Predictive modelling of the properties and toughness of polymeric materials Part II Effect of microstructural properties on the macroscopic response of rubber-modified polymers

The influence of microstructural properties on the macroscopic mechanical behaviour has been studied by finite element predictions of the response of different microstructures of polystyrene (PS) or polycarbonate (PC) containing voids or rubbery particles, subjected to unidirectional extension. The voids represent a low-modulus non-adhering dispersed phase. The rubbery inclusions, which are assumed to be pre-cavitated and perfectly adhering, idealise core-shell particles with a hard rubber shell and a soft non-adhering or pre-cavitated core. The predictions show that the inclusion properties strongly affect the averaged post-yield response of the heterogeneous systems. Especially the post-yield strain softening can be eliminated by the introduction of voids in PC or rubbery particles in PS. Since macroscopic strain softening is believed to be the main cause of catastrophical stress or strain localisations, the softening elimination is believed to be primarily responsible for toughness enhancement of the polystyrene or polycarbonate systems. The results and experiences are extrapolated in order to explain the influence of the absolute length scale of a sub-micron sized morphology on the macroscopic behaviour, especially toughness. Two potential sources of particle-size effects are presented that may result in a stabilised, and thus tougher, macroscopic mechanical response, i.e. the yield stress reduction near a surface or interface because of a locally enhanced mobility of the polymer segments, and the temporary excessive hardening because of a sufficiently small size of the yield zones which results in a reduced effective entanglement distance. The paper concludes with a discussion on the extension of this knowledge to all other, for the moment amorphous, polymers.     

Predictive modelling of the properties and toughness of polymeric materials Part I Why is polystyrene brittle and polycarbonate tough?

The brittleness of polystyrene (PS) and the toughness but notch sensitivity of polycarbonate (PC) have been studied by the detailed finite element analyses of the stress and strain fields in a notched tensile bar with a minor defect. The defect represented a flaw or imperfection, generated during the test specimen production. The large-strain mechanical responses of both materials were approximated by an accurate elasto-viscoplastic constitutive model with appropriate material parameters. It was assumed that failure occurs instantaneously once the dilative stress exceeds a certain critical craze-initiation stress. The analyses show that the unstable post-yield mechanical response of both materials results in localisation of stresses and strains near the defect at a very low macroscopic strain (0.16%). As a result, a strong dilative stress concentration is formed just below the surface of the defect. For the polystyrene specimen, the critical stress is reached at the defect. For the polycarbonate, however, the effect of the stress concentrating defect was counteracted by a higher craze-initiation stress and stronger strain hardening. The PC craze-initiation resistance, however, did not suffice to overcome the dilative stress concentration raised by the notch tip.     

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.     

Effective behavior of porous elastomers containing aligned spheroidal voids

The theoretical need to recognize the link between the basic microstructure of nonlinear porous materials and their macroscopic mechanical behavior is continuously rising owing to the existing engineering applications. In this regard, a semi-analytical homogenization model is proposed to establish an overall, continuum-level constitutive law for nonlinear elastic materials containing prolate/oblate spheroidal voids undergoing finite axisymmetric deformations. The microgeometry of the porous materials is taken to be voided spheroid assemblage consisting of confocally voided spheroids of all sizes having the same orientation. Following a kinematically admissible deformation field for a confocally voided spheroid, which is the basic constituent of the microstructure, we make use of an energy-averaging procedure to obtain a constitutive relation between the macroscopic nominal stress and deformation gradient. In this work, both prolate and oblate voids are considered. As a numerical example, we study macroscopic nominal stress components for a hyperelastic porous material consisting of a neo-Hookean matrix and prolate/oblate voids subjected to 3-D and plane strain dilatational loadings. In this numerical study, the relation between the relevant microstructural variables (i.e., initial porosity and void aspect ratio) for a rather large range of applied stretch is put into evidence for two types of loading. Finally, a finite element (FE) simulation is presented, and the homogenization model is assessed through comparison of its predictions with the corresponding FE results. The illustrated agreement between the results demonstrates a good accuracy of the model up to rather large deformations.     

Influence of heat treatment on the tensile properties and fracture behaviour of an aluminium alloy-ceramic particle composite

The tensile deformation and fracture behaviour of aluminium alloy 2124 reinforced with different amounts of silicon carbide particulates was studied, in the as-extruded and heat-treated conditions, with the objective of investigating the influence of heat treatment and composite microstructural effects on tensile properties and quasi-static fracture behaviour. Results indicate that for a given microstructural condition, the elastic modulus and strength of the metal-matrix composite increased with reinforcement content in the metal matrix. For a given volume fraction of reinforcement, the heat-treated composite exhibited significantly improved modulus and strength-ductility relationships over the as-extruded counterpart. The increased strength of the Al-SiC composite is attributed to the competing and synergistic influence of strengthening precipitates in the matrix metal, residual stresses generated due to intrinsic differences in thermal expansion coefficients between components of the composite and strengthening from constrained plastic flow and triaxiality in the ductile matrix due to the presence of brittle reinforcement. Fracture on a microscopic scale is initiated by cracking of the individual or clusters of SiC particles present in the microstructure. Particle cracking was dominant for the as-extruded composite microstructure. For both the as-extruded and heat-treated conditions, particle cracking increased with reinforcement content in the matrix. Final fracture of the composite resulted from crack propagation through the matrix between clusters. Although these composites exhibited limited ductility on a macroscopic scale, on a microscopic scale the fracture mechanism revealed features reminiscent of ductile failure.     

Finite Element Analysis and Experimental Measurement of Deformation Behavior for NiTi Plate With Stress Concentration Part Under Uniaxial Tensile Loading

Abstract: The aim of this study is to verify the effectiveness of ordinary phenomenological constitutive relation of NiTi shape memory alloy under mechanical loading at a constant temperature, sufficiently. First, finite element analysis is performed by using ordinary phenomenological constitutive relation for rectangular plate with double notch under tensile loading at a constant temperature. Next, uniaxial tensile loading is carried out for 50.5Ni49.5Ti rectangular plate with double notch. At the same time, macroscopic stress–strain curve and local strain distribution are measured by using in‐house measurement system on the basis of digital image correlation. As a result, it is found that the stress–strain curve obtained from finite element analysis is much different from those obtained experimental measurement, especially during stress‐induced martensite transformation. The result can be derived from the phenomena of local strain band behavior arising in NiTi under mechanical loading. The phenomenological constitutive model used in present finite element analysis is constructed under assumptions that the material has isotropic characteristics and shows homogeneous deformation. However, this experimental result suggests that the material itself has anisotropy microscopically. Furthermore, material shows unique inhomogeneous deformation. Also, there is possibility that these anisotropic characteristic and inhomogeneous deformation behaviour may derive from its microstructure. In future, to sufficiently describe the macroscopic stress–strain curve of NiTi we should take into consideration the material microstructure.     

Comparative study on local and global mechanical properties of 2024 T351, 2024 T6 and 5251 O friction stir welds

The effect of the microstructure heterogeneity on the global and local tensile properties of friction stir welded joints in 5251 (O temper) and AA2024 (T351 and T6 tempers) aluminium alloys has been investigated. Micro-tensile tests parallel to the welding direction have been carried out in the regions representative of the main microstructural zones. The digital image correlation technique (DIC) has been used during transverse tensile tests for mapping the strain distribution and to determine the local stress–strain curves. A 3-D finite elements model has been developed to predict the weld behaviour from the tensile curves of the individual regions of the weld.The tensile properties of the 5251 O weld are relatively homogeneous leading to high ductility and fracture in the base material. In contrast, the tensile properties of the various regions of the 2024 T351 and 2024 T6 welds are very heterogeneous and essentially controlled by the state of precipitation. The thermo-mechanically affected zone is the weakest region where the strain localises during a transverse tensile test. The 2024 T6 base material is stronger than the 2024 T351 alloy, leading to a more pronounced strain localisation during transverse tensile tests and a lower overall ductility. Local tensile data obtained by strain mapping are in good agreement with the curves obtained by micro-tensile tests, and these results can be safely used in a finite elements model to predict the behaviour of the overall weld assembly.     

Micromechanics of room and high temperature fracture in 6xxx Al alloys

The micromechanics of ductile fracture has made enormous progress in recent years. This approach, which was mostly developed in the context of structural integrity analysis, is becoming a key tool for materials scientists to optimize materials fracture properties and forming operations. Micromechanical models allow quantitatively linking fracture properties, microstructure features at multiple lengths scales, and manufacturing conditions. After briefly reviewing the state of the art, this paper illustrates the application of the micromechanics-based methodology by presenting the results of an investigation on the damage resistance of 6xxx Al produced by extrusion.The presence of coarse, elongated, particles is the key microstructural feature affecting the fracture behaviour of 6xxx Al. The detrimental elongated β-type particles are transformed into rounded α-type particles by heat treatment. In situ tensile tests revealed that, at ambient temperature, the α particles and the β particles oriented with the long axis perpendicular to the main loading direction undergo interface decohesion, while the β particles oriented perpendicular to the loading direction break into several fragments. At high temperatures, only interface decohesion is observed. Uniaxial tensile tests on notched and smooth round bars were performed on two different alloys, at different temperatures ranging between 20 °C and 600 °C, under different loading rates, while systematically varying the content in β versus α particles. The ductility increases with decreasing amount of β particles, increasing temperature and strain rates, and decreasing stress triaxiality.A viscoplastic extension of the Gurson model has been developed for capturing the complex hierarchy of damage mechanisms, coupled with viscoplastic and stress state effects. Three populations of voids are modelled while accounting for the different void nucleation mechanisms leading to different initial void aspect ratio. Proper modelling of the initial void aspect ratio and of its evolution with void growth was the key to predict the effect of the β → α conversion on ductility. The void coalescence criterion takes into account the presence of secondary voids resulting from particle fragmentation. The characteristics of particles entering the model were all measured experimentally. The temperature and rate dependent flow properties of the matrix material have been obtained by inverse modelling. The only fitting parameters are the critical stresses for void nucleation. The model is validated by comparing the predictions to the experimental data involving different relative proportion of α and β particles, temperature, loading rate and stress triaxiality. This type of model opens the path for an “alloy by design” strategy which relates end-use properties to upstream manufacturing operations.     

Strain and stress concentrations in ductile fracture by void nucleation growth and coalescence

Abstract

Plastic deformation in ductile metals is limited by a mechanism in which voids, nucleated at second–phase particles, grow and coalesce to form a crack. The results of a finite–element solution for a spherical elastic inclusion in a plastically deforming matrix are discussed. These results have been used in conjunction with experimental work using notched tensile specimens to generate multiaxial states of stress from which the local conditions leading to decohesion of the inclusion/matrix interface were determined. An important feature of these results is the statistical distribution of the interfacial strength. This distribution is bimodal, showing the presence of both weakly and strongly bonded particles. The latter have a modal strength of about seven times the initial yield stress and the weakly bonded particles behave as pre–existing voids. Experiments in plane and axisymmetric states of strain indicate that while the stress state is of relevance to ductile failure, the remote strain state is not. The absence of a macroscopic strain state effect is explained in terms of the statistical distribution of the voids nucleated from the population of randomly distributed inclusions. The stress and strain concentrations possible in local patches of high porosity have been investigated by a finite-element approach based on the mechanics of a dilating continuum to determine void growth in the porous aggregate and the local conditions at failure.

MST/69     

Stress relaxation of high impact polystyrene

The stress relaxation behaviour of high impact polystyrene has been correlated with the microstructural changes observed in tensile tests. The inhomogeneity of plastic deformation, manifested as stress whitening, has been measured using microhardness tests. This method has been found to be sensitive to the amount of crazing in the material. The stress relaxation behaviour changed at the onset of crazing, but did not change appreciably as the volume fraction of crazes increased. An analysis of the relaxation in terms of a site population model based on White's approach suggests the macroscopic stress relaxation is related to the crazes in the boundary regions between the stress whitened and unwhitened material.     

Fatigue damage of high performance concrete through a 2D mesoscopic lattice model

Based on high-resolution digital images of High Performance Concrete (HPC) microstructures, a two-dimensional mesoscopic lattice model which accounts for fatigue damage is proposed. Fatigue damage is introduced by considering the coupled effects of loading cycles and tensile strain on stiffness degradation of microstructural lattice elements under fatigue loading. The ultimate tensile strain is defined as the failure threshold value for microstructural lattice elements. Further, the effects of the lattice element properties (i.e. size and finite element type) and fatigue loading parameters (i.e. stress levels) on the damage mechanisms of the HPC microstructure are investigated and discussed. It is found that lattice truss elements 1 mm long are satisfactory, giving also their smaller computational requirements in comparison to beam counterparts, to investigate fatigue damage in the HPC microstructure. The numerical results of the present model are consistent with experimental observations.     

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.     

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.     

Characterisation and investigation of local variations in mechanical behaviour in cast aluminium using gradient solidification,Digital Image Correlation and finite element simulation

Due to design and process-related factors, there are local variations in the microstructure and mechanical behaviour of cast components. This work establishes a Digital Image Correlation (DIC) based method for characterisation and investigation of the effects of such local variations on the behaviour of a high pressure, die cast (HPDC) aluminium alloy. Plastic behaviour is studied using gradient solidified samples and characterisation models for the parameters of the Hollomon equation are developed, based on microstructural refinement. Samples with controlled microstructural variations are produced and the observed DIC strain field is compared with Finite Element Method (FEM) simulation results. The results show that the DIC based method can be applied to characterise local mechanical behaviour with high accuracy. The microstructural variations are observed to cause a redistribution of strain during tensile loading. This redistribution of strain can be predicted in the FEM simulation by incorporating local mechanical behaviour using the developed characterisation model. A homogeneous FEM simulation is unable to predict the observed behaviour. The results motivate the application of a previously proposed simulation strategy, which is able to predict and incorporate local variations in mechanical behaviour into FEM simulations already in the design process for cast components.     

A Numerical Study of the Development of Tensile Principal Stresses During Die Compaction

Although the average macroscopic stresses are compressive during die compaction, tensile stresses develop locally, and lead to fragmentation in low ductility particles. In this article, this phenomenon is analyzed using finite element discretization of an assembly of particles. The simulations show that there are two stages in die compaction, an early stage in which increased levels of tensile stresses develop in a number of particles located along discrete load transmission paths, and a second stage where the increasing homogeneity of the stress field leads to a decrease of the number of particles developing tensile principal stresses. A comparison between two scenarios with interparticle friction of w =0 and w =0.5 is presented. It is shown that: (1) high friction results in a higher number of particles under tensile stresses especially at low relative density, and (2) the peak fraction of material under high tensile stresses is double for w =0.5 compared to the frictionless case, and occurs at ~85% versus 88% for w =0.     

Modelling of Woven Fabrics with the Discrete Element Method

The mechanical behaviour of woven fabrics is dominated by the kinematics of the constituents on the microscopic scale. Their macroscopic response usually shows non-linearities which are due to the mobility of the interlaced yarns. The major deformation mechanisms of fabrics, i.e. the crimp interchange in case of biaxial tension and the trellising motion of the yarns in case of shear, reflect the dependency of the macroscopic material behaviour on the microstructural deformation mechanisms.
We present a novel modelling approach for woven fabrics which is capable to represent directly and locally the microstructure and its kinematics at yarn level. With only a small set of assumptions on the micro-scale the complex macroscopic material behaviour can be directly obtained. The proposed model uses the Discrete Element Method (DEM) for the representation of the fabric's microstructure. It is intrinsically dynamic since the equations of motion are solved numerically for every mass point through an application of an explicit finite difference technique. The model covers the full mobility of the fabric's microstructure while being efficiently enough to model macroscopic patches of the material.
With this model we can study the influence of the different material features of the micro-scale on the macroscopic material behaviour. With some further extensions accounting for coatings or embeddings, the range from pure fabrics to fabric reinforced membranes and composites can be covered. Problems related to large deformations and localization as well as damage can be addressed with the presented modelling approach.     

Mechanical and damage analysis along a flat-rolled wire cold forming schedule

Numerical simulation is used to study patented high-C steel flat-rolled wire cold forming processes. An elasto-plastic power law, identified from mechanical tests, is used into Forge2005? finite element (FEM) package in order to describe the material behaviour during wire drawing followed by cold rolling. A through-process approach has been favoured, transferring residual wire-drawing stresses and strain into the flat-rolling preform. This mechanical analysis, associated with a triaxiality study, points to dangerous areas where fracture may initiate due to high tensile stresses. Lema?tre’s isotropic damage criterion, including crack closure effect, a -1/3 cut-off value of stress triaxiality, and tension/compression damage asymmetry, has been used and has confirmed the previous analysis. A number of non-coalesced voids nucleated on inclusions have been observed in the Scanning Electron Microscopy (SEM), especially in high-deformation zones (“blacksmith’s cross”). Their evolution has been simulated in the FEM model using spherical numerical markers, which deform into oblate or prolate ellipsoids. The deformation-induced morphological evolution of voids observed in the SEM compares well with the geometrical evolution of the markers, which suggests that the morphologies observed do not result from micro-crack propagation, but from material transport of the nucleated voids.     

Ductile rupture in thin sheets of two grades of 2024 aluminum alloy

The damage and rupture mechanisms of thin sheets of 2024 aluminum alloy (Al containing Cu, Mn, and Mg elements) are investigated. Two grades are studied: a standard alloy and a high damage tolerance alloy. The microstructure of each material is characterized to obtain the second phase volume content, the dimensions of particles and the initial void volume fraction. The largest particles consist of intermetallics. Mechanical tests are carried out on flat specimens including U-notched (with various notch radii), sharply V-notched and smooth tensile samples. Stable crack growth was studied using “Kahn samples” and pre-cracked large center-cracked tension panels M(T). The macroscopic fracture surface of the different specimens is observed using scanning electron microscopy. Smooth and moderately notched samples exhibit a slant fracture surface, which has an angle of about 45° with respect to the loading direction. With increasing notch severity, the fracture mode changes significantly. Failure initiates at the notch root in a small triangular region perpendicular to the loading direction. Outside this zone, slant fracture is observed. Microscopic observations show two failure micromechanisms. Primary voids are first initiated at intermetallic particles in both cases. In flat regions, i.e. near the notch root of severely notched samples, void growth is promoted and final rupture is caused by “internal necking” between the large cavities. In slanted regions these voids tend to coalesce rapidly according to a “void sheet mechanism” which leads to the formation of smaller secondary voids in the ligaments between the primary voids. These observations can be interpreted using finite element simulations. In particular, it is shown that crack growth occurs under plane strain conditions along the propagation direction.