Abstract
After determining the values of the nonlocal moduli for longitudinal waves in an infinite space, Fourier transforms of the equations of axially symmetric longitudinal waves in an infinite circularly cylindrical rod are established and decoupled according to the Pochhammer procedure. Dispersion equation is obtained from the conditions of traction free surface of the rod, and compared with its classical counterpart. While the velocity of long waves coincides, as required, with that derived in the classical case, the velocity of short waves turns out to be about 36% less.
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Abbreviations
- a :
-
interatomic spacing
- a 1,a 2,a 3 :
-
coefficients defined by (2.12.5)
- c :
-
wave phase velocity
- d :
-
rod diameter
- h, l :
-
defined by (2.15)
- k :
-
wave number
- overbar:
-
denotes Fourier transform
- R :
-
rod radius
- r, r′ :
-
vector of the point of observation and of generic point, respectively
- u :
-
displacement in thex 1-direction
- u, w :
-
displacements in ther- andz-direction, respectively
- \(\bar u(r,k;\omega )\) :
-
double Fourier transform ofu(r, z, t)
- δ(r′−r):
-
Dirac delta function
- μ, λ:
-
Lamé constants
- μ′, λ′:
-
nonlocal moduli
- \(\bar \mu \prime (k),\bar \lambda \prime (k)\) :
-
Fourier transforms of 03BC;′ and λ′
- A :
-
wave length
- ϱ:
-
mass density
- τ11 :
-
normal stress in thex 1-direction
- τ rr , τ rz , τ zz :
-
stress components in polar coordinates,r, α,z
- θ:
-
dilatation
- ξ, β* :
-
defined by (2.20.2)
- ω:
-
wave frequency
- \(\bar \Omega\) :
-
notation defined by (2.13.2)
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Prepared with partial support of the University of Delaware.
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Nowinski, J.L. On a nonlocal theory of longitudinal waves in an elastic circular bar. Acta Mechanica 52, 189–200 (1984). https://doi.org/10.1007/BF01179616
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DOI: https://doi.org/10.1007/BF01179616