Numerical simulations of crack tip fields in polycrystalline plastic solids

In this paper, modes I and II crack tip fields in polycrystalline plastic solids are studied under plane strain, small scale yielding conditions. Two different initial textures of an Al-Mg alloy, viz., continuous cast AA5754 sheets in the recrystallized and cold rolled conditions, are considered. The former is nearly-isotropic, while the latter displays distinct anisotropy. Finite element simulations are performed by employing crystal plasticity constitutive equations along with a Taylor-type homogenization as well as by using the Hill quadratic yield theory. It is found that significant texture evolution occurs close to the notch tip which profoundly influences the stress and plastic strain distributions. Also, the cold rolling texture gives rise to higher magnitude of plastic strain near the tip.     

Influence of crack tip constraint on void growth in ductile FCC single crystals

Effect of constraint (stress triaxiality) on void growth near a notch tip in a FCC single crystal is investigated. Finite element simulations within the modified boundary layer framework are conducted using crystal plasticity constitutive equations and neglecting elastic anisotropy. Displacement boundary conditions based on mode I, elastic, two term K-T field are applied on the outer boundary of a large circular domain. A pre-nucleated void is considered ahead of a stationary notch tip. The interaction between the notch tip and the void is studied under different constraints (T-stress levels) and crystal orientations. It is found that negative T-stress retards the mechanisms of ductile fracture. However, the extent of retardation depends on the crystal orientation. Further, it is found that there exists a particular orientation which delays the ductile fracture processes and hence can potentially improve ductility. This optimal orientation depends on the constraint level.     

Constraint effects on ductile fracture processes near a notch tip under mixed-mode loading

In this work, the effect of constraint on ductile fracture process of microvoid growth and coalescence near a notch tip in a ductile material under mode I and mixed mode loading (involving modes I and II) is investigated. To this end, two sets of finite element simulations are carried out under two-dimensional plane strain conditions. In the first set, a modified boundary layer formulation is employed in which the mixed mode elastic K–T field is prescribed as remote boundary conditions. Several analyses are carried out corresponding to different values of T-stress and remote elastic mode-mixity. Next, ductile four point bend specimens subjected to mode I and mixed mode loading are considered. In both sets of simulations, the interaction between a notch tip and a pre-nucleated hole ahead of it is modelled. The background material is represented by the Gurson constitutive model and micro-void nucleation at uniformly distributed small scale particles is also taken into account. The accumulation of matrix plastic strain and porosity in the ligament between the notch tip and the hole as well as the growth of the hole are studied. Finally, the effect of crack tip constraint on the relationship between the fracture toughness and mode mixity is examined.     

The impact behavior of aluminum alloy 6061: Effects of notch severity

In this paper the influence of notch acuity and test temperature on the impact behavior of aluminum alloy 6061 is presented and discussed. Notch angles of 45°, 60°, 75° and 90° were chosen for a standard charpy impact test specimen containing two such notches positioned at right angles to the applied load. For a given angle of the notch the dynamic fracture toughness increased with an increase in test temperature. At a given test temperature, the impact toughness of a ductile microstructure decreased with an increase in notch severity. For the least severe notch dynamic fracture surfaces revealed the occurrence of localized mixed-mode deformation at the elevated temperature. An increase in notch severity resulted in essentially Mode-I dominated fracture at all test temperatures. The results are discussed in light of alloy microstructure, fracture mechanisms and deformation field ahead of the advancing crack tip.     

Numerical simulations of hole growth and ductile fracture initiation under mixed-mode loading

Numerical simulations of ductile fracture initiation caused by the interaction between a notch tip and a nearby hole under mixed-mode loading involving modes I and II are performed. Attention is restricted to plane strain, small-scale yielding conditions. The Gurson constitutive model that accounts for the ductile failure mechanisms of micro-void nucleation, growth and coalescence is employed within the framework of a finite deformation plasticity theory. The failure of the ligament connecting the notch tip and the hole by either microvoid coalescence or by intense plastic strain localization is modelled. The effect of mode-mixity on the notch tip deformation, hole growth and the critical value of J at fracture initiation is examined. The dominant failure mechanism which is operative near the notch tip for various extents of mixity of modes I and II is identified.     

Fracture toughness of short-fibre reinforced thermoplastics

The failure process of short-fibre reinforced thermoplastics is characterized by different energy dissipation mechanisms, especially by mode II debonding along the fibre/matrix interface, sliding of debonded regions, brittle or ductile matrix fracture and pull-out. It is assumed that these failure processes are acting within a certain zone ahead of the notch tip— the dissipation zone. The modes of energy dissipation are mainly affected by the matrix fracture mode (brittle or ductile) which is mainly determined by the loading rate or temperature conditions. On the basis of an energy principle and relationships for the different energy dissipation mechanisms, we propose theoretical expressions for the static and dynamic fracture toughness and we compare these with experimental results.     

Fracture of notched polycarbonate under hydrostatic pressure

The brittle fracture behaviour and plastic deformation of round-notched polycarbonate bars subjected to three-point bending under hydrostatic pressure have been studied. Below a certain critical pressure, the brittle fracture initiated from an internal craze nucleated at the tip of the local plastic zone ahead of the notch rooT. The position of the nucleation of the craze receded from the tip of the notch with increasing applied pressure. When the pressure was increased over a critical value, general yielding occurred by passage of the plastic zone across the notched cross-section, that is, the brittle to ductile transition took place. A qualitative analysis of the stress distribution within the plastic zone explains that the brittle to ductile transition under hydrostatic pressure occurs when the general yield takes place before a critical stress for brittle crack propagation is reached.     

A numerical investigation of ductile fracture initiation in a high-strength low-alloy steel

In this work, static and drop-weight impact experiments, which have been conducted using three-point bend fracture specimens of a high-strength low-alloy steel, are analysed by performing finite-element simulations. The Gurson constitutive model that accounts for the ductile failure mechanisms of microvoid nucleation, growth and coalescence is employed within the framework of a finite deformation plasticity theory. Two populations of second-phase particles are considered, including large inclusions which initiate voids at an early stage and small particles which require large strains to nucleate voids. The most important objective of the work is to assess quantitatively the effects of material inertia, strain rate sensitivity and local adiabatic temperature rise (due to conversion of plastic work into heat) on dynamic ductile crack initiation. This is accomplished by comparing the evolution histories of void volume fraction near the notch tip in the static analysis with the dynamic analyses. The results indicate that increased strain hardening caused by strain rate sensitivity, which becomes important under dynamic loading, plays a benign role in considerably slowing down the void growth rate near the notch tip. This is partially opposed by thermal softening caused by adiabatic heating near the notch tip.     

A critical shear displacement criterion for ductile–brittle transition under mixed mode I and II loading

Abstract

Micromechanisms producing ductile and brittle damage operate in parallel at a crack tip. The dominant mode of failure depends upon which of the two (ductile or brittle) damage parameters first reaches its critical value. This has been shown by a study of ductile–brittle transition behaviour in HY100 steel under mixed mode I and II loading. The transition from ductile to brittle behaviour in HY100 steel was found to be affected by mixed mode I and II ratio (ratio of imposed tensile and shear loading) in a manner such that with increasing shear the transition temperature decreased. In the present paper, a criterion is proposed based on the shear strain ahead of a notch tip, to predict the fracture behaviour at any given temperature and mixed mode ratio.     

Numerical simulations of fracture initiation in ductile solids under mode I, dynamic loading

In this paper, an overview of some recent computational studies by the authors on ductile crack initiation under mode I, dynamic loading is presented. In these studies, a large deformation finite element procedure is employed along with the viscoplastic version of the Gurson constitutive model that accounts for the micro-mechanical processes of void nucleation, growth and coalescence. A three-point bend fracture specimen subjected to impact, and a single edge notched specimen loaded by a tensile stress pulse are analysed. Several loading rates are simulated by varying the impact speed or the rise time and magnitude of the stress pulse. A simple model involving a semi-circular notch with a pre-nucleated circular hole situated ahead of it is considered. The growth of the hole and its interaction with the notch tip, which leads to plastic strain and porosity localization in the ligament connecting them, is simulated. The role of strain-rate dependence on ductile crack initiation at high loading rates, and the specimen geometry effect on the variation of dynamic fracture toughness with loading rate are investigated.     

Notch-texture strengthening mechanism in commercially pure titanium thin sheets

Simultaneous effects of notch and texture on strengthening mechanisms of rolled thin sheets of commercially pure titanium were investigated. The presence of notch led to the restriction of deformation systems and different fracture behaviors compared to un-notched specimens. The loss of material’s ability to accommodate plastic deformation at the notch tip with increase in rolling reductions changed the notch strengthening phenomenon to the notch weakening one. At medium levels of deformation, due to the simultaneous development of a triaxial stress state and strong basal texture at the notch tip, a new strengthening mechanism which is called “notch-texture strengthening mechanism” led to a significant enhancement of tear strength. However, the lack of stress triaxiality in un-notched tensile specimens and a strong basal texture component in other notched specimens reduced the impact of strengthening. It was found that the restriction of deformation systems due to the c-axis compression condition at the notch tip was responsible for this strengthening mechanism.     

NUCLEATION,BLUNTING OR PROPAGATION OF A NANOCRACK IN DISLOCATION-FREE ZONES OF THIN CRYSTALS

Nucleation, blunting and propagation of nanocracks in dislocation-free zones (DFZs) ahead of crack tips in ductile and brittle metals have been investigated by tensioning in situ with a TEM, and analysed using microfracture mechanics. The results show that in either ductile or brittle metals, many dislocations could be emitted from a loaded crack tip and a DFZ formed after equilibrium. The stress in the DFZ may be up to the cohesive strength of the material, and then a nanocrack is initiated in the DFZ or directly from the crack tip. In ductile metals, the nanocrack is blunted into a void or notch during constant displacement. In brittle metals, the nanocrack propagated as a cleavage microcrack rather than being blunted.     

Is stress concentration relevant for nanocrystalline metals?

Classical fracture mechanics as well as modern strain gradient plasticity theories assert the existence of stress concentration (or strain gradient) ahead of a notch tip, albeit somewhat relaxed in ductile materials. In this study, we present experimental evidence of extreme stress homogenization in nanocrystalline metals that result in immeasurable amount of stress concentration at a notch tip. We performed in situ uniaxial tension tests of 80 nm thick (50 nm average grain size) freestanding, single edge notched aluminum specimens inside a transmission electron microscope. The theoretical stress concentration for the given notch geometry was as high as 8, yet electron diffraction patterns unambiguously showed absence of any measurable stress concentration at the notch tip. To identify possible mechanisms behind such an anomaly, we performed molecular dynamics simulations on scaled down samples. Extensive grain rotation driven by grain boundary diffusion, exemplified by an Ashby-Verrall type of grain switching process, was observed at the notch tip to relieve stress concentration. We conclude that in the absence of dislocations, grain realignment or rotation may have played a critical role in accommodating externally applied strain and neutralizes any stress concentration during the process.     

Effects of thickness and texture on mechanical properties anisotropy of commercially pure titanium thin sheets

Simultaneous effects of thickness and texture on the anisotropy of mechanical properties and fracture behaviors of commercially pure titanium thin sheets were studied. The activation of different deformation systems, due to the split distribution of basal texture, led to mechanical properties anisotropy. The crack initiation and propagation energies, when the loading direction was parallel to the initial rolling direction, decreased with increasing thickness ranges from 0.25 to 1 mm. The changes of size, shape and distribution of dimples with increasing thickness confirmed the restriction of deformation systems and the development of triaxial stress state and plane-strain condition at the notch tip. However, in transverse-directed specimens, the energy release rate increased with increasing specimen thickness up to 0.75 mm and then decreased. The fractography of these specimens explained the simultaneous effects of thickness and texture on structural stability and high accommodated plastic deformation at the notch tip.     

NUCLEATION AND SHORT CRACK GROWTH IN FATIGUED POLYCRYSTALLINE COPPER

Surface evolution in polycrystalline copper specimens with a shallow notch has been studied in interrupted constant strain amplitude cyclic loading. The inhomogeneous strain distribution close to stress amplitude saturation leads to the formation of extrusions and intrusions along persistent slip bands within the grain and also in suitably oriented grain boundaries. Numerous primary cracks within a grain or at a grain boundary are nucleated. Some cracks can grow further either by linking with existing cracks or by nucleation of new elementary cracks ahead of the crack tip. Crack growth rates of individual cracks fluctuate considerably but for each strain amplitude, which results in a saturated plastic strain amplitude, a crack growth rate of an equivalent crack can be established. This crack growth rate was found to depend strongly on the plastic strain amplitude in agreement with the Manson-Coffin law.     

Size effect in fracture of unidirectional composite plates

Fracture of notched, unidirectionally reinforced composite plates with well-bonded ductile matrices is typically preceded by the formation of long, discrete plastic shear zones aligned in the fiber direction. Onset of fracture is associated with a critical tension stress in a certain small process zone ahead of the notch tip; the critical stress is often equal to the tensile strength of the unnotched composite plate. Length of the shear zones can be estimated by plastic limit-analysis, and the local tension stress ahead of the notch found by superposition of the stress caused by remotely applied loads with the stress induced by the shear tractions in the plastic zone, which is shown to be dominated by a logarithmic singularity. In as-fabricated boron-aluminum composites, this fracture mechanism was analyzed and confirmed by numerous experiments (G.J. Dvorak, J. Zarzour and Y. Benveniste, Engineering Fracture Mechanics 42, 501–517, 1992). Since the notch tip field is not described by a stress intensity factor, experimental notched strength data cannot be interpreted in terms of a single material property, such as toughness. An alternative scaling procedure is outlined for prediction of notched strength of wide plates on the basis of data obtained from small size specimens. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of elastoplastic sharp notches

In this paper the elastoplastic solutions with higher-order terms for apex V-notches in power-law hardening materials have been discussed. Two-term expansions of the plane strain and the plane stress solutions have been obtained. It has been shown that the leading-order singularity approaches the value for a crack when the notch angle is not too large. In plane strain cases the elasticity does not enter the second-order solutions when the notch opening angle is too small. For a large notch angle, the two-term expansions of the plane strain near-tip fields are described by a single amplitude parameter. The plane stress solutions generally contain the elasticity terms. The boundary layer formulations based on the small-strain plasticity theory confirm that a dominance zone exists ahead of the notch tip. Finite element results give good agreement to the asymptotic solutions under both plane strain and plane stress conditions. The second-order terms cannot improve the predictions significantly. The near-tip fields are dominated by a single parameter. Finite element calculations under the finite strain J ₂-flow plasticity theory revealed that the finite strains can only affect local characterization of the asymptotic solution. The asymptotic solution has a large dominance zone around the notch tip. For an apex notch bounded to a rigid substrate the leading-order singularity falls with the notch angle significantly more slowly than in the homogeneous material. It vanishes at the notch angle about 135° for all power-hardening exponents. The elasticity effects enter the second-order solutions when the notch angle becomes large enough. The tip fields are characterized by the hydrostatic stress and the shear stress ahead of the notch.     

On the fracture of small samples under higher order strain gradient plasticity

In this work we perform Finite Element simulations within the framework of large deformation elasto-viscoplasticity on a material that is sensitive to the gradients of plastic strain and incorporates a single intrinsic length scale parameter. Both small scale yielding simulations and those on a finite sized sample show that large stress enhancements can occur at the tip of a notch due to gradient effects. The amount of plastic strain and opening stress that can be expected at the notch tip depends on an interplay between the notch radius, specimen dimensions and boundary conditions. It is shown that cleavage can be the favored criterion for failure in even a ductile material when the notch radius is small compared to the intrinsic length scale. Moreover, for large intrinsic length scales, failure may not always initiate at a notch but may be triggered away from it due to the presence of a boundary impermeable to dislocations.     

Failure assessment of notched polycrystalline graphite under tensile-shear loading

The fracture load and the fracture initiation angle were experimentally measured for a V-notched specimen made of polycrystalline graphite under combined tensile-shear loading. The experimental results were obtained for several specimens with different notch angles and various notch tip radii. The experimental observations showed that for a constant notch tip radius, the fracture load in pure tensile loading conditions decreases as the notch angle increases. Moreover, for a constant notch angle, as the notch tip radius increases the fracture load in graphite specimens enhances in the entire domain between pure tensile and pure shear loading conditions. A recently developed failure criterion was then used to estimate the experimental values of the notch fracture resistance and the fracture initiation angle for the tested graphite specimens. The experimental results could be estimated very well by using the results of the proposed criterion.