Solution of a paradox related to the rigid bar pull-out problem in standard elasticity

This paper aims at modeling pull-out tests of a reinforcement bar from a concrete matrix. The considered system is an elastic cylinder, within which is embedded a rigid bar placed along the central axis. The pull-out test consists in extracting the bar while fixing the outer lateral boundary of the cylinder. Typically, the concrete matrix would be modeled as a Cauchy continuum (i.e. first gradient elastic continuum), while the slender reinforcement bar would be modeled either as an inner elastic cylinder, or – when its radius tends to zero – as a one-dimensional beam. However, we show that such a model yields a null total deformation energy for the concrete cylinder if the bar is modeled as a beam. This result is physically paradoxical, as the extraction of even the slenderest bar requires an energy input, which is transferred to the concrete matrix as deformation energy. In the present work, this paradox is solved by considering a strain-gradient deformation energy for the concrete matrix. It is shown that such a modeling approach allows deformation to occur even when the radius of the inner bar tends to zero. This result is found to be parametrized by a characteristic length which is physically justified through granular micromechanics. Numerical results are also obtained and hint at a possible way to identify the characteristic length experimentally in future works.