2. Benchmark solution#
2.1. Calculation method used for the reference solution#
Temperature field: |
exact analytical calculation. |
Thermomechanical calculation: |
thermoelastic stress field in the non-cracked bar given by an exact analytical expression |
displacement of the crack lips calculated from influence functions determined numerically by finite elements |
|
stress intensity factor calculated from the surface tensions released along the crack, using unlimited solid weight functions for a constant pressure distribution on the lips at intervals along the radius. |
2.2. Benchmark results#
Fourier number: \(\mathit{Fo}\mathrm{=}\frac{\kappa t}{{R}^{2}}\) (dimensionless time)
Biot number: \(\mathit{Bi}\mathrm{=}\frac{\mathit{hR}}{\lambda }\) (dimensionless exchange coefficient)
Expression of temperature as a function of r and t:
the eigenvalues \({\mu }_{n}\) are the solutions of the above equation in which \({J}_{0}\) and \({J}_{1}\) are the Bessel functions of the first kind of order 0 and 1.
The tables below summarize the temperature values (\(°C\)) for three specific radii and for three Fourier numbers:
\(\mathrm{F0}=\mathrm{0,001}\)
Ref. (10000 terms) |
|
\(r=0\) |
3,9968E-12 |
\(r=1\) |
2,2204E-13 |
\(r=2\) |
2,79689E+1 |
\(\mathrm{F0}=\mathrm{0,4}\)
Ref. (900 terms) |
|
\(r=0\) |
1.6230E-1 |
\(r=1\) |
6,2391E+0 |
\(r=2\) |
7,7365E+1 |
\(\mathrm{F0}=1\)
Ref. (900 terms) |
|
\(r=0\) |
9,8644E+1 |
\(r=1\) |
9.9018E+1 |
\(r=2\) |
9,9835E+1 |
Expression of axial stress in the uncracked bar as a function of \(r\) and \(t\) :
The table below summarizes the \({\sigma }_{\mathrm{zz}}(\mathrm{Pa})\) stress values for \(r=a\) (crack background) and for three Fourier numbers:
Ref. (900 terms) |
|
\(\mathrm{F0}=\mathrm{0,001}\) |
4,584029E+6 |
\(\mathrm{F0}=\mathrm{0,4}\) |
6,397099E+7 |
\(\mathrm{F0}=1\) |
8,200300E+5 |
Stress intensity factor (dimensionless) as a function of Fourier number
2.3. Uncertainty about the solution#
Less than \(\text{5 \%}\).
2.4. Bibliographical references#
J.M. ZHOU, T. TAKASE, and Y. IMAI: Opening and closing behavior of an external circular crack due to axisymmetrical heating. Engng.Fract.Mechs., 47, no. 4, 559-568, 1994.