Reflection of Waves in Micropolar Cubic Medium with Voids

1. Introduction

2. Equations
i Cubic Matrix
ii Micropolar cubic
iii Voids

3. Waves Propagation in Micropolar Cubic Material with Voids
(1) Formulation of the problem
(2) Solution of the problem
(3) Practical application

4. Reflection of Waves in Micropolar Cubic Material with Voids
(a) Formulation of the problem
(b) Solution of the problem
(c) Graphs, Results and Conclusions

Abstract

We studied the reflection of elastic plane waves in a micropolar cubic medium containing voids. We found that four distinct wave types propagate through this medium, each traveling at a different speed. The speed of the waves and reflection coefficients depend upon the materials constant as well as on the angle of propagation. We conducted numerical studies on a specific model. We graphically illustrated how different amplitude ratios vary with the angle of incidence. Our findings can be used to reproduce previously published results. We explicitly calculated the amplitude ratios.

Keywords: reflection, micropolar, cubic, plane, voids.

Reflection of Waves in Micropolar Cubic Medium with Voids (Part 2)

Introduction

In cubic material three waves propagate in a given direction. Knott was first who introduce the concept of reflection and transmission of waves in isotropic material. After that different researchers did work on reflection and transmission of waves in isotropic material with magnetic, rotational, voids and thermal effect. It observe that reflection of waves depends upon the angle of incident of the wave. It also depend upon the boundary conditions.

In this work first we discuss the propagation of waves in micropolar cubic material with voids and in second part we discuss the reflection of waves in micropolar cubic material in the presence of voids effect. We obtain four waves.

Othman et al. used the (G-N) theory to study deformation of micropolar thermoelastic solid with porousity and they used normal mode to find different analytical expressions and temperature distribution. Rajneesh et al. determined the steady state response of a semi-infinite micropolar elastic medium with voids. They found the different analytical expression by using fourier transformation. Lakes determined the torsional and bending rigidity for rod shaped specimens of dense polyurethane foam and syntactic foam.

Reflection of Waves in Micropolar Cubic Medium with Voids

In this research, we have obtained four waves that are, one longitudinal, two transverse and a 4^th wave due to voids. Amplitude of these waves is calculated numerically and also numerical results has been shown graphically after choosing particular micropolar cubic material.

Formulation of the problem

Constitutive relations for a micropolar cubic medium with voids are given as:

(1) $\begin{equation*}\sigma_{ij} = A_{ijkm} \gamma_{km}\end{equation*}$

(2) $\begin{equation*}\mu_{ij} = B_{ijkm} \kappa_{km}\end{equation*}$

Where, $A_{ijkm}$ and $B_{ijkm}$ are $4^{th}$ rank tensor. For cubic material such that

(3) $\begin{equation*}A_{ijkm} = A_2 \delta_{ij} \delta_{km} + A_3 \delta_{ik} \delta_{jm} + A_4 \delta_{im} \delta_{jk} + A \delta_{ij,km}\end{equation*}$

(4) $\begin{equation*}B_{ijkm} = B_2 \delta_{ij} \delta_{km} + B_3 \delta_{ik} \delta_{jm} + B_4 \delta_{im} \delta_{jk} + B \delta_{ij,km}\end{equation*}$

The deformation and micro tensor is explicitly defined by:

(5) $\begin{equation*}\gamma_{ij} = u_{(j,i)} - \epsilon_{ijk} \psi_k\end{equation*}$

(6) $\begin{equation*}\kappa_{ij} = \psi_{(j,I)}\end{equation*}$

The balance laws are:

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

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

where $\mu_{ji}, \mu_j, \rho, \psi$ are couple stress, stress tensor, component of displacement vector, Bulk mass density and microrotation vector.

We mull over a homogeneous and cubic material of an unbounded dimension with a Cartesian coordinate system $(x_1, x_2, x_3)$ . We consider a change in plane due to stress parallel to the $x_1x_2$ plane having the displacement vector $\mu$ and $\psi$ micro-rotation vector.