Three-Phase Lag Model on Thermoelastic Interaction in an Unbounded Fiber-Reinforced Anisotropic

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Printed in the United States of America Computational and Theoretical Nanoscience

Vol. 11, 1–6, 2014

Three-Phase Lag Model on Thermoelastic Interaction in an Unbounded Fiber-Reinforced Anisotropic

Medium with a Cylindrical Cavity

Ibrahim A. Abbas

Faculty of Science and Arts-Khulais, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia;

Faculty of Science, Department of Mathematics, Sohag University, Sohag, Egypt

The present investigation deals with the thermoelastic interaction in an unbounded fiber-reinforced anisotropic medium with a cylindrical cavity. The three-phase-lag (3PL) thermoelastic model and Green and Naghdi model (GNIII) are employed to study the thermophysical quantities. In the three- phase-lag model, the equation of heat conduction a containing fourth order derivative with respect to time. The cavity surface is subjected to uniform step in temperature of its internal boundary, which is assumed to be traction free. The numerical results of temperature, displacement, redial stress and hoop stress are obtained with the help of the finite element procedure. The effects of reinforcement are studied and the comparisons of the results for different thermoelastic models are made.

Keywords: Three-Phase Lag, Generalized Thermoelasticity, Anisotropic, Fiber-Reinforced, Finite Element Method.

1. INTRODUCTION

Fibre-reinforced materials have many applications in auto- motive fields, notably in modern bicycles and motorcycles, where its high strength-to-weight ratio is of importance.

Improved manufacturing techniques are reducing the costs and time to manufacture, such as laptops, fishing rods, archery equipment, stringed instrument bodies, racquet frames etc. Fiber-reinforced materials are used in a variety of structures due to their low weight and high strength.

The analysis of stress and deformation of fiber-reinforced materials has been an important subject of solid mechanics for the recentlly years. The mechanical and thermal behav- iors of fibre-reinforced anisotropic materials are modeled by the theory of linear thermoelasticity for anisotropic materials, with the preferred direction coinciding with the fibre direction. The theory of fibre-reinforced anisotropic materials has been extensively discussed in the literature, Belfield et al.¹ discussed the problem of stress in elas- tic plates reinforced by fibres lying in concentric circles.

Hashin and Rosen² gave the elastic moduli for fibre- reinforced materials. Sengupta and Nath³ studied the sur- face waves in fibre-reinforced anisotropic elastic media.

Singh⁴gave comments, this decoupling cannot be achieved by the introduction of the displacement potentials for wave propagation in fibre-reinforced anisotropic media.

Singh⁵ studied the wave propagation in thermally con- ducting linear fiber-reinforced composite materials with one relaxation time. Abbas,⁶ Othman and Abbas⁷ and Abbas and Abd-Alla⁸ studied the wave propagation in fibre-reinforced anisotropic media by numerical method.

In classical theory of thermoelasticity, Fourier’s heat con- duction theory assumes that the thermal disturbances prop- agate at infinite speed which is unrealistic from the phys- ical point of view. The theory of couple thermoelasticity was extended by Lord and Shulman.⁹ A more rigorous theory of thermoelasticity by introducing two relaxation times has been formulated by Green and Lindsay.¹⁰ The theory was extended for anisotropic body by Dhaliwal and Sherief.¹¹Three new thermoelastic theories have been pro- posed by Green and Naghdi^12–14Recently Raychoudhuri¹⁵ has introduced the three-phase-lag heat conduction equa- tion in which the Fourier with the introduction of three different phase-lags for the heat flux vector, the tem- perature gradient and the thermal displacement gradient.

Mukhopadhyay and Kumar¹⁶ have studied the effects of phase-lags on wave propagation in a thick plate. Also, Sub- sequently Kar and Kanoria¹⁷have employed the theory of thermoelasticity with three-phase-lag to discuss a thermal shock problem. The theory of self-reinforced materials has been discussed in the literature.^18–21 Tian et al!²² Abbas,²³

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Abbas and Abd-Alla,²⁴ and Abbas and Othman²⁵ applied the finite element method in different generalized thermoe- lastic problems. Recently, Bose et al.²⁶ Ono et al.²⁷ and Sajfert et al.²⁸studied other problems in waves.

The aim of present paper is to investigate the effects of phase lags and reinforcement on thermoelastic interac- tion in an unbounded fiber-reinforced anisotropic medium with a cylindrical cavity. In the investigation, we con- sider thermoelastic problem involving such circumferen- tially reinforced plates. The governing equations has been solved numerically using a finite element method. Numer- ical results for the field quantities are represented graphi- cally with some comparisons.

2. FORMULATION OF THE PROBLEM

By following Roychoudhuri,¹⁵ Abbas and Abd-Alla,⁸ the basic governing equations and the constitutive relations for three-phase lag model on generalized thermoelastic theory in a fiber-reinforced anisotropic medium whose preferred direction is that of a unit vectora, of the form:

The equation of motion

"_ij#j=$u¨_i# i#j=1#2#3 (1) Stress–strain–temperature relations

"_ij=%e_kk&_ij+ 2'_Te_ij+ ()a_ka_me_km&_ij+ a_ia_je_kk* + 2)'_L−'_T*)a_ia_ke_kj+ a_ja_ke_ki*+ +a_ka_me_km

×a_ia_j−+_ij)T−T_o*&_ij# i#j#k#m=1#2#3 (2) The energy equation has the form

)K_ij+ ,_-K_ij^∗*T˙_#ji+ ,_TK_ijT¨_#ji+ K_ij^∗T_#ji=

! 1+ ,_q .

.t+ ,_q² 2!

.² .t²

"

×)$c_eT¨+ T_o+_iju¨_i#j*# i#j=1#2#3 (3) When,_-=,_T=,_q=0, the Eqs. (1)–(3) correspond to the equations of thermoelasticity with Green and Naghdi model (GN). In the cylindrical coordinate system)r# /#z*

for the axially symmetric problem, the displacement vector possesses only the radial componentu=u)r#t*, wherer is the radial distance.a have components)0#1#0*for cir- cumferential reinforcement. The constitutive relations and field equations without body forces and heat sources in the present case are

."_rr .r + 1

r)"_rr−"_//*=$.²u

.t² (4)

"_rr=)%+ 2'_T*.u

.r+ )%+ (*u

r−+₁₁)T−T_o* (5)

"_//=)%+ (*.u

.r + )%+ 2(+ 4'_L−2'_T+ +*u r

−+₂₂)T−T_o* (6)

. .t

#

)K₁₁+ ,_-K₁₁^∗*.²T

.r² + )K₂₂+ ,_-K₂₂^∗*1 r

.T .r

$

+ ,_T .² .t²

! K₁₁.²T

.r² + K₂₂1 r

.T .r

"

+ K₁₁^∗.²T .r² + K₂₂^∗1

r .T

.r =

! 1+ ,_q .

.t+ ,_q² 2

.² .t²

".² .t²

×

!

$c_eT+ T_o+₁₁.u

.r + T_o+₂₂u r

"

(7) where +₁₁ =2)%+ '_T*(₁₁+ )%+ (*(₂₂# +₂₂ =2)%+ (*(₁₁+ )%+ 2(+ 4'_L−2'_T+ +*(₂₂!

For convenience, introducing the following non- dimensional variables

)r^′#u^′*=c0)r#u*# )t^′#,_-^′#,_T^′#,_q^′*=c²0)t#,_-#,_T#,_q*#

)"_rr^′#"_//^′ *=1

D)"_rr#"_//*# T^′=T−T_o T_o # 0=$c_e

K₁₁# c=

%D

$# D=%+ 2(+ 4'_L−2'_T+ + (8)

Governing equations reduce to the following form:

. .r

! B₁.u

.r + B₂u r −B₃T

"

+ )B₁−B₂*1 r

.u .r + )B₂−1*u

r²−)B₃−B₄*T r =.²u

.t² (9)

! 1₁+ 1₃ .

.t+ 1₅.² .t²

".²T .r² +

! 1₂+ 1₄ .

.t+ 1₆.² .t²

"1 r

.T .r

= .² .t²

! 1+ ,_q .

.t+ ,_q² 2

.² .t²

"!

T+ 1₇.u .r+ 1₈u

r

"

(10)

"_rr=B₁.u .r+ B₂u

r−B₃T (11)

"_//=B₂.u .r+ u

r−B₄T (12) where

)B₁#B₂#B₃#B₄*= 1

D)%+ 2'_T# %+ (#T_o+₁₁#T_o+₂₂*#

)1₁# 1₂*= 1

$c_ec₁²)K₁₁^∗#K₂₂^∗* )1₃# 1₄# 1₅# 1₆*=

!

1+ ,_-K₁₁^∗

$c_ec₁²#K₂₂ K₁₁+ ,_-K₁₁^∗

$c_ec²₁# ,_T#,_TK₂₂ K₁₁

"

# )1₇# 1₈*= 1

$c_e)+₁₁# +₂₂* 3. BOUNDARY CONDITIONS

From the preceding description, the boundary and initial conditions may be expressed as:

"_rr)1#t*=0# T)1#t*=T₁H)t*# t >0 (13)

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u=T=0 att=0# r≥1# .u .t =.T

.t =0 att=0 andu#T→0 whenr→ ' (14) whereH)t*denotes the Heaviside unit step function.

4. FINITE ELEMENT METHOD

The Finite element method is a powerful technique orig- inally developed for numerical solution of general prob- lems. In order to investigate the numerical solution of the generalized thermoelasticity based upon three-phase lags and Green and Naghdi models, the finite element method is adopted due to its flexibility in modeling layered struc- tures and its capability in obtaining full field numerical solution. In our case, the corresponding nodal values of displacement componentuand temperatureT are given as follows:

u=

&m i=1

N_iu_i)t*# T =

&m i=1

N_iT_i)t* (15) whereN are the shape functions andmdenotes the num- ber of nodes per element. Using the Galerkin procedure, the corresponding weighting functions are approximated by the same shape functions. Thus,

&u=&^m

i=1

N_i&u_i# &T =&^m

i=1

N_i&T_i (16) With Eqs. (15) and (16), u^′=u_#i and T^′ =T_#i can be expressed as

u^′=&^m

i=1

N_i^′u_i)t*# T^′=&^m

i=1

N_i^′T_i)t* (17)

&u^′=&^m

i=1

N_i^′&u_i# &T^′=&^m

i=1

N_i^′&T_i (18) Thus, the Eqs. (2) and (2) in its finite element equations form can be obtained as

&me e=1

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎝

* 0 0

H₂₁^e H₂₂^e +,u^e)4*

T^e)4*

- +

* 0 0

G^e₂₁ G^e₂₂ +

× ,u^e)3*

T^e)3*

- +

*M₁₁^e 0 M₂₁^e M₂₂^e

+,u¨^e

¨ T^e

-

+

*0 0

0 C₂₂^e +,u˙^e

˙ T^e

-

+

*K₁₁^e K₁₂^e 0 K₂₂^e

+,u^e T^e

-

= ,F₁^e

F₂^e -

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎠

(19)

wheremeis the total number of elements. In Appendix A, the coefficients in Eq. (19) are presented.

Symbolically, the Eq. (19) can be written as

H d^)4*+ G d^)3*+ Md¨+ Cd˙+ Kd=F^ext (20)

whereH,G,M,CandKare matrices andF^extis external force vector;d=2u T3^t!The backward difference method is used to determine the time derivatives of the unknown variables.

5. RESULTS AND DISCUSSION

With an aim to illustrate the problem, we use the fol- lowing physical constants for generalized fibre-reinforced thermoelastic materials.⁸

$=2660 kg/m³# %=5!65×10¹⁰N/m²# '_T=2!46×10¹⁰N/m²# '_L=5!66×10¹⁰ N/m²# (=−1!28×10¹⁰N/m²# +=220!90×10¹⁰N/m²#

(₁₁=0!017×10⁻⁴ deg⁻¹#T₁=1#

(₂₂=0!015×10⁻⁴ deg⁻¹#c_e=0!787×10³ J kg⁻¹ deg⁻¹# ,_-=0!1# ,_T=0!15# ,_q=0!3#

K₁₁=0!0921×10³ J m⁻¹s⁻¹ deg⁻¹#

K₂₂=0!0963×10³ J m⁻¹ s⁻¹ deg⁻¹#T_o=293 k# t=0!5 Using this data, the Figures 1–12 shows that the behav- iors of displacement u, temperature T, radial and hoop stresses"_rr,"_//for different values of the radial distance r. The results for temperature, displacement, radial stress and hoop stress has been carried out by takingT₁=1 and t=0!5! We have three group of graphs where we have three application:

Fig. 1. The variation distribution of temperature att=0!5!

Fig. 2. Distribution of displacement att=0!5!

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Fig. 3. Distribution of radial stress att=0!5!

Fig. 4. Distribution of hoop stress att=0!5!

Fig. 5. Distribution of temperature under three-phase lag model at t=0!5!

Fig. 6. Distribution of displacement under three-phase lag model at t=0!5!

Fig. 7. Distribution of radial stress under three-phase lag model att= 0!5!

Fig. 8. Distribution of hoop stress under three-phase lag model att= 0!5!

Fig. 9. Distribution of temperature att=0!5 with different values of,_q.

Fig. 10. Distribution of displacement att=0!5 with different values of,q.

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Fig. 11. Distribution of radial stress att=0!5 with different values of,_q.

Fig. 12. Distribution of hoop stress att=0!5 with different values of,q.

The first group (Figs. 1–4) exhibits the differences between the three-phase lags model (3PL) and Green and Naghdi model (GNIII). The lags have a significant effect on the speed of the wave propagation and all the fields.

From Figure 1 we deduce that under (3PL) model, the temperature vanish smoothly far from the nearest end of the medium which is more realistic than the correspon- dence result of (GNIII) model. Figure 2 is plotted for dis- placement u versus radial distance r. It is seen that the displacement corresponding have maximum magnitudes at the boundary to each model. It is also observed that as r increases the displacement change their signs then it approaches and ultimately becomes zero. Figure 3 displays a comparison of the radial stress in the context of the two models. Atr=1, the radial stress reduces to zero which agree with the boundary condition. Figure 4 is drawn to show the variation of hoop stress against radial distance r. Also, for each model the hoop stress have a maximum magnitudes at the boundary.

The second group (Figs. 5–8) represent the variations of the physical quantities under three-phase lags model with the effects of reinforcement such that (WRE) refer to reinforcement by the solid line (——) and (NRE) refer to absence reinforcement which given by dot line (...). As expect, the reinforcement has a great effect on the distri- bution of field quantities.

The third group (Figs. 9–12) demonstrate the behavior of the temperature, displacement, radial stress and hoop stress under three-phase lag model with reinforcement for

four different values of phase-lag of the heat flux when ),_-=0!1# ,_T=0!15*. The difference is more pronounced for increasing the phase-lag of the heat flux.

6. CONCLUSION

In this paper we have studied the thermoelastic interaction in an unbounded fiber-reinforced anisotropic medium with a cylindrical cavity under three-phase lag model and Green and Naghdi model. The problem has been solved numeri- cally using a finite element method. From the results, it is noted that the nature of variations of all the fields is qual- itatively similar for both the models. The other important facts are observed as follows.

(1) The reinforcement has a great effect on the distribution of field quantities.

(2) Three-phase lag model shows higher absolute values for all fields as compared to the model of Green and Naghdi model.

(3) The way of vanishing the temperature in three-phase lag model in comparison with Green and Naghdi model leads us to claim that the three-phase lag model is more realistic than that of Green and Naghdi model.

(4) We have found that, the phase-lag of the heat flux has significant effects on all the fields.

NOMENCLATURE

$ Mass density

T Temperature change of a material particle

u_i Displacement vector components T_o Reference uniform temperature of the

body e_ij Strain tensor

+_ij Thermal elastic coupling tensor

"_ij Stress tensor K_ij Thermal conductivity

c_e Specific heat at constant strain K_ij^∗ Material characteristic of the theory

,₄ Phase-lag of the thermal displacement gradient

,_T Phase-lag of the temperature gradient ,_q Phase-lag of the heat flux

(_ij Linear thermal expansion tensor

%,'_T Elastic parameters

(,+, ('_L−'_T) Reinforced anisotropic elastic parameters.

APPENDIX A

The coefficients appeared in Eq. (20) are given by H₂₁^e =1 ,_q²

22N3^T

!

1₇2N^′3+ 1₈ r 2N3

"

dr#

H₂₂^e=1 ,_q²

22N3^T2N3dr#

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G^e₂₁=1 ,_q2N3^T

!

1₇2N^′3+ 1₈ r 2N3

"

dr#

H₂₂^e =1

,_q2N3^T2N3dr#

M₁₁^e =1

2N3^T2N3dr# M₂₁^e=1 2N3^T

!

1₇2N^′3+ 1₈ r 2N3

"

dr#

M₂₂^e=1 2N3^T

!

2N3−1₅2N^′′3−1₆ r 2N^′3

"

dr#

C₂₂^e =1 2N3^T

!

−1₃2N^′′3−1₄ r 2N^′3

"

dr#

K₁₁^e =1 # 2N^′3^T

!

B₁2N^′3+ B₂ r 2N3

"

+ 2N3^T

!B₂−B₁

r 2N^′3+ 1−B₂ r² 2N3

"$

dr#

K₁₂^e =1 #

−2N^′3^TB₃2N3+ B₃−B₄

r 2N3^T2N3

$ dr#

F₁^e=2N3^T,¯(^r1# K₂₂^e =1 #

1₁2N^′3^T2N^′3−1₂

r 2N3^T2N^′3

$ dr#

F₂^e=2N3^Tq¯(^r1

where,represent the component of the traction, and¯ q¯rep- resents heat flux.

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Received: 8 February 2013. Accepted: 3 March 2013.