Two dimensional deformation in microstretch thermoelastic half space with microtemperatures and internal heat source

Abstract: The purpose of this paper is to study the two dimensional deformation due to internal heat source in a microstretch thermoelastic solid with microtemperatures (MTSM). A mechanical force is applied along the interface of fluid half space and microstretch thermoelastic half space. The normal mode analysis has been applied to obtain the exact expressions for component of normal displacement, microtemperature, normal force stress, microstress tensor, heat flux moment tensor, and couple stress for MTSM. The effect of internal heat source, micropolarity, and microstretch on the above components has been depicted graphically.


Introduction
The dynamical interaction between the thermal and mechanical has great practical applications in modern aeronautics, astronautics, nuclear reactors, and high-energy particle accelerators. Classical elasticity is not adequate to model the behavior of materials possessing internal structure. Furthermore, the micropolar elastic model is more realistic than the purely elastic theory for studying the response of materials to external stimuli. Suhubi (1964a, 1964b) developed a ABOUT THE AUTHOR The author, Praveen Ailawalia working as a professor at M M University, Sadopur, Ambala, Haryana (India) is actively involved in the field of thermoelasticity, micropolar elasticity. He has 17 years of teaching experience in different Universities and institutions. The author has more than 70 research publications in international journals of repute. He has guided four PhD students and four students are currently working with him. The author has discussed deformation in thermoelastic medium and micropolar elastic medium in many of his research papers. The research problem discussed in the paper helps in analyzing the behavior of a medium with temperature changes if the medium undergoes deformation due to an internal heat source. The results obtained in the paper may be applied to various geological problems which involves sources acting in the medium. The problem can be further discussed in case of mechanical sources applied on the free surface of the medium or along the interface of two different mediums.

PUBLIC INTEREST STATEMENT
Studying the two dimensional deformation due to internal heat source in a microstretch thermoelastic solid with microtemperatures is very useful in the study of earthquake engineering, seismology, and volcanic eruptions. It helps us to study the effect of a heat source in the medium and the deformation caused in the medium due to the heat source. Grot (1969) discussed a theory of thermodynamics of elastic bodies with microstructure whose microelements possess microtemperatures. Říha (1976) studied heat conduction in materials with microtemperatures. Iesan and Quintanilla (2000) studied a theory of thermoelasticity with microtemperatures. Iesan (2001) proposed the theory of micromorphic elastic solids with microtemperatures. Exponential stability in thermoelasticity with microtemperatures was studied by Casas and Quintanilla (2005). Scalia and Svanadze (2006) gave the solutions of the theory of thermoelasticity with microtemperatures. Magaña and Quintanilla (2006) discussed the time decay of solutions in one-dimensional theories of porous materials. Aouadi (2008) discussed some theorems in the isotropic theory of microstretch thermoelasticity with microtemperatures. Ieşan and Quintanilla (2009) discussed thermoelastic bodies with inner structure and microtemperatures. Scalia, Svanadze, and Tracinà (2010) studied basic theorems in the equilibrium theory of thermoelasticity with microtemperatures. Quintanilla (2011) discussed the growth and continuous dependence in thermoelasticity with microtemperatures. Steeb, Singh, and Tomar (2013) studied time harmonic waves in thermoelastic material with microtemperatures. Chiriţă, Ciarletta, and D'Apice (2013) studied the theory of thermoelasticity with microtemperatures. Singh, Kumar, and Kumar (2014) discussed a problem in microstretch thermoelastic diffusive medium. Kumar and Kaur (2014) studied the reflection and refraction of plane waves at the interface of an elastic solid and microstretch thermoelastic solid with microtemperatures (MTSM).
In the present problem, the authors have discussed deformation due to internal heat source and a mechanical force which is applied along the interface of fluid half space and microstretch thermoelastic half space with microtemperatures. The normal mode analysis has been applied to obtain the exact expressions for component of normal displacement, microtemperature, normal force stress, microstress tensor, heat flux moment tensor, and couple stress for MTSM. The effect of internal heat source, micropolarity, and microstretch on the above components has been depicted graphically.
The behavior of a thermo-microstretch isotropic material with microtemperatures without body forces, body couples, stretch force, heat sources, and first heat source moment is governed by the following equations given by Eringen (1990) and Ieşan (2007) as, The constitutive relations are, using Equations 4-9 in Equations 1-3, we get the equations, where = (3 + 2 + K) t 1 , 1 = (3 + 2 + K) t 2 , t 1 , t 2 are the coefficents of linear thermal expansion, and μ are Lame's constants, K, α, β, γ are the micropolar constants of the solid, 0 , 0 , 1 are the stretch constants, and j 0 , μ 1 , μ 2 , k 1 , k 2 , k 3 , k 4 , k 5 , k 6 are the constitutive coefficients. t ij is the component of stress tensor, m ij is the coupled stress tensor, * i is the microstress tensor, q ij is the first heat flux moment tensor, ⃗ u = (u i ) is the displacement vector, ⃗ = ( i ) is the microrotation vector, ⃗ w = (w i ) is the microtemperature vector and ϕ* is the scalar microstretch, ρ is the density, J is the microinertia, c * is the specific heat at constant strain, Q 1 is the internal heat source, K* is the thermal conductivity, and T is the thermodynamic temperature above reference temperature T 0 .
The equations of motion and stress components in fluid (Ewing, Jardetzky, & Press, 1957) are: (1) We consider a normal force of magnitude F 1 acting along the interface of microstretch thermoelastic medium with microtemperatures (medium I) occupying the region 0 ≤ z ≤ ∞ and a non-viscous fluid (medium II) in the region −∞ ≤ z ≤ 0 is shown in Figure 1.
For convenience, the following non-dimensional variables are used: Assuming the scalar potential functions ψ 1 (x, z, t), ψ 2 (x, z, t), ψ 3 (x, z, t), and ψ 4 (x, z, t) defined by the relation in non-dimensional form as, using above non-dimensional variables and relation given by Equation 17, Equations 10-14 reduce to (after dropping superscripts), x .

Analytic solution
The solution of the considered physical variable can be decomposed in terms of normal mode and can be considered in the following form, where ω is the complex frequency, a is the wave number in x-direction, and ̄i(z),̄ * (z),T(z), ,Q 1 (z) are the amplitudes of field quantities.
The physical constants for water are given by Ewing et al. (1957):

Discussion
The variation of normal displacement for MTSM, TSMWM, and TSMWS is similar in nature. These values decrease sharply in the entire range. The values of normal displacement for TSM increase in the range 0 ≤ x ≤ 2.3 and then the values approach zero with a straight curve. The variations of microtemperature for MTSM and TSMWS are opposite in nature which shows that microstructure has significant effect on microtemperature. The values of microtemperature for TSM are very less and lie in a very short range. These variations of normal displacement and microtemperature are shown in Figures 2 and 3, respectively.  also quite close to each other. The values for these medium (TSMWS and TSM) decrease sharply and then follow a straight curve to converge. With difference in magnitude, the variation of microstress tensor for MTSM and TSMWM is similar in nature. These values decrease uniformly and then approach to zero with increase in horizontal distance. The variation of microstress tensor is shown in Figure 5. Figure 6 shows that the variations of heat flux moment tensor are similar in nature for all mediums. There is difference in magnitude among all the solids which proves the effect of micropolarity and microstress in the medium. It is again observed that the values of heat flux moment tensor for TSM are very less and hence as compared to other medium, the variation lies in a very short range.
In the absence of stretch effect, the variation of couple stress is effected to a great extent as visible in Figure 7. The values increase in the range 0 ≤ x ≤ 4.0 and then show a constant behavior. The variations are sharper for TSMWS in comparison to MTSM.

Conclusion
Both micropolarity and stretch effect have a significant effect on the normal displacement, microtemperature, normal force stress, microstress tensor, heat flux moment tensor, and tangential couple stress. The values of all the quantities for a generalized TSM are less in magnitude as compared to the medium with micropolarity and stretch effect. Micropolarity does not show appreciable effect on microstress tensor but microstretch has a significant effect on couple stress. Such type of problems is very useful in the study of earthquake engineering, seismology, and volcanic eruptions. It helps us to study the effect of a heat source in the medium and the deformation caused in the medium due to the heat source.