid Mechanics

However, not only do all the basic types of partial differential equations appear in solid mechanics, but requirements for uniqueness and existence are essential to formulating the inverse problem, understanding the so-called paradox associated with certain wedge solutions, and then appreciating the difficulties with boundary conditions inherent in the governing hyperbolic equations of slip-line theory. There are only 15 weeks in a standard American academic term for which this text is designed. Therefore the solutions to classic elasticity problems presented in the intermediate chapters have been ruthlessly selected to meet one or more of the following criteria: a. to best demonstrate fundamental solution techniques particularly in two dimensions, b. to give insight as to the isotropic and deviatoric field requirements, c. to present questions, perhaps unanswered, concerning the theory and suggest unsolved problems that might excite student interest, d. to display particular utility for design, e. to serve as a benchmark in establishing the range where simpler strength-of-materials type analysis is adequate, or f. are useful in validating the more complicated numerical or experimental models necessary when closed-form solutions are not feasible. At one time the term “Rational Mechanics” was considered as part of the title of this text to differentiate it from others that cover much of the same material in much greater detail, but from the perspective of solving boundary value problems rather than visualizing the resulting fields so as to understand “how structures work.” The phrase “Rational Mechanics” is now old-fashioned but historically correct for the attitude adopted in this text of combining the elastic and plastic behavior as a continuous visual progression to collapse. Principles of Solid Mechanics

This book makes liberal use of footnotes that are more than just references. While texts in the humanities and sciences often use voluminous footnotes, they are shunned in modern engineering texts. This is, for a book on Rational Mechanics, a mistake. The intention is to excite students to explore this, the richest subject in applied science. Footnotes allow the author to introduce historical vignettes, anecdotes, less than reverent comments, uncertain arguments, ill-considered hypotheses, and parenthetical information, all with a different perspective than is possible in formal exposition. In footnotes the author can speak in a different voice and it is clear to the reader that they should be read with a different eye. Rational Mechanics is more than analysis and should be creative, fun, and even emotional. To close this preface on an emotional note, I must acknowledge all those professors and students, too numerous to list, at Princeton, Caltech, Delaware, and Buffalo, who have educated me over the years. This effort may serve as a small repayment on their investment. It is love, however, that truly motivates. It is, therefore, my family: my parents, Rowland and Jean; their grandchildren, Rowland, George, Kelvey, and Jean; and my wife, Martha Marcy, to whom this book is dedicated.

### Principles of Solid Mechanics – Table of Contents

PREFACE

INTRODUCTION

Types of Linearity

Displacements-Vectors and Tensors

Finite Linear Transformation

Symmetric and Asymmetric Components

Principal or Eigenvalue Representation

Field Theory

STRAIN AND STRESS

Deformation (Relative Displacement)

The Strain Tensor

The Stress Tensor

Components at an Arbitrary Orientation (Tensor Transformation)

Isotropic and Deviatoric Components

Principal Space and Octahedral Representation

Two-Dimensional Stress or Strain

Mohr’s Circle for a Plane Tensor

Mohr’s Circle in Three Dimensions

Equilibrium of a Differential Element

Other Orthogonal Coordinate Systems

Summary

STRESS-STRAIN RELATIONSHIPS (RHEOLOGY)

Linear Elastic Behavior

Linear Viscous Behavior

Simple Viscoelastic Behavior

Fitting Laboratory Data with Viscoelastic Models

Elastic-Viscoelastic Analogy

Elasticity and Plasticity

Yield of Ductile Materials

Yield (Slip) of Brittle Materials

STRATEGIES FOR ELASTIC ANALYSIS AND DESIGN

Rational Mechanics

Boundary Conditions

Tactics for Analysis

St. Venant’s Principle

Two-Dimensional Stress Formulation

Types of Partial Differential Field Equations

Properties of Elliptic Equations

The Conjugate Relationship between Mean Stress and Rotation

The Deviatoric Field and Photoelasticity

Solutions by Potentials

LINEAR FREE FIELDS

Isotropic Stress

Uniform Stress

Geostatic Fields

Uniform Acceleration of the Half-Space

Pure Bending of Prismatic Bars

Pure Bending of Plates

TWO-DIMENSIONAL SOLUTIONS FOR STRAIGHT AND CIRCULAR BEAMS

The Classic Stress-Function Approach

Airy’s Stress Function in Cartesian Coordinates

Polynomial Solutions and Straight Beams

Polar Coordinates and Airy’s Stress Function

Simplified Analysis of Curved Beams

Circular Beams with End Loads

Concluding Remarks

RING, HOLES AND INVERSE PROBLEMS

Lames Solution for Rings under Pressure

Small Circular Holes in Plates, Tunnels, and Inclusions

Harmonic Holes and the Inverse Problem

Harmonic Holes for Free Fields

Neutral Holes

Solution Tactics for Neutral Holes-Examples

Rotating Disks and Rings

WEDGES AND THE HALF-SPACE

Concentrated Loadings at the Apex

Uniform Loading Cases

Uniform Loading over a Finite Width

Nonuniform Loadings on the Half-Space

Line Loads within the Half-Space

Diametric Loadings of a Circular Disk

Wedges with Constant Body Forces

Corner Effects-Eigenfunction Strategy

TORSION

Elementary (Linear) Solution

St. Venant’s Formulation (Noncircular Cross-Sections)

Prandtl’s Stress Function

Membrane Analogy

Thin-Walled Tubes of Arbitrary Shape

Hydrodynamic Analogy and Stress Concentration

CONCEPTS OF PLASTICITY

Plastic Material Behavior

Plastic Structural Behavior

Plastic Field Equations

Example-Thick Ring

Limit Load by a “Work” Calculation

Theorems of Limit Analysis

The Lower-Bound Theorem

The Upper-Bound Theorem

Example-The Bearing Capacity (Indentation) Problem

ONE-DIMENSIONAL PLASTICITY FOR DESIGN

Plastic Bending

Plastic “Hinges”

Limit Load (Collapse) of Beams

Limit Analysis of Frames and Arches

Limit Analysis of Plates

Plastic Torsion

Combined Torsion with Tension and/or Bending

SLIP-LINE ANALYSIS

Mohr-Coulomb Criterion (Revisited)

Lateral “Pressures” and the Retaining Wall Problem

Graphic Analysis and Minimization

Slip-Line Theory

Purely Cohesive Materials (f = 0)

Weightless Materials (g = 0)

Retaining Wall Solution for f = 0 (EPS Material)

Comparison to the Coulomb Solution (f = 0)

Other Special Cases: Slopes and Footings (f = 0)

Solutions for Weightless Mohr-Coulomb Materials

The General Case

An Approximate “Coulomb Mechanism”

Note: Each chapter also contains a section of Problems and Questions