Access the full text.

Sign up today, get DeepDyve free for 14 days.

Jeyaraman, Poongkothai; Mahesh, Samydurai; Selvamani, Rajendran; Dimitri, Rossana; Tornabene, Francesco

Curved and Layered Structures
, Volume 9 (1): 11 – Jan 1, 2022

/lp/de-gruyter/multi-thermal-waves-in-a-thermo-diffusive-piezo-electric-functionally-A66VkplExC

- Publisher
- de Gruyter
- Copyright
- © 2022 Poongkothai Jeyaraman et al., published by De Gruyter
- eISSN
- 2353-7396
- DOI
- 10.1515/cls-2022-0010
- Publisher site
- See Article on Publisher Site

Curved and Layer. Struct. 2022; 9:105–115 Research Article Poongkothai Jayaraman, Samydurai Mahesh, Rajendran Selvamani*, Rossana Dimitri, and Francesco Tornabene Multi thermal waves in a thermo diffusive piezo electric functionally graded rod via refined multi-dual phase-lag model https://doi.org/10.1515/cls-2022-0010 pling effect of thermoelastic and electric fields plays cru- Received Jul 24, 2021; accepted Nov 29, 2021 cial role in piezo-thermoelastic materials and finds applica- tion as sensors/actuators for automatic fire control systems, Abstract: In the present work, a novel analytical model infra-red (IR) detectors and devices for intruder alarming, is provided for wave dispersion in a piezo-thermoelastic thermal imaging and geographical mapping [1]. They also diffusive functionally graded rod through the multi-phase have applications in petrochemical plants, military vessels, lag model and thermal activation. The plain strain model tunnels, underground construction, solar towers, chim- for thermo piezoelectric functionally graded rod is consid- neys, and boilers. The sensors/ actuators are often used ered. The complex characteristic equations are obtained by in the form of thin disk, hollow cylinder/rod and spheri- using normal mode method which satisfies the nonlinear cal shell structures for their best performance. When sen- boundary conditions of piezo-thermoelastic functionally sors/actuators designed by piezo-thermoelastic materials graded rod. The numerical calculations are carried out for are used in thermomechanical/electromechanical environ- copper material. The results of the variants stress, mechan- ment the distribution of the displacement or electric poten- ical displacement, temperature and electric distribution, tial could be controlled effectively by applying the concept frequency are explored against the geometric parameters of functionally grading of piezo-thermoelastic materials. and some special parameters graded index, concentration The advantage of functionally graded material (FGM) is, it constants are shown graphically. The observed results will has continuously graded properties due to spatially vary- be discuss elaborate. The results can be build reasonable ing microstructures produced by nonuniform distributions attention in piezo-thermoelastic materials and smart mate- of the reinforcement phase as well as by interchanging rials industry. the role of reinforcement and matrix (base) materials in a Keywords: Thermoelastic diffusion, generalized piezother- continuous manner. The continuous variation of proper- moelasticity, FG rod, RPL model, wave dispersion ties may offer advantages such as local reduction of stress concentration, increased efficiency and increased bonding strength [2–4]. Functionally graded materials and struc- tures have potential applications in space planes, rocket 1 Introduction engine components, dentals, orthopedic implants, armour plates, bullet-proof vests, thermal barrier coatings, opto- Piezo-thermoelastic materials are smart materials, which electronic devices, automobile engine components, nuclear responds to mechanical, thermal and electric loads and reactor components, turbine blades and heat exchanger [5– exhibits their coupling effect simultaneously. This cou- 10]. Literatures related to piezothermelastic materials, its functionally graded structures and their thermal diffu- *Corresponding Author: Rajendran Selvamani: Department of sion problems are presented here for understanding the mathematics, Karunya Institute of Technology and Sciences, Coim- importance of present work. Mindlin [11] carried out de- batore, 641114, Tamilnadu, India; Email: selvamani@karunya.edu tailed investigation on high frequency vibrations of a ther- Poongkothai Jayaraman: Department of Mathematics, Govern- mopiezoelectric plate by incorporating the coupling of elas- ment Arts College, Udumalpet, Tamil Nadu, India tic, electric and thermal fields in integral energy balance Samydurai Mahesh: Department of Mathematics, V.S.B. Engineer- equation and obtained a uniqueness theorem to estab- ing College, Karur, Tamil Nadu, India Rossana Dimitri, Francesco Tornabene: Department of Innova- lish various face and edge-conditions sufficient to assure tion Engineering, University of Salento, Lecce, Italy Open Access. © 2022 P. Jayaraman et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License 106 | P. Jayaraman et al. unique solutions of the two-dimensional equations. Chan- functionally graded material based on the dual-phase-lag drasekharaiah [12] developed a generalized linear theory theory to reveal the role of microstructural interactions for thermo piezoelectric materials by combining the thermo in the fast transient process of heat conduction. They ob- elasticity theory of Lebon [13] and the conventional thermo- tained exact expressions for the thermal wave speed in piezoelectricity theory of Mindlin [14]. He obtained an equa- one-dimensional, functionally graded material with differ- tion of energy balance and a theorem on the uniqueness ent geometries based on the dual-phase-lag and hyperbolic of solution and explained finite speed for thermal signals heat conduction theories. The distribution of transient tem- in linear thermopiezoelectric materials. Rao and Sunar [15] perature for various types of dynamic thermal loads was have used finite element method (FEM) to analyse the ther- also obtained. Abouelregal and Zenkour [23] carried out mal effect in thermopiezoelectric materials and its appli- detailed analysis on the vibrational characteristics of a func- cation to distributed dynamic measurement and active vi- tionally graded (FG) microbeam subjected to a ramp-type bration control of advanced intelligent structures by. It is heating in the context of the dual phase lag model. They found that the thermal effects have an impact on the per- reported the effect of ramping time on the lateral vibration, formance of a distributed control system. The degree of temperature, displacement, stress, moment and the strain impact may vary depending on the piezoelectric material, energy of the FG micro beam. the environment where the system operates, and the mag- The study of thermal diffusion problems in piezother- nitude of the feedback voltage. Selvamani et al. [16] studied moelastic materials has many applications in industrial the influence of hygro, thermal and piezo fields in a func- engineering, thermal power plants, submarine structures, tionally graded piezoelectric rod using three-dimensional pressure vessel, aerospace, chemical engineering and met- elasticity equation in linear form. For the formulation of allurgy. When piezothermoelastic materials are involved the problem, it was assumed that the stiffness and thermal in severe thermal environments, thermal stress and diffu- conductivity of the material transport via the radial coor- sion may cause damages and modifications in functioning dinate. The numerical results showed the effect of hygro, of the structure. Therefore it is necessary to analyse the thermal and piezo fields in the physical variables via grad- electric field and deformations induced by thermal loading ing values and moisture constants. Poongkothai et al. [17] in piezothermoelastic materials which will provide insight carried out analysis on thermo-electro effects on the disper- for proper design of functionally graded structures. Elha- sion of functionally graded piezo electric rod coupled with gary [24] has done extensive research on thermal diffusion inviscid uid fl via three-dimensional elasticity equation in in a two-dimensional thermoelastic thick plate subject to linear form. Dube et al. [18] considered the effect of pressure, laser heating within the context of the theory of generalized thermal and electrostatic excitation on an infinitely long, thermoelastic diffusion with one relaxation time. Kumar simply-supported, orthotropic, piezoelectric, flat panel in and Gupta [25] analysed reflected and transmitted waves re- cylindrical bending and offered exact solution. They used sulting from an interface between inviscid uid fl half-space Fourier series to expand displacement, electric potential and a thermoelastic diffusion solid half-space using dual- and temperature fields to satisfy boundary conditions at phase-lag heat transfer (DPLT) and dual-phase-lag diffu- the longitudinal edges. Ding et al. [19] reported numerical sion (DPLD) models. They demonstrated that the amplitude results for a spherically symmetric thermoelastic problem ratios and energy ratios of various reflected and transmitted of a functionally graded pyroelectric hollow sphere. Ootao waves are functions of angle of incidence, frequency of the and Tanigawa [20] supplied uniform thermal load to a func- incident wave and the thermoelastic diffusion properties of tionally graded thermopiezoelectric hollow sphere to study media concerned. Tripathi et al. [26] considered continuous the transient piezothermoelastic response. They obtained supply of axisymmetric heat and an internally generated one-dimensional solution for the temperature change in a heat to a two dimensional thick circular plate of infinite ex- transient state which explained the piezothermoelastic re- tent and finite thickness within the framework of classical sponse of a functionally graded thermopiezoelectric hollow coupled, L-S and G-L theory. They obtained exact solutions sphere. Wu and Huang [21] developed a modified Pagano for temperature distribution, displacement and the stress method for a three-dimensional (3D) coupled analysis of a components in the Laplace transform domain. Accounting simply-supported, doubly curved functionally graded (FG) internally generated heat is significant because engineer- piezo-thermo-elastic shells under thermal loads. They stud- ing components thick-walled pressure vessels, such as a ied effect of the material-property gradient index on the nuclear containment vessel, a cylindrical roller etc. gener- through-thickness distributions of various variables in the ates internal heat, which influences the performance of the thermal, electric, and mechanical fields. Akbarzadeh and component. Chen [22] investigated heat conduction in one-dimensional Multi thermal waves in a thermo diffusive piezo electric functionally graded rod | 107 Abbas et al. [27] investigated on deformation produced sion problems investigated so for in thermoelastic materi- in a micro polar thermoelastic diffusion medium due to ther- als have not included piezoelectric equation. The present mal source by the use of finite element method (FEM) in the paper is dedicated to investigation on thermal diffusion framework of Lord-Shulman (L-S) theory of thermoelastic- behaviour of a one dimensional functionally graded piezo- ity. They obtained the components of displacement, stress, thermoelastic cylindrical rod using refined multi-phase- microrotation, temperature change and mass concentra- lags (RPL) theory. The normal mode method will be used tion using numerical method and showed the impact of mi- to obtain the exact solution of the coupled thermo elastic cropolarity, diffusion and relaxation times on the obtained equations. The analytical expressions for these coupled components graphically. He et al. [28] studied the effect of equations would be the components of mechanical dis- moving heat source on the dynamic thermal and elastic placement, temperature, electric displacement, electric responses of a piezoelectric rod fixed at both ends in the potential, and stresses. Numerical results for these field context of Lord and Shulman generalized thermo-elastic quantities are tabulated and illustrated graphically. theory with one relaxation time. From the results it is found that the temperature, thermally induced displacement and stress of the rod are decrease at large source speed. Babaei 2 Basic equation and formulation of and Chen [29] investigated thermo-piezoelectricity problem of a one-dimensional (1-D), finite length, and functionally the plain stress graded medium excited by a moving heat source using the Lord and Shulman theory of generalized coupled thermo We consider a cylindrical rod problem. It is assumed to be elasticity. Ma and He [30] worked on the dynamic response made up of functionally graded (inhomogeneous) trans- of a piezoelectric-thermo elastic rod made of piezoelectric versely isotropic, thermo elastic medium within a uniform ⃒ ⃒ ⃒ ⃒ T−T ( 0 ) ceramics (PZT-4) subjected to a moving heat source within temperature ≪ 1 and initial concentration C , in ⃒ ⃒ the context of the fractional order theory of thermo elastic- the undistributed state occupying the domain R ≤ r ≤ ∞, ity. They investigated the effects of fractional order param- whose state can be expressed in terms of the space vari- eter and the velocity of heat source on the variations of the able r and the time t so that all the field functions vanish considered variables and the results showed that they have at infinity. We use cylindrical coordinates ( r, θ, z) are con- significant influence on the variations of the considered sidered with origin at θ = 0, and z – axis is setting along variables. with cylindrical axis. The density, elastic parameters and Othman et al. [31] studied the effect of the gravity field thermal conductivity of the material have been assumed to and the diffusion on a micro polar thermo elastic medium vary through the thickness according to simple power law with dependence on the temperature properties. Further in radial coordinate as under compressions were showed graphically in the presence and (︁ )︁ (︁ )︁ α α r r ρ = ρ , C = C , (1) 0 11 the absence of the gravity, the temperature-dependent prop- 11 a a (︁ )︁ (︁ )︁ (︁ )︁ α α α erties, the diffusion and the micro polar in the context of r r r 0 0 0 C = C , C = C , K = K 12 12 13 13 1 1 two types of Green-Naghdi (G-N) theory II and III. Zenk- a a a our [32] recently presented a refined multi-phase-lags the- Here ory to investigate the thermo elastic response of a gravitated 0 0 0 piezo-thermoelastic half-space. He obtained all fields like C = λ + 2µ , C = λ = C , 11 12 13 0 0 0 0 displacement, temperature, electric potential and thermo β = β = β , K = K = K 1 3 1 3 mechanical stress and demonstrated their dependency on where the exponent α represents the grading index of the the inclusion of gravity. Zenkour and Kutbi [33] developed a 0 0 0 material and λ, µ are Lame’s parameter’s, C , C , C , novel multi-phase-lag model to study the thermoelastic dif- 11 12 13 0 0 0 0 β , β , K , K are the homogeneous counterparts of the fusion behaviour of a one-dimensional half-space. For the 1 3 1 3 respective quantities. formation of the problem, they considered additional equa- The basic covering field equations and constitutive rela- tion for heat of mass diffusion and additional constitutive tions of generalized hexagonal piezo – thermo elastic with equation for the chemical potential. thermo diffusive for two-dimensional motion in r-z plane From the literature it is found that thermal diffu- are [34] sion is discussed elaborately in thermoelastic materials and piezoelectric materials. However in the case of piezo- 1. The Strain- displacement- relations thermoelastic materials thermal problems are is not dis- (︀ )︀ cussed comprehensively. This means, the thermal diffu- 2ε = u + u , i, j = 1, 2 (2) ij i,j j,i 108 | P. Jayaraman et al. 2. The Stress-Strain-Piezo-Temperature-Concentration thermal memories in which ι is the multi-phase-lag (MPL) of the heat ux, fl 0 ≤ ι < ι , and ι is the PL of temperature ( q ) θ θ σ = c ε − e E (3) ij ijkl kl kij k gradient. Here we apply classical thermo elasticity (CTE) (︀ )︀ − £ 𝛾 (T − T ) + b C theory is appeared by omitting all relaxation times, i.e., ij 0 ij ι = ι = ι . 0i D = e ε + ε E + p θ θ i ijk jk ij j i The L-S model will be appearing when ι = ι = ι = 0 0i ρT S = ρC T + aT C + 𝛾 T ε + p E 0 e 0 0 ij ij i i and ι = ι > 0. Also, The G-L model will be appearing 11 22 when ι = ι = ι = ι = 0 and ι = ι > 0. Where E = −ϕ , i, j, k = 1, 2, 3 01 02 03 11 i ,i So, the simple dual-phase-lag (SPL) model will be given 3. The Chemical-strain-temperature- diffusion relation by setting ι = ι , ι = ι = 0, ι = ι and omitting the 01 0 02 03 1j P = −b ε + bC − aT j, k, l = 1, 2, 3., (4) summations in Eq. (8). That is kj kl 4. The equation of motion 1 k £ = 1 + ι , £ = 1, (9) θ θ θ ∂t σ = ρu ¨ i, j = 1, 2, 3, (5) ∂ ∂ ij j £ = + ι , k = 2, 3, 4, 5 ∂t ∂t 5. The equation of Heat conduction in the RPL model Now, the refined multi-phase-lag (RPL) theory is given (︀ )︀ 1 1 by settingℜ = s = N > 1,ℜ = ℜ = N − 1. £ K T, = £ ρC T + aT C (6) 1 j 2 3 ( e 0 ) θ ij j q ,i (︀ )︀ That is + 𝛾 £ u + p E , i, j = 1, 2, 3 ij q i,j i i ∑︁ r i ι ∂ £ = 1 + , (10) 6. The equation of conservation of mass-diffusion r=1 r! ∂t (︃ )︃ s+1 s+1 (︂ )︂ ∑︁ ∂ ∂ j ∂ j q (︀ )︀ ∂c 1 2 £ = 1 + ι + £ = D P , i, j = 1, 2, 3 (7) s+1 ij ,j ∂t ∂t s=1 s + 1 ! ∂t θ ( ) ,i ∂t b (︂ )︂ ∂ m ∂ m k 1 2 £ = + + Where σ are the components of the stress tensor, ε ij ij 2 2 ∂r r ∂r r are the components of the strain tensor, e represent kij For N = 1, we set the components of piezoelectric tensor, D , E are the i i electric displacement, u(r, t) are the displacement 1 k £ = 1 + ι , £ = 1, (11) vector, T is the absolute temperature of the medium, θ θ θ ∂t T is the uniform temperature of the medium, c 2 ijkl ∂ ∂ 1 2 £ = £ = + ι , k = 2, 3, 4, 5 q q are the elastic parameter of the anisotropic medium 2 ∂t ∂t and 𝛾 is tensor moduli, K are the thermal conduc- ij ij The value of N may be reaches 5 or more according to the tivity components, P denotes the chemical potential refined multi-phase-lag (RPL) theory required. per unit mass, C is the concentration of the diffusive For an isotropic medium [38], we have material in the elastic body, D are the diffusion co- ij efficient, a is the thermo diffusive constant, b is the 𝛾 = β δ , b = β δ , D = Dδ , (12) 1 2 ij ij ij ij ij ij diffusive constant, C is the specific heat at constant C = C ε + C ε − £ (β (T − T ) + β C) 11 r 12 1 0 2 ijkl θ θ strain, S is the entropy per unit mass, ρ reference material of the density. We assume that the material Where C and C are the grading elastic parameters and 11 12 parameters satisfy the inequalities K, λ, µ , D, T, T , β , β are the material constants given by 1 2 C > 0. β = 3λ + 2µ α and β = 3λ + 2µ α , α is the co- ( ) ( ) c 1 t 2 t efficient of linear thermal expansion, α is the coefficient In the foregoing relations, the parameters i i of linear diffusion expansion D. Represents the diffusion £ i = 1, 2, 3 and £ i = 1, 2 can be expressed as ( ) ( ) coefficient and δ is Kronecker’s delta. ij r+1 r+1 ∑︁ ℜ −1 ∂ i ι ∂ 0 Then the constitutive Eqs. (4) and (6) can be expressed £ = 1 + ι + , (8) θ 0i r+1 ∂t r−1 r + 1 ! ∂t ( ) as (︃ )︃ s+1 s+1 ∑︁ j ι ∂ ∂ ∂ j q (︀ )︀ £ = 1 + ι + , q 1j s+1 σ = c ε + c ε − £ β (T − T ) + β C + e E (13) 11 r 12 1 0 2 s=1 ij θ θ ij ij ∂t ∂t (s + 1)! ∂t Here θ denotes the temperature change of a material P = −β ε + bC − aT (14) 2 kk particle, The parameters ι , ι , ι and ι represent the 0i 1j 0 Multi thermal waves in a thermo diffusive piezo electric functionally graded rod | 109 Now, let us consider a cylindrical rod whose coordinates are Also, The Electrostatics equations can be reduced to r, θ, z in the context of the MPL model. All the functions ( ) 2 2 ∂ u ∂ are depending on the time t and the coordinate r and will e + e + e T − N E (23) ( ) ( ) 31 15 15 11 ∂r∂z ∂r be assuming to vanish as r → ∞. ∂ ∂ T − T ( ) + e T − N E + p For the axisymmetric plane strain, The following dis- ( 33 33 ) 3 ∂z ∂z placement components u r, t are considered as; ( ) Further, The Heat conductive equation is reduced to (︂ )︂ i.e., u = u r, t , u = u = 0 (15) ( ) 1 2 3 2 ∂ T m ∂T 1 1 1 K£ + = £ [ρC T + aT C] (24) q e 0 ∂x r ∂r Also, The strain displacement relations are (︂ )︂ ∂u + β T £ + p E 1 0 q 3 ∂u u ∂x e = , e = , e = 0, (16) 11 22 33 ∂r r The Substitution of Eq. (14) into Eq. (17) yields the diffusion e = 0, e = 0, e = 0 13 12 23 equation as Then, the Equations of heat transport and heat of mass (︂ )︂ ∂C ∂ diffusion can be expressed as b£ = D bc − β e − aT , (25) ( ) ∂t ∂r 1 2 1 K£ ∇ θ = £ ρC T + aT C (17) q ( e 0 ) Where e = ε , dilatation. For our convenience, we intro- kk (︂ )︂ ∂u 2 duce the following non-dimensionless variables in the next + β T £ + p E 1 0 q 3 ∂x part {︀ }︀ (︂ )︂ ′ ′ ′ x , u = C η{x, u} ∂C D ∂ P £ = (18) {︀ }︀ {︀ }︀ 2 ′ ′ ′ ′ 2 ∂t b ∂x t , ι , ι , ι = ηC t , ι , ι , ι q 1 q 1 0i θ 0 1j θ Keeping in view the equalities of these material parameters β (T − T ) β C ′ 1 0 ′ 2 θ = , C = 2 2 ′ ′ and using relations (1) and (2). ρ C ρ C 0 0 σ σ u r Then the relation of the constitutive Eqs. (3) can be * 11 * 22 σ = , σ = , U = , x = , 11 22 0 0 a a C C reduced to: 11 11 [︃ ]︃ e = e = e , N = N , T = T − T 13 33 15 11 33 ( 0) (︁ )︁ ∂u u α ¯ r + C − 0 0 ∂x ¯ σ = c (︀ )︀ + e E (19) 11 11 13 1 ¯ ¯ £ β (T − T ) + β C 1 0 2 β (T − T ) β T β C ′ 1 0 1 0 ′ 2 θ = , β = , C = , (26) 1 1 2 2 ′ ′ λ + 2µ p C p C ⎡ ⎤ 0 0 ∂u u C + − (︁ )︁ ∂x λ + 2µ β C P (︃ )︃ ( ) α 2 2 0 ′ r ¯ ⎢ ⎥ 0 C = β = , P = ¯ ¯ 0 2 σ = σ = c + e E (20) ⎣ β (T − T ) ⎦ 22 33 11 1 0 33 1 ρ λ + 2µ β +β C ρC β T e 0 η = , ε ¯ = , e = e 13 33 K ρC λ + 2µ e ( ) σ = σ = σ = 0 All the governing equations, with above non-dimensionless 12 23 31 variables are reduced to C β β 1 2 ¯ 12 ¯ ¯ Here C = , β = , β = 0 0 1 0 2 0 ∂u C C C 11 11 11 e = , The Electric field displacement can be simplified as ∂x [︂ ]︂ D = e u + ω − N E (21) ( ,z ,x ) x 1 15 11 (︀ )︀ ∂U U ϑ 3 ′ ′ σ = X + S − £ θ + C + e E (27) 11 1 1 33 1 D = e u + e ω − N E + p T − T ,x ,z z ( ) ∂x x 3 31 33 33 3 0 [︂ ]︂ Also, using Eqs. (1–2) in Eqs. (3) simplifying, we get the (︀ )︀ ∂U U ϑ 3 ′ ′ equation of motion can be expressed as σ = X S + − £ θ + C + e E (28) 1 33 1 22 θ 1 ∂x x [︂ ]︂ k 3 £ [u] − £ (β (T − T ) + β C) (22) q 1 0 2 θ [︂ ]︂ ∂r (︀(︀ )︀)︀ k 3 ′ ′ [︂ (︂ )︂]︂ £ U − £ θ + C (29) [ ] 2 q θ 1 ¯ ¯ ∂E αE ∂ u ∂x 4 1z 1z + £ e ¯ + = ρ θ [︂ (︂ )︂]︂ ∂r r ∂t 2 ¯ ¯ ∂E αE ∂ U 4 1z 1z + £ e ¯ + = ∂x x ∂t 110 | P. Jayaraman et al. (︁ )︁ (︂ )︂ 2 ′ ′ [︀ ]︀ * * * * * * ∂ θ m ∂θ 1 1 1 ′ ′ ς e + ω + υ ς − φ = 0 1 2 £ + = £ θ + ε ¯ S C (30) q T 2 1 x ∂x ∂x * * * (︀ )︀ P = S C − e − S ω 2 4 2 + ε ¯ £ e + p E T q 3 Taking the divergence of Eq. (28) and using Eqs. (37) (︂ )︂ [︂ (︂ )︂]︂ 2 2 2 and (38). ∂ U ∂ ∂ * 5 * ς + £ ς + = p E (31) 1 2 3 2 2 We obtain ∂x∂z ∂x ∂z [︁ ]︁ (︁ )︁ 2 2 * * * * (︂ )︂ D − h (t) e − h (t) D ω + υ + φ = 0 (38) 1 2 (︀ )︀ ∂C ∂ 2 ′ ′ S £ − S C − eS − θ , (32) 3 4 1 2 ∂t ∂x And the equations of heat conduction, Electric displace- ment, and mass diffusion became ′ ′ P = S C − S θ − e (33) [︃ ]︃ 4 1 2 [︁ ]︁ h t e + ( ) 2 * 4 * D − h t ω + ε ¯ + h t φ = 0 (39) ( ) ( ) 3 T 6 Where S h t υ ( ) 2 3 (︂ )︂ 0 0 C aC ϑ ϑ 12 11 S = X , S = X , (34) 1 2 [︁ ]︁ 0 0 0 C β β 2 2 2 * * * * 11 1 2 S D − S h (t) ϑ − S D ω − D e + h (t) φ (40) 4 3 5 2 7 0 0 bC bC ϑ ϑ 11 11 S = X , S = X , 3 4 2 2 = 0 β ηD β 2 2 * * ς = (e + e ) , ς = (e T − N E) , 1 31 15 2 33 11 [︃ ]︃ [︃ ]︃ 2 2 D + S h t D + ( ) * * 4 9 * ∂T β T h t φ + ω + ϑ (41) ¯ ( ) E = , ε ¯ = 0 S h t h t ( ) ( ) ∂Z 4 9 10 C ηD 2 * + h t D e = 0 ( ) Where 3 The solution of the problem D = , (42) dx [︂ ]︂ In this section, we apply the normal mode analysis, which 1 d h m dh 2m h k t 1 t 2 t h = £ + + , gives exact solutions without any assumed restrictions on 1 q 2 2 h h dt dt h t t the displacement, temperature, dilation, concentration, h = £ h , ( ) stress, and chemical potential. It is applied to a wide range 2 t [︂ (︂ )︂]︂ of problem in different branches Othman et al. [35–37] ¯ ¯ dE αE d h * 4 1z 1z t φ = £ e ¯ + − We will firstly apply the following initial conditions: 13 dt t dt [︂ ]︂ [︂ ]︂ 1 1 ∂U ∂θ 1 2 h = £ h , h = £ h , ( ) ( ) U (x, 0) = , θ (x, 0) = , (35) 3 q t 4 q t 1 1 ∂t ∂t £ h £ h ( t ) ( t ) t=0 t=0 θ θ (︂ )︂ [︂ ]︂ [︂ ]︂ 1 dh ∂C ∂ϕ 2 t h = £ C x, 0 = , ς x, 0 = − ( ) ( ) 5 h dt ∂t ∂t t=0 t=0 [︂ (︂ )︂]︂ d α The appropriate solution that satisfied the above initial h = £ e ¯ E + , 6 13 1z ∂t t condition in terms of the normal mode. (︀ )︀ h = ε ¯ £ e + p E , 7 T q 3 Of the following forms: {︁ }︁ m e + e n ( 15 31) {︀ }︀ ′ ′ * * * * h = − U, θ , C , ς x, t = u , ω , υ , φ x h t (36) ( ) ( ) ( ) t N e m h = − , Where h t will be chosen such that the temperature, di- 9 t ( ) latation, displacement and concentration and their deriva- 2 2 N m N m n n 11 33 h = − = − , tives should be t = 0. 10 N N 33 11 Then we gets p E 3 z h = − * * e = Du (37) [︁(︁ )︁ (︁ )︁ ]︁ * ϑ * * * * The system of equations appeared in Eqs. (38–41) can be σ = X e + S m − ω + υ h t + φ ( ) 11 1 2 expressed in eight-order ordinary homogeneous differen- [︁(︁ )︁ (︁ )︁ ]︁ * * ϑ * * * * * * * * σ = σ = X S e + m − ω + υ h t + φ ( ) tial equations in the amplitude u x , ω x , υ x , φ x 22 33 1 2 ( ) ( ) ( ) ( ) Multi thermal waves in a thermo diffusive piezo electric functionally graded rod | 111 1 2 3 4 which can be written as: The relations between the parameters C , C , C , C , can j j j j (︃ )︃ be obtained by using Eqs. (46) into 8 6 {︁(︁ )︁ }︁ D + E D + 3 {︁ }︁ * * * * u , ω , υ , φ x = 0 (43) ( ) 2 3 4 1 4 2 C , C , C = {R , R , R } C (51) E D + E D + E j j j 1n 2n 3n j 2 1 0 Where Eq. (43) can be factored as (︀ )︀ 2 2 (︁ )︁(︁ )︁(︁ )︁(︁ )︁ h m m + ε ¯ S h − h 2 n n T 2 3 3 2 2 2 2 2 2 2 2 R = [︀ ]︀ (52) 1n D − K D − K D − K D − K (44) 2 1 2 3 4 S h + h h m − S h h ε ¯ ( ) 2 3 2 4 n 2 1 3 T {︁ }︁ 4 2 * * * * m − m ε ¯ h h + h + h + h h ( 2 4 1 3) 1 3 u x , ω x , υ x , φ x = 0 n n T ( ) ( ) ( ) ( ) [︀ ]︀ R = 2n S h + h h m − S h h ε ¯ ( 2 3 2 4) 2 1 3 n T (︀ )︀ where K , K , K , K are the roots with positive real parts 4 4 1 2 3 4 h m m h h + ε ¯ S h + h h 8 n n 10 11 T 4 9 7 8 [︀ ]︀ R = 3n of the characteristic equation. (S h + h h ) m − S h h ε ¯ 2 7 2 4 2 8 3 n T 8 6 4 2 K − E K + E K − E K + E = 0 (45) Now, the final form of the displacement temperature com- n 3 n 2 n 1 n 0 ponents is given by Where the coefficient E , i = 0, 1, 2, 3 are given by {︁ }︁ * * * θ , e , C, φ x, t (53) ( ) 8 6 4 2 K − E K + E K − E K + E = 0 (46) n 3 n 2 n 1 n 0 ∑︁ 1 −m x S h h h h = h t C {1, R , R , R } e 3 1 3 5 9 t ( ) 1n 2n 3n E = S − h j=1 4 2 (︀ )︀ S + ε ¯ S h h + S + h h 4 1 3 ( 4 9) 10 T 2 By integrating Eqs. (47) , we get +S h ε ¯ h + h + h 3 5 ( 3 1 3) ∑︁ E = 1 −m x S − h 4 2 u x, t = h t C R e (54) ( ) t ( ) j 4n j=1 ε ¯ S h + h h ε ¯ S − 1 + ε ¯ h h S + S ( ) ( ) T 3 2 3 T 2 T 2 4 2 4 (︀ )︀ +S h + h + S h 2 ( ) 4 1 3 3 5 h m m + ε ¯ S h − h − h 2 n T 2 3 3 6 E = R = −[︀ ]︀ (55) 4n S − h 4 2 2 S h + h h m − S h h ε ¯ ( ) 2 3 2 4 n 2 1 3 T ε ¯ S h + h h h (ε ¯ S + 1) + ε ¯ h h (S − S ) T 3 8 9 10 T 2 T 10 11 3 4 Then, we get the final result of plane stress, displacement +S (h − h ) + S h 4 1 3 3 8 and chemical potential E = S − h [︃ (︃ )︃]︃ 4 2 ∑︁ R − 1n 1 ϑ −m x σ = C X e (56) The complete solution of Eq. (44) is given by 11 j S h 1 + R + h 1 2 ( 2n ) n=1 {︁ }︁ * * * * u x , ω x , υ x , φ x (47) ( ) ( ) ( ) ( ) σ = σ 22 33 [︃ (︃ )︃]︃ {︁ }︁ ∑︁ ∑︁ S R − 1 1n 1 2 3 4 −m x 1 ϑ −m x n n = C X e = C , C , C , C , e j j j j h 1 + R + h 2 ( 2n ) j=1 n=1 ∑︁ i 1 −m x Where C , i, j = 1, 2, 3, 4 are the integration parameters ( ) D = C h 1 + R − h e 1 [ 9 (( 3n ) 10)] j j and m , n = 1, 2, 3, 4 are the positive roots of the j=1 n ( ) Characteristic equation ∑︁ 1 −m x D = C h − h 1 + R + h e [( ( ) )] 3 8 10 3n 11 8 6 4 2 m + E m + E m + E m + E = 0 (48) j=1 n 3 n 2 n 1 n 0 Here √︁ (︀ )︀ 3.1 The Boundary conditions 1 2 2 2 m = √︀ 2E + E − 2E E − 12E − 4E (49) 1 3 3 4 1 2 4 6E In this section we used following boundary conditions for √︃ (︀ √ )︀(︀ )︀ 4E E − 4 1 ± i 3 2E − 3E 1 obtaining required results. 2 4 1 √ (︀ √ )︀ m = 2,3 2 3 − 1∓ i 3 E Further, 3.1.1 Mechanical conditions √︁ 3 3 E = 108E + 12E + 8E − 36E E (50) 4 0 5 1 2 The time dependent periodic force with magnitude σ is √︁ 0 (︀ )︀ (︀ )︀ 2 2 2 2 E = 81E + 6E E 2E − 9E + E 4E − E assumed to be acting normal direction on the medium. 5 0 2 1 1 0 2 1 2 112 | P. Jayaraman et al. 0 10 −2 That is C = 0.0105 × 10 Nm , 0 6 −2 −1 T = 0.850 K, β = 2.3620 × 10 Nm deg , 0 1 σ 0, t = σ 0, t = p h t = −σ (57) 11 ( ) 33 ( ) 0 t ( ) 0 13 −1 ω = 1.9890 × 10 S , 0 2 −1 −1 −3 K = 0.3000 × 10 wm deg , ρ = 0.1910 Kg m , 1 0 0 −3 3.1.2 Thermal conditions k = 2, ε = 0.04162, a = 10 m, 10 −2 10 −2 λ = 7.76 × 10 Nm , µ = 3.86 × 10 Nm , A thermal boundary condition on the surface of the half – −1 −1 −3 K = 386 wm k , ρ = 8954 kgm , space subjected to thermal shock −5 −1 −4 3 −1 α = 1.78 × 10 k , α = 1.98 × 10 m kg . t c θ 0, t = θ h t = θ (58) ( ) ( ) 0 t 0 For suitability, the absolute values of the following thermo elastic variables have been adopted to represent the results; 3.1.3 Electric conditions θ = 10θ x, t , e ¯ = 10e x, t , u ¯ = 10 u x, t , {σ , σ } = ( ) ( ) ( ) 1 2 ¯ ¯ 10{σ , σ } x, t , C = 10C x, t , P = P x, t . Numer- ( ) ( ) ( ) 11 22 Also, the medium is free from the normal electric field E ical results are obtained for p = 1, θ = 10, ω = 1.95, 13 0 0 0 at x = 0, so that ω = 0.05, τ = τ = 0.1 and τ = τ = 0.05. 1 1j 0i θ Figure 1 and Figure 2 shows distribution of radial stress ∂ϕ E x, 0, t = − = φ h t = 0 (59) ( ) t ( ) in thermo-piezoelectric functionally graded rod against the 13 0 ∂z radius of rod with respect to various parameters such as graded index and concentration condition. In Figure 1 the 3.1.4 Concentration conditions radial stress for different values of graded index against the increasing values of radius of rod is observed. From A concentration can be applied on the surface of the half this, the distribution of radial stress is initially increased space x = 0 and taken the value C of in the normal direc- up to certain values of radius (r = 0.2) after decreased up tion to (r = 0.6) and follows unique nature for higher values ∂C C(0, t) = = C h t = 0 (60) of radius of rod in several values of graded index because ( ) 0 t ∂Z of their wavelength. The increasing values of the graded Therefore using equations [27] and (53) , (53) , (53) & (56) 1 2 3 index make quiet reasonable attendance in distribution of are the parameters C . It can be determinant by solving the following system ⎡ ⎤⎧ ⎫ ⎧ ⎫ 1 1 1 1 ⎪E ⎪ ⎪ θ ⎪ 1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥⎨ ⎬ ⎨ ⎬ R R R R E 0 ⎢ ⎥ 21 22 23 24 2 ⎢ ⎥ = (61) ⎣KR KR KR KR ⎦⎪E ⎪ ⎪ 0 ⎪ 31 32 33 34 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ S S S S E −σ 11 12 13 14 4 0 Where S = R − h (1 + R ) (62) 1n 1n 2 2n After determining the parameters, the final form of the phys- ical fields of the problem under the investigation can be Figure 1: Distribution of radial stress with the radius obtained. 4 Numerical discussion and results The purpose of numerical analysis and discussion consider the copper material. The Copper material properties are given according to the following values of parameters [4, 14, 17]. Figure 2: Distribution of radial stress with the radius 0 10 −2 0 10 −2 C = 0.4040 × 10 Nm , C = 0.212 × 10 Nm , 11 12 Multi thermal waves in a thermo diffusive piezo electric functionally graded rod | 113 radial stress is additionally noticed. In Figure 2 the radial stress for several values of concentration constant against the increasing values of radius of rod is detected. Since the spreading of radial stress is originally rise up to certain values of radius (r = 0.2) after decreased up to (r = 0.4) and monitors liner nature for upper values of radius of rod because of their wavelength. The increasing values of the concentration constant generates quiet reasonable attendance in spreading of radial stress is also perceived. The spread of radial stress against the radius of rod is Figure 5: Distribution of displacement stress with the radius via RPL observed for different thermoelasicity theories in Figure 3. ℜ = 4 From this observation the spread of radial stress is follows grown nature in small values of radius and follows dimin- distribution in thermopiezoelastic functionally graded rod ish nature for certain values of radius after follows linear against the radius of rod for different values of time param- nature for higher values of radius of rod in all theories of eter t in RPL model is detected in Figure 6 and Figure 7. The thermoelasicity. Additionally its observed RPL model gen- temperature distribution is follows diminish nature for in- erates excessive impact in distribution of radial stress. creasing values of radius of rod for different values of time parameter t in both RPL (ℜ = 2 and ℜ = 4) models. The temperature distribution creates sensible consideration for increasing value of in RPL model. Figure 3: Distribution of radial stress with the radius via-RPL ℜ = 1 Figure 6: Distribution of temperature with the radius via RPL ℜ = 2 The displacement stress in thermopiezoelastic func- tionally graded rod against the radius of rod for diverse val- ues of concentration constants in RPL model is detected in Figures 4 and 5. The displacement stress is follows diminish nature in minor values of radius of rod and follows unique nature for higher values of radius for diverse concentra- tion constants in both RPL (ℜ = 2 andℜ = 4) models. The displacement stress distribution makes reasonable atten- tion for increasing value of in RPL model. The temperature Figure 7: Distribution of temperature with the radius via RPL ℜ = 4 The distribution of electric displacement in ther- mopiezoelastic functionally graded rod against the radius of rod for different values of concentration constants in RPL model is noticed in Figures 8 and 9. The distribution of electric displacement follows oscillating nature for increas- ing values of radius for different concentration constants. The distribution of electric displacement creates reason- able attention for increasing value ofℜ in RPL model. The Figure 4: Distribution of displacement with the radius via RPL ℜ = 2 frequency in thermopiezoelastic functionally graded rod 114 | P. Jayaraman et al. 5 Conclusions The present work is generated analytical model for wave dispersion in a thermally activated chemico diffusive piezo- electric functionally graded rod through refined multi-dual phase-lag model. The normal mode method and suitable boundary conditions used to obtain the exact solution of the coupled thermopiezoelastic equations. The analytical Figure 8: Distribution of electric displacement with the radius via RPL ℜ = 2 expressions for these coupled equations would be the com- ponents of mechanical displacement, temperature, electric displacement, electric potential, and stresses. Numerical results for these field quantities are illustrated graphically. The exact findings from this graphically illustration are dis- cussed elaborate, for instance impacts of graded index, con- centration constants and different RPL models. The present results may be applicable to a broad range of piezothermoe- lastic materials and smart materials industry. Funding information: The authors state no funding in- volved. Figure 9: Distribution of electric displacement with the radius via RPL ℜ = 4 Author contributions: All authors have accepted responsi- bility for the entire content of this manuscript and approved against the radius of rod for different values of parameter t its submission. in RPL model is detected in Figures 10 and 11. The frequency distribution is follows growing nature for increasing values Conflict of interest: Francesco Tornabene and Rossana of radius of rod for different values of parameter t in both Dimitri, who are the co-authors of this article, are a current RPL (ℜ = 2 andℜ = 4) models. The temperature distribu- Editorial Board members of Curved an Layered Structures. tion creates sensible consideration for increasing value of This fact did not affect the peer-review process. The authors ℜ in RPL model. declare no other conflict of interest. References [1] Suresh S, Mortensen A. Fundamentals of Functionally Graded Ma- terials: Processing and Thermomechanical Behavior of Graded Metals and Metal-Ceramic Composites. London: IOM Communi- cations, LTD; 1998. [2] Tzou HS, Ye R. Pyroelectric and thermal strain effects of piezo- electric (PVDF and PZT) devices Mech Syst Signal Process. 1996;10(4):459-469. Figure 10: Distribution of frequency with the radius via RPL ℜ = 2 [3] Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG. Functionally Graded Materials: Design, Processing and Applications. London: Kluwer Academic Publishers; 1999. [doi:10.1016/j.mspro.2014.07.442]. [4] Paulino GH, Jin ZH, Dodds RH Jr. Failure of functionally graded materials. In: Comprehensive Structural Integrity. Elsevier Ltd; 2007. p. 607-644. [5] Müller E, Drašar C, Schilz J, Kaysser WA. Functionally graded materials for sensor and energy applications. Mater Sci Eng A. 2003 Dec;362(1-2):17-30. Figure 11: Distribution of frequency with the radius via RPL ℜ = 4 Multi thermal waves in a thermo diffusive piezo electric functionally graded rod | 115 [6] Niino M, Kisar K, Mori M. Feasibility study of FGM technol- Due to Laser Pulse. J Therm Stresses. 2014;37(12):1416-32. ogy in space solar power systems(SPSS), Mater Sci Forum. [25] Abbas AI, Kumar R, Kaushal S. Interaction due to thermal source 2005;492:163-8. in micropolar thermoelastic diffusion medium. J Comput Theor [7] Malinina M, Sammi T, Gasik M. Corrosion resistance of homoge- Nanosci. 2015;12(8):1780-6. neous and FGM coatings]. Mater Sci Forum. 2005;492–493:305- [26] He TH, Cao L, Li S. Dynamic response of a piezoelectric rod with 10. [doi:10.4028/www.scientific.net/MSF.492-493.305 thermal relaxation. J Sound Vib. 2007;306(3-5):897–907. [8] Xing A, Jun Z, Chuanzhen H, Jianhua Z. Development of an ad- [27] Babaei MH, Chen ZT. Transient thermo piezoelectric response vanced ceramic tool material – functionally gradient cutting ce- of a one-dimensional functionally graded piezoelectric medium ramics. Mater Sci Eng A. 1998;248(1-2):125–31. to a moving heat source. Arch Appl Mech. 2010;80(7):803-13. [9] Kawasaki A, Watanabe R. Thermal fracture behavior of [doi:10.1007/s00419-009-0342-x] metal/ceramic functionally graded materials. Eng Fract Mech. [28] Ma Y, He TH. Dynamic response of a generalized piezoelectric- 2002;69(14-16):1713-1728. thermoelastic problem under fractional order theory of ther- [10] Woodward B, Kashtalyan M. Performance of functionally graded moelasticity. Mech Adv Mater Struct. 2016;23(10):1173-80. plates under localised transverse loading. Compos Struct. [doi:10.1080/15376494.2015.1068397] 2012;94(7):2254-62. [doi:10.1016/j.compstruct.2012.02.012] [29] Othman MI, Hilal MI, Elmaklizi YD. The effect of gravity and [11] Mindlin RD. Equations of high frequency vibrations of thermo diffusion on micropolar thermoelasticity with temperature- piezoelectric crystal plates. Int J Solids Struct. 1974;10(6):625-37. dependent elastic medium under G–N theory. Mech Mech Eng. [doi:10.1016/0020-7683(74)90047-X] 2017;21(3):657–77. [12] Chandrasekharairh DS. A generalized linear thermoelasticity [30] Zenkour AM. Wave propagation of a gravitated piezo- theory for piezoelectric media. Acta Mech. 1988;71:39-49. thermoelastic half-space via a refined multi-phase-lags [13] Lebon G. A generalized theory of thermoelasticity. J Tech Phys. theory, mechanics of advanced materials and struc- 1982;23:37-46. tures. Mech Adv Mater Struct. 2020;27(22):1923-34. [14] Mindlin RD. On the equations of motion of piezoelectric crystals. [doi:10.1080/15376494.2018.1533057] In: Muskilishivili NI, Editor. Problems of continuum mechanics. [31] Zenkour AM, Kutbi MA. Multi thermal relaxations for thermodiffu- Philadelphia:SIAM, 1961. p.282-290. sion problem in a thermoelastic half-space Int J Heat Mass Transf. [15] Rao SS, Sunar M. Analysis of distributed thermopiezoelectric 2019;143:118568. [doi:1016/j.ijheatmasstransfer.2019.118568] sensors and actuators in advanced intelligent structures. AIAA J. [32] Sharma JN, Kumar M. Plane harmonic waves in piezo thermo 1993;31(7):1280-6. [doi:10.2514/3.11764] elastic materials. Indian J Eng Mater Sci. 2000;7(5):434-42. [16] Selvamani R, Poongkothai J, Jayakumar S. Hygro thermal vibra- [33] Othman MI, Hasona WM, Abd-Elaziz EM. Effect of rotation on tion characteristics of an axisymmetric piezoelctric functionally micro-polar generalized thermoelasticity with two-temperature graded rod. AIP Conf Proc. 2019;2166:020026. using a dual-phase-lag model. Can J Phys. 2014;92(2):149-158. [17] Poongkothai J, Selvamani R. Thermo electro environment effects [34] Othman MI. A Lord-Shulman theory under the dependence of in a dispersion of functionally graded thermo piezo electric rod the modulus of elasticity on the reference temperature in two coupled with inviscid fluid. AIP Conf Proc. 2019;2166:020027. dimensional generalized thermoelasticity. J Therm Stresses. [doi:10.1063/1.5131614] 2002;25(11):1027-45. [doi:10.1080/01495730290074621] [18] Dube GP, Kapuria S, Dumir PC. Exact piezothermoelastic solu- [35] Othman MI, Abbas IA. Effect of rotation on a magneto- tion of simply-supported orthotropic flat panel in cylindrical thermoelastic homogeneous isotropic hollow cylinder with en- bending. Int J Mech Sci. 1996;38(11):1161-77. [doi:10.1016/0020- ergy dissipation using finite element method. J Comput Theor 7403(96)00020-3]. Nanosci. 2015;12:2399-404. [doi:https://doi.org/10.1166/jctn. [19] Ding HJ, Wang HM, Chen WQ. Dynamic responses of a func- 2015.4039] tionally graded pyroelectric hollow sphere for spherically sym- [36] Sharma JN, Kumar PK, Mishra KC. Dynamic response of function- metric problems. Int J Mech Sci. 2003 Jun;45(6-7):1029-51. ally graded cylinders due to time-dependent heat flux. Mecca- [doi:10.1016/j.ijmecsci.2003.09.005] nica. 2016;51:139-154. [20] Ootao Y, Tanigawa Y. Transient piezothermoelastic anal- [37] Sharma JN, Kumar PK, Mishra C. Analysis of free vibrations in ysis for a functionally graded thermopiezoelectric hol- axisymmetric functionally graded thermo elastic cylinders. Acta low sphere. Compos Struct. 2007 Dec;81(4):540-54. Mech. 2014;225(6):1581-1594. [doi:10.1016/j.compstruct.2006.10.002] [38] Ebrahimi F, Dabbagh A. Magnetic field effects on ther- [21] Wu CP, Huang SE. Three-dimensional solutions of functionally mally affected propagation of acoustical waves in rotary graded piezo-thermo-elastic shells and plates using a modified double-nanobeam systems. Waves Random Complex Media. Pagano method. Comput Mater Contin. 2009;12(3):251-281. 2018;31(1):1–21. [doi:10.1080/17455030.2018.1558308]. [22] Akbarzadeh AH, Chen ZT. Heat conduction in one-dimensional [39] Pradhan SC, Phadikar JK. Small scale effect on vibration functionally graded media based on the dual-phase-lag theory. of embedded multilayered graphene sheets based on non- Proc Inst Mech Eng C J Mech Eng. Sci. 2013, 227(4), 744-759. local continuum models. Phys Lett A. 2009;373(11):1062–9. [doi:10.1177/0954406212456651] [doi:10.1016/j.physleta.2009.01.030]. [23] Abouelregal AE, Zenkour AM. Effect of phase lags on ther- [40] Reddy JN. Analysis of functionally graded plates. Int J Numer moelastic functionally graded microbeams subjected to Methods Eng. 2000;47(1-3):663–84. ramp-type heating. IJST-T Mech Eng. 2014;38(M2):321-35. [doi:10.22099/IJSTM.2014.2498] [24] Elhagary MA. A Two-Dimensional Generalized Thermoelastic Dif- fusion Problem for a Thick Plate Subjected to Thermal Loading

Curved and Layered Structures – de Gruyter

**Published: ** Jan 1, 2022

**Keywords: **Thermoelastic diffusion; generalized piezothermoelasticity; FG rod; RPL model; wave dispersion

Loading...

You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!

Read and print from thousands of top scholarly journals.

System error. Please try again!

Already have an account? Log in

Bookmark this article. You can see your Bookmarks on your DeepDyve Library.

To save an article, **log in** first, or **sign up** for a DeepDyve account if you don’t already have one.

Copy and paste the desired citation format or use the link below to download a file formatted for EndNote

Access the full text.

Sign up today, get DeepDyve free for 14 days.

All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.