Waves propagation in an arbitrary direction in heat conducting orthotropic elastic composites

Transcription

1 Rakenteiden Mekaniikka (Journal of Structural Mechanics) Vol. 46 No 03 pp. 4-5 Waves propagation in an arbitrary direction in heat conducting orthotropic elastic composites K. L. Verma Summary. Dispersion of thermoelastic harmonic waves propagating in an arbitrary direction in a layered heat conducting orthotropic elastic composite in the context of generalized thermoelasticity is studied. Considering three dimensional field euations of thermoelasticity and to obtain characteristic euation continuity of displacements temperature stresses and thermal gradient at the layers interfaces is employed and the corresponding sixteenth order characteristic determinant is examined. Particular results for the coupled and uncoupled thermoelasticity are obtained as special cases of the obtained results by taking thermal relaxation time and coupling constant eual to zero. Results of previous investigations are derived as particular cases. Finally numerical results are also obtained and represented graphically. Key words: composites orthotropic generalized thermoelasticity coupled thermal relaxation time Introduction dvanced high strength high modulus composite materials must undergo careful inspection to sort out manufacturing errors in-service degradation and defect formation due to the influence of elevated temperatures moisture cosmic radiation etc. The growing practical importance of such materials especially in thermal environment has stimulated many analytical studies. The exact dispersion relations for composite structures can be found in many good books [] and [] with many unidirectionally reinforced layers are at least of orthotropic characters and sometimes transversely isotropic. Wave types occurring in bounded layered anisotropic media are very complicated and in thermoelasticity the problem becomes even more complicated because in thermoelasticity solutions to both the heat conduction and thermoelasticity problems for all the layers are reuired. These solutions are also to satisfy the thermal and mechanical boundary and interface conditions. s a result conventional procedure for thermoelastic analysis of a multilayered medium results in having to solve system of two simultaneous euations for a large number of unknown constants as in Refs. [3-7]. By introducing thermal relaxation time constants into the heat conduction euation new generalized theories of thermoelasticity have been developed in an attempt to eliminate this paradox of infinite velocity of thermal propagation. Of all the nonclassical theories at present Lord and Shulman [8] Green and Lindsay [9] theories of 4

2 the generalized thermoelasticity are mainly considered for engineering applications. Various methods to study the isothermal elasticity problems in heat conducting medium are studied in Ref.[0]. Propagation of plane harmonic thermoelastic waves in an infinitely extended anisotropic solid is considered and investigated in []. In Ref. [] the governing euations of generalized thermoelasticity for anisotropic media were derived. This paper attempts to study the dispersion of harmonic waves propagating in an arbitrary direction in layered orthotropic elastic composite in the context of generalized thermoelasticity in Ref. [3]. Three dimensional field euations of thermoelasticity are considered and the corresponding sixteenth order characteristic determinant is examined. The purpose of this paper is to examine the dispersive effects in layered thermoelastic composites where the direction of the corresponding harmonic waves makes an arbitrary angle with respect to the layers. The results for the coupled and uncoupled thermoelasticity are obtained as particular cases of the obtained results by setting thermal relaxation time and the coupling constant eual to zero. Relevant results of previous investigations are deduced as special cases. similar type of the approach has also been used in Ref. [4] for the corresponding elastic material. Problem Formulation Consider a set of Cartesian coordinate system x ( x x x ) i = 3 in such a manner that 3 x - axis is normal to the layering. The basic field euations of generalized thermo- elasticity for an infinite generally anisotropic thermoelastic medium with one thermal relaxation time τ 0 at uniform temperature T 0 in the absence of body forces and heat sources are where σi = ρu i i = 3 () KT ρc( T τ T )= Tβ [ u τ u ] () i i e 0 0 i i 0 i σi = Cilekl βit (3) β = C θ (4a) i il kl and ρ is the density t is the time u i is the displacement in the x i direction K i are the thermal conductivities C e and τ 0 are respectively the specific heat at constant strain and thermal relaxation time σ i and e i are the stress and strain tensor respectively; β i are thermal moduli; θ i is the thermal expansion tensor; T is temperature; and the fourth order tensor of the elasticity C il satisfies the (Green) symmetry conditions: c il = c kli = c ilk = c ikl βi = β i θi = θ i. (4b) Here comma notations are used for spatial derivatives and superposed dot represents differentiation with respect to time. Strain-displacement relation is e u u i i i =. (5) 5

3 Specializing the euations (-5) for orthotropic media the governing euations () and () via (3) to (5) are where cu cu cu ( c c ) u ( c c ) u βt = ρu (6a) ( c c ) u c u c u c u ( c c ) u β T = ρu (6b) ( c c ) u ( c c ) u c u c u c u β T = ρu (6c) K T K T K T ρc ( T τ T ) e 0 = T [ β ( u τ u ) β ( u τ u ) β ( u τ u )] β = c α c α c α 3 3 β = c α c α c α 3 3 β = c α c α c α On considering euations (6) and (7) for each layer and at the interface between two layers the displacements temperature thermal stresses and temperature gradient must be continuous. (6d) (7) nalysis For harmonic waves propagating in an arbitrary direction the displacement components and temperature u u u3 and T are written as ( 3 3 ) ( u u u3 T) = ( U( x3) U( x3) U3( x3) U4( x3)) e ix l x l x l x ct (8) where x is the wave number c is the phase velocity (= ω /x) ω is the circular freuency l l and l 3 are the direction cosine defining the propagation direction. Flouet s theory reuires functions U ( = 34. ) to have the same periodicity as that of the layering. Hence the problem is reduced to that of one pair of layers where i ( l3 ) x3 ( U T)( x3) = Ue x α = 34 (9) α is an unknown ratio of the wave number components and U are constants.upon substitution from (9) into (8) and (6) we have Mmn( α ) Un = 0 m n = 34. (0a) Here M = ( l lc 66 α c55 ζ ) M = ( c c66) ll M3 = ( c3 c55) lα M4 = l M = ( lc 66 lc α c44 ζ ) M = ( c c ) l α

4 where M = β l 4 M = ( lc lc α c ζ ) M M = βα 34 3 = εω ζ liω 4 M = εω ζ l β i ω M 4 = εω ζ αβ i ω 43 3 M = ( l K l K α ωζ τ) (0b) 44 3 ρc cc e β T0 ζ = ω = ε = and τ = τ0 i ω. c K ρcc e The existence of nontrivial solutions for U ( = 3 4) demands the vanishing of the determinant in euations (0a) and yields the eighth degree polynomial euation α α α α = () where the coefficients 3 and 4 are given in ppendix-i. Euations (9) is rewritten as 8 i ( l3 ) x3 ( U U U T) = ( U U U U ) e x α. () = For each α =...8 using the relations (0) and expressing the displacements ratios as U U D ( α) U = = γ D( α ) U 3 D ( α) U = = δ D( α ) U 4 3 D ( α) = = Θ = 8 (3) D( α ) D( α ) D( α ) i = 3 (4) i are given in ppendix-ii. Therefore the solution is 8 i ( l3 ) x3 ( U U U T) ( γ δ θ ) U e x = α (5) 3 = In view of the continuity of the displacement components temperature tractions and temperature gradient across the interface of the two layers the following conditions must be satisfied: ( 3 = 0 ) = ( 3 = 0) ( 3 = 0 ) = ( 3 = 0) x = = σ ( x = 0) u x u x () () () () T x T x () () σ () () T x3 0 T x3 0 = = = (6) 7

5 where subscripts () and () refer to layers I and II respectively 0 and 0 are values of x 3 near zero. Because of periodicity of the deformation and thermoelastic stress fields additional conditions obtained are: u ( x = h ) = u ( x = h ) () () 3 3 T ( x = h ) = T ( x = h ) σ () () 3 3 ( x = h ) = σ ( x = h ) = 3; () () T ( x = h ) = T ( x = h ). (7) () () 3 3 Upon substitution of the displacement temperature stress and temperature gradient components into (6) and (7) sixteen linear homogeneous euations for sixteen () () () () constants U U... U 7 and U8 are obtained. For nontrivial solutions the determinant of the coefficients must vanish. This yields the following characteristic euation: det = 0 = 8. (8) B B The 8 8 matrices B and B are given in ppendix-iii. On simplifying euation (8) we have which implies that either or ( ) det det B B = 0 (9a) det = 0 (9b) ( B B ) det = 0. (9c) If euation (9a) holds true then the problem reduces to a free wave propagation in a single thermoelastic plate of thickness h and in this case [ B ] [ B ][ ] [ ] will not exist if is singular. On the hand if is nonsingular then exists and accordingly (9c) exists. In order to solve the problem numerically one has to solve (8) and to solve it is sufficient to consider either euation (9c) for heat conducting composite plates and euation (9b) to solve for free thermoelastic plate. 8

6 Particular cases Uncoupled thermoelasticity If the coupling constant ε = 0 then thermal and elastic fields decoupled from each other and from euation (0b) we have M4 = M4 = M43 = 0. In this case euation (0a) reduces to ( l K l K α ωζ τ)( α 6 Fα 4 Fα F ) =. (0) From the above euation (0) considering the factor α α α = () 6 4 F F F3 0 euation () corresponds to the characteristic euation in the uncoupled thermoelasticity where = c33c44c55 F = [( c c c c c ) c c c c ] l [( c c c c ) c c c c ] l ( c33c44 c33c55 c44c55) ζ F = [( c c c c ) c c c ] l [( c c c c c ) c c c c ] l [( c c ( c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c (c c c c c c ) ( c c ) ( ) c3) cc 3 cc33 c3] ll 3 55 c66c c66c55 c3 l ζ 6 c3c44 c3 cc33 cc55 c66c c33c66 l c33 c44 c55 ζ ( ){( ( ) ) ( ) F = c l c l ζ c l c c l ζ ζ [( c c c c ) c ] l l c c l c l Euation (4) reduces to }. D ( α ) = M ( α ) M ( α ) M ( α ) M ( α ) () D ( α ) = M ( α ) M ( α ) M ( α ) M ( α ) 3 3 D ( ) 0 3 α = D( α ) = M ( α ) M ( α ) M ( α ). (3) 33 3 Euations (3) correspond to the orthotropic materials purely elastic composites which are obtained and discussed by Yamada and Nasser [7] and on the hand second factor of euation (0) is 9

7 l K l K α ωζ τ= (4) 3 0. Euation (4) corresponds to the purely thermal wave which is clearly influenced by the thermal relaxation timeτ. Coupled Thermoelasticity In this is case when there is no thermal relaxation time i.e. τ 0 = 0 and hence τ=i/ω. Following above procedure we arrived at freuency euation in the coupled thermoelasticity. Numerical Results and Discussion Using characteristic euation (8) numerical results for phase velocity versus wave number are presented for the first few lower modes to indicate the dependence of dispersion upon the angle of propagation and thermal relaxation times. The material chosen for this purpose is luminum epoxy composite / Carbon steel as layer I (h =0.7) and Layer II (h = 0.3) respectively. Figure depicts the dispersion curves when the direction cosines of propagation are l =0.59 l =0.54 l 3 =0.799 whereas Figure and Figure 3 depict the dispersion curves when the direction cosines l=0.95 l =0.55 l 3 =0.834 and l =0.5 l =0.707 l 3 =0.696 in all these figures thermal relaxation time τ 0 =.0-7. It is observed that lowers modes are more influenced with changes in direction cosines whereas little variation is noticed in the upper modes. Further as l increases the phase velocity of lower modes decreases with wave number. Each of figure display wave speeds (coupled) corresponding to uasi-longitudinal uasi-transverse and uasi-thermal at zero wave number limits lower modes are found to highly influenced and phase velocity of higher modes have higher values and decreases as wave number increases. One of the thermoelastic modes seems to be associated with uick change in the slope of the mode. In anisotropic plates the distinction between mode types is somewhat artificial since the euation for thermal and elastic wave modes i.e. uasi-longitudinal and uasitransverse and shear horizontal modes will be generally be coupled with uasi-thermal wave modes. For wave propagation in the direction of symmetry some wave types revert to pure modes leading simple characteristic euation of lower order. Conseuence of elastic anisotropy in media is the loss of pure wave modes for general propagation direction. t zero wave number limits each figure displays wave speeds corresponding to one uasilongitudinal two uasitransverse and a uasithermal in generalized thermoelasticity and higher mode. It is apparent that the largest value corresponds to the uasi-longitudinal and the additional mode appears is a uasi-thermal mode. t low wave number limits modes are found to highly influence and it vary with the direction cosine. t relatively low values of the wave number little change is seen in these values. s wave number increases others high modes appear one of the modes seems to be associated with uick change in the slope of the mode. It is also observed that with change in direction cosines lower modes highly influenced whereas small variation is noticed in the high modes. Thus in generalized thermoelasticity at low 0

8 values of the wave number only the lower modes get affected at low values of wave number and the little change is seen at relatively high values of wave number. The low value region of the wave number is found to be of more physical interest in generalized thermoelasticity. Further as at high wave number limits small variation is observed and so the second sound effects are short lived. Quasi-longitudinal uasi-transverse (two) and uasi-thermal waves are found coupled with each other due to the thermal and anisotropic effects also wave-like behavior in the additional uasi-thermal modes. For uncoupled and coupled theory of thermoelasticity the results can be obtained from the present analysis by setting coupling terms and thermal relaxation times eual to zero..5 Phase Velocity [Non-dimensional] Wave Number [Non-dimensional] uasithermal uasitransverse uasitransverse uasilongitudinal Higher modes Figure. Phase velocity versus wave number for the direction cosinel = 0.59 l = 0.54 and l 3 = in generalized thermoelasticity

9 .5 Phase Velocity [Non-dimensional] Wave Number [Non-dimensional] uasithermal uasitransverse uasitransverse uasilongitudinal Higher modes Figure. Phase velocity versus wave number for the direction cosine l = 0.95 l = 0.55 and l 3 = in generalized thermoelasticity.5 Phase Velocity [Non-dimensional] Wave Number [Non-dimensional] uasithermal uasitransverse uasitransverse uasilongitudinal Higher modes Figure 3. Phase velocity verses wave number for the direction cosinel = 0.5 l = and l 3 = in generalized thermoelasticity.

10 ppendix-i. = c33 c44 c55 K3 = Bω εζ i ω PK c c c ( l l K ω τζ ) = Bω ζ i ω PK P( l K l ω τζ ) 3 = Bω εζ i ω PK P( l l K ω τζ ) = Bω εζ iω Pl ( lk ω τζ ) P = [( c c c c c ) c c c c ] l [( c c c c ) c c c c ] l -( c c c c c c ) ζ P = [( c c c c ) c c c ] l [( c c c c c ) c c c c ] l [( c c ( c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c ) c c c c c ] l l [(c c c c c c c c c c c ) l (c c c c c c c c c c c c c )] ζ ( c c c ) ζ { P = ( c l c l ζ ) [ ζ ( c ) l ( c c ) l ] ζ B [ c c c c ) c ] ll c l c l = c c β [ ] B = ( c c ) βζ ( c c c c ) φ c c -( c c c ) β l [ ( c c c c ) ββ c c ) β ( c c c c )] l B = [ β ( c c ) β c ) c c c ] l c c ( c c c c ) β c c ( c c cc66) β3 [( c3 c3c66 c44 cc55 c66c55 cc3) β3 ( c c c c c c c c c c c c 5] β {[ ( c c c c c c c c c c c c ) β c l ( c66) β3 ζ ( c44 c33) ζ ( c55 c3) βζ 3 l ζ cc66β3 ( c33c66 c44c55) β 4 ( c c c c ) ββ l ( c c ) β ( c c ) β ( c c ) ββ] l ζ βζ ] [ { } ] } [ B = ( c l c l ζ ) [ ζ ( c ) l ( c c ) l ] ζ [ c c c c ) c ] l l c l c l }

11 ppendix-ii D ( α ) = M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) D ( α ) = M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) D ( α ) = M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) M ( α ) D( α ) = M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( α ) M ( ) ppendix-iii = = = γ () = δ () 3 = θ () 4 = b c () () 5 55 = b c () () 6 44 = b () 7 3 = ( l α ) θ B E () () 8 3 = γ () = δ () 3 = θ () 4 = ηb c () () 5 55 ηb c = () () 6 44 = ηb () 7 3 = η( l α ) θ () () 8 3 = E k B E k () ( ) α = = iq l3 h h e Q= x h h ( ) E = e () iq( l3 ) h h α b = lδ α ( m) ( m) ( m) η = c c () () b = l δ α γ ( m) ( m) ( m) ( m) b = c l c l γ c α δ βθ ( m) ( m) ( m) ( m) ( m) ( m) ( m) ( m) c c c = m = ( m) ( m) ( m) 4

12 References [] W. M. Ewing W. S. Jardetsky and F. Press Elastic waves in Layered Media McGraw-Hill NY [] L. M. Brekhovskikh Waves in Layered Media cademic Press New York 960. [3] J. Padovan Thermoelasticity of nisotropic Half Space SME Journal of pplied Mechanics 4: [4] J. Padovan Thermoelasticity of nisotropic Generally Laminated Slabs subect to spatially Periodic Loads SME Journal of pplied Mechanics 4: [5] T. R. Tauchert and. Y. koz Stationary Temperature and stress Fields in an nisotropic Elastic Slab SME Journal of pplied Mechanics 4: [6] T. R. Tauchert Thermoelastic nalysis of Laminated Transversely isotropic Slabs Journal Thermal Stresses 3: [7] Y. Tanigawa H. Murakami and Y. Ootao Transient Thermal Stress nalysis of a Laminated Composite Beam Journal Thermal Stresses : [8] H. W. Lord and Y. Shulman generalized dynamical theory of thermoelasticity Journal of the Mechanics and Physics of Solids 5: [9]. E. Green and K.. Lindsay Thermoelasticity Journal of Elasticity : [0] B.. Boley High temperature structures and materials Pergamon Press oxford 964. [] D. K. Baneree Y. K. Pao Thermoelastic waves in nisotropy Solids Journal of the coustical Society of merica 56: [] R. S. Dhaliwal and H. H. Sherief Generalized Thermoelasticity for nisotropic Media Quarterly Journal pplied Mathematics 38: [3] M Yamada and Nasser S. N. Propagation Direction in Layered Transversely isotropic Elastic Composite Journal of Composite Materials 5: [4] K. L. Verma Propagation Direction in Layered Orthotropic Elastic Composites in Generalized Thermoelasticity Proceeding of third International Congress on Thermal Stresses K.L. Verma Department of Mathematics Government Post Graduate College Hamirpur (H.P.) INDI klverma@netscape.net kl.verma@rediffmail.com 5