A theoretical study of mechanical source in a hygrothermoelastic medium with an overlying non-viscous fluid

Praveen Ailawalia; Priyanka; Marin Marin

doi:10.1515/jmbm-2022-0286

Article Open Access

A theoretical study of mechanical source in a hygrothermoelastic medium with an overlying non-viscous fluid

Praveen Ailawalia , Priyanka and Marin Marin

Published/Copyright: August 30, 2023

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From the journal Journal of the Mechanical Behavior of Materials Volume 32 Issue 1

Abstract

This work demonstrates deformation in hygrothermoelastic medium with an overlying non-viscous fluid of uniform thickness. A constant mechanical force is applied along the fluid-layer. The normal-mode analysis technique is applied to solve the governing equations of the medium. The analytical expressions of displacement, moisture concentration, temperature distribution, and stresses are obtained for the hygrothermoelastic medium and depicted graphically for different values of fluid-layer depth. The finding of this work is that the values of the aforementioned physical quantities decrease with an increase in the fluid-layer, which justifies the decaying nature of waves with depth. The novelty of the problem is that no research has been carried out so far for analyzing hygrothermoelastic medium subjected to thermoelastic deformation.

Keywords: non-viscous; fluid-layer; normal mode; temperature distribution; hygrothermoelastic medium; moisture concentration

1 Introduction

The theory of dynamic thermoelasticity is being widely explored by researchers due to its applications in various disciplines of science. The classical coupled theory of thermoelasticity assumes the parabolic nature of heat conduction predicting infinite velocity of heat propagation that is practically impossible. During the last few decades, various theories of generalized thermoelasticity have been developed involving hyperbolic heat equation that admits a finite speed of thermal signals. Chandrasekharaiah and Srinath [1] considered the linear theory of thermoelasticity without energy dissipation for discussing thermoelastic interactions in homogeneous and isotropic media under the effect of a continuous point heat source. Baksi et al. [2] explored three-dimensional infinite rotating elastic medium for discussing heat sources in a magnetothermoelastic medium with thermal relaxation. Othman and Lotfy [3] formulated a two-dimensional problem of generalized magnetothermoelasticity in context of three theories of the generalized thermoelasticity subjected to temperature-dependent properties. The effect of reinforcement in a generalized thermoelastic medium subjected to mode-I crack was demonstrated by Lotfy [4]. Furthermore, fiber-reinforced thermoelastic medium subjected to magnetic field, gravity, and rotation in the context of three different theories of thermoelasticity was studied by Othman and Lotfy [5]. Ailawalia and Budhiraja [6] studied deformation in thermo-microstretch elastic medium underlying non-viscous fluid-layer subjected to an internal heat source. Marin et al. [7] introduced an additional degree of freedom to develop the mechanical behavior of a body in which the skeletal material is elastic and interstices are voids of material. Ailawalia and Sachdeva [8] demonstrated the effect of an internal heat source on deformation in a thermoelastic solid with microtemperatures. Khan et al. [9] examined third-grade magnetohydrodynamic fluid with variable thermal conductivity and chemical reactions over an exponentially stretching surface. Lotfy et al. [10] investigated the propagation of waves in semi-infinite semiconducting medium on the application of a magnetic field and hydrostatic initial stress. Lotfy et al. [11] observed the behavior of a semiconductor medium subjected to electromagnetic and Thomson effects. Khamis et al. [12] studied thermal-piezoelectric problem for photothermal process in a semiconductor medium. Alhejaili et al. [13] explored non-local excited semiconductors under the influence of the laser short-pulse effect.

Recently, coupling between heat, moisture, and deformation is being observed in engineering problems and various branches of science. Severe damage of material occurs when moisture and temperature interact with mechanical stresses. Such findings suggest that the combined interaction of moisture, deformation, and heat needs to be analyzed. Sih et al. [14] and Weitsman [15] developed coupled equations for hygrothermoelastic medium keeping in the principles of irreversible thermodynamics and continuum mechanics. Fluid-saturated porous media subjected to finite deformation under hygrothermoelastic theory was examined by Advani et al. [16]. A coupled micro-macromechanical approach was adopted by Aboudi and Williams [17] to examine hygrothermoelastic composites. The vibration characteristics of hygrothermoelastic laminated composite doubly curved shells were studied by Kundu and Han [18]. Hosseini and Ghadiri [19] discussed a two-dimensional problem in a coupled hygrothermoelastic medium. Lamba and Deshmukh [20] discussed the two unsteady-state responses of a finite long solid cylinder subjected to axisymmetric hygrothermal loading. Peng et al. [21] put forward a hygrothermal coupling model to study the effect of the phase delay of heat and moisture fluxes on hollow cylinders with convective surfaces. Rao et al. [22] demonstrated a micromechanical model for fiber-reinforced composites to investigate their hygrothermoelastic properties. Ailawalia et al. [23] obtained analytic expressions for surface wave propagation in hygrothermoelastic medium with hydrostatic initial stress. Brischetto and Torre [24] presented a novel 3D hygroelastic shell model for studying multilayered composite and sandwich structures under the influence of hygrometric loading. He et al. [25] discussed the behavior of stiffened metal doubly curved shallow shells with porous microcapsule coating under the exposure of hygrothermoelastic conditions. Verma et al. [26] analyzed the hygrothermal response in a hollow cylinder by applying the theory of uncoupled–coupled heat and moisture.

This study deals with the study of deformation in hygrothermoelastic medium with an overlying non-viscous fluid-layer of finite thickness h . The fluid-layer is subjected to a constant mechanical force. The displacement components, moisture concentration, temperature distribution, and stress components are evaluated and presented graphically to show the effect of the fluid-layer in the hygrothermoelastic medium. The same model has been discussed by Sharma and Sharma [27], in which they discussed the propagation of surface waves in a homogenous, isotropic, thermally conducting elastic solid underlying a layer of viscous liquid of finite thickness.

2 Basic equations

We consider a hygrothermoelastic half-space with an overlying non-viscous fluid-layer of depth h . A normal mechanical load of constant magnitude is acting on the surface of fluid-layer. A rectangular coordinate system ( x , y , z ) with z -axis pointing vertically downward is considered. The region z > 0 represents hygrothermoelastic medium (medium I), and the region − h ≤ z < 0 represents the non-viscous fluid-layer (medium II). Following Hosseini and Ghadiri Rad [19], governing equations in hygrothermoelastic medium without body forces and heat sources are given by:

(1) σ j i , j = ρ u ¨ i ,

(2) D T T , i i + D T m m , i i − T ˙ − α i j T T 0 ρ c u ˙ j , j = 0 ,

(3) D m m , i i + D m T T , i i − m ˙ − α i j m m 0 D m k m u ˙ j , j = 0 ,

where Eq. (1) represents the equation of motion in hygrothermoelastic medium, whereas Eqs. (2) and (3) represents the coupled heat conduction and moisture diffusion equations. Further, constitutive stress–strain relations [19] are given by:

(4) σ i j = C i j r s ε r s − α i j m m − α i j T T ,

where

α i j T = α T δ i j , α T = ( 3 λ + 2 μ ) γ T , α i j m = α m δ i j , α m = ( 3 λ + 2 μ ) γ m , C i j r s = 2 G ν 1 − 2 ν δ i j δ r s + G δ i r δ j s + G δ i s δ j r ,

where ε r s , σ i j , and u i are the components of strain, stress, and displacement, respectively; ρ is the density; D T is the temperature diffusivity; T is the temperature; m is the moisture concentration, T 0 is the reference temperature; D m is the diffusion coefficient of moisture; D T m and D m T are the coupled diffusivities; c is the heat capacity; m 0 is the reference moisture; k m is the moisture diffusivity; α i j T and α i j m are the material coefficients that arise due to coupling between stresses and temperature or moisture concentration, respectively; α T is the coefficient of linear thermal expansion; α m is the coefficient of moisture expansion; λ and μ are Lame’s constants.

Also, the governing equations of motion and stress-displacement relationships for the non-viscous fluid are expressed as (Ewing et al. [28]):

(5) λ f ∇ ( ∇ . u f → ) = ρ f u f → ¨ ,

(6) σ l k f = λ f u r , r f δ l k , l , k = 1 , 2 , 3 ,

where ρ f is the fluid density, λ f is Lame’s constant, and ∇ = i ˆ ∂ ∂ x + j ˆ ∂ ∂ y + k ˆ ∂ ∂ z .

3 Formulation of the problem

For a two-dimensional analysis of the problem, the waves are considered to be propagating in x − z plane. Therefore, displacement vector in the hygrothermoelastic medium may be considered as u → = ( u , 0 , w ) , where u = u ( x , z , t ) and w = w ( x , z , t ) . The reduced equations of motion (1)–(3) and constitutive relations (4) in two dimensions reduce to:

(7) 2 G ( 1 − ν ) 1 − 2 ν ∂ 2 u ∂ x 2 + G 1 − 2 ν ∂ 2 w ∂ x ∂ z + G ∂ 2 u ∂ z 2 − α m ∂ m ∂ x − α T ∂ T ∂ x = ρ ∂ 2 u ∂ t 2 ,

(8) G ∂ 2 w ∂ x 2 + G 1 − 2 ν ∂ 2 u ∂ x ∂ z + 2 G ( 1 − ν ) 1 − 2 ν ∂ 2 w ∂ z 2 − α m ∂ m ∂ z − α T ∂ T ∂ z = ρ ∂ 2 w ∂ t 2 ,

(9) D T Δ 2 T + D T m Δ 2 m − ∂ T ∂ t − α T T 0 ρ c ∂ ∂ t ∂ u ∂ x + ∂ w ∂ z = 0 ,

(10) D m Δ 2 m + D m T Δ 2 T − ∂ m ∂ t − α m m 0 D m k m ∂ ∂ t ∂ u ∂ x + ∂ w ∂ z = 0 ,

(11) σ z x = G ∂ u ∂ z + ∂ w ∂ x ,

(12) σ z z = 2 G ν 1 − 2 ν ∂ u ∂ x + 2 G ( 1 − ν ) 1 − 2 ν ∂ w ∂ z − α m m − α T T ,

where Δ 2 = ∂ 2 ∂ x 2 + ∂ 2 ∂ z 2 .

To facilitate numerical computation work, the introduction of the following dimensionless quantities is an important step:

(13) x ′ = 1 l x , z ′ = 1 l z , u ′ = 1 l u , w ′ = 1 l w , t ′ = D m l 2 t , m ′ = m , T ′ = T T 0 , σ i j ′ = σ i j G .

Using (13) in (7)–(11) and dropping primes, we obtain following non-dimensional equations in the hygrothermoelastic medium.

(14) 2 G ( 1 − ν ) 1 − 2 ν ∂ 2 u ∂ x 2 + G 1 − 2 ν ∂ 2 w ∂ x ∂ z + G ∂ 2 u ∂ z 2 − α m ∂ m ∂ x − α T T 0 ∂ T ∂ x = ρ D m 2 l 2 ∂ 2 u ∂ t 2 ,

(15) G ∂ 2 w ∂ x 2 + G 1 − 2 ν ∂ 2 u ∂ x ∂ z + 2 G ( 1 − ν ) 1 − 2 ν ∂ 2 w ∂ z 2 − α m ∂ m ∂ z − α T T 0 ∂ T ∂ z = ρ D m 2 l 2 ∂ 2 w ∂ t 2 ,

(16) D T T 0 Δ 2 T + D T m Δ 2 m − D m T 0 ∂ T ∂ t − α T T 0 D m ρ c ∂ ∂ t ∂ u ∂ x + ∂ w ∂ z = 0 ,

(17) D m Δ 2 m + D m T T 0 Δ 2 T − D m ∂ m ∂ t − α m m 0 D m 2 k m ∂ ∂ t ∂ u ∂ x + ∂ w ∂ z = 0 ,

(18) σ z x = ∂ u ∂ z + ∂ w ∂ x ,

(19) σ z z = 2 ν 1 − 2 ν ∂ u ∂ x + 2 ( 1 − ν ) 1 − 2 ν ∂ w ∂ z − α m m G − α T T T 0 G .

Now, introducing potential functions ϕ and ψ defined by:

(20) u = ∂ ϕ ∂ x − ∂ ψ ∂ z , w = ∂ ϕ ∂ z + ∂ ψ ∂ x ,

in Eqs. (14)–(16), we obtain the following equations in terms of potential functions:

(21) 2 G ( 1 − ν ) 1 − 2 ν Δ 2 ϕ − α m m − α T T 0 T = ρ D m 2 l 2 ∂ 2 ϕ ∂ t 2 ,

(22) G Δ 2 ψ = ρ D m 2 l 2 ∂ 2 ψ ∂ t 2 ,

(23) D T T 0 Δ 2 T + D T m Δ 2 m − D m T 0 ∂ T ∂ t − α T T 0 D m ρ c ∂ ∂ t ( Δ 2 ϕ ) = 0 ,

(24) D m Δ 2 m + D m T T 0 Δ 2 T − D m ∂ m ∂ t − α m m 0 D m 2 k m ∂ ∂ t ( Δ 2 ϕ ) = 0 .

4 Solution of the problem

The considered physical variables may be decomposed, and the solution may be assumed in terms of normal modes in the form:

(25) { ϕ , ψ , m , T } = { Φ , Ψ , P , Q } ( z ) exp { ω t + i b x } ,

where ω is the complex frequency and b is the wave number in x -direction. Using the aforementioned solution in (21)–(24), we obtain the following equations:

(26) [ a 11 D 2 + a 12 ] Φ − α m P − α T T 0 Q = 0 ,

(27) [ a 18 D 2 + a 19 ] Φ + [ a 16 D 2 + a 17 ] P + [ a 14 D 2 + a 15 ] Q = 0 ,

(28) [ b 11 D 2 + b 12 ] Φ + [ b 13 D 2 + b 14 ] P + [ b 15 D 2 + b 16 ] Q = 0 ,

(29) [ G D 2 + a 13 ] Ψ = 0 .

Eliminating the parameters Φ , P , and Q from Eqs. (26)–(28), we obtain a sixth-degree differential equation in terms of Φ , P , and Q as:

(30) [ p 11 D 6 + p 12 D 4 + p 13 D 2 + p 14 ] ( Φ , P , Q ) = 0 ,

where the values of p 11 , p 12 , p 13 , and p 14 are given in Appendix.

Using radiation conditions ϕ , ψ , T , and m → 0 as z → ∞ , the solution of Eq. (30) can be expressed in the form:

(31) ϕ = [ A 1 exp { − k 1 z } + A 2 exp { − k 2 z } + A 3 exp { − k 3 z } ] exp { ω t + i b x } ,

(32) m = [ B 1 exp { − k 1 z } + B 2 exp { − k 2 z } + B 3 exp { − k 3 z } ] exp { ω t + i b x } ,

(33) T = [ C 1 exp { − k 1 z } + C 2 exp { − k 2 z } + C 3 exp { − k 3 z } ] exp { ω t + i b x } .

Similarly, the solution of Eq. (29) may be expressed in the form:

(34) ψ = A 4 exp { − k 4 z } exp { ω t + i b x } ,

where k 1 2 , k 2 2 , and k 3 2 are the roots of Eq. (30) and k 4 2 = − a 13 G .

Also, the coupling constants B j and C j can be expressed in terms of A j ( j = 1 , 2 , 3 ) as:

(35) B j = R j A j , C j = S j A j ,

where the values of R j and S j are given in Appendix.

Applying a similar methodology, the solution for the fluid-layer is obtained as:

(36) ϕ f = [ E 1 exp { − k 5 z } + E 2 exp { k 5 z } ] exp { ω t + i b x } , ψ f = 0

where ϕ f and ψ f correspond to the potential functions for displacement components u f and w f of the fluid-layer and k 5 2 = b 2 + ω 2 f 11 .

5 Boundary conditions

This investigation deals with the deformation of a hygrothermoelastic medium with an overlying non-viscous fluid of depth h . On the surface of the fluid-layer, a mechanical force of magnitude F is applied leading to the following boundary conditions:

The stress components are continuous along the interface z = 0 :
σ z z = σ z z f , σ z x = σ z x f .
The interface z = 0 is thermally insulated:
∂ T ∂ z = 0 .
The moisture concentration satisfies the following boundary condition at z = 0 :
∂ m ∂ z = 0 .
The normal displacements are continuous along the interface z = 0 :
w = w f .
A mechanical force of magnitude F is applied along the surface z = − h :
σ z z f = − F exp { ω t + i b x } .

Using the aforementioned boundary conditions, we obtain the following non-homogenous system of six equations:

(37) l 1 A 1 + l 2 A 2 + l 3 A 3 + l 4 A 4 + l 5 E 1 + l 5 E 2 = 0 ,

(38) h 1 A 1 + h 2 A 2 + h 3 A 3 + h 4 A 4 = 0 ,

(39) k 1 S 1 A 1 + k 2 S 2 A 2 + k 3 S 3 A 3 = 0 ,

(40) k 1 R 1 A 1 + k 2 R 2 A 2 + k 3 R 3 A 3 = 0 ,

(41) k 1 A 1 + k 2 A 2 + k 3 A 3 − i b A 4 − k 5 E 1 + k 5 E 2 = 0 ,

(42) l 5 exp { k 5 h } E 1 + l 5 exp { − k 5 h } E 2 = F ,

where the values of l 1 , l 2 , l 3 , l 4 , l 5 , h 1 , h 2 , h 3 , and h 4 are given in Appendix.

The aforementioned non-homogenous system of six equations can be solved by Cramer’s rule using MATLAB, and the values of constants A n ( n = 1 , 2 , 3 , 4 ) , E 1 , and E 2 can be evaluated.

Using the expressions of ϕ and ψ given by expressions (31)–(34) in Eqs. (17)–(20), the displacement components, moisture concentration, temperature, and stresses in the hygrothermoelastic medium are obtained as:

(43) u = i b ∑ j = 1 3 A j exp { − k j z } + k 4 A 4 exp { − k 4 z } exp { ω t + i b x } ,

(44) w = − ∑ j = 1 3 k j A j exp { − k j z } + i b A 4 exp { − k 4 z } exp { ω t + i b x } ,

(45) m = ∑ j = 1 3 R j A j exp { − k j z } exp { ω t + i b x } ,

(46) T = ∑ j = 1 3 S j A j exp { − k j z } exp { ω t + i b x } ,

(47) σ z x = ∑ j = 1 4 h j A j exp { − k j z } exp { ω t + i b x } ,

(48) σ z z = ∑ j = 1 4 l j A j exp { − k j z } exp { ω t + i b x } ,

where A n = Δ n Δ ∗ ( n = 1 , 2 , 3 , 4 ) and Δ n , and Δ ∗ are the determinants of order 6 × 6 whose values are given in Appendix.

6 Particular case

Neglecting diffusion coefficients in the medium, i.e., α m = D T m = D m T = m 0 = 0 , the problem is reduced to a thermoelastic analysis in a thermoelastic medium with an overlying non-viscous fluid. The same problem has been discussed by Sharma and Sharma [27], in which they discussed the propagation of surface waves in a homogenous, isotropic, thermally conducting elastic solid underlying a layer of viscous liquid of finite thickness.

7 Numerical results

In order to verify the analytical results obtained in the previous section, we present a numerical example by taking wood slab as a porous material. The physical constants for the material are given by Chang and Weng [29] and Yang et al. [30], λ = 46.92 × 1 0 9 N/m 2 , μ = 24.17 × 1 0 9 N/m 2 , ρ = 370 kg/m 3 , ν = 0.33 , m 0 = 10 % , α m = 2.68 × 1 0 − 3 cm ∕ cm ( % H 2 O ) , T 0 = 28 3 ° K , α T = 31.3 × 1 0 − 6 cm ∕ cm ( K ° ) , k = 0.65 w/m ( K ° ) , c = 2,500 J/Kg ( K ° ) , k m = 2.2 × 1 0 − 8 kg/m sM , D m = 2.16 × 1 0 − 6 m 2 ∕ s , D T = k ρ c , D m T = 0.648 × 1 0 − 6 m 2 ( % H 2 O ) ∕ s ( K ° ) , and D T m = 2.1 × 1 0 − 7 m 2 ( K ° ) ∕ s ( % H 2 O ) . The physical constants for water (non-viscous fluid) are given by Ewing et al. [28], ρ f = 1.0 × 1 0 3 kg ∕ m 3 and λ f = 0.214 × 1 0 10 N ∕ m 2 .

The numerical results are obtained for normal displacement, normal force stress, moisture concentration, and temperature distribution against horizontal distance x for three different values of non-dimensional fluid-layer depth ( h = 0.1 , 1.0 and 10.0) and for l = 1 , t = 1 , b = 2.1 , and ω = ω 0 + i ξ ( ω 0 = − 2.5 , ξ = 1 ), at the surface z = 1 . The effect of non-viscous fluid depth on the quantities has been depicted graphically. The graphical results have been obtained using the numerical values in the code generated using MATLAB 7.0 software.

8 Discussions

The graphical results obtained can be discussed as follows.

Figure 1 shows the variation of normal displacement with horizontal distance x for three different values of non-dimensional fluid-layer depth taken as h = 0.1 , 1.0 , and 10.0. The value of normal displacement starts decreasing near the point of application of mechanical source. Furthermore, the variation is oscillatory for all values of h . The magnitude of oscillation decreases with an increase in the value of h .

Figure 1

Normal displacement w versus horizontal distance x .

Figure 2 depicts how normal stress varies with the horizontal distance x . This variation is observed for three values of non-dimensional fluid-layer depth as in the above point. Again, oscillatory behavior is observed in this variation. From this figure, it is clear that oscillations die out on increasing the depth of the fluid-layer. In contrast to the oscillations observed in the first case, here oscillations first decrease to zero and then increase.

$Figure 2 Normal force stress σ z z {\sigma }_{zz} versus horizontal distance x x .$

Figure 2

Normal force stress σ z z versus horizontal distance x .

Figure 3 represents the dependence of moisture concentration on the fluid-layer depth in oscillatory manner. The figure shows the same behavior as shown in Figure 2. This proves that variations in normal stress and moisture concentration are similar with difference in magnitude.

Figure 3

Moisture concentration m versus horizontal distance x .

Figure 4 displays the variation in temperature distribution on varying fluid-layer depth. Variation of temperature distribution as shown in Figure 4 is similar to the variation of normal displacement in Figure 1 but there is significant difference in magnitude.

Figure 4

Temperature distribution T versus horizontal distance x .

So, oscillatory behavior in variations is observed for all physical quantities. The common thing is that the values of all physical fields decrease with the increase in the depth of the fluid-layer.

9 Conclusion

The analytical results shows that four waves propagate in the medium.
Three waves, namely, longitudinal displacement wave, diffusion wave, and thermal waves, are coupled waves while the transverse displacement wave is independent.
Fluid-layer depth over the surface of hygrothermoelastic half-space affects the deformation in the medium.
As the depth of fluid-layer increases, the values of physical quantities decrease. This finding justifies the theoretical nature of waves that intensity of waves decays with depth.
The research problem is theoretical but it can be used for obtaining useful information for researchers working in the field of geophysics, oil extraction, navigation, hydrology, etc.
This research problem is a novel problem that gives exact expressions for normal displacement, normal stress, moisture concentration, and temperature distribution in a hygrothermoelastic medium with an overlying non-viscous fluid of finite thickness.

Funding information: This study is not funded by any agency.
Author contributions: All authors contributed to the study conception and design. Material preparation and analysis were performed by Praveen Ailawalia and Priyanka. The first draft of the manuscript was written by Marin Marin and Priyanka. The final draft was written by Praveen Ailawalia, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Conflict of interest: On behalf of all authors, the corresponding author states that there is no conflict of interest.

Appendix

f 11 = λ f l 2 ρ f D m 2 , f 12 = λ f G , a 11 = 2 G ( 1 − ν ) 1 − 2 ν , a 12 = − 2 G ( 1 − ν ) 1 − 2 ν b 2 + ρ D m 2 ω 2 l 2 , a 13 = − G b 2 + ρ D m 2 ω 2 l 2 , a 14 = D T T 0 , a 15 = − [ D T T 0 b 2 + ω T 0 D m ] , a 16 = D T m , a 17 = − D T m b 2 , a 18 = − α T ω D m T 0 ρ c , a 19 = α T ω D m T 0 b 2 ρ c , b 11 = − α m ω D m 2 m 0 k m , b 12 = α m ω D m 2 m 0 b 2 k m , b 13 = D m , b 14 = − [ D m b 2 + D m ω ] , b 15 = D m T T 0 , b 16 = − D m T T 0 b 2 , c 11 = a 11 b 15 , c 12 = a 12 b 15 + a 11 b 16 + α T T 0 b 11 , c 13 = a 12 b 16 + α T T 0 b 12 , c 14 = − α m b 15 + α T T 0 b 13 , c 15 = − α m b 16 + α T T 0 b 14 , e 11 = a 11 a 14 , e 12 = a 12 a 14 + a 11 a 15 + α T T 0 a 18 , e 13 = α T T 0 a 19 + a 12 a 15 , e 14 = − α m a 14 + α T T 0 a 16 , e 15 = − α m a 15 + α T T 0 a 17 , p 11 = c 11 e 14 − e 11 c 14 , p 12 = c 11 e 15 + c 12 e 14 − e 11 c 15 − e 12 c 14 , p 13 = c 12 e 15 + c 13 e 14 − e 12 c 15 − e 13 c 14 , p 14 = c 13 e 15 − e 13 c 15 , R j = − [ c 11 k j 4 + c 12 k j 2 + c 13 ] c 14 k j 2 + c 15 , S j = a 11 k j 2 + a 12 − α m R j α T T 0 , h j = − 2 i b k j ( j = 1 , 2 , 3 ) , h 4 = − ( k 4 2 + b 2 ) , l j = − 2 ν 1 − α ν b 2 + 2 ( 1 − ν ) 1 − α ν k j 2 − α m R j G − α T T 0 S j G ( j = 1 , 2 , 3 ) , l 4 = 2 i b k 4 ( 2 ν − 1 ) 1 − α ν , l 5 = f 12 ( b 2 − k 5 2 ) , Δ ∗ = l 1 l 2 l 3 l 4 l 5 l 5 h 1 h 2 h 3 h 4 0 0 k 1 S 1 k 2 S 2 k 3 S 3 0 0 0 k 1 R 1 k 2 R 2 k 3 R 3 0 0 0 k 1 k 2 k 3 − i b − k 5 k 5 0 0 0 0 l 5 exp ( k 5 h ) l 5 exp ( − k 5 h ) , Δ 1 = l 1 l 2 l 3 l 4 l 5 0 h 1 h 2 h 3 h 4 0 0 k 1 S 1 k 2 S 2 k 3 S 3 0 0 0 k 1 R 1 k 2 R 2 k 3 R 3 0 0 0 k 1 k 2 k 3 − i b − k 5 0 0 0 0 0 l 5 exp ( k 5 h ) F .

Similarly, Δ 2 is obtained by replacing the second column of Δ ∗ by 0 0 0 0 0 F .

Δ 3 is obtained by replacing the third column of Δ ∗ by 0 0 0 0 0 F .

Δ 4 is obtained by replacing the fourth column of Δ ∗ by 0 0 0 0 0 F .

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Received: 2022-07-05

Revised: 2023-03-17

Accepted: 2023-04-19

Published Online: 2023-08-30

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/jmbm-2022-0286

Keywords for this article

non-viscous; fluid-layer; normal mode; temperature distribution; hygrothermoelastic medium; moisture concentration

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