2. Benchmark solution#

2.1. Development of the analytical solution for CJS1#

We always have:

_images/Object_10.svg

where

_images/Object_11.svg

represents the confinement pressure.

Still to be determined

_images/Object_12.svg

.

Elastic phase:

By simply writing the elastic law, we have:

\({\sigma }_{\mathrm{xx}}^{0}={\sigma }_{\mathrm{xx}}^{0}+\lambda {\varepsilon }_{\mathrm{zz}}+(\lambda +2\mu ){\varepsilon }_{\mathrm{xx}}+\lambda {\varepsilon }_{\mathrm{xx}}\)

\({\sigma }_{\mathrm{zz}}={\sigma }_{\mathrm{zz}}^{0}+(\lambda +2\mu ){\varepsilon }_{\mathrm{zz}}+2\lambda {\varepsilon }_{\mathrm{xx}}\)

where here \(\lambda\) and \(\mu\) are the Lamé coefficients.

By eliminating \({\varepsilon }_{\mathrm{xx}}\) between these two equations, we find:

\({\sigma }_{\mathrm{zz}}={\sigma }_{\mathrm{zz}}^{0}+\frac{\mu (3\lambda +2\mu )}{(\lambda +\mu )}{\varepsilon }_{\mathrm{zz}}\)

Plastic phase:

We have:

_images/Object_18.svg

where

_images/Object_19.svg

represents the confinement pressure.

We deduce for the components of the diverter

_images/Object_20.svg

:

\({s}_{\mathrm{zz}}=2\left[\frac{1}{3}{I}_{1}-{\sigma }_{\mathrm{xx}}^{0}\right]\) and \({s}_{\mathrm{xx}}={\sigma }_{\mathrm{xx}}^{0}-\frac{1}{3}{I}_{1}\)

either: \({s}_{\mathrm{II}}=\sqrt{6}\left[{\sigma }_{\mathrm{xx}}^{0}-\frac{1}{3}{I}_{1}\right]\) and \(\mathrm{det}(\underline{\underline{s}})=2{\left[\frac{1}{3}{I}_{1}-{\sigma }_{\mathrm{xx}}^{0}\right]}^{3}\)

Therefore: \(h({\theta }_{S})={(1-\gamma )}^{1/6}\)

Moreover, when the deviatory mechanism criterion is reached: \({s}_{\mathrm{II}}h({\theta }_{S})+{R}_{m}{I}_{1}=0\)

Hence the relationship:

\({I}_{1}=\frac{\sqrt{6}{\sigma }_{\mathrm{xx}}^{0}}{\sqrt{\frac{2}{3}}-\frac{{R}_{m}}{{(1-\gamma )}^{1/6}}}\)

and finally, for the vertical constraint, we have:

\({\sigma }_{\mathrm{zz}}=\frac{\sqrt{6}{\sigma }_{\mathrm{xx}}^{0}}{\sqrt{\frac{2}{3}}-\frac{{R}_{m}}{{(1-\gamma )}^{1/6}}}-2{\sigma }_{\mathrm{xx}}^{0}\)

In addition, it can be calculated that the transition between the elastic and perfectly plastic states occurs for an axial deformation equal to:

\({\varepsilon }_{\mathrm{zz}}=\frac{\mu (3\lambda +2\mu )}{\lambda +\mu }\left[\frac{\sqrt{6}{\sigma }_{\mathrm{xx}}^{0}}{\sqrt{\frac{2}{3}}-\frac{{R}_{m}}{{(1-\gamma )}^{1/6}}}-2{\sigma }_{\mathrm{xx}}^{0}\right]\)

2.2. Benchmark results#

Constraints \({\sigma }_{\mathrm{xx}}\), \({\sigma }_{\mathrm{yy}}\), and \({\sigma }_{\mathrm{zz}}\) at points \(A\), \(B\), and \(C\).

2.3. Uncertainty about the solution#

Analytical solution for CJS1.