Anisotropic hardening model based on non-associated flow rule and combined nonlinear kinematic hardening for sheet materials

 A material model for more effective analysis of plastic deformation of sheet materials is presented in this paper. The model is capable of considering the following aspects of plastic deformation behavior of sheet materials: the anisotropy in yielding stresses in different directions by using a quadratic yield function (based on Hill’s 1948 model and stress ratios), the anisotropy in work hardening by introducing non-constant flow stress hardening in different directions, the anisotropy in plastic strains in different directions by using a quadratic plastic potential function and non-associated flow rule (based on Hill’s 1948 model and plastic strain ratios, r-values), and finally some of the cyclic hardening phenomena such as Bauschinger’s effect and transient behavior for reverse loading by using a coupled nonlinear kinematic hardening (so-called Armstrong-Frederick-Chaboche model). Basic fundamentals of the plasticity of the model are presented in a general framework. Then, the model adjustment procedure is derived for the plasticity formulations. Also, a generic numerical stress integration procedure is developed based on backward-Euler method (so-called multistage return mapping algorithm). Different aspects of the model are verified for DP600 steel sheet. Results show that the new model is able to predict the sheet material behavior in both anisotropic hardening and cyclic hardening regimes more accurately. By featuring the above-mentioned facts in the presented constitutive model, it is expected that more accurate results can be obtained by implementing this model in computational simulations of sheet material forming processes. For instance, more precise results of springback prediction of the parts formed from highly anisotropic hardened materials or that of determining the forming limit diagrams is highly expected by using the developed material model.

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.

On the definition of an kinematic hardening effect graph for sheet metal forming process simulations

The objective of this work is to develop a kinematic hardening effect graph (KHEG) which can be used to evaluate the effect of kinematic hardening on the model accuracy of numerical sheet metal forming simulations and this without the need of complex material characterisation. The virtual manufacturing process design and optimisation depends on the accuracy of the constitutive models used to represent material behaviour. Under reverse strain paths the Bauschinger effect phenomenon is modelled using kinematic hardening models. However, due to the complexity of the experimental testing required to characterise this phenomenon in this work the KHEG is presented as an indicator to evaluate the potential benefit of carrying out these tests. The tool is validated with the classic three point bending process and the U-channel width drawbead process. In the same way, the capability of the KHEG to identify effects in forming processes that do not include forming strain reversals is identified.

Deep drawing behaviour of ultrafine grained copper : modelling and experiment

Ultrafine grained materials produced by severe plastic deformation methods possess attractive mechanical properties such as high strength compared with traditional coarse grained counterparts and reasonable ductility. Between existing severe plastic deformation methods the Equal Channel Angular Pressing is the most promising for future industrial applications and can produce a variety of ultrafine grained microstructures in materials depending on route, temperature and number of passes during processing. Driven by a rising trend of miniaturisation of parts these materials are promising candidates for microforming processes. Considering that bi-axial deformation of sheet (foil) is the major operation in microforming, the investigation of the influence of the number of ECAP passes on the bi-axial ductility in micro deep drawing test has been examined by experiments and FE simulation in this study. The experiments have showed that high force was required for drawing of the samples processed by ECAP compare to coarse grained materials. The limit drawing ratio of ultrafine grained samples was in the range of 1.9–2.0 with ECAP pass number changing from 1 to 16, while a higher value of 2.2 was obtained for coarse grained copper. However, the notable decrease in tensile ductility with increase in strength was not as pronounced for bi-axial ductility. The FE simulation using standard isotropic hardening model and von Mises yielding criterion confirmed these findings.

Analytic study on pure bending of metal sheets

In this work, analytical models of pure bending are developed to simulate a particular type of bend test and to determine possible errors arising from approximations used in analyzing experimental data. Analytical models proposed for steels include a theoretical solution of pure bending and a series of finite element models, based on the von Mises yield function, are subjected to different stress and strain conditions. The results show that for steel sheets the difference between measured and calculated results of the moment-curvature behaviour is small and the numerical results from the finite element models indicate that experimental results obtained from the test are acceptable in the range of the pure bending operation. Further for magnesium alloys, which exhibit unsymmetrical yielding, the algorithm of the yield function with a linear isotropic hardening model is implemented by programming a user subroutine in Abaqus for bending simulations of magnesium. The simulations using the proposed user subroutine extract better results than those using the von Mises yield function.

Analysis of work hardening and recrystallization during the hot working of steel using a statistically based internal variable model

A mathematical model has been developed which describes the hot deformation and recrystallization behavior of austenite using a single internal variable: dislocation density. The dislocation density is incorporated into equations describing the rate of recovery and recrystallization. In each case no distinction is made between static and dynamic events, and the model is able to simulate multideformation processes. The model is statistically based and tracks individual populations of the dislocation density during the work-hardening and softening phases. After tuning using available data the model gave an accurate prediction of the stress–strain behavior and the static recrystallization kinetics for C–Mn steels. The model correctly predicted the sensitivity of the post deformation recrystallization behavior to process variables such as strain, strain rate and temperature, even though data for this were not explicitly incorporated in the tuning data set. In particular, the post dynamic recrystallization (generally termed metadynamic recrystallization) was shown to be largely independent of strain and temperature, but a strong function of strain rate, as observed in published experimental work.

A semianalytical Sachs model for the flow stress of a magnesium alloy

A semianalytical Sachs-type equation for the flow stress of magnesium-base alloys is developed using the Schmid law, power law hardening, and a sigmoidal increase in the twinning volume fraction with strain. Average Schmid factors were estimated from electron backscattered diffraction (EBSD) data. With these, the equation provides a reasonable description of the flow curves obtained in compression and tension for samples of Mg-3Al-1Zn cut in different orientations from rolled plate. The model illustrates the general importance of basal slip and twinning in magnesium alloys. The significance of prismatic slip in room temperature tension testing is also highlighted. This is supported with EBSD slip line trace analysis and rationalized in terms of a possible sensitivity of the critical resolved shear stress for prismatic (cross) slip to the stress on the basal plane.

Conversion of the hot torsion test results into flow curve with multiple regimes of hardening

The method of Fields and Backofen has been commonly used to reduce the data obtained by hot torsion test into flow curves. The method, however, is most suitable for materials with monotonic strain hardening behaviour. Other methods such as Stüwe’s method, tubular specimens, differential testing and the inverse method, each suffer from similar drawbacks. It is shown in the current work that for materials with multiple regimes of hardening any method based on an assumption of constant hardening indices introduces some errors into the flow curve obtained from the hot torsion test. Therefore such methods do not enable accurate prediction of onset of recrystallisation where slow softening occurs. A new method to convert results from the hot torsion test into flow curves by taking into account the variation of constitutive parameters during deformation is presented. The method represents the torque twist data by a parametric linear least square model in which Euler and hyperbolic coefficients are used as the parameters. A closed form relationship obtained from the mathematical representation of the data is employed next for flow stress determination. Two different solution strategies, the method of normal equations and singular value decomposition, were used for parametric modelling of the data with hyperbolic basis functions. The performance of both methods is compared. Experimental data obtained by FHTTM, a flexible hot torsion test machine developed at IROST, for a C–Mn austenitic steel was used to demonstrate the method. The results were compared with those obtained using constant strain and strain rate hardening characteristics.

Application of radial return mapping algorithm for finite strain elastoplasticity in a hybrid finite element and particle-in-cell model

The radial return mapping algorithm within the computational context of a hybrid Finite Element and Particle-In-Cell (FE/PIC) method is constructed to allow a fluid flow FE/PIC code to be applied solid mechanic problems with large displacements and large deformations. The FE/PIC method retains the robustness of an Eulerian mesh and enables tracking of material deformation by a set of Lagrangian particles or material points. In the FE/PIC approach the particle velocities are interpolated from nodal velocities and then the particle position is updated using a suitable integration scheme, such as the 4th order Runge-Kutta scheme[1]. The strain increments are obtained from gradients of the nodal velocities at the material point positions, which are then used to evaluate the stress increment and update history variables. To obtain the stress increment from the strain increment, the nonlinear constitutive equations are solved in an incremental iterative integration scheme based on a radial return mapping algorithm[2]. A plane stress extension of a rectangular shape J2 elastoplastic material with isotropic, kinematic and combined hardening is performed as an example and for validation of the enhanced FE/PIC method. It is shown that the method is suitable for analysis of problems in crystal plasticity and metal forming. The method is specifically suitable for simulation of neighbouring microstructural phases with different constitutive equations in a multiscale material modelling framework.

A heuristic model selection scheme for representing hot flow data using the hot torsion test results

The problem of "model selection" for expressing a wide range of constitutive behaviour adequately using hot torsion test data was considered here using a heuristic approach. A model library including several nested parametric linear and non-linear models was considered and applied to a set of hot torsion test data for API-X 70 micro-alloyed steel with a range of strain rates and temperatures. A cost function comprising the modelled hot strength data and that of the measured data were utilized in a heuristic model selection scheme to identify the optimum models. It was shown that a non-linear rational model including ten parameters is an optimum model that can accurately express the multiple regimes of hardening and softening for the entire range of the experiment. The parameters for the optimum model were estimated and used for determining variations of hot strength of the samples with deformation.

A physically based phenomenological model for deformation twinning in magnesium alloy

A partial differential equation is developed that captures the evolution of key characteristics of tensile twinning in magnesium base alloys. The objective is to provide a framework for ascertaining the effects of hardening – due to grain refinement, precipitation and dislocation substructure – on twin volume fraction, thickness and length. The model is developed with the help of observations made on alloy AZ31. It is shown that it is necessary to consider the nucleation of twins at locations where neighbouring twins impinge on the grain boundary. The model provides a reasonable approximation for the role of grain size on twinning. It predicts a period of low apparent work hardening following yielding and shows that this should be more extensive for finer grain sizes, in agreement with experiment. Finally, some predictions are made on the effect of changing the resistance to twinning.