Thermoelastic Constitutive Relations Research Articles

Magneto-thermoelastic nonlinear dynamic modeling of a rotating functionally graded shell

Research is conducted on the dynamic modeling of a rotating functionally graded (FG) cylindrical shell subjected to magnetic and temperature fields. Based on the elasticity theory and generalized Hooke's law on the physical neutral surface, nonlinear geometric equations and thermoelastic constitutive relations are determined. According to the Kirchhoff-Love theory, variational formulas of strain energies for deformation, temperature, and centrifugal force are obtained. Considering the rotational effect, the kinetic energy and its variational formula are derived. The electromagnetic force model incorporating magnetization effect of the ferromagnetic FG shell is established by utilizing the electromagnetic theory. Subsequently, the magneto-thermoelastic dynamic model of the rotating FG shell is developed by adopting the Hamilton's principle. The model can reveal the coupling mechanisms of the interaction and superposition of multi-physical fields. Finally, taking the primary resonance as example, detailed numerical analyses are performed to investigate the effects of different parameters on vibration response and dynamical stability.

Three-dimensional analysis of thermoelastic damping in couple stress-based rectangular plates with nonlocal dual-phase-lag heat conduction

Since thermoelastic damping (TED) plays a decisive role in the functioning of micro/nanoresonators, accurate assessment of its magnitude in these tiny systems is imperative. Successfully simulating the thermomechanical behavior of various mechanical elements requires consideration of heat transfer in all three principal directions. Furthermore, there is no denying the impact of size on the mechanical and thermal domains. In light of these reasons, the current study strives to create a three-dimensional (3D) size-dependent TED model for rectangular micro/nanoplates via the use of the modified couple stress theory (MCST) and nonlocal dual-phase-lag (NDPL) heat transfer model concurrently for the first time. To accomplish this, by applying the Galerkin method to the 3D NDPL-based heat equation, the relation of temperature field is derived. Moreover, the coupled thermoelastic constitutive relations are extracted by including the couple stress effect. Lastly, by exploiting the obtained relations in the energy dissipation approach, an analytical TED formula in the form of infinite trigonometric series is attained. To appraise the rightness of derived formulation, the method of model reduction is applied. Moreover, an elaborate convergence analysis is implemented to realize the number of sufficient terms of the provided solution that result in well-grounded outcomes. A rigorous simulation study is also accomplished on the role of 3D heat conduction, scale parameters of MCST and NDPL model, boundary constraints, geometry and material in the variations of TED. Findings imply the definite impact of heat conduction dimension on TED value in rectangular plates with a large ratio of thickness to length and width.

A new combined asymptotic-tolerance model of thermoelasticity problems for thin uniperiodic cylindrical shells

AbstractThe objects of consideration are thin linearly thermoelastic Kirchhoff–Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential direction (uniperiodic shells). The aim of this contribution is to formulate and discuss a new averaged mathematical model for the analysis of selecteddynamic thermoelasticity problemsfor the shells under consideration. This so-called combined asymptotic-tolerance model is derived by applying the combined modelling including the consistent asymptotic and the tolerance non-asymptotic modelling techniques, which are conjugated with themselves into a newprocedure. The starting equations are the well-known governing equations of linear Kirchhoff–Love theory of thin elastic cylindrical shells combined with Duhamel–Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation. For the periodic shells, the starting equations have highly oscillating, non-continuous and periodic coefficients, whereas equations of the proposed model have constant coefficients dependent also on a cell size.

A new combined asymptotic-tolerance model of thermoelasticity problems for thin biperiodic cylindrical shells

The objects of consideration are thin linearly thermoelastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss a new averaged mathematical model for the analysis of selected dynamic thermoelasticity problems for the shells under consideration. This so-called combined asymptotic-tolerance model is derived by applying the combined modelling including the consistent asymptotic and the tolerance non-asymptotic modelling techniques, which are conjugated with themselves into a new procedure. The starting equations are the well-known governing equations of linear Kirchhoff-Love theory of thin elastic cylindrical shells combined with Duhamel-Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation. For the periodic shells, the starting equations have highly oscillating, non-continuous and periodic coefficients, whereas equations of the proposed model have constant coefficients dependent also on a cell size.

Scale Effects on Thermoelastic Coupling Wave Propagation of Micro-Beam Resonator Using Nonlocal Stain Gradient and Generalized Thermoelasticity

Coupled thermoelastic analysis in the study of microscale solid mechanics has attracted considerable attention along with the miniaturization and functionalization of devices. Higher-order continuum mechanics theories have considered scale effects; however, the theories to predict thermoelastic performances of structures in micro-scale are still incomplete. The purpose of this paper is to study scale and thermal-induced effects by considering stress nonlocality and strain gradient elasticity in micro-structure resonators. The thermoelastic constitutive relations are established based on the nonlocal strain gradient (NSG) theory. The wave propagation problem of a Timoshenko micro-beam in conjunction with Green and Naghdi’s generalized thermoelastic (G–N) theory is investigated. The governing equations are derived and analytically solved by the eigenvalue method. Finally, the dispersion relation of a frequency spectrum is determined and the influences of scale parameters and coupled thermoelastic effect are discussed; in addition, the analysis results are compared with those under different size-dependent theories accordingly.

Mathematical modelling of thermoelasticity problems for thin biperiodic cylindrical shells

The objects of consideration are thin linearly thermoelastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss two new averaged mathematical models for the analysis of selected dynamic thermoelasticity problems for the shells under consideration: the non-asymptotictolerance and the consistent asymptotic models. The starting equations are the well-known governing equations of linear Kirchhoff-Love theory of thin elastic cylindrical shells combined with Duhamel–Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation in which the heat sources are neglected. For the microperiodic shells under consideration, the starting equations mentioned above have highly oscillating, non-continuous and periodic coefficients. The tolerance model is derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. It has constant coefficients depending also on a cell size. Hence, this model makes it possible to study the effect of a microstructure size on the global shell thermoelasticity (the length-scale effect). The consistent asymptotic model is obtained using the consistent asymptotic approach. It has constant coefficients being independent of the period lengths. Moreover, the comparison between the tolerance model for biperiodic shells proposed here and the known tolerance model for cylindrical shells with a periodic structure in the circumferential direction only (uniperiodic shells) is presented.

Dynamics of pull and release of graphene nanoribbons

The pull and release molecular dynamics simulations of graphene nanoribbons (GNR) lead to their length dependent several folded conformations in the ground states at finite temperatures. The folding occurs when the potential energy of the stretched GNR is sufficient to overcome the energy barriers between the flat GNR and its folded conformations. We explore the dynamics of GNRs during their pull and release as the flat GNR develops intrinsic ripples as soon as thermal fluctuations are introduced and rich mechanical phenomena are found even before the folding process starts. Nudged elastic band calculations were performed to understand the nature of energy barriers and the minimum energy paths between the flat and the rippled nanoribbons. The potential and the kinetic energies of nanoribbons show strong oscillatory characters with attenuation in amplitudes at the start and the end of the pulling process, but as soon as the release process starts the oscillatory nature becomes weak giving rise to small fluctuations. Both thermostatted and unthermostatted nanoribbons were studied with some important observations. Different equivalent continuum models were analyzed which can mimic these experiments. It has been found that the pull and release of the GNRs without any intrinsic rippling can be successfully simulated with a thin plate continuum model using Hooke’s law which does not account for geometric nonlinearities and therefore is aptly able to capture small deformations of the flat GNR. This relation fails when the GNR develops ripples and then undergoes through the pull and release experiments. The rippled GNR was modeled by introducing ripples in the flat continuum GNR by nonlinear curve fitting of the atomistic z-coordinates with a sinusoidal function. Then three different constitutive relations were tried and it was found that modeling GNRs as St. Venant–Kirchhoff materials, which assume a linear relation between the second Piola–Kirchhoff stress tensor and the Green-St. Venant strain tensor correctly reproduce the atomistic behavior before the folding starts. The temperature results were also reproduced by solving the energy balance equation where a thermoelastic constitutive relation between the second Piola–Kirchhoff stress tensor and the Green-St. Venant strain tensor subtracted by the thermal strain tensor was assumed.

Nonlinear steady-state responses of weakly bonded composite shell structure under hygro-thermo-mechanical loading

The nonlinear micromechanical finite element (FE) model is derived to compute the steady-state time-dependent deflections of the weakly connected layered composite panel. The distorted structural geometry is modelled mathematically via the nonlinear strain kinematics in the framework of two different displacement theories (constant and linear functions of the through-thickness stretching term). The thermo-elastic constitutive relation is invoked to count the modified composite properties due to the change in environment. Besides, a sub-laminate approach with end continuity conditions is adopted to model the loosely bonded layers. The equation of motion of shallow panel under the mechanical excitation and the environmental effect is expressed through Hamilton’s principle due to the environmental changes. The deflection parameters due to the steady-state loadings are predicted using the combinations of three different techniques, i.e., the FE, the direct iterative method and Newmark’s constant acceleration. The nonlinear model responses are compared with published data considering all of the strain terms to capture the actual structural distortion. The applicability of the micromechanical FE model is deliberated by taking the combined influences of geometry, property, end constraints and the debonding parameters (size, location and position). A few conclusions are dawn on the micromechanical model after the numerical experimentations.

Thermal vibration characteristics of pre/post-buckled bi-directional functionally graded tapered microbeams based on modified couple stress Reddy beam theory

This paper investigates the nonlinear vibration characteristics of pre- and post-buckled nonuniform bi-directional functionally graded (BDFG) microbeams subjected to nonlinear thermal loading. The formulations derived herein are based on thermoelastic constitutive relations of the higher-order Reddy beam theory in conjunction with the modified couple stress theory, von Karman nonlinear strains, and physical neutral plane concept, for the first time. Power-law model is employed to describe the continuous variation of temperature-dependent thermomechanical material properties of BDFG in both the thickness and length directions. Variable microstructure length scale parameter and Poisson’s ratio in both directions are considered. The variable coefficient nonlinear governing equations and associated boundary conditions of nonuniform BDFG microbeam in the thermal environment are derived utilizing the Hamilton principle. The obtained equations are discretized using the differential quadrature method accounting for various boundary conditions, and iterative Newton’s method is adopted to solve the resulting nonlinear algebraic equations. The developed model and solution procedure are validated by comparing the predicted results with those in the available literature. To this end, extensive parametric studies are carried out to explore the influence of linear and nonlinear temperature profiles, microstructure length scale parameter ratio, microstructure length scale parameter-to-thickness ratio, gradient indices in thickness and length directions, material-temperature dependency, and taperness parameters on the nonlinear vibration response and higher-order mode frequencies of tapered BDFG microbeams accounting different boundary conditions.

Tolerance and asymptotic modelling of dynamic thermoelasticity problems for thin micro-periodic cylindrical shells

The problem of linear dynamic thermoelasticity in Kirchhoff–Love-type circular cylindrical shells having properties periodically varying in circumferential direction (uniperiodic shells) is considered. In order to describe thermoelastic behaviour of such shells, two mathematical averaged models are proposed—the non-asymptotic tolerance and the consistent asymptotic models. Considerations are based on the known Kirchhoff–Love theory of elasticity combined with Duhamel-Neumann thermoelastic constitutive relations and on Fourier’s theory of heat conduction. The non-asymptotic tolerance model equations depending on a cell size are derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. The consistent asymptotic model equations being independent on a microstructure size are obtained by means of the consistent asymptotic approach. Governing equations of both the models have constant coefficients, contrary to starting shell equations with periodic, non-continuous and oscillating coefficients. As examples, two special length-scale problems will be analysed in the framework of the proposed models. The first of them deals with investigation of the effect of a cell size on the shape of initial distributions of temperature micro-fluctuations. The second one deals with study of the effect of a microstructure size on the distribution of total temperature field approximated by sum of an averaged temperature and temperature fluctuations.

Thermodynamic properties and thermoelastic constitutive relation for cubic crystal structures based on improved free energy

This study proposes new models for thermodynamic properties and thermoelastic constitutive relation of materials with cubic crystal structures. The motion equation of atoms in cubic crystal structures is decomposed into structural deformation and thermal vibration at first. Then based on the thermo-mechanical coupling mechanism, both the thermal and mechanical aspects of structural deformation are investigated. And the thermal vibration equation is built at structural deformation positions by considering the non-harmonic approximation of interatomic potentials. Further, the improved formula of free energy is established as a function of the structural deformation and thermal vibration frequencies, which includes the thermo-mechanical coupling and non-harmonic effects. And the thermodynamic properties, including internal energy, entropy, heat capacity and thermal expansion, are derived from the free energy. Besides, based on the multiplicative decomposition of deformation gradient, the thermoelastic constitutive relation is constructed at finite deformation for cubic crystal materials. Finally, numerical results of the thermodynamic properties, thermoelastic stress–strain relations and elastic constants for face-centered cubic metals Cu, Au and Ag from 0 K to the melting points at 0–50 GPa are provided by comparing with the experimental data to demonstrate the usability of the present models.

Nonlocal thermo-elastic constitutive relation of fibre-reinforced composites

The effective properties of composite materials have been predicted by various micromechanical schemes. For composite materials of constituents which are described by the classical governing equations of the local form, the conventional micromechanical schemes usually give effective properties of the local form. However, it is recognized that under general loading conditions, spatiotemporal nonlocal constitutive equations may better depict the macroscopic behavior of these materials. In this paper, we derive the thermo-elastic dynamic effective governing equations of a fibre-reinforced composite in a coupled spatiotemporal integral form. These coupled equations reduce to the spatial nonlocal peridynamic formulation when the microstructural inertial effects are neglected. For static deformation and steady-state heat conduction, we show that the integral formulation is superior at capturing the variations of the average displacement and temperature in regions of high gradients than the conventional micromechanical schemes. The approach can be applied to analogous multi-field coupled problems of composites.

Thermo-elastic material parameters identification using modified error in constitutive equation approach

This paper presents an identification procedure for anisotropic thermo-elastic heterogeneous material profile based on modified error in constitutive equation (MECE) approach. The inverse problem is posed as an optimization problem where the objective functional evaluates the difference in constitutive relation that associates kinematically admissible strain field to the statically admissible stress field. An additional term due to corruption in measurement data is included in the cost functional as a penalty form. While following standard MECE-based identification procedure, we have proposed a trace norm of the constitutive discrepancy functional that arises due to two dissimilar fields for material parameter update. In the process, we obtain explicit parameter update formula for general anisotropic thermo-elastic material. However, unlike elastic case, parameter update equations are nonlinear due to thermo-elastic constitutive relation. Finally, the potential of the proposed procedure in estimating anisotropic material parameters is illustrated through some large-scale parameter estimation problems.

Stress, deflection, and frequency analysis of CNT reinforced graded sandwich plate under uniform and linear thermal environment: A finite element approach

The frequency, deflection, and stress values of carbon nanotube reinforced sandwich plate structure is investigated numerically under the influence of the mechanical loading and thermal field. The effective properties of individual components of sandwich structure (face sheet and core layer) are assumed to be temperature‐dependent and modeled via thermoelastic constitutive relations. In addition, the distribution of carbon nanotube through the thickness of the face sheets are assumed to follow the rule based grading pattern. Further, the sandwich structural panel modeled using the extant high‐order shear deformation kinematic theory for the mathematical purpose. The desired structural responses (frequency, deflection and stress) obtained numerically through a generic finite element based computer code developed in MATLAB. The convergence criteria and the accuracy of the current high‐order finite element solutions have been assessed by solving a sufficient number of numerical examples. Lastly, the influences of structural parameters (core to face thickness ratios, carbon nanotube fractions, type of loading, aspect ratios, end constraints, length to thickness ratios, type of carbon nanotube gradings, and type of temperature gradings) on the transverse deflection and frequencies of carbon nanotube‐reinforced sandwich structure are obtained under the elevated thermal field and the subsequent conclusions discussed accordingly. POLYM. COMPOS., 39:3792–3809, 2018. © 2017 Society of Plastics Engineers

A Novel Constitutive Formulation for Rubberlike Materials in Thermoelasticity

The objective of this paper is to present a new framework to formulate thermoelastic constitutive relations for initially isotropic rubberlike materials undergoing finite deformations. The strain-energy function for incompressible materials is extended to include the effects of compressibility and temperature changes. The novelty of this framework is that only a few material functions and material parameters to be fitted with the experimental data are required, and these functions and parameters have clear physical meaning. In order to validate the proposed formulation, the Gent–Gent model for incompressible rubbers is chosen as an illustrative example. A new expression of the Helmholtz free energy of rubberlike materials, which takes into account the material compressibility and thermal effect, is then derived. In this generalized Gent–Gent model, only one material function and six material parameters are introduced. It is shown that the generalized Gent–Gent model can be used to predict the stress-strain behavior over the entire range of deformation. Even for incompressible materials, the strain-energy function in this paper is different from that given by Gent himself. The generalized Gent–Gent model can also adequately describe the thermal-mechanical coupling effect, in which thermoelastic inversion phenomena occur.

Theoretical model of double-walled carbon nanotube pullout from a composite matrix

A micromechanical model of double-walled carbon nanotube (DWCNT) pullout from a composite matrix is presented with the interfacial residual stress and van der Waals (vdW) forces taken into consideration. The interfacial residual stress induced by thermal expansion coefficient (TEC) mismatch is introduced via thermo-elastic constitutive relations. The influence of vdW interactions between two layers of DWCNT on the interfacial stress distributions of DWCNT and matrix is analyzed. The analytical expressions of interfacial shear stress and the axial stresses of DWCNT and matrix are derived, respectively. Furthermore, the influences of temperature change, interfacial friction coefficient, DWCNT aspect ratio, DWCNT volume fraction and the relative modulus between DWCNT and matrix are illustrated and discussed.

Finite thermoelasticity of constrained elastomers subject to biaxial loading

Well accepted descriptors of the thermoelastic behavior of elastomers undergoing finite strain remain elusive largely due to the continuing lack of appropriate multiaxial data. In this paper we present a new theoretical framework for inferring thermoelastic constitutive relations directly from biaxial data. In particular, we consider an experimentally natural decomposition of the motion into two parts – one due to traction-free uniform heating and one due to isothermal mechanical loading – and impose a mechanical incompressibility constraint on each motion. It is shown that, regardless of the order of the motions, one obtains a thermoelastic generalization of the classical result of Rivlin and Saunders for identifying response functions from (isothermal) biaxial data.

THERMOELASTIC EXCITATION OF MULTI-DIRECTIONAL FIBROUS COMPOSITE CYLINDERS

The thermoelastic response in multi-directional fibrous composite cylinders, transporting heat waves, has been examined by using the finite element method. A simple transformation-based multi-directional material model is evolved where, within a unit cell, a number nof unidirectional composite bundles can be dispersed in three orthogonal planes such that they maintain material axisymmetry. Thermoelastic constitutive relations for the present model are improvised from a set of established micromechanical theories for unidirectional composites. An explicit transient thermal analysis, covering all conventional modes of heat transfer, evaluates the temperature time history in composite structures with rotational symmetry. A dynamic structural analysis evaluates the transient stress and deformation patterns of the cylinder as affected by heat waves. The results depict the effects of different types of heat waves, moisture and internal coating on the thermoelastic response.

THERMOELASTTCITY OF UNIDIRECTIONALLY MICROPOLAR CORD-REINFORCED COMPOSITES

This article develops a unipolar thermoelastic constitutive relation for twisted cord-reinforced elastomeric composites. Depending on the cord twist about the axial direction, the associated torsional moment and extensional forces may be strongly coupled. Such coupling leads to simultaneous extensional and torsional thermal expansion. The generality of the scheme of formulation is such that large deformation / strain situations can be handled. This is achieved through the use of second Piola Kirchhoff stress and moment stress tensors and their associated kinematic measures. Note that such pairings are cord, i.e., particle following. In this context the provide for a consistent ongoing update of local and global configurational orientations.

Simple Solutions of the Free-Edge Stresses in Composite Laminates under Thermal and Mechanical Loads

Intense and localized interlaminar stresses generally occur in a narrow boundary region near the free edge of a multilayered anisotropic laminate under mechanical and temperature loads. Quantitative measures of interlaminar action across interfaces may be readily obtained through purely algebraic operations, even if nonlinear and inelastic material behavior becomes significant in the boundary region due to severe strain concentration. These measures are the limiting values of the Lekhnitskii stress functions F and $$ (and of the normal derivative of F) along interfaces and toward the interior region of the laminate. In the present work, they are used as the basis of an exceedingly simple and efficient method of interlaminar stress analysis that is potentially applicable to free-edge problems involving nonlinear thermoelastic constitutive relations. Example solutions are obtained for symmetric, four-layer, cross-ply and angle-ply laminates under a temperature load and two different types of strain loads, and the results are found to be in reasonable agreement with the existing numerical and analytical solutions based on elaborate analysis methods.