Journal of Mechanical Science and Technology 24 (2010) 337~342 www.springerlink.com/content/1738-494x
DOI 10.1007/s12206-009-1109-6
Dynamic problem of generalized thermoelastic diffusive medium† Rajneesh Kumar* and Tarun Kansal Department of Mathematics, Kurukshetra University, Kurukshetra-136 119, India (Manuscript Received January 21, 2009; Revised September 25, 2009; Accepted October 7, 2009) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract The equations of generalized thermoelastic diffusion, based on the theory of Lord and Shulman with one relaxation time, are derived for anisotropic media with rotation. The variational principle and reciprocity theorem for the governing equations are derived. The propagation of leaky Rayleigh waves in a viscous fluid layer overlying a homogeneous isotropic, generalized thermoelastic diffusive halfspace with rotating frame of reference is studied. Keywords: Wave propagation; Rotation; Viscous fluid; Phase velocity; Attenuation coefficient ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction The study of the interaction of elastic waves with fluidloaded solids has been recognized as a viable means for nondestructive evaluation of solid structures. Qi [1] studied the influence of viscous fluid loading on the propagation of leaky Rayleigh wave in the presence of heat conduction effects. The spontaneous movement of the particles from high concentration region to the low concentration region is defined as diffusion and it occurs in response to a concentration gradient expressed as the change in the concentration due to change in position. The thermo diffusion in elastic solids is due to coupling of fields of temperature, mass diffusion and that of strain in addition to heat and mass exchange with environment. This article deals with the generalized thermoelastic diffusion problem with one relaxation time for anisotropic media in rotating frame of reference. The propagation of leaky Rayleigh waves in a viscous fluid layer overlying a homogeneous isotropic, thermoelastic diffusive half-space with rotating frame of reference in the context of generalized theoryies of thermoelastic diffusion is investigated. The phase velocity and attenuation coefficient of these waves are presented graphically to depict the effect of viscosity.
2. Basic Equations The basic governing equations for an anisotropic, homogeneous elastic solid with generalized thermoelastic diffusion in †
This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin * Corresponding author. Tel.: +94 1612 0992, Fax.: + E-mail address:
[email protected] © KSME & Springer 2010
rotating frame of reference are:
σ ij , j + ρ Fi = ρ[ui + {Ω × (Ω × u)}i + (2Ω × u )i ], σ ij = cijkl ekl + aijT + bij C , −qi ,i = ρT0 S , − K ijT, j = qi + τ 0 qi , ρST0 = ρC E T − aij eij T0 + aCT0 , ∂α ij* P, j ∂xi
= C + τ 0C , P = bij eij + bC − aT ,
(1)
Here, the medium is rotating with angular velocity, Ω = Ωyˆ where yˆ is the unit vector along the axis of rotation
and the equations of motion include two additional terms: the centripetal acceleration Ω × (Ω × u) due to time-varying motion, and the Coriolis acceleration 2Ω × u . The remaining symbols have their usual similar meanings as in Aouadi [2]. For a coupled thermoelastic diffusion problem, τ 0 = τ 0 = 0. Equations (ii), (v) and (vii) of (1) can be rewritten as:
σ ij = d ijkl ekl + sijT + ς ij P, ρ S = κ T − sij eij + nP , C = ε P − ς kl ekl + nT ,
(2)
bij bkl bij , dijkl = cijkl − , ς ij = , b b b where 2 1 ρ CE a a + ,ε = , n = . κ= T0 b b b sij = aij +
abij
3. Variational principle The principle of virtual work with variation of displacements (the virtual work of body forces Fi , inertial forces ρ ui , centripetal forces ρ{Ω × (Ω × u)}i , Coriolis forces ρ (2Ω × u )i ,
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surface forces hi = σ ji n j and the virtual work of internal forces) for the elastic deformable body is written as
∫ ρ[ X
i
∫
∫
− ui ]δu i dV + hi δu i dA = σ ji δu i , j dV ,
V
A
δ ui =
V
X i = Fi − {Ω × (Ω × u )}i − (2Ω × u )i
(3)
We define a vector J connected with the entropy through the relation ρ S = − J i ,i . Using this relation and from eqns. (iiiiv) of (1) , we obtain (4)
Multiplying (4) by δ J i and integrating over the region of the body and using the divergence theorem with the aid of (ii) of eqn. (2) and above relation, we get
∫ Tn δ J dA − ∫ s Tδ e dV + n∫ Tδ PdV + δ (G + M ) = 0 j
ij
A
ij
V
κ
V
1 G= T dV , M = 2 2
∫
2
V
T0 d d2 ( + τ 0 2 ) J iδ J j dV Kij dt dt
∫
V
Now we introduce the vector function N defined as C = − Ni ,i . Using this relation and from eqns. (vi) of (1) , we
obtain 1
α ij*
(
d d2 + τ 0 2 ) N i + P, j = 0 dt dt
(5)
Multiplying (5) by δ Ni and integrating over the region of the body and using the divergence theorem with the aid of (iii) of eqn. (2) and above relation, we get
∫ Pn δ N dA − ∫ ς Pδ e dV + n∫ Pδ TdV + δ ( X + Y ) = 0 j
j
ij
A
X =
ij
V
ε 2
∫
P 2 dV , Y =
V
V
2 1 1 dNi 0 d Ni ( τ )δ N j dV + 2 α ij* dt dt 2
∫
Using above two derived equations, eqns. (2) and (3), we obtain the variational principle in the final form
∫
δ (W + G + M + X + Y + n PTdV ) = V
∫ ρ[ X − u ]δ u dV + ∫ h δ u dA − ∫ Pn δ N dA − ∫ Tn δ J dA i
i
i
where W =
i
A
and the above equation is reduced to the following relation d (W + G + M + X + Y + n PTdV ) = dt
∫
V
∫ ρ[ X − u ]u dV + ∫ h u dA − ∫ Pn N dA − ∫ Tn J dA
i
i
A
i
i
i
V
i i
A
i
A
i
i i
A
The uniqueness theorem can be proved on the basis of Aouadi [2].
4. Reciprocity theorem We shall consider a homogeneous anisotropic generalized thermoelastic diffusive body occupying the region V and bounded by the surface A . We assume that the stresses σ ij and the strains eij are continuous together with their first derivatives whereas ui ,T , C , P are continuous and have continuous derivatives up to the second order, for x ε V + A, t > 0 . The components of surface traction, the normal component of the heat flux and the normal component of the chemical flux at regular points of ∂V , are given by hi = σ ji n j , q = KijT, j ni , p = α ij* P, j n j
(6)
respectively. We denote by n j the outward unit normal of ∂V . To the system of field equation, we must adjoin boundary conditions and initial conditions. We consider boundary conditions as: ui (x, t ) = Ui (x, t ),T (x, t ) = φ (x, t ), P(x, t ) = ψ (x, t ), for all x ε A, t > 0 and homogeneous initial conditions: ui ( x,0) = ui (x,0) = T (x,0) = T (x,0) = P(x,0) = P (x,0) = 0, f
or all x ε V , t = 0
V
V
∂ui ∂T , etc. dT = ui dT , δ T = dT = TdT ∂t ∂t
i
T0 d d2 ( + τ 0 2 ) J i + T, j = 0 K ij dt dt
j
tual increment of the temperature δ T , etc. correspond to the increments occurring really in the body in question. Then
i
i
A
1 dijkl eij ekl dV . 2
∫ V
Now we assume that the virtual displacements δ ui , the vir-
We derive the dynamic reciprocity relationship for a generalized thermodiffusion bounded body V , which satisfies eqs. (1) and (2), the boundary conditions and the homogeneous initial conditions, and subjected to the action of body forces Fi (x, t ) , surface traction hi (x, t ) , centrifugal forces Ω×(Ω×u) and Coriolis forces 2Ω × u , the heat flux q (x, t ) and the chemical flux p (x, t ) . The Laplace integral transform is defined as ∞
f (x, s ) = L( f (x, t )) =
∫ f ( x, t ) e 0
− st
dt
(7)
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We now consider two problems where applied body forces, chemical potential and the surface temperature are specified differently. Let the variables involved in these two problems be distinguished by superscripts in parentheses. Thus, we have ui(1) , eij(1) , σ ij(1) , T (1) , P (1) ..... for the first problem and ui(2) , eij(2) , σ ij(2) , T (2) , P (2) ..... for the second problem. Each set of variables satisfies the system of Eqs. (1), (2), the boundary conditions and the homogeneous initial conditions. Using the strain-displacement relation, the assumption σ ij = σ ji and the divergence theorem with the aid of (i) of (1), (6) and (7), we obtain
∫
∫
∫
σ ij(1)eij(2) dV = hi(1)ui(2) dA − ρ s 2ui(1)ui(2) dV +
V
A
V
∫
ρ [ Fi(1) − {Ω × (Ω × u)}i(1) − (2Ω × su )i(1) ]ui(2) dV V
ized parts of the reciprocity theorem, we shall use the convolution theorem t
L−1{F ( s )G ( s )} =
∫
t
∫
f (t − ξ ) g (ξ )dξ = g (t − ξ ) f (ξ )d ξ
0
0
Applying inverse Laplace transform in first, second and generalized parts of the reciprocity theorem, we obtain these parts in final form.
5. Isotropic case By setting cijkm = λδ ijδ km + µ (δ ik δ jm + δ jkδ im ), K ij = K δ ij ,
α ij* = Dδ ij , aij = − β1δ ij , bij = − β 2δ ij ,
A similar expression is obtained for the integral
∫σ
(2) (1) ij eij dV ,
from which together with the above equation, it
V
follows that
∫[σ
(1) (2) ij eij
∫
−σ ij(2)eij(1) ]dV = [ hi(1)ui(2) − hi(2)ui(1) ]dA +
V
ρ
∫
A
[Yi(1)ui(2)
− Yi(2)ui(1) ]dV ,
V
Yi( g ) = Fi ( g ) − {Ω × (Ω × u )}i( g ) − (2Ω × su )i( g ) , g = 1, 2
Now multiplying eij(2) by the corresponding equation (i) of (2) for the first problem, eij(1) by the analogous equation for the second problem, subtracting and integrating over the region V and using the symmetry properties of dijkl , we obtain
∫[σ
(1) (2) ij eij
∫
−σ ij(2)eij(1) ]dV = sij [T (1) eij(2) − T (2) eij(1) ]dV +
V
∫
V
of force stress tensor in the viscous medium, u o = (u1o , u2o , u3o ) is the displacement vector in the viscous medium,
From above two equations, we get the first part of the reciprocity theorem. Following Aouadi [2],we can obtain the second part of reciprocity theorem.
∫
Eliminating integrals sij [T (1)eij(2) − T (2)eij(1) ]dV , V
∫
4 ∂ ∂ o ( K o + η )∇ (∇.u o ) − η ∇ × (∇ × u o ) = ρ ou 3 ∂t ∂t 2 ∂ ∂ π ij = ( K o − η )u ok , kδ ij + η (uio, j + u oj ,i ) 3 ∂t ∂t
Where ρ o is the fluid density at rest. π ij are the components
ς ij [ P (1)eij(2) − P (2) eij(1) ]dV
V
V
where λ and µ are Lame's constants, K is the coefficient of thermal conductivity, D is the diffusion coefficient, ( β1 , β 2 ) = (3λ + 2 µ )(α t ,α c ), α t and α c are the coefficients of linear thermal and diffusion expansion, in the Eqs. (1), we obtain the basic equations for isotropic, homogeneous elastic solid with generalized thermoelastic diffusion and from which the equations of motion in the rotating frame of reference, equation of heat conduction and equation of mass diffusion can be obtained as given in Kumar et al. [3] based upon LordShulman (L-S) and Coupled (CT) theories of thermoelasticity. Following Felher [4], the basic equations in a viscous medium are
ς ij [ P (1)eij(2)
−
P (2) eij(1) ]dV ,
∫
o co = K
ρo
is the sound velocity, K o and η are respec-
tively bulk modulus and fluid viscosity.
6. Formulation and solution of the problem We consider a viscous compressible fluid layer of thickness
n[ P (2)T (1) − P (1)T (2) ]dV
V
from first and second parts of the reciprocity theorems, we obtain the generalized parts of the reciprocity theorem in the Laplace transform domain. To invert the Laplace transform in first, second and general-
H overlying a homogeneous isotropic, generalized thermoe-
lastic diffusive half-space in rotating frame of reference. The origin of the Cartesian coordinate system ( x1, x2 , x3 ) is taken at any point on the plane surface (interface) and x3 -axis points vertically downwards into the solid half-space which is thus represented by x3 ≥ 0 . In this paper, the problem of symmetric with respect to x3 -axis is considered, so that, all vari-
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R. Kumar and T. Kansal / Journal of Mechanical Science and Technology 24 (2010) 337~342
ables are independent of x2 . We define the following dimensional quantities:
µ* =
gating in a viscous fluid layer overlying isotropic thermoelastic diffusive half-space with rotation as
π ij µ ηω1* * λ ,η * = ,λ = , π ij' = , 2 β1T0 β1T0 β1T0 ρ f c1
ρ o* =
R1M 1 − R2 M 2 − R3 M 3 − R4 M 4 = 0 ,
ρ oc12 o ' o ' (u1o , u3o ) w1* * ,(u1 , u3 ) = ,η1 = η *ρ o* , β1T0 c1
Where w1* and c1 are the characteristic frequency and the longitudinal wave velocity respectively. For viscous fluid, we have u1o =
∂φ o ∂ψ o o ∂φ o ∂ψ o − + , u3 = ∂x1 ∂x3 ∂x3 ∂x1
(8)
Where, φ o and ψ o are scalar velocity potential and vector velocity potential components along the x2 -axis. The remaining dimensionless quantities can be taken from Kumar et al. [3]. Using dimensionless quantities and (8) in the basic equations of both isotropic thermoelastic diffusive halfspace and viscous fluid and assuming the solution (u1 , u3 , T , C ,φ o ,ψ o ) = (1, L, S , R, E , F )U eιξ ( x1 + mx3 − ct ) where
ω , is the dimensionless phase velocity, ξ quency and ξ is the complex wave number; known parameter. 1, L, S , R are, respectively, ratios of displacements u1 , u3 , T and C with c=
ω is the frem is still un-
the amplitude respect to u1 ,
the expressions for u1 , u3 , T , C ,φ o and ψ o are obtained as 4
(u1 , u3 , T , C ) =
∑
Where Ri , M i , i = 1, 2,3, 4 are given in appendix. Case-I: In absence of rotation effect, we obtain the dispersion equation for leaky Rayleigh waves propagating in a viscous fluid layer overlying isotropic thermoelastic diffusive half-space. Case-II: For H → ∞ , we obtain the secular equation for leaky Rayleigh waves at solid-fluid interface x3 = 0 with rotation. Case-III: For H → 0 , we obtain the secular equation for Rayleigh waves propagating in isotropic thermoelastic diffusive half-space with rotation.
9. Numerical results and discussion We now represent for numerical results for copper material (thermoelastic diffusive solid), the physical data is given below: 10 -1 2 10 -1 2 λ = 7.76×10 kg·m ·s , µ = 3.86×10 kg·m ·s , T0 = 0.293×103 K, CE = 0.3831×103 J·kg-1·K-1, α t = 1.78×10-5 K-1, α c = 1.98×10-4 m3·kg-1,
a = 1.2×104 m2·s-2·K-1, b = 9×105kg·m5·s-2, τ 0 = 0.25s, τ 0 = 0.15s,D=0.85×10-8 g·m-3·s, ρ = 8.954×103 kg·m-3, K = 0.386×103 W·m-1·K-1 , Ω =12 rpm.
(1, ni , ki , fi ) Ai exp(ιξ ( x1 + mx3 − ct )),
i =1
φ o = [ A5 cos(ξ m5 x3 ) + A6 sin(ξ m5 x3 )]exp(ιξ ( x1 − ct )),
(9)
ψ o = [ A7 cos(ξ m6 x3 ) + A8 sin(ξ m6 x3 )]exp(ιξ ( x1 − ct )), where mi2 , i = 1,2,3,4 are the roots of the polynomial equation: m8 + A*m6 + B*m 4 + C *m 2 + D* = 0,
Where the coefficients A* , B* , C * and D* are given in the appendix. The roots m5 , m6 are also given in the appendix.
7. Boundary conditions The boundary conditions at x3 = − H and at solid-fluid interface x3 = 0 are as follows:
π 33 = π 31 = 0 at x3 = − H σ 33 = π 33 , σ 31 = π 31, u1 = u1o , u3 = u3o , T,3 = C,3 = 0 at x3 = 0 .
8. Derivation of secular equations Invoking the boundary conditions and using Eqs. (9), we obtain the secular equation for leaky Rayleigh waves propa-
For viscous fluid (kerosene and seawater), we take the numerical data from White [5]. In Figs. 1 and 2, the solid and dash lines correspond to the CT theory of thermoelastic diffusion for first and second modes for the case of kerosene (CT1K and CT2K), respectively. The diamond and square symbols, respectively, on these lines correspond to the CT theory of thermoelastic diffusion for first and second modes for the case of seawater (CT1S and CT2S). Similarly, the star and lower triangles, respectively, on these lines correspond to the L-S theory of thermoelastic diffusion for first and second modes for the case of kerosene (LS1K and LS2K). The triangle and circle symbols, respectively on solid and dash lines correspond to the L-S theory of thermoelastic diffusion for first and second modes for the case of seawater (LS1S and LS2S). From Fig. 1, we observe that in both the cases of kerosene and seawater, the values of phase velocity corresponding to CT1K, CT2K, LS1K and LS2K increase. The values of phase velocity for the second mode are higher than that of first mode in both the cases of CT and L-S theories. The values of phase velocity for CT theory are more as compared to L-S theory for both modes. If we compare both the cases, we find that the increase in the phase velocity is more in the case of seawater than that of kerosene. Fig. 2 shows that the values of attenua-
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341
References [1] Qi,Q., Attenuated leaky Rayleigh waves. J. Acoust. Soc. America 95 (1994) 3222-3231. [2] M. Aouadi, Generalized theory of thermoelastic diffusion for anisotropic media, J. Thermal Stresses 31 (2008) 270-285. [3] R. Kumar and T. Kansal, Effect of rotation on Rayleigh waves in an isotropic generalized thermoelastic diffusive half-space, Archives of mechanics, 60 (5) (2008) 421-443, 2008. [4] M.Fehler, Interactions of seismic waves with a viscous liquid layer, Bull. Seismological Society of America, 72 (1982), 55-72. [5] F. M.White, Fluid Mechanics-McGraw Hill Int., Eds., 1994.
Appendix Fig. 1. Variations of phase velocity w.r.t frequency. 0 (τ t10 ,τ 10 f ) = 1 − ιω (τ 0 ,τ ), Γ =
Ω
ω
, f 2 = 2ιΓc 2 ,
( f1 , f3 ) = (1, δ1 ) − c 2 (1 + Γ 2 ), f 4 = ζ 2τ t10c, f5 = ιξ * +cτ t10 , f 6 = cζ 1τ t10 , f7 = q3* − ιω −1c 2τ 10 f , I1 = q1 −
q3* , I 2 = q1* − f3q3* , I 3 = f5q3* + ιξ f7 + f 6 q2* , I 4 = f 6 − f5 , I 5 = q3* + q2* , I 6 = f 5 f 7 + f 6 q2* , I 7 = f7 + q2* , I8 = q1* − δ 2 q3* , I9 = ιξ − I 4 , I10 = f 4 q3* + q1* f 6 , I11 = f6 + f 4 , I12 = f 4 f 7 + q1* f 6 , I13 = f 7 + q3* f3 − q1* , I14 = I11 + f3 f 6 , I15 = q1* (ιξ + f5 ) − f 4 q2* , I16 = f5 + ιξ f3 + f 4 , I17 = f5 q1* − f 4q2* , I18 = ( f3 + 1)q2* + q1* , d1* = ιξ I 2 − I 3 − q1*I 4 − f 4 I 5 , d 2* = f3 I 3 + I 6 + q1*I 4 + f 4 I 7 , d3* = q1*I 9 − δ 2 I3 − f 4 I 5 , d 4* = δ 2 I 6 + f 4 I 7 + q1* I 4 , d5* = δ 2 I10 + f 4 I1 − q1* I11 , d 6* =
δ 2 I12 − f 4 I13 − q1*I14 , d 7* = ιξ q1* (δ 2 − 1), d8* = Fig. 2. Variations of attenuation coefficient w.r.t. frequency.
tion coefficient vibrate in all the cases. The values of attenuation coefficient for the case of seawater increase more as compared to the case of kerosene. For CT theory, the values increase slightly higher than that of L-S theory in both the cases of kerosene and seawater.
10. Conclusions The generalized model for thermoelastic diffusion equations for anisotropic media in the rotating frame of reference based on the theory of Lord and Shulman with one relaxation time is given. The propagation of leaky Rayleigh waves in a viscous fluid layer lying over a homogeneous isotropic thermoelastic diffusive half-space with rotating frame of reference has been considered. The phase velocity and attenuation coefficient of leaky Rayleigh waves are presented graphically.
δ 2 I15 + f 4 (q1* + q2* ) − q1* (ιξ + I16 ), d9* = δ 2 I17 + * f 4 I18 − q1* ( f3 f5 + I16 ), d10 = ιξ ( f1I1 − δ 2 I8 ) + δ1d1* − d 7* , * * d11 = f1d1* − δ1d 2* − δ 2 d3* − ιξ f 22 q3* + d 5* − d8* , d12 = * = δ 2 d 4* − f 22 I 3 − δ1 f3 I 6 − f1d 2* + d 6* − d9* , d13 * f3 ( I12 + f1I 6 − I17 ) + f 22 I 6 , d14 = ιξδ1I1 , ∆1i =
ξ I8 mi5 + ιξ q3* f 2mi4 + ι d3*mi3 − f 2 I 3mi2 + ι d 4*mi + f 2 I 6 , ∆ 2i = d 5*mi4 − ι f 2 I10 mi3 − d 6*mi2 + ι f 2 I12 mi − f3 I12 , ∆3i = d 7*mi6 + ξ q1* f 2 mi5 − d8*mi4 + ι f 2 I15mi3 + d 9*mi2 − ι f 2 I17 mi + f3 I17 , ∆ 4i = −ιξ I1mi6 + d1*mi4 + d 2*mi2 − f3 I 6 , * * * * * * A* = d10 / d14 , B* = d11 / d14 , C * = d12 / d14 , D* = * * −d13 / d14 , ni = ∆1i / ∆ 4i , ki = ∆ 2i / ∆ 4i ,
fi = ∆3i / ∆ 4i , i = 1, 2, 3,4. m5 = −1 +
3c 2f
3c12c 2 ιc , m6 = −1 + * − 4ιωη *c12 ξη
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di = ιλ * − (λ * + 2 µ * )(mi ni + ξ −1 (ki + fi )), hi = µ (ι ni − mi ), i = 1, 2,3, 4 *
ci = cos(ξ mi H ), si = sin(ξ mi H ), i = 5,6, R1 = R2 =
(m62 − 1) s5 , c5
R1 2ι m6 s6 2ι m6c6 , R3 = , R4 = , c5 c5 tan(ξ m5 H )
∆1 = [ R3c6 ∆ 2 = [ − R3s6
0 −η1* R3ξ −2ιξη1*m6
∆3 = [2ι m5 s5 η1* R3ξ ∆ 4 = [2ι m5c5 ∆i = [0 di
0 −1 0 0]Tr , 0 0 0] , Tr
0 −1 0 0 0] ,
0 −2ιξη1*m5 hi
Tr
0 −ι m6
−c −cni
0 ι m5 mi ki
Tr
0 0] , Tr
mi fi ] , i = 5,6,7,8
M 1 = det(∆1 ∆ 3
∆2
∆5
∆6
∆7
∆8 ),
M 2 = det(∆ 4
∆3
∆2
∆5
∆6
∆7
∆ 8 ),
M 3 = det(∆ 4
∆1 ∆ 2
∆5
∆6
∆7
∆8 ),
M 4 = det(∆ 4
∆1 ∆ 3
∆5
∆6
∆7
∆8 ),
Prof. Rajneesh Kumar Born on 08-081958, did M.Sc. (1980) from Guru Nanak Dev University, Amritsar (Punjab, India), M. Phil. (1982) from Kurukshetra University Kurukshetra (Haryana, India) and Ph. D. (1986) in Applied Mathematics from Guru Nanak Dev University, Amritsar (Punjab, India). Guided 52 M. Phil. students, 9 students awarded Ph.D. degree and 8 students are doing Ph.D. under my supervision. I have 200 publications in Journals of international repute. My area of research work is Continuum Mechanics (micropolar elasticity, thermoelasticity, poroelasticity, magnetoelasticity, micropolar porous couple stress theory, viscoelasticity, mechanics of fluid).
Mr. Tarun Kansal Born on 05-101982, did M.Sc. (2004) from Kurukshetra University, Kurukshetra (Haryana, India.). Presently doing research as a Senior Research Fellow (SRF-CSIR) in the Department of Mathematics, Kuru-kshetra University, Kurukshetra since Jan 8, 2007.
