Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity cosserat models
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.
Section snippets
Granular model
A granular beam of length L resting on two simple supports is modeled by a finite number of grains interacting together. Such a model could be presented by considering the microstructured granular chain comprising n + 1 rigid grains with diameter a (a = L/n) that are connected by n shear and rotational springs, as shown in Fig. 1. It is assumed that the elastic support springs are located at the center of each rigid grain. Each grain has two degrees of freedom which are denoted by Wi for the
Resolution of the difference equation
In this section, the exact solution for the fourth-order linear difference eigenvalue problem of Eq. (15) will be established (see the books of Goldberg [29] or Elaydi [30] for the general solution of linear difference equations). This approach, as detailed for instance by Elishakoff and Santoro [31, 32], has been used to analyze the error in the finite difference based probabilistic dynamic problems. Eqs. (15) and (16) restricted to the vibration terms, the linear fourth-order difference
Nonlocal approximate solutions - continuous approach
The fourth-order difference equations of Eq. (19) may be continualized in two general ways: the simplest approach is based on the polynomial expansions in which the finite differences operators are expanded with the Taylor approximation. This leads to a higher-order gradient Cosserat continuum theory. Another effective method considers a rational expansion based on the Padé approximation which could give better homogenized solution compared to the Taylor series (see for instance Duan et al. [24]
Discussion
The eigenfrequency results of the two branches are gathered together for all approaches (local, nonlocal and continuum ones) in Figs. 13 and 14. The results are reported as a function of mode number (p) for four typical grain number values () and the dimensionless parameters of and . It can be obtained from the figures that the results of Padé approximation are closer to the ones obtained by exact resolution. Another point
Conclusion
This paper investigates the macroscopic free vibration behavior of a discrete granular system resting on a Winkler elastic foundation. This microstructured system consists of uniform grains elastically connected by shear and rotation springs. It is shown that the discrete deflection equation of this granular system (Cosserat chain) is mathematically equivalent to the finite difference formulation of a shear deformable Bresse-Timoshenko beam resting on Winkler foundation. Next, the natural
Credit author statement
Massoumi S., Challamel N. and Lerbet J., Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity Cosserat models, submitted for publication, J. Sound Vibration, 2020.
Paper : The three authors have contributed to the formulation of the model, the mathematical description of the mechanical model and its numerical resolution.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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