Integral micromorphic model for band gap in 1D continuum

Abstract

The design of band gap metamaterials, i.e., metamaterials with the capability to inhibit wave propagation of a specific frequency range, has numerous potential engineering applications, such as acoustic filters and vibration isolation control. In order to describe the behavior of such materials, a novel integral micromorphic elastic continuum is introduced, and its ability to describe band gaps is studied in the one-dimensional setting. The nonlocal formulation is based on a modification of two terms in the expression for potential energy density. The corresponding dispersion equation is derived and converted to a dimensionless format, so that the effect of individual parameters can be described in the most efficient way. The results indicate that both suggested nonlocal modifications play an important role. The original local micromorphic model reproduces a band gap only in the special, somewhat artificial case, when the stiffness coefficient associated with the gradient of the micromorphic variable vanishes. On the other hand, the nonlocal formulation can provide band gaps even for nonzero values of this coefficient, provided that the penalty coefficient that enforces coupling between the micromorphic variable and nonlocal strain is sufficiently high and the micromorphic stiffness is sufficiently low.

Appendix A

The Maxwell–Rayleigh model (see, e.g., [36]) is physically motivated by the idea of an axially deforming continuous bar combined with continuously distributed local resonators. Each elementary bar segment of length $\,\text{ d }x$ and unit sectional area is connected by a spring of infinitesimal stiffness $K\,\text{ d }x$ to an infinitesimal mass $m\,\text{ d }x$, where K and m are given stiffness and mass densities. Denoting the mass density of the basic bar material as ${\hat{\rho }}$ and its elastic modulus as $\mu$, we can express the free energy density as

$$\begin{aligned} \psi = \frac{1}{2}\mu \varepsilon ^2 + \frac{1}{2}K\theta ^2 \end{aligned}$$

(42)

and the kinetic energy density as

$$\begin{aligned} \mathcal{E}_{kin} = \frac{1}{2}\rho _b{\dot{u}}^2 + \frac{1}{2}m({\dot{u}}+{\dot{\theta }})^2 \end{aligned}$$

(43)

where $\varepsilon =u'$ is the axial strain in the bar, u is the axial displacement in the bar and $\theta$ is the relative displacement of the local resonator (with respect to the bar). We adopt essentially the same notation as in [32], except that $\rho$ from [32] is replaced here by $\rho _b$ because it is not the total mass density of the system consisting of the bar and the added resonators, but only partial mass density of the bar, and k from [32] is replaced by K, to avoid confusion with the wave number.

The first terms on the right-hand sides of (42)–(43) are standard expressions valid for the elastic material of the bar under uniaxial stress, while the second terms represent the contributions of the added resonators. The relative displacement $\theta$ corresponds to the change of length of the spring and the sum $u+\theta$ is the absolute displacement of the added mass at the end of the resonator, which is why the kinetic energy depends on the square of the absolute velocity ${\dot{u}}+{\dot{\theta }}$.

Straightforward application of the Hamilton principle leads to the equations of motion

$$\begin{aligned} (\rho _b+m)\ddot{u} + m\ddot{\theta }= & {} \mu u'' \end{aligned}$$

(44)

$$\begin{aligned} m\ddot{u} + m\ddot{\theta }= & {} - K\theta \end{aligned}$$

(45)

and the harmonic ansatz

$$\begin{aligned} u(x,t)= & {} U\,\textrm{e}^{i(kx-\omega t)} \end{aligned}$$

(46)

$$\begin{aligned} \theta (x,t)= & {} \Theta \,\textrm{e}^{i(kx-\omega t)} \end{aligned}$$

(47)

yields the dispersion equation

$$\begin{aligned} \det \left( \begin{array}{cc} (\rho _b+m)\omega ^2-\mu k^2 &{} m\omega ^2 \\ m\omega ^2 &{} m\omega ^2-K \end{array}\right) =0 \end{aligned}$$

(48)

This can be rewritten as a quadratic equation in terms of the squared circular frequency,

$$\begin{aligned} \rho _b m\omega ^4 -[\mu mk^2+K(\rho _b+m)]\omega ^2 +\mu K k^2 =0 \end{aligned}$$

(49)

It would be easy to write the resulting dispersion relation analytically, and its structure would be similar to (24). However, formal equivalence with the micromorphic model analyzed in Sect. 3.1 is best seen if (49) is compared with the dispersion equation

$$\begin{aligned} \rho \eta \omega ^4-[\rho (k^2A+H)+\eta k^2(E+H)]\omega ^2 +k^4A(E+H)+k^2EH=0 \end{aligned}$$

(50)

obtained from (22) by setting $C^*(k)=E+H$ and $\alpha _0^*(k)=1$. Quadratic Eqs. (49) and (50) lead to identical dispersion diagrams if their coefficients satisfy the conditions

$$\begin{aligned} \frac{\mu mk^2+K(\rho _b+m)}{\rho _b m}= & {} \frac{\rho (k^2A+H)+\eta k^2(E+H)}{\rho \eta } \end{aligned}$$

(51)

$$\begin{aligned} \frac{\mu K k^2}{\rho _b m}= & {} \frac{k^4A(E+H)+k^2EH}{\rho \eta } \end{aligned}$$

(52)

To make these equalities valid for each $k^2\ge 0$, the model parameters must satisfy conditions

$$\begin{aligned} \frac{K(\rho _b+m)}{\rho _b m}= & {} \frac{ H}{\eta } \end{aligned}$$

(53)

$$\begin{aligned} \frac{\mu }{\rho _b }= & {} \frac{\rho A+\eta (E+H)}{\rho \eta } \end{aligned}$$

(54)

$$\begin{aligned} \frac{\mu K}{\rho _b m}= & {} \frac{EH}{\rho \eta } \end{aligned}$$

(55)

$$\begin{aligned} 0= & {} \frac{A(E+H)}{\rho \eta } \end{aligned}$$

(56)

On the left-hand sides, we have parameters of the Maxwell–Rayleigh model, and on the right-hand sides parameters of the local micromorphic model. Condition (56) can be satisfied only if $A=0$ or $E+H=0$, but the latter case would lead to $E=0$ and $H=0$ (since these parameters cannot be negative) and then it would be impossible to satisfy the other conditions for general values of the Maxwell–Rayleigh parameters. Therefore, we need to set $A=0$, and then condition (54) simplifies to

$$\begin{aligned} \frac{\mu }{\rho _b } = \frac{E+H}{\rho } \end{aligned}$$

(57)

We can also divide both sides of (55) by the corresponding sides of (53) (assuming that, except for A, no other parameters vanish), which leads to a simpler condition

$$\begin{aligned} \frac{\mu }{\rho _b+ m} = \frac{E}{\rho } \end{aligned}$$

(58)

We have ended up with three relatively simple conditions, namely (53), (57) and (58), which link four parameters of the Maxwell–Rayleigh model ($\rho _b$, m, $\mu$ and K) to four remaining parameters of the micromorphic model ($\rho$, $\eta$, E and H). If the conditions are satisfied, both models provide exactly the same dispersion diagram.

To make the parameter correspondence unique, it is natural to add the requirement that both models should give the same kinetic energy in the fundamental case of an undeformed body moving at constant speed, v. For the Maxwell–Rayleigh model, this occurs if $u(x,t)=vt$ and $\theta (x,t)=0$, which leads to kinetic energy density $(\rho _b+m)v^2/2$. For the micromorphic model, $u(x,t)=vt$ and $\chi (x,t)=0$ leads to kinetic energy density $\rho v^2/2$. Both expressions are the same if $\rho =\rho _b+m$, which is quite natural, since the total mass density of the Maxwell–Rayleigh model corresponds to the sum of the mass densities of the basic beam and of the added resonator mass. When $\rho$ is set equal to $\rho _b+m$, condition (58) reduces to $E=\mu$, which is also very natural, since $\mu$ is in fact the elastic modulus of the basic bar material. As a consequence, both models give the same free energy under uniform strain and static equilibrium. The remaining two conditions then permit identification of the non-standard parameters. If the Maxwell–Rayleigh model is considered as the primary one, parameters of the “equivalent” (in terms of the dispersion diagram) micromorphic model are

$$\begin{aligned} E= & {} \mu \end{aligned}$$

(59)

$$\begin{aligned} \rho= & {} \rho _b+m \end{aligned}$$

(60)

$$\begin{aligned} H= & {} \frac{\mu m}{\rho _b} \end{aligned}$$

(61)

$$\begin{aligned} \eta= & {} \frac{\mu m^2}{K(\rho _b+m)} \end{aligned}$$

(62)

$$\begin{aligned} A= & {} 0 \end{aligned}$$

(63)

The opposite parameter transformation is described by

$$\begin{aligned} \mu= & {} E \end{aligned}$$

(64)

$$\begin{aligned} \rho _b= & {} \frac{E\rho }{E+H} \end{aligned}$$

(65)

$$\begin{aligned} K= & {} \frac{EH^2\rho }{(E+H)^2\eta } \end{aligned}$$

(66)

$$\begin{aligned} m= & {} \frac{H\rho }{E+H} \end{aligned}$$

(67)

provided that $A=0$. A set of four positive parameters always transforms into a set of four positive parameters of the other model, and so there is no problem with parameter admissibility and the correspondence is bijective.

In summary, we have shown that the one-dimensional Maxwell–Rayleigh model in the form presented in [32] leads to the same dispersion diagrams as a special version of the local micromorphic model with parameter A set to zero, i.e., the only version of this particular model that leads to a band gap. Both models also share another feature: under static conditions, they reduce to the standard elastic continuum. In the one-dimensional static case, it is necessary to add external forces, otherwise one would get only the description of trivial equilibrium states at uniform strain. Therefore, we consider body forces b that are introduced via the energy of external forces characterized by density $\mathcal{E}_{ext}=-bu$ where u is the displacement. The Hamilton principle is then replaced by the Lagrange principle of minimum potential energy, which leads to the static equilibrium conditions

$$\begin{aligned} -(E+H)u''+H\chi '= & {} b \end{aligned}$$

(68)

$$\begin{aligned} -A\chi '' + H\chi -Hu'= & {} 0 \end{aligned}$$

(69)

for the micromorphic model and

$$\begin{aligned} -\mu u''= & {} b \end{aligned}$$

(70)

$$\begin{aligned} K\theta= & {} 0 \end{aligned}$$

(71)

for the Maxwell–Rayleigh model. It is immediately clear that (71) leads to $\theta =0$ and that (70) is the standard static equilibrium equation for a one-dimensional linear elastic continuum. This means that the enrichment will not be activated in statics, and the model cannot capture size effects related to high strain gradients. For the micromorphic model, Eqs. (68)–(69) are more complicated, but when parameter A is set to zero, Eq. (69) yields $\chi =u'$, and Eq. (68) then reduces to $-Eu''=b$, i.e., again to the standard equilibrium equation, with no enrichment.