Chapter 1 - Fundamentals
Published online by Cambridge University Press: 05 June 2012
Summary
To analyze waves in nonlinear materials, we must derive the equations governing the motion of such materials. In this chapter we define three types of motion: translation, rotation, and deformation. We define a quantity called the deformation gradient. This quantity is used to describe volume strain, longitudinal strain, and shear strain. Then stress and pressure are discussed. We introduce these concepts in three spatial dimensions and then specialize the results to one spatial dimension. The chapter closes with a derivation of two important laws of motion: the conservation of mass and the balance of momentum.
Index Notation
We begin by introducing a compact and convenient way to describe quantities in three spatial dimensions with a Cartesian coordinate system. We also show how to transform quantities between different Cartesian coordinate systems.
Cartesian Coordinates and Vectors
A Cartesian coordinate system has three straight coordinates that are mutually perpendicular to each other (see Figure 1.1). The coordinates are named x1, x2, and x3. The subscripts 1, 2, and 3 are called indices. When we let k represent any of these indices, we can refer to the coordinates by the compact notation xk. Each coordinate xk also has a shaded arrow that represents a unit coordinate vectorik. We use a boldface symbol to denote a vector. The length of each ik is equal to unity.
Now consider another vector v that is drawn with arbitrary angles to the three coordinates. We draw a box with sides parallel to ik and with v as a diagonal.
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- Publisher: Cambridge University PressPrint publication year: 1998