Concept of damping metamaterial

In order to establish a framework for the design of material systems with programmable damping properties, a fundamental design for a unit cell was established. The unit cell consists of a combination of two functional layers (Figure 1). The first layer consists of a flexible sealed cavity, here a bellow, with an orifice at the interface to the second layer. This first layer provides the elasticity of the material and the damping effect in one direction. The bottom layer consists of an auxetic porous structure, which expands in the vertical direction by a lateral strain. Through the vertical expansion of the structure, an integrated plug is lifted, which opens the orifice and thus serves as an openable valve for the cavity. Therefore, the bottom layer provides the trigger functionality of the material system to switch between states with different damping and stiffness.

Figure 1. Implementation of a programmable damping effect in microstructure based on unit cells. (a) Basic design of a unit cell with two functional layers, the sealed cavity or bellow layer (blue, top) and the auxetic layer (black, bottom). (b) Schematic of the different regimes that can be programmed into the material system.

This parametric model enables the material systems to trigger functionality, the stiffness as well as frequency-dependent damping properties to be customised depending on the requirements of the application (Figure 1b). In the next sections, two complementary numerical models to optimise the structures will be introduced and subsequently validated by experiments.

Analytical mechanical model

To design and analyse the general behaviour of a unit cell, a simplified mechanical model of the bellow layer is established as shown in Figure 2. It comprises the bellow, characterised by its height h, its diameter d (defining the cross-sectional area A), its inherent stiffness k, its inherent damping value b and its blowing rigidity c, which can also, as a first approximation, be used to include the compressibility of the fluid (Figure 2a).

Figure 2. Analytical model for the programmable damping material. (a) Schematic of the bellow layer with characteristic parameters for analytical model. Frequency-dependent results of dynamic stiffness (b) and loss angle (c) over orifice opening represented by fluid flow factor s.

The state of the orifice of the bellow is described by the parameter s, which quantifies the fluid flow at a certain internal pressure p. Small values of s thus describe an almost fully closed valve, whereas large values of s describe an open valve. The model is excited by an external harmonic force F over time t at a certain frequency f and the elongation x over time t is evaluated regarding the excitation, yielding a dynamic stiffness and a loss angle for each configuration.

The deformation behaviour of the bellow can be described in a first approximation using two equations:

(1) $$ \begin{align} F(t) = A\;p(t)+k\;x(t)+b\;\dot{x}(t),\end{align} $$

(2) $$ \begin{align}A\;x(t) = \frac{1}{c}\;p(t)+\int s\;p(t) dt. \end{align}$$

Equation (1) describes the normal equilibrium of force at the point of load application. The input force is opposed to the sum of the forces resulting from the mechanics of the bellow itself, that is, its stiffness and damping properties, and the current internal pressure acting on the cross-sectional area A.

Equation (2) describes the fluid volume inside the bellow, considering both its blowing rigidity c and the fluid flow through the valve, assuming that the fluid flow is proportional to the internal pressure p and the parameter s. The internal pressure p is defined as the relative internal pressure compared to the ambient pressure, therefore in the unloaded case p equals 0.

Assuming harmonic behaviour of all time-dependent values with frequency f and amplitude values of $\widehat{x}$ , $\widehat{p}$ , and $\widehat{F},$ respectively in complex notation yields equation (3) for the dynamic stiffness of the system by dividing the force amplitude by the stroke amplitude:

(3) $$ \begin{align}{k}_{\mathrm{dyn}} = \frac{\widehat{F}}{\widehat{x}} = k+\frac{A^2}{\frac{1}{c}+\frac{s}{i2\pi f\;}\kern0.24em }+i2\pi f\;b. \end{align}$$

From this equation, the maximum static stiffness is reached for a fully closed valve (s = 0) and is given by

(4) $$\begin{align}{k}_{\mathrm{dyn},\max} = {A}^2c+k,\end{align}$$

whereas the minimum static stiffness is reached for a fully open valve ( $s\to \infty$ ) and is given by

(5) $$\begin{align}{k}_{\mathrm{dyn},\min} = k.\end{align}$$

The damping of the system can be characterised by the loss angle $\varphi$ , which is the angle between force and stroke, and it can be calculated in the following way:

(6) $$\begin{align}\tan \varphi = \frac{A{c}^2 s\omega +b{c}^2{s}^2\omega +b{\omega}^3}{A^2c{\omega}^2+ k\omega +k{c}^2{s}^2}.\end{align}$$

Assuming the material damping characterised by the parameter $b$ can be neglected compared to the fluidic damping effect, the maximum loss angle $\varphi$ can be mathematically determined by equating its first derivative to 0, leading to the value of $s$ at which the highest damping occurs

(7) $$\begin{align}{s}_{\mathrm{maxdamp}} = \frac{\omega }{c}\sqrt{\frac{A^2c+k}{k}} = \frac{\omega }{c}\sqrt{\frac{k_{\mathrm{dyn},\max }}{k_{\mathrm{dyn},\min }}.}\end{align}$$

The maximum damping can therefore be achieved for large stiffness ratios. The resulting maximum loss angle can be calculated in the following way:

(8) $$\begin{align}{\varphi}_{\mathrm{max}} = {\tan}^{-1}\left(\frac{1}{2}\frac{\frac{A^2c+k}{k}-1}{\sqrt{\frac{A^2c+k}{k}}}\right).\end{align}$$

Exemplary curves of the dynamic stiffness and loss angle according to the above model are shown in Figure 2b,c to visualise the influence of the frequency and shifts of ${s}_{\mathrm{maxdamp}}$ . The shape of the stiffness versus frequency curve is similar for all excitation frequencies and exhibits two distinct regimes for extreme values of s, converging toward ${k}_{\mathrm{dyn},\min }$ for the fully open and ${k}_{\mathrm{dyn},\max }$ for fully closed system. However, a lateral shift of the transition toward larger values of s is observed with increasing frequency.

The loss angle reaches a maximum ${\varphi}_{\mathrm{max}}$ independently of the excitation frequency. The position of the peak of the loss angle ${s}_{\mathrm{maxdamp}}$ shifts to higher values of s with increasing excitation frequency.

Fluid-mechanics model

Based on the previous model, the critical fluid flow ${s}_{\mathrm{maxdamp}}$ can be used as a design parameter. However, in practice, this parameter is not accessible as it is dependent on the material topography and cannot be directly correlated to the magnitude of the lateral strain required. Therefore, besides the numerical model that can be used to make the principal design of the damping unit cells, we propose a practical approach based on fluid dynamics combined with a FEM simulation (Figure 3). It is used to determine the experimental shifts of the maximum damping depending on the frequency and the lateral strain and predict the lateral strains with maximum damping behaviour at a certain frequency.

Figure 3. Fluid-based model to predict the critical strain for different frequencies based on numeric simulations of the unit cell. (a) FEM model to simulate the strain-dependent cross-section for a specific parametrised unit cell. (b) Visualisation of the cross-section for $\varepsilon = 0$ and $\varepsilon >0$ . (c) Qualitative results of the fluid-mechanics prediction for the flow number ${\mathrm{FN}}_{\operatorname{norm},f}\left(\varepsilon \right)$ at two different frequencies.

Fluid flow is usually described by the Reynold’s number ${R}_{\mathrm{e}}$ .

(9) $$\begin{align}{R}_e = \frac{\rho \cdot v\cdot {s}_{\mathrm{char}}}{\unicode{x3bc} }.\end{align}$$

The Reynold’s number is a function of the density of the fluid $\rho$ , the viscosity of the fluid $\unicode{x3bc}$ , the fluid flow speed $v$ , the cross-sectional area A and the characteristic linear dimension ${s}_{char}$ . In the case of a programmable unit cell, the fluid flow speed is defined as

(10) $$\begin{align}v = \frac{\skew6\dot{V}}{A\left(\varepsilon \right)} = \frac{\dot{V_{\mathrm{norm}}\cdot f}}{A\left(\varepsilon \right)}.\end{align}$$

where $\skew6\dot{V}$ is the volume flow per time increment and is proportional to the frequency. The cross-sectional area $A$ is a function of the lateral strain $\varepsilon$ and is characteristic of a unit cell and the central design parameter.

In this work, we define a flow number FN proportional to the Reynold’s number.

(11) $$\begin{align}{\mathrm{FN}}_f\left(\varepsilon \right) = \frac{1}{A\left(\varepsilon \right)}f\propto {R}_e.\end{align}$$

The characteristic flow number for a unit cell can be determined by analysing the damping behaviour of a unit cell experimentally and measuring the critical lateral strain ${\varepsilon}_{\mathrm{crit},{f}_0}$ for which damping is highest at a frequency ${f}_0$ . Normalising the flow number by ${\mathrm{FN}}_{f_0}\left({\varepsilon}_{\mathrm{crit},{f}_0}\right)$ yields a design criterion for the strain at different frequencies.

(12) $$\begin{align}{\mathrm{FN}}_{\operatorname{norm},f}\left(\varepsilon \right) = \frac{1}{A\left(\varepsilon \right)}\cdot f\cdot \frac{1}{{\mathrm{FN}}_{f_0}\left({\varepsilon}_{\mathrm{crit},{f}_0}\right)}.\end{align}$$

$A\left(\varepsilon \right)$ depends on the design parameters of the unit cell and thus has to be computed using a time-dependent FEM simulation (Figure 3a,b). As shown in Figure 3c, the flow number ${\mathrm{FN}}_{\operatorname{norm},f}\left(\varepsilon \right)$ can be plotted as a function of $\varepsilon$ as long as the relationship between A and $\varepsilon$ is known. For target frequencies ${f}_n$ the normalised flow number ${\mathrm{FN}}_{\operatorname{norm},{f}_n}$ needs to be close to 1 to reach maximum damping properties and determine the ${\varepsilon}_{\mathrm{crit},{f}_n}$ .

The two previous models provide the tools to design a unit cell with programmable damping properties. They represent simplified models and cannot quantitatively predict the damping behaviour without any experiments, but still narrow down the design space and thus drastically reduce the development time for new applications.

Mechanical testing: Air-filled system and effect of lateral strain

The following sections showcase the functionality of exemplary unit cells and serve as validation for the aforementioned models. To validate the results from the numerical models, unit cells were fabricated using selective laser sintering (Figure 4). Mechanical experiments were performed under different loading conditions while the unit cells were subject to varying boundary conditions.

Figure 4. Test rig with macroscopic programmable damping material with three unit cells in a row. The testing was performed with an excitation of 2 mm stroke at 1 Hz.

First, only the bellow was excited by a 2 mm stroke at 1 Hz in two different states: with a completely open valve and with a completely closed valve using a wax seal. The force response was monitored over the course of 10 cycles to determine the equivalents of ${k}_{\mathrm{dyn}}$ and ${k}_{\mathrm{dyn},\max }$ from the mechanical model. A preload of 0.19 N was used to guarantee contact between the structure and the testing machine at all times. The hysteresis is narrow in both cases which means the damping is low. The stiffness of the unit cell is approximated based on a linear fit, ${k}_{\mathrm{dyn},\max }$ and ${k}_{\mathrm{dyn}}$ (cf. Figure 1), corresponding to a factor of about 3.5 increase in the stiffness. In both cases, the system response showed a small settling effect but stabilised after a few cycles. This showcases the general ability of the unit cell to cover a wide range of stiffness based on the lateral strain trigger (cf. Figure 1b).

To further describe the effects of the lateral strain trigger, one exemplary unit cell was modified with a further coating to smooth the surface roughness and was meticulously positioned in the tensile testing machine to qualitatively analyse the expected effects. The results are displayed in Figure 5 at a measurement rate of 1 Hz. The shape of the force-displacement curves within one deformation cycle strongly varied depending on the strain (Figure 5a). For both small and large strains, the loading and unloading curves are fairly linear with small hysteresis and the stiffness values are in good agreement with the results presented in the previous section. For intermediate strains of around 0.3%, the loading and unloading slopes are distinctly different and nonlinear, resulting in a large hysteresis (Figure 5a).

Figure 5. Measurement results of one exemplary macroscopic unit cell at different strains. (a) Force/stroke diagram for different strain conditions, steady state at 1 Hz. (b) Evaluation of linearised stiffness for different strain conditions. (c) Evaluation of loss angle for different strain conditions (squares) and results from the numerical model (crosses).

To evaluate the phenomenon quantitatively, the stiffness of the system is calculated using a linear fit between the minimum and maximum values of force and stroke whereas the loss angle is calculated using the hysteresis area $H$ , using the formula given in equation (13):

(13) $$\begin{align}\varphi = \arcsin \left(\frac{4H}{\pi}\frac{1}{\max (F)-\min (F)}\frac{1}{\max (x)-\min (x)}\right).\end{align}$$

The stiffness of the system decreases with increasing strain from 2.5 to 0.7 N/mm between 0 and 0.8% strain (Figure 5b). For strains above 0.8%, the stiffness plateaued at around 0.7 N/mm. The damping angle exhibited a bell-shaped curve with a peak value of around 38° at 0.3% strain with values close to 3° for strains of 0 and 0.2% and loss angles from 15° down to 5° for strains from 0.4% up to 1.6%.

To contrast the results from experiments and the mechanical model, the loss angle was derived from the mechanical model based on a set of parameters mimicking the macroscopic experimental demonstrator: h = 18 mm, d = 24 mm, b = 0.3 Ns m⁻¹, k = 0.7 N/mm, c = 0.88 N/cm⁵ (cf. Figure 2a). The experimentally determined stiffnesses ${k}_{\mathrm{dyn}}$ were used to calibrate the model and determine the flow factors s from simulations that are expected for the different strains (Table 1). For these flow factors s, both experimental and simulation data of the loss angle for an excitation frequency of 1 Hz are compared in Figure 5c. Experimental results and numerical predictions show a similar trend with a maximum in the loss angle between 0.2 and 0.4% strain. Deviations in the absolute values are observed especially around the peak which is likely caused by experimental inaccuracies in the manufacturing and alignment of the specimens.

Table 1. Comparison of experimental and numerical data for one exemplary macroscopic unit cell at different strains as shown in Figure 5b,c

Preliminary experiments showed that the trigger position and behaviour of air-filled unit cells are qualitatively in agreement with the models. However, manufacturing variations caused by the SLS process have a strong impact on the surface roughness and therefore the airflow within the cavities. Introducing water to the cells improved this and enabled carrying out experiments with a system of three unit cells to study the frequency-dependent behaviour in the next section.

Mechanical testing: Fluid-filled system

Additional mechanical experiments were performed on three parallel unit cells submerged in water to obtain an incompressible medium in the unit cell (Figure 6) to showcase the frequency-dependent behaviour of the unit cells. Figure 6a shows the force-displacement curves of the samples between 0 and 3% strain at 1 Hz with the same characteristic behaviour as previously observed for air-filled unit cells. The dynamic stiffness increased due to the addition of water and varied between 10 N/mm for low strains and 2.5 N/mm at high strains which corresponds to a 4-fold change in stiffness. An increase in the area of the hysteresis and loss angle was observed at about 1% strain for 1 Hz.

Figure 6. Measurement results of three macroscopic unit cells filled with water at different strains. (a) Force/stroke diagram for different strain conditions, steady state at 1 Hz. (b) Evaluation of linearised stiffness for different strain conditions and different frequencies. (c) Evaluation of loss angle for different strain conditions and different frequencies.

Measurements were repeated at frequencies of 2, 1 and 0.2 Hz and the dynamic stiffness, as well as loss angle, were analysed (Figure 6b,c). The dynamic stiffness decreases with increasing strain for all frequencies. It could be observed that the transition to the lower stiffness plateau ${k}_{\mathrm{dyn},\min }$ is shifted toward lower strains with lower frequencies which is in good agreement with the mechanical model. The loss angle exhibits a peak between 0 and 2% strain for all frequencies. A shift of the peak position to lower strains can be observed although the quantitative analysis of the peaks is challenging as the strain increments are quite large and the differences in the behaviour below 0.5 Hz are small. At 0.1 Hz it is not clear whether there is a maximum at all in this strain regime.

For validation of the fluid mechanics model, the cross-sectional area of the unit cells as a function of strain was calculated using a finite element study and the flow numbers ${\mathrm{FN}}_{\operatorname{norm},f}\left(\varepsilon \right)$ calculated and plotted in Figure 7. The frequency of 2 Hz was chosen as the reference and therefore, based on experimental results, the flow numbers normalised so that ${\mathrm{FN}}_{2\mathrm{Hz}}\left(2\%\right)$  = 1. The fluid mechanics model predicts decreasing ${\varepsilon}_{\mathrm{crit},{f}_n}$ with decreasing frequency ${f}_n$ to be ${\varepsilon}_{\mathrm{crit},1\mathrm{Hz}}\sim 1.1\%$ , ${\varepsilon}_{\mathrm{crit},0.5\mathrm{Hz}}\sim 0.5\%$ , ${\varepsilon}_{\mathrm{crit},0.2\mathrm{Hz}}\sim 0.1\%$ and ${\varepsilon}_{\mathrm{crit},0.1\mathrm{Hz}}<0.1\%$ . These results are in good agreement with the results from the experiments shown in Figure 6c.

Figure 7. Predictions based on the fluid-mechanics model for different frequencies from 0.1 to 2 Hz.

Microscopic fluid-filled system

Scalability of the unit cell principle was demonstrated with unit cells manufactured by two-photon polymerisation (Figure 8). This technique enables manufacturing the unit cells in a single process and out of a single raw material. In this case, the unit cells were neither filled with air nor water due to the manufacturing process but contain nonpolymerised resin as a damping fluid.

Figure 8. Microscopic proof of principle for damping unit cells. (a) Model of the two printed configurations and (b) results of microcompression tests of structures exhibiting two force regions.

Two different unit cell types were manufactured to mimic the macroscopic cells with two functional layers, either allowing or preventing fluid flow between the two layers (Figure 8a). Unit cells were arranged in arrays of three times three unit cells with a size of approximately $30\times 30\times 30\;\unicode{x3bc} {\mathrm{m}}^3$ and all unit cells were tested at the same time. Compression experiments revealed highly nonlinear deformation behaviour for both unit cells (Figure 8b): The open unit cell was generally softer than the closed unit cell. Both curves qualitatively show two different regimes: For small deformations, the unit cells stiffness is linear over a small range of displacement and then transitions into a second regime with a higher slope at a similar force of about 2 μN. The open cell exhibits a slope of 0.06 mN/μm in the first and 1.64 mN/μm in the second regime. The closed cell exhibits a slope of 0.23 mN/μm in the first and 2.01 mN/μm in the second regime. Investigation of the structures before and after compression in an SEM revealed no significant plastic deformation of the cells.

The dynamic properties as well as the strain trigger were not investigated as part of this research work but will need to be explored in future experiments with more sophisticated unit cell designs and testing equipment.