Effect of micro-randomness on macroscopic properties and fracture of laminates

Composite materials demonstrate a considerable extent of heterogeneity. A non-uniform spatial distribution of reinforcement results in variations of local properties of fibrous laminates. This non-uniformity not only affects effective properties of composite materials but is also a crucial factor in initiation and development of damage and fracture processes that are also spatially non-uniform. Such randomness in microstructure and in failure evolution is responsible for non-uniform distributions of stresses in composite specimens even under externally uniform loading, resulting, for instance, in a random distribution of matrix cracks in cross-ply laminates. The paper deals with statistical features of a distribution of carbon fibres in a transversal cross-sectional area in a unidirectional composite with epoxy matrix, based on various approaches used to quantify its microscopic randomness. A random character of the fibres’ distribution results in fluctuations of local elastic moduli in composites, the bounds of which depend on the characteristic length scale. A lattice model to study damage and fracture evolution in laminates, linking randomness of microstructure with macroscopic properties, is discussed. An example of simulations of matrix cracking in a carbon fibre/epoxy cross-ply laminate is given.     

Analysis of bone as a composite material

Bone is considered to be a composite material consisting of a high elastic modulus mineral ‘fibres’ embedded in a low elastic modulus organic matrix permeated with pores filled with liquids.Properties and distribution of phases are established or estimated from the experimental data. Theories of composite materials are applied to bone with various contents of phases, and the discussion summarizes possible modes of deformation under load. Elastic behaviour is attributed to the joint properties of collagen and hydroxyapatite. Plastic deformation may occur in the amorphous portions of both mineral and organic phases. Visco-elastic characteristics are credited to the flow of liquids and viscous deformation of gels and sols.     

Finite element modelling of thermally bonded bicomponent fibre nonwovens: Tensile behaviour

Nonwovens are polymer-based engineered textiles with a random microstructure and hence require a numerical model to predict their mechanical performance. This paper focuses on finite element (FE) modelling the elastic–plastic mechanical response of polymer-based core/sheath type thermally bonded bicomponent fibre nonwoven materials. The nonwoven fabric is treated as an assembly of two regions having distinct mechanical properties: fibre matrix and bond points. The fibre matrix is composed of randomly oriented core/sheath type fibres acting as load-transfer link between bond points. Random orientation of individual fibres is introduced into the model in terms of the orientation distribution function (ODF) in order to determine the material’s anisotropy. The ODF is obtained by analysing the data acquired with scanning electron microscopy (SEM) and X-ray micro computed tomography (CT). On the other hand, bond points are treated as a deformable bicomponent composite material composed of the sheath material as matrix and the core material as fibres having random orientations. An algorithm is developed to calculate the anisotropic material properties of these regions based on properties of fibres and manufacturing parameters such as the planar density, core/sheath ratio and fibre diameter. Having distinct anisotropic mechanical properties for two regions, the fabric is modelled with shell elements with thicknesses identical to those of the bond points and fibre matrix. Finally, nonwoven specimens are subjected to tensile tests along different loading directions with respect to the machine direction of the fabric. The force–displacement curves obtained in these tests are compared with the results of FE simulations.     

Probabilistic aspects of strength of short-fiber composites with planar fibre distribution

The study of the fracture of short-fibre composites must involve statistics as an integral part. Two components of composite strength, each with probabilistic aspects, are described in this paper: fibre crossover reinforcement, and fibre gap bridging before fracture. The fibre crossover density is proposed as a measure of mutual fibre strengthening. and simulations are performed to estimate this density. Several different fibre orientations are proposed which have identical elastic properties but different crossover densities, indicating that more information is required for strength prediction than for elastic property prediction. The crossover density is a random variable whose average increases roughly as a fibre length squared function, and whose coefficient of variation decreases with increasing fibre length. The phenomenon of fibres bridging microcrocks is also examined as a fracture mechanism for fibres whose length well exceeds their critical length. General probabilistic expressions are derived which give the distribution of the number of fibres bridging a gap perpendicular to the applied load. These formulae are applied to the distribution of strength of an aligned fibre system.     

On the bulk thermal properties of composite materials: Slender nearly-perfect conductors at dilute concentrations

The slender body theory developed by Russel and Acrivos[7] in their study of bulk elastic properties of composite materials is extended to cover the bulk thermal properties. The material considered consists of slender fibres embedded in a matrix and is assumed to be statistically homogeneous. Each phase is characterized by a scalar conductivity constant and the volume concentration of the fibres is assumed small enough so that interaction effects can be neglected. The increase in the thermal conductivity along the direction of the fibres is found to be of the order $\text{φ}{}_{\mathrm{f}}\text{K}{}^2\text{ln2/K}$, where $\text{φ}{}_{\mathrm{f}}$ is the volume fraction of the fibre and $\text{K}$ is the slenderness ratio of the inclusion. The cases of spatial random and planar random distribution of fibres are also mentioned.     

Modelling of the Mechanical Properties of Composite Materials at High Temperatures: Part 1. Matrix and Fibers

This paper is devoted to modelling the thermomechanical behaviour of charring composite materials at high temperatures. A multi-level model with an internal structure of unidirectional composite materials consisting of four structural levels is developed.With the help of this model, structural constitutive relations and expressions connecting elastic modules and strength characteristics of charring matrix and fibres with the properties of their internal phases are derived. Comparison of the model with experimental data for different types of matrices and fibres is conducted. Specific phenomena of the high-temperature behaviour of charring composites at high temperatures are analyzed.     

A numerical approach for determining the effective elastic symmetries of particulate-polymer composites

In this paper we deal with the problem of determining on the one hand the effective elastic properties of particulate-polymer composite materials and on the other hand the actual degree of symmetry of the resulting homogenised material. This twofold purpose has been accomplished by building a 2D as well as a 3D finite element model of the heterogeneous material and by using the strain-energy based numerical homogenisation technique. Both finite element models are able to reproduce with a good level of accuracy the real microstructure of the composite material by considering a random distribution of both particles and air bubbles (that are generated by the fabrication process). To assess the effectiveness of the proposed models, we present a numerical study to determine the effective elastic properties of the composite along with a comparison with the existing analytical and experimental results taken from literature and a sensitivity analysis in terms of the spatial distribution of the particles of the unit cell. Numerical results show that both models are able to provide the equivalent elastic properties with a very good level of accuracy when compared to experimental results and that the particulate-reinforced polymer composite could show, depending on the particles volume fraction and arrangement, an isotropic or a cubic elastic symmetry.     

Measuring the elastic properties of high-modulus fibres

The measurement of the elastic constants of several highly oriented thermoplastic polymer fibres is described. The method makes use of the hot-compaction process, developed and patented in this laboratory, which enables a solid section of highly oriented polymer to be produced from an aggregate of highly oriented fibres. As only a small fraction of the original fibre is melted and recrystallized during the process, the compacted materials offer a unique opportunity for measuring fibre properties in the bulk. An ultrasonic immersion technique is used to measure the elastic properties of the compacted materials, from which the properties of the polymer fibres are inferred. The experimentally determined fibre elastic properties have been compared with other oriented polymer materials to assess any similarities in elastic anisotropy between different methods for producing fibre orientation, and compared with theoretical upper limits for the fibre elastic properties based on theoretical estimates for the polymer crystal unit cell appropriately averaged for hexagonal symmetry using the aggregate model.     

防热复合材料高温力学性能

通过对高温环境下防热材料内部热化学烧蚀机理的分析,利用Eshelby等效夹杂方法研究了组元材料烧蚀-相变特性和高温力学性能的变化规律。假设材料热化学反应后的热解(热氧化)生成相介质统计均匀分布,考虑了烧蚀反应产生的气孔与固体相介质之间的相互作用,预报了单向纤维增强复合材料微结构与宏观性能之间的变化关系,并进行了数值计算。研究结果表明:单向复合材料纵向杨氏模量随温度升高而衰减,并与加温速率有关,典型热防护材料的高温力学性能的理论预报与实验数据进行比较,结果吻合较好,说明理论模型正确,为防热复合材料热结构分析奠定了基础。      

Elastic property prediction by finite element analysis with random distribution of materials for tungsten/silver composite

In the present numerical study, we introduce a finite element analysis for heterogeneous materials via a random distribution of materials to predict effective elastic properties. With this random distributing strategy, a large scale parametric analysis via finite element becomes feasible for the multi-phase heterogeneous solids. Taking a well-documented tungsten–silver bi-continuous material as an example, the numerical prediction provided here for the effective properties is checked by experimental testing data available in open publication. Discussions on the present finite element prediction and other approaches are also made by comparing with Hashin and Shtrikman (J Mech Phys Solids 11:127–140, 1963) bounds in the composite mechanics.     

Finite element simulation of low-density thermally bonded nonwoven materials: Effects of orientation distribution function and arrangement of bond points

A random and discontinuous microstructure is one of the most characteristic features of a low-density thermally bonded nonwoven material, and it affects their mechanical properties significantly. To understand their effect of microstructure on the overall mechanical properties of the nonwoven material, discontinuous models are developed incorporating random discontinuous structures representing microstructures of a real nonwoven material. Experimentally measured elastic material properties of polypropylene fibres are introduced into the models to simulate the tensile behaviour of the material for its both principle directions: machine direction and cross direction. Additionally, varying arrangements of bond points and schemes of fibres’ orientation distribution are implemented in the models to analyse the respective effects.     

Order and disorder in heterogeneous material microstructure: Electric and elastic characterisation of dispersions of pseudo-oriented spheroids

The paper deals with the electrical and elastic characterisation of dispersions of pseudo-oriented ellipsoids of rotation: it means that we are dealing with mixtures of inclusions of different eccentricities and arbitrary non-random orientational distributions. The analysis ranges from parallel spheroidal inclusions to completely random oriented inclusions. A unified theory covers all the orientational distributions between the random and the parallel ones. The electrical and micro-mechanical averaging inside the composite material is carried out by means of explicit results which allows us to obtain closed-form expressions for the macroscopic or equivalent dielectric constants or elastic moduli of the overall composite materials. In particular, this study allows us to affirm that the electrical behaviour of such a dispersion of pseudo-oriented particles is completely defined by one order parameter which depends on the given angular distribution. Moreover, the elastic characterisation of this heterogeneous material depends on two order parameters, which derive from the orientational distribution. The theory may be applied to characterise media with different shapes of the inclusions (i.e. spheres, cylinders or planar inhomogeneities) yielding a set of procedures describing several composite materials of great technological interest.     

Damage of short-fibre reinforced materials with anisotropy induced by complex fibres orientations

In this paper, a behaviour model for damageable elastoplastic materials reinforced with short fibres that have complex orientations is proposed. The composite material is seen as the assembly of the matrix medium and several linear elastic fibre media. Its macroscopic behaviour is computed thanks to an additive decomposition of the state potential, with no need to implement complex methods of homogenisation. A 4th-order tensor that depends on the characteristics of each fibre medium is introduced to model the anisotropic damage of the matrix material induced by the reinforcement, as well as the progressive degradation of the fibre–matrix interface. The division of short fibres into several families means that complex distributions of orientation or random orientation can be easily modelled. The model is tested for the case of a polyamide reinforced with different contents of short-glass fibres with distributed orientations and subjected to uniaxial tensile tests in different loading directions. The comparison of the results with experimental data (extracted from the literature) demonstrates the efficiency of the model.     

A novel method for the measurement of elastic moduli of fibres

The elastic modulus of fibres used in composite materials is a parameter of prime importance in the determination of overall mechanical behaviour. However, evaluation of Young’s modulus, E, of a fibre is a delicate operation given the small dimensions (diameter typically a few tens of microns), and therefore low forces involved in tensile testing. This article treats a novel method of modulus assessment involving the bending of fibres, clamped at one extremity, by forced vibrations. The fibre behaves as a beam, and when the forced oscillations approach (one of) the resonant frequency(ies) of the fibre, the bending amplitude increases. Classical beam theory allows evaluation of Young’s modulus from knowledge of resonant frequency, and fibre dimensions and density. Preliminary application of the technique using fibres of E-glass, having well known elastic characteristics, has given good results and shown its inherent potential. Subsequently, the technique developed was used on recycled fibres in order to obtain their Young’s modulus and to assess their loss of mechanical properties when compared to virgin fibres.     

The effect of mechanical defects on the strength distribution of elementary flax fibres

Flax fibres are finding non-traditional applications as reinforcement of composite materials. The mechanical properties of fibres are affected by the natural variability in plant as well as the damage accumulated during processing, and thus have considerable variability that necessitates statistical treatment of fibre characteristics. The strength distribution of elementary flax fibres has been determined at several fibre lengths by standard tensile tests, and the amount of kink bands in the fibres evaluated by optical microscopy. Strength distribution function, based on the assumption that the presence of kink bands limits fibre strength, is derived and found to provide reasonable agreement with test results.     

2D finite element analysis of thermally bonded nonwoven materials: Continuous and discontinuous models

Due to a random structure of nonwoven materials, their non-uniform local material properties and nonlinear properties of single fibres, it is difficult to develop a numerical model that adequately accounts for these features and properly describes their performance. Two different finite element (FE) models – continuous and discontinuous – are developed here to describe the tensile behaviour of nonwoven materials. A macro-level continuum finite element model is developed based on the classic composite theory by treating the fibrous network as orthotropic material. This model is used to analyse the effect of thermally bonding points on the deformational behaviour and deformation mechanisms of thermally bonded nonwoven materials at macro-scale. To describe the effects of discontinuous microstructure of the fabric and implement the properties of polypropylene fibres, a micro-level discontinuous finite element model is developed. Applicability of both models to describe various deformational features observed in experiments with a real thermally bonded nonwoven is discussed.     

A non-destructive X-ray microtomography approach for measuring fibre length in short-fibre composites

An improved method based on X-ray microtomography is developed for estimating fibre length distribution of short-fibre composite materials. In particular, a new method is proposed for correcting the biasing effects caused by the finite sample size as defined by the limited field of view of the tomographic devices. The method is first tested for computer generated fibre data and then applied in analyzing the fibre length distribution in three different types of wood fibre reinforced composite materials. The results were compared with those obtained by an independent method based on manual registration of fibres in images from a light microscope. The method can be applied in quality control and in verifying the effects of processing parameters on the fibre length and on the relevant mechanical properties of short fibre composite materials, e.g. stiffness, strength and fracture toughness.     

Fabrication of a carbon fibre/aluminium alloy composite under microgravity

Fabrication of a composite material with ultra low density and high stiffness under microgravity is the objective of the present investigation. The composite structure to be obtained is a random three-dimensional array of high modulus, short carbon fibres bounded at contact points by an aluminium alloy coated on the fibres. The material is highly porous and thus has a very low density. The motivation toward the investigation, simulation experiments, choice of the component materials and in-flight experiment during ballistic trajectory of a National Space Development Agency rocket are described herein. Supporting experimental evidence shows that the cohesion between the carbon fibre and the aluminium alloy is excellent, by which the achievement of desired properties of such composites seems probable.     

A combined experimental–numerical approach for generating statistically equivalent fibre distributions for high strength laminated composite materials

A technique is presented where actual experimental distributions, measured from a high strength carbon fibre composite, are considered in the development of a novel method to generate statistically equivalent fibre distributions for high volume fraction composites. The approach uses an adjusted measure of nearest neighbour distribution functions to define inter-fibre distances. The statistical distributions, characterising the resulting fibre arrangements, were found to be equivalent to those in the actual microstructure. Finite element models were generated and used to determine the effective elastic properties of the composite and excellent agreement was obtained. The algorithm developed is simple, robust, highly efficient and capable of reproducing actual fibre distributions for high strength laminated composite materials. It does not require further heuristic steps, such as those seen in fibre stirring/shaking algorithms, in order to achieve high volume fraction microstructures and provides a useful alternative to both microstructure reproduction and random numerical models.     

Non-Gaussian simulation of local material properties based on a moving-window technique

In this paper, a moving-window micromechanics technique, Monte Carlo simulation, and finite element analysis are used to assess the effects of microstructural randomness on the local stress response of composite materials. The randomly varying elastic properties are characterized in terms of a field of local effective elastic constitutive matrices using a moving-window technique based on a finite element model of a given digitized composite material microstructure. Once the fields are generated, estimates of the random properties are obtained for use as input to a simulation algorithm that was developed to retain spectral, correlation, and non-Gaussian probabilistic characteristics. Rapidly generated Monte Carlo simulations of the constitutive matrix fields are used in a finite element analysis to create a series of local stress fields associated with the random material sample under uniaxial tension. This series allows estimation of the statistical variability in the local stress response for the random composite. The identification of localized extreme stress deviations from those of the aggregate or effective properties approach highlight the importance of modeling the stochastic variability of the microstructure.