# The Bloch-Floquet method and solution based on Fourier series

Owing to the Bloch-Floquet method, a solution to the wave equation (3.1) is searched in the following form

where F(x) is periodic functions of co-ordinates, F(x) = F(x + 1,>), 1,, = /?i 1| + *p _{2}*2.

*P,P*

*2*

*=*0,±1,±2,li, Ь are translations vectors of the square lattice.

We present the function F(x) and properties of components G(x), p(x) in the form of Fourier series

where the coefficients б„,„_{2}, C„,„_{2} are defined in the following way

and the operator *(-)dS* denotes integration along the cell area *do, dS = dxdx**2**, So = 1*^{2} stands for the cell area.

Substituting relations (3.23), (3.24) into equation (3.1) and comparing the coefficients standing by the terms exp [/2л7^{_|} *{jx* +./2*2)]» *j,ji* = 0,±1,±2,..., we obtain the infinite system of linear algebraic equations for unknowns Л„_{|/|2}

The condition for existence of solution to the system (3.26) is defined by equating to zero the determinant of the matrix composed of their coefficients. It gives the dispersive relation for *со* and *p.* It should be emphasized that the given method does not employ boundary conditions (3.4) in the explicit form. Condition of ideal contact between fibers and matrix are “hidden” in equation (3.1) and in development (3.24) for F(x), where the fields of displacements and stresses are implicitly treated as continuous.

In the case of the numerical studies reported below, the dispersive relations are computed on a basis of truncation of system (3.26) assuming — y_{max} < *j,j _{2}* < Ушах- The number of employed equations is equal to (2j

_{max}+ l)

^{2}. The introduced truncation has physical interpretation: we neglect high frequencies.

In order to illustrate the effect of wave filtering, let us split the problem (3.23) into real *p _{R}* and imaginary part

*p,*of the vector

*p = p*

_{K}+ ip_{t}The imaginary part *ц/ = ц,* of the wave number presents the attenuation coefficient. The values *ц/* = 0 correspond to pass band, whereas *ц/ Ф* 0 correspond to stop band. Boundaries of the pass and stop bands are defined through condition

*il* = *nnjJ*sin^{4}^ + cos^{4}0 , *n* = 1,2,3,.... The corresponding length of waves are defined by the formula *L* = (2//и *)J*sin^{4} +cos^{4}ф.

# Numerical results

In order to compare solutions obtained with a help of the Fourier series with result obtained by other authors, we consider the non-homogeneous material composed of the matrix with properties GO = 1, pO = 1 and with voids, G<^{2}) = 0, p(^{2}) = 0, *A/L* = 0.4, c<^{2}) « 0.503.

**Figure 3.5 **Dispersive curves of composite with voids.

Dispersive curves are shown in Fig. 3.5. The dashed lines correspond to computations for ,/'_{max} = 1, whereas the solid lines correspond to y'_{max} = 2; circles refer to the results reported in work [376] and obtained by the Rayleigh method. The dispersive diagram is composed of two parts separated by the vertical dashed line. The first part corresponds to the orthogonal *(ф =* 0), and left part corresponds to the diagonal *(ф = n/4*) direction of the wave propagation. In the quasi-homogenous case (0) 0)

the solution is isotropic and does not depend on the angle *ф.* However, with increase of the frequency *со* the composite exhibits anisotropic properties. Brown color refers to the full stop band, where the signal propagation is not possible in any of directions. Increase of /_{max} increases accuracy of the numerical results. As an example of the low-contrast composite we consider material with aluminium matrix (G^^{1}) = 27.9

**Figure 3.6 **Dispersive curves of the composite “nickel-aluminium”.

GPa, = 2700 kg/m^{3}) and nickel fibers (G^{(2}> = 75.4 GPa, p<^{2}> = 8936 kg/m *c^ =* 0.35). The related dispersive curves are shown in Fig. 3.6.

The results obtained for *j _{mdX} =* 1 (dashed curves) and for

*j*

_{max}= 2 (solid curves) are close to each other which confirm the fast convergence of the solution. As it follows from computation of the acoustic branch of spectrum (Fig. 3.7), the homogenization method allows to account of the dispersion effect, though the good accuracy is achieved only on the interval of low frequencies.

**Figure **3.7 Acoustic branch of the composite “nickel-aluminium”, * =* 0.

In the case of high-contrast composite (epoxy matrix with G*^{1}* = 1.53 GPa, pO = 1250 kg/m^{3} and carbon fibers with G*^{2}' = 86 GPa, p*W =* 1800 kg/m^{3},

**Figure 3.8 **Acoustic branch of the carbon-epoxide plastic,

*c™* = 0.5), the solution based on the Fourier series converges more slowly (Fig. 3.8). The homogenization method yields qualitatively correct results up to the first

stop band (for *Цц1 = л/J*sin^{4}0 +cos^{4}(f> we get *v _{g}* « 0).