Probing hysteretic elasticity in weakly nonlinear materials

In this study, we propose an optimized NRUS measuring and data processing protocol dedicated to small bone samples specially designed to be used in a four-point bending mechanical fatigue test. Our goal was to assess the elastic and dissipative hysteretic nonlinear parameters repeatability with the proposed protocol using several classes of materials with weak, intermediate and high nonlinear properties. In the proposed data processing, the frequency shift Δf as a function of excitation drive amplitude is measured relatively to a reference resonance peak curve fο (obtained at the lowest excitation level) which is repeated before each excitation drive level. Our results show that the proposed correction may be an alternative to a stringent control of temperature (which could not be achieved in this study) by increasing significantly NRUS sensitivity. With our correction procedure, we measured relative resonant frequency shifts of 10^-5, which are below 10^-4, often considered as the limit to NRUS sensitivity under common experimental conditions. In our experiments, we identified external temperature fluctuation as one of the major source of resonance frequency variation. A variation of 0.1°C caused a frequency variation of 0.01%, which is similar to the expected nonlinear frequency shift for weakly nonlinear materials. In the absence of correction, the data could not be interpreted to support the existence of hysteretic nonlinear behavior in bone.