2. Benchmark solution

2.1. Calculation of the reference solution

Here we consider the case of a triaxial test for which the confinement stresses are applied in the directions \(x\) and \(z\) and for which the imposed deformation direction is the direction \(y\). It is also assumed that the parameter \(\eta\) is independent of the hardening parameter \(\gamma\), that is to say \({\phi }^{\mathit{end}}\mathrm{=}{\phi }^{\mathit{rup}}\mathrm{=}{\phi }^{\mathit{res}}\): it is then possible to calculate an analytical solution to the problem. The plasticity and flow criteria are written as:

We consider a situation of increasing loading for which the preceding equations can be written non-incrementally:

The elasticity relationships give:

that is to say:

with \({\sigma }_{3}^{0}\) and \({\sigma }_{1}^{0}\) the values of \({\sigma }_{1}\) and \({\sigma }_{3}\) at the start of loading. It therefore remains to calculate \({\sigma }_{1}\) as a function of \(\gamma\) using the plasticity criterion to obtain \(\gamma\), \({\sigma }_{1}\) and \({\varepsilon }_{3}\).

1st case: \(\gamma \mathrm{\le }{\gamma }^{\mathit{rup}}\)

By noting

and

where

,

,

and

are given in the reference documentation of the law of behavior, \(\gamma\) is a solution of the polynomial of degree 2:

,

with \(\gamma\) in the \(\mathrm{[}\mathrm{0,}{\gamma }^{\mathit{rup}}\mathrm{]}\) interval.

2nd case: \({\gamma }^{\mathit{rup}}\mathrm{\le }\gamma \mathrm{\le }{\gamma }^{\mathit{res}}\)

Using the notations from the reference documentation of the amended Hoek-Brown law for a, d, c and

, \(\gamma\) is a solution of the polynomial of degree 2:

3rd case: \({\gamma }^{\mathit{res}}\mathrm{\le }\gamma\)

In this case \({\sigma }_{1}\) is constant:

and

.

2.2. Benchmark results

Constraints \({\sigma }_{\mathit{xx}}({\sigma }_{3})\), \({\sigma }_{\mathit{yy}}({\sigma }_{1})\), and \({\sigma }_{\mathit{zz}}({\sigma }_{3})\) at point \(D\).

Move \({\varepsilon }_{\mathit{xx}}({\varepsilon }_{3})\) and \({\varepsilon }_{\mathit{yy}}({\varepsilon }_{1})\) to point \(D\).