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This paper is devoted to the study of disturbances due to impact and continuous strip thermal sources, tem- perature or temperature gradient input acting on the rigidly fixed and charge free(open circuit) surface of a homogeneous, transversely isotropic, thermally conducting, generalized piezothermoelastic half-space. The Laplace and Fourier transforms technique have been employed to solve the model consisting of partial dif- ferential equations and boundary conditions in the transformed domain. In order to obtain the results in the physical domain the quadratic complex polynomial characteristic equation corresponding to the associated system of coupled ordinary differential equations has been solved by using DesCartes’ algorithm with the help of irreducible Cardano’s method. The inverse transform integrals are evaluated by using numerical technique consisting of Fourier series approximation and Romberg integration. The temperature change, stresses and electric potential so obtained in the physical domain are computed numerically and presented graphically for cadmium selenide (CdSe) material. The study may find applications in smart structures, pie- zoelectric filters, resonators, transducers, sensing devices and vibration control.

Application of different types of loads on the surface of piezoelectric materials is an active research subject for engineers and scientists. Smart and intelligent structures are developed to enhance the performance of the structural components. In some cases, the load bearing substrates of these smart structures are made of composite materials. Ashida et al. [

The theory of coupling of thermal and strain fields gives rise to coupled thermoelasticity and was formulated by Duhamel [

Harinath [14,15] considered the problem of surface point and line source over a homogeneous isotropic generalized thermoelastic halfspace. Majhi [

The present paper deals with the distribution of temperature change, stresses and electric potential in a generalized piezo-thermoelastic (6 mm class) material halfspace due to impact and continuous strip thermal sources acting on its surface. A combination of the Laplace and Fourier integral transforms has been used to solve the problem in the transform domain. The results in the physical domain are attained with the help of a numerical technique for inverting the integral transforms [

We consider a homogeneous, transversely isotropic, thermally conducting generalized piezothermoelastic halfspace which is initially at uniform temperature. We takeaxis along the poling direction and also as sume that the medium is transversely isotropic in the sense that the planes of isotropy are perpendicular to the axis. We take origin of the co-ordinate system at any point on the plane surface and axis pointing vertically downward into the halfspace, which is thus represented by. It is assumed that an impact/continuous strip thermal source is acting at the rigidly fixed surface of the medium as shown in the

and are the electrical displacement and electric potential, respectively. The superposed dot denotes time

derivatives and coma notation is used for spatial derivatives.

Where we have defined and used the quantities

The primes have been suppressed for convenience. Here, and, are respectively, the coefficients of linear thermal expansion and thermal conductivity, in the direction orthogonal to the axis of symmetry and along the axis of symmetry; and are the mass density and specific heat at constant strain, respectively; is the thermoelastic coupling constant; is the characteristic frequency of the medium; is piezothermoelastic coupling constant; are elastic parameters; are piezoelectric constants;, are the electric permittivities perpendicular and along the axis of symmetry; is pyroelectric constant in direction; and are respectively denote stresses and electrical displacement;, are the thermal relaxation time parameters and is the longitudinal wave velocity in the medium. The symbol is Kronecker’s delta in which corresponds to the Lord-Shulman (LS) and refers to the Green-Lindsay (GL) theories of thermoelasticity. The thermal relaxation time parameters and satisfy the inequalities

in case of GL theory only. However, it has been proved by Strunin [

The following initial and regularity conditions are assumed to be satisfied:

In addition to above boundary conditions, the surface of the piezothermoelastic solid is subjected to time dependant strip thermal sources (impact or continuous) in the region and assumed to be rigidly fixed and charge free (open circuit). Therefore, the corresponding boundary conditions are given as Rigidly fixed and open circuit:

Temperature input (TI):

Temperature gradient (TG):

where, and, the prime has been suppressed. Here the function is a well behaved function of time and is defined as

where is a Heaviside unit step function, denotes the Dirac delta function.

In order to solve the problem we apply Laplace transform with respect to time ‘t’ and Fourier transform with respect to x defined by Churchill [

Upon operating Transformations (13) and (14) on the system of Equations (1) to (4), we obtain

where, , ,

The above coupled system of ordinary differential Equations (15-18) upon retaining that part of the solution which satisfies the radiation condition (j = 1, 2, 3, 4) leads to the following formal transformed solution

where, and are the amplitude ratios, obtained as

, , (20)

and the characteristic roots are given by the relations

Here the quantities F, and, are defined in the Appendix. Upon using Solution (19) in the Equations (5-8), the transformed stresses and electric displacement are obtained as

where

Upon applying integral transforms (13) and (14) to the boundary conditions (12) and using the Solution (22), we obtain a nonhomogeneous system of linear algebraic equations in the unknowns for each set of conditions, TI or TG.

After solving the above system of equations we obtain

where

and can be written from by replacing the permutation of suffixes (2, 3, 4) in and with (1, 3, 4), (1, 2, 4) and (1, 2, 3) respectively.

Thus the transformed solutions of various field functions such as displacements, temperature change, stresses, electric potential and electric displacement can be obtained from Equations (19) and (22) upon solving the values of from Equation (24) in case of thermal loads (TI/TG) under the considered electrical and mechanical conditions prevailing at the surface of the halfspace.

Due to the existence of damping term in Equations (1-4) the dependence of characteristic roots on the integral transform parameters and is complicated. Hence analytically inversion of integral transform is difficult and cumbersome because the isolation of and q is not easily possible. This difficulty, however, can be overcome if we use some approximate or numerical methods. Therefore, in order to obtain the solution of the instant problem in the physical domain, we invert the integral transforms in Equations (19) and (22) by using a numerical technique [

The expressions for various transformed field functions can formally be expressed as a function of, and of the form. Upon inverting the Fourier transform, we get

where and respectively, denote the even and odd parts of the function with respect to. For fixed values of, and, the function inside the braces in Equation (26) can be considered as a Laplace transform of some function. Using the inversion formula for Laplace transform [

where is an arbitrary real number greater than the real parts of the singularities of. Taking, the above Integral (27) takes the form

Expanding the function in Fourier series in the interval, the approximate Formula (28) becomes

where

is the discretisation error which can be made arbitrarily small by choosing large enough. Since the infinite series in Equation (29) can be summed up to a finite number (N) of terms, the approximate value of h (t) becomes

While using Formula (31) to evaluate, we also introduce a truncation error that must be added to the discretisation error to produce the total approximation error. In order to accelerate the process of convergence of the solution, the “Korrecktur” method is used to reduce the discretisation error and the algorithm is employed to reduce the truncation error. The Korrecktur formula provides us where and. Thus, the approximate value of becomes

where is an integer such that. We shall now describe the algorithm that is used to accelerate the convergence of the series in Equation (31). Let N be an odd natural number and let be the sequence of partial sums of Equation (31). We define the sequence by

, ,;

It can be shown that the sequence converges to faster than the sequence of partial sums (m = 1, 2, 3,). The actual procedure used to invert the Laplace transforms consists of using Equation (29) together with the -algorithm. The values of and are chosen according to the criteria outlined by Honig and Hirdes [

The last step in the inversion process is to evaluate the Integral (26). According to Bradie [

In order to illustrate and compare the theoretical results obtained in the previous sections, in the context of LS, GL and CT theories of thermoelasticity, we now present some numerical results. The material for the purpose of numerical calculations is taken as cadmium selenide (CdSe) having hexagonal symmetry (6 mm class) and belongs to the class of transversely isotropic material. The physical data for a single crystal of CdSe material is given below [

, , , , , , , , , , , , , , , , , , , , , , ,

The value of thermal relaxation time parameter has been estimated from the relation, see Chandrasekharaiah [

the existence of wave-front and finite speed of heat propagation. It is also revealed that the magnitude of temperature change due to impact TI is signifycantly large as compared to that for continuous one. The various curves are quite distinguishable due to significant effect of thermal relaxation time. It is also observed that temperature change has a non-zero value only in a particular region of the halfspace and outside that region its values almost vanish identically which means that no thermal disturbance can be felt outside that particular region. On comparing the results of temperature change for three different theories of thermoelasticity, it is observed that for both impact and continuous load.

It is observed from

The comparison of Figures 2-9 reveals that the magnitude of temperature change and electric potential interlace according as in case of TI load and these trends get reversed for TG load with the exception that the variation of electric potential follows the trend periodically in the latter case. The variations of vertical and shear stresses follow the inequalities for TI load and for TG load except that the inequalities get reversed for shear stress in the latter case.

The present analysis and the used values of parameters lead to following conclusions:

1) All the considered field parameters are noticed to be quite large near the vicinity of thermal sources and decrease with increasing epicentral distance to ultimately vanish at certain value of epicentral distance under both types of impact or continuous thermal loads (TI/TG). This ascertained the existence of wave fronts and hence finite speed of heat propagation.

2) The profiles of temperature change with epicentral distance show that this quantity has a non-zero value in certain region of the halfspace and almost identically zero outside that region. This means that no thermal disturbance is felt outside this particular region. Similar behavior is also noticed from the profiles of the other considered functions viz. stresses and electric displacement.

3) Significant effect of thermal relaxation times has been observed on the profiles of various considered functions in the CdSe material because all the profiles of considered functions are quite distinguishable. Hence the results for all the considered field parameters show the difference between the three different theories of thermoelasticity namely CT, LS and GL.

4) It is also observed that the magnitude of all the field functions due to impact thermal loads are quite large as compared to that in case of continuous one almost at a particular epicentral distance.

5) The shear stress development is very small as compared to the vertical stress for both types of thermal loads. It means that in addition to thermal wave, vertical stress wave carries the major portion of energy and meager amount propagate in the form of shear stress wave, which is consistent with the boundary conditions.

6) The temperature change and electric potential interlace according to the inequalities

for TI load and these trends get reversed for TG load with periodic variations in case of electric potential in the latter case.

7) The magnitudes of vertical and shear stresses obey the inequalities for TI load and for TG load with some variations in the magnitude of shear stress in the latter case.

The authors are thankful to the reviewers for their useful suggestions for the improvement of this work. The author (JNS) is also thankful to The CSIR New Delhi for providing financial assistance via scheme No. 25(0184) EMR-II.

The coefficients, in Equation (20) and, in equation (21) are obtained as

where

= Mass density

= Specific heat at constant strain

= Thermoelastic coupling constant

= Characteristic frequency

= Piezothermoelastic coupling constant

= Thermal conductivity along orthogonal to the axis of symmetry

= Thermal conductivity along the axis of symmetry

= Elastic parameters

= Piezoelectric constants

= Electric permittivity perpendicular to the axis of symmetry

= Electric permittivity along the axis of symmetry

= Pyroelectric constant

= Stresses

= Electrical displacement

= Electric potential

= Temperature change

= Kronecker’s delta

, Longitudinal wave velocity in the medium

, Displacement vector