Effective Coefficient Of Thermal Expansion Research Articles

Fibrous nanocomposites with interface stress: Hill’s and Levin’s connections for effective moduli

We study the macroscopic behavior of solids containing circular cylindrical nanoinclusions of the same size with surface effects prevailing at interfaces. The overall thermomechanical properties of these solids are shown to comply with two sets of exact connections. The first set, similar to Hill’s universal connections, provides two constraints among the three axisymmetric overall elastic moduli. The second set relates the effective coefficients of thermal expansion to the effective moduli, in analogy with Levin’s formula. In contrast to the classical results, the presence of surface effects makes both sets of connections dependent on the absolute size of the nanoinclusions.

Modeling of the thermopiezoelastic behavior of prismatic smart composite structures made of orthotropic materials

Asymptotic homogenization models for prismatic smart composite structures are derived and the effective elastic, piezoelectric, and thermal expansion coefficients are obtained. The actuation coefficients characterize the intrinsic transducer nature of active smart materials that can be used to induce strains and stresses in a coordinated fashion. Examples of such actuators employed with smart composite material systems are derived from piezoelectric, magnetostrictive and some other materials. The constituents of the smart structures are assumed to exhibit orthotropic characteristics. The original problem for the regularly non-homogeneous smart composite structure reduces to a system of three simpler types of problem, called unit cell problems. It is precisely these unit cell problems that enable the determination of the aforementioned coefficients. These effective coefficients are universal in nature and can be used to study a wide variety of boundary value problems associated with a smart structure of a given geometry.

Micromechanics of Smart Composite Plates with Periodically Embedded Actuators and Rapidly Varying Thickness

Asymptotic homogenization models for smart composite plates with periodically arranged embedded actuators and rapidly varying thickness are derived. The formulated models enable the determination of both local fields and effective elastic, actuation, thermal expansion, and hygroscopic expansion coefficients from three-dimensional local unit cell problems. The actuation coefficients, for example piezoelectric or magnetostrictive, characterize the intrinsic transducer nature of active smart materials that can be used to induce strains and stresses in a coordinated fashion. The theory is illustrated by means of examples pertaining to thin smart composite plates of uniform thickness, riband waferreinforced smart composite structures, and sandwich smart composite plates with honeycomb filler.

A numerical study on the coefficients of thermal expansion of fiber reinforced composite materials

In the present work, the effective coefficients of thermal expansion (CTE) of fiber reinforced composites are studied by micromechanical modeling using the finite element method. Representative unit cell models, cylinders which are embedded in cubes with unit dimensions, having different fiber volume fractions were produced using the finite element program ANSYS. Results of various finite element solutions for different types of composites were compared with the results of various analytical methods and with the available experimental results. Also, the expansion behavior of different material systems with respect to fiber content was determined numerically. All of the models and finite element analyses are in good agreement with the experimental data for longitudinal CTEs, however Rosen–Hashin and finite element results for transverse CTE are generally in much better agreement with the experimental data than the other methods for all materials investigated.

Evaluation of elastic and thermoelastic properties of lotus-type porous metals via effective-mean-field theory

Abstract The effective-mean-field (EMF) theory, consisting of Mori–Tanaka’s mean-field theory and Bruggeman’s effective medium approximation, was extended in order to calculate the coefficients of thermal expansion (CTE) of composite materials. The effective elastic constants and CTE of lotus-type porous metals, possessing cylindrical pores aligned unidirectionally, were evaluated with the EMF theory.

Analysis of Underground Pipelines Subjected to Frost Heaving Forces

This paper examined the fundamental problem of an interaction between a soil medium that experiences frost heave in a closed system and an underground pipeline. The coupled thermal transfer and structural analysis, considering the material nonlinearity of freezing soil, for an underground pipeline subjected to the low cyclic load was focused on the development of computational scheme by introducing the effective heat capacity concept and the effective thermal expansion coefficient into the study. The effective heat capacity model in the thermal transfer analysis took into consideration the phase-change effect in the frozen fringe of a soil medium. The comparative analyses between the theory and the actual performances were valuable in establishing a level of confidence in the application of introduced theory to the field. The numerical results in the paper illustrated the influence for the frost heave of a soil medium on the temperature-dependent development of stress fields on metallic underground pipe walls in South Korea.

Thermoelastic effect induced by ferroelastic domain switching

Temperature changes induced by externally applied mechanical stress were observed in the Pb3(PO4)2 ferroelastic crystal. These characteristic features can be explained by the thermoelastic effect as in other thermoelastic martensitic materials. In this study, the mechanical stress dependence of the thermoelastic effect was studied using a modified thermomechanical analysis system. The thermoelastic effect strongly depended on the mechanical stress and was enhanced by ferroelastic domain switching, responsible for the pseudoelastic behavior of ferroelastic materials. The ∂T∕∂σsin, the change of sample temperature with varying stress enhanced by pseudoelasticity, was ∼9.5×10−2°C∕MPa at 159 °C, ferroelastic temperature region under 1.2 MPa sinusoidal stress. This value is eight to nine times higher than the values obtained under bias stress. This result can be explained by a thermoelastic equation which includes the effective thermal expansion coefficient in the ferroelastic phase.

Thermal Expansion of BaO · 2B2O3 in the Vitreous and Crystalline States: I. Thermal Expansion and Density of Barium Diborate Prepared from Monolithic Glass of the Same Composition

The thermal expansion and density of vitreous and polycrystalline barium diborates prepared by crystallization of monolithic glasses of stoichiometric composition are investigated in the temperature range 20–700°C. The glass crystallizes in the form of the α and β modifications depending on the heat treatment conditions. The α phase nucleates on the surface of the monolithic sample and grows inward the sample in the form of textured layers. The β phase nucleates in the melt bulk and grows in the form of spherulites. Textured polycrystalline barium diborate in the form of the α phase possesses a pronounced anisotropy of thermal expansion with respect to the direction of crystalline-layer growth. This is associated with the negative expansion along one of the crystallographic axes. The effective temperature coefficients of thermal expansion in the temperature range 20–600°C increase in the order: α phase-glass-β phase. Upon heating above 690°C, the β phase transforms partially into the α phase. This is accompanied by a considerable increase in the volume. The densities of the glass and polycrystalline samples in the temperature range 20–700°C increase in the order: glass-α phase–β phase.

A Super-Precise CTE Evaluation Method for Ultra-Low-Expansion Glasses Using the LFB Ultrasonic Material Characterization System

A super-precise method of evaluating the coefficient of thermal expansion (CTE) of ultra-low-expansion glasses for future extreme ultra-violet lithography (EUVL) systems was developed using the line-focus-beam ultrasonic material characterization (LFB-UMC) system. Evaluation was demonstrated for two commercial glasses, TiO2-SiO2 glass (C-7971) and Li2O-Al2O3-SiO2 glass ceramic (Zerodur). For the C-7971 specimens, the sensitivity and resolution in the velocity measurement of leaky surface acoustic waves (LSAWs) for the CTE were estimated to be 4.40 (ppb/K)/(m/s) and ±0.77 ppb/K for ±2σ (σ: standard deviation). LSAW velocity differences caused by different TiO2 concentrations and distributions or striae in each specimen were successfully detected and evaluated. For the Zerodur specimens, LSAW velocity differences associated with the chemical compositions and crystallization conditions were observed among different ingots and specimens. This ultrasonic method is expected to be an extremely useful and effective CTE evaluation technology and to contribute to improving and developing EUVL-grade glass materials.

Micromechanical analysis of lattice blocks

Multiphase lattice blocks with periodic structure are analyzed by a continuum-based micromechanical approach. As a result, effective stiffness tensors, global initial yield surfaces, global damage thresholds, effective inelastic stress–strain responses and critical yielding temperatures of lattice blocks are established. Applications are given for various types of elastic and inelastic lattice blocks made of an aluminum alloy. Furthermore, a lattice block with negative effective Poisson’s ratios is considered, and two types of two-phase lattice blocks that are capable to produce negative effective coefficients of thermal expansion are presented.

Finite element simulation of the thermal properties of particulate and continuous network-reinforced metal-matrix composites

An analysis of the effective coefficient of thermal expansion (CTE) and thermal conductivity (TC) of metal-matrix composites was conducted, focusing on composites with potential for use in electronic packaging applications. Using the finite element method, these thermal properties were simulated on discontinuous SiC-particle and continuous three-dimensional SiC-network-reinforced Al-matrix composites. The calculated CTEs of particle-reinforced composites agree well with those predicted by the Kerner model. While these CTEs increase with increasing temperature, the composites with SiC networks were found to exhibit the opposite trend in the high-temperature region (above about 250 °C). The TCs of the three-dimensional SiC-network-reinforced composites are found to be much less sensitive to volume fraction of reinforcement than those of particle-reinforced composites. Continuous three-dimensional network-reinforced composites possess a higher TC than particulate composites, and their CTEs agree well with those of the electronic devices. Therefore, it is possible to use three-dimensional SiC-network-reinforced Al-matrix composites as electronic packaging materials to achieve a high heat dissipation rate and yet to minimize interfacial stresses resulting from a CTE mismatch.

Thermal expansion of Ti-containing hydrogenated amorphous carbon nanocomposite thin films

The effective coefficients of thermal expansion (CTE) of Ti-containing hydrogenated amorphous carbon (Ti–C:H) thin films were measured. Ti–C:H thin films with compositions ranging from nearly pure a-C:H to nearly pure TiC were deposited on Si(100) substrates. Effective CTEs were determined from temperature induced changes in the curvature of film/substrate assemblies. Measured effective CTE values for Ti–C:H are ∼5.7×10−6K−1, and show little dependence on the Ti composition.

Parametric based design of CFRP honeycomb sandwich cylinder for a spacecraft

This paper presents a new simple parameter based design method for designing a CFRP honeycomb sandwich cylinder for spacecraft applications. The parameters that constitute the design are aspect ratio, ply orientation, ply sequence, number of plies and effective coefficient of thermal expansion of the sandwich construction. A detailed design and analysis is carried out by simulating the loads and subsystems as lumped masses on the cylinder. A detailed study of various parameters and its effects towards optimal design for given dynamic and static constraints is presented well. This is very much helpful for easy understanding of significant parameters and its consequences for optimum design of CFRP honeycomb sandwich cylinder.

MULTILEVEL, MICROMECHANICAL MODEL FOR THERMAL ANALYSIS OF WOVEN-FABRIC COMPOSITE MATERIALS

A multilevel and multiscale approach was presented for woven-fabric composites, especially for plain-weave composites, to compute effective coefficients of thermal expansion (CTE) as well as thermal stresses at the fiber and matrix level. The approach consisted of three modules: fiber-strand, strand-fabric, and lamination modules. The present study developed the strand-fabric module for thermal analysis because other modules had already been developed previously. In addition, detailed finite-element analysis was conducted for plain-weave woven fabric composites. The finite-element models contained discrete representative fibers. Then the analytical results were compared to the finite-element analysis results for both CTEs and microthermal stresses. Two different composite materials (glass/epoxy and boron/epoxy) and different fiber volume fractions were considered in the study. Generally, the analytical results compared very well with the finite-element analysis results, which used more than 100,000 three-dimensional brick elements. Thermal stresses were computed at the fiber and matrix level; those stresses affect damage and failure of composites.

The effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics

We present a micromechanical analysis on the effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics in terms of their microstructural information. The overall behaviors of ferroelectric ceramics are profoundly influenced by the microstructural phenomena, where the macroscopic pyroelectric effect can be induced in an originally isotropic, thus non-pyroelectric ceramic composed of randomly oriented ferroelectric grains through poling, during which the polar axes of grains are switched by the applied electric field or mechanical stress. To analyze these complicated phenomena, we will first establish an exact connection between the effective thermal moduli and the effective electroelastic moduli of ferroelectric ceramics, and then combine the exact connection with the effective medium approximation to provide an estimate on the effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics in terms of the orientation distribution of grains and poling conditions, where the texture evolution as a result of domain switching during poling has been taken into account. Numerical results are presented and good agreements with known theoretical results and some experimental data are observed.

A study on the variation of effective CTE of printed circuit boards through a validated comparison between strain gages and Moire interferometry

The effective coefficient of thermal expansion (CTE) of printed circuit boards (PCBs) have a great deal of influence on the reliability of solder joints in microelectronic packages. In this paper we carryout a systematic characterization of nineteen circuit board samples using strain gages, further validated by moire/spl acute/ interferometry, to understand the impact of CTE variation on the reliability of solder joints. It is shown that the measured effective coefficient of thermal expansion varied in a wide range by as much as 5 PPM about the commonly used value of 17 PPM. This is shown to cause a significant reliability impact for a representative plastic ball grid array package assembly. The comparison between strain gage and moire/spl acute/ interferometry showed good overall correlation, but differed from each other by as much as 2.73 parts per million (PPM). The possible sources of error in each technique are identified and discussed.

A statistical micromechanics-based multi-scale framework for effective thermomechanical behaviours of particle reinforced composites

Particle-reinforced composite materials have been widely used as they can exhibit nearly isotropic material properties and are often easy to process. Some silicon carbide particle reinforced aluminium composites with high volume concentration of reinforcement exhibit excellent thermophysical properties and can be used in advanced electronic packaging. In this paper, a statistical micromechanics-based multi-scale material modelling framework is introduced to describe the macroscopic effective thermomechanical properties of the particle-reinforced composite. The formulation differs from most of the existing methods in that the interaction effects among the reinforcing particles are directly accounted for by considering pair-wise interaction and statistical information on particle distribution is included. The strain and stress concentration factor tensors that relate the local average strain and stress fields, respectively, to the corresponding global average fields are derived according to the theory of average fields. The effective coefficient of thermal expansion for the particle-reinforced composite material is derived. Comparisons of the prediction from the proposed framework to the results from other existing methods are presented. The results are expressed in analytical closed-form in terms of the thermal and mechanical properties of the two constituent phases and the volume fraction of particles. No parameter estimation or data fitting is required.

Packaging a fiber bragg grating with metal coating for an athermal design

This paper presents an athermal package to compensate for temperature deviation of a fiber Bragg grating (FBG) with a metal coating. The metal coating is electroless plated onto the optical fiber to function as a thermal compensator. From the cross section of a thermal compensator used for FBG athermal package, an optical fiber and a metal coating appear. Therefore, in theory, the two-phase constant model is applied in designing the metal coating thickness associated with the effective thermal expansion coefficient and the effective Young's modulus in the composite thermal compensator. As a result, the total variation of Bragg wavelength over the temperature ranging from 30 to 80/spl deg/C is around 0.05 nm in a new package based upon the concept of a bimaterial device. To the authors' knowledge, the configuration proposed in this paper is probably the simplest bimaterial package design there is.

Thermal expansion behaviour of aluminium/SiC composites with bimodal particle distributions

The thermal response and the coefficient of thermal expansion (CTE) of aluminium matrix composites having high volume fractions of SiC particulate have been investigated. The composites were produced by infiltrating liquid aluminium into preforms made either from a single particle size, or by mixing and packing SiC particulate of two largely different average diameters (170 and 16 μm, respectively). The experimental results for composites with a single particle size indicate that the hysteresis in the thermal strain response curves is proportional to the square root of the particle surface area per unit volume of metal matrix, in agreement with current theories. Instead, no simple relationship is found between the hysteresis and any of the system parameters for composites with bimodal particle distributions. On the other hand, the overall CTE is shown to be mainly determined by the composite compactness or total particle volume fraction; neither the particle average size nor the particle size distribution seem to affect the overall CTE. This result is in full agreement with published numerical results obtained from finite element analyses of the effective CTE of aluminum matrix composites. Our results also indicate that the CTE varies with particle volume fraction at a pace higher than predicted by theory.

Helical carbon nanotube arrays: thermal expansion

An earlier investigation [Comp. Sci. Technol. 62 (2002) 419] of the effective mechanical properties of large arrays of carbon nanotubes, assembled in helical array geometries of circular cross-section, is extended for the prediction of the effective thermal expansion coefficients (CTE) of the array. The analysis utilized anisotropic elasticity to examine the behavior of a layered cylinder with layers consisting of discontinuous carbon nanotubes embedded in a polymeric matrix and with collimation direction in each layer following a helical path prescribed by a linear variation in the tangent of the helix angle with radial position. The effective axial, transverse and shearing coefficients of thermal expansion of the array are determined in terms of the degree of twist, number of layers, properties of the carbon nanotube and surrounding polymer matrix.