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Elasticity of Solids

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Understanding Acoustics

Part of the book series: Graduate Texts in Physics ((GTP))

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Abstract

If we take a piece of solid matter that is initially at rest and apply equal and opposing forces to the sample, Newton’s 2nd Law of Motion guarantees that the sample will remain at rest because the net force on the sample is zero. If the sample is an elastic solid, then those forces will cause the solid to deform by an amount that is directly proportional to the applied forces. When the forces are removed the sample will return to its original shape and size. These are the characteristics that are required if we say the behavior of the solid is “elastic.”

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Notes

  1. 1.

    For gases, this is an important distinction (e.g., the adiabatic bulk modulus of air is 40% larger than the isothermal bulk modulus). The difference between the adiabatic and isothermal Young’s modulus can be expressed in terms of the material’s absolute (Kelvin) temperature, T; the (volumetric) coefficient of thermal expansion, α = (1/V) (∂V/∂T)p; mass density, ρ; and specific heat (per unit mass) at constant pressure, c p.

    $$ {E}_{\mathrm{ad}}=\frac{E_{\mathrm{iso}}}{1-{E}_{\mathrm{iso}} T{\alpha}^2/9\rho {c}_{\mathrm{p}}} $$

    In most solids, this is a small effect. At room temperature, E iso Tα 2/9ρ c p is about 0.44% for aluminum and 340 ppm for copper.

  2. 2.

    Simeon Denis Poisson (1781–1840).

  3. 3.

    If the diameter of a sound beam is much greater than its wavelength, the associated compressions and expansions do not allow the material to “squeeze out” because there is a compression both above and below that is trying equally hard to squeeze the material in the transverse direction.

  4. 4.

    In fluids, shearing forces diffuse, they are not restored elastically. See Sect. 9.4.

  5. 5.

    The Lame constants, λ and μ, also are taken as the two isotropic moduli in some elasticity calculations, although they are not listed in Table 4.1. The shear modulus, G, and μ are identical and \( \lambda =\frac{E\nu}{\left(1+\nu \right)\left(1-2\nu \right)}=\frac{G\left(E-2G\right)}{3G-E}=\frac{2 G\nu}{1-2\nu } \).

  6. 6.

    I will apologize for indulging in a bit of a fraud at this point by introducing piezoelectric solids. So far, we have addressed the elastic behavior of isotropic solids that require only two independent moduli to completely specify their elastic behavior. Piezoelectric solids are intrinsically anisotropic crystalline materials that not only require more than two elastic constants, but also require specification of the electrical impedance that provides the load across their electrodes, Z load. For example, a piezoelectric material’s stiffness will be different if the electrodes are electrically shorted together (i.e., Z load = 0) or left as an “open circuit” (i.e., Z load = ∞). Anisotropic elasticity will be addressed later in this chapter, but for more complex crystalline substances, there are necessarily more than two independent elastic constants. In practice, once the relevant constants have been identified for a specific deformation, they are incorporated into Hooke’s law in the same way as the elastic constants of an isotropic solid.

  7. 7.

    A ferroelectric ceramic behaves like a piezoelectric crystal. The difference is the piezoelectric behavior of the crystal is intrinsic and a ferroelectric material (usually a ceramic or polymer) only exhibits piezoelectric behavior after the material has been “polarized,” usually by application of a large electric field at elevated temperatures.

  8. 8.

    It is important to remember that in a textbook example it is convenient to define “some generic steel,” but properties of steel (e.g., modulus, yield strength, endurance limit, heat capacity, thermal conductivity , electrical conductivity, etc.) vary with alloy composition and temper. In a commercial design, the choice of the material is very important.

  9. 9.

    Tonpiltz is from the German ton (tone) and piltz (mushroom). Apparently, the piezoelectric stack is the stem and the head mass is the “singing mushroom” cap.

  10. 10.

    You will also want to employ some protective eyewear.

  11. 11.

    The moment of inertia \( I={\displaystyle \int \rho {y}^2 dA} \) is related to the square of the radius of gyration since the radius of gyration is equivalent to I for a material of unit mass density, if divided by cross-sectional area S. The square of the radius of gyration is also equivalent to the second moment of area.

  12. 12.

    In principle, the width could be kept constant and the thickness could be tapered so t(x) = [t(0)/L 1/3](L − x)1/3. Since spring steel is available in constant thickness sheets, it is frequently more convenient to provide a linear taper of the width.

  13. 13.

    Unlike the moving-coil electrodynamic loudspeaker in Fig. 2.12, the voice coil is wound around a laminated steel core, like the cores of electrical transformers, and is stationary. The piston is attached to an armature that supports several magnets that move due to the oscillatory magnetic forces produced by the alternating currents through the stationary coils. Such moving-magnet linear motors can be far more efficient than the moving coil version but at the price of reduced bandwidth.

  14. 14.

    The terms elastomeric and rubberlike will be used interchangeably in this chapter. While all rubbers are elastomers, not all elastomers are rubbers. The distinction is codified in the American Society for Testing and Materials (ASTM) Standard D 1566, which is based on the length of time required for a deformed sample to return to its shape after removal of the deforming force, as well as the extent of that recovery [24].

  15. 15.

    In mathematics and signal processing the Kramers–Kronig relations are known as Sokhotski–Plemelj theorem or the Hilbert transform.

  16. 16.

    For an expression like that in Eq. (4.63), it is actually easier to solve for the minimum of the inverse than to solve for the maximum:

    $$ \frac{d}{ d x}\left(\frac{1+{x}^2}{x}\right)=\frac{d}{ d x}\left(\frac{1}{x}+ x\right)=0 $$
  17. 17.

    The fact that these integrals are a consequence of linear response theory and causality can be proven by the use of complex integration of functions that are “analytic in the upper half-plane” and the Cauchy residue theorem.

  18. 18.

    Carbon black (soot) is a common material that is mixed with rubber to increase its stiffness and strength. The shear modulus of natural rubber (latex) can increase by an order of magnitude if it is mixed with carbon black.

  19. 19.

    Said another way, the molecular mobility is “frozen out.”

  20. 20.

    Recall from Sect. 2.5.2, the decibel i s defined in terms of a power or energy ratio (see Eq. 2.69). Since transmissibility, T, is a ratio of linear quantities, as defined in Eq. (2.96), it can be expressed as a decibel only when squared: dB = 10 log10 T 2 = 20 log10 T.

  21. 21.

    An automotive valve spring in an engine that operates at 2000 rpm will be stressed 240 million times during the operational lifetime of 2000 hours. A typical home refrigerator/freezer may last 15 years. (In America, refrigerators are replaced more often as an interior decorating choice than due to product failure.) If a refrigerator uses a spring that vibrates at 60 Hz, then the number of fully reversing stress cycles accumulated in 15 years of service would be 28 billion cycles.

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  1. Pine Grove Mills, PA, USA

    Steven L. Garrett

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Garrett, S.L. (2017). Elasticity of Solids. In: Understanding Acoustics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-49978-9_4

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