Plasticity theory of ductile fracture by void growth and coalescence

The axisymmetric cold forging is used in the automobile industry to manufacture the formed parts en masse. But, this process is often limited by the onset of surface cracking which reduces the quality of the final product. In this paper, the theoretical investigation of the initiation of the ductile fracture in axisymmetric upsetting processes is carried out using a simple method of analysis. The developed method is based on the theory of plasticity and a fracture criterion, in which the influence of the growth and coalescence of the cylindrical and spherical voids is considered. The cracking at the free surface of the cylindrical billet is modelled. Dedicated to Professor Theordor Lehmann, Bochum     

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     

Numerical prediction of creep crack growth in different geometries using simplified multiaxial void growth model

Abstract

This paper considers the prediction of creep crack growth (CCG) in different fracture mechanics geometries using finite element (FE) analysis based on a material independent simplified multiaxial failure strain model at the crack tip. The comparison is first made by modelling C(T) specimen tests under plane stress and plane strain conditions using creep properties of a C–Mn steel at 360°C. In addition, in order to examine CCG due to different geometries, a single edge notch specimen (SENT), centre cracked tension specimen (CCT) and three-point bending (3PB) specimen have been modelled and analysed. In all cases, it is found, depending on the geometry, that for this steel at low creep temperatures the applied load develops a high reference stress/yield stress (σ_ref/σ_y) ratio, which helps reduce constraint at the crack tip. The predictions are analysed under plane stress/plane strain loading conditions identifying the effects of geometry on cracking rates and the implications for predicting long term test or component failure times exceeding where the applied σ_ref/σ_y<<1.     

Application of fracture mechanics to creep crack growth

International Journal of Fracture -     

The application of fracture mechanics to creep crack growth

A review is made of creep cracking test results analysed using various parameters such as the stress intensity factor, net section stress, crack opening displacement and deformation energy rate. Theoretical predictions of creep crack growth rate on the basis of creep laws are discussed. Creep crack growth due to vacancy diffusion and condensation is considered. Analytical treatments are reviewed. A new solution is presented.     

On the creep crack growth prediction by a local approach

Classical methods to predict crack growth in structures are generally based on fracture mechanics concepts. For high-temperature applications, where creep (monotonic or cyclic) or thermal stresses are present, such classical approaches lead to large difficulties. An alternative method is to calculate as accurately as possible the actual local behaviour including viscoplasticity and creep damage effects. The different levels of the possible “local approaches” are briefly reviewed and discussed; the case of creep crack growth is then studied in detail, through the use of viscoplastic constitutive equations including creep damage effect. Both the creep damage and the hardening of the metal are supposed to be isotropic, characterized respectively by the following scalar internal variables: the Kachanov's damage variable D and the cumulated viscoplastic strain p. The evolution equation of creep damage is a differential non-linear one with non-linear cumulative effect. The local states of different mechanical fields ((σ, , D) and their redistribution, due to damage effect, are accurately investigated and illustrated by various numerical examples. Finally the approach is applied to the creep of initially cracked CT specimen.     

Ductile fracture by multiple void growth and interaction ahead of a notch tip in polycrystalline plastic solids

The objectives of this paper are to study the effects of plastic anisotropy and evolution in crystallographic texture with deformation on the ductile fracture behaviour of polycrystalline solids. To this end, numerical simulations of multiple void growth and interaction ahead of a notch tip are performed under mode I, plane strain, small scale yielding conditions using two approaches. The first approach is based on the Hill yield theory, while the second employs crystal plasticity constitutive equations and a Taylor-type homogenization in order to represent the ductile polycrystalline solid. The initial textures pertaining to continuous cast Al–Mg AA5754 sheets in recrystallized and cold rolled conditions are considered. The former is nearly-isotropic, while the latter displays pronounced anisotropy. The results indicate distinct changes in texture in the ligaments bridging the voids ahead of the notch tip with increase in load level which gives rise to retardation in porosity evolution and increase in tearing resistance for both materials.     

Particle toughening of polymers by plastic void growth

An analysis is given for the toughening of particle filled polymers assuming that plastic void growth around debonded, or cavitated particles is the dominant energy absorbing mechanism. The controlling parameter is the debonding (or cavitation) surface energy which triggers the growth of a plastic void around the particle which in turn enhances the toughness. Literature data are examined for particles ranging from 0.01 to 25 μm in radius in thermoset resins and it is found that the surface work for very small particles is the surface work of adhesion while for larger sizes it is some fraction of the matrix toughness. This larger debonding energy is found to be proportional to the particle radius. Large toughness increases are predicted, and observed, in the nano-range, i.e. 0.01 μm which are shown to require good particle dispersion and high matrix ductility. More modest increases are predicted at the micron scale but these are more robust.     

Mechanics of void growth and void collapse in crystals

The mechanics of void deformation in single crystals is studied in a fully three-dimensional setting, taking into account all twelve slip systems for fcc and bcc crystals. The increments of the local field variables are calculated analytically using Eshelby's approach for the three-dimensional inclusion problem. The collapse or growth of voids in three dimensions is investigated when a rate-dependent elastoplastic material is subjected to compression and tension at a high rate. The deformation of an initially spherical cavity is calculated incrementally, and, it is shown that, even under all-around uniform loading, a void deforms into a complicated shape which is defined by the structure and symmetry of the slip systems. An equivalent ellipsoid is used to approximate the deformed void shape at each incremental step, and the procedure is continued until the equivalent ellipsoid collapses into a crack or a needle, or it expands or shrinks in a self-similar manner. Several numerical examples are presented, and the numerical results are compared with the experimental observations, obtaining good correlation. The significant effects of loading rate on the material response are illustrated. It is shown that the material becomes stronger at higher loading rates. The three-dimensional final void geometry under various loading conditions is studied. The difference between void deformation mechanism under tension and compression is illustrated. The possible overall failure mechanism caused by collapse and/or growth of pre-existing cavities is discussed. From the comparison of the corresponding results with those of the two-dimensional case it is shown that the double-slip system in two dimensions can be used effectively to simulate the three-dimensional problem.     

Discrete modelling of ductile crack growth by void growth to coalescence

Ductile crack growth is analyzed by discrete representation of the voids growing near a blunting crack-tip. Coalescence of the nearest void with the crack-tip is modeled, followed by the subsequent coalescence of other discretely represented voids with the newly formed crack-tip. Necking of the ligaments between the crack-tip and a void or between voids involves the development of very large strains, which are included in the model by using remeshing at several stages of the plastic deformation. The material is here described by standard isotropic hardening Mises theory. For a very small void volume fraction the crack-tip tends to interact with one void at a time, while larger void volume fractions lead to simultaneous interaction of multiple voids on the plane ahead of the crack-tip. In some cases a change from one of these mechanisms to the other is seen during growth through the many voids represented here. First uniformly spaced voids of equal size are considered, but also a few computations for a random distribution of the void spacings or of the void sizes are carried out.     

Prediction of ductile fracture in axisymmetric tension by void coalescence

Fracture strains are predicted for ductile materials undergoing void growth and coalescence. The calculation scheme is based on Gurson–Tvergaard yield function and its associated flow rule. Fracture condition is identified by vanishing stress-carrying capacity of the material. The plastic flow parameters are all determined from experimental evidences for a variety of alloys. Comparison among predicted fracture strains and experimental ones is given for a wide range of conventional and superplastic materials as well as powder compacts. Finally an approximate fracture criterion is proposed.     

Ductile failure by void nucleation,growth and coalescence

A wide range of stress states has been generated using axisymmetric and plane strain notched tensile specimens which have been analysed by elastic-plastic finite element analysis. Observations of voids on metallographic sections have been combined with calculated deformation histories and the hole growth equations to determine the local conditions for hole nucleation and the strength of the particle-matrix interface. These calculations show that nucleation is a statistical event for which the radial stress at the particle/matrix interface has been determined. Subsequent calculations give the local conditions at void coalescence in the average material and in statistical inhomogeneities. The statistics of the inclusion distribution determine the size scale over which void coalescence must occur in order to create a crack-like defect.

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Résumé En utilisant des éprouvettes de traction entaillées axisymétriques soumises à état plan de déformation, analysées par éléments finis élasto-plastiques, on a pu élaborer une gamme importante d'états de tension. En combinant les observations de lacunes de section métallographique avec l'histoire des déformations calculées et des équations de croissance d'une cavité, on a déterminé les conditions locales qui président à la croissance de cavité et à la résistance de l'interface particule-matrice. Ces calculs montrent que l'apparition de cavités est un évènement statistique pour lequel la contrainte radiale à l'interface particule-matrice a pû être déterminé. Des calculs subséquents fournissent les conditions locales qui conduisent à la -coalescence de lacunes dans un matériau standard et pour des conditions d'inhomogénéité statistiquement réparties. La statistique de distribution des inclusions détermine l'échelle au delà de laquelle une coalescence de porosités peut se produire et crée un défaut assimilable `a une fissure.

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Nomenclature a semi-major axis of elliptical void, radius at minimum cross-section of notch tensile specimen - b semi-minor axis of elliptical void - E uniaxial elastic modulus - e matrix strain - H aggregate hardening rate - J intermediate variable in void equations - K intermediate variable in void equations - L intermediate variable in void equations - M eccentricity of elliptical void - n power hardening index - P(x) probability of nucleation at a radial stress x - r distance of void from specimen axis - R mean radius of elliptical void, tip radius of notch - s matrix stress deviator V volume - aggregate stress - matrix stress - ø yield function - p plastic component - -(overbar) effective, representative or equivalent value - 0 initial value, initial yield value in uniaxial tension - m mean value - y current yield value in uniaxial tension - remote value     

On the void growth in C-Mn structural steel during plastic deformation

A procedure has been established to measure the relationship between void growth, plastic strain and stress triaxiality during the fracture of a C-Mn structural steel. A calibration factor for void growth was determined using the smooth tensile specimen tests and pre-cracked side-grooved three-point bend bar tests. The factor was expected to be applicable over a rather wide range of the constraint generating geometries.The experimental void growth relation shows a real void growth rate in the structural steel which is smaller than that determined by the Rice-Tracey's void growth theory for stress triaxialities, larger than about 1.2. However, when the value is smaller than about 1.2, the void growth predicted by the Rice-Tracey's theory tends to slightly underestimate the void growth in the steel. The variations of the plastic strain and relative void volume have been measured and evaluated ahead of the blunting crack tip in the pre-cracked side-grooved three-point bend specimens at the point where the ductile tearing initiated. These results are related to theories for the level of triaxial stresses.

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Résumé On a mis sur pied une procédure pour mesurer la relaxation liant la croissance des lacunes, la dilatation plastique et l'état de triaxialité des contraintes au cours de la rupture d'un acier de construction au C-Mn. La croissance des lacunes a été étalonnée en recourant à des essais de traction sur éprouvettes lisses et à des essais de flexion en trois points de barreaux entaillés latéralement et préfissurés. Le facteur d'étalonnage tiré de ces essais devrait être applicable à une large gamme de géométries produisant une concentration de contraintes.L'expérience a montré que, dans l'acier de construction étudié, la croissance des lacunes est plus faible que ne le laisse prévoir la théorie de croissance de Rice-Tracey, dans le cas de triaxialité de contrainte a. 16 supérieure à environ 1,2. Par contre, lorsque cette triaxialité est inférieure à 1,2, la croissance de lacunes est légèrement sous-estimée par le théorie de Rice-Tracey.On a mesuré et évalué, dans les éprouvettes de flexion pré-fissurées, les variations de déformation plastique et de volume relatif de vide en avant de l'extrémité d'une fissure en cours d'ouvertures et ce dans la zone d'amorçage de l'arrachement ductile.Les résultats sont en relation avec les théories fixant le niveau de triaxialité d'une sollicitation.