ContentsIV ELASTICITY ii11 Elastostatics 111.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Displacement and Strain; Expansion, Rotation, and Shear . . . . . . . . . . 411.2.1 Displacement Vector and Strain Tensor . . . . . . . . . . . . . . . . . 411.2.2 Expansion, Rotation and Shear . . . . . . . . . . . . . . . . . . . . . 811.3 Stress and Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111.3.1 Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 111.3.2 Elastic Moduli and Elastostatic Stress Balance . . . . . . . . . . . . . 1411.3.3 Energy of Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 1611.3.4 Molecular Origin of Elastic Stress . . . . . . . . . . . . . . . . . . . . 1811.4 Young’s Modulus and Poisson’s Ratio fo r an Isotropic Material: A SimpleElastostatics Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2011.5 T2 Cylindrical and Spherical Coo rdinates: Connection Coefficients andComponents of Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111.6 T2 Solving the 3-Dimensional Elastostatic Equation in Cylindrical Coordi-nates: Simple Methods, Separation of Variables and Green’s Functions . . . 2711.6.1 Simple Methods: Pipe Fracture and Torsion Pendulum . . . . . . . . 2 711.6.2 Separation of Variables and Green’s Functions: Thermoelastic Noisein a LIGO Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.7 Reducing the Elastostatic Equations to One Dimension for a Bent Beam;Cantilever Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.8 Bifurcation, Buckling and Mountain Folding . . . . . . . . . . . . . . . . . . 4311.9T2 Reducing the Elastostatic Equations to Two Dimensions for a DeformedThin Plate: Stress-Polishing a Telescope Mirror . . . . . . . . . . . . . . . . 48iPart IVELASTICIT YiiElasticityVersion 1111.1.K, 03 January 2 011NOT YET WRITTEN: AN INTROD UCTION THAT SPELLS OUT THE CONTENTSOF THIS PART OF THE BOOK, AND THE IMPORTANCE OF ELASTICITY THEORYiiiChapter 11ElastostaticsVersion 1111.1.K, 3 January 20 11.Please send comments, suggestions, and errata via em ail to kip @caltech.edu or on pa per toKip Thorne, 350 - 17 C altech, Pasadena CA 91125Box 11.1Reader’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, and its most importantexample, decomposition of the strain tensor into expansion, rot ation, and shear(Sec. 11.2.2 and Box 11.2) will be encountered again in Part V (Fluid Mechanics)and Part VI (Plasma Physics).• Differentiation of vectors and tensors with the help of connection coefficients (Sec.11.5), will be used occasionally in Part V (Fluid Mechanics) and Part VI (PlasmaPhysics), and will be generalized to non-orthonormal bases in Part VII (GeneralRelativity) and used extensively there.• No other portions of this chapter are important f or subsequent Parts of this book.11.1 Ove rviewIn this chapter we consider static equilibria of elastic solids — for example, the equilibriumshape and internal strains of a cantilevered balcony on a building, deformed by the weightof people standing o n it.12From the point of view of continuum mechanics, a solid (e.g. a wooden board in thebalcony) is a substance that recovers its shape after the application and removal of any smallstress. In other words, after the stress is removed, the solid can be rotated and translated toassume its orig inal shape. Note the requirement that this be true for a ny stress. Many fluids(e.g. water) satisfy our definition as long as the applied 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 fo r a longtime.We shall confine our attention in this chapter to elastic solids, 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 s tress. 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 a n applied force F (Fig. 11.1a), its fractional change of length (its strain) isproportional to the force, ∆ℓ/ℓ ∝ F . In the language of stresses and strains (introducedbelow), Hoo ke’s law says that the longitudinal stress Tzz≡ (longitudinal force F per unitcross sectional area A of the rod) = F/A is proportional to the longitudinal strain Szz=∆ℓ/ℓ, with a proportiona lity constant E called Young’s modulus that is a property of thematerial from which the rod is made:FA≡ Tzz= ESzz≡ E∆ℓℓ. (11.1)zF +∆(a)ξ(b)Fig. 11.1: (a) Hooke’s one-dimensional law for a rod stretched by a force F : ∆ℓ/ℓ ∝ F . (b) The3-dimensional displacement vector ξ(x) ins ide the stretched rod.Hooke’s law will turn out to be one component of 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 approach to these tensors willfollow the geometric, frame-independent philosophy int roduced in Chap. 1. Some readers3may wish to review that philosophy and associated mathematics by rereading the “[N]”sections of Chap. 1.We begin in Sec. 11.2 by introducing, in a frame-independent way, the vectorial dis-placement field ξ(x) inside a stressed body and its gradient ∇ξ, which is the strain tensorS = ∇ξ. We then express the strain tensor as the sum of its irreducible tensorial parts: anexpansion Θ, a rotation R, and a shear Σ.In Sec. 11.3.1 we introduce the stress tensor for a deformed, isotropic, elastic material. InSec. 11.3.2, we discuss how such …
View Full Document
Unlocking...