Propagation Characteristics Of Shock Waves In One-Dimensional Granular Chain With Viscous Dissipation

Granular materials are composed of a large number of discrete,solid and macroscopic granules,which are a universal existence form of material and widely exist in many natural and artificial processes.It exhibits a very complex collective behavior,which is similar to the properties of the usual solid,liquid,gas,but also has its own unique properties.There are a large number of irreversible processes in the motion of the granular system,which results in the dissipation of the system energy.The strength of the dissipation will have a great impact on the dynamic behavior of the granular system.Due to the discrete nature of the granular structure itself and the characteristics of scale separation,the irreversible process exhibited by the granular system is extremely complex.The topic of impact into a granular medium is of considerable recent interest,and The phenomenon that objects are subjected to external shocks is common in many natural and artificial processes.In the past few decades,relevant experts and scholars at home and abroad have studied the shock wave formed in the elastic granular system from the aspects of theoretical analysis,numerical simulation and experimental operation,and revealed the physical laws of the shock wave propagation characteristics in the granular system.In view of the diversity and complexity of the internal structure of granular materials,little is known about the affected by the variation of granular system parameters on impact load.In this thesis,based on the previously established Hertz theory model and considering the existence of viscous dissipation related to the relative velocity between adjacent beads in granular systems,the effects of viscous dissipation and external load shock on the shock wave structure and its propagation characteristics are studied through theoretical analysis and numerical simulation.According to the magnitude of viscous dissipation and the intensity of the external impact in the granular chain,it is found that there are two different types of shock waves–monotonic tanh shock wave and oscillating shock wave.There is a critical value of viscosity?_c.It is found that in the shock front the beads oscillate and the maximum bead velocity is larger than the piston speed if the magnitude of the viscosity is less than?_c.On the contrary,the shock front is a discontinuous surface and the beads velocities are all equal to the piston speed.In theory,we established the dynamic coupled equations of one-dimensional viscoelastic granular chain.In the long wavelength approximation,the reduced perturbation method is used to obtain the Korteweg-deVries—Burgers(KdV—Burgers)equation which can describe the motion state of the system,and a class of shock wave special solutions that can explain the phenomena observed in numerical simulation is given.Besides,the critical point of transition between two different types of shock waves is obtained through theoretical analysis,which further verifies the correctness of the numerical simulation results.In addition,the wave generated by the external short-period piston impact excitation in a one-dimensional viscoelastic granular chains is studied numerically.In some cases,the wave looks like a shock wave,while in other cases,it may consist of several solitary waves or some nonlinear oscillating waves.These characteristics depend not only on granular material parameters,but also on impact parameters,such as granular material viscosity coefficient,piston impact velocity and the duration of impact.It is found that the longer the impact time is,the more obvious the shock wave structure is.If the impact time is short enough,the shock wave structure will immediately evolve into multiple solitary waves.The numerical results indicates that when the impaction period is short,the bead velocity will decay with time as a power law function,and the attenuation strength depends on parameters such as viscosity,piston impact velocity and bead radius.Finally,the scattering process of shock waves in a one-dimensional viscoelastic grain chain containing a single or multiple impurity beads is investigated.When there is only one impurity in the granular chain,there is only one scattering process in the granular chain.It is found that the amplitude of the beads oscillation increases with the increase of piston impact velocity,and decreases with the increase of viscosity coefficient.As the number of impurity beads changes,the number of scattering process will change.If the number of impurities increases gradually and larger than the critical value,two scattering processes will occur in the granular chain.This critical value depends on piston speed and viscosity coefficient.It is found that when the viscosity coefficient is large enough,the maximum amplitude of the reflected wave is not zero,but a constant.In this case the reflected wave may be a small amplitude pulse,even be a solitary wave.