Cylindrical Representative Volume Element Research Articles

Effective yield criterion for a porous single crystal accounting for void size effects

The growth of sub-grain voids in crystalline materials is affected by plastic anisotropy as well as void size effects. In this paper, an effective yield criterion is derived for a porous single crystal using homogenization theory and limit analysis. A two dimensional planar model of a single crystal containing a random distribution of cylindrical voids is assumed. The effective yield criterion is derived using plastic limit load analysis of a hollow cylindrical representative volume element, containing a concentric cylindrical void in a single crystalline matrix. A conventional strain gradient plasticity model with an embedded material length scale is assumed for the matrix, in order to account for the void size dependence of yielding. The yield loci and void growth rates predicted by the model under plane strain conditions are validated by comparison with numerical results obtained using finite elements, as well as prior analytical results for the special case of size independent matrix behavior. The model predictions for the size dependence of the yield stress as well as void growth under proportional loading conditions are discussed with reference of known results from experiments and lower scale dislocation dynamics simulations.

Investigating the elastic behavior of carbon nanocone reinforced nanocomposites

This paper deals with the evaluation of the effective mechanical properties of carbon nanocone centered composites using a 3D nanoscale representative volume element based on continuum mechanics. For extracting the effective material constants, the authors have taken the basis of theories of elasticity. The results constituting the effective Young's modulus of the composite and Poisson's ratio for different parameters stated above have been presented and validated with rule of mixtures. It can be clearly visualized from the results that the load-carrying capacities of carbon nanocones in the representative volume elements are quite significant and the same has been demonstrated with subsequent cases. Simulation-based modeling can show a considerable part in the improvement of carbon nanocone-based composites by providing results that help in appreciative of the performance of composites. Moreover, for a volume fraction of the CNC as 2.33% in a cylindrical representative volume element and a 19.2° apex angle of the cone, the stiffness of the composite can increase as many as 4.9 times of the matrix. Similarly for hexagonal and square, the increase is in terms of 4.3 and 3.01 times respectively. Cylindrical representative volume element is the best as it provides the maximum reinforcement in terms of effective Young's modulus of the composite. Carbon nanocone-based composites provide results that help in understanding the elastic behavior of composites.

Effect of interphase, curvature and agglomeration of SWCNTs on mechanical properties of polymer-based nanocomposites: Experimental and numerical investigations

In this study, the elastic modulus of single-walled carbon nanotubes (SWCNTs)/epoxy nanocomposite was studied using the 3D finite element method and compared with experimental results to investigate the effect of SWCNTs interphase, curvature, and agglomeration on the prediction of the elastic modulus. Nanocomposite specimens containing 0.1, 0.3, and 0.5 wt% SWCNTs were fabricated to obtain SWCNTs/epoxy elastic modulus. The elastic modulus increased until SWCNTs was incorporated up to 0.3 wt% and after that, the trend of increasing elastic modulus declined. TEM images showed that in higher contents of filler, there were some local SWCNTs agglomerations within the composites which caused a dropped in elastic modulus of specimens containing 0.5 wt% SWCNTs. Also, six different 3D representative volume element (RVE) of SWCNTs/epoxy including incorporated cylindrical, cylindrical with agglomeration, curved-cylindrical, cylindrical with interphase, cylindrical with interphase and agglomeration and curved-cylindrical with agglomeration SWCNTs in the epoxy matrix have been generated using Digimat-FE and their elastic modulus evaluated by Digimat-FE solver. The numerical results cleared that the simplest cylindrical RVE has the greatest discrepancy with experimental results which showed the necessity of consideration of three important parameters including SWCNTs interphase, curvature, and agglomerations. By considering SWCNTs interphase and agglomeration the difference of numerical and experimental results decreased so that in specimens containing 0.1 wt% SWCNTs the error was only 6.8%. Also, the best results obtained from RVE of curved-cylindrical with agglomeration in specimen containing 0.1 wt% SWCNTs with only 4.1% error which showed the importance of considering SWCNTs agglomeration and curvature for modeling of nanocomposites.

РОЗРАХУНКОВІ МОДЕЛІ ДЛЯ АНАЛІЗУ ЕФЕКТИВНИХ ХАРАКТЕРИСТИК ТРИВИМІРНИХ НАНОКОМПОЗИТІВ ІЗ СИСТЕМАМИ ВКЛЮЧЕНЬ

Для опису механічних властивостей тривимірних композитів та нанокомпозитів використовуються підходи, засновані на співвідношеннях механіки суцільного середовища. За припущеннями лінійної пружності розглянуті механічні властивості композиційних матеріалів та нанокомпозитів. Застосовуються умови ідеального контакту і некласичні умови Гуртіна-Мердока для опису нанорозмірного контакту між матрицею та включеннями на поверхні розділу, яка розглядається як еластична мембрана з власними пружними характеристиками та заданим поверхневим натягом. Розглянуті різні типи представницьких комірок. Для числової симуляції механічних властивостей композитних та нанокомпозитних матеріалів використовуються методи скінченних та граничних елементів. У тривимірному формулюванні здійснено порівняння результатів, отриманих при використанні методу скінченних елементів (гексагональний репрезентативний об’ємний елемент) і методу граничних елементів (циліндричний репрезентативний об’ємний елемент). Метод скінченних елементів застосовано для визначення напружено-деформованого стану різних репрезентативних об'ємних елементів тривимірних нанокомпозитів. Розглядаються матриці у вигляді куба та циліндра скінченних розмірів та неоднорідності у вигляді куль, сфер, циліндрів, волокон та трубок. Базовані на МСЕ розрахункові моделі поширено на опис ефективних пружних властивостей композитів з розподіленими нановключеннями з випадковою та впорядкованою орієнтаціями. Метод скінченних елементів застосовано для визначення НДС різних репрезентативних об'ємних елементів тривимірних нанокомпозитів. Досліджено вплив форми і відносних розмірів неоднорідностей та матриць репрезентативного об’єму на ефективні модулі пружності нанокомпозитів. Отримані моделі створюють інформативну базу для технологій синтезування нанокомпозитів з вдосконаленими деформаційними і міцнісними характеристиками.

Evaluation of elastic modulus in PE/CNT composites subjected to axial loads

In this paper, a range of matrix materials from polymer to metal matrix were considered which have extensive applications in various industries. The effective mechanical properties of CNT-based composites were evaluated using a 3-D cylindrical representative volume element (RVE). Continuum mechanics model was used for estimating the effective Young’s modulus in the axial direction of the RVE. The load transfer conditions between carbon nanotubes and matrix were modeled using a separated interfacial region. Numerical examples using FEM are presented which demonstrated that load carrying capacities of CNTs in a matrix were significant. With the addition of CNTs into a matrix at volume fractions of only about 2% and 5%, the stiffness of the composite was increase as high as 0.7 and 9.7 times for short and long CNTs, respectively. Also the effect of CNT addition to polyethylene matrix was studied. These simulation was performed using ABAQUS software and the obtained results were consistent with the experimental results reported in the literature.

Multiscale modeling of the elastic moduli of CNT-reinforced polymers and fitting of efficiency parameters for the use of the extended rule-of-mixtures

In this work, a bottom-up multiscale modeling approach is developed to estimate the effective elastic moduli of Carbon NanoTube (CNT)-reinforced polymer composites. The homogenization process comprises two successive steps, including an atomistic-based computational model and a micromechanics approach at the nano- and micro-scales, respectively. Firstly, the atomistic-based finite element model defines a cylindrical Representative Volume Element (RVE) that accounts for a carbon nanotube, the immediately surrounding matrix, and the CNT/polymer interface. The carbon-carbon bonds of the CNT are modeled using Timoshenko beams, whilst three-dimensional solid elements are used for the surrounding matrix. Through the application of four loading conditions, the RVEs are homogenized into transversely isotropic equivalent fibers by equating the associated strain energies. Secondly, the equivalent fibers are employed in a micromechanics approach to estimate the macroscopic response of non-dilute composites. This is performed using both the analytical Mori-Tanaka model and a computational RVE model with a hexagonal packing geometry. A wide spectrum of single- and multi-walled carbon nanotubes are studied, as well as two different polymeric matrices. Furthermore, the so-called efficiency parameters, imperative for the application of the simplified extended rule of mixtures, are characterized by polynomial expressions for practical filler contents. Finally, detailed parametric analyses are also provided to give insight into the sensitivity of the macroscopic response of CNT-reinforced polymer composites to microstructural features such as filler volume fraction, chirality or aspect ratio.

Multiscale modeling of regularly staggered carbon fibers embedded in nano-reinforced composites

Abstract This article deals with the multiscale modeling of stress transfer characteristics of nano-reinforced polymer composite reinforced with regularly staggered carbon fibers. The distinctive feature of construction of nano-reinforced composite is such that the microscale carbon fibers are packed in hexagonal array in the carbon nanotube reinforced polymer matrix (CNRP). We considered three different cases of CNRP, in which carbon nanotubes (CNTs) are: (i) aligned along the direction of carbon fiber, (ii) aligned radially to the axis of carbon fiber, and (iii) randomly dispersed. Accordingly, multiscale models were developed. First, molecular dynamics (MD) simulations and then Mori-Tanaka technique were used to estimate the effective elastic properties of CNRP. Second, a micromechanical three-phase shear lag model was developed considering the staggering effect of microscale fibers and the application of radial loads on the cylindrical representative volume element (RVE) of nano-reinforced composite. Our results reveal that the stress transfer characteristics of the nano-reinforced composite are significantly improved by controlling the CNT morphology, particularly, when they are randomly dispersed around the microscale fiber. The results from the developed shear lag model were also validated with the finite element shear lag simulations and found to be in good agreement.

Characterizing the strength and elasticity deviation in defective CNT reinforced composites

Abstract In this paper, the Multiscale 3D Representative Volume Element approach is proposed for modeling the elastic behavior of carbon nanotubes reinforced composites with vacancy defects. The Carbon Nanotube has been modeled using atomistic scale by Molecular Structural Mechanics. Space frame structure similar to 3D beams and point masses are employed to simulate the discrete geometrical constitution of the single walled carbon nanotube. The covalent bonds between carbon atoms found in the hexagonal lattices are assigned elastic properties using beam elements. The nodes are treated as Carbon atoms on which the point masses are applied. The vacancy defect is considered in the single wall carbon nanotubes. The matrix phase has been investigated using continuum mechanics approach at macro scale level. The carbon nanotube and matrix regions are connected by interfacial zone using beam elements. Using the proposed multi scale model, the deformations obtained from the simulations are used to predict the elastic modulus of the nanocomposite. The mechanical properties are evaluated for various Vacancy defect locations and number of defects. The influence of the vacancy defects on the chiral CNT reinforced composite is studied under an axial load condition. Numerical equations are used to extract the effective material properties for the cylindrical RVE with non-defective CNTs. The FEM results obtained for non-defective Carbon Nanotubes are consistent with analytical results for cylindrical Representative Volume Element, which validate the proposed model.

Mass transfer model of supercritical fluid degradation for carbon fiber composites

It is critical to study the mass transfer of supercritical fluid degradation for carbon fiber composites to investigate their degradation mechanism, design the reactor, and develop recycling processes. The mass transfer process of supercritical fluid degradation was described from two aspects: mass diffusion from outside to inside, and from inside to outside. Mass transfer model of supercritical fluid degradation was established based on a proposed concentric cylindrical representative volume element. The reaction kinetic parameters were incorporated into the mass transfer equation, and the concentration distribution of supercritical fluid, mass transfer rate, reaction order, and reaction rate constant during the carbon fiber composites degradation process were calculated. Relaxation time was incorporated into the mass transfer process, and the supercritical fluid concentration calculation method considering non-Fick effect was proposed. Finally, two pretreatment methods were adopted to speed up the mass transfer process.

A criterion for void coalescence in anisotropic ductile materials

Plastic anisotropy in the form of texture development and void shape evolution can significantly affect damage growth rates and overall strain to failure in ductile materials. A criterion for the onset of coalescence, which is the transition from void growth by diffuse plastic deformation to localized plasticity in the ligament connecting neighboring voids, is a critical component of any predictive model for ductile fracture. In this paper, a new micromechanics-based criterion for void coalescence, combining both forms of anisotropy above, is developed using homogenization and limit analysis of a hollow cylindrical representative volume element made of an orthotropic material of the Hill type. Two possible modes of coalescence, corresponding to necking instability and shear strain localization in the transverse inter-void ligament, are accounted for in the analysis. The final form of the coalescence criterion has an interesting symmetry with Gurson-type yield criteria for porous materials and is shown to be an improvement over existing models for the special case of isotropic matrix behavior. For validation of the analytical model, quasi-exact numerical coalescence loci are computed using a finite elements based limit analysis method for the special case of transversely isotropic materials. The analytical model is shown to be in good agreement with the numerical data, except for highly oblate void shapes approaching penny shape cracks. A heuristic modification for the model is proposed, which significantly improves the model predictions in that limit.

Evaluation of the effective mechanical properties of shape memory wires/epoxy composites using representative volume element

The mechanical properties of shape memory alloy/epoxy composites are evaluated using cylindrical representative volume element for long and short shape memory alloy wires considering the effect of interface between shape memory alloy wires and epoxy resin. The cohesive zone model is used to simulate the interface region between the shape memory alloy wire and the resin. An experimental pullout test is performed on the cylindrical specimens of resin containing single-shape memory alloy wire in order to achieve cohesive zone parameters. The finite element method is utilized to characterize the elastic properties of shape memory alloy/epoxy composites including axial modulus, transverse modulus, and Poisson’s ratio. The effect of length ratio between the shape memory alloy wire and the representative volume element on the mechanical properties of composite is investigated. Moreover, the critical aspect ratio for short shape memory alloy wires is calculated using shear-lag model.

On the study of elastic properties of CNT-reinforced composites based on element-free MLS method with nanoscale cylindrical representative volume element

This paper presents a micromechanical element-free method to study elastic properties of carbon nanotubes (CNTs) reinforced composites. A three-dimensional nanoscale cylindrical representative volume element (RVE) has been considered in the analysis, in which CNT and matrix are both modeled as elastic materials based on continuum mechanics. The proposed model is used to investigate the behaviors of CNT-reinforced composites with respect to their effective mechanical properties. The relevant formulas to extract the effective material constants for three loading cases are derived in detail, based on the elasticity theory. Besides, the meshless method based on the moving least-squares (MLS) approximation and element-free Galerkin (EFG) method is implemented to evaluate the established model. The computed results are compared with the available solutions obtained from other analytical and numerical methods. Moreover, these results show that the presence of CNTs can significantly improve the longitudinal and transverse characteristics of the composite materials.

Static Crack Propagation of Carbon Nanotube through Non-Bonded Interface of Nanocomposites

This study presents an analytical shear-lag model to illustrate the interface crack propagation of carbon nanotube (CNT) reinforced polymer-matrix composites (PMCs) using representative volume element (RVE). In the model, a 3D cylindrical RVE is picked to present the nanocomposite in which CNT/polymer chemically non-bonded interface is taken into consideration. In the non-bonded interface, the stress transfer of CNT is generally considered to be controlled by the combined contribution of mechanical interlocking, thermal residual stress, Poisson’s contraction and van der Waals (vdW) interaction. Since CNT/matrix interface becomes debonded due to crack propagation, vdW interaction which is a function of relative radial displacement of the CNT/matrix interface makes the modeling of the interface tricky and challenging. In order to solve this complexity, an iterative approach is proposed to calculate the vdW interaction for debonded CNT/matrix interface accurately. The analytical results aim to obtain the characteristics load displacement relationship in static crack propagation for CNT reinforced PMCs.

Multilevel approach for the local nanobuckling analysis of CNT-based composites

In the present paper, a multilevel approach for the local nanobuckling analysis of carbon nanotube (CNT) based composite materials is proposed and described. The approach comprises four levels, all of them at nanoscale. The first level aims to propose the potential that describes the interatomic forces between carbon atoms. In the second level, molecular dynamics simulations are performed to extract the elastic properties of the CNT. The third level aims to determine the stiffness of the material that surrounds the CNT (matrix), using the annular membrane analysis. In the fourth level, finite strip analysis of the CNT elastically restrained by the matrix is performed to calculate the critical strain at which the CNT buckles locally. In order to achieve accurate results and take the CNT-matrix interaction into account, the 3rd and 4th steps may be repeated iteratively until convergence is achieved. The proposed multilevel approach is applied to several CNTs embedded in a cylindrical representative volume element and illustrated in detail. It shows that (i) the interaction between the CNT and the matrix should be taken into account and (ii) the buckling at nanoscale is sensitive to several types of local buckling modes.

Analysis of elastic properties of carbon nanotube reinforced nanocomposites with pinhole defects

Due to their unique molecular structure, carbon nanotubes exhibit outstanding properties. They are regarded as ideal reinforcements of composites. In this paper, the effects of pinhole defects on mechanical properties are investigated for wavy carbon nanotubes based nanocomposites using 3-D Representative Volume Element with long carbon nanotubes. The carbon nanotubes are modeled as continuum hollow cylindrical shape elastic material with pinholes, having some curvature in its shape. These defects are considered on the single walled carbon nanotubes. The mechanical properties like Young’s modulus of elasticity are evaluated for various values of waviness index, as well as type and number of pinhole defects. The effects of interactions between both defects as well as their influence on the nanocomposites are studied under an axial loading condition. Numerical equations are used to extract the effective material properties for the different geometries of Representative Volume Elements with non-defective carbon nanotubes. The finite element method results obtained for non-defective carbon nanotubes are consistent with analytical results for cylindrical Representative Volume Elements, which validate the proposed model. It is observed that the presence of pinhole defects as well as waviness, can significantly reduces the effective reinforcement, when compared with nanotubes without pinhole defects and this reinforcement decreases with the increase of the number of pinhole defects.

Multiscale modeling for mechanical properties of carbon nanotube reinforced nanocomposites subjected to different types of loading

Multiscale modeling was presented for the nonlinear properties of polymer/single wall carbon nanotube (SWNT) nanocomposite under tensile, bending and torsional loading conditions. To predict the mechanical properties of both armchair and zigzag SWNTs, a finite element (FE) model based on the theory of molecular mechanics was used. For reducing the computational efforts, an equivalent cylindrical beam element was proposed, which has the unique advantage of describing the mechanical properties of SWNTs considering the nonlinearity of SWNT behavior. For a direct evaluation of the rigidities of the proposed equivalent beam, the data obtained through atomistic FE analyses of SWNT were fitted to six different equations, covering the three types of loading for both armchair and zigzag configurations. The proposed equivalent beam element was then used to build a cylindrical representative volume element (RVE) using which the effects of the interphase between SWNT and the polymer on the mechanical properties of RVE could be studied. It was found that while the interphase has a small effect on the nanocomposite stiffness, the ratio of (SWNT length)/(RVE length) dramatically affects the nanocomposite stiffness.

Effect of Pinhole Defects on the Elasticity of Carbon Nanotube Based Nanocomposites

Carbon nanotubes (CNTs) have been regarded as an ideal reinforcements of high-performance composites with enormous applications. In this paper, the effects of pinhole defect are investigated for carbon nanotube based nanocomposites using a 3D representative volume element (RVE) with long CNTs. The CNT is modeled as a continuum hollow cylindrical shape elastic material with pinholes in it. These defects are considered on the single wall (CNTs). The mechanical properties such as Young’s modulus of elasticity are evaluated for various pinhole locations and number of defects. The influence of the pinhole defects on the nanocomposite is studied under an axial load condition. Numerical equations are used to extract the effective material properties for the different geometries of RVEs with nondefective CNTs. The field-emission microscopy (FEM) results obtained for nondefective CNTs are consistent with the analytical results for cylindrical RVEs, which validate the proposed model. It is observed that the presence of pinhole defects significantly reduces the effective reinforcement when compared with nondefective nanotubes, and this reinforcement decreases with the increase in the number of pinhole defects. It is also found from the simulation results that the geometry of RVE does not have much significance on the stiffness of nanocomposites.

Hybrid Finite Element Formulation for Temperature Prediction in Carbon Nanofiber Based Composites

The temperature distribution in nanoscale cylindrical representative volume element (RVE) with centered carbon nanofiber (CNF) was modeled by the developed hybrid finite element approach, which used the axisymmetric Green’s function to approximate the interior field within the element, while the independent element boundary field with conventional shape functions was assumed to keep the continuity of field variables between adjacent elements. As a result, coupling with the constructed hybrid functional to link the independent interior and boundary fields, the solving linear system of equations with boundary integrals only were obtained. Numerical examples were given to show the accuracy of the proposed approach, and results were found in good agreement with those of ABAQUS. Also, it was obvious that the surface temperature of nanofiller almost kept constant during heat transfer, due to the large difference of thermal properties between nanofiller and matrix.

Plastic response of porous solids with pressure sensitive matrix

A practical straightforward procedure is suggested for constructing the yield surface of solids containing voids embedded in a pressure sensitive matrix. Derivation centers on expanding yield function in powers of porosity ratio f, with zero order term describing the void free matrix. The coefficients of that expansion are determined, in terms of stress invariants, from simple stress patterns (like spherical and cylindrical representative volume elements) at full yield, assuming perfectly plastic response. In this paper we concentrate on two families of matrices described by the Drucker–Prager and Schleicher pressure sensitivity. Limiting the power expansion up to second order terms we determine uniquely the two unknown expansion coefficients from spherical RVE under remote hydrostatic tension and compression. This approach does not employ kinematic fields, averaging methods or energy theorems, yet results are remarkably similar though not identical to available yield conditions. Key findings are supported by asymptotic expansions, at different levels of accuracy, for low hydrostatic pressure and low pressure sensitivity of matrix material.

Effect of Microstructure Evolution on the Overall Response of Porous-Plastic Solids

Ductile fracture is the macroscopic result of a micromechanical process consisting in void nucleation and growth to coalescence. While growing in size, voids also evolve in shape because of the non-uniform deformation field in the surrounding material; this shape evolution is either disregarded or approximately accounted for by constitutive laws for porous-plastic solids. To assess the effect of void distortion on the overall properties of a porous-plastic material prior to any coalescence-dominated event, we here present a micromechanical study in which the void-containing material is treated as a two-phase (matrix and inclusion) composite. A cylindrical representative volume element (RVE), featuring elliptic cross-section and containing a coaxial and confocal elliptic cylindrical cavity, is considered. In case of a matrix obeying flow theory of plasticity, the overall yield domain and the evolution laws for the volume fraction and aspect ratio of the void are obtained. Under assigned strain histories, these theoretical findings are then compared to finite element unit-cell simulations, in order to assess the capability of the proposed results to track microstructure evolution. The improvements with respect to the customarily adopted Gurson’s model are also discussed.