Research On The Modeling And Application Of Two-dimensional Coupled Granular-continuous Media

Granular media,ranging from dusts to floating ices or even to the meteorites,are the most common matter in our daily life.The dynamics of granular media composed of arrays of discrete particles has attracted considerable attention from both theoretical and practical points of view.Due to their nonlinear and discontinuous dynamics which coming from their highly nonlinear constitutive nature,the granular media have been shown rich nonlinear characteristics such as passively adaptive and tailorable dynamical properties.For example,granular media are capable of forming and transmitting a wide range of nonlinear stress waves,such as propagating solitary waves,traveling waves,a mixture of solitary and nonlinear shear waves,frequency bands or breathers,extreme nonlinear energy exchanges and “energy explosions”,and strongly non-reciprocal acoustics,and thus have been considered for various potential applications as novel engineering devices,e.g.nonlinear acoustic lenses,shock and energy absorbing layers,vibration mitigator or absorber,passive acoustic filters and acoustic switches.More particularly,granular media can also play an important role in the field of design and manufacture of ships and marine structures.To incorporate these remarkably versatile nonlinear dynamics of ordered granular media into practical applications,a natural step is to try to gain a better physical insight and understanding on the complex interactions of these highly discontinuous media with linearly elastic homogenous continua at their boundaries,which has been largely unexplored due to the complex nature of granular interactions in higher dimensions.Such a multi-dimensional interaction between a strongly nonlinear 2D granular medium and a linearly elastic medium poses distinct challenges,given the highly discontinuous,strongly nonlinear and transient nature of the granule-medium interactions,as well as the complexity stemming from the dispersive wave dynamics in both media.Moreover,rotational frictional effects in the granule-to-granule and granule-to-flexible boundary interactions further complicates the computational modeling.In this work,we study the highly complex interactions at the interface between the two-dimensional(2D)ordered granular medium and linearly elastic continua.2D numerical models of the aforementioned granular media accounting for axial Hertzian interactions,dissipation effects and particularly frictional effects have been constructed based on discrete element method(DEM),while the continua are modeled by finite element method(FEM).The coupled DEM-FEM models are then developed and examined.The remainder of this manuscript is mainly structured as follows:The wave propagation in 2D continuous solid medium is investigated based on FEM.By analyzing the distribution of energy density,stress and decomposing the displacement field,the propagating stress wave in the linear elastic solid can be explored.The propagation of longitudinal wave,stress wave,Rayleigh wave and even plane wave formed by superposition of longitudinal wave are clearly shown.The DEM model of 2D packed granular media are then discussed in detail.Apart from the highly discontinuous Hertzian granule-to-granule and granule-to-plate interactions,we also consider rotational and frictional effects in the granules;these effects render the acoustics of the granular-solid interface strongly nonlinear.The comparisons between the numerical results and the experimental data of a 2D granular system,as well as the energy conservation within the granular medium elucidate the validity and accuracy of the developed numerical model,which provides us with a reliable tool for precise modeling and predictive design of the dynamics of 2-D granular networks.An extended version of the granular system,which can be named as “force diode”,is then presented to highlight the possible nonlinear non-reciprocity in granular media.In addition,the nonlinear wave propagation in 2D dense packed granular system,which are significantly different from its counterpart in linear continuous solid,are discussed.The crucial role of friction effects on the dynamics of 2D granular media is also proved.Combining the advantages of DEM and FEM,the 2D coupled DEM-FEM model is developed and applied on solving the interaction problems between the discrete 2D granular media and continuous solid.The interaction forces that couple the granular medium to the continuous media are carefully analyzed by means of a computational algorithm of interrelated iterations and interpolations at successive time steps.Numerical results show that the coupled model can achieve good accuracy with lower computational costs than the standard FEM.Meanwhile,in order to overcome the instability caused by the standard Coulomb friction in tangential contact,the related stability analysis based on the nonlinear map of coupling terms during interaction is carried out.It is pointed out that the non-smooth nature of this Coulomb friction is the main source of numerical instability and a smoothed“Coulomb-tanh” friction model is then utilized.The convergence condition and the corresponding critical time step are then introduced and an adaptive algorithm which can adjust the time step based on the real-time contact status at each contact point is proposed to ensure the computational stability of the iterative algorithm of the coupled model.Convergence studies proved that this method can effectively overcome the instability problems brought by the traditional friction model,and ensure the convergence of iterative calculation at all times,which provides a powerful tool with validity,accuracy and robustness for the study of granular-solid interactions.Based on the presented 2D coupled DEM-FEM model,three complex granular-solid coupling systems are discussed.By taking full advantage of the nonlinear wave propagation characteristics in granular media as well as their highly nonlinear dynamics,the granular material can be packed as acoustic lens to generate and control “acoustic bullet” in its neighboring plane plate.Also,granular material can effectively dissipate energy so that it can be formed as damping layer to mitigate shock waves or as nonlinear dynamic absorber to suppress vibration.These numerical results have shown the various potential engineering applications of granular media.