Original Gurson Model Research Articles

Moving from theory to application: Evaluating the numerical implementation of void shape effects and damage delocalization in the modeling of ductile fracture in porous plastic metals

In this paper, we present a robust exploration of the Gologanu–Leblond–Devaux (GLD) model, an advanced iteration of the Gurson model, designed to predict ductile fractures in porous metals. Going beyond the limits of the original Gurson model, the GLD model accounts for cavity shape effects and non-local strain localization, marking a significant leap in fracture mechanics. We also present a comprehensive exposition of the GLD model and its non-local extension, establishing their compatibility with the concept of GSMs. Notably, we emphasize the uniqueness of solutions in the numerical implementation, underlining the imperative need for a meticulously devised mixed implicit/explicit algorithm. Furthermore, we set out to validate the GLD model through rigorous comparisons of our numerical simulations with experimental data. Employing a damage delocalization approach rooted in the natural logarithm of porosity, our study provides compelling evidence of the model’s performance. This approach mitigates issues observed with the original porosity rate, preventing excessive smoothing of porosity and maintaining the fidelity of stress–strain curves. In addition, we gave a profound theoretical elucidation of this phenomenon via Fourier’s analysis of porosity rate. Through this work, we not only enhance our understanding of ductile fracture behavior but also establish a robust numerical framework for its predictive modeling. The GLD model emerges as a powerful tool for the accurate analysis and prediction of fracture phenomena in porous materials, further advancing the field of materials science and engineering.

On the thermodynamics consistency of Gurson’s model and its computational implications

The aim of this paper is to investigate the thermodynamics consistency of Gurson’s model and notably its relation to the class of standard generalized materials. First, we briefly recall Gurson’s model in its original format and reanalyzed it in the thermodynamic framework of poroplasticity of saturated media. This allows to properly define the coupling between elasticity and plasticity and to demonstrate that Gurson’s model fits into the framework of generalized standard materials model, provided that the internal variables being the plastic strain and the Lagrangian plastic porosity (and not the Eulerian porosity as in the original Gurson model). In particular, by construction, the porosity evolution law is proved to be a full part of the thermodynamic formulation of the model with a generalized normality rule. The implications in terms of numerical implementation of Gurson’s model are then investigated; a new numerical scheme based on the time-implicit discretization of the Lagrangian porosity is notably proposed and discussed with respect to available algorithms.

A Lode-dependent Gurson model motivated by unit cell analyses

In this study, the effects of including a dependency on the third deviatoric stress invariant in the void evolution equation of the Gurson model are examined using unit cell calculations and imperfection band analyses. Finite element analyses of a unit cell model are conducted to approximate the behaviour of the material microstructure. The unit cell was modelled as a cube made from an elastic-plastic matrix governed by J2 flow theory with a spherical void located at the centre. The results of the unit cell calculations show a monotonic decrease in void growth when the stress state changes from generalized tension to generalized compression. To mimic the resulting evolution of the void volume fraction, an extension of the Gurson model based on the shear modification proposed by Nahshon and Hutchinson (2008) is proposed. This Lode-dependent void evolution term is further qualitatively assessed through comparisons with the unit cell simulations and through strain localization predictions using imperfection band analyses. The assessment demonstrates that the proposed modification of the void evolution equation is consistent with the evolution of the unit cell in the case of moderate and high stress triaxiality ratios. Furthermore, the imperfection band analyses exhibit a greater difference between the failure strain values in generalized tension and generalized compression using this Lode-dependent void evolution term compared to similar analyses that employ the original Gurson model or the shear-modified Gurson model. The Lode-dependent void evolution term thus renders the ductility predictions more consistent with previously reported studies based on unit cell calculations for the set of material parameters employed in this study.

Investigation and prediction of tearing failure during extrusion based on a modified shear damage model

In this study, a recent modified shear GTN model suitable for various triaxialities was implemented and validated, which combines the porous plasticity model with a damage mechanics concept. The original Gurson model was modified by coupling the shear damage and void volume fraction parameters into the yield surface. The stress update algorithm was developed via a user-material subroutine. With the help of representative volume elements (RVEs), a connection between the formation of macroscopic cracks and the microstructure was established. The modified GTN model was applied to predict the damage and tearing fracture. Additionally, the cohesive zone model was applied to the RVE to simulate the de-bonding of the grain interfaces from a microscopic perspective. Metallurgical inspection by means of optical and scanning electron microscopy was performed to explore the failure mechanism and validate the numerical computation results. Furthermore, the deformation of the shear band and the damage propagation mechanism during the extrusion process were discussed.

A model for ductile damage prediction at low stress triaxialities incorporating void shape change and void rotation

Ductile fracture at the high triaxiality regime is well-known to be controlled by void nucleation, growth and coalescence. However, under low stress triaxiality conditions and general three dimensional finite deformations, damage is still poorly predicted due to the complex loading state and microstructural changes under such a condition. Experimental results have revealed not only void growth, but also important void shape change and void rotation under shear-dominated loading. The ability of ductile damage models to predict both void growth with shape change and void rotation is thus crucial for complex loading applications. In the present study, a Gurson-like nonlinear homogenization-based model (namely GVAR) is proposed and compared with the constitutive models for elasto-plastic porous materials developed in Kailasam and Ponte Castañeda (1998) (VAR model) and Danas and Aravas (2012) (MVAR model). The proposed model is based on ad hoc modifications of the VAR model, to give sufficiently accurate results for void growth at both low and high stress triaxialities and keeping the functional form of the original Gurson model. The VAR and MVAR models were based on rigorous linear comparison composite (LCC) homogenization methods, which can describe the evolution of microstructure of porous materials, represented by the void volume fraction, the aspect ratios and the orientations of general ellipsoidal voids. The proposed GVAR model thus inherits these characteristics and provides a sufficiently accurate void growth formulation (and simple at the same time). In addition, the loading direction is not necessary aligned with the ellipsoidal void axes. These models are implemented in an object-oriented finite element (FE) code. The identification of model parameters and the assessment of the proposed model are then carried out via 3D periodic unit-cell computations subjected to different stress states. Comparative results show that the present model predicts relatively accurately the evolution of void volume fraction, void aspect ratios and void rotation for different initial void shapes, void volume fractions and under different stress triaxiality levels. A qualitative application to a tensile test on a notched round bar shows the efficiency of the model to predict microstructure evolution (i.e. voids volume, shape and orientation) in a real-scale model simulation. This model with few parameters to be identified is thus promising to predict damage under complex loading paths and ready to be applied to complex FE simulations.

Use of a modified Gurson model for the failure behaviour of the clinched joint on Al6061 sheet

ABSTRACTThis paper discusses the ductile fracture behaviour of the clinched joint on Alloy 6061 sheets. Failure behaviour of the clinched joint is associated with the nucleation, growth and coalescence of voids within the microstructure. Various corrections to the original Gurson model are proposed to allow for instability and final fracture of the material. This paper is concerned with the application of the modified Gurson–Tvergaard–Needleman damage developed by Ken Nahshon and Zhenyu Xue. The results of the tension tests are compared with those of numerical analysis to obtain the initial void volume fraction f0 and the shear damage coefficients kw. The modified Gurson–Tvergaard–Needleman model is used to describe the failure behaviour and the shear strength of the clinched joint. Good agreement is shown between the experiment results and the numerical results on the location of fracture and the maximum failure load.

Numerical integration of an advanced Gurson model for shear loading: Application to the blanking process

A new extension of the Gurson damage model has been proposed recently to predict ductile fracture under shear dominated loads. The aim of this work is to verify the ability of this approach to simulate, in an accurate way, the damage evolution in shearing processes. An implicit stress integration algorithm is then developed to implement the new model in a finite element code. The numerical procedure is checked through simulations of shear and uniaxial tension tests on a single elements. The extended Gurson damage model is tested and applied to the punching process to compare its predictive ability with the original approach. The obtained numerical results are in good agreement with experimental results of the punching process, showing better ductile fracture predictions compared to the original Gurson model.

Implementation and Validation of a Gurson Damage Model Modified for Shear Loading: Effect of Void Growth Rate and Mesh Size on the Predicted Behavior

The optimization of automotive security components requires good knowledge of the material state after fabrication, particularly with respect to damage that may have been done to the material by the manufacturing process. To achieve this, numerical simulation of the fabrication process is often undertaken. However, classical continuum damage models, like the Gurson [3] model, are not appropriate for the simulation of the blanking by punching operation because the material damage is primarily the result of shear stresses. This work is focused on the use and validation of a modified Gurson type damage model capable of modeling this process, which has recently been proposed by Nahshan [7]. After a brief description of the modification, the implementation and the validation of the modified Gurson model is detailed. A comparison between the original Gurson model and the modified model is presented in order to highlight the importance of the modification for a pure shear stress state and to show that the two models are equivalent for a purely hydrostatic stress state. It is also shown that the results from the modified model are dependent on the finite element mesh size.

APPLICATION OF DAMAGE MODEL WITH MATERIAL FLOW EVALUATION TO HIGH TEMPERATURE COMPRESSION PROCESSES

Mechanisms of ductile failure in a compression process are studied, based on micro-structural considerations, with analytical and experimental investigations performed on a cylindrical specimen. Geometrical softening is able to occur by void if the shear band is generated inside of materials. The original Gurson model is used to analyze tensile deformation, and a modified Gurson model is proposed, which incorporates damage accumulation under shear and compression. This modified Gurson model is also applied to the void closure of sintered material under compression loading. However, the compressed materials must be considered in both the void nucleation and closure, because the void is nucleated and closed at a specific angle. The results of individual analyses of nucleation and closure do not agree with observations, but the analysis considered here, of both the nucleation and closure of a void, corresponds well with experimental results.

Ductile shear failure or plug failure of spot welds modelled by modified Gurson model

For resistance spot welded shear-lab specimens, interfacial failure under ductile shearing or ductile plug failure are analyzed numerically, using a shear modified Gurson model. The interfacial shear failure occurs under very low stress triaxiality, where the original Gurson model would predict void nucleation and very limited void growth. Void coalescence would therefore be largely postponed. However, using the shear modification of the Gurson model, recently introduced by Nahshon and Hutchinson (2008) [1], failure prediction is possible at zero or even negative mean stress. Since, this shear modification has too large effect in some cases where the stress triaxiality is rather high, an extension is proposed in the present study to better represent the damage development at moderate to high stress triaxiality, which is known to be well described by the Gurson model. Failure prediction and tensile response curves for an interfacial shear failure or a ductile plug failure, are here compared when using either the original Gurson model, the shear modified model, or the extension to the shear modified model. The suggested extension makes it possible to use the shear modified model as a simple way of accounting for damage development under low triaxiality shearing, without further increasing the damage rate in regions of moderate to high stress triaxiality.

A modified Gurson model and its application to punch-out experiments

Recent experimental evidence has reiterated that ductile fracture is a strong function of stress triaxiality. Under high stress triaxiality loading, failure occurs as a result of void growth and subsequent necking of inter-void ligaments while under low stress triaxiality failure is driven by shear localization of plastic strain in these ligaments due to void rotation and distortion. The original Gurson model is unable to capture localization and fracture for low triaxiality, shear-dominated deformations unless void nucleation is invoked. A phenomenological modification to the Gurson model that incorporates damage accumulation under shearing has been proposed. Here we further extend the model and develop the corresponding numerical implementation method. Several benchmark tests are performed in order to verify the code. Finally, the model is utilized to model quasi-static punch-out experiments on DH36 steel. It is shown that the proposed modified Gurson model, in contrast to the original model, is able to capture the through-thickness development of cracks as well as the punch response. Thus, the computational fracture approaches based on the modified Gurson model may be applied to shear-dominated failures.

An analysis of Gurson model with parameters dependent on triaxiality based on unitary cells

Abstract The Gurson model is widely used in the continuum-mechanics framework to analyse the ductile fracture process promoted by the nucleation, growth, and coalescence of voids. Further works improved the original Gurson model by introducing two parameters, q 1 and q 2 , to adjust model predictions to the numerical results of a periodic array of cylindrical and spherical voids in hardening materials. This modified model is known as the Gurson–Tvergaard (GT) model. Commonly, these parameters are considered constants or dependent only on the material-hardening properties. However, there is evidence that these parameters also depend on the triaxiality of the stress field, as well as on initial porosity. In this work, a consistent fully implicit integration of the constitutive equations of the GT model, considering the q-parameters dependent on the triaxiality and the initial porosity of the stress field, is presented, and the corresponding consistent tangent operator is proposed. The model is validated by comparing the stress–strain behaviour, as well as the evolution of void volume fraction, of a voided cell and the equivalent cell of GT material with dependent parameters. The cases considered correspond to variable triaxiality stress fields, present in non-proportional loading conditions.

Effect of a shear modified Gurson model on damage development in a FSW tensile specimen

For a friction stir welded aluminum plate the resistance to ductile failure is studied by analyzing tensile test specimens cut out across the weldline. As the stress triaxiality is rather low in these tests, the Gurson material model is not expected to give a very accurate description of the void growth to coalescence. A recently proposed modified version of the Gurson model is used, in which an extra term in the damage evolution law allows for the prediction of failure even at zero or negative values of the mean stress. This modification of the Gurson model is purely phenomenological, such that the damage parameter does not really represent the void volume fraction. Various amounts of the additional damage evolution are compared with predictions of the original Gurson model. The analyses are carried out for different yield stress profiles transverse to the weld and for different specimen widths. It is found that the modification does provide additional damage development in the friction stir weld, which may help to fit experimental data. But the suggested modification depends strongly on the overall stress state, and may have a too strong effect in some cases where the stress triaxiality is rather high.

Modification of the Gurson Model for shear failure

Recent experimental evidence points to limitations in characterizing the critical strain in ductile fracture solely on the basis of stress triaxiality. A second measure of stress state, such as the Lode parameter, is required to discriminate between axisymmetric and shear-dominated stress states. This is brought into the sharpest relief by the fact that many structural metals have a fracture strain in shear, at zero stress triaxiality, that can be well below fracture strains under axisymmetric stressing at significantly higher triaxiality. Moreover, recent theoretical studies of void growth reveal that triaxiality alone is insufficient to characterize important growth and coalescence features. As currently formulated, the Gurson Model of metal plasticity predicts no damage change with strain under zero mean stress, except when voids are nucleated. Consequently, the model excludes shear softening due to void distortion and inter-void linking. As it stands, the model effectively excludes the possibility of shear localization and fracture under conditions of low triaxiality if void nucleation is not invoked. In this paper, an extension of the Gurson model is proposed that incorporates damage growth under low triaxiality straining for shear-dominated states. The extension retains the isotropy of the original Gurson Model by making use of the third invariant of stress to distinguish shear dominated states. The importance of the extension is illustrated by a study of shear localization over the complete range of applied stress states, clarifying recently reported experimental trends. The extension opens the possibility for computational fracture approaches based on the Gurson Model to be extended to shear-dominated failures such as projectile penetration and shear-off phenomena under impulsive loadings.

FEM Analysis of Ductile Crack Growth in Fracture Transition Region for Steels with Different Void Nucleation Frequency.

Local stress and strain fields in front of a crack have been computed by FEM analysis using a Gurson model which takes into account void nucleation and growth within the material. The original Gurson model has been revised to fit the actual material by introducing experimentally obtained deformation and fractographic data of the material. Calculation has been applied for the ductile-to-brittle transition regions up to the ductile crack growth stage in two steels with different transition behaviors. Higher magnitudes of local stress and strain as well as larger extensions have been shown for a steel which has less nucleated void volume fraction, resulting in higher resistance against the ductile crack growth. Local stress increases with lowering test temperature whereas local strain decreases. Insignificant temperature dependence of R-curves has been understood from the calculated J-integral values because of the cancellation of the stress increase and strain decrease. The calculated R-curves have been shown to be consistent with the experimentally observed ones.

Analytical study of a hollow sphere made of plastic porous material and subjected to hydrostatic tension-application to some problems in ductile fracture of metals

Abstract A solution to the problem stated in the title is given by treating the porous matrix as a homogeneous material obeying the Gurson-Tvergaard model. As a first application, it is shown that the flow stress under hydrostatic loading of a material containing two populations of voids with very different sizes is almost the same as that of a material with only one population of voids and the same total porosity. As a second application, a self-consistent method is used to derive a value for the parameter introduced by Tvergaard in the original Gurson model to account for the interactions between cavities. Other models are also considered and shown to fail to satisfy the self-consistency requirement, whatever the value chosen for the parameter characterizing interactions between voids.

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Recent experimental evidence has reiterated that ductile fracture is a strong function of stress triaxiality. Under high stress triaxiality loading, failure occurs as a result of void growth and subsequent necking of inter-void ligaments while under low stress triaxiality failure is driven by shear localization of plastic strain in these ligaments due to void rotation and distortion. The original Gurson model is unable to capture localization and fracture for low triaxiality, shear-dominated deformations unless void nucleation is invoked. A phenomenological modification to the Gurson model that incorporates damage accumulation under shearing has been proposed. Here we further extend the model and develop the corresponding numerical implementation method. Several benchmark tests are performed in order to verify the code. Finally, the model is utilized to model quasi-static punch-out experiments on DH36 steel. It is shown that the proposed modified Gurson model, in contrast to the original model, is able to capture the through-thickness development of cracks as well as the punch response. Thus, the computational fracture approaches based on the modified Gurson model may be applied to shear-dominated failures.