Meccanica (2011) 46: 413–421 DOI 10.1007/s11012-010-9322-z
O R I G I N A L A RT I C L E
Effect of rotation on plane waves at the free surface of a fibre-reinforced thermoelastic half-space using the finite element 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 fibrereinforced, rotating thermoelastic half-space proposed by Lord-Shulman is discussed. The problem has been solved numerically using a finite element method. Numerical results for the temperature distribution, the displacement components and the thermal stress are given and illustrated graphically. Comparisons are made with the results predicted by the coupled theory and the theory of generalized thermoelasticity with one relaxation time in the presence and absence of rotation and reinforcement. It is found that the rotation has a significant effect and the reinforcement has great effect on the distribution of field quantities when the rotation is considered. Keywords Rotation · Fibre-reinforced medium · Half-space · Thermal relaxation time · Finite element method M.I.A. Othman () Department of mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt e-mail:
[email protected] M.I.A. Othman Department of mathematics, Faculty of Science, Shaqra University, P.O. Box 1040, AL-Dawadmi 11911, Kingdom of Saudi Arabia I.A. Abbas Department of mathematics, Faculty of Science, Sohag University, Sohag, Egypt e-mail:
[email protected]
1 Introduction Fibre-reinforced composites are used in a variety of structures due to their low weight and high strength. The characteristic property of a reinforced composite is that its components act together as a single anisotropic unit as long as they remain in the elastic condition. Verma [1] discussed the problem of magnetoelastic shear waves in self-reinforced bodies. Chattopadhyay and Choudhury [2] investigated the propagation, reflection and transmission of magnetoelastic shear waves in a self-reinforced media. Chattopadhyay and Choudhury [3] studied the propagation of magnetoelastic shear waves in an infinite self-reinforced plate. Chattopadhyay and Michel [4] studied a model for spherical SH-wave propagation in self-reinforced linearly elastic media, In classical dynamical coupled theory of thermoelasticity, the thermal and mechanical waves propagate with an infinite velocity, which is not physically admissible. The theory of couple thermoelasticity was extended by Lord and Shulman (LS) [5] and Green and Lindsay [6] by including the thermal relaxation time in constitutive relations. During the last three decades a number of investigations [7–11] have been carried out using the aforesaid theories of generalized thermoelasticity. Note that in most of the earlier studies mechanical or thermal loading on the boundary surface was considered to be in the form of a shock. The exact solution of the governing equations of the generalized thermoelasticity theory for a coupled
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and nonlinear/linear system exists only for very special and simple initial and boundary problems. To calculate the solution of general problems, a numerical solution technique is used. For this reason the finite element method is chosen. The method of weighted residuals offers the formulation of the finite element equations and yields the best approximate solutions to linear and nonlinear boundary and partial differential equations. Applying this method basically involves three steps. The first step is to assume the general behavior of the unknown field variables in such a way as to satisfy the given differential equations. Substitution of these approximating functions into the differential equations and boundary conditions results in some errors, called the residual. This residual has to vanish in an average sense over the solution domain. The second step is the time integration. The time derivatives of the unknown variables have to be determined by former results. The third step is to solve the equations resulting from the first and second step by using a finite element algorithm program (see Zienkiewicz [12]). Abbas [13], Abbas and Abd-Alla [14] and Youssef and Abbas [15] applied the finite element method in different problems. In the present work, the (LS) theory is applied to study the influence of rein-forcement on the total deformation of rotating body and the interaction with each other. The problem has been solved numerically using a finite element method (FEM). Numerical results for the temperature distribution, displacement and the stress components are represented graphically.
2 Formulation of the problem and basic equations The constitutive equations for a fibre-reinforced linearly thermoelastic anisotropic medium with respect to the reinforcement direction a are σij = λekk δij + 2μT eij + α(ak am ekm δij + ai aj ekk )
fibre-direction as a ≡ (1, 0, 0). The strains can be expressed in terms of the displacement ui as 1 eij = (ui,j + uj,i ). 2 For plane strain deformation in the xy-plane, w = 0. Equation (1) then yields
Where σij are the components of stress; eij are the components 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 ), a12 + a22 + a32 = 1. We choose the
≡ 0,
(3)
σyy = A22 v,y + A12 u,x − γ (T − T0 ),
(4)
σzz = A12 u,x + λv,y − γ (T − T0 ),
(5)
σxy = μL (u,y + v,x ),
(6)
σzx = σzy = 0.
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 angular velocity = n, where n is a unit vector representing the direction of the axis of rotation. The displacement equation of motion in the rotating frame of reference has two additional terms (Schoenberg and Censor [17]): Centripetal acceleration, × ( × u) due to time-varying motion only and the Corioli’s acceleration 2 × u˙ where u is the dynamic displacement vector. These terms don’t appear in non-rotating media. The dynamic displacement vector is actually measured from a steady-state deformed position and the deformation is supposed to be small. The equation of motion in a rotating frame of reference in the context of Lord-Shulman’s theory is ρ[u¨ i + { × ( × u)}i + (2 × u) ˙ i ] = σij,j (i, j = 1, 2, 3).
(8)
The heat conduction equation
(1)
∂ ∂z
σxx = A11 u,x + A12 v,y − γ (T − T0 ),
+ 2(μL − μT )(ai ak ekj + aj ak eki ) + βak am ekm ai aj − γ (T − T0 )δij .
(2)
kT,ii =
∂2 ∂ + τ0 2 (ρCE T + γ T0 ui,i ). ∂t ∂t
(9)
Where ρ is the mass density, k is the thermal conductivity, CE is the specific heat at constant strain and τ0 is the relaxation time. From (3)–(6) we note that the third equation of motion in (8) is identically satisfied and the first two equa-
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tions become 2 ∂ u 2 ρ − u − 2 v˙ ∂t 2
(1) Thermal boundary condition that the surface of the half-space subjected to a thermal shock θ (0, y, t) = θ0 H (L − |y|)e−bt .
∂ 2u ∂ 2v ∂ 2u ∂T + B −γ + B , 2 1 ∂x∂y ∂x ∂x 2 ∂y 2 2 ∂ v 2 − v + 2 u˙ ρ ∂t 2 = A11
∂ 2v ∂ 2u ∂ 2v ∂T + B1 2 − γ , = A22 2 + B2 ∂x∂y ∂y ∂y ∂x
(10)
y = c1 ωy, ˜
u = c1 ωu, ˜
v = c1 ωv, ˜
t
= c12 ωt, ˜
= 2 , c1 ω˜ σij σij = , μT
τo
(11)
γ (T − T0 ) θ= , λ + 2μT
(12)
i = 1, 2,
∂ 2u ∂ 2v ∂ 2 u ∂T + D , + D − 2 1 ∂x ∂ y ∂x ∂x 2 ∂y 2
(13)
∂ 2v − 2 v + 2 u˙ ∂t 2 ∂ 2v ∂ 2u ∂ 2 v ∂T + D , + D − 2 1 ∂x∂y ∂y ∂y 2 ∂x 2
∂ 2θ ∂ 2θ + ∂x 2 ∂y 2 ∂u ∂v ∂2 ∂ + τ0 2 θ + ε + , = ∂t ∂x ∂y ∂t where (D11 , D22 , D1 , D2 ) = γ 2 T0
θ (x, y, 0) = 0,
∂θ (x, y, 0) = 0, ∂t
(18)
u(x, y, 0) = v(x, y, 0) = 0, ∂u(x, y, 0) ∂v(x, y, 0) = = 0. ∂t ∂t
(19)
3 Finite element method
= c12 ωτ ˜ o,
∂ 2u − 2 u − 2 v˙ ∂t 2
= D22
(17)
Initial conditions are:
T where ω˜ = ρCk E , c12 = λ+2μ . ρ In terms of the non-dimensional quantities defined in (12), the above governing equations reduce to (dropping the dashes for convenience)
= D11
(2) Mechanical boundary condition that the surface of the half-space is traction free σxx (0, y, t) = σxy (0, y, t) = 0.
where B1 = μL , B2 = α + λ + μL . For convenience, the following non-dimensional variables are used: ˜ x = c1 ωx,
(16)
(14)
(15)
(A11 ,A22 ,B1 ,B2 ) , λ+2μT
ε = ρCE (λ+2μT ) . Consider the boundary conditions at x = 0 as following
In order to investigate the propagation of plane waves in fibre-reinforced, rotating thermoelastic half-space proposed 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 flexibility in modelling layered structures and its capability in obtaining full field numerical solution. In order to solve the non-dimensional governing equations (13)–(15) using the finite element method (FEM), the weak formulations of these equations are derived. It is convenient to prescribe the set of independent test functions to consist of the displacement components u and v and the temperature θ . To obtain the weak formulation, the governing equations are multiplied by independent weighting functions and then are integrated over the spatial domain with the boundary. Applying integration by parts and making use of the divergence theorem reduces the order of the spatial derivatives and allows for the application of the boundary conditions. Using the well known Galerkin procedure, the unknown fields u, v and θ and the corresponding weighting functions are approximated by the same shape functions, which are defined piecewise on the elements. The last step towards the finite element discretization is to choose the element type and the associated shape functions. Eight nodes of quadrilateral elements are used. The unknown fields are approximated either by linear shape functions, which are defined by four corner nodes or by quadratic shape functions, which are defined by all of the eight nodes (twodimensional quadrilateral elements). On other hand
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the unknown fields are approximated either by linear shape functions, which are defined by three corner nodes or by quadratic shape functions, which are defined by all of the six nodes (two-dimensional triangular elements). The shape function is usually denoted by the letter N and is usually the coefficient that appears in the interpolation polynomial. A shape function is written for each individual node of a finite element and has the property that its magnitude is 1 at that node and 0 for all other nodes in that element. We assume that the master element has its local coordinates in the range [−1, 1]. In our case, the twodimensional quadrilateral elements are used, which had given by:
(ii) Quadratic shape functions 1 N1 = (1 − ξ )(1 − η)(−1 − ξ 4 1 N5 = (1 − ξ 2 )(1 − η), 2 1 N2 = (1 + ξ )(1 − η)(−1 + ξ 4 1 N6 = (1 + ξ )(1 − η2 ), 2 1 N3 = (1 + ξ )(1 + η)(−1 + ξ 4 1 N7 = (1 − ξ 2 )(1 + η), 2 1 N4 = (1 − ξ )(1 + η)(−1 − ξ 4 1 N8 = (1 − ξ )(1 − η2 ). 2
− η),
− η),
(21) + η),
+ η),
On the other hand, the time derivatives of the unknown variables have to be determined by Newmark time integration method (Reddy [18]).
4 Numerical results
(i) Sketch of linear shape functions
To study the effect of rotation and reinforcement on wave propagation, we use the following numerical values 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 ,
(ii) Sketch of quadratic shape functions
β = 0.32 × 109 N/m2 , ρ = 7800 kg/m2 ,
(i) Linear shape functions 1 N1 = (1 − ξ )(1 − η), 4 1 N2 = (1 + ξ )(1 − η), 4 1 N3 = (1 + ξ )(1 + η), 4 1 N4 = (1 − ξ )(1 + η). 4
(20)
θ0 = 1, b = 1, L = 1 and H is the heavyside unit step function. The field quantities, temperature, displacement components u, v and stress components σxx , σyy and σxy depend not only on space x and time t but also on the thermal relaxation time τ0 . Here all the variables are taken in nondimensional forms. Figures 1–6 exhibit the variation of the temperature, displacement components u, v and stress components σxx , σyy and
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Fig. 1 The temperature distribution for different values of at t = 0.1 and y = 0
Fig. 2 Horizontal displacement distribution u for different values of at t = 0.1 and y = 0
Fig. 3 Vertical displacement distribution v for different values of at t = 0.1 and y = 0
σxy with space x under the classical dynamical CD and Lord-Shulman’s theory (LS) i.e., when there is one thermal relaxation time (τ0 = 0.02) at t = 0.1 and
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Fig. 4 The distribution of stress component σxx for different values of at t = 0.1 and y = 0
Fig. 5 The distribution of stress component σyy for different values of at t = 0.1 and y = 0
Fig. 6 The distribution of stress component σxy for different values 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 observed from Fig. 1 that the rotation has no effect on the tem-
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Fig. 7 The temperature distribution for = 5 at t = 0.1 and y=0
Fig. 8 Horizontal displacement distribution u for = 5 at t = 0.1 and y = 0
Fig. 9 Vertical displacement distribution v for = 5 at t = 0.1 and y = 0
perature with respect to the CD and (LS) theories. Figure 2 shows that the rotation has decreasing effect on the displacement component u for 0.1 < x < 0.7, it is
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Fig. 10 The distribution of stress component σxx for = 5 at t = 0.1 and y = 0
Fig. 11 The distribution of stress component σyy for = 5 at t = 0.1 and y = 0
Fig. 12 The distribution of stress component σxy for = 5 at t = 0.1 and y = 0
clear that the curves under (LS) theory are greater than that of CD theory. Figure 3 shows that the rotation has increasing 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. 15 Vertical displacement distribution v for different values of t at = 5 and y = 1
Fig. 14 Horizontal displacement distribution u for different values of t at = 5 and y = 1
Fig. 16 The distribution of stress component σxx for different values of t at = 5 and y = 1
0 < x < 0.1 and decreasing effect for 0.1 < x < 1.0 in this region the curves under (LS) theory are greater than that of CD theory. Figures 4 and 5 show that the rotation has decreasing effect on the two components of stress σxx and σyy for 0 < x < 0.1 and increasing effect for 0.1 < x < 1.5. Figure 6 shows that the rotation has decreasing effect on the stress component σxy for 0 < x < 0.15 and increasing effect for 0.15 < x < 0.6. We have found that the rotation has a significant effect on the field quantities. Figures 7–12 show the variation of the temperature, displacement components u, v and stress components σxx , σyy and σxy with space x for = 5 and t = 0.1 under LordShulman’s theory (LS) with reinforcement (WRE) and without reinforcement (NRE). In Figs. 7–12 the solid line refer to the case with reinforcement and the dot line refer to the case without reinforcement i.e. (α = 0, β = 0 and (μL − μT ) = 0). It easily to see from
Figs. 7 and 8 that the reinforcement has small effect on the physical quantities temperature and displacement component u. Figure 9 shows that the reinforcement has decreasing effect on the displacement components v for 0 < x < 0.06 and increasing effect for 0.06 < x < 0.1. Figure 10 shows that the reinforcement has increasing effect on σxx for 0 < x < 0.1 and decreasing effect for 0.1 < x. Figure 11 shows that 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 increasing effect on σxy for 0 < x < 0.07 and decreasing effect for 0.07 < x < 0.15. Figures 13–18 show the variation of the temperature, displacement components u, v and stress components σxx , σyy and σxy with space x for = 5 and y = 1 under Lord-Shulman’s theory (LS) with reinforcement (WRE) for four different values of time
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Fig. 17 The distribution of stress component σyy for different values of t at = 5 and y = 1
Fig. 18 The distribution of stress component σxy for different values of t at = 5 and y = 1
(t = 0.05, 0.1, 0.15, 0.2). Figures 13 and 15 show that under the reinforcement the physical quantities temperature and displacement component v increasing as increasing time and decreasing as increasing x. Figure 14 shows that the displacement component u decreasing for 0 < x < 0.1 and increasing for 0.1 < x < 1.25 as increasing time t. Figures 16 and 17 show that the time has increasing effect on σxx and σyy for 0 < x < 0.1 and decreasing effect for 0.1 < x < 1.6. Figure 18 shows that the time has decreasing effect on σxy for 0 < x < 0.4.
References
5 Concluding remarks In this work the finite element method is used to study the problem of the effect of rotation on a thermal shock problem at the free surface of a fibre-reinforced thermoelastic half-space under the classical dynamical coupled theory CD and Lord-Shulman’s theory. We can obtain the following conclusions based on the above analysis: 1. There are significant differences for field quantities under the two theories due to essential differences between the coupled theory CD and the (LS) theory. 2. The rotation has a significant effect on the field quantities. 3. The reinforcement has a great effect on the distribution of field quantities when the rotation is considered.
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