Micromechanics of rubber–toughened polymers

A new micromechanical model is provided to account for the full interaction between rubber particles in toughened polymers. Three-dimensional large deformation elastic–plastic finite element analysis is carried out to obtain the local stress and strain fields and then a homogenization method is adopted to obtain the effective stress–strain relation. The dependence of the local stress and strain distributions and effective stress–strain relation on phase morphology and mechanical properties of rubber particles is examined under various transverse constraints. The profile for the effective yield surface is obtained at four different particle volume fractions. It is shown that stress triaxiality affects significantly the effective yield stress and the local stress concentrations. Rubber cavitation and matrix shear yielding are two coupled toughening mechanisms; which one occurs first depends on the properties of rubber particles and matrix and the imposed triaxiality. Rubber cavitation plays an important role in the toughening process under high tensile triaxial stresses. Axisymmetric modelling may underestimate, and two-dimensional plane-strain modelling may overestimate, the inter-particle interaction compared with three-dimensional modelling.     

Cavitation Instabilities in Plastics and Rubber-Modified Plastics

Spherical void expansion in plastics and rubber-modified plastics is investigated under radial traction conditions. The plastics are modeled as elastic-plastic pressure-sensitive materials and the rubbers are modeled as nonlinearly elastic materials. First, the growth of a spherical void in an infinite plastic matrix is investigated under remote radial traction conditions. The results show that the cavitation stress of the plastic decreases significantly as the pressure sensitivity increases. Then, the growth of a spherical void located at the center of a spherical rubber particle in an infinite plastic matrix is investigated under remote radial traction conditions. The results indicate that without any failure criteria for the rubber, the cavitation stress does not exist when the void is small and the rubber is characterized by high-order strain energy functions. However, when a failure criterion for the rubber is considered at a finite stretch ratio, the results show that the cavitation stress for the plastic with the rubber particle becomes close to that for the plastic without the rubber particle. This revised version was published online in August 2006 with corrections to the Cover Date.     

Cavitation instabilities between fibres in a metal matrix composite

Short fibre reinforced metal matrix composites (MMC) are studied here to investigate the possibility that a cavitation instability can develop in the metal matrix. The high stress levels needed for a cavitation instability may occur in metal–ceramic systems due to the constraint on plastic flow induced by bonding to the ceramics that only show elastic deformation. In an MMC the stress state in the metal matrix is highly non-uniform, varying between regions where shear stresses are dominant and regions where hydrostatic tension is strong. An Al–SiC whisker composite with a periodic pattern of transversely staggered fibres is here modelled by using an axisymmetric cell model analysis. First the critical stress level is determined for a cavitation instability in an infinite solid made of the Al matrix material. By studying composites with different distributions and aspect ratios of the fibres it is shown that regions between fibre ends may develop hydrostatic tensile stresses high enough to exceed the critical level for a cavitation instability. For cases where a void is located in such regions it is shown that unstable cavity growth develops when the void is initially much smaller than the highly stressed region of the material.     

On cavitation and macroscopic behaviour of amorphous polymer-rubber blends

Abstract

The macroscopic behaviour of rubber-modified polymethyl methacrylate (PMMA) was investigated by taking into account the microdeformation mechanisms of rubber cavitation. The dependence of the macroscopic stress–strain behaviour of matrix deformation on the cavitation of rubber particles was discussed. A phenomenological elastic-viscoplastic model was used to model the behaviour of the matrix material, while the rubber particles were modelled with the hyperelasticity theory. A two-phase composite material with a periodic arrangement of reinforcing particles of a circular unit cell section was considered. Finite-element analysis was used to determine the local stresses and strains in the two-phase composite. In order to describe the cavitation of the rubber particles, a criterion of void nucleation is implemented in the finite-element (FE) code. A comparison of the numerically predicted response with experimental result indicates that the numerical homogenisation analysis gives satisfactory prediction results.     

Three-dimensional elastoplastic finite element modelling of deformation and fracture behaviour of rubber-modified polycarbonates at different triaxiality

A periodic face-centred cuboidal cell model is provided to account for inter-particle interaction, and a particle-crack tip interaction model is developed to study the interaction between a blunting model I crack tip and the closest array of initially spherical rubber particles in an effective medium. Three-dimensional elastoplastic finite element analysis has been preformed to study the deformation and fracture behaviour of rubber-modified polycarbonates. The effective elastoplastic constitutive relation is derived by the method of homogenisation and local stress and strain distributions are obtained to explore the role of rubber cavitation in the toughening process at different stress triaxiality. 3D elastoplastic finite element results are compatible with experimental observations, that is, rubber particles can act as stress concentrators to initiate crazing or shear yielding in the matrix but they behave differently from voids at high triaxiality. Rubber cavitation plays an important role in the toughening process under high tensile triaxial stresses.     

Nylon-6/rubber blends

The stress field around a rubber particle and a cavitated particle in a nylon/rubber blend has been studied using an analytical and a finite element approach. Attention was paid to the influence of the mechanical properties of the dispersed phase and the applied stress state. The results show that the choice of the bulk modulus of the elastomer is crucial. It appeared that especially with a triaxial stress, the Von Mises stress increased strongly upon cavitation (a more than five-fold increase close to the particle) while the hydrostatic stress only increased slightly. Also, the stresses in particles in the neighbourhood of a cavity have been calculated. Stresses in particles lying in or close to the equatorial plane of the cavity were higher than stresses in the other particles. Therefore, propagation of cavitation is most likely to occur perpendicular to the applied stress.     

On cavitation and macroscopic behaviour of amorphous polymer-rubber blends

The macroscopic behaviour of rubber-modified polymethyl methacrylate (PMMA) was investigated by taking into account the microdeformation mechanisms of rubber cavitation. The dependence of the macroscopic stress–strain behaviour of matrix deformation on the cavitation of rubber particles was discussed. A phenomenological elastic-viscoplastic model was used to model the behaviour of the matrix material, while the rubber particles were modelled with the hyperelasticity theory. A two-phase composite material with a periodic arrangement of reinforcing particles of a circular unit cell section was considered. Finite-element analysis was used to determine the local stresses and strains in the two-phase composite. In order to describe the cavitation of the rubber particles, a criterion of void nucleation is implemented in the finite-element (FE) code. A comparison of the numerically predicted response with experimental result indicates that the numerical homogenisation analysis gives satisfactory prediction results.     

Cooperative cavitation in rubber-toughened polycarbonate

Particle cavitation in the stress-whitened zone ahead of a semicircular notch in polycarbonate blended with a core-shell rubber was characterized by transmission electron microscopy. Cavitation of rubber particles at five locations in the stress-whitened zone was correlated with the local stress and strain history. It was found that cavitation initiated some distance ahead of the notch when a mean stress condition was met. Initially, only a fraction of the particles cavitated and these were randomly distributed. Single cavitated particles grew into cavitated domains by cooperative cavitation of nearby particles until cavitation was arrested when shear yielding of the matrix provided an alternative mechanism for relief of strain energy. Far from the notch, where the stress state approached uniaxial tension, cavitated domains grew into linear arrays of cavitated particles. A mechanism of cooperative crazing in microlayer composites of polycarbonate and styrene/acrylonitrile copolymer was adapted to cooperative cavitation of core-shell rubber particles. It was proposed that cooperative cavitation of nearby particles occurred by impingement of a small plastic zone that formed at the equator of a cavitated particle.     

Dilatational bands in rubber-toughened polymers

A theory is advanced to explain the effects of rubber particle cavitation upon the deformation and fracture of rubber-modified plastics. The criteria for cavitation in triaxially-stressed particles are first analysed using an energy-balance approach. It is shown that the volume strain in a rubber particle, its diameter and the shear modulus of the rubber are all important in determining whether void formation occurs. The effects of rubber particle cavitation on shear yielding are then discussed in the light of earlier theories of dilatational band formation in metals. A model proposed by Berg, and later developed by Gurson, is adapted to include the effects of mean stress on yielding and applied to toughened plastics. The model predicts the formation of cavitated shear bands (dilatational bands) at angles to the tensile axis that are determined by the current effective void content of the material. Band angles are calculated on the assumption that all of the rubber particles in a band undergo cavitation and the effective void content is equal to the particle volume fraction. The results are in satisfactory agreement with observations recorded in the literature on toughened plastics. The theory accounts for observed changes in the kinetics of tensile deformation in toughened nylon following cavitation and explains the effects of particle size and rubber modulus on the brittle-tough transition temperature.     

A model for particle cavitation in rubber-toughened plastics

An energy-balance criterion for cavitation of rubber particles, which was proposed in an earlier paper [A. Lazzeri and C. B. Bucknall, J. Mater. Sci. 28 (1993) 6799], is developed by including a term for the energy stored in the matrix and released during expansion of the voids. The model relates the critical volume strain at cavitation to the radius of the rubber particle, and to the shear modulus, surface energy and failure strain of the rubber. The effects of temperature, strain rate and type of stress field upon cavitation behaviour and the resulting toughness of the two-phase polymer are discussed in terms of the model.     

基于周期性边界条件的炭黑填充橡胶复合材料力学行为的预测

在分析炭黑填充橡胶复合材料的宏观与细观特征之间联系的基础上,提出了具有随机分布形态的代表性体积单元,推导并应用了周期性细观结构的边界约束条件,建立了三维多颗粒夹杂代表性体积单元的数值模型,对炭黑填充橡胶复合材料的宏观力学行为进行了模拟仿真。研究表明,该模型通过周期性边界条件的约束保证了宏观结构变形场和应力场的协调性;计算得到的炭黑填充橡胶复合材料的弹性模量明显高于未填充橡胶材料,并随着炭黑颗粒所占体积分数的增加而增大;该模型对复合材料有效弹性模量的预测结果与实验结果吻合较好,而且比Bergstrom三维模型的预测结果更好,证实了该模型能够用于炭黑颗粒增强橡胶基复合材料有效性能的模拟分析。     

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part II: 3‐D non‐linear behaviour

In a companion paper, the effects of approximations in the flexural‐torsional stability analysis of beams was studied, and it was shown that a second‐order rotation matrix was sufficiently accurate for a flexural‐torsional stability analysis. However, the second‐order rotation matrix is not necessarily accurate in formulating finite element model for a 3‐D non‐linear analysis of thin‐walled beams of open cross‐section. The approximations in the second‐order rotation matrix may introduce ‘self‐straining’ due to superimposed rigid‐body motions, which may lead to physically incorrect predictions of the 3‐D non‐linear behaviour of beams. In a 3‐D non‐linear elastic–plastic analysis, numerical integration over the cross‐section is usually used to check the yield criterion and to calculate the stress increments, the stress resultants, the elastic–plastic stress–strain matrix and the tangent modulus matrix. A scheme of the arrangement of sampling points over the cross‐section that is not consistent with the strain distributions may lead to incorrect predictions of the 3‐D non‐linear elastic–plastic behaviour of beams. This paper investigates the effects of approximations on the 3‐D non‐linear analysis of beams. It is found that a finite element model for 3‐D non‐linear analysis based on the second‐order rotation matrix leads to over‐stiff predictions of the flexural‐torsional buckling and postbuckling response and to an overestimate of the maximum load‐carrying capacities of beams in some cases. To perform a correct 3‐D non‐linear analysis of beams, an accurate model of the rotations must be used. A scheme of the arrangement of sampling points over the cross‐section that is consistent with both the longitudinal normal and shear strain distributions is needed to predict the correct 3‐D non‐linear elastic–plastic behaviour of beams. Copyright © 2001 John Wiley & Sons, Ltd.     

Volume change and light scattering during mechanical damage in polymethylmethacrylate toughened with core-shell rubber particles

Mechanical damage was investigated in polymethylmethacrylate toughened with core-shell (hard core) rubber particles. During a tensile experiment, volume changes, light absorption, light scattering and a small strain elastic modulus were recorded. Light scattering was quantitatively related to the number of damaged particles and a fast partial unloading technique allowed determination of the non-elastic part of these changes in material properties. Experiments performed between 10^–5 and 10^–1s^{–1 and between 20 and 70 °C} showed time-temperature transitions. These appeared to be different for each property, and measurement of the activation energy for each parameter enabled microscopic damage mechanisms to be inferred. Three types of microstructural damage were observed: pure matrix plasticity at very low strain rates or high temperatures, rubber cavitation at correlated locations at medium strain rates and temperatures, and disordered cavitation, rubber tearing and matrix plasticity at high strain rates or low temperatures. The experimental mean stress triggering rubber cavitation was compared with the predicted value.     

Fracture behaviour of unmodified and rubber-modified epoxies under hydrostatic pressure

The fracture toughness and uniaxial tensile yield strengths of unmodified and CTBN-rubber-modified epoxies were measured under hydrostatic pressure. The purpose of these experiments was to learn how suppressing cavitation in rubber particles affects the deformation mechanisms and the fracture toughness of rubber-modified epoxy. It was found that the cavitation of CTBN-rubber could be suppressed at a relatively low pressure (between 30 and 38 M Pa). With cavitation suppressed, the rubber particles are unable to induce massive shearyielding in the epoxy matrix, and the fracture toughness of the rubber-modified epoxy is no higher than that of the unmodified epoxy in the pressure range studied. Unmodified epoxy shows a brittle-to-ductile transition in fracture toughness test. The reason for this transition is the postponement of the cracking process by applied pressure.Work performed while on a sabbatical leave at the University of Michigan.     

Ductility and toughenability study of epoxy resins under multiaxial stress states

The local strains in unmodified and rubber-modified epoxies under multiaxial stress states were examined. Matrix ductility was varied by using epoxide resins of different epoxide monomer molecular weights. The stress state was altered from a plane strain case to a plane stress case by varying the thickness of the test specimens. It was confirmed that, in the case of unmodified resins, the thinner specimens which experienced nearly uniaxial tensile stress exhibited much higher local strains at failure than the thicker counterparts which experienced highly triaxial tensile stress. Also, the cross-link density was reduced as monomer molecular weight increased, thus the increase in local plastic strain due to the stress state change also became greater. Furthermore, it was found that rubber modification markedly increased the plastic strain to failure, irrespective of the specimen dimensions, and that the extent of this plastic strain increased as cross-link density was lowered. These results are consistent with the concept that the cavitation of rubber particles relieves the initial multiaxial constraint in a thick specimen, induces a stress state closer to plane stress throughout the specimen, and consequently enables the matrix to deform to a larger extent. The results also show clearly that the toughenability of a matrix resin is not independent of the stress state and the matrix ductility.     

Fatigue of rubber-modified epoxies: effect of particle size and volume fraction

A change in crack-tip plastic zone/rubber particle interactions induces a transition in the fatigue crack propagation (FCP) behaviour of rubber-modified epoxy polymers. The transition occurs at a specific K level, K _T, which corresponds to the condition where the size of the plastic zone is of the order of the size of the rubber particles. At K>K _T, rubber-modified epoxies exhibit improved FCP resistance compared to the unmodified epoxy. This is because the size of the plastic zone becomes large compared to the size of the rubber particles and, consequently, rubber cavitation/shear banding and plastic void growth mechanisms become active. At K>K _T, both neat and rubber-modified epoxies exhibit similar FCP resistance because the plastic zone size is smaller than the size of the rubber particles and hence, the rubber cavitation/shear banding and plastic void growth mechanisms are not operating. As a result of these interactions, the use of smaller 0.2 m rubber particles in place of 1.5 m rubber particles results in about one order of magnitude improvement in FCP resistance of the rubber-modified system, particularly near the threshold regime. Such mechanistic understanding of FCP behaviour was employed to model the FCP behaviour of rubber-modified epoxies. It is shown that the near threshold FCP behaviour is affected by the rubber particle size and blend morphology but not by the volume fraction of the modifiers. On the other hand, the slope of the Paris-Erdogan power law depends on the volume fraction of the modifiers and not on the particle size or blend morphology.     

Physico-mechanical properties and water absorption of cement composite containing shredded rubber wastes

The main objective of this study was to investigate the potential utilisation of rubber waste in cementitious matrix, as fine aggregates, to develop lightweight construction materials. Composites containing different amounts of rubber particles, as partial replacement to cement by volume, were characterised by destructive and non-destructive testing. Five designated rubber contents varying from 10% to 50% by volume were used. The 28-days physical, mechanical and hydraulic transport properties of the cement composite were determined. Analyses included dry unit weight, elastic dynamic modulus, compressive and flexural strengths, strain capacity, and water absorption. Test results of the physico-mechanical behaviour indicated that the increase in rubber content decreases the sample unit weight with a large reduction in the strengths and elastic modulus values of the composites. Results have only shown that the introduction of rubber particles significantly increases the strain capacity of the materials. However, rubbers into cement paste enhances the toughness of the composite. Although the mechanical strengths were reduced, the composite containing 50% of rubber particles satisfies the basic requirement of lightweight construction materials and corresponds to “class II”, according to the RILEM classification. Test-results of the hydraulic transport properties revealed that the addition of rubber particles tends to restrict water propagation in the cement matrix and reduces water absorption of the composite. The decrease of the sorptivity-value is favourable to the durability of the specimen structures.     

Effect of test conditions on cavitation and failure during tensile loading of discontinuous metal matrix composites

Abstract

The effect of test temperature and reinforcement shape on cavitation is investigated for commercial purity Al–Al₂O₃ composite produced via either a powder route or squeeze infiltrationfollowed by extrusion. At room temperature, voids form at the reinforcement/matrix interface. Voids are favoured by reinforcement elongation and planar surfaces normal to the applied stress. Angularities themselves are not favoured sites for cavitation. Increasing the test temperature delays the onset of cavitation and angularities become favoured void nucleation sites. At higher temperature, necking begins to occur at much lower plastic strains, resulting in lower strains to failure, suggesting that void stability is reduced by increasing the temperature. Finite element calculations have been performed to investigate the component of the matrix stress which is responsible for the onset of cavitation in these materials. Results suggest that cavitation is stimulated by the attainment of a critical value of the hydrostatic stress at the interface, rather than of the interfacial normal stress.

MST/3029     

The importance of constraint relief caused by rubber cavitation in the toughening of epoxy

Many rubber-toughened epoxies are thought to derive the bulk of their toughness through the processes of rubber cavitation and plastic shear-yielding in the epoxy matrix. Constraint relief has been considered to be a key mechanism which allows extra plastic shear deformation to occur. The present work attempts to provide direct experimental evidence of the constraint relief effect by combining testing geometries that vary the degree of constraint with microscopic observations. The results show that the success of a rubber as a toughening agent for epoxies is closely related to its ability to cavitate. Evidence for local constraint relief is presented. Upon cavitation of the rubber, the stress state in a specimen with initial constraint is found to change to a plane stress state. The constraint relief circumvents or delays the crack initiation in the matrix, which allows more plastic deformation to occur.