Evolution of a primary pulse in the granular dimers mounted on a linear elastic foundation: An analytical and numerical study

In this exposition we consider the wave dynamics of a one-dimensional periodic granular dimer (diatomic) chain mounted on a damped and an undamped linear elastic foundation (otherwise called the on-site potential). It is very well known that periodic granular dimers support solitary wave propagation (similar to that in the homogeneous granular chains) for a specific discrete set of mass ratios. In this work we present the analytical investigation of the evolution of solitary waves and primary pulses in granular dimers when they are mounted on on-site potential with and without velocity proportional foundation damping. We invoke a methodology based on the multiple time-scale asymptotic analysis and partition the dynamics of the perturbed dimer chain into slow and fast components. The dynamics of the dimer chain in the limit of large mass mismatch (auxiliary chain) mounted on on-site potential and foundation damping is used as the basis for the analysis. A systematic analytical procedure is then developed for the slowly varying response of the beads and in estimating primary pulse amplitude evolution resulting in a nonlinear map relating the relative displacement amplitudes of two adjacent beads. The methodology is applicable for arbitrary mass ratios between the beads. We present several examples to demonstrate the efficacy of the proposed method. It is observed that the amplitude evolution predicted by the described methodology is in good agreement with the numerical simulation of the original system. This work forms a basis for further application of the considered methodology to weakly coupled granular dimers which finds practical relevance in designing shock mitigating granular layers.

Figure 1

Schematic of the granular dimer chain with no precompression subjected to linear on-site perturbations. The mass of the heavier beads is unity whereas the mass of the light beads is $\varepsilon$.

Figure 2

Response of the unperturbed granular dimer chain for impulse 20 and mass ratios (a) $\varepsilon =0.0615$, (b) $\varepsilon =0.0901$. Odd numbered beads are heavy beads.

Figure 3

Velocity profile of the beads of the perturbed granular dimer chain Eq. (2) with impulse 20 and mass ratio $\varepsilon =0.0901$. The values of the on-site stiffness used are ${\alpha }_1=0.1,{\alpha }_2=0.01$ and the damping is assumed to be absent.

Figure 4

Evolution of the primary pulse on the heavy beads of the auxiliary chain Eq. (10) with $\tilde{\delta }\left(t\right)=20,{\tilde{\alpha }}_1=1,{\tilde{\mu }}_1=0$, and $\tilde{\varepsilon }=0.1$. The simulation was performed on a system of 201 beads, and the on-site perturbation is started from the $51\text{st}$ bead.

Figure 5

Response of the unperturbed auxiliary chain Eq. (11) for contact index: $i=49,51,53$. The solution corresponds to the solitary wave solution of the auxiliary chain.

Figure 6

Response of Eq. (11) with (dashed lines) and without (solid lines) on-site perturbation. The parameter values used for the perturbed response are ${\tilde{\alpha }}_1=1,{\tilde{\mu }}_1=0,\tilde{\varepsilon }=0.1,\text{and}\tilde{\delta }=20$.

Figure 7

Evolution of the primary pulse amplitude on the beads of the perturbed granular dimers without on-site damping, i.e., ${\mu }_1={\mu }_2=0,{\alpha }_2=0.01$, and (a),(b) $\varepsilon ={\varepsilon }_{sw}^{\left(5\right)}=0.0615,{\alpha }_1$(0.0615, 0.3, 0.5, 1), (c),(d) $\varepsilon ={\varepsilon }_{sw}^{\left(2\right)}=0.3428,{\alpha }_1$(0.3428, 0.5, 1). Panels (a) and (c) represent the amplitude evolution on the heavy beads whereas panels (b) and (d) are for the corresponding light beads. Also the amplitude evolution in each panel corresponds to increasing on-site stiffness from top to bottom.

Figure 8

Comparison of the numerical solution (relative velocity) of Eq. (2) and the response from the method of maps $\left(\times \right)$ for the spatial evolution of the primary pulse on the beads of the perturbed granular dimers without on-site damping, i.e., ${\mu }_1={\mu }_2=0,{\alpha }_2=0.01$ corresponding to (a) $\varepsilon ={\varepsilon }_{sw}^{\left(5\right)}=0.0615,{\alpha }_1=0.0615$, (b) $\varepsilon ={\varepsilon }_{sw}^{\left(2\right)}=0.3428,{\alpha }_1=0.3428$.

Figure 9

Evolution of the primary pulse amplitude on the beads of the perturbed granular dimers without on-site damping, i.e., ${\mu }_1={\mu }_2=0,{\alpha }_2=0.01$, and (a),(b) $\varepsilon =0.125,{\alpha }_1$(0.125, 0.3, 0.5, 1), (c),(d) $\varepsilon =0.25,{\alpha }_1$(0.25, 0.5, 1). Panels (a) and (c) represent the amplitude evolution on the heavy beads whereas panels (b) and (d) are for the corresponding light beads. Also the amplitude evolution in each panel corresponds to increasing on-site stiffness from top to bottom.

Figure 10

Comparison of the numerical solution (relative velocity) of Eq. (2) and the response from the method of maps $\left(\times \right)$ for the spatial evolution of the primary pulse on the beads of the perturbed granular dimers without on-site damping, i.e., ${\mu }_1={\mu }_2=0,{\alpha }_2=0.01$ corresponding to (a) $\varepsilon =0.125,{\alpha }_1=0.125$, (b) $\varepsilon =0.25,{\alpha }_1=0.25$.

Figure 11

Evolution of the primary pulse amplitude on the beads of the perturbed granular dimers with on-site damping, i.e., ${\mu }_2={\alpha }_2=0.01$ and (a),(b) $\varepsilon ={\varepsilon }_{sw}^{\left(5\right)}=0.0615,{\alpha }_1={\mu }_1$(0.0615, 0.3, 0.5, 1); (c),(d) $\varepsilon ={\varepsilon }_{sw}^{\left(2\right)}=0.3428,{\alpha }_1={\mu }_1$(0.3428, 0.5, 1). Panels (a) and (c) represent the amplitude evolution on the heavy beads whereas panels (b) and (d) are for the corresponding light beads. The amplitude evolution in each panel corresponds to increasing on-site stiffness from top to bottom.

Figure 12

Comparison of the numerical solution (relative velocity) of Eq. (2) and the response from the method of maps $\left(\times \right)$ for the spatial evolution of the primary pulse on the beads of the perturbed granular dimers with on-site damping, i.e., ${\mu }_2={\alpha }_2=0.01$ corresponding to (a) $\varepsilon ={\varepsilon }_{sw}^{\left(5\right)}=0.0615,{\alpha }_1={\mu }_1=0.0615$ and (c) $\varepsilon ={\varepsilon }_{sw}^{\left(5\right)}=0.3428,{\alpha }_1={\mu }_1=0.3428$.