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A criterion is developed for the unhomogeneous yielding of materials containing arbitrarily oriented ellipsoidal voids. The criterion is built upon classical estimates for pure pressure and pure shear. A data-driven approach is then followed to incorporate the effects of void shape and orientation. A large number of micromechanical unit cell results are used to calibrate the yield criterion. A key feature of the criterion is that it predicts a significant reduction of the effective shear yield strength due to mere void inclination, with the reduction increasing with the void dimension perpendicular to the shear. The coupling between tension and shear deformation results in an apparent rotation of the yield surface, which provides a sound micromechanical basis for predicting void closure in shear among other new features. Once supplemented with evolution equations of relevant internal parameters, the resulting constitutive formulation will enable ductile failure simulations heretofore impossible to carry out on a sound physical basis for general loading conditions. [ABSTRACT FROM AUTHOR]

We present a multiphysics phase-field fracture model for thermo-elasto-plastic solids in the context of finite deformation and apply it to simulate the hot cracking phenomenon during metal additive manufacturing. The model is derived in a thermodynamically consistent manner, with the intercoupling mechanisms among elastoplasticity, phase-field crack and heat transfer comprehensively considered. It involves particularly coupled parameters among these materials physics, e.g. plasticity-dependent degradation function and fracture toughness, damage-dependent yield surface and thermal properties, and temperature-dependent elastoplastic properties and fracture strength. The finite element implementation of the coupled phase-field model is benchmarked with simulation results of a tensile test of an I-shape specimen, encompassing elastoplasticity, hardening, necking, crack initiation and propagation, in contrast to the related experimental results. The validated model is further employed to simulate the multiphysics hot cracking phenomenon in additive manufacturing in the context of both the effective powder-bed model and the powder-resolved model thanks to prior non-isothermal phase-field powder-bed-fusion simulations. Simulation results reveal certain key features of the hot crack and its dependency on process parameters like beam power and scan speed, which are helpful for the fundamental understanding of crack formation mechanisms and process optimization. [ABSTRACT FROM AUTHOR]

• High spatial and temporal resolution synchrotron X-ray CT enabled 4D imaging of void nucleation, growth, and coalescence with unprecedented details. • Large inclusion particles in SA508 steel led to large and prolate voids which, while striking, did not affect crack formation. • The final fracture was induced by densely populated carbide precipitates that led to small but closely spaced voids which quickly coalesced to form micro-cracks. • The results challenge established theories on ductile fracture and could potentially lead to new strategies for steel making. Samples of SA508 grade 3 nuclear pressure vessel ferritic steel were subjected to tensile straining whilst being simultaneously imaged in 3D in real time using high resolution, high frame rate time-lapse synchrotron computed tomography (CT). This enabled direct observation of void development from nucleation, through growth to coalescence and final failure validating many inferences made post-mortem or by theoretical models, as well as raising new points. The sparse, large inclusions were found to nucleate voids at essentially zero plastic strain (consistent with zero interfacial strength); these became increasingly elongated with straining. In contrast, a high density of small spherical voids were found to nucleate from the sub-micron cementite particles at larger strains (> 200%) only in the centre of the necked (high triaxiality) region. An interfacial strength approaching 2100 MPa was inferred and soon after their nucleation, these small voids coalesce to form internal microcracks that lead to the final failure of the specimen. Perhaps surprisingly, under these conditions of generally low triaxial constraint the large voids are simply cut across and appear to play no significant role in determining the final failure. The implications of these results are discussed in terms of ductile fracture behaviour and the Gurson model for ductile fracture. [Display omitted] [ABSTRACT FROM AUTHOR]

• TRC and DC Al strips subjected to thermal exposure at 300–600 °C. • The evolution of microstructure and mechanical properties during thermal exposure. • The thermal stability of grain morphology rationalized by cahn-lücke-stüwe model. • Mechanical properties predicted by strengthening model/multiscale fracture model. Al-Mn-Fe-Si strips were fabricated via both the twin-roll casting (TRC) and the more conventional route, direct-chill casting (DC). The two types of strips prepared were subjected to thermal exposure at a series of temperatures. Uniaxial tensile tests after the thermal exposure showed that while the DC strip presented a ∼74% decrease in the yield strength and ∼35% decrease in the ultimate tensile strength (UTS) after being exposed to 350 °C for 12 h, the TRC strip, in contrast, maintained its strength at temperatures up to ∼460 °C for the same duration. Systematic microstructure characterization revealed that the different thermal stability in the strength of the two types of strips arised from their distinct evolution in grain morphology and second phase particles during the thermal exposure. The calculation based on Cahn-Lücke-Stüwe (CLS) model suggested that due to the highly supersaturated solute atoms, at the beginning of the thermal exposure, the TRC strip experienced a strong solute drag which reduced the grain boundary migrating velocity to a value that is orders of magnitude smaller than that in the DC strip. With the progress of the thermal exposure, the solute atoms precipitated out, forming densely distributed second phase particles. For one thing, these particles stabilized the grain structure by inducing Zener pinning pressure which could be ten times higher than that in the DC strip, depending on the temperature. For another, they acted as dislocation obstacles and compensated for the strength loss owing to decreasing solution hardening. Both effects contributed to the TRC strip's fairly stable strength regarding thermal exposure below 460 °C. The present work could guide the direct application of the TRC strips in the industry. The results should also be helpful for the development of a fundamental framework for designing advanced TRC Al strips with improved mechanical properties at elevated temperatures. [ABSTRACT FROM AUTHOR]

The concept of energy release rate proposed by Griffith originally used for elastic brittle material was extended for ductile material by Irwin and Orowan using the sum of the surface energy and the work of plastic dissipation as fracture resistance. This fracture resistance, however, depends on the load type and specimen geometry, which hinders its wide application. In contrast, in this paper, a finite crack growth energy release rate is proposed by assuming a finite crack growth, which is conveniently applicable to finite element analysis. The effect of the plastic dissipation on the crack growth is considered in the driving force. Elastic-plastic finite element analysis is used to apply the present energy release rate. Compared to the experiment results, it is shown that for a fixed finite crack growth, a constant critical energy release rate can well predict the stable crack growths and residual strengths of sheets with single and various collinear cracks subjected to different types of loads. The present finite crack growth energy release rate is conceptually simple, has a solid physical foundation, and would be appealing for ductile crack growth analysis. [ABSTRACT FROM AUTHOR]

In ductile materials containing micro-voids, diffuse plasticity or localisation of plastic strain in narrow bands bridging the ligaments between voids can both occur as precursors to failure. In particular, localisation of plastic strain leads to coalescence through the formation of void sheets or plastic collapse of the ligament. If yielding is defined as the point in macro stress space at which the macroscopic plastic dissipation becomes large, either diffuse plasticity or localisation can cause yielding. Appropriate combinations of triaxiality T and Lode parameter L can cause localisation to occur earlier, and thereby modify the yield surface significantly. This is more likely to happen at high values of porosity. The competition between the two modes of yielding has been captured by a recently proposed multi-surface yield framework (Keralavarma, S. M., 2017, "A multi-surface plasticity model for ductile fracture simulations," Journal of the Mechanics and Physics of Solids, 103, pp. 100–120), where the competition between the Gurson criterion for yielding by diffuse plastic flow and a criterion for localized yielding within discrete coalescence bands leads to a piecewise smooth yield locus with sharp vertices. In the present work, we generate yield surfaces computationally by using a voided, cuboidal unit cell and a computational homogenisation framework that allows for both macro deformation gradient and macro Cauchy stress control. The basic aim is to see how the multi-surface framework of yield compares with macro yield loci generated computationally using a formulation where both finite deformations and void shape changes are allowed. We show, for spherical, prolate and oblate initial voids, that localisation inevitably hastens macro yield and adds sharp vertices to the yield locus, for a wide range of L and T. The multi-surface framework, at least for spherical initial voids, is remarkably successful in capturing this competition. [ABSTRACT FROM AUTHOR]

• Micromechanisms of crack advance in heterogeneous ductile materials are identified. • Microstructure & material parameter effects on these micromechanisms are quantified. • The experimentally observed microstructure-fracture correlations are rationalized. • The results presented herein provide guidelines to increase fracture toughness. The objective is to identify the micromechanisms of ductile crack advance, and isolate the key microstructural and material parameters that affect these micromechanisms and fracture toughness of ductile structural materials. Three dimensional, finite element, finite deformation, small scale yielding calculations of mode I crack growth are carried out for ductile material matrix containing two populations of void nucleating particles using an elasto-viscoplastic constitutive framework for progressively cavitating solid. The larger particles or inclusions that result in void nucleation at an early stage are modeled discretely while smaller particles that require large strains to nucleate voids are homogeneously distributed. The size, spacing and volume fraction of inclusions introduce microstructure-based length-scales. In the calculations, ductile crack growth is computed and fracture toughness is characterized. Several features of crack growth behavior and dependence of fracture toughness on microstructural and material parameters observed in experiments, naturally emerge in our calculations. The extent to which the microstructural and material parameters affect the micromechanisms of ductile crack advance and, hence, the macroscopic fracture toughness of the material is discussed. The results presented provide guidelines for microstructural engineering to increase ductile fracture toughness, for example, the results show that for a material with small inclusions, increasing the mean inclusion spacing has a greater effect on fracture toughness than for a material with large inclusions. [ABSTRACT FROM AUTHOR]

Abstract Voids can be observed at various scales in ductile materials, frequently of sizes lower than the grain size, leading to porous single crystals materials. Two local deformation modes of porous ductile (single crystals) materials have been identified and referred to as void growth and void coalescence, the latter being characterized by strong interactions between neighboring voids. A simple semi-analytical coalescence criterion for porous single crystals with periodic arrangement of voids is proposed using effective isotropic yield stresses associated with a criterion derived for isotropic materials. An extension of the coalescence criterion is also proposed to account for shear with respect to the coalescence plane. Effective yield stresses are defined using Taylor theory of single crystal deformation, and rely ultimately on the computation of average Taylor factors. Arbitrary sets of slip systems can be considered. The coalescence criterion is assessed through comparisons to numerical limit-analysis results performed using a Fast-Fourier-Transform based solver. A good agreement is observed between the proposed criterion and numerical results for various configurations including different sets of slip systems (Face-Centered-Cubic, Hexagonal-Close-Packed), crystal orientations, void shapes and loading conditions. The competition between void growth and void coalescence is described for specific conditions, emphasizing the strong influence of both crystal orientation and void lattice, as well as their interactions. [ABSTRACT FROM AUTHOR]

Abstract This research work proposes and validates a damage regularisation model for shell elements used in large-scale simulations. The model evaluates the ratio of bending to membrane loading in the elements based on the through-thickness gradient of the through-thickness plastic strain. The Cockcroft–Latham failure criterion is adopted, whose parameters are modified according to the length-to-thickness ratio of the shell elements in order to reduce the mesh dependency. This regularisation scheme is validated against experimental component tests on a double-chamber profile in an AA6005-T6 recrystallised aluminium alloy under quasi-static and impact loading conditions. The results show that the model is able to accurately predict fracture initiation under all tested conditions. [ABSTRACT FROM AUTHOR]

The paper discusses from first principles all aspects relevant to the plasticity of porous materials. Emphasis is laid on unhomogeneous yielding, defined as the process of yielding and plastic flow under gradient-free macroscopically nonuniform deformation. The nonuniformity is represented by strain localization in one or more bands of finite thickness. A universal feature of all intrinsic yield criteria is their dependence upon the normal and shear tractions resolved on the band. When specialized to isotropy, a Mohr–Coulomb criterion and a Rankine–Tresca criterion emerge as two extremes. The latter is an ideal that typifies the yield behavior of porous materials under arbitrary loadings. The general theory stands for a finite number of bands or yield systems. Its overall structure bears some features of crystal plasticity, but with dependence upon the resolved normal stress. The evolution of microstructural parameters can be given in general terms, being solely based on the kinematic constraints of unhomogeneous yielding and matrix incompressibility. Throughout the paper, the competition with homogeneous yielding, heretofore taken for granted, is analyzed with or without strain and strain-rate hardening effects. We close by discussing the thermodynamic consistency of this new class of constitutive relations and a link to strain-gradient theories. [ABSTRACT FROM AUTHOR]

This paper presents a unit cell model based on the observation that ductile fracture occurs when plastic flow is localized in a band. The unit cell consists of three void containing material units stacked in the direction normal to the localization plane. Localization takes place in the middle material unit while the two outer units undergo elastic recovery after failure occurs. Thus a failure criterion is established as when the macroscopic effective strain of the outer material units reaches the maximum value. Analyses are conducted to demonstrate the effect of the voids existing outside the localization band. Comparisons of the present model with several previous models suggest that the present model is not only easy to implement in finite element analysis but also more suitable to robustly determine the failure strain. A series of unit cell analyses are conducted for various macroscopic stress triaxialities and Lode parameters. The analysis results confirm that for a fixed Lode parameter, the failure strain decreases exponentially with the stress triaxiality and for a given stress triaxiality, it increases as the stress state approaches the generalized tension and generalized compression. The analysis results also reveal the effect of the stress state on the deformed void shape within and near the localization band. [ABSTRACT FROM AUTHOR]

Ductile fracture through void growth and coalescence depends significantly on the plastic anisotropy of the material and on void size, as shown by experiments and/or numerical simulations through several studies. Macroscopic (homogenized) yield criteria aiming at modeling nanoporous materials have been proposed only for the growth regime, i.e. non-interacting voids. The aim of this study is thus to provide a yield criterion for nanoporous materials relevant for the coalescence regime, i.e. when plastic flow is localized between voids. Through homogenization and limit analysis, and accounting for interface stresses at the void-matrix interface, analytical coalescence criterion is derived under the following conditions: axisymmetric loading, orthotropic material obeying Hill’s plasticity, cylindrical voids in cylindrical unit-cell. Incidentally, an orthotropic extension of the existing isotropic modeling of interface stresses through limit analysis is described and used. The proposed coalescence criterion is then extended to account for combined tension and shear loading conditions. Numerical limit analyses have been performed under specific conditions / materials parameters to get supposedly exact (up to numerical errors) results of coalescence stress. A good agreement between the analytical coalescence criteria derived in this study and numerical results is found for elongated spheroidal voids, making them usable to predict the onset of void coalescence in ductile fracture modeling of nanoporous materials. [ABSTRACT FROM AUTHOR]

Tensile properties and fracture characteristics of high purity polycrystalline nickel were investigated using micro-tensile specimens. The material response to the applied load was found to be sensitive to both geometry and sample size. The post-tests examination of the fracture surfaces revealed that the mechanisms leading to failure were associated with the nucleation, growth and coalescence of voids. These mechanisms were studied using finite element implementation of the Gurson–Tvergaard–Needleman (GTN) model. A physical meaning of the model parameters was proposed and validated against experimental data. The model provided good predictions of the failure mode, but did not capture the variability observed for the micro-tensile specimens. The factors such as machining process, surface roughness, and local variations in the microstructure were most likely responsible for these differences. [ABSTRACT FROM AUTHOR]

We explore the possibility of engineering the crack path by controlling a material's microstructure in order to increase its crack growth resistance. Attention is confined to a specific type of microstructure that is encountered in a variety of structural metals and alloys - second phase particles distributed in a ductile matrix. The type of controlled microstructure modeled is characterized by various sinusoidal distributions of particles with fixed mean particle spacing. Three dimensional, finite deformation small scale yielding calculations of mode I crack growth are carried out for such controlled microstructures using an elastic-viscoplastic constitutive relation for a progressively cavitating solid. The results show that appropriately engineered sinusoidal distributions of particles can give fracture toughness values 2 to 3 times greater than a random distribution of particles with the same mean particle spacing. Tearing modulus values can be increased by a factor of 1.5 to 2. The greatest crack growth resistance generally occurs when the amplitude and the wavelength of the sinusoidal distribution are increased together. When the amplitude and the wavelength of the sinusoidal distribution do not increase together the crack can jump from one crest (or trough) to the next crest (or trough) which tends to reduce the crack growth resistance. Fracture surface roughness statistics are also calculated. In contrast to the essentially universal value for random distributions of particles, the value of the computed Hurst exponent is found to depend on the amplitude and the wavelength of the sinusoidal profile. A correlation is found between the computed fracture toughness values and values of characteristic length scales of the fracture surface roughness. [ABSTRACT FROM AUTHOR]

Building on an analogy to ductile fracture mechanics, we investigate the energetic cost of debris particle creation during adhesive wear. Macroscopically, Reye proposed in 1860 that there is a linear relation between frictional work and wear volume at the macroscopic scale. Earlier work suggested a linear relation between tangential work and wear debris volume also exists at the scale of a single asperity, assuming that the debris size is proportional to the micro contact size multiplied by the junction shear strength. However, the present study reveals deviations from linearity at the microscopic scale. These deviations can be rationalized with fracture mechanics and imply that less work is necessary to generate debris than what was assumed. Here, we postulate that the work needed to detach a wear particle is made of the surface energy expended to create new fracture surfaces, and also of plastic work within a fracture process zone of a given width around the cracks. Our theoretical model, validated by molecular dynamics simulations, reveals a super-linear scaling relation between debris volume (V d) and tangential work (W t): V d ∼ W t 3 / 2 in 3D and V d ∼ W t 2 in 2D. This study provides a theoretical foundation to estimate the statistical distribution of sizes of fine particles emitted due to adhesive wear processes. [ABSTRACT FROM AUTHOR]

The gap test is a new type of fracture test developed in 2020, in which the end supports of a notched beam are installed with gaps that close only after the elasto-plastic pads next to notch introduce a desired constant crack-parallel compression σ x x (also called the T-stress). The test uses the size effect method to identify how such a compression alters the material fracture energy, G f , and the characteristic size c f of the fracture process zone (FPZ). In 2020, experiments showed that a moderate σ x x doubled the G f of a quasibrittle material (concrete) and a high σ x x reduced its G f to almost zero. A preliminary study by Nguyen et al. (2021) showed that the gap test can be extended to plastic-hardening polycrystalline metals. A generalized scaling law with an intermediate asymptote for large-scale yielding in small structures was derived, and limited tests of aluminum alloy showed its applicability. In this study, geometrically scaled gap tests of notched three-point bend fracture specimens of aluminum are conducted at three different levels of σ x x. An extended structural strength scaling law that captures the transition from the micrometer-scale FPZ through millimeter-scale yielding zone (YZ) to large-scale structures which follow linear elastic fracture mechanics (LEFM) is derived and then applied to analyze the effect of σ x x. Presented here are the gap tests of aluminum alloy, in which three different levels of σ x x are applied to scaled notched four-point-bend beams of depths D = 12, 24, 48 and 96 mm. Using an extended size effect law for plastic-hardening metals, it is found that, at crack-parallel stress σ x x ≈ − 40 % of the yield strength, the critical J-integral value gets roughly quadrupled, not only because of the well-known enlargement of the hardening YZ whose width is of millimeter scale, but also because of the increase of the FPZ width of micrometer scale. These results can be reproduced neither by line crack models, including the LEFM, cohesive crack and phase-field models, nor by peridynamic and various nonlocal models that ignore the tensorial nature of the material stress at the crack tip. The crack band models, being able to represent an FPZ of finite width and a YZ whose size evolves depending on σ x x , can capture the effect of crack-parallel stresses provided that a realistic 3D tensorial damage constitutive model is used. Here, Bai–Wierzbicki's model is shown to capture the σ x x effect on the G f and J c r qualitatively. [ABSTRACT FROM AUTHOR]

Ductile fracture through growth and coalescence of intergranular cavities is a failure mode observed experimentally in many metallic alloys used in industrial applications. Simulation of this fracture process in polycrystalline aggregates requires modeling of the plastic yielding of porous boundaries. However, classical yield criteria for porous materials such as the Gurson–Tvergaard–Needleman model and its current extensions cannot account for the complex coupling between loading state, crystallographic orientations, void shape and material behavior at grain boundaries. In order to bridge this modeling gap, two yield criteria for intergranular ductile void growth are proposed. The first one is a GTN-like model derived from limit-analysis which, once calibrated, accounts for spherical voids at the interface of rate-independent crystals. The second one is developed from a variational approach and predicts yielding in viscoplastic crystals containing intergranular ellipsoidal cavities. Both models are validated against a wide database of numerical limit-analysis of porous bi-crystals using a FFT solver. Satisfying agreements are obtained, paving the way to microstructure-informed intergranular ductile fracture simulation. The interplay between plastic yielding inside grains and along grain boundaries is finally studied based on the proposed yield criteria. [ABSTRACT FROM AUTHOR]

Void coalescence in columns (or necklace coalescence) is a computationally confirmed and physically observed mechanism of void link-up in metal alloys and polymers that has received little attention in the literature. Here, analytical treatment of the phenomenon proceeds from first principles of limit analysis and homogenization theories. A cylindrical unit cell embedding a cylindrical void of finite height is considered under axially symmetric loading. Two types of trial velocity fields are used in seeking an upper bound to the yield criterion corresponding to the particular regime of coalescence in columns. For each type, exact expressions of the overall yield criterion are obtained, albeit in implicit form when using continuous fields. Upon comparison with other modes of yielding allowing for void growth and coalescence in layers, an actual effective yield domain is obtained so as to ascertain regimes of stress state and microstructural states where void coalescence in columns prevails. The predictions are also assessed against finite element based limit analysis. [ABSTRACT FROM AUTHOR]

We present the first mixed mode phase-field model of ductile fracture. The contribution of crack opening and shearing deformations to the propagation of a crack is expressed by introducing two phase fields. Constitutive relations are then introduced to couple and distinguish these phase fields. Special attention is given to the maximum shear stress and its effect on the development of fractures. The proposed model is validated by tensile testing experiments found in the literature on Al 2024 T-351. Model predictions of the crack path and load-displacement response are compared to a plane-strain tension experiment, a round bar tension experiment and a notched round bar tension experiment. The model is shown to accurately capture the slant and cup-cone crack paths as well as force-displacement curves. [Display omitted] [ABSTRACT FROM AUTHOR]

Strain localization is often a precursor to the ductile failure of materials. This paper investigates plastic strain localization phenomena in the case of random microstructures, namely cubic cells made of an elastic-perfectly plastic matrix embedding distribution of identical non-overlapping spherical voids. The consideration of random microstructures allows for a better representation of the interaction between voids and greater diversity of failure modes than single-void (or unit) cells. The cells are simulated by means of the finite element (FE) method for proportional stress loading paths with controlled stress triaxiality and Lode parameters. Strain localization is detected by Rice's criterion computed at the cell level. This criterion is shown to accurately capture the onset of localization and the type of failure mode, either extension or shear banding. Moreover, the influence of the loading orientation, i.e. the orientation of the principal axes of the applied stress tensor with respect to the microstructure cube, is systematically studied. Significant anisotropy of failure behavior is observed, especially in the case of single void unit cells, which can be attributed to the intrinsic anisotropy of the simulation cells. Finally minimal failure strain values at localization with respect to all loading orientations are found. A zone of reduced ductility is observed under generalized shear loading conditions. • Ductile fracture is studied by simulating the failure of random microstructures. • Rice's criterion allows to detect localization and shear/extension failure modes. • A strong anisotropy of failure behavior is observed when rotating the loading. • This intrinsic anisotropy is stronger in single void cells than in random cells. • Minimal failure strain (T,L) response shows low ductility under generalized shear. [ABSTRACT FROM AUTHOR]

The variation of ductile crack growth resistance and fracture surface roughness with loading rate is modeled under mode I plane strain, small scale yielding conditions. Three-dimensional calculations are carried out using an elastic–viscoplastic constitutive relation for a progressively cavitating solid with two populations of void nucleating second phase particles. Larger inclusions that result in void nucleation at an early stage are modeled as discrete void nucleation sites while smaller particles that require large strains to nucleate voids are homogeneously distributed. The calculations are carried out for two values of density of the larger inclusions, 3.6% and 7.1%, and for prescribed loading rates K ̇̇ I ranging from 1 × 10 5 MPa m s − 1 to 5 × 10 7 MPa m s − 1 . The ductile fracture mode is found to undergo a transition from one that can be regarded as growth of a dominant main crack at the lower loading rates to one dominated by damage nucleation and micro-cracking ahead of the main crack at the higher loading rates. The values of J IC , the tearing modulus, T R , the total plastic dissipation and the plastic dissipation in the fracture process region are all found to increase with increasing loading rate. However, the ratio of plastic dissipation in the fracture process region to total plastic dissipation decreases with increasing prescribed loading rate. The fracture surfaces are found to display two self-affine regimes, with a Hurst exponent β ≈ 0.60 at small length scales and with β ≈ 0.45 at larger length scales. The multi-fractal spectra indicate multi-affine behavior in most cases but a range of loading rates and length scales exhibiting mono-affine behavior is also found. Parameters characterizing the fracture surface statistics, including the length scale at which a transition from a power law tail to an exponential tail occurs, are related to the mode of crack growth/damage accumulation. A linear relation is found between the values of J IC and T R and the saturation value of the correlation function of the surface roughness for K ̇̇ I < 3 × 10 7 MPa m s − 1 . [ABSTRACT FROM AUTHOR]

Ductile failure by void growth in an elasto-plastic material subjected to non-proportional loading, involving step changes in the stress triaxiality or the Lode parameter, is investigated using periodic unit cell model simulations. The equivalent strains to failure by the onset of void coalescence, defined as the localization of plasticity along a band of voids at the micro-scale, are determined as a function of the loading path parameters. The observed trends for the ductility under non-proportional loading are found to be inconsistent with the predictions of a continuum damage mechanics model based on the attainment of a constant critical damage variable, even if the model has been calibrated to predict accurate results for the ductility under proportional loading. It is shown that a recently developed failure criterion based on the onset of plastic instability in a porous material, coupled with a micromechanics-based void growth law, predicts the loading path dependence of failure under non-proportional loading, in better agreement with the cell model simulation results than the continuum damage model. In particular, the triaxiality dependence of the ductility is found to be primarily due to the hydrostatic stress dependence of void growth, while the Lode dependence is primarily due to the instability-based failure criterion with a relatively minor effect of the Lode parameter on void growth. Implications of these findings on the current modeling approaches to ductile failure under shear dominated loading are discussed. • Ductile failure under non-proportional loading is studied using unit cell simulations. • Effect of changes in both the stress triaxiality and Lode parameter are investigated. • Failure loci are inconsistent with a critical damage variable based failure criterion. • A plastic instability based criterion predicts the correct shapes of the failure loci. • The Lode effect on ductility is shown to be primarily due to the failure criterion. [ABSTRACT FROM AUTHOR]

Three different micromechanically-based computational models for fracture in porous ductile solids are compared and assessed. Model A is a unit cell model of a porous ductile solid comprising a uniform periodic distribution of voids subjected to normal macroscopic loading. Models B and C, on the other hand, are unit-cell type models that represent an imperfection band governed by a doubly periodic array of voids separating two non-porous outer blocks. The outer blocks have a finite size in Model B and are semi-infinite in Model C. The non-porous material surrounding the voids, and the material of the outer blocks in Model B and Model C, are considered as an elasto-plastic isotropic material. Numerical simulations are performed for a wide range of macroscopic stress states. For each model, various criteria for determining the onset of ductile failure are evaluated to demonstrate their impact on the failure predictions. The results show that the failure loci strongly depend on the computational model and failure criterion employed. Thus, these three models cannot be used interchangeably – neither to investigate failure mechanisms nor to develop or calibrate fracture models – and an unambiguous failure criterion must be chosen. [ABSTRACT FROM AUTHOR]

Abstract: The objective of this work is to derive an analytical solution for the problem of a second gradient plastic circular thin-walled tube in combined tension and torsion loading conditions. Explicit expressions for the ordinary and higher-order stresses, strain rate and its gradient, and displacement are obtained. For comparison purpose, we also present the analytical solution for the same problem when the thin-walled tube is made of classical (von Mises) plastic metals. The newly micromorphic model based solution reduces to that of the (ordinary) von Mises one when the characteristic length scale approaches zero. [Copyright &y& Elsevier]

Abstract: The effect of the stress state on the localization of plastic flow in a Levy–von Mises material is investigated numerically. A unit cell model is built with a spherical central void that acts as a defect triggering the onset of flow localization along a narrow band. Periodic boundary conditions are defined along all boundaries of the unit cell. Shear and normal loading is applied such that the macroscopic stress triaxiality and Lode parameter remain constant throughout the entire loading history. Due to the initially orthogonal symmetry of the unit cell model the deformation-induced anisotropy associated with void shape changes, both co-rotational and radial loading paths are considered. The simulation results demonstrate that the macroscopic equivalent plastic strain at the onset of localization after monotonic proportional loading decreases in stress triaxiality and is a convex, non-symmetric function of the Lode parameter. In addition to predicting the onset of localization through unit cell analysis, an analytical criterion is proposed for monotonic proportional loading which defines an open convex envelope in terms of the shear and normal stresses acting on the plane of localization. [Copyright &y& Elsevier]

Abstract: Three dimensional calculations of ductile fracture under mode I plane strain, small scale yielding conditions are carried out using an elastic-viscoplastic constitutive relation for a progressively cavitating solid with two populations of void nucleating second phase particles. Larger inclusions that result in void nucleation at an early stage are modeled discretely while smaller particles that require large strains to nucleate voids are homogeneously distributed. Full field solutions are obtained for eight volume fractions, ranging from 1% to 19%, of randomly distributed larger inclusions. For each volume fraction calculations are carried out for seven random distributions of inclusion centers. Crack growth resistance curves and fracture surface roughness statistics are calculated using standard procedures. The crack growth resistance is characterized in terms of both J IC and the tearing modulus T R . For all volume fractions considered, the computed fracture surfaces are self-affine over a size range of nearly two orders of magnitude with a microstructure independent roughness exponent of 0.53 with a standard error of 0.0023. The cut-off length of the scale invariant regime is found to depend on the inclusion volume fraction. Consideration of the full statistics of the fracture surface roughness revealed other parameters that vary with inclusion volume fraction. For smaller values of the discretely modeled inclusion volume fraction ( ), there is a linear correlation between several measures of fracture surface roughness and both J IC and T R . In this regime crack growth is dominated by a void-by-void process. For greater values of the discretely modeled inclusion volume fraction, crack growth mainly involves multiple void interactions and no such correlation is found. [Copyright &y& Elsevier]

Abstract: We perform an optimal-scaling analysis of ductile fracture in metals. We specifically consider the deformation up to failure of a slab of finite thickness subject to monotonically increasing normal opening displacements on its surfaces. We show that ductile fracture emerges as the net outcome of two competing effects: the sublinear growth characteristic of the hardening of metals and strain-gradient plasticity. We also put forth physical arguments that identify the intrinsic length of strain-gradient plasticity and the critical opening displacement for fracture. We show that, when J c is renormalized in a manner suggested by the optimal scaling laws, the experimental data tends to cluster—with allowances made for experimental scatter—within bounds dependent on the hardening exponent but otherwise material independent. [Copyright &y& Elsevier]

Abstract: The damage that occurs in shear deformations in view of anisotropy evolution is investigated. It is widely believed in the mechanics research community that damage (or porosity) does not evolve (increase) in shear deformations since the hydrostatic stress in shear is zero. This paper proves that the above statement can be false in large deformations of simple shear. The simulation using the proposed anisotropic ductile fracture model (macro-scale) in this study indicates that hydrostatic stress becomes nonzero and (thus) porosity evolves (increases or decreases) in the simple shear deformation of anisotropic (orthotropic) materials. The simple shear simulation using a crystal plasticity based damage model (meso-scale) shows the same physics as manifested in the above macro-scale model that porosity evolves due to the grain-to-grain interaction, i.e., due to the evolution of anisotropy. Through a series of simple shear simulations, this study investigates the effect of the evolution of anisotropy, i.e., the rotation of the orthotropic axes onto the damage (porosity) evolution. The effect of the evolutions of void orientation and void shape onto the damage (porosity) evolution is investigated as well. It is found out that the interaction among porosity, the matrix anisotropy and void orientation/shape plays a crucial role in the ductile damage of porous materials. [Copyright &y& Elsevier]

Abstract: A modified-JIC test on CT (compact tension) specimens of an alloy (Ti-Modified 4330 steel) was carried out. The microstructure (primary and secondary inclusions) in the fracture process zone and fracture surface are reconstructed with a microtomography technique. The zig-zag fracture profile resulting from nucleation of microvoid sheets at the secondary population of inclusions is observed. Embedding the experimentally reconstructed microstructure into the fracture process zone, the ductile fracture process occurring at different length scales within the microstructure is modeled by a hybrid multiresolution approach. In combination with the large scale simulation, detailed studies and statistical analysis show that shearing of microvoids (the secondary population of voids) determines the mixed mode zig-zag fracture profile. The deformation in the macro and micro zones along with the interaction between them affects the fracture process. The observed zig-zag fracture profile in the experiment is also reasonably captured. Simulations can provide a more detailed understanding of the mechanics of the fracture process than experiments which is beneficial in microstructure design to improve performance of alloys. [Copyright &y& Elsevier]

Abstract: A combined 3D finite element simulation and experimental study of interaction between a notch and cylindrical voids ahead of it in single edge notch (tension) aluminum single crystal specimens is undertaken in this work. Two lattice orientations are considered in which the notch front is parallel to the crystallographic direction. The flat surface of the notch coincides with the (010) plane in one orientation and with the plane in the other. Three equally spaced cylindrical voids are placed directly ahead of the notch tip. The predicted load–displacement curves, slip traces, lattice rotation and void growth from the finite element analysis are found to be in good agreement with the experimental observations for both the orientations. Finite element results show considerable through-thickness variation in both hydrostatic stress and equivalent plastic slip which, however, depends additionally on the lattice orientation. The through-thickness variation in the above quantities affects the void growth rate and causes it to differ from the center-plane to the free surface of the specimen. [Copyright &y& Elsevier]

Abstract: A numerical procedure based on the Boundary Element Method with internal cells and dedicated to the simulation of the ductile tearing of thin metal sheets is presented. Plasticity is handled with an integral formulation based on the initial strain approach involving a discretization of the planar domain. Time integration is performed in an implicit way for the local strain–stress relationships while the global algorithm relies on an explicit formulation. Damage is represented by the scalar parameter of the uncoupled local damage model of Rice and Tracey. Within the scope of our applications, the cracks propagate along paths a priori known. As damage spreads, boundary elements are gradually released. Elastoplastic problems with large yielding zones are solved and compared to reference solutions. At last, the ductile tearing of a specimen is addressed. The calibration of the critical damage parameter leads to numerical results in good agreement with the experimental ones. [Copyright &y& Elsevier]

Abstract: The predictive capabilities of the shear-modified Gurson model [Nielsen and Tvergaard, Eng. Fract. Mech. 77, 2010] and the Modified Mohr–Coulomb (MMC) fracture model [Bai and Wierzbicki, Int. J. Fract. 161, 2010] are evaluated. Both phenomenological fracture models are physics-inspired and take the effect of the first and third stress tensor invariants into account in predicting the onset of ductile fracture. The MMC model is based on the assumption that the initiation of fracture is determined by a critical stress state, while the shear-modified Gurson model assumes void growth as the governing mechanism. Fracture experiments on TRIP-assisted steel sheets covering a wide range of stress states (from shear to equibiaxial tension) are used to calibrate and validate these models. The model accuracy is quantified based on the predictions of the displacement to fracture for experiments which have not been used for calibration. It is found that the MMC model predictions agree well with all experiments (less than 4% error), while less accurate predictions are observed for the shear-modified Gurson model. A comparison of plots of the strain to fracture as a function of the stress triaxiality and the normalized third invariant reveals significant differences between the two models except within the vicinity of stress states that have been used for calibration. [Copyright &y& Elsevier]

Abstract: This paper develops an accurate and computationally efficient homogenization-based continuum plasticity-damage (HCPD) model for macroscopic analysis of ductile failure in porous ductile materials containing brittle inclusions. Example of these materials are cast alloys such as aluminum and metal matrix composites. The overall framework of the HCPD model follows the structure of the anisotropic Gurson–Tvergaard–Needleman (GTN) type elasto-plasticity model for porous ductile materials. The HCPD model is assumed to be orthotropic in an evolving material principal coordinate system throughout the deformation history. The GTN model parameters are calibrated from homogenization of evolving variables in representative volume elements (RVE) of the microstructure containing inclusions and voids. Micromechanical analyses for this purpose are conducted by the locally enriched Voronoi cell finite element model (LE-VCFEM) [Hu, C., Ghosh, S., 2008. Locally enhanced Voronoi cell finite element model (LE-VCFEM) for simulating evolving fracture in ductile microstructures containing inclusions. Int. J. Numer. Methods Eng. 76(12), 1955–1992]. The model also introduces a novel void nucleation criterion from micromechanical damage evolution due to combined inclusion and matrix cracking. The paper discusses methods for estimating RVE length scales in microstructures with non-uniform dispersions, as well as macroscopic characteristic length scales for non-local constitutive models. Comparison of results from the anisotropic HCPD model with homogenized micromechanics shows excellent agreement. The HCPD model has a huge efficiency advantage over micromechanics models. Hence, it is a very effective tool in predicting macroscopic damage in structures with direct reference to microstructural composition. [Copyright &y& Elsevier]

Machine learning techniques are increasingly used to extract important physical information from a broad range of materials and to identify their processing-structure–property relationships. In this work, a neural network framework is coupled with a crystal plasticity finite element based approach to predict the deformation and ductile failure behaviour of porous FCC single crystals when subjected to multi-axial loading conditions. The work relies on 3D unit cells with a centrally located spherical void to represent the microstructure of the porous single crystal material, and on a crystallographic slip-based crystal plasticity constitutive model to describe its deformation behaviour. Stress–strain data generated by unit cell finite element simulations are relied upon to construct the neural network model. Different strategies for the neural network input and output variables and parameters are first explored so as to optimise performance and accuracy. Both proportional and non-proportional loading conditions resulting from a constant and a varying stress triaxiality during deformation, respectively, are considered. The optimum neural network strategy is shown to successfully predict the behaviour of the porous single crystal under both proportional and non-proportional loading, albeit with the void behaviour at high stress triaxialities better described than that at low triaxialities. The results also reveal that the use of tensorial quantities for both stresses and strains as input and output neural network quantities is more suitable as a form of data representation for multiaxial loading conditions than uniaxial equivalent stress and strain quantities. It was also found that the inclusion of prior knowledge as neural network input quantities in the form of reference stress–strain solutions for the void-free single crystal considerably improves the predictive capabilities of the proposed data-driven approach, even when only a very limited number of training cases was used. [ABSTRACT FROM AUTHOR]

Abstract: The paper addresses the possibility to project both mechanical and thermal phenomena pertinent to the fracture process zone into a cohesive zone. A wider interpretation of the notion cohesive zone is thereby suggested to comprise not only stress degradation due to micro-cracking but also heat generation and energy transport. According to our experience, this widening of the cohesive zone concept allows for a more efficient finite element simulation of ductile fracture. The key feature of the formulation concerns the thermo-mechanical cohesive zone model, evolving within the thermo-hyperelastoplastic continuum, allowing for the concurrent modelling of both heat generation, due to the fracture process, and heat transfer across the fracture process zone. This is accomplished via thermodynamic arguments to obtain the coupled governing equation of motion, energy equation, and constitutive equations. The deformation map is thereby defined in terms of independent continuous and discontinuous portions of the displacement field. In addition, as an extension of the displacement kinematics, to represent the temperature field associated with the discontinuous heat flux across the fracture interface, a matching discontinuous temperature field involving the interface (or band) temperature is proposed. In the first numerical example, concerning dynamic quasi-brittle crack propagation in a thermo-hyperelastoplastic material, we capture the initial increase in temperature close to the crack surface due to the energy dissipating fracture process. In the second example, a novel application of ductile fracture simulation to the process of high velocity (adiabatic) cutting is considered, where some general trends are observed when varying the cutting velocity. [Copyright &y& Elsevier]

Abstract: A material force method is proposed for evaluating the energy release rate and work rate of dissipation for fracture in inelastic materials. The inelastic material response is characterized by an internal variable model with an explicitly defined free energy density and dissipation potential. Expressions for the global material and dissipation forces are obtained from a global balance of energy–momentum that incorporates dissipation from inelastic material behavior. It is shown that in the special case of steady-state growth, the global dissipation force equals the work rate of dissipation, and the global material force and J-integral methods are equivalent. For implementation in finite element computations, an equivalent domain expression of the global material force is developed from the weak form of the energy–momentum balance. The method is applied to model problems of cohesive fracture in a remote K-field for viscoelasticity and elastoplasticity. The viscoelastic problem is used to compare various element discretizations in combination with different schemes for computing strain gradients. For the elastoplastic problem, the effects of cohesive and bulk properties on the plastic dissipation are examined using calculations of the global dissipation force. [Copyright &y& Elsevier]

Special interface elements that account for ductile failure by the nucleation and growth of voids to coalescence are used to analyse crack growth. In these elements the stress component tangential to the interface is accounted for, as determined by the requirement of compatibility with the surrounding material in the tangential direction. Thus, the present interface description incorporates the important effect of stress triaxiality on damage evolution, which is not part of the usual cohesive zone models. The interface elements have been used previously for mode I loading conditions, but are here extended to cover non-symmetric mixed mode loading conditions for crack growth along an interface between dissimilar elastic–plastic solids. Crack growth resistance curves are calculated, and the dependence of the interface fracture toughness on the degree of mode mixity is studied. [Copyright &y& Elsevier]

A criterion for ductile rupture is derived using Rice's theory for macroscopic strain localization, and a constitutive relation for porous plastic solids that accounts for inhomogeneous yielding at the mesoscale. At the microscopic scale, it is assumed that failure occurs by void coalescence along a band of voids. An approximate, parameter-free closed form expression for the failure criterion is derived as a function of a single scalar damage variable –the porosity– and the macroscopic stress state, characterized by the stress triaxiality and Lode parameters. For practical applications, an uncoupled approach is developed whereby the failure criterion is supplemented with a damage-free plasticity model and a loading path dependent damage evolution law. The predictive capabilities of the approach are illustrated by comparisons with finite element cell model studies. In particular, the influence of strain hardening is investigated in some detail. [ABSTRACT FROM AUTHOR]

• Effects of quench rate on microstructure and mechanical properties are studied. • Transmission electron microscopy investigations are conducted. • Tensile tests on smooth and V-notch specimens and Kahn tear tests are performed. • The influence of stress state on plastic flow and fracture are investigated. • The nanostructure model NaMo was modified to account for precipitate free zones. The yielding, plastic flow and fracture of age hardenable aluminium alloys depend on the quench rate to room temperature after the solution heat-treatment at elevated temperature and before the artificial ageing. We investigate three AlMgSi alloys with different grain structure and crystallographic texture experimentally to determine the effects of quench rate (either water-quenching or air-cooling) on the precipitate microstructure and the mechanical properties, i.e., yield stress, work hardening and ductility. Tensile tests on smooth and V-notch specimens and Kahn tear tests are performed to study the influence of stress state on plastic flow and fracture. In addition, finite element simulations of the mechanical tests are performed for one of the alloys to investigate the validity of an extension of the Gurson model to high-exponent anisotropic plasticity. Transmission electron microscopy investigations show that the alloys and their precipitation microstructure are differently affected by the quench rate. Common for the three alloys is that the precipitate free zones around dispersoids and grain boundaries become larger, and the yield strength of the alloys becomes lower, after air-cooling than after water-quenching. The nanostructure model NaMo was modified to account for precipitate free zones, and was able to predict both the precipitation parameters and the tensile yield strength of all tempers and materials with a reasonable degree of accuracy, except in one case. In this case, the inhomogeneous precipitation in the material is too complex to be captured by the inherent precipitation model in NaMo. Due to the lower yield strength and higher work-hardening rate after air-cooling, the failure strain is increased for the smooth and V-notch tensile tests. The crack propagation energy, calculated from the Kahn tear tests, is markedly affected by the quench rate and the effect is different depending on the grain structure and plastic anisotropy, caused by the crystallographic texture. The anisotropic porous plasticity model used in the finite element simulations is able to precisely capture the fracture initiation in all the specimen geometries of the considered alloy, whereas the crack propagation energies of the Kahn tear tests are slightly overestimated. [ABSTRACT FROM AUTHOR]

• A unified dynamic fracture model which accounts for shear bands and cracks simultaneously is presented. • In this coupled thermo-mechanical problem, fracture is modeled by the phase field method. • A set of thermal-conductivity degradation functions are proposed, and derived from a novel micromechanics analytical approach using spherical harmonics. • It is shown that the thermal conductivity across cracks must be degraded to satisfy crack Neumann boundary conditions. • The standard quadratic degradation function in phase field has no physical basis and may poorly estimate the true heat transfer behavior. Dynamic loading of polycrystalline metallic materials can result in brittle or ductile fracture depending on the loading rates, geometry and material type. At high strain rates, mechanical energy due to plastic deformation may lead to significant temperature rise and shear localization due to thermal softening. These shear bands reduce the stress bearing capacity of the material and act as a precursor to ductile fracture (e.g. cracks that develop rapidly on top of a shear band). Understanding the heat transfer physics in thermo-mechanical problems when cracks are developed, is a fundamental science that has not been rigorously addressed. In particular, reliable damage models, that degrade the thermal-conductivity in continuum crack descriptions, are necessary to capture the correct heat transfer physics across fracture surfaces but are currently absent. In this paper, a novel set of isotropic thermal-conductivity degradation functions is derived based on a micro-mechanics void extension model of Laplace's equation. The key idea is to employ an analytical homogenization process to find the effective thermal-conductivity of an equivalent sphere with expanding spherical void. The closed form solution is obtained by minimization of the flux differences at the outer surfaces of the two problems, which can be achieved using the analytical solution of Laplace's equations, so called spherical-harmonics. A unified model, which has been developed in McAuliffe and Waisman (2015) to account for the simultaneous formation of shear bands and cracks is used herein as a numerical tool to investigate the behavior of the aforementioned thermal-conductivity model. In this unified model, the phase-field method is used to model crack initiation and propagation and is coupled to a temperature dependent visco-plastic model that captures shear bands. Two benchmark problems are presented to show the necessity of such physics-based degradation function in dynamic fracture problems. [ABSTRACT FROM AUTHOR]

• Plasticity and fracture of heat treated aluminum sheet is characterized. • New cruciform specimen is proposed that has proportional loading paths to fracture. • Fracture locus is determined by hybrid experimental-numerical method. • Plasticity modeling is critical for the fracture locus; Yld2004-18p criterion is used. The ductile fracture of an AA6111 aluminum sheet after a thermal cycle typical of auto-body paint-baking is investigated with the hybrid experimental-numerical method. The plastic flow of the material is examined by uniaxial tension, plane-strain tension, disk-compression and notched-tension experiments, that are used to calibrate the Yld2004-18p anisotropic yield criterion and the combined Swift-Voce hardening model. Then, the fracture behavior under equibiaxial and plane-strain tension, as well as uniaxial tension and shear, is characterized using a specially-developed cruciform specimen, along with center-hole and shear specimens, respectively. The cruciform fracture specimen proposed here contains two shallow hemispherical depressions (dimples) in the test-section, to initiate fracture. For the fracture characterization, special emphasis is put on specimen design, so that the stress states developed at the neighborhood of the fracture initiation point remain proportional throughout the loading history. In all experiments, the surface strain fields are measured by a stereo-type digital image correlation system. This information is used to validate finite element simulations of the fracture experiments. It is found that the Yld2004-18p model provides a better agreement with experiments than von Mises does, which underscores the sensitivity of the hybrid method to the plasticity models adopted. Once validated, these simulations are used to obtain the fracture loci in terms of two stress-state metrics, i.e., the stress triaxiality and Lode angle parameter. [ABSTRACT FROM AUTHOR]