Prediction Of Strain Localization Research Articles

Determination of nonlocal kernels in nonlocal models for solving strain localization problems

ABSTRACT The strain localization during the damage and failure of materials can be accurately predicted by nonlocal continuum theories. Yet, the results of theoretical predictions are directly affected by the nonlocal kernel formulation. In this study, the correlation between the nonlocal kernel formulation and the strain localization simulation results is revealed. First, based on the nonlocal damage theory, a novel free energy functional was proposed with strain and damage parameters as independent variables, and a constitutive relation considering nonlocality and elastic damage was derived within the framework of continuum thermodynamics. Then, one-dimensional governing equations were obtained through the dual-linearization method. Finally, strain localization predictions obtained with several nonlocal kernel formulations were presented, and the relationship between the nonlocal kernels and the solution was analyzed, thereby determining an appropriate nonlocal kernel function for the strain localization problem.

Flow stress curves for 980MPa- and 1.5GPa-class ultra-high-strength steel sheets weakened under high-stress triaxiality

The weakening behavior of ultra-high-strength steel(UHSS) remains unknown, even though it is a requirement for the accurate prediction of strain localization and ductile fracture in automobile applications. In this study, we, therefore, revealed the stress–strain curves of UHSSs up to their ductile fracture under different stress triaxialities for the first time. Smooth and notched tiny round-bar specimens cut from UHSSs with 1.6 mm thickness were subjected to tensile tests, during which the forces and neck diameters of the specimens were measured. Notably, the evaluated flow stresses in the 1.5 GPa-class notched UHSS specimens exhibited weakening of up to 3%. This was in contrast to the 980 MPa-class UHSS specimens, which did not exhibit notch-induced weakening. The absence of weakening during stress measurements, as confirmed by synchrotron X-ray diffraction (XRD) analysis, suggested that weakening was caused by material damage. However, the weakening behavior did not follow the previously developed damage models based on microvoid formation. The volume fraction of the microvoids, as observed by X-ray computer tomography, was extremely small (at 0.2%) and cannot account for the 3% material weakening indicated by the flow stress measurement results. A new damage mechanism, associated with unobservable small lattice vacancies, was implied in UHSSs deformation under high stress triaxialities.

Study on the influence of the yield surface shape in the hole expansion test

The hole expansion test has become popular since it allows to study the formability of metallic sheets, in particular the onset of necking in stretch flanging areas. The accurate prediction of strain localization and consequent fracture requires a proper description of the plastic behavior, particularly the anisotropic yield function. In the context of strain localization prediction, the yield criterion adopted plays an important role, particularly when using an associated flow rule, since the direction of the plastic strain tensor is modelled by the normal to the yield surface. In this work, the parameters of an advanced yield criterion are calibrated considering a wide set of experimental data, which includes results from uniaxial and biaxial tension tests. This enables establishing yield surfaces with similar shape in the plane defined by the stress components in the rolling and transverse directions and a null shear component in the same plane. However, their shape changes slightly when considering non-null values for that shear component. The numerical simulations of the hole expansion test demonstrate the impact of these slight differences on the thickness strain distribution and, consequently, in the instant and location of the necking.

Numerical modeling for accurate prediction of strain localization in hole expansion of a steel sheet

The hole expansion of a low carbon steel sheet shows an interesting feature that localized thinning and subsequent crack initiation are observed inside the specimen, and not at the hole edge as is typically expected. The present work investigated a numerical modeling approach to predict this localization behavior within the framework of a finite element (FE) analysis. Plastic anisotropy of the sheet was taken into account using the anisotropic yield functions Yld2000-2d and Yld2004-18p for the plane stress and three-dimensional elements, respectively. Careful examination of the FE model revealed that the influence of the out-of-plane stress is very small, suggesting that shell elements can be efficiently used in the analysis. The influence of friction was also found to be negligibly small. However, the constitutive description exhibited a significant influence in that even a slight change in the yield function parameters resulted in a considerable difference in the prediction. For this reason, several sets of parameters were obtained based on the different material properties, and their influences on the hole expansion simulation were analyzed. In particular, the prediction accuracy could be greatly improved when the yield function parameters were optimized such that the flow stresses and plastic strain rate ratios in uniaxial and plane strain states were well captured.

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.

Influence of yield surface curvature on the macroscopic yielding and ductile failure of isotropic porous plastic materials

Numerical unit cell models of an approximative representative volume element for a porous ductile solid are utilized to investigate differences in the mechanical response between a quadratic and a non-quadratic matrix yield surface. A Hershey equivalent stress measure with two distinct values of the yield surface exponent is employed as the matrix description. Results from the unit cell calculations are further used to calibrate a heuristic extension of the Gurson model which incorporates effects of the third deviatoric stress invariant. An assessment of the porous plasticity model reveals its ability to describe the unit cell response to some extent, however underestimating the effect of the Lode parameter for the lower triaxiality ratios imposed in this study when compared to unit cell simulations. Ductile failure predictions by means of finite element simulations using a unit cell model that resembles an imperfection band are then conducted to examine how the non-quadratic matrix yield surface influences the failure strain as compared to the quadratic matrix yield surface. Further, strain localization predictions based on bifurcation analyses and imperfection band analyses are undertaken using the calibrated porous plasticity model. These simulations are then compared to the unit cell calculations in order to elucidate the differences between the various modelling strategies. The current study reveals that strain localization analyses using an imperfection band model and a spatially discretized unit cell are in reasonable agreement, while the bifurcation analyses predict higher strain levels at localization. Imperfection band analyses are finally used to calculate failure loci for the quadratic and the non-quadratic matrix yield surface under a wide range of loading conditions. The underlying matrix yield surface is demonstrated to have a pronounced influence on the onset of strain localization.

Hardening effects on strain localization predictions in porous ductile materials using the bifurcation approach

The localization of deformation into planar bands is often considered as the ultimate stage of strain prior to ductile fracture. In this study, ductility limits of metallic materials are predicted using the Gurson–Tvergaard–Needleman (GTN) damage model combined with the bifurcation approach. Both the GTN constitutive equations and the Rice bifurcation criterion are implemented into the finite element (FE) code ABAQUS/Standard within the framework of large plastic strains and a fully three-dimensional formulation. The current contribution focuses on the effect of strain hardening on ductility limit predictions. It is shown that the choice of void nucleation mechanism has an important influence on the sensitivity of the predicted ductility limits to strain hardening. When strain-controlled nucleation is considered, varying the hardening parameters of the fully dense matrix material has no effect on the porosity evolution and, consequently, very small impact on the predicted ductility limits. For stress-controlled nucleation, the porosity evolution is directly affected by the strain hardening characteristics, which induce a significant effect on the predicted ductility limits. This paper also discusses the use of a micromechanics-based calibration for the GTN q-parameters in the case of strain-controlled nucleation, which is also shown to allow accounting for the hardening effects on plastic strain localization.

Investigation of localized necking in substrate-supported metal layers: Comparison of bifurcation and imperfection analyses

Localized necking is often considered as precursor to failure in metal components. In modern technologies, functional components (e.g., in flexible electronic devices) may be affected by this necking phenomenon, and to avoid the occurrence of strain localization, elastomer substrates are bonded to the metal layers. This paper proposes an investigation of the development of localized necking in both freestanding metal layers and elastomer/metal bilayers. Finite strain versions of both rigid–plastic flow theory and deformation theory of plasticity are employed to model the mechanical response of the metal layer. For the elastomer, a neo-Hookean constitutive law is considered. Localized necking is predicted using both bifurcation (whenever possible) and Marciniak–Kuczynski analyses. A variety of numerical results are presented, which pertain to the prediction of localized necking in freestanding metal layers and metal/substrate bilayers. The effects of the constitutive framework and the presence of an elastomer substrate on strain localization predictions have been specifically highlighted. It is demonstrated that the addition of an elastomer layer can retard significantly the occurrence of localized necking. It is also demonstrated that the results of the Marciniak–Kuczynski analysis tend towards the bifurcation predictions in the limit of a vanishing size for the geometric imperfection.

Ductility limit prediction using a GTN damage model coupled with localization bifurcation analysis

Because the localization of deformation into narrow planar bands is often precursor to material failure, several approaches have been proposed to predict this phenomenon. In this paper, the Gurson–Tvergaard–Needleman (GTN) elastic–plastic–damage model for ductile materials is considered. A large-strain version of this constitutive model is coupled with the Rice localization criterion, which is based on bifurcation theory, to investigate strain localization. The resulting loss of ellipticity condition is then used to determine ellipticity loss diagrams (ELDs) associated with strain paths that are those typically applied to metals under biaxial stretching. A sensitivity analysis is conducted with respect to the model parameters on a representative selection of ductile materials. The analysis shows that the damage parameters have a significant impact on the predicted ELDs, which confirms the predominant role of damage-induced softening in triggering plastic flow localization with the adopted constitutive description combined with the bifurcation approach. As a consequence of this high sensitivity, it appears that the proper identification of damage parameters is a key issue for accurate plastic flow localization predictions using the GTN model coupled with bifurcation theory. The effect of the dense matrix hardening parameters on the strain localization predictions of the voided aggregate, although found much smaller in the whole, is more noticeable for the plane strain tension loading path or, more generally, when the critical hardening modulus required for localization is not strongly negative.

Experimental Validation of Finite Element Analysis of Human Vertebral Collapse Under Large Compressive Strains

Osteoporosis-related vertebral fractures represent a major health problem in elderly populations. Such fractures can often only be diagnosed after a substantial deformation history of the vertebral body. Therefore, it remains a challenge for clinicians to distinguish between stable and progressive potentially harmful fractures. Accordingly, novel criteria for selection of the appropriate conservative or surgical treatment are urgently needed. Computer tomography-based finite element analysis is an increasingly accepted method to predict the quasi-static vertebral strength and to follow up this small strain property longitudinally in time. A recent development in constitutive modeling allows us to simulate strain localization and densification in trabecular bone under large compressive strains without mesh dependence. The aim of this work was to validate this recently developed constitutive model of trabecular bone for the prediction of strain localization and densification in the human vertebral body subjected to large compressive deformation. A custom-made stepwise loading device mounted in a high resolution peripheral computer tomography system was used to describe the progressive collapse of 13 human vertebrae under axial compression. Continuum finite element analyses of the 13 compression tests were realized and the zones of high volumetric strain were compared with the experiments. A fair qualitative correspondence of the strain localization zone between the experiment and finite element analysis was achieved in 9 out of 13 tests and significant correlations of the volumetric strains were obtained throughout the range of applied axial compression. Interestingly, the stepwise propagating localization zones in trabecular bone converged to the buckling locations in the cortical shell. While the adopted continuum finite element approach still suffers from several limitations, these encouraging preliminary results towards the prediction of extended vertebral collapse may help in assessing fracture stability in future work.

Issues in connection to the prediction of strain localization in metal sheets

The current paper deals with the prediction of strain localization in metal sheets. Especially the effect of strain rate dependency on the necking phenomenon is treated. The ability of FEM to predict strain localization during simulations of sheet forming operations is investigated. An approach for the determination of the strain rate material parameters, based on an experimentally determined FLD0, is suggested. The current approach is demonstrated for three sheet materials, and from the results the approach can be judged as promising.