MAGNETO-VISCOELASTIC PLANE WAVES IN ROTATING MEDIA IN THE GENERALIZED THERMOELASTICITY II

A study is made of the propagation of time-harmonic magneto-thermoviscoelastic plane waves in a homogeneous electrically conducting viscoelastic medium of Kelvin-Voigt type permeated by a primary uniform external magnetic field when the entire medium rotates with a uniform angular velocity. The generalized thermoelasticity theory of type II (Green and Naghdi model) is used to study the propagation of waves. A more general dispersion equation for coupled waves is derived to ascertain the effects of rotation, finite thermal wave speed of GN theory, viscoelastic parameters and the external magnetic field on the phase velocity, the attenuation coefficient, and the specific energy loss of the waves. Limiting cases for low and high frequencies are also studied. In absence of rotation, external magnetic field, and 
viscoelasticity, the general dispersion equation reduces to the dispersion equation for coupled thermal dilatational waves in generalized thermoelasticity II (GN model), not considered before. It reveals that the coupled thermal dilatational waves in generalized thermoelasticity II are unattenuated and nondispersive in contrast to the thermoelastic 
waves in classical coupled thermoelasticity (Chadwick (1960)) which suffer both attenuation and dispersion.


Formulation of the problem and the basic equations
We consider an infinite, homogenous, isotropic, thermally and electrically conducting viscoelastic solid permeated by a primary magnetic field B 0 = (B 1 ,B 2 ,B 3 ). The viscoelastic medium is characterized by the density ρ, Lame's constants λ, µ, and viscoelastic parameters λ , µ , and is uniformly rotating with an angular velocity Ω = Ω w, where w is the unit vector representing the direction of the axis of rotation. The displacement equation of motion in the rotating frame of reference has two additional terms: centripetal acceleration Ω × ( Ω × u) due to the time-varying motion only and the Coriolis acceleration 2 · Ω × • u, where u is the dynamic displacement vector. These terms do not appear in a nonrotating medium. The dynamic displacement vector is actually measured from a steady-state deformed position and the deformation is assumed to be small. The displacement equations of motion in a viscoelastic solid of Kelvin-Voigt type with increase of temperature θ above the reference temperature θ 0 are where J × B is the electromagnetic body force, J is the current density, B = B 0 + b is the total magnetic field, b = (b x ,b y ,b z ) is the perturbed magnetic field which is assumed to be small so that the products with b and u and their derivatives can be neglected for linearization of the field equations, v = (3λ + 2µ)α 0 , α 0 is the coefficient of linear thermal expansion of the solid, and the dots represent the derivatives with respect to time t. The coupled heat conduction equation of the theory of thermoelasticity (type II) without energy dissipation proposed by Green and Naghdi [14] is (2.2) where c v is the specific heat of the solid at constant volume, ρ is the density of the medium, T 0 is the initial reference temperature, k * (> 0) is a material constant characteristic of the theory, Q is the external rate of heat supply per unit mass, and ∆ is the dilatation so that ∆ = div u. The finite thermal wave speed is (k * /ρc v ) 1/2 . In the present problem Q = 0, so that the heat conduction equation becomes The electromagnetic field is governed by Maxwell's equations with the displacement current and charge density neglected [5] ∇ × H = J, where B = µ e H and µ e is the magnetic permeability. The generalized Ohm's law is where the time-independent part of Ω × u is neglected, σ is the electrical conductivity, ∂ u/∂t is the particle velocity of the medium, and the small effect of temperature gradient on J is also ignored.

Plane wave solutions and dispersion relation
We consider the propagation of plane waves in the rotating medium in the x-direction so that all quantities are proportional to exp[i(kx − ωt)], where (ω/2π) is the wave frequency and (2π/k) is the wave length. We will assume that ω is real, but k may be complex. The analysis will be carried out without any discussion of the time-independent stresses and displacements that are caused by the centrifugal force and other possible body forces. We look for time-varying dynamic solutions, and as such, the time-independent part of the centripetal acceleration as well as all body forces will be neglected. However, the time-dependent part of the electromagnetic body force will be taken into consideration. In view of the above assumptions, we write all the field quantities in the form where p 0 , q 0 , r 0 ; j 1 , j 2 , j 3 ; b 1 , b 2 , b 3 ; Ω 1 , Ω 2 , Ω 3 , and T 0 are all constants.
S. K. Roy Choudhuri and M. Banerjee (Chattopadhyay) 1823 It follows from (2.4c) that div b = 0 which implies that b x = 0, since initially b = 0. Also, it follows from (2.4a) that µ e J = ∇ × b so that Thus the term J × B in (2.1) can be replaced by J × B 0 given by (3.7). Substituting (3.1) and (3.2) into (2.3), we find (3.8b) The Replacing B by primary magnetic field B 0 , (2.5) takes the form Making use of (3.1) and (3.9) and neglecting the product terms, (3.10) with J = (J x ,J y ,J z ) yields Eliminating J from (3.6) and (3.11), we get (3.14) We next put (3.1), (3.2), (3.3), (3.4), and (3.5) into (2.1) and suppress the factor exp[i(kx − ωt)] throughout the subsequent discussion to obtain the following equations: We next rewrite (3.13) and (3.14) in order to obtain their final forms Since b = (0,b y ,b z ) and b-field is normal to x-axis, we then choose the y-axis and the z-axis such that b-field is along the y-axis. Invoking the additional assumption Ω 1 = Ω 2 = 0 and Ω 3 = Ω = 0 and considering that r 0 ≡ 0 provided that µk 2 − ρω 2 = 0 (evident from (3.17)) so that B 3 ≡ 0, we set the applied and perturbed magnetic fields as (B 1 ,B 2 ,0) and (0,b 2 ,0), respectively. This leads to the following three homogenous equations with three unknowns p 0 , q 0 , and b 2 as S. K. Roy Choudhuri and M. Banerjee (Chattopadhyay) 1825 Elimination of p 0 , q 0 , b 2 gives the dispersion equation It follows from the dispersion equation that the significant effects of the rotation, viscoelasticity, and the thermal field on the phase velocity Re(ω/k) are reflected through the terms involving Ω, λ , and µ and the term containing α through k * , characteristic of GN theory.
Introducing the above result and notations, (3.25) takes the form This equation indicates the influence of the rotation and the thermal field through c T , T , and viscoelastic parameters s 1 and s 2 on the phase velocity. In the absence of rotation (Ω 0 = 0) and viscoelasticity (s 1 = 0, s 2 = 0) with R L = B 2 1 /ρc 2 1 µ e = 0, the dispersion relation (3.28) reduces to In this case, the phase velocity is broken up into two factors. The first factor corresponds to s 2 ξ 2 − χ 2 = 0, which leads to a transverse elastic wave (unaffected by thermal field in absence of rotation as expected).
S. K. Roy Choudhuri and M. Banerjee (Chattopadhyay) 1827 The other factor leads to The first factor corresponds to quasistatic oscillations of the electromagnetic field, not coupled with the displacement field, Parkus [20]. The second factor of (3.31) corresponds to dispersion equation (not considered earlier so far) for purely thermoelastic waves (GN model) leading to in contrast to the equation derived by Chadwick [3] in classical coupled thermoelasticity theory. The roots of this equation are real, indicating that purely thermoelastic waves in thermoelasticity of type II (GN model) are unattenuated (without energy dissipation) , not yet considered but are subject to dispersion. The roots of (3.32) are The phase speed of the thermoelastic waves in GN model of thermoelasticity is C E,T p = χc 1 /ξ = c 1 / M 1 ± N 1 = V E ,V T corresponding to +ve and −ve signs. Setting T = 0 leads to V E = c 1 , (for those materials for which c T > 1) which is the elastic dilatational wave speed and V T = c 1 c T = k * /ρc v = finite thermal wave speed of GN model. Thus V E corresponds to modified elastic dilatational wave speed and v T corresponds to the modified thermal wave speed modified by c T which is the nondimensional thermal wave speed of GN model, a characteristic of the theory and the thermoelastic coupling constant T . Clearly v E < v T , implying that the modified elastic wave follows the modified thermal wave for those materials for which k * > ρc v c 1 2 . Equation (3.28) represents a more general dispersion relation in the sense that it incorporates the effects of rotation, viscoelasticity and the finite thermal wave speed c T , thermoelastic coupling T , and external magnetic field R H . Also, it shows that if the primary magnetic field has a transverse component, the longitudinal and transverse components of the displacement vector are linked together.
1828 Magneto-viscoelastic plane waves in rotating media As (3.28) is very complicated, we consider the following limiting cases in order to examine the effects of the rotation, viscoelasticity and the thermal wave speed c T , thermoelastic coupling T , and external magnetic field R H on the phase velocity, on attenuation coefficient of waves, and also on specific energy loss.

Low-frequency region (χ 1)
In this case, the wave frequency ω is much smaller than ω * . We consider this case with finite electrical conductivity (σ = 0, v H = 0). Thus when χ = 0, ξ 2 = 0 so that we can write ξ 2 = iφχ + 0(χ 2 ), where φ is to be determined. We substitute ξ 2 into (3.28), retain the terms containing χ 4 , and then equate the coefficient of χ 4 to zero in order to obtain an equation for φ as The root of (4.1) corresponds to one kind of slow wave because Thus for the low frequency, the rotation, the thermal field k * , T , and the viscoelasticity have no influence on the phase velocity in the case of finite conductivity. The equation for φ is linear so that its only root φ = (1 + R H )/ H . This corresponds to only one slow wave influenced by the electromagnetic field. This fact was not noticed in the works by Choudhuri and Debnath [12], which reveals two kinds of slow waves. Then the phase velocity can be found from the result where R 2 m = 1 + R H = 1 + v 2 A /c 2 1 and v A is the Alfvèn wave velocity. It follows from (4.3) that there exists a magneto-elastic wave. It follows from the real and imaginary parts of ξ that the phase velocity is The attenuation factor is (4.5) The phase speed and attenuation factor are independent of c T , viscoelastic parameters, the thermal wave speed, and thermoelastic coupling T to the order of (χ) for χ 1.
However considering terms of 0(χ 2 ) for χ 1, we obtain from the general dispersion equation (3.28) that where (4.7) It follows from the real and imaginary parts of ξ that the phase velocity is and the attenuation factor is (4.10) This confirms that the phase speed and the attenuation factors both are affected by rotation, viscoelastic parameters, finite thermal wave speed c T , the thermoelastic coupling T , the external magnetic field, and the electromagnetic parameter H .

High-frequency region (χ 1)
This case corresponds to the case of wave frequency ω, much larger than ω * . Dividing the dispersion equation (3.28) by χ 7 and neglecting all terms involving the second and higher powers of (1/χ), (3.28) becomes Thus the effect of rotation, viscoelastic parameters s 1 , s 2 , and the electromagnetic parameter H on the phase velocity is observed to the first order of (1/χ) whereas without viscoelastic effect, (5.2) Thus no effect of rotation and thermal parameters k * , T on the phase velocity is observed, but however electromagnetic parameter H affects it.
1830 Magneto-viscoelastic plane waves in rotating media To the first order of (1/χ) for (χ 1), the phase velocity c p and the attenuation coefficient factor a f are as follows: Now dividing (3.28) by χ 7 , retaining the terms of the order of (1/χ) 2 for (χ 1) and neglecting the higher powers of (1/χ), we obtain Clearly, the roots of this dispersion equation with complex coefficients are complex, indicating that the coupled magneto-thermo-viscoelastic waves undergo attenuation and dispersion. Both the phase speed and the attenuation coefficients of the coupled waves are influenced by rotation, viscoelastic parameters s 1 , s 2 , external magnetic field and the finite wave speed c T , characteristic of GN theory, and the thermoelastic coupling constant T to the second order of (1/χ) for large frequency.
In absence of viscoelastic effects, the above equation gives where where It follows from the real and the imaginary parts of ξ that the phase velocity is and the attenuation factor is The results (5.10) and (5.11) are similar to (5.4) and (5.6) reported by Choudhuri and Debnath [12]. The results (5.10) and (5.11) correspond to the phase speed and attenuation factor of the coupled magneto-thermoelastic wave in a rotating medium in thermoelasticity of type II (GN model), not considered so far. These are clearly influenced by c T , rotation, T , and the external magnetic field. It is important to observe that rotation does exert influence on both the phase velocity and the attenuation factor for high frequencies to the second order of (1/χ). Also both the phase speed and the attenuation factor are modified by the applied magnetic field, thermal parameters k * , T through the term L for high frequency. This fact was not noticed for the case of low frequency up to the order of (χ 2 ).

Specific energy loss
Making reference to Kolsky [15], the specific energy loss (∆W/W) is defined as the ratio of the energy dissipated per stress cycle to the total vibrational energy and is given by To the second order of χ for χ 1, the specific energy loss from (4.8) and (4.9) is given by Therefore, the specific energy loss is affected by rotation, finite thermal wave speed c T , the thermoelastic coupling T , the external magnetic field, and the electromagnetic parameter.
1832 Magneto-viscoelastic plane waves in rotating media To the first order of 1/χ for χ 1, the specific energy loss is obtained from (5.3) and (5.4) in the form Equation (6.3) shows that the specific energy loss is independent of any field parameters in the case of high frequency up to the first order of (1/χ).
However, to the second order of (1/χ), the expression for the specific energy loss in absence of viscoelastic effect is obtained from (5.10) and (5.11) in the form This result confirms that the specific energy loss is affected by the rotation to the second order of (1/χ) for the case of high frequency and depends on thermal parameters T , finite thermal wave speed c T of GN theory of thermoelasticity of type II, electromagnetic parameter H , and the transverse magnetic field.

Discussion
(1) Magneto-thermo-viscoelastic-dilatational shear waves in generalized thermoelasticity II undergo both attenuation and dispersion in contrast to the purely coupled thermoelastic waves in generalized thermoelasticity II (without thermal energy dissipation) which suffer no attenuation and dispersion.
(2) The coupled waves are influenced by the finite thermal wave speed of GN theory, rotation, viscoelastic parameters, and the external magnetic field, and by the fact that due to the presence of the transverse magnetic field R H and rotation, the longitudinal and transverse motions are linked together.
(3) For low frequency (χ 1, χ being the ratio of the frequency to some standard frequency ω * ), the rotation, viscoelastic parameters s 1 and s 2 , and the thermal field have no effect on the phase velocity to the first order of χ corresponding to only one slow wave influenced by the electromagnetic field R H only. But to the second order of χ, the phase velocity, attenuation coefficient, and the specific energy loss are affected by rotation, viscoelastic parameters and depend on the finite thermal wave speed c T , the thermoelastic coupling T , electromagnetic parameter, and the transverse magnetic field R H .
(4) For large frequency, rotation, viscoelastic parameters, and the electromagnetic parameter H influence the phase velocity and attenuation coefficient to the order of (1/χ). However, to the order of (1/χ) 2 , the phase speed and attenuation coefficients of the waves are affected by rotation, viscoelastic parameters s 1 , s 2 , the finite thermal wave speed c T of GN theory, the transverse magnetic field R H , and the thermoelastic coupling T .
(5) It reveals also that to the order of (1/χ), the specific energy loss is independent of any field parameters and is a constant. Without viscoelastic effect, the specific loss is however affected by rotation, finite thermal wave speed c T , external magnetic field R H , and the electromagnetic parameter H to the second order of (1/χ) in the case of magnetothermoelastic waves in rotating media in generalized thermoelasticity II, not studied before.