Implement the Laplace transform to convert viscoelastic functions of asphalt mixtures

Abstract

The relaxation modulus of an asphalt mixture is adopted for evaluation of pavement structural soundness. However, it is difficult to perform a strain-controlled relaxation test on an asphalt mixture to obtain the relaxation modulus. The aim of this paper is to obtain the relaxation modulus expression in the Laplace transform domain and then perform a Laplace transform to derive the relaxation modulus from creep compliance. A uniaxial static creep test was conducted on four types of asphalt mixtures at 20 °C, 30 °C, and 40 °C and under a load of 0.3 MPa. The Prony series and power law were used to model creep compliance with good fitting of R2 > 0.95. Because the Laplace transform of the power law was a gamma function, the Prony series was used to simulate creep compliance and then applied to interconverting the viscoelastic functions. The Simpson and Gauss methods were used to convert the relaxation modulus from creep compliance based on a convolution formula and the calculation results from the two methods were shown to be equivalent. Based on the Laplace transform formula for creep compliance and the relaxation modulus, an equation for calculating the relaxation modulus in the Laplace domain was proposed. Numerical inversion of the Laplace transform, using the fixed Talbot method, was implemented to compute the relaxation modulus in the time domain and the results were compared with those derived by using the Gauss method. The average error is less than 10−4 and the computational speed of the fixed Talbot method is a factor of 25 faster than that of the Gauss method, demonstrating the high precision and high efficiency of the Laplace transform method for computing the relaxation modulus. The method developed in this study provides an effective tool for interconverting viscoelastic functions of asphalt mixtures.