Wave transmission in 2D nonlinear granular-solid interfaces, including rotational and frictional effects

Abstract

We study the highly complex wave transmission at the interface between a two-dimensional (2D) hexagonally structured granular medium and a linearly elastic thin plate; we refer to this system as the “granular-solid interface”. By applying an impulsive excitation at the free end of the granular medium we study the nonlinear acoustics at the interface. A computational model is developed, where the thin plate under the plane-stress assumption is discretized by finite-elements (FEs), whereas the granular medium by discrete-elements (DEs). Apart from the highly discontinuous Hertzian granule-to-granule and granule-to-plate interactions, we also take into account rotational and frictional effects in the granules; these effects render the acoustics of the granular-solid interface strongly nonlinear and highly discontinuous. The interaction forces coupling the granular medium to the plate are computed by means of an algorithm of interrelated iterations and interpolations at successive time steps. Since frictional effects may yield numerical instabilities, our approach incorporates the continuous “Coulomb–tanh” friction model, whose efficacy is verified through convergence studies. By formulating appropriate theoretically predicted convergence criteria, we show that the stability of the algorithm depends on the time step, the mesh size of the FE model, and the frictional model parameters. Accordingly, convergence is ensured by introducing a self-adaptive time step scheme, which is informed by theoretical convergence criteria. An application of the algorithm for a specific granular-solid interface demonstrates its validity, accuracy and robustness. Wave transmission through the discrete–continuum interface is drastically delayed by the granular medium, which, inflicts significant “softening” to the nonlinear acoustics. Moreover, there is strong nonlinear wave dispersion and energy localization in the granular medium, resulting in highly reduced wave transmission to the plate. Moreover, these nonlinear acoustical features are tunable with the applied shock (or input energy). The model and results presented in this work apply to a broad class of nonlinear discrete–continuum interfaces, with broad applications, e.g., shock/blast mitigation, granular containers with flexible boundaries and acoustic non-reciprocity.

Appendices

Appendix 1: The computational algorithm and its numerical stability

The computational algorithm is outlined in the flow chart of Fig. 13.

Fig. 13

Flow chart of the computational algorithm for computing the converged interaction forces applied to the plate in the time interval $0 \le \tau \le T$; $\left( {{\varvec{N}},{\varvec{f}}} \right)_{s + 1}^{\left( j \right)}$ denotes the j-th iteration of the interaction force vector, and ${\varvec{F}}_{{{\varvec{s}} + 1}}$ the converged force vector at the time instant $\tau_{s + 1}$

An iterative scheme computes the converged global interaction force vector ${\varvec{F}}_{s + 1}^{C}$ between the granular medium and the plate at the time instant $\tau = \tau_{s + 1}$ once the solution at the previous time instant $\tau = \tau_{s}$ has been determined (where the notation of the main text holds throughout). The iteration scheme is divided into two distinct phases. In the first phase we consider exclusively the response of the thin plate. To compute the response of the plate in the first iteration ($j = 1$) of the flow chart of Fig. 13 at time instant $\tau_{s + 1}$, we select the global force vector applied to the plate as ${\varvec{F}}_{s + 1}^{\left( 1 \right)} = {\varvec{F}}_{{\varvec{s}}}^{C}$. Then, the first iteration of the response of the plate ${\varvec{x}}_{s + 1}^{\left( 1 \right)}$ at the time instant $\tau_{s + 1}$, is computed by solving Eq. (11) using the $\beta$-Newmark method based on the assumption of constant acceleration between successive time steps,

$$\dot{{\varvec{x}}}_{s + 1}^{\left( 1 \right)} = \dot{{\varvec{x}}}_{s}^{C} + \left[ {\left( {1 - \gamma } \right){\ddot{\varvec{x}}}_{s}^{C} + \gamma {\ddot{\varvec{x}}}_{s + 1}^{\left( 1 \right)} } \right] \Delta \tau$$

(15a)

$${\varvec{x}}_{s + 1}^{\left( 1 \right)} = {\varvec{x}}_{s}^{C} + \dot{\varvec{x}}_{s}^{C} \Delta \tau + \left[ {\frac{1 - 2\beta }{2} {\ddot{\varvec{x}}}_{s}^{C} + \beta {\ddot{\varvec{x}}}_{s + 1}^{\left( 1 \right)} } \right]\Delta \tau^{2}$$

(15b)

$$\varvec{M}{\ddot{\varvec{x}}}_{s + 1}^{\left( 1 \right)} + \varvec{K}{\varvec{x}}_{s + 1}^{\left( 1 \right)} = {\varvec{F}}_{s + 1}^{\left( 1 \right)}$$

(15c)

with $\beta = 1/4$ and $\gamma = 1/2$. Note that we use the converged values for the response variables at the previous time step $\tau_{s}$, and that the equations above are linear with respect to the response “status” vectors of the plate, ${\varvec{x}}_{s + 1}^{\left( 1 \right)}$, $\dot{\varvec{x}}_{s + 1}^{\left( 1 \right)}$ and ${\ddot{\varvec{x}}}_{s + 1}^{\left( 1 \right)}$, at the current time instant $\tau_{s + 1}$. Once the first iteration for the status vectors ${\varvec{x}}_{s}^{\left( 1 \right)}$, $\dot{\varvec{x}}_{s}^{\left( 1 \right)}$ and ${\varvec{x}}_{s + 1}^{\left( 1 \right)}$ have been evaluated, the first iteration of the displacement vector ${\varvec{x}}^{\left( 1 \right)} \left( \tau \right)$ of the plate can be evaluated over the entire time interval $\tau \in \left[ {\tau_{s} ,\tau_{s + 1} } \right]$ as follows:

$$\begin{array}{*{20}c} {{\varvec{x}}^{\left( 1 \right)} \left( \tau \right) = {\varvec{x}}_{s}^{C} + \left( {\tau - \tau_{s} } \right)\dot{\varvec{x}}_{s}^{C} + \frac{{\left( {\tau - \tau_{s} } \right)^{2} }}{{\Delta \tau^{2} }}\left( {{\varvec{x}}_{s + 1}^{\left( 1 \right)} - {\varvec{x}}_{s}^{C} - \dot{\varvec{x}}_{s}^{C} \Delta \tau } \right)} \\ \end{array}$$

(16)

This expression is based on the assumption of (approximately) constant acceleration between successive time steps. Once the response of the thin plate in the entire interval $\tau \in \left[ {\tau_{s} ,\tau_{s + 1} } \right]$ is determined, the responses of the rigid layers at the free boundary of the plate are also available, which act as a moving boundary for the DE system modeling the granular medium. This completes the first phase of the first iteration, as it evaluates the first iterate of the plate response.

In the second phase of the first iteration (i.e., still for $j = 1$), we compute the response of the granular medium at the time instant $\tau_{s + 1}$. To this end, the DE Eqs (1) are solved numerically using the fourth order Runge–Kutta method. To this end, the DE system (1) is transformed into a set of first-order differential equations,

$$\dot{\varvec{u}}_{b}^{\left( 1 \right)} = {\varvec{v}}_{b}^{\left( 1 \right)}$$

(17a)

$$\dot{\varvec{v}}_{{\varvec{b}}}^{\left( 1 \right)} = \varvec{L}\left( {{\varvec{u}}_{b}^{\left( 1 \right)} ,\varvec{v}_{b}^{\left( 1 \right)} ,{\varvec{x}}^{\left( 1 \right)} \left( \tau \right),\dot{\varvec{x}}^{\left( 1 \right)} \left( \tau \right)} \right)$$

(17b)

or,

$$\dot{\varvec{w}}_{b}^{\left( 1 \right)} = {\varvec{G}}\left( {{\varvec{w}}_{b}^{\left( 1 \right)} ,{\varvec{x}}^{\left( 1 \right)} \left( \tau \right),\dot{\varvec{x}}^{\left( 1 \right)} \left( \tau \right)} \right) ,\quad \tau \in \left[ {\tau_{s} ,\tau_{s + 1} } \right]$$

(17c)

where ${\varvec{u}}_{b}^{\left( 1 \right)}$ denotes the first iteration of the displacement vector of all granules in the time interval $\tau \in \left[ {\tau_{s} ,\tau_{s + 1} } \right]$, ${\varvec{v}}_{b}^{\left( 1 \right)} =$ $\dot{{u}}_{b}^{\left( 1 \right)}$, ${\varvec{w}}_{b}^{\left( 1 \right)} \equiv \left( {{\varvec{u}}_{b}^{\left( 1 \right)} ,{\varvec{v}}_{b}^{\left( 1 \right)} } \right)$, and ${\varvec{L}}\left( \cdot \right),{ }$ ${\varvec{G}}\left( \cdot \right)$ are highly discontinuous functions, given that the interaction forces between granules can be Hertzian, frictional and viscous (due to structural damping of the material of the granules). The DE system (17c) is then numerically solved by applying the fourth order Runge–Kutta method,

This computation yields the status vectors of the first iteration of the granular medium at $\tau = \tau_{s + 1}$. Following that, the second iteration of the interaction forces at $\tau_{s + 1}$ is computed by applying Eqs (2) and (5) with the first iteration of the plate and granular mediums responses. Hence, we compute the second iterates of the interaction force vectors applied to the plate, namely, ${\varvec{F}}_{s + 1}^{\left( 2 \right)} \equiv \left( {{\varvec{N}}_{s + 1}^{\left( 2 \right)} ,{\varvec{f}}_{s + 1}^{\left( 2 \right)} } \right)$ at time instant $\tau_{s + 1}$. This competes the second phase of the first iteration at $\tau_{s + 1}$. Then, the previous iteration scheme can continue with the second iteration at $\tau_{s + 1}$, setting $j = 2$, and repeating the two previous phases, and so on.

This iterative scheme generates a nonlinear map that relates the j-th iterates of the interaction forces at the rigid layers on the plate to the $(j + 1)$-th iterates, or in symbolic form, $\left[ {{\varvec{N}}_{s + 1}^{\left( j \right)} ,{\varvec{f}}_{s + 1}^{\left( j \right)} } \right] \to \left[ {{\varvec{N}}_{s + 1}^{{\left( {j + 1} \right)}} ,{\varvec{f}}_{s + 1}^{{\left( {j + 1} \right)}} } \right]$ at the current time step $\tau_{s + 1}$. With increasing number of iterations the interaction forces are expected to converge (the conditions for convergence are discussed below), by satisfying the following two convergence criteria,

$$\left| {f_{k,s + 1}^{{\left( {j + 1} \right)}} - f_{k,s + 1}^{\left( j \right)} } \right| < abstol; \left| {N_{k,s + 1}^{{\left( {j + 1} \right)}} - N_{k,s + 1}^{\left( j \right)} } \right| < abstol$$

(18a)

$$\left| {f_{k,s + 1}^{{\left( {j + 1} \right)}} - f_{k,s + 1}^{\left( j \right)} } \right|/\left| {f_{k,s + 1}^{\left( j \right)} } \right| < reltol; \left| {N_{k,s + 1}^{{\left( {j + 1} \right)}} - N_{k,s + 1}^{\left( j \right)} } \right|/N_{k,s + 1}^{\left( j \right)} < reltol$$

(18b)

where $abstol$ denotes the absolute tolerance, and $reltol$ the relative tolerance. As a reminder, $N_{k,s + 1}^{\left( j \right)}$ and $f_{k,s + 1}^{\left( j \right)}$ denote the k-th elements of the global normal and tangential interaction force vectors on the plate, ${\varvec{N}}_{s + 1}^{\left( j \right)}$ and ${\varvec{f}}_{s + 1}^{\left( j \right)}$ at time instant $\tau_{s + 1}$. The interaction force vectors are considered to have converged if either (18a) and/or (18b) is satisfied for all nodes $k$ at the rigid layers of the plate (cf. Fig. 3). Supposing that convergence has been achieved at the J-th iteration, we denote the converged interaction force vectors at time instant $\tau_{s + 1}$ by $\left( {{\varvec{N}}_{s + 1}^{C} ,{\varvec{f}}_{s + 1}^{C} } \right) \equiv \left( {{\varvec{N}}_{s + 1}^{\left( J \right)} ,{\varvec{f}}_{s + 1}^{\left( J \right)} } \right)$. Once the interaction forces on the plate have converged, all the response status vectors at the current time step $\tau_{s + 1}$ will have converged as well, and the computation proceeds to the next time step $\tau_{s + 2}$, where the outlined iteration scheme is repeated until the entire time interval $T$ of the simulation is exhausted.

Clearly, central to the stability and convergence of the computational algorithm is the stability of the previous global map $\left[ {{\varvec{N}}_{s + 1}^{\left( j \right)} ,{\varvec{f}}_{s + 1}^{\left( j \right)} } \right] \to \left[ {{\varvec{N}}_{s + 1}^{{\left( {j + 1} \right)}} ,{\varvec{f}}_{s + 1}^{{\left( {j + 1} \right)}} } \right]$ at an arbitrary—say the (j + 1)th iteration. To ensure the convergence of the $\beta$-Newmark method, the time step increment $\Delta \tau$ should be sufficiently small so that variations of the interaction forces due to the time delay of the wave propagation at a contact point at $\tau_{s + 1}$ has negligible effects on the responses at the other contact points at the current time step $\tau_{s + 1}$. Hence, since the interaction forces at a contact point of the plate have an approximately local effect, the next iteration of the interaction forces at that contact point should also be determined only by the local response at the contact point (i.e., we assume that $\Delta \tau$ is small enough so that “coupling” effects between different contact points may be neglected). Accordingly, the global map $\left[ {{\varvec{N}}_{s + 1}^{\left( j \right)} ,{\varvec{f}}_{s + 1}^{\left( j \right)} } \right] \to \left[ {{\varvec{N}}_{s + 1}^{{\left( {j + 1} \right)}} ,{\varvec{f}}_{s + 1}^{{\left( {j + 1} \right)}} } \right]$ can be decomposed into individual 2D local maps $\left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right) \to \left( {N_{k,s + 1}^{{\left( {j + 1} \right)}} ,f_{k,s + 1}^{{\left( {j + 1} \right)}} } \right)$ at each contact point $k$ at the current time step. Clearly, the global map is stable if all 2D local maps are stable (i.e., for every contact point $k$ on the plate).

Note that the (j + 1)th iterate of each interaction force component at the contact point $k$ of the plate at time instant $\tau_{s + 1}$ can be expressed in explicit form as follows,

$$N_{k,s + 1}^{{\left( {j + 1} \right)}} = - \frac{4}{3}E^{*} \sqrt {R^{*} } \left[ {u_{k,p,n,s + 1}^{\left( j \right)} - u_{k,b,n,s + 1}^{\left( j \right)} } \right]_{ + }^{3/2}$$

(19a)

$$f_{k,s + 1}^{{\left( {j + 1} \right)}} = - \mu \left| {N_{k,s + 1}^{{\left( {j + 1} \right)}} } \right|\tanh \left[ {k_{s} \left( {v_{k,p,t,s + 1}^{\left( j \right)} - v_{k,b,t,s + 1}^{\left( j \right)} } \right)} \right]$$

(19b)

where referring to the notation introduced in Fig. 3, the subscripts $p,b$ denote plate and contacting granule, respectively, whereas the subscripts $n,t$ denote normal and tangential components, respectively. Hence, in (19a,b) the variables $u_{k,p,n,s + 1}^{\left( j \right)}$ and $u_{k,b,n,s + 1}^{\left( j \right)}$ $\left( {u_{k,p,t,s + 1}^{\left( j \right)} {\text{ and }}u_{k,b,t,s + 1}^{\left( j \right)} } \right)$ denote the j-th iterates of the normal (tangential) components of the deformations of the plate and the contacting granule at time instant $\tau_{s + 1}$, respectively. In addition, $v = \dot{u}$, represents the corresponding velocity. Furthermore, $N_{k,s + 1}^{{\left( {j + 1} \right)}}$ and $f_{k,s + 1}^{{\left( {j + 1} \right)}}$ denote the normal and tangential forces applied at the kj-th contact point of the plate at the current time instant $\tau_{s + 1}$. As the small rigid layers at the boundary of the plate are assumed to be flat and massless, the dissipative term in the normal force is omitted, while $E^{*} = E_{b} /\left( {1 - \nu_{b}^{2} } \right)$ and $R^{*} = R_{b}$.

Relations (19a, b) are in scalar form, with the positive directions defined in Fig. 3. To evaluate the stability of the 2D local map $\left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right) \to \left( {N_{k,s + 1}^{{\left( {j + 1} \right)}} ,f_{k,s + 1}^{{\left( {j + 1} \right)}} } \right)$ it is necessary to examine its $\left( {2 \times 2} \right)$ Jacobian matrix,

$$\frac{{\partial \left( {N_{k,s + 1}^{{\left( {j + 1} \right)}} ,f_{k,s + 1}^{{\left( {j + 1} \right)}} } \right)}}{{\partial \left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right)}} = \frac{{\partial \left( {N_{k,s + 1}^{{\left( {j + 1} \right)}} ,f_{k,s + 1}^{{\left( {j + 1} \right)}} } \right)}}{{\partial \left( {u_{k,p,n,s + 1}^{\left( j \right)} - u_{k,b,n,s + 1}^{\left( j \right)} ,v_{k,p,t,s + 1}^{\left( j \right)} - v_{k,b,t,s + 1}^{\left( j \right)} } \right)}}\left[ {\frac{{\partial \left( {N_{k,s + 1}^{{\left( {j + 1} \right)}} ,f_{k,s + 1}^{{\left( {j + 1} \right)}} } \right)}}{{\partial \left( {u_{k,p,n,s + 1}^{\left( j \right)} - u_{k,b,n,s + 1}^{\left( j \right)} ,v_{k,p,t,s + 1}^{\left( j \right)} - v_{k,b,t,s + 1}^{\left( j \right)} } \right)}}} \right]$$

(20)

and compute its eigenvalues. The first multiplicative term in (20) can be explicitly evaluated by relations (19a, b). To evaluate the second multiplicative term, we consider the discretized differential equations at successive time steps to evaluate the sensitivities of the responses of the plate and granules at the current time step. The response of the plate at the j-th iteration and time instant $\tau_{s + 1}$ subject to the interaction forces at the same iteration can be computed by solving Eq. (11):

$$\dot{\varvec{x}}_{s + 1}^{\left( j \right)} = \dot{\varvec{x}}_{s}^{C} + \frac{1}{2}{\ddot{\varvec{x}}}_{s}^{C} \Delta \tau + \frac{1}{2}{\ddot{\varvec{x}}}_{s + 1}^{\left( j \right)} \Delta \tau$$

(21a)

$${\varvec{x}}_{s + 1}^{\left( j \right)} = {\varvec{x}}_{s}^{C} + \dot{\varvec{x}}_{s}^{C} \Delta \tau + \frac{1}{4}{\ddot{\varvec{x}}}_{s}^{C} \Delta \tau^{2} + \frac{1}{4}{\ddot{\varvec{x}}}_{s + 1}^{\left( j \right)} \Delta \tau^{2}$$

(21b)

$${\ddot{\varvec{x}}}_{s + 1}^{\left( j \right)} = \left( {{\varvec{M}} + \frac{1}{4}{\varvec{K}}\Delta \tau^{2} } \right)^{ - 1} {\varvec{F}}_{s + 1}^{\left( j \right)} - \left( {{\varvec{M}} + \frac{1}{4}{\varvec{K}}\Delta \tau^{2} } \right)^{ - 1} {\varvec{K}}\left( {{\varvec{x}}_{s}^{C} + \dot{\varvec{x}}_{s}^{C} \Delta \tau + \frac{1}{4}{\ddot{\varvec{x}}}_{s}^{C} \Delta \tau^{2} } \right)$$

(21c)

We note at this point that the normal and tangential displacements at the $k - th$ contact point can be linearly related to the plate displacement vector ${ \varvec{x}}_{s + 1}^{\left( j \right)}$ as follows:

$$u_{k,p,n,s + 1}^{\left( j \right)} = {\varvec{T}}_{k,n} {\varvec{x}}_{s + 1}^{\left( j \right)}$$

(22a)

$$u_{k,p,t,s + 1}^{\left( j \right)} = {\varvec{T}}_{k,t} {\varvec{x}}_{s + 1}^{\left( j \right)}$$

(22b)

$${\varvec{T}}_{k} = \left( {{\varvec{T}}_{k,n} ,{\varvec{T}}_{k,t} } \right)^{T}$$

(22c)

Note that $u_{k,p,n,s + 1}^{\left( j \right)}$ and $u_{k,p,t,s + 1}^{\left( j \right)}$ are the components of $\varvec{x}_{s + 1}^{\left( j \right)}$ at the normal and tangential DOFs at the node of the k-th contact point, respectively; also, ${\varvec{T}}_{k,n}$ and ${\varvec{T}}_{k,t}$ are sparse vectors whose only non-zero terms, with value equal to unity, are located at the normal and tangential driving DOFs, respectively, for the k-th contact point. Since the corresponding components of the force vector ${\varvec{F}}_{s + 1}^{\left( j \right)}$ are $N_{k,s + 1}^{\left( j \right)}$ and $f_{k,s + 1}^{\left( j \right)}$, It follows that:

$$\begin{array}{*{20}c} {\frac{{\partial {\varvec{F}}_{s + 1}^{\left( j \right)} }}{{\partial \left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right)}} = {\varvec{T}}_{k}^{T} } \\ \end{array}$$

(23)

Therefore, the plate component of the second multiplicative term in Eq. (20) could be computed by the chain rule:

$$\frac{{\partial \left( {u_{k,p,n,s + 1}^{\left( j \right)} ,u_{k,p,t,s + 1}^{\left( j \right)} } \right)}}{{\left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right)}} = \frac{{\partial \left( {u_{k,p,n,s + 1}^{\left( j \right)} ,u_{k,p,t,s + 1}^{\left( j \right)} } \right)}}{{\partial {\varvec{x}}_{s + 1}^{\left( j \right)} }} \cdot \frac{{\partial {\varvec{x}}_{s + 1}^{\left( j \right)} }}{{\partial {\ddot{\varvec{x}}}_{s + 1}^{\left( j \right)} }} \cdot \frac{{\partial {\ddot{\varvec{x}}}_{s + 1}^{\left( j \right)} }}{{\partial {\varvec{F}}_{s + 1}^{\left( j \right)} }} \cdot \frac{{\partial {\varvec{F}}_{s + 1}^{\left( j \right)} }}{{\partial \left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right)}}$$

(24a)

$$\frac{{\partial \left( {v_{k,p,n,s + 1}^{\left( j \right)} ,v_{k,p,t,s + 1}^{\left( j \right)} } \right)}}{{\partial \left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right)}} = \frac{{\partial \left( {\dot{u}_{k,p,n,s + 1}^{\left( j \right)} ,\dot{u}_{k,p,t,s + 1}^{\left( j \right)} } \right)}}{{\partial \dot{{\varvec{x}}}_{s + 1}^{\left( j \right)} }} \cdot \frac{{\partial \dot{\varvec{x}}_{s + 1}^{\left( j \right)} }}{{\partial {\ddot{\varvec{x}}}_{s + 1}^{\left( j \right)} }} \cdot \frac{{\partial {\ddot{\varvec{x}}}_{s + 1}^{\left( j \right)} }}{{\partial {\varvec{F}}_{s + 1}^{\left( j \right)} }} \cdot \frac{{\partial {\varvec{F}}_{s + 1}^{\left( j \right)} }}{{\partial \left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right)}}$$

(24b)

Substituting relations (21), (22) and (23) into Eqs. (24a, b), we derive the following expressions:

$$\frac{{\partial \left( {u_{k,p,n,s + 1}^{\left( j \right)} ,u_{k,p,t,s + 1}^{\left( j \right)} } \right)}}{{\partial \left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right)}} = \frac{1}{4} \Delta \tau^{2} {\varvec{T}}_{k} \left( {{\varvec{M}} + \frac{1}{4}{\varvec{K}}\Delta \tau^{2} } \right)^{ - 1} {\varvec{T}}_{k}^{T}$$

(25a)

$$\frac{{\partial \left( {v_{k,p,n,s + 1}^{\left( j \right)} ,v_{k,p,t,s + 1}^{\left( j \right)} } \right)}}{{\partial \left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right)}} = \frac{1}{2} \Delta \tau {\varvec{T}}_{k} \left( {{\varvec{M}} + \frac{1}{4}{\varvec{K}}\Delta \tau^{2} } \right)^{ - 1} {\varvec{T}}_{k}^{T}$$

(25b)

Since we assumed that the effect of the interaction forces is nearly local, it is logical to assume that due to the symmetry of the rigid layers the normal forces cannot generate tangential responses, while the tangential forces cannot generate normal responses at the current time step. This yields the following further approximations:

$$\begin{array}{*{20}c} {\frac{{\partial u_{k,p,n,s + 1}^{\left( j \right)} }}{{\partial f_{k,s + 1}^{\left( j \right)} }} \approx \frac{{\partial v_{k,p,n,s + 1}^{\left( j \right)} }}{{\partial f_{k,s + 1}^{\left( j \right)} }} \approx \frac{{\partial u_{k,p,t,s + 1}^{\left( j \right)} }}{{\partial N_{k,s + 1}^{\left( j \right)} }} \approx \frac{{\partial v_{k,p,t,s + 1}^{\left( j \right)} }}{{\partial N_{k,s + 1}^{\left( j \right)} }} \approx 0} \\ \end{array}$$

(26)

Substituting (26) to relations (25a, b), we derive the following closed form simplified diagonalized sensitivity matrix for the responses at the contact points on the plate:

$$\frac{{\partial \left( {u_{k,p,n,s + 1}^{\left( j \right)} ,v_{k,p,t,s + 1}^{\left( j \right)} } \right)}}{{\partial \left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right)}} = Diag\left[ { \frac{1}{4} \Delta \tau^{2} {\varvec{T}}_{k,n} \left( {{\varvec{M}} + \frac{1}{4}{\varvec{K}}\Delta \tau^{2} } \right)^{ - 1} {\varvec{T}}_{k,n}^{T} ,\frac{1}{2}\Delta \tau {\varvec{T}}_{k,t} \left( {{\varvec{M}} + \frac{1}{4}{\varvec{K}}\Delta \tau^{2} } \right)^{ - 1} {\varvec{T}}_{k,t}^{T} } \right]$$

(27)

Similarly, by imposing the previous assumptions we can compute the corresponding sensitivities $\frac{{\partial \left( {u_{k,b,n,s + 1}^{\left( j \right)} ,v_{k,b,t,s + 1}^{\left( j \right)} } \right)}}{{\partial \left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right)}}$ of the contacting boundary granules. However, as the mass of a single granule is much larger than the mass of the finite element of the plate attached to the contacting rigid layer, it is logical to assume that the response of the rigid layer should be much more sensitive with respect to the interaction forces compared to the response of the granule. Hence, as an additional approximation, we may neglect the sensitivities related to the granule responses, thus simplifying Eq. (20) as follows:

$$\frac{{\partial \left( {N_{k,s + 1}^{{\left( {j + 1} \right)}} ,f_{k,s + 1}^{{\left( {j + 1} \right)}} } \right)}}{{\partial \left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right)}} \approx \frac{{\partial \left( {N_{k,s + 1}^{{\left( {j + 1} \right)}} ,f_{k,s + 1}^{{\left( {j + 1} \right)}} } \right)}}{{\partial \left( {u_{k,p,n,s + 1}^{\left( j \right)} ,v_{k,p,t,s + 1}^{\left( j \right)} } \right)}} \cdot \frac{{\partial \left( {u_{k,p,n,s + 1}^{\left( j \right)} ,v_{k,p,t,s + 1}^{\left( j \right)} } \right)}}{{\partial \left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right)}}$$

(28)

As a final step, the Jacobian matrix of the local nonlinear map at contact point $k$ on the rigid layer of the plate, $\left( {N_{k,s + 1}^{\left( j \right)} ,f_{k,s + 1}^{\left( j \right)} } \right) \to \left( {N_{k,s + 1}^{{\left( {j + 1} \right)}} ,f_{k,s + 1}^{{\left( {j + 1} \right)}} } \right)$ is approximated in closed form by substituting the relations (19a, b) and (27) into (28). As the Jacobian matrix is semi-negative definite for arbitrary interaction forces, according to the Banach fixed point theorem the local map has a unique fixed point (i.e., it is guaranteed to converge to a solution), and its eigenvalues are approximately evaluated in closed form as follows:

$$\lambda_{k1,s + 1} = - \frac{1}{2} E^{*} \sqrt {R^{*} } \Delta \tau^{2} {\varvec{T}}_{k,n} \left( {{\varvec{M}} + \frac{1}{4}{\varvec{K}}\Delta \tau^{2} } \right)^{ - 1} {\varvec{T}}_{k,n}^{T} \left( {u_{k,p,n,s + 1}^{\left( j \right)} - u_{k,b,n,s + 1}^{\left( j \right)} } \right)_{ + }^{1/2}$$

(29a)

$$- \frac{1}{2} \mu k_{s} \left| {N_{k,s + 1}^{\left( j \right)} } \right| \Delta \tau { T}_{k,t} \left( {{\varvec{M}} + \frac{1}{4}{\varvec{K}}\Delta \tau^{2} } \right)^{ - 1} {\varvec{T}}_{k,t}^{T} {\text{cosh}}\left[ {k_{s} \left( {v_{k,p,t,s + 1}^{\left( j \right)} - v_{k,b,t,s + 1}^{\left( j \right)} } \right)} \right]^{ - 2}$$

(29b)

Accordingly, the local map is stable if the moduli of both eigenvalues are smaller than unity, which gives the conditions (27a, b).

Appendix 2

Below we provide links for the following animations for the acoustics of:

The monolithic plate of Fig. 11a subject to a uniform impulse of $1\,\upmu {\text{s}}$ and $20000\;{\text{N/m}}$ intensity https://uofi.box.com/s/13ia09hrmnsctw2tl5xm47p6te78qzby

The granular-solid interface of Fig. 11c with uniform initial velocities $v_{0} = 0.3414\,{\text{m/s}}$ of its five left granules, corresponding to the same energy input to case (i) for the monolithic plate https://uofi.box.com/s/tvxsuqehqfhgs1w8ype4s89tcwve0qly

The granular-solid interface of Fig. 11c with uniform initial velocities $v_{0} = 0.5\,{\text{m/s}}$ and $v_{0} = 1.0\;{\text{m/s}}$ of its five left granules https://uofi.box.com/s/xi8hp6zzz61ofanvy6iq102t8e00h9lv, https://uofi.box.com/s/5ck6xjbnhac76t6bf0d044zdwikgxot2