Thermoelastic Composites Research Articles

Asymptotic modeling of steady vibrations of thin inclusions in a thermoelastic composite

Asymptotic modeling of steady vibrations of thin inclusions in a thermoelastic composite

Mean field interaction model accounting for the spatial distribution of inclusions in elastic-viscoplastic composites

A cluster interaction model has been proposed to account for the spatial distribution and morphology of particles when estimating the effective properties of elastic and thermoelastic composites (Molinari and El Mouden, 1996). In the present paper this approach is extended to elastic-viscoplastic composites. To this end the tangent linearization of the non-linear viscoplastic law and the concept of additive interaction equation are used. Although the extension is formulated for the non-linear case, first applications are considered for linear viscoelastic composites, a situation rich enough to evaluate the interest of the cluster interaction approach. Results of the model are compared to numerical homogenization for periodic unit cells with two cubic configurations.

Robust topology optimization of thermoelastic metamaterials considering hybrid uncertainties of material property

Metamaterials with extreme thermoelastic properties (i.e. thermal expansion and stress) offer significant value in applications that require maintaining thermal geometric stability, e.g., aerospace equipment. However, material uncertainty has a great impact on the thermoelastic properties of such multiphase composites. Thus, a robust topology optimization design for the thermoelastic composites under hybrid uncertainties of material property is investigated. The distribution of base materials in the periodic unit cell is achieved by multi-phase solid isotropic material with penalization method and the equivalent properties of composites are evaluated by the numerical homogenization method. To obtain a clear material interfaces in the unit cell, a filtering method named the element perimeter control method is proposed. Considering the uncertainties of material property, the linear thermal expansion coefficients of two constituent materials are assumed to be a random variable and an interval variable, respectively. A robust topology optimization model based on Polynomial-Chaos-Chebyshev-Interval method is proposed, where the Polynomial chaos and Chebyshev interval functions are integrated to perform uncertainty analysis. The robust objective function is formulated by a combination of average mean, average and bandwidth of standard deviation of the objective value. The effectiveness of the proposed robust topology optimization methodology is illustrated with several numerical examples.

Topological optimization of thermoelastic composites with maximized stiffness and heat transfer

The topological optimization of composite structures is widely used while tailoring materials to achieve the required engineering physical properties. In this paper, the problem of topological optimization of the microstructure of a composite aimed at the construction of a material with most effective values of the bulk modulus of elasticity and thermal conductivity taking into account competing mechanical and thermal properties of the materials included in the composite is defined and solved. A two-phase composite consists of two base materials, one of which has a higher Young modulus but lower thermal conductivity, while the other has a lower Young modulus but higher thermal conductivity. A new class of problems for composites containing material pores or technological inclusions of different shapes is considered. Effective thermoelastic properties are obtained using the asymptotic homogenization method. A modified solid isotropic material with penalization (SIMP) model is used to regularize the problem. The problem of the isoperimetric constraints is solved by the method of moving asymptotes (MMA). The influence on optimal topology of the composite in the presence of two competing materials, and optimality criteria using linear weight functions are investigated. Pareto spaces that provide deep understanding of how these goals compete in achieving optimal topology are constructed.

The Two-scale Finite Element Errors Analysis for One Class of Thermoelastic Problem in Periodic Composites

The thermoelastic composites are widely used in some kinds of complex environments. In this paper, the two-scale finite element method for one class of thermal mechanical problem in periodic structure is analyzed. The two-scale finite element method and the approximate errors for one class of coupling thermoelastic problem are obtained.

Effects of superimposed eigenstrains on the overall thermoelastic moduli of composites

This study investigates the effects that an initial local eigenstrain field, when superimposed on the thermal eigenstrain field, has on the overall thermal expansion coefficients and heat capacities of thermoelastic composites. The study can also be seen as an investigation into how a local residual stress field affects these overall moduli, as initial eigenstrains are generally a source of residual stresses. The approach taken is thermodynamic. Expressions that include the superimposed eigenstrain field are developed for the overall moduli within the framework of small strain thermoelasticity with temperature dependent materials. These expressions, which are written in terms of the concentration tensors and residual fields (stress and strain fields given rise to by the eigenstrains under zero overall stress and strain, respectively), contain correction terms that are absent in the expressions developed within linear thermoelasticity. Taking into account the temperature dependence of the constituent moduli is shown to be essential to capture the effects of the superimposed eigenstrain field. A Ti–6Al–4V/ZrO2 composite is investigated for which the correction terms are found to be negligible for the heat capacities but significant for the thermal expansion coefficients. This suggests that, for applications with large temperature changes, using the linear-thermoelasticity-based expressions can affect the accuracy of the estimates of the overall moduli, and therefore the accuracy of thermostructural analyses of composite structures. The proposed expressions can be of use to estimate the overall thermoelastic moduli in contexts in which the strains remain small, temperature changes are large, and superimposed eigenstrains may be present.

Exact Solution of Laminated Orthogonal Hyperbolic Shells of Thermoelastic with Four Simply Supported

The exact solution of laminated Orthogonal Hyperbolic shell of thermoelastic composite is very important to investigate it’s the mechanical properties. Therefore, many studies were completed in this paper. Firstly, the non-homogeneous state equation in generic coordinates is derived for thermoelastic composites including thermal gradient relationships by the basic equations and state space method. Secondly, an augmented dimension constitutive relationship was obtained by combining thermal gradient relationships into constitutive relationships of composite constitutive relationships based on the theory of symplectic variable and precise time-integration with dimension expanding method. Lastly, uniting state space method and the augmented dimension constitutive relationships, a homogeneous state equation which can be solely solved was deduced by eliminating the stresses in plane. The homogeneous state equation can simplify greatly the solution process of the laminated Orthogonal Hyperbolic shell, improve the calculation accuracy. What’s more, it can be discrete straightly in finite element format. So, the homogeneous state equation, derived in the paper, can supply a new method for the exact solution of laminated structures with complicated boundary condition in practice.

Exact results for the local fields and the effective moduli of fibrous composites with thickly coated fibers

Fibrous composites that consist of thickly coated cylindrical fibers embedded in a matrix are considered. All of the three phases of the composite are assumed to be transversely isotropic. The study consists of three parts. In the first part exact relations are derived for the influence functions that connect applied uniform overall fields to the induced local fields in piezomagnetoelectric systems. We consider the case of coated fibers with a concentrically circular cross section, and contrast the derived relations with the more limited ones that could be obtained in the case of coated fibers with an arbitrary cross section. The derivation is based on the ability to create uniform strain, electric and magnetic fields in the composite by the application of certain mechanical, electric, magnetic and thermal loadings. In the second part of the study, exact microstructure-independent connections are derived for a subgroup of the effective moduli of the homogenized piezomagnetoelectric composite which exhibits overall transverse isotropy. In the third part of the study, the derived exact connections between the effective moduli are reduced to the setting of thermoelasticity; allowing the coating to be thin and highly stiff or highly compliant, we make contact with the exact connections derived lately in the literature for two-phase fibrous thermoelastic composites with surface-stress-type and spring-type imperfect interfaces.

Multi-site micromechanics of composite materials with temperature-dependent constituents under small strain and finite thermal perturbation assumptions

This study presents mean-field based micromechanics models to predict the effective thermoelastic properties, namely, elasticities, thermal expansions and heat capacity, of thermoelastic composite materials with temperature-dependent constituents under finite temperature changes and small strain assumptions. First, the Helmholtz potential density for small strain finite thermoelasticity has been presented. The fundamental solution based on Green’s function of elasticity problem has been used to derive the general expressions for the elastic and thermal strains concentration tensors with temperature-dependent material properties. A family of mean-field based micromechanics models (multi-site Eshelby dilute model, Mori–Tanaka model and self-consistent model) has been presented. The models are general enough to account for the morphology and topology textures of the microstructure of a thermoelastic composite. Numerical examples based on the multi-site Mori–Tanaka model are used to quantify the differences of the effective properties based on the small strain finite thermoelasticity in comparison to the linear thermoelasticity. The predictions of the multi-site Mori–Tanaka micromechanics model are also compared to those of the VAMUCH, a finite element based micromechanics model.

Finite element modeling of porous thermoelastic composites with account for their microstructure

The paper discusses approaches to the determination of the effective moduli of porous anisotropic thermoelastic composite materials based on the effective moduli method, modeling of the representative volumes with account for microstructure and finite element technologies of solving static thermoelastic problems for heterogeneous bodies. The paper formulates a set of boundary-value problems of thermoelasticity with boundary conditions of the first kind that enables computing all effective stiffness moduli, thermal conductivity moduli and coefficients of thermal stresses for porous anisotropic material. The technology of the finite element method is described for the numerical solution of boundary-value thermoelastic problems in the representative volume of a porous material. A range of the simulation methods for the inner structure of the representative volumes in cubic finite element lattice is represented by the way of giving properties of the porous material to a certain number of finite elements. The methods considered are the simple random method, the method that supports the connectivity of the skeleton for the porosity up to 90%, the Witten-Sander method that enables us to obtain a cluster from the pores of fractal type, and the growing from the plane method that forms several clusters from the pores in the vicinity of one of the edges of the cubic lattice. The example considered is the model of porous silicon that is taken as a material of cubic symmetry. The computation results allow analyzing the influence of various structures of the representative volumes on the effective moduli. The comparison of the computed effective characteristics of porous silicon with a range of known analytical and numerical results has been carried out. The results of the test computations have demonstrated that the effective material properties of porous thermoelastic composites can strongly depend on the models of their representative volumes.

Micromechanical modeling of coupled viscoelastic–viscoplastic composites based on an incrementally affine formulation

This study proposes a micromechanical modeling of inclusion-reinforced viscoelastic–viscoplastic composites, based on mean-field approaches. For this, we have generalized the so-called incrementally affine linearization method which was proposed by Doghri et al. (2010a) for elasto-viscoplastic materials. The proposal provides an affine relation between stress and strain increments via an algorithmic tangent operator. In order to find the incrementally affine expression, we start by the linearization of evolution equations at the beginning of a time step around the end time of the step. Next, a numerical integration of the linearized equations is required using a fully implicit backward Euler scheme. The obtained algebraic equations lead to an incrementally affine formulation which is form-similar to linear thermoelasticity, therefore known homogenization models for linear thermoelastic composites can be applied. The proposed method can deal with general viscoelastic–viscoplastic constitutive models with an arbitrary number of internal variables. The semi-analytical predictions are validated against finite element simulations and experimental results.

Wave propagation and transient heat transfer in thermoelastic composites

We revisit the pioneering works by Ene [1] and Francfort [2] which give the macroscopic equivalent model for wave propagations and transient heat transfers in thermoelastic composite media, by using the method of asymptotic expansions. The obtained model (denoted as model I in the following) shows a similar structure to the classical thermoelastic system which couples the inertial wave equation with the transient heat equation. By analyzing the dimensionless description at the heterogeneity scale and by estimating the dimensionless numbers herein, we show that two very different characteristic frequencies are present. At the higher one, the obtained model (model II) describes wave propagation with memory effects whereas macroscopic transient heat transfers are absent. At the lower frequency, model III describes transient heat transfers while the composite material undergoes a quasi-static deformation. Model III is similar to model I where the inertial term is neglected. Model II is not reducible to model I.

Effective specific heats of multi-phase thermoelastic composites

This paper studies the effective properties of multi-phase thermoelastic composites. Based on the Helmholtz free energy and the Gibbs free energy of individual phases, the effective elastic tensor, thermal-expansion tensor, and specific heats of the multi-phase composites are derived by means of the volume average of free-energies of these phases. Particular emphasis is placed on the derivation of new analytical expressions of effective specific heats at constant-strain and constant-stress situations, in which a modified Eshelby's micromechanics theory is developed and the interaction between inclusions is considered. As an illustrative example, the analytical expression of the effective specific heat for a three-phase thermoelastic composite is presented.

Effect of residual interface stress on effective thermal expansion coefficient of particle-filled thermoelastic nanocomposite

The surface/interface energy theory based on three configurations proposed by Huang et al. is used to study the effective properties of thermoelastic nanocomposites. The particular emphasis is placed on the discussion of the influence of the residual interface stress on the thermal expansion coefficient of a thermoelastic composite filled with nanoparticles. First, the thermo-elastic interface constitutive relations expressed in terms of the first Piola-Kirchhoff interface stress and the Lagrangian description of the generalized Young-Laplace equation are presented. Second, the Hashin’s composite sphere assemblage (CSA) is taken as the representative volume element (RVE), and the residual elastic field induced by the residual interface stress in this CSA at reference configuration is determined. Elastic deformations in the CSA from the reference configuration to the current configuration are calculated. From the above calculations, analytical expressions of the effective bulk modulus and the effective thermal expansion coefficient of thermoelastic composite are derived. It is shown that the residual interface stress has a significant effect on the thermal expansion properties of thermoelastic nanocomposites.

Variational estimates for the effective response and field statistics in thermoelastic composites with intra-phase property fluctuations

In this work, variational estimates are provided for the macroscopic response, as well as for the first and second moments of the stress and strain fields, in thermoelastic composites with non-uniform distributions of the thermal stress and elastic moduli in the constituent phases. These estimates are obtained in terms of a ‘comparison composite’ with uniform phase properties depending on the first and second moments of a certain combination of the given intra-phase thermal stresses and modulus field distributions. Under certain hypotheses, these estimates can be shown to lead to upper and lower bounds for the free energy of the composite, which reduce to standard results when the intra-phase fluctuations vanish. An illustrative application is given for rigidly reinforced composites with a non-uniform distribution of the thermal stress in the matrix phase.

On thermoelastostatics of composites with nonlocal properties of constituents I. General representations for effective material and field parameters

A theory of thermoelastic composites with nonlocal properties of constituents is analyzed for multiphase elastic solids of arbitrary geometry and material symmetry. Due to their generality, one uses the nonlocal integral models because the gradient models are usually derived as approximations of corresponding integral models in the immediate (infinitely closed) vicinity of the point being considered. One explores a simplified theory for linear (macroscopically) elasticity, which differs from the classical local theory in the stress–strain constitutive relation only, whereas the equilibrium and compatibility equations remain unaltered. One obtains the new representation of the effective modulus and compliance through the mechanical influence function which does not explicitly depend (as opposed to its local counterpart) on the elastic operators of constituents. The representations for the effective eigenstrains and eigenstresses through either the mechanical influence functions or transformation influence functions are presented. The effective strain energy and potential energy are expressed in terms of only average values of the state variables and the effective properties. Representations of both the first and second statistical moments of stress and strain fields in the constituents are also performed. Many of the results were obtained as the straightforward generalizations of their local counterparts because the methods used for obtaining the mentioned results widely exploit the Hill’s (1963) condition which holds for any compatible strain field and equilibrium stress field not necessarily related to each other by a specific stress–strain relation.

Distribution of local fields in composites

The behavior of composites depends crucially on the distribution of local fields. The ways to characterize and find such distributions are poorly understood. In this paper, an attempt is made to present some simple approximate formulas for distributions of the local fields in variational problems. They are given in an explicit form for thermo-elastic composites and heat-conducting composites. The accuracy is tested for the case of heat conductivity of chess-board structures.

Dispersion and attenuation in thermoelastic multisize particulate composites

This paper deals with the problem of multiple scattering by a random distribution of spherical solid particles in a solid. The material properties of both media are taken as thermoelastic. The radii of the inclusions may be different. The self-consistent method in its variant of the effective medium is used to find the dispersion and attenuation of quasi-elastic, quasi-thermal and shear waves. The single scattering problem required by this technique is solved approximately by means of the Galerkin method applied to an integral equation using the Green function. Numerical results display a characteristic resonance phenomena which appears in the interval where the results are approximately valid, that is, for very long waves down to wavelengths about twice the largest diameter of the spheres. Examples are shown, for composites with two sets of inclusions, which have either a very similar or dissimilar size. Comparisons are made with the elastic counterpart. Among the material properties, the mass density ratio, inclusion to matrix, seems to play an important and simple role. Frequency intervals are distinguished and shown to depend on that ratio, where the attenuation and dispersion of quasi-elastic and P-waves are either very close to each other or not at all. The same applies to shear waves in either composite. The mass density ratio also displays a simple monotonic decreasing behaviour as a function of the frequency at the first attenuation maximum and velocity minimum. These results may be of interest for the nondestructive testing characterization of particulate composites.

Scale Effects in Infinitesimal and Finite Thermoelasticity of Random Composites

A method for homogenization of linear or non-linear (finite strain) thermoelastic random composites is presented. With the help of variational principles, the method leads to hierarchies of scale-dependent bounds on constitutive responses, which are illustrated in several cases through computational mechanics. This, in turn, provides estimates of the minimal size of the Representative Volume Element (RVE). The effects of mismatch between the composite's phases as well as the temperature difference on the minimal size of the RVE are also considered. The results of linear and non-linear thermoelasticity theories are compared.

Thermoelastic composites with columnar microstructure and cylindrically anisotropic phases. Part I: Exact results

A wide class of composite materials, which are in this paper referred to as having columnar microstructure, posses the microgeometrical characteristic that the constituent phases are homogeneous along one and only one direction. This class includes as an important subclass all the composites consisting of a homogeneous matrix reinforced by aligned parallel continuous homogeneous fibers. In the present work, considering a transversely isotropic composite with columnar microstructure and with cylindrically anisotropic phases, a number of exact results are established for effective thermoelastic moduli by modelling the composite as a nested composite cylinder assemblage. The results obtained in this work extend those of Hashin and Rosen [Z. Hashin, B.W. Rosen, The elastic moduli of fiber-reinforced materials, ASME J. Appl. Mech. 31 (1964) 223–232] and Hashin [Z. Hashin, Thermoelastic properties and conductivity of carbon/carbon fiber composites, Mech. Mater. 8 (1990) 293–308].