Ductile fracture: Experiments and computations

Numerous criteria have been developed for ductile fracture (DF) prediction in metal plastic deformation. Finding a way to select these DF criteria (DFCs) and identify their applicability and reliability, however, is a non-trivial issue that still needs to be addressed in greater depth. In this study, several criteria under the categories of ‘uncoupled damage criterion’ and the ‘coupled damage criterion’, including the continuum damage mechanics (CDM)-based Lemaitre model and the Gurson-Tvergaard-Needleman (GTN) model, are investigated to determine their reliability in ductile failure prediction. To create diverse stress and strain states and fracture modes, different deformation scenarios are generated using tensile and compression tests of Al-alloy 6061 (T6) with different sample geometries and dimensions. The two categories of criteria are coded into finite element (FE) models based on the unconditional stress integration algorithm in the VUMAT/ABAQUS platform. Through physical experiments, computations and three industrial case studies, the entire correlation panorama of the DFCs, deformation modes and DF mechanisms is established and articulated. The experimental and simulation results show the following. (1) The mixed DF mode exists in every deformation of concern in this study, even in the tensile test of the round bar sample with the smallest notch radius. A decrease of stress triaxiality (η-value) leads to a reduction in the accuracy of DF prediction by the two DFC categories of DFCs, due to the interplay between the principal stress dominant fracture and the shear-stress dominant factor. (2) For deformations with a higher η-value, both categories of DFCs predict the fracture location reasonably well. For those with a lower or even negative η-value, the GTN and CDM-based criteria and some of the uncoupled criteria, including the C&L, Ayada and Oyane models, provide relatively better predictions. Only the Tresca and Freudenthal models can properly predict the shear dominant fracture. The reliability sequence of fracture moment prediction is thus the GTN model, followed by the CDM-based model and the uncoupled models. (3) The applicability of the DFCs depends on the use of suitable damage evolution rules (void nucleation/growth/coalescence and shear band) and consideration of several influential factors, including pressure stress, stress triaxiality, the Lode parameter, and the equivalent plastic strain or shear stress. These parameters determine the deformation mode (shear dominant or maximum principal stress dominant deformation) and, further, the DF mechanism (dimple fracture/shear fracture/mixed fracture).     

Numerical simulation of fracture mode transition in ductile plates

Fracture mode of ductile solids can vary depending on the history of stress state the material experienced. For example, ductile plates under remote in-plane loading are often found to rupture in mode I or mixed mode I/III. The distinct crack patterns are observed in many different metals and alloys, but until now the underlying physical principles, though highly debated, remain unresolved. Here we show that the existing theories are not capable of capturing the mixed mode I/III due to a missing ingredient in the constitutive equations. We introduce an azimuthal dependent fracture envelope and illustrate that two competing fracture mechanisms, governed by the pressure and the Lode angle of the stress tensor, respectively, exist ahead of the crack tip. Using the continuum damage plasticity model, we demonstrate that the distinctive features of the two crack propagation modes in ductile plates can be reproduced using three dimensional finite element simulations. The magnitude of the tunneling effect and the apparent crack growth resistance are calculated and agree with experimental observations. The finite element mesh size dependences of the fracture mode and the apparent crack growth resistance are also investigated.     

Influences of initial porosity,stress triaxiality and Lode parameter on plastic deformation and ductile fracture

Local mechanical properties in aluminum cast components are inhomogeneous as a consequence of spatial distribution of microstructure,e.g.,porosity,inclusions,grain size and arm spacing of secondary dendrites.In this work,the effect of porosity is investigated.Cast components contain voids with different sizes,forms,orientations and distributions.This is approximated by a porosity distribution in the following.The aim of this paper is to investigate the influence of initial porosity,stress triaxiality and Lode parameter on plastic deformation and ductile fracture.A micromechanical model with a spherical void located at the center of the matrix material,called the representative volume element(RVE),is developed.Fully periodic boundary conditions are applied to the RVE and the values of stress triaxiality and Lode parameter are kept constant during the entire course of loading.For this purpose,a multi-point constraint(MPC)user subroutine is developed to prescribe the loading.The results of the RVE model are used to establish the constitutive equations and to further investigate the influences of initial porosity,stress triaxiality and Lode parameter on elastic constant,plastic deformation and ductile fracture of an aluminum die casting alloy.     

Effects of the stress state on plasticity and ductile failure of an aluminum 5083 alloy

The experimental and numerical work presented in this paper reveals that stress state has strong effects on both the plastic response and the ductile fracture behavior of an aluminum 5083 alloy. As a result, the hydrostatic stress and the third invariant of the stress deviator (which is related to the Lode angle) need to be incorporated in the material modeling. These findings challenge the classical J₂ plasticity theory and provide a blueprint for the establishment of the stress state dependent plasticity and ductile fracture models for aluminum structural reliability assessments. Further investigations are planned to advance, calibrate and validate the new plasticity and ductile fracture models.     

Effects of stress invariants and reverse loading on ductile fracture initiation

Recent research studies on ductile fracture of metals have shown that the ductile fracture initiation is significantly affected by the stress state. In this study, the effects of the stress invariants as well as the effect of the reverse loading on ductile fracture are considered. To estimate the reduction of load carrying capacity and ductile fracture initiation, a scalar damage expression is proposed. This scalar damage is a function of the accumulated plastic strain, the first stress invariant and the Lode angle. To incorporate the effect of the reverse loading, the accumulated plastic strain is divided into the tension and compression components and each component has a different weight coefficient. For evaluating the plastic deformation until fracture initiation, the proposed damage function is coupled with the cyclic plasticity model which is affected by all of the stress invariants and pervious plastic deformation history.For verification and evaluation of this damage-plasticity constitutive equation a series of experimental tests are conducted on high-strength steel, DIN 1.6959. In addition finite element simulations are carried out including the integration of the constitutive equations using the modified return mapping algorithm. The modeling results show good agreement with experimental results.     

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.     

Predicting ductile fracture of low carbon steel sheets: Stress-based versus mixed stress/strain-based Mohr–Coulomb model

Two distinct implementations of the Mohr–Coulomb failure model are used in conjunction with a non-associated quadratic plasticity model to describe the onset of fracture in low carbon steel sheets. The stress-based version corresponds to the original Mohr–Coulomb model in stress space. For the mixed stress/strain-based version, the Mohr–Coulomb failure criterion is first transformed into the space of stress triaxiality, Lode angle parameter and equivalent plastic strain and then used as stress-state dependent weighting function in a damage indicator model. Basic fracture experiments including tensile specimens of different notch radii and a punch test are performed to calibrate the material parameters of the respective models. Subsequently, the models are used to predict the crack initiation in a Hasek test and during the stamping of an anticlastic structure. Unlike for the calibration experiments, the loading history during stamping is highly non-linear. Both models can be calibrated with similar accuracy, but the strain-based model predicts the instant of onset of fracture with greater accuracy in the stamping experiment which is an advantage of the empirical damage accumulation rule.     

A coupled elastoplastic-damage constitutive model with Lode angle dependent failure criterion

A coupled elastoplastic-damage constitutive model with Lode angle dependent failure criterion for high strain and ballistic applications is presented. A Lode angle dependent function is added to the equivalent plastic strain to failure definition of the Johnson–Cook failure criterion. The weakening in the elastic law and in the Johnson–Cook-like constitutive relation implicitly introduces the Lode angle dependency in the elastoplastic behaviour. The material model is calibrated for precipitation hardened Inconel 718 nickel-base superalloy. The combination of a Lode angle dependent failure criterion with weakened constitutive equations is proven to predict fracture patterns of the mechanical tests performed and provide reliable results. Additionally, the mesh size dependency on the prediction of the fracture patterns was studied, showing that was crucial to predict such patterns.     

Modeling of ductile fracture: Application of the mechanism-based concepts

This paper summarizes our recent studies on modeling ductile fracture in structural materials using the mechanism-based concepts. We describe two numerical approaches to model the material failure process by void growth and coalescence. In the first approach, voids are considered explicitly and modeled using refined finite elements. In order to predict crack initiation and propagation, a void coalescence criterion is established by conducting a series of systematic finite element analyses of the void-containing, representative material volume (RMV) subjected to different macroscopic stress states and expressed as a function of the stress triaxiality ratio and the Lode angle. The discrete void approach provides a straightforward way for studying the effects of microstructure on fracture toughness. In the second approach, the void-containing material is considered as a homogenized continuum governed by porous plasticity models. This makes it possible to simulate large amount of crack extension because only one element is needed for a representative material volume. As an example, a numerical approach is proposed to predict ductile crack growth in thin panels of a 2024-T3 aluminum alloy, where a modified Gologanu–Leblond–Devaux model [Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing nonspherical voids – Case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754; Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing nonspherical voids – Case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Tech. 116, 290–297; Gologanu, M., Leblond, J.B., Perrin, G., Devaux, J., 1995. Recent extensions of Gurson’s model for porous ductile metals. In: Suquet, P. (Ed.) Continuum Micromechanics. Springer-Verlag, pp. 61–130] is used to describe the evolution of void shape and void volume fraction and the associated material softening, and the material failure criterion is calibrated using experimental data. The calibrated computational model successfully predicts crack extension in various fracture specimens, including the compact tension specimen, middle crack tension specimens, multi-site damage specimens and the pressurized cylindrical shell specimen.     

Modeling of ductile fracture: Significance of void coalescence

In this paper void coalescence is regarded as the result of localization of plastic flow between enlarged voids. We obtain the failure criterion for a representative material volume (RMV) in terms of the macroscopic equivalent strain (E_c) as a function of the stress triaxiality parameter (T) and the Lode angle (θ) by conducting systematic finite element analyses of the void-containing RMV subjected to different macroscopic stress states. A series of parameter studies are conducted to examine the effects of the initial shape and volume fraction of the primary void and nucleation, growth, and coalescence of secondary voids on the predicted failure surface E_c(T, θ). As an application, a numerical approach is proposed to predict ductile crack growth in thin panels of a 2024-T3 aluminum alloy, where a porous plasticity model is used to describe the void growth process and the expression for E_c is calibrated using experimental data. The calibrated computational model is applied to predict crack extension in fracture specimens having various initial crack configurations and the numerical predictions agree very well with experimental measurements.     

A ductile damage criterion at various stress triaxialities

The paper discusses the effect of stress triaxiality on the onset and evolution of damage in ductile metals. A series of tests including shear tests and experiments on smooth and pre-notched tension specimens was carried out for a wide range of stress triaxialities. The underlying continuum damage model is based on kinematic definition of damage tensors. The modular structure of the approach is accomplished by the decomposition of strain rates into elastic, plastic and damage parts. Free energy functions with respect to fictitious undamaged configurations as well as damaged ones are introduced separately leading to elastic material laws which are affected by increasing damage. In addition, a macroscopic yield condition and a flow rule are used to adequately describe the plastic behavior. Numerical simulations of the experiments are performed and good correlation of tests and numerical results is achieved. Based on experimental and numerical data the damage criterion formulated in stress space is quantified. Different branches of this function are taken into account corresponding to different damage modes depending on stress triaxiality and Lode parameter. In addition, identification of material parameters is discussed in detail.     

On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3)

Strain and damage interactions during tearing of a ductile Al-alloy with high work hardening are assessed in situ and in 3D combining two recently developed experimental techniques, namely, synchrotron laminography and digital volume correlation. Digital volume correlation consists of registering 3D laminography images. Via simultaneous assessments of 3D strain and damage at a distance of 1-mm ahead of a notch root of a thin Compact Tension-like specimen, it is found that parallel crossing slant strained bands are active from the beginning of loading in a region where the crack will be slanted. These bands have an intermittent activity but are stable in space. Even at late stages of deformation strained bands can stop their activity highlighting the importance of plasticity on the failure process rather than damage softening. One void is followed over the loading history and seen to grow and orient along the slant strained band at very late stages of deformation. Void growth and strain are quantified. Gurson–Tvergaard–Needleman-type simulations using damage nucleation for shear, which is based on the Lode parameter, are performed and capture slant fracture but not the initial strain fields and in particular the experimentally found slant bands. The band formation and strain distribution inside and outside the bands are discussed further using plane strain simulations accounting for plastic material heterogeneity in soft zones.     

New true-triaxial rock strength criteria considering intrinsic material characteristics

A reasonable strength criterion should reflect the hydrostatic pressure effect, minimum principal stress effect,and intermediate principal stress effect. The former two effects can be described by the meridian curves, and the last one mainly depends on the Lode angle dependence function. Among three conventional strength criteria, i.e.Mohr–Coulomb(MC), Hoek–Brown(HB), and Exponent(EP) criteria, the difference between generalized compression and extension strength of EP criterion experience a firstly increase then decrease process, and tends to be zero when hydrostatic pressure is big enough. This is in accordance with intrinsic rock strength characterization. Moreover, the critical hydrostatic pressure I_c corresponding to the maximum difference of between generalized compression and extension strength can be easily adjusted by minimum principal stress influence parameter K. So, the exponent function is a more reasonable meridian curves, which well reflects the hydrostatic pressure effect and is employed to describe the generalized compression and extension strength.Meanwhile, three Lode angle dependence functions of L_(MN),L_(WW), and L_(YMH), which unconditionally satisfy the convexity and differential requirements, are employed to represent the intermediate principal stress effect. Realizing the actual strength surface should be located between the generalized compression and extension surface, new true-triaxial criteria are proposed by combining the two states of EP criterion by Lode angle dependence function with a same lode angle. The proposed new true-triaxial criteria have the same strength parameters as EP criterion. Finally, 14 groups of triaxial test data are employed to validate the proposed criteria. The results show that the three new true-triaxial exponent criteria,especially the Exponent Willam-Warnke criterion(EPWW)criterion, give much lower misfits, which illustrates that the EP criterion and L_(WW) have more reasonable meridian and deviatoric function form, respectively. The proposed new true-triaxial strength criteria can provide theoretical foundation for stability analysis and optimization of support design of rock engineering.     

EVALUATION OF DUCTILE FRACTURE CRITERIA IN A GENERAL THREE-DIMENSIONAL STRESS STATE CONSIDERING THE STRESS TRIAXIALITY AND THE LODE PARAMETER

This paper is concerned with evaluation of various ductile fracture criteria in a general three-dimensional stress state of stress triaxiality, the Lode parameter and the equiva- lent plastic strain to fracture. Evaluation is carried out by comparing fracture loci constructed by fracture criteria to experimental results of A12024-T351. Comparison demonstrates that the Modified Mohr-Coulomb criterion and a newly proposed criterion provide sufficient predictabil- ity of fracture strain. Moreover, evaluation is emphasized on the predicted cut-off value for stress triaxiality. The evaluation demonstrates that the Cockcroft-Latham, Brozzo, Oh, Ko-Huh and the new criteria coupled a reasonable cut-off value for ductile materials.     

含主应力轴旋转的土体本构关系研究进展

主应力轴旋转对岩土工程的影响日益受到人们的重视．本文从纯主应力轴旋转、纯应力洛德角变化、多种因素变化、排水与不排水等方面较全面地描述了含主应力轴旋转情况下土体的基本变形特性．对当前较有影响的含主应力轴旋转的土体本构模型（基于一般应力空间的土体本构模型、运动硬化模型、边界面模型、土体弹塑性应力应变关系的完全应力增量表述等等）进行了较为系统的评述．提出了合主应力轴旋转的实验研究中存在的核心问题与建立合主应力轴旋转的土体本构关系的根本途径．      

Numerical simulation of 3D ductile cracks formation using recent improved Lode-dependent plasticity and damage models combined with remeshing

Damage to fracture transition has become a popular topic in the ductile fracture scientific community. Indeed, the transition from a damage continuous approach to a discontinuous fracture is not straightforward both from mechanical and numerical points of view. In the present study, a new improved Lode dependent phenomenological coupled damage model is used to investigate the ductile fracture in different mechanical tests. The remeshing and elements erosion techniques are employed to propagate the ductile cracks in 3D models using Forge® finite element code. This code is based on a mixed velocity–pressure formulation using the MINI element P1+/P1$P1+/P1$. In addition, the plasticity behavior is modeled by a Lode-dependent plasticity criterion. Applications to different mechanical tests at different loading configurations, using identified damage model parameters, show good agreement in terms of fracture prediction between experimental and numerical results.     

Comparing Two Damage Models Under Shear Stress

Recently, a new macro-mechanical model for ductile damage was presented by Wierzbicki, Xue et al., trying to address the role assumed by Lode angle and pressure sensitivity. The new model requires several experimental tests to identify all its parameters, but should outperform the “standard” formulations at high triaxiality levels. In this work a comparison between the Lemaitre’s Continuous Damage Mechanics model and the Wierzbicki’s one is proposed. After calibration of both models, the displacement field measured using the Digital Image Correlation technique in a large shear test case is compared with results of Finite Element simulations obtained using the two damage models considered. Results are not conclusive but show that the new damage model is quite accurateas rupture criterion, but damage evolution history deviates to some extent from the experimentally observed behavior.     

Modelling of dynamic ductile fracture and application to the simulation of plate impact tests on tantalum

A model of dynamic damage by void nucleation and growth is proposed for elastic-viscoplastic materials sustaining intense loading. The model is dedicated to ductile materials for which fracture is caused by microvoiding. The material contains potential nucleation sites where microvoids are generated when the local pressure overcomes the nucleation pressure. A probability density function is adopted to describe the fluctuation of the nucleation pressure within the material. The void growth is described by using a hollow sphere model where micro-inertia effects are accounted for. The matrix weakening due to void growth is also included.The model has been first tested under uniaxial deformation. When the strain rate is assumed constant, the pressure inside the material has nearly a linear response up to a maximum. An analytical expression for the maximum pressure is proposed.Finite element simulations of plate impact tests have been carried out and compared to experiments on tantalum. From simulations based on the proposed model, an increase of the spall strength is observed with higher shock intensities. Therefore, the relationship between the velocity pullback and spall strength usually assumed in the literature (based on the acoustic approach) seems to be inadequate. Velocity profiles are simulated for different flyer thicknesses and different impact velocities with close agreement with experiments.