Numerical and Analytical Approaches for Calculating the Effective Thermo-Mechanical Properties of Three-Phase Composites

The aim of this article is to envisage the effective thermo-mechanical properties of three phase composites made up of coated unidirectional cylindrical fibers using homogenization techniques. The main focus is on square arrangements of cylindrical fibers in composite. The numerical approach is based on the micro-mechanical unit cell modeling technique using finite element method (FEM) with appropriate boundary conditions and it allows the extension to composites with arbitrary geometrical inclusion configurations, providing a powerful tool for fast calculation of their effective properties. The results obtained from the numerical technique are compared with those obtained by means of the analytical asymptotic homogenization method (AHM) for different fiber volume fractions. In order to analyze the interphase effect, the effective properties are compared with the results obtained from some theoretical approaches reported in the literature.     

THERMOELASTICITY OF A REGULARLY NONHOMOGENEOUS THIN CURVED LAYER WITH RAPIDLY VARYING THICKNESS

A regularly nonhomogeneous (composite), anisotropic, thin curved layer with rapidly oscillating material parameters and thickness is considered for the case when mean thickness and period scale have small magnitudes of the same order. A three-dimensional thermoelasticity problem for this layer is reduced to a homogenized shell model by means of an asymptotic homogenization method for periodic structures. The effective thermoelastic and thermal material parameters of this shell are expressed in terms of solutions for auxiliary local problems in the cell of periodicity. Using the solution of the boundary-value problem for the homogenized shell and the solutions of the local problems, one can obtain a three-dimensional microstructure of the stresses, displacements and temperature with a high accuracy

This general model is applied to the derivation of thermoelastic and thermal constitutive equations for network periodic shells. The relations obtained lay the foundation for a new continuous model of thermoelasticity and heat conductivity for network periodic shells and plates.     

Effective elastic properties of planar SOFCs: A non-local dynamic homogenization approach

The focus of the article is on the analysis of effective elastic properties of planar Solid Oxide Fuel Cell (SOFC) devices. An ideal periodic multi-layered composite (SOFC-like) reproducing the overall properties of multi-layer SOFC devices is defined. Adopting a non-local dynamic homogenization method, explicit expressions for overall elastic moduli and inertial terms of this material are derived in terms of micro-fluctuation functions. These micro-fluctuation functions are then obtained solving the cell problems by means of finite element techniques. The effects of the temperature variation on overall elastic and inertial properties of the fuel cells are studied. Dispersion relations for acoustic waves in SOFC-like multilayered materials are derived as functions of the overall constants, and the results obtained by the proposed computational homogenization approach are compared with those provided by rigorous Floquet–Bloch theory. Finally, the influence of the temperature and of the elastic properties variation on the Bloch spectrum is investigated.     

Coupled heat equation in thermoelasticity with temperature dependent moduli

Abstract

New and consistent expressions for the coupled heat equation are developed within the framework of small-strain thermoelasticity for both Fourier and Cattaneo–Vernotte conduction models. These expressions place no restrictions on the changes in temperature, allow for the temperature dependence of the thermoelastic moduli, and include all the coupling terms as functions of the thermoelastic moduli and their derivatives. As applications, (i) an extended Lord–Shulman-type model is derived that takes into account the temperature dependence of the thermoelastic moduli, and (ii) the equations underpinning the experimental technique of thermoelastic stress analysis are revisited.     

ANALYTICAL TREATMENT OF AXISYMMETRICAL THERMOELASTIC FIELD WITH KASSIR'S NONHOMOGENEOUS MATERIAL PROPERTIES AND ITS ADAPTATION TO BOUNDARY VALUE PROBLEM OF SLAB UNDER STEADY TEMPERATURE FIELD

The method of an analytical development of thermoelastic problems for a medium with Kassir's nonhomogeneous material properties is developed. For isothermal problems of such a nonhomogeneous body, an analytical method of development has already been proposed by M. K. Kassir under the assumption that the shear modulus of elasticity G is changed arbitrarily with the variable z of the axial coordinate according to the relation G(z) = G_oz^m. However, the analytical procedure for the thermoelastic field has not been established. In this article, introducing the thermoelastic displacement potential function, the analytical method of development for the axisymmetrical thermoelastic field is established. As an illustrative example, we consider the thermoelastic problem of a slab. Assuming that the shear modulus of elasticity G, the thermal conductivity λ, and the coefficient of linear thermal expansion α vary with the variable [zcirc] of the dimension-less axial coordinate according to the relation G([zcirc]) = G_o[zcirc]^m, λ([zcirc]) = λ₀[zcirc]^l, α([zcirc]) = α₀[zcirc]ⁿ, the axisymmetric temperature solution in a steady state for the slab is obtained and the associated thermal stress components are evaluated theoretically. Numerical calculations are carried out for several cases, taking into account the variety of the nonhomogeneous material properties. Numerical results are shown graphically.     

The phase change thermoelastic analysis of biological tissue with variable thermal properties during cryosurgery

Abstract

In this work, the coupling phase change heat transfer process and thermal stress behavior of biological tissue during cryosurgery are studied in the context of a generalized thermoelastic theory. The nonlinear governing equations are constructed while considering the variable thermal properties and solved by a time-domain finite element method based on the effective heat capacity formulation. A 2-D tumor and normal tissue model is adopted for simulating the freezing process in cryosurgery. The effects of temperature-dependent thermal properties and relaxation time on the responses of biological tissue are discussed and illustrated graphically.     

Practical Thermal Multi–Scale Analysis for Composite Materials–Mechanical-Orientated Approach

This paper describes a thermal multi-scale formulation for composite materials based on a mechanical homogenization approach. The presented formulation evaluates the effective macroscopic thermal conductivity of the composite materials and also the microscopic heat flux field by scaling down to the micro-scale level. The effective thermal conductivity of the composite materials was calculated by applying the homogenization theory over the unit cell. The uniqueness of the presented multi-scale analysis related to the elastic problems solved at the microscopic scale (unit cell). This method has the advantage of applying periodic boundary conditions and uniform macroscopic temperature gradient over the unit cell. The proposed thermal multi-scale analysis was verified and its efficiency was demonstrated on large scale problem.     

Mechanical and thermal couplings in helical strands*

Abstract

A generalized Timoshenko rod model is developed for helical strands and helically reinforced cylinders. The thermomechanical constitutive law has five effective elastic moduli, and two thermal coefficients, which can be obtained with the finite element method, or partly from analytic solutions. The model predicts nonclassical bending and thermoelastic behavior of helical strands. First, bending–shearing coupling is explicitly captured, which leads to non-planar bending under a transverse shear force, or a bending moment. Second, torsion and thermal expansion are coupled due to structural chirality. The dispersion relation of harmonic thermoelastic waves is governed by four non-dimensional parameters: two thermoelastic coupling constants, one chirality parameter and the Fourier number. The quasi-longitudinal and the quasi-torsional waves (“quasi” meaning the longitudinal mode is always coupled with a small torsional motion, and vice versa, due to chirality) are dispersive and damped, and dependent on temperature. The adiabatic-isothermal transition of the wave propagation is determined by the Fourier number.     

The effect of the tortuosity of fluid phases on the phase velocity of Rayleigh wave in unsaturated porothermoelastic media

Abstract

Based on the frame of elastic theory for unsaturated porous medium, considering the influence of the thermal effect, the effects of the tortuosity of fluid phases on the phase velocity of Rayleigh wave in unsaturated porous media are studied. Firstly, the thermoelastic wave equations for three-phase porous media are established by taking into account the effect of the tortuosity of fluid phases. Secondly, through a theoretical derivation, the dispersion equation of Rayleigh wave for unsaturated porothermoelastic media is given by introducing the potential functions. Finally, the variations of the phase velocity of Rayleigh wave are analyzed with numerical examples. The results demonstrate that the phase velocity of Rayleigh wave is not just related to the frequency but also affected by the tortuosity of pore-fluid. The effect of the tortuosity of pore-liquid on the phase velocity of Rayleigh wave is more significant than that of the tortuosity of pore-gas on the phase velocity of Rayleigh wave.     

ON THE PROPAGATION OF THERMOELASTIC WAVES IN MEDIA WITH LONG-RANGE COHESIVE FORCES

Abstract

After establishing the fundamental equations of nonlocal coupled thermoelasticity in Fourier space, nonlocal longitudinal thermoelastic waves in an infinite space are analyzed. Identification of the elastic dispersion equation obtained with its counterpart derived in lattice dynamics leads to values of the nonlocal elastic moduli. Similar identification of the velocity of thermal waves with its counterpart in second sound theory yields the values of the nonlocal thermal moduli. A numerical example involving solid helium is given.     

ANALYSIS OF LAMINATE METAL MATRIX COMPOSITES

This article describes an integrated micro/ macro mechanical study of the thermo-elastic-uiscoplastic behavior of unidirectional metal matrix composites. The microme-chanical analysis of the elastic moduli are based on the representative unit cell approach with comparisons also drawn with the composite cylinder assemblage model. These “ homogenization ” results are later incorporated into the vanishing fiber diameter model develop the macroscopic stress-strain relations. The finite element method is employed for the analysis in conjunction with the Bodner and Partom thermoviscoplas-tic constitutive model for the associated macroscopic analysis and a “ smeared” element approach. Comparisons are made against experimental and analytical results available in the literature whenever feasible.     

Coupled 1-D stress and thermal waves in temperature dependent nonlinear elastic and viscoelastic media*

Abstract

A general analysis is formulated for the closed loop coupled thermal and displacement viscoelastic 1-D wave problem. The proper inclusion of the highly temperature sensitive viscoelastic material properties renders the problem nonlinear, even though the displacements and material properties are considered to obey linear relations. In the present article. the previous analysis is enlarged and reformulated by (a) the inclusion of nonlinear elastic and viscoelastic constitutive relations as formulated in Hilton, (b) the addition of thermal waves to the displacement waves, and by (c) temperature dependent material density and viscoelastic moduli and compliances. The wave problem studied here is of significant importance in modeling, material characterization, determination of instantaneous moduli, nonlinear analytical solution protocols and the nonlinear interaction of temperature, material properties, and wave motions. Analytical and numerical solution protocols are presented and evaluated.     

Dynamic Behavior of Thermal Stress in a Functionally Graded Material Thin Film Subjected to Thermal Shock

In the present article, a one-dimensional dynamic thermoelastic problem in a functionally graded material (FGM) thin film subjected to a thermal shock loading is analyzed. An exact analytical solution is obtained by employing techniques of the space-variable transformation and Laplace transform, in the case where variations in material properties of the FGM thin film are expressed as exponential functions of the space-variable. Numerical calculations for time histories of thermal stresses have been carried out. Obtained numerical results reveal that the thermal stress oscillation is unsteady when the mechanical impedance depends on the space-variable, whereas it is monotonic and periodic when the impedance is independent of the space-variable. The factor which causes the different behavior of the thermal stress oscillations is also investigated based on the analytical and numerical results.     

Interfacial stresses and fracture analysis of a two-phase composite under time-dependent heat flux

An analysis of a two-phase composite component under time-dependent heat flux is presented. The fundamental thermoelastic solution is obtained in terms of complex potentials via the technique of the analytical continuation in order to satisfy the continuous conditions on the interface. The hereditary integral associated with the Kelvin–Maxwell model is applied to simulate the thermoviscoelastic properties while a thermorheologically simple material is considered. Based on the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real-time solution can then be solved numerically by taking inverse Laplace transform. Some typical examples of interface stresses induced by various time-dependent heat flux are discussed. Finally, the solution of a crack embedded in the bi-material subjected to a uniform heat flux is also discussed.     

DERIVATION OF SYSTEMS OF FUNDAMENTAL EQUATIONS FOR A THREE-DIMENSIONAL THERMOELASTIC FIELD WITH NONHOMOGENEOUS MATERIAL PROPERTIES AND ITS APPLICATION TO A SEMI-INFINITE BODY

A method of analytical development of three-dimensional thermoelastic problems for a medium with nonhomogeneous material properties is developed in this article. Assuming that the shear modulus elasticity G, the thermal conductivity lambda, and the coefficient of linear thermal expansion alpha vary with the power product form of axial coordinate variable z and introducing two kinds of displacement functions and the thermoelastic displacement function, the system of fundamental differential equations for such a three-dimensional field is established. As an illustrative example, we consider the thermoelastic problem of a semi-infinite body. The three-dimensional temperature solution in a steady state is obtained and the associated components of thermal displacement and stress are evaluated theoretically. Numerical calculations are carried out for several cases taking into account the variety of the nonhomogeneous material properties of G, lambda, and alpha, and these results are shown graphically.     

THERMOELASTIC PROPERTIES OF COMPOSITES CONTAINING ELLIPSOIDAL INHOMOGENEITIES

This article studies the thermal stresses and the effective thermoelastic properties of composites containing ellipsoidal inhomogeneities. The cluster scheme developed recently by A. Molinari and M. El Mouden in The Problem of Elastic Inclusions at Finite Concentration, Int. J. Solids Struct, vol. 33, pp. 3131 - 3150, 1996, for the case of elastic inclusions embedded in an isotropic elastic matrix, is generalized to the case of ellipsoidal thermoelastic inclusions embedded in an anisotropic thermoelastic matrix. The shape, spatial distribution, and orientation of the inhomogeneities are taken into account in our scheme. The theoretical results for a composite of S_iO₂ particles in a Kerimid matrix are in good agreement with experimental measurements.     

Propagation of thermoelastic waves in unsaturated porothermoelastic media

Abstract

In the present work, the characteristics of bulk wave propagation in an unsaturated porothermoelastic medium are studied taking into consideration of the thermal effect. The wave equations of thermoelastic waves of the problem are established by the mass balance equations, generalized Darcy law, momentum balance equations and generalized non-Fourier heat conduction law of the three-phase medium composed of solid, liquid and gas in unsaturated soils. The dispersion equations of bulk waves in unsaturated porothermoelastic media are derived using the potential functions. Compressional, thermal and shear waves with various speeds are analyzed numerically. It can be found that the thermal expansion coefficient has great influence on the wave speeds of P1 wave and thermal wave. The thermal conductivity only affects the speed of the thermal wave. A rise in the internal temperature of the unsaturated porothermoelastic medium can cause an increase of the wave speeds of compressional, thermal and shear waves.     

Thermoelasticity and P-wave simulation based on the Cole-Cole model

Abstract

The classical Zener model of thermoelasticity can be represented by a mechanical (or viscoelastic) model based on two springs and a dashpot, commonly called standard-linear solid, whose parameters depend on the thermal properties and a relaxation time, and yield the isothermal and adiabatic velocities at the low- and high-frequency limits. This model differs from the more general Lord-Shulman theory of thermoelasticity, whose low-frequency velocity is the adiabatic one. These theories are the basis of thermoelastic attenuation in inhomogeneous media, with heterogeneities much smaller than the wavelength, such as Savage’s theory of thermoelastic dissipation in a medium with spherical pores. In this case, the shape of the relaxation peak differs from that of the Zener and Lord-Shulman models. In these effective homogeneous media, the anelastic behavior of real materials can better be described by using a stress-strain relation based on fractional derivatives. In particular, wave propagation (dispersion and attenuation) is well described by a Cole-Cole stress-strain equation, as illustrated by the agreement with Savage’s theory. We propose a time-domain algorithm based on the Grünwald-Letnikov numerical approximation of the fractional time derivative involved in the time-domain representation of the Cole-Cole model. The spatial derivatives are computed with the Fourier pseudospectral method. We verify the results by comparison with the analytical solution, based on the Green function. The numerical example illustrates wave propagation at an interface separating a porous medium and a purely solid phase.     

THERMOELASTIC STABILITY OF ORTHOTROPIC CIRCULAR PLATES

In this article the thermoelastic buckling of a circular orthotropic composite plate is discussed. The plate is assumed to be geometrically perfect. The equilibrium and stability equations, derived via variational formulations, are used to determine the prebuckling forces and the buckling temperatures. The equations are based on the Love-Kirchhoff hypothesis and Sanders' nonlinear strain-displacement relation. Critical buckling temperatures associated with the uniform temperature rise, gradient through-the-thickness temperature, and linear temperature variation along the radius are obtained. The results are validated for the first type of loading with the known data in literature.     

Control of Thermally Induced Vibration in a Composite Beam with Damping Effect

ABSTRACT

This article deals with control of thermally induced vibration in a composite beam with damping effect. The beam consists of a central thermoelastic structural layer and two outer piezothermoelastic layers. The thermoelastic vibration in the beam is suppressed by the superposition of electroelastic vibration. The electroelastic vibration is repeatedly excited through application of electric potential differences across the piezoelectric layers. The initiation and termination times of the applied electric pulses are determined by a method similar to the optimization procedure developed previously by the authors. The amplitudes of the applied electric pulses are determined using a control method based on speed feedback. Numerical results for an aluminum/PZT ceramic beam are illustrated graphically.