In the metal sheet-forming process, localised thinning immediately before failure is a rapid and intensive process. This paper discusses such intensive thinning and elaborates on the importance of the hardening behaviour at large plastic strains. In particular, for the reliable prediction of intensive sheet-thinning evolution, an accurate determination of the hardening behaviour is important; however, the identification cannot be performed using classical uniaxial tests owing to necking. Therefore, this paper describes the development of a new shear specimen that provides enough data for hardening curve characterisation up to an equivalent plastic strain value that is twice that in a tensile test. After characterisation, based on the inverse identification procedure, two verification cases confirmed the identified data. Both cases are typically considered stochastic processes, and they exhibit a large scatter in the experimental testing: Necking evolution in a uniaxial case and localisation evolution of a notched specimen. However, the verification analyses confirm that both processes are deterministic, and a good prediction could be reached exclusively by a precise identification of the hardening behaviour at large plastic strains.

Purpose - The purpose of this study, which is designed for the implementation of models in the implicit finite element framework, is to propose a robust, stable and efficient explicit integration algorithm for rateindependent elasto-plastic constitutive models.Design/methodology/approach - The proposed automatic substepping algorithm is founded on an explicit integration scheme. The estimation of the maximal subincrement size is based on the stability analysis.Findings - In contrast to other explicit substepping schemes, the algorithm is self-correcting by definition and generates no cumulative drift. Although the integration proceeds with maximal possible subincrements, high level of accuracy is attained. Algorithmic tangent stiffness is calculated in explicit form and optionally no analytical second-order derivatives are needed.Research limitations/implications - The algorithm is convenient for elasto-plastic constitutive models, described with an algebraic constraint and a set of differential equations. This covers a large family of materials in the field of metal plasticity, damage mechanics, etc. However, it cannot be directly used for a general material model, because the presented algorithm is convenient for solving a set of equations of a particular type.Practical implications - The estimation of the maximal stable subincrement size is computationally cheap. All expressions in the algorithm are in explicit form, thus the implementation is simple and straightforward. The overall performance of the approach (i.e. accuracy, time consumption) is fully comparable with a default (built-in) ABAQUS/Standard algorithm.Originality/value - The estimated maximal subincrement size enables the algorithm to be stable by definition. Subincrements are much larger than those in conventional substepping algorithms. No error control, error correction or local iterations are required even in the case of large increments.

The paper presents a derivation of the consistent tangent operator (CTO) for the class of iterative constitutive integration algorithms. The derivation is based on a variational analysis of algorithmic equations with respect to the strain dependency of state variables in each iteration. Such treatment results in a recursive formulation, where the value of CTO is updated based on its old value. The proposed formulation is demonstratively applied to the cutting-plane algorithm (CPA), for which it is known that a closed form solution for CTO does not exist.

The paper focuses on the modelling of springback within a formed stainless steel sheet.The main subject of this work is the construction of a constitutive model which simultaneously considers sheet anisotropy, damage evolution, and stiffness degradation in material during forming.The developed model is based on the Gurson-Tvergaard-Needleman damage model, which is adequately extended by the implementation of the anisotropic Hill48 plasticity and Mori-Tanaka's approach to stiffness degradation.Considering the established relationships, some material parameters that are included in the model are characterised by the corresponding measurements.The experimental validation of the developed constitutive model is performed on a springback test, which consists of bending and releasing rectangular stainless steel specimens that were previously plastically prestrained to a different degree, either in the rolling or transverse direction.A comparison of the proposed modelling approach to the classical approach by using the Hill48 model clearly indicates that the simultaneous modelling of material phenomena, especially the coupling of stiffness degradation with anisotropic plasticity, can be the true key to obtaining a more accurate prediction of the springback in sheet-metal-forming applications.

The paper presents a derivation of the consistent tangent operator (CTO) for the cuttingplane algorithm (CPA). For a class of plasticity models that are suitable to be integrated using CPA, an explicit recursive expression is analytically derived and is updated in each iteration of the CPA integration procedure to yield the final value of the CTO when the CPA is converged. (C) 2014 Elsevier B.V. All rights reserved.

The paper deals with constitutive modeling of highly anisotropic sheet metals and presents FEM based earing predictions in a round cup drawing simulation of highly anisotropic aluminum alloys where more than four ears occur. For that purpose the BBC2008 yield criterion, which is a plane-stress yield criterion formulated in the form of a finite series, is used. Thus defined criterion can be expanded to retain more or less terms, depending on the amount of given experimental data. To be used in sheet metal forming simulations the constitutive model, derived in accordance with the associated flow theory of plasticity, has been implemented in a general purpose finite element code ABAQUS/Explicit via VUMAT subroutine, considering alternatively different number of parameters in the BBC2008 yield criterion, where possible number of parameters are any multiple of number 8. For the integration of the constitutive model the explicit NICE (Next Increment Corrects Error) integration scheme has been used. The CPU time consumption for an explicit deep drawing simulation, which is based on the developed constitutive model, has been proven to be, due to effectiveness of the used integration scheme, fully comparable to the performance experienced when the simulation is performed with ABAQUS built-in constitutive models and implicit integration schemes. Two aluminum alloys, namely AA5042-H2 and AA2090-T3, have been considered for a validation of the constitutive model. The respective BBC2008 model parameters have been identified for both alloys with a developed numerical procedure, based on a minimization of the specified cost function. For both materials, the simulation predictions based on the BBC2008 model prove to be in very good agreement with the experimental results. Further, in order to show the flexibility of the BBC2008 model in modeling of highly anisotropic sheet metal response, we have introduced a highly anisotropic fictitious material which yields, according to the theory, twelve ears in cup drawing. As it is shown in the paper the BBC2008 model is able to predict twelve ears in cup drawing simulation with the formulation containing 16 parameters for anisotropy description only. The flexibility and accuracy of the constitutive model together with the robust identification and integration procedure guarantee the applicability of the BBC2008 yield criterion in industrial applications.

This paper presents a methodology based on the NICE integration scheme [1, 2] for simple and rapid numerical implementation of a class of plasticity constitutive models. In this regard, an algorithm is purposely developed for the implementation of newly developed advanced constitutive models into explicit finite element framework. The methodology follows the organization of the problem state variables into an extended form, which allows the constitutive models' equations to be organized in such a way, that the algorithm can be optionally extended with minimal effort to integrate also evolution equations related to a description of other specific phenomena, such as damage, distortional hardening, phase transitions, degradation etc. To confirm simplicity of the program implementation, computational robustness, effectiveness and improved accuracy of the implemented integration algorithm, a deep drawing simulation of the cylindrical cup is considered as the case study, performed in ABAQUS/Explicit. As a fairly complex considered model, the YLD2004-18p model [3, 4] is first implemented via external subroutine VUMAT. Further, to give additional proof of the simplicity of the proposed methodology, a combination of the YLD2004-18p model and Gurson-Tvergaard-Needleman model (GTN) is considered. As demonstrated, the implementation is really obtained in a very simple way.

The article introduces, as a result of further development of the first-order scheme NICE, a simple and efficient higher-order explicit numerical scheme for the integration of a system of ordinary differential equations which is constrained by an algebraic condition (DAE). The scheme is based on the truncated Taylor expansion of the constraint equation with order h of the scheme being determined by the highest exponent in the truncated Taylor series. The integration scheme thus conceived will be named NICE h , considering both principal premises of its construction. In conjunction with a direct solution technique used to solve the boundary value problem, the NICE h scheme is very convenient for integrating constitutive models in plasticity. The plasticity models are defined mostly by a system of algebraic and differential equations in which the yield criterion represents the constraint condition. To study the properties of the new integration scheme, which, like the forward-Euler scheme, is characterised by its implementation simplicity due to the explicitness of its formulations, a damage constitutive model (Gurson–Tvergaard–Needleman model) is considered. The general opinion that the implicit backward-Euler scheme is much more accurate than the thus-far known explicit schemes is challenged by the introduction of the NICE h scheme. The accuracy of the higher-order explicit scheme in the studied cases is significantly higher than the accuracy of the classical backward-Euler scheme, if we compare them under the condition of a similar CPU time consumption.

The paper deals with the integration of elasto-plastic constitutive models using recently developed NICE integration scheme [1],[2]. The emphasis is put on the stability of the integration, since this issue was not sufficiently addressed in previous publications of the NICE.Nonlinear boundary value problems are nowadays typically solved numerically using finite element method (FEM) with implicit "static" (e.g. ABAQUS/Standard) or explicit "dynamic" approach (e.g. ABAQUS/Explicit). The NICE scheme was primarily developed for the integration of elasto-plastic constitutive models within explicit integration of a given boundary value problem, as a replacement for traditionally used backward-Euler scheme. The simplicity of the implementation, more than satisfactory accuracy and low time consumption of calculation, certainly outperforms the properties of other available schemes, including properties of the backward-Euler scheme. The only open issue regarding the NICE scheme is its conditional stability, which originates from the integration of evolution equations in a "forward" manner, whereas the backward-Euler scheme exhibits unconditional stability.The aim of this paper is to derive stable time increment for the NICE scheme and to show, that for practical quasi-static applications it is much larger than the stable time increment size given for the integration of dynamic boundary value problem equations.

Springback, a phenomenon that is governed by elastic strain recovery after the removal of forming loads, is of great concern in sheet metal forming. There is no doubt that in this regard, physically reliable numerical modelling of the forming process and predictions of springback obtained by respective computer simulations are crucial for controlling this problem. Unfortunately, by currently available approaches, springback still cannot be adequately predicted in general. In this paper, a new constitutive model is proposed which considers simultaneously sheet anisotropy, damage evolution and strain path-dependent stiffness degradation during sheet metal forming. For parameter identification of the built constitutive model, a particular experimental procedure is developed and an optimization procedure is employed to solve the inverse problem that arises. The proposed approach to constitutive modelling is validated in the end by a simulation of the springback in the formed HSS steel sheet. The simulation results, which prove to be in good agreement with the experimental ones, lead to the conclusion that accurate modelling only of anisotropic yielding is not enough to accurately predict the springback phenomenon; the constitutive model should also include the strain path-dependent change of the elastic moduli.

The paper deals with constitutive modelling of highly anisotropic sheet metals. It presents FEM based caring predictions in cup drawing simulation of highly anisotropic aluminium alloys where more than four ears occur. For that purpose the BBC2008 yield criterion, which is a plane-stress yield criterion formulated in the form of a finite series, is used. Thus defined criterion can be expanded to retain more or less terms, depending on the amount of given experimental data. In order to use the model in sheet metal forming simulations we have implemented it in a general purpose finite element code ABAQUS/Explicit via VUMAT subroutine, considering alternatively eight or sixteen parameters (8p and 16p version). For the integration of the constitutive model the explicit NICE (Next Increment Corrects Error) integration scheme has been used. Due to the scheme effectiveness the CPU time consumption for a simulation is comparable to the time consumption of built-in constitutive models. Two aluminium alloys, namely AA5042-H2 and AA2090-T3, have been used for a validation of the model. For both alloys the parameters of the BBC2008 model have been identified with a developed numerical procedure, based on a minimization of the developed cost function. For both materials, the predictions of the BBC2008 model prove to be in very good agreement with the experimental results. The flexibility and the accuracy of the model together with the identification and integration procedure guarantee the applicability of the BBC2008 yield criterion in industrial applications.

The paper focuses on a reliable identification of the hardening curve of sheet metal after the onset of necking by means of a purposely developed shear test. The main comparative advantage of the test is its simple implementation needing no further development of the experimental equipment used in standard tensile testing. In addition, in order to achieve more flexible hardening curve description from those given by the classical analytical hardening laws, the paper presents the hardening curve approximation based on the cubic spline technique. The identified hardening curves are validated and compared to the analytically extrapolated ones obtained by a continuation of classical hardening laws for large plastic strain. It is shown, that due to large difference in response the analytical extrapolation is seriously disputable.

Precise integration of a constitutive law is required whenever a boundary value problem is to be solved. In this paper a new explicit scheme for the integration of elasto-plastic constitutive models, that consist of algebraic yield condition and differential evolution equations, is presented. The scheme was primarily developed for the integration of constitutive models in boundary value problems which are solved with a direct solving technique (like in Abaqus/Explicit). Although the scheme is as fast and as simple for implementation as unstable explicit forward-Euler scheme, it can reach the same level of accuracy as commonly used backward-Euler scheme. Two examples are shown in the paper, the first one being simple for demonstration purposes only, whereas in the second one the scheme is also implemented in the Abaqus/Explicit user subroutine VUMAT. The obtained results confirm that the scheme is accurate and stable. It can be concluded in consequence, that the scheme is very handy for implementation of complex elasto-plastic constitutive laws into finite element software.

In accordance with the great importance given to the subject of stiffness degradation, in particular with regard to metal forming, this work experimentally investigates the anisotropic elastic properties of plastically prestrained cold-rolled sheet metal (stainless steel EN 1.4301, also AISI 304). From the experiments performed, two main conclusions regarding stiffness degradation can be extracted. First, since under specific stretching the intensity of the normalized Young’s moduli degradation in both directions remains approximately similar, it may be concluded that the potential initial elastic anisotropy tends to be preserved during loading. Second, as the evidenced stiffness degradation has proved to be strongly correlated with the stretching direction of the sheet metal, it can be concluded that the stiffness evolution in the cold rolled sheet steel is path dependent. These interesting discoveries also provide some answers for modelling the kinetic damage evolution laws in damage mechanics.

The paper focuses on a reliable identification of the hardening curve of sheet metal after onset of necking with the developed shear test. The main comparative advantage of the test is its simple procedure for which no further development of experimental equipment regarding standard tensile test is needed. The only modification of the standard tensile test refers to a purposely designed symmetric shear specimen which enables achievement of large shear strains during stable loading condition. In addition the paper presents the hardening curve approximation based on the cubic spline techniques. Such approach is very convenient for use because splines are smooth and are able to approximate accurately local curvatures without global estimation being influenced. From a comparison between the analytically extrapolated and identified hardening curves it is shown that using analytical extrapolation at large strains is seriously flawed as a result of the considerable differences between the curves.

The article presents a simple but efficient numerical scheme for the integration of non‐linear constitutive equations, in which the principal reason for the inaccuracy of the classical explicit schemes, for example forward‐Euler scheme, is effectively eliminated. In the newly developed explicit scheme, where there is no need for iteration, the implementation simplicity of the forward‐Euler scheme and accuracy of the approach, which is using the backward‐Euler scheme to integrate the constitutive equations, are successfully combined. Computational performance of the proposed next increment corrects error (NICE) integration scheme, particularly regarding the accuracy and the CPU time consumption, is first analysed on a case of complex loading of a material point. When comparing it to the forward‐Euler, backward‐Euler, trapezoidal and midpoint integration schemes, it turns out that because of its capability of a fast and relatively accurate integration of the constitutive equations, the NICE scheme is very convenient for the integration of constitutive models, where a direct solution technique is used to solve a boundary value problem. Although the deduction of the new integration scheme is general, its implementation for shell applications needs particular care. Namely, in order to satisfy the zero normal stress condition during the whole integration, a through‐thickness strain increment has to be adequately chosen in each integration step. The NICE scheme, which was also implemented into ABAQUS/Explicit via User Material Subroutine (VUMAT) interface platform, has been additionally compared with the ABAQUS/Explicit default integration scheme (backward‐Euler) and forward‐Euler scheme. Two loading case‐studies, namely the bending of a square plate and the stretching of a specimen including the onset of necking, are considered with two constitutive models—the von Mises and GTN material model being adopted. Generally, the NICE scheme has demonstrated to be advantageous in cases, where reasonable accuracy and very fast integration of the constitutive model is demanded, which is mostly the case in engineering computations with a direct solution method, for example explicit dynamics and metal forming process simulations. Copyright © 2009 John Wiley & Sons, Ltd.

The aim of this work is to show first, how the springback of a steel sheet drawn part is affected by the stiffness degradation, as it results from the damage evolved during forming process, and second, to build a respective modeling approach to take this degradation into account. For the consideration of the orthotropic elastic properties degradation we develop an approach, based on the Mori-Tanaka theory, where damage is considered by inclusion of ellipsoidal cavities. The respective void shape evolution is proposed to be identified with the measurements of elastic modulus in two perpendicular directions during the uniaxial tensile test of a flat specimen at different loading stages. The proposed approach is coupled with the Gurson-Tvergaard-Needleman (GTN) plastic potential, though it could be substituted by almost any other continuum damage model. At the end the presented approach is experimentally validated by a simple springback test, developed by authors. A very good agreement between by calculation predicted and measured springback amount is observed.

The paper presents a simple but efficient new numerical scheme for the integration of nonlinear constitutive equations. Although it can be used for the integration of a system of algebraic and differential equations in general, the scheme is primarily developed for use with the direct solution methods for solving boundary value problems, e.g. explicit dynamic analysis in ABAQUS/Explicit. In the developed explicit scheme, where no iteration is required, the implementation simplicity of the forward-Euler scheme and the accuracy of the backward-Euler scheme are successfully combined. The properties of the proposed NICE scheme, which was also implemented into ABAQUS/Explicit via User Material Subroutine (VUMAT) interface platform, are compared with the properties of the classical forward-Euler scheme and backward-Euler scheme. For this purpose two highly nonlinear examples, with the von Mises and GTN material model considered, have been studied. The accuracy of the new scheme is demonstrated to be at least of the same level as experienced by the backward-Euler scheme, if we compare them on the condition of the same CPU time consumption. Besides, the simplicity of the NICE scheme, which is due to implementation similarity with the classical forward-Euler scheme, is its great Advantage.

Springback, the phenomenon that is governed by strain recovery after removal of forming loads, is of great concern in sheet metal forming. There is no doubt that physically reliable numerical modelling of the forming process and predictions of springback obtained by respective computer simulations are crucial to control this problem. Unfortunately, by currently available approaches springback still cannot be adequately predicted in general. This paper is an attempt of building a corresponding constitutive model, which will simultaneously consider sheet anisotropy, damage evolution and strain path dependent stiffness degradation during metal forming. First, for the identification of the parameters in the built constitutive model a particular experimental procedure is deliberately developed. To solve the arisen inverse problem an optimization procedure is employed. The proposed approach to constitutive modelling is validated in the end by a simulation of the springback in the formed HSS steel. The simulation results prove to be in good agreement with the experimental ones. From the performed comparisons it is clearly indicated, that only simultaneous modelling of material properties can be the true key to obtain accurate prediction of springback in sheet metal forming.