Combined Isotropic-kinematic Hardening Research Articles

Cyclic hardening and softening behavior of the low yield point steel BLY160: Experimental response and constitutive modeling

The ability to model the cyclic behavior of low yield point steel accurately is essential for its application to seismic design. In this study, uniaxial tests of monotonic and strain controlled cyclic loading are conducted to investigate the mechanical properties of the low yield point steel BLY160. As observed in the test, work-hardening is an important hysteretic characteristic, which significantly influences the energy dissipation capacity of the steel. However, the cyclic softening in both stiffness and stress evolve as the plastic strain accumulates, which is manifested in the decrease of both the stress amplitude and the elastic domain of the stress–strain hysteresis loops. This change in the shape of the cyclic loops is called the “flattening effect” in this paper. To describe the cyclic hardening and softening effect by using one unified set of material parameters, a more sophisticated model based on the concept of internal variables is proposed as an extension of the combined isotropic-kinematic hardening model. The novel idea of a “transformation zone” with the introduction of an additional internal variable is proposed to synchronize the updates of isotropic and kinematic hardening such that the simultaneous change in the stress amplitude and the elastic region can be reproduced. Note that the cyclic softening develops in three stages: incubation, growth and the stabilization period. The logistic function, which presents a curve with an “S” shape, is first used to trace the evolution of the softening. Finally, the model parameters are identified by the test data, and the accuracy of the proposed model is verified by comparing the simulated performance and the experimental response under different loading histories. The results demonstrate that the extended model can elaborately describe the evolutional processes of cyclic hardening and softening by incorporating characteristics, such as the nonlinear transient behavior from elastic to plastic status, early yield under reversed loading and strain history dependence.

Comparison of Three Methods for Material Hardening Parameter Identification under Cyclic Tension-Compression Loadings: Roll Levelling Case Study

The roll levelling is a forming process used to remove the residual stresses and imperfections of metal strips by means of plastic deformations. The process is especially important to avoid final geometrical errors when coils are cold formed or when thick plates are cut by laser. In the last years, and due to the appearance of high strength materials such as Ultra High Strength Steels, machine design engineers are demanding a reliable tool for the dimensioning of the levelling facilities. In response to this demand, Finite Element Analysis is becoming an important technique able to lead engineers towards facilities optimization through a deeper understanding of the process.In this scenario, the accuracy and quality of the simulation results are highly dependent on the accuracy of the implemented material model. During roll levelling process, the sheet metal is subjected to cyclic tensile-compressive deformations, therefore a proper constitutive. model which considers the phenomena that occurs during cyclic loadings, such as the Bauschinger effec, work hardeningt and the transient behaviour, is needed. The prediction of all these phenomena which affect the final shape of the product are linked to the hardening rule.In the present paper, the roll levelling simulation of a DP1000 steel is performed using a combined isotropic-kinematic hardening formulation introduced by Chaboche and Lemaitre since its simplicity and its ability to predict the Bauschinger effect. The model has been fitted to the experimental curves obtained from a cyclic tension-compression test, which has been performed by means of a special tool developed to avoid the buckling of the specimen during compressive loadings. The model has been fitted using three different material hardening parameter identification methodologies which have been compared.

A duality-based path-following semismooth Newton method for elasto-plastic contact problems

A Fenchel dualization scheme for the one-step time-discretized contact problem of quasi-static elasto-plasticity with combined kinematic–isotropic hardening is considered. The associated path is induced by a coupled Moreau–Yosida/Tikhonov regularization of the dual problem. The sequence of solutions to the regularized problems is shown to converge strongly to the optimal displacement–stress–strain triple of the original elasto-plastic contact problem in the space-continuous setting. This property relies on the density of the intersection of certain convex sets which is shown as well. It is also argued that the mappings associated with the resulting problems are Newton—or slantly differentiable. Consequently, each regularized subsystem can be solved mesh-independently at a local superlinear rate of convergence. For efficiency purposes, an inexact path-following approach is proposed and a numerical validation of the theoretical results is given.

A new bounding-surface plasticity model for cyclic behaviors of saturated clay

A new combined isotropic–kinematic hardening rule is proposed based on the concept of the generalized homological center and the generalization of Masing’s rule. The key point of the new hardening rule is that the unloading event can be treated as if it were virgin loading through taking the stress reversal point as the new generalized homological center of the bounding surface. Therefore, a new simple bounding-surface plasticity model with three important features for the cyclic behaviors of saturated clay is developed. Firstly, according to the movement of the generalized homological center, the model can harden not only isotropically but also kinematically to account for the anisotropy and memory the particular loading events. Secondly, the continuous cyclic loading is divided into the first loading, unloading and reloading processes and they are treated differently when calculating the hardening modulus to describe the soil responses accurately. The third feature is taking the generalized homological center as the mapping origin in the mapping rule to reflect the plastic flow in the unloading event. The behaviors of saturated clay for the monotonic and cyclic stress-controlled and strain-controlled triaxial tests are simulated by the model. The prediction results show an encouraging agreement with the experimental data.

Modelling of Fretting Wear under Partial Slip Conditions Using Combined Isotropic-Kinematic Hardening Plasticity Model

This paper presents finite element modelling of fretting wear under partial slip conditions using combined isotropic-kinematic hardening plasticity model with the emphasized to investigate the cyclic-plasticity behaviour predicted under fretting condition. The model is based on two-dimensional (2D) cylinder-on-flat contact configuration of titanium alloy, Ti-6Al-4V. A number of wear profiles at specific number of wear cycle (6000th, 60000th, 150000th and 300000th) are simulated. Contact pressure, tangential stress, shear stress, equivalent plastic strain, tangential plastic strain and also shear plastic strain are gathered and analysed. It is found that the plastic strain response of the combined isotropic-kinematic hardening plasticity model is slightly higher compare to linear kinematic hardening plasticity model [1].

An elastoplastic model with combined isotropic–kinematic hardening to predict the cyclic behavior of stiff clays

This paper presents a kinematic hardening model for describing some important features of natural stiff clays under cyclic loading conditions, such as closed hysteretic loops, smooth transition from the elastic behavior to the elastoplastic one and changes of the compression slope with loading/unloading loops. The model includes two yield surfaces, an inner surface and a bounding surface. A non-associated flow rule and a kinematic hardening law are proposed for the inner surface. The adopted hardening law enables the plastic modulus to vary smoothly when the kinematic yield surface approaches the bounding surface and ensures at the same time the non-intersection of the two yield surfaces. Furthermore, the first loading, unloading, and reloading stages are treated differently by applying distinct hardening parameters. The main feature of the model is that its constitutive equations can be simply formulated based on the consistency condition for the inner yield surface based on the proposed kinematic hardening law; thereby, this model can be easily implemented in a finite element code using a classic stress integration scheme as for the modified Cam Clay model. The simulation results on the Boom Clay, natural stiff clay, have revealed the relevance of the model: a good agreement has been obtained between simulations and the experimental results from the tests with different stress paths under cyclic loading conditions. In particular, the model can satisfactorily describe the complex case of oedometric conditions where the deviator stress is positive upon loading (compression) but can become negative upon unloading (extension).

A micromechanics approach to homogenizing elasto-viscoplastic heterogeneous materials

The variational asymptotic method for unit cell homogenization (VAMUCH) has emerged as a general-purpose micromechanics code capable of predicting the effective properties of heterogeneous materials and recovering the local fields. The objective of this paper is to propose a micromechanics approach enabling VAMUCH to homogenize elasto-viscoplastic heterogeneous materials. An affine formulation of the constitutive relations for an elasto-viscoplastic constituent, which exhibits viscoplastic anisotropy and combined isotropic–kinematic hardening, is derived. The weak form of the problem is derived using an asymptotic method, discretized using finite elements, and implemented into VAMUCH. The new features of VAMUCH are validated with examples such as homogenizing binary, fiber-reinforced, and particle-reinforced composites. VAMUCH is found to be capable of handling various microstructure, complex material models, complex loading conditions, and complex loading paths. More sophisticated material models can be implemented into it.

Multiobjective Optimization of Single Point Incremental Forming: Comparison between Isotropic and Combined Hardening Behavior

Single point incremental forming (SPIF) of sheet metal is a promising process to produce small batch production and prototyping. This process consists of a controlled process of displacement performed on a three-axis CNC milling machine. In former work, the most critical factors which affected single point incremental forming process were found to be formed shape, tool size, material type, material thickness and incremental step size. The present work is focused on an optimization strategy of (SPIF) process determined by a numerical study based on finite element analyses (FEA) according to a Box-Behnken Design of Experiments. Two types of hardening behaviour laws of material are used: isotropic and combined isotropic-kinematic hardening behaviour. To do so, a set of numerical simulations are carried out for an aluminum truncated cone as geometry of a benchmark model. The simulation results include some decisions about the mechanical resistance and geometrical quality of the parts such as the thickness distribution and the magnitude of springback. In this paper, the main objective is to present an overview of multiobjective design optimization of process parameters in single point incremental forming operation in order to minimize the sheet thinning rate and the springback simultaneously. In this investigation, the steps of optimization procedure include the using of Box-Behnken experimental design for sample producing, response surface model for coarse fitting and a developed Multiobjective Genetic Algorithm (MOGA) for exact solving of fitness functions. The results show that these methods are able to determine all the best possible compromise with respect several antagonistic objectives as well as generate the approximate Pareto optimal solutions. So these will make it possible to choose the appropriate process parameters according to the objectives functions to be minimized and consequently the improvement of the products formed by the process of incremental forming.

Material Modelling and Springback Analysis for Multi-stage Rotary Draw Bending of Thin-walled Tube Using Homogeneous Anisotropic Hardening Model

The aim of this paper is to compare several hardening models and to show their relevance for the prediction of springback and deformation of an asymmetric aluminium alloy tube in multi-stage rotary draw bending process. A three-dimensional finite-element model of the process is developed using the ABAQUS code. For material modelling, the newly developed homogeneous anisotropic hardening model is adopted to capture the Bauschinger effect and transient hardening behaviour of the aluminium alloy tube subjected to non-proportional loading. The material parameters of the hardening model are obtained from uniaxial tension and forward-reverse shear test results of tube specimens. This work shows that this approach reproduces the transient Bauschinger behaviour of the material reasonably well. However, a curve-crossing phenomenon observed for this material cannot be captured by the homogeneous anisotropic hardening model. For comparison purpose, the isotropic and combined isotropic-kinematic hardening models are also adopted for the analysis of the same problem. The predictions of springback and cross-section deformation based on these models are discussed.

Continuum Damage Mechanics (CDM) Based Local Approach to the Sheet-Bulk Metal Formability Prediction

Since sheet-bulk metal forming processes inherit properties of both sheet and bulk metal forming processes, their analysis requires on one side following certain methods conventionally devised in these process classes analyses whereas on the other side leaving certain customs out. For instance, inherent anisotropy of the rolled sheet has to be taken into account whereas due to non-vanishing out of plane stress component, analysis with thin shells using the plane stress state assumption is no more applicable. Similarly, methods based on necking instabilities, i.e. forming limit diagrams, which are typically used in sheet metal formability assessment; fall short in sheet-bulk metal formability prediction. In the present study, we propose a local approach to fracture, more specifically a phenomenologically based Lemaitre variant CDM model, devised frequently in bulk metal forming analysis, as an alternative. For this purpose, a combined nonlinear isotropic-kinematic hardening plasticity with Hill48 type initial anisotropy is fully coupled with isotropic damage. Together with the concept of effective stress and equivalent strain principle, quasi-unilateral damage evolution is used, where the energetic contribution of the compressive stress state to the damage driving force is scaled with a so-called crack closure parameter, . For the quasi unilateral damage evolution is inactive whereas for it is fully active which completely suppresses the development of damage under compressive stress states. The framework devises state coupling between elasticity and damage and kinematic coupling between plasticity and damage which increases the relative effect of on the eventual damage development. To this end, a direct extension to the finite strains for metal forming analysis is realized using a corotational formulation and the developed framework is implemented as a VUMAT subroutine for ABAQUS Explicit. For evaluation of the predictive capability of the model, teeth forming process results for DC04 reported in Soyarslan et al. 2011, An Experimental and Numerical Assessment of Sheet-Bulk Formability of Mild Steel DC04, Journal of Manufacturing Science and Engineering, Vol. 133 6, (2011) S. (061008) 1-9, are used. Mechanical material characterization studies are realized using a hybrid experimental-numerical procedure. This methodology relies on minimizing the difference between the experimentally handled global clamp force demand diagrams and the diagrams from the simulations at the complete range of the experiments involving fracture. As known finite element solutions with softening material models are prone to pathological mesh dependence. For this fact, a crack band method is used where the minimum element size, as a controlling parameter of the localization size, is also fitted through the characterization studies and identically used in the process simulations. The simulations show that a correct prediction of the zone and time of fracture is possible for the selected process whereas since the teeth formation process is mainly a compressive process, once the quasi-unilateral damage development is not used, i.e. for , a premature crack prediction is recorded which is not compatible with the experimental findings.

Measurements of Bauschinger effect and transient behavior of a quenched and partitioned advanced high strength steel

In recent decades, the needs for new advanced high strength steels (AHSS) with high ductility and strength have rapidly increased to achieve the targets of more fuel-efficient and safer vehicles in automotive industry. However, several undesirable phenomena are experimentally observed during the forming of such materials, particularly with complex loading and large plastic deformation. Springback is one of the most important problems that should be compensated in sheet metal forming process. In this paper, we investigated the hardening behavior of a Q&P (quench and partitioning) steel designated by QP980CR, which is a new third generation advance high strength steel, from the Baosteel Group Corp. in Shanghai, China. The uni-axial tensile and cyclic simple shear tests were conducted. The uni-axial tensile tests were performed on the specimens at 0°, 45° and 90° to rolling direction (RD). The flow stress and transverse strain evolution were obtained in view of the digital image correlation (DIC) measurement. The plastic anisotropy was optimized from the uni-axial tensile tests and thereafter incorporated into the simulations of cyclic simple shear tests. The cyclic simple shear tests were conducted with three prestrains to measure the Bauschinger effect, transient behavior and permanent softening, and to determine the material parameters of the combined isotropic-kinematic hardening model.

An extended homogenous yield function based anisotropic hardening model for description of anisotropic hardening behavior of materials

Except for the Bauschinger effect and permanent softening, work hardening stagnation and cross-effect are often observed deformation behaviors for sheet metals subjected to strain path changes. These complex deformation behaviors are assumed to have great effect on the sheet forming process and the following spring back. To constitutively, and more precisely model these plastic behaviors, extensive models have been successfully developed. Particularly, the combined isotropic-kinematic hardening model is powerful to capture the plastic behavior of sheet metals during forward and reverse loading. However, there is strong coupling between anisotropic yielding and anisotropic hardening with the kinematic hardening. Recently, Barlat et al. (2011) [61] proposed a homogenous yield function based anisotropic hardening model to describe the Bauschinger effect, which is not formulated with the kinematic hardening rule. In the present work, the original anisotropic hardening model is extended to capture the often observed cross-effect during multi-stage loading and work hardening stagnation during reverse loading. Although only four coefficients are newly introduced in the extended model, it can capture well the main trends of work hardening stagnation and cross-effect. The capability of this extended model is demonstrated with applications to two materials, namely, high strength steel and mild steel DC06.

Springback Prediction Using the Split-Ring Test Based on a Combined Anisotropic Hardening Model

Material model which could describe the Bauschinger effect is essential to accurately predict springback since strain path reverse is quite common in sheet metal forming. In this paper, a combined isotropic-kinematic hardening law which can capture the Bauschinger effect, transient behavior and permanent softening was used to model the hardening behavior. Also, the non-quadratic anisotropic yield function, Yld2000-2d, was chosen to describe the anisotropy. An inverse identificat- ion method was carried out to calibrate the material parameters by using uni-axial tension and Bauschinger simple shear tests. Experiments of cylindrical cup drawing and following split-ring tests were performed. Similar processes were carried out by numerical simulation, and springback predicted by present model was compared with experiments and that of vonMises model. The result shows that the current model significantly improves the accuracy of springback prediction.

Comparison Between Two Experimental Procedures for Cyclic Plastic Characterization of High Strength Steel Sheets

In finite element analysis of sheet metal forming the use of combined isotropic-kinematic hardening models is advisable to improve stamping simulation and springback prediction. This choice becomes compulsory to model recent materials such as high strength steels. Cyclic tests are strictly required to evaluate the parameters of these constitutive models. However, for sheet metal specimens, in case of simple axial tension-compression tests, buckling occurrence during compression represents a serious drawback. This is the reason why alternative set-ups have been devised. In this paper, two experimental arrangements (a cyclic laterally constrained tension-compression test and a three-point fully reversed bending test) are compared so as to point out the advantages and the disadvantages of their application in tuning the well-known Chaboche’s hardening model. In particular, for tension-compression tests, a new clamping device was specifically designed to inhibit compressive instability. Four high strength steel grades were tested: two dual phases (DP), one transformation induced plasticity (TRIP) and one high strength low alloy material (HSLA). Then, the Chaboche’s model was calibrated through inverse identification methods or by means of analytical expressions when possible. The proposed testing procedure proved to be successful in all investigated materials. The achieved constitutive parameters, obtained independently from the two experimental techniques, were found to be consistent. Their accuracy was also been assessed by applying the parameter set obtained from one test to simulate the other one, and vice versa. Clues on what method provides the better transferability are given.

Experimental investigation and constitutive modeling for the hardening behavior of 5754O aluminum alloy sheet under two-stage loading

The two-stage loading tests of 5754O aluminum alloy sheet were carried out. In the first loading stage, the uniaxial tensile tests of big sheets were carried out. Then small specimens were cut off from the pre-strained big sheets along different directions. And in the second loading stage, the uniaxial tensile tests of the small specimens were performed. From the experimental results, it is found that the initial yield stress of each specimen in the second loading stage decreases when the strain path changes. In addition, when the strain path changes the transient effect appeared and no obvious permanent softening was observed. In this study, in order to describe the hardening behavior of 5754O aluminum alloy sheet under two-stage loading, the Chaboche type combined isotropic–kinematic hardening models were adopted with Yld2000-2d and Hill48 as yield functions. It is proven that no permanent softening can be described with Chaboche type model in two-stage loading. Three methods for determining the parameters of the hardening models were developed in order to establish accurate isotropic–kinematic hardening model describing the hardening behavior of 5754O aluminum alloy sheet under two-stage loading. The established constitutive models were implemented into the commercial FEM code ABAQUS as a user material subroutine (UMAT) for numerical simulations. By comparing the experimental and simulated results of the two-stage loading tests, the isotropic–kinematic hardening models describing the hardening behavior of 5754O aluminum alloy sheet under two-stage loading were accurately determined. Also, the influences of the characterization method of Hill48 yield function on the accuracy of the resulting hardening models were discussed. It is also found that the established isotropic–kinematic hardening model describing the hardening behavior under two-stage loading can describe reasonably the springback profile of the three point bending of the pre-strained specimen.

Non-associated flow rule with symmetric stiffness modulus for isotropic-kinematic hardening and its application for earing in circular cup drawing

Under a standard derivation, the stiffness modulus for the non-associated flow rule is asymmetric since its plastic potential (for the plastic strain increment under the normality rule) differs from the plastic yield stress function (to define the elastic range). A new formulation was developed in this work, which leads to the symmetric stiffness modulus for the non-associated flow rule, under the framework of the combined isotropic-kinematic hardening law for generalization purposes. As for its application, the non-quadratic Yld2000-2d (Barlat et al., 2003) function (and Hill’s (1948) function for comparison) was utilized to validate the formulation for earing in circular cup drawing of AA2090-T3 and AA5042 sheets, which successfully represented 6 and 8 ears, respectively.

Shot peening and peen forming finite element modelling – Towards a quantitative method

Peen forming is commonly used on aluminium alloys in the aerospace industry for wing skin shaping. Numerous analytical, numerical, and experimental studies have been made to better understand the effects of various peening parameters on the final material state and to predict deformed shapes, but conclusions were often limited to trends. The purpose of this study is therefore to develop and verify experimentally quantitative numerical tools for peen forming applications by studying the simple case of peening an Almen-sized AA-2024 aluminium strip in an Almen holder. The first step consisted in improving an existing random dynamic model by determining optimal dimensions. The AA-2024 target mechanical behaviour was characterized experimentally and a combined isotropic-kinematic hardening law was selected to model the material behaviour. The dynamic impact model and material constitutive law provided good prediction of peening-induced stresses in thick AA-2024 for two shot velocities. The sequence-sensitive aspect of the forming process was also investigated and a new shell-based finite element model was proposed. Numerical and experimental results for three shot velocities were compared to evaluate the validity of this numerical simulation method and promising agreement was observed.

Combined Model to Describe Both the Mechanical and Microstructural Transients under a Reversal of Strain

A strain reversal produces microstructural and mechanical transients that cannot be described with conventional models applicable under monotonic deformation conditions. State-of-the-art models incorporating plastic strain path effects are polarized into continuum constitutive models of cyclic plasticity and multiscale constitutive approaches extrapolating crystallographic hardening models for monocrystals to polycrystals. The present work proposes a model midway between these two approaches that describes the mechanical transient through a combined kinematic-isotropic hardening model and the microstructural transient and its effect on the subsequent static recrystallization kinetics. The incorporation of an effective strain concept makes the link among the different submodels and allows for the first time an accurate reproduction with a reduced number of parameters both for mechanical and microstructural experimental data during the transient following a strain reversal.

A model of one-surface cyclic plasticity and its application to springback prediction

This paper presents an elasto-plastic constitutive model based on one-surface plasticity, which can capture the Bauschinger effect, transient behavior, permanent softening, and yield anisotropy. The combined isotropic–kinematic hardening law was used to model the hardening behavior, and the non-quadratic anisotropic yield function, Yld2000-2d, was chosen to describe the anisotropy. This model is closely related to the anisotropic non-linear kinematic hardening model of Chun et al. [2002. Modeling the Bauschinger effect for sheet metals, part I: theory. International Journal of Plasticity 18, 571–95.]. Different with the model, the current model captures in particular permanent softening with a constant stress offset as well as the Bauschinger effect and transient behavior under strain path reversal. Inverse identification was carried out to fit the material parameters of hardening model by using uni-axial tension/compression data. Springback predicted by the resulting material model was compared with experiments and with material models that do not account for permanent softening. The results show that the resulting material model has a good capability to predict springback.

Spring-Back Evaluation of Stretch Bending Process Based on Chaboche Combined Isotropic-Kinematic Hardening Laws

We present a study on spring-back prediction in the stretching bending process using the Chaboche model combined isotropic-kinematic hardening law and Mises yielding criterion, and a material user subroutine (VUMAT, UMAT) program was developed base on the ABAQUS interface for the model. The effects of different hardening law on the spring-back in the stretch forming process was also analyzed and compared. The simulation results show that the combined isotropic-kinematic hardening law has the better spring-back prediction compared with the pure isotropic and kinematic hardening law in the stretch forming process, which is verified by the experimental results.