A very fast temperature increase, produced by a nonuniform heat generation, induces in a simply supported, isotropic, cylindrical rod both longitudinal and flexural vibrations. This paper presents an analytical method to study these vibrations and determine the stresses they provoke. The proposed procedure relies on three main steps: an exact solution for the temperature field is first obtained, by means of Fourier–Bessel expansions; quasistatic thermal stresses are then computed as a function of the calculated temperature distribution, making use of the thermoelastic displacement potential and of the solution to the equivalent isothermal two-dimensional stress problem; finally, longitudinal and flexural vibrations excited by an equivalent thermal force and thermal bending moment are determined using the mode-summation method. The influence of thermal shock duration on the maximum value of the longitudinal dynamic stress and of the ratio between the characteristic thermal time and structural response time on the dynamic bending deflection is analyzed and discussed. Finally, a comparison between the analytical model and experimental measurements is presented. The analytical model described in this paper allows the complete evaluation, within the linear elastic domain, of quasistatic and dynamic thermal stresses induced in an isotropic cylindrical rod by rapid internal heating.

1.
Boley
,
B. A.
, 1955, “
Thermally Induced Vibrations of Beams
,”
J. Aeronaut. Sci.
0095-9812,
23
, pp.
179
181
.
2.
Boley
,
B. A.
, 1972, “
Approximate Analyses of Thermally Induced Vibrations of Beams and Plates
,”
ASME J. Appl. Mech.
0021-8936,
39
, pp.
212
216
.
3.
Boley
,
B. A.
, and
Barber
,
A. D.
, 1957, “
Dynamic Response of Beams and Plates to Rapid Heating
,”
ASME J. Appl. Mech.
0021-8936,
24
(
3
), pp.
413
416
.
4.
Murozono
,
M.
, 1996, “
Thermally Induced Bending Vibrations of Internally Heated Beams in Air
,”
J. Therm. Stresses
0149-5739,
19
, pp.
649
670
.
5.
Blandino
,
J. R.
, and
Thornton
,
E. A.
, 2001, “
Thermally Induced Vibration of an Internally Heated Beam
,”
ASME J. Vibr. Acoust.
0739-3717,
123
, pp.
67
75
.
6.
Burgreen
,
D.
, 1962, “
Thermoelastic Dynamics of Rods, Thin Shells, and Solid Spheres
,”
Nucl. Sci. Eng.
0029-5639,
12
, pp.
203
217
.
7.
Burgreen
,
D.
, 1967, “
Thermoelastic Dynamics of a Pulse Reactor
,”
Nucl. Sci. Eng.
0029-5639,
30
, pp.
317
327
.
8.
Bargmann
,
H.
, 1973, “
Dynamic Response of External Targets Under Thermal Shock
,” CERN Technical Note No. LAB II∕BT∕Int∕73-3.
9.
Sievers
,
P.
, 1974, “
Elastic Stress Waves in Matter Due to Rapid Heating by an Intense High-Energy Particle Beam
,” CERN Technical Note No. LAB II∕BT∕74-2.
10.
Lessen
,
M.
, 1956, “
Thermoelasticity and Thermal Shock
,”
J. Mech. Phys. Solids
0022-5096,
5
, pp.
57
61
.
11.
Lessen
,
M.
, 1959, “
Thermoelastic Waves and Thermal Shock
,”
J. Mech. Phys. Solids
0022-5096,
7
, pp.
77
84
.
12.
Chadwick
,
P.
, and
Sneddon
,
I. N.
, 1958, “
Plane Waves in an Elastic Solid Conducting Heat
,”
J. Mech. Phys. Solids
0022-5096,
6
, pp.
223
230
.
13.
Chadwick
,
P.
, 1962, “
On the Propagation of Thermoelastic Disturbances in Thin Plates and Rods
,”
J. Mech. Phys. Solids
0022-5096,
10
, pp.
99
109
.
14.
Bargmann
,
H.
, 1974, “
Recent Developments in the Field of Thermally Induced Waves and Vibrations
,”
Nucl. Eng. Des.
0029-5493,
27
, pp.
372
385
.
15.
Elsener
,
K.
, 2000, “
General Description of the CERN Project for a Neutrino Beam to Gran Sasso (CNGS)
,” CERN AC Note No. 2000-03.
16.
Mura
,
T.
, 1956, “
Dynamical Thermal Stresses Due to Thermal Shock
,”
Faculty of Engineering, Meiji University
, Research Reports.
17.
Bertarelli
,
A.
, 2003, “
An Analytical Model to Study Transient Thermal Stresses in Graphite Target Rods Hit by Off-Axis Beam for CNGS Facility
,” Technical Note No. EST-ME-2003-06.
18.
Bertarelli
,
A.
, and
Kurtyka
,
T.
, 2004, “
Dynamic Thermo-Mechanical Phenomena Induced in Isotropic Cylinders Impacted by High Energy Particle Beam
,”
Proceedings of the VIII International Conference on Structures Under Shock and Impact (SUSI)
,
N.
Jones
and
C. A.
Brebbia
, eds.,
Wessex Institute of Technology
, pp.
33
43
.
19.
Dallocchio
,
A.
,
Bertarelli
,
A.
, and
Kurtyka
,
T.
, 2006, “
A New Analytical Method to Evacuate Transient Thermal Stresses in Cylindrical Rods Hit by Proton Beams
,”
Proceedings of the Tenth European Particle Accelerator Conference (EPAC06)
,
Edinburgh, Scotland
.
20.
Wilfinger
,
R.
, 2005, “
Proton-Induced Thermal Stress-Wave Measurements for ISOLDE and CNGS
,” Ph.D. thesis, Vienna University of Technology, Geneva.
21.
Kalbreier
,
W.
,
Middelkoop
,
W. C.
, and
Sievers
,
P.
, 1974, “
External Target at the SPS
,” CERN Technical Note No. Lab II∕BT∕74-1.
22.
Graff
,
K. F.
, 1991,
Wave Motion in Elastic Solids
,
Dover
,
New York
, pp.
116
121
.
23.
Boley
,
B. A.
, and
Weiner
,
J. H.
, 1997,
Theory of Thermal Stresses
,
Dover
,
New York
, pp.
30
44
.
24.
1972,
Handbook of Mathematical Functions
,
M.
Abramowitz
and
I. A.
Stegun
, eds.,
Dover
,
New York
, pp.
370
372
.
25.
Goodier
,
J. N.
, 1937, “
On the Integration of the Thermo-Elastic Equations
,”
Philos. Mag.
0031-8086,
23
, pp.
1017
1032
.
26.
Timoshenko
,
S.
, and
Goodier
,
J. N.
, 1970,
Theory of Elasticity
,
3rd ed.
,
McGraw-Hill
,
New York
, pp.
476
481
and
132
135
.
27.
Thomson
,
W. T.
, 1993,
Theory of Vibration With Applications
,
4th ed.
,
Chapman and Hall
,
London
, pp.
100
101
and
345
349
.
28.
Suhubi
,
E. S.
, 1964, “
Longitudinal Vibrations of a Circular Cylinder Coupled With a Thermal Field
,”
J. Mech. Phys. Solids
0022-5096,
12
, pp.
69
75
.
29.
Blevins
,
R. D.
, 2001,
Formulas for Natural Frequency and Mode Shape
,
Krieger
,
Malabar, FL
, pp.
107
108
.
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