Conduction in a Hollow Cylinder
Note
The project file for this example is available to be viewed/run in FLAC3D.[1] The project’s main data files are shown at the end of this example.
A hollow cylinder of infinite length is initially at a constant temperature of 0°C. The inner radius of the cylinder is exposed to a constant temperature of 100°C, and the outer radius is kept at 0°C. The problem is to determine the temperatures and thermally induced stresses in the cylinder when the equilibrium thermal state is reached.
Nowacki (1962) provides the solution to this problem in terms of the temperatures and radial, tangential, and axial stresses at the steadystate thermal state:
where: 
\(T\) 
= 
temperature; 
\(r\) 
= 
radial distance from the cylinder center; 

\(a\) 
= 
inner radius of the cylinder; 

\(b\) 
= 
outer radius of the cylinder; 

\(T_a\) 
= 
temperature at the inner radius; 

\(\sigma_r\) 
= 
radial stress; 

\(\sigma_t\) 
= 
tangential stress; 

\(\sigma_a\) 
= 
axial stress; 

\(m\) 
= 
\(3 K \alpha_t \over {\lambda + 2 G}\); 

\(\lambda\) 
= 
\(K  {2 \over 3} G\); 

\(K\) 
= 
the bulk modulus; 

\(G\) 
= 
the shear modulus; and 

\(\alpha_t\) 
= 
the linear thermalexpansion coefficient. 
The analytical solutions for temperature and stresses are programmed as FISH functions in the FLAC3D data file. The analytical and numerical results can then be compared directly in tables.
The following properties are prescribed for this example:
Geometry 

inner radius of cylinder (\(a\)) 
1.0 m 
outer radius of cylinder (\(b\)) 
2.0 m 
Material Properties 

density (\(\rho\)) 
2000 kg/m^{3} 
specific heat (\(C_p\)) 
880.0 J/kg °C 
thermal conductivity (\(k\)) 
4.2 W/m °C 
linear thermalexpansion coefficient (\(\alpha_t\)) 
5.4 × 10^{6}/ °C 
shear modulus (\(G\)) 
28.0 GPa 
bulk modulus (\(K\)) 
48.0 GPa 
A thin quartersection of the cylinder is modeled with FLAC3D. Figure 1 shows the FLAC3D grid. A constanttemperature boundary of 100°C is specified for the inner radius of the model; the temperature at the outer radius is specified to be 0°C.
The material in this problem is elastic. Therefore, an uncoupled analysis is appropriate. The thermal calculation is performed first to reach the equilibrium heatflux state, then the thermally induced mechanical stresses are calculated.
Numerical and analytical results are compared in Figure 2 through Figure 5. The figures show plots of tables for temperature and stress distributions through the cylinder at steady state. In each figure, the analytical values are plotted as crosses, and the numerical values are plotted as dashed lines. The plotted values are normalized, with temperature normalized by dividing by \(T_a\) and stress normalized by dividing by \(m G T_a\). Figure 2 shows the temperature distribution at steady state for the numerical and analytical solutions. The agreement is very good, with an error of less than 0.1%. A comparison of results for radial, tangential, and axial stress distributions at steady state are provided in Figure 3, Figure 4, and Figure 5, respectively.
Reference
Nowacki, W. Thermoelasticity. New York: AddisonWesley (1962).
Data File
; Heating of a hollow cylinder
; coupled and uncoupled thermal/mechanical calculations
model new
model largestrain off
fish automaticcreate off
model title ...
'Heating of a hollow cylinder  thermal ... mechanical calculations'
model configure thermal
;  main computation 
zone create cylindricalshell point 1 (2,0,0) point 2 (0,0.1,0) ...
point 3 (0,0,2) dimension (1,1,1,1) size 10 1 12
zone face skin
;  mechanical model
zone cmodel assign elastic
zone property bulk 48e9 sh 28e9
zone face apply velocityz 0 range group 'Bottom'
zone face apply velocityx 0 range group 'West1'
zone face apply velocityy 0 range group 'North' or 'South'
;  thermal model
zone thermal cmodel isotropic
zone thermal property conductivity 4.2 expansion 5.4e6 specificheat 880
zone initialize density 2000
zone face apply temperature 100 range group 'West2'
zone face apply temperature 0 range group 'East'
model save 'cylinit'
; uncoupled analysis
model mechanical active off
model thermal active on
model solve
model mechanical active on
model thermal active off
model solve
model thermal active on
model save 'cylucpl'
; coupled analysis
model restore 'cylinit'
model mechanical active on
model thermal active on
model mechanical substep 10000
model mechanical follower on
model solve thermal ratio 1e5 mechanical convergence 1
model save 'cylcpl'
program return
Endnotes
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