1. Reference problem#
1.1. Geometry#
External annular crack in a semi-infinite cylindrical bar
We’ll take \(a/R=\mathrm{0,5}\) and \(H/R\ge 5\).
\(a=1m\)
\(R=2m\)
\(H=10m\)
1.2. Material properties#
The material is isotropic linear thermoelastic.
Young’s module |
\(E=2E11\mathrm{Pa}\) |
Poisson’s ratio |
\(\nu =\mathrm{0,3}\) |
Linear expansion coefficient |
\(\alpha =1E-5C{°}^{-1}\) |
Thermal conductivity |
\(\lambda =50W/\mathrm{m.C}°\) |
Thermal diffusivity |
\(\kappa =\lambda /\rho {C}_{p}=\mathrm{0,5}{m}^{2}/s\) |
Heat exchange coefficient |
\(h=250W/{m}^{2}C°\) |
We will choose \(h\) such as \(\mathrm{Bi}=\mathrm{hR}/\lambda =10.\) |
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1.3. Boundary conditions and loading#
Mechanical boundary conditions
\(\mathrm{UX}={U}_{r}=0\) on the axis of revolution \(r=0\)
\(\mathrm{UY}={U}_{z}=0\) on the ligament \(0\le r\le a\)
Resulting axial force zero at the upper edge; this boundary condition will be translated by a set of \((n-1)\) linear relationships \(\mathit{UY}(1)\mathrm{=}\mathit{UY}(2)\mathrm{=}\mathrm{...}\mathrm{=}\mathit{UY}(n)\) between the longitudinal displacements of the \(n\) nodes of the upper edge (free axial expansion, preservation of the flatness of the cross section of the bar).
Conditions of unilateral contact on the lip of the crack in order to manage its closure.
Thermal boundary conditions
Zero heat flow on the axis of revolution \(\mathrm{AB}\) (by symmetry)
Zero heat flow on ligament \(\mathrm{OA}\) (by symmetry) and on crack \(\mathrm{AC}\).
Convection flow \(\frac{\partial T}{\partial r}=h({T}_{\mathrm{ext}}-T)\) at edge \(r=R\), \({T}_{\mathrm{ext}}\) denoting the temperature of the outside environment.
Thermal load
The temperature of the outside environment undergoes an instantaneous \({T}_{\mathrm{ext}}=\mathrm{T0}\ast H(t)\) scale where \(H(t)\) is Heaviside’s unit—scale function. Given the boundary conditions the temperature does not vary according to \(z\). We will take \({T}_{0}=100°C\) in order to obtain the closure of the lip, in the vicinity of the skin of the room, at the beginning of the thermal shock.
1.4. Initial conditions#
Mechanical initial conditions
Zero movements, deformations and stresses at all points.
Initial thermal conditions
Initial temperature zero at all points.