1. Reference problem#

1.1. Geometry#

Straight section of a thick tube with a

internal radial crack

The material is standard isotropic linear thermoelastic.

Young’s module	\(E=1000\mathrm{MPa}\)
Poisson’s Ratio	\(\nu =\mathrm{0,3}\)
Linear expansion coefficient	\({\alpha }_{T}=1E-6\)
Elastic limit	\({\sigma }_{0}=1\mathrm{MPa}\) (used to define the initial stress field created by the auto-fretting process under the hypothesis of a previous Von Mises elastoplastic type behavior)

Boundary conditions (for half a room in region \(y\ge 0\))

Block \(\mathrm{UY}=0\) on segment \(\mathrm{AB}\) and on ligament \(\mathrm{DE}\) (symmetry).

Linear relationship \(\mathrm{UX}(A)+\mathrm{UX}(E)=0\) (to block horizontal translation)

Loading

Loading no. 1:	radial traction \({\sigma }_{\mathrm{rr}}({r}_{2})={\sigma }_{0}\) on the external face; this mechanical loading produces the same \({K}_{I}\) as an internal pressure acting simultaneously on the internal radius \({r}_{1}\) and on the lips of the crack, without taking into account the non-linearity of contact.

Loading #2:	thermal loading equivalent to autofrettage defined as follows:





	In these formulas, \(\rho\) designates the maximum radius of the zone that has undergone autofrettage, \({T}_{1}\) the temperature at radius \({r}_{1}\) and \({T}_{\rho }\) the temperature at radius \(r=\rho\) in the thick, non-cracked tube. In the application referred to here, \(\rho ={r}_{2}\) is taken, which corresponds to the autofretting of the entire section of the thick tube, and contact nonlinearity is not taken into account. A negative \(K\) is expected for positive temperatures (compression of the uncracked tube).

Loading #3:	linear combination loading no. 2 \(\text{+}\) \(\alpha\) * loading no. 1, \(\alpha\) (\(\ne \alpha T\)!) designating the mechanical load factor; contact nonlinearity is taken into account here, which implies an incremental application of the mechanical load.