A rubber gasket has shear modulus and Poisson's ratio

PHYSICS · Subjective · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Hooke's law for an isotropic solid

Problem

A rubber gasket has shear modulus G=0.5 MPaG = 0.5\,\mathrm{MPa} and Poisson's ratio ν=0.4999\nu = 0.4999. It is placed in a fluid and subjected to a uniform confining pressure of 5 MPa5\,\mathrm{MPa} (equal compressive stress on all faces, no shear). Someone argues: "rubber has an everyday Young's modulus E=3G≈1.5 MPaE = 3G \approx 1.5\,\mathrm{MPa}, which is very small, so under 5 MPa5\,\mathrm{MPa} of pressure the fractional volume change is about ΔV/V≈−p/E≈−3.3\Delta V/V \approx -p/E \approx -3.3, that is, the gasket collapses to a small fraction of its original volume." Find the correct fractional volume change ΔV/V\Delta V/V under this hydrostatic pressure, and explain what is wrong with the quoted argument.

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