In the present work, we present a two-dimensional soft sphere granular system, with inclination, that models the transition from an amorphous solid to the crystalline phase by shear cycles induced by cyclic deformations of the boundary. To simulate an effective temperature, the system is subjected to vibration. Under these conditions, the system exhibits a controlled transition to hexagonal order, where the crystallization rate and extent depend critically on the shear frequency. The study focuses on the analysis of the effect of the shear frequency in phase change, and introduces a dimensionless shear frequency \(\tilde{f} = f_s \tau _r\) , where \(\tau _r = \sqrt{m/k}\) is the intrinsic relaxation timescale of the particles, to identify the regimes in which mechanical annealing is effective. The soft granular particles used are polyacrylamide hydrogel spheres, with an estimated Young’s modulus of the order of \(10^4\) Pa, consistent with previous measurements for single-network polyacrylamide gels. Hexagonal order is measured in terms of the sixth-bond orientational order parameter \(\psi '_{6}\) . By following the temporal evolution of this parameter, we find that low shear frequencies on the order of \(10^{-3}\) Hz (i.e., \(\tilde{f} \ll 1\) ) favor the growth of hexagonal grains, while higher frequencies tend to reduce hexagonal order, leading to an unstable structure. Additionally, we characterize particle dynamics through autocorrelation measurements in the time series of \(\psi '_{6}\) using Fourier spectral analysis (FA). For all cases with non-zero shear frequency, the power spectra follow a power law, \(P(f) \propto 1/f^{\beta }\) with \(\beta > 1\) , indicating non-stationarity. In contrast, for the static (0 Hz in shear frequency) case, the power spectrum is flat ( \(\beta \approx 0.04\) ), suggesting stationary white noise behavior in the time series.
