A projectile impact drives a nearly rectangular compressive longitudinal stress pulse…

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Fluids & Continuum Physics · Superposition, boundaries and interference

Bending, Ringing, Breaking and Flowing, Ch Elastic waves in a solid and at its surface

Problem

A projectile impact drives a nearly rectangular compressive longitudinal stress pulse into the front face of a steel plate of thickness

d = 25\,\mathrm{mm}

. The pulse has peak compressive stress of magnitude

\sigma_0 = 950\,\mathrm{MPa}

and duration

\tau = 400\,\mathrm{ns}

, and it travels normal to the faces of the plate. The back face of the plate is free, with only air (

\rho_{\rm air} = 1.2\,\mathrm{kg\,m^{-3}}

, sound speed

340\,\mathrm{m\,s^{-1}}

) beyond it. For the steel take

\rho = 7850\,\mathrm{kg\,m^{-3}}

and

c_L = 5.86\,\mathrm{km\,s^{-1}}

. Assume the pulse is a plane one-dimensional stress wave and that the plate is thick enough (

d

much greater than the pulse's spatial length) that only the first reflection at the back face matters. Steel of this grade fractures (spalls) wherever the net tensile stress exceeds

\sigma_{\rm spall} = 780\,\mathrm{MPa}

. ((a)) Find the stress reflection coefficient at the back face, and hence the peak stress (magnitude and sign) immediately after the pulse has reflected. ((b)) As the pulse turns round at the back face as tension, the reflected wave travels back into the plate through the rest of the incident pulse, which is still arriving. Treating both as rectangular blocks of stress, find at what depth below the back face, and how long after the front of the pulse first reaches it, the net stress first becomes tensile, and its value there. Does the plate spall, and roughly how thick is the scab that separates? ((c)) Find the fraction of the pulse's energy that is reflected back into the steel, and hence the fraction transmitted into the air.

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