This paper describes a simplified model for predicting the axial displacement, stress, and strain in pipes subjected to internal shock waves. This model involves the neglect of radial and rotary inertia of the pipe, so its predictions represent the spatially averaged or low-pass–filtered response of the tube. The simplified model is developed first by application of the physical principles of conservation of mass and momentum on each side of the shock wave. This model is then reproduced using the mathematical theory of the Green's function, which allows other load and boundary conditions to be more easily incorporated. Comparisons with finite element simulations demonstrate that the simple model adequately captures the tube's axial motion, except near the critical velocity corresponding to the bar wave speed $E/ρ$. Near this point, the simplified model, despite being an unsteady model, predicts a time-independent resonance, while the finite element model predicts resonance that grows with time.

## References

1.
Beltman
,
W.
,
Burscu
,
E.
,
Shepherd
,
J.
, and
Zuhal
,
L.
,
1999
, “
,”
ASME J. Pressure Vessel Technol.
,
121
(
3
), pp.
315
322
.10.1115/1.2883709
2.
Beltman
,
W.
, and
Shepherd
,
J.
,
2002
, “
,”
J. Sound Vib.
,
252
(
4
), pp.
617
655
.10.1006/jsvi.2001.4039
3.
Shepherd
,
J. E.
, and
Akbar
,
R.
,
2009
, “
Forces Due to Detonation Propagation in a Bend
,” Graduate Aeronautical Laboratories California Institute of Technology, Pasadena, CA, Technical Report No. FM2008-002.
4.
Shepherd
,
J. E.
, and
Akbar
,
R.
,
2009
, “
Piping System Response to Detonations. Results of ES1, TS1 and SS1 Testing
,” Graduate Aeronautical Laboratories California Institute of Technology, Pasadena, CA, Technical Report No. FM2009-001.
5.
Karnesky
,
J.
,
Damazo
,
J. S.
,
Chow-Yee
,
K.
,
Rusinek
,
A.
, and
Shepherd
,
J. E.
,
2013
, “
Plastic Deformation Due to Reflected Detonation
,”
Int. J. Solids Struct.
,
50
(
1
), pp.
97
110
.10.1016/j.ijsolstr.2012.09.003
6.
Timoshenko
,
S. P.
,
1934
,
Theory of Elasticity
, 1st ed.,
McGraw-Hill
,
New York
.
7.
Kolsky
,
H.
,
1963
,
Stress Waves in Solids
,
Dover
,
New York
.
8.
Haberman
,
R.
,
2004
,
Applied Partial Differential Equations
, 4th ed.,
Pearson Education
9.
Schiffner
,
K.
, and
Steele
,
C. R.
,
1971
, “
Cylindrical Shell With an Axisymmetric Moving Load
,”
AIAA J.
,
9
(
1
), pp.
37
47
.10.2514/3.6122