Abstract: Abstract This work proposed the use of the Carrera Unified Formulation (CUF) for the vibration and buckling analysis of structures subjected to thermal loads. In detail, the variation of natural frequencies for progressively large thermal loads is investigated. Here, particular attention is focused on the study of buckling thermal loads as degenerate cases of the vibration analysis and on the mode aberration caused by thermal stresses. From this standpoint, the use of CUF for the development of high-order beam and plate models is fundamental. Indeed, Lagrange-like (LE) polynomials are considered for developing the kinematic expansion and Layerwise (LW) theories are employed to characterize the complex phenomena that may appear in composite structures. A linearized formulation to study the natural frequencies variation as a function of the progressive increasing thermal loadings is adopted. Different isotropic and laminated composite structures have been analyzed and compared with the Abaqus solution to validate the presented methodology and provide some benchmark solutions. In addition, a parametric study was conducted to evaluate the stacking sequence and thickness effect in the vibration modes and thermal buckling loads. The results document the excellent accuracy and reliability of the presented methodology and show the potentialities of this numerical tool able to analyze cases that are difficult to study experimentally.

Abstract: Abstract Problems related to the change in state of a PCM offer a great contribution in the field of heat transfer, describing melting, freezing, and sublimation. While most articles already exist on the mathematical aspects of this type of problem that account for constant thermal conductivity, there is still insufficient modeling and solutions involving convection driven by heated fluid and temperature-dependent thermal conductivity, which are presently being considered. In connection with this, the moving boundary problems become highly non-linear and their exact treatments are restricted. To deal with non-linearity, the numerical solution of the problem is obtained via the LWC technique. Convergence analysis of the non-linear model is extensively presented. Numerical codes are validated against analytical results and found to be in strong agreement. Numerical data of the solidification of the alloy is presented for the validation of the current model. It is found that the rate of freezing increases as the value of the dimensionless parameter b increases. The impact of other parameters on freezing is discussed in detail. Furthermore, in the Stefan problem with convection and temperature-dependent thermal conductivity, the rate of freezing is faster than the standard problem, and in the Stefan problem with convection and fixed thermal conductivity.

Abstract: Abstract In this article, a two-dimensional problem is studied in a homogeneous isotropic thermoelastic medium with double porosity in the context of nonlocal thermoelasticity and the problem is treated in the presence of the Green–Naghdi model of type-III. Constitutive relations and field equations are derived and the state space approach is employed to solve this problem. Thermal shock is applied on the boundary of the surface. From these relations and expressions, we have calculated stress components, displacement components, and temperature for different values of distance and time numerically and plotted graphically to demonstrate and compare theoretical results. The influences of void parameters, locality, and nonlocality on the various physical quantities are also analyzed. Some particular and special cases are also deduced from this study and compared with the existing results.

Abstract: AbstractThis article analyzes the vibrational behavior of the double-layered micro-nanosphere to simulate the drug delivery mechanism in the biological structure under the influence of the temperature environment and the viscoelastic substrate for the polar lipid fraction E (PLFE) liposome isotropic material model. In order to obtain the micro-nano structural equations of the double-layer spherical shell, the displacement-strain relations of the shell have been expressed according to the first-order shear deformation theory and the non-local strain gradient theory. The partial differential equations of motion have been obtained by applying Hamilton’s principle. The system of linear couple equations have been solved using the Galerkin method. After validating the model with the results of the articles available in the literature to determine the accuracy of the presented model, numerical results are presented to investigate the effects of various parameters such as the radius of curvature ratio to length, damping coefficient, Kelvin-Voight damping coefficient, boundary conditions and temperature on vibration frequency response are provided and discussed. Results of the current research indicates that natural frequency decreases as the damping coefficient related to the viscous effect between the liposome bilayers increase. Results of the current research can be used as a benchmark for drug delivery applications.Keywords: Bilayer doubly curved micro-nanospheredrug deliveryliposomenonlocal strain gradient theory Disclosure statementThe authors declare that there is no conflict of interest.Additional informationFundingThis work is based upon research funded by Iran National Science Foundation (INSF) under project No. 99022724.

Abstract: Thermal buckling and vibration behaviors of functionally graded (FG) sandwich microplates are investigated by using a unified higher-order shear deformation theory and stochastic collocation (SC) method. Uniform and linear distributions are considered for thermal effect and lognormal distributions are used to characterize the variability of the materials properties. The governing equations are derived by using Hamilton’s principle and solved by Ritz’s approach. To demonstrate the effectiveness and accuracy of the current model, the results from SC are compared with those from Monte Carlo Simulation. The effects of boundary conditions, temperature distribution, thickness-to-length ratio, material scale characteristics and power-law index on the fundamental frequencies and critical buckling temperature of the FG sandwich microplates are investigated. The FG sandwich microplates’ stochastic analysis provides some new findings that may be utilized as references in the future.

Abstract: Abstract Viscoelastic materials (elastomers, resins, and polycrystalline metals) are of significant importance due to their numerous applications in biology, civil, and mechanical engineering alongwith other scientific disciplines. Viscoelastic materials possess strong relation with temperature and diffusion phenomenon and their mechanical properties exhibit memory dependence. With this view, fractional order strain is considered to study thermoviscoelastic diffusion interactions in the context of hyperbolic two-temperature-three-phase lag model. The investigation is carried out for an infinite, homogeneous, isotropic medium containing a spherical cavity. Initially, the medium is held quiescent. The boundary of the cavity is subjected to continuous concentration and ramp type thermal load in stress-free state. The problem is solved in Laplace transformed domain. The aim of this article is to analyze the behavior of field variables under thermoelastic models with different fractional order strain, ramping, and viscosity parameters. Physical data of Copper material is considered for numerical inversion technique and obtained results are plotted graphically. Here it is revealed that fractional order strain parameters lead to lower perturbations of stress and viscous effects result in higher temperature. One temperature and hyperbolic two temperature theories produce similar solutions. Ramp type thermal loading at the boundary significantly influences the temporal profile.

Abstract: The present article is related to propagation and reflection of elastic wave through a nonlocal generalized thermo-diffusive semiconducting elastic solid. The non-local theory is employed to study the wave behavior. Three phase lag model with higher order fractional order derivative is incorporated to discuss heat propagation through the medium, in addition with two phase lags diffusion equation. The Helmholz vector rule is applied to decompose the system into longitudinal and transverse components. The frequency dispersion relation indicates the presence of four coupled longitudinal and one un-coupled shear vertical wave. The speed of the waves is plotted against angular frequency for local and nonlocal medium. The cutoff frequency of the waves is also depicted graphically. The longitudinal P-wave is taken to be an incident wave at the free surface of the solid to compute the reflection coefficients. The influences of fractional order and nonlocal parameters on amplitude ratios are also studied. The effect of these parameters is found to be significant. The results are proved in the context of energy conservation. The results obtained from the current investigation are very useful for scientists working on problems of geophysics and various fields of mechanics.

Abstract: Abstract This research article presents an analysis of the behavior of an initially stressed monoclinic piezothermoelastic half-space based on memory-dependent three-phase lags heat transfer law (MPS) underlying a thermoelastic half-space (TS) with subject to the hyperbolic two-temperature (H2T), classical two-temperature (C2T), and without two-temperature (W2T) models. By applying the normal mode approach, calculate the amplitude ratios and further utilize them to obtain the waves’ energy ratios and interaction energy. The effect of memory-dependent derivatives (MDD), initial stresses, various kernel functions, and two temperature factors on the variation of energy ratios with incidence angle are graphically shown. The findings of this study have the potential to optimize material design, enhance seismic imaging techniques, improve thermal management in machinery, develop renewable energy systems, and facilitate materials characterization across diverse industries.

Abstract: Abstract In this work, a FGM thick wall tube under both thermal and mechanical loadings is studied by referring to the Mori-Tanaka method. The studied FGM tube is assumed to be made of two distinct linear elastically deformable materials equipped with unique volume fractions. Specially, its material parameters are firstly evaluated in the scheme of the Mori-Tanaka method which can more accurately depict the effective material properties of FGMs composites. Later, we derive the ordinary differential equation (ODE) of the displacement along radial direction, based on which we determine the approximate analytical results of displacement and later derive explicit forms of stress components along all the radial, axial and circumferential directions. After comparison, we found that the derived analytical results agree well with that obtained through numerical method. Moreover, for the same researching problem we found the Mori-Tanaka method could outperform the Voigt method. Further, the results are valid for materials with different Poisson’s ratios rather than constant Poisson’s ratios usually used in the existing references. Finally, parametric studies are also conducted by exploring the variations of the displacement and stress components affected by different volume fractions and distinct thermal conductivities and expansion coefficients.

Abstract: In the framework of thermo-electro-elasticity, the present paper investigates the singular behaviors of interface corners, interface cracks, composite wedges and spaces for one-dimensional hexagonal quasicrystal. The stress function and temperature variation can be described as the exponential form with a view to stress and heat flux singularities. Based on the Stroh formalism, the analytical expressions of singular orders of stress and heat flux are easily established by simple multiplication of the crucial matrix. Numerical examples of the singular orders are given for some general cases including single, bi-material, and tri-material wedges and spaces under different boundary conditions. Numerical results show that the geometry structures, material properties, boundary conditions, and heat conduction coefficients have great influences on singularities, but thermal moduli have no effect on singularities.

Abstract: Abstract This research analyzes the influence of temperature changes on the vibration of single-walled carbon nanotubes (SW-CNTs) composite joined conical-cylindrical shells. The governing dynamic equations of temperature-dependent CNTs with initial thermomechanical stresses are established using the Love shell assumptions and classical shell theory. The initial thermomechanical stresses are derived from the linear membrane approach method. Two possibilities are assumed for the calculation of temperature change: a uniform temperature distribution and steady-state heat transfer by conduction through the thickness of the shell. The initial thermomechanical stresses are determined using the linear membrane approach. The generalized differential quadrature (GDQ) method is used to solve the equations after combining it with continuity conditions between the conical part and the cylindrical part and various boundary conditions. After validating the natural frequency and the different types of temperature distribution with the studies of other researchers, the effects of semi vortex of the cone, the volume fraction, and the type of distribution on the temperature rise are given as the results. The type of temperature distribution has the greatest influence among the parameters.

Abstract: Beam, plate, and shell structures manufactured from functionally graded materials (FGMs) are becoming increasingly prevalent in practice. The manufacture of these structures is not free from flaws, one of which is that the thickness distribution of the material does not change continuously. This is the first study to apply two distinct analytical solutions to examine the static and free vibration responses of imperfect FG nanobeams in terms of material distribution and temperature influence. The calculation formulas are based on the novel shear strain theory, which, despite being simple, accounts for both bending and shear strain, rendering the calculation procedure highly convenient. Either of the two precise solutions is computed using direct integrals and can be computed for various boundary conditions, which is not possible with Navier’s solutions. The results of the verification indicate that the approaches given in the article guarantee the needed precision. This study also analyzes the effect of several geometrical, material, nonlocal, and imperfection characteristics on the static bending and free oscillation response of imperfect FG nanobeams.

Abstract: Abstract The motive of this work is to figure out the impact of hyperbolic two temperature on the forced vibrations of thermodiffusive axisymmetric half-space in the framework of fractional thermoelastic diffusion with three-phase lags. The governing equations are simplified by introducing potential functions and solved by applying the Laplace and Hankel transform techniques. The boundary surface is restricted by axisymmetric thermal, mechanical, and mass concentration loads. The expressions for field quantities in closed form are obtained analytically. To convert the field quantities into physical space, a mathematical inversion process is employed. The numerical calculation is performed for copper material. The numerically computed results are presented graphically to portray the different physical effects. The results are compared with thermoelastic diffusion with single-phase lag and dual-phase lag.

Abstract: Abstract The dynamic thermomechanical analysis on stiffened composite plates with damages are investigated in this article based on the extended layerwise method (XLWM) and three-dimensional finite elements method; a thermomechanical extended-layerwise/Solid-element method (TELW/SE) is established. In TELW/SE, the thermomechanical extended-layerwise method (TELW) and thermomechanical three-dimensional solid elements are used to simulate plate and stiffeners, respectively. The final governing equations are assembled by using the interface conditions to ensure the compatibility of displacements and temperature, together with the internal force equilibrium. Several typical damages in stiffened composite plates can be described accurately in the proposed method, such as delamination, transverse crack and stiffener/plate interface debonding.

Abstract: Abstract A complex slowness vector governs the propagation of plane attenuated waves in a thermoelastic medium with arbitrary anisotropy. This vector is resolved into a propagation vector and attenuation vector, which may have different directions to represent the inhomogeneous propagation of attenuated waves. The attenuation vector is resolved further to identify the homogeneous attenuation and inhomogeneous attenuation. An incidence of attenuated wave at the plane boundary of thermoelastic medium results in four reflected inhomogeneous waves. A generalized form of Snell’s law relates the slowness vectors of reflected waves to the slowness vector of incident wave. The complex slowness vector of each reflected wave is resolved to define its phase velocity, propagation direction, attenuation angle and attenuation coefficient. At thermoelastic boundary, the conservation of incident energy is achieved through a contribution from the interactions of reflected waves among themselves as well as with incident wave. A numerical example is considered to analyze the propagation characteristics of reflected waves at insulated or isothermal boundary for inhomogeneous propagation of incident wave.

Abstract: Abstract Joule heat induced by electric current may well lead to thermal stress and contribute to structural failure and crack evolution in a range of modern materials. In this paper, we use complex variable methods to obtain the thermoelastic fields around an elliptic cavity in a plate subjected to remote electric and thermal loads. Of particular interest is the fact that we consider the temperature dependency of the corresponding thermoelastic coefficients. We introduce an analytical method which is verified by the fact that it leads to results which reduce to the classical analytical results in the literature when the thermoelastic coefficients are taken to be temperature-independent. We find that when the elliptic cavity is reduced to the representation of a linear crack, the corresponding stress intensity factor increases to a greater extent under the temperature-dependent assumption versus the corresponding classical results. Our findings are significant in forming the basis for the analysis of the evolution of crack growth under severe temperature gradients such as those occurring as a result of Joule heat loading.

Abstract: Kinematically nonlinear coupled thermoelasticity of the FGM cylindrical shell is investigated under heat shock. The energy and equations of motion are solved simultaneously as a system of equations for an FG cylindrical shell. The classical theory of coupled thermoelasticity is used to solve the problem. The first-order shear deformation theory for the shell is considered. Also, the terms thermal coupling and rotational inertia are included in the solution. The finite element method is employed to solve the problem numerically in the space domain and the Newmark method in the time domain. Temperature distribution across the shell thickness is assumed to be linear. The radial displacement for different values of the power law index is plotted in terms of time. The occurrence of thermal buckling is examined.

Abstract: The homogeneous isotropic micropolar porous thermoelastic plate under memory-dependent derivatives (MDD) has been taken into consideration to investigate the wave propagation regarding Lord-Shulman (LS) and Green-Lindsay (GL) theories. The governing equations are non-dimensionalized and solved using normal mode analysis. The dispersion equations for both symmetric and anti-symmetric modes of wave propagation are obtained in the assumed model. The comparisons between the variation of phase velocity and attenuation coefficient corresponding to LS and GL theories are shown graphically using MATLAB software. The amplitudes of dilatation, volume fraction, micro-rotation and temperature distribution are computed analytically and presented in the form of graphs for LS and GL theories in the presence of MDD. Various particular cases are discussed in detail.

Abstract: Abstract The theory of wave propagation in non-isothermal porous rocks has been introduced in geophysics in recent years by combining the single-phase-lag (SPL) Lord-Shulman (LS) model of heat conduction with Biot’s poroelasticity theory. However, the theory of SPL thermoporoelasticity is inadequate in describing the lagging behavior of thermal relaxation for wave dissipation due to fluid and heat flow effects. We address this problem by incorporating a dual-phase-lags (DPL) model of heat conduction into thermoporoelasticity, utilizing analytical solutions and numerical simulations. The DPL model involves two lagging times: the (macroscopic) heat-flux lagging time τq from the LS model and an additionally introduced lagging time τT of temperature gradient that characterizes the fluid phase. A plane-wave analysis predicts four propagation waves, namely, fast P, slow P, thermal (T), and shear (S). We calculate wavefield snapshots by using a finite-difference solver for the DPL thermoelastic equations and provide further insight into the physics of two lagging times for porous rocks. The simulations show that the DPL model induces higher thermal attenuation and larger velocity dispersion compared to the SPL model, especially at high frequencies. The influence of fluids is crucial for wave propagation within thermoporoelastic media.

Abstract: Abstract The research problem aims to investigate the effect of heat and moisture on the propagation of plane waves in a micropolar hygro-thermoelastic medium. The governing equations of a hygro-thermoelastic micropolar medium are developed and solved to determine the velocity equation. The plane-wave solution reveals that the medium is propagated by two coupled transverse displacement waves, namely Coupled Transverse Micropolar wave (CTM-wave) and Coupled Transverse Displacement wave (CTD-wave). Additionally, three coupled longitudinal waves are observed: thermal diffusion TD-wave, longitudinal displacement P-wave, and moisture diffusion mD-wave. To assess the characteristics of these waves in the micropolar hygrothermal medium, the speed and distance of propagation are calculated for each wave type. This analysis provides information on how fast and how far the P-wave, TD-wave, mD-wave, CTM-wave, and CTD-wave can travel in the medium. Furthermore, the research problem determines the equations for the coefficient of reflection and energy ratio when a coupled plane wave is incident on the medium. These coefficients quantify the amount of reflected energy compared to the incident energy and provide insights into the wave behavior at the interface of the medium. Finally, the variations in energy ratio and reflection coefficient are depicted graphically, allowing for a visual representation of how these quantities change with different parameters or conditions. These graphs provide a comprehensive understanding of the wave behavior and the effects of heat and moisture on the propagation characteristics in the micropolar hygrothermal medium.

Abstract: In this study, we present a general numerical solution to the thermoelastic problem of interacting elliptical inhomogeneities in an infinite matrix material. Our solution provides a framework to study microstructural responses to the external mechanical, thermal, electric, and magnetic fields simultaneously. The proposed semi-analytical method, which enables us to reduce the computational cost and calculate the interactions between many inclusions, is based on conformal mapping, series expansion of the corresponding complex potentials and boundary conditions. The temperature, stress, strain, and displacement functions are derived explicitly in the matrix material and inclusions. The thermal loadings tested in this study include temperature variations and a uniform heat flow. The model is validated against multiple analytical calculations and benchmark numerical problems. The method is ready to apply for simulations with many inclusions.

Abstract: Abstract Analysis and numerical results for the axisymmetric vibrations of non-uniform two-dimensional functionally graded circular plates subjected to two-directional temperature distribution are presented on the basis of the exact neutral surface and Mindlin plate theory. The mechanical properties of the plate material are supposed to be temperature-dependent and graded in thickness as well as in radial direction. Based on the thermal boundary conditions, a classical solution of the two-directional heat conduction equation is obtained by the separation of variables method. Equations for thermoelastic equilibrium and axisymmetric motion of the plate are derived in the framework of Hamilton’s principle. The numerical solution of these equations is carried out by the generalized differential quadrature method to compute the thermally induced displacements and natural frequencies using MATLAB. The lowest three values are reported as the frequency parameter for the first three modes of clamped and simply-supported plates. To show the influence of the exact neutral surface, taper parameters, material graded index, heterogeneity and density parameter, numerical results are presented. Mode shapes are plotted for the specified plates.

Abstract: Abstract There is a mathematical analogy between the poroelastic and thermoelastic behavior of the compressional P wave, whose dissipation is due to the coupling between the elastic deformation and the phenomenon of pressure and heat diffusion, represented by Darcy’s law and heat conduction, respectively. This attenuation effect is more pronounced in heterogeneous media, where conversion of the fast P wave to the slow (Biot) and thermal diffusive modes occurs at material interfaces. Specifically, the problem is to obtain the P-wave properties of a porous medium due to temperature gradients between the solid and fluid phases. Then, we consider a simplified one-dimensional porous medium and obtain the quality factor, Q, and phase velocity as a a function of frequency, based on an isostress condition and using Gassmann equation and the Kramers Kronig relations.The model allows for the incorporation of a relaxation time to simulate a proper wave-like behavior at high frequencies, avoiding in this way infinite velocities. Moreover, we perform a complete analysis varying the different parameters, namely, the heat conduction, the specific heat, the thermal expansion, the types of solid and fluid and the pore size.

Abstract: Abstract The in-situ railway repair welding process consists of multiple weld passes, which makes it significantly different from other rail welding processes. In this study, finite element simulations of repair welding are performed to predict the resulting microstructure and residual stresses. To accurately simulate the material behavior, the modeling includes phase transformation kinetics, cyclic hardening plasticity, transformation induced plasticity, and multi-phase homogenization. More specifically, four different homogenization methods are investigated: isostrain, isostress, self-consistent and linear mixture rule. The performance of the material modeling is demonstrated by simulating multiple weld passes using a classical three-bar welding experiment. Based on the results, the self-consistent method and linear mixture rule are used in a 3D full-scale railhead repair weld simulation, in which the former generates a more realistic mechanical response. The immense computational cost associated with 3D full-scale, full-detail multi-pass welding simulations is addressed by exploring different model reduction schemes. From this study, a 2D generalized plane strain model, extended with out-of-plane axial and bending stiffness, is found to replicate the full-scale model at a mere fraction of the computational cost. Finally, the longitudinal residual stress distribution obtained from the reduced model is shown to correlate well with experimental measurements.

Abstract: Abstract The present work is focused on investigating the axisymmetric deformation in a thick circular plate with a heat source in modified couple stress thermoelastic (MCT) under void, diffusion, and phase lags impacts. The new set of governing equations is formulated and solved by using Laplace and Hankel transform techniques after converting the system of equations into dimensionless form and using the potential functions. The plate is subjected to a ramp-type heat source along with a thermal source, mass concentration source, and source over volume fraction field of voids (SVFV). For application, particular types of sources are taken to demonstrate the utility of the problem. The physical quantities like displacements, temperature field, mass concentration, chemical potential, and volume fraction field are determined analytically in the closed form. A numerical mathematical technique is employed to determine the resulting quantities for the original region and displayed in form of graphs to examine the different physical effects. The problem is validated by comparing results with those obtained by different authors. The results provide a motivation to investigate thermal conducting modified couple stress elastic solid under the physical impact of mass diffusion, porosity and phase lags as a new class of material. The present work is very much expected to be useful for various deformation and vibration problems in geophysics, material science, engineering, and biothermoelastic materials.

Abstract: Abstract During the charging process of a lithium-ion battery, heat generation and volume changes occur, leading to internal stresses between the active plates and current collectors due to thermal and diffusivity mismatch. Excessive stresses can result in electrode fracture, causing mechanical and electrical failures of the battery. This paper presents analytical solutions for the temperature and concentration distribution inside the lithium-ion battery during galvanostatic charging, and the associated thermal and diffusion induced stresses of layered electrode are also predicted. Numerical analysis demonstrates the importance of considering the generated heat during electrochemical charging when assessing the stress state of the electrodes. To prevent thermal runaway, it is crucial to ensure efficient heat dissipation during high-current charging. The maximum stress is observed at the lateral surfaces of the active plates, highlighting the significance of thermal stress in reducing the diffusion induced stress. Furthermore, employing a current collector with a higher coefficient of thermal expansion and lower elastic modulus can mitigate the overall stress on the active plate. This paper offers a valuable theoretical model for parametric studies and optimization of charging strategies for lithium-ion batteries.

Abstract: The study presents a theoretical modeling and analysis for the purpose of designing of acoustic devices and sensors with high performance. The propagation of Rayleigh-type waves in a piezoelectric substrate guided with fiber-reinforced layer is studied. Lord-Shulman (L-S) theory is used for thermoelasticity for both layer and substrate. Direct Sturm-Liouville method is employed to obtain the dispersion relation using traction free, thermally insulated, electrically open and short boundary conditions. The effect of different material parameters (e.g., reinforce, thickness ratio, and amplitude of thermal displacement) on the velocity (phase and group) of the considered wave are shown graphically by taking suitable numerical examples. The effect of specific loss factor is presented through graph. The present study may be utilized for better optimization of surface acoustic wave devices and sensors.

Abstract: Abstract In this paper we consider the three-phase-lag model of heat conduction that involves second-order effects in phase lag of the heat flux vector. This model leads to a fourth-order in time equation of Moore–Gibson–Thompson type. We use the thermodynamic restrictions derived from the compatibility of the constitutive equation with the Second Law of Thermodynamics to study the properties of the solutions of the initial boundary value problems associated with the model in concern. In this connection we establish a series of well-posedness results concerning the related solutions like: uniqueness, continuous data dependence, exponentially stability or domain of influence. Furthermore, based on the thermodynamic restrictions, we show that the thermal model in question admits damped in time propagating waves as well as exponentially decaying standing modes. We also show that when the thermodynamic restrictions are not fulfilled, then wave solutions appear that cause the energy blows up as time goes to infinity.