A concrete dam wall (density ) stands at rest

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Strain and stress in a deformed solid

Problem

A concrete dam wall (density ρ=2400 kg m−3\rho = 2400\,\mathrm{kg\,m^{-3}}) stands at rest, holding back the water behind it; the only body force is gravity, fi=−ρgf_i = -\rho g vertically. Separately, a steel rod (density ρ=7850 kg m−3\rho = 7850\,\mathrm{kg\,m^{-3}}) is struck end-on by a hammer, so that a compressive stress pulse of magnitude 60 MPa60\,\mathrm{MPa} builds up over a length of 2 mm2\,\mathrm{mm} along the rod. ((a)) For the dam wall, estimate the size of the body-force term in Cauchy's equation of motion, in N m−3\mathrm{N\,m^{-3}}, taking g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}}. Since the wall is at rest, what must the stress-gradient term equal, and what is the inertia term? ((b)) For the struck rod, estimate the size of the stress-gradient term ∂jσij\partial_j\sigma_{ij} (in N m−3\mathrm{N\,m^{-3}}) from the given pulse, and hence the inertia term ρ ∂2ui/∂t2\rho\,\partial^2u_i/\partial t^2 expressed as an acceleration (in m s−2\mathrm{m\,s^{-2}}). Compare the body-force term ρg\rho g for steel with the stress-gradient term, and state which term of Cauchy's equation may be neglected in each of the two situations.

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