Stress-Wave Reflection Example

Problem

A steel plate is 20 mm thick. The impact contact duration is estimated at 15 microseconds. Is the stress-wave effect significant?

Given

• Plate thickness: t = 20 mm = 0.020 m
• Steel wave speed: c ~ 5400 m/s (3D dilatational)
• Contact duration: t_c = 15 microseconds

Solution

Wave transit time through thickness:
  t_wave = t / c = 0.020 / 5400 = 3.7 microseconds

Response parameter:
  D = t_c / t_wave = 15 / 3.7 = 4.1

Since D > 1, the stress wave has reflected multiple
times during the contact. The through-thickness
stress state is approximately equilibrated.

However, for the in-plane wave (span 200 mm):
  t_inplane = 0.200 / 5400 = 37 microseconds
  D_inplane = 15 / 37 = 0.41

The in-plane wave has NOT reached the supports
during the contact — the in-plane response
is local and inertial.

Interpretation

The through-thickness response is quasi-static (multiple reflections), but the in-plane response is inertial (supports not reached). This is a mixed response — the through-thickness stress is equilibrated, but the global bending has not developed. A shell model may be adequate for the through-thickness, but the inertial effects must be captured in the in-plane response.