  Oscillatory shear has been widely used to study the rheological properties of
suspensions under unsteady shear. Furthermore, recent works have shown that
oscillatory flows can improve the flowability of dense suspensions. While most
studies have been done under constant volume we here study oscillatory shear
flows of a two-dimensional suspensions using a normal pressure-controlled
set-up. To characterise the rheology, we introduce both a complex macroscopic
friction coefficient $\mu^{*}$, following the convention of the complex
viscosity $\eta^{*}$, and a shear-rate averaged viscous number $J'$. The
rheology and microstructure of dense suspensions are studied by systematically
varying both the strain magnitude $\gamma_0$ and $J'$ using numerical
simulations. We study both suspensions composed of frictional ($\mu_p=0.4$) or
frictionless ($\mu_p=0$) particles and find that the critical values, as $J'
\to 0$, of both the complex macroscopic friction and the number of contacts,
both total and sliding, decrease with decreasing $\gamma_0$. For suspensions
composed of frictional particles we also find that the critical (i.e.~the shear
jamming) packing fraction $\phi_c$ increase with decreasing $\gamma_0$. In both
cases, frictional and frictionless, we find that the rheological response
approaching the shear jamming turns from a viscous to an elastic response as
$\gamma_0$ is lowered below $\sim0.33$.
