Fracture In Ductile Materials Research Articles

An elastoplastic phase-field model for quasi-static fracture of nickel-based super-alloys

In the present study, an elastoplastic phase-field model of quasi-static fracture in ductile materials is proposed in the variational framework for J2 plasticity with isotropic hardening, which is suitable to describe the quasi-static behavior of metals as investigated in the performed experiments. These contributions include: (1) the free energy functions for coupling elastic response, plastic yielding and damage evolution are established. (2) The new elastic and plastic energy degradation functions are constructed to quantitatively describe the relationship between energy release and phase-field evolution of elastoplastic materials. (3) Damage evolution and plastic yielding criteria are derived. (4) From a numerical point of view, we derive the governing equations and the corresponding weak forms and the overall solution procedure for the phase-field model is given via the use of a return-mapping algorithm. This phase-field model was validated by a series of tensile experiments on Inconel 718 nickel-based super-alloys standard specimens. In order to compare the simulation results with the experimental results more comprehensively, the digital image correlation (DIC) technique is applied to experimentally investigate the specimen deformation information. In addition, to verify the potential of the model to capture complex cracks, we performed Nooru-Mohamed tests. The numerical simulation results are in good agreements with the results of previous experimental work.

Application of ductile fracture model for the prediction of low cycle fatigue in structural steel

The complexity of the loading conditions makes it challenging to accurately predict the damage evolution and resulting ductile fracture in materials and structures. This paper presents an application method to predict fracture initiation for low cycle fatigue of 4-point bending in structural steel SS275. The swift isotropic hardening law was used in the numerical simulation to account for large strain deformation. The damage indicator framework with the Hosford-Coulomb (HC) model was adopted to characterize the damage accumulation and fracture behavior. Uniaxial tension tests on specimens with various fracture modes were conducted to determine the material properties required for calibration of the plasticity model and fracture model param1eters. Non-linear finite element analyses were performed to identify fracture model parameters based on the loading path of fracture initiation. The presented HC model was applied to an extremely low fatigue test (less than 100 cycles) and demonstrated good accuracy in predicting fracture initiation for complex loading scenarios.

The simulation of 3D hypervelocity spallation using a hydrocode PAGOSA with FLIP+MPM

In this work, a hydrocode PAGOSA with FLIP+MPM is presented and exercised to investigate the fracture in ductile material. The merit of PAGOSA with FLIP+MPM to solve the advection problem is first illustrated by a solid piston periodically moving in a sealed tube with air. Furthermore, the ability of PAGOSA with FLIP+MPM to capture the fracture in material is shown by a simple stretching fracture in ductile material. In both of two benchmark problems, the PAGOSA results and analytical solutions are also presented for comparison. Then PAGOSA with FLIP+MPM is used to model complex spall in ductile material, which is a challenging problem in engineering applications. The convergences of PAGOSA with FLIP+MPM—based on both the mesh size and the marker density—are investigated by monitoring free surface velocity. To further show the ability of PAGOSA with FLIP+MPM to predict fracture in ductile material, the numerical results are compared with the experimental results and other published numerical results. Moreover, the effect of the spall parameter in PAGOSA with FLIP+MPM on the numerical simulation is also analyzed by investigating the free surface velocity. Finally, the effects of the peak compressive stress, the tensile strain rate, and the loading rate on the spallation are further investigated. The numerical results show that PAGOSA with FLIP+MPM can improve PAGOSA's performance when applied to predicting fracture in ductile materials, and is robust enough to accurately predict spall fracture in ductile materials.

An improved damage-coupled viscoplastic model for predicting ductile fracture in aluminum alloy at high temperatures

Predicting the ductile fracture in aluminum alloys during hot deformation is important for controlling product quality. To this end, this paper proposes an improved damage-coupled viscoplastic model for predicting the ductile fracture in aluminum alloys at high temperatures. For this model, a dislocation-based viscoplastic model is adopted, and a novel high-temperature damage model is further proposed. The proposed damage model considers the temperature and strain rate effects and also the nonlinear characteristics and stress triaxiality effect of damage evolution. The influence of stress triaxiality is evaluated through notched tensile tests conducted at different temperatures. To describe the actual material degradation, the damage variable is coupled with yield criteria according to continuous damage mechanics. Furthermore, the determination of model parameters based on the results of the uniaxial and notched tensile tests at different temperatures is considered in detail. The proposed constitutive model is applied to analyze damage evolution in a hot tensile process and to predict the fracture occurring during an isothermal bulging test, by employing the user-defined material subroutine VUMAT. The experimental and simulation results showed good agreement, thereby proving the validity of the proposed model for predicting the ductile fracture in aluminum alloys during hot deformation. This study introduces a novel damage model for predicting the ductile fracture in materials, such as aluminum alloys and certain metallic materials, dominated by tensile deformation at high temperatures.

A possible link between brittle and ductile failure by viewing fracture as a topological defect

A continuum model of fracture that describes, in principle, the propagation and interaction of arbitrary distributions of cracks and voids with evolving topology without a “fracture criterion” is developed. It involves a “law of motion” for crack tips, primarily as a kinematical consequence coupled with thermodynamics. Fundamental kinematics endow the crack tip with a topological charge. This allows the association of a kinematical conservation law for the charge, resulting in a fundamental evolution equation for the crack-tip field and, in turn, the crack field. The vectorial crack field degrades the elastic modulus in a physically justified anisotropic manner. The mathematical structure of this conservation law allows an additive “free” gradient of a scalar field in the evolution of the crack field. We associate this naturally emerging scalar field with the porosity that arises in the modeling of ductile failure. Thus, porosity-rate gradients affect the evolution of the crack field, which then naturally degrades the elastic modulus, and it is through this fundamental mechanism that spatial gradients in porosity growth affect the strain energy density and the stress-carrying capacity of the material and, as a dimensional consequence related to fundamental kinematics, introduce a length scale in the model. A key result of this work is that brittle fracture is energy-driven while ductile fracture is stress-driven. Under overall shear loadings where the mean stress vanishes or is compressive, the shear strain energy can still drive shear fracture in ductile materials.

Unifying the effects of in and out-of-plane constraint on the fracture of ductile materials

Effects of plastic constraint on the fracture of materials have been studied extensively. Often in such studies, the plastic constraint is divided into in-plane and out-of-plane directions and each treated separately. Such a separation adds considerable complexity to the engineering structural integrity assessment analyses. Despite previous suggestions for unifying the effects of constraint in a single parameter, the current engineering assessments have not been updated due to lack of direct experimental validation of such parameters. In this study, we directly measured the effects of in-plane and out-of-plane constraints, for the first time, in the form of plastic zone around the crack using advanced experimental techniques. The measurement of constraints in four specimens with different levels of in and out of plane constraints, allowed us to show and relate the interdependency of in and out of plane constraints. The tests were carried out using synchrotron X-ray tomography with in-situ loading. Attenuation contrast between the constituents of the metal matrix composite material used allowed the tomograms to be analysed using digital volume correlation which calculated the full-field displacement within the samples. The displacement fields were used via a finite element framework to calculate the energy release rate in the form of the J-integral along the crack fronts. The measured plastic zone sizes, dependant on the combined level of in plane and out of plane constraints, were used successfully to rank the J-Integral at fracture of the samples. It was therefore proved the level of plastic constraint can be quantified by using the size of the plastic zone as without separating it into two components thus simplifying the treatment of constraint in structural analyses significantly.

Dynamic response of ductile materials containing cylindrical voids

The fracture of ductile materials is often the result of the nucleation, growth and coalescence of microscopic voids. In this thesis, we mainly focus on the dynamic void evolution in porous media containing cylindrical voids. This study covers a problem that is of particular interest in many areas of research (e.g. development of shock mitigation devices for civil or military applications). Owing to the development of additive manufacturing, the processing of porous material with cylindrical voids is an option to create lightweight materials having interesting properties in terms of energy dissipation. Therefore, our work aims at describing the dynamic response of architectured materials such as honeycomb structures. In dynamic loading, microvoids sustain an extremely rapid expansion which generates strong acceleration of particles in the vicinity of cavities. These micro-inertia effects are known to play a significant role in the macroscopic response and the development of damage in porous media. In fact, the overall macroscopic stress is found to be the sum of two contributions: a static term (micro-inertia independent term) and a dynamic term (micro-inertia dependent term), the latter being related to the microstructure (e.g. size and aspect ratio of voids). In our work, a cylindrical shell is adopted as a Representative Volume Element (internal and external radii a and b, length 2l) for the porous material. The static term is derived from a yield function available in the literature. The dynamic stress is evaluated analytically using a trial velocity field for cylindrical voids combined with the multi-scale approach developed in the literature in LEM3. It is shown that the dynamic stress is scaled by the mass density, two characteristic lengths of the voids, the porosity, the macroscopic strain rate tensor and the time derivative of the strain rate tensor. An important outcome of the model is the differential lengthscale effect which exists between in-plane and out of plane components of the macroscopic stress. Namely, it is observed for axisymmetric loading that in-plane dynamic stress components are only related to the void radius a while the out of plane stress component is linked to a and the length of the RVE, l. In the thesis, we present the dynamic response of the porous medium when subjected to various loading conditions: spherical loading, axisymmetric plane strain loading, uniaxial loading and biaxial loading. While for plane strain loading under quasi static condition, the overall axial stress is spherical, in dynamic conditions, the inertia contribution hinders the overall stress tensor from being spherical. Another important result of the proposed theory is the effect of the void length, which does not exist in quasi static conditions where the overall response is solely modulated by the porosity. The case of thin cylinders under dynamic loading reveals a peculiar damage kinetics. In fact, the damage developed in such porous materials results from an increase of the void radius and a reduction of the external radius. The void collapse for uniaxial as well as for biaxial loadings are new observations. The analytical model predictions are validated based on comparisons with finite element calculations (Abaqus/Explicit).

A new total‐Lagrangian smooth particle hydrodynamics approximation for the simulation of damage and fracture of ductile materials

SummarySmooth particle hydrodynamics (SPH) is gaining popularity for the simulation of solids subjected to machining, wear, and impacts. Its attractiveness is due to its abilities to simulate problems, involving large deformations resulting from the absence of mesh as well as the development of the total‐Lagrangian version of SPH (TLSPH) to solve tensile instabilities and an hourglass control algorithm to reduce rank‐deficiency problems. However, when TLSPH is used with continuum damage mechanics, nonphysical residual forces can emerge and lead to instabilities. To solve this issue, the “pseudospring” method that scales the kernel function with damage was proposed by Chakraborty and Shaw. This method preserves local linear momentum and is stable for elastic solids but can generate instabilities when solids damage after experiencing high plastic strains. In this contribution, we present a new TLSPH approximation that conserves both linear and angular momenta, which is suitable for the simulation of damage and fracture of ductile materials. Using single‐edge notched samples and tensile tests, we show that this approximation is stable whatever the stress/strain state and is in good agreement with finite element simulations and experimental results.

An efficient phase-field model for fatigue fracture in ductile materials

Fatigue fracture in ductile materials, e.g. metals, is caused by cyclic plasticity. Especially regarding the high numbers of load cycles, plastic material models resolving the full loading path are computationally very demanding. Herein, a model with particularly small computational effort is presented. It provides a macroscopic, phenomenological description of fatigue fracture by combining the phase-field method for brittle fracture with a classic durability concept. A local lifetime variable is obtained, which degrades the fracture resistance progressively. By deriving the stress-strain path from cyclic material characteristics, only one increment per load cycle is needed at maximum. The model allows to describe fatigue crack initiation, propagation and residual fracture and can reproduce Paris behaviour.

Effects of residual stresses and localised strain-hardening on the fracture of ductile materials

The strain history of a ductile material can affect its apparent initiation fracture toughness as well as its resistance to subsequent crack growth. Using neutron diffraction or synchrotron X-ray diffraction in conjunction with digital image correlation, it is possible to map both the total strain around the tip of a propagating crack, and the elastic strain component. Separating the elastic and irreversible strain components allows us to experimentally investigate the effects that residual stresses and prior strain-hardening have on crack propagation. This article presents the results of two experiments using these techniques. In the first experiment, it was shown that residual stresses in a 7xxx-series aluminium alloy affect not only the initiation of fracture but also fracture propagation. During crack propagation, residual stress relaxation occurs due to extension of the crack and plastic deformation in the region ahead of the crack tip. This affects the material’s apparent crack growth resistance in a non-linear manner and hence changes its fracture stability. In a second experiment, it was shown that the plastic zone ahead of a crack in a ductile material (in this case a ductile ferritic steel) can be modified by localised strain-hardening prior to fracture. Localised strain-hardening using an indentation tool was observed to cause small increases in the specimens’ peak load capacity and its energy absorption prior to the onset of unstable tearing. Using synchrotron diffraction, it was shown that is because hardening a region ahead of the crack tip causes stress redistribution to this region and away from the crack tip itself during fracture loading. These findings can inform how ductile and semi-ductile fracture is treated in structural integrity assessment procedures, particularly in the presence of residual stress. They also suggest that new localised peening treatments to improve the fracture resistance of structures in certain areas may be possible.

A phase-field formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects

Phase-field models have been a topic of much research in recent years. Results have shown that these models are able to produce complex crack patterns in both two and three dimensions. A number of extensions from brittle to ductile materials have been proposed and results are promising. To date, however, these extensions have not accurately represented strains after crack initiation or included important aspects of ductile fracture such as stress triaxiality. This work introduces a number of contributions to further develop phase-field models for fracture in ductile materials. These contributions include: a cubic degradation function that provides a stress–strain response prior to crack initiation that more closely approximates linear elastic behavior, a derivation of the governing equations in terms of a general energy potential from balance laws that describe the kinematics of both the body and phase-field, introduction of a yield surface degradation function that provides a mechanism for plastic softening and corrects the non-physical elastic deformations after crack initiation, a proposed mechanism for including a measure of stress triaxiality as a driving force for crack initiation and propagation, and a correction to an error in the configuration update of an elastoplastic return-mapping algorithm for J2 flow theory. We also present a heuristic time stepping scheme that facilitates computations that require a relatively long load time prior to crack initiation. A number of numerical results will be presented that demonstrate the effects of these contributions.

Micro‐mechanical analyses of the effect of stress triaxiality and the Lode parameter on ductile damage and failure

AbstractThe paper deals with the effect of different stress states on plastic deformations, damage and fracture of ductile materials. To be able to model these effects a continuum damage model has been introduced taking into account the dependence of stress‐state on the constitutive equations. The model is based on the introduction of damaged and fictitious undamaged configurations. All parameters appearing in the constitutive equations are stress‐state‐dependent which can be characterized by the stress intensity, the stress triaxiality and the Lode parameter. Only experiments are not adequate enough to determine all constitutive parameters. Thus, additional series of three‐dimensional micro‐mechanical simulations of representative volume elements have been performed to get more insight in the complex damage mechanisms. These simulations cover a wide range of stress triaxialities and Lode parameters in tension, shear and compression domains. After all, the results from the micro‐mechanical simulations are used to suggest the damage equations and to identify corresponding parameters. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Research on Several Fracture Forms and Controlling Parameters of Ductile Materials

The experimental procedures of mixed mode fracture of several ductile materials are investigated. The position and direction of crack initiation are determined according to a new method. The variations of fracture caused by different physical mechanisms are analyzed. Through studying the changing regularity of different fracture mechanisms, it is believed that the main ways of the fracture in ductile materials can be divided into three groups. They are traction fracture and two different types of shear fracture. Void nucleation, expansion and coalescing are the dominant mechanism of traction fracture. The formation and development of localized shear bands are the dominant mechanism of the two different types of shear fracture. Localized large plastic deformation will cause damage within the material. The fundamental factor, which causes the occurrence of fracture in the material, is a certain stress parameter at the dangerous point has reached the critical fracture value of the material. Based on those phenomena listed above, several fracture controlling parameters for different fracture forms have been discussed, and several new parameters, which affect various fracture forms, have been proposed.

A micromechanical constitutive model for dynamic damage and fracture of ductile materials

This paper proposes a detailed theoretical analysis of the development of dynamic damage in plate impact experiments for the case of high-purity tantalum. Our micro-mechanical model of damage is based on physical mechanisms (void nucleation and growth). The model is aimed to be general enough to be applied to a variety of ductile materials subjected to high tensile pressure loading. In this respect, the work of Czarnota et al. (J Mech Phys Solids 56:1624–1650, 2008) has been extended by introducing the concept of nucleation law and by entering a nonlinear formulation of the elastic response based on the Mie-Gruneisen equation of state. This later aspect allows us to consider high impact velocities. All model parameters are directly assessed by experimental measurements to the exception of the nucleation law which is characterized by the way of an inverse identification method using three free-surface velocity profiles (at low, intermediate and high impact velocities). It is shown that the nucleation law can be consistently determined in the range of operating pressures. The nucleation law being identified, the development of internal damage happens to be a natural outcome of the modelling. The model is applied to predict damage development and free-surface velocity profiles for various test conditions. The variety and the quality of results support the physical basis (in particular micro-inertia effects) upon which the proposed model of dynamic damage is based.

Dynamic void nucleation and growth in solids: A self-consistent statistical theory

We present a framework for a self-consistent theory of spall fracture in ductile materials, based on the dynamics of void nucleation and growth. The constitutive model for the material is divided into elastic and “plastic” parts, where the elastic part represents the volumetric response of a porous elastic material, and the “plastic” part is generated by a collection of representative volume elements (RVEs) of incompressible material. Each RVE is a thick-walled spherical shell, whose average porosity is the same as that of the surrounding porous continuum, thus simulating void interaction through the resulting lowered resistance to further void growth. All voids nucleate and grow according to the appropriate dynamics for a thick-walled sphere made of incompressible material. The macroscopic spherical stress in the material drives the response in all volume elements, which have a distribution of critical stresses for void nucleation, and the statistically weighted sum of the void volumes of all RVEs generates the global porosity. Thus, macroscopic pressure, porosity, and a distribution of growing microscopic voids are fully coupled dynamically. An example is given for a rate-independent, perfectly plastic material. The dynamics of void growth gives rise to a rate effect in the macroscopic material even though the parent material is rate independent.

Simplified model for prediction of dynamic damage and fracture of ductile materials

A simplified model for prediction of dynamic damage and fracture of ductile material has been proposed. The plastic flow of matrix and the void growth are dealt with separately after we show that two separated loading sub-surfaces and corresponding normality rules for matrix and damage exist. The equation of the loading sub-surface for the matrix plastic flow is derived by the means of introducing a mapping damage-free solid of matrix material, whose constitutive relation is supposed to have been determined via Hopkinson bar experiments. Based on the results of recovered experiments the law of damage evolution is phenomenally established. The model has been applied to predict some spall experiments carried out on tantalum, and results show it predicts the experiments very well.

Numerical model of ductile fracture kinetics: comparison of results of 2-D simulations to experimental data

Implementation of kinetic description of evolution of damage in material is necessary for specific cases, when total duration of the whole process and duration of damage development are comparable, e.g. for hypervelocity acceleration and impact. To provide this capability, isotropic model of kinetics of ductile fracture of structure materials was developed on basis of NAG model (Nucleation and Growth) and introduced into elastic-plastic DMK and MIMOZA 2-D codes. Special experiment was conducted: Steel 3 cylinder sample was loaded by sliding detonation of the high explosive layer of variable thickness that produced the axial cone-like cavern. Clear and distinctive data obtained in this simple experiment could be recommended for testing of 2-D numerical codes intending to model dynamic ductile fracture of material. Numerical simulations of this experiment were performed using DMK and MIMOZA codes implementing developed isotropic model of kinetics of ductile fracture. Simulated dimensions of the axial cone-like cavern are in good agreement with experimental data.

Energy dissipation mechanisms in ductile fracture

The objective is to investigate energy dissipation mechanisms that operate at different length scales during fracture in ductile materials. A dimensional analysis is performed to identify the sets of dimensionless parameters which contribute to energy dissipation via dislocation-mediated plastic deformation at a crack tip. However, rather than using phenomenological variables such as yield stress and hardening modulus in the analysis, physical variables such as dislocation density, Burgers vector and Peierls stress are used. It is then shown via elementary arguments that the resulting dimensionless parameters can be interpreted in terms of competitions between various energy dissipation mechanisms at different length scales from the crack tip; the energy dissipations mechanisms are cleavage, crack tip dislocation nucleation and also dislocation nucleation from a Frank–Read source. Therefore, the material behavior is classified into three groups. The first two groups are the well-known intrinsic brittle and intrinsic ductile behavior. The third group is designated to be extrinsic ductile behavior for which Frank–Read dislocation nucleation is the initial energy dissipation mechanism. It is shown that a material is predicted to exhibit extrinsic ductility if the dimensionless parameter bρ disl 1/2 ( b is Burgers vector, ρ disl is dislocation density) is within a certain range defined by other dimensionless parameters, irrespective of the competition between cleavage and crack tip dislocation nucleation. The predictions compare favorably to the documented behavior of a number of different classes of materials.

The flow theory of mechanism-based strain gradient plasticity

The flow theory of mechanism-based strain gradient (MSG) plasticity is established in this paper following the same multiscale, hierarchical framework for the deformation theory of MSG plasticity in order to connect with the Taylor model in dislocation mechanics. We have used the flow theory of MSG plasticity to study micro-indentation hardness experiments. The difference between deformation and flow theories is vanishingly small, and both agree well with experimental hardness data. We have also used the flow theory of MSG plasticity to investigate stress fields around a stationary mode-I crack tip as well as around a steady state, quasi-statically growing crack tip. At a distance to crack tip much larger than dislocation spacings such that continuum plasticity still applies, the stress level around a stationary crack tip in MSG plasticity is significantly higher than that in classical plasticity. The same conclusion is also established for a steady state, quasi-statically growing crack tip, though only the flow theory can be used because of unloading during crack propagation. This significant stress increase due to strain gradient effect provides a means to explain the experimentally observed cleavage fracture in ductile materials [J. Mater. Res. 9 (1994) 1734; Scripta Metall. Mater. 31 (1994) 1037; Interface Sci. 3 (1996) 169].

Crack Tip Fields with Strain Gradient Effects

In this paper, a kind of strain gradient theory proposed by Chen and Wang (2001) is briefly introduced and is used to analyze the crack tip field in homogeneous materials and interface crack tip field with the help of finite element methods. Differences are found between the two kinds of crack tip fields. For the crack tip field in homogeneous materials, the whole field consists of a strain gradient dominance zone, a classical plasticity field and a K field. While for the interface crack tip field, there are only the strain gradient dominance zone and a K field and the classical plasticity field hardly exists. The reason will be given in the present paper. For both kinds of crack tip fields, the effective stresses near the crack tip are much higher than the counterparts in the classical field, which provides an explanation to the experimental observation of cleavage fracture in ductile materials (Elssner et al., 1994).