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Effect of thermal loading due to laser pulse on thermoelastic porous medium under G-N theory
Accepted Manuscript Effect of Thermal Loading due to Laser Pulse on Thermoelastic Porous Medium under G-N Theory Mohamed I.A. Othman, M. Marin PII: DOI: Reference:
S2211-3797(17)31447-X https://doi.org/10.1016/j.rinp.2017.10.012 RINP 990
To appear in:
Results in Physics
Received Date: Accepted Date:
4 August 2017 4 October 2017
Please cite this article as: Othman, M.I.A., Marin, M., Effect of Thermal Loading due to Laser Pulse on Thermoelastic Porous Medium under G-N Theory, Results in Physics (2017), doi: https://doi.org/10.1016/j.rinp.2017.10.012
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Effect of Thermal Loading due to Laser Pulse on Thermoelastic Porous Medium under G-N Theory 1,2
Mohamed I. A. Othman and M. Marin 1
3
Department of Mathematics, Faculty of Science, P.O. Box 44519, Zagazig University, Zagazig, Egypt 2
Department of Mathematics, Faculty of Science, Taif University 888, Taif City, Saudi Arabia 3
Department of Mathematics, Transilvania University of Brasov, Romania Email: [email protected], [email protected]
Abstract: The aim of this paper is to study the wave propagation of generalized thermoelastic medium with voids under the effect of thermal loading due to laser pulse with energy dissipation. The material is a homogeneous isotropic elastic halfspace and heated by a non-Gaussian laser beam with the pulse duration of 0.2 ps. A normal mode method is proposed to analyse the problem and obtain numerical solutions for the displacement components, stresses, temperature distribution and the change in the volume fraction field. The results of the physical quantities have been illustrated graphically by comparison between both types II and III of Green-Naghdi theory for two values of time, as well as with and without void parameters. Keywords: Laser pulse; Voids; Energy dissipation; Green-Naghdi theory; Wave propagation; Thermoelasticity. 1. Introduction Green and Naghdi [1-3] developed a generalized theory of thermoelasticity which involves thermal displacement gradient as one of the constitutive variables in contrast to the classical coupled thermoelasticity which includes temperature gradient as one of the constitutive variables. An important feature of this theory is that it does not accommodate dissipation of thermal energy. On this theory the characterization of material response to a thermal phenomenon is based on three types of constitutive response functions. The nature of those three types of constitutive response functions is such that when the respective theories are linearized, type I is same as classical heat conduction equation (based on Fourier’s law), whereas type II, the internal rate of production of entropy is taken to be identically zero, implying no dissipation of thermal energy. This model is known as the theory of thermoelasticity without energy dissipation. Type III involves the previous two models as special cases, and admits dissipation of energy in general, in this model, introducing the temperature gradient 1
and thermal displacement gradient as the constitutive variables. Othman and Abbas [4] investigated the problem of generalized thermoelsticity of thermal shock in a nonhomogeneous isotropic hollow cylinder with energy dissipation. Kumar and Kumar [5] studied the propagation of wave at the boundary surface of transversely isotropic thermoelastic material with voids and isotropic elastic half-space. Abbas and Othman [6] used G-N theory to study the effect of rotation on thermoelstic waves in a homogeneous isotropic hollow cylinder. Excitation of thermoelastic waves by a pulsed laser in solid is of great interest due to extensive applications of pulsed laser technologies in material processing and non-destructive detecting and characterization. When a solid is illuminated with a laser pulse, absorption of the laser pulse results in a localized temperature increase, which in turn causes thermal expansion and generates a thermoelastic wave in the solid. In ultra-short pulsed laser heating, two effects become important. One is the non-Fourier’s effect in heat conduction which is a modification of the Fourier heat conduction theory to account for the effect of mean free time (thermal relaxation time) in the energy carrier’s collision process. Consideration of the non-Fourier effect also eliminates the paradox of the infinite heat propagation speed by Joseph and Preziosi [7, 8]. The other is the dissipation of the stress wave due to coupling between temperature and strain rate, which causes transform of mechanical energy associated with the stress wave to the thermal energy of the material. Othman and Kumar [9] investigated the reflection of magneto-thermoelasticity waves with temperature dependent properties in generalized thermoelasticity. Theory of elastic materials with voids is one of the most important generalizations of the classical theory of elasticity. This theory is concerned with elastic materials consisting of a distribution of small voids in which the void volume is included among the kinematic variables. Practically, this theory is useful for investigating various types of geological and biological materials for which elastic theory is inadequate. Nunziato and Cowin [10] developed a nonlinear theory of elastic material with voids. Cowin and Nunziato [11]
investigated the theory of linear elastic materials with voids. Youssef [12] used the state-space approach to two-temperature generalized thermoelasticity without energy dissipation of medium subjected to moving heat source. Marin et al. [13] studied an extension of the domain of influence theorem for anisotropic thermoelastic material with voids. Marin et al. [14] investigated the localization in time of solutions for 2
thermoelastic micropolar materials with voids. Othman et al. [15] studied the effect of diffusion on two-dimensional problem of generalized thermoelasticity with GreenNaghdi theory. In the present paper the wave propagation of the components of displacement, stress, temperature distribution and change in the volume fraction field in an isotropic homogeneous thermoelastic solid with voids subjected to thermal loading due laser pulse under (G-N III) was studied. The results performed and presented graphically with concluding remarks are given. 2. Formulation of the problem and basic equations We consider a homogeneous, isotropic, thermoelastic material with voids in the undeformed temperature T 0 , with a half space ( x ≥ 0 ). The rectangular Cartesian coordinate system ( x , y , z ) have originated on the surface (z = 0). For two dimensional problem we assume the dynamic displacement vector as u = (u,v, 0). All quantities considered will be a function of the time variable t, and of the coordinates x and y. According to Cowin and Nunziato [11], the field equations and constitutive
relations for linear homogenous, isotropic generalized thermoelastic solid with voids without body forces, heat sources and extrinsic equilibrated body force can be expressed in the context of (G-N III), as; µ u i, jj + (λ + µ ) u j ,ij + b φ,i − β T ,i = ρui ,
(1)
α φ,kk − b u k,k − ξ1φ − ω0 φ + m T = ρ ψ φ
(2)
kT ,ii + k *T,ii − mT 0 φ = ρ C e T + β T 0 ui ,i − ρ Q
(3)
σ ij = λ u k,k δ ij + µ (u i , j + u j ,i ) + b φ δ ij − β T δ ij ,
(4)
∂u
where e ij = 12 ( ∂∂xui + ∂x j ) are the components of strain tensor, ( i , j = 1,2, 3 ), λ , µ are the j
i
Lame' constants, α , b , ξ1 , ω0 , m , ψ are the material constants due to presence of voids, T
is the temperature distribution, β = (3λ + 2µ )αt such that αt is the coefficient of
thermal expansion, ρ is the density, C E is the specific heat, k is the thermal conductivity, k * is the material constant characteristic of the theory, T 0 is the reference temperature chosen so that (T −T 0 ) T 0 << 1 , φ is the change in volume fraction field, σ ij are the components of stress tensor, δ ij is the Kronecker delta, t is the time variable, a comma denotes material derivatives and Q is the heat input of the 3
laser pulse. 3. The boundary conditions The coefficients M n (n = 1, 2,3, 4) have to be determined such that the boundary conditions on the surface
y =0
take the form:
σ xx (x ,0, t ) = p1 (x ,t ) = − p1∗ exp[i (a x − ω t )], σ xy (x , 0, t ) = 0,
∂φ ∂y
= 0,
∂θ ∂y
=0
(5)
where p1 (x , t ) is an arbitrary function of x , t and p1∗ is the magnitude of the constant stress applied to the boundary. The plate surface is illuminated by laser pulse given by the heat input [16] Q (x , y ,t ) =
I 0γ ∗ t 2π r 2t 02
exp( −
x
2
r
2
−
t ) exp( − γ ∗ y ) t0
(6)
where I 0 is the energy absorption, t 0 is the pulse rise time ( t 0 = 0.2 ps ), r is the beam radius and γ ∗ is the absorption depth of heating energy. The components of stress tensor are; σ xx = λ [
∂u ∂v ∂u + ] + 2µ +b φ − βT ∂x ∂y ∂x
(7)
σ yy = λ [
∂u ∂v ∂v + ] + 2µ +b φ − βT ∂x ∂y ∂y
(8)
σ zz = λ [
∂u ∂v + ]+b φ − βT ∂x ∂y
(9)
∂u ∂v ] + ∂y ∂x
(10)
σ xy = µ [
(11)
σ xz = σ yz = 0
Using Eqs. (7)-(11) in Eqs. (1)-(3) we have; λ + µ ∂e b ∂φ β ∂T ρ ∂ 2u ) + − = µ ∂x µ ∂x µ ∂x µ ∂t 2 λ + µ ∂e b ∂φ β ∂T ρ ∂ 2v ∇2 v + ( ) + − = µ ∂y µ ∂y µ ∂y µ ∂t 2 ∇2 u + (
(∇ 2 −
(12) (13)
ξ1 ω0 ∂ ρ ψ ∂ 2 b m − − )φ − e + T = 0 α α ∂t α ∂t 2 α α
k ∇2T + k *
(14)
∂ 2 ∂φ ∂ 2T ∂ 2e ∂ ∇ T − m T0 = ρ Ce + β T0 −ρ Q 2 2 t ∂t ∂t ∂ ∂t ∂t
For simplifications we shall use the following non-dimensional variables:
4
(15)
σ ij ω* P ω*2ψ θ (x ′, y ′,u ′,v ′) = 1 (x , y , u ,v ), σ ij′ = , θ′ = , φ ′ = 1 φ , (t ′, τθ′ ,τ q′ ) = ω1* (t,τθ , τ q ), P1′ = 1 , T0 c1 µ µ c12 I 0′ =
2 ω1* λ + 2µ c I 0 , γ ∗′ = 1 γ ∗ , c12 = ( ) , and ω 1* = ρ C e c 1 * ρ µ c1 K ω1
(16)
We introduce the displacement potentials q1 ( x , y , t ) and q 2 ( x , y , t ) which relate to
displacement components as: u =
∂ q 1 ∂q 2 , + ∂x ∂y
v =
∂q 1 ∂q 2 , − ∂y ∂x
e = ∇ 2q 1 ,
∂u ∂v − = ∇ 2q 2 ∂y ∂ x
(17)
Using Eqs. (16) and (17) in Eqs. (12)-(15), we obtain (after suppressing primes) [(λ + 2 µ )∇ 2 − ρc12
[∇2 −
[∇ 2 − (
∂2 ∂t 2
]q1 +
bc12
ω1∗2ψ
(18)
φ − βT 0θ = 0,
ρc12 ∂ 2 ]q 2 = 0 µ ∂t 2
(19)
ρc12ψ ∂ 2 ω c2 ∂ ξ c2 m ψT 0 bψ ) − ( 0 1 ) − 1 1 ]φ − ( )∇ 2q1 + ( )θ = 0, 2 ∗ ∂t ∗2 α α α ∂t αω1 αω1
(20)
[( K + K ∗ω1∗
m c 14 ∂ φ ∂ ) ∇ 2 − ρ c e c 12 ]θ − β c 12 ∇ 2 q 1 − = − Q 0 f exp( − γ ∗ y ), ∂t ψ ω1∗ 3 ∂ t
where Q 0 =
I 0 γ ∗ µρω1*2 2π r 2T 0t 02
,
f = (1 −
(21)
t x2 t ) exp(− − ). t0 r 2 t0
4. The solution of the problem
The solution of the considered physical variables can be decomposed in terms of normal mode analysis in the following form [q1 , q 2 , φ ,θ , σ ij ](x , y ,t ) = [q1∗ , q 2∗ , φ ∗ ,θ ∗ , σ ij∗ ]( y ) exp[i (a x − ω t )],
(22)
where q1∗ , q 2∗ , φ ∗ ,θ ∗ , σ ij∗ are the amplitudes of the field quantities, i = − 1 , a and ω are wave number and frequency respectively. Using Eq. (22), Eqs. (18)-(21) take the form; (a1D 2 − a2 ) q1* + a3 φ * − a4 θ * = 0,
(23)
(D2 − a5 )q 2∗ = 0,
(24)
(D 2 − a6 ) φ * − (a7 D 2 − a 2 )q1∗ + a8 θ * = 0,
(25)
(a9 D 2 − a10 ) θ ∗ + a11φ ∗ − a12 (D 2 − a 2 )q1∗ = −Q 0 f
where
D=
d , dy
∗
exp( − γ ∗ y ),
a1 = λ + 2µ , a2 = a1a 2 − ρc12ω 2 , a3 =
5
bc12
ψω1∗2
(26) , a4 = βT 0 ,
a5 = a 2 −
ρc12ω 2 , µ
a6 = a 2 −
a11 =
i ω ω 0 c 12
α ω1∗
i ω mc14
+
ξ 1c 12 α ω 1∗ 2
−
bψ ρψ c 12 ω 2 m ψ T 0 a = K − i ωω ∗ K ∗ , a = a 2 + ρc c 2 , , a8 = 10 1 e 1 , a7 = , 9 α α α
, a12 = βc12 , f
ψω13
∗
= f exp[ − i (a x − ω t )].
Eliminating θ * and φ * between Eqs. (23), (25) and (26), we get the following sixth order differential equation for q1∗ ( y ) [D6 − A D 4 + B D2 − F ]q1∗ ( y ) = − Q 0 L1 f ∗ exp(−γ ∗ y ).
(27)
In a similar manner we arrive [D6 − A D 4 + B D2 − F ]θ ∗ ( y ) = − Q 0 L2 f ∗ exp( −γ ∗ y ),
(28)
[D6 − A D 4 + B D2 − F ]φ ∗ ( y ) = − Q 0 L3 f ∗ exp(−γ ∗ y ),
(29)
where
a13 = a1a6 + a2 − a3a7 ,
a17 = a2a11 + a3a12 a 2 ,
a14 = a2a6 − a3a7 a 2 ,
a18 = a3a9 ,
a15 = a4a6 − a3a8 ,
a19 = a3a10 − a4a11 ,
a16 = a1a11 + a3a12 ,
a a + a13a18 + a 4 a16 , A = 1 19 a1a18
a a + a15a17 a γ ∗4 − a13γ ∗2 + a14 a a +a a +a a +a a a γ ∗ 2 − a15 , L1 = 4 , , L2 = 1 B = 18 14 19 13 17 4 16 15 , F = 19 14 a1a18 a1a18 a1a18 a1a18
L3 =
a4 [a8 (a1γ ∗2 − a2 ) − a4 a7 (γ ∗2 − a 2 )] . a1a18
Eq. (27) can be factored as (D 2 − k 12 )(D 2 − k 22 )(D 2 − k 32 )q1∗ ( y ) = − Q 0 L1 f
∗
exp( −γ ∗ y ),
(30)
where k i2 , (i = 1, 2, 3) are the roots of the characteristic equation of Eq. (27). The general solution of Eqs. (27)-(29) and Eq. (24), bound at y → ∞, is given by 3
q1 (x , y , t ) = ∑ M n exp(− k n y − i ωt + iax ) + Q 0 N 1L1 f exp(−γ ∗ y ),
(31)
n =1
(32)
q 2 (x , y , t ) = M 4 exp(− k 4 y − i ωt + iax ), 3
θ (x , y , t ) = ∑ H 1n M n exp(− k n y − i ωt + iax ) + Q 0 N 1L 2 f exp(−γ ∗ y ),
(33)
n =1 3
φ (x , y ,t ) = ∑ H 2 n M n exp(− k n y − i ωt + iax ) + Q 0 N 1L3 f exp(−γ ∗ y ),
(34)
n =1
where k 42 = a5 , are the roots of the characteristic equation of Eq. (23), M n (n =1,2,3,4) are some coefficients, N1 =
−1
γ
∗6
− Aγ
∗4
+Bγ
∗2
−F
,
H 1n
a k 4 −a k 2 + a = 1 n 13 n 14 a 4 k n2 − a15
, H 2n
a H − a k 2+ a = 4 1n 1 n 2 , a3
(n = 1, 2, 3)
Using Eqs. (31), (32) and (17) the displacement components take the form: 3
u (x , y , t ) = [ ∑ iaM n exp(−k n y ) − k 4M 4 exp(−k 4 y )]exp[i (ax − ωt )] − Q0N 1L1 ( n =1
6
2x )f exp(−γ ∗ y ), r2
(35)
3
v (x , y ,t ) = [∑ −k n M n exp(−k n y ) − iaM 4 exp(−k 4 y )]exp[i (ax − ωt )] − Q0γ ∗N 1L1f exp(−γ ∗ y ).
(36)
n =1
To get the components of stress, substituting from Eq. (16) in Eq. (7)-(11), and then using Eqs. (33)-(36); 3 σxx (x , y ,t ) = [ ∑ H 3n M n exp(−k n y ) + N 2k 4M 4 exp(−k 4 y )]exp[i (ax − ωt )] +Q0f 1 exp(−γ ∗ y ), n =1 3
σ yy (x , y ,t ) = [ ∑ H 4n M n exp(−k n y ) − N 2 k 4 M 4 exp(−k 4 y )]exp[i (ax − ωt )] + Q 0 f 2 exp(−γ ∗ y ),
(37) (38)
n =1
3
σ z z ( x , y , t ) = ∑ H 5 n M n exp( − k n y ) exp[i (ax − ωt ) + Q 0 f 3 exp( − γ ∗ y ),
(39)
n =1
3
σ xy (x , y , t ) = [ ∑ H 6n M n exp(−k n y ) + N 3M 4 exp(−k 4 y )]exp[i (ax − ω t )] + Q0f 4 exp (−γ ∗ y ). n =1
where
b1 =
λ + 2µ , µ
b2 =
λ , µ
b3 =
b c 12
µψ
ω 1∗ 2
, b4 =
βT µ
0
,
(40)
N 2 = iak 4 (b2 − b1 ), N 3 = k 42 +a2,
−2 4x 2 − 2 4x 2 f 1 = N 1f [ L1b1 ( ) + b 2 γ ∗ 2 L1 + b 3 L 3 − b 4 L 2 ], f 2 = N 1f [ L1b 2 ( ) + b1γ ∗2 L1 + b 3 L 3 − b 4 L 2 ], + + 2 4 r r r2 r4 f4 =
4N 1L1x γ ∗f r
2
,
−2 4x 2 f 3 = N 1f [ L1b 2 ( + + γ ∗ 2 ) + b 3 L 3 − b 4 L 2 ], r2 r4
H 3n = − a 2b1 + b2k n2 + b3H 2n − b4H 1n ,
H 4n = − a 2b 2 + b1k n2 + b3H 2n − b4 H 1n , H 5n = − a 2b 2 + b2 k n2 + b3H 2n − b 4 H 1n ,
H 6 n = − 2iak n .
Using the expressions of the variables considered into the above boundary conditions (5), the following equations satisfied by the coefficients M n (n = 1, 2,3, 4) ; 3
∑ H 3n M n + N 2 = − p1∗
(42)
n =1 3
∑ H 6n M n + N 3 = 0
(43)
n =1 3
∑ −k n H 2n M n = 0
(44)
n =1 3
∑ −k n H 1n M n = 0
(45)
n =1
Applying the inverse of matrix method on the system of four equations (42)-(45), the coefficients M n (n = 1, 2,3, 4) can be obtained. M 1 H 31 M 2 = H 61 M 3 − k 1H 21 M 4 − k 1H 11
H 32 H 62
H 33 H 63
− k 2 H 22
− k 3 H 23
−k 2 H 12
− k 3 H 13
N2 N3 0 0
−1
− p1∗ 0 . 0 0
7
(46)
5. Particular and special cases 5.1 Neglecting the voids
Neglecting the void parameters effect, i.e. ( α = b = ξ1 = ω0 = m = ψ = 0 ). Putting ( α = b = ξ1 = ω0 = m = ψ = 0 ) in Eqs. (23)-(26) we get (a1D 2 − a2 ) q1∗ − a4 θ * = 0,
(47)
(D2 − a4 ) q 2∗ = 0,
(48)
(a9 D 2 − a10 ) θ ∗ − a12 (D 2 − a 2 )q1∗ = − Q 0 f
∗
exp(− γ ∗ y ).
(49)
Eliminating q1∗ and θ ∗ between the Eqs. (47) and (49) we obtain the following fourth order differential equation satisfied by q1∗ and θ ∗ [D 4 − A1 D 2 + B1 ]{q1* ( y )} = − L 4Q 0 f ∗ exp(−γ ∗ y ),
(50)
[D 4 − A1 D 2 + B1 ]{θ ∗ ( y )} = − L5Q 0 f ∗ exp(−γ ∗ y ),
(51)
The solutions of Eqs. (50), (51) and (48) are 2
(q 1 , θ )( x , y , t ) = ∑ (1, h1n ) R n exp( − rn y + iax − i ω t ) + ( L 4 , L 5 ) L 6Q 0 f exp( − γ ∗ y ),
(52)
q 2 (x , y , t ) = R 3 exp(− r3 y + iax − i ωt ).
(53)
n =1
The expressions for the displacement components and the stresses in the generalized thermoelastic medium with thermal loading due to laser pulse in the context of the (G-N type III) theory, are 2
u ( x , y , t ) = [ ∑ iaR n exp( − rn y ) − r3 R 3 exp( − r3 y )]exp[i (ax − ωt )] − Q 0 L 6 L 4 ( n =1
2x ) f exp( −γ ∗ y ), r2
2
v ( x , y , t ) = [ ∑ − rn R n exp(− rn y ) − i aR 3 exp(− r3 y )]exp[i (ax − ωt )] − Q 0γ ∗ L 4 L 6 f exp(−γ ∗ y ),
(54) (55)
n =1
2
σ x x ( x , y , t ) = [ ∑ h 2 n R n exp( − rn y ) +N 4 R 3 exp( − r3 y )] exp[i (a x − ω t )] + Q 0 f 5 exp( − γ ∗ y ),
(56)
n =1
2
σ yy (x , y , t ) = [ ∑ h3n R n exp(− rn y ) −N 4 R 3 exp(−r3 y )]exp[i (a x − ω t )] + Q 0 f 6 exp(−γ ∗ y ),
(57)
n =1 2
σ zz (x , y ,t ) =
∑ h 4 n R n exp( − rn
y ) exp[i (a x − ω t )] + Q 0 f 7 exp( − γ ∗ y ),
(58)
n =1 2
σ xy (x , y ,t ) = [ ∑ h5n R n exp(− rn y ) +N 5 R3 exp(−r3 y )]exp[i (a x − ω t )] + Q0 f 8 exp(−γ ∗ y ),
(59)
n =1
where rn2 (n = 1, 2, 3) are the roots of the characteristic equation of Eq. (50) and (48),
8
a22 =
1 , A1 = a22 (a2a9 + a1a10 + a4a12 ), B1 = a22 (a2a10 + a4a12a 2 ), L 4 = a4a22 , a1a9
h1n =
a1rn2 − a2 −1 , L6 = , h2 n = − b1a 2 + b 2 rn2 − b 4 h1n , h3n = − b 2a 2 + b1rn2 − b 4 h1n , ∗ 4 a4 γ − A1γ ∗2 + B1
L5 = a22 (a1γ ∗ −a2 ),
2
h4n = − b2a2 + b2rn2 − b4 h1n , h5 n = −2ia rn , N 4 = i a r3 (b2 − b1 ), f 5 = L6 f [L4b1 ( −2 + 4x ) + b2γ ∗2 L4 − b4 L5 ], 2 4 r
f 7 = L 6 f [L 4b 2 (
f8 =
−2 r
4xL 6 L 4 f γ ∗ r
2
2
+
4x 2 r
4
) + b 2γ ∗2 L 4 − b 4 L5 ],
f 6 = L 6 f [ L 4b 2 (
−2 r
2
+
r
4x 2 r
4
) + b1γ ∗2 L 4 − b 4 L 5 ],
, N 5 = r32 + a 2 .
In order to determine the coefficients Rn (n = 1,2,3) , we use the boundary conditions in Eq. (5) and by using the inverse of matrix method as follows: h22 R1 h21 R = h h52 2 51 R −r h 3 1 11 −r2 h12
N4 N5 0
−1
−p∗ 1 0 . 0
(60)
5.2 Generalized thermoelasticity theory without energy dissipation (G-N type II):
putting (K ∗ = 0) in the heat equation. 6. Numerical analysis and discussion
For numerical computations, magnesium material was chosen for purposes of numerical evaluations [17]. The constants of the problem were taken as; T 0 = 298 K ,
λ = 2.17 ×1010N / m2,
C E = 1.04×103 J / Kg deg,
µ = 3.278×1010 N / m2,
ω1* = 3.58 ×1011 / s ,
ρ = 1.74 × 103 Kg / m 3 ,
αt =1.78×10−5N / m2,
β = 2.68×106 N / m2 deg,
K = 1.7 ×102W / m deg.
The voids parameters and the laser pulse parameters are ψ =1.753×10−15m2 , α = 3.688×10−5N , ξ1 =1.475×1010 N / m2 , b =1.13849×1010N / m2 , m = 2×106N / m2 deg,
ω0 = 0.0787 ×10−3 N / m 2s ,
r =100 µm, γ ∗ = 0.5m−1,
I0 =105J ,
t0 = 0.2p.sec
The comparisons were carried out for: p1∗ = 10,
a = 1.7,
ω = b0 + i ξ , b0 = 0.3, ξ = 0.7,
x = 10,
(t = 0.3, 0.9)
and
0 ≤ y ≤ 5.
The comparisons have established for two cases: (i) With two values of time
(t = 0.3, 0.9) in
the context of (G-N type II) theory and (G-N
type III) theory in the presence laser pulse with the pulse duration (t0 = 0.2 p.sec). (ii) In the presence and absence void parameters in the context of (G-N type II) theory 9
and (G-N type III) theory in the presence laser pulse with the pulse duration (t0 = 0.2 p.sec), at (t = 0.9).
The computations are carried out on the surface plane x = 10. The numerical technique outlined above is used for the distribution of the real part of the non-dimensional displacements u , v , the non-dimensional temperature θ , the distributions of nondimensional stresses σ xx , σ xy and change in the volume fraction field φ with distance y for the problem.
Figs. 1-6 are plotted to show the variation of the above quantities against the distance y , in both of the (G-N type II) theory and (G-N type III) theory of the two values of time
(t = 0.3, 0.9),
where the solid lines represent results for the (G-N type III)
theory for (t = 0.3), the large dash line represents results for the (G-N type III) theory for
(t = 0.9),
the small dash line represents results for the (G-N type II) theory at
(t = 0.3),
the small dash line with dot represents results for the (G-N type II) theory at
(t = 0.9).
In Fig. 1, the displacement component u is plotted against the distance y , it is
observed that the displacement u for the (G-N type II) theory is greater than that of the (G-N type III) theory and it is clear that the values of solutions for
(t = 0.9)
are
greater than that for (t = 0.3). Fig. 2 investigates the displacement component v against the distance y , the values of v increases with the increase of the time parameter t , the values in the case of (G-N type III) theory are greater than in the case of (G-N type II) theory. Fig. 3 explains the distribution of temperature θ against the distance y , in this figure, the values of θ in the case of (G-N type III) theory are smaller than in the case of (G-N type II) theory. Figs. 4 and 5 depict the distributions of stress components σ xx and σ xy in the context of both (G-N type III) theory and (GN type II) theory; we see that the values of solutions for (t = 0.3) are smaller than that
for
(t = 0.9).
From this figure, the time parameter has a decreasing effect. Fig. 6
expresses the distribution of change in the volume fraction field φ against the distance y , it is observed that the change in the volume fraction field φ for the (G-N type II)
theory is greater than that of the (G-N type III) theory and it is clear that the values of solutions for (t = 0.3) are smaller than that for
(t = 0.9).
Figures 7-11 show the behavior of the physical quantities against distance y in 2D 10
during
(t = 0.9),
with and without voids effect, where the solid lines represent results
for the (G-N type III) theory to voids effect, the large dash line represents results for the (G-N type III) theory without voids effect, the small dash line represents results for the (G-N type II) theory with voids effect and the small dash line with dot represents the results for the (G-N type II) without voids effect. From this figure, the void parameters have a significant role in the distribution of all physical quantities in the problem. Fig. 7 depicts the distribution of the displacement components u , the values of the displacement u for presence voids are small compared to those for absence voids in the range 0 ≤ y ≤ 0.4 and great on the range 0.4 ≤ y ≤ 3.4, while the values are the same for all cases at y ≥ 3.4. Fig. 8 shows the displacement component
v against the distance
y , the
values of v for presence voids are great compared to
those for absence voids. Fig. 9 clarifies the distribution of the temperature θ is decreasing in the all cases (with and without voids) and the values of the temperature θ for presence voids are small compared to those for absence voids. Fig. 10
displays that the distribution of the stress component σ xx always begin from negative values in the all cases and the values of the stress component σ xx for presence voids are large compared to those for absence voids in the range 0 ≤ y ≤ 0.6 ; small in the range 0.6 ≤ y ≤ 1.6, while the values are the same for the cases at y ≥ 1.6. Fig. 11 explains the distribution of tangential stress σ xy versus y . The values of σ xy are increasing for the presence void parameters. In this figure the values of σ xy satisfying the boundary conditions of the problem. Figs. 12-13 are giving 3D surface curves for the physical quantities i.e., the tangential stress σ xy and the change in the volume fraction field φ , for the thermal loading due to laser pulse on thermoelastic medium with voids in the context of the (G-N type II) theory and (G-N type III) theory. These figures are very important to study the dependence of these physical quantities on the vertical component of distance. 7. Conclusion
According to the above results, we can conclude that: 1.
The void parameters and the different values of time in the current model have significant effects on all the fields. 11
2.
The value of all physical quantities converges to zero with an increase in distance y
3.
and all functions are continuous.
The comparison of different theories of thermoelasticity, i.e. both types of II and III of (G-N) theory are carried out.
4.
The analytical solutions based upon normal mode analysis for thermoelastic problem in solids have been developed and uzed.
5.
The deformation of a body depends on the nature of the applied forces and thermal loading due to laser pulse as well as the type of boundary conditions.
References [1]
A.E. Green, P.M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proceedings of the Royal Society London A 432 (1991) 171-194.
[2]
A.E. Green, P.M. Naghdi, On undamped heat waves in an elastic solid, Journal of Thermal Stresses 15 (1992) 253-264.
[3]
A.E. Green, P.M. Naghdi, Thermoelasticity without energy dissipation, Journal of Elasticity 31 (1993) 189-208.
[4]
M.I.A. Othman, I.A. Abbas, Generalized thermoelsticity of thermal shock problem in a non-homogeneous isotropic hollow cylinder with energy dissipation, International
Journal of Thermophysics 33 (2012) 913-923. [5]
R. Kumar, R. Kumar, Propagation of wave at the boundary surface of transversely isotropic thermoelastic material with voids and isotropic elastic half-space, Applied Mathematics and Mechanics (English Edition) 31 (2010) 1153-1172.
[6]
I.A. Abbas, M.I.A. Othman, Effect of rotation on thermoelstic waves with GreenNaghdi theory in a homogeneous isotropic hollow cylinder, International Journal of Industrial Mathematics 1 (2009) 121-134.
[7]
D.D. Joseph, L. Preziosi, Heat waves, Reviews of Modern Physics 61 (1989) 4173.
[8]
D.D. Joseph, L. Preziosi, Heat waves, Reviews of Modern Physics 62 (1990) 375391.
[9]
M.I.A. Othman, R. Kumar, Reflection of magneto-thermoelasticity waves with temperature dependent properties in generalized thermoelasticity, International Communications in Heat and Mass Transfer 36 (2009) 513-520.
[10] J.W. Nunziato, S.C. Cowin, A non-linear theory of elastic materials with voids, Arch. Ration. Mech. Anal. 72 (1979) 175-201. 12
[11] S.C. Cowin, J.W. Nunziato, Linear theory of elastic materials with voids, Journal of Elasticity 13 (1983) 125-147. [12] H.M. Youssef, State-space approach to two-temperature generalized thermoelasticity without energy dissipation of medium subjected to moving heat source, Applied Mathematics and Mechanics (English Edition) 34 (2013) 63-74.
[13] M. Marin, R.P. Agarwal, M.I.A. Othman, Localization in time of solutions for thermoelastic micropolar materials with voids, Computers, Materials & Continua 40 (2014) 35-48. [14] M. Marin, M.I.A. Othman, I.A. Abbas, An extension of the domain of influence theorem for anisotropic thermoelastic material with voids, Journal of Computational and Theoretical Nanoscience 12 (2015) 1594-1598. [15] M.I.A. Othman, S.Y. Atwa, R.M. Farouk, The effect of diffusion on twodimensional problem of generalized thermoelasticity with Green-Naghdi theory, International Communications in Heat and Mass Transfer 36 (2009) 857-864. [16] H.M. Al-Qahtani, S. K. Datta, Laser-generalized thermoelastic waves in an anisotropic infinite plate: Exact analysis, Journal of Thermal Stresses 31 (2008) 569-583. [17] R.S. Dhaliwal, A. Singh, Dynamic coupled thermoelasticity, Hindustan Publ. Corp, New Delhi, 1980.
Figure 1. Variation of displacement component u with horizontal distance y .
13
Figure 2. Variation of displacement component v with horizontal distance y .
Figure 3. Variation of temperature θ with horizontal distance y .
Figure 4. Variation of normal stresses σ xx with horizontal distance y .
14
Figure 5. Variation of tangential stresses σ xy with horizontal distance y .
Figure 6. Variation of volume fraction field distribution φ with horizontal distance y .
Figure 7. Variation of displacement component u with horizontal distance y .
15
Figure 8. Variation of displacement component v with horizontal distance y .
Figure 9. Variation of temperature θ with horizontal distance y .
Figure 10. Variation of normal stresses σ xx with horizontal distance y .
16
Figure 11. Variation of normal stresses σ xy with horizontal distance y .
Figure 12. (3D) The component of stress σ xy against both components of distance based on the (G-N type III) theory at t = 0.3.
Figure 13. (3D) Distribution of volume fraction field φ against both components of distance based on the (G-N type III) theory at t = 0.3.
17
Highlight 1) The comparison of different theories of thermoelasticity, i.e. both types of II and
III
of (G-N) theory. 2) Analytical solutions with normal mode for thermoelastic problem in solids have been developed. 3) The laser pulse has a significant effect on the field quantities. 4) The void parameters and the different values of time have important effect.
18
S2211-3797(17)31447-X https://doi.org/10.1016/j.rinp.2017.10.012 RINP 990
To appear in:
Results in Physics
Received Date: Accepted Date:
4 August 2017 4 October 2017
Please cite this article as: Othman, M.I.A., Marin, M., Effect of Thermal Loading due to Laser Pulse on Thermoelastic Porous Medium under G-N Theory, Results in Physics (2017), doi: https://doi.org/10.1016/j.rinp.2017.10.012
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Effect of Thermal Loading due to Laser Pulse on Thermoelastic Porous Medium under G-N Theory 1,2
Mohamed I. A. Othman and M. Marin 1
3
Department of Mathematics, Faculty of Science, P.O. Box 44519, Zagazig University, Zagazig, Egypt 2
Department of Mathematics, Faculty of Science, Taif University 888, Taif City, Saudi Arabia 3
Department of Mathematics, Transilvania University of Brasov, Romania Email: [email protected], [email protected]
Abstract: The aim of this paper is to study the wave propagation of generalized thermoelastic medium with voids under the effect of thermal loading due to laser pulse with energy dissipation. The material is a homogeneous isotropic elastic halfspace and heated by a non-Gaussian laser beam with the pulse duration of 0.2 ps. A normal mode method is proposed to analyse the problem and obtain numerical solutions for the displacement components, stresses, temperature distribution and the change in the volume fraction field. The results of the physical quantities have been illustrated graphically by comparison between both types II and III of Green-Naghdi theory for two values of time, as well as with and without void parameters. Keywords: Laser pulse; Voids; Energy dissipation; Green-Naghdi theory; Wave propagation; Thermoelasticity. 1. Introduction Green and Naghdi [1-3] developed a generalized theory of thermoelasticity which involves thermal displacement gradient as one of the constitutive variables in contrast to the classical coupled thermoelasticity which includes temperature gradient as one of the constitutive variables. An important feature of this theory is that it does not accommodate dissipation of thermal energy. On this theory the characterization of material response to a thermal phenomenon is based on three types of constitutive response functions. The nature of those three types of constitutive response functions is such that when the respective theories are linearized, type I is same as classical heat conduction equation (based on Fourier’s law), whereas type II, the internal rate of production of entropy is taken to be identically zero, implying no dissipation of thermal energy. This model is known as the theory of thermoelasticity without energy dissipation. Type III involves the previous two models as special cases, and admits dissipation of energy in general, in this model, introducing the temperature gradient 1
and thermal displacement gradient as the constitutive variables. Othman and Abbas [4] investigated the problem of generalized thermoelsticity of thermal shock in a nonhomogeneous isotropic hollow cylinder with energy dissipation. Kumar and Kumar [5] studied the propagation of wave at the boundary surface of transversely isotropic thermoelastic material with voids and isotropic elastic half-space. Abbas and Othman [6] used G-N theory to study the effect of rotation on thermoelstic waves in a homogeneous isotropic hollow cylinder. Excitation of thermoelastic waves by a pulsed laser in solid is of great interest due to extensive applications of pulsed laser technologies in material processing and non-destructive detecting and characterization. When a solid is illuminated with a laser pulse, absorption of the laser pulse results in a localized temperature increase, which in turn causes thermal expansion and generates a thermoelastic wave in the solid. In ultra-short pulsed laser heating, two effects become important. One is the non-Fourier’s effect in heat conduction which is a modification of the Fourier heat conduction theory to account for the effect of mean free time (thermal relaxation time) in the energy carrier’s collision process. Consideration of the non-Fourier effect also eliminates the paradox of the infinite heat propagation speed by Joseph and Preziosi [7, 8]. The other is the dissipation of the stress wave due to coupling between temperature and strain rate, which causes transform of mechanical energy associated with the stress wave to the thermal energy of the material. Othman and Kumar [9] investigated the reflection of magneto-thermoelasticity waves with temperature dependent properties in generalized thermoelasticity. Theory of elastic materials with voids is one of the most important generalizations of the classical theory of elasticity. This theory is concerned with elastic materials consisting of a distribution of small voids in which the void volume is included among the kinematic variables. Practically, this theory is useful for investigating various types of geological and biological materials for which elastic theory is inadequate. Nunziato and Cowin [10] developed a nonlinear theory of elastic material with voids. Cowin and Nunziato [11]
investigated the theory of linear elastic materials with voids. Youssef [12] used the state-space approach to two-temperature generalized thermoelasticity without energy dissipation of medium subjected to moving heat source. Marin et al. [13] studied an extension of the domain of influence theorem for anisotropic thermoelastic material with voids. Marin et al. [14] investigated the localization in time of solutions for 2
thermoelastic micropolar materials with voids. Othman et al. [15] studied the effect of diffusion on two-dimensional problem of generalized thermoelasticity with GreenNaghdi theory. In the present paper the wave propagation of the components of displacement, stress, temperature distribution and change in the volume fraction field in an isotropic homogeneous thermoelastic solid with voids subjected to thermal loading due laser pulse under (G-N III) was studied. The results performed and presented graphically with concluding remarks are given. 2. Formulation of the problem and basic equations We consider a homogeneous, isotropic, thermoelastic material with voids in the undeformed temperature T 0 , with a half space ( x ≥ 0 ). The rectangular Cartesian coordinate system ( x , y , z ) have originated on the surface (z = 0). For two dimensional problem we assume the dynamic displacement vector as u = (u,v, 0). All quantities considered will be a function of the time variable t, and of the coordinates x and y. According to Cowin and Nunziato [11], the field equations and constitutive
relations for linear homogenous, isotropic generalized thermoelastic solid with voids without body forces, heat sources and extrinsic equilibrated body force can be expressed in the context of (G-N III), as; µ u i, jj + (λ + µ ) u j ,ij + b φ,i − β T ,i = ρui ,
(1)
α φ,kk − b u k,k − ξ1φ − ω0 φ + m T = ρ ψ φ
(2)
kT ,ii + k *T,ii − mT 0 φ = ρ C e T + β T 0 ui ,i − ρ Q
(3)
σ ij = λ u k,k δ ij + µ (u i , j + u j ,i ) + b φ δ ij − β T δ ij ,
(4)
∂u
where e ij = 12 ( ∂∂xui + ∂x j ) are the components of strain tensor, ( i , j = 1,2, 3 ), λ , µ are the j
i
Lame' constants, α , b , ξ1 , ω0 , m , ψ are the material constants due to presence of voids, T
is the temperature distribution, β = (3λ + 2µ )αt such that αt is the coefficient of
thermal expansion, ρ is the density, C E is the specific heat, k is the thermal conductivity, k * is the material constant characteristic of the theory, T 0 is the reference temperature chosen so that (T −T 0 ) T 0 << 1 , φ is the change in volume fraction field, σ ij are the components of stress tensor, δ ij is the Kronecker delta, t is the time variable, a comma denotes material derivatives and Q is the heat input of the 3
laser pulse. 3. The boundary conditions The coefficients M n (n = 1, 2,3, 4) have to be determined such that the boundary conditions on the surface
y =0
take the form:
σ xx (x ,0, t ) = p1 (x ,t ) = − p1∗ exp[i (a x − ω t )], σ xy (x , 0, t ) = 0,
∂φ ∂y
= 0,
∂θ ∂y
=0
(5)
where p1 (x , t ) is an arbitrary function of x , t and p1∗ is the magnitude of the constant stress applied to the boundary. The plate surface is illuminated by laser pulse given by the heat input [16] Q (x , y ,t ) =
I 0γ ∗ t 2π r 2t 02
exp( −
x
2
r
2
−
t ) exp( − γ ∗ y ) t0
(6)
where I 0 is the energy absorption, t 0 is the pulse rise time ( t 0 = 0.2 ps ), r is the beam radius and γ ∗ is the absorption depth of heating energy. The components of stress tensor are; σ xx = λ [
∂u ∂v ∂u + ] + 2µ +b φ − βT ∂x ∂y ∂x
(7)
σ yy = λ [
∂u ∂v ∂v + ] + 2µ +b φ − βT ∂x ∂y ∂y
(8)
σ zz = λ [
∂u ∂v + ]+b φ − βT ∂x ∂y
(9)
∂u ∂v ] + ∂y ∂x
(10)
σ xy = µ [
(11)
σ xz = σ yz = 0
Using Eqs. (7)-(11) in Eqs. (1)-(3) we have; λ + µ ∂e b ∂φ β ∂T ρ ∂ 2u ) + − = µ ∂x µ ∂x µ ∂x µ ∂t 2 λ + µ ∂e b ∂φ β ∂T ρ ∂ 2v ∇2 v + ( ) + − = µ ∂y µ ∂y µ ∂y µ ∂t 2 ∇2 u + (
(∇ 2 −
(12) (13)
ξ1 ω0 ∂ ρ ψ ∂ 2 b m − − )φ − e + T = 0 α α ∂t α ∂t 2 α α
k ∇2T + k *
(14)
∂ 2 ∂φ ∂ 2T ∂ 2e ∂ ∇ T − m T0 = ρ Ce + β T0 −ρ Q 2 2 t ∂t ∂t ∂ ∂t ∂t
For simplifications we shall use the following non-dimensional variables:
4
(15)
σ ij ω* P ω*2ψ θ (x ′, y ′,u ′,v ′) = 1 (x , y , u ,v ), σ ij′ = , θ′ = , φ ′ = 1 φ , (t ′, τθ′ ,τ q′ ) = ω1* (t,τθ , τ q ), P1′ = 1 , T0 c1 µ µ c12 I 0′ =
2 ω1* λ + 2µ c I 0 , γ ∗′ = 1 γ ∗ , c12 = ( ) , and ω 1* = ρ C e c 1 * ρ µ c1 K ω1
(16)
We introduce the displacement potentials q1 ( x , y , t ) and q 2 ( x , y , t ) which relate to
displacement components as: u =
∂ q 1 ∂q 2 , + ∂x ∂y
v =
∂q 1 ∂q 2 , − ∂y ∂x
e = ∇ 2q 1 ,
∂u ∂v − = ∇ 2q 2 ∂y ∂ x
(17)
Using Eqs. (16) and (17) in Eqs. (12)-(15), we obtain (after suppressing primes) [(λ + 2 µ )∇ 2 − ρc12
[∇2 −
[∇ 2 − (
∂2 ∂t 2
]q1 +
bc12
ω1∗2ψ
(18)
φ − βT 0θ = 0,
ρc12 ∂ 2 ]q 2 = 0 µ ∂t 2
(19)
ρc12ψ ∂ 2 ω c2 ∂ ξ c2 m ψT 0 bψ ) − ( 0 1 ) − 1 1 ]φ − ( )∇ 2q1 + ( )θ = 0, 2 ∗ ∂t ∗2 α α α ∂t αω1 αω1
(20)
[( K + K ∗ω1∗
m c 14 ∂ φ ∂ ) ∇ 2 − ρ c e c 12 ]θ − β c 12 ∇ 2 q 1 − = − Q 0 f exp( − γ ∗ y ), ∂t ψ ω1∗ 3 ∂ t
where Q 0 =
I 0 γ ∗ µρω1*2 2π r 2T 0t 02
,
f = (1 −
(21)
t x2 t ) exp(− − ). t0 r 2 t0
4. The solution of the problem
The solution of the considered physical variables can be decomposed in terms of normal mode analysis in the following form [q1 , q 2 , φ ,θ , σ ij ](x , y ,t ) = [q1∗ , q 2∗ , φ ∗ ,θ ∗ , σ ij∗ ]( y ) exp[i (a x − ω t )],
(22)
where q1∗ , q 2∗ , φ ∗ ,θ ∗ , σ ij∗ are the amplitudes of the field quantities, i = − 1 , a and ω are wave number and frequency respectively. Using Eq. (22), Eqs. (18)-(21) take the form; (a1D 2 − a2 ) q1* + a3 φ * − a4 θ * = 0,
(23)
(D2 − a5 )q 2∗ = 0,
(24)
(D 2 − a6 ) φ * − (a7 D 2 − a 2 )q1∗ + a8 θ * = 0,
(25)
(a9 D 2 − a10 ) θ ∗ + a11φ ∗ − a12 (D 2 − a 2 )q1∗ = −Q 0 f
where
D=
d , dy
∗
exp( − γ ∗ y ),
a1 = λ + 2µ , a2 = a1a 2 − ρc12ω 2 , a3 =
5
bc12
ψω1∗2
(26) , a4 = βT 0 ,
a5 = a 2 −
ρc12ω 2 , µ
a6 = a 2 −
a11 =
i ω ω 0 c 12
α ω1∗
i ω mc14
+
ξ 1c 12 α ω 1∗ 2
−
bψ ρψ c 12 ω 2 m ψ T 0 a = K − i ωω ∗ K ∗ , a = a 2 + ρc c 2 , , a8 = 10 1 e 1 , a7 = , 9 α α α
, a12 = βc12 , f
ψω13
∗
= f exp[ − i (a x − ω t )].
Eliminating θ * and φ * between Eqs. (23), (25) and (26), we get the following sixth order differential equation for q1∗ ( y ) [D6 − A D 4 + B D2 − F ]q1∗ ( y ) = − Q 0 L1 f ∗ exp(−γ ∗ y ).
(27)
In a similar manner we arrive [D6 − A D 4 + B D2 − F ]θ ∗ ( y ) = − Q 0 L2 f ∗ exp( −γ ∗ y ),
(28)
[D6 − A D 4 + B D2 − F ]φ ∗ ( y ) = − Q 0 L3 f ∗ exp(−γ ∗ y ),
(29)
where
a13 = a1a6 + a2 − a3a7 ,
a17 = a2a11 + a3a12 a 2 ,
a14 = a2a6 − a3a7 a 2 ,
a18 = a3a9 ,
a15 = a4a6 − a3a8 ,
a19 = a3a10 − a4a11 ,
a16 = a1a11 + a3a12 ,
a a + a13a18 + a 4 a16 , A = 1 19 a1a18
a a + a15a17 a γ ∗4 − a13γ ∗2 + a14 a a +a a +a a +a a a γ ∗ 2 − a15 , L1 = 4 , , L2 = 1 B = 18 14 19 13 17 4 16 15 , F = 19 14 a1a18 a1a18 a1a18 a1a18
L3 =
a4 [a8 (a1γ ∗2 − a2 ) − a4 a7 (γ ∗2 − a 2 )] . a1a18
Eq. (27) can be factored as (D 2 − k 12 )(D 2 − k 22 )(D 2 − k 32 )q1∗ ( y ) = − Q 0 L1 f
∗
exp( −γ ∗ y ),
(30)
where k i2 , (i = 1, 2, 3) are the roots of the characteristic equation of Eq. (27). The general solution of Eqs. (27)-(29) and Eq. (24), bound at y → ∞, is given by 3
q1 (x , y , t ) = ∑ M n exp(− k n y − i ωt + iax ) + Q 0 N 1L1 f exp(−γ ∗ y ),
(31)
n =1
(32)
q 2 (x , y , t ) = M 4 exp(− k 4 y − i ωt + iax ), 3
θ (x , y , t ) = ∑ H 1n M n exp(− k n y − i ωt + iax ) + Q 0 N 1L 2 f exp(−γ ∗ y ),
(33)
n =1 3
φ (x , y ,t ) = ∑ H 2 n M n exp(− k n y − i ωt + iax ) + Q 0 N 1L3 f exp(−γ ∗ y ),
(34)
n =1
where k 42 = a5 , are the roots of the characteristic equation of Eq. (23), M n (n =1,2,3,4) are some coefficients, N1 =
−1
γ
∗6
− Aγ
∗4
+Bγ
∗2
−F
,
H 1n
a k 4 −a k 2 + a = 1 n 13 n 14 a 4 k n2 − a15
, H 2n
a H − a k 2+ a = 4 1n 1 n 2 , a3
(n = 1, 2, 3)
Using Eqs. (31), (32) and (17) the displacement components take the form: 3
u (x , y , t ) = [ ∑ iaM n exp(−k n y ) − k 4M 4 exp(−k 4 y )]exp[i (ax − ωt )] − Q0N 1L1 ( n =1
6
2x )f exp(−γ ∗ y ), r2
(35)
3
v (x , y ,t ) = [∑ −k n M n exp(−k n y ) − iaM 4 exp(−k 4 y )]exp[i (ax − ωt )] − Q0γ ∗N 1L1f exp(−γ ∗ y ).
(36)
n =1
To get the components of stress, substituting from Eq. (16) in Eq. (7)-(11), and then using Eqs. (33)-(36); 3 σxx (x , y ,t ) = [ ∑ H 3n M n exp(−k n y ) + N 2k 4M 4 exp(−k 4 y )]exp[i (ax − ωt )] +Q0f 1 exp(−γ ∗ y ), n =1 3
σ yy (x , y ,t ) = [ ∑ H 4n M n exp(−k n y ) − N 2 k 4 M 4 exp(−k 4 y )]exp[i (ax − ωt )] + Q 0 f 2 exp(−γ ∗ y ),
(37) (38)
n =1
3
σ z z ( x , y , t ) = ∑ H 5 n M n exp( − k n y ) exp[i (ax − ωt ) + Q 0 f 3 exp( − γ ∗ y ),
(39)
n =1
3
σ xy (x , y , t ) = [ ∑ H 6n M n exp(−k n y ) + N 3M 4 exp(−k 4 y )]exp[i (ax − ω t )] + Q0f 4 exp (−γ ∗ y ). n =1
where
b1 =
λ + 2µ , µ
b2 =
λ , µ
b3 =
b c 12
µψ
ω 1∗ 2
, b4 =
βT µ
0
,
(40)
N 2 = iak 4 (b2 − b1 ), N 3 = k 42 +a2,
−2 4x 2 − 2 4x 2 f 1 = N 1f [ L1b1 ( ) + b 2 γ ∗ 2 L1 + b 3 L 3 − b 4 L 2 ], f 2 = N 1f [ L1b 2 ( ) + b1γ ∗2 L1 + b 3 L 3 − b 4 L 2 ], + + 2 4 r r r2 r4 f4 =
4N 1L1x γ ∗f r
2
,
−2 4x 2 f 3 = N 1f [ L1b 2 ( + + γ ∗ 2 ) + b 3 L 3 − b 4 L 2 ], r2 r4
H 3n = − a 2b1 + b2k n2 + b3H 2n − b4H 1n ,
H 4n = − a 2b 2 + b1k n2 + b3H 2n − b4 H 1n , H 5n = − a 2b 2 + b2 k n2 + b3H 2n − b 4 H 1n ,
H 6 n = − 2iak n .
Using the expressions of the variables considered into the above boundary conditions (5), the following equations satisfied by the coefficients M n (n = 1, 2,3, 4) ; 3
∑ H 3n M n + N 2 = − p1∗
(42)
n =1 3
∑ H 6n M n + N 3 = 0
(43)
n =1 3
∑ −k n H 2n M n = 0
(44)
n =1 3
∑ −k n H 1n M n = 0
(45)
n =1
Applying the inverse of matrix method on the system of four equations (42)-(45), the coefficients M n (n = 1, 2,3, 4) can be obtained. M 1 H 31 M 2 = H 61 M 3 − k 1H 21 M 4 − k 1H 11
H 32 H 62
H 33 H 63
− k 2 H 22
− k 3 H 23
−k 2 H 12
− k 3 H 13
N2 N3 0 0
−1
− p1∗ 0 . 0 0
7
(46)
5. Particular and special cases 5.1 Neglecting the voids
Neglecting the void parameters effect, i.e. ( α = b = ξ1 = ω0 = m = ψ = 0 ). Putting ( α = b = ξ1 = ω0 = m = ψ = 0 ) in Eqs. (23)-(26) we get (a1D 2 − a2 ) q1∗ − a4 θ * = 0,
(47)
(D2 − a4 ) q 2∗ = 0,
(48)
(a9 D 2 − a10 ) θ ∗ − a12 (D 2 − a 2 )q1∗ = − Q 0 f
∗
exp(− γ ∗ y ).
(49)
Eliminating q1∗ and θ ∗ between the Eqs. (47) and (49) we obtain the following fourth order differential equation satisfied by q1∗ and θ ∗ [D 4 − A1 D 2 + B1 ]{q1* ( y )} = − L 4Q 0 f ∗ exp(−γ ∗ y ),
(50)
[D 4 − A1 D 2 + B1 ]{θ ∗ ( y )} = − L5Q 0 f ∗ exp(−γ ∗ y ),
(51)
The solutions of Eqs. (50), (51) and (48) are 2
(q 1 , θ )( x , y , t ) = ∑ (1, h1n ) R n exp( − rn y + iax − i ω t ) + ( L 4 , L 5 ) L 6Q 0 f exp( − γ ∗ y ),
(52)
q 2 (x , y , t ) = R 3 exp(− r3 y + iax − i ωt ).
(53)
n =1
The expressions for the displacement components and the stresses in the generalized thermoelastic medium with thermal loading due to laser pulse in the context of the (G-N type III) theory, are 2
u ( x , y , t ) = [ ∑ iaR n exp( − rn y ) − r3 R 3 exp( − r3 y )]exp[i (ax − ωt )] − Q 0 L 6 L 4 ( n =1
2x ) f exp( −γ ∗ y ), r2
2
v ( x , y , t ) = [ ∑ − rn R n exp(− rn y ) − i aR 3 exp(− r3 y )]exp[i (ax − ωt )] − Q 0γ ∗ L 4 L 6 f exp(−γ ∗ y ),
(54) (55)
n =1
2
σ x x ( x , y , t ) = [ ∑ h 2 n R n exp( − rn y ) +N 4 R 3 exp( − r3 y )] exp[i (a x − ω t )] + Q 0 f 5 exp( − γ ∗ y ),
(56)
n =1
2
σ yy (x , y , t ) = [ ∑ h3n R n exp(− rn y ) −N 4 R 3 exp(−r3 y )]exp[i (a x − ω t )] + Q 0 f 6 exp(−γ ∗ y ),
(57)
n =1 2
σ zz (x , y ,t ) =
∑ h 4 n R n exp( − rn
y ) exp[i (a x − ω t )] + Q 0 f 7 exp( − γ ∗ y ),
(58)
n =1 2
σ xy (x , y ,t ) = [ ∑ h5n R n exp(− rn y ) +N 5 R3 exp(−r3 y )]exp[i (a x − ω t )] + Q0 f 8 exp(−γ ∗ y ),
(59)
n =1
where rn2 (n = 1, 2, 3) are the roots of the characteristic equation of Eq. (50) and (48),
8
a22 =
1 , A1 = a22 (a2a9 + a1a10 + a4a12 ), B1 = a22 (a2a10 + a4a12a 2 ), L 4 = a4a22 , a1a9
h1n =
a1rn2 − a2 −1 , L6 = , h2 n = − b1a 2 + b 2 rn2 − b 4 h1n , h3n = − b 2a 2 + b1rn2 − b 4 h1n , ∗ 4 a4 γ − A1γ ∗2 + B1
L5 = a22 (a1γ ∗ −a2 ),
2
h4n = − b2a2 + b2rn2 − b4 h1n , h5 n = −2ia rn , N 4 = i a r3 (b2 − b1 ), f 5 = L6 f [L4b1 ( −2 + 4x ) + b2γ ∗2 L4 − b4 L5 ], 2 4 r
f 7 = L 6 f [L 4b 2 (
f8 =
−2 r
4xL 6 L 4 f γ ∗ r
2
2
+
4x 2 r
4
) + b 2γ ∗2 L 4 − b 4 L5 ],
f 6 = L 6 f [ L 4b 2 (
−2 r
2
+
r
4x 2 r
4
) + b1γ ∗2 L 4 − b 4 L 5 ],
, N 5 = r32 + a 2 .
In order to determine the coefficients Rn (n = 1,2,3) , we use the boundary conditions in Eq. (5) and by using the inverse of matrix method as follows: h22 R1 h21 R = h h52 2 51 R −r h 3 1 11 −r2 h12
N4 N5 0
−1
−p∗ 1 0 . 0
(60)
5.2 Generalized thermoelasticity theory without energy dissipation (G-N type II):
putting (K ∗ = 0) in the heat equation. 6. Numerical analysis and discussion
For numerical computations, magnesium material was chosen for purposes of numerical evaluations [17]. The constants of the problem were taken as; T 0 = 298 K ,
λ = 2.17 ×1010N / m2,
C E = 1.04×103 J / Kg deg,
µ = 3.278×1010 N / m2,
ω1* = 3.58 ×1011 / s ,
ρ = 1.74 × 103 Kg / m 3 ,
αt =1.78×10−5N / m2,
β = 2.68×106 N / m2 deg,
K = 1.7 ×102W / m deg.
The voids parameters and the laser pulse parameters are ψ =1.753×10−15m2 , α = 3.688×10−5N , ξ1 =1.475×1010 N / m2 , b =1.13849×1010N / m2 , m = 2×106N / m2 deg,
ω0 = 0.0787 ×10−3 N / m 2s ,
r =100 µm, γ ∗ = 0.5m−1,
I0 =105J ,
t0 = 0.2p.sec
The comparisons were carried out for: p1∗ = 10,
a = 1.7,
ω = b0 + i ξ , b0 = 0.3, ξ = 0.7,
x = 10,
(t = 0.3, 0.9)
and
0 ≤ y ≤ 5.
The comparisons have established for two cases: (i) With two values of time
(t = 0.3, 0.9) in
the context of (G-N type II) theory and (G-N
type III) theory in the presence laser pulse with the pulse duration (t0 = 0.2 p.sec). (ii) In the presence and absence void parameters in the context of (G-N type II) theory 9
and (G-N type III) theory in the presence laser pulse with the pulse duration (t0 = 0.2 p.sec), at (t = 0.9).
The computations are carried out on the surface plane x = 10. The numerical technique outlined above is used for the distribution of the real part of the non-dimensional displacements u , v , the non-dimensional temperature θ , the distributions of nondimensional stresses σ xx , σ xy and change in the volume fraction field φ with distance y for the problem.
Figs. 1-6 are plotted to show the variation of the above quantities against the distance y , in both of the (G-N type II) theory and (G-N type III) theory of the two values of time
(t = 0.3, 0.9),
where the solid lines represent results for the (G-N type III)
theory for (t = 0.3), the large dash line represents results for the (G-N type III) theory for
(t = 0.9),
the small dash line represents results for the (G-N type II) theory at
(t = 0.3),
the small dash line with dot represents results for the (G-N type II) theory at
(t = 0.9).
In Fig. 1, the displacement component u is plotted against the distance y , it is
observed that the displacement u for the (G-N type II) theory is greater than that of the (G-N type III) theory and it is clear that the values of solutions for
(t = 0.9)
are
greater than that for (t = 0.3). Fig. 2 investigates the displacement component v against the distance y , the values of v increases with the increase of the time parameter t , the values in the case of (G-N type III) theory are greater than in the case of (G-N type II) theory. Fig. 3 explains the distribution of temperature θ against the distance y , in this figure, the values of θ in the case of (G-N type III) theory are smaller than in the case of (G-N type II) theory. Figs. 4 and 5 depict the distributions of stress components σ xx and σ xy in the context of both (G-N type III) theory and (GN type II) theory; we see that the values of solutions for (t = 0.3) are smaller than that
for
(t = 0.9).
From this figure, the time parameter has a decreasing effect. Fig. 6
expresses the distribution of change in the volume fraction field φ against the distance y , it is observed that the change in the volume fraction field φ for the (G-N type II)
theory is greater than that of the (G-N type III) theory and it is clear that the values of solutions for (t = 0.3) are smaller than that for
(t = 0.9).
Figures 7-11 show the behavior of the physical quantities against distance y in 2D 10
during
(t = 0.9),
with and without voids effect, where the solid lines represent results
for the (G-N type III) theory to voids effect, the large dash line represents results for the (G-N type III) theory without voids effect, the small dash line represents results for the (G-N type II) theory with voids effect and the small dash line with dot represents the results for the (G-N type II) without voids effect. From this figure, the void parameters have a significant role in the distribution of all physical quantities in the problem. Fig. 7 depicts the distribution of the displacement components u , the values of the displacement u for presence voids are small compared to those for absence voids in the range 0 ≤ y ≤ 0.4 and great on the range 0.4 ≤ y ≤ 3.4, while the values are the same for all cases at y ≥ 3.4. Fig. 8 shows the displacement component
v against the distance
y , the
values of v for presence voids are great compared to
those for absence voids. Fig. 9 clarifies the distribution of the temperature θ is decreasing in the all cases (with and without voids) and the values of the temperature θ for presence voids are small compared to those for absence voids. Fig. 10
displays that the distribution of the stress component σ xx always begin from negative values in the all cases and the values of the stress component σ xx for presence voids are large compared to those for absence voids in the range 0 ≤ y ≤ 0.6 ; small in the range 0.6 ≤ y ≤ 1.6, while the values are the same for the cases at y ≥ 1.6. Fig. 11 explains the distribution of tangential stress σ xy versus y . The values of σ xy are increasing for the presence void parameters. In this figure the values of σ xy satisfying the boundary conditions of the problem. Figs. 12-13 are giving 3D surface curves for the physical quantities i.e., the tangential stress σ xy and the change in the volume fraction field φ , for the thermal loading due to laser pulse on thermoelastic medium with voids in the context of the (G-N type II) theory and (G-N type III) theory. These figures are very important to study the dependence of these physical quantities on the vertical component of distance. 7. Conclusion
According to the above results, we can conclude that: 1.
The void parameters and the different values of time in the current model have significant effects on all the fields. 11
2.
The value of all physical quantities converges to zero with an increase in distance y
3.
and all functions are continuous.
The comparison of different theories of thermoelasticity, i.e. both types of II and III of (G-N) theory are carried out.
4.
The analytical solutions based upon normal mode analysis for thermoelastic problem in solids have been developed and uzed.
5.
The deformation of a body depends on the nature of the applied forces and thermal loading due to laser pulse as well as the type of boundary conditions.
References [1]
A.E. Green, P.M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proceedings of the Royal Society London A 432 (1991) 171-194.
[2]
A.E. Green, P.M. Naghdi, On undamped heat waves in an elastic solid, Journal of Thermal Stresses 15 (1992) 253-264.
[3]
A.E. Green, P.M. Naghdi, Thermoelasticity without energy dissipation, Journal of Elasticity 31 (1993) 189-208.
[4]
M.I.A. Othman, I.A. Abbas, Generalized thermoelsticity of thermal shock problem in a non-homogeneous isotropic hollow cylinder with energy dissipation, International
Journal of Thermophysics 33 (2012) 913-923. [5]
R. Kumar, R. Kumar, Propagation of wave at the boundary surface of transversely isotropic thermoelastic material with voids and isotropic elastic half-space, Applied Mathematics and Mechanics (English Edition) 31 (2010) 1153-1172.
[6]
I.A. Abbas, M.I.A. Othman, Effect of rotation on thermoelstic waves with GreenNaghdi theory in a homogeneous isotropic hollow cylinder, International Journal of Industrial Mathematics 1 (2009) 121-134.
[7]
D.D. Joseph, L. Preziosi, Heat waves, Reviews of Modern Physics 61 (1989) 4173.
[8]
D.D. Joseph, L. Preziosi, Heat waves, Reviews of Modern Physics 62 (1990) 375391.
[9]
M.I.A. Othman, R. Kumar, Reflection of magneto-thermoelasticity waves with temperature dependent properties in generalized thermoelasticity, International Communications in Heat and Mass Transfer 36 (2009) 513-520.
[10] J.W. Nunziato, S.C. Cowin, A non-linear theory of elastic materials with voids, Arch. Ration. Mech. Anal. 72 (1979) 175-201. 12
[11] S.C. Cowin, J.W. Nunziato, Linear theory of elastic materials with voids, Journal of Elasticity 13 (1983) 125-147. [12] H.M. Youssef, State-space approach to two-temperature generalized thermoelasticity without energy dissipation of medium subjected to moving heat source, Applied Mathematics and Mechanics (English Edition) 34 (2013) 63-74.
[13] M. Marin, R.P. Agarwal, M.I.A. Othman, Localization in time of solutions for thermoelastic micropolar materials with voids, Computers, Materials & Continua 40 (2014) 35-48. [14] M. Marin, M.I.A. Othman, I.A. Abbas, An extension of the domain of influence theorem for anisotropic thermoelastic material with voids, Journal of Computational and Theoretical Nanoscience 12 (2015) 1594-1598. [15] M.I.A. Othman, S.Y. Atwa, R.M. Farouk, The effect of diffusion on twodimensional problem of generalized thermoelasticity with Green-Naghdi theory, International Communications in Heat and Mass Transfer 36 (2009) 857-864. [16] H.M. Al-Qahtani, S. K. Datta, Laser-generalized thermoelastic waves in an anisotropic infinite plate: Exact analysis, Journal of Thermal Stresses 31 (2008) 569-583. [17] R.S. Dhaliwal, A. Singh, Dynamic coupled thermoelasticity, Hindustan Publ. Corp, New Delhi, 1980.
Figure 1. Variation of displacement component u with horizontal distance y .
13
Figure 2. Variation of displacement component v with horizontal distance y .
Figure 3. Variation of temperature θ with horizontal distance y .
Figure 4. Variation of normal stresses σ xx with horizontal distance y .
14
Figure 5. Variation of tangential stresses σ xy with horizontal distance y .
Figure 6. Variation of volume fraction field distribution φ with horizontal distance y .
Figure 7. Variation of displacement component u with horizontal distance y .
15
Figure 8. Variation of displacement component v with horizontal distance y .
Figure 9. Variation of temperature θ with horizontal distance y .
Figure 10. Variation of normal stresses σ xx with horizontal distance y .
16
Figure 11. Variation of normal stresses σ xy with horizontal distance y .
Figure 12. (3D) The component of stress σ xy against both components of distance based on the (G-N type III) theory at t = 0.3.
Figure 13. (3D) Distribution of volume fraction field φ against both components of distance based on the (G-N type III) theory at t = 0.3.
17
Highlight 1) The comparison of different theories of thermoelasticity, i.e. both types of II and
III
of (G-N) theory. 2) Analytical solutions with normal mode for thermoelastic problem in solids have been developed. 3) The laser pulse has a significant effect on the field quantities. 4) The void parameters and the different values of time have important effect.
18