Wave Propagation in Micropolar Cubic Material with Voids

Reflection of Waves in Micropolar Cubic Medium (Remaining Part)

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Reflection of Waves in Micropolar Cubic Medium with Voids (Previous Part)

Wave Propagation in Micropolar Cubic Material with Voids

Formulation of the problem

Equation of motion

(1)   \begin{equation*} \sigma_{ij,j} = \rho \ddot{u}_i \end{equation*}

Field equation

(2)   \begin{equation*}\sigma_{ij} = A_{ijkm} \gamma_{km} + \phi(\beta_1 \delta_{i1} \delta_{j1} + \beta_2 \delta_{i2} \delta_{j2})\end{equation*}

For micropolar

(3)   \begin{equation*}\mu_{(ji,j)} + \epsilon_{ijk} \sigma_{jk} = \rho J_{ij}(\ddot{\psi}_j)\end{equation*}

Wave Propagation in Micropolar Cubic Material with Voids

The modified void equation

(4)   \begin{equation*}\alpha \phi_{,ii} - \omega_0 \phi - \gamma \dot{\phi} - \beta u_{i,i} = \rho \kappa \ddot{\phi}\end{equation*}

Solution of the problem

Where we consider a x_1 x_2 plane having the displacement vector \mu_i=(\mu_1,\mu_2,0)  and micro-rotation vector \psi = (0,0,\psi_3)

(5)   \begin{equation*}(u_i, \psi_3, \varphi) = (A_d_i, kB, \varphi_0) e^{ik(x_1 p_1 + x_2 p_2 - vt)} \quad (i = 1, 2)\end{equation*}

For i=1 equation (1) give us

(6)   \begin{equation*}(D_1 - \rho v^2) A_d_1 k^2 + (A_5 p_1 p_2) A_d_2 k^2 - (iA_6 p_2) Bk - i(\beta_1 p_1) \varphi_0 k = 0\end{equation*}

 Similarly for i=2, equation (1) become,

(7)   \begin{equation*}(A_5 p_1 p_2) A_d_1 k^2 + (D_2 - \rho v^2) A_d_2 k^2 + (iA_6 p_1) Bk - i(\beta_2 p_2) \varphi_0 k = 0\end{equation*}

For the micropolar equation using (3)

(8)   \begin{equation*}(iA_6 p_2)A_{d_1} k - (iA_6 p_1)A_{d_2} k + [k^2 j(D_3-\rho v^2)+6A_6]B = 0\end{equation*}

For void equation, we used (4)

(9)   \begin{equation*}(i\beta p_1)A_{d_1} k + (i\beta p_2)A_{d_2} k + [k^2 \kappa(D_4-\rho v^2)+\omega_0-iv\gamma k] \varphi_0 = 0\end{equation*}

Where

A_6=A_3-A_4, A_5=A_2+A_3, A=A_1-A_2-A_3-A_4

For  non-trivial solution, equations  (6) to (9)  implies

(10)   \begin{equation*}\begin{vmatrix}(D_1-\rho v^2)k^2 & A_5 p_1 p_2 k^2 & -iA_6 p_2 k & -i(\beta_1 p_1)k \\A_5 p_1 p_2 k^2 & (D_2-\rho v^2)k^2 & iA_6 p_1 k & -i(\beta_2 p_2)k \\iA_6 p_2 k & -iA_6 p_1 k & k^2 j(D_3-\rho v^2)+6A_6 & 0 \\ip_1 \beta k & ip_2 \beta k & 0 & k^2 \kappa(D_4-\rho v^2)+\omega_0-iv\gamma k\end{vmatrix} = 0\end{equation*}

Solving this determinant, we have following fourth-order secular equation.

\xi^4 + K \xi^3 + L \xi^2 + M \xi + N = 0

where \xi = \rho v^2. Solving this determinant we have four roots. So there exist four waves.

Reflection of Waves in Micropolar Cubic Material with Voids

Since there exist four waves (QLD-wave, QCLD-wave, QCTM-wave and QTD-wave due to voids), against the incident  QLD-wave.

Formulation of the problem

Using free boundary conditions we have

\sigma_{i2}=0, m_{23}=0 and \phi_{,2}=0 at x_{2}=0

For i=1 in first boundary condition becomes

(11)   \begin{equation*}A_3 u_{(2,1)} + A_4 u_{1,2} - A_6 \psi_3 = 0\end{equation*}

For i=2 in first boundary condition we have

(12)   \begin{equation*}A_1 u_{(2,2)} + A_2 u_{1,1} + \beta_2 \varphi = 0\end{equation*}

 For micropolar equation we used second boundary condition

(13)   \begin{equation*}B_3 \psi_{3,2} = 0\end{equation*}

for void equation we have,

(14)   \begin{equation*}A_1 u_{2,22} + A_5 u_{1,12} + A_4 u_{2,11} - A_6 \psi_{3,1} - \rho \frac{\partial^2 u_2}{\partial t^2} = 0\end{equation*}

Let the components of displacement and microrotation vectors are following.

(15)   \begin{equation*}u_1^{(\alpha)} = R^{(\alpha)} d_1^{(\alpha)} e^{i\eta_{\alpha}}\end{equation*}


(16)   \begin{equation*}u_2^{(\alpha)} = G^{(\alpha)} R^{(\alpha)} d_1^{(\alpha)} e^{i\eta_{\alpha}}\end{equation*}


(17)   \begin{equation*}\psi_3^{(\alpha)} = ik_{\alpha} H^{(\alpha)} R^{(\alpha)} d_1^{(\alpha)} e^{i\eta_{\alpha}}\end{equation*}


(18)   \begin{equation*}\varphi^{(\alpha)} = \varphi_0^{(\alpha)} k_{\alpha} R^{(\alpha)} d_1^{(\alpha)} e^{i\eta_{\alpha}}\end{equation*}

Where \eta_{\alpha} = k_{(\alpha)} (x_1 p_1^{(\alpha)} + x_2 p_2^{(\alpha)} - v_{\alpha} t) and \alpha=0,1,2,3,4

Using equations (6)-(8) , we derived the following values:

    \[G^{(\alpha)} = \frac{-[p_1^{(\alpha)} (D_1^{(\alpha)} - \rho v_{\alpha}^2) + A_5 p_1^{(\alpha)} p_2^{(\alpha)2} - i(\beta_1 p_1^{(\alpha)2} + \beta_2 p_2^{(\alpha)2}) \varphi_0]}{[p_2^{(\alpha)} (D_2^{(\alpha)} - \rho v_{\alpha}^2) + A_5 p_1^{(\alpha)2} p_2^{(\alpha})]}\]

    \[H^{(\alpha)} = \frac{A_6 G^{(\alpha)} p_1^{(\alpha)} - A_6 p_2^{(\alpha)}}{jk_{\alpha}^2 (D_3^{(\alpha)} - \rho v_{\alpha}^2) + 6A_6}\]

where

    \[D_1^{(\alpha)} = A_1 (p_1^{(\alpha)})^2 + A_4 (p_2^{(\alpha)})^2\]


    \[D_2^{(\alpha)} = A_4 (p_1^{(\alpha)})^2 + A_1 (p_2^{(\alpha)})^2\]


    \[D_3^{(\alpha)} = \frac{B_4 (p_1^{(\alpha)})^2 + B_4 (p_2^{(\alpha)})^2}{j}\]


    \[D_4^{(\alpha)} = \frac{\alpha (p_1^{(\alpha)})^2 + \alpha (p_2^{(\alpha)})^2}{\kappa}\]

Now by using the components of displacement and microrotation vector in equations (11)-(14)

(19)   \begin{equation*}A_3 ik_{\alpha} G^{(\alpha)} R^{(\alpha)} d_1^{(\alpha)} p_1^{(\alpha)} e^{i\eta_{\alpha}} + A_4 ik_{\alpha} R^{(\alpha)} d_1^{(\alpha)} p_2^{(\alpha)} e^{i\eta_{\alpha}} - A_6 ik_{\alpha} H^{(\alpha)} R^{(\alpha)} d_1^{(\alpha)} e^{i\eta_{\alpha}} &= 0 \end{equation*}


(20)   \begin{equation*}A_1 ik_{\alpha} G^{(\alpha)} R^{(\alpha)} d_1^{(\alpha)} p_2^{(\alpha)} e^{i\eta_{\alpha}} + A_2 ik_{\alpha} R^{(\alpha)} d_1^{(\alpha)} p_1^{(\alpha)} e^{i\eta_{\alpha}} + \varphi_0^{(\alpha)} \beta_2 k_{\alpha} R^{(\alpha)} d_1^{(\alpha)} e^{i\eta_{\alpha}} &= 0 \end{equation*}


(21)   \begin{equation*}B_3 (ik_{\alpha})^2 R^{(\alpha)} H^{(\alpha)} d_1^{(\alpha)} p_2^{(\alpha)} e^{i\eta_{\alpha}} &= 0 \end{equation*}


(22)   \begin{equation*}(D_2 - \rho v^2) k_{\alpha}^2 G^{(\alpha)} R^{(\alpha)} d_1^{(\alpha)} e^{i\eta_{\alpha}} + A_5 k_{\alpha}^2 R^{(\alpha)} d_1^{(\alpha)} p_1^{(\alpha)} p_2^{(\alpha)} e^{i\eta_{\alpha}} - A_6 k_{\alpha}^2 H^{(\alpha)} R^{(\alpha)} d_1^{(\alpha)} p_1^{(\alpha)} e^{i\eta_{\alpha}} &= 0\end{equation*}

Now by using Snell’s law in the equations (19)-(22). Where Snell’s law is

    \[k_0 p^{(0)} = k_1 p^{(1)} = k_2 p^{(2)} = k_3 p^{(3)} = k_4 p^{(4)}\]


    \[k_0 v_0 = k_1 v_1 = k_2 v_2 = k_3 v_3 = k_4 v_4\]

Using Snell’s law in equation (19)

    \[\frac{[A_3 G^{(1)} p_1^{(1)} + A_4 p_2^{(1)} - A_6 H^{(1)}]k_1 R^{(1)} d_1^{(1)}}{[A_3 G^{(0)} p_1^{(0)} + A_4 p_2^{(0)} - A_6 H^{(0)}]k_0 R^{(0)} d_1^{(0)}} + \frac{[A_3 G^{(2)} p_1^{(2)} + A_4 p_2^{(2)} - A_6 H^{(2)}]k_2 R^{(2)} d_1^{(2)}}{[A_3 G^{(0)} p_1^{(0)} + A_4 p_2^{(0)} - A_6 H^{(0)}]k_0 R^{(0)} d_1^{(0)}} +\]

    \[\frac{[A_3 G^{(3)} p_1^{(3)} + A_4 p_2^{(3)} - A_6 H^{(3)}]k_3 R^{(3)} d_1^{(3)}}{[A_3 G^{(0)} p_1^{(0)} + A_4 p_2^{(0)} - A_6 H^{(0)}]k_0 R^{(0)} d_1^{(0)}} + \frac{[A_3 G^{(4)} p_1^{(4)} + A_4 p_2^{(4)} - A_6 H^{(4)}]k_4 R^{(4)} d_1^{(4)}}{[A_3 G^{(0)} p_1^{(0)} + A_4 p_2^{(0)} - A_6 H^{(0)}]k_0 R^{(0)} d_1^{(0)}}\]

    \[= -1\]

Let

    \[a_{11} = \frac{k_1 [A_3 G^{(1)} p_1^{(1)} + A_4 p_2^{(1)} - A_6 H^{(1)}] d_1^{(1)}}{k_0 [A_3 G^{(0)} p_1^{(0)} + A_4 p_2^{(0)} - A_6 H^{(0)}] d_1^{(0)}}\]


    \[a_{12} = \frac{k_2 [A_3 G^{(2)} p_1^{(2)} + A_4 p_2^{(2)} - A_6 H^{(2)}] d_1^{(2)}}{k_0 [A_3 G^{(0)} p_1^{(0)} + A_4 p_2^{(0)} - A_6 H^{(0)}] d_1^{(0)}}\]


    \[a_{13} = \frac{k_3 [A_3 G^{(3)} p_1^{(3)} + A_4 p_2^{(3)} - A_6 H^{(3)}] d_1^{(3)}}{k_0 [A_3 G^{(0)} p_1^{(0)} + A_4 p_2^{(0)} - A_6 H^{(0)}] d_1^{(0)}}\]


    \[a_{14} = \frac{k_4 [A_3 G^{(4)} p_1^{(4)} + A_4 p_2^{(4)} - A_6 H^{(4)}] d_1^{(4)}}{k_0 [A_3 G^{(0)} p_1^{(0)} + A_4 p_2^{(0)} - A_6 H^{(0)}] d_1^{(0)}}\]

Now we have

(23)   \begin{equation*}a_{11} \frac{R^{(1)}}{R^{(0)}} + a_{12} \frac{R^{(2)}}{R^{(0)}} + a_{13} \frac{R^{(3)}}{R^{(0)}} + a_{14} \frac{R^{(4)}}{R^{(0)}} = -1\end{equation*}

Similarly using Snell’s law in equation (20)

    \[\frac{[A_1 G^{(1)} p_2^{(1)} + A_2 p_1^{(1)} + (\beta_2^{(0)} \varphi_0^{(0)})/i]k_1 R^{(1)} d_1^{(1)}}{[A_1 G^{(0)} p_2^{(0)} + A_2 p_1^{(0)} + (\beta_2^{(0)} \varphi_0^{(0)})/i]k_0 R^{(0)} d_1^{(0)}} + \frac{[A_1 G^{(2)} p_2^{(2)} + A_2 p_1^{(2)} + (\beta_2^{(2)} \varphi_0^{(2)})/i]k_2 R^{(2)} d_1^{(2)}}{[A_1 G^{(0)} p_2^{(0)} + A_2 p_1^{(0)} + (\beta_2^{(0)} \varphi_0^{(0)})/i]k_0 R^{(0)} d_1^{(0)}} +\]

    \[\frac{[A_1 G^{(3)} p_2^{(3)} + A_2 p_1^{(3)} + (\beta_2^{(3)} \varphi_0^{(3)})/i]k_3 R^{(3)} d_1^{(3)}}{[A_1 G^{(0)} p_2^{(0)} + A_2 p_1^{(0)} + (\beta_2^{(0)} \varphi_0^{(0)})/i]k_0 R^{(0)} d_1^{(0)}} + \frac{[A_1 G^{(4)} p_2^{(4)} + A_2 p_1^{(4)} + (\beta_2^{(4)} \varphi_0^{(4)})/i]k_4 R^{(4)} d_1^{(4)}}{[A_1 G^{(0)} p_2^{(0)} + A_2 p_1^{(0)} + (\beta_2^{(0)} \varphi_0^{(0)})/i]k_0 R^{(0)} d_1^{(0)}}\]

    \[= -1\]

Let

    \[a_{21} = \frac{k_1 [A_3 G^{(1)} p_1^{(1)} + A_4 p_2^{(1)} - A_6 H^{(1)}] d_1^{(1)}}{k_0 [A_3 G^{(0)} p_1^{(0)} + A_4 p_2^{(0)} - A_6 H^{(0)}] d_1^{(0)}}\]


    \[a_{22} = \frac{k_2 [A_3 G^{(2)} p_1^{(2)} + A_4 p_2^{(2)} - A_6 H^{(2)}] d_1^{(2)}}{k_0 [A_3 G^{(0)} p_1^{(0)} + A_4 p_2^{(0)} - A_6 H^{(0)}] d_1^{(0)}}\]


    \[a_{23} = \frac{k_3 [A_3 G^{(3)} p_1^{(3)} + A_4 p_2^{(3)} - A_6 H^{(3)}] d_1^{(3)}}{k_0 [A_3 G^{(0)} p_1^{(0)} + A_4 p_2^{(0)} - A_6 H^{(0)}] d_1^{(0)}}\]


    \[a_{24} = \frac{k_4 [A_3 G^{(4)} p_1^{(4)} + A_4 p_2^{(4)} - A_6 H^{(4)}] d_1^{(4)}}{k_0 [A_3 G^{(0)} p_1^{(0)} + A_4 p_2^{(0)} - A_6 H^{(0)}] d_1^{(0)}}\]

Now we have

(24)   \begin{equation*}a_{21} \frac{R^{(1)}}{R^{(0)}} + a_{22} \frac{R^{(2)}}{R^{(0)}} + a_{23} \frac{R^{(3)}}{R^{(0)}} + a_{24} \frac{R^{(4)}}{R^{(0)}} = -1\end{equation*}

To study the effect of  voids on wave propagation, we use a micropolar cubic material (crystal)

MaterialStiffnessDensity Micropolar constantsVoids Parameters
A_1A_2A_3A_4B_3B_4    J\alpha\omega\gamma
Crystal13.9713.753.22.27.870.0560.0470.0190.011.280.1

Conclusion

As with the increase in the angle of propagation, the amplitude ratio of the waves are also increasing, so reflection of the waves is just increasing with the presence of voids effect in micropolar cubic material .

Reflection of Waves in Micropolar Cubic Medium with Voids

Reflection of Waves in Micropolar Cubic Medium with Voids

Wave Propagation in Micropolar Cubic Material with Voids

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