Fractional derivatives and recurrent neural networks in rheological modelling – part I: theory

The aim of this paper was to develop a general approach based on fractional time derivatives and recurrent neural networks to model the rheological behaviour of asphalt materials. The paper focuses on elastic and viscoelastic material characteristics. It consists of two parts. In this first part, the theoretical aspects of modelling are discussed. A brief introduction into the theory of rheological elements based on fractional time derivatives is provided. The fractional differential equation of a general rheological element (base element) is developed from which a huge variety of other rheological elements can be derived, e.g. fractional Newton, Kelvin and standard solid elements. A new approach is presented for solving the fractional differential equations. Artificial neural networks are developed to compute the stress–strain–time behaviour of fractional rheological elements in a numerical efficient way. The approach is tested and verified. The second part of this work will appear later. It will be focused on applications of the new theoretical work to pavement engineering problems.