Wave propagation in phononic materials based on the reduced micromorphic model by one-sided Fourier transform

A one-dimensional problem of wave propagation in phononic materials is solved under the reduced micromorphic model introduced recently. An efficient technique is used for the solution, based on one-sided Fourier transform. This allows obtaining an exact solution in closed form, which can be utilized to check approximate solutions obtained by other methods. The results are confirmed numerically by the method of finite differences. They illustrate the existence of frequency band gaps.


Introduction
Materials with periodic structures, i.e., phononic materials, are obtained by assembling in a suitable manner individual element with particular shape and size, in such a way so as to exhibit targeted mechanical properties. Such materials have witnessed increasing interest and attracted a great attention in the past two decades due to their ability to exhibit phenomena which are not found in usual materials, e.g., inhibition of certain frequencies under wave propagation. They are presently an important component in science and technology, having extensive applications as smart materials in intelligent microstructures in acoustic and vibration engineering. Some of the potential applications of phononic advanced materials include the design and construction of acoustic filters and transducers and advanced materials for noise control. An overview of phononic advanced materials is found in [1] [1][2][3].
Modelling of phononic materials requires consideration of media with microstructures. These are naturally multiscale materials, where many physical phenomena would be exhibited at different space and time scales. This is in line with the concept of the micromorphic continuum mechanics. In the micromorphic mechanics, a continuum has an inner structure, which has its own state variables.
Many researchers have paid attention to studying the new phenomena results from the development of the new models in continuum mechanics. The new models seem to be more complicated due to the complexity of the structure of the materials, increasing the degree of freedom and increasing the material parameters. So, some authors strove to introduce precisely description model to capture the new phenomena at the nano-scale measure. The most popular model is the micromorphic model introduced by Eringen [4,5]. The original micromorphic theory [6] represents the dynamic balance of elastic materials with 12 equations of motion that describe 3-displacements, 6-strains, and 3-rotations of the material's microstructure. The constitutive equations of isotropic-linear elastic materials depend on 18 material coefficients in the context of the original micromorphic model. The identification of these material coefficients for various material was a great challenge over the past decades.
Recently, micromorphic models have shown great success in modeling the mechanics of advanced materials, e.g., phononic and metamaterials [7][8][9][10][11]. For instance, the band gap structures of acoustic metamaterials have been developed based on a relaxed linear elastic micromorphic model [12][13][14]. It was revealed that acoustic metamaterials can stop or attenuate the propagation of waves at certain frequency domains [15][16][17]. Furthermore, Chen and Wang [18] discussed the size-effect on the band structures of nanoscale phononic crystals. The transfer-matrix method which used in optics is developed to compute the band structures of a nanoscale layered phononic crystal based on the nonlocal elastic continuum theory. Wu et al. [19] used the finite element method and the transfer-matrix method to study the propagation of elastic waves in one-dimensional phononic crystals with functionally graded materials. Yan et al. [20] studied time harmonic wave propagation in nanoscale periodic layered piezoelectric materials. Recently, alternative versions of the original micromorphic theory have been developed, e.g., the relaxed micromorphic model and the reduced micromorphic model [21]. Muhammad et al. [22] study the transition and topological interface modes with topological phases for 1D phononic crystals consisting of circular aluminum beams. The same authors [23] study here 1D topological phononic crystals with interface states produced by an exchange of wave mode polarization and geometric phases, using the spectral element method a e-mail: amrramadaneg@yahoo.com (corresponding author) b e-mail: moustafa_aboudina@hotmail.com c e-mail: afghaleb@gmail.com with Timoshenko beam model for flexural wave propagation. Recent work in this field study different aspects of wave propagation in media with periodic microstructures [24][25][26][27].
The reduced micromorphic model was developed by eliminating redundant microstructural degrees of freedom. The reduced micromorphic model depends on eight material coefficients only. These material coefficients were related to the material microstructure [28].
In this paper, a first attempt is undertaken to solve a boundary-value problem based on a new reduced micromorphic model suggested by Shaat [21], in which the author demonstrates the effectiveness of his reduced model and its ability to account for many features of wave propagation and bandgap in phononic materials. An efficient technique is presented, based on the "one-sided Fourier transform", to solve a problem of wave propagation in a slab of phononic material to describe the effect of microstructure on wave propagation in phononic materials. This technique was used earlier [29,30] in problems of elasticity and thermoelasticity for rectangular domains. It allows to obtain an analytic solution in closed form, in contrast to other methods, for example the Laplace transform which needs numerical inversion in most cases of interest. This exact solution can be used to check the validity of approximate solutions obtained by other methods. Moreover, the method can deal with cases where the behavior of the solution for large times is not known a priori, and with arbitrary initial and boundary conditions, a case that finds difficulty when using Fourier cosine or sine transforms, and spectral methods. The obtained numerical results reveal the existence of frequency band gaps as expected. The obtained results are confirmed numerically by using the finite difference method.
No attempt is undertaken to compare the present results with those obtained by other methods.

Reduced micromorphic model: review
In the context of the reduced micromorphic model, the deformation of material is describable using the following kinematical variables [21]: where ε i j are the components of the infinitesimal strain tensor, and s i j are the components of a microstrain tensor that represent the deformation of the material's microstructure, γ i j are the components of the coupling tensor that takes into account the difference between the microstrain s i j and the macrostrain ε i j fields, χ i jk are the components of the gradient of the microstrain tensor s i j , which is symmetric over the last two indices. Tensors τ i j and m i jk are introduced to capture the effects of microstrain on the macro-scale deformation of multiscale materials. According to the reduced micromorphic model, the constitutive equations have the form: and the equations of motion are: with the natural boundary conditions: Here λ m and μ m are microscopic Lamé moduli representing the stiffness of the material's microstructure, e.g., grain or unit cell [21], λ and μ are the elastic moduli of a material confined between two-unit cells, λ c and μ c are two elastic moduli that adjust the coupling between the microscopic stiffness and the macroscopic stiffness of the material, 1 and 2 are two length scales, ρ is the mass density of the macro-scale material, ρ m is the mass density of the material particle, and J is a micro-inertia, f i and H jk are the body forces and body higher-order-moments, respectively, t i and m jk stand for the external surface force and couple applied to the medium. The governing equations for the displacement and the microstrain fields are: the non-vanishing components of kinematic relations, Eqs. (1)-(3) are: where for brevity, we replace u 1 by u and s 11 by s. In view of Eq. (9), Eqs. (5)-(7) take the form: and the field Eqs. (8) yield: showing the coupling between the macroscopic mechanical displacement and the microstrain: (i) effect of microstrain on elastic wave propagation through the coefficient β 2 , which seems to be similar to the effect of temperature in classical thermoelasticity; (ii) effect of macroscopic displacement on microstrain through the coefficient β 4 . The coefficients appearing in Eqs. (11) are given in terms of the different macroscopic and microstructure material parameters by the expressions: It is easy to verify that the system of partial differential Eqs. (11) has two sets of characteristic curves dx ± √ β 1 dt 0 corresponding to the coupled elastic wave travelling with speed √ β 1 , in addition to two degenerate characteristics dx 0. Details will be omitted for the sake of brevity. On the characteristics with positive slope the following condition holds: which may be useful when integrating by the method of characteristics. The problem will be solved in a thick slab 0 ≤ x ≤ , and t > 0, occupied by a material with specified material parameters, under the following initial and boundary conditions where u 0 and s 0 are the amplitudes of the applied waves, a and b are parameters to be chosen during the numerical calculations. The exponential introduced in the boundary conditions is for convenience purposes only and can be removed altogether if sufficient conditions for the existence of Fourier transform are secured.
In the present paper, we use a different technique based on the one-sided Fourier transform to find the solution for the system of partial differential Eqs. (11), with initial and boundary conditions (13). It has shown good efficiency in solving rather general types of initial-boundary-value problems of systems of PDE's, although not quite popular in the literature (see [45,46], for example). This transform has been successfully used in the study of analytic signals [47] from which negative frequencies are filtered-out. It provides a natural way for investigating frequency band gaps as will become clear from the upcoming results. Through a change of the unknown function, the method has the flexibility to deal with cases where the behavior of the solution for large times is not known a priori, and with arbitrary initial and boundary conditions. Again, it can deal with a multitude of boundary conditions. Let us define the integral transform of a function f (x, t) as follows: It is to be noted that obtaining the complex conjugate of function F amounts simply to changing the sign of ω in the above definition.
Substitute Euler's formula e iωt cos(ωt) + i sin(ωt) (15) into the definition (14) to get: Thus, the real and the imaginary parts of the function F(x, ω) are: The inverse expressions for Eqs. (17)(18) are given by: The following formula for the transform of the second time derivative is easily derived, and will be used in the sequel: Apply the suggested technique to the system of PDE's (11) to obtain the system of ordinary differential equations for the transformed functions U and S: Here a is a constant to control the attenuation of the incident wave, b is a constant to determine the location of the hump on the curve below which represents the integral in Eqs. (23) for concrete values of parameters a 0.1 and b 1.0 (Fig. 2).
For bounded solutions, α(ω) must be real. The expressions for λ(ω) depend on the choice of the material parameters for the considered material. The two functions U (x, ω) and S (x, ω) are obtained as: and The involved coefficients in the above expressions are not all independent. In fact, the functions U (x, ω) and S(x, ω) in Eqs. (27) and (28) are solutions of the field Eq. (22) in the ω− domain.
Apply boundary conditions (23) to obtain the solution of system (22) in the form: and S(x, ω) S(0, ω)(cos(α(ω)x) − cot(α(ω) ) sin(α(ω)x)), The functions in (29) and (30) have simple poles at the zeros of the equation sin[α(ω) ] 0. We have represented on Fig. 3 the function U (0.01, ω)/U (0, ω) to show the type of behavior of the kernels in the integrals yielding the solution. In particular, there is a denumerable number of frequencies at which this function does not exist. Now, it is required to invert the two functions U (x, ω) and S(x, ω) using Eqs. (19) or (20) as: or else Both choices are equally good. We shall opt for the former choice for definiteness. The integrals in Eqs. (31)(32) or (33)(34) are evaluated numerically.

Finite difference method
For the numerical usage, we consider a domain in (x, t) plane and discretized by the steps x h and t k. The values of the unknown functions at any point (x, t) (ih, jk) are denoted by f x i , t j f i, j . The following expressions are the finite-difference representations of the various partial derivatives [41] and [42]: Substitute relations (40) into Eq. (11) to get: with 4h β 4 , the initial and boundary conditions (13), they take the form: where N and M are the number of nodes along x and t axes respectively. The applied boundary conditions are discretized as follows: and g t j s 0 1 − cos bt j e −at j , 0 ≤ t ≤ 2π, 0, t > 2π.

Numerical results and discussion
Here we consider a concrete phononic material with given values of the physical parameters. The solution will be obtained by the above-mentioned method under the given boundary conditions. Comparison between the present results and those obtained numerically by finite-differences will be exposed. Other types of boundary conditions may be treated equally well. The space and time steps for the numerical scheme are taken as h 1 and k 0.005.

Worked application
The material constants for the matrix and inclusion are The parameters used for the reduced micromorphic model are listed on the following table: Phononic material constants We have calculated these two functions for all considered times and at many locations. Figures 5 and 6 for the wave propagation of displacement and microstrain calculated numerically at the location x 2 have shown good agreement with the analytical solution based on the one-sided Fourier transform, thus confirming the validity of the obtained results. One notices a decrease in amplitude in both the displacement and the microstrain as the main hump reaches successive locations at increasing distances from the boundary, as compared to the imposed boundary functions. Curve fitting is used to show that each of these two amplitudes obeys an exponential decrease law with the spatial coordinate x. Comparison with the case of classical elasticity allows reaching a conclusion that such a decrease in displacement is conditioned by the damping effect of microstructure. Figures 7 and 8 show the distributions of stresses as functions of time at the location x 2. It appears that these two functions, as well as the displacement and microstrain, are characterized by an "overdamping" which takes the form of damped oscillations about the zero value for large times.
Figures 9, 10, 11 and 12 are 3-D plots for displacement, microstrain, stress tensor and couple stress tensor as functions of (x, t) illustrating the wave propagation of the boundary regime.
The calculations have shown good stability of the numerical scheme with respect to time. Ten thousand steps of time were used during the calculations.

Band gap phenomenon
The analytical solution by one-sided Fourier transform given above, through the transformed functions, has enabled us to reveal the existence of band gaps which forbid wave propagation for certain frequency bands. In fact, calculations show that the condition of non-negativity of the function α(ω) guarantying the existence of bounded solutions to the problem is satisfied everywhere on the real axis of frequencies, except when 0.053 ≤ ω ≤ 0.057. When this constraint goes into the right-hand sides of Eqs. (29) and (30), it produces a denumerable set of band gaps. This is illustrated on Fig. 13. different methods were used to solve the problem: An analytical method based on the use of one-sided Fourier transform, and a finite-difference numerical scheme. Both methods yielded identical results. The one-sided Fourier transform method has provided the opportunity to obtain an exact solution in closed form, to be used to verify the efficiency of approximate solutions obtained by other methods. The method allows more flexibility in dealing with different initial and boundary conditions, and can be used, in contrast to other methods, in case when the behavior of the unknown function at large values of time is unknown a priori. The transformed functions for the displacement and the microstrain as obtained from the analytical solution were then used to put in evidence the existence of a frequency band gap, whose location is bound to change if other boundary conditions are used. We have given plots to show how the wave energy is distributed among the frequencies at different locations. Although the elastic wave bandgap is usually limited to the high frequency range [48], our work does not prevent the existence of such bandgaps at lower frequencies. It is believed that the used type of boundary conditions will be decisive in this respect. Future work in progress concentrates of the extension of the present model to two-dimensional wave propagation problems.
Funding Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).
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