0210.1 - Part III ELASTICITY 1 Chapter 10 Elastostatics...

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Unformatted text preview: Part III ELASTICITY 1 Chapter 10 Elastostatics Version 0210.1, 08 January 2003 Please send comments, suggestions, and errata via email to [email protected] and [email protected], or on paper to Kip Thorne, 130-33 Caltech, Pasadena CA 91125 10.1 Introduction In this chapter we consider static equilibria of elastic solids. Our first concern is to generalize Hooke’s law, derived initially for weights hanging on thin wires, to accommodate three dimensional stress. The most appropriate mathematical machinery for continuum mechanics involves tensors and their derivatives. In particular, 3-dimensional deformations and the stresses they produce will be described by a second-rank strain tensor and a second-rank stress tensor . These will allow us to describe deformations and stresses in three dimensional solids of irregular shape. We will then develop a formalism to derive stresses from strains and vice versa and will illustrate it by application to the problems of thermoelastic noise in gravitational-wave detectors, large telescope mirrors, cantilever bridges, and mountains. This standard approach to elasticity is our first example of a common (some would com- plain far too common) approach to a physics problem, namely to linearize it. Linearization may be acceptable when the distortions are small. However, under strong loading, elas- tic media may become neutrally stable to small displacements which can therefore grow to large amplitude. A classical result of elasticity theory, due originally to Euler, is that when an elastic solid is compressed, there comes a point where stable equilibria can disappear. For an applied force in excess of this maximum, the solid will buckle . This is an example of bifurcation , a phenomenon, as has subsequently been realized, that is common to many physical systems. We will illustrate bifurcation using a strut under a compressive load. We will encounter other examples of bifurcation when we study fluids, in Part IV of the book. 10.2 Strain; Expansion, Rotation, and Shear From the point of view of continuum mechanics, a solid is a substance that recovers its shape after the application and removal of any small stress. That is to say, after the stress 1 2 is removed, the solid can be rotated and translated to assume its original shape. Note the requirement that this be true for any stress. Many fluids (e.g. water) are effectively incompressible. This means that they satisfy our definition as long as the applied stress is isotropic; however, they will deform permanently under a shear stress. Other materials (for example, the earth’s crust) are only elastic for limited times, but undergo plastic flow when a stress is applied for a long time....
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0210.1 - Part III ELASTICITY 1 Chapter 10 Elastostatics...

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