Anisotropic Hardening Research Articles

Experimental validation of numerical model for asymmetric deep drawing of DP780 steel sheet using digital image correlation

A validated numerical model for in-plane stress/strain prediction is essential in understanding the deformation behaviour of sheet metal forming process, in particular, asymmetric deep drawing of advanced high strength steel sheet. In this work, the Yld2000-2d anisotropic yield criterion integrated with the initial homogeneous anisotropic hardening model was employed to describe the complex material behaviours of DP780 steel as well as the adoption of Yoshida chord model for elastic modulus degradation. Digital image correlation technique was utilized to measure the in-plane strain and shape deviation of the developed P- channel. The validity of the numerical model was assessed by comparing the predicted strain distribution and twist springback with the measured results. It indicates that the developed numerical model based on the selected constitutive models is acceptable for the deformation analysis, although the predicted discrepancies still exist.

Application of the virtual fields method to the identification of the homogeneous anisotropic hardening parameters for advanced high strength steels

In the present paper, an inverse problem solution so called the virtual fields method (VFM) is implemented to identify the parameters of the homogeneous anisotropic hardening (HAH) model, a distortional plasticity-based model that describes the material plastic behavior when subjected to strain path changes. The framework of the identification method that combines the formulation of the yield condition, the constitutive stress–strain relation and the principle of virtual work is presented. For validation purpose, the proposed identification method was first attempted on finite element (FE) generated data for a forward-reverse simple shear test to investigate its capability in retrieving the input constitutive parameters. The influence of noise was also evaluated. Then, the identification method was applied to a selection of advanced high strength steel (AHSS), namely DP600, TRIP780 and TWIP980, sheet specimens, subjected to a small number of forward-reverse simple shear cycles. The material constitutive parameters were identified using the VFM based on which shear stress–strain curves were calculated and compared with their experimental counterparts. Good agreement was found between the calculated and the experimental curves despite the larger discrepancies observed in the reverse loading paths. To adjust these discrepancies, the original HAH model was modified with respect to the permanent softening related state variables. After modification, the model was simplified with only one state variable related to permanent softening. It was found that the discrepancies observed in the reverse loading paths were reduced with the modified HAH model.

The anisotropic elasto-plastic analysis using a natural neighbour RPIM version

In this work, the natural neighbour radial point interpolation method (NNRPIM), an advanced discretization meshless technique, is used to study several solid mechanics benchmark examples considering an elasto-plastic approach. The NNRPIM combines the radial point interpolators (RPI) with the natural neighbour geometric concept. The nodal connectivity and the background integration mesh depend entirely on the nodal discretization and are both achieved using mathematic concepts, such as the Voronoi diagram and the Delaunay tessellation. The obtained interpolation functions, used in the Galerkin weak form, are constructed with the RPI and possess the delta Kronecker property, which facilitates the imposition of the natural and essential boundary conditions. Due to the organic procedure employed to impose the nodal connectivity, the obtained displacement and the stress field are smooth and accurate. Since the scope of this work is to extend and validate the NNRPIM in the anisotropic elasto-plastic analysis, it is used as nonlinear solution algorithm, the modified Newton–Raphson initial stiffness method. The efficient “backward-Euler” procedure is considered to return the stress to the Hill anisotropic yield surface. The elasto-plastic algorithm of solution is described. Several benchmark two-dimensional problems are solved, considering anisotropic elasto-plastic materials with anisotropic hardening, and the obtained solutions are compared with finite-element solutions, showing that the meshless approach developed is efficient and accurate.

Assessment of anisotropic hardening models for conventional deep drawing processes

Assessing the predictive capabilities of recent advanced constitutive modelling approaches for processes with industrial complexity is a challenging task. Real process conditions such as blankholder pressure distribution, friction and tool elasticity sensitively affect experimental observations, making the isolation of constitutive effects difficult. A systematic approach is proposed in this work to assess the performance of anisotropic hardening models with the least possible disturbance from process conditions. Two deep drawing examples were used for these purpose (“cross die” and “lackfrosch”) in conjunction with a mild steel (DC05). Optically measured strain distributions have been compared to corresponding simulations, which have been calibrated to accurately match the measured blank draw-in. The effect of initial yield locus shape as well as anisotropic hardening effects have been discussed.

Evolutionary anisotropy and flow stress in advanced high strength steels under loading path changes

Chaboche kinematic hardening, Yoshida–Uemori, and homogenous anisotropic hardening (HAH) models were evaluated in terms of their prediction capability of flow stress and evolutionary r-value under loading path changes. The first two models are based on kinematic hardening, whereas the HAH model is based on distortional hardening. Continuous compression–tension (CT) tests with pre-strains for dual-phase 780 and transformation-induced plasticity (TRIP) 780 steel sheets were conducted. For the initial anisotropy described by the Yld2000-2d model, uniaxial tension tests for three material orientations and a biaxial test were performed. During the CT tests, both stress and r-value variations with plastic strain were measured. All the models were implemented into the finite element software ABAQUS using user material subroutines. Experimentally, both sheets exhibited typical anisotropic hardening features such as the Bauschinger effect, transient hardening, and long-range softening. The TRIP 780 exhibited small work-hardening stagnation. In addition, the evolutionary r-value during the second loading was significant. The three models could predict most of the features in the stress–strain curves during the second loading. However, only the HAH model could predict the experimentally observed r-value evolution. This evolutionary r-value predicted by the HAH model could be explained by distortion of the yield locus during the first loading and its rapid recovery toward the initial yield locus shape.

Anisotropic UH model for soils based on a simple transformed stress method

SummaryA simple method called anisotropic transformed stress (ATS) method is proposed to develop failure criteria and constitutive models for anisotropic soils. In this method, stress components in different directions are modified differently in order to reflect the effect of anisotropy. It includes two steps of mapping of stress. First, a modified stress tensor is introduced, which is a symmetric multiplication of stress tensor and fabric tensor. In the modified stress space, anisotropic soils can be treated to be isotropic. Second, a TS tensor is derived from the modified stress tensor for the convenience of developing anisotropic constitutive models to account for the effect of intermediate principal stress. By replacing the ordinary stress tensor with the TS tensor directly, the unified hardening model is extended to model the anisotropic deformation of soils. Anisotropic Lade's criterion is adopted for shear yield and shear failure in the model. The form of the original model formulations remains unchanged, and the model parameters are independent of the loading direction. Good agreement between the experimental results and predictions of the anisotropic unified hardening model is observed. Copyright © 2016 John Wiley & Sons, Ltd.

Evaluation of Springback for DP980 S Rail Using Anisotropic Hardening Models

The effect of anisotropic hardening models on springback of an S-rail part was investigated. Two advanced constitutive models based on distortional and kinematic hardening, which captured the Bauschinger effect, transient hardening, and permanent softening during strain path change, were implemented in a finite element (FE) code. In-plane compression–tension tests were performed to identify the model parameters. The springback of the S-rail after forming a 980 MPa dual-phase steel sheet sample was measured and analyzed using different hardening models. The comparison between experimental and FE results demonstrated that the advanced anisotropic hardening models, which are particularly suitable for non-proportional loading, significantly improved the springback prediction capability of an advanced high strength steel.

Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths

This paper aims to identify the mechanisms associated to the transient hardening behaviour of dual phase steels under strain path changes, and to capture the observed material behaviours with appropriate constitutive models. First, three DP steel sheets with different amounts of martensite were tested under monotonic and various strain path changes. Second, microstructural analysis of the materials before and after strain path change were performed by means of SEM, TEM, and EBSD. The contribution of texture evolution on the mechanical behaviour was also assessed using the visco-plastic self-consistent (VPSC) polycrystal plasticity model. Transient hardening behaviour and permanent softening were observed in the tension–tension tests for all the studied DP steels. These behaviours were explained by the development of strain gradients during the first load resulting from strain accommodation incompatibilities between the ferrite and martensite phases. For the purpose of describing the macroscopic material behaviours, the enhanced homogeneous anisotropic hardening (HAH) model (Barlat et al., 2014) integrated with the Yld2000-2d anisotropic yield function were adopted for constitutive modelling. The simulation results were discussed in view of the microstructure evolution.

Influence of material models on theoretical forming limit diagram prediction for Ti–6Al–4V alloy under warm condition

Forming limit diagram (FLD) is an important performance index to describe the maximum limit of principal strains that can be sustained by sheet metals till to the onset of localized necking. It offers a convenient and useful tool to predict the forming limit in the sheet metal forming processes. In the present study, FLD has been determined experimentally for Ti–6Al–4V alloy at 400 °C by conducting a Nakazima test with specimens of different widths. Additionally, for theoretical FLD prediction, various anisotropic yield criteria (Barlat 1989, Barlat 1996, Hill 1993) and different hardening models viz., Hollomon power law (HPL), Johnson–Cook (JC), modified Zerilli–Armstrong (m-ZA), modified Arrhenius (m-Arr) models have been developed. Theoretical FLDs have been determined using Marciniak and Kuczynski (M–K) theory incorporating the developed yield criteria and constitutive models. It has been observed that the effect of yield model is more pronounced than the effect of constitutive model for theoretical FLDs prediction. However, the value of thickness imperfection factor (f0) is solely dependent on hardening model. Hill (1993) yield criterion is best suited for FLD prediction in the right hand side region. Moreover, Barlat (1989) yield criterion is best suited for FLD prediction in left hand side region. Therefore, the proposed hybrid FLD in combination with Barlat (1989) and Hill (1993) yield models with m-Arr hardening model is in the best agreement with experimental FLD.

Accurate characterization of biaxial stress-strain response of sheet metal from bulge testing

A new method considering anisotropic deformation is developed to accurately calculate principal stresses and strains as well as the actual thickness at the specimen pole in circular bulge testing combined with digital image correlation techniques. The new method takes into account the elastic dilatancy and the bending effect when calculating the thickness at the specimen pole, the ratio of specimen thickness to the radius of curvature on the specimen outer surface and non-balanced biaxial curvatures in principal directions when calculating the effective stress at the specimen pole, which have been not considered in previous methods from literature including the ISO Standard (ISO 16808: 2014). The accuracy of the proposed calculation method is validated by circular hydraulic bulge testing for a 1.1 mm thick Al–Mg alloy and a 1.0 mm thick dual-phase steel. Accuracy is improved in comparison to previous methods described in the literature, which underestimate the thickness at the specimen pole and lead to an overestimation of stresses by up to 3% for the tested materials. In addition, attributed to the consideration of anisotropic deformation in the proposed method, the plastic strain ratio (also known as the rb-value) in biaxial deformation is obtained as a function of equivalent plastic strain, which can be used to calibrate advanced anisotropic hardening models that allow for deformation-induced evolution of the yield surface shape.

Developments of New Sheet Metal Forming Technology and Theory in China

Developments of new sheet metal forming technology and theory in China are reviewed in detail in this paper. Advances of crystal plasticity on the deformation mechanism of Mg alloy are firstly described, especially its applications on the prediction of sheet forming process. Then, a new macroscopic constitutive model is introduced, which possesses an enhanced description capacity of tension/compression anisotropy and anisotropic hardening. In order to take into account the twinning process of hexagonal close-packed material, a modified hierarchical multi-scale model is also established with adequate accuracy in a shorter computational time. The advanced forming limit of sheet metal, mainly about aluminum alloy, is also investigated. Besides the above theory developments, some new sheet metal forming technologies are reviewed simultaneously. The warm forming technology of Mg alloy is discussed. New processes to form sheet parts and to bend tubes are proposed by using hard granules. On the other hand, a new kind of ultra-high-strength steel based on typical 22MnB5 by introducing more residual austenite and Cu-rich phase to increase the elongation and strength and its novel forming method that integrates hot stamping and quenching participation are proposed. Progresses in sheet hydroforming, press forging and electromagnetic forming of sheet metal parts are also summarized.

Constitutive modelling of high strength titanium alloy Ti-6Al-4 V for sheet forming applications at room temperature

• A rigours set of experiments were performed to measure the initial yield surface of Ti-6Al-4V at room temperature under different-stress states. • A constitutive equation based anisotropic hardening model (HAH) model combined with the new determined yield function was proposed and implemented into the FEA- Abaqus implicit solver. • The proposed model showed ability to accurately predict the cyclic hardening response of Ti-6Al-4V, and therefore presents a promising approach for the prediction of deformation behaviour of Ti-6Al-4V alloy in sheet metal forming processes. To enable the design and optimisation of forming processes at room temperature the material behaviour of Ti-6Al-4 V needs to be accurately represented in numerical analysis and this requires an advanced material model. In particular, an accurate representation of the shape and size of the yield locus as well as its evolution during forming is important. In this study a rigorous set of experiments on the quasi-static deformation behaviour of a Ti-6Al-4 V alloy sheet sample at room temperature was conducted for various loading conditions and a constitutive material model developed. To quantify the anisotropy and asymmetry properties, tensile and compression tests were carried out for different specimen orientations. To examine the Bauschinger effect and the transient hardening behaviour in – plane tensile – compression and compression – tensile tests were performed. Balanced biaxial and plane strain tension tests were conducted to construct and validate the yield surface of the Ti-6Al-4 V alloy sheet sample at room temperature. A recently proposed anisotropic elastic-plastic constitutive material model, so-called HAH, was employed to describe the behaviour, in particular for load reversals. The HAH yield surface is composed of a stable component, which includes plastic anisotropy and is distorted by a fluctuating component. The key of the formulation is the use of a suitable yield function that reproduces the experimental observations well for the stable component. Meanwhile, the rapid evolution of the material structure must be captured at the macro - scale level by the fluctuating component embedded in the HAH model. Compared to conventional hardening equations, the proposed model leads to higher accuracy in predicting the Bauschinger effect and the transient hardening behaviour for the Ti-6Al-4 V sheet sample tested at room temperature.

A crystal plasticity model of a formation of a deformation band structure

The formation of deformation bands with the typically alternating sign of the misorientation across their boundaries is interpreted as spontaneous deformation instability caused by anisotropy of hardening. To analyse the nature of the fragmentation, a model of a rigid-plastic crystal domain deformed by symmetric double slip in a plane-strain compression is considered. The basic reason for the deformation band existence is that a local decrease in number of active slip systems in the bands is energetically less costly than a homogeneous deformation by multislip. However, such model of the bands predicts their extreme orientation and their width tends to zero. This trend is modified by hardening caused by a build up of the band boundaries and by a dislocation bowing (Orowan) stress. The model provides an explanation of observed orientation of the bands, their width and the significant change in the structural morphology seen as the band reorientation occurs at large strains. The predictions are in a favourable agreement with the available observations.

Parameter identification of the homogeneous anisotropic hardening model using the virtual fields method

In this paper, the virtual fields method is applied to identify the constitutive parameters of the homogeneous yield function-based anisotropic hardening model. The procedure, previously developed for isotropic hardening law, is extended to the anisotropic case in this study. For validation purpose, a finite element model is developed to check the feasibility of the proposed methodology in retrieving the target constitutive parameters. The identification results are critically discussed. The suitability of this finite element model is also assessed with a view to future experiments.

Anisotropic viscoelastic–viscoplastic continuum model for high-density cellulose-based materials

A continuum material model is developed for simulating the mechanical response of high-density cellulose-based materials subjected to stationary and transient loading. The model is formulated in an infinitesimal strain framework, where the total strain is decomposed into elastic and plastic parts. The model adopts a standard linear viscoelastic solid model expressed in terms of Boltzmann hereditary integral form, which is coupled to a rate-dependent viscoplastic formulation to describe the irreversible plastic part of the overall strain. An anisotropic hardening law with a kinematic effect is particularly adopted in order to capture the complex stress–strain hysteresis typically observed in polymeric materials. In addition, the present model accounts for the effects of material densification associated with through-thickness compression, which are captured using an exponential law typically applied in the continuum description of elasticity in porous media.Material parameters used in the present model are calibrated to the experimental data for high-density (press)boards. The experimental characterization procedures as well as the calibration of the parameters are highlighted. The results of the model simulations are systematically analyzed and validated against the corresponding experimental data. The comparisons show that the predictions of the present model are in very good agreement with the experimental observations for both stationary and transient load cases.

Comparison of two models for anisotropic hardening and yield surface evolution in bcc sheet steels

Abstract The purpose of the current work is the investigation and comparison of aspects of the material behavior predicted by two models for anisotropic, and in particular cross, hardening in bcc sheet steels subject to non-proportional loading. The first model is the modified form (Wang et al., 2008) of that due to Teodosiu and Hu (1995, 1998). In this (modified) Teodosiu-Hu model (THM), cross hardening is assumed to affect the yield stress and the saturation value of the back stress. The second model is due to Levkovitch and Svendsen (2007) and Noman et al. (2010). In the Levkovitch-Svendsen model (LSM), cross hardening is assumed to affect the flow anisotropy. As clearly demonstrated in a number of works applying the THM (e.g., Boers et al., 2010; Bouvier et al., 2005, 2003; Hiwatashi et al., 1997; Li et al., 2003; Thuillier et al., 2010; Wang et al., 2008) and the LSM (e.g., Clausmeyer et al., 2014, 2011b; Noman et al., 2010), both of these are capable of predicting the effect of cross hardening on the stress-deformation behavior observed experimentally in sheet steels. As shown in the current work, however, these two models differ significantly in other aspects, in particular with respect to the development of the yield stress, the back stress, and the yield surface. For example, the THM predicts no change in the shape of the yield surface upon change of loading path, in contrast to the LSM and crystal plasticity modeling of bcc sheet steels (Peeters et al., 2002). On the other hand, the LSM predicts no hardening stagnation after cross hardening as observed in experiments, in contrast to the THM. Examples are given.

A non-associated plasticity model with anisotropic and nonlinear kinematic hardening for simulation of sheet metal forming

A material model for more thorough analysis of plastic deformation of sheet materials is presented in this paper. This model considers the following aspects of plastic deformation behavior of sheet materials: (1) the anisotropy in yield stresses and in work hardening by using Hill’s 1948 quadratic yield function and non-constant stress ratios which leads to different flow stress hardening in different directions, (2) the anisotropy in plastic strains by using a quadratic plastic potential function and non-associated flow rule, also based on Hill’s 1948 model and r-values, and (3) the cyclic hardening phenomena such as the Bauschinger effect, permanent softening and transient behavior for reverse loading by using a coupled nonlinear kinematic hardening model. Plasticity fundamentals of the model were derived in a general framework and the model calibration procedure was presented for the plasticity formulations. Also, a generic numerical stress integration procedure was developed based on backward-Euler method, so-called multi-stage return mapping algorithm. The model was implemented in the framework of the finite element method to evaluate the simulation results of sheet metal forming processes. Different aspects of the model were verified for two sheet metals, namely DP600 steel and AA6022 aluminum alloy. Results show that the new model is able to accurately predict the sheet material behavior for both anisotropic hardening and cyclic hardening conditions. The drawing of channel sections and the subsequent springback were also simulated with this model for different drawbead configurations. Simulation results show that the current non-associated anisotropic hardening model is able to accurately predict the sidewall curl in the drawn channel sections.

Modelling of anisotropic hardening behavior for the fracture prediction in high strength steel line pipes

With the aim of improving prediction capabilities of FE simulation of anisotropic materials, a new plasticity model is proposed, based on Hill(48) criterion. The model provides a 3D description of the anisotropic behavior to represent the ductile hardening in all directions up to fracture; a Lode angle effect is also introduced, which allows a description of shear stress states. The model has been implemented into a commercial finite element code and calibrated with dedicated experimental laboratory methodology for high grade steel pipelines. A preliminary validation of the plasticity model, on a full scale a pipe burst test, is also provided.

Measurement and modeling of simple shear deformation under load reversal: Application to advanced high strength steels

In this paper, the stress–strain behavior under load reversal of advanced high strength steel (AHSS) sheet samples was measured using a modified simple shear (SS) apparatus. The forward–reverse loading behavior was characterized for three different grades of AHSS; namely, DP, TRIP and TWIP steels. For comparison purpose, compression–tension (CT) tests were also carried out for the same materials. For all the cases, a typical complex anisotropic hardening behavior, including the Bauschinger effect, transient strain hardening with high rate and permanent softening, was observed during load reversal. No premature localization and sheet buckling occurred in these experiments, which have been major technical hurdles in CT tests. For example, an engineering shear strain of over 40%, which corresponds to an effective strain of roughly 0.2, at reversal was achieved for DP980 although the uniform elongation of this material in uniaxial tension is only 5%. A recently developed distortional hardening model (HAH) was employed to reproduce the SS stress–strain curves. Using the coefficients determined with these SS data, the CT behavior was predicted with the HAH model and compared with experimental results. This complementary study indicated that the constitutive model determined from the SS flow curves satisfactorily reproduced the CT hardening behavior. As an application, finite element simulations of springback were carried out for 2D draw-bending of a strip sheet.

Исследование влияния анизотропии поверхностного пластического упрочнения на распределение остаточных напряжений в полых и сплошных цилиндрических образцах

surface plastic hardening, roller burnishing process, hydraulic shot blasting procedure, deformation anisotropy, solid and hollow cylindrical samples, residual stresses, mathematical model, experimental data For the calculation of residual stresses and plastic strains in hollow and solid surface-hardened cylindrical specimens we suggest the mathematical models, which take into account both the cases of hardening leading to the isotropy (hydraulic shot blasting procedure) and to the anisotropy (roller burnishing process) of plastic strains in the surface layer. The introduced mathematical model has a hardening anisotropy parameter which ties the axial and circumferential components of the residual plastic strains tensor. We use the determined axial and/or circumferential components of the residual stresses tensor as the input information. Also we use the following assumptions: smallness of the off-diagonal elements of the residual plastic strains tensor and residual stresses tensor, plastic incompressibility of material, absence of the secondary plastic strains of the material in the compression area of the surface layer. The boundary value problems of the hardened layer stress-strain state estimation after the hardening for the hollow and solid cylindrical specimens are solved and the solution is given in the paper. We give the method for the mathematical model parameter identification under the condition of selfequilibrated residual stresses and the method for the experimental determination of residual stresses by the circles and strips approach. Also, the cycle of the hardening experiments for the hollow and solid specimens from the 40Kh steel (having different proportions of internal and outer diameters) in the roller burnishing process and hydraulic shot blasting modes was performed with the determination of the residual stresses. The mathematical model adequacy is verified through the comparisons with experimental data; good agreement of the calculated and experimental data is demonstrated. The calculated numerical values of the hardening anisotropy parameter are given. The fact that the procedure of surface anisotropic hardening (roller burnishing process) leads to the strong layering of the epures of axial and circumferential residual stresses in depth of the hardened layer is established. Contrastingly, the case of the isotropic hardening (hydraulic shot blasting) leads to almost coinciding epures. Using the experimental and calculated data we have shown that the hardening of tool roller type leads to the larger absolute values of the compressing stresses, which by more than 30% exceeds the stresses after the hardening of hydraulic shot blasting type for the specimens of the same geometry. For the main results of the research we have given the necessary data in the table and also we have presented the epures for the residual stresses distribution. © PNRPU