Elasticity

Tensor, Dyadic, and Engineering Approaches

Pei Chi Chou, Nicholas J. Pagano, Nicholas J. Pagano

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486136141
• Topic: Technology & Engineering
• Subtopic: Civil Engineering
• Index: Technology & Engineering

Exceptionally clear text treats elasticity from engineering and mathematical viewpoints. Comprehensive coverage of stress, strain, equilibrium, compatibility, Hooke's law, plane problems, torsion, energy, stress functions, more. 114 illustrations. 1967 edition.

1

Analysis of Stress

1.1. Introduction

In this chapter we shall define stress and discuss two important aspects of stress, namely, the state of stress at a point, which leads to the transformation of stress equations, and the equations governing the variation of the stress components in space. Emphasis will be placed on the physical nature of stress in contrast to the material of Chapter 9, where we are more concerned with the mathematical structure of tensor quantities. The two-dimensional case is used for the detailed derivation of equations, and most of the three-dimensional equations are presented without derivation. A graphical representation of the transformation of stress components is given in addition to the analytical equations. The equations of equilibrium are derived from the engineering point of view by taking a small element as a free body.

Most of the material in this chapter, which deals essentially with two-dimensional problems, is usually covered in elementary courses, such as strength of materials. It is included here and discussed in detail in order to clarify the basic nature of the stress tensor. The notations and sign conventions adopted, where possible, are consistent with those used in the general approach in Chapter 9, and, in general, they follow the most accepted ones in engineering literature.

The theory of stress presented in this chapter is applicable to any continuum, e.g., elastic or plastic solids, viscous fluids, regardless of the mechanical properties of the material.

1.2. Body Forces, Surface Forces, and Stresses

When a body is subjected to an applied load system, internal forces are induced in the body. The behavior of the body, i.e., the changes in its dimensions (its deformation) or in some cases, its eventual failure, is mainly a function of the internal force distribution, which in turn depends upon the external force system. The response of a body to an external force system is conveniently studied by grouping forces into two categories, body forces and surface forces. Body forces are associated with the mass of the body and are distributed throughout the volume of a body; they do not result from direct contact with other bodies. Gravitational, magnetic, and inertia forces are all body forces. They are specified in terms of force per unit volume, i.e., body force intensity. The x, y, and z components of body force intensity are given the symbols F_x, F_y, and F_z. Many authors use the term “body force,” with units of force per unit volume, to mean “body force intensity.” The exact meaning of the term is always clear from the context in which the term is used. Surface forces result from physical contact between two bodies, or more subtly, they may represent the force which an imaginary surface within a body exerts on the adjacent surface.

If an imaginary cutting plane is assumed to pass through a body as shown in Fig. 1.1 and part I is analyzed as a free body, we observe that surface forces P₁ and P₂ are held in equilibrium (assuming the body is in equilibrium) by the force exerted on part I by part II. This force, however, is distributed over the entire plane; that is, any elementary area ΔA is subjected to a force ΔF; consequently, the average force per unit area is

Fig. 1.1 External Surface Force...

Table of contents

• DOVER CLASSICS OF SCIENCE AND MATHEMATICS
• Title Page
• Copyright Page
• Dedication
• Preface
• Table of Contents
• Introduction
• 1 - Analysis of Stress
• 2 - Strain and Displacement
• 3 - Stress-Strain Relations
• 4 - Formulation of Problems in Elasticity
• 5 - Two-Dimensional Problems
• 6 - Torsion of Cylindrical Bars
• 7 - Energy Methods
• 8 - Cartesian Tensor Notation
• 9 - The Stress Tensor
• 10 - Strain, Displacement, and the Governing Equations of Elasticity
• 11 - Vector and Dyadic Notation in Elasticity
• 12 - Orthogonal Curvilinear Coordinates
• 13 - Displacement Functions and Stress Functions
• Index
• CATALOG OF DOVER BOOKS