Hannes

Hello, thank you for the welcome and for providing the description of the stress wave and the subtleties (danke). Coming back to the most simple case, could you or anybody indicate the systematic mathematical way to derive the impact solution to the wave equation of the displacements? This is now my main question. The relations below, being in agreement with your description, at the same time satisfy the wave equation for the displacements and thus appear to be a successful guess of the correct solution. The impact stress wave is described by Heavisides function, theta(x>0) = 1 and theta(x<0) = 0: sigma = - rho c v theta(ct - x) after the impact (0 < t < L/c) and sigma = - rho c v theta(2L - ct - x) after the first reflection (L/c < t < 2L/c). This implies for the displacements relative to the unstressed rod (u = 0 for zero stress): u = rho c v/E (ct - x) theta(ct - x) after the impact (0 < t < L/c) and u = rho c v/E (2L - ct - x) theta(2L - ct - x) after the first reflection (L/c < t < 2L/c). Best regards, Hannes

The impact should be elastic, so the rod may bounce off. However, I am only interested in a solution of the initial stress wave before it reaches the end of the rod.

Hello all, the problem: A rod of length L, cross sectional area A, Young's modulus E and density rho moves with velocity v, hitting with its end on a rigid plane. Questions: 1. What is the maximum impact stress sigma? 2. How propagates the elastic stress wave in the rod (time and coordinate dependence)? An answer to question 1 could be sketched in this way: The impact stress is sigma = rho c v, where the wave velocity c is the square root of E/rho. This follows from the impuls change of the rod dp in a time interval dt, (i) dp = sigma A dt, substituting (ii) dp = rho A v c dt. Some explanations: Equation (i) invokes the stress sigma at the impacting cross section which is equal to the impuls change dp per time and area of the rod. Equation (ii) calculates the impuls change dp = rho dV v, where dV = A c dt is the increase of the compressed stress wave volume with nearly zero velocity at the impacting side of the rod and v is the velocity of the decreasing uncompressed region outside the stress wave. I am looking for the solution procedure for question 2. Any hints, a book or internet links are appreciated.