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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111.3.2 Elastic Mo duli and Elastostatic Stress Balance . . . . . . . . . . . . . 1411.3.3 Energy of Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 1611.3.4 Molecular Origin of Elastic Stress . . . . . . . . . . . . . . . . . . . . 1711.4 Young’s Modulus and Poisson’s Ratio for an Isotropic Material: A SimpleElastostatics Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2011.5T2 Cylindrical and Spherical Coordinates: Connection Coefficients andComponen ts of Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111.6T2 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 ........ 2711.6.2 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 EquationstoTwoDimensionsforaDeformedThin Plate: Stress-Polishing a Telescope Mirror . . . . . . . . . . . . . . . . 48iPart IVELASTICITYiiChapter 11ElastostaticsVersion 1011.1.K, 14 January 2009.Please send comments, suggestions, and errata via email to [email protected] or on paper toKip Thorne, 350-17 Caltech, 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, rotation, and shear(Sec. 11.2.2 and Box 11.2) 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.11.5), 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 for subsequent Parts of this book.11.1 OverviewIn this chapter we consider static equilibria ofelasticsolids—forexample,theequilibriumshape and in ternal strains of a cantilev ered balcony on a building, deformed by the weightof people standing on it.12From the point of view of continuum mechanics, a solid (e.g. a wooden b o ard 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 original shape. Note the requirement that this be true for any 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 for a longtime.We shall confine our attention in this chapter to elastic solids,whichdeformwhilethestress is applied in such a way that the magnitude of the deformation (quantified by atensorial strain)islinearlyproportionaltotheapplied,tensorial stress.Thislinear,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. 11.1a), its fractional change of length (its strain) isproportional to the force, ∆!/! ∝ F .Inthelanguageofstressesandstrains(introducedbelow), Hooke’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=∆!/!,withaproportionalityconstantE called Young’s modulus that is a property of thematerial from which the rod is made:FA≡ Tzz= ESzz≡ E∆!!. (11.1)zF +6(a)j(b)Fig. 11.1: (a) Hook e’s one-dimensional law for a rod stretched by a force F :∆!/! ∝ F .(b)The3-dimensional displacement v ector ξ(x)insidethestretchedrod.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 introduced 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)insideastressedbodyanditsgradient∇ξ,whichisthestraintensorS = ∇ξ.Wethenexpressthestraintensorasthesumofitsirreducible tensorial parts:anexpansion Θ, a rotation R,andashear 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 a material resists volume change (an expansion-type strain)by developing an opposing isotropic stress, with a stress/strain ratio that is equal to the bulkmodulus K;andhowthematerialalsoresistsashear-typestrainbydevelopinganopposingshear stress with a stress/strain ratio equal to twice the shear modulus 2µ.Wethencomputethe elastic force density inside the material, as the divergence of the sum of these t wo elasticstresses, and we formulate the law of
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