Analytical expressions for the effective coefficients of fibre-reinforced composite materials under the influence of inelastic distortions

We aim to deduce analytic expressions for the homogenised coefficients that describe the mechanical behaviour of a uniaxially fibre-reinforced composite material consisting of two solid constituents undergoing inelastic distortions, one representing the extracellular matrix, and the other representing the inclusions that model the fibres. While our work is mathematical in nature, our underlying goal is to explore questions related to biology, as biological systems, such as soft and hard tissues, can change their properties in response to various internal and external factors. One of our key motivations is to tackle the computational complexities involved in determining the effective, macroscopic properties of such biological systems. This requires addressing the interactions across different scales of the so-called cell and homogenised problems. To achieve this, we begin by formulating the governing equations that describe the dynamics of the composite’s constituents, provided by the balance of linear momentum and the law for the evolution of the inelastic distortions. We then employ the asymptotic homogenisation technique to derive the individual cell-level problems and the homogenised macroscopic equations. This process also yields expressions for effective coefficients that capture the overall behaviour of the composite material. In a first step towards our investigations and to illustrate the capabilities of our approach, we consider the case in which the composite under study possesses a fibre-reinforced structure. Together with additional hypotheses, we concentrate on the calculation of the effective properties using complex variable methods. Finally, after obtaining general formulae, we focus on providing numerical results.