Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity cosserat models

Abstract

The present study theoretically investigates the free vibration problem of a discrete granular system. This problem can be considered as a simple model to rigorously study the effects of the microstructure on the dynamic behavior of the equivalent continuum structural model. The model consists of uniform grains confined by discrete elastic interactions, to take into account the lateral granular contributions. This repetitive discrete system can be referred to discrete Cosserat chain or a lattice elastic model with shear interaction. First for the simply supported granular beam resting on Winkler foundations, due to the critical frequencies which concern the nature of the dynamic results, the natural frequencies are exactly calculated, starting from the resolution of the linear difference eigenvalue problem. The natural frequencies of such a granular model are analytically calculated for whatever modes. It is shown that the difference equations governed to the discrete system converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation (also classified as a continuous Cosserat beam model) for an infinite number of grains. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations. This continuous gradient elasticity Cosserat model is obtained from a polynomial or a rational expansion of the pseudo-differential operators, stemming from the continualization process. Scale effects of the granular chain are captured by the continuous gradient elasticity model. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. The results clarify the dependency of the beam dynamic responses to the beam length ratio.

Introduction

In order to adapt a standard continuum theory to granular materials, it is necessary to introduce the independent rotational degrees of freedom (DOF) in addition to the conventional translational ones. This helps to describe accurately the relative movements between the microstructure and the average macroscopic deformations. One may obtain higher-order gradient continua with additional degrees of freedom. One may also obtain Cosserat modeling that consequently leads to non-classical continuum or polar continuum theories (Cosserat type theories, e.g. Cosserat and Cosserat [1]; Nowacki [2]). Voigt [3] was the pioneer of developing this concept who first showed the existence of couple –stress in materials. Cosserat continuum theories belong to the larger class of generalized continua which introduce intrinsic length scales into continuum mechanics via higher-order gradients or additional degrees of freedom (Eringen [4, 5], Forest [6]). Feng [7] analyzed the behavior of the granular medium considering normal, shear and rotation interactions. In contrast, the classical continuum mechanics ignores the rotational interactions among particles and neglects the size effect of material particles. Schwartz et al. [8] studied dispersive analysis of a granular medium with normal and shear interactions, neglecting the rotational elastic interactions.

On the other hand, in beam analysis, the Bresse-Timoshenko model takes into account both beam shear flexibility and rotatory inertia (Bresse [9] and Timoshenko [10, 11]). The effects of shear and rotational inertia can be significant in case of calculating eigenfrequencies for short beams, or in case of sufficiently small shear modulus. The Bresse-Timoshenko beam model is also a generalization of the Euler-Bernoulli model and admits a kinematics with two independent fields, a field of transverse displacement and a field of rotation. Timoshenko pointed out that the effects of cross-sectional dimensions on the beam dynamic behavior and frequencies could be significant. Timoshenko [10, 11] calculated the exact eigenfrequencies for such a beam with two degrees of freedom resting on two simple supports. Several lattice models have been developed based on microstructured Timoshenko beam models in order to go further in understanding the microstructure effects in presence of shear (see Ostoja-Starzewski [12] and Attar et al. [13]). The static and dynamic properties of a Cosserat-type lattice interface is studied by Vasiliev et al. [14]. Calculation of eigenfrequencies for a Bresse-Timoshenko beam with various boundary conditions and elastic interaction with a rigid medium is obtained by Wang and Stephens [15], Manevich [16] or Elishakoff et al. [17] (see more recently Elishakoff [18] and Challamel and Elishakoff [19]). Bresse-Timoshenko beam theory is merely a one-dimensional Cosserat continuum medium by considering two independent translational and rotational degrees of freedom (Rubin [20] and Exadaktylos [21]). Thus, there is a fundamental link between these two continuum theories. In this paper, it is noticed the relation between Cosserat discrete theories and the continuum ones.

The present study focuses on the vibration of a granular beam with both bending and shear granular interactions. The granular beam is assumed to interact elastically with a rigid elastic support, a discrete elastic foundation labeled as a discrete Winkler foundation (Winkler [22]). Note that the governing difference equations of the present model coincide with the ones of Cosserat granular model of Pasternak and Mühlhaus [23] in the absence of elastic foundation, but differ from the ones of the discrete shear model studied by Duan et al. [24]. However, for some specific bending/ shear interaction modeling, the model developed by Bacigalupo and Gambarotta [25] can be mathematically reformulated with the difference equation presented in this paper.

This paper is arranged as follows: First, a discrete granular beam model is introduced from a geometrical and a mechanical point of view. The grain interaction and material parameters are defined in detail. Then from the dynamic analysis of the lattice beam model, the deflection equations of the finite granular beam are derived. This fourth-order linear difference equation is solved by using the exact resolution of the difference equation. For an infinite number of grains, the deflection equation of a continuous beam (a fourth-order linear differential equation) is obtained asymptotically. Next, the eigenfrequencies of the discrete granular model and the continuous one, are obtained and compared as well. In the end, two asymptotic continualization methods are used to investigate continuous beam from the discrete lattice problem. With this aim, the polynomial expansion of Taylor and the rational expansion of Padé (with involved pseudo-differential operators) are used to derive new enriched beam models. These two nonlocal continualization approaches with the introduction of the gradient terms engage the neighbor influences which allow the passage from discrete result to continuous ones and simultaneously capture the length effect.