An investigation of the possible inflation of stimulated Brillouin backscattering (SBS) due to ion kinetic effects is presented using electromagnetic particle simulations and integrations of three-wave coupled-mode equations with linear and nonlinear models of the nonlinear ion physics. Electrostatic simulations of linear ion Landau damping in an ion acoustic wave, nonlinear reduction of damping due to ion trapping, and nonlinear frequency shifts due to ion trapping establish a baseline for modeling the electromagnetic SBS simulations. Systematic scans of the laser intensity have been undertaken with both one-dimensional particle simulations and coupled-mode-equations integrations, and two values of the electron-to-ion temperature ratio (to vary the linear ion Landau damping) are considered. Three of the four intensity scans have evidence of SBS inflation as determined by observing more reflectivity in the particle simulations than in the corresponding three-wave mode-coupling integrations with a linear ion-wave model, and the particle simulations show evidence of ion trapping.