Internal Necking Research Articles

A constitutive model for elastoplastic solids containing primary and secondary voids

In many ductile metallic alloys, the damage process controlled by the growth and coalescence of primary voids nucleated on particles with a size varying typically between 1 and 100 μm, is affected by the growth of much smaller secondary voids nucleated on inclusions with a size varying typically between 0.1 and 3 μm. The goal of this work is first to quantify the potential effect of the growth of these secondary voids on the coalescence of primary voids using finite element (FE) unit cell calculations and second to formulate a new constitutive model incorporating this effect. The nucleation and growth of secondary voids do essentially not affect the growth of the primary voids but mainly accelerate the void coalescence process. The drop of the ductility caused by the presence of secondary voids increases if the nucleation strain decreases and/or if their volume fraction increases and/or if the primary voids are flat. A strong coupling is indeed observed between the shape of the primary voids and the growth of the second population enhancing the anisotropy of the ductility induced by void shape effects. The new micromechanics-based coalescence condition for internal necking introduces the softening induced by secondary voids growing in the ligament between two primary voids. The FE cell calculations were used to guide and assess the development of this model. The use of the coalescence condition relies on a closed-form model for estimating the evolution of the secondary voids in the vicinity of a primary cavity. This coalescence criterion is connected to an extended Gurson model for the first population including the effect of the void aspect ratio. With respect to classical models for single void population, this new constitutive model improves the predictive potential of damage constitutive models devoted to ductile metal while requiring only two new parameters, i.e. the initial porosity of second population and a void nucleation stress, without any additional adjustment.

Rupture mechanisms in combined tension and shear—Micromechanics

A micromechanics model based on the theoretical framework of plastic localization into a band introduced by Rice is developed. The model consists of a planar band with a square array of equally sized cells, with a spherical void located in the centre of each cell. The periodic arrangement of the cells allows the study of a single unit cell for which fully periodic boundary conditions are applied. The micromechanics model is applied to analyze failure by ductile rupture in experiments on double notched tube specimens subjected to combined tension and torsion carried out by the present authors. The stress state is characterized in terms of the stress triaxiality and the Lode parameter. Two rupture mechanisms can be identified, void coalescence by internal necking at high triaxiality and void coalescence by internal shearing at low triaxiality. For the internal necking mechanism, failure is assumed to occur when the deformation localizes into a planar band and is closely associated with extensive void growth until impingement of voids. For the internal shearing mechanism, a simple criterion based on the attainment of a critical value of shear deformation is utilized. The two failure criteria capture the transition between the two rupture mechanisms successfully and are in good agreement with the experimental result.

Rupture mechanisms in combined tension and shear—Experiments

An experimental investigation of the rupture mechanisms in a mid-strength and a high-strength steel were conducted employing a novel test configuration. The specimen used was a double notched tube specimen loaded in combined tension and torsion at a fixed ratio. The effective plastic strain, the stress triaxiality and the Lode parameter were determined in the centre of the notch at failure. Scanning electron microscopy of the fractured surfaces revealed two distinctively different ductile rupture mechanisms depending on the stress state. At high stress triaxiality the fractured surfaces were covered with large and deep dimples, suggesting that growth and internal necking of voids being the governing rupture mechanism. At low triaxiality it was found that the fractured surfaces were covered with elongated small shear dimples, suggesting internal void shearing being the governing rupture mechanism. In the fractured surfaces of the high-strength steel, regions with quasi-cleavage were also observed. The transition from the internal necking mechanism to the internal shearing mechanism was accompanied by a significant drop in ductility.

Proposal and use of a void model for the simulation of inner fracture defect in drawing

In a previous paper [Komori K. Acta Mater 1999;47:3069], a new model of void coalescence, which was derived with reference to the Thomason model of void coalescence based on internal necking, was proposed and proved to be effective in the simulation of the drawing of copper. In this study, the simulation of the drawing of four kinds of carbon steel is performed using our proposed model. First, the two void shapes and two void configurations in our previous study are unified into one void shape and one void configuration to improve the proposed model. Then the analysis and the experiment of drawing are performed, and the validity of our proposed model is clarified by comparing the analytical results with the experimental results.

The effect of roll and clad sheet geometry on the necking instability during rolling of clad sheet metals

Experimental evidence is presented showing that the rolling strain required to produce internal necking in clad sheet depends both on the volume fraction of each constituent in the clad and the geometry of the rolls. This result was not predicted by earlier localization analysis that approximates the stress/strain field in rolling with that of plane strain compression. The evolution of hardness during rolling and complementary finite element results point to redundant shearing of the soft phase as the reason for this behavior. Such redundant shear differentially strengthens the softer component of the clad and reduces the induced tensile stress in the harder component, delaying the localization. Therefore, unwanted strain localization in rolling of clad sheets can be delayed significantly by the use of small radius rolls in configurations such as a Sendzimir mill.

An approach to predict free surface fracture in bulk forming

This work presents a unified approach to predict surface strains at failure in bulk forming processes. The approach does not deal with a specific process but rather with prescribed strain and stress paths. The material to be processed is assumed to possess an initial void volume fraction that grows and colaesces with straining, ending by fracture. The predictions are based on a formulation for voided solids according to the Gurson-Tvergaard yield function adapted to include orthotropic anisotropy. The incident of fracture is characterized by shear band formation within the ligaments of the matrix material among spheroidal voids as described by McClintock. The results are represented by a straight line plot of tensile limit strain versus the compressive strain for different loading paths. These limit curves are shown to be dependent on the initial void fraction, hardening, and anisotropy of the matrix matrial. Alloys with lower initial void fractions as well as those of higher hardening show better workability. The model is applied to predict bulk formability curves for steels AISI 1040 and 1045, Aluminum AI 7075-T6, and copper, based on the proper selection of micromechanical parameters for these alloys. The validity of the model is ensured through fairly favorable comparison with experimentally determined limit curves. The current failure conditions are suitable to predict the experimental dual slope fracture line that may exist for some alloys such as cold-drawn steel AISI 1045 and aluminum 2024-T6 by considering two mechanisms of failure: internal necking in the ligament material between voids, followed by transition to shear band formation.

Proposal and use of a void model for the simulation of shearing

In our previous paper [Acta Mater. 47(10) (1999) 3069–3077], a new model of void coalescence, which was derived with reference to Thomason's model of void coalescence based on internal necking, was proposed and proved to be effective in the analysis of multipass drawing. In this paper, the simulation of shearing is performed using our proposed model. First, the two void shapes and two void configurations in our previous study are unified into one void shape and one void configuration to improve the proposed model. Then the analysis and the experiment of shearing are performed, and the validity of our proposed model is clarified by comparing the analytical results with the experimental results.

Failure analysis of anisotropic sheet-metals using a non-local plastic damage model

A ductile damage model which takes account of void shape effect and anisotropy of the matrix material is integrated into an explicit finite element framework to predict the damage evolution which occurs under crash or stamping process. The void coalescence failure mechanism by internal necking is also considered by using a modified Thomason's plastic limit-load model. The pathological mesh dependence has been overcome by a non-local approach where the Laplacians of the effective plastic strain and of the porosity are incorporated. In this paper, the implicit gradient formulation constitutes an additional partial differential equation of the Helmholtz type which is solved separately by a boundary element method (BEM). The presented formulation is applied to a mild-steel, a Ni-based sheet-metal (INCO 718) and an aluminium alloy (AA 5182). The effectiveness of the numerical predictions is verified carrying out a set of experimental tests with a flat-headed punch.

Instability of an anisotropic power-law fluid in a basic state of plane flow

Initiation of cylindrical structures by buckling or necking in an anisotropic power-law fluid is treated for general plane flow. The principal axis of anisotropy, x′, in the stiffest direction in shortening or extension may be viewed as the trace of a foliation or lamination. Plane-flow constitutive relations between components of rate of deformation, D ′ x x and D ′ x y , and of deviatoric stress, s ′ x x and s ′ x y , for the fluid are D ′ x x = B ( Y ′ 2 ) [ ( n − 1 ) / 2 ] s ′ x x and D ′ x y = a 2 B ( Y ′ 2 ) [ ( n − 1 ) / 2 ] s ′ x y , where Y ′ 2 = ( s ′ x x ) 2 + a 2 ( s ′ x y ) 2 is an anisotropic invariant, a 2 is the anisotropy parameter, and n is the stress exponent. We determine the rate of amplification of wavelength components in the deflection of the foliation, θ, from a mean orientation parallel to x. Linearly independent, or non-interacting normal modes have a periodic, band-like form θ ( x , y ) ≅ ∂ ζ / ∂ x = − ( λ A ) sin [ λ ( x − v y ) ] , where ζ is the height of a foliation trace above its mean plane, v=tan β, where β is the angle between the normal to mean foliation and the axial surface, positive clockwise, and L=2π/ λ is the foliation-parallel wavelength. Evolution of a component may be followed through a finite bulk deformation provided θ remains ≪1. The growth rate of slope, λA, is independent of L. Components with axial plane normal to the foliation ( β=0) are strongly amplified in foliation-parallel shortening. If n>>1, internal necking (boudinage) occurs in foliation-parallel extension for components with axial plane inclined at a large angle to the foliation normal. In combined shortening and shear, the most rapidly growing component has an axial plane that dips steeply in the direction of shear. For n>1, maximum instability occurs for combined foliation-parallel shear and shortening rather than pure shortening. Weak instability is present in foliation-parallel shear. This anisotropic nonlinear fluid approximates the behavior of an isotropic power-law medium containing preferentially oriented but anastomosing slip surfaces, or that of a rock in which a stiffer component of lenticular form is embedded in a softer matrix.

Prediction of Central Bursting in Drawing and Extrusion of Metals

In this work central bursting in drawing and extrusion of metals is investigated. The analysis is based on a modified stress distribution within the die zone due to Shield (Shield, R. T., 1955, J. Mech. Phys. Solids, 3, pp. 246–258) together with Gurson–Tvergaard’s yield function (Tvergaard, V., 1981, Int. J. Fract., 17, pp. 389–407) and its associated flow rule for voided solids. The effects of hardening and evolution of void shape on void growth are considered. Various fracture criteria are employed to predict the process conditions at which central bursting occurs. The first criterion is due to Avitzur (Avitzur, B., 1968, ASME J. Eng. Ind., 90, pp. 79–91 and Avitzur, B., and Choi, J. C., 1986, ASME J. Eng. Ind., 108, pp. 317–321), the second and simplest criterion is based on vanishing mean stress while a suggested third criterion depends on the current value of the void volume fraction. Two other criteria which are basically due to Thomason’s internal necking condition (Thomason, P. F., 1990, Ductile Fracture of Metals, Pergamon, Oxford) as well as McClintock’s shear band formation criterion are applied (McClintock, F. A., Kaplan, S. M., and Berg, C. S., 1966, Int. J. Fract. Mech., 2, p. 614, and McClintock, F. A., 1968, in Ductility, ASM, Metals, Park, OH). The critical process conditions are predicted and compared with the available experimental data. Comparison showed that predictions based on the vanishing mean stress and the current void volume fraction criteria are closer to experiments than those based on Thomason’s internal necking and McClintock criteria.

A model for ductile fracture based on internal necking of spheroidal voids

A ductile fracture criterion based on internal necking of the intervoid matrix ligament is proposed. The criterion assumes the existence of an array of spheroidal voids subjected to extensional growth according to Gurson–Tvergaard flow function. It takes into account the influence of matrix strain hardening, void shape evolution as well as the softening due to existence of voids. Comparison among the proposed criterion and other criteria existing in the literature indicates its appropriateness for axisymmetric and plane strain conditions for unnotched and notched bars. The predictions of the proposed criterion are shown to be valid for a large variety of alloys for which the fracture strains and the micro-mechanical data were mostly identified experimentally.

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.

Prediction of ductile fracture using isotropic and kinematic hardening rules including void nucleation

ABSTRACTA formulation combining kinematic translation and isotropic expansion for a yield surface based on Gurson–Tvergaard function is used to describe void growth. By adopting a criterion of internal necking of the ligaments between voids, fracture strains for tensile bars made of conventional alloys and powder metal compacts—with micromechanical parameters mostly identified from experiments—are predicted according to kinematic, isotropic and mixed hardening models. Fracture strains predicted by the kinematic‐hardening model are in closer agreement with experiments whereas those estimated according to isotropic‐hardening model are overestimated. The consideration of either step‐like or continuous void nucleation models indicates its great influence on fracture strains and emphasizes a further need to quantify the statistical parameters involved in these models.

Influence of microstructure-driven strain localization on the ductile fracture of metallic alloys

Most models in the literature that are used to understand and predict ductile failure, ductility or fracture toughness implicitly consider that the plastic response of the material is homogeneous at the level of the microstructure. However, in a number of situations, plasticity is intrinsically localized, because the microstructure is spatially heterogeneous, or because the loading leads to strain localization on the microscale, such as in fatigue loading, or because the dynamics of plastic flow are unstable, such as in Portevin–Le Chatelier instabilities. This paper presents, from analysing various examples, the influence of these strain localization phenomena on the ductile fracture behaviour of metallic alloys. Examples to illustrate these effects will be chosen from, firstly, internal necking between cavities growing to ductile fracture, secondly, damage mechanisms from fatigue persistent slip bands, thirdly, the effect of Portevin–Le Chatelier bands on toughness and, fourthly, damage mechanisms in alloys with precipitate-free zones at grain boundaries.

Damage Identification for Anisotropic Sheet-Metals Using a Non-Local Damage Model

A new damage model which takes account of void shape effect and anisotropy of the matrix material is integrated into the explicit finite element framework to predict the damage evolution which occurs under crash or stamping process. The void coalescence failure mechanism by internal necking is also considered by using a modified Thomason’s plastic limit-load model. An inverse method is developed to identify the material parameters by correlating experimental and numerical measurements with tensile tests on sheet-metal strips. The pathological mesh dependence has been overcome by a non-local approach where the evolution equation for the porosity is modified by an additional term containing a characteristic internal length. In this paper the evaluation of the Laplacian is based on a least square approximation of the internal state variable around each integration points. The presented formulation is applied to an Ni-based sheet-metal (INCO 718) and an aluminium alloy (AA 5182). The effectiveness of the numerical predictions is verified carrying out a set of experimental tests with a flat-headed punch.

Application of an extended void growth model with strain hardening and void shape evolution to ductile fracture under axisymmetric tension

Abstract The version of the Gurson model adapted by Tvergaard is extended by fitting, based on numerical results proposed in the literature, the tuning parameters q1 and q2 in order to better capture the effects of the strain hardening and void shape evolution. The model is completed by two types of analytical void coalescence criteria, one for internal necking (Thomason 1968 and 1990) and one for void sheet (McClintock 1968). The validity of the proposed formulation is realized through comparison with more elaborate finite element models found in the literature. Using an approximate description of the variation of the stress triaxiality in the minimum section of an axisymmetric necking zone, the void growth rates and fracture strains are predicted. These are evaluated for several ductile alloys for which the experimental micro-mechanical data are reported in the literature.

Micromechanics of coalescence in ductile fracture

Significant progress has been recently made in modelling the onset of void coalescence by internal necking in ductile materials. The aim of this paper is to develop a micro-mechanical framework for the whole coalescence regime, suitable for finite-element implementation. The model is defined by a set of constitutive equations including a closed form of the yield surface along with appropriate evolution laws for void shape and ligament size. Normality is still obeyed during coalescence. The derivation of the evolution laws is carefully guided by coalescence phenomenology inferred from micromechanical unit-cell calculations. The major implication of the model is that the stress carrying capacity of the elementary volume vanishes as a natural outcome of ligament size reduction. Moreover, the drop in the macroscopic stress accompanying coalescence can be quantified for many initial microstructures provided that the microstructure state is known at incipient coalescence. The second part of the paper addresses a more practical issue, that is the prediction of the acceleration rate δ in the Tvergaard–Needleman phenomenological approach to coalescence. For that purpose, a Gurson-like model including void shape effects is used. Results are presented and discussed in the limiting case of a non-hardening material for different initial microstructures and various stress states. Predicted values of δ are extremely sensitive to stress triaxiality and initial spacing ratio. The effect of initial porosity is significant at low triaxiality whereas the effect of initial void shape is emphasized at high triaxiality.

The Localized Internal Necking Process between Microvoids in a Plastic Solid

The Localized Internal Necking Process between Microvoids in a Plastic Solid

Macro and Microscopic Deformation Behaviors of Materials with Artificial Inclusions.

The influence of artificial inclusions on tensile properties of iron-base sintered material with artificial inclusion is sudied experimentally and numerically. Stress and strain around inclusion have been examined by FEM. The microscopic observation of the initiation and growth of voids leading to macroscopic necking and subsequent ductile fracture has been carried out. The ultimate tensile strength depends on the interparticle spacing;which is function of volume fraction and particle size of inclusion. Initiation of a void at the interface between matrix and inclusion depends on the sort and size of inclusion. The onset of numerically predicted microscopic necking, internal necking, prior to the experimentally observed macrosoopic necking relates to the distance between inclusions which is depend on both of the volume fraction and grain size of inclusion.

Fracture Mechanism of Ferritic Ductile Cast Iron in Extremely Low Cycle Fatigue.

Low cycle fatigue tests were carried out under push-pull loading conditions using Ferritic Ductile Cast Iron (FDI). In order to clarify the fatigue fracture mechanism of FDI in an Extremely Low Cycle Fatigue (ELCF) regime, an observation of the longitudinal sections of the partially fatigued specimens were performed. The fatigue process in the ELCF regime can be classified into three stages : (a) Generation of micro voids, caused by the debonding of the graphite-matrix interface. (b) Growth of the micro voids, perpendicular to the loading axis. (c) Formation of an internal macro crack, due to the coalescence of micro voids inside the material. Two types of microvoid coalescence processes have been observed. The first, due to the growth of internal micro-cracks and the second, due to internal necking of the matrix. The fatigue process of the FDI in the ELCF regime was accompained by the static fracture mechanism.