Equivalent Coefficient Of Thermal Expansion Research Articles

Homogenization of Thermal Properties in Metaplates.

Three-dimensional metamaterials endowed with two-dimensional in-plane periodicity exhibit peculiar thermoelastic behaviour when heated or cooled. By proper design of the unit cell, the equivalent thermal expansion coefficient can be programmed and can also reach negative values. The heterogeneity in the third direction of such metamaterials also causes, in general, a thermal-induced deflection. The prediction of the equivalent thermal properties is important to design the metamaterial suitable for a specific application. Under the hypothesis of small thickness with respect to the global in-plane dimensions, we make use of asymptotic homogenization to describe the thermoelastic behaviour of these metamaterials as that of an equivalent homogenous plate. The method provides explicit expressions for the effective thermal properties, which allow for a cost-effective prediction of the thermoelastic response of these metaplates.

Thermal expansion properties of double concave arrow honeycomb structure with double materials

This article proposes a kind of double concave double-arrow cell. The equivalent thermal expansion coefficient is obtained by using the geometric relation of deformation. The theoretical results agree well with the finite element results. The effect of material matching and geometry size on the equivalent thermal expansion coefficient is discussed. Based on the proposed two-material double-arrow cell, a two-material honeycomb structure is constructed. The thermal expansion coefficient, equivalent elastic modulus and Poisson’s ratio of the structure under thermal load are numerically discussed. It is found that the thermal expansion coefficient of the structure increases with the increase of the thermal expansion coefficient of the material, and the thermal expansion coefficient of the up arrow has a greater effect on the thermal expansion performance of the structure than that of the down arrow. Compared with the single material honeycomb structure, the double-material concave honeycomb structure does not produce the deformation mode of surrounding diffusion, and its longitudinal expansion is smaller. It shows that zero expansion can be achieved by adjusting the material and geometric parameters, and this structure provides a new idea for the design of other zero expansion structures.

Numerical study of thermal stress of a building material based on clay and date palm fibers

This work aims to present, through a numerical study, the results of the thermal stress that appears within a construction material based on soil stabilized by date palm fibers (PDF) in a dry climate region. These plant fibers constituting an abundant and renewable natural waste have always been favored to develop and improve the thermal performance of building materials, nevertheless the problem of daily temperature variations in these regions leads to cracks that may cause deformations and damage of building walls. The simulation begins with the generation of 3D structures and then homogenization techniques and finite element calculations (FE) are used to estimate the equivalent coefficient of thermal expansion of the composite. This coefficient varies from 5.4.10-6 per °C for pure clay to 5.9.10-6 per °C for a fraction of 30% of date palm fibers, showing that the content of date palm fibers in adobe and clay brick acts on the thermal stress and stability of traditional construction buildings.

Large-scale flexible-resonators with temperature insensitivity employing superoleophobic substrates.

Whispering gallery mode polymer resonators are becoming competitive with devices made of other materials, however, the inherent thermal sensitivity of the materials and the small size limit their applications, such as high-precision optical gyroscope. Here, a method is proposed for fabricating large-scale NOA65 resonators with quality factors greater than 105 on a chip employing superoleophobic. The sandwich structure as the core layer of resonator is used to present the flexible remodeling characteristics, the surface roughness remains below 1 nm when the diameter changes by more than 25%. Importantly, theoretical and experimental results show that under the tuning action of external pressure, the equivalent thermal expansion coefficient of the resonator gradually approaches the glass sheet on both sides with the variation of 2 × 10-4 /°C∼0.9 × 10-4 /°C, and the corresponding temperature response range of 0.12 nm/°C∼-0.056 nm/°C shows the promise of temperature insensitivity resonators on a chip.

Calculation of Iron Loss with Stress in Stator Core by Shrinkage Tolerance

This paper presents a method for calculating the iron loss of the stator core of an electric motor owing to the stress generated by shrink fit tolerance in the production process. Shrink fit is used to fix the frame and stator core, shaft, and rotor core in a motor, and the corresponding condition varies depending on the material. Three finite element analysis steps were performed to reduce the iron loss owing to stress during shrinking of a frame and stator core. In step 1, the shrink fit process was applied to a frame and stator core, considering the manufacturing tolerances through three-dimensional modeling. Thermal-structure finite element analysis was performed to apply the same conditions as those in the shrink fit process. The shrink fit was expanded by applying heat to the frame, followed by natural cooling to calculate the contact stress between the frame and the stator core. In step 2, the same contact stress as that in step 1 was derived using structural analysis through two-dimensional modeling of the frame and stator core without tolerances. The contact stress was calculated by applying the equivalent thermal expansion coefficient of the frame, and it was confirmed that the manufacturing tolerance and maximum stress intensity are linearly related. In step 3, electromagnetic analysis was performed at the rated operating point of the 2.2-kW induction motor using the model obtained in step 2. The magnetic flux density distribution of the stator core was derived via electromagnetic analysis and the iron loss, including the stress distribution, realized in step 2. The iron losses obtained under different conditions, including the stress of the stator core owing to the shrink fit tolerance, were compared, and the effectiveness of the shrink fit tolerance required to achieve a motor with high efficiency was evaluated.

A novel series of mechanical metamaterials with sign-changing coefficient of thermal expansion and their parameter analysis

Mechanical metamaterials show great application prospects in many fields such as aerospace, biomedical engineering, energy and transportation. The development of mechanical metamaterials with multiple peculiar properties is an important prerequisite for the development of intelligent sensors with excellent performance and the realization of multifunctional integration of structures. In this paper, a series of novel mechanical metamaterials are designed by adding rhomboid negative thermal expansion units into re-entrant hexagonal honeycombs, which have different signs of the coefficient of thermal expansion when the temperature rises or falls, and the analytical formula of their equivalent thermal expansion coefficient are derived. Parameter analysis and numerical simulations show that the metamaterial can realize the sign-changing properties of thermal expansion coefficient without entering the range of large geometric deformation under specific constituting materials and geometric parameters. In addition, this design idea is extended to three-dimensional form.

Analysis on the evolution of frost heaving pressure of penetrating crack considering water content and migration

Frozen heaving failure of fractured rock mass is commonly encountered in engineering in cold regions, which is chiefly caused by the frost heaving pressure arising from the water–ice phase change in the crack. To explore the evolution of frost heaving pressure in penetrating elliptical crack considering water content and water migration, a new theoretical model embodying the frost heaving pressure evolutionary character was established by introducing freezing ratio function. The equivalent thermal expansion coefficient was used to analyze the evolution process of frost heaving pressure under the effect of water–ice phase change, which was then verified. It was found that the evolution process of frost heaving pressure can be divided into three stages: free expansion stage of water–ice phase change, rapid growth stage of frost heaving pressure, and stable stage of frost heaving pressure. Subsequently, the influences of rock thermal expansion effect, properties of rock and ice, and water content of crack on the frost heaving pressure were investigated. The results indicate that the impact of rock thermal expansion on frost heaving pressure is extremely slight, which is negligible. Comparing with the properties of rock, the properties of ice show significant effects on the frost heaving pressure, particularly the Poisson ratio of ice. In the case of identical water migration ratio, the peak frost heaving pressure increases linearly with the water content of crack.

Tunable Coefficient of Thermal Expansion of Composite Materials for Thin-Film Coatings

In most engineering applications, the coefficients of thermal expansion (CTEs) of different materials in integrated structures are inconsistent, especially for the thin-film multilayered coatings. Therefore, mismatched thermal deformation is induced due to temperature variation, which leads to an extreme temperature gradient, stress concentration, and damage accumulation. Controlling the CTEs of materials can effectively eliminate the thermally induced stress within the layered structures and thus considerably improve the mechanical reliability and service life. In this paper, randomly distributed fibers are incorporated into the matrix material and thus utilized to tune the material CTE from the macroscopical viewpoint. To this end, finite element (FE) modeling is proposed for fiber-reinforced matrix composites. In order to overcome the challenges of creating numerical models at a mesoscale, the random distribution of fibers in three-dimensional space is realized by proposing a fiber growth algorithm with the control of the in-plane and out-of-plane angles of fibers. The homogenization method is adopted to facilitate the FE simulations by using the representative volume element (RVE) of composite materials. Periodic boundary conditions (PBC) are applied to realize the prediction of the equivalent CTE of macroscopic composite materials with randomly distributed fibers. In the established FE model, the random distribution of carbon fibers in the matrix makes it possible to tune the CTE of the composite material by considering the orientation of fibers in the matrix. The FE predictions show that the volume fraction of carbon fibers in the composite materials is found to be crucial to macroscopic CTE, but results in minor variations in Young’s modulus and shear modulus. With the developed ABAQUS plug-in program, the proposed tuning method for CTE is promising to be standardized for industrial practice.

Equivalent Modulus and Mass Loss of C/C-SiC Gradient Matrix Composites at High Temperature

Abstract The oxidation at high temperature of carbon fiber reinforced carbon-silicon carbide (C/C-SiC) gradient matrix composites results in structural damages and performance degradation. Oxidation kinetics model of C/C-SiC gradient matrix composites was built. A representative volume element (RVE) was established according to the microstructure of the composites. The evolution law of the structure characterization of the composites and the mass loss model were derived from the oxidation kinetics model. And equivalent elastic modulus with oxidation time at different temperatures and coefficient of thermal expansion(CTE) at different temperatures were calculated by RVE. The results show that the equivalent elastic modulus decreases with oxidation time and increases rapidly after the crack heals. The higher the temperature, the lower decreasing rate. And the equivalent CTE increases with temperature, while the increasing rate decreases with temperature. The calculated result of mass loss is consistent with the previous research.

Multi-material 3D double-V metastructures with tailorable Poisson's ratio and thermal expansion

Natural materials have achieved excellent comprehensive mechanical properties and fascinating synergistic functionalities through composite architectures and optimum materials distribution within a single tissue during natural selection/adaptation. Producing three-dimensional (3D) architectures with multi-materials has been a major challenge in artificial materials. Herein, 3D double-V hierarchical (DVH) lattices with multi-materials were designed and fabricated using interlocking assembly strategy. Theoretical analysis, numerical simulation and experimental verification were conducted to investigate the mechanical response of the multi-material metastructures, especially effective Poisson's ratio and coefficient of thermal expansion (CTE). The results revealed that the equivalent Poisson's ratio of the 3D DVH lattices can be tuned by changing not only the geometric parameters but also the relative stiffness of the two V segments. Poisson's ratio of the multi-material metamaterials has the minimum/maximum value at transition point and tends to zero with the increase or decrease of the relative stiffness. The equivalent CTE of concave 3D DVH lattices can be tuned by changing not only the geometric parameters but also the relative CTE and relative stiffness of the two V segments from magnified positive to near zero or large negative. Coupled negative Poisson's ratio and CTE, which are hardly found in natural materials, are easily achieved by the architected DVH composite lattices. This work provides an innovative approach for the design, fabrication and characterization of 3D architected cellular materials with simultaneously tailorable auxetic behavior and thermal expansion, which are promising in advanced engineering applications.

Numerical simulation and experimental validation of bending and curling behaviors of liquid crystal elastomer beams under thermal actuation

The exceptional actuation properties of liquid crystal elastomers (LCEs) have made these materials highly attractive for various emerging applications such as soft robotics and artificial muscles. The large strain gradients occurring under thermal stimuli induce bending and curling of initially flat LCE films. Due to the complex physics behind the spontaneous deformation in nematic liquid crystal elastomers, there is no single universal finite element-based method for the simulation of the behaviors of LCE actuators. In this work, we developed a simple layered 2D model for modeling and simulation of the bending and curling characteristics of LCE beams based on the gradient of the temperature-dependent equivalent thermal expansion. The appropriate parameters were derived by measuring the radius of curvature of the LCE film aligned unidirectionally at one surface produced on a rubbed Kapton film. It was found that in a large range of thicknesses (12–134 μm) of the LCE beams, the equivalent thermal expansion coefficients tend to approach a similar value. It was demonstrated and experimentally validated that the thermal expansion model is very effective in predicting the nonlinear curling behavior of LCE beams of various thicknesses. Remarkably, the model is also capable of simulating the rolling behavior of LCE beams with tapered thickness variation. The proposed method offers good flexibility in terms of the geometric shape and expansion parameters, computational efficiency, and accuracy.

Sunlight irradiation and wind effect on the interlaminar stresses of the organic solar cell

The organic solar cell has attracted much interest due to its high power conversion efficiency and low cost. This paper studies the interlaminar stresses between the working layer and the substrate of the organic solar cell. The effects of solar irradiation and wind speed have been considered as well. The multilayered film model and the thin film–substrate model are employed separately in reaction to the different magnitude of the film and substrate thickness. Both models straightforwardly show the dangerous stress areas at the two ends of the working layer. Numerical examples reveal that the interlaminar stress increases as the solar irradiation increases while decreases with the wind speed increasing. A thicker working layer of the organic solar cell results in larger interlaminar stresses. The critical value of sunlight irradiation for varying external environment is predicted. The critical value of sunlight irradiation at the wind speed of Vf = 15 m/s increases by nearly 20%, compared with that of the wind speed Vf = 5 m/s. In addition, the effect of the equivalent thermal expansion coefficient of the working layer on the interlaminar stress is also explored.

Study of designable coefficient of thermal expansion in cellular structures

In this work, a model of cellular structures in thermal fields has been established based on methods of material mechanics, structural mechanics and thermodynamics. Cellular structures have different deformation behaviors in thermal environment with different coefficients of thermal expansion (CTE), the desirable equivalent coefficient of thermal expansion (ECTE) of cellular structures can be achieved by introduce members with different CTEs. Three different configurations are proposed to study the relationship between ECTE and geometrical configuration. The results show that: desirable ECTE can be obtained by designing configuration shape of cellular structures, inclined angle of members and positions of different CTE materials.

Evolution of interfacial contact during low pressure rotary friction welding: A finite element analysis

Development of rotary friction welding (RFW) is of critical importance for extending the RFW technology to join large-scale structures, such as rods/pipes for generator rotors and nuclear power plants. In this paper, finite element (FE) analysis is employed to analyze the thermal-mechanical process during low-pressure rotary friction welding (RFW), and the dynamic evolution process of interface contact is illustrated. Low-pressure RFW process of D50Re steel rod with a diameter of 100 mm is investigated by using the fully coupled thermal-mechanical FE analysis. In the analysis process, the two work pieces are regarded as two contacting entities, and a new method based on equivalent thermal expansion (TE) coefficient is proposed in order to study the dynamic contact behavior during low-pressure RFW. Three simulation cases, which consider no TE, real TE coefficient and equivalent TE coefficient respectively, are performed. It is shown from the comparison between the simulation results and the experimental data that, the equivalent TE coefficient is viable approach in the FE analysis to improve the prediction on the axial shortening and the welding torque. It was demonstrated that significant incomplete interfacial contact occurs during the friction stage in low-pressure RFW. The formation and expansion of the contact zone is in a thermo-mechanical fully coupled manner. Since the temperature is not uniform across the friction interface, the contact area is first formed in the high temperature area, and the expansion of the contact area occurs subsequently. When the temperature at the interface is uniform, almost full contact would be achieved over the friction interface. The modeling and simulation method proposed in this paper is helpful for understanding the underlying physical phenomenon during RFW process. If experimental tests can be extended to determine the equivalent TE coefficient, the accuracy and the reliability of the proposed simulation method in this paper will be further improved.

Warpage Analysis of Fan-Out Panel-Level Packaging Using Equivalent CTE

In this study, we developed a simple method to obtain the equivalent coefficient of thermal expansion (CTE) by comparing the results of the finite element method (FEM) with the experimental results. This equivalent CTE was used to effectively describe the final mechanical/chemical shrinkage behavior after post-molding curing (PMC). During fan-out panel-level packaging, the chemical/molding shrinkage, curing and temperature cool down processes can cause excessive warpage of the panel structure. When this warpage becomes excessively large, the subsequent processing is substantially more difficult. Thus, it is important to be able to accurately predict structural warpage before manufacturing. FEMs are commonly used to estimate panel warpage during molding, however, the shrinkage behavior of the epoxy resin during molding and the material properties after curing are difficult to fit, and a much experimentation is usually required to obtain the temperature-dependent material properties of the molding compound and parameters required for the empirical equation, e.g., PVTC (Pressure Volume Temperature Cure). Furthermore, the simulation results from these empirical equations are often inconsistent with the experimental results. However, our results showed that under various geometric conditions, the equivalent CTE remained almost constant, and thus can be used in FEM as a reliable material property to predict the warpage after a panel PMC process.

The effect of graphene on the deflections of multiscale composites under thermo-mechanical loading

The present investigation aims to shed light on an important aspect of graphene reinforced multiscale composites that may appear in the form of reduced dimensional stability of the structure when exposed to thermal gradients. The study consists of two main parts: an extensive review of the related literature and a thermo-mechanical analysis of a laminated composite beam made of three-phases: arbitrary oriented Graphene Nanoplatelets (GNPs) embedded in an epoxy resin (ER) polymer matrix, reinforced with E-Glass fibers.The review of related literature reveals that the problem of finding the equivalent coefficient of thermal expansion (CTE) for multiscale composites is not as easy as the conventional composites, the trend is different, and more complicating parameters are involved. The analysis includes the predictions of the effective thermo-elastic properties of the multiscale composite laminate, based on Halpin-Tsai theory and hierarchy modeling, and the formulation of the beam deflection governing equation, based on Timoshenko nonlinear (large deflection) beam theory. Different GNPs weight fractions, different E-glass fiber volume fractions, and three lamination layups are investigated. It is found that the addition of GNPs increases the beam deflections when exposed to temperature gradients and that the laminate longitudinal CTE is the main factor responsible for this effect.

Numerical investigation of the WC re-precipitation impact on the residual stress state in WC20 wt.-%Co hardmetal

In WC-Co hardmetals severe micro stresses form between the carbide and binder phases during the cooling stage of the liquid phase sintering process. The main cause of these micro residual stresses is the difference in the coefficients of thermal expansion in each of the constituent phases. There have also been discussions as to whether or not the solution and re-precipitation of elemental tungsten may be an additional driver for the temperature-dependent formation of residual stresses [1, 2]. To further investigate the influence of the re-precipitation effect on the residual stress state in hardmetals, a numerical approach combining the diffusion modelling capabilities of thermocalc/DICTRA [3] is applied in conjunction with temperature-dependent nonlinear mechanical analysis utilizing the finite element method. This combination allows for a qualitative estimate of the re-precipitation effect on the final residual stress state in hardmetals. Additionally, the temperature dependence of the residual stress state in hardmetals is investigated, with and without consideration of the solution and re-precipitation of W and C in the binder.A detailed 2.5D representative volume element based on EBSD images of a WC20wt.-%Co hardmetal has been derived for use in mechanical FE simulation. In the mechanical analysis, the cobalt binder is modelled as temperature-dependent elastic-viscoplastic isotropic material with isotropic hardening. Carbides are modelled as purely orthotropic elastic materials with temperature-dependent description of the elastic constants. The DICTRA results are applied as additional internal loads to the mechanical model via two equivalent coefficients of thermal expansion. A comparison of model results obtained with and without re-precipitation effect permits the impact of the re-precipitation effect on the residual stress state to be examined qualitatively.Finally, the model results are compared to experimental measurements of the residual stress state in WC20wt.-%Co hardmetal specimens, obtained via neutron diffraction.

Elastic and elastic-plastic analysis of multilayer thin films filled with heterogeneous materials

Due to the mismatch between the coefficients of thermal expansion (CTE) of two adjacent films, the residual stress was growing up during thermal cycling. The aim of this work is to extend the Stoney equation for the multilayer thin films with heterostructure (voids filled with gas or other solids) or unsmooth interface. The general theoretical models were built for elastic and elastic-plastic deformation in the multilayer films with void region filled with other solid or gas. The proposed closed solution (CS) was simplified for analyzing the micro/nano devices with the micromachined multilayered multilayer films structure that thin films locate on a much thicker substrate. One model of through silicon via (TSV) has been built and analyzed. Based on the finite element method (FEM) and the initial CS, a modified CS is built up. The influence of the location and thickness of void, CTE and Young’s modulus (YM) on the normal stress of the thin films was analyzed by the simplified CS and FEM. Based on the FEM and CS the linear and coupled relationship has been set up. With the FEM analysis, the equivalent CTE and YM influenced by the void can be described by equation. The difference of critical temperature for the film from elastic deformation to plastic deformation was studied.

Determination of thermo-elastic parameters for dynamical modeling of 2.5D C/SiC braided composites

An approach on determining elastic parameters and Coefficient of thermal expansion (CTE) of a 2.5-dimensional (2.5D) braided composites is proposed in this paper, by adopting mesoscopic mechanics integrated Finite element (FE) modeling. According to the geometric features of meso-structure, Representative volume cell (RVC) models of composite for predicting thermo-elastic parameters are established. On the basis of the models, homogenized parameter prediction is carried out in three steps: Firstly, equivalent elastic properties is predicted subject to periodic displacement boundary conditions; secondly, the equivalent thermal modulus is determined by using the periodic non-adiabatic temperature boundary conditions; thirdly, using the obtained elastic parameters and thermal modulus to calculate the equivalent coefficient of thermal expansion. A multiscale finite element analysis is conducted: The thermo-elastic parameters of yarn is calculated using the RVC model; subsequently, the equivalent parameters of the yarn are substituted into the RVC of 2.5D braided C/SiC composites, to predict the thermo-elastic parameters. Results indicate that the CTE determined by homogenized parameter prediction of 2.5D C/SiC composites shows good agreements with the experimental results. At last, the refined FE model and the equivalent homogeneous model are employed to verify the effectiveness of the predicted parameters in terms of effective modeling. After the comparative analysis on thermal modal data between refined model and equivalent model, the acquired results demonstrate the effectiveness of the proposed method in determining thermo-elastic parameters.

二维周期性单胞网格结构的等效热弹性分析

The thermo-elasticity of 2D lattice structures with periodic unit cells was studied. The lattice structure was homogenized as a pseudo-membrane (PM) structure and the equivalent thermal expansion coefficients (TECs) of the PM were derived. The TECs were expressed as explicit functions of the geometrical and physical parameters of the links in unit cells. Simultaneously, the elastic properties were re-defined based on the new geometry of the unit cell under thermal load. Numerical results were given to show the difference between the deformations of the structure under different thermo-mechanical loads, and a uniform pressure was applied on the top surface of the cantilever beam with or without thermal loads (the temperature increment was positive, negative or null, respectively). In simulation, the deformations of the lattice structure beams (with different sizes of unit cells) and the corresponding PM beams were calculated numerically. The theoretical solution of the beam deformation was also given with the elasticity of the PM beam. Differences between the solutions verify the correctness of the derived equivalent parameters. Results show the validity of the PM method for equivalent thermo-mechanical analysis of 2D lattice structures with periodic unit cells.