Static bending of granular beam: exact discrete and nonlocal solutions

• Published in: Meccanica (Milan) 2022-08, Vol.57 (8), p.2043-2066
• Main Authors: Massoumi, Sina, Challamel, Noël, Lerbet, Jean, Wautier, Antoine, Nicot, François, Darve, Félix
• Format: Article
• Language: English
• Subjects: Automotive Engineering; Bending; Boundary conditions; Chains; Civil Engineering; Classical Mechanics; Continuum modeling; Degrees of freedom; Difference equations; Differential equations; Discrete element method; Engineering; Exact solutions; Life Sciences; Mathematical models; Mechanical Engineering; Nonlocal elasticity; Polynomials; Size effects
• ISSN: 0025-6455; 1572-9648
• DOI: 10.1007/s11012-022-01531-7

This study is an attempt towards a better understanding of the length scale effects on the bending response of the granular beams. To this aim, a unidimensional discrete granular chain composed of a finite number of rigid grains is studied. It is assumed that shear and rotational interactions exist at the rigid grain interfaces. This granular model can be classified also as a discrete Cosserat chain with two independent degrees of freedom (DOF) for each grain (the deflection and the rotation). Subsequently, such a discrete model permits to introduce the size effect (grain dimension) in the bending formulation of a microstructured granular beam. It is shown that the bending deformation solutions of this chain asymptotically converge towards the continuum beam model of Bresse–Timoshenko (neglecting the length scale). The exact solutions of this granular model subjected to a uniform distributed loading, are investigated for various boundary conditions which are defined at the grain level. Accordingly, a twin numerical problem is studied to compare the exact analytical results with the numerical ones simulated by discrete element method (DEM). Eventually, through the continualization of the coupled difference equations system governing the discrete beam, a nonlocal elasticity Cosserat continuum model is obtained. The process of continualization consists in approaching the difference equations by differential equations applied either by the polynomial or the rational development in which a length scale appears. It is shown that both the granular model and the nonlocal beam model give very close and eventually coincident results.