Plane waves reflection in micropolar transversely isotropic generalized thermoelastic half-space

Purpose

The purpose of this paper is to study the reflection of plane periodic wave’s incident on the surface of generalized thermoelastic micropolar transversely isotropic medium.

Methods

Plane wave propagation is studied to calculate complex velocities of the four waves from the complex roots of a quartic equation. The complex velocity of the attenuated wave in the medium is resolved to calculate its propagation (phase) velocity and quality factor of attenuation. Reflection of waves at the free surface of micropolar transversely isotropic generalized thermoelastic elastic half-space has been discussed.

Results

Numerical examples calculate the amplitude ratios of reflected waves at the free surface of the micropolar transversely isotropic generalized thermoelastic elastic half-space to evince the effects of anisotropy and theories of generalized thermoelasticity.

Conclusions

It is concluded from the present study that there are four quasi wave propagates in of generalized thermoelastic micropolar transversely isotropic medium viz., quasi-longitudinal displacement (qLD) wave, quasi-transverse displacement (qTD) wave, quasi- transverse microrotational (qTM) wave and quasi thermal (qT) waves.

Depending upon the mechanical properties, the material of the earth has been classified as elastic, viscoelastic, sandy, granular, microstructure, etc. Some parts of the earth may supposed to be composed of material possessing micropolar/granular structure instead of continuous elastic material.

To explain the fundamental departure of microcontinuum theories from the classical continuum theory, the former is a continuum model embedded with microstructures to describe the microscopic motion or a non-local model to describe the long-range material interaction. This extends the application of the continuum model to microscopic space and short-time scale. Micromorphic theory [1,2] treats a material body as a continuous collection of a large number of deformable particles, with each particle possessing finite size and inner structure. Using assumptions such as infinitesimal deformation and slow motion, micromorphic theory can be reduced to Mindlin’s microstructure theory (1964). When the microstructure of the material is considered rigid, it becomes the micropolar theory [3].

Eringen’s micropolar theory is more appropriate for geological materials like rocks and soil since this theory takes into account the intrinsic rotation and predicts the behavior of material with inner structure. The linear theory of micropolar thermoelasticity was developed by [4] and [5] to include thermal effects and is known as micropolar coupled thermoelasticity.

Inspite of these studies, no attempt has been made to study the reflection of waves in transversely isotropic micropolar generalized thermoelastic medium. Thus, the present study of the effect of anisotropy in a reflection of waves at the free surface of transversely isotropic micropolar generalized thermoelastic medium has its due importance in engineering and geophysical problems where the situation so demands.

We analyze the reflection of waves in transversely isotropic micropolar generalized thermoelastic medium. This study has many applications in various field of science and technology, namely, atomic physics, industrial engineering, thermal power plants, submarine structures, pressure vessel, aerospace, chemical pipes and metallurgy.

The basic equations in dynamic theory of the plain strain of a homogeneous and micropolar transversely isotropic generalized thermoelastic solid in the absence of body forces, body couples and heat sources are given by the following:

(a) Constitutive relations

for Lord and Shulman (L-S) theory

for Green and Lindsay (G-L) theory

The deformation and wryness tensor are defined by the following:

(b) Balance laws

where and are the stress tensor, couple stress tensor and thermal conductivity tensor, respectively; is the heat flux vector; is the entropy; T is the absolute temperature; is the specific heat at constant strain; and are the thermal relaxation times; is the bulk mass density; is the microinertia; and are the components of displacement vector and microrotation vector, respectively; is the thermal elastic coupling tensor; are characteristic constants of material where satisfies the symmetric properties

and and . Tensor does not possess this property; it forms a pseudo-tensor, inversion of the coordinate system changing its sign. In a centrosymmetric body, all components of vanish.

We have used appropriate transformations following [6] on the set of Equations 1 to 5 to derive equations for micropolar thermoelastic transversely isotopic medium and restricted our analysis to the two-dimensional problem.

We consider a homogeneous, centrosymmetric, micro polar thermoelastic transversely isotropic medium initially in an undeformed state and at uniform temperature . We take the origin of coordinate system on the top plane surface and -axis pointing normally into the half-space, which is thus represented by . We consider plane waves such that all particles on a line parallel to -axis are equally displaced, so all partial derivatives with respect to the variable would be zero. Therefore, we take and so that the field equations and constitutive relations reduce to the following:

where are the coefficients of linear thermal expansion, we have used the notations for the material constants.

For L-S theory ; for G-L theory, ; and for coupled thermoelasticity theory, . The thermal relaxation times and satisfy the inequality for G-L theory only. However, it has been proved by [7] that the inequalities are not mandatory for and to follow.

For further considerations, it is convenient to introduce the dimensionless variables defined by the following:

where is the characteristic frequency of the material, and is the longitudinal wave velocity of the medium.

Let denotes the unit propagation vector, and k are the phase velocity and the wave number of the plane waves propagating in plane, respectively. We seek plane-wave solution of the equations of motion of the form

Making use of Equation 18 and the values of and T from Equation 19 in the Equations 11 to 14, we obtain four homogeneous equations in four unknowns and , which for the non-trivial solution yield

where

The complex coefficient implies that four roots of this equation may be complex. The complex phase velocities of the quasi-waves, given by , will be varying with the direction of phase propagation. The complex velocity of a quasi-wave, i.e. defines the phase propagation velocity and attenuation quality factor for the corresponding wave. Therefore, four waves propagating in such a medium are attenuating. The same directions of wave propagation and attenuation vector of these waves make them homogeneous wave. These waves are called quasi-waves because polarizations may not be along the dynamic axes. The waves with velocity may be named as quasi-longitudinal displacement (qLD) wave, quasi-transverse displacement (qTD) wave, quasi-transverse microrotational (qTM) wave and quasi thermal wave (qT) that are propagating with the descending phase velocities, respectively.

Reflection at the free surface

We consider a homogeneous micropolar generalized thermoelastic transversely isotropic half-space occupying the region . Incident qLD or qTD or qTM or qT wave at the interface will generate reflected qLD, qTD, qTM and qT waves in the half-space . The total displacements, microrotation and temperature distribution are given by the following:

where

is the angular frequency. Here, subscripts 1,2,3, and 4 denote the quantities corresponding to incident qLD, qTD, qTM and qT wave, respectively, whereas the subscripts 5,6,7 and 8, respectively, denote the corresponding reflected waves and

The expressions for are obtained from the expressions for given in Equation 21 on substituting the values for and .

For incident qLD wave, ; for incident qTD wave, ; for incident qTM wave, ; for incident qT wave, ; for reflected qLD wave, ; for reflected qTD wave, ; for reflected qTM wave, ; and for reflected qT wave, .

We assume that the boundaries of the half-space are stress-free thermally insulated. Therefore, the appropriate boundary conditions at the surface are as follows: (a) Vanishing of the normal stress

(b) Vanishing of the tangential stress

(c) Vanishing of the tangential couple stress

(d) Vanishing of the temperature gradient

where h is the surface heat transfer coefficient;

corresponds to thermally insulated boundaries and refers to isothermal boundaries.

The boundary conditions given by Equations 27 to 30 must be satisfied for all values of and , so we have

Then from Equations 23, 24 and 31, we have

which corresponds to the Snell’s law in the present case and

Here, , , and i.e. the angle of incidence is equal to the angle of reflection in micropolar transversely isotropic generalized thermoelastic solid so that the velocities of reflected waves are equal to their corresponding incident waves, i.e. , , and .

Making use of Equations 15 to 18, 22, 31 and 32 in the boundary conditions given by Equations 26 to 29, we obtain four simultaneous equations as follows:

where

In case of incident qLD wave, . Dividing set of Equation 33 throughout by , we obtain a system of four non-homogeneous equations in four unknowns which can be solved by Gauss elimination method and we have

In case of incident qTD wave, and thus we have

In case of incident qTM wave, and thus we have

In case of incident qT wave, and thus we have

where

and can be obtained by replacing, respectively, the 1st, 2nd,..., 4th columns of by .

In order to illustrate the theoretical results obtained in the preceding sections, we now present some numerical results. For numerical computation, we take the values for relevant parameters for micropolar transversely isotropic generalized thermoelastic medium as follows:

Following [8], we take the non-dimensional values for aluminium epoxy-like composite as follows:

Figures 1,2,3,4 give the graphical representation for the variations of amplitude ratios of reflected qLD, qTD, qTM and qT waves when four types of waves viz. qLD, qTD, qTM and qT are incident at the free surface to compare the results in two cases, (a) the waves incident from MTIT and (b) the waves incident from MIT medium. Figure 1 represents graphically the variations of amplitude ratios and in case of incident qLD wave. Figures 2,3,4 show similar cases for incident qTD, qTM and qT waves, respectively. Here, and are the amplitude ratios of reflected qLD, qTD, qTM and qT waves, respectively. These variations are shown for two theories of thermoelasticity, viz. L-S and G-L. In these figures, the solid and broken curves without center symbol correspond to the case of L-S theory, while solid and broken curves with center symbol correspond to the case of G-L theory.

Incident qLD wave

It is evident from Figure 1a that the amplitude ratio of reflected qLD wave first increases sharply, then oscillates within the interval and decreases with further increase in frequency for MTIT. However, for MIT, its value initially oscillates and then increases a little to become a constant at the end with increase in frequency. This behavior is noticed for both cases of L-S and G-L theories.

Figure 1b,c indicates the variations of amplitude ratios and of reflected qTD and qTM waves, which show that for the case of G-L theory and MTIT, their value increases with increase in frequency, while in the case of L-S theory, its value starts varying with initial increase then becomes constant for some time and then increases again with increase in frequency. Similar variations are noticed for MIT, except for G-L theory, where its value tends to decrease at the end. It is depicted from Figure 1d that the value of amplitude ratio goes on increasing with increase in frequency in all the cases.

Incident qTD wave

The variations in the amplitude ratio of various reflected wave for incident qTD wave are shown in Figure 2. It is depicted from Figure 2a that the value of amplitude ratio of sharply increases to a peak value and oscillate to become constant. Similar variations are noticed for the case of G-L theory with slight difference in their amplitude. However, for MIT, its value initially oscillates with varying amplitude and then flattens to become zero at the end for both cases of L-S and G-L theories.

It can be seen from Figure 2b that for L-S theory, the value of amplitude ratio for MTIT initially oscillates with a hump in the interval and then decreases. While for G-L theory, its value initially oscillates and then decreases to attain a constant value with increase in frequency. For MIT and for both theories of thermoelasticity, their values start with initial oscillation to become constant. It is evident from this figure that the the amplitude ratio gets increased due to anisotropy.

Figure 2c,d shows the variations of amplitude ratio and within the interval , oscillate arbitrarily with different amplitude and then become constant with increase in frequency. The similar variations are depicted for all the curves, except for MIT and G-L theory where the value of amplitude ratio decreases with increase in frequency, while the value of amplitude ratio increases with increase in frequency.

Incident qTM wave

Figure 3 illustrates the variations of amplitude ratios of with frequency for incident qTM wave. It can be seen from these figures that the variation pattern of the amplitudes are almost similar with difference in their peak values. Their values show a hump within an interval and after that they tend to attain a constant value. The amplitude ratio of the first two waves gets increased due to anisotropy, while for the remaining, their values show oscillatory nature. The amplitude ratios and have higher values for L-S theory as compared to those for G-L theory, while the remaining amplitudes, initially the values are higher for L-S theory and reverse behavior is noticed afterwards.

Incident qT wave

The variations in amplitude ratio of various reflected waves for incident qT wave are shown in Figure 4. The amplitude ratio sharply decreases to become constant for MIT, while for the case of MTIT, its value sharply increases then sharply decrease to become constant at the end. Slight differences in their amplitudes have been observed. The variations of and are shown in Figure 4b,c. It can be seen from these figures that the values oscillate within the interval , showing the peaks of different amplitudes. After this interval, the values for all the cases become steady.

Figure 4d shows the variations in the value of , which indicates that anisotropy as well as angle of incidence shows a significant impact on it throughout the whole range. The behavior of is oscillatory within the range . The amplitude ratio first increases from small value to a maximum by executing small oscillation and ultimately decreases to become steady. The value for the case of L-S theory is higher as compared to those for G-L theory. Anisotropy shows a greater impact on as compared to the relaxation times.

Propagation of waves in a micropolar transversely iso tropic generalized thermoelastic half-space have been discussed. The amplitude ratios have been computed and plotted graphically for L-S and G-L theories of thermoelasticity. It is concluded from the figures that the value of amplitude ratios and shows sharp oscillation at initial frequencies for incident qLD and qT waves as compared to qTM and qTD incident waves. An appreciable effects of anisotropy and relaxation time are noticed on amplitude ratios of various reflected waves.

The authors declare that they have no competing interests.

RK formulated the problem. RRG drafted the manuscript and aligned the manuscript sequentially. She also carried out the numerical computations and interpret them graphically. Both authors read and approved the final manuscript.

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