A two-scale homogenization framework for nonlinear effective thermal conductivity of laminated composites

In this study, we formulate the effective temperature-dependent thermal conductivity of laminated composites. The studied laminated composites consist of laminas (plies) made of unidirectional fiber-reinforced matrix with various fiber orientations. The effective thermal conductivity is obtained through a two-scale homogenization scheme. A simplified micromechanical model of a unidirectional fiber-reinforced lamina is formulated at the lower scale. Thermal conductivities of fiber and matrix constituents are allowed to change with temperature. The upper scale uses a sublaminate model to homogenize temperature-dependent thermal conductivities of only a representative lamina stacking sequence in laminated composites. The effective thermal conductivity of each lamina, in the sublaminate model, is obtained using the simplified micromechanical model. The thermal conductivities from the micromechanical and sublaminate models represent average nonlinear properties of fictitiously homogeneous composite media. Interface conditions between fiber and matrix constituents and within laminas are assumed to be perfect. Experimental data available in the literature are used to verify the proposed multi-scale framework. We then analyze transient heat conduction in the homogenized composites. Temperature profiles, during transient heat conduction, in the homogenized composites are compared to the ones in heterogeneous composites. The heterogeneous composites, having different fiber arrangements and sizes, are modeled using finite element (FE) method.     

Effective thermal properties of viscoelastic composites having field-dependent constituent properties

This study introduces a micromechanical model for predicting effective thermal properties (linear coefficient of thermal expansion and thermal conductivity) of viscoelastic composites having solid spherical particle reinforcements. A representative volume element (RVE) of the composites is modeled by a single particle embedded in the cubic matrix. Periodic boundary conditions are imposed to the RVE. The micromechanical model consists of four particle and matrix subcells. Micromechanical relations are formulated in terms of incremental average field quantities, i.e., stress, strain, heat flux and temperature gradient, in the subcells. Perfect bonds are assumed along the subcell’s interfaces. Stress and temperature-dependent viscoelastic constitutive models are used for the isotropic constituents in the micromechanical model. Thermal properties of the particle and matrix constituents are temperature dependent. The effective coefficient of thermal expansion is derived by satisfying displacement and traction continuity at the interfaces during thermo-viscoelastic deformations. This formulation leads to an effective time–temperature–stress-dependent coefficient of thermal expansion. The effective thermal conductivity is formulated by imposing heat flux and temperature continuity at the subcells’ interfaces. The effective thermal properties obtained from the micromechanical model are compared with analytical solutions and experimental data available in the literature. Finally, parametric studies are also performed to investigate the effects of nonlinear thermal and mechanical properties of each constituent on the overall thermal properties of the composite.     

Time-dependent response of active composites with thermal, electrical, and mechanical coupling effect

The mechanical and physical properties of materials change with time. This change can be due to the dissipative characteristic of materials like in viscoelastic bodies and/or due to hostile environmental conditions and electromagnetic fields. We study time-dependent response of active fiber reinforced polymer composites, where the polymer constituent undergoes different viscoelastic deformations at different temperatures, and the electro-mechanical and piezoelectric properties of the active fiber vary with temperatures. A micromechanical model is formulated for predicting effective time-dependent response in active fiber composites with thermal, electrical, and mechanical coupling effects. In this micromechanical model limited information on the local field variables in the fiber and matrix constituents can be incorporated in predicting overall performance of active composites. We compare the time-dependent response of active composites determined from the micromechanical model with those obtained by analyzing the composites with microstructural details. Finite element (FE) is used to analyze the composite with microstructural details which allows quantifying variations of field variables in the constituents of the active composites.     

A multi-scale model for coupled heat conduction and deformations of viscoelastic functionally graded materials

An integrated micromechanical-structural framework is presented to analyze coupled heat conduction and deformations of functionally graded materials (FGM) having temperature and stress dependent viscoelastic constituents. A through-thickness continuous variation of the thermal and mechanical properties of the FGM is approximated as an assembly of homogeneous layers. Average thermo-mechanical properties in each homogeneous medium are computed using a simplified micromechanical model for particle reinforced composites. This micromechanical model consists of two isotropic constituents. The mechanical properties of each constituent are time–stress–temperature dependent. The thermal properties (coefficient of thermal expansion and thermal conductivity) of each constituent are allowed to vary with temperature. Sequentially coupled heat transfer and displacement analyses are performed, which allow analyzing stress/strain behaviors of FGM having time and temperature dependent material properties. The thermo-mechanical responses of the homogenized FGM obtained from micromechanical model are compared with experimental data and the results obtained from finite element (FE) analysis of FGMs having microstructural details. The present micromechanical-modeling approach is computationally efficient and shows good agreement with experiments in predicting time-dependent responses of FGMs. Our analysis forecasts a better design for creep resistant materials using particulate FGM composites.     

Fully coupled heat conduction and deformation analyses of nonlinear viscoelastic composites

This study presents an integrated micromechanical model-finite element framework for analyzing coupled heat conduction and deformations of particle-reinforced composite structures. A simplified micromechanical model consisting of four sub-cells, i.e., one particle and three matrix sub-cells is formulated to obtain the effective thermomechanical properties and micro–macro field variables due to coupled heat conduction and nonlinear thermoviscoelastic deformation of a particulate composite that takes into account the dissipation of energy from the viscoelastic constituents. A time integration algorithm for simultaneously solving the equations that govern heat conduction and thermoviscoelastic deformations of isotropic homogeneous materials is developed. The algorithm is then integrated to the proposed micromechanical model. A significant temperature generation due to the dissipation effect in the viscoelastic matrix was observed when the composite body is subjected to cyclic mechanical loadings. Heat conduction due to the dissipation of the energy cannot be ignored in predicting the factual temperature and deformation fields within the composite structure, subjected to cyclic loading for a long period. A higher creep resistant matrix material or adding elastic particles can lower the temperature generation. Our analyses suggest that using particulate composites and functionally graded materials can reduce the heat generation due to energy dissipation.     

Responses of viscoelastic polymer composites with temperature and time dependent constituents

This study formulates a concurrent micromechanical model for predicting effective responses of fiber reinforced polymer (FRP) composites, whose constituents exhibit thermo-viscoelastic behaviors. The studied FRP composite consists of orthotropic unidirectional fiber and isotropic matrix. The viscoelastic material properties for the fiber and matrix constituents are allowed to change with the temperature field. The composite microstructures are idealized with periodically distributed square fibers in a matrix medium. A unit-cell model, consisting of four fiber and matrix subcells, is generated to obtain effective nonlinear thermo-viscoelastic responses of the composites. A time-integration algorithm is formulated to link two different thermo-viscoelastic constitutive material models at the lowest level (homogeneous fiber and matrix constituents) to the effective material responses at the macro level, and to transfer external mechanical and thermal stimuli to the constituents. This forms a concurrent micromechanical model, which is needed as the material properties of the constituents depend on the temperature field. Consistent tangent stiffness matrices are formulated at the fiber and matrix constituents and also at the effective composite level to improve prediction accuracy. The thermo-viscoelastic responses obtained from the concurrent micromodel are verified with available experimental data. Detailed finite element (FE) models of the FRP microstructures are also generated using 3D continuum elements for several fiber volume fractions. Thermo-viscoelastic responses of the concurrent micromodel are also compared to the ones of the detailed FRP microstructures.     

Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities

Summary A micromechanical framework is proposed to investigate effective mechanical properties of elastic multiphase composites containing many randomly dispersed ellipsoidal inhomogeneities. Within the context of the representative volume element (RVE), four governing micromechanical ensemble-volume averaged field equations are presented to relate ensemble-volume averaged stresses, strains, volume fractions, eigenstrains, particle shapes and orientations, and elastic properties of constituent phases of a linear elastic particulate composite. A renormalization procedure is employed to render absolutely convergent integrals. Therefore, the micromechanical equations and effective elastic properties of a statistically homogeneous composite are independent of the shape of the RVE. Various micromechanical models can be developed based on the proposed ensemble-volume averaged constitutive equations. As a special class of models, inter-particle interactions are completely ignored. It is shown that the classical Hashin-Shtrikman bounds, Walpole's bounds, and Willi's bounds for isotropic or anisotropic elastic multiphase composites are related to the noninteracting solutions. Further, it is demonstrated that the Mori-Tanaka methodcoincides with the Hashin-Shtrikman bounds and the noninteracting micromechanical model in some cases. Specialization to unidirectionally aligned penny-shaped microcracks is also presented. An accurate, higher order (in particle concentration), probabilistic pairwise particle interaction formulation coupled with the proposed ensemble-volume averaged equations will be presented in a companion paper.     

Effective in situ material properties of micron-sized SiO2 particles in SiO2 particulate polymer composites

Several analytical models exist for determination of the Young’s modulus and coefficient of thermal expansion (CTE) of particulate composites. However, it is necessary to provide accurate material properties of the particles as input data to such analytical models in order to precisely predict the composite’s properties, particularly at high particle loading fractions. In fact, the constituent’s size scale often presents a technical challenge to accurately measure the particles’ properties such as Young’s modulus or CTE. Moreover, the in situ material properties of particles may not be the same as the corresponding bulk properties when the particles are embedded in a polymer matrix. To have a better understanding of the material properties and provide useful insight and design guidelines for particulate composites, the concept of “effective in situ constituent properties” and an indirect method were employed in this study. This approach allows for the indirect determination of the particle’s in situ material properties by combining the experimentally determined composite and matrix properties and finite element (FE) models for predicting the corresponding composite properties, then backing out the effective in situ particle properties. The proposed approach was demonstrated with micron-size SiO₂ particle reinforced epoxy composites over a range of particle loading fractions up to 35 vol.% by indirectly determining both the effective Young’s modulus and the effective CTE of the particles. To the best of our knowledge, this study is the first published report on the indirect determination of both the Young’s modulus and the CTE of micron size particles in particulate composites. Similar results on Young’s modulus of micron-size SiO₂ particles measured from nano-indentation testing are encouraging.     

Influence of thermal residual stresses on the composite macroscopic behavior

The influence of the thermal residual stress on the deformation behavior of a composite has been analyzed with a new micromechanical method. The method is based on secant moduli approximation and a new homogenized effective stress to characterize the plastic state of the matrix. It is found that the generated thermal residual stresses after cooling and their influence on the subsequent deformation behavior depends significantly on the aspect ratio of the inclusions. With prolate inclusions, the presence of thermal residual stresses generate a higher compressive hardening curves of the composite, but it is reversed with oblate inclusions. For particle reinforced composite, thermal residual stresses induce a tensile hardening curve higher than the compressive one and this is in agreement with experimental observations.     

Study on the Thermal Expansion and Thermal Cycling of AlNp／Al Composites

The AIN particle reinforced aluminum matrix composites with 50% volume fraction were fabricated by squeeze-casting technology.The thermal expansion behavior and its response to thermal cycling were studied between 20℃ and 400℃.Compared with four theoretical models,the measured CTEs of the composite lie within the elastic bounds derived by Schapery′s analysis .Schapery′s model and Kerner′s model agree well with the CTEs of the composites at lower temperature and elevated temperature,respectively.Strain hysteresis was observed between heating and cooling curves during cycling.This was attributed primarily to the anelastic behavior of the matrix induced by matrix residual stresses.     

Automatic generation of 2D micromechanical finite element model of silicon–carbide/aluminum metal matrix composites: Effects of the boundary conditions

Two-dimensional finite element (FE) simulations of the deformation and damage evolution of Silicon–Carbide (SiC) particle reinforced aluminum alloy composite including interphase are carried out for different microstructures and particle volume fractions of the composites. A program is developed for the automatic generation of 2D micromechanical FE-models with randomly distributed SiC particles. In order to simulate the damage process in aluminum alloy matrix and SiC particles, a damage parameter based on the stress triaxial indicator and the maximum principal stress criterion based elastic brittle damage model are developed within Abaqus/Standard Subroutine USDFLD, respectively. An Abaqus/Standard Subroutine MPC, which allows defining multi-point constraints, is developed to realize the symmetric boundary condition (SBC) and periodic boundary condition (PBC). A series of computational experiments are performed to study the influence of boundary condition, particle number and volume fraction of the representative volume element (RVE) on composite stiffness and strength properties.     

Finite-element modeling of initial matrix failure in CFRP under static transverse tensile load

The failure of transversely loaded unidirectional CFRP has been investigated by the use of mechanical and thermo-mechanical test methods and finite-element analysis. The case considered here is characterized by a high interfacial strength between fiber and matrix, so that matrix failure governs the fracture process of the composite. On the basis of the experimental results, the parabolic and other failure criteria were applied to the FE calculations. The failure dependence of the resin on the actual stress state could be described. Furthermore, the influence of thermal residual stresses on the initial matrix failure has been investigated, and the actual stiffnesses and thermal expansion changes of the epoxy resins and the composites as a function of temperature have been determined experimentally. The results of the mechanical and thermo-mechanical tests performed on the pure resins and on the composites were incorporated into a finite-element analysis and compared with the transverse tensile properties of the composite laminates. In the FE analysis, the local fiber-volume fraction was varied over a wide range in order to investigate its influence on the thermal residual stresses and transverse composite strength. The results could explain the low strain to failure of transverse laminates under tensile loading.     

Effects of Interphase Damage and Residual Stresses on Mechanical Behavior of Particle Reinforced Metal-Matrix Composites

Mechanical behavior of aluminum matrix composites reinforced with SiC particles are predicted using an axisymmetric micromechanical finite element model. The model aims to study initiation and propagation of interphase damage subjected to combination of thermal and uniaxial loading. Effects of manufacturing process thermal residual stresses and interphase de-bonding are considered. The model includes a square Representative Volume Element (RVE) from a cylindrical unit cell representing a quarter of SiC particle surrounded by Al-3.5wt.%Cu matrix. Suitable boundary conditions are defined to include effects of combined thermal and uniaxial tension loading on the RVE. An appropriate damage criterion with a linear relationship between radial and shear stresses for interphase damage is introduced to predict initiation and propagation of interphase de-bonding during loading. A damage user subroutine is developed and coupled to the finite element software to model interphase damage. Overall Stress-strain behavior of particulate metal-matrix composite by considering residual stresses is compared with experimental data to estimate interphase strength. Effects of thermal residual stresses in elastic, de-bonding and plastic zones of composite system are discussed in details. Furthermore, parametric study results show high influence of interphase strength on the overall mechanical behavior of composite material.     

Microstructure-guided numerical simulations to predict the thermal performance of a hierarchical cement-based composite material

This paper presents a microstructure-guided numerical homogenization technique to predict the effective thermal conductivity of a hierarchical cement-based material containing phase change material (PCM)-impregnated lightweight aggregates (LWA). Porous inclusions such as LWAs embedded in a cementitious matrix are filled with multiple fluid phases including PCM to obtain desirable thermal properties for building and infrastructure applications. Simulations are carried out on realistic three-dimensional microstructures generated using pore structure information. An inverse analysis procedure is used to extract the intrinsic thermal properties of those microstructural components for which data is not available. The homogenized heat flux is predicted for an imposed temperature gradient from which the effective composite thermal conductivity is computed. The simulated effective composite thermal conductivities are found to correlate very well with experimental measurements for a family of LWA-PCM composites considered in the paper. Comparisons with commonly used analytical homogenization models show that the microstructure-guided simulation approach provides superior results for composites exhibiting large property contrast between phases. By linking the microstructure and thermal properties of hierarchical materials, an efficient framework is available for optimizing the material design to improve thermal efficiency of a wide variety of heterogeneous materials.     

Thermal conductivity of multiphase particulate composite materials

Hollow particle filled composites, called syntactic foams, are used in weight sensitive structural applications in this paper. In this paper, homogenization techniques are used to derive estimates for thermal conductivity of hollow particle filled composites. The microstructure is modeled as a three-phase system consisting of an air void, a shell surrounding the air void, and a matrix material. The model is applicable to composites containing coated solid particles in a matrix material and can be further expanded to include additional coating layers. The model is successful in predicting thermal conductivity of composites containing up to 52% particles by volume. Theoretical results for thermal conductivity are validated with the results obtained from finite element analysis and are found to be in close agreement with them. A simplified approximation of the theoretical model applicable to thin shells is also validated and found to be in good agreement with the corresponding finite element results. The model is applicable to a wide variety of particulate composite materials and will help in tailoring the properties of particulate composites as per the requirements of the application.     

A micromechanical characterization of graphite-fiber/epoxy composites containing a heterogeneous interphase region

An improved micromechanical model based on the method of cells is introduced in order to describe three-phase, continuous-fiber composite materials containing a heterogeneous interphase region. The model's capability represents a significant improvement over that of the previous version (which is applicable to a homogeneous interphase) in that additional microstress information is obtained within the interphase region. A critical assessment of the model demonstrates that the predictions are consistent with data reproduced by using other micromechanical models. The study includes a parametric simulation in which the effective properties and the mechanical stresses associated with model graphite-fiber/epoxy composites are predicted as a function of the dimensions and Young's modulus of the interphase. Three different interphases are modeled such that the Young's modulus varies between that of the fiber and the matrix according to a generalized parabolic function of the radial coordinate. The parabolic functions are specified such that two of the model composites possess an interphase whose effective Young's modulus is above that of the matrix. The third interphase is specified such that its effective Young's modulus is below that of the matrix. The data indicate that the interphase dimensions and the functional form describing the interphase Young's modulus significantly influence the composite microstresses. These data may be used to help identify optimum material combinations during composite material synthesis.     

A theoretical approach for the thermal expansion behavior of the particulate reinforced aluminum matrix composite Part I A thermal expansion model for composites with mono-dispersed spherical particles

Microstructural observation revealed that the increase in the volume fraction of SiC particles lowers the coefficient of thermal expansion (CTE) of the composite, and the CTE of the metal matrix composites is proportional to the size of the Si phase. To analyze the thermal expansion behavior of aluminum matrix composites, a new model for the CTE of the mono-dispersed binary composite on the basis of Ashelby's cutting and welding approach was proposed. In the theoretical model, it was considered that during cooling relaxation of residual stresses could create an elasto-plastic deformation zone around a SiC or Al₂O₃ particle in the matrix. The size of reinforced particles and other metallurgical factors of the matrix alloy and composite were also considered. In this model, the interacting effect between the reinforced hard particle and the soft matrix is considered by introducing the influence of the elasto-plastic deformation zone around a particle, which is distinguished from the previous models. It was revealed that the CTE of the composite are influenced by the particle volume fraction, the elastic modulus and Poisson's ratio as well as the elasto-plastic deformation zone size and the particle size.     

Micromechanical analysis of layered systems of MMCs subjected to bending––effects of thermal residual stresses

A three-dimensional finite element micromechanical model is presented to study the effects of manufacturing process thermal residual stresses on the mechanical behavior of layered systems of metal matrix composites subjected to four point bending. The presented model contains layered systems, consisting of layers of monolithic titanium alloy (IMI834) and unidirectional fiber reinforced titanium metal matrix composite (SiC/Ti). A representative volume element (RVE) was defined and appropriate boundary conditions were imposed to apply bending and temperature change simultaneously on the model. In an agreement with experimental data, the model is able to predict asymmetric behavior of the composite in tension and compression on the bottom and top surfaces of the beam. This is due to the existence of a high level of thermal residual stresses arising from cool-down from manufacturing temperature. As a result of this asymmetric behavior, the neutral axis of the beam during bending moves from the mid-surface through the compressive part of the beam.     

Thermal residual stresses in metal matrix composites: A review

Recently, metal matrix composites (MMCs) have generated a considerable interest in the materials field because of their attractive physical and mechanical properties. However, during the fabrication of MMCs, thermal residual stresses are reportedly developed in the matrix as a result of the mismatch of the thermal expansion coefficients between the reinforcement and the matrix. It is well established that these residual stresses have a significant effect on the composite properties. For example, due to the presence of thermal residual stresses, it is almost never possible to achieve the maximum elastic response of the composites. In addition, yield stress and fracture toughness of the composites are significantly affected by thermal residual stresses. In this paper, a critical review of the published literature on thermal residual stresses in MMCs and their effect on composite properties are presented. Also, experimental and numerical techniques that are currently available to measure and estimate thermal residual stresses are reviewed and discussed.