Governing Field Equations Research Articles

Eringen’s nonlocal elasticity theory for the analysis of two temperature generalized thermoelastic interactions in an anisotropic medium with memory

PurposeThe purpose of this paper is to investigate the thermoelastic interactions in a homogeneous, transversely isotropic infinite medium with a spherical cavity in the context of two temperature Lord-Shulman (2TLS) generalized theory of thermoelasticity considering Eringen’s nonlocal theory and memory dependent derivative (MDD). Memory-dependent derivative is found to be better than fractional calculus for reflecting the memory effect which leads us to the current investigation.Design/methodology/approachThe governing field equations of the problem are solved analytically using the eigenvalue approach in the transformed domain of Laplace when the cavity’s boundary is being loaded thermomechanically. Using MATLAB software the numerical solution in real space-time domain is obtained by Stehfest method.FindingsNumerical results for the different thermophysical quantities are presented in graphs and the effects of delay time parameter, non-local parameter and two temperature parameters are studied thereafter. The outcomes of this study convince that the displacement u, conductive temperature ϕ, thermodynamic temperature θ are concave upward whereas radial stress τrr is concave downward for every choice of delay time parameter ω, two temperature parameter η and non-local parameter “ζ”. As a specific instance of our findings, the conclusions of an equivalent problem involving integer order thermoelasticity theory can be obtained, and the corresponding results of this article can be readily inferred for isotropic materials.Originality/valueThe novelty of this research lies in the adoption of generalized thermoelastic theory with memory dependent derivative and Eringen’s nonlocality for analyzing the thermoelastic interactions in an infinite body with spherical cavity by employing eigenvalue approach. It has applications to many thermo-dynamical systems.

Dynamical interactions of the magnetic field in a binary mixture of interacting thermoelastic solids

A more focused scientific approach in the context of a mixture of thermoelastic solids would entail investigating the basic ideas guiding the behavior of the complex materials, including constitutive model development, multi-scale modeling at various length scales, and design optimization for thermoelastic solids with their precise properties. The present work is set up to study the effect of the magnetic field as an external effect on the mixture of two interacting thermoelastic solids. The fundamental mathematical equations were shown and solved by the analytical harmonic wave solution to give the solution of the descriptive functions of the discussed problem. The displacements and stresses for the mixture of the two interreacted thermoelastic solids with the mutual temperature were clarified in the 2-D and 3-D. The results are analyzed and discussed. The obtained solution for the functions is finite everywhere. The solutions for the quantities of practical interest are represented graphically with different choices of the wave number of the harmonic wave as the magnetic field with time. In addition to the numerous physical factors in the governing field equations as depicted in the figures, boundary conditions also play a role in the current damping phenomenon.

The Effect of Gravitational Decoupling on Constraining the Mass and Radius for the Secondary Component of GW190814 and Other Self-bound Strange Stars in f(Q) Gravity Theory

Inspired by the conundrum of the gravitational event GW190814, which brings to light the coalescence of a 23 M ⊙ black hole with a yet-to-be-determined secondary component, we look to modeling compact objects within the framework of gravity by employing the method of gravitational decoupling. We impose a quadratic equation of state (EOS) for the interior matter distribution, which in the appropriate limit reduces to the MIT bag model. The governing field equations arising from gravitational decoupling bifurcate into the and sectors, leading to two distinct classes of solutions. Both families of solutions are subjected to rigorous tests, qualifying them to describe a plethora of compact objects, including neutron stars, strange stars, and the possible progenitor of the secondary component of GW190814. Using observational data of mass–radius relations for compact objects LMC X-4, Cen X-3, PSR J1614–2230, and PSR J0740+6620, we show that it is possible to generate stellar masses and radii beyond 2.0 M ⊙ for neutron stars. Our findings reveal that the most suitable and versatile model in this framework is the quadratic EOS, which accounts for a range of low-mass stars and typical stellar candidates describing the secondary component of GW190814.

Teleparallel Robertson-Walker Geometries and Applications

In teleparallel geometries, the coframe and corresponding spin connection are the principal geometric objects and, consequently, the appropriate definition of a symmetry is that of an affine symmetry. The set of invariant coframes and their corresponding spin connections that respect the full six dimensional Lie algebra of Robertson–Walker affine symmetries are displayed and discussed. We will refer to such geometries as teleparallel Robertson–Walker (TRW) geometries, where the corresponding derived metric is of Robertson–Walker form and is characterized by the parameter k=(−1,0,1). The field equations are explicitly presented for the F(T) class of teleparallel TRW spacetimes. We are primarily interested in investigating the k≠0 TRW models. After first studying the k=0 models and, in particular, writing their governing field equations in an appropriate form, we then study their late time stability with respect to perturbations in k in both the cases of a vanishing and non-vanishing effective cosmological constant term. As an illustration, we consider both quadratic F(T) theories and power-law solutions.

Spreading of a thin droplet on a soft substrate

A thin liquid droplet spreads on a soft viscoelastic substrate with arbitrary rheology. Lubrication theory is applied to the governing field equations in the liquid and solid domains, which are coupled through the free boundary at the solid–liquid interface, to derive a set of reduced equations that describe the spreading dynamics. Fourier transform techniques and the finite difference method are used to construct a solution for the dynamic liquid–gas and solid–liquid interface shapes, as well as the macroscopic contact angle. Substrate properties affect the spreading dynamics through the contact angle and internal droplet flow fields, and these mechanisms are revealed. Increased substrate softness increases the spreading rate, whereas increased viscoelasticity decreases the spreading rate. For the case of a purely elastic substrate, the spreading power-law exponent recovers Tanner's law in the rigid limit and increases with substrate softness.

On Flamant–Boussinesq problem with dynamical flexoelectric effect and micro-inertia effect in dielectrics subjected to dynamical wave loading

This work investigates the Flamant–Boussinesq problem for a half-space made of a homogeneous and isotropic dielectric material. The dynamical flexoelectric effect and the dynamical flexocoupling between displacement and polarization, due to mechanical and electrical states, are taken in consideration. The mechanical loading is taken as a wave of a decaying behavior in time at the surface of a half-space, while the electric potential is considered in an open circuit with no charge on the terminals. The first strain gradient theory of elasticity is used as a mathematical frame in the problem formulation. The equation of motion for the representative volume element additionally accounts for the micro-inertia effect because of the intrinsic structure of the dielectrics at the nanoscale. The governing equations and the boundary conditions for homogeneous, isotropic dielectric material are presented with reference to previous work, using a variational technique for internal energies and external forces. An analytical harmonic wave solution is obtained for the problem under consideration, involving different coupling parameters arising from the mechanical and electrical loadings. The results are analyzed and discussed. The solutions for the quantities of practical interest are represented graphically with different choices of material parameters and flexocoupling parameters. The solution is finite everywhere. The existing damping phenomenon arises, not only from the various physical parameters in the governing field equations as shown in the figures, but also through the boundary conditions.

Special Cosserat rods with rate-dependent evolving natural configurations

We present a nonlinear, geometrically exact, and thermodynamically consistent framework for modeling special Cosserat rods with evolving natural configurations. In contrast to the common usage of the point-wise Clausius–Duhem inequality to embody the Second Law of Thermodynamics, we enforce the strictly weaker form that the rate of total entropy production is non-decreasing. The constitutive relations between the state variables and applied forces needed to close the governing field equations are derived via prescribing frame indifferent forms of the Helmholtz energy and the total dissipation rate and requiring that the state variables evolve in a way that maximizes the rate of total entropy production. Due to the flexibility afforded by enforcing a global form of the Second Law, there are two models obtained from this procedure: one satisfying the stronger form of the Clausius–Duhem inequality and one satisfying the weaker global form of the Clausius–Duhem inequality. Finally, we show that in contrast to other viscoelastic Cosserat rod models introduced in the past, certain quadratic strain energies in our model yield both solid-like stress relaxation and creep.

An Analysis of the Photo-Thermoelastic Waves Due to the Interaction between Electrons and Holes in Semiconductor Materials under Laser Pulses

In this paper, the interaction between holes and electrons in semiconductor media is analyzed based on the existing mathematical–physical model. The elasto-thermodiffusion (ETD) theory, according to photothermal (PT) transport processes, has been used to study the model under the impact of the non-Gaussian laser pulse. A one-dimensional (1D) electronic/thermoelastic deformation is described, in detail, by the governing field equations. The governing field equations are taken in non-dimensional forms. The governing equations are established based on coupled elasticity theory, plasma diffusion equations, and moving equations. To determine the physical field quantities in this problem analytically in the Laplace domain, some boundary conditions are taken at the free surface of the semiconductor medium. The inversion of the Laplace transform is implemented using a numerical method to obtain the complete solutions in the time domain for the basic physical fields involved. The effects of the phase lag (relaxation time) of the temperature gradient, phase lag of the heat flux, and laser pulses are graphically obtained and discussed in comparison to silicon and germanium semiconductor materials. The wave behavior of the main fields in the semiconductors, according to optoelectronics and the thermoelastic processes, is obtained and graphically represented.

Thermal Conductivity and Thermophoretic Impacts of Micropolar Fluid Flow by a Horizontal Absorbent Isothermal Porous Wall with Heat Source/Sink

Boundary layer analysis is invoked to clarify the aspects of variable thermal conductivity and thermophoretic forces on a steady state of MHD micropolar fluid flow in the existence of a uniform transverse magnetic field along an isothermal horizontal plate. The micropolar pattern permits the rotational freedom degrees that lead to couple stresses and a non symmetric stress tensor. The initiated PDEs governing the case pattern are mutated into a non-dimensional system due to proper transformations. The transformed mathematical governing equations are solved by implementing a very potent computer algebra software MATLAB code. The plotted graphs analyzed the attitude of multiple physical aspects involving factors on the flow attitude of micropolar velocity and angular velocity and temperature. Through the involved factors, the couple stress, skin friction and Nusselt number are manifested and interpreted amply. A new outcome for drag force and heat gradient experienced by the key factors is portrayed. Augmentation in Ω results in the thermophoretic forces that encapsulate the mass transmission. The local Nusselt number strengthened as the thermal conductivity, heat absorption factors or wall suction velocity were improved, and weakened due to the existence of viscous dissipation or heat generation impacts. As a particular case, the governing field equations of a classical Newtonian liquid are given by dropping the micropolar parameter impacts.

Time-nonlocal six-phase-lag generalized theory of thermoelastic diffusion

Lagging behavior is also expected in the process of mass diffusion similar to the process of heat transport during the ultra fast transient (D. Y. Tzou, Macro to microscale heat transfer: The lagging behavior. John Wiley and Sons, Ltd., 2015). We therefore propose a novel multi-field coupled mathematical model for a linear, isotropic and homogeneous thermoelastic diffusive continuum by taking into account the modified Fourier’s law of heat conduction, the modified Fick’s law of mass diffusion and extending the concept of phase lag from heat conduction to mass diffusion. The modified Fourier’s law includes temperature gradient and thermal displacement gradient among the constitutive variables with three different phase lags for the heat flux vector, the temperature gradient and the thermal displacement gradient. The modified Fick’s law includes chemical potential gradient and the chemical potential displacement gradient among the constitutive variables with three different phase lags for the mass flux vector, the chemical potential gradient and the chemical potential displacement gradient. The Fourier’s law of heat conduction is replaced by a fractional order approximation of the modified Fourier’s law whereas the Fick’s law of mass diffusion is replaced by a fractional order approximation of the modified Fick’s law. The proposed model includes some of the existing models in the field of thermoelastic diffusion as special cases. A variational principle is obtained and uniqueness of solution of the governing field equations under suitable conditions is proved. Finally, a reciprocity theorem is established for the proposed generalized thermoelastic diffusion model.

An Effective Algorithm for 2D Marine CSEM Modeling in Anisotropic Media Using a Wavelet Galerkin Method

The marine controlled-source electromagnetic method (MCSEM) has attracted considerable attention as an approach to explore marine oil and gas resources and geological structures. This study presents a new wavelet Galerkin method (WGM) to solve the forward modeling problem of 2D MCSEM data incorporating conductivity anisotropy. The method uses Daubechies wavelets that may be differentiated based on the need to solve the governing field equations of MCSEM. A quasi-minimal residual method was adopted by combining an incomplete LU preconditioner to solve the WGM equations. The numerical results were compared with the analytical solution and those obtained by the finite difference and element methods. The results show that the proposed WGM is superior to the finite element and difference methods in terms of computing time and memory requirements. This algorithm can be applied to solve the forward modeling problem of MCSEM. The conductivity anisotropy of the background medium affects the MCSEM response more than the reservoir anisotropy. The match between the modeled results and measured data for the simplified real model demonstrates the necessity for using the anisotropic model to interpret data. Although this study used the proposed algorithm for 2D models, it may also be used for 3D models.

Reflection at non-free boundary of a micropolar piezoelectric half-space

This paper presents a study on plane waves propagating in a micropolar piezoelectric material with transverse isotropy. The plane wave solutions of the governing field equations suggest that there exists three coupled plane waves propagating in a micropolar piezoelectric medium. Reflection of a plane wave incident at a non-free surface of a micropolar piezoelectric half-space is considered. The analytical expressions of the reflection coefficients and energy ratios are obtained. A quantitative example is set up to compute the speeds, reflection coefficients and energy ratios of reflected waves. The speeds, energy ratios and reflection coefficients are illustrated graphically against the propagation angle. The numerical results show that the boundary parameters connected to the mechanical nature of the non-free surface affect significantly the energy ratios and reflection coefficients.

Energy shares of plane harmonic wave at an interface between two microstructured flexoelectric half-spaces

In the present work, we consider a problem of reflection and transmission of plane waves at an interface between two different isotropic dielectric half-spaces by considering the effects of flexoelectricity, micro-inertia and strain gradient elasticity. The plane wave solutions of the governing field equations for such a medium shows the existence of two quasi-plane waves. An incident plane wave at an interface between two dissimilar half-spaces generates two reflected and two transmitted waves. By using the continuous conditions at the interface, the relations between amplitude ratios of reflected and transmitted waves are derived in terms of a transfer matrix. A quantitative example of the model is set up to evaluate numerically the amplitude and energy ratios of various reflected and transmitted waves. The numerical results are validated by the law of energy conservation. The numerical results are illustrated graphically against the angle of incidence to show the effects of flexoelectricity, micro-inertia and strain gradients on the amplitude and energy ratios.

A discrete adjoint based level set topology optimization method for stress constraints

This paper proposes a new methodology for computing boundary sensitivities in level set topology optimization using the discrete adjoint method. The adjoint equations are constructed using the discretized governing field equations. The objective function is differentiated with respect to the boundary point movement for computing boundary sensitivities using the discrete adjoint equations. For this purpose, we present a novel approach where we perturb the boundary implicitly by locally modifying the level set function around a given boundary point. These local perturbations are combined with the derivatives of the objective function with respect to the volume fractions of individual elements to compute boundary sensitivities. This enables the circumvention of smoothing or interpolation methods typically used in level set topology optimization to compute sensitivities; and improves the accuracy of the sensitivities and the convergence characteristics. We demonstrate the effectiveness of our method in the context of stress minimization and stress constrained topology optimization problems for orthogonal bracket design under multiple load cases.

Variational principle, uniqueness and reciprocity theorems for higher order time-fractional four-phase-lag generalized thermoelastic diffusion model

In the present paper, a variational principle is derived for the recently introduced higher order time-fractional four-phase-lag generalized thermoelastic diffusion model for a linear, isotropic and homogeneous thermoelastic diffusive continuum. Then uniqueness of the solution for the governing field equations of the considered model is proved under suitable conditions. Finally a reciprocity theorem is obtained for the said model.

Numerical study of temperature dependent thermal conductivity and homogeneous-heterogeneous reactions on Williamson fluid flow

Understanding the flow nature of non-Newtonian fluids need more studies in recent days in terms of their various aspects. Further, heat transfer analysis is an inspiring topic in non-Newtonian flows because of its dominant role in technology. With these incentives, a theoretical investigation is performed to study the time-dependent flow of a non-Newtonian fluid model with focus on heat and mass transfer. In this study, we proposed the mathematical model for the flow of Williamson fluid by incorporating the impact of infinite shear rate of viscosity. With the assistance of rheological expression of Williamson fluid, we construct the governing field equations. For physical relevance we analyzed the impacts of homogeneous-heterogeneous reactions on the flow field. The system of governing partial differential equations is transformed into a set of ordinary differential equations by adopting the non-dimensional variables. This non-dimensional ruling problem together with physical boundary conditions are tackled numerically by utilizing Runge–Kutta Fehlberg scheme via MATLAB software. The physical behavior of friction factor, reduced Nusselt number, dimensionless velocity, temperature and concentration distributions for distinct estimations of leading parameters is examined with the assistance of graphs.

Wave propagation modeling in periodic elasto-thermo-diffusive materials via multifield asymptotic homogenization

A multifield asymptotic homogenization technique for periodic thermo-diffusive elastic materials is provided in the present study. Field equations for the first-order equivalent medium are derived and overall constitutive tensors are obtained in closed form. These lasts depend upon the micro constitutive properties of the different phases composing the composite material and upon periodic perturbation functions, which allow taking into account the effects of microstructural heterogeneities. Perturbation functions are determined as solutions of recursive non homogeneous cell problems emanated from the substitution of asymptotic expansions of the micro fields in powers of the microstructural characteristic size into local balance equations. Average field equations of infinite order are also provided, whose formal solution can be obtained through asymptotic expansions of the macrofields. With the aim of investigating dispersion properties of waves propagating inside the medium, proper integral transforms are applied to governing field equations of the homogenized medium. A quadratic generalized eigenvalue problem is thus obtained, whose solution characterizes the complex valued frequency spectrum of the first-order equivalent material. The validity of the proposed technique has been confirmed by the very good matching obtained between dispersion curves of the homogenized medium and the lowest frequency ones relative to the heterogeneous material. These lasts are computed from the resolution of a quadratic generalized eigenvalue problem over the periodic cell subjected to Floquet–Bloch boundary conditions. An illustrative benchmark is conducted referring to a Solid Oxide Fuel Cell (SOFC)-like material, whose microstructure can be modeled through the spatial tessellation of the domain with a periodic cell subjected to thermo-diffusive phenomena.

Analytical solution for the fully coupled static response of variable stiffness composite beams

An analytical solution is presented for the 3D static response of variable stiffness non-uniform composite beams. Based on Euler-Bernoulli theory, a set of governing differential equations are obtained, in which four degrees of freedom are fully coupled. For the variable stiffness beam, the governing field equations have variable coefficients reflecting the stiffness variation along the beam. Using the direct integration technique, the general analytical solution is derived in the integral form and the closed-form expressions of the obtained solutions are presented employing a series expansion approximation. The series expansion representation enables the proposed approach to be applicable for variable stiffness composite beams with arbitrary span-wise variation of properties. As an alternative solution, the Chebyshev collocation method is applied to the proposed formulation to verify the results obtained from the analytical solution. A number of variable stiffness composite beams made by fibre steering with various boundary conditions and stacking sequences are considered as the test cases. The static response are presented based on the analytical solution and Chebyshev collocation method and excellent agreement is observed for all test cases. The proposed model presents a reliable and efficient approach for capturing the complicated behaviour of variable stiffness non-uniform composite beams.

Effects of rotation, voids and diffusion on characteristics of plane waves in a thermoelastic material

PurposeThe purpose of this paper is to study the effects of rotation, voids and diffusion on characteristics of plane waves in a thermoelastic material.Design/methodology/approachLord and Shulman generalization of linear thermoelasticity is used to study the plane waves in a rotating thermoelastic material with voids and diffusion. The thermoelastic solid is rotating with a uniform angular velocity. The problem is specialized in two dimensions to study wave propagation. The plane harmonic solutions of governing field equations in a plane are obtained.FindingsA velocity equation is obtained which indicates the propagation of five coupled plane waves in the medium. Reflection of an incident plane wave from stress-free surface of a half-space is also considered to obtain the amplitude ratios of various reflected waves. A numerical example is considered to illustrate graphically the effects of rotation, frequency, void and diffusion parameters on speeds and amplitude ratios of plane waves.Originality/valueThe present problem covers the combined effects of rotation, voids and diffusion on characteristics of plane waves in linear thermoelastic material in the context of Lord and Shulman (1967) and Aouadi (2010) theories, which are not studied in literature yet.

Generalized Thermo-Elastodynamics for Polycrystalline Metallic Thin Films in Response to Ultrafast Laser Heating

A generalized thermo-elastodynamic model applicable to investigating single- and polycrystalline metallic films for their coupled thermal-mechanical responses to ultrafast laser heating is presented. Built upon the two-temperature model, the governing field equations feature hyperbolic electron and lattice energy transports that admit finite propagation speed and incorporate a term of energy dissipation to the generation of thermally induced mechanical disturbance. Electron energy transport in polycrystalline thin film is investigated in-depth by considering temperature dependence and size effect due to film surface and grain boundary scattering. Numerical results generated using the generalized one-dimensional model for a single-crystalline gold film are favorably examined against published experimental data for validation. Several averaged grain diameters are considered to explore the coupled thermal-mechanical responses in polycrystalline gold films of different thicknesses to better understand the dynamics and size effect on energy transport subject to pulsed femtosecond laser.