Modelling the properties of rubber-modified epoxy polymers

A finite-element model for rubber particles in a polymeric matrix has recently been proposed which is based upon a collection of spheres, each consisting of a sphere of rubber surrounded by an annulus of matrix. We have used this model to investigate in detail the stress distributions in and around a rubber particle, or a void, in a matrix of epoxy polymer. We have deduced the bulk modulus of the rubber-toughened epoxy and considered the implications of the stress distributions on the observed toughening micromechanisms. Of particular concern has been the effects of the volume fraction and the properties of the rubber phase.     

Heterogeneous and homogenized models for predicting the indentation response of particle reinforced metal matrix composites

In this study, three-dimensional heterogeneous and homogenized finite element models are used to predict the indentation response of particle reinforced metal matrix composites (PRMMCs). The matrix is assumed to have elasto-plastic behavior whereas the particles (uniform in size and spherical in shape) are assumed to be harder than the matrix, and possess linear elastic behavior. The particles (25 % by volume) are randomly distributed in the metal matrix. Two modeling approaches are used. In the first approach, the PRMMC is fully replaced by an equivalent homogenous material, and its material properties are obtained through homogenization using representative volume element approach under periodic boundary conditions. In second approach, a small cubical volume under the indenter is modeled as heterogeneous material with randomly distributed particles, whereas the remaining domain is assigned equivalent material properties obtained through homogenization. The elastic material properties obtained through simulations are found within Hashin–Shtrikman bounds. A suitable size cubical volume consisting of heterogeneities under the indenter is established by considering different cubical volumes so as to capture the actual indentation response. The simulations are also carried out for different particle sizes to establish a suitable particle size. These simulations show that the second modeling approach yields harder indentation response as compared to first modeling approach due to the local particle concentration under the indenter.     

Non-destructive ultrasonic evaluation of CaC03-filled polypropylene mouldings

The dynamic elastic moduli and Poisson's ratio of calcium carbonate-filled polypropylene mouldings have been determined for a continuous range of filler loadings of between 0 and 40% volume fraction. Measurements were made non-destructively on bulk samples in the form of plates 2 cm × 2 cm square and 3 mm thick, at 5 MHz. The composition dependences of the elastic properties are compared with the Hashin and Shtrikman bounds for two-phase materials. All data were in close correspondence with the lower bound, suggesting that the fill particles were well dispersed, relatively free of agglomeration, well bonded to the matrix, and that particle sizes were much less than /4, i.e. much less than 100 ym, where is the wavelength of compressional waves.     

Non-local constitutive equations for functionally graded materials

In this paper, composites with a graded distribution of heterogeneities are considered. The heterogeneities vary in statistically non-uniform fashion since in a finite layer (or region) properties such as local volume fraction vary gradually. In order to study this class of composites, a procedure of analysis which leads to the effective constitutive non-local operator of the medium is proposed. For two-phase composites, an approximation of Hashin–Shtrikman type for this operator has been obtained in real space and this has been developed explicitly in the case of laminates.     

Elastic constants and thermal expansion of aluminum-SiC metal-matrix composites

The elastic behavior and the thermal expansivity of metal-matrix composites have been investigated using ultrasonic velocity and strain gage measurements. The composites used in this study consisted of three aluminum alloys reinforced with different concentrations of SiC particles. The results show that the elastic constants increase and the coefficients of thermal expansion decrease with particle content. The results also show that the behavior of elastic constants with reinforcement can be best represented by the calculations of the upper and lower bounds of Hashin and Shtrikman. The behavior of thermal expansion, however, agrees with bounds developed by Schapery. In addition, both properties are found to be related through a model linking the strain to the elastic and thermal stresses in the composite. This relationship gives promise for the nondestructive characterization of the composites using these measurements.     

An analytical model to study the effective stiffness of the composites with periodically distributed sphere particles

In order to analytically study the overall elastic stiffness of the composite containing periodically dispersed sphere particles, a new micro-mechanics model is developed in this paper. Three kinds of typical particle packing arrangements in the form of simple cubic lattice, body-centered cubic lattice and face-centered cubic lattice are considered and compared. The special characteristics of regular distribution are fully considered by incorporating the necessary geometrical symmetry conditions into strain Green’s function. It is found that particle arrangement obviously affects the macroscopic elastic response of such the kind of composite. Moreover, most of the predictions by the present model are in good agreement with the FEM computations. The effective Young’s modulus of BCC composite the effective shear modulus of SC composite are not in the range of the Hashin–Shtrikman bounds. The present model is also useful to verify some other numerical results mainly obtained by the unit-cell model, for instance, damage variables, matrix plasticity, etc.     

Bounds on the overall elastic and instantaneous elastoplastic moduli of periodic composites

Bounds on the overall elastic and instantaneous elastoplastic moduli of composites with periodic microstructures are found using the extremum principles of Hashin and Shtrikman (1962) and the analytic solution of Nemat-Nasser et al. (1982). The bounds contain terms which depend on the geometric properties of the constituent materials and the corresponding interaction effects. Examples are presented for composites whose constituents are elastically isotropic having isotropic and kinematic plastic hardening responses.     

Toughening of polystyrene by natural rubber-based composite particles: Part III Fracture mechanisms

Impact testing has allowed the toughness of PS blends to be correlated with the morphology of the dispersed rubber phase, which was a natural rubber (NR) in particle form, coated with a shell of polystyrene (PS) or polymethylmethacrylate (PMMA). PS subinclusions were also introduced into the NR core. The impact resistance of the prepared PS blends began to rise steeply at a particle content of about 18 wt %. Transmission electron microscopy (TEM) in combination with osmium tetroxide staining techniques, allowed direct analysis of the crazing and cavitation processes in the composite natural rubber particle-toughened PS blends. Bulk samples were studied at high and slow deformation speeds. Different deformation mechanisms were effective, depending on the location of the observed stress-whitened zone relative to the notch tip. The apparent fracture mechanisms in rubber-toughened PS blends were also studied by scanning electron microscopy. PS blends containing polydisperse natural rubber-based particles or monodisperse poly(n-butylacrylate)-based particles, and commercial high-impact polystyrene, were compared. This revised version was published online in November 2006 with corrections to the Cover Date.     

Measurement and prediction of thermal properties of alkali-activated fly ash/slag binders at elevated temperatures

This paper presents a comprehensive experimental study of thermal properties of various alkali-activated binders at ambient and elevated temperatures. The binders were prepared using alkali-activated low calcium fly ash/ground granulated blast-furnace slag at ratios of 100/0, 90/10, 50/50 and 0/100 wt%. These binders can be considered as a composite of solid, water and air. Accordingly, a three-phase model is applied to predict thermal conductivity of the binders at ambient temperature. At elevated temperatures, the Hashin–Shtrikman model is used to estimate the bounds of thermal conductivity for alkali-activated binders containing of fly ash. To validate the above models, a transient plane source measurement technique was applied to measure the thermal conductivity and heat capacity at temperatures ranging from 23 to 600 °C. Data generated is then utilised to develop analytical expressions for estimating thermal properties as a function of temperature. The simplified relationships can be used for estimating the fire resistance of structural elements made from alkali-activated cementitious materials.     

Second phase volume fraction and rubber particle size determinations in rubber-toughened polymers: A simple stereological approach and its application to the case of high impact polystyrene

The determination of the volume fraction and of the particle size distributionF(R) of the rubbery phase is necessary in many rubber-toughened polymers. High impact polystyrene is a good model system in which such a determination has to be performeda posteriori. The more commonly used procedures, i.e. phase-separation methods, analysis of micrographs and indirect mechanical measurements, all present drawbacks and difficulties. A simple stereological method, in which some of these difficulties are avoided, is proposed for the determination of and of some features ofF(R) from the analysis of micrographs. The validity of the method is tested by means of a wide numerical simulation. The proposed method, together with a standard phase-separation procedure and with indirect, mechanical measurements, is tested experimentally on two series of high impact polystyrene and the collected data are compared and discussed. The results of this investigation suggest that approximate views relating elastic properties of rubber-toughened materials only to the rubber particle volume fraction, however this parameter has been measured, not considering the size, the morphology and/or the structure of the rubber particles, are possibly questionable.     

Exact second-order estimates of the self-consistent type for nonlinear composite materials

This work is concerned with the application of a general procedure for estimating the effective behavior of nonlinear composites to obtain new estimates of the self-consistent type. The procedure makes use of a linear comparison thermoelastic composite with the tangent moduli of the nonlinear phases, evaluated at appropriately chosen estimates for the average strain in the phases. Unlike the classical incremental and secant modulus self-consistent estimates for nonlinear composites, the new self-consistent estimates are found to satisfy all known bounds, including some recently established bounds of the Hashin–Shtrikman type. Also, the new self-consistent estimates are found to be exact to second order in the contrast, and thus, in agreement with recently established small-contrast asymptotic expansions for nonlinear composites. In addition, for composites with incompressible, isotropic phases and statistically isotropic microstructures, the new self-consistent estimates are found to depend on the determinant of the strain, thus giving different predictions under uniaxial and simple shear loading conditions.     

Low-temperature behaviour of epoxy-rubber particulate composites

Toughness and mechanical property data are presented for a carboxyl-terminated acrylonitrile butadiene (CTBN) rubber-modified epoxy resin in the temperature range 20 to – 110° C. A toughening model based on ultimate strain capability and tear energy dissipation of the rubber, present as dispersed microscopic particles in an epoxy matrix, is used to explain the suppression of composite toughness (G _Ic ) below – 20° C. The toughness loss is attributed to a glass transition in the rubber particles, and to a secondary transition in the epoxy resin, both occurring in the range – 40 to – 80° C. Strain-tofailure and modulus measurements on bulk rubber-epoxy compounds, formulated to simulate rubber particle compositions, confirm a decrease in rubber ductility coincident with the onset of composite toughness loss. An increase in rubber tear energy associated with its transition to a rigid state can explain the observation that even at low temperatures composite toughness generally remains significantly higher than that of pure epoxy. Although the low-temperature epoxy transition reduces molecular mobility in the matrix phase, residual ductility in, and energy dissipation by, the rubber particles determine the extent of composite toughness suppression. The low-temperature data bear out the particle stretching-tearing model for toughening.     

Rubber-modified bone cement

A precursor rubber-toughened polymethyl-methacrylate (PMMA) powder developed by ball-milling was incorporated into a series of test bone cements, with different combinations of PMMA powder, rubber-toughened PMMA powder, MMA monomer and benzoyl peroxide (BPO) initiator. The resulting microstructures were characterized by electron microscopy and measurements made of the tensile properties, fracture mechanics parameters and curing features. It is demonstrated that rubber-toughened PMMA powder additions give a significant increase in elongation and fracture toughness, with a reduction in setting time.     

Comparison of three separated flow models

 An improved stochastic separated flow (ISSF) model developed by the present authors is compared with two other widely used trajectory models, the deterministic separated flow (DSF) model and the stochastic separated flow (SSF) model, in numerical simulations of gas–particle flows behind a backward-facing step. The DSF and ISSF models are found to need only 250 computational particles to obtain a statistically stationary solution of mean and fluctuating velocities of the particles, while the SSF model requires as many as 10,000 computational particles. Apart from comparing the sensitivity of required computational particles for different models, prediction capability of different models on mean velocities, fluctuating velocities and re-circulation region are also compared in this paper. Predicted results of streamwise mean velocity of particle phase agree well with experimental data for all the three models. For the mean fluctuating velocity of the particle phase, predictions using the ISSF model agree well with experiment data, while the DSF and the SSF models have a significant difference. Only the SSF and the ISSF models are capable of predicting re-circulation regions of the particle phase. As a comparison, the ISSF model has a distinct advantage over the other two models both in terms of accuracy and efficiency. Received 20 October 2001 / Accepted 5 February 2002     

Modelling of overall plastic deformation in rubber-toughened polymers

Summary. In this paper, we provide a constitutive model for overall (macroscopic) plastic deformation behavior in a rubber-toughened polymer blend. A porous plasticity theory is employed as a basis for the constitutive modeling. In our investigation, the polycarbonate (PC) is chosen as a matrix material of polymer blend. First, the true uniaxial stress-strain relation for PC, which is an important part of the constitutive model, is carefully measured. Secondly, finite element analyses of neck propagation in a tensile specimen of PC are performed to test the efficiency of the introduction of the accurately measured true stress-strain relation into the model. Then, in order to investigate local and average deformation behavior of the matrix material (PC) around cavitated rubber particles in polymer blend, an axisymmetric unit cell analysis is carried out. Finally, finite element analyses of the neck propagation in a tensile specimen of a rubber-toughened PC are performed, and the numerical results are compared to experimental results. It is revealed that the present constitutive model has the ability to well reproduce the behavior of a rubber-toughened polymer blend with rather small volume fraction of rubber particles, which is up to about 10%. However, for blends with larger volume fraction of the rubber particles, the discrepancy between the computational and the experimental results increases. Several possibilities of enhancing the model are discussed.     

Rubber-modified polymer composites

The effect of rubber modification on the mechanical properties of a polymer composite consisting of polymethyl methacrylate (PMMA) beads embedded in a PMMA matrix was studied. The synthetic rubber used, a styrene-butadiene copolymer (SBR), was dissolved after mastication into the methyl methacrylate monomer, thus ensuring that rubber dispersion takes place in the matrix phase. The results obtained show that the mechanical properties of the rubber-modified material produced by the technique described are greatly dependent on the total rubber content, since it affects the form and particle size of the dispersed phase.     

Dynamic crack propagation behaviour of rubber-toughened poly(methyl methacrylate)

Dynamic crack propagation has been studied in detail for a series of transparent rubber-toughened samples of poly(methyl methacrylate) using a combination of high-speed photography and the optical method of transmitted caustics. The dynamic stress intensity factor has been measured as a function of rubber content, crack length and loading rate. The dynamic stress intensity factor is found to increase significantly as the rubber content increases, which is consistent with the improvement in impact behaviour found on the addition of rubber particles. It is proposed that the toughening takes place through crack tip blunting caused by localized shear yielding induced by the presence of the rubber particles.     

Elastic field and effective moduli of periodic composites with arbitrary inhomogeneity distribution

This paper develops a semi-analytic model for periodically structured composites, of which each period contains an arbitrary distribution of particles/fibers or inhomogeneities in a three-dimensional space. The inhomogeneities can be of arbitrary shape and have multiple phases. The model is developed using the Equivalent Inclusion Method in conjunction with a fast Fourier Transform algorithm and the Conjugate Gradient Method. The interactions among inhomogeneities within one computational period are fully taken into account. An accurate knowledge of the stress field of the composite is obtained by setting the computational period to contain one or more structural periods of the composite. The effective moduli of the composite are calculated from average stresses and elastic strains. The model is used to analyze the stress field and effective moduli of anisotropic composites that have cubic symmetry. It shows that the bulk and shear moduli predicted by the present model are well located within the Hashin-Shtrikman bounds. The study also shows that the stress field of the composite can be significantly affected by the distribution of inhomogeneities even though the effective moduli are not affected much.