Computers, Materials & Continua DOI:10.32604/cmc.2021.015563
Article

Rotational Effect on the Propagation of Waves in a Magneto-Micropolar Thermoelastic Medium

A. M. Abd-Alla*,1, S. M. Abo-Dahab2, M. A. Abdelhafez1 and A. M. Farhan3,4

1Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt
2Department of Mathematics, Faculty of Science, South Valley University, 83523, Egypt
3Department of Physics, Faculty of Science, Jazan University, Jazan, Saudi Arabia
4Department of Physics, Faculty of Science, Zagazig University, Zagazig, Egypt
*Corresponding Author: A. M. Abd-Alla. Email: mohmrram@yahoo.com
Received: 28 November 2020; Accepted: 14 February 2021

Abstract: The present paper aims to explore how the magnetic field, ramp parameter, and rotation affect a generalized micropolar thermoelastic medium that is standardized isotropic within the half-space. By employing normal mode analysis and Lame’s potential theory, the authors could express analytically the components of displacement, stress, couple stress, and temperature field in the physical domain. They calculated such manners of expression numerically and plotted the matching graphs to highlight and make comparisons with theoretical findings. The highlights of the paper cover the impacts of various parameters on the rotating micropolar thermoelastic half-space. Nevertheless, the non-dimensional temperature is not affected by the rotation and the magnetic field. Specific attention is paid to studying the impact of the magnetic field, rotation, and ramp parameter of the distribution of temperature, displacement, stress, and couple stress. The study highlighted the significant impact of the rotation, magnetic field, and ramp parameter on the micropolar thermoelastic medium. In conclusion, graphical presentations were provided to evaluate the impacts of different parameters on the propagation of plane waves in thermoelastic media of different nature. The study may help the designers and engineers develop a structural control system in several applied fields.

Keywords: Magnetic field; micropolar; thermoelastic; rotation; ramp parameter; normal mode analysis

Nomenclature

1  Introduction

Because it is relevant in several applications, researchers have paid due attention to generalized thermoelasticity. Its theories include governing equations of the hyperbolic type and disclose the thermal signals’ finite speed. In a standardized thermoelastic isotropic half-space, Abd-Alla et al. [1] investigated how wave propagation is affected by a magnetic field. Abouelregal [2] discussed the improved fractional photo-thermoelastic system for a magnetic field- subjected rotating semiconductor half-space. Singh et al. [3] studied reflecting plane waves from a solid thermoelastic micropolar half-space with impedance boundary circumstances. Said [4] studied the propagation of waves in a two-temperature micropolar magnetothermoelastic medium for a three-phase-lag system. The authors of [5] explored the impact of rotation on a two-temperature micropolar standardized thermoelasticity utilizing a dual-phase lag system. Rupender [6] studied the impact of rotation on a micropolar magnetothermoelastic medium because of thermal and mechanical sources. Bayones et al. [7] investigated the impact of the magnetic field and rotation on free vibrations within a non-standardized spherical within an elastic medium. Bayones et al. [8] discussed the thermoelastic wave propagation within the half-space of an isotropic standardized material affected by initial stress and rotation. Kalkal et al. [9] investigated reflecting plane waves in a thermoelastic micropolar nonlocal medium affected by rotation. Lotfy [10] explored two-temperature standardized magneto-thermoelastic responses within an elastic medium under three models. Zakaria [11] discussed the effect of hall current on a standardized magneto-thermoelasticity micropolar solid under heating of the ramp kind. Ezzata et al. [12] studied the impacts of the fractional order of heat transfer and heat conduction on a substantially long hollow cylinder that conducts perfectly. Morse et al. [13] employed Helmholtz’s theorem. The authors of [14] investigated the transient magneto-thermoelasto-diffusive interactions of the rotating media with porous in the absence of the dissipation of energy influenced by thermal shock. Kumar et al. [15] studied the axisymmetric issue within a thermoelastic micropolar standardized half-space. The authors of [16] studied the free surface’s reflection of the thermoelastic rotating medium reinforced with fibers with the phase-lag and two temperatures. Deswal et al. [17] studied the law of fractional-order thermal conductivity within a two-temperature micropolar thermo-viscoelastic. Using radial ribs, the authors of [18] examined the improved rigidity of fused circular plates. The authors of [19] investigated the motion equation for a flexible element with one dimension utilized for the dynamical analysis of a multimedia structure.

The paper aims to study the thermoelastic interactions within an elastic standardized micropolar isotropic medium in the presence of rotation affected by a magnetic field. The authors could obtain accurate solutions of the measured factors using the Lame’s potential theory and the normal mode analysis. They could obtain and present in graphs the numerical findings of the distributions of stress, displacement, and temperature, for a crystal-like magnesium material.

2  Constitutive Relations and Field Equations

Constitutive relations and field equations are considered within a generalized micropolar magneto-thermoelastic medium in the presence of rotation as:

(i) The Constitutive Relations

σij=λur,rδij+μ(ui,j+uj,i)+k(uj,i-εijrϕr)-β1θδij,(1)

mij=αϕr,rδij+βϕi,j+γϕj,i(2)

(ii) Motion’s Stress Equation

(λ+μ)∇(∇.u→)+(μ+k)∇2u→+k(∇xϕ→)-β1∇θ+f→=ρ(∂2u→∂t2)+Ω→x(Ω→xu→)+2Ω→xu→.(3)

(iii) Motion’s Couple Stress Equation

(α+β+γ)∇(∇.ϕ→)-γ∇x(∇xϕ→)+k∇xu→-2kϕ→=ρJ∂2ϕ→∂t2.(4)

(iv) The Equation of Thermal Conductivity

K1*(1+K*∂∂t)∇2θ=ρcE(θ¨+β1T0e¨(5)

For these equations, the authors use the summation convention and utilize the comma to signify material derivative.

3  Formulating the Problem

Take into account an isotropic standardized micropolar generalized magneto-thermoelastic in the presence of rotation. The rectangular Cartesian coordinate scheme (x,y,z) is used in which the half-space surface plays the role of the plane z = 0 and the z − axis points vertically inwards. There are two extra terms for the displacement motion equation within the rotating frame, i.e., the Coriolis acceleration 2Ωxu̇→ because of the moving frame of reference and the centripetal acceleration Ω→x(Ω→xu→) because of the motion that varies over time. The analysis is restricted to the x − z plane. Therefore, the quantities within the medium do not depend on the variable y. The angular velocity Ω→, displacement vector u→, magnetic field vector H→ and micro-rotation vector ϕ→ have the components

u→=u(x,z,t),w→=w(x,z,t),Ω→=(0,Ω,0),H→=(0,H0,0),ϕ→=(0,ϕ2,0).(5a)

J→=curlh→,(6a)

curlE→=-μe∂h→∂t,(6b)

divh→=0,(6c)

divE→=0,(6d)

E→=-μe(∂u→∂t×H→),(6e)

h→=curl(u→xH→).(6f)

in which u→ represents the displacement vector, H→ represents the magnetic field, μe represents the magnetic permeability, J→ represents the density of the electric current, E→ represents the electric intensity, and h→ represents the perturbed magnetic field over the principal magnetic field. We apply the initial magnetic field vector H→ in Cartesian coordinates (x,y,z) to Eq. (1)

u→=(u,0,w),H→=(0,H0,0),E→=-μe(-H0∂w∂t,0,H0∂u∂t),h→=(-H0w,0,H0u),j→=-H0(∂u∂x+∂w∂z),f→=μe(j→xH→).(7)

where f→ is the Lorentz force.

Combining Eqs. (3), (4) and Eqs. (5a), (7) provides

(λ+μ)∂∂x(∂u∂x+∂w∂z)+(μ+k)∇2u-k∂ϕ2∂z-β1∂θ∂x+μeH02(∂2u∂x2+∂2w∂x∂z)=ρ(∂2u∂t2-Ω2u+2Ω∂w∂t),(8)

(λ+μ)∂∂z(∂u∂x+∂w∂z)+(μ+k)∇2w-k∂ϕ2∂x+β1∂θ∂z+μeH02(∂2w∂z2+∂2u∂x∂z)=ρ(∂2w∂t2-Ω2w-2Ω∂u∂t),(9)

γ∇2ϕ2+k(∂u∂z-∂w∂x)-2kϕ2=ρj∂2ϕ2∂t2.(10)

The following form expresses Eq. (5) of thermal conductivity given expression (5a)

(K1*(1+K*∂∂t)∇2θ=∂2∂t2(ρcEθ+β1*T0e).(11)

Substituting (5a) into (1) and (2), we can express the arising stress as

σxx=(λ+2μ+k)∂u∂x+λ∂w∂z-β1θ,(12)

σzz=(λ+2μ+k)∂w∂z+λ∂u∂x-β1θ,(13)

τzx=μ∂w∂x+(μ+k)∂u∂z-kϕ2,(14)

mzy=γ∂ϕ2∂z,mxy=γ∂ϕ2∂x.(15)

where

These variables that are non-dimensional are employed to change the previous equations into non-dimensional structures

(x′,z′)=ω*c1(x,z),t′=ω*t,(u′,w′)=ρω*c1T0β1(u,w),θ′=θT0,Ω′=Ωω*,σij′=σijT0β1,ϕ2′=ρc12T0β1ϕ2,mij′=ω*c1T0β1mij.(16)

where

Eqs. (8)–(11) concerning previous the variables that are non-dimensional decline to (dropping the primes)

(λ+μ)∂∂x(∂u∂x+∂w∂z)+(μ+k)∇2u-k∂ϕ2∂z-β1ρc12β1e∂θ∂x+μeH02(∂2u∂x2+∂2w∂x∂z)=ρc12(∂2u∂t2-Ω2u+2Ω∂w∂t),(17)

(λ+μ)∂∂z(∂u∂x+∂w∂z)+(μ+k)∇2w+k∂ϕ2∂z-β1ρc12β1e∂θ∂z+μeH02(∂2w∂z2+∂2u∂x∂z)=ρc12(∂2w∂t2-Ω2w-2Ω∂u∂t),(18)

γc12∇2ϕ2+kω*2(∂u∂z-∂w∂x-2ϕ2)=ρj∂2ϕ2∂t2,(19)

[K1*+K*ω*∂∂t]ω*2T0c12∇2θ=ω*2∂2∂t2(ρcET0θ+β1*T02β1eρc12e(20)

Utilizing Helmholtz decomposition [13] and providing the potential ϕ and ψ→ through the following equation

u→=∇ϕ+∇→∧ψ→,ψ→=(0,-ψ,0).(21)

Using Eq. (5a), the components of displacement u and w take the following form

u=∂ϕ∂x+∂ψ∂z,w=∂ϕ∂z-∂ψ∂x.(22)

where

ϕ(x,z,t) and ψ(x,z,t) represent scalar potential functions and ψ→ represents the vector of the potential function

Substituting Eq. (22) into Eqs. (17)–(20) gives us

(∇2-∂2∂t2+Ω2)ϕ-(1+β1∂∂t)θ+2Ω∂ψ∂t=0,(23)

(a0+a2+4∂∂t)ψ-2Ω∂ϕ∂t-a0ϕ2=0,(24)

(∇2-2a3-a4∂2∂t2)ϕ2+a3∇2ψ=0,(25)

({(a1+2a2)+4∂∂t}-4∂2∂t2)θ-4a5∂2∂t2∇2ϕ=0.(26)

where

4  Normal Mode Analysis

For resolving the governing equations, the authors measured physically the decomposed variable concerning the normal modes in the following form:

(u,w,ϕ,ψ,θ,σij,ϕ2)(x,z,t)=(u*,w*,ϕ*,ψ*,θ*,σij*,ϕ2*)(z)e(ωt+imx).(27)

in which ω represents the complex time constant (frequency), i represents the imaginary unit, m represents the wave number in the x-direction, and u*,w*,ϕ*,ψ*,θ*,σij* and ϕ2* represent the functions’ amplitudes.

Using (27), Eqs. (23)–(25) and Eq. (20) are, as follows

(D2-A1)ϕ*(z)-A2θ*(z)+A3ψ*(z)=0,(28)

(D2-B1)ψ*(z)-B2ϕ*(z)-B3ϕ2*(z)=0,(29)

(D2-C1)ϕ2*(z)+a3(D2-m2)ψ*(z)=0,(30)

(D1(D2-m2)-D3)θ*(z)-D2(D2-m2)ϕ*(z)=0.(31)

in which

Eliminating ψ*(z),ϕ*(z),ϕ2*(z) and θ*(z) from Eqs. (28)–(31), the differential equation become

(D8-PD6+QD4-RD2+S)(ψ*(z),ϕ*(z),ϕ2*(z),θ*(z))=0.(32)

in which

in which ε3=1D1.

Eq. (27) is represented, as follows

((D2-λ12)(D2-λ22)(D2-λ32)(D2-λ42))(ψ*(z),ϕ*(z),ϕ2*(z),θ*(z))=0.(33)

in which λn2 (n = 1, 2, 3, 4) represent the roots of the characteristic Eq. (32).

λ1=123((3e1-6e2)),λ2=123((3e1+6e2)),λ3=123((3e1-6e2)),λ4=123((3e1+6e2)).(34)

where

The solution of Eq. (28) is written as

ψ*(z)=∑n=14Mn(m,ω)e-λnz,(35)

ϕ*(z)=∑n=14Mn′(m,ω)e-λnz,(36)

ϕ2*(z)=∑n=14Mn″(m,ω)e-λnz,(37)

θ*(z)=∑n=14Mn‴(m,ω)e-λnz.(38)

in which Mn,Mn′,Mn″ and M″′n represent some parameters that depend on m and ω. Substitution of Eqs. (35)–(38) into Eqs. (28)–(31) gives the following relations

ϕ2*(z)=∑n=14S1nMn(m,ω)e-λnz,(39)

ϕ*(z)=∑n=14S2nMn(m,ω)e-λnz,(40)

θ*(z)=∑n=14S3nMn(m,ω)e-λnz.(41)

where

5  Applications

Take into account a magneto-thermoelastic micropolar solid having a half-space that rotates z≥0. Mn′s, as constants, are defined by imposing the proper boundary settings.

5.1 Thermal Boundary Conditions

The authors apply a heat shock of the ramp kind to the isothermal boundary of the plane z = 0, as follows

θ(x,0,t)=ϕ*δ(x)H(t).(42)

in which δ(x) represents Dirac-delta function and ϕ* represents a constant temperature. Moreover, H(t) represents a Heaviside function defined as

H(t)={0t≤0tt00≺t≤t01t≥t0(43)

In this equation, t0 represents a ramp parameter that shows the time required for raising the heat.

5.2 Mechanical Boundary Conditions

The boundary plane z = 0 is free of traction. In terms of mathematics, the boundary conditions take the following form

σzz(x,0,t)=0,(44)

τzx(x,0,t)=0,(45)

mzy(x,0,t)=0.(46)

Applying Eqs. (16)–(27) make the former boundary conditions represented, as follows

θ*(z)=R1,(47)

σzz*(z)=0,(48)

τzx*(z)=0,(49)

mzy*(z)=0.atz=0(50)

where R1=ϕ*1-e-ω0t0ω2.

Utilizing the non-dimensional quantities identified in (16), expressing stress Eqs. (12)–(15) and displacement Eq. (22), as well as the relations (44)–(46) becomes:

u*(z)=∑n=14(imS2n-λn)Mn(m,ω)e-λnz,(51)

w*(z)=∑n=14(-λn-im)Mn(m,ω)e-λnz,(52)

σzz*(z)=∑n=14S4nMn(m,ω)e-λnz,(53)

τzx*(z)=∑n=14S5nMn(m,ω)e-λnz,(54)

mzy*(z)=∑n=14S6nMn(m,ω)e-λnz.(55)

in which

A four-equation non-homogeneous structure is yielded by the boundary conditions (47)–(50), supported by expressions (51)–(55) and (38) as follows

[S31S32S33S34S41S42S43S44S51S52S53S54S61S62S63S64][M1M2M3M4]=[R1000](56)

The expressions of Mn, (n = 1, 2, 3, 4) attained by resolving the structure (56), when replaced in Eq. (38) and Eqs. (51)–(55), give these expressions of field variables

u(x,z,t)=[(imS21-λ1)M1e-λ1z+(imS22-λ2)M2e-λ2z+(imS23-λ3)M3e-λ3z+(imS24-λ4)M4e-λ4z]e(ωt+imx),(57)

w(x,z,t)=-[(λ1S21+im)M1e-λ1z+(λ2S22+im)M2e-λ2z+(λ3S23+im)M3e-λ3z+(λ4S24+im)M4e-λ4z]e(ωt+imx),(58)