Rule Of Mixtures Research Articles

Free Vibration and Buckling Analyses of Functionally Graded Plates With Graphene Platelets Reinforcement

Abstract While existing research has focused on using graphene platelets (GPLs) as reinforcement for homogeneous matrices, this study proposes a new nanocomposite for plate structures consisting of GPLs incorporated into a conventional functionally graded matrix with the aim of enhancing their overall stiffness. The performance of such plates is evaluated via free vibration and buckling analyses in the present study. Note that the matrix phase is graded continuously with the power law distribution across the plate's thickness, whereas various GPL dispersion patterns along the thickness are studied. The material properties of the typical functionally graded matrix are determined by the rule of mixture, and then those of the composite are estimated by the modified Halpin–Tsai model as well as the rule of mixture. Based on Hamilton's principle and the novel four-unknown refined plate theory (RPT4), the governing equations of the plate are developed. The Navier-type solution scheme is then adopted to get the critical buckling load and natural frequency of the nanocomposite plate. Numerical findings are examined to evaluate the novel nanocomposite plate model, and a parametric study is also conducted. In addition, high-accurate results are provided via the Navier-type solution here as benchmark solutions for further studies on functionally graded material structures reinforced by GPLs.

An experimental and analytical investigation to determine thermal conductivity of epoxy-filler composites for space applications

Thermal conductivity of epoxy-filler composite based Thermal Interface Materials (TIMs) is of utmost importance for thermal management systems used in space applications. Previous studies have shown that depending on the filler particle mixed into the epoxy, thermal conductivity of the composite can either be increased or decreased. Towards that, we have selected four different filler particles (micron sized) – aluminium, copper, zinc and silver to enhance the thermal conductivity of the base epoxy. Thermal conductivity of these 4 epoxy-filler composites has been determined experimentally using an indigenous developed setup at the temperatures ranging from 4.2 K to 323 K. The values have also been reported at various volume fractions (up to 15 %). In addition, after each preparation as the filler particles are mixed mechanically into the epoxy, they are randomly distributed, and this can affect the composite’s thermal conductivity (for the same volume fraction). The experimental determination of thermal conductivity for all the possible distributions is a time consuming, and cumbersome task. In the literature, thermal conductivity values are usually reported for few possible distributions. Therefore, we have developed a novel analytical model to predict all the possible values of thermal conductivity for a specific volume fraction. The predicted values compare well with our experimental data reported in this work at temperatures ranging between 338 K (above room temperature) to 4.2 K (liquid helium temperature). The predicted values are also in good agreement with the experimental values of different fillers such as red mud, pine wood dust and glass fiber etc. from the literature. Additionally, in comparison to other theoretical models like Lewis-Nielsen, Rule of mixture, and Poisson’s distribution, the present model predicts the thermal conductivity values more precisely. This work will be significant in the designing of components where heat transfer plays an important role towards the safety of the component.

EVALUATION OF PHYSICAL AND MECHANICAL PROPERTIES OF BN REINFORCED AL7079 METAL MATRIX COMPOSITES

Metal matrix composites (MMCs) are regarded as viable alternatives to conventional materials such as metals, plastics, and ceramics in structural applications due to their properties, including lightweight, high specific stiffness, elevated elastic modulus, enhanced specific strength, and excellent wear resistance. Among the many metal composites, aluminum metal matrix composites (MMCs) have emerged as sophisticated engineering materials for several prospective applications in engineering industries due to their superior qualities compared to typical aluminum alloys. Ceramic particles are the most extensively utilized reinforcements in MMCs due to their superior wear resistance, thermal stability, and exceptional bonding with the matrix. This research work has been conducted to develop ceramic particle-reinforced aluminum matrix composites and to evaluate their physical and mechanical properties. Al7079 alloy and Boron Nitride (BN) are selected as the matrix and reinforcement for the study respectively. The stir casting technique was utilized to fabricate the composites.Al7079-BN composites are prepared by varying the percentage of BN reinforcement particles from 0 to 8% by volume, with an increment of 2%. Test specimens were machined from the produced composites for the evaluation of microstructural, physical, mechanical properties according to ASTM standards. The microstructural analysis of the produced samples was conducted using scanning electron microscopy (SEM). The density of the composite was assessed empirically using Archimedes principle and theoretically through the rule of mixture. Microstructural analysis reveals a homogeneous distribution of BN particles throughout the matrix without any agglomeration, and it also demonstrates excellent bonding. The density of the composites decreased by approximately 4.6% with the incorporation of BN reinforcement up to 8%, attributed to the lower density of BN particles. The mechanical properties such as hardness, tensile strength, Youngs modulus, and compressive strength of the composite increases significantly.

Reliable longitudinal electrical conductivity characterisation of unidirectional CFRP tapes

Induction welding is a promising assembly method for carbon fibre reinforced thermoplastic composites. In this process, an alternating electromagnetic field induces eddy currents in a composite susceptor. Accurate determination of the electrical conductivity of the carbon fibre network is essential for the development of physics-based simulations of the process. This work focuses on characterisation of the longitudinal electrical conductivity of unidirectionally reinforced paek using the six-probe method. Special attention is given to validating the boundary conditions and assumptions underlying the experimental procedure. The experimental results show that the longitudinal electrical conductivity consistently adheres to the rule of mixtures; nevertheless, achieving this finding proves sensitive to the care put into the procedural execution. Additionally, the study provides a valuable dataset for various commercially available tapes relevant to the aerospace industry.

Analysis of Thermodynamic Models for Liquid-Vapor Equilibrium: Evaluating Accuracy and Applicability

This article investigates the liquid-vapor equilibrium of four binary refrigerant systems: R134a + R290, R152a + R1234ze, R152a + R1243zf, and R1243zf + R134a. The study employs three thermodynamic models for accurate predictions: the Peng-Robinson equation with the classical mixture rule of van der Waals (vdW) and the Wilson model, the PC-SAFT equation, and the PR-MC-WS-NRTL model. Activity coefficients are determined using the Peng-Robinson equation with vdW and the Wilson model. The PC-SAFT equation and the PR-MC-WS-NRTL model are also applied to model the data. The calculated results show good agreement with reference data. Favorable agreements exist between the calculated results and the reference data, with relative errors remaining below (0.15 and 0.42) % for the molar fraction and the pressure, respectively. This research provides valuable insights into the accuracy and applicability of different thermodynamic models in predicting liquid-vapor equilibrium within refrigerant systems.

Nonlinear dynamic characteristics of smart FG-GPLRC sandwich varying thickness truncated conical shell with internal resonance for first three order modes

This paper examines the 1:1:1 internal resonant nonlinear dynamic characteristic of the simply supported varying thickness functionally graded graphene platelets reinforced composite (FG-GPLRC) smart truncated sandwich conical shell subject to the combined effects of transverse load and in-plane force. The truncated smart sandwich conical shell is composed of an FG-GPLRC varying thickness core and two magneto-electro-elastic face layers, whose material properties and constitutive relations are individually identified by the rule of mixture, improved Halpin-Tsai approach and generalized Hooke's law. Utilizing the first-order shear deformation theory (FSDT), von Karman's geometrical nonlinearity, Hamilton's principle and Galerkin technique, the 3DOF dimensionless nonlinear dynamic formulations for the truncated smart FG-GPLRC conical shell are established. The multiple-scale technique is applied to developing the averaged equations for the truncated smart FG-GPLRC conical shell under combined resonance. The frequency-response and force-response curves, Poincare maps, phase portraits, time history diagrams, bifurcation and maximum Lyapunov exponent diagrams can be portrayed by the nonlinear equation solver and Runge-Kutta approach. The effects of the damping and tuning parameters, transverse and in-plane forces on the 1:1:1 internal resonant nonlinear dynamic characteristic of truncated smart varying thickness FG-GPLRC sandwich conical shell are examined.

An efficient moving-mesh strategy for predicting crack propagation in unidirectional composites: Application to materials reinforced with aligned CNTs

This paper presents an efficient numerical approach for reproducing the process of crack propagation inside unidirectional composites subjected to general loading conditions, with special reference to epoxy materials enhanced with embedded aligned CNTs. This approach involves a traditional FE framework improved by the Moving Mesh (MM) technique based on the Arbitrary Lagrangian-Eulerian (ALE) formulation and the Interaction Integral Method (M−integral). The MM serves as a powerful numerical tool to simulate the discrete crack advance with minimal remeshing, thus reducing the computational complexities. Instead, the M−Integral method, formulated for generally anisotropic materials, has been employed to extract the mixed-mode Stress Intensity Factors (SIFs), which are necessary to define the crack onset condition and propagation direction on the basis of the modified Maximum Hoop Stress Criterion. The proposed strategy includes the extended rule of mixtures to evaluate the homogenized elastic properties of nano-reinforced composites. The validity of the proposed methodology has been assessed through comparisons with experimental data and numerical results available in the literature.

Free vibration and buckling characteristics of temperature dependent FGCNT reinforced plates in finite element framework of a non-polynomial axiomatic approach

In the present work, a generalized finite element formulation is established and implemented for the free vibration and buckling characteristics for temperature dependent carbon nanotube (CNT) reinforced plate. The uniform and functionally graded distribution of CNT is assumed in the plate and the plate is modeled in non-polynomial axiomatic framework. The extended rule of mixture is employed for computing the equivalent material parameters which are temperature dependent. Further, an eight-noded C0 continuous element with seven degree of freedom is adopted for discretization of the plate domain. The convergence and comparison studies are carried out to ensure the accuracy and effectiveness of the proposed formulation. As temperature conditions significantly influence structures integrity, on that account, thorough parametric studies are performed to examine the influence of volume fraction (VCNT ), span to thickness ratio ( a / h ), CNT distribution and boundary conditions on the FG-CNTR plate for the different temperature conditions. The results demonstrate that the efficient use of CNT distribution and boundary conditions can consequently improve free vibration and buckling characteristics of the temperature dependent FG-CNTR plates.

The transient dynamic wave process in unidirectional composites

The propagation of nonstationary waves in unidirectional composite material is simulated by finite-element analysis. The composite material is explicitly formulated, i.e., discrete, stiff, elastic inclusions uniformly distributed in a viscoelastic matrix are modeled. An original procedure rooted in the linear Boltzmann–Volterra equations accounts for the viscoelastic effects in the matrix behavior. The chosen approach involves finite-element simulations of the transient wave process in two indicative configurations to study the effects attributed to wave scattering by the inclusions. The transient wave propagation in the unidirectional composite material and a viscoelastic specimen without inclusions are compared. Peculiarities in the numerical results due to the finite-element algorithm employed are decoupled from the output that reveals purely mechanical phenomena. For this purpose, the transient wave propagation in a one-dimensional elastic continuum, for which the exact analytical solution is well known, is effectively obtained by finite-element analysis. Furthermore, based on the unidirectional composite a homogenized model is defined by applying the mixture rule. An additional comparison between the composite containing discrete inclusions and the analogous homogenized material provides a basis to assess the output of models obtained by implementing a self-consistent scheme in a broader context.

Prediction and validation of flanking sound transmission in cross-laminated timber junctions with resilient interlayers

Propelled by its small carbon footprint, low weight and high stiffness, cross-laminated timber (CLT) has experienced a significant commercial growth in the last decade. However, these latter two features potentially contribute to poor acoustic performance. A critical factor is flanking sound, in which vibrational energy is transmitted between two elements connected through a common junction, driving the need to include vibration isolation solutions such as resilient interlayers in the junction design. In this work, the vibration reduction index across CLT junctions is predicted with an analytical wave approach for semi-infinite thin plates. Each CLT panel is modelled as an orthotropic or isotropic equivalent single layer (ESL). Multiple options to determine the ESL properties are compared, including the law of mixtures and the modified gamma method. The interlayer is considered to be a thick, flexible waveguide, whose out-of-plane motion is governed by shear. The prediction model is validated experimentally with on-site measurements for a vertical T-junction with a polyurethane foam interlayer. The predictions are moderately accurate in the entire frequency range, with deviations below 10 dB across all 1/3 octave bands. Depending on the layer configuration, the orthotropy of the ESL can significantly influence the predictions.

Fabrication of hierarchical layered structure in Zr/Ti composite via multi-step heat treatments and its effect on the mechanical properties

The strength and ductility of metallic structural materials are two pivotal properties generally considered mutually contradictory. In the past few decades, numerous heterogeneous materials have been designed and fabricated to overcome this trade-off. Biomimetic-designed hierarchical layered (HL) materials represent a promising approach. In the present study, HL Zr/Ti materials with three hierarchical levels were fabricated via multi-step heat treatments. Scanning Electron Microscopy (SEM), Transmission electron microscope (TEM), Electron Backscatter Diffraction (EBSD), Transmission Kikuchi Diffraction (TKD), and Energy Dispersive X-ray Spectroscopy (EDS) were employed to observe the evolution of the HL microstructure. The mechanical properties of the HL Zr/Ti materials were assessed through hardness and tensile tests, analyzing the influence of microstructural changes in the diffusion layer on the overall mechanical performance. Our results demonstrated that the HL structure could introduce extra strength beyond the rule of mixtures (ROM) due to multiple-scaled hetero-deformation induced (HDI) strengthening. Additionally, the refinement of the second-level layer thickness effectively inhibited crack propagation. This research provides new insights into developing multi-level HL materials with enhanced mechanical properties through cost-effective and scalable manufacturing processes.

Wear behavior of CoCrFeNi high-entropy alloy prepared by mechanical alloying

Equiatomic quaternary CoCrFeNi powder was prepared from the individual metals by mechanical alloying of the mixed powders at room temperature. The as-milled powder contained particles of irregular shape with an average particle size of 0.62 μm, and exhibited a dual-phase composition of the face-centered cubic and the body-centered cubic structures. The as-milled powder was consolidated by hydraulic pressing at room temperature and sintered at 950 °C in argon and vacuum. The compacted material exhibited a single-phase composition of the face-centered cubic structure alone and a chemical composition close to equiatomic with uniform distribution of all constituents, indicating that the desired CoCrFeNi high-entropy alloy (HEA) had been formed. The microhardness of the vacuum-sintered CoCrFeNi HEA was measured to be 118±7 HV, exceeding that calculated by the rule of mixture. The coefficient of friction was determined by pin-on-disk dry sliding studies at room temperature and found to be 0.65 and 0.44 for the applied loads of 25 for 35N, respectively, and the corresponding wear rates were 4.6×10-4 and 9.1×10-4 mm3/N·m. XPS studies on the worn alloy surfaces showed the formation of oxidized tribo-layers, composed of predominantly Cr2O3. Overall, this study has demonstrated that mechanical alloying followed by simple furnace sintering at a moderate temperature is able to yield an abrasion-resistant CoCrFeNi HEA suitable for structural applications.

Low-velocity impact behavior analysis on composite double curved-shell reinforced with graphene platelet

ABSTRACT This research investigates the low-velocity impact behavior of a spherical rigid body on a composite shell reinforced with graphene platelets (GPLs), in which GPL have been uniformly dispersed and randomly arranged within each individual layer. The weight proportion of GPL has been altered across the thickness. The modified Halpin Tsai model is employed to analyze the Young's modulus and the effective material characteristics of a composite reinforced with graphene platelets (GPLRC). The mass density and Poisson's ratio of the viscoelastic multilayer composite are determined using the rule of mixture. The material parameters of each layer are defined according to the Kelvin–Voigt model. The governing equations have been derived by applying the first order shear deformation shell theory, Hamilton's principle, and modifying the Hertz contact law. The PDE problems have been transformed into ODE equations through the utilization of the Galerkin method. Subsequently, the ordinary equations have been solved utilizing the Newmark method. A parametric study is conducted to examine the effects of various factors, such as fraction weight, GPL distribution pattern, impactor mass, radius, viscoelastic modulus, and initial velocity, on the low velocity impact response of a composite shell reinforced with GPL.

Prediction of Tensile Properties of Natural Fiber Reinforced PP Hybrid Composites by Facca-Kortschot-Yan (FKY) and Palsule Equations

Abstract Natural fiber reinforced polymer hybrid composites are advanced materials for commercial and engineering applications. Rule of hybrid mixtures has been applied to develop Facca-Kortschot-Yan (FKY) Equation and Palsule equation for predicting tensile strength and modulus of these hybrid polymer composites. This study predicts the values of tensile strength and modulus of a few natural fibers reinforced polypropylene hybrid composites by these two equations and compares them with the reported experimental values.

Achievement of comprehensive wear and thermal property of cold spayed CuCrZr coating via gradient structure design

To address the inherent challenge of optimizing both strength and conductivity of the CuCrZr alloy, cold spraying is employed to enhance its surface with a comprehensive range of performance attributes. A novel gradient structure, featuring a SiC/CuCrZr layer at the top and an AlN/CuCrZr layer at the bottom, successfully fulfills the comprehensive requirements for wear resistance and thermal conductivity. Results show that the composite coating is composed of extensively deformed CuCrZr particles and undeformed ceramic particles and its microhardness and thermal conductivity closely adhere to the rule of mixtures. Additionally, the gradient coating shows the equivalent wear resistance to the SiC-CuCrZr coating. This study demonstrates the remarkable versatility of cold spraying in fabricating seamless gradient coating, while achieving predictable and controllable variations in microhardness and thermal conductivity across the coating’s thickness.

On the nonlinear mechanics of hybrid skew aerospace structures

This study proposes a functionally graded (FG) nanocomposite hybrid skew plate element which is fabricated from a polymeric matrix and is reinforced with two well-known nanoscale reinforcements, namely, carbon nanotubes (CNTs) and graphene platelets (GPLs). The Mori–Tanaka approach, the Halpin-Tsai scheme and standard rule of mixtures are employed to estimate the effective mechanical properties of the hybrid plates. To model this structural element, Reddy's third-order shear deformation theory (TSDT) is used. By considering the von-Kármán assumptions and using the Hamilton's principle, the nonlinear static bending and free vibration governing equations are determined. Then the isogeometric analysis (IGA) is employed to establish the structural stiffness and mass matrices of the skew hybrid plates. After validating the correctness of the presented solution procedure with the previously published results, the effects of various variables such as number of layers, distribution pattern of reinforcements (CNTs and GPLs) and their volume fractions, skew angle and width-to-thickness ratio are investigated. It is found that with considering nonlinear von-Kármán relationship in the strain field, the distributions of σxx and σyy about the mid-plane are no longer symmetric.

Effect of bioceramic inclusions on gel-cast aliphatic polymer membranes for bone tissue engineering applications: An in vitro study.

Polylactic acid (PLA) has been extensively used in tissue engineering. However, poor mechanical properties and low cell affinity have limited its pertinence in load bearing bone tissue regeneration (BTR) devices. Augmenting PLA with β-Tricalcium Phosphate (β-TCP), a calcium phosphate-based ceramic, could potentially improve its mechanical properties and enhance its osteogenic potential. Gels of PLA and β-TCP were prepared of different % w/w ratios through polymer dissolution in acetone, after which polymer-ceramic membranes were synthesized using the gel casting workflow and subjected to characterization. Gel-cast polymer-ceramic constructs were associated with significantly higher osteogenic capacity and calcium deposition in differentiated osteoblasts compared to pure polymer counterparts. Immunocytochemistry revealed cell spreading over the gel-cast membrane surfaces, characterized by trapezoidal morphology, distinct rounded nuclei, and well-aligned actin filaments. However, groups with higher ceramic loading expressed significantly higher levels of osteogenic markers relative to pure PLA membranes. Rule of mixtures and finite element models indicated an increase in theoretical mechanical strength with an increase in β-TCP concentration. This study potentiates the use of PLA/β-TCP composites in load bearing BTR applications and the ability to be used as customized patient-specific shape memory membranes in guided bone regeneration.

Constitutive Modeling, Processing Map Optimization, and Recrystallization Kinetics of high-grade X80 pipeline steel

Hot compression tests were used to examine the hot deformation behavior and of high-grade pipeline X80 steel at strain rates of 0.001-1 s-1 and temperatures of 950-1100 ºC. Two approaches were used to model material flow curves. The first used the Zener-Hollomon parameter, Z, to show the impact of deformation temperature and strain rate on flow stress, using the Arrhenius constitutive equation. The second used a rule of mixture to represent the material's flow curve, considering work hardening and dynamic recrystallization flow stresses. The first method was accurate at low Z and low strains, while the second method accurately displayed peak stress, was suitable for large strains, and predicting flow curves at industrial strain rates. The microstructure of X80 steel exhibits strain rate dependence under hot deformation conditions, with finer and more equiaxed microstructures observed at higher strain rates. These findings align with processing map predictions for stable deformation behavior, suggesting potential for optimized hot working processes. The static recrystallization kinetics showed that the Avarmi exponent, n, ranges from 1 to 2 for strains of 0.2 and 0.35 at 950°C, but higher values were observed under hot deformation conditions.

Atomic-scale insights into the effect of layer thickness ratio on the tensile properties of heterostructured nano-aluminum

Gradient-structured alloys with lamellar microstructures show extraordinary potential in breaking the strength-ductility trade-off dilemma. When partial grain size exceeds the Hall-Petch limit, the deformation mechanism is not fully understood. In this study, the competing relationship between grain boundary (GB) softening and hetero-deformation-induced (HDI) strengthening in laminated nano-grained aluminum was investigated under uniaxial tensile loading. In the soft domains, the high activity of grain boundaries results in pronounced GB softening. However, when the volume fraction of the soft layer exceeds 50 %, the flow stress surpasses the predicted values of the rule of mixtures (ROM), demonstrating a significant HDI strengthening. A detailed analysis of microstructural evolution is conducted, including strain gradients, GB migration, and dislocation distribution. The effects of mechanical incompatibility between soft and hard layers on the distribution of plastic strain and overall material strength are elucidated in gradient structures. The study provides critical theoretical insights into the design and development of high-performance gradient nanocrystalline aluminum alloys, particularly in applications where a balance between strength and ductility is of paramount importance.

Multiscale dynamic behavior of imperfect hybrid matrix/fiber nanocomposite nested conical shells with elastic interlayer

This investigation delves into the free vibration characteristics of coupled nested conical shells (CNCSs) made of porous composite materials. These two conical shells are connected by a mid-layer of elastic springs. The composite materials used in the shells consist of epoxy, nanofillers, and fibers. Two types of nanofillers are considered: Graphene Nanoplatelets (GNPs) and Carbon Nanotubes (CNTs), while E-glass fiber is used as the fiber. The nanofillers are distributed in four different patterns within the shell section. Porosity is uniformly distributed along the shell section and characterized by a coefficient. The rule of mixtures is employed to ascertain the equivalent material properties of the hybrid materials, while the Chamis approach is utilized for three-phase materials. First-order shear deformation theory (FSDT) and Donnell's theory are utilized for modeling the conical shells. The governing equations of motion are established through Hamilton's principle are solved using the generalized differential quadrature method (GDQM). Seven different boundary conditions (BCs) are considered to encompass the full range of BCs for CNCSs and four type of BCs for single truncated conical shell (STCS). The solution's accuracy is verified, and the effects of various parameters on the natural frequency parameter (NFP) of the shell are investigated, such as BCs, circumferential wave number (n), nanofillers pattern, semi-vertex angle, nanofillers angle, and mid-layer stiffness. Initially, a comprehensive investigation into the vibration behavior of a STCS is presented, followed by an analysis of the NFP of the CNCSs. The results demonstrate that the stiffness of the elastic mid-layer significantly influences the NFP of the system. The orientation of the nanofillers in the shell can increase or decrease the NFP. Additionally, the relationship between mode number and n depends on the type of BCs of the shells.