Reuss Model Research Articles

Evaluating the effect of variable fiber content on mechanical properties of additively manufactured continuous carbon fiber composites

Fiber volume fraction is a driving factor in mechanical properties of composites. Micromechanical models are typically used to predict the effective properties of composites with different fiber volume fractions. Since the microstructure of 3D-printed composites is intrinsically different than conventional composites, such predictions need to be evaluated for 3D-printed composites. This investigation evaluates the ability of the Voigt, Reuss, and Halpin–Tsai models to capture the dependence of modulus and strength of 3D-printed composites on varying fiber volume fraction. Tensile coupons were printed with continuous carbon fiber-reinforced Onyx matrix using a Markforged Mark Two printer. Specimens were printed at five different volume fractions with unidirectional fibers oriented at either [Formula: see text] to obtain longitudinal, shear, and transverse properties, respectively. It is shown that the Voigt model provides an excellent fit for the longitudinal tensile strength and a reasonable fit for the longitudinal modulus with varied fiber content. For the transverse direction, while the Reuss model fails to capture the transverse modulus trend, the Halpin–Tsai model provides a reasonable fit as it incorporates more experimental parameters. Like conventional composites, addition of fibers degrades the transverse strength, and the transverse strength decreases with increasing fiber volume fraction. The shear modulus variation with fiber content could not be fitted reasonably with either Halpin–Tsai model or Reuss model.

Determination of the macroscopic residual stress: Modeling and experimental evaluation of texture and local grain-interaction

When the texture is present in the sheet, it is extremely difficult to test the residual stress based on the X-ray diffraction method. Based on the two extreme traditional Voigt and Reuss models, this paper proposes two specific corresponding local models (local Voigt and local Reuss) for the measurement of the plane residual stress of cold rolled sheet texture materials. The residual stress test results of CuNiSi specimen using these two local models show that the proposed local models are credible and can accurately characterize the residual stress in the specimen. In addition, based on the X-ray diffraction sin2ψ method of the classic residual stress test, a weighted average residual stress calculation method is proposed for cold-rolled texture materials. The dψ-sin2ψ method was used to analyze the effect of texture on residual stress. The results show that the copper texture C{211}〈111〉 in CuNiSi specimen has the effect of concentrating the residual compressive stress in the region. The stress increased by about 50 MPa. However, the effect of brass texture B{011} 〈211〉 and R texture R{011} 〈211〉 on residual stress was not obvious.

Interaction between Fe66Cr10Nb5B19 metallic glass and aluminum during spark plasma sintering

In the area of metal matrix composites, novel reinforcing options are currently being evaluated. Particles of amorphous alloys present an interesting possibility to reinforce soft metals. In the present work, the interaction between Fe66Cr10Nb5B19 metallic glass and aluminum during spark plasma sintering (SPS) was studied for the first time. In order to trace the phase and microstructural changes upon sintering, mixtures containing 20 vol% and 50 vol% of metallic glass were subjected to SPS at 500–570 °C. After SPS at 500 °C, no reaction layer between the metallic glass particles and aluminum was observed. After SPS at 570 °C, a reaction layer containing Fe2Al5 and FeAl3 formed around the Fe-based cores. The Vickers hardness of composites obtained from mixtures containing 20 vol% Fe66Cr10Nb5B19 at 540 °C was 75 HV and increased to 280 HV after sintering at 570 °C due to the formation of thicker reaction layers at the interface. The hardness of the composite sintered from the mixture containing 20 vol% Fe66Cr10Nb5B19 at 570 °C was between the values predicted by Reuss and Voigt models. Comparison of results of SPS of the powder mixtures with those of SPS of a pre-compacted pellet and electric current-free annealing suggests that local heating at the interface caused by interfacial resistance may be an important factor influencing the reaction advancement at the interface and the formation of Al-containing intermetallics.

Nonlinear oscillations of composite conical microshells with in-plane heterogeneity based upon a couple stress-based shell model

Abstract The current investigation deals with the nonlinear oscillation response of conical microshells made of a composite material with functional graded (FG) in-plane heterogeneity is studied in the presence of the size dependency. To accomplish this purpose, various types of homogenization schemes including Voigt model, Reuss model, Mori-Tanaka model, and Hashin-Shtrikman bounds model are employed. The size-dependent characteristics are taken into consideration on the basis of the modified couple stress theory of elasticity within the framework of the higher-order shear deformation shell theory. The couple stress-based differential equations of motion are constructed via the Hamilton's principle. An efficient numerical solution methodology adopting generalized differential quadrature (GDQ) method together with the pseudo-arc technique is put to use to obtain the modified couple stress-based nonlinear frequency of homogenized FG composite conical microshells. It is demonstrated that by increasing the value of the maximum shell deflection, the couple stress type of size effect plays more important role in the nonlinear vibration response of FG composite conical microshells. Additionally, it is indicated by changing the boundary conditions from simply supported one to clamped one, the influence of couple stress size dependency decreases.

Polylactic Acid/Polycaprolactone Blends: On the Path to Circular Economy, Substituting Single-Use Commodity Plastic Products.

Circular economy comes to break the linear resource to waste economy, by introducing different strategies, two of them being: using material from renewable sources and producing biodegradable products. The present work aims at developing polylactic acid (PLA), typically made from fermented plant starch, and polycaprolactone (PCL) blends, a biodegradable polyester, to study their potential to be used as substitutes of oil-based commodity plastics. For this, PLA/PCL blends were compounded in a batch and lab scale internal mixer and processed by means of injection molding. Tensile and impact characteristics were determined and compared to different thermoplastic materials, such as polypropylene, high density polyethylene, polystyrene, and others. It has been found that the incorporation of PCL into a PLA matrix can lead to materials in the range of 18.25 to 63.13 megapascals of tensile strength, 0.56 to 3.82 gigapascals of Young’s modulus, 12.65 to 3.27 percent of strain at maximum strength, and 35 to 2 kJ/m2 of notched impact strength. The evolution of the tensile strength fitted the Voigt and Reuss model, while Young’s modulus was successfully described by the rule of mixtures. Toughness of PLA was significantly improved with the incorporation of PCL, significantly increasing the energy required to fracture the specimens. Blends containing more than 20 wt% of PCL did not break when unnotched specimens were tested. Overall, it was found that the obtained PLA/PCL blends can constitute a strong and environmentally friendly alternative to oil-based commodity materials.

Couple stress-based nonlinear buckling analysis of hydrostatic pressurized functionally graded composite conical microshells

The present study deals with the size-dependent nonlinear buckling characteristics of conical microshells made of functionally graded (FG) composite materials under uniform hydrostatic pressure based upon the modified couple stress theory of elasticity. Accordingly, a modified couple stress-based shell model within the framework of the higher-order shear deformation shell theory and von Karman geometrical nonlinearity is constructed. Using the virtual work's principle in conjunction with the adjacent equilibrium criterion, the non-classical governing differential equations are established. The material properties of FG composite conical microshells are estimated on the basis of different homogenization schemes. To solve the size-dependent nonlinear problem, the generalized differential quadrature discretization pattern together with the Galerkin technique is employed. It is seen that among various types of homogenization scheme, the Voigt and Reuss models represent, respectively, the overestimated and underestimated critical buckling pressures. Also, it is found that for a FG composite conical microshell with higher semi-vertex angle, the influence of the material property gradient index on the nonlinear critical buckling pressure diminishes. In addition, it is observed that the couple stress size dependency plays more important role in the nonlinear buckling behavior of FG composite conical microshells with lower ratio of R1/h. These patterns are the same for all types of boundary conditions.

On the surface elastic-based shear buckling characteristics of functionally graded composite skew nanoplates

Abstract The prime objective of the present investigation is to predict the shear buckling characteristics of skew nanoplates made of a functionally graded material (FGM) in the presence of surface stress effect. For this purpose, the Gurtin-Murdoch surface theory of elasticity is applied to the higher-order shear deformation plate theory within the framework of the oblique coordinate system. Different types of the homogenization scheme including Reuss model, Voigt model, Mori-Tanaka model, and Hashin-Shtrikman bounds model are taken into consideration in order to extract the effective mechanical properties of FGM skew nanoplates. The Ritz method using Gram-Schmidt shape functions is utilized to obtain the surface elastic-based shear buckling loads of FGM skew nanoplates. It is indicated that by increasing the value of the index associated with the material property gradient, the significance of the surface stress type of size effect on the shear buckling behavior of a FGM skew nanoplate improves. Moreover, by changing the boundary conditions from simply supported ones to clamped ones, the influence of the skew angle on the surface elastic-based shear buckling load of a FGM skew nanoplate increases. Also, it is illustrated that by increasing the width to thickness ratio of a skew nanoplate, the free surface area increases which results in to enhance the effect of surface residual stress on its shear buckling characteristics.

An efficient two-scale 3D FE model of the bone fibril array: comparison of anisotropic elastic properties with analytical methods and micro-sample testing.

In this study, 3D finite element analyses (FEA) are conducted to quantify the orthotropic elastic properties and investigate the load transfer mechanism of bone at the sub-lamellar level. Three finite element (FE) unit cells with periodic boundary conditions are presented to model a two-scale microstructure of bone including a mineralized collagen fibril (MCF), the extrafibrillar matrix (EFM) and the resulting fibril array (FAY) under arbitrary loading. The axial and transverse elastic properties of the FAY computed by FEA are calibrated with unique experimental results on ovine micro-samples showing a coherent fibril orientation. They are then systematically compared with those calculated using analytical methods including the basic Voigt, Reuss and shear-lag models, the Mori-Tanaka scheme and the upper and lower bounds by Hashin and Shtrikman. The predicted axial strain ratios between the two-scales are discussed with respect to a recent small-angle X-ray scattering and wide-angle X-ray diffraction study. Beyond apparent elastic properties, the FE models provide stress distributions at both hierarchical levels, confirm the shear lag mechanisms within the MCF and between MCF and EFM and identify potential damage sites under arbitrary loading conditions. A comprehensive sensitivity analysis shows that mineral volume fraction in the fibril array is the dominant parameter on the axial and transverse elastic moduli, while the MCF volume fraction in FAY is the most sensitive variable for the ratio of axial versus transverse elastic modulus followed by the elastic moduli of hydroxyapatite and collagen. The FE model of the FAY developed and calibrated in the current study represents an anatomically realistic, experimentally validated and computationally efficient basis for investigating the apparent yield, post-yield and failure behaviors of lamellar bone in future research.

Size-dependent shear buckling response of FGM skew nanoplates modeled via different homogenization schemes

The size effects on the shear buckling behaviors of skew nanoplates made of functionally graded materials (FGMs) are presented. The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness. To estimate the associated effective material properties, various homogeniza-tion schemes including the Reuss model, the Voigt model, the Mori-Tanaka model, and the Hashin-Shtrikman bound model are used. The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates. The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem. It is found that the significance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index. Also, by increasing the skew angle, the critical shear buckling load of an FGM skew nanoplate enhances. This pattern becomes a bit less significant for a higher value of the material property gradient index. Furthermore, among various homogenization models, the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads, and the difference between them reduces by increasing the aspect ratio of the skew nanoplate.

Explicit nonlinear and plastic analysis model of steel–concrete–steel sandwich panels

This paper presents an explicit nonlinear and plastic analysis model (ENPAM) for steel–concrete–steel (SCS) panels. This model considers the behavior of concrete under biaxial stresses and the post-yield stress-strain relationship of steel plates. A simplified measure is utilized for the simulation of concrete. The elastic–plastic stiffness of steel plates is derived from the Prandtl–Reuss model. ENPAM fully embodies the stress redistribution during the plastic flow after steel plate yielding. The model's effectiveness is demonstrated based on comparisons with experimental data obtained for SCS panels subjected to pure shear and shear with a small compression. A finite element model is also established to verify the effectiveness of ENPAM for the load cases of shear with large compression.

Fabrication and Characterization of Metal-High Entropy Alloy Composites

The scope of the present study is focused on high-entropy alloy (HEA), a new type of alloy system-reinforced aluminium metal matrix composite (AMMC) development. The novel metal–metal composite is fabricated through powder metallurgy by reinforcing prepared AlCoCrCuFe HEA particles with aluminium. Headed for identifying the effect of HEA on aluminium, weight percentage of HEA is varied from 0 to 15% with an equal interval of 3%. By using X-ray diffractometer, phase constituents of HEA and Al base material are analysed besides the fact that the scanning electron microscope is utilized for morphology analysis of HEA particles and AMMC. Density of the aluminium increases gradually with the increase in HEA content due to higher density of reinforcements, and maximum micro-hardness of 71.3 HV is attained for 15% HEA reinforcement addition which is 37.6 HV for unreinforced aluminium. Among the various predictive methodologies, the density and hardness of Al–AlCoCrCuFe metal–metal composite are found close to the values predicted by the Reuss and Voigt model, respectively. Decrease in wear rate was found for aluminium MMC with the increase in HEA reinforcement addition.

Determination of Diffraction Elastic Constants of Gamma-Double-Prime in Polycrystalline Ni-Base Superalloys

In this paper we introduce a new method to measure diffraction elastic constants of gamma double prime phase in Ni-base superalloys using in-situ neutron diffraction. During aging heat treatment of the alloys, stress is applied so that gamma double prime variants can be selected, which facilitates accurate fitting of the diffraction peaks. Then the Miller Indices reflections from different gamma double prime variants can be used to separate the overlapping gamma and gamma double prime diffraction peaks. The measured values agree with the elastic constants calculated using Kroner and Reuss models.

Analysis of Polymethylmethacrylate Destruction on High-Speed Loading

A system of equations is presented for describing the shock-compressed state of polymethylmethacrylate in the approximation of uniaxial deformation. With the use of the Prandtl–Reuss model, the mechanism of transition of the brittle failure of polymethylmethacrylate to the brittle-plastic one at high speeds of the striker and high times of its effect on this mechanism has been determined.

Particle Dispersibility and sound propagation of ultrasounds for magnetic soft composites

We measured the sound velocity for magnetic soft composites consisting of a polymeric matrix of polyurethane and a magnetic particle of carbonyl iron or barium ferrite, and investigated the effect of the particle dispersibility on the sound velocity at 10 MHz for the composites. Morphological observations and dynamic viscoelastic measurements at 1 Hz were also carried out, and the results revealed that the carbonyl iron was dispersive particle and the barium ferrite was aggregative particle in the polyurethane matrix. The sound velocity for polyurethane elastomer without magnetic particles was 1444 ± 12 m/s and it decreased with the volume fraction of magnetic particles. The sound velocity of magnetic composites with barium ferrite obeyed the Reuss model at whole volume fractions, however, magnetic composites with carbonyl iron exhibited an under deviation from the Reuss model at volume fractions above ~0.053. It was also found that the longitudinal modulus was insensitive to the creation of the particle network and sensitive to the aggregation of magnetic particles. The threshold seen in the longitudinal modulus was different from that in the nonlinear parameter of viscoelasticity, indicating that the ultrasound of 10 MHz detects microscopic heterogeneity of particle dispersion.

A revision of results for standard models in elasto-perfect-plasticity theory

We consider two most studied standard models in the theory of elasto-plasticity in arbitrary dimension $$d\ge 2$$ , namely, the Hencky model and the Prandtl–Reuss model subjected to the von Mises condition. There are many available results for these models—from the existence and the regularity theory up to the relatively sharp identification of the plastic strain in the natural function/measure space setting. In this paper we shall proceed further and improve some of known estimates in order to identify sharply the plastic strain. More specifically, we rigorously improve the integrability of the displacement and the velocity (which was known only under a nonnatural assumption that the Cauchy stress is bounded), show the BMO estimates for the stress and finally also the Morrey-like estimates for the plastic strain. In addition, we shall provide the whole theory up to the boundary. As an immediate consequence of such improved estimates, we provide a sharper identification of the plastic strain than that known up to date. In particular, in two dimensional setting, we show that the plastic strain can be point-wisely characterized in terms of the stresses everywhere although the stress is possibly discontinuous and thus the natural duality pairing in the space of measures could be violated.

Comparison Between Biphasic and Triphasic Model for Predicting the Elastic Modulus of Concrete

The concrete is a composite material which, on the scale of the microstructure, can be considered as consisting of three phases: the matrix, the inclusions and the interface transition zone. The latter has features that 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 are 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 (HδS) models. Many experimental results are considered in the confrontation. So the model developed 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.

A FIRST LEVEL APPROXIMATION MODEL OF THE DYNAMIC ELASTICITY OF A PARTIALLY BIORESORBABLE COMPOSITE INTERNAL FIXATION SYSTEM

The objective of this work is to develop a first level approximation hybrid model that predicts the rate of reduction of the elastic modulus for a composite internal fixation plate. The first level model combines a modified composite theory in which the bioresorbable constituent moduli and volume fractions are time-dependent, with predictions from a Finite Element (FE) model, but ignores viscoelastic effects. The FE model is used to predict relationships between geometrical constraints (material interfaces), while a theoretical combined elastic modulus is determined using the Reuss model. The composite internal fixation device design comprises Poly-L-lactide and Hydroxyapatite (HA/PLLA) and Titanium. The geometrical relationships established by the FE static model are combined with the theoretical mathematical model in order to predict the rate of decrease of the elastic modulus of the composite device. The first level model predicts that the composite plate modulus decreases at a rate that appropriately compensates for the increasing modulus of the healing bone during fracture healing. This suggests that the proposed composite fixation plate has a strong potential for improved fracture healing through the reduction of stress shielding while the fracture heals, and the elimination of stress shielding after fracture healing.

Spherical harmonics analysis based on the Reuss model in elastic macro strain and stress determination by powder diffraction

In this paper a new approach to macro strain/stress analysis by generalized spherical harmonics is presented. It consists of expanding the stress tensor weighted by texture in a series of generalized spherical harmonics with the ground state of expansion specific to the classical Reuss model of an isotropic polycrystal. Like previously reported models having a ground state of hydrostatic type [Popa & Balzar (2001). J Appl Cryst. 34, 187–195] and of Voigt type [Popa et al. (2014). J Appl Cryst. 34, 154–159], the actual model is appropriate for use with Rietveld refinement.

Review on micromechanics of nano‐ and micro‐fiber reinforced composites

This article deals with the prediction of thermomechanical properties of fiber reinforced composites using several micromechanics models. These include strength of material approach, Halpin–Tsai equations, multi‐phase mechanics of materials approaches, multi‐phase Mori–Tanaka models, composite cylindrical assemblage model, Voigt–Reuss models, modified mixture rule, Cox model, effective medium approach and method of cells. Several composite systems reinforced with short and long, aligned, random and wavy reinforcements were considered. In addition, different aspects such as fiber‐matrix interphase, fiber‐matrix interfacial thermal resistance, fiber geometry, and multiple types of reinforcements were considered to model the composites systems. The current study also presents some important preliminary concepts and application of developed micromechanics models to advanced nanocomposites such as carbon nanotube reinforced composite. Main contribution of the current work is the investigation of several analytical micromechanical models, while most of the existing studies on the subject deal with only one or two approaches considering few aspects. POLYM. COMPOS., 39:4243–4274, 2018. © 2017 Society of Plastics Engineers

Pattern decomposition for residual stress analysis: a generalization taking into consideration elastic anisotropy and extension to higher-symmetry Laue classes

The residual stress state of ion-conducting layers (yttria stabilized zirconia) and protective hard coatings (α-aluminium oxide, titanium carbonitride) was investigated using X-ray diffraction techniques. Its evaluation within the tetragonal, trigonal and cubic phases present was performed by a whole powder pattern decomposition procedure according to Pawley. Going beyond previous work, the applied refinements directly include the influence of elastic anisotropy on the residual stress results. Starting from the single-crystal elastic coefficients, the X-ray elastic constants according to the Voigt and Reuss models were calculated. Finally, the Neerfeld–Hill model as a generalization was implemented to introduce thehkl-specific X-ray elastic constants for calculating the residual stress magnitude within the least-squares minimization routine. It was possible to resolve the residual stress state in stacked layers of different chemical and phase composition and to reproduce the results obtained by the classical χ- and ω-inclination sin2Ψ techniques.