Meccanica (2011) 46: 413–421DOI 10.1007/s11012-010-9322-z

O R I G I NA L A RT I C L E

Effect of rotation on plane waves at the free surfaceof a fibre-reinforced thermoelastic half-space using the finiteelement method

Mohamed I.A. Othman · Ibrahim A. Abbas

Received: 2 July 2009 / Accepted: 25 May 2010 / Published online: 3 August 2010© Springer Science+Business Media B.V. 2010

Abstract The propagation of plane waves in fibre-reinforced, rotating thermoelastic half-space proposedby Lord-Shulman is discussed. The problem has beensolved numerically using a finite element method. Nu-merical results for the temperature distribution, thedisplacement components and the thermal stress aregiven and illustrated graphically. Comparisons aremade with the results predicted by the coupled the-ory and the theory of generalized thermoelasticity withone relaxation time in the presence and absence of ro-tation and reinforcement. It is found that the rotationhas a significant effect and the reinforcement has greateffect on the distribution of field quantities when therotation is considered.

Keywords Rotation · Fibre-reinforced medium ·Half-space · Thermal relaxation time · Finite elementmethod

M.I.A. Othman (�)Department of mathematics, Faculty of Science,Zagazig University, P.O. Box 44519, Zagazig, Egypte-mail: m_i_othman@yahoo.com

M.I.A. OthmanDepartment of mathematics, Faculty of Science,Shaqra University, P.O. Box 1040, AL-Dawadmi 11911,Kingdom of Saudi Arabia

I.A. AbbasDepartment of mathematics, Faculty of Science, SohagUniversity, Sohag, Egypte-mail: ibrabbas7@yahoo.com

1 Introduction

Fibre-reinforced composites are used in a variety ofstructures due to their low weight and high strength.The characteristic property of a reinforced compos-ite is that its components act together as a singleanisotropic unit as long as they remain in the elasticcondition. Verma [1] discussed the problem of magne-toelastic shear waves in self-reinforced bodies. Chat-topadhyay and Choudhury [2] investigated the prop-agation, reflection and transmission of magnetoelasticshear waves in a self-reinforced media. Chattopadhyayand Choudhury [3] studied the propagation of mag-netoelastic shear waves in an infinite self-reinforcedplate. Chattopadhyay and Michel [4] studied a modelfor spherical SH-wave propagation in self-reinforcedlinearly elastic media, In classical dynamical coupledtheory of thermoelasticity, the thermal and mechani-cal waves propagate with an infinite velocity, whichis not physically admissible. The theory of couplethermoelasticity was extended by Lord and Shulman(LS) [5] and Green and Lindsay [6] by including thethermal relaxation time in constitutive relations. Dur-ing the last three decades a number of investigations[7–11] have been carried out using the aforesaid theo-ries of generalized thermoelasticity. Note that in mostof the earlier studies mechanical or thermal loading onthe boundary surface was considered to be in the formof a shock.

The exact solution of the governing equations ofthe generalized thermoelasticity theory for a coupled

414 Meccanica (2011) 46: 413–421

and nonlinear/linear system exists only for very spe-cial and simple initial and boundary problems. To cal-culate the solution of general problems, a numericalsolution technique is used. For this reason the finiteelement method is chosen. The method of weightedresiduals offers the formulation of the finite elementequations and yields the best approximate solutionsto linear and nonlinear boundary and partial differen-tial equations. Applying this method basically involvesthree steps. The first step is to assume the general be-havior of the unknown field variables in such a way asto satisfy the given differential equations. Substitutionof these approximating functions into the differentialequations and boundary conditions results in some er-rors, called the residual. This residual has to vanish inan average sense over the solution domain. The secondstep is the time integration. The time derivatives of theunknown variables have to be determined by formerresults. The third step is to solve the equations result-ing from the first and second step by using a finite el-ement algorithm program (see Zienkiewicz [12]). Ab-bas [13], Abbas and Abd-Alla [14] and Youssef andAbbas [15] applied the finite element method in dif-ferent problems.

In the present work, the (LS) theory is applied tostudy the influence of rein-forcement on the total de-formation of rotating body and the interaction witheach other. The problem has been solved numericallyusing a finite element method (FEM). Numerical re-sults for the temperature distribution, displacementand the stress components are represented graphically.

2 Formulation of the problem and basic equations

The constitutive equations for a fibre-reinforced lin-early thermoelastic anisotropic medium with respectto the reinforcement direction a are

σij = λekkδij + 2μT eij + α(akamekmδij + aiaj ekk)

+ 2(μL − μT )(aiakekj + ajakeki)

+ βakamekmaiaj − γ (T − T0)δij . (1)

Where σij are the components of stress; eij are thecomponents of strain; λ,μT are elastic constants;α,β, (μL − μT ) and γ are reinforcement parameters,T is the temperature above reference temperature T0

and a ≡ (a1, a2, a3), a21 + a2

2 + a23 = 1. We choose the

fibre-direction as a ≡ (1,0,0). The strains can be ex-pressed in terms of the displacement ui as

eij = 1

2(ui,j + uj,i). (2)

For plane strain deformation in the xy-plane, ∂∂z

≡ 0,w = 0.

Equation (1) then yields

σxx = A11u,x + A12v,y − γ (T − T0), (3)

σyy = A22v,y + A12u,x − γ (T − T0), (4)

σzz = A12u,x + λv,y − γ (T − T0), (5)

σxy = μL(u,y + v,x), σzx = σzy = 0. (6)

Where

A11 = λ + 2(α + μT ) + 4(μL − μT ) + β,

A12 = α + λ, A22 = λ + 2μT .(7)

Since the medium is rotating uniformly with an angu-lar velocity � = n, where n is a unit vector repre-senting the direction of the axis of rotation. The dis-placement equation of motion in the rotating frame ofreference has two additional terms (Schoenberg andCensor [17]): Centripetal acceleration, � × (� × u)

due to time-varying motion only and the Corioli’s ac-celeration 2� × u where u is the dynamic displace-ment vector. These terms don’t appear in non-rotatingmedia. The dynamic displacement vector is actuallymeasured from a steady-state deformed position andthe deformation is supposed to be small.

The equation of motion in a rotating frame of refer-ence in the context of Lord-Shulman’s theory is

ρ[ui + { × ( × u)}i + (2 × u)i] = σij,j

(i, j = 1,2,3). (8)

The heat conduction equation

kT,ii =(

∂

∂t+ τ0

∂2

∂t2

)(ρCET + γ T0ui,i). (9)

Where ρ is the mass density, k is the thermal conduc-tivity, CE is the specific heat at constant strain and τ0

is the relaxation time.From (3)–(6) we note that the third equation of mo-

tion in (8) is identically satisfied and the first two equa-

Meccanica (2011) 46: 413–421 415

tions become

ρ

[∂2u

∂t2− 2u − 2v

]

= A11∂2u

∂x2+ B2

∂2v

∂x∂y+ B1

∂2u

∂y2− γ

∂T

∂x, (10)

ρ

[∂2v

∂t2− 2v + 2u

]

= A22∂2v

∂y2+ B2

∂2u

∂x∂y+ B1

∂2v

∂x2− γ

∂T

∂y, (11)

where B1 = μL, B2 = α + λ + μL.For convenience, the following non-dimensional

variables are used:

x′ = c1ωx, y′ = c1ωy,

u′ = c1ωu, v′ = c1ωv,

t ′ = c21ωt, τ ′

o = c21ωτo,

′ =

c21ω

, θ = γ (T − T0)

λ + 2μT

,

σ ′ij = σij

μT

, i = 1,2,

(12)

where ω = ρCE

k, c2

1 = λ+2μT

ρ.

In terms of the non-dimensional quantities definedin (12), the above governing equations reduce to (drop-ping the dashes for convenience)

∂2u

∂t2− 2u − 2v

= D11∂2u

∂x2+ D2

∂2v

∂x ∂ y+ D1

∂2u

∂y2− ∂T

∂x, (13)

∂2v

∂t2− 2v + 2u

= D22∂2v

∂y2+ D2

∂2u

∂x∂y+ D1

∂2v

∂x2− ∂T

∂y, (14)

∂2θ

∂x2+ ∂2θ

∂y2

=(

∂

∂t+ τ0

∂2

∂t2

)[θ + ε

(∂u

∂x+ ∂v

∂y

)], (15)

where (D11,D22,D1,D2) = (A11,A22,B1,B2)λ+2μT

,

ε = γ 2T0ρCE(λ+2μT )

.Consider the boundary conditions at x = 0 as fol-

lowing

(1) Thermal boundary condition that the surface of thehalf-space subjected to a thermal shock

θ(0, y, t) = θ0H(L − |y|)e−bt . (16)

(2) Mechanical boundary condition that the surface ofthe half-space is traction free

σxx(0, y, t) = σxy(0, y, t) = 0. (17)

Initial conditions are:

θ(x, y,0) = 0,∂θ(x, y,0)

∂t= 0, (18)

u(x, y,0) = v(x, y,0) = 0,

∂u(x, y,0)

∂t= ∂v(x, y,0)

∂t= 0.

(19)

3 Finite element method

In order to investigate the propagation of plane wavesin fibre-reinforced, rotating thermoelastic half-spaceproposed by Lord-Shulman by finite element method,the (FEM) Zienkiewicz and Taylor [12], Reddy [18]and Cook et al. [19] are adopted due to its flexibil-ity in modelling layered structures and its capabil-ity in obtaining full field numerical solution. In or-der to solve the non-dimensional governing equations(13)–(15) using the finite element method (FEM), theweak formulations of these equations are derived. Itis convenient to prescribe the set of independent testfunctions to consist of the displacement componentsu and v and the temperature θ . To obtain the weakformulation, the governing equations are multipliedby independent weighting functions and then are in-tegrated over the spatial domain with the boundary.Applying integration by parts and making use of thedivergence theorem reduces the order of the spatialderivatives and allows for the application of the bound-ary conditions. Using the well known Galerkin pro-cedure, the unknown fields u, v and θ and the corre-sponding weighting functions are approximated by thesame shape functions, which are defined piecewise onthe elements. The last step towards the finite elementdiscretization is to choose the element type and the as-sociated shape functions. Eight nodes of quadrilateralelements are used. The unknown fields are approxi-mated either by linear shape functions, which are de-fined by four corner nodes or by quadratic shape func-tions, which are defined by all of the eight nodes (two-dimensional quadrilateral elements). On other hand

416 Meccanica (2011) 46: 413–421

the unknown fields are approximated either by lin-ear shape functions, which are defined by three cor-ner nodes or by quadratic shape functions, which aredefined by all of the six nodes (two-dimensional tri-angular elements). The shape function is usually de-noted by the letter N and is usually the coefficient thatappears in the interpolation polynomial. A shape func-tion is written for each individual node of a finite el-ement and has the property that its magnitude is 1 atthat node and 0 for all other nodes in that element.We assume that the master element has its local co-ordinates in the range [−1,1]. In our case, the two-dimensional quadrilateral elements are used, whichhad given by:

(i) Sketch of linear shape functions

(ii) Sketch of quadratic shape functions

(i) Linear shape functions

N1 = 1

4(1 − ξ)(1 − η),

N2 = 1

4(1 + ξ)(1 − η),

N3 = 1

4(1 + ξ)(1 + η),

N4 = 1

4(1 − ξ)(1 + η).

(20)

(ii) Quadratic shape functions

N1 = 1

4(1 − ξ)(1 − η)(−1 − ξ − η),

N5 = 1

2(1 − ξ2)(1 − η),

N2 = 1

4(1 + ξ)(1 − η)(−1 + ξ − η),

N6 = 1

2(1 + ξ)(1 − η2),

N3 = 1

4(1 + ξ)(1 + η)(−1 + ξ + η),

N7 = 1

2(1 − ξ2)(1 + η),

N4 = 1

4(1 − ξ)(1 + η)(−1 − ξ + η),

N8 = 1

2(1 − ξ)(1 − η2).

(21)

On the other hand, the time derivatives of the unknownvariables have to be determined by Newmark time in-tegration method (Reddy [18]).

4 Numerical results

To study the effect of rotation and reinforcement onwave propagation, we use the following numerical val-ues for the physical constants [15]

λ = 7.59 × 109 N/m2,

μT = 1.89 × 109 N/m2,

μL = 2.45 × 109 N/m2,

α = −1.28 × 109 N/m2,

β = 0.32 × 109 N/m2,

ρ = 7800 kg/m2,

θ0 = 1, b = 1, L = 1 and H is the heavyside unit stepfunction.

The field quantities, temperature, displacementcomponents u, v and stress components σxx , σyy andσxy depend not only on space x and time t but alsoon the thermal relaxation time τ0. Here all the vari-ables are taken in nondimensional forms. Figures 1–6exhibit the variation of the temperature, displacementcomponents u, v and stress components σxx , σyy and

Meccanica (2011) 46: 413–421 417

Fig. 1 The temperature distribution for different values of att = 0.1 and y = 0

Fig. 2 Horizontal displacement distribution u for different val-ues of at t = 0.1 and y = 0

Fig. 3 Vertical displacement distribution v for different valuesof at t = 0.1 and y = 0

σxy with space x under the classical dynamical CDand Lord-Shulman’s theory (LS) i.e., when there isone thermal relaxation time (τ0 = 0.02) at t = 0.1 and

Fig. 4 The distribution of stress component σxx for differentvalues of at t = 0.1 and y = 0

Fig. 5 The distribution of stress component σyy for differentvalues of at t = 0.1 and y = 0

Fig. 6 The distribution of stress component σxy for differentvalues of at t = 0.1 and y = 0

y = 0 for three different values of ( = 0,5,9) i.e.in the absence and presence of rotation. We observedfrom Fig. 1 that the rotation has no effect on the tem-

418 Meccanica (2011) 46: 413–421

Fig. 7 The temperature distribution for = 5 at t = 0.1 andy = 0

Fig. 8 Horizontal displacement distribution u for = 5 att = 0.1 and y = 0

Fig. 9 Vertical displacement distribution v for = 5 at t = 0.1and y = 0

perature with respect to the CD and (LS) theories. Fig-ure 2 shows that the rotation has decreasing effect onthe displacement component u for 0.1 < x < 0.7, it is

Fig. 10 The distribution of stress component σxx for = 5 att = 0.1 and y = 0

Fig. 11 The distribution of stress component σyy for = 5 att = 0.1 and y = 0

Fig. 12 The distribution of stress component σxy for = 5 att = 0.1 and y = 0

clear that the curves under (LS) theory are greater thanthat of CD theory. Figure 3 shows that the rotation hasincreasing effect on the displacement component v for

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Fig. 13 The temperature distribution for different values of t at = 5 and y = 1

Fig. 14 Horizontal displacement distribution u for differentvalues of t at = 5 and y = 1

0 < x < 0.1 and decreasing effect for 0.1 < x < 1.0in this region the curves under (LS) theory are greaterthan that of CD theory. Figures 4 and 5 show that therotation has decreasing effect on the two componentsof stress σxx and σyy for 0 < x < 0.1 and increas-ing effect for 0.1 < x < 1.5. Figure 6 shows that therotation has decreasing effect on the stress compo-nent σxy for 0 < x < 0.15 and increasing effect for0.15 < x < 0.6. We have found that the rotation has asignificant effect on the field quantities. Figures 7–12show the variation of the temperature, displacementcomponents u, v and stress components σxx , σyy andσxy with space x for = 5 and t = 0.1 under Lord-Shulman’s theory (LS) with reinforcement (WRE) andwithout reinforcement (NRE). In Figs. 7–12 the solidline refer to the case with reinforcement and the dotline refer to the case without reinforcement i.e. (α = 0,β = 0 and (μL − μT ) = 0). It easily to see from

Fig. 15 Vertical displacement distribution v for different valuesof t at = 5 and y = 1

Fig. 16 The distribution of stress component σxx for differentvalues of t at = 5 and y = 1

Figs. 7 and 8 that the reinforcement has small effecton the physical quantities temperature and displace-ment component u. Figure 9 shows that the reinforce-ment has decreasing effect on the displacement com-ponents v for 0 < x < 0.06 and increasing effect for0.06 < x < 0.1. Figure 10 shows that the reinforce-ment has increasing effect on σxx for 0 < x < 0.1and decreasing effect for 0.1 < x. Figure 11 showsthat the reinforcement has decreasing effect on σyy

for 0 < x < 0.17 and increasing effect for 0.17 < x.Figure 12 shows that the reinforcement has increasingeffect on σxy for 0 < x < 0.07 and decreasing effectfor 0.07 < x < 0.15.

Figures 13–18 show the variation of the tempera-ture, displacement components u, v and stress com-ponents σxx , σyy and σxy with space x for = 5 andy = 1 under Lord-Shulman’s theory (LS) with rein-forcement (WRE) for four different values of time

420 Meccanica (2011) 46: 413–421

Fig. 17 The distribution of stress component σyy for differentvalues of t at = 5 and y = 1

(t = 0.05,0.1,0.15,0.2). Figures 13 and 15 showthat under the reinforcement the physical quantitiestemperature and displacement component v increas-ing as increasing time and decreasing as increas-ing x. Figure 14 shows that the displacement com-ponent u decreasing for 0 < x < 0.1 and increasingfor 0.1 < x < 1.25 as increasing time t . Figures 16and 17 show that the time has increasing effect onσxx and σyy for 0 < x < 0.1 and decreasing effect for0.1 < x < 1.6. Figure 18 shows that the time has de-creasing effect on σxy for 0 < x < 0.4.

5 Concluding remarks

In this work the finite element method is used to studythe problem of the effect of rotation on a thermalshock problem at the free surface of a fibre-reinforcedthermoelastic half-space under the classical dynam-ical coupled theory CD and Lord-Shulman’s theory.We can obtain the following conclusions based on theabove analysis:

1. There are significant differences for field quantitiesunder the two theories due to essential differencesbetween the coupled theory CD and the (LS) the-ory.

2. The rotation has a significant effect on the fieldquantities.

3. The reinforcement has a great effect on the distri-bution of field quantities when the rotation is con-sidered.

Fig. 18 The distribution of stress component σxy for differentvalues of t at = 5 and y = 1

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