Poisson's Ratio

One of the many new problems that Russ Hobbie and I added to the 4th edition of Intermediate Physics for Medicine and Biology deals with Poisson’s ratio. From Chapter 1:

Problem 25 Figure 1.20, showing a rod subject to a force along its length, is a simplification. Actually, the cross-sectional area of the rod shrinks as the rod lengthens. Let the axial strain and stress be along the z axis. They are related to Eq. 1.25, s_z = E ε_z. The lateral strains ε_x and ε_y are related to s_z by s_z = - (E/ν) ε_x = -(E/ν) ε_y, where ν is called the “Poisson’s ratio” of the material.
(a) Use the result of Problem 13 to relate E and ν to the fractional change in volume ΔV/V.
(b) The change in volume caused by hydrostatic pressure is the sum of the volume changes caused by axial stresses in all three directions. Relate Poisson’s ratio to the compressibility.
(c) What value of ν corresponds to an incompressible material?
(d) For an isotropic material, -1 ≤ ν ≤ 0.5. How would a material with negative ν behave?
Elliott et al. (2002) measured Poisson’s ratio for articular (joint) cartilage under tension and found 1 ≤ ν ≤ 2. This large value is possible because cartilage is anisotropic: Its properties depend on direction.

The citation is to a paper by Dawn Elliott, Daria Narmoneva and Lori Setton, “Direct Measurement of the Poisson’s Ratio of Human Patella Cartilage in Tension,” in the Journal of Biomechanical Engineering, Volume 124, Pages 223–228, 2002. (Apologies to Dr. Narmoneva, whose name was misspelled in our book. It is now corrected in the errata, available at the book website.)

As hinted at in our homework problem, a particularly fascinating type of material has negative Poisson’s ratio. Some foams expand laterally, rather than contract, when you stretch them; see Roderic Lakes, “Foam Structures with a Negative Poisson’s Ratio,” Science, Volume 235, Pages 1038–1040, 1987. A model for such a material is shown in this video. Lakes’ website contains much interesting information about Poisson’s ratio. For instance, cork has a Poisson’s ratio of nearly zero, making it ideal for stopping wine bottles.

Simeon Denis Poisson (1781–1840) was a French mathematician and physicist whose name appears several times in Intermediate Physics for Medicine and Biology. Besides Poisson’s ratio, in Chapter 9 Russ and I present the Poisson equation in electrostatics, and its extension the Poisson-Boltzmann equation governing the electric field in salt water. Appendix J reviews the Poisson probability distribution. Finally, Poisson appeared in this blog before, albeit as something of a scientific villain, in the story of Poisson’s spot. Poisson is one of the 72 names appearing on the Eiffel Tower.