Integration through transients approach to the $\mu (\mathcal{I})$ rheology

This work generalises the granular integration through transients formalism introduced by Kranz et al. [Phys. Rev. Lett. 121, 148002 (2018)] to the determination of the pressure. We focus on the Bagnold regime, and provide theoretical support to the empirical $\mu (\mathcal{I})$ rheology laws, that have been successfully applied in many granular flow problems. In particular, we confirm that the interparticle friction is irrelevant in the regime where the $\mu (\mathcal{I})$ laws apply.