Mori-Tanaka Micromechanical Model Research Articles

Parametric study of CNT waviness effect in CNT/polymer nanocomposites using multiscale finite element modeling

The effect of CNT waviness on the elastic properties of CNT-reinforced polymer nanocomposites is investigated using multiscale finite element modeling. CNT and the interphase are modeled at the nanoscale as discrete structures of beams and trusses, and the polymer matrix is treated as a microscale continuum medium. Using the model, a parametric study of the effect of CNT waviness is carried out. The effect of CNT waviness is also calculated using a three-phase Mori-Tanaka micromechanical model, and the results are compared. The model is validated with molecular dynamics simulation results and experimental data available in the literature.

Micromechanical modeling for the prediction of time-dependent effective properties and macroscopic coupled field response of polymer reinforced piezoelectric composites

In the time domain, a new straightforward micromechanical modeling is developed to predict the time-dependent behavior and the macroscopic coupled response of polymer piezoelectric composite materials. The modeling is based on the dynamic electroelastic Green’s tensor allowing the derivation of convolution localization equations in Stieltjes space. The Mori-Tanaka micromechanical model is adopted to derivate the dynamic electroelastic localization tensor. The presented modeling leads to an expression of the time-dependent effective electroelastic properties directly in the time domain, without an iteration procedure, through convolution products in Stieltjes space. It allows the prediction of the time-dependent behavior and the coupled macroscopic response while taking into account the constituent properties, volume fractions, and shape of inclusions. In implementing the expression of the effective behavior, one has to go through the computation of direct and inverse tensorial convolution products. A trapezoidal numerical scheme is adopted for this sake. Many numerical results are presented, and for the sake of validation, a comparison is made with the ones obtained based on the Laplace transform approach as well as with experimental and numerical results.

Effective nonlinear responses of three-phase magnetoelectric composites

A computational method, dubbed simplified unit-cell micromechanics model, is generalized and applied to establish the effective nonlinear responses of three-phase magnetoelectric composites that are composed of two distinct magnetostrictive and piezoelectric phases embedded in elastic polymer matrices. The nature of nonlinear constitutive behavior of each constituent is expected to significantly influence the overall responses of the composites. To obtain the effective nonlinear responses, a mathematical linearization is first introduced to perform the constitutive linearization for the nonlinear materials, and the resulting constitutive equations are then unified and nested into the micromechanics model followed by iterations in order to minimize errors from the linearization process. For the purpose of comparison, we also reformulate the well-established Mori–Tanaka micromechanics model insofar as its mathematical structure is aligned with that of the simplified unit-cell model. Numerical results are first validated against limited experimental measurements available in literature. Parametric studies are then conducted in order to reveal the effect of phase constitutive laws, volume fractions, and geometries on the overall nonlinear responses of there-phase magnetoelectric composites. The contributions of this work complement those of earlier studies that prevalently devoted to two-phase magnetoelectric composites and linear magneto-electro-elastic coupled responses only.

Computational Modeling of Polymer Matrix Based Textile Composites.

A simple approach to the multiscale analysis of a plain weave reinforced composite made of basalt fabrics bonded to a high performance epoxy resin L285 Havel is presented. This requires a thorough experimental program to be performed at the level of individual constituents as well as formulation of an efficient and reliable computational scheme. The rate-dependent behavior of the polymer matrix is examined first providing sufficient data needed in the calibration step of the generalized Leonov model, which in turn is adopted in numerical simulations. Missing elastic properties of basalt fibers are derived next using nanoindentation. A series of numerical tests is carried out at the level of yarns to promote the ability of a suitably modified Mori–Tanaka micromechanical model to accurately describe the nonlinear viscoelastic response of unidirectional fibrous composites. The efficiency of the Mori–Tanaka method is then exploited in the formulation of a coupled two scale computational scheme, while at the level of textile ply the finite element computational homogenization is assumed, the two-point averaging format of the Mori–Tanaka method is applied at the level of yarn to serve as a stress updater in place of another finite element model representing the yarn microstructure as typical of FE based multiscale approach. Several numerical simulations are presented to support the proposed modeling methodology.

Prediction of the macroscopic behavior of multi-layered elastic composites

In this work, elastic fiber reinforced composite materials consisting of several layers are considered. Each layer is assumed to be a unidirectional fiber reinforced composite. The interface between layers is assumed to be perfect, which means the displacement and the traction through interfaces are continuous. To predict the overall behavior of the considered composite a two step homogenization procedure is considered. In the first step, effective properties of fiber reinforced layers are predicted based on the Mori-Tanaka micromechanical model and in the second one, a general matrix method is constructed to obtain the macroscopic properties of multi-layered composites. Analytical expressions of effective properties, for reinforced composites constituted with arbitrary N layers, are derived. The combination of the Mori-Tanaka method with the general matrix method allowed obtaining the effective properties of reinforced elastic multi-layered composites.

Designing an interpenetrating network of silane-functionalized nanocomposites for enhanced particle dispersity and interfacial bonding strength

We design an interpenetrating network of silane-functionalized nanocomposites to enhance particle dispersity and interfacial bonding strength in nanocomposites. To qualitatively measure the enhanced particle dispersity in the nanocomposites, we carry out rheological characterizations and suggest a simple particle dispersity index from the viscoelastic responses. Yield stresses of the nanocomposites are obtained by fitting with the Carreau-Yasuda model. After comparing the yield stresses and elastic moduli of the nanocomposites during photo-polymerization, we confirm that interphase regions significantly affect the interfacial bonding and mechanical strengths. Prediction of elastic moduli of the nanocomposites by the Mori-Tanaka micromechanics model agrees well with the experimental data, verifying that a good interpenetrating network exists between the nanoparticles and matrix.

A Coupled Mori–Tanaka model and FEM RVE approach for the geometrical nonlinear dynamic response of the FG-CNTRC plate based on a novel shear strain function using isogeometric finite element procedure

In this study, Mori–Tanaka micromechanical model has been coupled with finite element representative volume element (RVE) approach to model functionally graded CNT reinforced (FG-CNTRC) plate. The coupled model also termed as extended Mori–Tanaka (E-MT) effectively determines the mechanical properties of the CNT reinforced composite plates. The E-MT model has the provision to consider different types orientation of the CNT fibers in matrix phase with the incorporation of second order orientation tensor matrix. The solution of the E-MT model has been propagated to the FG-CNTRC structural model to obtain the geometrically nonlinear response. The nonlinear governing equations of motions are obtained using the Green–Lagrange strain field and solved using nonlinear Newmark’s time integration method. The effect of various parameters like aspect ratio of CNT reinforcement, orientation of the CNT fibers, span thickness ratio of the plate, different type of edge constraint, and the gradation of the CNT fiber across the thickness of the plate have been studied in this paper.

Analysis of nonlinear piezomagnetism for magnetostrictive terfenol-D composites

This study presents two micromechanics formulations to investigate the nonlinear piezomagnetism for magnetostrictive Terfenol-D composites subjected to large magnetic driving fields. A simplified unit-cell micromechanics method that hypothesizes the composite microstructure forming a periodic array is developed, and the well-known Mori-Tanaka micromechanics model is reformulated for nonlinear composite problems. This work elucidates the nature of the approximation in applying the unit-cell and Mori-Tanaka micromechanics models to magnetostrictive composites insofar as the concentration tensors are concerned. A first moment secant linearization but conventional tangent linearization is employed for nonlinear magneto-elastic constitutive laws in order to obtain a unified and linearized constitutive equation. Later, a homogenization process on the nonlinear responses of magnetostrictive composites is performed via numerical iterations in order to minimize errors from the linearization process. Different composites comprising of unidirectional Terfenol-D fiber, particle, and lamina reinforcements are studied, respectively. For nonlinear piezomagnetic strain responses simulated in this work, the predictions given by the unit-cell and Mori–Tanaka schemes are in good agreement with the experimental data available in the literature. Numerical results are also presented to illustrate the performance of each micromechanics model for common composite connectivity, i.e., 1–3, 0–3, and 2–2 types. Depending on Terfenol-D geometry, volume fraction, and applied magnetic field, the magnetostrictive composites could exhibit various degrees of nonlinear piezomagnetism. The presented micromechanics models can further be served as composite constitutive equations to analyze smart composite structures such as beams commonly utilized in sensor and actuator applications.

Characterizing the mechanical properties of nanocomposites with aligned graphene

AbstractThis study fabricated nanocomposites with aligned graphene platelets and examined their mechanical properties. Both reduced graphene oxide (RGO) and functionalized graphene (FG) were used in this study. Graphene platelets were incorporated into epoxy matrix through mechanical mixing, followed by sonication for homogeneous dispersion. To align the graphene platelets, before curing, an electric field was applied to the epoxy–graphene compound. Real‐time microscopic observation revealed that under an electric field, the graphene platelets rotated, translated, and eventually formed a chain‐like microstructure along the electric field. Tensile tests were conducted on the nanocomposites with aligned graphene, and failure mechanisms were identified using scanning electrical microscopy of the fracture surfaces. For comparison, nanocomposites with randomly oriented graphene were prepared and then subjected to tensile loading. Moreover, the moduli of the graphene nanocomposites were characterized using the modified Mori–Tanaka micromechanical model. The experimental results indicated that the nanocomposites with aligned graphene exhibited higher tensile moduli and tensile strength than those with randomly oriented graphene. Moreover, the nanocomposites with FG had superior interfacial properties and thus superior mechanical properties compared with those with RGO.

APPLICATION OF THE MORI-TANAKA METHOD TO DESCRIBE THE RATE-DEPENDENT BEHAVIOR OF UNIDIRECTIONAL FIBROUS COMPOSITES

This paper examines the possibility of using the Mori-Tanaka micromechanical model describe the rate dependent behavior of the polymer matrix based fibrous composites. The generalized Leonov model is adopted to capture the time and rate dependent character of the selected matrix, while fibers are assumed elastic. The performance of the Mori-Tanaka method is tested against the finite element simulations carried out in the framework of first-order homogenization. For simplicity, the periodichexagonal array model is chosen to represent the microstructural arrangement of fibers in the yarn cross-section. To match the predictions provided by the two approaches a suitable modification to the original Mori-Tanaka method is proposed. An extensive parametric study is presented to illustrate a considerable improvement of the predictive capability of the modified Mori-Tanaka method.

Evaluating the effective creep properties of graphene-reinforced polymer nanocomposites by a homogenization approach

The creep modulus of graphene nanoplatelet (GNP)-reinforced epoxy nanocomposites is estimated using a homogenization approach based on the Mori-Tanaka micromechanical model. Formation of the agglomerated GNPs and existence of the interphase region between graphene nanoparticles and polymer matrix are considered in the micromechanical analysis. The predictions from micromechanical simulations show good agreement with the experimental data existing in the open literature. At high graphene content, the local agglomeration of GNPs and interfacial region must be considered to obtain acceptable estimations. Uniform dispersion of graphene nanoparticles into the epoxy nanocomposites is found to significantly enhance the creep modulus. At time 30 min, the addition of 0.1 vol% GNPs can increase the nanocomposite creep modulus about 31.6%. However, the GNP agglomeration presents worse efficiencies in improving the nanocomposite creep properties. The graphene/epoxy nanocomposites with thicker and stiffer interphases show higher creep moduli. Furthermore, the epoxy nanocomposites reinforced by the aligned GNPs exhibit much better creep properties than the randomly dispersed-reinforced epoxy nanocomposites. When the graphene volume fraction is 1%, the 30 min creep modulus of nanocomposite reinforced with aligned GNPs can be increased about 17% as compared to the randomly dispersed GNP-filled nanocomposite.

Effects of nano-sized silica particles on the off-axis creep performance of unidirectional fiber-reinforced polymer hybrid composites

A multi-step homogenization approach is presented to predict the off-axis creep response of hybrid polymer matrix composites (HPMCs) reinforced with unidirectional carbon fibers and silica nanoparticles. The first step deals with evaluating the viscoelastic properties of silica nanoparticle-polymer nanocomposite using the Mori-Tanaka micromechanical model. Two essential features affecting the behavior, including silica nanoparticle agglomeration and interphase region generated due to the interaction between the nanoparticle and polymer are taken into account. In the second step, the off-axis viscoelastic behavior of HPMCs is extracted from the homogenized nanocomposite and carbon fiber properties using a unit cell-based micromechanical model. Some comparative studies between the predictions and available experiment are directed to verify the homogenization process. All the model predictions are in good agreement with the experimental data. The results indicate that with increasing the fiber off-axis angle from 0° to 90°, the presence of silica nanoparticles leads to a reduction in the HPMC creep compliance. Also, the proposed multi-step homogenization approach is applied to investigate the effects of volume fraction, size and agglomeration degree of silica nanoparticles; thickness and material properties of the interphase region; and off-axis angle and volume fraction of the carbon fiber on the HPMC creep response.

Micromechanics-based constitutive modeling of magnetostrictive 1–3 and 0–3 composites

This study presents micromechanics models for the analysis of magneto-thermo-mechanical coupled hysteresis responses for magnetostrictive composites, comprising of an elastic polymer reinforced by magnetostrictive continuous fibers and particles, respectively, subjected to a cyclic magnetic field loading under various circumstances. Constitutive relation for the magnetostrictive reinforcements that can describe nonlinear strain and magnetic flux density in a variety of thermal environments is employed, while the polymeric matrix is assumed to be a linear elastic material. A simplified unit-cell micromechanics model is developed for the magnetostrictive composites whose microstructures are idealized with periodically distributed arrays of cubic representative unit cells. To obtain the overall nonlinear responses of the magnetostrictive composites, linearized micromechanical relations are first used to provide trial solutions followed by iterative schemes in order to correct errors from linearizing the nonlinear responses. The micromechanical estimations are validated by experimental data available in literature. For comparison, the Mori-Tanaka micromechanics model is further reformulated for the magnetostrictive composites. The presented micromechanical formulations can reveal the effects of constituent volume fraction, applied prestress, temperature, the geometry of the magnetostrictive reinforcement, and the direction of applied magnetic field on the overall performance of the magnetostrictive composites under periodic magnetic field loadings.

The effect of nanoparticle conglomeration on the overall conductivity of nanocomposites

This study provides a multi-step analytical model to investigate the synergistic effects of carbon black nanoparticles and micro-scale carbon fibers on the electrical conductivity of the polymer matrix multi-scale nanocomposites. In the first step, the homogenized electrical conductivity of the nanoparticle-polymer nanocomposites containing the nanoparticle agglomeration is calculated using the percolation like electrical network model. The second step presents the details of Mori-Tanaka micromechanical model to predict the effective electrical conductivity of the multi-scale nanocomposites composed of the micro-fibers embedded in the nanoparticle-enriched polymer as the matrix phase. The influences of different parameters such as the nanoparticle agglomerate diameter to tunneling distance ratio, fiber aspect ratio, intrinsic electrical conductivity, volume fraction, and polymer potential barrier height on the electrical resistivity of the multi-scale nanocomposites are investigated. Based on the comparative studies, the model predictions are in good agreement with the available experimental results. It is found that the electrical resistivity decreases with the addition of multi-scale fillers. The results also suggest that the tunneling distance between the agglomerates plays a dominant role in the electrical conductivity of the multi-scale nanocomposites.

Vibration of smart laminated carbon nanotube-reinforced composite cylindrical panels on elastic foundations in hygrothermal environments

This paper studies vibration characteristics of smart laminated carbon nanotube-reinforced composite cylindrical panels resting on elastic foundations affected by hygrothermal environmental conditions. Effective material properties are estimated by the Mori-Tanaka micromechanics model. The motion equations for the vibration problem of the laminated composite panel are derived from the lamination theory and the first order shear deformation theory and take into consideration of the effects of elastic foundations, piezoelectricity, and temperature and moisture variations. Natural frequencies of panel vibration under various mechanical boundary conditions are computed using wave propagation approach. Parametric studies with the effects of boundary conditions, elastic foundations, panel geometry, carbon nanotube reinforcement, piezoelectricity, and hygrothermal conditions on frequencies are presented. Adding more ending supports, elastic foundations, and increase of nanotube volume fraction have increasing influence on the frequencies, while increase of panel length and decrease of its thickness, bonding piezoelectric materials, and increase of temperature and moisture lead to the decrease of frequencies.

Numerical investigation on normal and oblique ballistic impact behavior of functionally graded plates

This paper presents a numerical model to determine the ballistic performance of functionally graded plates under normal and oblique impact. Functionally graded plates consist of metal (Al 6061) and ceramic (SiC) components. The volume fraction of the two components varying throughout the thickness of the functionally graded plates were tailored according to a power-law. The Mori-Tanaka micro-mechanics model was used for determining the local material properties in the graded region of the plates. In the ballistic analyses, the deformation and damage conditions of the functionally graded plates having metal-rich (n = 0.1), linear (n = 1.0) and ceramic-rich (n = 10.0) material compositions were investigated at five different projectile impact angles such as 0°, 15°, 30°, 45°, 60°. In addition, ballistic limit velocity, which is an important parameter in armor design, has been determined at different projectile impact angles of functionally graded plates. It was found that the material composition gradient as well as projectile impact angle played a significant role on the ballistic performance of the plates.

Effect of graphene nanosheet dispersion on diffusion-induced stresses in layered sn-based nanocomposite electrode for lithium-ion batteries

An analytical method is proposed to evaluate the diffusion-induced stresses (DISs) in a layered electrode consisting of a current collector and two graphene nanosheet (GNS)-reinforced nanocomposite active plates for lithium-ion batteries. The main focus is placed on investigating the dispersion effect of GNSs within the Tin (Sn)-based nanocomposite active plates on the DISs of the layered electrode. Three types of GNS dispersion, including aligned, randomly distributed, and agglomerated state are considered in the analysis. The effective material properties of the Sn-based nanocomposites reinforced by different GNS volume fractions are predicted using the Mori-Tanaka micromechanical model. It is found that the DISs in the nanocomposite electrodes are very sensitive to the GNS dispersion type. Aligning the GNSs within the Sn-based nanocomposite active plates can reduce the peak stresses in both current collector and active plate. So, from the mechanical viewpoint of designing an electrode, alignment of GNSs within the nanocomposite active plates is an optimized condition. However, agglomeration of GNSs may increase the stress in the whole electrode. Also, the effects of amount and dispersion type of GNSs as well as the thickness ratio of current collector to active plate on the DISs and the curvature of the bilayer Sn-based nanocomposite electrode for the lithium-ion batteries are extensively discussed. Addition and alignment of the GNSs within the Sn nanocomposite active plate can significantly decrease the peak curvature of the bilayer electrode.

A novel model to predict the stiffness and strength of unidirectional glass/epoxy composites at different strain rates

In the present research, a novel rate-dependent micromechanical model was presented to predict the stiffness and strength of unidirectional glass/epoxy composites. To predict the strain-rate dependent stress–strain behavior of glass fibers, using the Maxwell model with the aid of semi-empirical relations, a new viscoelastic constitutive model was proposed. Moreover, to predict the strain-rate dependent ultimate strength of brittle glass fibers, the elasto-plastic strain-rate dependent Cowper-Symonds material model was simplified by deleting the plastic terms. The strain-rate dependent mechanical properties of the polymer were also investigated by using the modified Goldberg model. Then, by modifying the Mori-Tanaka micromechanical model, a rate dependent micromechanical model was developed to predict the effective elastic properties (stiffness and strength) of unidirectional fibrous composites at arbitrary strain rates. The present model was called the strain-rate dependent Mori-Tanaka micromechanical model. As inputs, the present model just needs the viscoelastic and viscoplastic properties of fibers and polymer. Therefore, the present model reduces the necessary experimental data to predict the rate-dependent mechanical properties of unidirectional composites. For verification of the present model, the results were compared with experimental data and very good consistency in predicting the rate-dependent behavior of unidirectional composites was observed.

Effects of nanotube agglomeration on wave dynamics of carbon nanotube-reinforced piezocomposite cylindrical shells

Dynamics of wave propagation in carbon nanotube (CNT)-reinforced piezocomposite cylindrical shells affected by nanotube agglomeration is investigated in this study for the first time by developing an analytical approach incorporating existing theories and models. The Mori-Tanaka micromechanics model in combination of the first-order shear deformation shell theory and wave propagation solution are employed to determine wave propagation characteristics of piezocomposite cylindrical shells reinforced with agglomerated CNTs. The effects of both partial and complete nanotube agglomerations on the effective elastic properties and wave dynamics are examined within various axial and circumferential wave numbers for different wave modes by solving an eigenvalue problem. It is found that nanotube agglomeration leads to the reduction of wave phase velocity as a result of decrease in the effective elastic properties. The developed methodology in this study can be used for analysis of the data of structural health monitoring by the non-destructive testing (NDT) in estimating the degree of nanotube agglomeration in nanocomposites.

Free vibration analysis of smart laminated carbon nanotube-reinforced composite cylindrical shells with various boundary conditions in hygrothermal environments

Abstract Free vibration analysis is carried out for piezoelectric coupled carbon nanotube (CNT)-reinforced composite cylindrical shells with the influences of various boundary conditions and hygrothermal environmental conditions for the first time. A simple and effective non-iterative mathematical method is used to calculate the natural frequencies. The equilibrium equations of motion are obtained based on the first-order shear deformation shell theory with the coupling effects of piezoelectricity, temperature, and moisture, respectively based on the Maxwell equation, the Fourier heat conduction equation, and the Fickian moisture diffusion equation. The Mori-Tanaka micromechanics model is used to estimate the resulting material properties for a composite shell reinforced with CNTs. Presented methodology and attained results are validated with the existing results in the literature. The effects of the boundary conditions, lamination stacking sequence, volume fraction and orientation of CNTs, piezoelectricity, and geometry of the shell on the natural frequencies of the shell are investigated. A moderate effect of temperature/moisture variation on the natural frequencies is also observed. It is found that the influence of structural boundary conditions is more significant at higher CNT volume fractions and for thicker and shorter shells, and the piezoelectricity effect is more obvious at higher circumferential mode. The model and results presented in this study can be utilized to determine vibration characteristics of smart CNT-reinforced composites subjected to hygrothermal loading as well as mechanical loading.