Crystal Elasticity

6Tension–Compression in a Cylindrical Rod

6.1. Tension–compression without transverse deformation

Consider a cylindrical rod of length L, radius R, density ρ and Young’s modulus E; assume that it is vibrating longitudinally and at first only its longitudinal deformation ε_x is considered, and the corresponding displacement u, at any point x (-L/2≤x≤+L/2). Then the following relation can be written:

[6.1]

The energies corresponding to deformation T and displacement V can be written as:

[6.2]

[6.3]

The vibration resonance is characterized by a harmonic function of displacement whose general form is f(x,y,z,t)=f(x,y,z)sin(2πF_Lt), where F_L is the resonance frequency. The integration over a period T=1/F_L therefore yields:

[6.4]

In what follows, this relation is systematically used at any speed in the displacement energy. The spatial Lagrangian can be written in a timeindependent form:

[6.5]

Hamilton’s principle or Lagrange’s least action principle yields:

[6.6]

The integration by parts ...