Hexagonal Representative Volume Element Research Articles

Viscoelastic damage evaluation of the axon.

In this manuscript, we have studied the microstructure of the axonal cytoskeleton and adopted a bottom-up approach to evaluate the mechanical responses of axons. The cytoskeleton of the axon includes the microtubules (MT), Tau proteins (Tau), neurofilaments (NF), and microfilaments (MF). Although most of the rigidity of the axons is due to the MT, the viscoelastic response of axons comes from the Tau. Early studies have shown that NF and MF do not provide significant elasticity to the overall response of axons. Therefore, the most critical aspect of the mechanical response of axons is the microstructural topology of how MT and Tau are connected and construct the cross-linked network. Using a scanning electron microscope (SEM), the cross-sectional view of the axons revealed that the MTs are organized in a hexagonal array and cross-linked by Tau. Therefore, we have developed a hexagonal Representative Volume Element (RVE) of the axonal microstructure with MT and Tau as fibers. The matrix of the RVE is modeled by considering a combined effect of NF and MF. A parametric study is done by varying fiber geometric and mechanical properties. The Young’s modulus and spacing of MT are varied between 1.5 and 1.9 GPa and 20–38 nm, respectively. Tau is modeled as a 3-parameter General Maxwell viscoelastic material. The failure strains for MT and Tau are taken to be 50 and 40%, respectively. A total of 4 RVEs are prepared for finite element analysis, and six loading cases are inspected to quantify the three-dimensional (3D) viscoelastic relaxation response. The volume-averaged stress and strain are then used to fit the relaxation Prony series. Next, we imposed varying strain rates (between 10/sec to 50/sec) on the RVE and analyzed the axonal failure process. We have observed that the 40% failure strain of Tau is achieved in all strain rates before the MT reaches its failure strain of 50%. The corresponding axonal failure strain and stress vary between 6 and 11% and 5–19.8 MPa, respectively. This study can be used to model macroscale axonal aggregate typical of the white matter region of the brain tissue.

An integrated macro/micro-scale approach for in situ evaluation of matrix cracking in the polymer matrix of cryogenic composite tanks

Crack propagation in the matrix of composites is one of the major factors of propellant leakage in liner-less cryogenic composite tanks. In this paper, based on multi-scale finite element analysis strategy, an integrated macro/micro-scale (IMM) approach is developed to predict the failure of composite matrix by phase-averaging stress under cryogenic condition. In this model, the failure criterion of conventional maximum principal stress is improved by updating in situ strengths that is obtained by computing the stress concentration factors in a hexagonal representative volume element (RVE) under compression or transverse tension. Compared to other failure criteria, the predictions from this IMM approach agree with experimental results and can reflect the mismatch in coefficients of thermal expansion between different constituents. In addition, the impacts of materials systems and helical layups on the matrix cracking in composite tanks are also evaluated. This cross-scale method of IMM provides a powerful numerical tool for the reliability analysis of cryogenic composite tanks in complex working environments.

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.

Shape and cell wall slenderness effects on the stiffness of wood cell aggregates in the transverse plane

The present study investigates a homogenization method in the framework of finite element method to determine the effective stiffness properties of wood cell aggregates in the transverse plane. For this purpose, square and regular hexagonal representative volume elements are chosen to mimic the wood cells. Thereafter, simulation experiments are conducted to understand how different cell shapes and cell wall slenderness, which is cell wall thickness to height ratio, affect the stiffness properties in the transverse plane. The comparison between analytical and computational homogenization results show that square cells have higher elastic moduli than the ones computed for hexagonal cells whereas shear modulus of both cell shapes have more or less the similar values. This can be explained due to the effective deformation mechanisms under different loading conditions. Thus, the present study provides an effective stiffness estimation tool and insight for wood cell aggregates.

Modeling and Analysis the Effect of Helical Carbon Nanotube Morphology on the Mechanical Properties of Nanocomposites Using Hexagonal Representative Volume Element

Carbon nanotubes (CNTs) are the ultimate reinforcing materials for the development of an entirely new class of composites. However, they have the complicated shapes and do not usually appear as straight reinforcements when introduced in polymer matrices. This decreases nanotube’s effectiveness in enhancing the matrix mechanical properties. In this paper, nanostructure having hexagonal representative volume element (RVE), theory of elasticity of anisotropic materials and finite element method (FEM) are used to investigate the effect of helical CNT morphology on effective mechanical properties of nanocomposites. CNT with different helical angles are modeled to estimate the nanocomposite mechanical properties. The results of helical nanotube models are compared with the effective mechanical properties of nanocomposites reinforced with straight nanotubes.

A constituent-behavior-motivated model for damage in fiber reinforced composites

The aim of the work is to predict the onset and the progression of damage in the continuous fiber reinforced composite laminates taking into consideration the different mechanical behavior of the constituents in the composite under different loading conditions. The stresses in the constituents are obtained using a representative volume element (RVE) approach. Finite element analysis is carried out to extract the stresses in the constituents and the overall effective properties of the composite material. To consider the realistic behavior of the matrix of the composite material, a modified Drucker–Prager failure criterion is used. Direct test data on the individual constituents are used in this approach to model the failure of each of the constituents separately using the extracted stresses in the individual constituents. Failure envelope plotted for a unidirectional fiber reinforced composite material using the proposed model clearly shows that the initiation of damage is different in different constituents of the composite material. This is more realistic compared to the popularly used macroscopic criteria such as Tsai–Wu criterion. Propagation of failure in fiber reinforced composite is captured using a simple progressive damage model. It is also found that the hexagonal RVE has more predictive capabilities than the square RVE since the average distance between the fiber may play a major role in the strength prediction of composites.

Hexagonal Representative Volume Element for Modeling and Analysis of Mechanical Properties of Carbon Nanotube Reinforced Composites

Carbon nanotubes (CNTs) possess extremely high stiffness, strength and resilience, and may provide the ultimate reinforcing materials for the development of nanocomposites. In this paper, nanostructure is modeled as a linearly elastic composite medium, which consists of a homogeneous matrix having hexagonal representative volume elements (RVEs) and homogeneous cylindrical nanotubes. Formulas to extract the effective material constants from solutions for the RVE under axial as well as lateral loading conditions are derived based on the continuum mechanics approach. Numerical examples using the FEM are presented, which demonstrate that the load carrying capacities of the CNTs in a matrix are significant. For the RVEs having long carbon nanotube, better values of stiffness in axial direction are found as compared to stiffness in the lateral direction. Also, It is found that the square RVEs tend to overestimate the effective Youngs moduli of the CNT-based composites, and the hexagonal RVEs may be the preferred models for obtaining more accurate results.

Effective properties of nodular cast-iron: A multi-scale computational approach

A numerical approach based on a computational constitutive multi-scale model is used in this work to predict the effective Young’s modulus and Poisson’s ratio of a pearlitic nodular cast iron. The Representative Volume Element (RVE) is defined based on a set of micrographs acquired from an optical device. For each micrograph, two RVE with different shape, rectangular and hexagonal, are defined. The volume fraction of graphite and metal matrix and the boundary of each object were identified on each RVE by using a procedure of image enhanced and segmentation. The set of RVE obtained was meshed with triangular finite elements. The numerical results obtained with rectangular and hexagonal RVE are compared with results obtained by means of an analytical expression. The influence of graphite fractions, aspect ratios and nodularity are investigated. The results show that graphite fraction has the largest influence on the effective modulus E. This feature is independent of the external shape of the RVE. Only one multi-scale model show a good agreement with the analytical expression in the effective modulus E. Also, the influence of the graphite volume fraction on the effective Poisson’s ratio is investigated by means of multi-scale simulation.

Effect of carbon nanotube orientation on the mechanical properties of nanocomposites

Carbon nanotubes (CNTs) have been regarded as ideal reinforcements of high-performance composites with enormous applications. In this paper, nano-structure is modeled as a linearly elastic composite medium, which consists of a homogeneous matrix having hexagonal representative volume elements (RVEs) and homogeneous cylindrical nanotubes with various inclination angles. Effects of inclined carbon nanotubes on mechanical properties are investigated for nano-composites using 3-D hexagonal representative volume element (RVE) with short and straight CNTs. The CNT is modeled as a continuum hollow cylindrical shape elastic material with different angles. The effect of the inclination of the CNT and its parameters is studied. Numerical equations are used to extract the effective material properties for the hexagonal RVE under axial as well as lateral loading conditions. The computational results indicated that elastic modulus of nano-composite is remarkably dependent on the orientation of the dispersed SWNTs. It is observed that the inclination significantly reduces the effective Young’s modulus of elasticity under an axial stretch. When compared with lateral loading case, effective reinforcement is found better in axial loading case. The effective moduli are very sensitive to the inclination and this sensitivity decreases with the increase of the waviness. In the case of short CNTs, increasing trend is observed up to a specific value of waviness index. It is also found from the simulation results that geometry of RVE does not have much significance on stiffness of nano-structures. The results obtained for straight CNTs are consistent with ERM results for hexagonal RVEs, which validate the proposed model results.

Effect of waviness on the mechanical properties of carbon nanotube based composites

Carbon nanotubes have been regarded as ideal reinforcements of high performance composites with enormous applications. In this paper, effects of wavy carbon nanotubes on mechanical properties such as elasticity and strength aspects are investigated for nanocomposites using a 3-D representative volume element with long and short wavy carbon nanotubes. The carbon nanotube is modeled as a continuum hollow cylindrical shape elastic material with some curvature in its shape. The effect of the waviness of the carbon nanotubes and its parameters are studied. Numerical equations are used to extort the effective material properties for the different geometries of representative volume elements, under axial loading for straight carbon nanotubes. It is observed that waviness significantly reduces the effective reinforcement of the nanocomposites. The effective moduli are very sensitive to the waviness and this sensitivity decreases with the increase of the waviness. In the case of short carbon nanotubes, increasing trend is observed up to a specific value of waviness index. Tensile strength analysis shows a decreasing trend of the strength against the increasing the values of waviness indices. Geometry of the RVE is also found to be affecting the strength of the carbon nanotube reinforced composites. Simulation is applied on the straight carbon nanotubes with the hexagonal representative volume element, and the results are found to be consistent with the rule of mixtures results, which validate the proposed model.

Modelling and analysis of mechanical behaviour of carbon nanotube reinforced composites

In this paper, the mechanical properties of carbon nanotube reinforced composites are evaluated based on the continuum mechanics approach using a hexagonal representative volume element (RVE) under lateral loading conditions. It is paramount in such analyses that the correct boundary conditions be imposed, such that they simulate the actual deformation within the composite. Numerical equations are used to evaluate the mechanical properties from numerical solutions for the hexagonal RVEs under the lateral load case. An extended rule of mixtures is applied to evaluate the effective axial Young’s modulus, for validation of the proposed model. For the RVEs having long carbon nanotube, better values of stiffness in axial direction are found as compared to stiffness in the lateral direction. Also, the comparative evaluation of both square and hexagonal RVEs with short carbon nanotubes is presented here. The short carbon nanotube is not found to be as effective as long carbon nanotube in reinforcing the composite.

Evaluation of the Mechanical Properties of CNT Based Composites Using Hexagonal RVE

Carbon nanotubes (CNTs) possess extremely high stiffness, strength, and resilience, and may provide ultimate reinforcing materials for the development of nanocomposites. In this paper, the effective material properties of CNT-based composites are evaluated based on the continuum mechanics using a hexagonal representative volume element (RVE). Numerical equations are used to extract the effective material properties from numerical solutions for the hexagonal RVEs under axial loading case. An extended rule of mixtures for estimating effective Young’s modulus in the axial direction of the RVE is applied. It has been observed that the addition of the CNTs in a matrix at volume fractions of only about 3.6%, the stiffness of the composite is increased by 33% for long CNT at Et/Em=10, whereas not much improvement in stiffness has been noticed in the case of short CNTS at Et/Em=10. Effectiveness of composites is evaluated in terms of various dimensions such as thickness, diameter, and length of CNT. These results suggest that short CNTs in a matrix may not be as effective as long CNTs in reinforcing a composite. The simulation results are consistent with the experimental ones reported in literature. Also, the comparative evaluation of all three types of RVEs is presented here.