Research Papers

Dynamic Response of Rapidly Heated Cylindrical Rods: Longitudinal and Flexural Behavior

[+] Author and Article Information
Alessandro Bertarelli, Tadeusz Kurtyka

Mechanical and Materials Engineering Group, Technical Support Department,  European Organization for Nuclear Research (CERN), CH-1211 Geneva 23, Switzerland

Alessandro Dallocchio

Mechanical and Materials Engineering Group, Technical Support Department,  European Organization for Nuclear Research (CERN), CH-1211 Geneva 23, Switzerland; Department of Mechanical Engineering,  Polytechnic University of Turin, Corso Duca degli Abruzzi 24, 10129 Turin, Italy

J. Appl. Mech 75(3), 031010 (Apr 08, 2008) (13 pages) doi:10.1115/1.2839901 History: Received February 27, 2007; Revised September 14, 2007; Published April 08, 2008

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.

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

Energy distribution on target rod due to proton beam impact

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Figure 2

Temperature T¯ as a function of time t¯ with eccentricity η¯=0.6

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Figure 3

Quasistatic in-plane stresses σ¯r, σ¯θ, τ¯rθ and axial stress σ¯z0 at zero-axial strain as a function of r¯ (t=τ, η¯=0.6)

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Figure 4

Equivalent thermal bending moment M¯x(t¯)(t¯=(t−τ)∕td)

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Figure 5

Equivalent dynamic excitations (qualitative plot)

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Figure 6

Dynamic flexural response w∕ws at z=L∕2 (r=R, φ=3π∕2, η=0.6)

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Figure 7

Scaled dynamic bending stress (σfd∕σfs and σfd∕σref) at z=L∕2 (r=R, θ=3π∕2, η=0.6R)

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Figure 8

Dynamic axial stress scaled to the reference stress σzd∕σref as a function of time at different location along the cylinder

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Figure 9

Scaled dynamic axial stress σzd∕σref along rod length at different instants

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Figure 10

Thermal shock longitudinal dynamic response H=σzd(0.5,τ+t0∕4)∕σref as a function of τ0=τ∕t0

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Figure 11

Variation of the ratio of dynamic maximum deflection to static maximum deflection with parameter B at z=L∕2

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Figure 12

Variation of the ratio of maximum dynamic bending stress to static bending stress with parameter B at z=L∕2

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Figure 13

Scaled global axial stress σztot∕σref and quasistatic axial stress σz0∕σref as a function of time at r=R, θ=3π∕2, η=0.6R (logarithmic scale)

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Figure 14

Thermoelastic coupling term as a function of time at r=R, θ=π∕2

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Figure 15

Flexural displacement at the rod center z=L∕2; comparison between analytical model and experimental data




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