Fractional model of concrete hereditary viscoelastic behaviour

Abstract

The evaluation of creep effects in concrete structures is addressed in the literature using different predictive models, supplied by specific codes, and applying the concepts of linear viscoelastic theory with ageing. The expressions used in the literature are mainly based on exponential laws, which are introduced in the integral expression of the Boltzmann principle; this approach leads to the need of finding approximated numerical solutions of the viscoelastic response. In this study, the hereditary fractional viscoelastic model is applied to concrete elements, underlining the convenience of using creep or relaxation functions expressed by power laws. The full reciprocal character of creep and relaxation, described through the power laws, is shown in Laplace domain, and the basic theorems of linear viscoelastic theory are re-written. In order to simplify the solution of fractional expressions, the Grünwald–Letnikov approach is followed and a discretization of fractional operators is applied through a simple matrix procedure. Two cases are examined: The first is the application of the fractional model to the creep laws supplied by international codes, finding the power law which approximates in the best way the creep curve given by the B3 predictive model. The second one is the application to the experimental data of an actual concrete mixture, for which the corresponding power law is found and introduced in the fractional model. A best-fitting procedure is applied to obtain the two parameters of power law in order to describe the hereditary character of concrete creep; afterwards, ageing is interpreted through the variation of coefficients with time. A third application on a concrete beam shows the suitability of the proposed model to the estimation of the delayed deformations of actual concrete structures.