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)

Field equation

(2)

For micropolar

(3)

**Wave Propagation in Micropolar Cubic Material with Voids**

The modified void equation

(4)

### Solution of the problem

Where we consider a plane having the displacement vector and micro-rotation vector

(5)

For i=1 equation (1) give us

(6)

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

(7)

For the micropolar equation using (3)

(8)

For void equation, we used (4)

(9)

Where

, ,

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

(10)

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

where . 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

, and at

For in first boundary condition becomes

(11)

For in first boundary condition we have

(12)

For micropolar equation we used second boundary condition

(13)

for void equation we have,

(14)

Let the components of displacement and microrotation vectors are following.

(15)

(16)

(17)

(18)

Where and

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

where

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

(19)

(20)

(21)

(22)

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

##### Using Snell’s law in equation (19)

Let

Now we have

(23)

##### Similarly using Snell’s law in equation (20)

Let

Now we have

(24)

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

Material | Stiffness | Density | Micropolar constants | Voids Parameters | |||||||

J_{ } | |||||||||||

Crystal | 13.97 | 13.75 | 3.2 | 2.2 | 7.87 | 0.056 | 0.047 | 0.019 | 0.01 | 1.28 | 0.1 |

**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 .