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.     

Particle cavitation in rubber toughened epoxies: the role of particle size

Rubber toughened epoxies are used in a wide range of applications including adhesives when toughness is a crucial property. It is well known that the cavitation of the rubber particles is an important process to optimise the toughness of such materials. This article describes the development of a predictive model to describe the dependence of rubber particle cavitation on particle size. The model is developed using a combination of experimental observations and finite element simulations. Predictions have been obtained for both uniaxial loading conditions and the triaxial loading conditions expected ahead of a crack. The model has been extended to consider the cavitation of nano-sized ‘rubber’ particles.     

焊接接头中的裂端应力三轴性

采用弹塑性有限元方法研究了平面应变条件下裂位于焊缝中心不同强度匹配和裂纹深度焊接接头的裂端应力三轴性。结果发现,与均匀材料相比,同等载荷水平下,高匹配接头裂端应力三轴性降低,低匹配接头裂端应力三轴性升高;随裂纹变浅,应力三轴性降低,强度匹配的影响也更为显著。从焊缝和母材塑性变形与应变硬化交互作用方面进行了解释。     

Criteria for cavitation of rubber particles: Influence of plastic yielding in the matrix

The principles of the cavitation criteria for rubber particles in polymeric matrices are briefly reviewed. Although these criteria are based on a linear elastic analysis, it is shown that it is possible to extend them to take into account the elastic-plastic behaviour of the matrix. In this objective, the representative volume element of a periodic material was meshed and computations were performed using a finite element method. The results reported in this paper focus mainly on cavitation under uniaxial tension and examine the influence on the hydrostatic stress in the rubber particles of different parameters such as the volume fraction of rubber, the plastic behaviour of the matrix or the ratio of the elastic moduli. In all cases, plastic yielding in the matrix leads to saturation of the hydrostatic stress in the rubber phase. It is also shown that the history of cavitation barely influences the progression of plasticity in the matrix.     

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.     

Fracture of austenitic steel subject to a wide range of stress triaxiality ratios and crack deformation modes

The stress triaxiality ratio (hydrostatic pressure divided by von Mises equivalent stress) strongly affects the fracture behaviour of materials. Various fracture criteria take this effect into consideration in their effort to predict failure. The dependency of the fracture locus on the stress triaxiality ratio has to be investigated in order to evaluate these criteria and improve the understanding of ductile fracture.This was done by comparing the experimental results of austenitic steel specimens with a strong variation in their stress triaxiality ratios. The specimens had cracks with varying depths and crack tip deformation modes; tension, in-plane shear, and out-of-plane shear. The crack growth in fracture mechanics specimens was compared with the failure of standard testing specimens for tension, upsetting and torsion. By associating the experimental results with finite element simulations it was possible to compare the critical plastic equivalent strain and stress triaxiality ratio values at fracture. In the investigated triaxiality regime an exponential dependency of the fracture locus on the stress triaxiality ratio was found.     

Experimental study of cavity growth in ductile rupture

Composite materials made of a steel matrix and spherical alumina particles were prepared to study the growth of cavities nucleated from Al₂O₃ inclusions during deformation at room temperature. Two materials containing different volume fractions of Al₂O₃ particles, i.e. ƒ = 0.5% and 2%, were investigated. Axisymmetric notched specimens were employed to determine the effect of stress triaxiality on cavity growth rate. These specimens were calculated by finite element method. They were predeformed at room temperature and subsequently broken at − 196°C. The experimental results are in broad agreement with the theoretical results derived from Rice and Tracey model. In particular it is observed that the cavity growth is proportional to the local strain obtained from the finite element calculations. Moreover the effect of stress triaxiality intervenes exponentially as predicted by the Rice and Tracey model. However, the theoretical proportionality factor in front of the exponential term is lower than the experimental one. The reasons for this discrepancy, especially the effect of interactions between neighbouring inclusions which are not taken into account in this model, are briefly discussed.     

Theoretical and numerical modelling of creep crack growth in a carbon-manganese steel

This paper presents an analytical and numerical study of time dependent crack growth at elevated temperatures. A triaxiality dependent damage model is used to represent the multiaxial creep ductility of the material and an analytical model to predict steady state crack growth in terms of the fracture parameter C^∗, designated the NSW-MOD model, is presented. This model is an enhancement of the earlier NSW model for creep crack growth as it accounts for the dependence of stress and strain on angular position around the crack tip. Elastic-creep and elastic-plastic-creep finite element analyses are performed for a cracked compact tension specimen and the crack propagation rate in the specimen is predicted. It is found that in general the NSW-MOD model gives an accurate estimate of the crack growth rate when compared to the finite element predictions and experimental data for a carbon-manganese steel. However, crack growth rates predicted from the finite element analysis at low values of C^∗ may be higher than those predicted by either the NSW or NSW-MOD model. This enhanced level of crack growth may be associated with the non-steady state conditions experienced at the crack tip.     

Stress state and stress‐induced microstructural evolution around the crack tip of G115 steel after dwell‐fatigue crack propagation

Many components operate under creep‐fatigue loading causing an increasing need to learn the knowledge about creep‐fatigue crack growth (CFCG). The stress and microstructural evolution around the crack tip of G115 steel after CFCG were investigated. According to the finite element simulations, the variations of equivalent stress and stress triaxiality in the crack tip zone are presented. Furthermore, the evolutions of martensitic laths, dislocations, and precipitates were systematically studied through electron backscatter diffraction and transmission electron microscopy observations. The laths at the crack tip or under a larger hold time are wider than those remote from the crack tip or under a shorter dwell time. Meanwhile, the dislocation densities reduced significantly at the crack tip or under a larger hold time. The different variations of laths' width and dislocation densities resulted from the different stress triaxiality and creep strain. W‐rich Laves and Cu‐rich phases appeared during CFCG. The Laves phase coarsened rapidly because of the stress‐accelerated diffusion.     

Stress structure and deformation behavior of mode I three-dimensional crack in elastic-plastic state

The effects of load and geometry on the stress structure of a Mode I three-dimensional crack are investigated by means of finite element method. The functions of plastic deformation and stress triaxiality constraint, during the failure process, are then analyzed. It is found that three regions, namely the plane strain similar zone ZI, the plane stress similar zone ZIII and the transition layer between them ZII, exist in front of the crack tip; three-dimensional deformation behavior is different from that in two-dimensional states even in the ZI and ZIII zones. It is also revealed that the failure form and position of a Mode I three-dimensional crack will be determined by both plastic strain and stress triaxiality.     

Finite element simulation of fracture behaviour for aged duplex stainless steels

In this paper, the local approach model developed by Gurson–Tvergaard has been applied to simulate both the crack initiation and the crack growth of aged duplex stainless steel. The parameters of the Gurson–Tvergaard model have been obtained, from axisymmetric notched specimen testing, as a function of the ageing time at 400°C, the ferrite content of the steel and the stress triaxiality. After that, to simulate the fracture of CT specimens, finite element (FE) calculations have been effected in order to obtain the stress triaxiality value at each point on the process zone ahead of the crack tip of these specimens. The adequate damage parameters concerning triaxiality are determined from the ones obtained at the notched specimens, in order to be used in FE simulations of fracture behaviour. With them, the corresponding J−Δa curves have been simulated as representative of both the crack initiation and crack propagation stages, and compared with experimental results in order to validate the methodology proposed.     

修正GTN模型及其在预测硅钢冷轧边裂中的应用

考虑剪应变对微孔洞损伤演化的影响, 对GTN损伤模型的损伤演化机制进行修正, 建立了适用于不同应力三轴度水平的损伤模型. 结合隐式应力更新算法和显式有限元计算, 采用VUMAT子程序实现了修正GTN模型在有限元软件ABAQUS中的数值计算. 通过模拟纯剪切和剪切-拉伸两组试样的损伤演化和断裂行为, 验证了修正GTN模型在不同应力三轴度承载条件下的有效性. 运用修正GTN损伤模型模拟含边部缺口的带钢在轧制过程中裂纹的萌生和扩展行为, 模拟结果与实验相一致, 表明该模型可有效地用于带钢缺陷在轧制过程中扩展行为的分析和预测. 模拟和实验结果表明, 带钢边部缺口在轧制过程中, 缺口前沿和后沿均会萌生裂纹, 且后沿裂纹扩展更为明显.     

Multiscale analysis of stiffness and fracture of nanoparticle-reinforced composites using micromechanics and global–local finite element models

A combined micromechanics analysis and global–local finite element method is proposed to study the interaction of particles and matrix at the nano-scale near a crack tip. An analytical model is used to obtain the effective elastic modulus of nanoparticle-reinforced composites, then a global–local multi-scale finite element model with effective homogeneous material properties is used to study the fracture of a compact tension sample. For SiO₂ particle-reinforced epoxy composites with various volume fractions, the simulation results for effective elastic modulus, fracture toughness, and critical strain energy release rate show good agreement with previously published experimental data. It is demonstrated that the proposed parametric multi-scale model can be used to efficiently study the toughness mechanisms at both the macro and nano-scale.     

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.     

预制裂纹角度对橡胶拉伸断裂影响的有限元分析

初始裂纹的形态影响着裂纹尖端的应力场和扩展方向,进而决定着橡胶材料的使用寿命。目前人们关于预制裂纹试样拉伸断裂的研究主要集中在直裂纹,很少涉及预制裂纹角度的改变对橡胶拉伸断裂的影响。文中应用ANSYS有限元分析软件计算拉伸状态下含不同裂纹角度橡胶试样裂纹尖端的等效应力值和撕裂能的大小,判断裂纹是否扩展及扩展方向,并对橡胶试样进行拉伸验证试验测试。结果表明,在拉伸断裂过程中,裂纹尖端的应力值和撕裂能随着初始预制裂纹角度的增大而增大,裂纹尖端形状均由初始的尖点变成圆弧状;含不同裂纹角度橡胶试样的拉伸断裂形貌与裂纹预测扩展方向基本一致,验证了有限元分析的正确性。     

Effects of short cracks produced at blunted pre-crack tip on stress and strain distributions

The present work investigates problems: (1) How are the plastic strain and the stress (triaxiality) re-distributed after a short crack initiated, extended and blunted at the pre-crack tip? (2) How do the above changes put a crucial effect on the triggering of the cleavage fracture? Based on the previous observations of configuration changes and fracture surfaces of pre-crack tips, Finite element method (FEM) simulations of a short crack initiated, extended and blunted at a pre-crack tip and calculations of distributions of stress, strain and triaxiality are carried out for 3PB pre-cracked HSLA steel specimens tested at -130^°C. The results reveal that: as long as the fatigue pre-crack is only blunted, in its vicinity a region where the accumulated strain is sufficient to nucleate a crack, and a region where the stress (triaxiality) is sufficient to propagate a crack nucleus are separated by a distance. The nucleated crack cannot be propagated and the cleavage fracture cannot be triggered. While a short crack produced at the fully blunted fatigue pre-crack, the strain retains, the stress (triaxiality) is rebuilt. An initiated and significantly extended and then blunted short crack makes a tip configuration, which on one hand is much sharper than that of the fully blunted original pre-crack tip, on other hand is wide enough to spread its effects into the high stress covered region. This sharpened crack tip configuration re-builds a ‘sharper’ distribution of stress (triaxiality) and makes two regions metioned above closer. Finally the two regions overlap each other and a cleavage crack can be initiated and propagated at a distance ahead of the blunted fatigue pre-crack.     

Mechanical heterogeneity and validity of J-dominance in welded joints

Fracture criterion of the J-integral finds wide application in the integrity evaluation of welded components, but there exist some confused problems such as the dependence of the fracture toughness on the strength mis-matching and specimen geometry which need to be clarified. It is rough and unsuitable to attribute the variation of J-integral fracture parameter simply to the effect of mechanical heterogeneity. In the present paper, a two-dimensional finite element method is employed to analyze the distribution and variation of crack tip field of welded joints with different strength mis-matching in four kinds of specimen geometry, and then the validity of J-dominance in welded joints is investigated. It is found that the crack tip field of mis-matched joint is different from that of either the weld metal or base metal of which the joint is composed, but it is situated between those of weld metal and base metal. Under the plane strain, there is obvious difference in stress triaxiality for different strength mis-matched joints. The validity of J-dominance in welded joint can not be obtained by comparing whether the stress triaxiality meets that required by the HRR solution because of the existence of mechanical inhomogeneity. By ascertaining if the stress triaxiality of welded joint near the crack tip is dependent of specimen geometry, the conclusion can be arrived at: for plane stress the validity of J-dominance is valid, whilst for plane strain the validity of J-dominance is lost. Based on the above, attempt has been made to point out that the influence of mechanical heterogeneity on the fracture toughness of weldment arises from the variation of constraint intensity-crack tip stress triaxiality. Compared with the effect of mechanical heterogeneity on the stress triaxiality, the losing of validity of J-dominance in mis-matched joint under plane strain may play a more critical role in the variation of J-integral fracture parameter of weldment.     

On the numerical implementation of a shear modified GTN damage model and its application to small punch test

Gurson-type models have been widely used to predict failure during sheet metal forming process. However, a significant limitation of the original GTN model is that it is unable to capture fracture under relatively low stress triaxiality. This paper focused on the fracture prediction under this circumstance, which means shear-dominated stress state. Recently, a phenomenological modification to the Gurson model that incorporates damage accumulation under shearing has been proposed by Nahshon and Hutchinson. We further calibrated new parameters based on this model in 22MnB5 tensile process and developed the corresponding numerical implementation method. Lower stress triaxiality were realized by new-designed specimens. Subsequently, the related shear parameters were calibrated by means of reverse finite element method and the influences of new introduced parameters were also discussed. Finally, this shear modified model was utilized to model the small punch test (SPT) on 22MnB5 high strength steel. It is shown that the shear modification of GTN model is able to predict failure of sheet metal forming under wide range of stress state.     

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.     

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。