Porous Plasticity Model Research Articles

Failure analysis based on microvoids damage model for DP600 steel on in-situ tensile tests

Considering the effect of shear loading, an original GTN damage model is extended by combining the damage mechanics concept and the porous plasticity model. The extended model in terms of various stress triaxiality was deduced in detail. The model is extended by coupling two damage parameters into the yield function. The implementation is achieved by the development of the user material subroutine. Internal damage evolution was observed by scanning electron microscopy (SEM). Parameters were determined from experimental data and previous studies. The damage model was applied to in-situ tensile tests to compare its predictability in various specimens. The obtained numerical results were in good agreement with experimental results, showing better ductile fracture predictions.

Unit cell simulations and porous plasticity modelling for recrystallization textures in aluminium alloys

The well-known Gurson model has been heuristically extended to incorporate effects of matrix anisotropy on the macroscopic yielding of porous ductile solids. Typical components of recrystallization textures for aluminium alloys were used to calibrate the Barlat Yld2004-18p yield criterion using a full-constraint Taylor homogenization method. The resulting yield surfaces were further employed in unit cell simulations using the finite element method. Unit cell calculations are invoked to investigate the evolution of the approximated micro structure under pre-defined loading conditions and to calibrate the proposed porous plasticity model. Numerical results obtained from the unit cell analyses demonstrate that anisotropic plastic yielding has great impact on the mechanical response of the approximated micro structure. Despite the simplifying assumptions that underlie the proposed constitutive model, it seems to capture the overall macroscopic response of the unit cell. However, to further enhance the numerical predictions, the model should be supplemented with a void evolution expression that accounts for directional dependency, and a void coalescence criterion in order to capture the last stages of deformation.

Numerical Modelling of Damage Evolution in Ingot Forging

The ingot forging process is numerically simulated applying both the Shima-Oyane porous plasticity model as a coupled damage model and the uncoupled normalized Cockcroft & Latham criterion. Four different cases including two different lower die angles (120o and 180o) and two different sizes of feed (400mm and 800mm) are analysed. Comparison of the simulation results with recommendations in literature on ingot forging, indicates the normalized Cockcroft & Latham damage criterion to be the most realistic of the two.

Semi-implicit finite strain constitutive integration of porous plasticity models

Two porous plasticity models, Rousselier and Gurson–Tvergaard–Needleman (GTN), are integrated with a new semi-implicit integration algorithm for finite strain plasticity. It consists of using relative Green–Lagrange during the iteration process and incremental frame updating corresponding to a polar decomposition. Lowdin׳s method of orthogonalization is adopted to ensure incremental frame-invariance. In addition, a smooth replacement of the complementarity condition is used. Since porous models are known to be difficult to integrate due to the combined effect of void fraction growth, stress and effective plastic strain evolution, we perform a complete assessment of our semi-implicit algorithm. Semi-implicit algorithms take advantage of different evolution rates to enhance the robustness in difficult to converge problems. A detailed description of the constitutive algorithm is performed, with the key components comprehensively exposed. In addition to the fully detailed constitutive algorithms, we use mixed finite strain elements based on Arnold׳s MINI formulation. This formulation passes the inf–sup test and allows a direct application with porous models. Isoerror maps for two common initial stress states are shown. In addition, we extensively test the two models with established benchmarks. Specifically, the cylindrical tension test as well as the butterfly shear specimen are adopted for validation. A 3D tension test is used to investigate mesh dependence and the effect of a length scale. Results show remarkable robustness.

Combining porous plasticity with Coulomb and Portevin-Le Chatelier models for ductile fracture analyses

For large void volume fraction, the Rousselier porous plasticity model transforms naturally into a coalescence model, in the whole range of stress triaxiality, in agreement with the necessary kinematic coalescence condition (NKCC). It enables to model slant fracture in notched tensile or cracked specimens. Nevertheless, void coalescence is not the single mechanism involved in slant fracture. That is why it is necessary to combine porous plasticity with other models. In this paper, the Coulomb fracture model and the Portevin-Le Chatelier (PLC) model (or dynamic strain aging: DSA) are formulated at the slip system scale. The Coulomb model combines the resolved normal and shear stresses for each slip plane and direction. For DSA, we postulate that each slip system has its own history of dislocation pinning and unpinning by solute atoms. The models are fully coupled in the framework of classical polycrystalline plasticity. A Reduced Texture Methodology (RTM) is used to provide the computational efficiency needed for numerical applications. The RTM approach involves a significant reduction of the number of representative crystallographic orientations. The models are applied to a notched tensile specimen taken from a 6260 aluminum alloy thin-walled extrusion. Fractographic examinations show a combination of dimples and large smooth areas on the slant fracture surface (mixed fracture). It highlights the need for combined fracture models. The PLC model gives very sharp oscillations of the macroscopic plastic strain rate, associated with moving plastic strain rate bands. It leads to a significant reduction of ductility compared to porous plasticity alone.

Numerical assessment of an anisotropic porous metal plasticity model

The objective of this paper is to perform numerical assessment of a micromechanical model of porous metal plasticity developed previously by the authors. First, upper bound estimates for the yield loci are computed using homogenization and limit analysis of a spheroidal representative volume element containing a confocal spheroidal void, neglecting elasticity. Unlike in the development of the analytical model, the computational limit analysis is performed without recourse to approximations so that the obtained yield loci are rigorous upper bounds for the true criterion. Next, the model’s macroscopic dilatancy at incipient plastic flow is compared against that of the numerical limit analysis approach. Finally, finite-element calculations, with elasticity included, are presented for transversely isotropic porous unit-cells loaded axisymmetrically. The effective stress–strain response as well as evolution of the unit-cell porosity and void aspect ratio are compared with theoretical predictions.

Modeling of Ductile Fracture at Engineering Scales: A Mechanism-Based Approach

ABSTRACTThis paper summarizes the work we conducted in recent years on modeling plastic response of metallic alloys and ductile fracture of engineering components, with the emphasis on the effect of the stress state. It is shown that the classical J2 plasticity theory cannot correctly describe the plasticity behavior of many materials. The experimental and numerical studies of a variety of structural alloys result in a general form of plasticity model for isotropic materials, where the yield function and the flow potential are expressed as functions of the first invariant of the stress tensor and the second and third invariants of the deviatoric stress tensor. Several mechanism-based models have been developed to capture the ductile fracture process of metallic alloys. Two of such models are described in this paper. The first one is a cumulative strain damage model where the damage parameter is dependent on the stress triaxiality and the Lode parameter. The second one is a modification to the Gurson-type porous plasticity models, where two damage parameters, representing void damage and shear damage respectively, are coupled into the yield function and flow potential. These models are shown to be able to predict fracture initiation and propagation in various specimens experiencing a wide range of stress states.

A multiscale phenomenological constitutive model for strain rate dependent tensile ductility in nanocrystalline metals

A variational multiscale constitutive model that accounts for strain rate dependent ductility of nanocrystalline materials during intergranular failure has been presented. The presented model is an extension of the previous work [1], in which a nanocrystalline material is modelled as two-phase with grain interior being modelled using crystal plasticity theory while grain boundary affected zone using porous plasticity model which accounts for ductile damage due to void growth and coalescence. The model capability of capturing the strain rate dependent deformation and failure has been demonstrated through validations against uniaxial test data taken from literature. The validated results show a good agreement between experimental and simulated response.

On the extension of the Gurson-type porous plasticity models for prediction of ductile fracture under shear-dominated conditions

One of the major drawbacks of the Gurson-type of porous plasticity models is the inability of these models to predict material failure under low stress triaxiality, shear dominated conditions. This study addresses this issue by combining the damage mechanics concept with the porous plasticity model that accounts for void nucleation, growth and coalescence. In particular, the widely adopted Gurson–Tvergaard–Needleman (GTN) model is extended by coupling two damage parameters, representing the volumetric damage (void volume fraction) and the shear damage, respectively, into the yield function and flow potential. The effectiveness of the new model is illustrated through a series of numerical tests comparing its performance with existing models. The current model not only is capable of predicting damage and fracture under low (even negative) triaxiality conditions but also suppresses spurious damage that has been shown to develop in earlier modifications of the GTN model for moderate to high triaxiality regimes. Finally the modified GTN model is applied to predict the ductile fracture behavior of a beta-treated Zircaloy-4 by coupling the proposed damage modeling framework with a recently developed J2–J3 plasticity model for the matrix material. Model parameters are calibrated using experimental data, and the calibrated model predicts failure initiation and propagation in various specimens experiencing a wide range of triaxiality and Lode parameter combinations.

Constitutive modeling of void-growth-based tensile ductile failures with stress triaxiality effects

In most metals and alloys, the evolution of voids has been generally recognized as the basic failure mechanism. Furthermore, stress triaxiality has been found to influence void growth dramatically. Besides strain intensity, it is understood to be the most important factor that controls the initiation of ductile fracture. We include sensitivity of stress triaxiality in a variational porous plasticity model, which was originally derived from hydrostatic expansion. Under loading conditions rather than hydrostatic deformation, we allow the critical pressure for voids to be exceeded so that the growth due to plasticity becomes dependent on the stress triaxiality. The limitations of the spherical void growth assumption are investigated. Our improved constitutive model is validated through good agreements with experimental data. Its capacity for reproducing realistic failure patterns is also indicated by a numerical simulation of a compact tensile (CT) test.

Prediction of the Sandia Fracture Challenge using a shear modified porous plasticity model

Blind analyses of the localization and ductile fracture behavior of an arbitrarily shaped coupon, performed for the Sandia Fracture Challenge, are presented. These analyses were performed using a shear-modified Gurson porous plasticity model that was calibrated using standard mechanical test data. The blind predictions were found to be in good agreement with experimental data, thus increasing confidence in the calibration methodology where test data is not available. In addition to the analyses submitted to the Challenge, additional non-blind analyses examining the role of both as-machined versus nominal coupon geometry and the selected value of initial porosity were conducted. It was found that by capturing the as-machined, out of tolerance, geometry in numerical calculations the fracture mode was altered from tension to shear. This agrees well with the experimental observation that all in-tolerance coupons failed in a tensile mode while all out-of-tolerance coupons failed in a shear mode. Additionally, it was found that by modifying the initial void volume fraction to a lower value still within the range that the calibration methodology provided, calculations capturing this as-machined geometry were able to closely match experimental results.

A phenomenological variational multiscale constitutive model for intergranular failure in nanocrystalline materials

We present a variational multiscale constitutive model that accounts for intergranular failure in nanocrystalline fcc metals due to void growth and coalescence in the grain boundary region. Following previous work by the authors, a nanocrystalline material is modeled as a two-phase material consisting of a grain interior phase and a grain boundary affected zone (GBAZ). A crystal plasticity model that accounts for the transition from partial dislocation to full dislocation mediated plasticity is used for the grain interior. Isotropic porous plasticity model with further extension to account for failure due to the void coalescence was used for the GBAZ. The extended model contains all the deformation phases, i.e. elastic deformation, plastic deformation including deviatoric and volumetric plasticity (void growth) followed by damage initiation and evolution due to void coalescence. Parametric studies have been performed to assess the model's dependence on the different input parameters. The model is then validated against uniaxial loading experiments for different materials. Lastly we show the model's ability to predict the damage and fracture of a dog-bone shaped specimen as observed experimentally.

The Effect of Random Voids in the Modified Gurson Model

The porous plasticity model (usually referred to as the Gurson–Tvergaard–Needleman model or modified Gurson model) has been widely used in the study of microvoid-induced ductile fracture. In this paper, we studied the effects of random voids on the porous plasticity model. Finite-element simulations were conducted to study a copper/tin/copper joint bar under uniaxial tension using the commercial finite-element package ABAQUS. A randomly distributed initial void volume fraction with different types of distribution was introduced, and the effects of this randomness on the crack path and macroscopic stress–strain behavior were studied. It was found that consideration of the random voids is able to capture more detailed and localized deformation features, such as different crack paths and different ultimate tensile strengths, and meanwhile does not change the macroscopic stress–strain behavior. It seems that the random voids are able to qualitatively explain the scattered observations in experiments while keeping the macroscopic measurements consistent.

Multiscale Modeling of Nanocrystalline Materials: A Variational Approach

AbstractThis paper presents a variational multi‐scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc‐material is modeled as a two‐phase material consisting of a grain interior (GI) phase and a grain boundary (GB) phase. A rate‐independent isotropic porous plasticity model is employed to describe the GB phase, whereas a crystal‐plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the GI phase. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor‐type homogenization. It is shown that the proposed model is able to capture the inverse Hall‐Petch effect. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Ultrasonic-assisted manufacturing processes: Variational model and numerical simulations

We present a computational study of ultrasonic assisted manufacturing processes including sheet metal forming, upsetting, and wire drawing. A fully variational porous plasticity model is modified to include ultrasonic softening effects and then utilized to account for instantaneous softening when ultrasonic energy is applied during deformation. Material model parameters are identified via inverse modeling, i.e. by using experimental data. The versatility and predictive ability of the model are demonstrated and the effect of ultrasonic intensity on the manufacturing process at hand is investigated and compared qualitatively with experimental results reported in the literature.

Dilatational viscoplasticity of polycrystalline solids with intergranular cavities

We propose constitutive models for polycrystalline aggregates with intergranular cavities and test them against full-field numerical simulations. Such conditions are prevalent in many engineering applications and failure of metallic components (e.g. HIPing and other forming processes, spallation under dynamic loading conditions, etc.), where the dilatational effects associated with the presence of cavities must be accounted for, and standard polycrystalline models for incompressible plasticity are not appropriate. On the other hand, it is not clear that the use of porous plasticity models with isotropic matrix behavior is relevant, particularly, when large deformations can lead to significant texture evolution and therefore to strong matrix anisotropy. Of course, finite strains can also lead to significant changes in the porosity and pore shape, resulting in additional anisotropy development. In this work, we make use of ‘variational linear-comparison’ homogenization methods to develop constitutive models simultaneously accounting for texture of the matrix, porosity and average pore shape and orientation. The predictions of the models are compared with full-field numerical simulations based on fast Fourier transforms to study the influence of different microstructural features (e.g. overall porosity, texture of the matrix phase, single-crystal anisotropy, etc.) and type of loading (triaxiality) on the dilatational viscoplastic behavior of voided polycrystals. The results are also compared with the predictions of isotropic-matrix porous plasticity models to assess the effect of the possible matrix anisotropy in textured samples.

Constitutive modeling of strain rate effects in nanocrystalline and ultrafine grained polycrystals

We present a variational two-phase constitutive model capable of capturing the enhanced rate sensitivity in nanocrystalline (nc) and ultrafine-grained (ufg) fcc metals. The nc/ufg-material consists of a grain interior phase and a grain boundary affected zone (GBAZ). The behavior of the GBAZ is described by a rate-dependent isotropic porous plasticity model, whereas a rate-independent crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The scale bridging from a single grain to a polycrystal is done by a Taylor-type homogenization. It is shown that the enhanced rate sensitivity caused by the grain size refinement is successfully captured by the proposed model.

Void growth and coalescence in ductile solids with stage III and stage IV strain hardening

State of the art ductile fracture models often rely on simple power laws to describe the strain hardening of the matrix material. Power laws do not distinguish between the two main stages of hardening observed in polycrystals, referred to as stage III and stage IV hardening, and which emerge from the evolution of the dislocation substructure. The aim of this study is to couple a physics based strain hardening law including these two stages to a micromechanics based ductile damage model. One of the main motivations is that, the stage IV constant hardening rate stage, occurring only at large strain, will be attained in most ductile failure problems if not at the overall level of deformation, at least locally around the growing voids. Furthermore, proper modelling of the stage III involving dislocation storage and recovery terms and the transition to stage IV provides a link with the underlying physical mechanisms of deformation and with the microstructure. First, in order to evaluate the effects of the stage III and stage IV hardening on void growth and coalescence, an extensive parametric study is performed on two-dimensional (2D) axisymmetric finite element (FE) unit cell calculations, using a Kocks–Mecking type hardening law. The cell calculations demonstrate that accounting for the stage IV hardening can have a profound effect on delaying void coalescence and increasing the ductility. The magnitude of the recovery term during stage III has also a significant effect on the void growth rate. Then, the Kocks–Mecking law is incorporated into the Gologanu–Leblond–Devaux (GLD) porous plasticity model supplemented by two different versions of the Thomason void coalescence criterion. The predictions of the damage model are in good agreement with the results of the FE calculations in terms of the stress–strain curves, the evolution of void shape and porosity, as well as the strain value at the onset of void coalescence.

A variational multiscale constitutive model for nanocrystalline materials

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior phase and a grain boundary effected zone (GBAZ). A rate-independent isotropic porous plasticity model is employed to describe the GBAZ, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The constitutive models of both phases are formulated in a small strain framework and extended to finite deformation by use of logarithmic and exponential mappings. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization where a log-normal grain size distribution is assumed. It is shown that the proposed model is able to capture the inverse Hall–Petch effect, i.e., loss of strength with grain size refinement. Finally, the predictive capability of the model is validated against experimental results on nanocrystalline copper and nickel.

On stress-state dependent plasticity modeling: Significance of the hydrostatic stress, the third invariant of stress deviator and the non-associated flow rule

It has been shown that the plastic response of many materials, including some metallic alloys, depends on the stress state. In this paper, we describe a plasticity model for isotropic materials, which is a function of the hydrostatic stress as well as the second and third invariants of the stress deviator, and present its finite element implementation, including integration of the constitutive equations using the backward Euler method and formulation of the consistent tangent moduli. Special attention is paid for the adoption of the non-associated flow rule. As an application, this model is calibrated and verified for a 5083 aluminum alloy. Furthermore, the Gurson–Tvergaard–Needleman porous plasticity model, which is widely used to simulate the void growth process of ductile fracture, is extended to include the effects of hydrostatic stress and the third invariant of stress deviator on the matrix material.