Differential Constitutive Equations Research Articles

Coupled responses of thermomechanical waves in functionally graded viscoelastic nanobeams via thermoelastic heat conduction model including Atangana–Baleanu fractional derivative

Accurately characterizing the thermomechanical parameters of nanoscale systems is essential for understanding their performance and building innovative nanoscale technologies due to their distinct behaviours. Fractional thermal transport models are commonly utilized to correctly depict the heat transfer that occurs in these nanoscale systems. The current study presents a novel mathematical thermoelastic model that incorporates a new fractional differential constitutive equation for heat conduction. This heat equation is useful for understanding the effects of thermal memory. An application of a fractional-time Atangana–Baleanu (AB) derivative with a local and non-singular kernel was utilized in the process of developing the mathematical model that was suggested. To deal with effects that depend on size, nonlocal constitutive relations are introduced. Furthermore, in order to take into consideration, the viscoelastic behaviour of the material at the nanoscale, the fractional Kelvin–Voigt model is utilized. The proposed model is highly effective in properly depicting the unusual thermal conductivity phenomena often found in nanoscale devices. The study also considered the mechanical deformation, temperature variations, and viscoelastic characteristics of the functionally graded (FG) nanostructured beams. The consideration was made that the material characteristics exhibit heterogeneity and continuous variation across the thickness of the beam as the nanobeam transitions from a ceramic composition in the lower region to a metallic composition in the upper region. The complicated thermomechanical features of simply supported viscoelastic nanobeams that were exposed to harmonic heat flow were determined by the application of the model that was constructed. Heterogeneity, nonlocality, and fractional operators are some of the important variables that contribute to its success, and this article provides a full study and illustration of the significance of these characteristics. The results that were obtained have the potential to play a significant role in pushing forward the design and development of tools, materials, and nanostructures that have viscoelastic mechanical characteristics and graded functions.

Analysis of thermo-mechanical incompatibility of fullerenes with polymer matrices of composites

Unique strength and rigidity properties of carbon nanotubes and fullerenes do them very perspective reinforcing admixtures into composites produced on the basis of polymeric and metallic matrices. It is established that even relatively small their dopes into composites can essentially enlarge physico-mechanical characteristics of polymers. At the same time, comparatively low value of the linear thermal expansion coefficient of carbon nanomolecules results in the possibility of the emergence of the internal mechanism of the additional intrastructural stresses origination in the polymer conditioned due to thermomechanical incompatibility of the composite fractions. To examine this possibility, the theoretical modelling of the thermomechanical deforming of the considered heterogeneous structures under thermal change action was carried out. In doing so, the fullerenes were simulated as a thin elastic shell with adduced (effective) thickness, elasticity modulus, Poissonn’s coefficient and linear thermal expansion coefficient, the polymer matrix was supposed to be a homogeneous elastic medium with the prescribed parameters of thermoelasticity. The system of constitutive ordinary differential equations was deduced, which described the thermoelastic stress-strainedstate of the elastic fragment chosen for consideration. These equations solutions were deduced in the closed form for the case when the system temperature changed in. Fenolformaldegid, epoxy rezin, polycarbonate, polyamide, polystirol, polyester, and polypropylene, possessing increased values of the linear thermal expansion coefficients, were chosen as the matrix materials. With these initial data, the values of the fullerene radial displacements were calculated for two magnitudes of the sphere radius. It is demonstrated that owing to the high value of the fullerene effective elasticity modulus, the thermal deflections of its wall poorly depend on the properties of the encompassing matrix and practically are equal to its free thermal displacements. The thermal stresses of the polymer matrices have the largest values in the zones of the interface surfaces and they decrease proportionally to cube of the radial coordinate, so, the damages provoked by them have localized character.

Analytical and numerical investigation of viscous fluid-filled spherical slip cavity in a spherical micropolar droplet

<abstract><p>This article presents an analytical and numerical investigation on the quasi-steady, slow flow generated by the movement of a micropolar fluid drop sphere of at a concentrical position within another immiscible viscous fluid inside a spherical slip cavity. Additionally, the effect of a cavity with slip friction along with the change in the micropolarity parameter on the movement of the fluid sphere is introduced. When Reynolds numbers are low, the droplet moves along a diameter that connects their centres. The governing and constitutive differential equations are reduced to a computationally convenient form using appropriate transformations. By using the resulting linear partial differential equations for the stream functions and using the method of separation variables, we can obtain their solutions. General solutions for velocity fields are found using spherical coordinate systems, which are based on the concentric point of the cavity; this allows to obtain solutions to the Navier-Stokes equations internal and external to the spherical droplet. The vorticity-microrotation boundary condition is used in regard to the micropolar droplet case in a viscous fluid. The normalised drag forces acted upon the micropolar drop are illustrated via graphs and tables for diverse values of the viscosity ratio and drop-to-wall radius ratio, with the change of the spin parameter that attaches the microrotation to vorticity. The correction wall factor is shown to increase with an increase in the drop-to-wall radius ratio, when moving from the gas bubble case to the solid sphere case, with an increase in the micropolarity parameter, and with an increase in the slip frictional resistance. This study is relevant due to its potential uses in a variety of biological, natural, and industrial processes, including the creation of raindrops, the investigation of blood flow, fluid-fluid extraction, the forecasting of weather conditions, the rheology of emulsions, and sedimentation phenomena.</p></abstract>

Physics-informed few-shot deep learning for elastoplastic constitutive relationships

Elastoplastic modeling is essential for accurately predicting material behavior in various engineering applications. However, existing approaches to developing intelligent models for elastoplasticity face limitations due to the scarcity of available data, particularly in cases where experiments cannot support full training of these models. To address this challenge, we propose a physics-informed few-shot learning framework that incorporates classic elastoplasticity theory as prior knowledge. Instead of directly solving specific constitutive ordinary differential equations, we utilize general mechanical inequations as physical regularization to confine the optimization space. Our framework is model-agnostic in the uniaxial scenario and free from complicated numerical implementation, preserving the advantages of the data-driven paradigm in terms of efficiency and simplicity. In the multiaxial scenario, the framework can automatically calibrate multiple interwoven material parameters of the underlying constitutive model, facilitating the generation of ample multiaxial data for downstream learning. To further improve convergence, we introduce a novel training strategy. A numerical experiments with only 32 pieces of training data validate the effectiveness of the developed framework. The trained uniaxial model exhibits exceptional generalization capabilities, achieving 44.9% higher accuracy than the purely data-driven model. For multiaxial relationships, our framework demonstrates remarkable efficiency in calibrating material constants compared to conventional machine or manual methods, and the trained model accurately reproduces highly nonlinear multiaxial elastoplastic curves. Our work addresses the pressing need for accurate elastoplastic modeling in the absence of large datasets and provides a promising new solution that can significantly improve generalization capabilities.

New insights into the extended and generalised PTT constitutive differential equations: weak flows

This work presents a comparison between the PTT-X (extended Phan-Thien and Tanner (PTT)) and the generalised PTT (gPTT) viscoelastic models. The PTT-X model was derived based on a combination of reptation and network theories, allowing in this way a microstructural justification for the kernel function. The gPTT model is based on the network theory, with an empirical kernel function for the rate of destruction of junctions, that proved to be effective fitting experimental rheological data for polymer melts and solutions. A review on the background of both models is provided and the two models are then compared considering simple flows. This comparison allows one to attribute in some way a microstructural nature to the parameters involved in the gPTT model. Also, a new analytical solution is derived for the Poiseuille flow of the PTT-X model.

Limits of Performance of Polyurethane Blowing Agents

A MATLAB program was developed to simulate urethane-forming reactions by solving over a dozen differential equations, energy balance, mass balance, and constitutive equations simultaneously. The simulation program was developed for half a decade to simulate the basic kinetics of polyurethane reactions and more complex phenomena that cannot be obtained in laboratories. In the current investigation, the simulation is applied to determine the limits of the performance of polyurethane foam formation. n-pentane, cyclohexane, and methyl formate were used as physical blowing agents, and water was used as a chemical blowing agent. The simulation code increases the accuracy of the results and makes the foam performance process less time- and money-consuming. Specifically, the MATLAB code was developed to study the impact of physical and chemical blowing agents at different loadings on the performance of rigid polyurethane foams. Experimental data were used to validate the simulation results, including temperature profiles, height profiles, and the tack-free time of urethane foam reactions. The simulation results provide a window for the proper type and the optimum amount range of different physical and chemical blowing agents.

A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials

Viscoelastic materials are widely used as devices for vibration control in modern engineering applications. They exhibit both viscous and elastic characteristic when undergoing deformation. They are mainly characterized by three time-dependent mechanical properties such as creep, stress relaxation and hysteresis. Among them, stress relaxation is one of the most important features in the characterization of viscoelastic materials. This phenomenon is defined as a time-dependent decrease in stress under a constant strain. Due to the inherent nonlinearity shown by the material response over a certain range of strain when viscoelastic materials are subjected to external loads, nonlinear rheological models are needed to better describe the experimental data. In this study, a singlenonlinear differential constitutive equation is derived froma nonlinear rheological model composed of a generalized nonlinear Maxwell fluid model in parallel with a nonlinear spring obeying a power law for the prediction of the stress relaxation behavior in viscoelastic materials. Under a constant strain-history, the time-dependent stress is analytically derived in the cases where the positive power law exponent, α ˂ 1 and α ˃1. The Trust Region Method available in MATLAB Optimization Toolbox is used to identify the material parameters. Significant correlations are found between the experimental relaxation data taken from literature and exact analytical predictions. The obtained results show that the developed rheological model with integer and non-integer orders nonlinearities accurately describes the experimental relaxation data of some viscoelastic materials.

On the role of tensor interpolation in solving high-WI viscoelastic fluid flow

The high Weissenberg number (Wi) problem (HWNP) has long been a challenge of viscoelastic fluid flow simulation. This Letter points out that the tensor interpolation method during solving the differential constitutive equations is the main origin of the loss of symmetric positive-definite (SPD) property of the conformation tensor, which is the trigger of the HWNP. Instead of component-based interpolation, we propose a tensor-based interpolation method for the conformation tensor, which is essentially SPD, and the results show that this method is very effective in dealing with the HWNP by significantly improving the numerical accuracy on the invariants of conformation tensor as well as greatly improving the SPD property of the conformation tensor. Moreover, the high-order total variation diminishing schemes can also be easily constructed and applied to solve high-Wi viscoelastic fluid flow under the proposed framework without adding artificial diffusion.

Mixed convective eyring-powell ferro magnetic nanofluid flow suspension towards a stretching surface with buoyancy effects through numerical analysis

This article examines the impact of buoyancy on the magnetic Eyring-Powell nanofluid flow toward a stretching surface. Coupled similarity equations are created from the governing flow equations. For the particular instance of pure fluid flow, the numerically computed self-similar results are matched with the available literature and found to be in acceptable harmony. The shooting approach was used to arrive at numerical computations to the constitutive ordinary differential equations. The impacts of different fluid flow parameters, nano concentration parameters and heat transfer, are shown graphically for both aiding and opposing flows. It has been discovered that for both aiding and opposing problems, the skin friction is less affected by the buoyant force brought on by temperature differences. Under buoyancy, the rate of heat transfer increments for aiding flow problem while it declines for opposing flow.

Impact of exothermic chemical reaction on MHD unsteady mixed convective flow in a rectangular porous cavity filled with nanofluid

The rectangular enclosure filled with nanofluid has several technical applications including building thermal management systems and photovoltaic thermal management systems. This investigation theoretically investigates the transport features of the time-dependent, mixed convective, electrically conducting nanofluid flow inside a cavity saturated with permeable medium. The impact of a uniform magnetic field applied in the transverse direction to the fluid flow and Arrhenius kinetics theory based exothermic chemical reaction are considered. The Buongiorno's model is adopted in the present analysis to characterize the nanofluid. The constitutive partial differential equations are solved by using the Marker and Cell (MAC) technique. The impact of the nondimensional numbers on transport phenomena of the present study is elucidated graphically. The outcomes disclose that the pertinent parameters such as the Reynolds number (Re), Hartmann number (Ha), Darcy number (Da), Lewis number (Le), Richardson number (Ri), thermophoresis parameter , Brownian motion parameter , and Frank-Kamenetskii number (Fk) play a main role on regulating the transport characteristics of the present study. Results demonstrate that the isotherms increase with escalating Frank–Kamenetskii number values. Higher Richardson number values improve the heat transfer inside the enclosure.

CONSTRUCTION OF THE EXACT SOLUTION OF BLOOD FLOW OF OLDROYD-B FLUIDS THROUGH ARTERIES WITH EFFECTS OF FRACTIONAL DERIVATIVE MAGNETIC FIELD AND HEAT TRANSFER

In this paper, we construct the exact solution of blood flow of Oldroyd-B fluids to represent the non-Newtonian characteristics of biofluids and to study the unsteady flow of blood and heat transfer through arterial segment with external magnetic field applied normal to the flow direction in the presence of thermal radiation and body acceleration. In the study, we investigate the influence of Caputo time-fractional parameter, heat transfer, external body acceleration, and magnetic field on the blood velocity and temperature distribution in a straight circular cylindrical vessel. The constitutive partial differential equation and temperature equation governing blood flow in the arterial wall are solved using Laplace and finite Hankel transforms. In addition, Gaver Stehfest’s algorithm is used for the inverse Laplace transform. The results obtained are interpreted graphically and discussed from the physiological point of view. The graphs show that the rate of heat transfer and blood velocity become less with the increase in the external magnetic field strength which is very good in regulating blood flow as well as temperature distribution during treatment. Based on the graphical representations, blood velocity and temperature distribution both decrease in ascending order of the fractional parameter as memory effect. Further, owing to the dissipation of energy caused by blood viscoelasticity and magnetic field effect, during pulsatile flow of blood, the heat transfer rate at the wall of the artery is enhanced. The significance of this study can be found in the application fields such as biomedical engineering, medicine and pathology.

Line contact lubricated by a fluid described by non-Newtonian Giesekus model

Abstract Modelling hydrodynamic lubrication helps engineers and material scientists designing lubricants that minimize viscous energy losses without compromising the durability of engine components. This type of modelling requires selecting lubricant rheological constitutive equations that provide realistic information on fluid behaviour within tribological contact zones. Due to complexity of differential constitutive equations the modelling effort was limited to generalized Newtonian or Oldroyd-B fluids. In this paper, we present more realistic description of hydrodynamic lubrication using the Giesekus fluid capturing both viscous shear thinning and the development of elastic stresses at high Weissenberg numbers. The problem analysis is done using a modified regular perturbation method with respect to the small mobility parameter $\alpha $. The perturbation analysis modification is caused by the fact that the problem contains several a priori unknown boundaries. The actual perturbation solutions for functions of pressure and gap, as well as the coordinate of exit from the contact and the exit film thickness, are obtained as linear functions of $\alpha $. The effects of the model parameters such as the mobility factor, relaxation time and the polymer contribution to viscosity on pressure, shear and extensional viscosities within the contact zone are analysed. It is shown that the presence of polymers helps to increase the gap and reduce the viscous losses associated with hydrodynamic friction. This result is consistent with empirical observations of fuel economy enhancement for polymer-containing lubricants.

Unsteady natural convection flow due to fractional thermal transport and symmetric heat source/sink

Unsteady natural convection flow of viscous fluids in a circular cylinder, due to a generalized fractional thermal transport is analytically studied. The considered mathematical model is based on a new fractional differential constitutive equation of the thermal flux suitable to describe the thermal memory effects. To develop the mathematical model, the time-fractional Caputo-Fabrizio derivative is used. The generalized constitutive equation becomes equivalent to the classical Fourier's law for the zero value of the fractional order of derivative. Analytical solutions for the fluid temperature and velocity are determined using the Laplace and finite Hankel transforms. The influence of the memory parameter on heat transfer and fluid motion is discussed by numerical simulations and graphical illustrations.

Simulation of Switched-Mode Power Conversion Circuits With Extended Impedance Method

Efficient simulation of switched-mode power conversion circuits is necessary for the analysis and optimization of power electronics. This paper introduces the extended impedance method (EIM) to the simulation and analysis of general switched-mode power converter circuits. By using the EIM method, the conventional impedance concept can be extended to involve more linear time-variant (LTV) and nonlinear devices, such as MOSFET switches and diodes, which are commonly used in switched-mode power conversion circuits. EIM models a circuit network from the component level in the frequency domain, rather than from the system-level constitutive differential equation in the time domain. Compared with the conventional averaging method, EIM can reveal more high-frequency features of a switched-mode converter circuit, in particular, those around the switching instants. A buck converter is taken as a representative example to validate the EIM based analysis. Both the simulation and experimental results prove that EIM provides precise numerical results, which match the real waveform, as the commercial simulators, e.g. PSpice and ADS, do. Moreover, it largely reduces the computational complexity towards more efficient and more flexible steady-state analysis of switched-mode power electronics.

A piecewise recursive semi-analytical method for effective evaluation of heterogeneous aging viscoelastic materials

A novel semi-analytical method in the time domain is proposed to predict the effective properties of heterogeneous aging viscoelastic materials. By virtue of a series expansion in a discretized time interval, variables and time varying constitutive coefficients can temporally be described more precisely, and a piecewise variable separation is realized. Consequently, a set of recursive constitutive equations are derived from a second-order differential constitutive equation with time-varying coefficients, and a spatially and temporally coupled effective evaluation is decoupled into a series of recursive spatial effective evaluations which can be tackled with Mori–Tanaka method (M–T), self-consistent method (SC), or other well developed skills, providing time-varying effective strain/stress. A recursive adaptive algorithm is stressed, so as to ensure stable temporal solution accuracy for different step sizes, and avoid the temporal inaccuracy caused either by the relatively lower truncation order or improper step size. Four numerical examples are provided to verify the effectiveness of the proposed approach by comparing the prediction results with reference solutions.

Homotopic solutions of an unsteady magnetohydrodynamic flow of Casson nanofluid flow by a vertical cylinder with Brownian and viscous dissipation effects

The present study investigates the mathematical and computational framework of an unsteady magnetohydrodynamic (MHD) Casson nanofluid flow subjected to convective heat transfer enhancement in a vertically moving cylinder. The two main important aspects of nanosized particles movement in nanofluids are the Brownian motion and thermophoresis force which have substantial impacts on the thermophysical characteristics of nanofluids. The mathematical formulation regarding the assumed diversified and control parameters are modeled in constitutive partial differential equations (PDEs). The boundary layer theory is used to reduce the nonlinearity of subsequent dimensionless partial differential equations. The homotopic solution of these dimensionless PDEs is obtained via an analytical algorithm. The impact of sundry influential physical parameters comprises flow and heat transfer individualities. Computational results of the homotopic scheme toward several governing parameters of interest, such as the skin friction, heat, and mass transfer at the diverse condition and dependency of different parameters are offered by means of plotted graphs and elaborated.

On Nonlocal Vertical and Horizontal Bending of a Micro-Beam

The vertical and horizontal bending of micro-beams subjected to axial compressive and transverse concentrated loadings is a common lateral deformation in micro-/nano-engineering that plays a significant role in the design and optimization of micro-/nano-devices. The present study aims to investigate it using the nonlocal theory. For this purpose, the simplified mathematical model is developed, and the nonlocal differential constitutive equation is applied. Since the mechanical properties of micro-beams are different from those of macro-beams, the non-classical nonlocal bending moment is introduced to improve the classical bending formulation in order to adapt to the vertical and horizontal bending of micro-beams. The effects of the external load, external size, structural stiffness, and internal characteristic scale on the vertical and horizontal bending deformation including the midpoint deflection and critical compression are presented. The present analytical model and results are validated by the finite element method. It is shown that the critical compression decreases with increasing the internal characteristic scale. Moreover, the midpoint deflection varies remarkably with respect to the axial and transverse loadings, structural stiffness, internal characteristic scale, and external size. An obvious nonlocal scale effect is found, in which the internal characteristic scale cannot be neglected compared with the external size. Besides, a threshold value of the structural stiffness is determined, and the connotation of the nonlocal interaction requires that the structural stiffness shall not be lower than that threshold. A mutual restriction between structural stiffness and external loadings is observed in the vertical and horizontal bending. In particular, it is further proved that the classical continuum mechanics can not be used in micro-/nano-scaled mechanics through a strange phenomenon that is contrary to mechanical common sense in the calculation example. The study is expected to be beneficial to the design and application of micro-beams subjected to the vertical and horizontal bending.

Themo-bioconvection of gyrotactic microorganisms in a polymer solution near a perforated Riga plate immersed in a DF medium involving heat radiation, and Arrhenius kinetics

In modern era, thermo-migration of microorganisms is an appealing research topic in bio-nanotechnology, bio engineering, and biomedical. In this context, a mathematical model describing thermo-bioconvection of Sutterby nanofluid flow including motile gyrotactic microorganisms near a perforated Riga plate under the physical impacts of heat radiation, and Arrhenius kinetics associated with binary chemical reaction is formulated and simulated here. The Darcy-Forchheimer (DF) law is applied to determine the porosity of porous media. The Grinberg term is taken for the Lorentz force owing to the parallel Riga plate wall. Appropriate translations are discharged to turn the constitutive partial differential equations (PDEs) into ordinary differential equations (ODEs), that are numerically computed by opting the Runge–Kutta-Fehlberg method (RKF-45) along with shooting strategy. The physical insights of various controlling variables on the transport profiles, Sherwood number, Nusselt number, and microorganisms density number are exemplified through requisite graphs and tables. It must be admitted that with enlarging Darcy number, the nanofluid velocity declines, while Forchheimer number has opposite consequence on it. The motile microorganisms density sharply decreases for improving values of activation parameter. The present modeling would provide preliminary guidances in a variety of biotechnological and industrial applications.

A Comparison between the Extended and the Generalised Phan-Thien and Tanner Constitutive Differential Equations

A Comparison between the Extended and the Generalised Phan-Thien and Tanner Constitutive Differential Equations

Understanding Thermal Lagging Behaviors in Thermoelectric Elements With the Dual-Phase-Lag Model

Abstract This study investigates the heat transport mechanism in semiconductor elements within a homogeneous thermoelectric cooling system using the dual-phase-lag (DPL) model. The thermal lagging behavior is analyzed and explored during the energy transport process. The coupled energy and constitutive partial differential equations are solved simultaneously to reduce the complexity of the high-order spatial and time derivatives. This approach simplifies the mathematical solution process and reduces numerical instabilities when compared to the conventional methodology in which either the temperature or heat flux is solved individually with a single equation. The effect of the thermal lagging behavior on energy transport is examined and compared to results by using the Cattaneo–Vernotte model. Furthermore, the phase-lag behavior on the temperature and heat flux profiles is investigated in detail. This study provides perceptive information for engineering applications in which the microscale heat transport phenomenon plays a significant role during the design process. Adding the dual-phase-lag model to the traditional heat diffusion model is a complementary option for engineers in the thermoelectric industry.