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 strength of dislocation interactions and their effect on latent hardening in pure Magnesium

This study is dedicated to the quantification of latent hardening and its effect on the plasticity of pure hexagonal magnesium. To this end, discrete dislocation dynamics simulations are used to (1) extract latent hardening parameters coupling different slip systems, and to (2) assess the validity of two existing constitutive models linking slip system strength to dislocation densities on all slip systems. As hexagonal materials deform via activation of different slip modes, each with different mobilities and lattice friction stress, the effects of the latter on latent hardening evolution are also investigated. It is found that the multi-slip formulation proposed by Franciosi and Zaoui gives accurate predictions when multiple interactions are involved while the formulation suggested by Lavrentev and Pokhil systematically overestimates the flow stress. Similar to FCC materials, it is also found that collinear interactions potentially contribute the most to latent hardening. Basal/pyramidal c + a interactions are found to be very strong, while interactions involving second-order pyramidal c + a primary dislocations appear to be the weakest ones. Finally, the latent hardening parameters, extracted from the discrete dislocation dynamics simulations, are used in polycrystal simulations and the impact of finely accounting for latent hardening on predictions of the macroscopic anisotropic response is shown to be of significant importance.

Direct current electric potential in an anisotropic half-space with vertical contact containing a conductive 3D body

Detailed studies of anomalous conductors in otherwise homogeneous media have been modelled. Vertical contacts form common geometries in galvanic studies when describing geological formations with different electrical conductivities on either side. However, previous studies of vertical discontinuities have been mainly concerned with isotropic environments. In this paper, we deal with the effect on the electric potentials, such as mise-`a-la-masse anomalies, due to a conductor near a vertical contact between two anisotropic regions. We also demonstrate the interactive effects when the conductive body is placed across the vertical contact. This problem is normally very difficult to solve by the traditional numerical methods. The integral equations for the electric potential in anisotropic half-spaces are established. Green’s function is obtained using the reflection and transmission image method in which five images are needed to fit the boundary conditions on the vertical interface and the air-earth surface. The effects of the anisotropy of the environments and the conductive body on the electric potential are illustrated with the aid of several numerical examples.

The mechanical characterization of low work hardening, high strength steel

One of the major challenges in assessing the mechanical properties of recovery annealed steel is the strain localization that occurs almost immediately on the formation of the first Lüders band, such that no or limited propagation of the Lüders band occurs along the tensile coupon. The stress raiser associated with the geometry of the standard tensile coupon means that this plastic deformation is often completely outside the standard extensometers on the coupon. Hence, no strain is measured during the test. While this is not important for assessing the tensile strength of the steel, it does mean that the strain related properties, such as the elastic limit of the steel, cannot be measured using standard testing techniques.This work addresses this issue by examining three techniques for ensuring that the strain occurs inside the extensometer. It is shown that the best technique is the extended extensometer, where the gauge length covers slightly more than the tensile coupon parallel length. While this leads to some variation in the width of the material being measured, compensation can be be made by adjusting the strain to correct the Young's Modulus.This technique has direct implications not just for recovery annealed steels, but for other high strength, low work hardening materials such as ultrafine ferrite. A particular requirement of these high strength steels in structural applications is a high elastic limit; hence, measurement of the strain related properties for these high strength materials must be considered vital in their mechanical assessment.

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.

Transmission electron microscopy characterization of the bake-hardening behavior of transformation-induced plasticity and dual-phase steels

The effect of prestraining (PS) and bake hardening (BH) on the microstructures and mechanical properties has been studied in transformation-induced plasticity (TRIP) and dual-phase (DP) steels after intercritical annealing. The DP steel showed an increase in the yield strength and the appearance of the upper and lower yield points after a single BH treatment as compared with the as-received condition, whereas the mechanical properties of the TRIP steel remained unchanged. This difference appears to be because of the formation of plastic deformation zones with high dislocation density around the “as-quenched” martensite in the DP steel, which allowed carbon to pin these dislocations, which, in turn, increased the yield strength. It was found for both steels that the BH behavior depends on the dislocation rearrangement in ferrite with the formation of cell, microbands, and shear band structures after PS. The strain-induced transformation of retained austenite to martensite in the TRIP steel contributes to the formation of a complex dislocation structure.