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.

Background

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.

Basic equations

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:

• Constitutive relations

  tkl=AklmnEmn+GklmnΨmn−βkl(T+τ1Ṫ),
  mkl=BlkmnEmn+GmnlkΨmn,
  qk=KklT,k,

for Lord and Shulman (L-S) theory

for Green and Lindsay (G-L) theory

The deformation and wryness tensor are defined by the following:

• Balance laws

  tkl,k=ρül,
  mkl,k+ϵlmntmn=ρTϕ̈l,
  qk,k=ρT0η̇,

where tkl,mkl, and Kkl are the stress tensor, couple stress tensor and thermal conductivity tensor, respectively; qk is the heat flux vector; η is the entropy; T is the absolute temperature; C∗ is the specific heat at constant strain; τ0 and τ1 are the thermal relaxation times; ρ is the bulk mass density; 횥 is the microinertia; ul and ϕk are the components of displacement vector and microrotation vector, respectively; βkl is the thermal elastic coupling tensor; Aklmn,Gklmn,Blkmn are characteristic constants of material where Aklmn,Blkmn satisfies the symmetric properties

and βkl=βlk and Kkl=Klk . Tensor Gklmn 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 Gklmn vanish.

Formulation of the problem

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 To . We take the origin of coordinate system on the top plane surface and x3 -axis pointing normally into the half-space, which is thus represented by x3≥0 . We consider plane waves such that all particles on a line parallel to x2 -axis are equally displaced, so all partial derivatives with respect to the variable x2 would be zero. Therefore, we take u→=(u1,0,u3),ϕ→=(0,ϕ2,0) and ∂/∂x2=0 so that the field equations and constitutive relations reduce to the following:

where β1=A11α1+A13α3,β3=A31α1+A33α3,K1=A56−A55,K2=A66−A56,X=K2−K1, α1,α3 are the coefficients of linear thermal expansion, we have used the notations 11→1,33→3,12→7,13→6, 23→5 for the material constants.

For L-S theory τ1=0 no=1 ; for G-L theory, τ1>0 no=0 ; and for coupled thermoelasticity theory, no=0,τ1=τo=0 . The thermal relaxation times τ0 and τ1 satisfy the inequality τ1≥τ0>0 for G-L theory only. However, it has been proved by [ ⁷ ] that the inequalities are not mandatory for τ0 and τ1 to follow.

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

where ω∗ is the characteristic frequency of the material, and c1 is the longitudinal wave velocity of the medium.

Plane-wave propagation

Let p→(p1,0,p3) denotes the unit propagation vector, c and k are the phase velocity and the wave number of the plane waves propagating in x1x3 plane, respectively. We seek plane-wave solution of the equations of motion of the form

Making use of Equation ¹⁸ and the values of u1,u3,ϕ2 and T from Equation ¹⁹ in the Equations ¹¹ to ¹⁴ , we obtain four homogeneous equations in four unknowns ū1,ū3,ϕ̄2 and T̄ , 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 ci,i=1,2,3,4 , will be varying with the direction of phase propagation. The complex velocity of a quasi-wave, i.e. ci=cR+ıcI, defines the phase propagation velocity Vi=(cR2+cI2)/cR and attenuation quality factor Qi−1=−2cI/cR 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 Vi(i=1,2,3,4) 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 x3>0 . 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 x3>0 . The total displacements, microrotation and temperature distribution are given by the following:

u1=∑j=18AjeıEj,u3=∑j=18BjeıEj,ϕ2=∑j=18CjeıEj,T=∑j=18DjeıEj,

where

Ej=ω[t−(x1sinej−x3cosej)/cj],j=1,2,3,4,

Ej=ω[t−(x1sinej+x3cosej)/cj],j=5,6,7,8,

ω

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

Bj=Λ1iΛiAj,Cj=Λ2iΛiAj,Dj=Λ3iΛiAj,(j=1,…,8),Λi=d7ω2−ξ2a2ξa5ξa7ξa3ω2−d11−ξ2a60ξa40a9+a8ξ2,Λ1i=ξ2a11ξa5ξa7ξa12ω2−d11−ξ2a60ξa130a9+a8ξ2,Λ2i=ξ2a11d7ω2−ξ2a2ξa7ξa12ξa30ξa13ξa4a9+a8ξ2,Λ3i=ξ2a11d7ω2−ξ2a2ξa5ξa12ξa3ω2−d11−ξ2a6ξa13ξa40.

The expressions for ai,i=1,2,....15 are obtained from the expressions for ai,i=1,2,....15 given in Equation ²¹ on substituting the values for p1 and p2 .

Figure 1

For incident qLD wave, p1=sine1,p3=−cose1 ; for incident qTD wave, p1=sine2,p3=−cose2 ; for incident qTM wave, p1=sine3,p3=−cose3 ; for incident qT wave, p1=sine4,p3=−cose4 ; for reflected qLD wave, p1=sine5,p3=cose5 ; for reflected qTD wave, p1=sine6,p3=cose6 ; for reflected qTM wave, p1=sine7,p3=cose7 ; and for reflected qT wave, p1=sine8,p3=cose8 .

Boundary condition

We assume that the boundaries of the half-space are stress-free thermally insulated. Therefore, the appropriate boundary conditions at the surface x3=0 are as follows:

• Vanishing of the normal stress

  t33=0,

• Vanishing of the tangential stress

  t31=0,

• Vanishing of the tangential couple stress

  m32=0,

• Vanishing of the temperature gradient

  ∂T∂y+hT=0=0,

where h is the surface heat transfer coefficient;

Figure 2

The boundary conditions given by Equations ²⁷ to ³⁰ must be satisfied for all values of x1 and t , so we have

Then from Equations ²³ , ²⁴ and ³¹ , we have

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

Figure 3

Figure 4

Here, e1=e5 , e2=e6 , e3=e7 and e4=e8 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. c1=c5 , c2=c6 , c3=c7 and c4=c8 .

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, A2=A3=A4=0 . Dividing set of Equation ³³ throughout by A1 , 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, A1=A3=A4=0 and thus we have

In case of incident qTM wave, A1=A2=A4=0 and thus we have

In case of incident qT wave, A1=A2=A3=0 and thus we have

where

and △ip(i=1,2,..,4)(p=1,...,4) can be obtained by replacing, respectively, the 1st, 2nd,..., 4th columns of △ by [−A1p,−A2p,−A3p,−A4p]T .

Numerical results and discussion

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 |Z1|,|Z2|,|Z3| and |Z4| in case of incident qLD wave. Figures ² , ³ , ⁴ show similar cases for incident qTD, qTM and qT waves, respectively. Here, |Z1|,|Z2|,|Z3| and |Z4| 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.