Revisiting thermorheological simplicity in crumb rubber modified bitumen using fractional Maxwell modelling

Abstract

The linear viscoelastic behaviour of highly modified bitumen such as crumb rubber modified bitumen (CRMB) is commonly characterised using time-temperature superposition (TTSP), although its validity depends on the assumption of thermorheological simplicity. This assumption is often violated in highly modified bitumen, reducing the reliability of conventional master curve construction and increasing experimental effort. This study evaluates a three-parameter Fractional Maxwell Model for characterising five CRMBs containing 5%, 10%, 15%, 22%, and 30% CR by weight of base bitumen. Frequency sweep tests were conducted using a dynamic shear rheometer at 10, 30, 50, and 70 °C, and model parameters were determined through nonlinear optimisation. The fractional Maxwell model accurately reproduced the measured storage and loss moduli, with \(\mathrm{R}^{2}\) values above 0.96 and normalised root mean square errors (NRMSE) below 5% in 50 of the 53 fitted datasets. Model-reconstructed master curves closely matched experimentally shifted TTSP master curves for unaged and short-term aged conditions, with \(\mathrm{R}^{2}\) values above 0.989 in log space and NRMSE below 3.1%. After long-term ageing, the model retained high accuracy for CRMB-22 (\(\mathrm{R}^{2} = 0.998\)), but deviations increased for the other bitumen, with NRMSE reaching 10.3% for CRMB-30. The fractional order parameter (\(\alpha \)) increased approximately linearly with temperature, while its temperature sensitivity decreased with increasing CR content. Interpolated \(\alpha \) values at 30 and 50 °C differed from fitted values by less than 0.06, indicating that fractional modelling may reduce the number of test temperatures required, particularly for low-CR bitumen.