Propagation of Elastic Stress Waves simuletd with FEM

 Japanese All Rights Reserved by Dr. Shinozuka, Jun. Unauthorized duplication shall be a violation of applicable laws.

When a bar collides with a bar, stress wave propagates like a wave in these bars. A longitudinal stress wave only can be considered if the diameter of a bar is small.

A stress increment depends on a mass density, a speed of stress wave and a change in a particle velocity of the bar. Of course, the change in the particle velocity depends on the impact speed. An elastic wave speed (sonic speed of the bulk material ) depends on a mass density, Young's modulus and Poisson's ratio. Assuming that the impact speed is slow and an impact stress, which is obtained by integrating the stress increment with respect to time, is lower than the yield stress of the bar, an elastic stress wave propagates from the impacted surface into these bars. In this case, only an elastic deformation occurs. When an elastic wave reaches at the end of a bar, it reflects on the surface. If the surface be the free and the stress is compressive stress, the reflected stress is tensile stress. The stress in a region where the transmitted and reflected stress overlap is zero. On the other hand, if the surface is the fixed surface and stress is compressive stress, compressive stress reflects. The stress in a region where the transmitted and reflected stress overlap is compressive stress, the magnitude of the stress becomes twice that of the transmitted stress.

The following animations show some results simulated by using of a FEA. Simulations were performed with a dynamic and thermo elastic plastic FEM software (RDynFem) that I have developed.

In this simulation, an elastic bar (input bar) collides with a stationary elastic bar (output bar). The material properties of the bars are the same and the diameter of bars are the same. If the length of bars is the same, the input bar stops, while the output bar flies at the impact speed after the elapsed time that the elastic wave travels twice the length of the bar. No stress remains since the compressive stress is canceled by the tensile stress. If the length of output bar is longer than that of input bar, the input bar stops, while the output bar flies at the impact speed too, after the elapsed time that the elastic wave travels twice the length of the input bar. We can see that compressive stress and tensile stress appears alternately in the output bar. The position that the tensile stress is generated by the interferences of stresses first does not depend on the length of the output bar. The distance from the surface to the position is equal to the length of the input bar.

When the length of output bar is the same as that of input bar.

When the length of output bar is twice as long as that of input bar.

When the length of output bar is three times as long as that of input bar.

This animation shows distribution of effective stress.

By the way, the situation of stress propagation is as follows, when the right hand surface of the output bar is fixed.

When the length of output bar is the same as that of output bar.

When the length of output bar is three times as long as that of input bar.

You can see the stress reflected at the fixed surface doubles (you can see the color of the right had side of the bar becomes deep blue), though the stress becomes zero for free surface. Then the input bar begin to move in opposite direction, though it stops when the right hand side of the output bar is free.

These animation show the longitudinal stress distribution.

These results were simulated with FEM software "RDynFem" that has been developed by Dr. Shinozuka, Jun.