On the axially-symmetric steady wave propagation in elastic circular rods

Julián Adem
1954 Quarterly of Applied Mathematics  
In the first part of this report we find an exact solution for the problem of steady wave propagation in an isotropic, elastic circular bar of infinite length, free of stress on its lateral surface and loaded by a harmonic body force, parallel to its axis, whose amplitude is a Dirac delta function. The solution of this problem gives at the same time the solution for a semi-infinite bar with a special prescribed load on its boundary plane. In the second part, we find a solution for the
more » ... ite bar in which the conditions at the boundary plane are prescribed in terms of functions which give implicitly the stresses and displacements. Finally, using a frequency of interest in current ultrasonic experimental work, we develop a numerical example and compare the result with the case of a low frequency. 1. Infinite bar with the body force &{z)e~""i. We consider an infinite circular bar of a perfectly elastic isotropic material, free of stresses on the lateral surface and loaded by the body force 8(z)e~"", where S(z) is the Dirac delta function defined by 5(z) = 0 for 2^0, f 5(2) dz = 1, J -CO and co is a positive number. The problem is that of determining the displacement at all points of the rod for the steady case (i.e. the time dependence for stresses and displacements is The general solution. For an elastic isotropic medium the equation of motion is where u is the displacement vector; X is the body force vector, X and /1 elastic constants and p the density. Fig. 1 Using cylindrical coordinates (r, 6, 2) as shown in Fig. 1 , we have axial symmetry (i.e. the solution is independent of 6). ♦
doi:10.1090/qam/63912 fatcat:zrlsk7mzfvfwhkqs6vjglwlhre