Axisymmetric Tension Research Articles

Stress triaxiality and Lode angle parameters driven phase field coupled finite deformation plasticity formulation of ductile fracture

The effect of stress triaxiality and Lode angle parameters on crack initiation and propagation is investigated using the phase field coupled elasto-plasticity formulation. To accomplish this, a multi-invariants, i.e., first invariant of stress tensor and second and third invariants of deviatoric stress tensor, dependent finite deformation hyperelasto-plasticity is coupled with phase field theory of ductile fracture. The novel ideas include a proposition of a phase field coupled multi-invariants dependent yield function by including the Hosford equivalent stress in the Drucker–Prager yield function in order to accurately predict the ductile response of the material prior to initiation of the failure and the postulation of the threshold energy in the phase field evolution equation defining a measure of the ductility of materials. The measure of ductility is reported in the form of damage initiation surface in terms of threshold plastic energy as a function of stress triaxiality and normalized Lode angle parameters using the Mohr–Coulomb fracture initiation criterion. The model parameters are calibrated based on the available experimental results in the literature considering the stress triaxiality and Lode angle parameters dependent responses of different specimens. The crack initiation and propagation paths predicted by the proposed model under the different states of stress triaxiality and Lode angle parameters, such as axisymmetric tension, plane strain condition, and axisymmetric compression, match with those predicted experimentally in the literature.

Experimental study on post-fire mechanical properties and fracture behavior of Q690 steel

Fire is one of the most dangerous disasters for high-strength steel structures. After a fire event, the deterioration of mechanical properties of high-strength steel together with the complex stress state generated by external loading may lead to an unexpected failure or collapse of high-strength steel structures. Therefore, it is very critical to evaluate the post-fire behavior of high-strength steel considering stress state effects. This study conducted a series of tests on the Q690 steel specimens after being exposed to high temperatures ranging from 200 to 1000°C with different cooling methods including air-cooling and water-cooling methods. The tested specimens including smooth round specimens, notched round specimens, and grooved plate specimens were designed to consider the failure mode in axisymmetric stress state and plane strain state. To validate the accuracy of the test results, the mechanical properties such as yield strength, ultimate strength, elastic modulus, and ultimate strain were first compared with other test results reported in the literature. The post-fire ultimate strength under different stress states was obtained directly from the test results. Then, the post-fire equivalent plastic strain at fracture of different specimens was obtained from test data together with the finite element analysis results and analyzed under different stress triaxialities and Lode angle parameters. According to this study, an increase in stress triaxiality can result in an increase in ultimate strength but a decrease in the equivalent plastic strain at fracture for Q690 steel after fire. When the exposed temperature was not higher than 800°C, the post-fire strength of the plane strain specimen (Grooved Plane Specimen) was higher than that of the axisymmetric tension specimen (Notched Round Specimen) under a similar stress triaxiality; however, for their post-fire equivalent plastic strain at fracture, the results were opposite.

Dynamic Axisymmetric Tension of a Thin Round Ideally Rigid-Plastic Layer

Dynamic Axisymmetric Tension of a Thin Round Ideally Rigid-Plastic Layer

Dynamic Axisymmetric Tension of a Thin Round Ideally Rigid-Plastic Layer

We consider the stress-strain state that occurs during dynamic tension of a homogeneous round layer of an incompressible ideally rigid-plastic material that obeys the Mises–Genka criterion. The upper and lower bases are stress-free, and the radial velocity is set on the lateral boundary. The possibility of thickening or thinning of the layer is taken into account, which simulates neck formation and further development of the neck. Two characteristic tension modes are revealed. First one is associated with a rather high rate of removal of the side boundary of the layer from the center, the second one is associated with acceleration. In the second case, we have carried out an analysis using the method of asymptotic integration, which makes it possible to approximately find the parameters of the stress-strain state.

On the role of shape and distribution of secondary voids in the mechanism of coalescence

Ductile fracture of metals at room temperature has been studied extensively. Since localization of plastic flow is often viewed as a precursor to ductile fracture, several computational studies analyzing the influence of stress triaxiality and the Lode parameter on the critical strain for onset of localization have been reported. The implications of strain localization in materials containing randomly distributed voids for the mode of failure, that is, internal necking or shear banding have also been discussed. Apart from the influence of stress state, it is also of interest to model the effect of fine-scale microstructural attributes on the mode of failure and, hence, on material’s ductility. Therefore, in the present work, the mechanism of ductile fracture of an elastic–plastic solid containing two populations of pre-existing voids is modeled numerically. In the undeformed state, all the voids belonging to each family are assumed to have identical shape and size. Two different periodic spatial arrangements of primary voids are considered such that, in the absence of secondary voids, one of the configurations leads to internal necking whereas the other one exhibits a tendency for shear localization as the mode of coalescence for the larger voids. Small size secondary voids are then introduced explicitly and the role of shape and distribution of these sub-microscopic voids in the mechanism of coalescence and final failure is illustrated under plane strain as well as axisymmetric tension with a superposed hydrostatic stress. Depending on the microstructural attributes of the secondary voids, the mode of primary void coalescence is observed to shift from internal necking to void-sheeting and vice-versa. The implications of this shift in the mode of void coalescence on mesoscopic ductility is examined and a comparison of numerical predictions is made with the available experimental results.

Stress-state-dependency of post-necking hardening rule and its influence on ductile fracture prediction

This paper presents the investigation of the stress-state-dependency of the post-necking hardening rule of the Q355B structural steel and its influence on fracture prediction. The comparisons of the testing result and the simulation based on the Mises model with the stress-state-independent hardening rule indicate that triaxiality dependency under axisymmetric tension is negligible, while the reduced yield stress increases with the rising triaxiality under plane strain. In addition, the overestimated predicted force based on the Mises model is continuously ramping up along with the plastic deformation under the shear-dominant condition. It can be considered by the reduced post-necking hardening rate under the requirement of convexity of the yield surface. The stress-state-dependent hardening model (SDH model) is herein proposed including the modified triaxiality-dependent Hosford yield criterion and state-dependent post-necking hardening rule. The comparative studies of the proposed and conventional models show similar calculated fracture strains under high triaxiality, but considerably larger fracture strain under low triaxiality from the proposed SDH model, since the reduced post-necking hardening rate intensifies the localization of plastic deformation. Using the proposed pressure-dependent maximum shear stress (PMSS) fracture model, the calibrated fracture parameters based on the data from SDH the model produce better fracture simulations under various stress states, especially the complex condition in multi-hole plate (MHP) specimen. It emphasizes the necessity of the consideration of stress-state-dependency in the post-necking hardening rule for accurate fracture prediction.

Uncoupled fracture model of Q690 high‐strength steel based on stress triaxiality and Lode angle parameter

AbstractIn order to accelerate the popularization of high‐strength (HS) steel, the fracture property of Chinese Q690 HS steel was investigated through experiments, numerical simulation, and model establishment in this paper. A three‐dimensional fracture dataset of Q690 steel based on “stress triaxiality‐Lode angle parameter‐fracture strain” was obtained by a hybrid method of experiment and simulation, illustrating that the ductility of Q690 steel under plane strain is lower than that under axisymmetric tension. Eight existing uncoupled fracture models were calibrated, and the Xue‐W model shows reasonable applicability on Q690 steel. To improve the accuracy of fracture prediction, a stress triaxiality and Lode angle parameter dependent uncoupled ductile fracture model was newly proposed, and a simplified version that can be conveniently calibrated without specially designed specimen was provided for engineering application. The proposed model and the simplified version were verified, respectively, and both can achieve satisfactory prediction on the fracture behavior of Q690 HS steel.

The effect of material orientation on void growth

In this work, we have brought to light the effect of material orientation on void growth. For that purpose, we have performed finite element calculations using a cubic unit-cell model with a spherical void at its center and subjected to periodic boundary conditions. The behavior of the material is described with an elastic isotropic, plastic orthotropic constitutive model with yielding defined by Yld2004-18p criterion (Barlat et al., 2005). We have used the multi-point constraint subroutine developed by Dakshinamurthy et al. (2021) to enforce constant values of macroscopic stress triaxiality T and Lode parameter L in calculations that have been carried out for different stress states resulting from the combination of T=0.33, 1 and 2, with L=−1, 0 and 1 (axisymmetric tension, generalized shear and axisymmetric compression, respectively). Firstly, we have performed numerical simulations in which the loading directions are collinear with the orthotropy axes of the material, so that the principal directions of macroscopic stress and strain are parallel. Investigation of the cases for which the minor loading axis coincides either with the rolling, the transverse or the normal direction, has shown that the initially spherical void turns into an ellipsoid whose rate of growth and eccentricity depend on both stress state and material orientation. A key result is that for specific material orientations the anisotropy switches the effect of Lode parameter on void growth, reversing the trends obtained for isotropic von Mises materials. Secondly, we have carried out calculations using a novel strategy which consists of including angular misalignments within the range 0∘≤θ≤90∘, so that one loading direction is parallel to one of the symmetry axes of the material, and θ is the angle formed between the other two loading directions and the second and third orthotropy axes. In fact, to the authors’ knowledge, these are the first unit-cell calculations ever reported in which the material is modeled using a macroscopic anisotropic yield function with prescribed misalignment between loading and material axes and, at the same time, the macroscopic stress triaxiality and the Lode parameter are controlled to be constant during loading. The finite element calculations have shown that the misalignment between loading and material axes makes the void and the faces of the unit-cell to rotate and twist during loading. Moreover, the main contribution of this work is the identification of an intermediate value of the angle θ for which the growth rate of the void reaches an extreme value (minimum or maximum), so that the numerical results indicate that material orientation and angular misalignment can be strategically exploited to control void growth, and thus promote or delay localization and fracture of anisotropic metal products. The conclusions of this research have been shown to be valid for three different materials (aluminum alloys 2090-T3, 6111-T4 and 6013) and selected comparisons have also been performed using two additional yield criteria (CPB06ex2 and Yld2011-27p).

Plastic energy-based analytical approach to predict the mechanical response of two-phase materials with application to dual-phase steels

A composite made of two phases is considered with a perfect disorder of the phases, and isotropic behavior. The strain hardening behavior of such a composite is modeled under axisymmetric tension. The approach is based on using the strain hardening behavior of the two constituent phases together with a relation between the plastic energy of the two phases. The newly developed analytical model was applied to several dual-phase steel alloys and on iron-silver composite metal. These case studies revealed that the equal-power approach reproduces faithfully the strain hardening behavior of the composite, together with the strain partitioning between the two phases, in good agreement with experiments.

Micro-Mechanisms and Modeling of Ductile Fracture Initiation in Structural Steel after Exposure to Elevated Temperatures

This paper aims to (1) study ductile fracture behavior, and (2) provide a computational tool for predicting fracture initiation in ASTM A572 Gr. 50 structural steels under axisymmetric tension loading are heated to elevated temperatures and cooled down in air and in water. Employing the post-fire test results reported in the literature for A572 Gr. 50 steels, this paper carries out coupon-level finite element (FE) simulations to capture the stress and strain fields and explore the micro-mechanism of post-fire fracture in ASTM A572 Gr. 50 steels, respectively. Numerical results show that the effects of the experienced temperature and cooling method on fracture parameters are more significant for the steels cooled after being heated to temperatures from 800 °C to 1000 °C than those from 500 °C to 700 °C, due to microstructural changes during the cooling process. Air-cooled and water-cooled specimens show an improvement and a significant reduction in ductility, respectively. A modified void growth model (VGM) is proposed by introducing two additional temperature-dependent functions, through which the effects of elevated temperature and cooling method on fracture behavior are quantitatively analyzed. Limitations of this study are also discussed.

A modified micromechanics framework to predict shear involved ductile fracture in structural steels at intermediate and low-stress triaxialities

This paper employs a micromechanics framework to investigate the mechanisms and to predict the ductile fracture of structural steels at intermediate and low-stress triaxialities and under shear involved stress states. Unit cell-based micromechanical analyses are carried out to provide insights into the combined effects of triaxiality, Lode parameter, and shear stress component on the micro-mechanisms of ductile fracture. Based on the micromechanical analyses, an existing fracture model is modified to incorporate the influence of the shear stress component. Experimental investigations are carried out on three axisymmetric tension specimens, and a series of shear specimens made of Chinese Q460 steels to achieve uniaxial tension and shear dominated loading conditions, respectively. Finite element analyses are performed to evaluate the stress and strain fields in the tension and shear specimens. The modified model is calibrated by combining the experimental results with micromechanical analyses. Validation studies show that the predicted fracture initiation by the model agrees well with experimental results, implying a good performance of the proposed micromechanics framework for the prediction of the shear-dominated ductile fracture at intermediate and low-stress triaxialities.

Micromechanics-based identification of a ductile fracture model for three structural steels

From an engineering point of view, it is beneficial to reduce the number of mechanical tests required to calibrate the plasticity and fracture models of a structural material. In this study, the ductile fracture model for three high-strength steels is identified based on unit cell simulations, metal and porous plasticity modelling, and strain localization analysis combined with a single uniaxial tensile test per material. Finite element simulations of a unit cell model with a spherical void are performed with the matrix material described by metal plasticity and used to calibrate the parameters of the porous plasticity model. Strain localization analyses are conducted using the imperfection band approach with metal plasticity outside and porous plasticity inside the imperfection band. These simulations are first used to determine the nucleation rate in the porous plasticity model giving the experimentally obtained fracture strain in uniaxial tension, and then to compute the fracture locus under proportional loading in generalized axisymmetric tension. By combining the fracture locus with a simple damage accumulation rule and metal plasticity, finite element simulations of ductile fracture in tensile tests on smooth and notched specimens of the three steels are performed. Comparison of the predicted results with existing experimental data shows that the fracture model gives satisfactory estimates of ductility for a wide range of stress triaxiality ratios in steels of different strengths. This study shows the potential of micromechanical analyses in the calibration of fracture models for engineering applications.

Fundamental problem in optimizing the biaxial testing specimen

Most of the elements in structural design are subjected to the complex loading conditions that need to be predicted in advance for a reliable design. The failure prediction of these structural elements is based on the failure theories that employ the experimental data of the uniaxial, the biaxial and the shear tests. Among these tests, the biaxial testing is used to model the behavior of the structure loaded in more than one direction. In this study, we discover an interesting fact for the biaxial tension specimen that the thickness reduction of the measuring region cannot increase the stress to a high level. This behavior of the biaxial testing is different from the uniaxial testing where the thickness reduction of the measuring region always increases its stress to a higher level. Therefore, we further investigate the biaxial testing specimen via related fundamental problem, i.e., an axisymmetric plate with varying thickness under axisymmetric tension. We perform the shape optimization based on our analytical solution and the FEM results. For an infinite axisymmetric disk, the optimized design should include a low slope diverging section connecting the inner to outer region. On the contrary, the larger inner measuring region for finite axisymmetric disk would result in higher corresponding stress. These conclusions can be used as a guideline for the design of a biaxial testing specimen.

Numerical study of ductile failure under non-proportional loading

This paper investigates two numerical methods for predicting the initiation of ductile failure under moderately and strongly non-proportional loading paths. Two distinct phenomena are considered as indicators for the initiation of ductile failure: (i) the localization of deformation into a narrow band and (ii) the coalescence of microscopic voids. Recent experimental data in the literature from various axisymmetric tension tests on a high-strength steel are used to calibrate and validate the two methods. In the first method, which is based on the imperfection band approach, strain localization analyses are conducted using the deformation history extracted from finite element simulations of the tension tests. In the second method, axisymmetric unit cells are utilized to evaluate the onset of void coalescence using the stress history extracted from the same finite element simulations of the experiments. The various uniaxial tension tests yield different moderately and strongly non-proportional loading paths that are used to evaluate the predictive capabilities of the two methods. The numerical results are further used to discuss the similarities and differences between the two methods. Both the strain localization analyses and the coalescence analyses are found capable of predicting the initiation of failure in the experiments with good accuracy; however, the coalescence analyses are generally in closer agreement to the experiments.

Hydrogen-microvoid interactions at continuum scale

Experimentally, it has been shown that hydrogen can either enhance the internal necking failure or induce internal shearing failure of microvoids. In this study, the numerical investigation of hydrogen-microvoid interactions under the framework of hydrogen enhanced localized plasticity mechanism reveals that the actual effect of hydrogen depends on the stress state as well as on the hydrogen trapping effect. Hydrogen enhanced internal necking failure is observed over the entire stress space at a low level of trapping effect. While such failure is still observed in the high triaxiality regime at a high level of trapping effect, hydrogen induced internal shearing failure is observed in the low triaxiality regime. A hydrogen induced internal shearing failure criterion is proposed, and the failure loci corresponding to the low and high levels of trapping effect are constructed. The hydrogen induced internal shearing failure locus is found to be approximately independent of stress triaxiality while the hydrogen enhanced internal necking failure locus maintains similar triaxiality dependency as in the absence of hydrogen. The loss of ductility, in terms of reduction in failure strain and dimple size, is more pronounced in the case of hydrogen induced internal shearing failure. Subsequent study of the Lode effect reveals that plane strain tension is the most critical case for hydrogen induced internal shearing failure while axisymmetric tension is the most critical for hydrogen enhanced internal necking failure.

Strain hardening model of TWIP steels with manganese content

High Mn austenitic Fe–Mn–C steel is a superior material which exhibits excellent strain hardening behavior, with high strength and good ductility as a result of deformation-induced twinning. In this paper, a dislocation-based strain hardening model is developed by considering the effect of the Mn content on the mechanical behavior of TWIP steel. The Mn content varies in the range of 22–26wt%. In the model, the stacking fault energy (SFE) is considered to be dependent on the composition of the TWIP steel and the plastic strain. The driving force for twinning, which is controlled by the SFE, dominates the mechanical twin nucleation probability. The dislocation density evolution, which controls the strain hardening behavior, is influenced by the twinning volume fraction. The mechanical behaviors of TWIP steels with different Mn contents are tested in axisymmetric tension, and the microstructure evolution is observed in-situ by means of light-optical microscopy. The twinning volume fraction is determined on the basis of the observed twin area fraction. The predictions of the proposed model are verified by comparison with experimental data and the influence of the Mn content of the TWIP steel on the mechanical behavior is also analyzed.

Ductile fracture of Q460 steel: Effects of stress triaxiality and Lode angle

The ductile fracture characteristics of Chinese Q460 high strength structural steel under quasi-static condition were studied by using mechanical tests of four types of notched specimens. The influence of stress state on fracture mechanism of the material was investigated by observing the fracture surfaces of all test specimens using the Scanning Electron Microscope. Meanwhile, corresponding numerical simulations were conducted to collect the critical stress and strain at notch for all test specimens. The effects of stress triaxiality and Lode angle parameter, which were found to be the key parameters governing the ductile fracture of metallic material in many studies, on fracture strain of the Q460 structural steel were investigated. The analysis results show that different fracture mechanisms were observed in different stress triaxiality regions. At high stress triaxialities, Q460 steel exhibits a typical mechanism of “void nucleation, growth and coalescence”. When stress triaxiality equals to zero, a shear fracture mechanism was observed. At low stress triaxiality values, fracture develops as a combination of shear and void growth modes. In addition, the ductility of Q460 structural steel under pure shear or plane strain is lower than that under axisymmetric tension, especially at low stress triaxiality.

Void coalescence in ductile solids containing two populations of voids

Many metallic alloys contain a primary and a secondary population of voids, the difference in size reaching up to orders of magnitude. The present study extends Thomason’s and Benzerga−Leblond’s criteria for the onset of coalescence of primary voids to incorporate the effects of a secondary population. Plastic limit load analysis is performed within a finite element (FE) framework by using three dimensional (3D) cubic unit cells. The macroscopic stress state of the unit cells corresponds to axisymmetric tension with Σ22>Σ11=Σ33. The extended versions of both criteria are able to successfully predict the critical stress for the onset of coalescence.

Correlation between strength differential effects in the plastic flow of the matrix and the rate of damage growth in porous polycrystals

In this paper, we show that there is a strong correlation between the strength differential (SD) effects in the plastic flow of the matrix, which arise from its dependence on the third stress invariant, void evolution, and ultimately the ductility of porous metallic polycrystals. For this purpose, detailed micromechanical finite-element analyses of three-dimensional unit cells are carried out. The plastic flow of the matrix is described by a criterion that accounts for strength-differential effects induced by shear deformation mechanisms of the constituent grains through a macroscopic parameter, k; only if there is no SD, k is zero, and the von Mises criterion is recovered. Numerical analyses are conducted for macroscopic proportional tensile loadings corresponding to fixed values of the stress triaxiality (ratio of the mean stress to the second stress invariant). It is shown that for the same macroscopic loading, the local plastic strains and the local stress distribution are strongly dependent on the sign of the parameter k. This in turn has a huge impact on damage accumulation, and ultimately affects the ductility of the porous polycrystals. Specifically, for axisymmetric loadings at third stress invariant positive, the rate of void growth is the slowest in the material with k negative, while the reverse holds true for equibiaxial tension (third stress invariant negative). Consequently, the ductility in axisymmetric tension at third-stress invariant positive is also markedly different from that in equibiaxial tension (third-stress invariant negative).

Influence of the Lode parameter and the stress triaxiality on the failure of elasto-plastic porous materials

This work makes use of a recently developed “second-order” homogenization model to investigate failure in porous elasto-plastic solids under general triaxial loading conditions. The model incorporates dependence on the porosity and average pore shape, whose evolution is sensitive to the stress triaxiality and Lode parameter L. For positive triaxiality (with overall tensile hydrostatic stress), two different macroscopic failure mechanisms are possible, depending on the level of the triaxiality. At high triaxiality, void growth induces softening of the material, which overtakes the intrinsic strain hardening of the matrix phase, leading to a maximum in the effective stress–strain relation for the porous material, followed by loss of ellipticity by means of dilatant shear localization bands. In this regime, the ductility decreases with increasing triaxiality and is weakly dependent on the Lode parameter, in agreement with earlier theoretical analyses and experimental observations. At low triaxiality, however, a new mechanism comes into play consisting in the abrupt collapse of the voids along a compressive direction (with small, but finite porosity), which can dramatically soften the response of the porous material, leading to a sudden drop in its load-carrying capacity, and to loss of ellipticity of its incremental constitutive relation through localization of deformation. This low-triaxiality failure mechanism leads to a reduction in the ductility of the material as the triaxiality decreases to zero, and is highly dependent on the value of the Lode parameter. Thus, while no void collapse is observed at low triaxiality for axisymmetric tension (L=-1), the ductility of the material drops sharply with decreasing values of the Lode parameter, and is smallest for biaxial tension with axisymmetric compression (L=+1). In addition, the model predicts a sharp transition from the low-triaxiality regime, with increasing ductility, to the high-triaxiality regime, with decreasing ductility, as the failure mechanism switches from void collapse to void growth, and is in qualitative agreement with recent experimental work.