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A rule of mixtures is employed and modified to examine the effective elastic properties of a unidirectional composite lamina reinforced with nanostructure-hybrid fibers. Such fiber system is designated when nanostructure such as nanowires or carbon nanotubes are radially grown on the surface of primary fiber. When combines with matrix, a complex three-phase composite with enhanced elastic properties is expected. Herein, the applicability of this simple micromechanics method to reliably estimate the effective properties of such advanced novel composite material is assessed. The results demonstrated that the method is capable of modeling the effects on elastic properties of a composite due to the presence of nanostructure. However, in light of published experimental data and other micromechanics results, the proposed method is found to be at best, applicable for a composite that has a very low fiber volume fraction only except when an axial Young’ modulus is predicted.

Previously reported strength prediction models for polycrystalline Ni-base superalloys tend to underestimate the overall strength. This is primarily due to neglecting the rule of mixtures for hardening by γ/γ′ and inaccurate estimation of other strengthening factors such as solid solution hardening of γ matrix and grain boundary strengthening. To address these issues, a series of single-crystal tie-line modeling alloys with γ′ size suitable for validating the strong pair-coupling and Orowan looping mechanisms were prepared, in an attempt to develop more reliable γ′ hardening models. Critical resolved shear stress values were measured on these modeling alloys at 650 °C using compression tests, which allowed the distinction between strength contribution from rule of mixture and interfacial strength. Compared with the experimentally obtained interfacial strengths, the predicted results by classical models could not reflect the strength increment expected at higher γ′ volume fraction. Hence, a modified version of the classical equations has been proposed here for improved predictability.

The accurate prediction on the response of the composite material is difficult to achieve due to the complexity of its mechanical properties. However, such complexity can be understood well using the micromechanical analysis. One of the most important goals of micromechanical analysis is to predict the failure and strength of the composite material on the basis of the geometries and properties of the matrix and the fibers. This study aims to develop a simplified micromechanical model that predicts the direct tensile strength of a randomly oriented short Carbon Fiber Reinforced Concrete (CFRC) using the modified rule of mixtures based on the assumptions that CFRC will fail by fiber failure mode and with the perfect interfacial bond between the matrix and the fibers. PAN-based High Tensile short carbon fibers distributed randomly in 3D with low fiber volume fractions (Vf) of 0.10%, 0.15%, 0.20%, 0.25% and 0.30% were used in this study. Fiber lengths (Lf) of sizes 19mm, 30mm, and 38mm were used. The designed compressive strength considered for each fiber volume fraction and fiber length was 21MPa, 28MPa, and 35MPa. There were three samples of specimens considered for each case. Each case was tested for its cylindrical compressive strength and its direct tensile strength. Test results showed that the tensile strength of CFRC was optimum at Vf = 0.10% and Lf = 38mm. Finally, good agreement has been observed between the experimental tensile strength and predicted tensile strength using the micromechanical model.

This chapter explains the mechanical behaviour of composites. It begins by introducing the different forms of reinforced plastics including fibrous and particulate composites. It describes the types of reinforcement that are commonly employed and the use of thermosets and thermoplastics as matrix materials. It then concentrates on the analysis of continuous fibre composites. These are introduced initially using the Rule of Mixtures before being treated as a single ply and later as multi-ply laminates. Analysis is presented in the form of matrix algebra which is set out through a series of example calculations. These show how properties such as modulus and Poisson’s ratio may be computed for symmetrical and asymmetrical laminates, and then used to evaluate stress and strain. The effects of fatigue and impact on reinforced plastics are briefly considered at the end of the chapter.

This work investigates the impact of carbon black (CB) as a porogenic agent and conductive additive in the preparation of electrically conductive nanoporous carbon gels. For this, a series of materials were prepared by the polycondensation of resorcinol/formaldehyde mixtures in the presence of increasing amounts of carbon black. The conductivity of the carbon gel/CB composites increased considerably with the amount of CB, indicating a good dispersion of the additive within the carbon matrix. A percolation threshold of ca. 8 wt.% of conductive additive was found to achieve an adequate &ldquo, point to point&rdquo, conductive network. This value is higher than that reported for other additives, owing to the synthetic route chosen, as the additive was incorporated in the reactant&rsquo, s mixture (pre-synthesis) rather than in the formulation of the electrodes ink (post-synthesis). The CB strongly influenced the development of the porous architecture of the gels that exhibited a multimodal mesopore structure comprised of two distinct pore networks. The microporosity and the primary mesopore structure remained rather unchanged. On the contrary, a secondary network of mesopores was formed in the presence of the additive. Furthermore, the average mesopore size and the volume of the secondary network increased with the amount of CB.

Fiber Metal Laminates (FMLs) are hybrid composite materials consisting of alternating metal layers bonded to fiber-reinforced prepreg layers. ARALL, CARALL and GLARE belong to this new family of materials. GLARE is the most successful FML up to now and is currently being used for the construction of primary aerospace structures. In this study, the elastic buckling response of rectangular GLARE FMLs subjected to shearing stresses is investigated using the finite element method and eigenvalue buckling analysis. Simply supported, clamped and mixed boundary conditions are considered. Using validated FEM models, the buckling coefficient-aspect ratio diagrams of seven GLARE grades are obtained along with the diagrams of three UD glass-epoxy composites and monolithic 2024-T3 aluminum. The rule of mixtures is evaluated and found to be a simple method to estimate approximately the elastic buckling stress of the GLARE plates. An approximate formula is derived for the estimation of the critical buckling coefficient of the GLARE plates using the buckling coefficients of their constituents.

The results of a predictive analysis of the mock-up of a reactor containment building, based on computer numerical simulation, are presented in this paper and compared with the results obtained in the context of the VeRCoRs project during the 2015–2018 experimental campaign. The analysis is based on the Serial-Parallel Rule of Mixtures theory applied on a three-dimensional finite element model of the building. All the singular parts of the structure (two buttresses, the main penetrations, the gusset, etc.) and the complete prestressing tendons system are included in the structural model. The non-linear constitutive models used to describe the behaviour of the concrete, reinforcing steel and the tendons are formulated. A description of the Serial-Parallel Rule of Mixtures algorithm is also given. The structural response computed for the successive pressure tests applied on the mock-up are compared with the monitored structural behaviour. Finally, results for a simulation considering the aging of the tendon system during 40 years are shown and discussed.

This work deals with free vibration and buckling analysis of Functionally Graded single walled Carbon nanotube reinforced composite plate. The material properties were computed by the extended rule of mixtures. The non-dimensional frequency parameter was found by generating eigenvalues and eigenvectors solved by using ABAQUS. The subspace method was used for generating Eigenvalues of buckling for the plate. This work was intended to study the effects of varying temperature, volume fractions of CNT and CNT grading on non-dimensional natural frequencies and buckling parameters. The results of this study were validated from the results available in literature. It was found that the CNT volume fraction and CNT grading type has significant effect on non-dimensional frequency parameters. Similarly, ample effect of these parameters were observed for buckling of plate.

The evolution of the bone-implant interface reflects the implant osseointegration and bond strength, thereby determining the overall implant stability in the jawbone. Quantitative ultrasound represents a promising alternative technique to characterize the interfacial integrity, precisely due to the fact that those waves propagate essentially along the bone-implant interface, and are therefore influenced by its state. This study reports a numerical investigation of ultrasonic wave propagation for a commercial implant-jawbone system in which the thickness and mechanical properties of the interfacial layer (corresponding to the interphase) are systematically varied through the application of a rule of mixtures, in order to mimic the evolution from a dominantly soft tissue - like medium up to a fully healed bone. A simple figure of merit is devised in terms of an RMS-like (root mean square) factor based on the implant displacements, that evolves continuously and significantly with the bone “healing” process, thereby providing unequivocal information on the nature of the investigated bone-implant interface. The results show that the wave propagation pattern is primarily dictated by the impedance mismatch rather than by the interface thickness. This study validates the concept of quantitative ultrasonic testing as a sensitive alternative to the widespread resonant frequency analysis, thereby opening the way for future sensitivity analyses that will address more refined bone-implant interface pathologies such as those observed in the clinical realm.

The present research focused on the polymer hybrid composite fabrication and its characterization. Kevlar fibers (also called Aramid fibers, KF) are mixed with Sansevieria trifasciata fiber (snake plant leaf fibers, STF) to improve the performance of the epoxy matrix. Former fiber is synthetic fiber and the later is natural fibers are combined proportionately by the rule of mixtures KF and STF treated fiber systems. Wet-hand layup was used to organize systems with weight ratios of KF/STF for treated, viz. 1:0; 0.5:0.5; 0.75:0.25; 0.25:0.75; 0:1 (typically named as A, B, C, D, and E systems from the left). It was found that tensile strength for system-D (treated) was found improvement due to the fact that dust-free, rough, and improved surface area. Impact strength was found significant for the system-D when compared with others. The interface and voids at the fracture surface were improved for the systems C and D which were observed from the SEM images. Chemical resistance found good all the samples except carbon tetrachloride due to the hit of carbon atoms which consequently imparted erosion of the fiber out of the matrix.

International audience; In the present work, a commercial high solids epoxy-based paint used for the corrosion protection of steel was characterized by electrochemical impedance spectroscopy (EIS). The aim was to analyze the influence of water absorption on the physical structure of the paint, through the impedance data analysis. First, the water uptake was followed by gravimetric measurements on free-standing films as a function of the exposure time to a 0.5 M NaCl solution. Gravimetric measurements indicated a mass loss, linked to a release of a plasticizer from the formulation. This compound was dosed in the NaCl solution. The water volume fraction was also determined from the high-frequency part of the impedance diagrams, employing a linear rule of mixtures. The water fraction values obtained from the two methods were in acceptable agreement. The glass transition temperature (Tg) was determined by differential scanning calorimetry for the dry coating as well as for the coating after different immersion times in the NaCl solution. The use of the dielectric permittivity formalism allowed the mean relaxation times to be obtained from the impedance data analysis. The real part of the electrical conductivity was also considered. The variation of these parameters values as a function of the exposure time to the NaCl solution, was correlated with the Tg variation. They accounted for plasticization at the beginning of immersion (water uptake) and antiplasticization for longer exposure times (plasticizer release). For the studied system, it was possible to extract parameters representative of the molecular mobility from the impedance spectra analysis and to link them to the modification of the coating’s physical structure.

A hollow fiber composite provides more flexibility to tailor the required material properties when compared to solid fiber composite. In the present work, an attempt has been made to steady the micromechanical thermal behavior of a hollow fiber composite using a finite element software ANSYS, and the thermal conductivities of the composite material are determined. The inner and outer diameters of hollow fiber are selected for the volume fractions of void, fiber and matrix equal to 0.2, 0.44 and 0.36, respectively. Finite element models are validated with rule of mixtures for longitudinal conductivity and further extended for the prediction of transverse thermal conductivity. Effect of fiber–matrix interface debond and fiber randomness on thermal conductivity of the composite is studied for a range of values of conductivity ratio of fiber to matrix. From the present study, it is observed that transverse thermal conductivity is influenced by debond and randomness, and deviation in this property from that of a regular model is more with more mismatch in the properties of constituents.

This work explores the possibility of tailoring the thermal conductivity and thermal expansion of rare earth monosilicates through the introduction of multiple rare earth cations in solid solution. Six rare earth monosilicates are studied: Sc2SiO5, Y2SiO5, Nd2SiO5, Dy2SiO5, Er2SiO5, and Yb2SiO5. Four equimolar binary cation mixtures and a high entropy five-cation equimolar mixture were characterized. Thermal expansion was measured up to 1200 ˚C with X-Ray Diffraction (XRD) and bulk thermal conductivity was measured by Hot Disk technique. The linear coefficient of thermal expansion (CTE) of mixed-cation systems followed a rule of mixtures, with average linear CTE between 6 - 9x10-6 /˚C. Scandium monosilicate showed a lower linear CTE value as well as a notably lower degree of CTE anisotropy than other rare earth monosilicates. Thermal conductivity was found to decrease below rule of mixtures values through increasing heterogeneity in rare earth cation mass and ionic radii, as expected for the thermal conductivity of solid-solutions. The high entropy mixture RE2SiO5 (RE=Sc, Y, Dy, Er, and Yb) shows a thermal conductivity of 1.06 W/mK at room temperature, demonstrating that high entropy rare earth silicates are strong candidates for novel dual-purpose thermal and environmental barrier coatings.

A systematic study has been undertaken of the reinforcement of poly (vinyl alcohol) (PVA) by hexagonal boron nitride (hBN) nanosheets (BNNSs) produced by the liquid-phase exfoliation of hBN crystals. Three types of BNNSs with different geometries were prepared, two of which had similar lateral dimensions and two that had a similar aspect ratio (length/thickness). PVA nanocomposites with different loadings of the three types of BNNSs were prepared and this enabled the effect of BNNS volume fraction and geometry upon the mechanical properties such as Young's modulus, yield stress and breaking strength, to be determined. Although the Raman scattering from hBN is relatively weak compared with that from graphene, it was shown that Raman spectroscopy could be used to both evaluate the distribution of the BNNSs in the nanocomposites and follow stress transfer from the polymer matrix to the BNNSs. It was found that the reinforcement of the polymer could be modelled using a combinations of the rule of mixtures and modified shear lag theory. The highest level of reinforcement was found for the BNNSs with the largest aspect ratio although there was evidence of a decline in the level of reinforcement at the highest loadings of all types of BNNSs, as the result of agglomeration of the nanosheets.

The volume fraction of the carbon nanotube (CNT) plays a key role in ensuring the performance of the CNT reinforced polymer composite, especially under the severe vibration, which leads to the resonance and failure of the composite structure. In this paper, a bottom-up multiscale method is adopted to study the effect of the CNT volume fractions on the nonlinear vibration of the poly (methyl methacrylate) (PMMA)/CNT composite. According to the molecular simulation, the longitudinal, transverse and shear moduli of the PMMA/CNT nanocomposites are found to increase with the increasing CNT volume fractions. Substituting the simulated moduli into the extended rule of mixtures (EROM), the efficiency parameters of the PMMA/CNT composite with various CNT volume fractions are derived based on a homogenization approach. The derived efficiency parameters are used in the functionally graded (FG)-based EROM to obtain the expressions of the longitudinal, transverse and shear moduli of the macroscopic composite plate, so as to obtain the constitutive equation for the nonlinear vibrations of the FG-based PMMA/CNT composite plate. The subsequent meshless simulation results demonstrate that the natural frequencies of the FG-based composite plate increase with the increasing volume fractions, whereas the ratios of the nonlinear to linear frequencies decrease. Using the bottom-up multiscale analysis, the macroscopic vibration responses are analyzed for the PMMA/CNT composites with the CNT volume fractions up to 9.0%, which provides a paradigm of the design of the PMMA/CNT composite by considering the CNT volume fraction effect.

We present a combination of the Finite Element Method (FEM), the Particle Finite Element Method (PFEM), and the Discrete Element Method (DEM) for modeling and analyzing the failure of reinforced concrete structures under impulsive wave forces originating from free-surface flows in critical water hazards. The free-surface water flow is modeled with the PFEM, while the structural behavior and the fractures induced by the water forces in the structure are modeled with a coupled FEM–DEM technique. The concrete material behavior is simulated with a standard isotropic damage model. The reinforcing bars are modeled by a rule of mixtures procedure, for simplicity. The possibilities of the new integrated PDFEM approach for predicting the evolution of free-surface tsunami-type waves and their devastating effect on constructions are validated with experiments on the failure of reinforced concrete plates under large impacting waves, performed in a laboratory facility in Japan. This work has been supported by the Spanish Government program FPU: FPU16/02697. The authors gratefully acknowledge the received support.

Commercial production of high strength steel plates by the quenching and tempering (Q&T) route requires control of alloy design and heat treatment parameters to achieve the desired strength and toughness through thickness. Plates with different thicknesses (up to approximately 100 mm) are produced for applications in the energy and power or lifting and excavation sectors. For thick plate the difference in cooling rate through thickness affects the as-quenched microstructure with martensite, auto-tempered martensite and lower and/or upper bainite being present. The different as-quenched microstructures can show a different response to tempering which affects the final strength and toughness.In this study the starting microstructure of a low alloy 0.17 wt% C Q&T steel has been varied using isothermal heat treatment at 430 °C to create mixed martensite and lower bainite microstructures (nominally 25:75; 50:50 and 75:25 percentages). The effects of tempering at 600 °C for times between 0.5 and 16 hours on the carbide precipitates and hardness of the mixed microstructures have been investigated and compared to the tempering response of single phase (martensite and lower bainite) microstructures. It has been found that the hardness decrease due to tempering is larger in the martensitic structure than the bainitic structure due to more rapid carbide coarsening. The as-quenched hardness of the mixed microstructures can be predicted by a rule of mixtures using the single phase properties. The tempering response of the mixed microstructures is discussed.

Rule of mixture models are usually used in the tensile properties prediction of polymer composites reinforced with synthetic fibres. They are less utilized for natural fibre/polymer composites due to natural fibres physical and mechanical properties variability which reduces rule of mixture model's prediction values accuracy compared to the experimental values. This had led to studies conducted by various researchers to improve the existing rule of mixture models to give a better reflection of the true natural fibres properties and enhance the rule of mixture models prediction accuracy. In this paper, rule of mixture model's utilization includes the existing rule of mixture models as well as proposed rule of mixture models which have one or more factors incorporated into existing rule of mixture models for natural fibre/polymer composites tensile properties prediction are reviewed.

Industrial printed circuit boards (PCB)s are composites with complex geometrical features. In this study, a homogenization scheme was proposed to model bending, i.e., the primary deformation mode of industrial PCBs. Four-point bend tests were conducted on two types of industrial PCBs to measure their bending stiffness in the longitudinal and the lateral directions, respectively. The experimental results showed significant anisotropy and non-uniformity in the mechanical properties of the PCBs. PCB modelling was performed using the finite element method and a zoning approach. Three micromechanics models, namely Cox-Krenchel model, modified Cox Lamina model and Pan model, were adopted. These models were compared to the classical rule of mixtures (ROM). Results showed that ROM and the micromechanics models gave the upper and the lower bounds, respectively, of the stiffness and overall response of the PCB in bending. Among the micromechanics model, Pan model gave the best prediction of elastic stiffness, at accuracy of more than 80%.

The performance grading (PG) tests use aged asphalt specimens to simulate the in situ performance of asphalt binder. The main purpose of this study is to examine the existing micromechanical models, such as the inverse rule of mixtures, Hashin-Shtrikman, generalized self-consistent scheme (GSCS), and 4-phase, to determine the dynamic shear modulus of aged asphalt mastics. The Wicket and the vGP (van Gurp-Palmen) plots were advisable tools for checking the PG test results consistency and Time-Temperature superposition principle validation on three different asphalt mastics. The generalized logistic sigmoidal model (GLSM) yielded well-fitted master curves of modulus and phase angle based on the models’ predictions for full linear viscoelastic (LVE) characterization. The 4-phase model had the best modulus predictability even better than the well-known 2S2P1D (2 Spring-2 Parabolic element-1 Dashpot) model that followed by the inverse rule of mixtures, Hashin-Shtrikman, and GSCS models. All of the models had an excellent phase angle predictability that was comparable with 2S2P1D model predictions. It is recommended to combine the typical PG tests of asphalts, a well-defined micromechanical model like 4-phase, and a well-known rheological master curve model such as GLSM for estimating the LVE properties of asphalt mastics promisingly.

In this paper, the dynamic response of functionally graded viscoelastic hollow cylinder subjected to thermo-mechanical loads is studied using the meshless local Petrov-Galerkin method. The material is assumed to be graded in the radial direction with aluminum as viscoelastic constituent and alumina as the elastic constituent. The macroscopic viscoelastic properties are evaluated using the rule of mixtures and the inverse rule of mixtures, whilst that of the cylinder is computed by employing the Mori-Tanaka homogenization scheme. A systematic parametric study is carried out to bring out the influence of material gradient index, the viscoelastic properties and the boundary conditions on the dynamic/transient response of the cylinder.

Biodegradable and ecologically friendly polymer materials attract great attention of many scientific groups in the world as they fit well in the sustainable development policy and are considered to be “a right thing to do” by the general public. Such polymers can be modified by the addition of different fillers, favorably of natural origin. In the paper we provide a comparison between composites based on two biodegradable polymers: poly(lactic acid)—biodegradable, natural stock polymer and poly(butylene succinate)—biodegradable polymer produced from fossil based materials. For each polymer we have prepared a series of composites with different fibres (natural: hemp and flax, and manmade: Cordenka) and different filler loadings. To fully characterize obtained materials thermal, mechanical and surface free energy measurements were performed, completed with morphology observations and an attempt to compare the experimental data for tensile measurements with values obtained using the modified rule of mixtures. The tensile results calculated using the modified rule of mixture for below 30% fibre loading are found to be fitting the experimental data. Composites mechanical properties and morphology were strongly affected by the type of fibre used and its loading, however thermal properties remained almost unchanged. In specific, Cordenka fibres tend to form bunches which presence greatly influences the mechanical properties but still our studies have shown clear advantage of manmade Cordenka fibres over the hemp and flax fibres when considering distribution and fibre–polymer interaction.

Ni-14 at.% W/SiC nanocomposite coatings containing 0-6 vol.% of submicron size (0.35 μm) SiC particles were obtained using pulsed electrodeposition technique on mild steel substrate. The coatings were subsequently characterized for uniformity of SiC distribution, grain size and mechanical properties (hardness and elastic modulus) using SEM, XRD and nanoindentation techniques, respectively. It was observed that the SiC particulates were uniformly distributed within the coating. The coating matrix (Ni-W) also exhibited nanograin size. Tribological behavior of Ni-14 W/SiC nanocomposite was analyzed under dry sliding wear test conditions using pin on disk method. The interrelationship between SiC content, interparticle spacing, mechanical properties and wear behavior was analyzed. The results suggest that whereas hardness and modulus of nanocomposite coatings varied with SiC content as per Rule of Mixtures, wear mechanism can be best explained on the basis of Inverse Rule of Mixtures.

A base wash procedure was used to partially remove the oxygen functional groups from the graphene oxide (GO) flakes to prepare the base-washed GO (BwGO). It is found that the base wash treatment does not alter the physical properties (size, thickness etc.) of the GO significantly but the chemical composition is changed. Nanocomposites were prepared by incorporating the BwGO flakes obtained in a poly(vinyl alcohol) (PVA) matrix. It was found the storage modulus of the nanocomposites is enhanced from 4.4 GPa to 6.5 GPa with 5 wt% of BwGO. This is in agreement with the micromechanical estimation obtained by using Raman spectroscopy that follows the interfacial stress transfer from the matrix to the BwGO fillers. The lower effective modulus of BwGO than GO as calculated using the classical ‘rule of mixtures’ is due to the loss of functional groups on GO that serve as a surfactant to prevent the flakes from re-agglomerating. An agglomeration factor ηa is therefore proposed and a concept of ‘effective volume fraction’ is introduced to quantify and evaluate the level of agglomeration of fillers in nanocomposites, which can be otherwise difficult to visualize optically. It is found that the removal of the functional groups causes the flakes to re-agglomerate, and reduces the ‘effective volume fraction’ by about 10–20%.

Present research deals with the shear buckling behaviour of composite skew plates reinforced with aligned single walled carbon nanotubes (CNTs). Distribution of CNTs across the thickness of the skew plate are assumed to be uniform or functionally graded. Two different types of shear loads are considered. The case of rectangular shear which produces pure shear and the case of skew shear which results in a combined uniform shear and uniaxial tension/compression. Suitable for moderately thick plates, first order shear deformation plate theory is used to estimate the displacement field of the plate. The equivalent properties of the composite media are obtained by means of the refined rule of mixtures approach which contains efficiency parameters to capture the size dependent properties of the CNTs. With the aid of the Hamilton principle, transformation of the orthogonal coordinate system to an oblique one and the conventional Ritz method whose shape functions are constructed according to the Gram–Schmidt process, the stability equations of the plate are established and solved for two different types of loading, namely rectangular and skew shear loads. As shown, through introduction of a proper functionally graded pattern, i.e., FG-X pattern, the buckling load of the plate may be increased, significantly.

The intent of this paper is to firstly perform the vibration analysis of the functionally graded carbon nanotube reinforced composites (FG-CNTRC) circular, annular and sector plates with arbitrary boundary conditions by means of the semi-analytical method which is proposed by the author’s team. In the material model, the distribution of the carbon nanotubes is uniform or functionally graded along with the thickness direction of structures and four types of the CNTs distribution are studied in this paper. The refined rule of mixtures approach containing the efficiency parameters is adopted to determine the properties of the composite media. The admissible displacement functions of the FG-CNTRC circular, annular and sector plates are uniformly expanded as the modified Fourier series which embodies a standard cosine Fourier series and several auxiliary functions which are introduced to eliminate the limit of the boundary conditions. On this foundation, the first-order shear deformation elasticity theory is employed to construct the energy expression of the FG-CNTRC circular, annular and sector plates. Then the Ritz-variational energy method is used to decide the natural frequencies and the associated mode shapes. To examine the convergence, accuracy, stability and efficiency of the computational model, the comprehensive studies are conducted. Based on that, some crucial parametric studies covering the influence of the geometrical parameters, CNTs distributions, volume fraction of CNTs and boundary conditions are also investigated in detail.

A new material tailoring method for spherical and cylindrical vessels made of functionally graded materials (FGMs) is presented. It is assumed that the FG material is composed of an Al–SiC metallic-matrix composite. A uniform ratio of in-plane shear stress to yield strength [ $$\varphi \left( r \right) $$ ] is used as the design criterion to utilize the maximum capacity of the vessel. The aim is to find a distribution of SiC particles in the radial direction, i.e., $$f\left( r \right) $$ , that achieves a uniform index $$\varphi \left( r \right) =\hbox {const}.$$ through the wall thickness of the internally pressurized spherical or cylindrical vessel. Both the Mori–Tanaka and rule-of-mixtures homogenization schemes are used to express the effective elastic module and Poisson’s ratio. Moreover, the strength of the composite is expressed based on the rule of mixtures. Besides, finite element simulation is carried out to verify the accuracy of the analytical solution. The effects of input parameters such as the internal pressure, strength of the SiC particles, ratio of in-plane shear stress to effective yield strength, and choice of homogenization scheme on the tailored distribution of the SiC volume fraction in the radial direction are also investigated.

The present investigation deals with the large amplitude response of shallow thick circular arches subjected to thermal, mechanical, or thermomechanical loads. The structure is resting on a three-parameter elastic foundation containing the Winkler springs, the shear Pasternak layer, and hardening nonlinear springs. The arch is made of a through-the-thickness functionally graded material. Except for Poisson’s ratio, the other properties of the arch are assumed to be temperature and position dependent. Each property is estimated according to the rule of mixtures in terms of the volume fraction of the constituents. The case of uniform temperature rise and uniform radial lateral load is considered. The governing equations of the arch are obtained with the aid of the third-order shear deformation arch theory and the von Karman type of geometrical nonlinearity. The developed coupled and nonlinear equilibrium equations are solved using the two-step perturbation technique for the case of simply supported arches. Closed-form expressions are developed for nonlinear bending equilibrium path of shallow arches under thermal and mechanical loads. Results are provided for different power law indices, geometrical parameters, and the foundation stiffness.

The mechanical integrity of the separator is critical to the reliable operation of a battery. Due to its minimal thickness, compression experiments with a single/a few layers of separator are difficult to perform. In this work, a capacitance based displacement set-up has been developed for the measurement of the through thickness direction (TTD) compression stress-strain behavior of the separator and the investigation of its interaction with the electrode. The experiments were performed for a stack of two layers of Celgard 2400 separator, NMC cathode, and separator/NMC cathode/separator stack in both dry and wet (i.e. submersed in dimethyl carbonate DMC) conditions. The experimental results reveal that the separator compression modulus can be significantly affected by the presence of DMC. The iso-stress based rule of mixtures was used to compute the compressive stress-strain curve for the stack from that of the separator and NMC layer. The computed curve agreed with the experimental curve reasonably well up to about 0.16 strain but deviated significantly to a softer response at higher strains. The results suggest that, in the stack, the TTD compressive deformation of the separator is influenced by the NMC cathode.

The characterization of composite materials, particularly those reinforced with vegetable fibres, presents additional challenges, like the influence of the manufacturing process, fibre-matrix adhesion problems, progressive stiffness reduction or material degradation due to environmental elements (moisture, heat and UV) exposure. The present work studies the suitability of inverse characterization, based on modal properties, to determine the mechanical properties of jute fibre-reinforced composite plates, subjected to environmental exposure to simulate outdoors application conditions. With this method both the elastic and the damping properties of composite specimens are found, which highlights its great usefulness in the prediction of mechanical properties. It was found that this non-destructive method can better estimate material properties than the rule of mixtures, reflecting manufacturing, processing conditions and also the influence of moisture absorption and UV degradation on the mechanical properties of these natural fibre composites. This method is an important tool since it can be performed in the final parts where the results account for all external influences.

In this work, nonlocal nonlinear analysis of functionally graded plates subjected to static loads is studied. The nonlocal nonlinear formulation is developed based on the third-order shear deformation theory (TSDT) of Reddy (1984, 2004). The von Karman nonlinear strains are used and the governing equations of the TSDT are derived accounting for Eringen’s nonlocal stress-gradient model (Eringen, 1998). The nonlinear displacement finite element model of the resulting governing equations is developed, and Newton’s iterative procedure is used for the solution of nonlinear algebraic equations. The mechanical properties of functionally graded plate are assumed to vary continuously through the thickness and obey a power-law distribution of the volume fraction of the constituents. The variation of the volume fractions through the thickness have been computed using two different homogenization techniques, namely, the rule of mixtures and the Mori–Tanaka scheme. A detailed parametric study to show the effect of side-to-thickness ratio, power-law index, and nonlocal parameter on the load-deflection characteristics of plates have been presented. The stress results are compared with the first-order shear deformation theory (FSDT) to show the accuracy of nonlocal nonlinear TSDT formulation.

The buckling and free vibration of initially stressed functionally graded cylindrical shell reinforced with non-uniformly distributed graphene platelets (GPLs) are investigated using the state-space formulation based on three-dimensional elasticity theory. The shell is under an axial initial stress and composed of multilayers with GPLs uniformly dispersed in each individual layer but its weight fraction changing layer-by-layer along the thickness direction. The modified Halpin-Tsai model and rule of mixtures are employed to evaluate the effective elastic properties of the GPL-reinforced shell. Analytical buckling and frequency solutions are obtained for simply supported shells. Numerical results are presented for functionally graded GPL-reinforced cylindrical shells with five GPL dispersion patterns (GPL-UD, GPL-V, GPL-A, GPL-X, and GPL-O). The effects of GPL weight fraction, dispersion pattern, geometry, and size as well as the influence of initial stress on the buckling and free vibration characteristics of the shell are discussed in detail. It is found that the addition of a small amount of GPLs significantly increases the critical buckling stress and natural frequencies. The GPL-X pattern outperforms other patterns for thin composite shells while the uniform pattern GPL-UD works better for thick composite shells.

In this work, a micromechanical model of tensile behavior of fibrous composites was formulated by taking into account of the stress-strain behavior of fiber and matrix, the directional arrangement of fibers in the composites, and the relative space occupied by the constituent fibers and matrix in the composites. A simple and easy-to-use expression was obtained for predicting the tensile stress of composites as a function of tensile stresses of fiber and matrix, fiber and matrix volume fractions, and fiber orientation factor. The model was validated with a set of biocomposites prepared by using nettle and poly(lactic acid) fibers and employing compression moudling technology. The model was also compared with the well-known "rule of mixtures" equation using the experimental results. While the "rule of mixtures" equation was found to highly overestimate the tensile stress, the model predicted the experimental results satisfactorily.

This paper experimentally investigates adhesive fracture resistance and crack path selection in adhesive joints containing well-defined localized interfacial defects. Several systematic patterns of localized interfacial defects were created on base-acid treated aluminum adherends by physical vapor deposition of copper through a mask. Adhesive joints were prepared using a commercially available, structural epoxy adhesive and the effect of localized interface defects on the performance of adhesive joints was studied. Under mode-I loading conditions, the presence of localized weak interfaces influenced the fracture energy of a propagating debond over a considerable distance. For a crack tip approaching a given weak interface pattern, a falling or reverse R-curve type trend was observed. Within the same DCB specimen, as the crack tip advanced beyond the patterned region, a rising R-curve type trend was observed as fracture energy increased with increasing crack lengths, which were recorded visually and were also inferred using compliance and crack length relationship. In addition, the mode-I fracture energy was found to scale with the area fraction of the weak interfaces according to a rule of mixtures. For the adhesive system and the joint geometry used in this study, it was observed that cohesive failure in the adhesive layer can be obtained even in the presence of exceptionally weak interfaces (similar in size to those detected in mode-I loading) when loaded under mixed mode conditions (achieved through asymmetric loading of DCB-like specimens), that tended to steer the crack away from the interface. When loaded with the opposite mode mixity direction, the crack tip propagated through the defects, though the fracture energy did not exhibit similar R-curve type trends as observed in mode-I tests. The results offer insights into the interaction of propagating cracks in adhesive layers and their interactions with discrete, localized defects, which could lead to improvements in surface preparations and bond integrity or even to joint designs having intentionally placed defects useful in controlled disassembly or for other purposes.

The purpose of this paper is to investigate the non-linear bending behavior of functionally graded shell structures reinforced by carbon nanotubes (FG-CNTRC) using non-linear double directors shell model and membrane enhancement. This model induces a high order variation of the displacement field along the thickness direction and imposes a zero transverse shear stresses condition in the top and bottom surfaces. The effective material properties of FG-CNTRC are estimated by the modified rule of mixtures using some efficiency parameters. Uniformly (UD) and three graded distributions, known as FG-V, FG-O and FG-X, are considered to show the effects of carbon nanotubes (CNTs) profiles and their volume fractions on non-linear deflections of various shapes of shell structures. To illustrate these effects, a set of numerical examples including beams, plates, cylindrical panels and hemispherical shells are presented. The effects of some geometrical parameters are also examined. A convergence studies are carried out to validate the proposed non-linear model, then non-linear results for FG-CNTRC shell structures are provided leading hence to outline the applicability and the efficiency of the present model.

A custom built reusable cell for in situ lithiation and mechanical deformation studies while in an X-ray tomograph was demonstrated, and the strain and volume changes of a composite graphite anode were computed from 3D X-ray microcomputed tomography data via Digital Volume Correlation (DVC). The test anode was a composite electrode comprised of a porous compliant matrix, graphite as the Li+ host material, 5-μm ZrO2 marker particles for use with DVC, and active carbon black to enhance electrical conductivity. The composite electrodes were hot-pressed to control their porosity, and in turn the mechanical integrity of the resulting material. This composite anode was included in a half-cell and lithiated in situ while in a tomograph, and intermittent 3D data were collected at different lithiation levels up to full gravimetric capacity. Strain measurements by DVC demonstrated relatively uniform expansion of the freestanding electrode with average normal strains in the three directions varying by 20%, while the internal shear strains were found to be negligible. The average experimental strains were about 75% of the theoretical value, as estimated by the rule of mixtures, which implies that ~25% of the normal strains in graphite, due to lithiation, are accommodated by the surrounding matrix.

A new method for prediction of elastic constants in randomly oriented fiber composites is proposed. It is based on mechanics of composites, the rule of mixtures and total mass balance tailored to the spectral element mesh composed of 3D brick elements. Selected elastic properties predicted by the proposed method are compared with values obtained by another theoretical method. The proposed method is applied for simulation of Lamb waves in glass-epoxy composite plate reinforced by randomly oriented fibers. Full wavefield measurements conducted by the scanning laser Doppler vibrometer are in good agreement with simulations performed by using the time domain spectral element method.

In the current work, three dimensional finite element analyses have been performed to predict the directional strength in terms of Young’s modulus of carbon nanotube based composite and the results are compared with the Extended Rule of Mixtures (ERM). Square based prismatic Representative Volume Element (RVE) has been considered to model the SWCNT reinforced nanocomposite. In this study, reinforced CNT in RVE has been idealized as solid annular tube with hemispherical cap at its both ends. The parameters which are considered for these analyses are modulus ratio (ratio of Young’s modulus of CNT and matrix), height fraction (ratio of CNT height and RVE length), aspect ratio (ratio of CNT height and diameter) and volume fraction of CNT. Further, different CNT diameters are also being considered in order to investigate their effects on effective Young’s modulus of the composite. It is observed from the study that higher values of height fraction, modulus ratio, volume fraction and aspect ratio of CNT are beneficial to enhance the effective Young’s modulus of the composite. Variation of CNT diameter does not have significant effect for high modulus ratio. However, CNT with smaller diameter leads to comparatively higher effective Young’s modulus of nanocomposite in case of low modulus ratio.

A multistep model is proposed for calculating the tensile modulus values of polymer/carbon nanotube (CNT) nanocomposites (PCNTs) based on the modified rule of mixtures, assuming a percolated network of nanoparticles. In the first step, the network of nanoparticles is considered as a new phase with a novel volume fraction and Young's modulus. Then, the volume fraction of the filler network in the PCNTs is correlated to the density of the network. Also, the percolation of the nanoparticles is related to the aspect ratio of the nanoparticles. Finally, a new model is proposed based on the modified rule of mixtures (the Riley model) of the properties of the filler network. The predictions of the proposed model are compared with experimental results and the roles of the nanoparticles and network properties in the modulus values of nanocomposites are determined. The proposed model presents acceptable predictions when compared with the experimental data. Moreover, the density and modulus of the filler network, as well as the aspect ratio and diameter of the nanoparticles was found to directly affect the moduli of the nanocomposites.

Ti-6Al-4V is one of the most widely used materials for successful applications in aerospace, automotive, and biomedical industries due to its high specific strength, excellent mechanical and thermal properties, and outstanding corrosion resistance. These properties make Ti-6Al-4V very difficult to be machined using conventional manufacturing processes. The electrical discharge machining (EDM) is a very viable solution to machine Ti-6Al-4V. However, the EDMed surface usually undergoes phase transformation during the EDM process due to the rapid heating and cooling in EDM. In this paper, the commercial FEM software package Abaqus is used to investigate the effects of EDM conditions on the crater size, phase transformation in HAZ, and the residual stress distribution in EDMing of Ti-6Al-4V. The phase transformations of Ti-6Al-4V alloy during EDM are modeled based on phase transformation kinetics and flow stress is described using the rule of mixtures. Experiments are conducted to validate FEM model, which are then used to predict crater size, phase transformation and residual stress for different discharge energy levels.

The effects of annealing temperature, thermal cycle times, precipitates, and Nd content on the thermal expansion behavior of as-extruded Mg-xNd (x=2, 4) binary alloys were investigated. It demonstrated that the coefficient of thermal expansion (CTE) of alloys decreased with the increasing of Nd content. The annealing treatment eliminated partial residual stress inside the alloy, which led to the CTEs of annealed samples higher than that of the unannealed samples at low temperature. However, it was opposite at high temperature after multiple thermal cycles due to the amount of Mg41Nd5 phase increased after annealing treatment. The results of the calculated cumulative residual strain of the Mg-Nd alloys with thermal cycling indicated that the alloys annealed at 325 ℃ has the best thermal stability. The rule of mixtures (ROM) and Kerner model were proposed to predict the CTEs of Mg-Nd alloys. It was found that the prediction of the Kerner model was agrees well with the experimental data of 1st thermal cycle of samples without internal stress.

Functionally graded porous structures (FGPSs) are gaining interest in the biomedical sector, specifically for orthopaedic implants. In this study, the compressive behaviour of seven different FGPSs comprised of Face Centred Cubic (FCC) and the Octet truss unit cells (OCT) were analysed. The porosity of the structures were graded in different directions (radially, longitudinally, laterally and longitudinally & radially) by varying the strut diameters or by combining the two types of unit cells. The structures were manufactured by laser power bed fusion and compression tests were performed. Radially and laterally porous graded structures were found to outperform uniform porous structures with an increase in stiffness of 13.7% and 21.1% respectively. The experimental and finite element analysis (FEA) results were in good agreement with differences in elastic modulus of 9.4% and yield strength of 15.8%. A new FEA beam model is proposed in this study to analyse this type of structures with accurate results and the consequent reduction of computational time. The accuracy of the Kelvin-Voight model and the rule of mixtures for predicting the mechanical behaviour of different FGPSs was also investigated. The results demonstrate the adequacy of the analytical models specifically for hybrid structures and for structures with smooth diameter transitions.

Heterogeneous materials can show size-dependent behaviour in which the bending modulus depends on sample size. In fibre composite materials the interaction between fibre and matrix can lead to such a size effect.Then the effective modulus calculated by the rule of mixtures can either underestimate or overestimate the bending modulus of the laminate, depending on the fibre/matrix material mismatch, the microstructural morphology (fibre distribution) and the laminate thickness. In this work, the bending behaviour of a laminate comprising unidirectional fibre composite plies is considered using Euler-Bernoulli beam theory and the influence of size on the bending modulus is investigated. The effective bending modulus of each ply is calculated and used to formulate the overall bending modulus of the laminate. The results show that the laminate bending modulus depends on the number of plies, the number of fibres through the thickness of each ply, the fibre spacing and radius, and the mismatch of fibre and matrix material properties in each ply. Our analysis shows that accounting for the ply microstructure (fibre spacing and radius and number of fibres per ply) can lead to a 10% difference in the predicted bending modulus in a three ply laminate, when there are less than four fibres through the thickness in each ply.

In the present work, a high-strength composite steel structure developed by imparting laser surface hardening treatment on a low-carbon low-Manganese automotive steel sheet has been comprehensively analysed. The layered composite steel structure has been successfully developed by re-engineering the surface of the steel using diode laser hardening treatment up to a depth of 250–300 µm through its thickness. Hardness at the treated surface improved by 150% to that of its base due to formation of a mixture of hard phases constituting martensite and bainite along with retained ferrite. Indeed, coupling EBSD technique with Weibull distribution of various phase fractions determined by image quality helped analyse microstructure effectively. The tensile property of the layered composite steel sheet was found to yield significant improvements in both Yield Strength (YS) (40–44%) and Ultimate Tensile Strength (UTS) (19–21%) due to sandwich effect of composite layer constituting hardened layer and soft base accomplished by a strengthening mechanism associated with rule of mixtures concept. In fact, Young’s modulus of the composite steel sheet, determined from slope of tensile stress–strain diagram was found to be convergent with ultrasonic test result. Additionally, the crystallographic texturing effects of the hardened layer and untreated base measured using standard XRD technique re-confirmed their influence on Young’s Modulus and r-bar values obtained.

This paper presents a theoretical analysis for the frequency response of thermally pre/post buckled composite pipes reinforced with carbon nanotubes (CNTs). The vibrational behavior of the uniformly heated nanocomposite pipe surrounded by a nonlinear elastic foundation is analytically studied. The nanocomposite media is introduced using a refined rule of mixtures that is calibrated with the available data on the basis of molecular dynamics simulations. Distribution of CNTs across the radius of composite pipes can be uniform or functionally graded (FG) and properties are considered to be temperature-dependent. A higher-order shear deformation model and the von Karman assumption are employed to extract the nonlinear equations of motion. The system of partial differential equations is established by means of Hamilton’s principle. A two-step perturbation method is utilized to carry out closed-form solutions for these nonlinear equations. The natural frequencies are obtained for the immovable simply-supported boundary conditions. A comparison study is provided for the case of isotropic homogeneous pipes in which good agreement is obtained. The novel numerical results are presented to show the influences of geometric characteristics, CNT distribution pattern, foundation stiffness, and the CNT volume fraction upon the frequency response of the nanocomposite pipe subjected to uniform temperature rise loading.

This work explores the multiobjective design optimization of symmetric and non-symmetric sandwich plates with ceramic-metal-ceramic functionally graded (FG) core, along with ceramic faces (if included), under thermo-mechanical loadings. The FG core is made of two functionally graded material (FGM) layers, each with the volume fractions of the constituent phases defined by a power-law function through-thickness, with the interface between them metal-rich and the core outer surfaces ceramic-rich. In line with symmetric or non-symmetric sandwich plates, the design variables involve the thickness of each FGM layer of the core and the index of its power-law function, along with the thickness of each ceramic face. The thermal and mechanical problems are fully coupled using a layerwise mixed least-squares model with multi-field independent variables, including, most importantly, displacements, temperature and transverse stresses. The FG core z-continuous effective properties are also fully described by high-order z-expansions, as demonstrated adopting the rule of mixtures. The multiobjective optimization problem is solved by Direct MultiSearch derivative-free method, minimizing mass, transverse displacement and the stress field, measured by Tsai-Hill failure criteria, all together. Numerical results provide optimal symmetric and non-symmetric sandwich plates with ceramic-metal-ceramic FG core, considering distinct constituent materials and including three-dimensional approximate solutions for validation.

This research investigates the free vibrations of graphene platelet reinforced composite (GPLRC) cylindrical shells. Composite cylindrical shell is composed of a number of layers, each layer of composite shell has a different amount of graphene reinforcement, which leads to functionally graded (FG) distribution of the reinforcements. For this purpose, the first-order shear deformation theory of shells and Donnell kinematic relations have been used. To estimate the elastic properties of the composite material, Halpin-Tsai micro-mechanical relationship has been used where the effects of nanofillers is also included. On the other hand to obtain the mass density and Poisson's ratio of the media, the simple rule of mixtures approach is adopted. The Hamilton principle has been used to derive the governing equations for the free vibrations of shells as well as the boundary conditions. The obtained equations are five in number which are coupled in terms of displacements and rotations. Suitable for simply supported boundary conditions of the shell, Fourier expansions have been used as the conventional Navier solution. An analytical solution is provided to obtain the natural frequencies as well as the numbers of their modes. The results of this study at first are compared with the known data for isotropic homogeneous shells and after that novel results are provided for FG-GPLRC cylindrical shells. The results of the present study show that the weight fraction and the distribution of GPLs are two important factors on the natural frequencies of the shell, so that with increasing the weight fraction of GPLs, the natural frequencies of the shell are strongly increased. Also when the inner and outer layers of the shell have the maximum amount of reinforcements, maximum frequencies are achieved.

Polypropylene self-reinforced composites (PP-SRCs) are advantageous in terms of recycling, impact resistance, and price; however, a deeper understanding of their mechanical properties is required to expand their commercialization. Here, an in-depth study was conducted to understand the effect of changing PP-tape as a function of the draw ratio on the mechanical properties of PP-SRC. As the draw ratio increased, PP-tape showed higher orientation and thus significantly improved the tensile strength and Young's modulus. The tensile properties of PP-SRC showed the same trend; this was successfully predicted by employing the rule of mixtures as they were dominated by the reinforcing effect of PP-tape. Meanwhile, the impact resistance of PP-SRC improved with increasing draw ratio, but the T-peel adhesion deteriorated. This occurred because the failure mode at higher draw ratios transitioned from tape fracture to delamination, resulting in enhanced energy absorption ability, and the decrease in polymer chain mobility hindered interfacial inter-diffusion.

This paper presents the fire rating performance of load-bearing thin-walled system sheathed with Functionally Graded Material (FGM) board under standard ISO834 fire through numerical simulation. FGMs are one of the new classes of advanced composite materials that possess continuous variation of material properties within a given direction. The composition of the FGM sheathing is defined by the volume fractions of the constituents’ materials (metal/ceramic) based on the power-law (P-FGM) material function. The general rule of mixtures is then used to predict the thermo-mechanical properties of the FGM sheathing. Fire rating analyses for the Cold-Formed Steel (CFS) wall system were conducted under steady-state conditions where the elastic buckling load from bifurcation analysis under fire conditions was first obtained. Then using the RIKS ON algorithm, collapse analysis was performed in a time frame until failure deformation occurred. The effect of non-uniform temperature on mechanical and thermal properties of the wall stud was included at each time frame in both elastic and collapse analysis. From the FE analysis using ABAQUS, it was observed that the use of FGM as a sheathing material for fire protection increases the failure time for all load ratios compared with the traditional gypsum board. The increase in failure time has a significant implication for improving the safety of occupants during fire scenarios. The study also shows that the novel composite material (FGM board), if properly designed, can lead to an alternative fire protection material in the thin-walled system.