ContentsIII ELASTICITY ii10 Elastostatics 110.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110.2 Displacement and Strain; Expansion, Rotation, and Shear . . . . . . . . . . 410.2.1 Displacement Vector and Strain Tensor . . . . . . . . . . . . . . . . . 410.2.2 Expansion, Rotation and Shear . . . . . . . . . . . . . . . . . . . . . 810.3 Cylindrical and Spherical Coordinates: Connection Coefficients and Compo-nents of Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110.4 Stress and Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1610.4.1 Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1610.4.2 Elastic Moduli and Elastostatic Stress Balance . . . . . . . . . . . . . 1910.4.3 Energy of Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 2110.4.4 Molecular Origin of Elastic Stress . . . . . . . . . . . . . . . . . . . . 2210.4.5 Young’s Modulus and Poisson’s Ratio for Isotropic Material . . . . . 2410.5 Solving the 3-Dimensional Elastostatic Equation; Thermoelastic Noise in Grav-itational Wave Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.6 Reducing the Elastostatic Equations to One Dimension for a Bent Beam;Cantilever Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.7 Reducing the Elastostatic Equations to Two Dimensions; Deformation ofPlates — Keck Telescope Mirror . . . . . . . . . . . . . . . . . . . . . . . . . 4010.8 Bifurcation - Mountain Folding . . . . . . . . . . . . . . . . . . . . . . . . . 4 4iPart IIIELASTICIT YiiChapter 10ElastostaticsVersion 0610.1.K, 03 January 2007. NOTE: cross references in this chapter to Chaps.1, 2, 3 are to new 2006 versions, placed on the course website in January 2007.Please send comments, suggestions, and errata via e mail to k [email protected] or on paperto Kip Thorne, 130-33 Caltech, Pasadena C A 91125Box 10.1Reader’s Guide• This chapter relies heavily on the geometric view of Newtonian physics (includingvector and tensor analysis) laid out in the sections of Chap. 1 labeled “[N]”.• Chapter 11 (Elastodynamics) is an extension of this chapter; to understand it, thischapter must be mastered.• The idea of the irreducible tensorial parts of a tensor, a nd its most importantexample, decomposition of the strain tensor into expansion, rotation, and shear(Sec. 10.2.2 and Box 10.2) will be encountered again in Part IV (Fluid Mechanics)and Part V (Plasma Physics).• Differentiation of vectors and tensors with the help of connection coefficients (Sec.10.3), will be used occasionally in Part IV (Fluid Mechanics) and Part V (PlasmaPhysics), and will be generalized to non-orthonormal bases in Part VI (GeneralRelativity) and used extensively there.• No other portions of this chapter are important f or subsequent Parts of this book.10.1 IntroductionIn this chapter we consider static equilibria of elastic solids — fo r example, the equilibriumshape and internal strains of a cantilevered balcony on a building, deformed by the weight12of people standing on it.From the point of view of continuum mechanics, a solid (e.g. a wooden board in thebalcony) is a substance that recovers its shape a fter the application and removal of any smallstress. In o ther words, after the stress is removed, the solid can be rotated and t ranslated toassume its original shape. Note the requirement that this be true for any stress. Many fluids(e.g. water) satisfy our definition as long as t he a pplied stress is isotropic; however, they willdeform 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 longtime.We shall confine our attention in this chapter to elastic so l i ds, which deform while thestress is applied in such a way that the magnitude of the deformation (quantified by atensorial strain) is linearly proportional to the applied, tensorial stress. This linear, three-dimensional stress-strain relationship, which we shall develop and explore in this chapter,generalizes Hooke’s famous one-dimensional law (originally expressed in the concise Latinphrase “Ut tensio, sic vis”). In English, Hooke’s law says that, if an elastic wire or rod isstretched by an applied force F (Fig. 10 .1a), its fractional change of length (its strain) isproportional to the force, ∆ℓ/ℓ ∝ F . In the language of stresses and strains (introducedbelow), Hooke’s law says that the longitudinal stress Tzz≡ (longitudinal fo rce F per unitcross sectional area A of the rod) = F/A is proportional to the longitudinal strain Szz=∆ℓ/ℓ, with a proportionality constant E called Young’s modulus that is a property o f thematerial from which the rod is made:FA≡ Tzz= ESzz≡ E∆ℓℓ. (10.1)zF +∆(a)ξ(b)Fig. 10.1: (a) Hooke’s one-dimensional law for a rod stretched by a force F : ∆ℓ/ℓ ∝ F . (b) The3-dimensional displacement vector ξ(x) inside the stretched rod.Hooke’s law will turn out to be one component o f the three-dimensional stress-strainrelation, but in order to understand it deeply in that language, we must first develop a deepunderstanding of the strain tensor and the stress tensor. Our a pproach to these tensors willfollow the geometric, fra me-independent philosophy introduced in Chap. 1. Some readers3may wish to review that philosophy and associated mathematics by rereading Secs. 1.3, 1.5,1.9, and 1.11.We begin in Sec. 10.2 by introducing, in a frame-independent way, the vectorial displace-ment field ξ(x) inside a stressed body and its gradient ∇ξ which is the strain tensor S = ∇ξ.We then express the strain tensor as the sum of its irreducible tensorial parts: an e xpan s ionΘ, a rotation R, and a s hear Σ. Because elasticity theory entails computing gradients ofvectors and tensors, and practical calculations are often best performed in cylindrical orspherical coordinate systems, we present a mathematical digression in Sec. 10.3 — an intro-duction to how one can perform practical calculations of gradients of vectors and tensors inthe orthonormal bases associated with curvilinear coordinate systems, using t he concept ofa connection coefficient (the directional derivative of one basis vector field along another).In Sec. 10.3 we also …
View Full Document
Unlocking...