J. Comp. & Math. Sci. Vol.2 (4), 675-682 (2011)
Generalized Thermoelastic Waves in Laminated Structures K. L. VERMA Department of Mathematics, Government Post-Graduate College Hamirpur -177005, H. P. India ABSTRACT Small-amplitude thermoelastic wave propagation in heat conducting laminated structures subject to an imposed pre-strain, at a given homogeneous state of temperature is analyzed in the context of generalized theory thermoelasticity with one thermal relaxation time. The analysis is carried forward for transversely isotropic materials and using appropriate boundary conditions, 12 x 12 matrix for two-layer structure is obtained. A numerical technique is adopted to avoid numerical instabilities and to represent the dispersion curves graphically at different values of thermal relaxation time. Keywords: Generalized thermoelasticity, laminated, waves, amplitude, thermal relaxation time.
1. INTRODUCTION The growing application of composite materials demands reliable, nondestructive testing methods in order to evaluate the mechanical behavior of new composite materials, as well as for the insitu inspection of existing structures. Layered structures are finding an increasing interest in engineering applications. Consequently, efficient and robust computational tools are required for the analysis of such structural models. It is a well known fact that most materials undergo volumetric variations when subjected to temperature variations.
The stresses induced by these variations are called thermally induced stresses, or simply thermal stresses. Thermoelasticity is the study of the influence of temperature of an elastic body on the stress and strain distributions in the body and also of the inverse effect of deformation on the temperature distribution. The study of thermoelastic problems has always been an important branch in solid mechanics Nowacki9, Nowinski,10. The classical theory of thermoelasticity is based on the Fourier’s Heat Conduction law, due to this reason; the theory describes a finite propagation speed for elastic waves whereas it predicts an
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K. L. Verma, J. Comp. & Math. Sci. Vol.2 (4), 675-682 (2011)
infinite speed of propagation for the thermal disturbance. The second part, which is clearly unrealistic, arises from the fact that the Fourier’s Heat Conduction law assumes that thermal disturbances travel at an infinite speed through a medium. This implies that if a heat source is suddenly applied at any point in a body, the thermal disturbance induced by the heat source is instantly felt at every other point in the body. Generalized Thermoelasticity refers to the development of thermoelasticity theory that takes into account the finite propagation speed of thermal disturbance as a wave and its effect on the coupled thermalmechanical field. Several models have been developed to incorporate the hyperbolic heat conduction equation into thermoelasticity theory. In 1967, Lord and Shulman8 proposed the first known generalized theory which used the Maxwell-Cattaneo equation to describe a heat flux vector in an elastic half space. This theory has been extended to anisotropic solids by Dhaliwal and Sherief4. The literature dedicated to hyperbolic thermoelastic models is quite large and its detailed review can be found in Chandrasekharaiah2, 3. Hawwa and Nayfeh7 studied the propagation of harmonic waves in a laminated composite consisting of an arbitrary number of layered anisotropic plates. Verma11, 12; Verma and Hasebe13-15 have studied various thermoelastic problems of infinite plates in the context of in generalized thermoelasticity. Gei et al.5,6 have studied thermoelastic small-amplitude wave propagation problems. In this paper generalized thermoelastic small-amplitude wave propagation in laminated structures subject to an imposed pre-strain, is analyzed in the
context of generalized theory of thermoelasticity. Analysis is carried forward for transversely isotropic materials and using appropriate boundary conditions a 12 x 12 matrix for two-layer structure is obtained, whose vanishing determinant represent the dispersion equation. A numerical technique is adopted to avoid numerical instabilities and to the dispersion curves plotted at different values of thermal relaxation time. Special cases of coupled and classical cases are discussed. 2. GOVERNING EQUATIONS The basic field equations of generalized thermoelasticity used for this study, for an infinite generally anisotropic thermoelastic medium at uniform temperature T0 in the absence of body forces and heat sources are
∂τ ij (u) ∂ 2 ui + f i = ρ 2 , i = 1, 2,3 (1) ∑ ∂x j ∂t j =1 3
KijT,ij − ρCe (T& +τ0T&&) = T0 βij ( u&i, j + τ 0u&&i, j ) (2) where τij = Cijkl ekl − βijT , β ij = Cijklα kl
(3)
the summation convention is implied; ρ is the density, t is the time, ui
is the
displacement in the x i direction, Kij are the thermal conductivities, Ce and τ0 are respectively the specific heat at constant strain, and thermal relaxation time, σ ij and
eij
are
the
stress
and
strain
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
tensor
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K. L. Verma, J. Comp. & Math. Sci. Vol.2 (4), 675-682 (2011)
respectively; β ij are thermal moduli; α ij is the thermal expansion tensor; T is temperature above the uniform temperature T0; and the fourth order tensor of the elasticity C ijkl satisfies the (Green)
wave, both complex numbers, but in such a way that the frequency, kv be real. A j
( j = 1, 2...6) are
unknown
whereas the six complex roots α 2 = −α1 ,
α 4 = −α 3 , α 6 = −α 5 are the solutions of
symmetry conditions:
the characteristic equation
c ijkl = c klij = c ijlk = c jikl , and αij = α ji ,
∆a 6 + B1α 4 + B2α 2 + B3 = 0
βij = β ji , K ij = K ji .
K1T,11 + K 2 T,22 − ρ Ce(T& + τ 0 T&&) = T0 [ β1 (u&,1 + τ 0 u&&,1 ) + β 2 (u&,2 + τ 0 u&&,2 )]
(6)
where
When equations (1)-(3) are specialized to transversely isotropic materials in reference to our problem, we have from (1) and (2)
σ 11 = c 11 e 11 + c 12 e 22 − β 1T σ 22 = c 12 e 11 + c 22 e 22 − β 2 T σ 12 = 2 c 66 e 12
constants,
B1 = [−(c1 β 2 + c1 )c2ω1∗τζ 2 -( Pζ 2 + J ) K + c1 c2 ] B2 = [((ζ 2 − 1) β 2 + 2 β c3 − c1 )ε1 + ( Pζ 2 + J )]ω1∗τζ
(4a)
(7)
− (ζ 2 − 1)(ζ 2 − c2 ) K − ( Pζ 2 + J ) B3 = [(ζ 2 − 1 − ε1 )ω1∗τζ + (1 − ζ 2 )](c2 − ζ 2 )
(4b)
3. ANALYSIS Consider the displacement and temperature fields associated with a sinusoidal small-amplitude thermoelastic traveling wave propagating along direction x1 for a transversely isotropic material, are characterized by the following expressions
(u1, u2 , T ) =
{1, f (α j ), g(α j )} Aj eα jkx2 (5) eik ( x1−vt ) ∑ −α j kx2 j =1 +{1, − f (α ), g(α )} A e j j j +3 3
where k and v are, respectively, the wavenumber and the speed of propagation of the
P = c1 + c2 , J = c32 − c22 − c1 , ∆ = c1 c2 K The functions f (α j ) and g (α j )
f (α j ) =
M11(αk )M33(αk ) − M13 (αk )M31(αk ) M13(αk )M32 (αk ) − M12 (αk )M33(αk )
M (α )M (α ) − M32 (αk )M11(αk ) g(αj ) = 31 k 12 k M13(αk )M32 (αk ) − M12 (αk )M33 (αk )
(8)
where
M11 = 1 + c2α 2 − ζ 2 , M 12 = c3α , M13 = 1, M 22 = c2 + c1α 2 − ζ 2 , M 23 = βα ,M 31 = ε1ω1∗ζ 2τ , M 31 = ε1ω1∗ζ 2τβα , M 33 = 1 + K α 2 − ω1∗ζ 2τ
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
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678
c2 =
K. L. Verma, J. Comp. & Math. Sci. Vol.2 (4), 675-682 (2011)
c66 c c +c β 2T , c1 = 22 ,c3 = 12 66 , ε1 = 1 0 , c11 c11 c11 ρCec11
ρv2
Cc ζ2 = , ω1∗ = e 11 , τ =τ0 + i , ω c11 K1
β= The
[ u 1 , u 2 , T , σ 2 2 , σ 1 2 , T ′]T r = F ( x 2 ) A e ik ( x1 − vt )
where (10)
β2 K ,K = 2 β1 K1 Aj
constants
αj , (j =1,2,...6)
and
the
coefficients
the
functions
f (α j ) and g (α j ) are, in general, different from layer to layer. (i) the mechanical boundary conditions under incremental plane strain constraint conditions in the transverse direction
u3 = 0, σ j 3 = 0 , j = 1,2
(11a)
(ii) at a boundary subject to dead load:
σ 12 = σ 22 = T ′ = 0,
(11b)
−
σ + i 2 = σ i−2 , T ′ = T ′ , +
the
vector of the arbitrary constants defined in( ) and the matrix F ( x 2 ) can be factorized in F ( x 2 ) = Q G ( x 2 ) where 1 1 1 1 1 1 f (α ) f (α ) f (α ) − f (α ) − f (α ) − f (α ) 2 3 1 2 3 1 g(α1) g(α2 ) g(α3 ) g(α1) g(α2 ) g(α3) (12) Q= D12 D13 D11 D12 D13 D11 D21 −D21 −D22 −D23 D22 D23 D32 D33 −D31 −D32 −D33 D31
where
D1k = (c3 − c 2 ) + c1α k γ k + iξ −1 β 2δ k D2 k = c 2 (α k + γ k ) , D3k = α k δ k ,
σ mn =
(13)
σ mn , m, n = 1,2 and iξ
D1 j +3 = D1 j , D2 j +3 = − D2 j , D3 j + 3 = − D3 j , j = 1, 2,3 G( x2 ) = diag(ekα1x2 ,ekα2 x2 ,ekα3 x2
(iii) at an interface between two layers +
T A = [ A1 , A 2 , A3 , A 4 , A5 , A 6 ] is
(11d)
(14)
,e−kα1x2 ,e−kα2 x2 ,e−kα3x2 )
_
u + i = ui− , T = T , i = 1,2
(11c)
Here the subscript (+) and (-) designates the interface between two layers, and T ′ = T, x2 is the temperature gradient. Through (5) and (3) the boundary conditions (11) must be imposed, and an eigen value problem for the complex propagation velocity is obtained, which can be solved numerically. In order to sketch the adopted numerical procedure, a matrix notation is employed. For the i-th layer
Now, the three terms of G ( x2 ) which have positive real part of the exponent may trigger numerical instability due to the exponential function. To avoid this problem we further decompose G ( x2 ) in
G ( x2 ) = G ∗ ( x2 ) A ∗
(15)
where G ∗ ( x2 ) = diag(ekα1 ( x2 − x2 ) , ekα2 ( x2 − x2 ) ,e kα3 ( x2 − x2 ) lh
lh
lh
, e− kα1 ( x2 − x2 ) , e− kα2 ( x2 − x2 ) , e− kα3 ( x2 − x2 ) ) uh
uh
uh
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
K. L. Verma, J. Comp. & Math. Sci. Vol.2 (4), 675-682 (2011)
A ∗ = diag (e k α 1 x 2 , e k α 2 x 2 , e k α 3 x 2 ,
679
(16)
shortcoming associated with positive exponents in G ∗ which are responsible of bad conditioning of the problem.
is a matrix of constants. The real parts of the exponents in (16) are always less or equal zero when the boundary conditions are imposed. The expression (11d) can be written as
As an illustration, solving the eigen value problem for a two-layer structure means to impose vanishing determinant of the following 12 x 12 matrix for a two-layer structure.
[u1 , u2 , T , σ 22 , σ 12 , T ′]T = F ∗ ( x2 ) A ∗ Aeik ( x1 − vt )
F4∗−16, j (0) M 0 ∗1 ∗2 det F ( h1 ) M − F ( h2 ) = 0 0 M F4∗−26, j ( h1 + h2 )
lh
lh
lh
e − kα 1 x2 , e − kα 2 x2 , e − k α 3 x2 ) uh
uh
uh
(17)
∗
where vector A A collects a new set of constants and the matrix F ∗ ( x2 ) = QG ∗ ( x2 ) has no longer the
j = 1, 2...6 .
1.5
Phase Velocity
1.2
0.9
0.6
0.3
0
2
4
6
8
10
Wave number
Figure 1 Phase velocity dispersion curves for thermal relaxation times
= .000000028
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
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K. L. Verma, J. Comp. & Math. Sci. Vol.2 (4), 675-682 (2011) 1.5
Phase Velocity
1.2
0.9
0.6
0.3
0
2
4
6
8
10
Wave number
Figure 2 Phase velocity dispersion curves for thermal relaxation times = .00000028
1.5
Phase Velocity
1.2
0.9
0.6
0.3
0
2
4
6
8
10
Wave number
Figure 3 Phase velocity dispersion curves for thermal relaxation times
= .00000028
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
K. L. Verma, J. Comp. & Math. Sci. Vol.2 (4), 675-682 (2011)
4. NUMERICAL RESULTS In this section, the phase velocity dispersion curves are plotted as functions of the product of frequency and plate thickness. Two layers of equal thickness h are considered for computation NaF material, for which, the basic physical data is given in Banerjee and Pao14, and the graphs are represented in Figs. 1-3 for thermal relaxation times 2.8 × 10−8 sec s,
2.8 ×10−7 sec s, and 2.8×10−6 secs respectively. From figures it is observed that results pertaining for a given materials exhibit that the coupling, which makes the waves always dispersive, introduces only a weak effect. Further, each of figure display coupled three wave speeds corresponding to quasilongitudinal, quasi-transverse and quasithermal at zero wave number limits, for the higher value wave numbers higher modes appear in both cases with wave number increases. One of the thermoelastic modes seems to be associated with quick change in the slope of the mode. Lower modes are found to highly influence by the thermal relaxation times at low values of wave number. If we take thermal relaxation time equal to zero, results reduces for coupled thermoelasticity. Further if we set the coupling constant ε 1 equal to zero then obtained results reduces for classical case. 5. CONCLUSIONS Analysis for small-amplitude thermoelastic waves superimposed upon a given homogeneous state of temperature and deformation in a multi laminated structure in the context of generalized theory of thermoelasticity is presented. Particularly
681
elastic layers perfectly bonded to each other and deformed in plane strain have been considered, in a fully generalized theory of thermoelasticity. Obtained results pertaining for a given materials exhibit that the coupling, which makes the waves always dispersive, introduces only a weak effect, in the sense that the imaginary part of the propagation velocity turns out to be small, when compared to the real part. Effects of thermal relaxation time are also observed. Within a range of parameters, including the case of null pre-strain, results can be studied to demonstrate to show that a particular value of frequency exists such that the propagation speed becomes independent of the temperature coupled thermoelasticity and classical case are also the special case of the analysis. REFERENCES 1. Banerjee, D. K. and Pao, Y.K., “Thermoelastic waves in anisotropy solids”, J. Acoust. Soc. Am. 56, 14441453, (1974). 2. Chandrasekharaiah D. S., “Thermoelasticity with second sound.- A review”, Applied Mechanics Review 39:3 355-376, (1986). 3. Chandrasekharaiah D. S., “Hyperbolic thermoelasticity.- A review of recent literature”, Applied Mechanics Review, 51:12 705-729, (1998). 4. Dhaliwal, R. S. and Sherief, H. H.Generalized thermoelasticity for anisotropic media”, Q. Appl. Math., 38, 1, (1980). 5. Gei, M., Bigoni, D. and Franceschini, G.: “Thermoelastic small-amplitude wave propagation in nonlinear elastic multilayers”. Mathematics and Mechanics of Solids, 9,555-568, (2004).
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K. L. Verma, J. Comp. & Math. Sci. Vol.2 (4), 675-682 (2011)
Gei, M., Bigoni, D. and Franceschini, G, Propagation of thermoelastic waves in Layered structures, XVI Congresso AIMETA 1-10. Hawwa M. A. and Nayfeh A. H., “The general problem of thermoelastic waves in anisotropic periodically laminated composites”, Composite Engineering. 51499-1517, (1995). Lord H. W. and Shulman Y. A generalized dynamical theory of thermoelasticity J. Mech. Phys. Solid. 15: 299 (1967). Nowacki W., Thermoelasticity - Int. Ser. Monographs in Aeronautics and Astronautics PWN Warsaw. (1962). Nowacki W., Dynamic problems of thermoelasticity Noordhoff. International Publishing, Leyden, the Netherlands. (1975). Verma K. L, “Thermoelastic vibrations of transversely isotropic plate with thermal relaxations”. International Journal of Solids and Structures,
38, 8529-8546, (2001). 12. Verma K. L, “On the propagation of
waves in layered anisotropic media in generalized thermoelasticity”, International Journal of Engineering Science, 40 (18), 2077-2096, (2002). 13. Verma K. L. and Hasebe N., “Wave propagation in transversely isotropic plates in generalized thermoelasticity”. Arch. Appl. Mech. 72:6-7, 470-482, (2002). 14. Verma K. L. and Hasebe N., “On the flexural and extensional thermoelastic waves in orthotropic with thermal relaxation times,” Journal of Applied Mathematics, 1, 69–83, (2004). 15. Verma K. L., “Propagation Of Generalized Thermoelastic Waves In Laminated Structures”, 7th International Conference on Thermal Stresses, National Taiwan University of Science and Technology Taipei, TAIWAN Proceedings, pp. 707-710, June 4th to June 7th, (2007).
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