Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 2, pp. 399-422, Warsaw 2012 50th Anniversary of JTAM THERMOELASTIC DISTURBANCES IN A TRANSVERSELY ISOTROPIC HALF-SPACE DUE TO THERMAL POINT LOAD Kishori L. Verma Government Post Graduate College, Department of Mathematics, Hamirpur, India e-mail: kl.verma@rediffmail.com; klverma@netscape.net The objective of this paper is to study disturbances due to thermal point load in a homogeneous transversely isotropic half-space in genera- lized thermoelasticity. A combination of the Fourier and Hankel trans- form technique is applied to obtain the solutions to governing equations. Cagniared’s technique is used to invert the transformed solutions for small times. Theoretically obtained results, for temperature, stresses are computed numerically for a zinc material. It is found that variations in stressesandtemperature aremoreprominentat small times anddecrease with passage of time.Theg results obtained theoretically are represented graphically at different values of thermal relaxation times. Keywords: transversely isotropic, generalized thermoelasticity,Cagniard technique, thermal point load Nomenclature T0 – uniform temperature Cij – elastic parameters λ,µ – thermal conductivity ρ – density of medium Ce – specific heat at constant strain τ0 – thermal relaxation time K3,K1 – coefficients of thermal conductivities α3,α1 – coefficients of linear thermal expansions ε1 – thermoelastic coupling constant VR – Rayleigh waves velocity T – temperature v – velocity of compressional waves 400 K.L. Verma L−1 – inverse Laplace transform δ(x) – Dirac delta function 1. Introduction Thermoelasticity theory, Chadwick (1960, 1979) and Nowacki (1962, 1975), of thermal disturbances has aroused considerable interest in the last centu- ry, but systematic research started only after thermal waves – called second sound – were first measured in materials like solid helium, bismuth and so- dium fluoride. Thus, the thermoelasticity theories, which admit a finite speed for thermal signals, have been receiving a lot of attention for the past thirty years. In contrast to the conventional coupled thermoelasticity theory based on a parabolic heat equation, Biot (1956), which predicts an infinite speed for the propagation of heat, these theories involve a hyperbolic heat equation and are referred to as generalized thermoelasticity theories. The Lord and Shulman (1967) theory introduces a single time constant to dictate the relaxation of thermal propagation as well as the rate of change of strain rate and the rate of change of heat generation, and obtained a wave- type heat equation by postulating a new law of heat conduction to replace the classical Fourier law for isotropic bodies. Later, the theory was developed and extended to anisotropic solids by Dhaliwal and Sherief (1980). These thermoelastic models are based on hyperbolic-type equations for temperature, and are closely connected with the theories of second sound, which view heat propagation as a wave-like phenomenon. Themajority of the work by Chandrasekharaiah (1986, 1998) in this field has been devoted to various aspects of linear thermoelastic models considering isotropic materials, very little work has been done considering materials which are anisotropic in nature. Hence, the study of thermo-mechanical interactions and thermoelastic disturbances in anisotropicmaterials is justifiedand is of great importanceand practical use in engineeringapplications especially in thecontext of generalized theory of thermoelasticity. Verma (1999) and Verma and Hasabe (2002) studied thermoelastic pro- blems by considering equations for transversely isotropic heat conducting pla- tes with thermal relaxations times. Harinath (1975, 1980) considered the pro- blems of surface point and line source over a homogeneous isotropic ther- moelastic halfspace in thermoelasticity. De Hoop (1959) modified and used a method originally presented byCagniard (1962) to solve the disturbances that are generated by an impulsive, concentrated load applied along a line on the Thermoelastic disturbances in a transversely isotropic... 401 free surface of a homogeneous isotropic elastic half-space. Nayfeh and Nasser (1972) developed the displacements and temperature fields in a homogeneous isotropic generalized thermoelastic halfspace subjected on the free surface to an instantaneously applied heat source using the Cagniard-De Hoop method (Cagniard, 1962). In this paper, using a combination of the Laplace and Hankel transforms, thegoverning equations of transversely isotropic thermoelastic solidhalf-space, which are subjected to thermal point load on its free surface are solved. The resulting equations are then inverted using the Cagniard-DeHoopmethod for small times. The results obtained theoretically have been verified numerically and illustrated graphically for a single crystal of zinc. 2. Formulation of the problem We consider thermal and elastic wave motion of small amplitude in homoge- neous heat conducting transversely isotropic elastic solids with thermal rela- xation, at a uniform temperature T0, and considering the plane of isotropy is perpendicular to z-axis.We take z-axis pointing normally into the half space, which is thus represented by z­ 0. The disturbance is caused by a suddenly applied thermal point source on the free surface of the initially undisturbed elastic solid. This source is acting in the direction of z-axis at the origin of the cylindrical coordinate system (r,θ,z) which is any point of the plane bo- undary z =0. The problem is axi-symmetric with respect to the z =0. The governing equations of motion and heat conduction for the displacement vec- tor u(r,z,t) = (u,0,w) and temperature T(r,z,t) for such a medium in the absence of the heat source and the body forces in the context of generalized theory of linear thermoelasticity are given by [ C11 ( ∂2 ∂r2 +r−1 ∂ ∂r −r−2 ) +C44 ∂2 ∂z2 −ρ ∂2 ∂t2 ] u+(C13+C44) ∂2w ∂r∂z =β1 ∂T ∂r [ C44 ( ∂2 ∂r2 +r−1 ∂ ∂r ) +C33 ∂2 ∂z2 −ρ ∂ 2 ∂t2 ] w+(C13+C44) ∂2u ∂r∂z =β3 ∂T ∂z (2.1) K1 ( ∂2 ∂r2 +r−1 ∂ ∂r ) T +K3 ∂2T ∂z2 −ρCe ( ∂ ∂t + τ0 ∂2 ∂t2 ) T =β1T0 ( ∂2u ∂r∂t +r−1 ∂u ∂t +β ∂2w ∂z∂t + τ0 ∂3u ∂r∂t2 +r−1 ∂u ∂t +β ∂3w ∂z∂t2 ) where 402 K.L. Verma β1 =(C11+C12)α1+C13α3 β3 =2C13α1+C33α3 β= β3 β1 (2.2) Cij are being the isothermal parameters, Ce and τ0 are the specific heat at constant strain and thermal relaxation time, respectively. K3,K1 and α3,α1 are the coefficients of thermal conductivities and linear thermal expansions respectively, along and perpendicular to the axis of symmetry. If we take C11 =C33 =λ+2µ C44 =2µ C13 =λ K3 =K1 =K α1 =α3 =αt β1 =β3 =(3λ+2µ)αt (2.3) then equations (2.1) reduce to the corresponding form for an isotropic body, with Lamé’s parameters λ, µ, thermal conductivity K and the coefficients of linear thermal expansion αt. We define the dimensionless quantities r′ = w∗ v r z′ = w∗ v z t′ =w∗t τ ′0 =w ∗τ0 w ′ = ρw∗v β1T0 w T ′ = T T0 k= K3 K1 c1 = C33 C11 c2 = C44 C11 c3 = C13+C44 C11 ε1 = β21T0 ρCeC11 (2.4) where k1 = K1/(ρCe) and v = √ C11/ρ are the thermal diffusivity and the velocity of compressional waves in the x-direction, respectively. Here ε1 is the thermoelastic coupling constant. Introducing above quantities (2.4) into equations (2.1), we obtain (on sup- pressing the primes throughout) ( ∂2 ∂r2 +r−1 ∂ ∂r −r−2+ c2 ∂2 ∂z2 − ∂2 ∂t2 ) u+ c3 ∂2w ∂r∂z = ∂T ∂r [ c2 ( ∂2 ∂r2 +r−1 ∂ ∂r ) + c1 ∂2 ∂z2 − ∂2 ∂t2 ] w+ c3 ∂2u ∂r∂z =β ∂T ∂z ( ∂2 ∂r2 +r−1 ∂ ∂r ) T +k ∂2T ∂z2 − ( ∂ ∂t + τ0 ∂2 ∂t2 ) T = ε1 ( ∂ ∂t + τ0 ∂2 ∂t2 )(∂u ∂r +r−1u+β ∂w ∂z ) (2.5) The boundary conditions at the surface z = 0 are σrz =σzz =0 hT + ∂T ∂z = Q0δ(r)f(t) 2πr (2.6) Thermoelastic disturbances in a transversely isotropic... 403 where σrz, σzz are thermal stresses, Q0 is a constant and δ(r) is the Dirac delta function, h is Biot’s heat transfer coefficient and f(t) is an arbitrary single-valued finite and continuous function of time and must have only one numerical value. Equations (2.6) may also be written as (c3− c2) ∂u ∂x + c1 ∂w ∂z −βT =0 ∂u ∂z + ∂w ∂x =0 hT + ∂T ∂z =Q∗0δ(x)f(t) (2.7) where Q∗0 = vQ0/T0. The condition at infinity requires that the solutions be bounded as z becomes large. Finally, the initial conditions are such that the medium is at rest for t< 0. 3. Solution of the problem The condition at infinity requires that the solutions be bounded as z becomes large. Finally, the initially conditions are such that the medium is at rest for t< 0. Apply the Laplace transform with respect to time and the Hankel trans- form with respect to r to the system of equations (2.5) to (2.7). The appro- priate solution of the resulting equation is then constructed and subsequently inverted. The Laplace and the exponential Fourier transforms are defined re- spectively as φ(r,z,p) = ∞ ∫ 0 φ(r,z,p)e−pt dt φ̂(q,z,p)= ∞ ∫ 0 rφ(q,z,p)Jn(qr) dr (3.1) where n=1 in the case of u(r,z,p) and n=0 for w(r,z,p) and T(r,z,p) to equation (3.1)2, we obtain û′′ = 1 c2 [(q2+p2)û−qT̂ + c3qŵ′] ŵ′′ = 1 c1 [(c2q 2+p2)û− c3qû′+βT̂ ] T̂ ′′ = 1 k [(q2+ τp2)T +ε1τp 2(qû+βŵ)] (3.2) where τ = τ0+p −1. 404 K.L. Verma The system of equations (3.2) can be written as d dz W(q,z,p) =A(q,p)W(q,z,p) (3.3) where W = [ U U ′ ] A= [ O I A2 A1 ] A1 =         0 c3q c2 0 − c3q c1 0 β c1 0 ε1τp 2β k         U =    û ŵ T̂    A2 =         q2+p2 c1 0 − q c2 0 c2q 2+p2 c1 0 ε1τp 2q k 0 q2+ τp2 k         (3.4) O=    0 0 0 0 0 0 0 0 0    I=    1 0 0 0 1 0 0 0 1    To solve equation (3.3), we have W(q,z,p)=X(q,p)exp(mz) So that A(q,p)W(q,z,p) = mW(q,z,p), which leads to the eigenvalue pro- blem. The characteristic equation corresponding to the matrix A is given by det(A−mI)= 0 (3.5) on expansion we have m6−λ1m4+λ2m2−λ3 =0 (3.6) where λ1 = Pq2+Jp2 c1c2 + q2+ τp2 k + ε1τp 2β 2 kc1 λ2 = { k(q2+p2)(c2q 2+p2)+(Pq2+Jp2)(q2+ τp2) +ε1τp 2q2[p2β 2 +(c1−2c3β+β 2 )] } 1 kc1c2 λ3 =(c2q 2+p2)[(q2+p2)(q2+ τp2)+ε1τp 2q2] 1 kc1c2 P = c1+ c 2 2+ c 2 3 J = c1+ c2 (3.7) Thermoelastic disturbances in a transversely isotropic... 405 The eigenvalues of thematrix A are the characteristic roots ±mi (i=1,2,3) of equation (3.6). We assume that real parts of are positive. The eigenvector X(q,p) corresponding to the eigenvalue m can be determined by solving the homogeneous equation (A−mI)X(q,p)=0 (3.8) The set of eigen-vectors Xi(q,p) (i=1,2, . . . ,6) may be obtained as Xi(q,p)= [ Xi1(q,p) Xi2(q,p) ] (3.9) where Xi1(q,p)=    −q aimi bi    Xi2(q,p)=    −qmi aim 2 i bimi    Xj1(q,p)=    −q −aimi bi    Xj2(q,p)=    qmi aim 2 i −bimi    j= i+3 (3.10) and ai = 1 ∆i [c2βm 2 i +(c3−β)q2−p2β] bi = 1 ∆i [(q2+p2− c2m2i)(c2q2+p2−c1m2i)+ c23q2m2i ] ∆i =(c1− c3β)m2i − c2q2−p2 (3.11) Thus the solution to (3.3) is given by W(q,z,p) = 3 ∑ i=1 [BiXi(q,p)exp(miz)+Bi+3Xi+3(q,p)exp(−miz)] (3.12) where Bi, (i=1,2, . . . ,6) are arbitrary constants. Equation (3.12) represents the general problem in the axi-symmetric case of generalized homogeneous transversely isotropic thermoelasticity by employing the eigenvalue approach. The displacements, temperature, stresses and temperature gradient in the transformeddomainwhich satisfy the radiation conditions canbewritten from equations (3.1) and (3.2) as û=−q[B4exp(−m1z)+B5exp(−m2z)+B6exp(−m3z)] ŵ=−[B4a1m1exp(−m1z)+B5a2m2exp(−m2z)+B6a3m3exp(−m3z)] 406 K.L. Verma T̂ =B4b1exp(−m1z)+B5b2exp(−m2z)+B6b3exp(−m3z) σ̂zz = 3 ∑ i=1 [(c3− c2)q2+aic1m2i −βbi]Bi+3exp(−m3z) (3.13) σ̂rz = c2 2 3 ∑ i=1 Bi+3mi(1+ai)exp(−m3z) T̂ ′ =− 3 ∑ i=1 Bi+3mibiexp(−m3z) where T̂ ′ = dT̂/dz. Applying transforms (3.1)1,2 to the boundary conditions, and above rela- tions σ̂zz =0 σ̂rz =0 hT̂ + T̂ ′ = −Q∗qf̂ 2π at z=0 (3.14) we obtain 3 ∑ i=1 [(c3− c2)q2+aic1m2i −βbi]Bi+3 =0 3 ∑ i=1 Bi+3mi(1+ai)=0 3 ∑ i=1 (h−mi)biBi+3 = −Q∗qf̂ 2π (3.15) For a stress free thermally insulated boundary (heat transfer coefficient h→ 0), and for a stress free isothermal boundary (h→∞). Solving equations (3.15) for B4,B5 and B6, we get B4 =− Q∗q 2π∆∗p { m3(1+a3)[(c3− c2)q2−βb2+a2c1m22] −m2(1+a2)[(c3− c2)q2−βb3+a3c1m23] } B5 = Q∗q 2π∆∗p { m3(1+a3)[(c3− c2)q2−βb1+a1c1m21] −m1(1+a1)[(c3− c2)q2−βb3+a3c1m23] } B6 =− Q∗q 2π∆∗p { m2(1+a2)[(c3−c2)q2−βb1+a1c1m21] −m1(1+a1)[(c3− c2)q2−βb2+a2c1m22] } (3.16) where we have taken f̂(p)= 1/p, and Thermoelastic disturbances in a transversely isotropic... 407 ∆∗ =(h1−h2m1)b1{m3(1+a3)[(c3− c2)q2−βb2+a2c1m22] −m2(1+a2)[(c3− c2)q2−βb3+a3c1m23]} − (h1−h2m2)b2{m3(1+a3)[(c3− c2)q2−βb1+a1c1m21] −m1(1+a1)[(c3− c2)q2−βb3+a3c1m23]} − (h1−h2m3)b3{m2(1+a2)[(c3− c2)q2−βb1+a1c1m21] −m1(1+a1)[(c3− c2)q2−βb2+a2c1m22]} (3.17) Thus a formal solution to equations (2.5) is given by (u,w,T) =L−1    ∞ ∫ −∞ 3 ∑ k=1 (a1k,a2k,a3k)Jn(qr)e −mkz dq    (3.18) where L−1 designate the inverse Laplace transform and where we have set (i=1,2,3) a1i =−qBi+3 a2i =−qmiBi+3 a3i = biBi+3 (3.19) 4. Inversion of transforms To obtain the solution, we use the Cagniard (1962) method to. This method consists of recasting each integral in (3.18) into the Lapalace transform of a known function, thus allowing one to write down the inverse transform by inspection. Mathematically, this procedure is based on De-Hoop (1959), Ca- gniard (1962) and Fung (1965) a rather elementary observation that L−1    pn 2π ∞ ∫ t0 f(t)e−pt dt−pn−1f(0)−pn−2f ′(0)− . . .−f(n−1)(0)    = dnf(t) dtn H(t− t0) (4.1) and L−1    1 2πpn ∞ ∫ t0 f(t)e−pt dt    = ∫ 1 ∫ 2 · · · ∫ n f(t)H(t− t0) dt n=0,1,2, . . . (4.2) For this technique to apply, it is therefore essential that we obtain an explicit expression for mk and that we isolate the Laplace transform parameter p as 408 K.L. Verma shown in (4.1) and (4.2). To this end,we observe that equation (3.6) pertain to the coupled, dilatational, distortional and thermal waves. To find the explicit expression, we seek for solution to (3.6) for small values of the thermoelastic coupling constant ε1. Assuming that ε1 is sufficiently small, we find that m2j =m 2 j0+ε1m 2 j1+ . . . j=1,2,3 (4.3) where m2j0 are given m210,m 2 20 = Pq2+Jp2± √ Pq2+Jp2−4c1c2(c2q2+p2)(p2+q2) kc1c2 mN230 = q2+ τp2 k (4.4) m2j1 = τp2{(c2q2+p2)q2−m2j0[(c1−2c3β+β 2 )q2+β 2 p2− c2β 2 m2j0]} kc1c2(m 2 j0−m2i0)(m2j0−mN2k0) i 6= j 6= k=1,2,3 In view Jn(ξ)= √ 2 πξ cos [ ξ− ( n+ 1 2 )π 2 ] =Re { √ 2 π exp [ −i ( ξ− ( n+ 1 2 ))π 2 ]} (4.5) ofWatson (1945), formal solution (3.18) can be we written as (u,w,T) =L−1    Re [ ∞ ∫ 0 ( 3 ∑ i=1 a∗1i, 3 ∑ i=1 a∗2i, 3 ∑ i=1 a∗3i ) exp(−iqr−mkz) dq ]    (4.6) and a∗1i = a1i √ 2q πr exp ( i π 4 ) (a∗2i,a ∗ 3i)= √ 2q πr exp ( i 3π 4 ) (a2i,a3i) (4.7) Due to existence of the damping term in temperature field equation (2.5)3, isolation of p is impossible. However, this isolation of p may be achieved for small time, i.e. if we assume p to be large. Hence, an expansion in the inverse power of p followed by the change of variable q= pη, reduces mNk0 and m 2 k1 to m10 = pα10 m20 = pα20 mN30 = pα30+ 1 2 kα30 m2j1 = p 2 ( α2j1+ α∗2j1 p ) j=1,2,3 (4.8) Thermoelastic disturbances in a transversely isotropic... 409 and α210,α 2 20 = [ Pη2+J± √ (Pη2+J)2−4c1c2(η2+1)(c2η2+1) ] 1 2c1c2 α230 = η2+ τ0 k α2ji = τ0η 2(c2η 2+1)− α2j0 αjik [(c1−2c3β+β 2 )η2+β 2− c2β 2 α2j0] αjik = kc1c2(α 2 j0−α2i0)(α2j0−α2k0) i 6= j 6= k=1,2,3 α∗ 2 11 =α 2 11 ( τ0+ 1 k ( α210−α230 )) α∗221 =α 2 21 [ τ0+ 1 k (α220−α230) ] α∗231 =α 2 31 { τ0+[(α 2 10+α 2 20)−2α230] c1c2 α312 } − τ0 kα312 [(c1−2c3β+β 2 )η2+β 2−2c2β 2 α230] (4.9) Special Case: We take f(t) = H(t), the unit step function so that the surface of the half-space is subjected to a thermal source of magnitude Q∗0 and f(p) = 1/p. Substitution of equations (4.1) to (4.9)2 in equations (3.13) and then into equation (3.3) yields (u,w,T) =L−1 { 3 ∑ i=1 ui, 3 ∑ i=1 wi, 3 ∑ i=1 T i } (4.10) and uk =Re ∞ ∫ −∞ ( √ pA1k+ √ 1 p B1k)exp[−p(zαk0+iηr)] dη wk =Re ∞ ∫ −∞ ( √ pA2k + √ 1 p B2k)exp[−p(zαk0+iηr)] dη Tk =Re ∞ ∫ −∞ ( √ pA3k+ √ 1 p B3k)exp[−p(zαk0+iηr)] dη (4.11) Aij and Bij are given in the Appendix A and B. 410 K.L. Verma 5. Singularities of the integrals In order to evaluate integrals (4.10) and (4.11), we consider a complex va- riable and distort the path of integration in the η-plane. The integrals are hexa-valued functions of η, when the choice of signs in α10, α20 and α30 is unrestricted, and these representations require a six-leaved Riemann surface. However, at all points of the path of integration, we have confined to the leaf of the Riemann sheet defined by Re(αj0)­ 0, (j=1,2,3) everywhere due to our choice that Re(mk)­ 0, and these are called the upper leaf. The possible singular points of the integrals are. a)Branchpoints.Thebranchpoints aregivenby(discriminantofEq. (4.9)1) √ (Pη2+J)2−4c1c2(η2+1)(c2η2+1)=0 αk0 =0 k=1,2,3 (5.1) and αk0 =0 for k=1,2,3 provide η=±i, η=± i √ c2 , η=±i √ τ0 (5.2) For an isotropic medium, it reduces to η = ±i, η = ±iv1/v2, η = ±i √ τ0 which are same as obtained by Nayfeh and Nasser (1972) and Sharma (1986) , where v1 and v2 are the velocities of dilatational and distortional waves. Again first equation of (5.1) is quadratic equation in η2 and has real roots if the discriminant of this equation is positive. Further, if PJ > 2c1c2(c2+1) P 2 > 4c1c 2 2 (5.3) then equation (5.1) cannot have positive roots in η2. Therefore, assume that equation (5.1) is hold and its discriminant is positive, thus thequartic equation has only pure imaginary pure roots. Physically, it is justified since we do not want the solution assumed to break down for points of the real η-axis. Otherwise, the waves of somewavelengths which correspond to these singular points of the real η-axis are propagated with amplitudes which are linear functions of depth in themedium.The corresponding branch points are of the type η=±iη0. b)Poles.Other singular points of the integrands are its poles,whichare given by (α210−α220)(α220−α230)(α230−α210)= 0 αk0 =0 ∆ ′(η)= 0 (5.4) Thermoelastic disturbances in a transversely isotropic... 411 Equation (5.4)1 provides α 2 10 = α 2 20 = α 2 30. This does not hold true as Re(αk0) ­ 0 and α10 6= α20 6= α30. Therefore, this yields no singularities. The poles of (5.4)2 coincide with branch points (5.2). Now to find poles gi- ven by (5.4)3, on taking η= i/V , rationalizing and simplifying it, reduces to Eq. (45) of Verma (2001), giving phase velocity for isothermal Rayleigh wa- ves in a transversely isotropic half-space in thermoelasticity. It can be easily verified (see Abubakar, 1961) that on the assumption P > Jc2, only one ro- ot of the resulting equation (see Eq. (45), Verma (2001)), satisfies (5.4)3 on the upper leaf of the Riemann surface, and that is the root which lies in the range 0 < V 2 < c2. Let it be V 2 R, where VR is the Rayleigh waves veloci- ty in uncoupled theory of thermoelasticity, which is the same as obtained by Verma (2001). Thus, on the assumption made, the singularities of integrands (4.9)4,5,6, which lie on the upper leaf of the Riemann surface are η=±i η=± i√ c2 η=±i√τ0 η=±iη0 η=± i Vr (5.5) In the special case of τ0 < 1 and V 2 R = 0.1834 for a zinc crystal, the path of integration is along the real axis. Tomake the functions of η single-valued in the complex plane of integration, we make a cut joining the singularities i/ √ c2 and −i/ √ c2 in the η-plane. First, we consider one of integrals (4.9)3−6, say u1(x,z,p) = 1 π Im ∞ ∫ z/ √ c2 (A11 p + B11 p2 )dη dt e−pt dt (5.6) Using the equation, we get u1(x,z,t) =Re [ t ∫ 0 A11H ( t− z√ c2 )∂η ∂t dt+ t ∫ 0 dt t ∫ 0 B11H ( t1− z √ c2 )∂η ∂t dt ] (5.7) Similarly u2(x,z,t) =Re { t ∫ 0 A12H ( t− z√ c1 )∂η2 ∂t dt + t ∫ 0 [ t ∫ 0 B12H ( t1− z √ c1 )∂η2 ∂t1 dt1 ] dt } (5.8) 412 K.L. Verma u3(x,z,t) =Re { t ∫ 0 A13H ( t−z √ τ0 k )∂η3 ∂t dt + t ∫ 0 [ t ∫ 0 B13H ( t1−z √ τ0 k )∂η3 ∂t1 dt1 ]} Thus, we have u(x,y,t)= 3 ∑ k=1 Re { t ∫ 0 (t+skz) ∂ηk ∂t dt+ t ∫ 0 [ t ∫ 0 B1kH(t1−skz) ∂ηk ∂t1 dt1 ] dt } (5.9) where s1 =1/ √ c2, s2 =1/ √ c1, s3 = √ τ0/k are the slowness of the transverse dilatational and the thermal waves, respectively. Similarly w(x,z,t) = 3 ∑ k=1 Re { t ∫ 0 A2kH(t−skz) ∂ηk ∂t dt + t ∫ 0 [ t ∫ 0 B2kH(t1−skz) ∂ηk ∂t1 dt1 ] dt } (5.10) T(x,z,t)= 3 ∑ k=1 Re [ A3kH(t−skz) ∂ηk ∂t + t ∫ 0 B3kH(t−skz) ∂ηk ∂t dt ] where ηk, k = 1,2,3 can be determined from t = αk0z+ iηkx. Also when the thermoelastic coupling constant ε1 vanishes, then the temperature field vanishes as well. 6. Numerical results and discussions The results obtained theoretically for temperature and stresses are computed numerically for a single crystal of zinc for which the physical data is given as ε1 =0.0221 c1 =0.385 c2 =0.2385 c3 =0.549 k=1.0 β=0.9 τ0 =0.02 T0 =296K c11 =1.628 ·1011Nm−2 ρ=7.14 ·103kmg−3 w∗ =5.01 ·1011 s−1 Thermoelastic disturbances in a transversely isotropic... 413 The computations were carried out for four values of time, namely τ =0.05, 0.1, 0.2, 0.5 at the surface z=0 and for a stress free heat transfer coefficient h → 0. The results for thermal stresses with respect to distance are shown in Figs. 1-4, respectively. From the figures, it is observed that the negative values of stresses are due to compression by a point load at the surface, and they increase in magnitude with the passage of time. The temperature also decreases fromapositive valuewith the passage of time.Also the variations of all these quantities are more prominent at small times and decrease with the passage of time. This established the fact that the second sound effect is short lived. All these quantities vanish whenwemove away from the heat source at a certain distance at all times, which shows the existence of the wave front and ascertain the fact that the generalized theory of thermoelasticity admits a finite velocity of heat. Fig. 1. Variation of the transverse stress near the surface with distance and time Fig. 2. Variation of the normal stress near the surface with distance and time 414 K.L. Verma Fig. 3. Variation of the horizontal stress near the surface with distance and time Fig. 4. Variation of the temperature near the surface with distance and time Appendix A z,k=1,2 A1k =−ℜ1 √ 2η πr exp ( i π 4 ) Nk0(η) α2k1 2αk0 A13 =−ℜ1 √ 2η πr exp ( i π 4 ) N30(η) α231 2α30 zexp ( −z 2kα30 ) ℜ1 = −Q∗ε1η2 2π∆′10 A2k =−ℜ2 √ 2η πr exp ( i 3π 4 ) Nk0(η)αk0 α2k1 2 z,k=1,2 A23 =−ℜ2 √ 2η πr exp ( i 3π 4 ) N30(η)α30 α231 2 zexp ( −z 2kα30 ) ℜ2 = −Q∗ε1η 2π∆′10 A3k =−ℜ3 √ 2η πr exp ( i 3π 4 ){ Nk0(η)bk0(η)+ε1 [ Nk0(η)bk1(η)+ Nk1bk0(η) −Nk0(η)bk0(η) α∗2k1 2αk0 z− ( Nk2bk0(η)− Nk0(η)bk0(η)Z(η) α2k1 2αk0 z )]} Thermoelastic disturbances in a transversely isotropic... 415 A33 =−ℜ3 √ 2η πr exp ( i 3π 4 ){ N30(η)b30(η)+ε1 [ N30(η)b31(η)+ N31b30(η) −N30(η)b30(η) (α∗231 2 − α 2 31 2kα30 ) z− (N32b30(η) −N30(η)b30(η)Z(η)) α231 2α30 z ]} exp ( −z 2kα30 ) Appendix B k=1,2 B1k =ℜ3η √ 2η πr exp ( i π 4 ){ Nk0(η)+ε1 [ Nk1(η)− Nk0(η) α∗2k1 2αk0 z − ( Nk2− Nk0(η)Z(η) α2k1 2αk0 z )]} k=1,2 B13 =ℜ3η √ 2η πr exp ( i π 4 ){ N30(η)+ε1 [ N31(η)− N30(η) (α∗231 2 − α231 2kα30 ) z +N30(η)Z(η) α231 2α30 z )]} exp ( −z 2kα30 ) ℜ3 = −Q∗η 2π∆′10 B2k =ℜ3 √ 2η πr exp ( i 3π 4 ){ Nk0(η)ak0(η)αk0+ε1 [ Nk1(η)ak0(η)αk0 + ( ak00(η)αk0+ak0(η) α2k1 2αk0 ) Nk0(η)− Nk0(η)ak0(η) α∗2k1 2 z −[(Nk2(η)ak0(η)+ Nk0(η)a∗k0(η))αk0− Nk1(η)Z(η)ak0(η)αk0] α2k1 2αk0 z )]} B23 =ℜ3 √ 2η πr exp ( i 3π 4 ){ N30(η)a30(η)α30+ε1 [ N31(η)a30(η)α30 + ( a300(η)α30+α30(η) α231 2α30 ) N30(η)−N30(η)a30(η)α30 (α∗231 2 − α 2 31 4kα30 ) − [ N30(η) ( (a∗30(η)+a40(η))α30+ α30(η) 2kα30 ) −N30(η)α30(η)Z(η) α231 2 z ]]} exp ( −z 2kα30 ) 416 K.L. Verma B3k =ℜ3 √ 2η πr exp ( i 3π 4 ){ [Nk2(η)− Nk0(η)Z(η)]bk0(η) +ε1 [ Nk2(η)bk1(η)+ Nk3(η)bk0(η)+Nk0(η)b ∗ k1(η) −[Nk0(η)bk1(η)+ Nk1(η)bk0(η)]Z(η)− Nk0(η)bk0(η)Z1(η) −[Nk2(η)bk0(η)− Nk0(η)bk0(η)Z(η)] α2k1 2αk0 z− ( Nk4(η)bk0(η) −Nk2(η)bk0(η)Z(η)− Nk0(η)bk0(η)Z3(η) α2k1 2αk0 z )]} B33 =ℜ3 √ 2η πr exp ( i 3π 4 ){ [N30(η)b32(η)− N30(η)b32(η)Z(η)] +ε1 [ N30(η)[b ∗ 31(η)+ b33(η)]+ N31(η)b32(η)+ N32(η)b30(η) −[N30(η)b32(η)− N30(η)b30(η)Z(η)] (α∗231 2 − α 2 31 4kα30 ) −N30(η)b30(η) α∗231 4kα30 z− ( N30(η)b34(η)− N30(η)b32(η)Z(η) −N30(η)b30(η)Z3(η) α231 2α30 z )]} exp ( −z 2kα30 ) where i, j and k are in the cyclic order (i,j,k=1,2,3, and i 6= j 6= k) and Ni0(η) =Tj00(η)Sk0(η)−Sj0(η)Tk00(η) Ni1(η) =Tj0(η)Sk0(η)+Sk00(η)Tj00(η)− [Tk0(η)Sj0(η)+Sj00(η)Tk00(η)] N12(η)=S31(η)T200(η)−S20(η)T301(η) N22(η)=S10(η)T301(η)−S31(η)T100(η) N32(η)=T ∗ 10(η)S20(η)+S ∗ 200(η)T100(η)− [T∗20(η)S10(η)+S∗100(η)T200(η)] N13(η)=T20(η)S31(η)+S ∗ 301(η)T200(η)+S30(η)T ∗ 20(η) −[T∗30(η)S20(η)+S200(η)T301(η)+S∗200(η)T300(η)] N25(η)=T302(η)S300(η)+S10(η)T ∗∗ N30(η)+S100(η)T301(η) −[T10(η)S32(η)+S∗302(η)T100(η)+S31(η)T∗10(η)] N14(η)=S32(η)T200(η)−S20(η)T302(η) N24(η)=S10(η)T302(η)−S32(η)T100(η) N15(η)=T ∗ 20(η)S31(η)+S ∗ 302(η)T200(η)+S32(η)T20(η) −[T301(η)S∗200(η)+S200(η)T302(η)+S20(η)T∗∗N30(η)] N23(η)=T ∗ 30(η)S10(η)+S100(η)T301(η)+S ∗ 100(η)T300(η) −[T∗10(η)S30(η)+S31(η)T10(η)+S∗301(η)T100(η)] Thermoelastic disturbances in a transversely isotropic... 417 Sk0(η)= Dk0(η)αk0 ∆k(η) Sk00(η)= Lk0(η) ∆k(η) S∗k00(η)= L∗k0(η) ∆k(η) S31(η)= D′30(η)k∆3(η)− (c1− c3β)D30(η)α30 k∆2k(η) S32(η)= 1 ∆3(η) [D′′30(η)k∆3(η)− (c1−c3β)D′30(η) ∆3(η) + (c1− c3β)2D30(η)α30 k∆23(η) ] S∗301(η)= 1 ∆3(η) [L∗30(η)+L40(η)]k∆3(η)− (c1−c3β)L30(η) k∆3(η) S∗302(η)= 1 ∆3(η) [[L∗40(η)+L50(η)]k∆3(η)− [L∗30(η)+L40(η)](c1−c3β)L30(η) k∆3(η) + L30(η)(c1 −c3β)2 k 2 ∆23(η) ] Dk0(η)= c2βα 2 k0− (c3−β)η2−β ∆Nk0(η)= (c1− c3β)α2k0− c2η2−1 D′30(η) = D30(η) 2kβα30 + c2β 2 α30 k D′′30(η)= c2β 2 2kβα30 (Lk0(η),L ∗ k0(η)) = (Dk0(η) 2αk0 +c2βαk0− Dk0(η)(c1− c3β)αk0 ∆k(η) ) (α2k1,α ∗2 k1) L40(η)= { 1 2kα30 ( 2c2β− D30(η) ∆3(η)α 2 30 ) − c1−c3β k∆3(η) [D30(η) 2α30 +α30 ( c2β− D30(η) ∆3(η) )]} α231 L∗40(η)= 1 2k 2 α230 ( D30(η) ∆3(η)α 2 30 − c2β ) + (D30(η) ∆3(η) − c2β )( c1− c3β 2k 2 ∆3(η)α30 − (c1− c3β)2α30 k 2 ∆23(η) ) (ak0(η),a ∗ k0(η)) = D10(η) ∆k(η) (α2k1,a ∗2 k1) (ak00(η),a ∗ k00(η)) = ( c2β− Dk0(η) ∆k(η) (c1− c3β) )( α2k1 ∆k(η) , a∗2k1 ∆k(η) ) (a40(η),a ∗ 40(η))= (c2β 2 k − D30(η) ∆3(η)k (c1− c3β) )( α231 ∆3(η) , a∗231 ∆3(η) ) 418 K.L. Verma a400(η)=−c2β+ D30(η) ∆3(η) (c1− c3β)2 ( 1− 1 k ) a∗400(η)= a400(η) α∗231 ∆3(η) a50(η)= (D30(η)(c1 − c3β) ∆3(η) − c2β 2 )α231(c1− c3β) ∆23(η)k 2 a500(η)= ( c2β− D30(η) ∆3(η)k 2 (c1− c3β)3 ) α231 ∆3(η) − (c2β 2 k − D30(η) ∆3(η)k (c1− c3β) )(c1− c3β)3 ∆23(η)k α231 + (D30(η)(c1− c3β)3 ∆23(η)k 2 − c2β 2 (c1− c3β) k 2 ) (c1− c3β)α231 bk0(η)= Ek0(η) ∆k(η) (bk1,b ∗ k1)= Fk0(η)−bk0(η)(c1 − c3β)(α2k1,α∗2k1) ∆k(η) b31(η) = F30(η)− b30(η)(c1− c3β) ∆k(η) α231 b32(η) = (−2c1c2 k − b30(η)(c1− c3β) ∆k(η)k ) α231 b33(η) = −2c1c2 k − b32(η)(c1 − c3β)− b31(η)(c1− c3β) kα231 b34(η) = (F30(η)(c1− c3β)2 ∆23(η)k 2 + c1c2 k −E′30(η) c1− c3β k∆3(η) ) 1 ∆3(η) (c2β 2 k − D30(η) ∆3(η)k (c1− c3β) )( α231 ∆3(η) , a∗231 ∆3(η) ) b∗31(η) = b31(η) α∗231 α231 b∗33(η)= b33(η) α∗231 α231 b35(η) = [b31(η)(c1− c3β)2 ∆23(η)k 2 − c1− c3β k∆3(η) (c1c2 k 2 − b32(η)(c1− c3β) ) − c1− c3β k∆23(η) (F30(η)(c1− c3β)2 ∆23(η)k 2 − E′30(η) k∆3(η) + c1c2 k 2 )] α231 Ek0(η)=Gk0(η)Hk0(η)+H ′ k0(η) E ′ 30(η) = E30(η) k Thermoelastic disturbances in a transversely isotropic... 419 E′′30(η) = c1c2 k 2 Fk0(η)= c3η 2− c1GNk0(η)− c2Hk0(η) Gk0(η)= η 2+1− c2α2k0 Hk0(η)= c2η2+1− c1α2k0 H′k0(η)= c3η 2α2k0 Z(η)= ∆′20(η) ∆′10(η) Z1(η) = ∆′21(η)−2(η)∆11(η) ∆′10(η) Z2(η)= ∆′30(η)− Z(η)∆′20(η) ∆′10(η) Z3(η) = ∆′31(η)− ∆′ 30 (η) ∆′ 10 (η) ∆′11(η)+2Z(η)[Z(η)∆ ′ 11(η)−∆′21(η)]−∆′11(η)Z2(η) ∆′10(η) ∆′10(η)=− 3 ∑ j=1 Nj0(η)aj0(η)αj0 ∆′20(η)=− [ 2 ∑ j=1 [Nj0(η)a ∗ j0(η)+ Nj2(η)aj0(η)]αj0 +N30(η) ( [a∗30(η)+a40(η)]α30+ a30(η) 2kα30 ) ] ∆′30(η)=− [ 2 ∑ j=1 [Nj2(η)a ∗ j0(η)+ Nj4(η)aj0(η)]αj0 +N30(η) ( [a∗40(η)+a50(η)]α30+ a40(η)+a ∗ 30(η) 2kα30 − a30(η)α ∗2 31 4kα230 ) ] ∆′11(η)=− [ 2 ∑ j=1 [ Nj1(η)aj0(η)αj0+ ( aj00(η)αj0+aj0(η) α2j1 2αj0 ) Nj0(η) ] ] ∆′21(η)=− [ 2 ∑ j=1 [ Nj3(η)aj0(η)αj0+ ( aj00(η)αj0+aj0(η) α2j1 2αj0 ] ) Nj2(η) ] ] + 2 ∑ j=1 [ Nj1(η)a ∗ j0(η)α10+ Nj0(η) ( a∗j00(η)αj0+a ∗ j0(η) α2j1 2αj0 +aj0(η) α∗2j1 2αj0 )] +N30(η) [ a40(η)+a ∗ 30(η) α231 2α30 +[a∗300(η)+a400(η)]α30 + a300(η) 2kα30 + ( α∗231 2α30 − α 2 31 4kα230 ) a30(η) ] +N31(η) ( a40(η)+a ∗ 30(η)α30 + α30 2kα30 ) + N32(η)a30(η)α30 420 K.L. Verma ∆′31(η)=− [ 2 ∑ j=1 [ Nj3(η)a ∗ j0(η)αj0 + ( a∗j00(η)αj0+ a∗j0(η)α 2 j1+aj0(η)α ∗2 j1 2αj0 ) Nj2(η) ] ] + 2 ∑ j=1 [ Nj5(η)aj0(η)α10+ Nj4(η) ( aj00(η)αj0+aj0(η) α2j1 2αj0 ) +Nj0(η)a ∗ j0(η) α∗2j1 2αj0 ] +N30(η) [ N500(η)+a ∗ 400(η)α30+[a ∗ 40(η)+a50(η)] α231 2α30 + a400(η)+a ∗ 300(η) 2kα30 + ( α∗231 2α30 − α231 4kα230 ) a30(η) ] +N31(η) ( a50(η)+a ∗ 40(η)α30 + a40(η)+a ∗ 30(η) 2kα30 − a30(η)α ∗2 31 4kα230 ) + N32(η) ( a50(η)+a ∗ 40(η)α30+ a30(η) 2kα30 ) References 1. Abubakar Iya, 1961,Disturbance due to a line source in a semi-infinite trans- versely isotropic elastic medium,Geophysical Journal, 6, 337-359 2. Biot M.A., 1956, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics, 27, 240-253 3. Cagniarad I., 1962,Reflection and Refraction of Progressive Seismic Waves, Trans. by E. Flinn and C.Dix., McGraw-Hill, NewYork 4. Chadwick P., 1960, Progress in Solid Mechanics, R. Hill and I.N. Sneddon (Edit.), North Holland Publishing Co. 5. Chadwick P., 1979, Basic properties of plane harmonic waves in a presented heat conducting elastic material, J. of Thermal Stresses, 2, 193-214 6. Chandrasekharaiah D.S., 1986, Thermoelasticity with second sound – A review,Appl. Mech. Rev., 39, 355-376 7. Chandrasekharaiah D.S., 1998, Hyperbolic thermoelasticity. A review of recent literature,Applied Mech. Rev., 51, 705-729 8. Dhaliwal R.S., Sherief H.H., 1980,Generalized thermoelasticity for aniso- tropic media,Q. Appl. Math., 38, 1-8 9. Fung Y.C., 1965, Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, NJ Thermoelastic disturbances in a transversely isotropic... 421 10. Harinath K.S., 1975, Surface line sources over a generalized thermoelastic half-space, Ind. J. Pure Appl. Math., 8, 1347-1351 11. Harinath K.S., 1980, Surface line sources over a generalized thermoelastic half-space, Ind. J. Pure Appl. Math., 11, 1210-1216 12. HoopDeA.T., 1959,Amodification ofCagniard’smethod for solving seismic pulse problems,Appl. Sci. Res., 8, 349-356 13. Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermo- elasticity, J. Mech. Phys. Solids, 15, 299-309 14. Nayfeh A.H., Nasser S.N., 1972, Transient thermoelastic waves in a half- space with thermal relaxations, J. Appl. Math. Phys., 23, 50-67 15. NowackiW., 1962,Thermoelasticity, Int. Ser.Monographs inAeronautics and Astronautics, PWNWarsaw 16. NowackiW., 1975,DynamicProblems ofThermoelasticity, Leyden:Noordhoff 17. Sharma J.N., 1986, Transient generalized thermoelastic waves in a transver- sely isotropic half-space,Bulletin of the Polish Academy of Sciences, 34, 11/12, 631-646 18. VermaK.L, 2001,Thermoelasticvibrationsof transversely isotropicplatewith thermal relaxations, Int. Journal of Solids and Structures, 38, 8529-8546 19. VermaK.L,HasebeN., 1999,On the propagationofGeneralized thermoela- stic vibrations in plates,Quart. J. of Polish Acad. of Sci., Engineering Trans- actions, 47, 299-319 20. Verma K.L, Hasebe N., 2002, Wave propagation in transversely isotropic plates in generalized thermoelasticity,Arch. Appl. Mech., 72, 470-482 21. WatsonG.N., 1945,Theory ofBessel Function, 2nded.,CambridgeUniversity Press Termosprężyste zaburzenia w poprzecznie izotropowej półprzestrzeni wywołane punktowym obciążeniem termicznym Streszczenie Celem pracy jest zaprezentowanie zaburzeń wywołanych punktowym obciąże- niem termicznymprzyłożonymdo jednorodnej, poprzecznie izotropowej półprzestrze- ni w ogólnym sformułowaniu zagadnieniu termosprężystości.Dowyznaczenia równań układu zastosowano kombinację transformaty Fouriera i Hankela. Przy odwracaniu tak otrzymanych transformat użytometody Cagniarda dla krótkich przedziałów cza- sowych.Rezultaty analizypodkątemwyznaczenia temperatury i naprężeńotrzymano 422 K.L. Verma w drodze symulacji numerycznej dla przypadku cynku jako materiału badawczego. Wykazano, że oscylacje poziomu naprężeń i temperatury są szczególnie wyraźne dla krótkich przedziałów czasu i gasną z jego upływem.Wyniki badań zilustrowano gra- ficznie dla różnych czasów relaksacji termicznej. Manuscript received June 3, 2011; accepted for print July 17, 2011