An analysis of the imperfection sensitivity of square elastic-plastic plates under axial compression

For a simply supported elastic-plastic square plate under axial compression the post-bifurcation behaviour and the sensitivity to initial imperfections are investigated. An exact asymptotic expansion is given for the initial post-bifurcation behaviour of a perfect plate compressed into the plastic range. The imperfection sensitivity is studied through an asymptotic analysis of the behaviour of the hypoelastic plate that results from neglecting the effect of elastic unloading. The results of the asymptotic analyses are compared with results of a numerical incremental solution by means of a combined finite element—Rayleigh Ritz method. The paper considers the effect of different in-plane boundary conditions and the effect of various degrees of strain hardening.     

Necking of biaxially stretched elastic-plastic circular plates

The necking of an elastic-plastic circular plate under uniform radial tensile loading is investigated both within the framework of the three-dimensional theory and within the context of the plane-stress approximation. Attention is restricted to axisymmetric deformations of the plate. The material behavior is described by two different constitutive laws. One is a finite-strain version of the simplest flow-theory of plasticity and the other is a finite-strain generalization of the simplest deformationtheory, which is employed as a simple model of a solid with a vertex on its yield surface. For an initially uniform plate made of an incompressible material, bifurcation from the uniformly stretched state is studied analytically. The regimes of stress and moduli where the governing axisymmetric three-dimensional equations are elliptic, parabolic or hyperbolic are identified. The plane-stress local-necking mode emerges as the appropriate limiting mode from the bifurcation modes available in the elliptic regime. In the elliptic regime, the main qualitative features of the bifurcation behavior are revealed by the plane-stress analysis, although three-dimensional effects delay the onset of necking somewhat. For the deformation theory employed here, the first bifurcation modes are encountered in the parabolic regime if the hardening-rate is sufficiently high. These bifurcations are not revealed by a plane-stress analysis. For a plate with an initial inhomogeneity, the growth of an imperfection is studied by a perturbation method, by a plane-stress analysis of localized necking, and by numerical computations within the framework of the three-dimensional theory. When bifurcation of the corresponding perfect plate takes place in the elliptic regime, the finite element results show that the plane-stress analysis gives reasonably good agreement with the numerical results. When bifurcation of the corresponding perfect plate first occurs in the parabolic regime, then a bifurcation of the imperfect plate is encountered, that is, the finite element stiffness matrix ceases to be positive definite.     

Ductile fracture by cavity nucleation between larger voids

A mechanism of ductile fracture involving the interaction of relatively large voids with small-scale voids is studied by a computational model. The larger voids are described as circular cylindrical holes arranged in a doubly periodic array in the initial state. In the matrix material between these voids the nucleation and growth of much smaller voids is accounted for by using approximate constitutive equations for a ductile, porous medium. The computations show bands of highly localized straining and void growth, initiating at the surfaces of larger voids and growing into the matrix material, until the bands connect two neighbouring voids. The materials are analysed both under plane strain conditions and under conditions approximating those in a round tensile bar. The failure strains obtained under different principal stress ratios show rather good agreement when plotted against a measure of the stress-triaxiality.     

An analysis of ductile rupture in notched bars

Ductile fracture in axisymmetric and plane strain notched tensile specimens is analyzed numerically, based on a set of elastic-plastic constitutive relations that account for the nucleation and growth of microvoids. Final material failure by void coalescence is incorporated into the constitutive model via the dependence of the yield function on the void volume fraction. In the analyses the material has no voids initially; but as the voids nucleate and grow, the resultant dilatancy and pressure sensitivity of the macroscopic plastic flow influence the solution significantly. Considering both a blunt notch geometry and a sharp notch geometry in the computations permits a study of the relative roles of high strain and high triaxiality on failure. Comparison is made with published experimental results for notched tensile specimens of high-strength steels. All axisymmetric specimens analyzed fail at the center of the notched section, whereas failure initiation at the surface is found in plane strain specimens with sharp notches, in agreement with the experiments. The results for different specimens are used to investigate the circumstances under which fracture initiation can be represented by a single failure locus in a plot of stress triaxiality vs effective plastic strain.     

Instability and failure of internally pressurized ductile metal cylinders

Forlong, ductile, thick-walled tubes under internal pressure instabilities and final failure modes are studied experimentally and theoretically. The test specimens are closed-end cylinders made of an aluminum alloy and of pure copper and the experiments have been carried out for a number of different initial external radius to internal radius ratios. The experiments show necking on one side of the tubes at a stage somewhat beyond the maximum internal pressure. All tubes, except for one aluminum alloy tube, failed by shear fracture under decreasing pressure. The aluminum alloy tubes exhibited localized shear deformations in the neck region prior to fracture and also occasionally surface wave instabilities. The numerical investigation is based on an elastic-plastic material model for a solid that develops a vertex on the yield surface, using representations of the uniaxial stress-strain curves found experimentally. In contrast to the simplest flow theory of plasticity this material model predicts shear band instabilities at a realistic level of strain. A rather sharp vertex is used in the material model for the aluminum alloy, while a more blunt vertex is used to characterize copper. The theoretically predicted bifurcation into a necking mode, the cross-sectional shape of the neck, and finally the initiation and growth of shear bands from the highly strained internal surface in the neck region are in good agreement with the experimental observations.     

Emulsions of Heavy Crude Oils. II. Viscous Responses and Their Influence on Emulsion Stability Measurements

The stability of 30 heavy crude oil emulsions was studied in a parallel-plate laboratory coalescer (DC field). Particularly, viscous responses and their influence on the emulsion stability measurements were investigated. In addition to highlighting previous results from the same experimental setup and discussing these based on recent experience, new results at different temperatures and volume fractions of water were presented. A new semi-empirical model for the characteristic time of the destabilization process was presented. The electrical forces were modelled with a point-dipole approximation and the hydrodynamic resistance to droplet transport was modelled with an empirical term including the logarithmic viscosity of the oil phase. The new model clearly performed much better than the previous model, particularly for very viscous crude oils. Studies of the performance of industrial electrocoalescers have showed that simple electrostatic theory can potentially explain complex separation phenomena when the resistance to the coalescence step is reduced by an efficient demulsifier. The ultimate goal is to build a model for both the laboratory setup and the industrial coalescer so that laboratory experiments can be used to predict the behavior of the industrial process.     

Effect of stress-state and spacing on voids in a shear-field

Unit cell model analyses are carried out for a material with a periodic array of voids, subject to shear loading. Thus the focus is on ductile fracture in conditions of low stress triaxiality. It has been shown recently that voids in shear are flattened out to micro-cracks, which rotate and elongate until interaction with neighboring micro-cracks gives coalescence, so that the failure mechanism is very different from that under tensile loading. In the present studies the plane strain unit cell has fully periodic boundary conditions, so that any combination of the stress components in the overall average stress state can be prescribed. This also allows for studies of the effect of different initial void spacing in the two in-plane coordinate directions. The stress states considered are essentially simple shear, with various levels of tensile stresses or compressive stresses superposed, i.e. low positive stress triaxiality or even negative stress triaxiality. For high aspect ratio unit cells a clear localization band is found inside the cell, which actually represents several parallel bands, due to periodicity. In the materials represented by a low aspect ratio unit cell localization would also occur after that the maximum shear stress has been passed, but this is not shown when periodicity is enforced. The effect of superposed tensile or compressive stresses is found to be bigger for high aspect ratio unit cells than for low aspect ratios.     

Effect of pure mode I,II or III loading or mode mixity on crack growth in a homogeneous solid

Crack growth is analyzed numerically under combined mode I, II and III loading, or under loading in one of these modes alone. The solid is a ductile metal modelled as elastic–plastic, and the fracture process is represented in terms of a cohesive zone model. The analyses are carried out for conditions of small-scale yielding, with the elastic mixed mode solution applied as boundary conditions on the outer edge of the region analyzed. For pure mode I loading crack growth continued far beyond the maximum fracture toughness shows that the predicted subsequent steady-state toughness is well below the maximum. The reason for this is discussed in terms of the local stress and strain fields around the tip. For pure mode II or mode III loading it is shown that there is no maximum before the steady-state. Also results for different mixed mode conditions are presented and discussed in relation to the results for loading in only one mode. Most of the results are based on assuming that the peak tractions for tangential separation are equal to that for normal separation, but it is shown that a relatively smaller peak traction for tangential separation may significantly affect the predictions.