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Influence of initial stress and gravity on refraction and reflection of SV wave at interface between two viscoelastic liquid under three thermoelastic theories

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Abstract

This approach is to examine the impact of magnetic field, viscosity, gravity and initial stress on SV wave while traveling through the interface of two visco-thermoelastic liquid layers. The basic equations in context of three theories have been discussed to drive results for refracted thermal and P waves and reflected thermal, SV and P waves. After using the boundary conditions the amplitude ratios have been computed in matrix form.

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Abbreviations

P :

Initial stress

\(\overrightarrow {\acute\omega }\) :

Local rotation

\({\acute{s}}_{11}\) :

Principal or incremental stress component along x axis

\({\acute{s}}_{22}\) :

Principal or incremental stress components along y axis

\({\acute{s}}_{21}\) :

Shear stress component

S 11 :

Normal initial stress component along x axis

S 22 :

Normal initial stress component along y axis

ρ :

Density of medium

u :

Component of displacement along x axis

v :

Component of displacement along y axis

F 1 :

Lorent’z force along x axis

F 2 :

Lorent’z force along y axis

 \(\gimel\), μ :

Lame’s constant

d xy :

Shear strain component

d xx :

Principal strain component

α t :

Coefficient of linear thermal expansion

K :

Bulk modulus

T :

Absolute temperature

τ 1 :

Relaxation time

c e :

Specific heat per unit mass

ϑ :

Thermal conductivity

\({\mathbf{\mathcal{B}}}\) :

Electric intensity vector

°:

Perturbed magnetic field vector

\(\varvec{H}\) :

Magnetic field vector

\(\varvec{J}\) :

Electric current density vector

\(\in_{0}\) :

Electric permeability

μe :

Magnetic permeability

R H :

Magnetic pressure number

c 1 :

Elastic wave velocity

c 2 :

Rotational wave velocity

ζ :

Initial stress parameter

ω :

Circular frequency

k :

Wave number

ɛ T :

Thermoelastic coupling constant

τ θ :

Phase lag of gradient of temperature

d :

Strain component

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Authors and Affiliations

  1. Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan

    A. A. Khan & A. Afzal

Authors
  1. A. A. Khan
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  2. A. Afzal
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Correspondence to A. A. Khan.

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Technical Editor: Cezar Negrao.

Appendix

Appendix

$$m_{1} = - \frac{A}{4} - \frac{1}{2}\sqrt p - \frac{1}{2}\sqrt {\left( {q - z} \right)}$$
$$m_{2} = - \frac{A}{4} - \frac{1}{2}\sqrt p + \frac{1}{2}\sqrt {\left( {q - z} \right)}$$
$$m_{3} = - \frac{A}{4} + \frac{1}{2}\sqrt p - \frac{1}{2}\sqrt {\left( {q - z} \right)}$$
$$m_{4} = - \frac{A}{4} + \frac{1}{2}\sqrt p + \frac{1}{2}\sqrt {\left( {q - z} \right)}$$
$$m_{5} = \frac{1}{{2c_{2}^{2} }}\left( {g - \sqrt {\left( g \right)^{2} - 4k^{2} c^{2} \alpha c_{2}^{2} + 4k^{2} c_{2}^{4} } } \right)$$
$$m_{6} = \frac{1}{{2c_{2}^{2} }}\left( {g + \sqrt {\left( g \right)^{2} - 4k^{2} c^{2} \alpha c_{2}^{2} + 4k^{2} c_{2}^{4} } } \right)$$
$$m_{7} = \frac{2}{3}k \eta \omega$$
$$p = \frac{{A^{2} }}{4} - \frac{2B}{3} + \frac{{2^{{\frac{1}{3}}} \left( {B^{2} + 12D - 3AC} \right)}}{3h} + \frac{1}{{3*2^{{\frac{1}{3}}} }}h$$
$$q = \frac{{A^{2} }}{2} - \frac{4B}{3} - \frac{{2^{{\frac{1}{3}}} \left( {B^{2} + 12D - 3AC} \right)}}{3h} - \frac{1}{{3*2^{{\frac{1}{3}}} }}h$$
$$h = \left( {O + \sqrt {\left( { - 4\left( {B^{2} + 12D - 3AC} \right)^{3} + \left( O \right)^{2} } \right)} } \right)^{{\frac{1}{3}}}$$
$$O = 2B^{3} + 27A^{2} D - 72BD - 9ABC + 27C^{2}$$
$$l_{1} = K\left( { - k^{2} + m_{2}^{2} } \right) - \frac{2}{3}\eta k^{2} \omega - \frac{4}{3}\eta m_{2}^{2} i\omega - \gamma b_{2} - \tau_{1} i\omega b_{2}$$
$$l_{2} = K\left( { - k^{2} + m_{4}^{2} } \right) - \frac{2}{3}\eta k^{2} \omega - \frac{4}{3}\eta m_{4}^{2} i\omega - \gamma b_{4} - \tau_{1} i\omega b_{4}$$
$$l_{3} = - \frac{2}{3}k \eta \omega$$
$$b_{1} = \frac{\rho }{\gamma \tau }\left( { - c_{1}^{2} k^{2} \left( {1 + R_{H} + \zeta - \frac{i4\eta \omega }{3}} \right) + m_{1}^{2} c_{1}^{2} \left( {1 + R_{H} + \zeta - \frac{i4\eta \omega }{3} - gm_{1} } \right)} \right)$$
$$b_{2} = \frac{\rho }{\gamma \tau }\left( { - c_{1}^{2} k^{2} \left( {1 + R_{H} + \zeta - \frac{i4\eta \omega }{3}} \right) + m_{2}^{2} c_{1}^{2} \left( {1 + R_{H} + \zeta - \frac{i4\eta \omega }{3} - gm_{2} } \right)} \right)$$
$$b_{3} = \frac{\rho }{\gamma \tau }\left( { - c_{1}^{2} k^{2} \left( {1 + R_{H} + \zeta - \frac{i4\eta \omega }{3}} \right) + m_{3}^{2} c_{1}^{2} \left( {1 + R_{H} + \zeta - \frac{i4\eta \omega }{3} - gm_{3} } \right)} \right)$$
$$b_{4} = \frac{\rho }{\gamma \tau }\left( { - c_{1}^{2} k^{2} \left( {1 + R_{H} + \zeta - \frac{i4\eta \omega }{3}} \right) + m_{4}^{2} c_{1}^{2} \left( {1 + R_{H} + \zeta - \frac{i4\eta \omega }{3} - gm_{4} } \right)} \right)$$

All variables for second medium are same with dashes.

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Khan, A.A., Afzal, A. Influence of initial stress and gravity on refraction and reflection of SV wave at interface between two viscoelastic liquid under three thermoelastic theories. J Braz. Soc. Mech. Sci. Eng. 40, 208 (2018). https://doi.org/10.1007/s40430-018-1103-y

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