TABLE OF CONTENTS As we indicated in Section 2-2-9, the equilibrium equations contain more unknowns than equations so that in order to complete the formulation, additional equations must be developed. These fall into two categories. One category is the strain-displacement equations, which will be developed in this chapter. These equations give relationships between the displacements that a body undergoes and the strains that result in the body. Both strain and displacement are geometric concepts and therefore have no obvious connection with the concept of stress. The remaining category of equations is the stress-strain equations or, more generally, constitutive equations. These equations provide the necessary link between stress and strain, and their development will be taken up in Chapter 4.
We begin by deriving relationships between the strains and the displacement components and then show that the strains are not all independent, but are related through the compatibility equations. We then show that the strain-displacement relationships completely describe the state of strain at a point in a body. The concept of principal strains is then developed, along with several other useful concepts concerning strain. We close the chapter by deriving the strain-displacement and compatibility equations in polar coordinates.
When loads are applied to a body made up of a nonrigid material, the shape of the body changes and we say that the body deforms. What we mean is that the material points of the body change position relative to one another so that some points move closer together while others...
