Wave propagation in an initially stressed transversely isotropic thermoelastic half-space

1 Introduction

Several efforts are made to remove the “so-called paradox”, inherent in the classical coupled dynamical theory of thermoelasticity [1]: that the thermal signal propagates with an infinite speed. An extended thermoelasticity theory introducing one thermal relaxation time in the thermoelastic process was proposed by Lord and Shulman [2] and the temperature-rate dependent theory of thermoelasticity – which takes into account two relaxation times – was developed by Green and Lindsay [3]. Chandrasekharaiah [4], Hetnarski and Iganazack [5] in their recent surveys, reviewed the theory proposed by Green and Naghdi [6–9] as an alternate way for formulating the propagation of heat. This theory is capable of incorporating thermal pulse transmission in a consistent manner and makes use of general entropy law rather than the usual entropy inequality. The characterization of thermoelastic material response is based on three types of constitutive functions: type I, type II, and type III. When the theory of type I is linearized, a parabolic equation of heat conduction arises. Here, we focus on the theory of type II (a limiting case of type III), which does not admit energy dissipation. Following the Green-Naghdi (GN) theory of thermoelasticity without energy dissipation, further research work was conducted on the wave propagation in isotropic generalized thermoelastic solids (e.g., Quintanilla [10]; Taheri et al. [11]; Puri and Jordan [12]; Roychoudhuri and Byopadhyay [13]; Lazzari and Nibbi [14]; Quintanilla [15]).

Initial stresses may develop in a medium for several reasons, e.g., temperature variation during processing, rapid quenching, slow creep processes, differential external forces, gravity variations, etc. The Earth, in particular, is assumed to be under high initial stresses. Dey et al. [16, 17] studied the propagation of waves in a thermoelastic medium under initial stresses. Gupta and Gupta [18] discussed the reflection of waves in an initially stressed fiber-reinforced transversely isotropic medium. Based on this, the present paper deals with the propagation of waves in an initially stressed transversely isotropic medium in the context of the GN thermoelasticity theory of types II and III. This study may have applications in various fields of science and technology, including atomic physics, aerospace and industrial engineering (thermal power plants, submarine structures, pressure vessels, chemical pipes).

2 Basic equations

The constitutive relations and balance laws for a general anisotropic (with a center of symmetry) initially stressed thermoelastic medium, in the absence of body forces, are given by Green and Naghdi [9] and Montanaro [19] as follows:

– Constitutive relation:

– Balance law:

– Equation of heat conduction:

where ρ is the mass density, t_ij are the components of stress tensor, u_i is the mechanical displacement vector, eij=(ui,j+uj,i)2 are the components of the infinitesimal strain, T is the temperature change of a material particle, T₀ is the reference uniform temperature of the body, P=-t₁₁ is the normal initial stress and ω_ij=(u_j,i-u_i,j)/2 is the rotation tensor. Moreover, K_ij is the thermal conductivity tensor, Kij* denotes a characteristic material constant tensor, β_ij=C_ijklα_kl denotes the thermal elastic coupling tensor, α_kl denotes the coefficient of linear thermal expansion, c^* is the specific heat at constant strain and C_ijkl is the elasticity matrix. The various material tensors introduced obey the following symmetry properties Cijkl=Cklij=Cjikl, Kij*=Kji*, Kij=Kji, βij=βji. A comma notation is used for spatial derivatives and a superimposed dot denotes time differentiation.

3 Problem formulation

Following Slaughter [20], an appropriate transformation is applied to Eq. (1), in order to derive the governing equations for an initially stressed transversely isotropic medium, when our analysis is restricted to two dimensions. The origin of the coordinate system (x₁, x₂, x₃) is taken at the free surface of the half-space. The x₁–x₂ plane is chosen to coincide with the free surface and the x₃ axis is then normal to the half-space (x₃≥0). We consider plane waves such that all particles on a line parallel to x₂-axis are equally displaced. Therefore, all field quantities will be independent of the x₂ coordinate. Then, the component of the displacement vector is of the form

and the solutions are independent of x₂, i.e., ∂/∂x₂≡0. Thus, the governing differential equations for such a medium reduce to:

where β₁=C₁₁α₁+C₁₃α₃, β₃=C₃₁α₁+C₃₃α₃ and we have also used the notation 11→1,13→5,33→3 for the material constants.

To proceed further, it is convenient to introduce the non-dimensional quantities defined by

where L, t_o, T_o are parameters having dimension of length, time and temperature, respectively.

4 Solution of the problem

Let p→=(p1, 0, p3) denote the unit propagation vector, with c and ξ denoting respectively the phase velocity and the wave number of plane waves propagating in the x₁–x₃ plane. Then by seeking for plane wave solutions of the equations of motion of the form

We introduced Eqs. (8) and (9) into Eqs. (5)–(7) to obtain three homogeneous equations in three unknowns. Non-trivial solutions of the resulting system of equations can be derived when the following condition is fulfilled

where

A=f10,B=-f5f10-f1f10+f6f8+f3f7+f9, D=f1f5f9-f2f4f9,

C=-f5(f9+f1f10-f3f7)-f1(f9-f6f8)-f2(f4f10+f6f7)+f3f4f8,

f1=p12d1+p32d3-d13Pp32/2,f2=p1p3d3-d13Pp1p3/2,                     f3=ip1d4,f4=p1p3d2-d13Pp1p3/2,

f5=p12d3+p32d5+d13Pp12/2,f6=ip3d6,f7=ip7d11,f8=ip3d12,                                        f9=iωp12-d8p12+iωk¯p32-d9p32

f10=ε1,d1=C11to2ρL2,d2=(C13+C55/2)to2ρL2,d3=C55to22ρL2,d4=β1Toto2ρL2, d5=C33to2ρL2,d6=β3Toto2ρL2,

k¯=d7=k3k1,d8=k1*tok1,d9=k3*tok1,ε1=d10=ρC*L2k1to,d11=β1L2k1t0,d12=β3L2k1t0,d13=t02ρL2.

The roots of this equation give three values of c². The corresponding three positive values of c will then be the propagation velocities of the three possible waves. The waves with velocities c₁, c₂, c₃ correspond to three types of quasi waves. We name these waves as quasi-longitudinal displacement (qLD) wave, quasi thermal wave (qT), and quasi transverse displacement (qTD) wave.

5 Reflection waves

Consider a homogeneous initially stressed transversely isotropic half-space, in the context of G-N thermoelasticitytheory of types II and III, occupying the region x₃>0. Incident qLD or qT or qTD waves at the interface will generate reflected qLD, qT and qTD waves in the half-space x₃>0. The displacements and temperature distributions are given by

where

with ω denoting the angular frequency. Here the subscripts 1, 2, 3 denote, respectively, the quantities corresponding to incident qLD, qT and qTD waves, whereas the subscripts 4, 5 and 6 denote, respectively, the corresponding reflected waves, with

rj=∧1j∧j,  sj=∧2j∧j,

∧j=|ξ2(f5-cj2)ξf6cj2ξ3f8ξ2(f9+f10cj2)|, ∧1j=|ξ2f4ξf6cj2ξ3f7ξ2(f9+f10cj2)|, ∧2j=|ξ2f4ξ2(f5-cj2)cj2ξ3f7cj2ξ3f8|.

For incident waves:

• qLD-wave:p₁=sin e₁, p₃=cos e₁,

• qT-wave:p₁=sin e₂, p₃=cos e₂,

• qTD-wave: p₁=sin e₃, p₃=cos e₃,

For reflected waves:

• qLD-wave: p₁=sin e₄, p₃=cos e₄,

• qT-wave: p₁=sin e₅, p₃=cos e₅,

• qTD-wave: p₁=sin e₆, p₃=cos e₆.

It is further noted that e₁=e₄, e₂=e₅, e₃=e₆, that is, the angle of incidence is equal to the angle of reflection in generalized thermoelastic transversely isotropic media, so that the velocities of reflected waves are equal to their corresponding incident waves, i.e., c₁=c₄, c₂=c₅, c₃=c₆.

6 Boundary conditions

The boundary conditions at the thermally insulated surface x₃=0 are given by

where

The wave numbers ξ_j, j=1,2,…,6, and the apparent velocity

c_j, j=1,2,…,6, are connected through the relation

at the surface x₃=0. In order to satisfy the boundary conditions given by Eqs. (3), (15) may also be re-written as

Making use of Eqs. (8), (14), (15) and (16), along with the thermally insulated boundary conditions given by Eq. (3), we obtain

where

A1j={aj1+rjaj2-tjaj3,      j=1,2,3,aj1-rjaj2-tjaj3,j=4,5,6,,A2j={bj1+rjbj2,j=1,2,3,-bj1+rjbj2,j=4,5,6,,A3j={tjcj1,j=1,2,3,-tjcj1,j=4,5,6,,

where

aj1=-iωC13C11sinejcj, aj2=iωC33C11cosejcj, aj3=iβ3T0sjC11,  bj1=iωC55cosej2C11cj, bj2=iωC55sinej2C11cj,  cj1=ωcosejcj.

– Incident qLD-wave:

In the case of incident qLD-wave, A₂=A₃=0. Dividing the set of Eqs. (17) throughout by A₁, we obtain a system of three non-homogeneous equations in three unknowns which can be solved by using the Gauss elimination method to obtain

– Incident qT-wave:

In the case of incident qT-wave, A₁=A₂=0, and thus we have

– Incident qTD-wave:

In the case of incident qTD-wave, A₁=A₂=0, and thus we have

where Δ=|Aii+3|3×3 and Δip (i=1,2,3,p=1,2,3) can be obtained by replacing, respectively, the first, second and third column of Δ by [-A1p -A2p -A3p]T, where [ ]Tdenotes the transpose of the matrix.

7 Numerical results and discussion

In order to illustrate the theoretical analysis given in the preceding sections, we now present some numerical results. The following relevant physical constants for Cobalt material are taken from Dhaliwal and Singh [21] for a thermoelastic transversely isotopic material:

C11=3.071×1011Nm-2,C12=1.650×1011Nm-2,C13=1.027×1011Nm-2,C33=3.581×1011Nm-2,C55=1.51×1011Nm-2,  β1=7.04×106Nm-2K,β1=6.98×106Nm-2K,ρ=8.836×103Kgm-3,K1=6.90×102 Wm-1K,  K3=7.01×102 Wm-1K,K1*=1.313×102 Wsec,K3*=1.54×102 Wsec,c*=4.27×102JKgK,T=298 K.

The variations of amplitude ratio of the reflected qLD, qTD and qT waves, for incident qLD, qTD and qT waves at the free surface are shown graphically in order to compare the results obtained in the two cases: i.e., incident waves for a transversely isotropic medium in the context of thermoelasticity with energy dissipation (ISTIWED) and the standard case for isotropic thermoelastic (ISIWED) waves. In Figure 1, the graphical representation is given for the variation of amplitude ratios ∣Z₁∣, ∣Z₂∣ and ∣Z₃∣ for incident qLD-wave. Figures 2 and 3, respectively, represent similar situations, when qTD and qT waves are incident.

Figure 1

The variation of amplitude ratios of ∣Z_i∣ (i=1, 2, 3) with frequency for incident qLD-wave.

Figure 2

The variation of amplitude ratios of ∣Z_i∣ (i=1, 2, 3) with frequency for incident qT-wave.

Figure 3

The variation of amplitude ratios of ∣Z_i∣ (i=1, 2, 3) with frequency for incident qTD-wave.

Here ∣Z₁∣, ∣Z₂∣ and ∣Z₃∣ are the amplitude ratios of reflected qLD, qTD and qT waves, respectively. These variations are shown for two angles of incidence viz, θ=30°, 45°. In these figures the solid curves lines correspond to the case of ISTIWED, while broken curves correspond to the case of ISIWED. Moreover, the curves without the center symbol correspond to the case when θ=30°, and those with the center symbol (–o–o–) correspond to the case of θ=45°.

8 Conclusion

Reflection of waves from the free surface of an initially stressed transversely isotropic medium in the context of the G-N thermoelasticity theory of types II and III has been discussed. The appreciable effect of anisotropy and angle of incidence is depicted on amplitude ratios for various reflected waves. It can be concluded from the graphs that the amplitude ratio ∣Z₁∣ exhibits higher values because of anisotropy for all three types of incident waves (viz., qLD, qTD, qT), whereas the amplitude ratios ∣Z₂∣, ∣Z₃∣ shows oscillating behavior.