2. Reference solution

2.1. General case

The JOINT_MECA_ENDO law of behavior was developed as part of Ilaria Fontana’s thesis defended in March 2022. The thesis manuscript contains all the theoretical elements of the law.

The aim here is to reproduce a tensile test and a shear test under imposed constant normal stress. We recall that the parameters of the law are: normal and tangential stiffness \({K}_{n}\) and \({K}_{t}\), the coefficient of friction and, the coefficient of friction \(\mathrm{\mu }\), cohesion \(C\), and maximum shear stress \(T\), two parameters that respectively control the cohesion peak and the shear peak \({B}_{n}\), \({B}_{t}\), two parameters that allow to control the evolution of the stress peak in shear \({m}_{1}\) and \({m}_{2}\) and a parameter for regularizing the post-peak phase \({D}_{1}\) (which makes it possible in particular to ensure that there is no snap-back). It is important to note that \({K}_{n}\), \({K}_{t}\), \(\mathrm{\mu }\), \(C\), and \(T\) are « engineering » parameters that can be identified on experimental tests. However, \({B}_{n}\), \({B}_{t}\), \({m}_{1}\), \({m}_{2}\) and \({D}_{1}\) are numerical parameters that make it possible to ensure the good mathematical properties of the law and to control the softening phase. The values of \({B}_{n}\) and \({B}_{t}\) are determined from equations 2.48 and 2.49 of the thesis manuscript. In the current version of the law, the parameters \({m}_{1}\) and \({m}_{2}\) are set to \({m}_{1}=3\) and respectively at and \({m}_{2}=\mathrm{0,5}\), thus making it possible to have a « good shape » for the post-peak phase. Regarding the parameter \({D}_{1}\), it is considered that it must be ten times greater than its minimum value \({D}_{1\mathit{min}}\) defined by the non-snap-back condition described on page 47 of the manuscript. Note that other conditions on the parameters must be respected: a condition on \(C\) and \(T\) described by equation 2.50, a condition on \({K}_{n}\) and \({K}_{t}\) described by equation 2.24 and finally it is necessary to check that \({B}_{n}\) and \({B}_{t}\) are positive.

The Python script presented in the appendix makes it possible to calculate the numerical parameters of the law from the engineering parameters. It is possible to use this script in the opposite direction, i.e. to calculate the engineering parameters from the numerical parameters defined in advance. In the current version of the test case, a Python function is introduced at the start of each command file to calculate the numerical parameters of the law from the so-called « engineer » parameters.

For the shear test, the reference solution is given by equations 2? 38 and 2.39 of the thesis manuscript. For the traction test, the reference solution is given by equations 2.44 and 2.45 of the thesis manuscript. Each of these equations is expressed as a function of the damage variable alpha.