Calibration procedures for a computational model of ductile fracture

A recent extension of the Gurson constitutive model of damage and failure of ductile structural alloys accounts for localization and crack formation under shearing as well as tension. When properly calibrated against a basic set of experiments, this model has the potential to predict the emergence and propagation of cracks over a wide range of stress states. This paper addresses procedures for calibrating the damage parameters of the extended constitutive model. The procedures are demonstrated for DH36 steel using data from three tests: (i) tension of a round bar, (ii) mode I cracking in a compact tension specimen, and (iii) shear localization and mode II cracking in a shear-off specimen. The computational model is then used to study the emergence of the cup-cone fracture mode in the neck of a round tensile bar. Ductility of a notched round bar provides additional validation.     

Quasi-static and dynamic fracture initiation toughness of Ti/TiB layered functionally graded material under thermo-mechanical loading

Quasi-static and dynamic fracture initiation toughness of Ti/TiB layered functionally graded material (FGM) is investigated using a three point bend specimen. The modified split Hopkinson pressure bar (SHPB) apparatus in conjunction with induction coil heating system is used during elevated temperature dynamic loading experiments. A simple and accurate technique has been developed to identify the time corresponding to the load at which the fracture initiates. A series of experiments are conducted at different temperatures ranging from room temperature to 800 °C, and the effect of temperature and loading rate on the fracture initiation toughness is investigated. The material fracture toughness is found to be sensitive to temperature and the fracture initiation toughness increases as the temperature increases. Furthermore, the fracture initiation toughness is strain rate sensitive and is higher for dynamic loading as compared to quasi-static loading.     

Effect of pre-strain on fracture toughness of ductile structural steels under static and dynamic loading

The ductile fracture process consists of void nucleation, growth and coalescence. The whole ductile process can be divided into two successive steps: (I) the initial state to void nucleation, followed by (II) void growth up to void coalescence. Based on this suggestion, resistance to ductile fracture could be divided into the resistance to stage I and stage II, and accordingly the whole fracture toughness could be regarded to be due to contributions from stages I and II. The fracture toughness contributed from the two steps is, respectively, denoted as void nucleation-contributed fracture toughness and void growth-contributed fracture toughness. The effect of plastic pre-strain on the fracture toughness of ductile structural steels under static and dynamic loading (4.9 m/s) within the ductile fracture range was evaluated by summing contributions due to void nucleation-contributed and void growth-contributed fracture toughness. The effect of strain rate on fracture toughness was also investigated by the same means. The results show that both plastic pre-strain and high-speed loading decrease the void nucleation-contributed fracture toughness while their effects on the void growth-contributed fracture toughness depend on the variations in strength and ductility. Moreover, fracture toughness of structural steels generally decreases with increasing strain rate.     

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

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

Development of stress-modified fracture strain for ductile failure of API X65 steel

The present paper proposes ductile failure criteria in terms of true fracture strain (the equivalent strain to fracture) as a function of the stress triaxiality (defined by the ratio of the hydrostatic stress to the equivalent stress) for the API X65 steel. To determine the stress-modified fracture strain, smooth and notched tensile bars with four different notch radii are tested, from which true fracture strains are determined as a function of the notch radius. Then detailed elastic–plastic, large strain finite element analyses are performed to estimate variations of stress triaxiality in the tensile bars, which leads to true fracture strains as a function of the stress triaxiality, by combining them with experimental results. Two different failure criteria are proposed, one based on local stress and strain information at the site where failure initiation is likely to take place, and the other based on averaged stress and strain information over the ligament where ductile fracture is expected. As a case study, ligament failures of API X65 pipes with a gouge are predicted and compared with experimental data.     

Modified transformation formulae between fracture toughness and CTOD of ductile metals considering pre-deformation effects

Strengthening action and degrading action are proposed to describe influences of plastic deformation on fracture toughness in ductile metals. Strengthening effect represents the shield operation of dislocation from crack and degrading effect displays the emergence of damage. In order to demonstrate two effects, the existing transform equation between fracture toughness and crack-tip opening displacement is modified based on a novel damage variable. The modified transform formulae can help to find the maximum of fracture toughness for metals and keep safer performance for structural components.     

Fractal aspects of ductile and cleavage fracture surfaces

The aim of this paper is to evaluate and interpret the three-dimensional variational fractal dimension of a ductile and a cleavage fracture surface. The fracture surface is acquired by fracturing Charpy impact and static loaded specimens of a low alloy steel in ductile-to-brittle transition temperature range, and reconstructed by a stereoscopic technique. The three-dimensional variational method for measuring fractal dimension is improved by shifting algorithm and tested on the Takagi surface using the local fractal dimension. We find very good fractal behaviour in the ductile area, however, fractal characteristics in the cleavage area are less noticeable. The results are discussed in thermodynamical terms and promote the idea that fractal behaviour reflects the quasi-static process and that the fracture mechanisms in the ductile fracture are independent of strain rate (at least up to 10³ s⁻¹).     

Evaluation of ductile fracture in sheet metal forming using the ellipsoidal void model

The ductile fracture in the simulation of sheet-metal-forming processes is evaluated by the ellipsoidal void model previously proposed by the author. In the present study, the simulation and experiment of the hole expansion test are performed using six types of metals. For an alloy, the relationship between prestrain and hole expansion ratio calculated using the ellipsoidal void configuration and ellipsoidal void shape and that calculated using the ellipsoidal void configuration and circular void shape agree with the relationship obtained experimentally. For a pure metal, the relationship between prestrain and hole expansion ratio calculated using the average void configuration and average void shape agrees with that obtained experimentally. Furthermore, the method of introducing prestrain to an as-rolled sheet is proposed, and the prestrain in this sheet is estimated.     

Spallation caused by the diffusion and agglomeration of vacancies in ductile metals

In this paper, the spallation process for the ductile metals under plane shock loading is discussed in theory. By employing the phase transition theory and non-equilibrium theory, the spallation process may be understood as a result of the diffusion and agglomeration of the generated vacancies. Through the detailed theoretical analysis, the following important points are concluded: (1) the spalling temperature, a new concept, is proposed first and the appearance of spallation critical behavior is proved; (2) the quantitative grain size, tensile strain rate and temperature dependence of both the damage evolution rate and the void growth velocity is obtained; (3) the existence of a characteristic size for the voids and a characteristic stress at the void boundary is discovered first, and their magnitude depend on the vacancy excitation energy and the average volume of one vacancy; (4) the temperature of metal near the growing void is found to be high, possibly causing the metal to melt, and it decreases quickly with the distance away from the void; (5) the area of the plastic zone, surrounding one formed spherical void, is clarified; (6) the viewpoint is put forward that the void growth may arise from the agglomeration of vacancies rather than the emission of dislocations when the shocking temperature approaches spalling temperature. Most of the above theoretical results are novel and obtained first.     

Evaluation of directional mesh bias in concrete fracture simulations using continuum damage models

In the present comparative study, we investigate the influence of directional mesh bias on the results of failure simulations performed with isotropic and anisotropic damage models. Several fracture tests leading to curved crack trajectories are simulated on different meshes. The isotropic damage model with a realistic biaxial strength envelope for concrete is highly sensitive to the mesh orientation, even for fine meshes. The sensitivity is reduced if the definition of the damage-driving variable (equivalent strain) is based on the modified von Mises criterion, but the corresponding biaxial strength envelope is not realistic for concrete. The anisotropic damage models used in this study capture reasonably well arbitrary crack trajectories even if the biaxial strength envelope remains close to typical experimental data. Their superior performance can be at least partially attributed to their ability to capture dilatancy under shear, which is revealed by a comparative analysis of the behavior of individual models under shear with restricted or free volume expansion.     

Identification of ductile damage and fracture parameters from the small punch test using neural networks

This paper presents a method for the identification of deformation, damage and fracture properties of ductile materials. The small punch test is used to obtain the material response under loading. The resulting load displacement curve contains information about the deformation and failure behavior of the tested material. The finite element method is used to compute the load displacement curve depending on the parameters of the Gurson-Tvergaard-Needleman damage law. Via a systematic variation of the material parameters a data base is built up, which is used to train neural networks. This neural network can be used to predict the load displacement curve of the SPT for a given material parameter set. The identification of the material parameters is done by using a conjugate directions algorithm, which minimizes the error between an experimental load displacement curve and one predicted by the network function. The identified material parameters are validated by independent tests on notched tensile specimens. Furthermore, these parameters can be used to compute the crack growth in fracture specimens, which finally leads to a prediction of classical fracture toughness parameters.     

Meshfree Galerkin method for a rotating Euler-Bernoulli beam

In this article, we solve the free vibration problem of a rotating Euler-Bernoulli beam using the meshfree Galerkin method. Radial basis functions are used for interpolation. An improved formulation of the rotating Euler-Bernoulli beam free vibration problem with the Galerkin method is explained for the first time, which results into a symmetric stiffness matrix and gives a significant computational advantage over the formulation given in the existing literature. A conventional hp-version of Galerkin finite element method is used for comparison. Results are obtained at different non-dimensional rotating speeds of a rotating beam. Results show excellent agreement with the existing literature.     

An FEM study on crack tip blunting in ductile fracture initiation

Ductile fracture is initiated by void nucleation at a characteristic distance (I_c) from the crack tip and propagated by void growth followed by coalescence with the tip. The earlier concepts expressed I_c in terms of grain size or inter-particle distance because grain and particle boundaries form potential sites for void nucleation. However, Srinivas et al. (1994) observed nucleation of such voids even inside the crack tip grains in a nominally particle free Armco iron. In an attempt to achieve a unified understanding of these observations, typical crack-tip blunting prior to ductile fracture in a standard C(T) specimen (Mode I) was studied using a finite element method (FEM) supporting large elasto-plastic deformation and material rotation. Using a set of experimental data on Armco iron specimens of different grain sizes, it is shown that none of the locations of the maxima of the parameters stress, strain and strain energy density correspond to I_c. Nevertheless, the size of the zone of intense plastic deformation, as calculated from the strain energy density distribution ahead of the crack tip in the crack plane, compares well with the experimentally measured I_c. The integral of the strain energy density variation from the crack tip to the location of void nucleation is found to be linearly proportional to J_IC. Using this result, an expression is arrived at relating I_c to J_IC and further extended to CTOD_c.     

Numerical study on the effect of prestrain history on ductile fracture resistance by using the complete Gurson model

During installation and operation, pipeline steels may suffer from plastic pre-deformation (prestrain) due to accidental loading, cold bending or ground movement. The plastic prestrain history not only modifies steels’ yield and flow properties but also influences their fracture performance. This paper focuses on the effect of plastic prestrain history on ductile fracture resistance. Single edge notched tension (SENT) specimens have been selected for the numerical study and the crack is assumed to exist before a prestrain history was applied. The complete Gurson model has been applied to simulate the ductile fracture behaviour. The results show that prestrain history can reduce the fracture resistance significantly and neither the history-independent resistance model nor material-memory resistance model existing in the literature can be used to describe the prestrain history effect. Based on the numerical results an approximate history-dependent resistance model is proposed. The results also suggest that it is important to take the prestrain effect into consideration in future structural integrity assessment procedure for pipelines.     

Anisotropic ductile fracture of Al 2024 alloys

The anisotropic fracture of the 2024-T351 aluminium alloy is investigated using a micromechanics-based damage model accounting for the effect of the void aspect ratio and void distribution. The 2024-T351 Al alloy contains precipitation free bands in which most void nucleating particles are located. The presence of these bands, which are parallel to the rolling direction, primarily controls the distribution of damage and overall fracture anisotropy. The primary void nucleating particles also present a preferential elongation in the rolling direction. These key microstructural features have been determined using quantitative characterisation methods. The effects of void shape and void spacing on the fracture behaviour are elucidated by means of FE cell calculations. FE simulations of cylindrical notched round bars loaded in different orientations are made and compared with experimental data, allowing a better understanding of the damage process as well as the limitations of the modelling approach.     

Constitutive modeling of void shearing effect in ductile fracture of porous materials

Many solids, including geomaterials and commercially available metallic alloys, can be considered as a porous media. The Gurson-like model has been proposed to describe plastic deformation for such type of materials. It has attracted a great deal of attention and various modifications to this model have been proposed. The constitutive equations of Gurson-like model are governed by the first and second stress invariants and the current void volume fraction of the material. Tvergaard and Needleman included void nucleation, growth and coalescence to Gurson model in a phenomenological way [Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 1984;32(1):157–69] – thus suggesting the so called GTN model. Meanwhile, little attention was given to the dependence of the damage evolution on the third stress invariant. McClintock et al. [McClintock FA, Kaplan SM, Berg CA. Ductile fracture by hole growth in shear bands. Int J Fract Mechan 1966;2(4):614–27] proposed damage model based on the void evolution in localized shear banding. In the present paper, a separate internal damage variable which differs from the conventional void volume fraction is introduced. The GTN model is further extended to incorporate the void shearing mechanism of damage, which depends on the third stress invariant. Numerical aspects are addressed concerning the integration of the proposed constitutive relations. A unit cell is studied to illustrate the intrinsic mechanical behavior of the modified model. Computations of the deformation in axisymmetric and transverse plane strain tension are also performed. Realistic crack modes in these simulations are achieved for the modified GTN model.     

Modelling of fracture toughness data in the ductile to brittle transition temperature region by statistical analysis

A fracture toughness database for a ferritic 22NiMoCr37 steel forging for 12.5, 25, 50 and 100 mm thick specimens tested at nine different temperatures has been analysed statistically. The method employed uses a statistical procedure based on competing risks to evaluate the fracture toughness and quantify the probability of cleavage fracture as a function of temperature, specimen thickness and ductile crack growth. This paper describes the application of the competing risks statistical methods to the fracture toughness database obtained from the joint European Project.     

Derivation of separation laws for cohesive models in the course of ductile fracture

The paper addresses the determination of the traction-separation law of the cohesive model on a micromechanical basis. For this task, a specific failure mechanism, i.e. ductile damage consisting of void nucleation, growth and coalescence, is investigated. An approach already described in the literature is to transfer the deformation behaviour of the simplest representative volume element, i.e. a single voided unit cell, to the cohesive interface. After reviewing the existing approach, its main drawback, namely that the unit cell contains both, deformation and damage of a material point whereas the cohesive model should contain the material separation only, is addressed. A new approach is presented, in which the behaviour of a unit cell is partitioned in its elasto-plastic deformation and damage, and only the damage contribution is applied as the traction-separation law for the cohesive model. Instead of modelling the voided unit cell, a single element with Gurson type plastic potential for the damage has been employed as a reference for the behaviour at the microscale. A study with fracture specimens, C(T) and M(T), made of an engineering Aluminium alloy shows that the new approach exhibits a better transferability than the existing one.     

Simulation of ductile tearing by the BEM with 2D domain discretization and a local damage model

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.     

Experimental and numerical investigation on ductile fracture mechanism of aluminium alloy using new modified model

In the present paper, the ductile fracture of aluminium alloy 5052P-H34 is studied by experiments and simulations. Then, the extension of the damage growth model, which captures both tension as well as shear, was employed in the present paper, and a modified Rousselier model was proposed. A stress integration algorithm based on the general backward Euler return algorithm was developed and implemented into finite element (FE) models in the ABAQUS/Explicit platform. The shear coefficient was calibrated by a FE analysis based on an inverse calibration procedure combined with the physical experiments. The predictive capability of this model was studied by comparing the experiments with the simulations, and the validity of this model was verified. The results show that the modified Rousselier model can give more accurate results for both tension and shear failure.