CIRCULAR CRESTED THERMOELASTIC WAVES IN HOMOGENEOUS ISOTROPIC PLATES

The propagation of circularly crested thermoelastic waves in a homogeneous isotropic cylindrical plate subjected to stressfree and isothermal conditions is investigated in the context of conventional coupled thermoelasticity (CT), Lord-Shulman (LS), Green-Lindsay (GL), and Green-Nagdhi (GN) theories of thermoelasticity. The secular equation for the circular plate in closed form and isolated mathematical conditions for symmetric and skew symmetric wave mode propagation in completely separate terms are derived. It is shown that the motion for SH modes gets decoupled from the rest of the motion and remains unaffected due to thermomechanical coupling and thermal relaxation effects. The phase velocities for SH modes have also been obtained. It is noticed that the rest of the motion for circular crested waves is again governed by Rayleigh-Lamb-type secular equations. The secular equations for these plate and Lamé modes are also obtained and discussed for different regions. The results for coupled and uncoupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. At short wavelength limits, the secular equations for symmetric and skew symmetric waves in stressfree insulated and isothermal circular plate reduces to Rayleigh surface wave frequency equations. Finally, the numerical solution is carried out for aluminium-epoxy composite material, and dispersion curves for symmetric and skew-symmetric wave modes are presented to illustrate and compare the theoretical results. The theory and numerical computations are found to be in close agreement.     

PROPAGATION OF GENERALIZED VISCOTHERMOELASTIC RAYLEIGH–LAMB WAVES IN HOMOGENEOUS ISOTROPIC PLATES

The present paper is aimed at studying the thermoelastic interaction in an infinite Kelvin–Voigt-type viscoelastic, thermally conducting plate. The upper and lower surfaces of the plate are subjected to stress-free, thermally insulated or isothermal conditions. The coupled dynamic thermoelasticity and generalized theories of thermoelasticity, namely, Lord and Shulman's, Green and Lindsay's, and Green and Nagdhi's are employed to understand the thermomechanical coupling and thermal and mechanical relaxation effects. Secular equations for the plate in closed from and isolated mathematical conditions for symmetric and skew-symmetric wave mode propagation in completely separate terms are derived. In the absence of mechanical relaxations (viscous effect), the results for generalized and coupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. In the absence of thermomechanical coupling, the analysis for a viscoelastic plate can be deduced from the present one. The various forms and regions of Rayleigh–Lamb-type secular equation have been obtained and discussed in addition to Lame modes, decoupled shear horizontal (SH) modes, and thin-plate results. At short-wavelength limits, the secular equations for symmetric and skew-symmetric waves in a stress-free insulated and stress-free isothermal plate reduce to the Rayleigh surface wave frequency equation. The amplitudes of temperature and displacement components during symmetric and skew-symmetric motion of the plate have been computed and discussed. Finally, the numerical solution is carried out for copper material. The dispersion curves, and amplitudes of temperature change and displacements for symmetric and skew-symmetric wave modes are presented to illustrate and compare the theoretical results.     

Effect of Thermoelastic Coupling on Thermally Induced Plate Parametric Vibrations

Vibration control for linear elastic plate due to time-dependent in-plane forces in the presence of raised temperature is presented. Transverse inertia, modified terms corresponding to adiabatic deformations and a viscous external damping are taken into account. The coupled partial differential equations describing the transverse plate motion and the electric potential are used. The two piezoelectric layers are placed symmetrically as actuators. When voltages of equal magnitude but opposite phase are applied to the upper and the lower piezoelectric layers of the plate, induced strains are resulting in flexure action. The piezoelectric sensor is used to measure plate deformations. The derived coupled equations are applicable to stability analysis of plate parametric vibration. The control law is derived using the Liapunov approach with a functional as a sum of the modified mechanical plate energy and the energy of electric field. Influence of the thermoeleastic effect and feedback gain factor on stability domain is shown.     

Generalized thermoelasticity of a piezoelectric layer

In the present research, the response of a one-dimensional piezoelectric layer is investigated using the generalized thermoelasticity theory of Lord and Shulman. The layer is subjected to thermal shock on one surface. Three coupled equations, namely, motion equation, energy equation and Maxwell equation in terms of displacement, temperature, and electric potential are established. Using the proper transformation, the mentioned equations are given in a dimensionless form. These equations are discretized by means of the generalized differential quadrature method and traced in time by means of the Newmark time marching scheme. Numerical examples are provided to show the propagation and reflection of thermal, mechanical and electrical waves in a layer. It is shown that under the Lord and Shulman theory, temperature propagates with a finite speed, similar to mechanical displacement wave. However, the electric displacement and potential propagate with infinite speed.     

Propagation of Rayleigh Surface Waves in Microstretch Thermoelastic Continua Under Inviscid Fluid Loadings

The present article is aimed at an investigation of the propagation of generalized Rayleigh surface waves in a homogeneous, isotropic, microstretch thermoelastic solid half-space underlying an inviscid liquid half-space or layer of finite thickness, in the context of classical (coupled) and non-classical (generalized) theories of thermoelasticity. The secular equations in close form and isolated mathematical conditions are derived for generalized Rayleigh waves in the considered composite structure after obtaining general wave solutions of the model. The fluid overlying the solid half-space has been successfully modeled as thermal load in addition to normal (hydrostatic pressure) one. Some special cases of dispersion equations have also been deduced and discussed. The analytic expressions for the amplitudes of displacement, microstretch, microrotation and temperature change at the interfacial surface during the Rayleigh wave propagation are also derived. The results have been deduced and compared with the relevant publications available in the literature at the appropriate stages of this work. Finally, the analytical developments have been illustrated numerically for aluminum–epoxy-like material half-space under the action of inviscid liquid (water) half-space or layer of finite thickness. The computer simulated results in respect of phase velocity, attenuation coefficient, specific loss factor of energy dissipation and relative frequency shift due to fluid loadings are presented graphically in normalized form to observe their distinctions from those in the context of the well established theory of coupled thermoelasticity.     

SOLUTION FOR THERMAL AND MECHANICAL WAVES IN A PIEZOELECTRIC PLATE BY THE METHOD OF CHARACTERISTICS

The object of this article is to study the one-spatial dimensional thermal and mechanical waves in a piezoelectric infinite plate subjected to thermal, electric, and mechanical loadings. Based on the coupled generalized theory of piezothermoelasticity along with the modified Fourier law, the governing equations are expressed by a set of first-order partial differential equations with stress, particle velocity, electric field intensity, heat flow, and temperature as the unknown variables. This system of equations is analyzed by the method of characteristics. The numerical calculations are carried out for a traction-free PZT-4 plate subjected to impulsive surface heating. Graphical displays are utilized to present the outcome of the procedures.     

SHEAR DEFORMATION EFFECTS ON THERMALLY INDUCED INSTABILITY OF LAMINATED PLATES

A theoretical investigation of dynamics for linear elastic moderately thick structures due to uniform space- and time-dependent temperature fields is presented. The structures aredescribed by partial differential equations including a transverse inertia term for a general deformation state with interlaminar shear strains. A viscous model of external damping with a constant proportionality coefficient is assumed to describe a dissipation of the structural energy in the transverse motion. The particular problem of dynamic cylindrical bending due to the temperature with Gaussian and harmonic distributions is analyzed in detail. Dynamic thermal buckling of both symmetric alternating cross-ply composite plates and antisymmetric angle-ply composite laminates is investigated. The influence of neglecting the shear effects on dynamic stability regions is shown.     

Wave propagation in a generalized thermoelastic plate using eigenvalue approach

In the present work, we obtain a dispersion relation for Rayleigh–Lamb wave propagation in a plate of thermoelastic material. For this aim, we consider the theory of generalized thermoelasticity with one relaxation time. The thickness of the plate is taken to be finite and the faces of the plate are assumed to be isothermal and free from stresses. We obtain the analytical solution for the temperature, displacement components, and stresses using an eigenvalue approach. Finally, we derive a dispersion relation for the plate in closed form taking into account isothermal boundary conditions for wave mode propagation. To obtain the phase velocity and attenuation coefficients of propagating wave mode, we use the function iteration numerical scheme to solve the complex dispersion relation. The phase velocity and attenuation coefficients for the first five modes of waves are represented graphically for Lord–Shulman and classical coupled dynamical theories.     

Propagation Characteristics of Elasto-Thermodiffusive Surface Waves in Semiconductor Material Half-Space

The paper is aimed at an investigation of the propagation of elasto-thermodiffusive (ETN) surface waves in homogenous isotropic, thermally conducting, semiconductor material half-space with relaxation of heat and charge carrier fields. Secular equations, in isolated mathematical conditions and compact form, for the thermoelastic diffusive surface waves in semiconducting material half-space are derived. Some particular forms of the general secular equation are also deduced and investigated. The secular equations for thermoelastic (ET) and elastodiffusive (EN) surface waves have been obtained and discussed as special cases. The surface displacements during the wave propagation have also been obtained and discussed. The paths of surface particles of ETN, ET, and EN surface waves are found to be elliptical in nature. Numerical solution of various secular equations and other relevant relations is carried out for silicon (Si) semiconductor material with the help of functional iteration numerical technique. In order to illustrate and compare the analytical results, the dispersion curves, attenuation coefficient, and specific loss profiles of the waves are computed and presented graphically.     

Analysis of Thermoelastic Waves in Functionally Graded Hollow Spheres Based on the Green-Lindsay Theory

In this article, the Green–Lindsay theory of thermoelasticity is employed to study the thermoelastic response of functionally graded hollow spheres. This generalized coupled thermoelasticity theory admits the second sound phenomena and depicts a finite speed for temperature wave propagation. The materials of the hollow sphere are assumed to be graded through its thickness in the radial direction while a symmetric thermal shock load is applied to its boundary. The Galerkin finite element method via the Laplace transformation is used to solve the coupled form of governing equations. A numerical inversion of the Laplace transform is employed to obtain the results in time domain. Using the obtained solution, the temperature, displacement, radial stress, and hoop stress waves propagation are studied. Also the material distribution effects on temperature, displacement and stresses are investigated. Finally, the obtained results for the Green–Lindsay theory are compared with the results of classical thermoelasticity theory.     

Laser-Generated Thermoelastic Waves in an Anisotropic Infinite Plate: Exact Analysis

In recent years, the study of thermoelastic waves generated by lasers has been undertaken by several researchers because the technique provides an efficient non-contact technique for generation and detection of ultrasonic waves. Laser-generated ultrasonic waves have diverse applications ranging from material characterization to defect characterization. Transient ultrasonic guided waves generated by a pulsed laser in anisotropic infinite plate are investigated in this article. An exact analytical method is adopted for this purpose. The governing equations and boundary conditions are first transformed from spatial-time domain into wavenumber-frequency domain using Fourier Transform. After solving these equations and satisfying the boundary conditions in the wavenumber-frequency domain, the Cauchy's residue theorem is used to get the response in the spatial domain and then the numerical integration is used to eventually obtain the response in time domain. Results for dispersion and transient guided waves in infinite silicon nitride (Si₃N₄) plates are presented. Numerical results show that pulsed laser excites mainly the lowest Lamb modes, namely, the lowest symmetric (S ₀) and antisymmetric (A₀) modes. They also show that the transient response is dominated by the antisymmetric mode A₀ which shows dispersion characteristic. This study provides a quantitative model for laser generated ultrasonic waves in an anisotropic plate and can be used for non-destructive evaluation.     

A NUMERICAL TECHNIQUE FOR DYNAMIC COUPLED THERMOELASTICITY PROBLEMS WITH RELAXATION TIMES

A numerical method is proposed for Green and Lindsay' s dynamic thermoelasticity problems. The semidiscrete equations resulting from standard finite element formulations are mathematically manipulated so that the-resulting coupled heat equation and elastic equations both become symmetric. Such process allows for taking advantage of the traditional uncoupled thermal-structural solution procedures. Unconditionally stable time integration schemes are recommended to solve these matrix equations. The proposed technique is very effective and versatile in solving both finite speed thermal wave and stress wave propagation problems     

Generalized thermo-electro-elasticity of a piezoelectric disk using Lord-Shulman theory

Abstract

Current investigation deals with the generalized thermoelastic response of a finite hollow disk made of a piezoelectric material. The constitutive equations of the piezoelectric media are reduced to a two dimensional plane-stress state. To capture the finite speed of temperature wave, the single relaxation time theory of Lord and Shulman is used. Three coupled differential equations in terms of radial displacement, electric potential, and temperature change are obtained. These equations are written in a dimensionless presentation. With the aid of the differential quadrature method (DQM) a time-dependent algebraic system of equations is extracted. The Newmark time marching scheme is applied to trace the temporal evolution of temperature change, electric potential, radial displacement, stresses, and electric displacement. Numerical results demonstrate that radial displacement and temperature waves propagate with finite speed while the electric potential propagates with infinite speed.     

The transient thermo-electro-elastic fields in a piezoelectric plate with a crack

This paper is concerned with a thin plate made by a piezoelectric ceramic material and containing a crack perpendicular to its surfaces. It is assumed that the transient thermal stress is set up by the application of a heat flux as a function of the time and position along the crack edge and the heat flow by convection from the plate surfaces. The plate is also subjected to mechanical and electric loadings. The exact analytical formulae are obtained for transient thermo-electro-elastic fields in the plate. The exact analytical solutions for the stress and electric displacement intensity factors and crack-opening displacement are obtained. Numerical examples show, among others, a dependence of the stress and electric displacement intensity factors on the thermal and elastic, piezoelectric and dielectric constants of the piezoelectric materials.     

弹性地基处周期性圆形钢筋混凝土管道振动特性分析

针对弹性地基处周期性钢筋混凝土管道的波动特性,基于声子晶体理论和Flugge壳体理论,建立了圆形管道环径向轴对称波动微分方程,利用传递矩阵法建立了相邻胞元间的传递矩阵,数值分析了周期性管道结构的能带特性。结果表明,振动波在传播过程中存在禁带域和通带域,弹性地基对弯曲波在特定频率范围内的传播具有抑制作用,长度比的变化对周期性圆形混凝土管道禁带的幅值、宽度和个数影响显著,因此可通过调整结构尺寸参数改变结构中波的传播特性。     

Waves in Pyroelectrics

The propagation of waves in an infinite pyroelectric medium is studied in this paper. The governing equations for the pyroelectrics are written in three versions which are (1) the version containing heat flow, (2) the stiffened version and (3) the compact version. The results obtained by using versions (1), (2) and (3) are the same. Four characteristic wave velocities are found, three being analogous to those of elastic waves and the fourth wave, which is predominantly a temperature disturbance, corresponding to the heat pulse known as the second sound. It is found that all the velocities and attenuation coefficients are related to the wave normal and the material constants. The effect of the electric properties on the wave propagation is considered, which indicates that the temperature wave is not sensitive to pyroelectricity and piezoelectricity in the discussed case. The effects of relaxation time τ are estimated by several different values, and the results indicate that τ has very little effect on the mechanical velocities; whereas it plays a large role on the velocity and attenuation of temperature wave. The effects of the terms containing τ on the attenuation are researched in detail in one-dimensional case for clarity.     

GL Model on Reflection of P and SV Waves from the Free Surface of Thermoelastic Diffusion Solid Under Influence of the Electromagnetic Field and Initial Stress

This article is devoted to estimating the influence of magnetic field, electric field and initial stress in an elastic solid half-space under thermoelastic diffusion. The governing equations in the xz-plane are solved taking into consideration the Green–Lindsay (GL) model. The Reflection of dilatational (P) wave and Shear Vertical (SV) wave split into four waves: namely, P wave, thermal wave, mass diffusion wave and SV wave. The reflection phenomena of P and SV waves from the free surface of an elastic solid with thermoelastic diffusion under influence of magnetic field, electric field and initial stress is considered. The expressions for the reflection coefficients for the four reflected waves are obtained. These reflection coefficients are found to depend upon the angle of incidence θ of P and SV waves, thermoelastic diffusion, magnetic field, electric field and initial stress and other material parameters. The numerical values for the reflection coefficients are calculated analytically and presented graphically for varying values of these parameters.     

Transient Piezothermoelasticity of a Two-Layered Composite Hollow Cylinder Constructed of Isotropic Elastic and Piezoelectric Layers Due to Asymmetrical Heating

This article is concerned with the theoretical treatment of transient piezothermoelastic problem involving a two-layered hollow cylinder constructed of isotropic elastic and piezoelectric layers due to asymmetrical heat supply. The transient two-dimensional temperature is analyzed by the method of Laplace transformation. By using the exact solutions for piezoelectric hollow cylinder and isotropic hollow cylinder, the theoretical analysis of transient piezothermoelasticity is developed for a two-layered composite hollow cylinder under the state of plane strain. As an example, numerical calculations are carried out for an isotropic elastic hollow cylinder made of steel, bonded to a piezoelectric layer of cadmium selenide. Some numerical results for the temperature change, the stress and the electric potential distributions in a transient state are shown in figures. Furthermore, the influence of thickness of the piezoelectric layer or the isotropic elastic layer upon the temperature change, stresses and electric potential is investigated.     

Fundamental solutions in piezoelectricity. Penny-shaped crack solution

The problem of electroelasticity for piezoelectric materials is considered. For axially symmetric states three potentials are introduced, which determine the displacements, the electric potentials, the stresses, the components of the electric field vector and the electric displacements in a piezoelectric body. These fundamental solutions are utilized to solve the penny-shaped crack problem. Two cases of boundary—value problems are considered, namely the permeable and impermeable crack boundary conditions. Exact solutions are obtained for elastic and electric fields. The main results are the stress intensity factor for singular stress and the electric displacement intensity factor. The numerical results are presented graphically to show the influence of applied mechanical and electrical loading on the analyzed quantities and to clarify the effect of anisotropy of piezoelectric materials. It is show that the influence of anisotropy of the materials on these fields is significant.     

Coupled thermoelasticity analysis of FGM plate integrated with piezoelectric layers under thermal shock

Abstract

Based on theory of piezoelectricity and using generalized coupled thermoelasticity, transient response of a simply supported functionally graded material rectangular plate embedded in sensor and actuator piezoelectric layers under applied electric field and thermal shock is studied. Thermoelastic properties of the plate vary continuously along the thickness direction according to exponential functions and Poisson ratio is assumed to be constant. Applying Fourier series state space technique to the basic coupled thermoelastic differential equations results in the ordinary differential equations which are solved analytically by using Laplace transform. Validation of the present approach is assessed by comparing the numerical results with the available results in literature. In parametric study, effect of the relaxation time, applied voltage and temperature and time history of the thermoelastic response of FGM plate attached to piezoelectric layers are investigated.