## 2 Materials and Methods
### Yield Surface  
The plastic strain is controlled by a Mohr-Coulomb yield surface, which is characterized by a stress-dependent friction angle as follows:  
\[\varphi_{pk}=\varphi_{b}+R\log\left(\frac{S}{\sigma_{min}}\right), \tag{14}\]  
where \(\varphi_{b}\) is the basic friction angle of smooth, planar, unweathered rock surfaces; \(R\) is the equivalent roughness of the rockfill; and \(S\) is the equivalent strength of rockfill particles. Due to its nature, the rockfill is considered to be a cohesionless material. This equation has been derived from [32; 33; 34], starting from the equation valid for the rock joints and noting the similarities between the non-linear strength envelopes of the rock joints and rockfill. These similarities are due to the fact that they both have \"contact points\", i.e., highly stressed contacting asperities due to which failure drives the behavior under deviatoric stress.  
The equivalent roughness is a parameter that considers the degree of interlocking of the rockfill particles: the higher that the sharpness and the lower that the porosity are, the higher that \(R\) is and, in turn, \(\varphi_{pk}\) as a result. Here below, we report the approximation of the chart reported in [31] that has been used in the present model:  
\[R=15-(34-5.5r_{i})\,n, \tag{15}\]  
where \(n\) is the porosity after compaction, and \(r_{i}\) is the roundness index, with \(r_{i}=0\) for partly rounded/smooth blocks, \(r_{i}=1\) for angular/rough blocks, and \(r_{i}=2\) for very sharp, angular/very rough blocks. The parameter \(R\) can be reviewed as the structural part of the shear strength of the rockfill.  
The equivalent strength accounts for the effect of the particle dimensions on the breakage phenomenon: the higher the average dimension of the particle, the higher the breakage is, the lower \(S\) is, and, consequently, \(\varphi_{pk}\) is lower as a result. The chart proposed in [34] for triaxial tests has been fitted with the following equation:  
\[S=\sigma_{c}\left\{\frac{1}{4}\arctan\left[\frac{11}{5}\left(\frac{17}{25}- \log d_{50}\right)\right]+\frac{11}{20}\right\}, \tag{16}\]  
where \(\sigma_{c}\) is the uniaxial compressive strength of the rock, and \(d_{50}(\text{mm})\) is the particle size with 50% passing that has been chosen as the parameter to define the average dimension of the rockfill particles. It is clear that this is a simplified approach, as the grain size distribution cannot be summed up with a single parameter, and at least, another parameter should be used like, for example, the coefficient of uniformity \(C_{u}\), which plays an important role in the breakage of the rockfill. This coefficient is defined as the ratio between the grain size with 60% passing \(d_{60}\) and that with 10% passing \(d_{10}\): a large uniformity coefficient indicates a rockfill composed of a wide range of particle sizes, while a value of \(C_{u}\) near to one characterizes a rockfill with a narrow range of particle sizes. However, the shear strength differences between a well-graded and a poor-graded rockfill with equal \(d_{50}\) are mainly considered by the different porosities after compaction attainable, and this affects the structural component of the strength, being independent of any possible changes of \(S\), so the effect of \(C_{u}\) is somehow already included in \(R\).