Rule Of Mixtures Research Articles

Experimental and numerical investigation on the elastic properties of luffa–cenosphere‐reinforced epoxy hybrid composite

AbstractEstimating the elastic characteristics of natural fiber‐reinforced polymer composites such as luffa fiber reinforced with epoxy is challenging. The structure of luffa cylindrica is complex, like a three‐dimensional natural fibrous mat, netting‐like structure. The multiscale modeling of such structures is the challenge to be addressed. The prime objective of this work is to determine the specific elastic properties of luffa–cenosphere‐reinforced epoxy (LCE) composite, considering the effect of filler volume fractions. Furthermore, multiscale modeling techniques, such as representative volume elements (RVEs) of finite element techniques with chopped, unidirectional, plain, and twill weaving fiber arrangements, were employed. The longitudinal modulus, transverse modulus, shear modulus, and Poisson's ratio were predicted through these modeling approaches. However, experimental and analytical methodologies, including the rule of mixture and Halpin–Tsai, were considered to validate the finite element analysis results. The elastic characteristics of LCE composite were therefore shown to be enhanced by increasing filler volume fraction. However, the cenosphere's 20% volume fraction has the highest elastic properties as determined by analytical, experimental, and computational models. Analytical and finite element simulation results were compared with the experimental results, and based on the findings, the most suitable (unidirectional, chopped, plain, and twill weaving) RVE was identified for finite element modeling of LCE composite for the evaluation of elastic properties. Results from practical approaches and the RVE twill weaving model showed good agreement, with less than 1% error, compared to the other analytical and finite element methods.Highlights NFCs are gaining ground in polymer composites. Overcoming challenges in modeling of luffa fiber inside epoxy matrix. The study uses multiscale modeling with diverse fiber arrangements. Experimental and analytical methods used to confirm FEA results. Increased cenosphere volume fraction boosts LCE composite properties.

Buckling and post-buckling of anisogrid lattice-core sandwich plates with nanocomposite skins

This research offers an in-depth examination of the buckling and post-buckling responses observed in sandwich plates distinguished by the presence of an anisogrid lattice core and face layers composed of functionally graded carbon nanotube-reinforced composites (FG-CNTRC). To model this structure, a global continuous model for the lattice core is employed, while the displacement field is approximated using the First-Order Shear Deformation Theory (FSDT). Geometrically nonlinear strain relations based on the von-Kármán assumptions are incorporated to accurately capture the structural response. The homogenization of the FG-CNTRC skins is achieved through the modified rule of mixture, enabling estimations of effective material properties. To separate pre-buckling and post-buckling phases, we implement the adjacent-equilibrium criterion. Solving the linear (pertaining to buckling) and nonlinear (related to post-buckling) equations is based on the Generalized Differential Quadrature (GDQ) technique alongside an iterative method grounded in the displacement control strategy. Our analysis explores the influence of core and skin characteristics on the stability of these sandwich plates. By examining critical buckling thresholds, post-buckling equilibrium paths, and the underlying mechanisms driving these responses, we contribute insights into the structural stability and performance of such composite structures. This study not only enhances our understanding of buckling behavior in anisogrid lattice core sandwich plates but also holds promise for diverse engineering applications where structural stability is paramount.

Geometrical and material optimization of the functionally graded doubly-curved shell by metaheuristic optimization algorithms

This paper presents the simultaneous geometrical and material optimization of functionally graded doubly-curved shells under static load and free vibration. The weak form of governing equations is derived from third-order shear and normal deformation theory. Bending and free vibration analyses were conducted using the finite element method. The functionally graded material (FGM) is composed of metal and ceramic, and Voigt’s rule of mixtures is applied to capture the characteristics of FGMs throughout the shell thickness direction. In this regard, two multi-objective optimization algorithms (i.e., multi-objective particle swarm optimization and multi-objective cuckoo search optimization algorithms are employed. The objective functions combine minimizing the maximum normalized displacement and mass and maximizing the first dimensionless frequency. The results show that where minimizing the mass of the FG shell structure was considered an objective function, the value of the power index is equal to zero for almost all studied cases.

Modeling and analysis of a macro-fiber piezoelectric bimorph energy harvester operating in d33-Mode using Timoshenko theory

Piezoelectric bimorph cantilevered beams are frequently employed in energy harvesting applications. The use of macro-fiber composites, which combine piezoelectric fibers with rectangular cross-sections and interdigitated electrodes, has significantly enhanced their performance by enabling operation in the 33-mode of piezoelectricity, thereby increasing conversion efficiency. In this article, the Timoshenko beam theory is applied to overcome the limitations of the Euler-Bernoulli theory in modeling transverse vibrations of harvesters with low slenderness ratios (length to thickness ratios). The mixture rule formulation is applied to determine the equivalent properties of the macro-fiber composite (MFC) structure. The electromechanical properties of a representative volume element (RVE) bounded by two successive interdigitated electrodes are coupled to the overall electro-elastic dynamic model of the structure using the Timoshenko theory. Frequency response functions for the power and tip vibration displacement of the MFC bimorph beam are determined from the model. Experimental data from the literature are used to validate the improved results predicted by the model developed in this article. A comparative analysis between the electromechanical responses predicted using a model based on Timoshenko and a model based on Euler-Bernoulli theory is conducted showing agreement, for harvesters with large slenderness ratios and discrepancies of up to 30% for low slenderness ratios. It has also been shown that the model based on Euler-Bernoulli theory overestimates both the mechanical and the electrical responses for low slenderness ratios as the shear and rotary inertia effects are not negligible in this case.

Multi-factors effects analysis of nonlinear vibration of FG-GNPRC membrane using machine learning

Functionally graded graphene nanoplatelet reinforced composite (FG-GNPRC) have exhibited significant potential for the development of high-performance and multifunctional structures. In this paper, we present a machine learning (ML) assisted uncertainty analysis of nonlinear vibration of FG-GNPRC membranes under the influence of multi-factor coupling. Effective medium theory (EMT), Mori-Tanaka (MT) model and rule of mixture are utilized to evaluate the effective material properties of the composite membrane. Governing equations are derived via an energy method with the frameworks of the hyperelastic membrane theory, Neo-Hookean constitutive model and the couple dielectric theory. Randomly generated inputs after data pre-processing are fed into governing equations, which are solved by numerical methods for outputs. Three ML models, including artificial neural network (ANN), support vector regression (SVR) and AutoGluon-Tabular (AGT), are adopted to capture the complex relationship between the systematic inputs (i.e., structural dimensions, attributes of GNPs and pores, external electric field, etc.), frequency ratio and dimensionless amplitude of FG-GNPRC membranes. The results demonstrate that all three ML models demonstrate exceptional computational efficiency, and AGT presents higher prediction accuracy compared to the other two models. Based on the Shapley additive explanations (SHAP) approach, the effects of uncertainties of system parameters and multi-factor coupling on the nonlinear vibration of the FG-GNPRC membrane are analyzed. It is found that the uncertainty of structural parameters has the greatest impact on the nonlinear vibration of FG-GNPRC membranes, particularly when the membrane is subjected to a voltage of 10 V and smaller stretching ratio.

Study on fatigue performance of asphalt mixture in service life based on accelerated loading test

To further study the fatigue damage evolution law of asphalt mixture in service period, the accelerated pavement testing (APT) of graded gravel base asphalt pavement and in-situ pavement was conducted. The full-scale test road of graded gravel base asphalt pavement was built, and the mechanical response sensors and temperature sensors were embedded. The fatigue damage test of the full-scale test road was conducted at 15°C through the accelerated loading test (ALT) equipment. The mechanical responses under the repeated wheel loadings were captured by the strain sensor at the bottom of the asphalt layer, and the deflections of the pavement under different cycles of wheel load were tested by Falling Weight Deflectometer (FWD). The fatigue damage model of asphalt mixture was established, and the accuracy of the model was verified by the accelerated loading fatigue test of in-situ pavement. The results show that asphalt layer bears an alternating effect of tensile and compressive strain, and when the wheel load is reduced or the driving speed increases, the tensile strain at the bottom of the asphalt layer decreases. As an evaluation index of fatigue damage of asphalt layer, the residual modulus ratio can accurately characterize the cumulative degree of fatigue damage. The modulus of asphalt layer shows an inverse S-type curve with the repeated wheel loadings. Based on the quantitative analysis of the whole process of fatigue damage of asphalt layer, a nonlinear damage evolution model of modulus attenuation of asphalt mixture was established, which would provide a reference for the study of pavement service performance attenuation law in different climatic environments and structural compositions in the future.

Macro-microscopic study on the crack resistance of polyester fiber asphalt mixture under dry-wet cycling and neural network prediction

Asphalt mixture is a composite material with a complex multiphase dispersed system, and its macroscopic mechanical behavior is inherently related to the microstructure characteristics. To investigate the low-temperature crack resistance degradation law of polyester fiber asphalt mixture under different dry-wet cycling conditions, six polyester fiber contents (0%, 0.3%, 0.35%, 0.4%, 0.45%, and 0.5%) were dry-blended into SMA-13 asphalt mixture to prepare fiber-reinforced SCB specimens with pre-notches. Semi-circular bending tests were conducted to test the crack resistance performance of the fiber asphalt mixture after 0, 2, 4, 6, and 8 dry-wet cycles. The experimental results show that the crack resistance performance of specimens with different fiber contents decreases with the increase of dry-wet cycling times, and the influence of salt-dry-wet coupling is greater than that of water-dry-wet coupling. Under the same conditions, the crack resistance index (CRI) increases first and then decreases with the increase of polyester fiber content, reaching its maximum at a polyester fiber content of 0.4%. When the fiber is added at 0.5%, the agglomeration of polyester fibers restricts the low-temperature performance of the specimen. In addition, the DIC technology is used to analyze the trend of horizontal strain variation of specimens after different dry-wet cycles. The results indicate that with the increase of the cycle period, the strain concentration area becomes more apparent, and in the destruction stage, the full-field horizontal strain Exx in the crack concentration zone gradually increases, while the crack resistance decreases. Finally, addressing the limitations of traditional BP neural networks in solving nonlinear problems, a particle swarm optimization algorithm is proposed to optimize the BP neural network, and a PSO-BP neural network model is constructed. Through the analysis of evaluation indicators, it is found that the PSO-BP neural network has improved generalization ability compared to the traditional BP neural network model, and overfitting is reduced, making it an effective tool for predicting the low-temperature performance of polyester fiber asphalt mixtures.

Multifield and higher-order analysis of sandwich smart curved beams made of graphene origami auxetic metamaterial

This article is organized to investigate effect of material and geometric characteristics of graphene origami as well as multifield loading on the electro-magneto-elastic bending results of the sandwich smart nanocomposite curved beam. The analysis is performed using the higher-order shear deformation theory accounting out of plane normal strain. The constitutive relations are extended in the cylindrical coordinate in the presence of thermal, electrical, and magnetic loads. The governing equations are derived using the principle of virtual work with accounting electric and magnetic works. The effective material properties of graphene origami reinforced composite core are evaluated through Halpin–Tsai micromechanical models and rule of mixture for modulus of elasticity, Poisson’s ratio, and density. The analytical solution is presented for simply supported boundary conditions based on trigonometric functions. A verification test is performed for validation of the formulation, solution procedure, and numerical results. The stress and deformation components are evaluated with changes of multifield loading and characteristics of graphene origami along the thickness direction. The results show a significant effect of electrical, thermal, and magnetic loading on the stress and deformation results.

A nonlocal higher-order shear deformation approach for nonline ar static analysis of magneto-electro-elastic sandwich Micro/Nano-plates with FG-CNT core in hygrothermal environment

In this study, the nonlocal static analysis of Magneto-Electro-Elastic (MEE) sandwich micro/nano-plates with Functionally Graded Carbon Nanotubes (FG-CNTs) core in a hygrothermal environment is studied. The nonlocal static formulation is developed based on Reddy's High-order Shear Deformation theory (HSDT). The von Kármán nonlinear strains are used and the governing equations of the HSDT are derived accounting for Eringen's nonlocal stress-gradient model. The governing equations are solved using the Galerkin method, which involves expanding the displacement field based on orthogonal functions. The variation of the volume fractions through the thickness of FG-CNTs core layer has been computed using two different homogenization techniques: the rule of mixtures and the Mori–Tanaka scheme. A detailed parametric study to show the characteristics of the FG-CNTs core, nonlocal parameters, thermal and moisture changes, types of CNT reinforcement and volume fraction, electric and magnetic potentials, and geometric parameters. The results are compared with the first-order shear deformation theory (FSDT) and classical plate theory to show the accuracy of nonlocal High-order Shear Deformation theory(NHSDT) formulation.

Nonlinear aero-thermo-elastic flutter analysis of stiffened graphene platelets reinforced metal foams plates with initial geometric imperfection

Due to its lightweight design and high load-bearing capacity, the stiffened plate structure is extensively applied in the passive control of aircraft flutter. However, the evolution mechanism of the nonlinear response of stiffened plates within the unstable region remains unclear. In particular, there is a lack of research on the nonlinear aero-thermo-elastic properties of stiffened plates with initial geometric imperfection. Inspired by this, this article investigates the stability and post-instability behavior of stiffened geometrical imperfect graphene platelets reinforced metal foams (GPLRMF) plates under combined thermal load and aerodynamic pressure. Graphene platelets (GPLs) and foam are distributed evenly or unevenly in the plates, while the material properties are calculated by modified Halpin-Tsai model and rule of mixture. Considering initial geometric imperfection, the models of plates and stiffeners are established by first-order shear deformation (FOSD) plate and Timoshenko beam theories, respectively. Using the linear piston theory to simulate supersonic aerodynamic pressure. The stability boundary of stiffened plates is determined by eigenvalue analysis. The Newton-Raphson approach is used to simulate the aero-thermo-elastic static response of stiffened plates, and the dynamic post-flutter behavior of the stiffened plate is determined through the Runge-Kutta method. Finally, the effects of initial geometric imperfection, material parameters (foam distribution type, porosity coefficient, GPLs weight fraction and GPLs distribution type), and stiffeners (size, position, and stiffening scheme) on the thermal aeroelastic behavior of stiffened GPLRMF plates are revealed. It can be found that the presence of initial geometric imperfection will significantly change the nonlinear flutter behavior of the stiffened plates.

State-Space Formulation for Buckling and Free Vibration of Axially Functionally Graded Graphene Reinforced Nanocomposite Microbeam under Axially Varying Loads.

This paper focuses on the size-dependent free vibration and buckling behaviors of the axially functionally graded (AFG) graphene platelets (GPLs) reinforced nanocomposite microbeams subjected to axially varying loads (AVLs). With various axial grading patterns, the GPL nano-reinforcements are distributed throughout the polymer matrix against microbeam length, and the improved Halpin-Tsai micromechanics model and the rule of mixture are adopted to evaluate the effective material properties. Eigenvalue equations of the microbeams governing the static stability and vibration are derived based on the modified couple stress Euler-Bernoulli beam theory via the state-space method, and are analytically solved with the discrete equilong segment model. The effects of axial distribution patterns, weight fraction, and geometric parameters of GPLs, as well as different types of AVLs, on the size-dependent buckling load and natural frequency are scrutinized in detail. The results show that the synchronized axial distributions of GPLs and AVLs could improve the buckling resistance and natural frequency more powerfully.

Dielectric and viscoelastic properties of 3D-printed biobased materials

This study aimed to develop new biobased materials containing plasticized cellulose acetate (CA) by Fused Filament Fabrication (FFF). We investigated the influence of CA on the viscoelastic/dielectric properties and the influence of 3D printing on the dielectric properties of the polymer blends. A microstructural analysis showed that the blends had a strong heterogeneous morphology, as the two phases formed a fibrillar and lamellar structure. CA strongly increased the blend viscosity at 175 °C, multiplying by 100 times the complex viscosity between neat PLA and the blend containing 40% of CA by weight (CA-40). The addition of CA to the blends increased the dielectric constant (ε') and dielectric loss (ε''), as well as the alternative current electrical conductivity (σAC). The dielectric constant of CA (ε'CA) was proportional to its percentage in the blend, showing a behavior analogous to a rule of mixture. Moreover, the activation energy of the electrical conductivity (σDC) measured at T > 124 °C decreased with increasing CA content and was associated with the enhancement of the ionic conductivity provided by the CA and its plasticizer. A decrease in the α-relaxation temperature and the associated activation energy of the PLA was also linked to the presence of the CA plasticizer. Finally, 3D printing greatly decreased both ε', ε'' and σAC due to the internal voids induced by the 3D printing process. The porosity was measured at 12% for neat PLA and between 20% and 25% for the blends. These results showed the advantage provided by FFF technology in the production of PLA:CA blends with controlled dielectric properties, thereby favoring the use of these new materials in key dielectric areas.

A strength optimization approach for manufacturable 3D printed variable stiffness laminates with a strain-based approximation

The 3D printed continuous carbon fiber reinforced polymers (CFRP) are promising lightweight structures with outstanding mechanical performance. In this work, we propose a structure-manufacture integrated optimization approach for strength maximization of variable stiffness 3D printed CFRP composites by tailoring the steered printing paths. The rule of mixture is modified to evaluate material properties and strength of the printed laminates considering the effect of fiber volume fraction. To make this approach applicable to CFRP with asymmetric layups, a strain-based two-level convex approximation for the failure index based on the Tsai–Wu failure criterion is proposed, which can take the tension-bending coupling effect into consideration. The local failure indices are aggregated using the p-norm formula to reduce the size of the strength optimization model. In order to guarantee the manufacturability of the optimal design, the curvature of the steered printing fiber paths is constrained using a hybrid control method within the approach. The numerical cases show that the strength enhancement of the 3D printed variable stiffness laminates depends on the load, where an improvement between 35% and 46% can be achieved under the tension. Moreover, the proposed method can not only converge stably, but also be capable of optimizing structures with complex geometry.

Curing Law and Strength-Age Prediction Model of a Dense Polyurethane Mixture under the Influence of Moisture and Temperature

Curing Law and Strength-Age Prediction Model of a Dense Polyurethane Mixture under the Influence of Moisture and Temperature

Free vibration analysis of functionally graded graphene platelet-reinforced metal foam doubly curved panel

In this research, free vibration characteristics of functionally graded metal foam doubly curved panels reinforced with graphene platelets and with porosities have been surveyed. Halpin Tsai's approach is utilized for extracting the effective Young modulus of porous metal foam nanocomposite and also the effective density of nanocomposite porous doubly curved shell panel is estimated by using an extended rule of mixture. The FSDT hypothesis is utilized for determining the displacement field and the Finite element and Hamilton principle are utilized for deriving the mass and stiffness matrices of the structure. Finally, the influences of several variables such as porosity distribution, porosity coefficient, GPL dispersion pattern, the weight fraction of Nanofillers, and span angles on the free vibrations characteristics of doubly curved shell panels with FG porosities and reinforced by graphene platelet have been reported in detail.

Nonlinear thermal buckling analysis of temperature-dependent porous annular and circular microplates reinforced by graphene platelets by using isogeometric analysis method

In this research, by using isogeometric analysis (IGA) method, nonlinear thermal buckling analysis of porous circular and annular microplates reinforced by graphene platelets (GPLs) under uniform temperature rise is performed by taking both the temperature dependence of the material properties and the size effects into account. The closed-cell metal foam scheme combined with the Halpin-Tsai micromechanics model and the rule of mixture is utilized to identify the effective material properties of the porous nanocomposites for three GPL dispersion patterns and three porosity distributions. With the aid of the virtual work principle, the classical and the non-classical motion equilibrium formulation based on the higher-order shear deformation theory in conjunction with the modified strain gradient (MSG) theory for the nonlinear temperature-dependent buckling behaviour are established. The higher-order NURBS basis functions capable of accomplishing easily the smoothness of arbitrary continuity order are successfully utilized to solve the nonlinear governing equations by satisfying the C2-continuity of the displacement field required by the MSG theory. The thermal buckling behaviour is assessed in detail by using various parameters such as the thickness-to-radius ratio, the porosity coefficient, the GPL weight fraction and the material length scale as well as the porosity distribution scheme and the GPL dispersion pattern. The results show the significance of incorporating the temperature dependency of the material properties when analyzing the thermal buckling behaviour and present the combination scheme of porosity and GPL distributions for enhancing the thermal buckling resistance.

Calculation model for bearing capacity of steel-CFRP composite pipeline under internal pressure

The Carbon Fiber Reinforced Polymer (CFRP)-wrapped steel pipeline system is emerging as an alternative for the repair and long-distance transportation of oil and gas. This system is recognized as a composite material. Under internal pressure, the pipe predominantly undergoes circumferential stretching. This study derives the tensile constitutive model of the steel-CFRP composite material using uniaxial tensile tests and the rule of mixtures. Subsequently, utilizing this newly calibrated constitutive model, a computational model to assess the internal pressure capacity of the steel-carbon fiber composite pipe was established. Finally, a comparison between the theoretical outcomes of the computational model and the actual internal pressure test results revealed a high degree of correlation. This holds substantial significance for the design and practical implementation of this novel composite pipeline system.

Transmission of Lamb wave in a micro-mechanically piezoelectric fiber-reinforced composite plate

The present work addresses the propagation characteristics of Lamb wave in a piezoelectric fiber-reinforced composite material plate. The considered plate is micro-mechanically modeled using the analytical techniques of the Strength of Materials and Rule of Mixtures. Secular equations are deduced for symmetric and antisymmetric modes of Lamb wave by employing suitable boundary conditions. The data of two distinct PFRC materials are taken into account, viz. PZT-5A-epoxy and CdSe-epoxy with different fiber volume fractions to investigate the latent characteristics of Lamb wave propagation. The amplitudes of dilatation and electrical potential of considered material are examined numerically and manifested graphically for both kinds of modes, i.e., symmetric and antisymmetric modes. The obtained graphical results are also compared for both kinds of modes. A comparative study has been carried out to exhibit the concealed characteristics of Lamb wave propagation for various distinct cases of considered structure for both symmetric and antisymmetric modes. The determined analytical results are in well agreement with the results present in the existing literature.

Development of an equiatomic octonary TiNbTaZrMoHfWCr super-high-entropy alloy for biomedical applications

A super-high-entropy alloy (SHEA) with ΔSmix ≥ 2.0R (where R is the gas constant) was designed to produce metallic materials with superior mechanical properties to conventional alloys. As an alternative to conventional quinary high-entropy alloys (HEAs), herein, octonary SHEAs for biomedical applications (BioSHEAs) are proposed for the first time, and the TiNbTaZrMoHfWCr BioSHEA was fabricated. Arc-melted BioSHEA exhibited an extremely high yield strength of 1953 ± 84 MPa, which was approximately 550 MPa higher than that of the quinary TiNbTaZrMo BioHEA. This yield strength is considerably higher than that estimated by the rule of mixtures for pure metals, confirming the achievement of significant solid-solution strengthening induced by a supermulticomponent solid solution composed of elements with different atomic radii. Its biocompatibility was comparable to that of pure Ti and the quinary BioHEA, and superior to that of SUS316L. This study demonstrates the validity of a novel entropy-based guideline for increasing the mixing entropy to achieve metallic materials with ultrahigh strength.

Impact of surface/interface elasticity on the propagation of torsional vibration in piezoelectric fiber-reinforced composite and anisotropic medium

This work has explored surface/interface theory-based torsional wave propagation in a piezoelectric fiber-reinforced composite (PFRC) layer on top of a functionally graded elastic substrate. PFRC is assumed to be composed of piezoelectric fiber and epoxy matrix. Micro-mechanical investigations have been performed to determine the coefficients of PFRC using strength of materials (SM) and rule of mixtures (RM) techniques. The dispersion relation for the torsional wave under nonclassical boundary conditions is determined using surface/interface theory. The effects of surface/interface, fiber volume fractions, and heterogeneities on the phase velocity and the mode shapes of the wave have been demonstrated graphically.