Viggo Tvergaard

Analysis of the cup-cone fracture in a round tensile bar

Abstract Necking and failure in a round tensile test specimen is analysed numerically, based on a set of elastic-plastic constitutive relations that account for the nucleation and growth of micro-voids. Final material failure by coalescence of voids, at a value of the void volume fraction in accord with experimental and computational results, is incorporated in this constitutive model via the dependence of the yield condition on the void volume fraction. In the analyses the material has no voids initially; but high voidage develops in the centre of the neck where the hydrostatic tension peaks, leading to the formation of a macroscopic crack as the material stress carrying capacity vanishes. The numerically computed crack is approximately plane in the central part of the neck, but closer to the free surface the crack propagates on a zig-zag path, finally forming the cone of the cup-cone fracture. The onset of macroscopic fracture is found to be associated with a sharp “knee” on the load deformation curve, as is also observed experimentally, and at this point the reduction in cross-sectional area stops.

THE RELATION BETWEEN CRACK GROWTH RESISTANCE AND FRACTURE PROCESS PARAMETERS IN ELASTIC-PLASTIC SOLIDS

CKA~K growth initiation and subsequent resistance is computed for an elastic-plastic solid with an idealized traction separation law specified on the crack plane to characterize the fracture process. The solid is specified by its Young’s modulus, E, Poisson’s ratio, v, initial tensile yield stress, (or, and strain hardening exponent, N. The primary parameters specifying the traction-separation law of the fracture process are the work of separation per unit area, To. and the peak traction, 6. Highly refined calculations have been carried out for resistance curves. K,(Arr), for plane strain, mode I growth in small-scale yielding as dependent on the parameters characterizing the elastic-plastic properties of the solid and its fracture process. With K,, = [El-,/( I ~ v’)] ’ 2 as the intensity needed to advance the crack in the absence ofplasticity, K,J& is presented in terms of its dependence on the two most important parameters, d/nr and N, with special emphasis on initiation toughness and steady-state toughness, Three applications of the results are made : to predict toughnesss when the fracture process is void growth and coalescence, to predict the role of plasticity on interface toughness for similar materials bonded together, and to illuminate the role of plasticity in enhancing toughness in dual-phase solids. The regime of applicability of the present model to ductile fracture due to void growth and coalescence, wherein multiple voids interact within the fracture process zone, is complementary to the regime of applicability of models describing the interaction between a single void and the crack tip. The two mechanism regimes are delineated and the consequence of a transition between them is discussed.

Material Failure by Void Growth to Coalescence

Publisher Summary This chapter describes the material failure by coalescence of microscopic voids. The voids nucleate mainly at second phase particles, by decohesion of the particle-matrix interface or by particle fracture, and subsequently the voids grow because of plastic straining of the surrounding material. The growth of voids to coalescence by plastic yielding of the surrounding material involves so large geometry changes that finite strain formulations of the field equations are a necessary tool. A convected coordinate formulation of the governing equations is used. Convected coordinates are introduced, which serve as particle labels. The convected coordinate net can be visualized as being inscribed on the body in the reference state and deforming with the material. It is found that after nucleation, cavities elongate along the major tensile axis and that two neighboring cavities coalesce when their length has grown to the order of magnitude of their spacing. This local failure occurs by the development of slip planes between the cavities or simply necking of the ligament. A detailed micromechanical study of shear band bifurcation that accounts for the interaction between neighboring voids and the strongly nonhomogeneous stress distributions around each void has been carried out, and are also elaborated in this chapter.

The influence of plasticity on mixed mode interface toughness

Calculations are reported for the mixed mode toughness of an interface joining an elastic-plastic solid to a solid which does not yield plastically. A potential function of the components of the crack face displacements is used to generate the tractions along the interface where the fracture processes causing separation occur. The two main parameters characterizing this potential are the work of separation per unit area and a peak normal stress. This description of the interface separation process is embedded within the continuum description as a boundary condition on the interface linking the adjoining solids. Small-scale yielding in plane strain is considered with the remote field specified by the magnitude and phase of the mixed mode stress intensity factors. Crack growth resistance curves are computed for a range of the most important nondimensional material parameters and for various combinations of remote mixed mode loading. Particular emphasis is placed on the ratio of the steady-state interface toughness to the “intrinsic” work of separation as it depends on plastic yielding and on the combination of modes 1 and 2. Plasticity enhances the interface toughness for all modes of loading, but substantially more so in the presence of a significant mode 2 component of loading than in near-mode 1 conditions.

Effect of fibre debonding in a whisker-reinforced metal

Abstract For a whisker-reinforced metal matrix composite, failure by decohesion of the fibre-matrix interface is analysed. The unit-cell model applied for the numerical analyses represents a periodic array of aligned fibres, where neighbouring fibres are somewhat shifted relative to one another. The debonding behaviour at the interface is represented in terms of a cohesive zone model that describes decohesion by normal separation as well as decohesion by tangential separation. The predicted failure initiates by void formation near the sharp edges at the fibre ends, and subsequently fibre pull-out is described. The effect of friction during pull-out is included in the model.

An analysis of ductile rupture in notched bars

Abstract 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.

An analysis of ductile rupture modes at a crack tip

An elastic-Viscoplastic model of a ductile, porous solid is used to study the influence of the nucleation and growth of micro-voids in the material near the tip of a crack. Conditions of small scale yielding are assumed, and the numerical analyses of the stress and strain fields are based on finite strain theory, so that crack tip blunting is fully accounted for. An array of large inclusions or inclusion colonies, with a relatively low strength, results in large voids near the crack tip at a rather early stage, whereas small second phase particles in the matrix material between the inclusions require large strains before cavities nucleate. Various distributions of the large inclusions, and various critical strains for nucleation of the small scale voids between the inclusions, are considered. Localization of plastic flow plays an important role in determining the failure path between the crack tip and the nearest larger void, and the path is strongly sensitive to the distribution of the large inclusions. Values of the J-integral and the crack opening displacement at fracture initiation are estimated, together with values of the tearing modulus during crack growth, and these values are related to experimental results.

Influence of void nucleation on ductile shear fracture at a free surface

Abstract A n approximate continuum model of a ductile, porous material is used to study the influence of the nucleation and growth of micro-voids on the formation of shear bands and the occurrence of surface shear fracture in a solid subject to plane strain tension. Bifurcation into diffuse modes is analysed for a plane strain tensile specimen described by these constitutive relations, which account for a considerable plastic dilatancy due to void growth and for the possibility of non-normality of the plastic flow law. In particular, bifurcation into surface wave modes and the possible influence of such modes triggering shear bands is investigated. For solids with initial imperfactions such as a surface undulation, a local material inhomogeneity on an inclusion colony, the inception and growth of plastic flow localization is analysed numerically. Both the formation of void-sheets and the final growth of cracks in the shear bands is described numerically. Some special features of shear band development in the solid obeying non-normality are studied by a simple model problem.

Analysis of tensile properties for a whisker-reinforced metal-matrix composite

Abstract A micro-mechanical model is used to analyse the tensile properties of a ductile metal reinforced by short fibres. It is assumed that fibre alignment has been obtained during processing, and the numerical analysis for a representative unit cell also relies on the assumption of a periodic array of fibres. The special unit cell model developed here represents neighbouring fibres shifted relative to one another. Comparison of results for rigid fibres with results accounting for fibre elasticity shows that apart from the initial elastic range the overall stress-strain behaviour is dominated by matrix plasticity. The analyses focus on uniaxial tension in the fibre direction, but also the effect of stress triaxiality is investigated. Quite good agreement is obtained between the model predictions and published experimental results. The sensitivity of the overall stress-strain behaviour to the fibre volume fraction, the fibre length, and fibre spacings is studied by a number of analyses.

MICROMECHANICAL MODELS FOR GRADED COMPOSITE MATERIALS

Abstract Elastic response of selected plane-array models of graded composite microstructures is examined under both uniform and linearly varying boundary tractions and displacements, by means of detailed finite element studies of large domains containing up to several thousand inclusions. Models consisting of piecewise homogeneous layers with equivalent elastic properties estimated by Mori-Tanaka and selfconsistent methods are also analysed under similar boundary conditions. Comparisons of the overall and local fields predicted by the discrete and homogenized models are made using a C/SiC composite system with very different Youngs moduli of the phases, and relatively steep composition gradients. The conclusions reached from these comparisons suggest that in those parts of the graded microstructure which have a well-defined continuous matrix and discontinuous second phase, the overall properties and local fields are predicted by Mori-Tanaka estimates. On the other hand, the response of graded materials with a skeletal microstructure in a wide transition zone between clearly defined matrix phases is better approximated by the self-consistent estimates. Certain exceptions are noted for loading by overall transverse shear stress. The results suggest that the averaging methods originally developed for statistically homogeneous aggregates may be selectively applied, with a reasonable degree of confidence, to aggregates with composition gradients, subjected to both uniform and nonuniform overall loads.