Thermoelastic Waves Research Articles

Reflection of Plane Waves in Nonlocal Fractional-Order Thermoelastic Half Space

The problem of plane waves in nonlocal fractional-order thermoelasticity has been studied. We have considered the x-y plane for the governing equation of nonlocal fractional thermoelasticity and solved these governing equations to calculate the equation in terms of frequency. This frequency shows that three sets of waves exist, in which two are coupled and one is uncoupled. The reflection coefficient of plane waves for classical theory and LS theory has been calculated. The effect of phase speeds, specific losses, and attenuation coefficients with respect to the frequency and nonlocal parameter for the two theories (LS theory and the classical theory of thermoelasticity) has been studied numerically for all propagating waves, and the same has been plotted graphically and explained thoroughly.

Two phase local/nonlocal thermo elastic waves in a graphene oxide composite nanobeam subjected to electrical potential

AbstractThe present article studies the two phase local/nonlocal deformation of thermo electrical composite nanobeam reinforced with graphene oxide powder (GOP). The controlling equation of motion and boundary conditions are derived via two phase elasticity in differential form in conjunction with Hamilton's principle and Euler‐Bernoulli beam theory. The effect of a thermal and piezo electric field is incorporated with the governing equations by convoluting field quantities and displacement components. Then, Navier method is adopted to observe the influence of simply supported boundary conditions on the dynamics of the beam. Comparison studies are performed to display the accuracy and efficiency of this analytical model. Further, the effects of GOP weight fraction, two phase parameter, external electric voltage and temperature difference on the dimensionless frequency of GOP reinforced piezo thermo elastic composite beams are thoroughly investigated and highlighted through tables and dispersion curves. The results obtained from the present work gained the important role of GOP and thermo‐ piezoelectricity in predicting the vibration behaviors of two phase local/ nonlocal nanostructures.

Plane wave propagation in a fiber-reinforced thermoelastic rotating medium with variable thermal conductivity under modified Green–Lindsay model

The present investigation is concerned with the reflection of thermoelastic plane waves in an anisotropic, rotating, fiber-reinforced medium having variable thermal conductivity. Modified Green–Lindsay theory (MGL) is used to formulate the problem. It is shown that there exist three kinds of coupled quasi-waves, namely, (quasi longitudinal displacement), (quasi thermal) and (quasi transverse displacement). Using appropriate boundary conditions, the reflection coefficients, phase velocities and energy ratios of these reflected waves have been calculated numerically and shown graphically to exhibit the effects of rotational and fiber-reinforcement parameters. The characteristics of MGL model are discussed by comparing the numerical results procured for the present model with those procured in the case of Green–Lindsay (GL) model. It has been shown that during reflection phenomenon, the sum of energy ratios is equal to unity at each angle of the incident.

Laser ultrasonic imaging of submillimeter defect in a thick waveguide using entropy-polarized bilateral filtering and minimum variance beamforming

Recent quantum leap in far-field laser techniques has advanced noncontact implementation of nondestructive ultrasonic imaging, in pursuit of enhanced accessibility, detectability and practicability. Nevertheless, when laser-generated thermoelastic waves are extended to thick waveguides, they manifest fairly low signal-to-noise ratios (SNRs), along with severe wave diffusion, consequently lowering image resolution and contrast. With these motivations, a laser-ultrasonics imaging approach is developed, in conjunction with i) entropy-polarized bilateral filtering (Entropy-P-BF) for signal denoising, and ii) minimum variance (MV) beamforming for defect imaging, targeting at precise characterization of a submillimeter defect (with its characteristic dimension being smaller than the wave diffraction limit) in a thick waveguide. The entropy-polarized bilateral filtering denoises laser-induced ultrasonic wave signals via a two-dimensional convolution, the weight matrices of which are continuously updated according to local noise and uncertainty. With an elevated SNR, MV beamforming subsequently conducts an apodized beamforming to image the defect. Experimental validation is conducted by imaging a void-type defect, 0.7 mm only in its diameter, in a jet aero-engine turbine disk. Results prove that the developed approach is capable of characterizing a submillimeter defect accurately in a thick waveguide with thickness ∼25 times the wavelength of laser-induced shear wave, regardless of a fairly low SNR (<1dB).

Generalized thermoelastic waves in a homogeneous anisotropic plate with voids

AbstractIn this contribution, the thermoelastic Lamb waves in the homogeneous, transversely isotropic, plate with voids have been investigated in the context of classical and non‐classical theories of thermoelasticity. To investigate the problem, the fundamental governing equations have been developed in a unified way. On the solutions of these governing equations, the thermal and mechanical stress‐free boundary conditions, as well as constraints due to voids, have been imposed. The solutions of governing equations indicate that there exists a coupled system of wave motion called thermal waves (T‐mode), void wave motion (V‐mode), and elastic waves (E‐mode). However, the shear horizontal component of elastic wave decouples from the system is unaffected by elasto‐poro‐thermal coupling. For the symmetric and anti‐symmetric families of vibrations, the secular equation for the wave motion along with its special cases has been derived. To discover the wave characteristics, the secular equation has been solved using the numerical‐functional iteration method in MATLAB software, and the findings have been manifested by plotting the graphs. It has been concluded that the presence of voids affects the magnitudes of phase velocity, attenuation coefficient, and specific loss. This work may be helpful for geophysicists and engineers in the field of earthquake engineering. The developers of modern facilitating things such as acoustic touch screens, liquid viscosity sensors, etc. which are based on Lamb‐type wave propagation may also get benefitted.

Simulation of thermoelastic wave propagation in 3-D multilayered half-space media

SUMMARY Thermal phenomena are common in both the generation and propagation of seismic waves. To investigate the propagation characteristics of thermoelastic waves generated by point sources in a 3-D multilayered half-space, herein, we develop a method that can solve problems involving coupled mechanical and thermal motions. The method first transforms the governing equations into two sets of first-order ordinary differential equations with a unified form in the frequency–wavenumber domain and then solves them by a generalized reflection and transmission coefficient method. To verify the correctness of our method and the corresponding computer program, we compare our solutions with analytical solutions. We further apply the method to different multilayered models, and the numerical results clearly demonstrate the reflection and transmission phenomena of thermoelastic waves within the interlayer. Furthermore, the generated thermal and seismic signals are synchronous, and the two P waves predicted by generalized thermoelastic theory can be converted into each other. The records on the ground show that thermal motions influence our observation of direct waves, especially in the presence of high thermal conductivity subsurface media, it can significantly impact the propagation of both body and surface waves, making surface seismic observations more complex than those predicted by elastic theory.

Dual-function thermoelastic cloak based on coordinate transformation theory

• Transformation theory is extended to thermoelastic framework for dual-function cloak design. • Two transformation schemes with asymmetric or symmetric elastic tensors are given. • Invisibility of different cloaks is validated by FEM and observed in both temperature and displacement fields. In past years, various cloaks have attracted the attention of many scientists but they usually remain confined to a single function. Cloaks combining multiple functions, however, are desirable. In this paper, we design thermoelastic cloaks with dual functionality by the coordinate transformation technique. The transformation of the thermoelastic wave equations for cloaking and explicit expressions for the required material parameters are established. Symmetrization of the elastic tensor is applied using Norris’ gauge matrix to emphasize the feasibility of the designs. Two different operating conditions, transient elastic wave propagation and steady heat transfer, are adopted in numerical calculations for the designed cloaks. It is shown that, on one hand, from the perspective of observer, whether the cloak is symmetrized or not, its external response is identical and invisibility is reliable. On the other hand, it is destined that the symmetrization of the cloak would be accompanied by the increase of internal displacement, which may need to be paid attention to in the actual design. In addition, the temperature effects of both cloaks are consistent. The work in this paper may pave a way toward the realization of thermoelastic cloaks, thus broadening the research scope for metamaterials.

Band structure analysis of Green-Naghdi thermoelastic wave propagation in a GPLs/CNTs-reinforced metamaterial with energy dissipation

In this paper, the band structure analysis of the thermoelastic wave propagation in a phononic crystal (PC), which it is reinforced by graphene platelets (GPLs) and carbon nanotubes (CNTs), is presented. The reinforced PC beam is made of periodic unit-cells including a metal section and a GPLs/CNTs-reinforced epoxy section. The Green-Naghdi theory of the generalized coupled thermoelasticity with energy dissipation and a new modified micromechanical model to predict the mechanical properties of the reinforced section are employed in the analysis. An analytical solution and the transfer matrix method are adopted to compute the band structures. The frequency band-gaps are obtained and the effects of the GPLs and CNTs volume fractions on the width and location of the band-gaps are analyzed and discussed in details. The results show that the GPLs influence the frequency band-gaps and the band structures of the thermoelastic wave propagation in the reinforced PC beam much more remarkably than the CNTs. The proposed analytical solution method, the modified micromechanical model and the presented results can be employed for designing the reinforced PC structures by the GPLs, CNTs or both of them and also for tuning the band-gaps in the future researches.

Modeling of Delayed Thermo Elastic Waves in a Polygonal Ring Reinforced with Graphene Platelets

Modeling of Delayed Thermo Elastic Waves in a Polygonal Ring Reinforced with Graphene Platelets

Analytical Solution for Thermoelastic Stress Wave Propagation in an Orthotropic Hollow Cylinder

The problem of thermoelastic stress wave propagation in an orthotropic hollow cylinder is investigated using analytical methods. The fully coupled classical theory of thermoelasticity is used to extract the equations for an orthotropic cylinder. To solve the boundary value problem, heat conduction equation and equation of motion are divided into two different sets of equations, the first set consists of uncoupled equations with considering boundary conditions and the second set comprises coupled ones with initial conditions. Finite Hankel transform (Fourier-Bessel expansion) is utilized to solve the problem with respect to radial variable. Two different cases, pure mechanical load and pure thermal load, were studied numerically to show the effect of considering the thermomechanical coupling term in the heat conduction equation. To show the effect of considering the coupling term in the heat conduction equation, the temperature history is plotted for the pure mechanical load case, where the temperature rises without applying any thermal load. By applying boundary conditions on the inner surface of the cylinder, initiation of the stress waves from the inner surface of the cylinder, propagation through the thickness in the radial direction and reflection from the outer surface were observed in the plotted figures.

Reflection of temperature rate–dependent coupled thermoelastic waves on a non-local elastic half-space

Based upon the Green–Lindsay (GL) theory of thermoelasticity and the Eringen non-local elasticity theory, we describe the new constitutive relations and the field equations for generalized non-local thermoelastic model. The propagation of harmonically time-dependent thermoelastic plane waves of assigned frequency is studied in a homogeneous, isotropic, non-local infinite elastic solid by employing these equations. Reflection phenomenon of thermoelastic waves at a stress-free thermally insulated or isothermal flat boundary of a non-local thermoelastic solid half-space is also studied in case of incident coupled longitudinal wave. The amplitude ratios of the reflected waves to that of incident wave are determined analytically and numerically by considering an appropriate material. The numerical results are presented graphically to highlight the effects of various parameters of interest.

Nonlocal thermoelastic waves inside nanobeam resonator subject to various loadings

The present article focuses on the new meticulous model based on the postulate of memory-dependent derivatives to analyze the thermo-mechanical interactions inside the nano-beam-based machined resonators. Also, the size effect on dynamic responses of thermoelastic vibrations of homogeneous and isotropic nano-beam is considered. The fundamental expressions are formulated in the frame of non-local generalized thermoelasticity with paired relaxation times by operating the results of Euler-Bernoulli beam theory, non-local effect, and memory-dependent derivative. The proposed model is applied to study the nano-beam-based machined resonator subjected to the ramp-type heating and exponentially decaying time-dependent load. Closed-form solutions of the physical fields are examined by applying the Laplace transform mathematical mechanism. However, the coherence of the new thermal conductivity framework, a collation has been bestowed among the results obtained in the presence or absence of the memory-dependent derivative; also, the size effect is analyzed on the significant parameters of nano-beam such as deflection, temperature, displacement as well as bending moment. Moreover, the prominent influence of the distinct affecting parameters such as constituents of memory-dependent derivative (kernel function and time delay) and ramping time parameter with an applied load on the physical fields have been investigated with the help of quantitative results.

FEM–BEM coupling for the thermoelastic wave equation with transparent boundary conditions in 3D

We consider the thermoelastic wave equation in three dimensions with transparent boundary conditions on a bounded, not necessarily convex domain. In order to solve this problem numerically, we introduce a coupling of the thermoelastic wave equation in the interior domain with time-dependent boundary integral equations. Here, we want to highlight that this type of problem differs from other wave-type problems that dealt with FEM–BEM coupling so far, e.g., the acoustic as well as the elastic wave equation, since our problem consists of coupled partial differential equations involving a vector-valued displacement field and a scalar-valued temperature field. This constitutes a nontrivial challenge which is solved in this paper. Our main focus is on a coercivity property of a Calderón operator for the thermoelastic wave equation in the Laplace domain, which is valid for all complex frequencies in a half-plane. Combining Laplace transform and energy techniques, this coercivity in the frequency domain is used to prove the stability of a fully discrete numerical method in the time domain. The considered numerical method couples finite elements and the leapfrog time-stepping in the interior with boundary elements and convolution quadrature on the boundary. Finally, we present error estimates for the semi- and full discretization.

The Propagation of Thermoelastic Waves in Different Anisotropic Media Using Matricant Method

Thermoelasticity is a generalization of classical theories of elasticity and thermal conductivity and describes a wide range of phenomenon. The theory can precisely predict the propagation of thermoelastics waves in case of an isotropic medium. However, the propagation of thermoelastic waves in the anisotropic medium is not fully understood. In this case, the theory of elasticity employs an approximate theory of temperature stress which does not take into consideration the interactions of temperature and deformations. In this paper, an analytical study has been carried out by using method of matricant to investigate the propagation of longitudinal elastic and heat waves in the anisotropic medium of a monoclinic, trigonal, hexagonal, and cubical crystal systems. In this article, a solution to the problem of the propagation of thermal waves and the propagation of a thermal wave along z -axis has been obtained. The attenuation coefficient and phase velocity of thermal waves for various materials are determined. Specifically, the problem of propagation of heat waves in one dimension has been solved.

A deep learning approach based on the physics-informed neural networks for Gaussian thermal shock-induced thermoelastic wave propagation analysis in a thick hollow cylinder with energy dissipation

This paper proposes a novel approach based on the physics-informed neural networks (PINNs) as an efficient deep learning approach to solve a broad range of the coupled partial differential equations (PDEs). For the first time, the PINN methodology is successfully applied to the thermoelastic wave propagation analysis with energy dissipation in a thick hollow cylinder based on the Green-Naghdi (GN) theory of coupled thermoelasticity. The assumed thick hollow cylinder is subjected to Gaussian thermal shock loading applied on the inner bounding surface. This research presents a PINN architecture with self-adaptive weights, which does not suffer from the curse of dimensionality as opposed to the numerical methods. An in-depth discussion is provided on the method and its capabilities to address PDE-related issues. Moreover, the PINN-based parametric studies are performed to show the effects of the materials coefficients on the thermoelastic wave propagation and dynamic behaviors of the fields’ variables. Numerous examples of thermoelastic wave propagation analysis with and without energy dissipation are solved via this approach. The framework generates satisfactory results in terms of accuracy and stability, as summarized in the paper.

Thermoelastic wave propagation damping in a hollow FG-GPLRC cylinder with the spinning motion

This analysis deals with generalized thermoelasticity via Lord–Shulman’s theory for hollow cylinders reinforced with graphene platelets (GPL). The viscosity effect is also considered for applied materials. The second-order correlation homogenization techniques are utilized to obtain the equivalent thermo-mechanical properties. The hollow cylinder with spinning motion is modeled based on the linear thermoelastic constitutive law. The GPLs are functionally distributed along the cylinder’s radius based upon a power-law function. The Kelvin–Voigt type of viscosity behavior which is a differential state of viscous materials, is employed. The energy equation is attained based on the coupled and generalized framework. After obtaining the motion and energy equations in dimensionless form, the generalized differential quadrature (GDQ) and Newmark time marching methods are implemented to extract the temporal evolution and wave propagation of temperature, displacement, and stresses that occurred in the structure. After validating the responses with the available theoretical article, some numerical results are demonstrated to examine the effect of thermo-mechanical properties on the thermoelastic behavior of chosen nanocomposite cylinder.

Effects of homojunction on the reflected and transmitted waves at the interface between two thermoelastic semiconductor half spaces

It is systematically investigated that the influence of interface effect resulted from one homojunction on the reflected and transmitted thermoelastic waves between two bonded homogeneous isotropic thermoelastic semiconductor half spaces in this paper. Considering the geometrical and physical characteristics of homojunction, it is approximately treated as a two-dimensional isotropic imperfect interface without geometrical thickness mathematically but with elastic, thermal and semiconductor properties depicted through a dimensionless characteristic length factor physically. Considering the coupling effect of mechanical displacement, temperature change and charge carrier concentration perturbation, the imperfect interface boundary condition is established based on the generalized Green-Naghdi theory, the surface and interface elastic theory and the physical property of homojunction. The algebraic equation resulted from this imperfect interface boundary condition is solved to obtain the reflected and transmitted coefficients of thermoelastic waves expressed in the energy flux ratio, which are verified by the principle of energy conservation. The numerical calculation results demonstrate that the increase of geometrical thickness of homojunction and the angular frequency of thermoelastic waves can enhance the interface imperfect property and arise more evident influence of interface effect. Meanwhile, the interface effect has a more evident influence on the thermoelastic waves propagating along the interface normal direction. The physical mechanism revealed through the establishment of mathematical model provides a theoretical basis for using PN junction to regulate the propagation properties of thermoelastic waves in thermoelastic semiconductors.

Automated design of phononic crystals under thermoelastic wave propagation through deep reinforcement learning

This article presents a novel concept of deep reinforcement learning (DRL) to facilitate the reverse design of layered phononic crystal (PC) beams with anticipated band structures focusing on the band structure analysis of thermoelastic waves propagating. To this end, we define the reverse design of phononic crystals (PCs) as a game for the DRL agent. To achieve the desired band structure, the DRL agent needs to obtain the topological system of PC. We trained a DRL agent called deep deterministic policy gradient (DDPG). An environment is developed and used to simulate the reverse design of layered PCs with the acquisition of a reward function. The presented reward function encourages the agent to achieve the desired bandgaps. The trained DDPG agent can maximize the game’s score by attaining the desired bandgap. The presented concept allows the user to instantly generate the design parameters through the trained DDPG agent without unnecessary search over the design space. We demonstrated that the DRL agent could perform very well for the automated design of PCs with hundred design cases.

Thermoelastic Plane Waves in Materials with a Microstructure Based on Micropolar Thermoelasticity with Two Temperature and Higher Order Time Derivatives

The study of the effect of the microstructure is important and is most evident in elastic vibrations of high frequency and short-wave duration. In addition to deformation caused by temperature and acting forces, the theory of micropolar thermoelasticity is applied to investigate the microstructure of materials when the vibration of their atoms or molecules is increased. This paper addresses a two-dimensional problem involving a thermoelastic micro-polar half-space with a traction-free surface and a known conductive temperature at the medium surface. The problem is treated in the framework of the concept of two-temperature thermoelasticity with a higher-order time derivative and phase delays, which takes into consideration the impact of microscopic structures in non-simple materials. The normal mode technique was applied to find the analytical formulas for thermal stresses, displacements, micro-rotation, temperature changes, and coupled stress. The numerical results are graphed, and the effect of the discrepancy indicator and higher-order temporal derivatives is examined. There are also some exceptional cases that are covered.

Thermoelastic wave propagation due to local thermal shock on the functionally graded media

The present study is based on investigating the effect of locally applied thermal shock on the ceramic side of the functionally graded (FG) domain using nonlinear Lord-Shulman generalized thermoelastic theory. The constituent materials and therefore their properties in the two-dimensional FG domain are considered to be graded as a function of horizontal or vertical coordinates according to the power law. The finite element (FE) form of the nonlinear coupled differential equation is solved using the Picard method at each time-step in the framework of the Newmark time-integration scheme. The gradation of the material properties due to FGM is incorporated using graded finite elements in the FE formulation. The numerical results concerned with studying the effect of volume fraction index and orientation of the gradation of material properties across the FG domain are presented. It is observed from the study that both the gradation of material properties, as well as the direction of orientation of material gradation, have a substantial effect on the propagation of the thermal and elastic waves in the domain. Moreover, the thermal waves appear to be more dominant than the elastic waves at the time of application of thermal shock, which may also alter as time progresses.