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5 - The linear elastic solid approximation. Basic equations and boundary value problems in the linear theory of elasticity

Published online by Cambridge University Press:  05 June 2014

Grigory Isaakovich Barenblatt
Affiliation:
University of California, Berkeley
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Summary

The mathematical theory of elasticity based on the idealization presented in this chapter is a remarkable classical branch of the mechanics of continua, which has advanced far in the more than 200 years it has been studied. In this chapter a concise presentation of the fundamentals of this theory will be given, bearing in mind readers who have not met it before, and it will also serve as a preparation for the next chapter, where we discuss the mathematical modeling of fracture phenomena, which nowadays is the principal area of attention.

The fundamental idealization

A crucially important property of a deformable solid continuum is that it is possible for it to possess non-trivial stress distributions even when the body is at rest, i.e. when the velocity is everywhere equal to zero.

The theory of elasticity as a science is older than fluid mechanics. Its basic law, which was developed to a fundamental model, was formulated by Robert Hooke more than 300 years ago, in the article Hooke (1678).

Readers already know the formulation of Hooke's law for an elastic rod. A rod is an elastic body whose length l is substantially larger than its cross-sectional size s and which has a constant cross-section area S (see Figure 5.1, taken from the book of Galileo Galilei (1638)). Let us take the longitudinal direction of the rod as the x1 axis of a system of orthonormal Cartesian coordinates.

Type
Chapter
Information
Flow, Deformation and Fracture
Lectures on Fluid Mechanics and the Mechanics of Deformable Solids for Mathematicians and Physicists
, pp. 79 - 100
Publisher: Cambridge University Press
Print publication year: 2014

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