Nonlocal Heat Conduction Research Articles

Elasto-Thermodiffusion in a Slim Strip Revisited with New Definition of Nonlocal Heat Conduction

Elasto-Thermodiffusion in a Slim Strip Revisited with New Definition of Nonlocal Heat Conduction

Thermal analysis of 3D viscoelastic micropolar nanofluid with cattaneo-christov heat via exponentially stretchable sheet: Darcy-forchheimer flow exploration

This article presents a comprehensive study on the thermal behavior of a three dimensional viscoelastic micropolar nanofluid under the inspiration of Cattaneo-Christov heat conduction model. The nanofluid is subject to an exponentially stretchable sheet and the flow is characterized by Darcy-Forchheimer formulation. Similarity variables are used to convert viscoelastic nanofluids PDEs into ODEs. MATLAB platform with bvp4c command is used to solve resulting ODE. The impact of involving parameters such that velocity ratio, Schmidt number, Prandtl number, Peclet number, Forchheimer number, buoyancy ratio parameter, Brownian motion, Heat source, Magnetic parameter on velocity profiles, rotation, temperature concentration and density profiles via graphs and tables. Temperature curve is improved up for magnetic parameters, temperature exponent parameter and thermopherosis parameter while significance loss is noted for increasing value of Prandtl number Pr and relaxation time λT. This research bears significance in applications related to advanced heat exchangers, microfluidic systems, and other emerging technologies where the interplay of nanofluids, viscoelasticity, and non-local heat conduction is of paramount importance.

Electromagnetic thermo-viscoelastic response of piezoelectric rods considering memory dependent effects

Based on the theory of nonlocal elasticity and nonlocal heat conduction, a new dual-phase-lag heat conduction model with memory dependent effect is proposed in this article to explore the thermodynamic behavior of functionally graded rotating piezoelectric rods under the action of moving heat sources. Assuming that the material properties of functionally graded piezoelectric rods vary exponentially along the length direction, the end of the rod is rigid and fixed without voltage. Use Laplace transform to transform the problem into the spatial domain and perform analytical solutions, then use inverse Laplace transform to obtain the time-domain solution. Numerical solutions were performed for dimensionless displacement, temperature, electric potential, and stress, and the variation patterns of the physical quantities involved were described in graphical form. The calculation provides the effects of functional gradient non-uniformity index, thermal nonlocal parameters, kernel function, and viscoelastic parameters on the physical quantities involved.

Nonlocal dual-phase-lag thermoelastic damping analysis in functionally graded sandwich microbeam resonators utilizing the modified coupled stress theory

Functionally graded (FG) sandwich structures stand as one of the most representative composite structures owing to the thermal resistance and the energy absorption property in non-unfiorm thermal environment. Additionally, accurately estimating thermoelastic damping (TED) is of great importance for the design of high-performance micro/nano-resonators. Nevertheless, the classical TED models fail on the micro/nano-scale structures due to without considering the influences of the spatial size-dependent effects related to heat transfer and elastic deformation. The nonlocal heat conduction model and modified coupled stress theory are responsible for the size-dependent effects. To address this issue, present study aims to conduct the size-dependent TED model of FG sandwich microbeam resonators for TED analysis by incorporating the nonlocal dual-phase-lag (NDPL) heat conduction model and the modified coupled stress theory (MCST). It is assumed that the FG sandwich microbeam resonators consist of a ceramic core and FG surfaces. The energy equation and the transverse motion equation are formulated, and then, the analytical solution is solved by complex frequency method. Exact and closed-form expressions for TED can be obtained through the complex frequency method. The results are validated, and the parameter effects of the nonlocal thermal parameter, the material length-scale parameter, the power-law index and the vibration modes on the TED are analyzed. The results show that the energy dissipation can be reduced by the size-dependent effects resulting in improving the quality factor of microstructures. It is expected that these results may provide a theoretical basis for predicating TED in the design of FG sandwich micro/nano-resonators with high quality factor in extreme heat transfer environment.

Numerical solutions of nonlocal heat conduction technique in tumor thermal therapy

The heat transport that takes place in living tissue during magnetic tumor hyperthermia is described in this study using the nonlocal bioheat model in spherical coordinates. In magnetic fluid hyperthermia, it is crucial to regulate the therapeutic temperature. This paper suggests a hybrid numerical approach that employs the Laplace transforms, change of variables, and modified discretization techniques, coupled with nonlocal hyperbolic shape function, to tackle the present problem. This study investigates the impacts of nonlocal parameter and the disparity in thermophysical properties between diseased and healthy tissue. A graph is displayed to represent the numerical temperature results. The validity of the numerical findings is demonstrated by comparing them with the results reported in previous literature.

Dynamic behavior of nanoplate on viscoelastic foundation based on spatial-temporal fractional order viscoelasticity and thermoelasticity

The nanodevices often work in a thermal environment and lead to the viscoelastic and thermoelastic coupling which becomes more complicated when further coupled with the size effects. A spatial-temporal fractional order model is proposed to study the dynamic behavior of thermoelastic nanoplate in the present work. The integer order differential is extended to the fractional order differential with the integer order differential as a special case. This extension makes the modeling of small-scale effects of mechanical behavior and generalized heat conduction more flexible. Non-local strain gradient elasticity and non-local heat conduction are used to describe the spatial non-local effects while the fractional order differential to describe the complicated history-dependent effects or the temporal nonlocal effects. The standard solid viscoelastic model and generalized heat conduction model with the fractional order differentials lead to the fractional-order differential governing equations. The fractional order differential equation is solved by the Laplace transform method, and the analytical solution of the response is expressed in terms of the Mittag-Leffler function. In order to verify the reliability of the analytical solution, the numerical solution is also provided for comparison. Based on the numerical results of dynamic response under the step load, the effects of thermoelastic fractional order parameters, viscoelastic fractional order parameters, and small-scale parameters are discussed. The new spatial-temporal fractional order model is a natural extension of the traditional integer order model which can be recovered from the present model.

Non-classical theory of electro-thermo-elasticity incorporating local mass displacement and nonlocal heat conduction

A non-classical local gradient theory of nonferromagnetic thermoelastic dielectrics is presented, incorporating both the local mass-displacement process and the heat-flux gradient effect. The process of local mass displacement is related to the changes in material microstructure. The nonlocal heat conduction law is also addressed in the model. Thus, the generalized relationship between the higher-grade heat and entropy fluxes is adopted. The gradient-type constitutive relations and governing equations are derived using the fundamental principles of continuum mechanics, non-equilibrium thermodynamics, and electrodynamics. Due to the contribution of higher-grade flux, the nonlocal law of heat conduction is obtained. The constitutive relations for isotropic materials with the corresponding additional material constants are derived. To illustrate the local gradient theory and to show the electro-thermo-mechanical coupling effect in isotropic materials, a straightforward problem is analytically solved for a layered non-piezoelectric structure under non-uniform temperature distribution. The analytical results reveal that the thermal polarization effect can also be pronounced in isotropic materials. To illustrate the model considering the effect of nonlocal heat conduction, the propagation of spherical thermoelastic harmonic waves in a homogeneous and isotropic elastic medium with non-classical heat conduction law is studied.

Thermoelastic response of an elastic rod under the action of a moving heat source based on fractional order strain theory considering nonlocal effects

In this paper, based on fractional order strain theory and nonlocal heat conduction theory, the thermoelastic response of an elastic rod rigidly fixed at both ends and subjected to a moving heat source is investigated. The nonlocal fractional order strain control equation is established, and the analytical expressions of dimensionless temperature, displacement, and stress are solved using the Laplace integral transformation and numerical inverse transformation. The distribution law of physical quantities such as temperature, displacement, and stress when the fractional parameters, heat source moving speed, and thermal nonlocal parameters change is obtained. The results show that strain parameters and thermal nonlocal parameters have less influence on thermal–mechanical waves, while heat source velocity has a significant influence on thermal–mechanical waves.

A Buckling Analysis of Thermoelastic Micro/Nano-Beams Considering the Size-Dependent Effect and Non-Uniform Temperature Distribution.

Thermoelastic buckling of micro/nano-beams subjected to non-uniform temperature distribution is investigated in this paper. The mechanical governing equation is derived based on the surface effect and mechanical non-local effect. The non-local heat conduction model is used to predict temperature distribution in micro/nano-beams. Therefore, the obtained analytical solution can be used to analyze the influence of both the mechanical and thermal small scale effects on buckling of thermoelastic micro/nano-beams. In numerical simulations, a critical thickness is proposed to determine the influence region of both mechanical and thermal small scale effects. The influence of a small scale effect on buckling of micro/nano-beams must be considered if beam thickness is less than the critical thickness. In the influence region of a small scale effect, a surface effect has strong influence on the size-dependent buckling behavior, rather than mechanical and thermal non-local effects. Moreover, combined small scale effects, i.e., a surface effect and both mechanical and thermal non-local effects, lead to a larger critical load. Additionally, the influence of other key factors on buckling of the micro/nano-beams is studied in detail. This paper provides theoretical explanation to the buckling behaviors of micro/nano-beams under a non-uniform temperature distribution load.

A reduced-order peridynamic model for predicting nonlocal heat conduction in nanocomposites

A reduced-order peridynamic model is developed for predicting transient heat conduction in nanocomposites. Peridynamics is utilized to describe the nonlocal heat conduction process in the nanoscale. An implicit numerical scheme is constructed for numerical discretization, and the proper orthogonal decomposition is employed to establish the reduced-order model. The performance of the proposed method is verified quantitatively with benchmark problems. Discontinuities, such as cracks and interfacial thermal resistance, can be successfully captured with the computation time significantly reduced. The reduced-order peridynamics provides an alternative solution to balance the flexibility of peridynamics and the urgency of computation efficiency.

A nonlocal theory of heat transfer and micro-phase separation of nanostructured copolymers

Directed self-assembly of block copolymer (BCP) has shown promise as a nano-patterning technology for various applications such as microelectronics, semiconductor manufacturing and biotechnology. In this paper, we present a thermodynamically consistent nonlocal model of phase transformation and heat transfer for BCP via the continuum theory of mixtures. Our model is developed based on the mass and energy balances of each phase and a set of microforce balances to describe the dynamics of microphase separation during self-assembly and the size and memory effects of heat conduction in BCP thin films. We derive constitutive relations for coupling the phase field and temperature through upscaling the self-consistent field theory of polymers, and present the connections between the derived nonlocal heat conduction model and the phonon Boltzmann transport equation. We provide a finite element solution of the proposed model that includes unconditionally stable time-stepping schemes, and we perform various numerical experiments that showcase different features of the model. Furthermore, we conduct three-dimensional numerical experiments of laser zone annealing process to qualitatively investigate the effect of laser velocity on the self-assembly behavior of BCP thin films. Our results show that the proposed model can simulate essential features of thermally induced phase responses of BCPs, consistent with experimental observations and high-fidelity simulations.

Wave propagation in a non-local magneto-thermoelastic medium permeated by heat source

In this paper, we consider the new concept of non-local heat conduction equation to generalized magneto-thermoelastic problem of two dimensional isotropic and homogeneous half-space in presence of heat-flux at the boundary surface. By using the harmonic plane waves, the governing equations are transformed to the vector matrix differential equation which is then solved by eigenvalue method. The analytical closed form solutions for displacement component, temperature distribution and stress components have been made and comparisons are also illustrated graphically with the theory of non-local dual-phase-lag (NLDPL) and non-local Lord-Shulman (NLLS) theory for different values of physical parameters. The significant effects of non-local variables as well as phase lagging parameters on displacements, temperature distribution and stress components are studied by means of graphically and concluding remarks are drawn.

Investigation on buckling of Timoshenko nanobeams resting on Winkler-Pasternak foundations in a non-uniform thermal environment via stress-driven nonlocal elasticity and nonlocal heat conduction

This study investigates the size-dependent buckling behavior of Timoshenko nanobeams resting on Winkler-Pasternak foundations in a non-uniform thermal environment. A non-uniform temperature distribution is established through nonlocal heat conduction. Subsequently, the equivalent thermal load due to the obtained temperature distribution and boundary constraints is derived using the governing equations of the axial thermal deformation of the beams based on the stress-driven nonlocal elastic model. To obtain the critical buckling load of the nanobeams, the quadrature element method is used to numerically resolve the eigenvalue problem. In the numerical simulation section, we have presented a series of examples to analyze the effects of length-to-height ratios, nonlocal scale parameters, and Winkler-Pasternak foundation parameters on the buckling loads of the nanobeams under various boundary conditions. Moreover, we examine the effect of comprehensively considering both the elastic and thermal nonlocality on the thermal loads and finally mechanical buckling loads.

The Effect of a Nonlocal Thermoelastic Model on a Thermoelastic Material under Fractional Time Derivatives

This article develops a novel nonlocal theory of generalized thermoelastic material based on fractional time derivatives and Eringen’s nonlocal thermoelasticity. An ultra-short pulse laser heats the surface of the medium’s surrounding plane. Using the Laplace transform method, the basic equations and their accompanying boundary conditions were numerically solved. The distribution of thermal stress, temperature and displacement are physical variables for which the eigenvalues approach was employed to generate the analytical solution. Visual representations were used to examine the influence of the nonlocal parameters and fractional time derivative parameters on the wave propagation distributions of the physical fields for materials. The consideration of the nonlocal thermoelasticity theory (nonlocal elasticity and heat conduction) with fractional time derivatives may lead us to conclude that the variations in physical quantities are considerably impacted.

Correct and Stable Algorithm for Numerical Solving Nonlocal Heat Conduction Problems with Not Strongly Regular Boundary Conditions

For a nonlocal initial-boundary value problem for a one-dimensional heat equation with not strongly regular boundary conditions of general type, an approximate difference scheme with weights is constructed. A correct and stable algorithm for the numerical solving of the difference problem is proposed. It is proven that the difference scheme with weights is stable and its solution converges to the exact solution of the differential problem in the grid L2h-norm. Stability conditions are established. An estimate of the numerical solution with respect to the initial data and the right-hand side of the difference problem is given.

Microstructure size-dependent contact behavior of a thermoelectric film bonded to an elastic substrate with couple stress theory

Thermoelectric films linked to elastic substrates are typical engineering structures and have applications in heat recovery systems. This article uses the nonlocal heat conduction and couple stress theory to investigate the electric-thermal-elastic behavior of the thermoelectric film placed on the elastic half-plane. The internal characteristic lengths of both the thermoelectric film and the elastic substrate are introduced to reveal the scale effects. The coupling governing integro-differential equations in terms of the interface shear stress and tensile stress are derived for the thermoelectric film and the elastic substrate. The numerical results show how the distribution of the tensile and shear stress intensity factors is affected by the stiffness ratio of the substrate to the thermoelectric film, the internal characteristic lengths of the thermoelectric film and the elastic substrate, and the thickness to length ratio of the thermoelectric film. Compared with the classical model for a thicker thermoelectric film and stiffer elastic substrate, the present study demonstrates the necessity of taking into account the microstructure lengths of the thermoelectric film and the elastic substrate.

Mathematical modelling in nonlocal Mindlin’s strain gradient thermoelasticity with voids

A nonlocal theory for thermoelastic materials with voids based on Mindlin’s strain gradient theory was derived in this paper with some qualitative properties. We have also established the size effect of nonlocal heat conduction with the aids of extended irreversible thermodynamics and generalized free energy. The obtained system of equations is a coupling of three equations with higher gradients terms due to the length scale parameters ϖ and l. This poses some new mathematical difficulties due to the lack of regularity. Based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions to the one dimensional problem. By an approach based on the Gearhart-Herbst-Prüss-Huang theorem, we prove that the associated semigroup is exponentially stable; but not analytic.

Thermoelastic damping in the size-dependent micro/nanobeam resonator with nonlocal dual-phase-lag heat conduction

Precisely calculating quality factor based on thermoelastic damping (TED) is of great importance for the design and optimization of micro/nano-resonators. With the consideration of size-dependent effects in the thermal and mechanical fields and the non-Fourier (NF) heat conduction effect, analytical expressions of TED in the micro/nanobeam resonator are proposed by adopting the theories of the modified-couple-stress (MCS) model and the nonlocal dual-phase-lag (DPL) model. TED models expressed in the series form and the explicit form are both developed basing on the energy-definition approach. The differences among TED models are discussed. In the simulation, silicon and gold, which are two representative materials with typical intrinsic length-scale quantities and phase-lag times, are selected for illustration. Additionally, the nonlocal thermal length-scale parameter refers to the mean-free-path (MFP) of energy carriers, and the MCS mechanical length-scale parameter is not constant but proportional to the beam thickness. TED results obtained by the classical model and the previous DPL model are also shown for comparison. The peak phenomena of TED spectra including the peak damping and the peak frequency are investigated according to the present simple model. Results reveal that TED in micro/nanobeam resonators is of dependence observably on the effect of nonlocal DPL-NF heat conduction associated with the thermal field, and the quality factor is improved by the MCS mechanical size-dependent effect.

Nonlocal thermoelasticity: Transient heat conduction effects on the linear and nonlinear vibration of single-walled carbon nanotubes

This novel study presents a framework to investigate the effects of transient heat conduction on the nonlinear deflection and vibration of single-walled carbon nanotubes (SWCNTs) based on Eringen’s nonlocal elasticity and nonlocal heat conduction theories. The first phase of this research involves modifying the hyperbolic heat conduction theory by including a nonlocality term, in which the scale of energy transport is taken into account. The next phase includes a thermoelasticity solution for carbon nanotubes (CNTs), which is developed by Eringen’s nonlocal theory and implementing the obtained temperature. The differential quadrature method (DQM) is used to solve the transversely induced nonlocal heat conduction. We have shown the effect of thermal waves on the mode shapes and nonlinear vibration of a SWCNT by considering phase lag parameters. All boundaries are considered under a temperature-jump at the nanometer scale to consider the boundary phonon scattering. The effects of several parameters, including time-dependent boundary conditions, time lag, nonlocal parameter, length and radius, and end supports on the frequency response of SWCNTs are investigated.

Thermoelastic analysis of nanobar based on nonlocal integral elasticity and nonlocal integral heat conduction

The thermoelastic analysis is extremely important due to the highly integrated characteristics in micro- and nano-electro-mechanical systems. In this paper, thermal flux driven (TFD) and temperature gradient driven (TGD) two-phase local/nonlocal heat conduction integral models (NHCIM) are developed to describe size-dependent heat conduction. Combining TFD- and TGD-NHCIMs with strain- and stress-driven two-phase local/nonlocal elastic integral models, four types of nonlocal thermoelastic integral models are developed to predict the thermoelastic behavior of microstructures. The nonlocal thermoelastic integral models are applied to analyze the thermoelastic behavior of clamped-clamped microrod with prescribed boundary temperature and heat flux for each end. Analytical solutions for the distribution of temperature, displacement and thermal stresses are derived and expressed explicitly with several unknown constants, which can be determined by the standard and constitutive boundary conditions. The effects of nonlocal parameters on the distribution of temperature, displacement and thermal stress are investigated numerically and systematically. The numerical results show that nonlocal thermoelastic models based on the combination between strain- and stress-driven two-phase local/nonlocal elastic models respectively with TGD- and TFD-NHCIMs would lead to consistent size-dependent response when increasing the elastic and thermal nonlocal length-scale parameters at the same time, while those based on the combination between strain- and stress-driven two-phase nonlocal elastic models respectively with TFD- and TGD-NHCIMs would lead to inconsistent prediction.