PSI - Issue 23

Viacheslav Mokryakov et al. / Procedia Structural Integrity 23 (2019) 143–148 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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An original numerical method for calculating axisymmetric Pochhammer – Chree waves was presented in [19]. In particular, as an example, the behavior of a longitudinal compression shock wave is considered. It is shown that the longitudinal compression of the material on the rod axis can be several times greater than on the rod surface. This work is focused on the stress fields in rods at wave propagation. The stress-strain state in an infinite cylindrical rod under the conditions of a longitudinal axisymmetric Pochhammer – Chree wave is considered. Particular attention is paid to the localization of the maximal stress values (tension and shear). 2. Stress calculation We consider an infinite rod of radius a . The material of the rod has a density ρ, Young's modulus E , Poisson’s ratio ν. The surfac e of the rod is free from stress. We also intro duce the following notation: ω is the wave frequency; γ is the w ave number; c = ω/γ is the phase velocity of the elastic wave; 0 c E   is the velocity of infinitely long waves in an infinite rod;   1 2 c      is the velocity of P-waves; 2 c    is the velocity of S-waves in an infinite medium; c R is the velocity of Rayleigh waves. Also we will use the following expressions:  In addition, we write down the known identities for the Bessel functions, which we will use in the following:         1 1 2 n n n nJ x x J x J x     ,       1 1 2 n n n J x J x J x      The general solution of the axisymmetric Pochhammer-Chree equation is often presented in a complex form [6]:             1 1 0 0 ; i z t i z t r z u AhJ hr C J r e u A J hr C J r ie                       where A and C are coefficients determined by boundary conditions. To get the stress fields, we need to convert the general solution in a complex form into a real one. To do this, you need to combine the real and imaginary parts of the solution into a linear combination by entering the phase φ:             1 1 0 0 sin ; cos r z u AhJ hr C J r z t u A J hr C J r z t                         The dispersion relation is obtained from the boundary conditions: 0; 0 rr rz r a r a       So                   , , , 2 2 2 0 0 1 0 1 2 2 sin rr r r z z r r r u u u r u A h J hr Ah J hr AhJ hr r C J r C J r r z t                                         2 2 , , 1 1 2 cos rz r z z r u u Ah J hr C J r z t              Equating r = a, we get                     2 2 2 0 1 0 1 2 2 1 1 2 2 A h h J ha hJ ha a C J a J a a Ah J ha C J a                              Hence                         2 2 1 1 2 2 2 1 0 1 2 2 0 J a J a a J a A C h J ha h h J ha h a J ha                  The last equality is the dispersion relation, i.e. the relationship of the frequency ω with the wavenumber γ (or with the phase velocity c , replacing c   ). 2 2 1 1 1 c c h   , 2 2 1 1 c c 2    , 2 1 1 c h c H c     2 2 2 2 , 2 2 1 c c     2 2 c 2 2 K