Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 3, pp. 1067-1079, Warsaw 2017 DOI: 10.15632/jtam-pl.55.3.1067 EIGENVALUE APPROACH TO NANOBEAM IN MODIFIED COUPLE STRESS THERMOELASTIC WITH THREE-PHASE-LAG MODEL INDUCED BY RAMP TYPE HEATING Rajneesh Kumar Kurukshetra University, Department of Mathematics, Kurukshetra, India e-mail: rajneesh kuk@rediffmail.com Shaloo Devi Himachal Pradesh University, Department of Mathematics and Statistics, Shimla, India e-mail: shaloosharma2673@gmail.com This article deals with the study of a thermoelastic nanobeam in a modified couple stress theory subjected to ramp-typeheating.Themathematicalmodel is prepared for the nanobe- am in thermoelastic three-phase-lag.TheLaplace transformand the eigenvalue approachare used to find the displacement component, lateral deflection, temperature change and axial stress of the thermoelastic beam. The general algorithm of the inverse Laplace transform is developed to compute results numerically. The comparison of three-phase-lag, dual-phase- lag andGN-III (1993)models are represented, and their illustration is depicted graphically. This study finds the applications in engineering, medical science, sensors, etc. Keywords: modified couple stress thermoelastic, eigenvalue approach, nanobeam 1. Introduction Cosserat and Cosserat (1909) developed a mathematical model for a couple stress theory in which kinematical quantities are thedisplacement andmaterialmicrorotation.Yang et al. (2002) proposed a modified couple stress theory in which the couple stress tensor was symmetric and required only one material length parameter to capture the size effect which was caused by micro-structure. Various authors studied different problems in a modified couple stress theory (2008, 2011-2015). Tzou (1995a,b, 1997) proposed a dual-phase-lagmodel bymodifying the classical fourier law by an approximation with two different time translations: a phase-lag of the heat flux τq and a phase-lag of the temperature gradient τθ. Tzou (1995b) supported that model by experimental results.A review of five theories of thermoelasticity was given byHetnarski and Ignaczak (1999). Roychoudhuri (2007) developed a three-phase-lag model for a thermoelastic material. In that model, theFourier law of heat conductionwasmodifiedby introducing three different phase-lags for the heat flux vector, temperature gradient and thermal displacement component gradient. Quintanilla and Racke (2008) investigated stability of the three-phase-lag heat conduction equ- ation and the relations among three material parameters. Kumar et al. (2012) studied wave propagation in an anisotropic viscoelastic mediumwith the three-phase-lagmodel of thermoela- sticity. Sur andKanoria (2014) examined vibration of a gold nanobeam induced by a ramp-type laser pulse under the three-phase-lag model. The significance of using the eigenvalue approach is to reduce the problem on the vector- -matrix differential equation to algebraic eigenvalue problems. Thus the solutions for the field variables are obtained by determining eigenvalues and the corresponding eigenvectors. In this approach, the physical quantities are directly involved in the formulation of the problem and, 1068 R. Kumar, S. Devi as such, the boundary and initial conditions can be applied directly. The problem ofmicropolar thermoelasticity without energy dissipation by employing the eigenvalue approach was studied by Kumar et al. (2007). Zang and Fu (2012) constructed a new beam model for a viscoelastic micro-beam based on amodified couple stress theory. Abouelregal and Zenkour (2014) investigated the problem of an axially moving microbeam subjected to a sinusoidal pulse heating and an external transverse excitation with one relaxation time using the Laplace transform. Zenkour and Abouelregal (2015) investigated the problem of a thermoviscoelastic orthotropic continuumwith a cylindrical hole and variable thermal conduc- tivity under three-phase-lag model and solved the physical quantities by the Laplace transform technique. The effects of hall current and rotation in a modified couple stress theory subjected to the ramp type loading in the context of theory of generalized thermoelastic diffusionwas pre- sented by Kumar and Devi (2015). Thermoelastic interaction in a thermally conducting cubic crystal subjected to the ramp-type heating was investigated by Abbas et al. (2015). Reddy et al. (2016) discussed the problem of functionally graded circular plates with themodified couple stress theory by using the finite element method. On the basis of global local theory, a model for the composite laminated Reddy plate of a newmodified couple-stress theory was developed by Chen andWang (2016). Zenkour and Abouelregal (2016) discussed vibration of functionally gradedmicrobeams by using the Green-Naghdi thermoelasticity theory (1993) and the Laplace transform. The present investigation deals with a thermoelastic nanobeam in themodified couple stress theory induced by the ramp-type heating in the three-phase-lag model. The non-dimensional equations are written in form of the Laplace transform which is solved by the eigenvalue ap- proach. The expressions of the displacement component, lateral deflection, temperature change and axial stress are computed numerically and then represented graphically. Particular cases of interest are deduced from the present investigation. 2. Basic equations Following Yang et al. (2002) and Roychoudhuri (2007), the constitutive relations, equations of motion and the equation of heat conduction in the modified couple stress generalized thermo- elasticity with three-phase-lag model in the absence of body forces are: — constitutive relations tij =λekkδij +2µeij − 1 2 ekijmlk,l−β1Tδij mij =2αχij χij = 1 2 (ωi,j +ωj,i) ωi = 1 2 eipquq,p (2.1) — equations of motion ( λ+µ+ α 4 ∆ ) ∇(∇·u)+ ( µ− α 4 ∆ ) ∇2u−β1∇T = ρü (2.2) — equation of heat conduction with three-phase-lag [ K∗ ( 1+τν ∂ ∂t ) +K ∂ ∂t ( 1+τT ∂ ∂t )] ∆T = ( 1+τq ∂ ∂t + τ2q 2 ∂2 ∂t2 )( ρce ∂2T ∂t2 +T0β1 ∂2 ∂t2 (∇·u) ) (2.3) where tij are components of the stress tensor, λ and µ are Lame’s constants, δij is Kronecker’s delta, eij =(ui,j+uj,i)/2 are components of the strain tensor, eijk is the alternate tensor,mij are components of the couple-stress, β1 = (3λ+2µ)αt. Here αt are coefficients of linear thermal expansion and diffusion, respectively, T is temperature change, χij is symmetric curvature, ωij = (uj,i − ui,j)/2 are components of rotation, ωi is the rotational vector, α is the couple Eigenvalue approach to nanobeam in modified couple stress thermoelastic... 1069 stress parameter and u = [u1,u2,u3] is the displacement component, ρ is density, ∆ is the Laplacian operator, ∇ is del operator. K is the coefficient of thermal conductivity, K∗ is the material characteristic constant of the theory, ce is the specific heat at a constant strain, T0 is the reference temperature assumed to be such thatT/T0 ≪ 1. τT , τq and τν are the phase lags of the temperature gradient, of the heat flux and of the thermal displacement component gradient, respectively, such that τν <τT <τq. 3. Formulation of the problem Consider a homogeneous, isotropic, rectangular modified couple stress thermoelastic beam ha- vingdimensions of length (0¬x¬L),width (−d/2¬ y¬ d/2) and thickness (−h/2¬ z¬h/2) (Fig. 1). Let us take thex-axis along length of the beam, the y-axis alongwidth and z-axis along thickness, representing the axis ofmaterial symmetry.Therefore, anyplane cross-section initially perpendicular to the axis of the beam remains plane and perpendicular to the neutral surfa- ce during bending. According to the Euler-Bernoulli theory for a small deflection in a simple bending problem, the displacement components are given by u(x,y,z,t) =−z ∂w ∂x v(x,y,z,t) = 0 w(x,y,z,t) =w(x,t) (3.1) where w(x,t) is lateral deflection of the beam and t is time. The one-dimension stress compo- nent tx, with the aid of equations (2.1)1 and (3.1), yields (3.2)tx =−(λ+2µ)z ∂2w ∂x2 −β1T (3.2) Fig. 1. Schematic figure of the beam The flexural moment of the cross-section of the beam is given by M =Mσ+Mm = d ( h/2 ∫ −h/2 txz dz+ h/2 ∫ −h/2 mxy dz ) (3.3) whereMσ andMm are components of the bendingmoment due to the classic stress and couple stress tensors, respectively. Making use of the value of tx and mxy from (3.2) and (2.1)2 in (3.3), with the aid of (3.1), yield M =− [ (λ+2µ) dh3 12 +αA ]∂2w ∂x2 −β1d h/2 ∫ −h/2 Tz dz (3.4) Following Rao (2007), the equation of transverse motion of the beam is given by ∂2M ∂x2 −ρA ∂2w ∂t2 =0 (3.5) whereA= dh is the cross-sectional area of the beam. 1070 R. Kumar, S. Devi For a very thin beam, assuming that the temperature increment varies in terms of the sin(pz) function along thickness of the beam, where p=π/h as T(x,z,t) =T1(x,t)sin(pz) (3.6) Substituting the value ofM from (3.4) into equation (3.5), with the aid of equation (3.6), yields [ (λ+2µ) dh3 12 +αA ]∂4w ∂x4 +β1d ∂2T1 ∂x2 h/2 ∫ −h/2 z sin(pz) dz+ρA ∂2w ∂t2 =0 (3.7) Multiplying heat conduction equation (2.3), after using equation (3.1), by z and integrating themwith respect to the interval (−h/2,h/2), and with the use of equation (3.6), we obtain [ K∗ ( 1+ τν ∂ ∂t ) +K ∂ ∂t ( 1+ τT) ∂ ∂t )](∂2T1 ∂x2 −p2T1 ) = ( 1+ τq ∂ ∂t + τ2q 2 ∂2 ∂t2 )( ρce ∂2T1 ∂t2 − β1T0p 2h3 24 ∂4w ∂x2∂t2 ) (3.8) To facilitate solution, the following dimensionless quantities are introduced x′ = x L (z′,u′,w′)= (z,u,w) h (τ ′ν,τ ′ T ,τ ′ q, t ′)= (τν,τT ,τq, t)ν L T ′1 = β1T1 E (M ′,M ′T)= (M,MT) dEh2 t′x = tx E ν2 = E ρ K∗ = ce(λ+2µ) 4 (3.9) Making use of equation (3.9) in (3.7) and (3.8), after surpassing the primes, we obtain ∂4w ∂x4 +a1 ∂2T1 ∂x2 +a2 ∂2w ∂t2 =0 [ a3 ( 1+ τν ∂ ∂t ) + ∂ ∂t ( 1+ τT ∂ ∂t )](∂2T1 ∂x2 −a4T1 ) = ( 1+ τq ∂ ∂t + τ2q 2 ∂2 ∂t2 )( a5 ∂2T1 ∂t2 −a6 ∂2w ∂x2∂t2 ) =0 (3.10) where a1 = 2dEL p2 [ (λ+2µ)dh 3 12 +αA ] a2 = ρAν2L2 (λ+2µ)dh 3 12 +αA a3 = K∗L Kν a4 = p2 L2 a5 = ρceνL K a6 = β21T0νp 2h3 24KE 4. Problem solution The Laplace transform is defined as L{f(t)}= ∞ ∫ 0 e−stf(t) dt= f(s) (4.1) where s is the Laplace transform parameter. Eigenvalue approach to nanobeam in modified couple stress thermoelastic... 1071 Applying the Laplace transform defined by equation (4.1) to equations (3.10) and (3.11), gives d4w dx4 +a1 d2T1 dx2 +a2s 2w=0 d2T1 dx2 −a4T1 = a7s 2T1−a8s 2d 2w dx2 (4.2) where τq =1+ τqs+ τ2q 2 s2 τν =1+ τνs τT =1+ τTs a7 = a5τq a3τν +sτT a8 = a6τq a3τν +sτT The set of equations (4.2) can be written as d2v dx2 =−a9w+a10v−a11T1 d2T1 dx2 =−a12v+a13T1 (4.3) where d2w dx2 = v a9 = a2s 2 a10 = a1a8s 2 a11 = a1(a4+a7s 2) a12 = a8s 2 a13 = a4+a7s 2 The system of equations (4.4) can be written in a matrix form as DV(x,s) =AV(x,s) (4.4) where V= [ U DU ] U=    w v T1    A= [ O I A1 O ] A1 =    0 1 0 −a9 a10 −a11 0 −a12 a13    (4.5) andD= d/dz, I is the identity matrix of the order 3,O is a null matrix of the order 3. We take the solution to equation (4.4) as V(x,s) =Xr(x,s)e λz (4.6) such that A(x,s)V(x,s) =λV(x,s) (4.7) which leads to the eigenvalue approach. The characteristic equation of the matrix A can be written as λ6−G1λ 4+G2λ 2−G3 =0 (4.8) where G1 = a10+a13 G2 = a9+a10a13−a11a12 G3 = a9a13 The characteristic roots of equation (4.6) are also the eigenvalues of thematrixA. The eigenvec- torsX(x,s) corresponding to the eigenvalue λr can be determined by solving the homogeneous equations (A−λI)X(x,s)=0 (4.9) 1072 R. Kumar, S. Devi The set of eigenvectors Xr(x,s) may be obtained as Xr(x,s)= [ Xr1(x,s) Xr2(x,s) ] Xr1(x,s)=    br cr dr    Xr2(x,s)=λrXr1(x,s) for λ=λr, r=1,2,3 and Xj(x,s)= [ Xj1(x,s) Xj2(x,s) ] Xj1(x,s)=    br cr dr    Xj2(x,s)=λjXj1(x,s) for j= r+4, λ=−λr, r=1,2,3 and br =−a11 cr =−a11λ 2 r dr =λ 4 r −a10λ 2 r +a9 The solution to equation (4.6) reduces to V= 3 ∑ r=1 BrXr(x,s)e −λrx+ 3 ∑ r=1 Br+3Xr+3(x,s)e λrx (4.10) whereBi (i=1, . . . ,6) are arbitrary constants. Thus, the field quantities can be written as (w,v,T1)(x,s)= 3 ∑ r=1 (br,cr,dr)Bre −λrx+ 3 ∑ j=1 (bj+3,cj+3,dj+3)Bj+3e λjx (4.11) 5. Initial and boundary conditions Both initial and boundary conditions should be considered to solve the problem. The initial conditions of the problem are taken in the form as w(x,t) ∣ ∣ ∣ t=0 = ∂w(x,t) ∂t ∣ ∣ ∣ ∣ ∣ t=0 =0 T1(x,t) ∣ ∣ ∣ t=0 = ∂T1(x,t) ∂t ∣ ∣ ∣ ∣ ∣ t=0 =0 (5.1) Let us consider a nanobeamwith both ends are simply supported w(0, t) =0 ∂2w(0, t) ∂x2 =0 w(L,t) = 0 ∂2w(L,t) ∂x2 =0 (5.2) We consider the side of the nanobeam x = 0 being thermally loaded by ramp-type heating incidents into the surface of the nanobeam T1(0, t) = g0        0 t¬ 0 t/t0 0 t0 (5.3) where t0 is a non-negative constant called the ramp type parameter and g0 is a constant. We also assume that the other side of the nanobeam x=L is thermally insulated, and there is no variation of temperature on it, which thismeans that the following relationwill be satisfied dT1(L,t) dx =0 (5.4) Eigenvalue approach to nanobeam in modified couple stress thermoelastic... 1073 Applying the Laplace transform defined by equation (4.1) to boundary conditions (5.2)-(5.4), we obtain w(0,s)= 0 d2w(0,s) dx2 =0 T1(0,s)= g0 ( 1−e−st0 t0s2 ) w(1,s)= 0 d2w(1,s) dx2 =0 dT1(1,s) dx =0 (5.5) The values of displacement u and axial stress tx are then obtained u(x,s)= z ( 3 ∑ i=1 λibiBie −λix− 3 ∑ i=1 λibi+3Bi+3e λix ) tx(x,s)=− [ 3 ∑ i=1 (λ+2µ E zλ2ibi+di sin(pz) ) Bie −λix + 3 ∑ i=1 (λ+2µ E zλ2ibi+3+di+3 sin(pz) ) Bi+3e λix ] (5.6) Making use of the value of w and T1 from (4.11) in boundary conditions (5.5), with the aid of equations (5.6), after some calculations, we find the expressions of the displacement component, lateral deflection, temperature change and axial stress of the beam as (u,w)(x,s)= 3 ∑ i=1 (zλi,1)biBie −λix+ 3 ∑ i=1 (−λi,1)bi+3Bi+3e λix (T1, tx)(x,s)= 3 ∑ i=1 (di,Mi)Bie −λix+ 3 ∑ i=1 (di+3,Mi+3)Bi+3e λix (5.7) where Bi = ∆i ∆ i=1, . . . ,6 and 3 ∑ i=1 Mi =− (λ+2µ E zλ2ibi+di sin(pz) ) 3 ∑ i=1 Mi+3 =− (λ+2µ E zλ2ibi+3+di+3 sin(pz) ) ∆=          b1 b2 b3 b4 b5 b6 b1e −λ1 b2e −λ2 b3e −λ3 b4e λ2 b5e λ2 b6e λ2 b1λ 2 1 b2λ 2 2 b3λ 2 3 b4λ 2 1 b5λ 2 2 b6λ 2 3 b1λ 2 1e −λ1 b2λ 2 2e −λ2 b3λ 2 3e −λ3 b4λ 2 1e −λ1 b5λ 2 2e −λ2 b6λ 2 3e −λ3 d1 d2 d3 d4 d5 d6 −d1λ1e −λ1 −d2λ2e −λ2 −d3λ3e −λ3 −d4λ1e λ1 d5λ2e λ2 d6λ3e λ3          where ∆i (i = 1, . . . ,6) are obtained by replacing the i-th column with [0,0,0,0,g0((1 − e−st0)/(t0s 2)),0]T in∆i. 6. Particular cases (i) Dual-phase-lag model IfK∗ = τν =0, in equations (5.7), we obtain the corresponding results formodified couple stress thermoelastic materials with the dual-phase-lag model of thermoelasticity. 1074 R. Kumar, S. Devi (ii) GN-III model In the absence of τν = τT = τq = τ 2 q = 0 in equations (5.7), we obtain the corresponding results for modified couple stress thermoelastic materials with energy dissipation in the context of GN-III theory of thermoelasticity. (iii) If we take α=0 in equations (5.7), we obtain the corresponding results for thermoelastic materials with the three-phase-lag model of thermoelasticity. Our results in a special case are similar to those obtained by Sur andKanoria (2014). 7. Inversion of the Laplace transform We have obtained solutions for the displacement component, lateral deflection, temperature change and axial stress in the Laplace transform domain (x,s). We shall now briefly outline the numerical inversion method used to find the solution in the physical domain. Let f(s) be the Laplace transform of a function f(t). To obtain the solution of the problem in the physical domain, we invert the Laplace transform by using themethod described by Kumar (2016). 8. Numerical results and discussion We have chosen gold (Au) as the material for numerical computations. The physical data for gold are given by Sur and Kanoria (2014): λ = 198GPa, µ = 27GPa, αt = 14.2 · 10 −6K−1, ρ = 1930kg/m3, T0 = 0.293 · 10 3K, ν = 0.44, K = 200W/(mK), ce = 130J/(kgK), α = 2.5kgm/s2, t = 1.5s, τν = 0.02s, τT = 0.03s, τq = 0.04s, g0 = 1, t0 = 0.2, L = 1, d=1, h=10. Numerical computations have been carried out with the help of MATLAB software. By using this software, thedisplacement component, lateral deflection, temperature change, thermal stress, bendingmoment and axial stress with respect to distance are computed numerically and shown graphically in Figs. 2-7. In Figs. 2-4, the small dash line (- - - ) corresponds to the three- -phase-lag model (TPL), small dash line with the centre symbol (- - ∗ - -) corresponds to the dual-phase-lagmodel (DPL) and a small dash line with the centre symbol (- - ◦ - -) corresponds to the GN-III model respectively. Similarly, Figs. 5-7, the small dash line (- - -) corresponds to t0 = 0.2, small dash line with the centre symbol (- - ∗ - -) corresponds to t0 = 0.4, the small dash line with the centre symbol (- - ◦ - -) corresponds to t0 =0.6. Figure 2a shows the variation of the displacement component with respect to length of the beam for different models. The behavior and variation are similar for all the cases but with differences in their magnitudes. However, the values of the GN-III model are greater than compared to DPL and TPL models. Figure 2b depicts the variation of lateral deflection with respect to length of the beam for the three-phase-lag, dual-phase-lag and GN-III models. It is observed that the lateral deflection decreases for smaller values of length and oscillates for higher values of length for all TPL, DPL andGN-III models. Figure 3a presents the variation of axial stress with respect to length of the beam for TPL, DPL, GN-III models. As seen in the figure, the axial stress decreases smoothly in the whole region for all cases. Also, it is noticed that the axial stress has a large value for the three- -phase-lag (TPL) and dual-phase-lag (DPL) thermoelastic beams as compared to that for the GN-III thermoelastic beam. Figure 3b shows the variation of temperature change with respect to length of the beam for different thermoelastic (TPL, DPL, GN-III) models. It is observed from the figure that the behavior and variation are oscillatory in nature with fluctuating values in all the cases. Eigenvalue approach to nanobeam in modified couple stress thermoelastic... 1075 Fig. 2. (a) Displacement component u and (b) lateral deflectionw with respect to distance x for different phase lag theories of thermoelasticity Fig. 3. (a) Axial stress T x and (b) temperature change T with respect to distance x for different phase lag theories of thermoelasticity Fig. 4. BendingmomentM with respect to distance x for different phase lag theories of thermoelasticity Figure 4 shows the variation of bending moment with respect to length of the beam for different thermoelastic models. It is clearly seen in the figure that the value of bendingmoment decreases with a decrease in the value of length for all the cases of phase lag theories of ther- moelasticity. Also, the value of bendingmoment is higher in the range 0¬x¬ 0.25 for t0 =0.2 and smaller for t0 =0.4, 0.6 in the remaining range. 1076 R. Kumar, S. Devi Figure 5a shows the variation of lateral deflection with respect to length of the beam for different values of the ramp type parameter. Initially, the lateral deflection decreases with a difference in the ramp type parameter up to x ¬ 0.3 and then remains stable in the range 0.3 < x ¬ 0.8. Figure 5b presents the variation of temperature change with respect to length of the beam for different values of the ramp type parameter. The behavior and variation are oscillatory in nature for all the cases of the ramp type parameter. Fig. 5. (a) Lateral deflectionw and (b) temperature change T with respect to length x for different values of the ramp type parameter Figure 6a shows the variation of axial stress with respect to length of the beam for different values of the ramp type parameter. The value of axial stress decreases monotonically with an increase in length. Also, the value of axial stress for t0 =0.2 is greater than that the ramp type parameter t0 =0.4, 0.6. Figure 6b depicts the variation of displacement component with respect to length of the beam for different values of the ramp type parameter. The displacement component increases with an increase in length. Fig. 6. (a) Axial stress T x and (b) displacement component u with respect to distance x for different values of the ramp type parameter Figure 7a represents the variation of bending moment with respect to length of the beam for different values of the ramp type parameter. The bendingmoment smoothly decreases with a decrease in the value of length. The value of the bendingmoment is greater for t0 =0.2 up to the value of x=0.25 in comparison with that for t0 =0.4, 0.6, but shows opposite behavior in the remaining range. Figure 7b depicts the variation of thermal stress with respect to length of the beam for different values of the ramp type parameter. The oscillatory behavior is shown for Eigenvalue approach to nanobeam in modified couple stress thermoelastic... 1077 all the cases of ramp type parameters. It is clear from the figure that the value of thermal stress for t0 =0.6 is less in the range 0¬ x¬ 0.5 but reversed behavior is shown for t0 = 0.2, 0.4 in the considered region. Fig. 7. (a) Bending momentM and (b) thermal stress t z with respect to distance x for different values of the ramp type parameter 9. Conclusions In the present study, the effects of three-phase-lag, dual-phase-lag and GN-III on the displa- cement component, lateral deflection, axial stress, bending moment and temperature change are derived numerically and presented graphically. The effect of the ramp type parameter is shown graphically for lateral deflection, bendingmoment, displacement component, axial stress, thermal stress and temperature change. The Euler Bernoulli beam assumption and the Laplace transform technique are used to write the basic governing equations in form of vector-matrix differential equations which are then calculated by the eigenvalue approach. A numerical tech- nique has been adopted to determine solutions in the physical domain. It is observed from the obtained figures that the displacement component increases with a increase in length for all the three-phase-lag, dual-phase-lag and GN-III models but opposite behavior is observed for the axial stress. It is also noticed that the values of displacement component and axial stress for the three-phase-lag thermoelastic model is greater in comparison with the dual-phase-lag and GN-IIImodels.The lateral deflection and temperature change are oscillatory in nature butdiffer in theirmagnitude values for all the cases. Themethodused in the present study is applicable to awide range ofmathematical problems in the field of thermodynamics, thermoelasticity and co- uple stress theory. This study also find various applications to appliedmathematics, mechanical engineering, geophysical and industrial sectors. References 1. 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