Reuss Model Research Articles

Understanding mechanical characteristics of cellulose nanocrystals reinforced PHEMA nanocomposite hydrogel: in aqueous cyclic test

Cellulose nanocrystals (CNC) can be embedded within hydrogels to form tough and strong nanocomposite materials, which possess biomimetic properties from hydrogels including good biocompatibility, permeability and flexible mechanical characteristics. There are many potential applications for these strong nanocomposite hydrogels in medical devices, such as wound dressing or super absorbents. Whereas, the research on the mechanical properties of CNC reinforced nanocomposite remains at superficial level, and their nonlinear mechanical responses are rarely investigated in previous reports. Mechanical characteristics of CNC reinforced poly(2-hydroxyethyl methacrylate) (PHEMA) nanocomposite hydrogels, in terms of stress–strain correlations, fracture mechanism, and cyclic stretching responses, have been investigated in this work. Experimental results show that the modulus of the nanocomposite hydrogel tends to increase with increasing CNC content. Theoretical foundation for analysing the mechanical properties of hydrogels based on Mooney–Rivlin hyperelastic model, Voigt model and Reuss model has been developed and validated, which provides the prediction of the mechanical responses of CNC reinforced nanocomposite hydrogel to tension, especially the nonlinear responding behaviour.

First-principles calculations of typical anisotropic cubic and hexagonal structures and homogenized moduli estimation based on the Y-parameter: Application to CaO, MgO, CH and Calcite CaCO3

X-ray method to test the material properties and to obtain elastic constants is commonly based on the Reuss model and Kroner model. Y parameter has been turned out to be an effective method to estimate elastic properties of polycrystalline material. Since Y-parameters of cubic polycrystalline material based on the certain uniform stress (Reuss model) has not been given, our work aims to complete this part of the theoretical analysis, which can effectively compare elastic constants measured by the X-ray diffraction method. The structural and the elastic properties of cubic structures (CaO and MgO) and hexagonal structures (CH and Calcite CaCO3) are investigated by the density functional theory method. And then the credibility of Y parameters for determing elastic moduli of cubic structures is proved and elastic properties in typical crystallographic planes of [100], [110] and [111] are also calculated. Meanwhile, Young’s moduli of CH and Calcite structure are 58.08GPa and 84.549GPa, which are all close to references. Elastic properties of cubic and hexagonal structures under various pressures are calculated and the surface constructions of elastic moduli are drawn, showing the anisotropy at various directions. The crystal structure investigated in this work are typical of some primary or secondary components of Hardened Cements Pastes and their homogenized elastic properties are needed in a hierarchical multi-scale modeling, such as the one developed by some of the authors of this paper.

Study on Stress Development in the Phase Transition Layer of Thermal Barrier Coatings.

Stress development is one of the significant factors leading to the failure of thermal barrier coating (TBC) systems. In this work, stress development in the two phase mixed zone named phase transition layer (PTL), which grows between the thermally grown oxide (TGO) and the bond coat (BC), is investigated by using two different homogenization models. A constitutive equation of the PTL based on the Reuss model is proposed to study the stresses in the PTL. The stresses computed with the proposed constitutive equation are compared with those obtained with Voigt model-based equation in detail. The stresses based on the Voigt model are slightly higher than those based on the Reuss model. Finally, a further study is carried out to explore the influence of phase transition proportions on the stress difference caused by homogenization models. Results show that the stress difference becomes more evident with the increase of the PTL thickness ratio in the TGO.

X-ray diffraction analysis of residual stresses in textured ZnO thin films

Residual stresses are commonly generated in thin films during the deposition process and can influence the film properties. Among a number of techniques developed for stress analysis, X-ray diffraction methods, especially the grazing incidence set-up, are of special importance due to their capability to analyze the stresses in very thin layers as well as to investigate the depth variation of the stresses. In this contribution a method combining multiple {hkl} and multiple χ modes of X-ray diffraction stress analysis in grazing incidence set-up is used for the measurement of residual stress in strongly textured ZnO thin films. The method improves the precision of the stress evaluation in textured samples. Because the measurements are performed at very low incidence angles, the effect of refraction of X-rays on the measured stress is analyzed in details for the general case of non-coplanar geometry. It is shown that this effect cannot be neglected if the angle of incidence approaches the critical angle. The X-ray stress factors are calculated for hexagonal fiber-textured ZnO for the Reuss model of grain-interaction and the effect of texture on the stress factors is analyzed. The texture in the layer is modelled by Gaussian distribution function. Numerical results indicate that in the process of stress evaluation the Reuss model can be replaced by much simpler crystallite group method if the standard deviation of Gaussian describing the texture is less than 6°. The results can be adapted for fiber-textured films of various hexagonal materials.

Longitudinal Compression Behavior of Functionally Graded Ti–6Al–4V Meshes

The compressive deformation behavior in the longitudinal direction of graded Ti–6Al–4V meshes fabricated by electron beam melting was investigated using experiments and finite element methods (FEM). The results indicate that the overall strain along the longitudinal direction is the sum of the net strain carried by each uniform mesh constituent and the deformation behavior fits the Reuss model well. The layer thickness and the sectional area have no effect on the elastic modulus, whereas the strength increases with the sectional area due to the edge effect of each uniform mesh constituent. By optimizing 3D graded/gradient design, meshes with balanced superior properties, such as high strength, energy absorption and low elastic modulus, can be fabricated by electron beam melting.

Study of the influence of the interfacial transition zone on the elastic modulus of concrete using a triphasic model

Concrete can be considered as a three-phase composite material composed of cement matrix, aggregate particles, and interface transition zone (ITZ). Generally, the ITZ presents particular characteristics that can reduce the properties of concrete and therefore limits its performance. Thus, with such complex structures, this zone is the weakest zone of the composite. The evaluation of the effective behavior of composite using predictive models requires a consideration of this zone. In this context, an approach based on the model of double inclusion and on the Mori–Tanaka theory to predict the elastic modulus of concrete has been used. This approach will be compared with some analytical biphasic model such as Reuss model, Voigt model, the Voigt and Reus combined models, and Hashin and Shtrikman models. Many experimental results are considered in the confrontation. So the developed model predicts very satisfactorily the elastic modulus of the concrete unlike other models in which a discrepancy in the results is demonstrated in the majority of cases.

Numerical evaluation on mechanical properties of short-fiber-reinforced metal matrix composites: Two-step mean-field homogenization procedure

In this paper, the two-step mean-field homogenization procedures including the Mori–Tanaka (M–T) and interpolative double inclusion (D-I) models in the first-step homogenization procedure and the Voigt and Reuss models in the second-step homogenization procedure, are introduced and implemented to predict the effective elastic properties of short-fiber-reinforced metal matrix composites. To take into account the effect of fiber orientation, the detailed implementation of the fiber orientation transformation and fiber orientation averaging procedure is described. Compared with the effective elastic properties of short-fiber-reinforced metal matrix composites predicted by the RVE based FE homogenization method and measured from the uniaxial tensile experiments, the two-step mean-field homogenization procedures: D-I/Reuss, M–T/Reuss, D-I/Voigt and M–T/Voigt would provide the acceptable predictions on the effective elastic properties of short-fiber-reinforced metal matrix composites, and the D-I/Reuss and M–T/Voigt two-step mean-field homogenization procedures provide more tight bounds (D-I/Reuss as the lower bound and M–T/Vogit as the upper bound). For more accurate estimate of the effective elastic properties of short-fiber-reinforced metal matrix composites, a modified two-step mean-field homogenization procedure is proposed and validated.

Problem of elastic anisotropy and stacking faults in stress analysis using multireflection grazing-incidence X-ray diffraction

Multireflection grazing-incidence X-ray diffraction (MGIXD) was used to determine the stress- and strain-free lattice parameter in the surface layer of mechanically treated (polished and ground) tungsten and austenitic steel. It was shown that reliable diffraction stress analysis is possible only when an appropriate grain interaction model is applied to an anisotropic sample. Therefore, verification of the X-ray stress factors (XSFs) was accomplished by measuring relative lattice strains during anin situtensile test. The results obtained using the MGIXD and standard methods (χ and ω geometries) show that the Reuss and free-surface grain interaction models agree with the experimental data. Moreover, a new interpretation of the MGIXD results was proposed and applied for the first time to measure the probability of stacking faults as a function of penetration depth for a polished and ground austenitic sample. The XSF models verified in the tensile test were used in the analysis of residual stress components.

Small strain elasto-plastic multiphase-field model

A small strain plasticity model, based on the principles of continuum mechanics, is incorporated into a phase-field model for heterogeneous microstructures in polycrystalline and multiphase material systems (Nestler et al., Phys Rev 71:1---6, 2005). Thereby, the displacement field is computed by solving the local momentum balance dynamically (Spatschek et al., Phys Rev 75:1---14, 2007) using the finite difference method on a staggered grid. The elastic contribution is expressed as the linear approximation according to the Cauchy stress tensor. In order to calculate the plastic strain, the Prandtl---Reuss model is implemented consisting of an associated flow rule in combination with the von Mises yield criterion and a linear isotropic hardening approximation. Simulations are performed illustrating the evolution of the stress and plastic strain using a radial return mapping algorithm for single phase system and two phase microstructures. As an example for interface evolution coupling with elasto-plastic effects, we present crack propagation simulations in ductile material.

Notched strength prediction of glass fiber reinforced composite based on fracture toughness analysis

The notched strength prediction of normal glass mat composite and needle punched glass mat composite based on fracture mechanics was carried out in this study. The prediction constitutes of three parts including establishing the relation between characteristic distance and notch strength ratio (a ratio of notch strength to unnotched strength), modulus calculation in thickness direction and fracture toughness measurement. The characteristic distance was determined by point stress criterion and calculated from finite element analysis. After contrastive analysis from a mass of tensile results on notched specimens with an open hole, a relation between characteristic distance and notch sensitivity was established. Modulus calculation was calculated according to the classic theory of “Rule of Mixtures” and “Reuss Model”. After that, the results of modulus calculation together with fracture toughness results were employed to establish the relation between notch strength and characteristic distance according to elastic fracture mechanics. The characteristic distance was used in this part as a parameter of a crack. Finally, from the two equations between notch strength and characteristic distance, the notched strength of composite with an open hole can be predicted. The predicting results were compared and analyzed with experiment results.

Nonlinear dynamics of SMA-fiber-reinforced composite beams subjected to a primary/secondary-resonance excitation

In this paper, nonlinear free vibration and primary/secondary resonance analyses of shape memory alloy (SMA) fiber reinforced hybrid composite beams with symmetric and asymmetric lay-up are investigated. The simplified Brinson constitutive model and cosine phase transformation kinetics are utilized to simulate the behavior of the SMA materials and calculate the recovery stress. In order to predict the behavior of the smart laminated beam, Euler–Bernoulli beam theory and nonlinear von-Karman strain field are employed. Two types of micromechanical models, namely Voigt and Reuss models are considered. The Galerkin procedure together with the elliptic function and multi timescales method is adopted to obtain analytical solutions for the nonlinear free vibration and primary/secondary response phenomena. Numerical results reveal that some of the geometrical and physical parameters such as the SMA volume fraction, the amount of prestrain in the SMA fiber, orientation of composite fiber, vibration amplitude and temperature are important factors affecting the free vibration characteristic in the pre/post-buckled region, and primary and secondary resonance of the laminated beams reinforced with SMA fibers. The analytical solutions and results are reported for the first time and can serve as benchmark for researchers to validate their numerical and analytical methods in the future.

Forming Simulation of Thermoplastic Pre-Impregnated Textile Composite

Authors : Masato Nishi, Tetsushi Kaburagi, Masashi Kurose, Tei Hirashima, Tetsusei Kurasiki Abstract : The process of thermoforming a carbon fiber reinforced thermoplastic (CFRTP) has increased its presence in the automotive industry for its wide applicability to the mass production car. A non-isothermal forming for CFRTP can shorten its cycle time to less than 1 minute. In this paper, the textile reinforcement FE model which the authors proposed in a previous work is extended to the CFRTP model for non-isothermal forming simulation. The effect of thermoplastic is given by adding shell elements which consider thermal effect to the textile reinforcement model. By applying Reuss model to the stress calculation of thermoplastic, the proposed model can accurately predict in-plane shear behavior, which is the key deformation mode during forming, in the range of the process temperature. Using the proposed model, thermoforming simulation was conducted and the results are in good agreement with the experimental results.

有限要素法による織物強化熱可塑性樹脂のプレス成形解析

In this paper, we propose a new FE model of a carbon fiber reinforced thermoplastic (CFRTP) in order to capture the deformation during a thermoforming process because the thermoforming process of CFRTP has increased its presence in the automotive industry for its wide applicability to the mass production car. The proposed model can describe temperature dependent non-linear bending property of CFRTP by a set of elements which consists of two shell elements with membrane elements in between them. The membrane elements represent temperature dependent anisotropic in-plane behavior by calculating stress contributions of the textile reinforcement and thermoplastic in a parallel system. By applying Reuss model to the stress calculation of thermoplastic, the in-plane shear behavior which is the key deformation mode during forming can be accurately predicted. FE model is constructed based on the results of three point bending and bias-extension experiments which are conducted in the range of the process temperature. Thermoforming simulations are presented and compared to experimental results. Simulated outline and shear angle are in good agreement with experimental results. It will be shown by sensitivity study that the effect of the temperature plays an important role in deformation during a non-isothermal forming process.

Effect of fibre coating and geometry on the tensile properties of hybrid carbon nanotube coated carbon fibre reinforced composite

Hierarchically structured hybrid composites are ideal engineered materials to carry loads and stresses due to their high in-plane specific mechanical properties. Growing carbon nanotubes (CNTs) on the surface of high performance carbon fibres (CFs) provides a means to tailor the mechanical properties of the fibre–resin interface of a composite. The growth of CNT on CF was conducted via floating catalyst chemical vapor deposition (CVD). The mechanical properties of the resultant fibres, carbon nanotube (CNT) density and alignment morphology were shown to depend on the CNT growth temperature, growth time, carrier gas flow rate, catalyst amount, and atmospheric conditions within the CVD chamber. Carbon nanotube coated carbon fibre reinforced polypropylene (CNT-CF/PP) composites were fabricated and characterized. A combination of Halpin–Tsai equations, Voigt–Reuss model, rule of mixture and Krenchel approach were used in hierarchy to predict the mechanical properties of randomly oriented short fibre reinforced composite. A fractographic analysis was carried out in which the fibre orientation distribution has been analyzed on the composite fracture surfaces with Scanning Electron Microscope (SEM) and image processing software. Finally, the discrepancies between the predicted and experimental values are explained.

The micromechanics modeling piezoelectric composite materials by method of homogenization Mori Tanaka: study of piezo-composite SiO2/BaTiO3

In this work, the behavior of piezoelectric composite has been studied in terms of prediction, in order to contribute to understanding the piezoelectricity, perroelectricity phenomenon of these materials. The materials with the piezoelectric intrinsic properties have been added fillers to matrix to obtain the piezoelectric composites materials. The Mori-Tanaka model has been used to predict the mechanical behavior, which is part of the electromechanical formulation. The use of Mori-Tanaka model is based on its compatibility with the composites at high fiber volume content. The results from the Mori-Tanaka model have been compared to Voigt and Reuss models. The models have been simulated using programming language to determine stress-strain behavior.

Ion-beam modifications of mechanical and dimensional properties of silicon carbide

3C-SiC single crystal epitaxial layers, 6H-SiC single crystal plates and α-SiC Hexoloy sinters were irradiated with 4.0 MeV Xe or 4.0 MeV Au ions at room temperature. Mechanical and dimensional evolutions of silicon carbide are studied by means of nano-indentation and step-height measurements which are correlated with Rutherford backscattering spectrometry and channelling (RBS/C) data in single crystals. Irradiation with Xe ions induces a total lattice disorder related to amorphization for a fluence of 1 × 1015 cm−2 in both polytypes. When complete amorphization is reached, around the same values of Young's modulus (∼350 GPa) and Berkovich hardness (∼27 GPa) are found in both polytypes. The out-of-plane expansion increases with irradiation dose and the saturation value measured in the amorphous layer (normalized to the projected range of the ions) is close to 20–25%. Modifications of macroscopic properties are mainly governed by the disordered fraction of the material in a two-step damage process. A similar behaviour of material evolution is found for the cubic and hexagonal polytypes, either in single crystals or sintered polycrystalline samples. Calculations of Young's modulus and volume swelling are carried out using the analytical (Reuss and Voigt) models of homogenization.

Mechanical Properties of End-Linked PEG/PDMS Hydrogels

Poly(ethylene glycol) (PEG)/polydimethylsiloxane (PDMS) hydrogels were synthesized by cross-linking norbornene end-functionalized polymers with a tetrafunctional thiol using thiol–norbornene chemistry. The swelling capacity and mechanical properties, including the Young’s modulus (E) and fracture toughness (Gc), of the hydrogels were characterized and quantified as a function of the volume fractions of PEG and PDMS. E and Gc increased simultaneously with the volume fraction of PDMS. The moduli of the hydrogels were quantitatively described and predicted as a function of the volume fraction ratio of PEG to PDMS using the Voigt and Reuss models. The fracture toughness was well described by the Lake–Thomas theory at low volume fractions of PDMS. As the volume fraction of PDMS increased, PDMS not only controlled the swelling capacity of the hydrogels but also contributed to hydrogel toughness.

STUDY ON HALF-WIDTH AND MAXIMUM X-RAY INTENSITY OF DIFFRACTION LINE OF COLD-ROLLED AUSTENITIC STAINLESS STEEL AS A FUNCTION OF ARIMUTH ANGLE ψ

This research studies on the maximum x-ray counts and half-width of the x-ray diffraction line of textured austenitic stainless steel. The experimental stress-independent direction almost agrees with the theoretical value, determined from the Voigt, Reuss and Kroner models. The result of loading test for two perpendicular SUS316L austenitic stainless steel specimens shows that the broadness and maximum intensity of diffraction line in the rolling direction respectively get the lowest and highest values around the stress-independent direction. In the traverse direction, the broadness and maximum intensity of diffraction line vary slightly as in the case of the isotropic materials.

The Constitutive of Binary Medium for Soils Based on Voigt Model and Reuss Model

In order to study the structure of natural sedimental clays, the theoretical framework of breakage mechanics for soils and the linary medium model are proposed by Shen Zhujiang based on compatibility of deformation. However, the deformation between structural blocks and soft bands is compatible, and the stress between them is continuous. In this paper, two new constitutive equations are derived based on Voigt model and Reuss model respectively so as to provide theoretical bases for FEM and the test of mechanical properties for soils. At last, a numerical example is given to make clear the breakage laws of structural blocks.

Analysis of Stress-Strain Relationships and Line Broadening in Polycrystalline Materials Having Tetragonal Crystal System with [001] Fibre Texture by X-Ray Diffraction

The stress and strain relationships for polycrystalline materials having tetragonal crystal system with [001] fibre texture, which are required for stress analysis by X-ray diffraction (XRD), were formulated within the framework of the Reuss model of elasticity. In formulating the relationships, two types of orientations of constituent crystallites in the specimen, which contribute to XRD, were shown to exist by taking into account the Laue classes of the crystallites. The formulae representing strains measurable by XRD were found to be different, depending on which the Laue class, 4/mmm or 4/m, the specimen belongs to, even though both Laue classes are in the tetragonal crystal system. The formula obtained for the Laue class 4/mmm shows that the profile observed by XRD is formed in a symmetrical shape so that the strain can be the average value of those for the two different types of crystallites. On the other hand, the formula for the Laue class 4/m shows that the profile is formed in an asymmetrical shape so that the strain can be determined by the magnitudes of the structure factors in the two different types of crystallites. Additionally, it is shown that some of the Debye-Scherrer lines change their widths with the measurement direction. The order of magnitude of such a line broadening is discussed for two materials, BaTiO3 and MgF2.