Isotropic-kinematic Hardening Law Research Articles

Instrumented indentation for determining stress and strain levels of pre-strained DC01 sheets

In classical mechanical testing, determining the stress and strain states of complex shape specimens turns out to be very challenging, especially in the most deformed regions. In the latter case, when not enough material is present, non-destructive testing is needed for determining the corresponding mechanical state. In this study, a novel approach, based on the local quasi-non-destructive instrumented indentation technique (IIT) coupled with the inverse analysis technique (IAT), is used to determine the stress and strain levels of pre-strained DC01 specimens using monotonic and non-monotonic loading paths. Monotonic and cyclic shear tests as well as Marciniak tests, in both plane and equi-biaxial strain states, are used to apply different pre-strain levels. For monotonic loading paths, i.e. monotonic shearing and Marcianiak tests, the applied methodology showed very satisfying results in determining the stress and strain states of the pre-strained specimens, especially for pre-strain values close to the representative indentation strain values. A maximum stress error of less than 15% was obtained in the case of the highest applied pre-strain value (29.04%). The purely isotropic Voce hardening law was adopted for determining the hardening behavior of the studied specimens in both cases: as-received and pre-strained. In the case of cyclic shear, a mixed isotropic-kinematic hardening law is to be used to determine the stress and strain states of the pre-strained specimens. Adopting a mixed isotropic-kinematic Chaboche hardening law led to satisfying results where the estimation errors of the applied pre-strain value and corresponding yield stress didn't exceed 6% and 1.3%, respectively. The obtained results show that the proposed indentation-based method can be very useful in estimating the work-hardening level and corresponding plastic deformation of pre-strained parts under various loading paths

A new hybrid phase-field model for modeling mixed-mode cracking process in anisotropic plastic rock-like materials

Mixed-mode cracking is widely encountered in rock-like geomaterials under compression-controlled loadings. Some pending issues still need to be addressed. This work focuses on a new hybrid phase-field fracture method for cracking behavior of anisotropic plastic materials. Several new approaches are introduced. Firstly, a unified decomposition method of elastic strain energy driving crack growth is proposed by combining the advantages of classical spectral and volumetric–deviatoric decomposition algorithms. Then a new damage field evolution criterion is established, allowing to considering a nonlinear interaction between two basic crack propagation modes. The contribution of plastic strain energy to cracking process is investigated by considering an anisotropic yield criterion and combined isotropic-kinematic hardening laws. The validation of the established phase-field model is assessed by a series of representative examples, showing different types of cracking patterns. Sensitivity analysis of some key parameters involved in the model is also carried out to clarify the respective impacts on the progressive cracking patterns. Finally, the effectiveness of the proposed approach is validated by comparison with experimental results.

Application of different simulation approaches to numerically optimize high-frequency mechanical impact (HFMI) post-treatment process

The weld seam is generally the weak point of welded mechanical parts subject to fatigue loading. For this issue, a post-weld mechanical surface process called high-frequency mechanical impact (HFMI) was developed. This process combines both mechanical effects and a weld geometry improvement by generating compressive residual stresses and making a smoother transition between the base plate and the weld. Benefits of the process are statistically proven by numerous fatigue test results. Finite-element method (FEM) based numerical simulations of the process have been developed to estimate the material state after treatment. Good agreements with experimental results were obtained. Until present, rebounds of the pin between each primary impact have not been overlooked by such simulations. To discuss their eventual effects, signal of strain gauges glued on the pin was processed. A typical impact pattern of the pin kinetic during HFMI treatment could be identified and then implemented in a pre-existing FEM model. Numerical simulations were conducted using recently developed non-linear combined isotropic-kinematic hardening law with strain-rate dependency according to Chaboche und Ramaswamy–Stouffer models. These hardening laws were calibrated for S355 J2 mild steel. The simulation procedure was performed for flat specimen and representative butt weld joint.

Analysis of the stress and directional dependent Bauschinger-effect of sheet metals

The increasing application of lightweight metals in combination with a growth in the complexity of components provides new challenges to the numerical modeling of sheet materials. The choice of the material model and the associated mapping of the hardening behavior are of substantial importance for a realistic process prediction and the following spring back calculation in particular. The implementation of the Bauschinger-effect via isotropic-kinematic hardening laws can lead to a substantial improvement of the prediction quality. It is commonly known, that an accurate prognosis of sheet metal forming processes requires the consideration of the semi-finished products anisotropy. However, the identification of the Bauschinger-effect, which describes the reduction of the yield stress after a load reversal, and corresponding numerical models, is usually done under a specific stress state and in one direction of the sheet. Considering the vast variety of stress states and loading directions occurring in a forming operation, the anisotropic behavior of the Bauschinger-effect under uniaxial stress and its evolution during shearing is analyzed. Using a miniaturized tension-compression test, the cyclic hardening of the mild steel DX56 and the high strength steel DP600 and the aluminum alloy AA6016 is characterized in 0°, 45° and 90° to the rolling direction. By the identification of an isotropic-kinematic hardening law in combination with the anisotropic flow criterion Yld2000-2d the model’s ability to extrapolate the hardening behavior is evaluated. In the last step, the transferability of parameters from the uniaxial stress state to results from a modified shear test is analyzed.

A study on spring-back in U-draw bending of DP350 high-strength steel sheets based on combined isotropic and kinematic hardening laws

In this article, a numerical model for predicting spring-back in U-draw bending of DP350 high-strength steel sheet was presented. First, the hardening models were formulated based on combined isotropic–kinematic hardening laws, along with the traditional pure isotropic and kinematic hardening laws. A simplified method was proposed for determining the material parameters. Comparison of stress–strain curves of uniaxial tests at various pre-strains predicted by the numerical models and experiment showed that the combined isotropic–kinematic hardening model could accurately describe the Bauschinger effect and transient behavior subjected to cyclic loading conditions. Then, a finite element model was created to simulate the U-draw bending process using ABAQUS. Simulations were then conducted to predict the spring-back of DP350 high-strength steel in U-draw bending with geometry provided in the NUMISHEET’2011 benchmark problems. It was shown that the predictions of spring-back using the proposed model were in good agreement with the experimental results available in the literature. Finally, the effects of various tool and process parameters such as punch profile radius, die profile radius, blank holding force, and punch-to-die clearance on the spring-back were investigated. The simulation results suggested the significance of tool and process parameters on the final shape of the formed parts influenced by the spring-back.

Return mapping algorithm in principal space for general isotropic elastoplasticity involving multi-surface plasticity and combined isotropic-kinematic hardening within finite deformation framework

The compatibility with complicated elastoplasticity and efficiency of the constitutive integration algorithm both significantly influence the performance of finite element analysis for engineering practical problems. In this work, a numerical integration algorithm in principal space is proposed for general isotropic elastoplastic constitutive models that involve multi-surface plasticity with corners in the yield surface and combined isotropic-kinematic hardening law as well as nonlinear elasticity within the framework of finite deformation. For the multi-surface plasticity, a strategy, which uses the mid-direction of two plastic flow directions at a corner as the border of critical regions, is proposed to predict the yield functions activated in the return mapping iterations, making the prediction procedure simpler. By making use of the relative stress, the combined isotropic-kinematic hardening law is incorporated into the numerical integration algorithm in principal space. The consistent tangent operator is also derived. Besides, the fully implicit return mapping algorithm based on representation theorem is employed. The expressions of the first and second derivatives of yield/potential function, which are frequently evaluated in the algorithm, maintain a simple form and reduce the computational cost. Solution of finite element practical problems demonstrates that compatibility and efficiency of the constitutive integration algorithm are improved while accuracy is retained.

A multi-scale modelling of 3rd generation advanced high strength steels to account for anisotropic evolution of yield surface and plastic potential

Multi-scale models were integrated to account for anisotropic mechanical behaviour of the 3rd generation advanced high strength steels (3GAHSS). The micromechanical response of individual phases of the 3GAHSS in non-proportional loading was characterized based on the atomistically informed dislocation dynamics (DD) simulations. The characterized mechanical properties were utilized for the development of the crystal plasticity model to properly account for the cross-loading effect on the interactions of the dislocations. To predict anisotropic evolution of yield surface and plastic potential in non-proportional loading of the 3GAHSS, 3D representative volume elements (3D RVEs) were developed based on the microstructural information from the EBSD data. As for the practical applications such as predictions of formability and spring-back, an advanced phenomenological model was also developed based on a combined type nonlinear isotropic-kinematic hardening law and two-yield surface plasticity.

Numerical and experimental study of an interference fitted joint using a large deformation Chaboche type combined isotropic–kinematic hardening law and mortar contact method

In this paper the framework of a combined Chaboche type kinematic and isotropic hardening model is explained for large deformation with Henky strain description to evaluate strain ratcheting of interference fitted double shear lap joints near the hole based on the works of Montáns et al. [doi:10.1016/j.mechrescom.2012.03.001] and Henann and Anand [doi:10.1016/j.ijplas.2008.11.008]. The described elasto-plasticity formulation is in the context of hyper-elasticity and based on the dual multiplicative decomposition of the deformation gradient into elastic and plastic parts. The decomposition of plastic part into energetic and dissipative part is considered and a new rheological model is suggested for extension of one term Armestrong–Frederick (A–F) model to multi-component A–F (Caboche) model in finite regime. A brief and simple numerical integration scheme is suggested for integrating the presented kinematic hardening. The suggested integration algorithm reduces the complexity of large deformation and for convenience, allows embedding small strain subroutines. To solve the system of nonlinear equations of the constitutive model, the successive substitution method is suggested for simplifying the solution and reducing the computational cost appreciably. In order to avoid non-smooth interaction between the pin and the hole, and also unrealistic plastic deformation, a well-established and an accurate frictional mortar contact formulation is used and combined with the material nonlinearity. The whole numerical formulations are coded in 3D finite element code in FORTRAN programming language and the code is optimized so that it can be efficient for ratcheting simulation over cycles. The approach shows stability and robustness in convergence due to consistent linearization of all contact and internal forces even in the presence of contact, geometry and high material nonlinearity. To evaluate the efficiency and accuracy of the presented formulations, the ratcheting data from simulation is compared with the experimental ratcheting data from strain gauges attached to the studied double shear lap joints.

Development of a Global Indicator to Assess Mechanical Tests

Full-field measurement methods have emerged in the last years and these methods are characterized by directly providing displacement and strain fields for all points over the specimen surface. Thus, the design of heterogeneous tests can be performed for material parameter identification purposes since the inhomogeneous strain fields can be measured. However, (i) no defined criterion yet exists for designing new heterogeneous tests, (ii) it is rather difficult to compare and rate different tests and (iii) a quantitative way to define the best test for material behavior characterization of sheet metals has yet to be proposed. Due to this, the goal of this work is the development of a global indicator able to assess mechanical tests. The proposed indicator quantifies the strain state range, the deformation heterogeneity and the strain level achieved in the test, based on a continuous evaluation of the strain field up to rupture. This global indicator was applied to rank some classical tests, such as uniaxial tensile, simple shear, plane strain and biaxial tensile tests. These tests were carried out numerically by reproducing the virtual behavior of DC04 mild steel. A constitutive model composed by the non-quadratic Yld2004-18p yield criterion combined with a mixed isotropic-kinematic hardening law and a macroscopic rupture criterion was used. The performance of the tests was compared with the indicator and a ranking was established. The results obtained show that biaxial tension is the test providing more information for the mechanical behavior characterization of the material. It was also verified that plane strain test presents a better performance than simple shear and uniaxial tensile tests.

Material parameter identification within an integrated methodology considering anisotropy, hardening and rupture

The main focus of the present work is the development of a global procedure for a complete characterization of sheet metal behavior, including anisotropy, evolution of hardening and rupture. This efficient procedure involves the mechanical characterization, at room temperature and in quasi-static conditions, of the material up to large strains by evaluating the local strain field. The initial anisotropy, hardening and rupture of DC04 mild steel are characterized using a complex phenomenological model composed by the non-quadratic Yld2004-18p yield criterion combined with a mixed isotropic-kinematic hardening law and a macroscopic rupture criterion. For this purpose, an inverse methodology for material parameter identification is implemented based on several experimental tests considered as homogeneous and also, a mixed experimental-numerical approach to calibrate the rupture criterion is performed. In order to validate the obtained results, a deep drawing test, leading either to full drawing or rupture of the blank, is carried out. Digital image correlation is used to acquire additional data information in the deep drawing test. Strain fields at several stages of the deep drawing experiment are evaluated leading to a better comparison between the experimental observations and the results of the numerical simulation.

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.

Consistency condition of isotropic–kinematic hardening of anisotropic yield functions with full isotropic hardening under monotonously proportional loading

For the combined type isotropic–kinematic hardening law to account for the Bauschinger, transient and permanent softening behaviors observed in reverse loading, formulations have been initially developed for isotropic yield functions, mainly based on the von Mises criterion, and then later extended for anisotropic yield functions. Among the efforts to introduce anisotropic yield functions to the combined type hardening formulation, however, some inconsistency has been encountered in manipulating the kinematic hardening law, especially for the nonlinear type law, even though their von Mises yield function versions have been consistent. Therefore, theoretical clarification and clearance of such inconsistency were attempted in this work by imposing the following consistency condition: the combined type hardening law is expected to behave the same as the full isotropic hardening for monotonously proportional loading, regardless of anisotropic yield functions which are coupled with the combined type hardening law. An example to account for the anisotropic hardening of an anisotropic yield function utilizing the combined type hardening law, but for which the consistency condition was partially released, was also demonstrated.

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.

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.

Springback evaluation of friction stir welded TWB automotive sheets

Springback behavior of automotive friction stir welded TWB (tailor welded blank) sheets was experimentally investigated and the springback prediction capability of the constitutive law was numerically validated. Four automotive sheets, aluminum alloy 6111-T4, 5083-H18, 5083-O and dual-phase DP590 steel sheets, each having one or two different thicknesses, were considered. To represent mechanical properties, the modified Chaboche type combined isotropic-kinematic hardening law was utilized along with the non-quadratic orthogonal anisotropic yield function, Yld2000-2d, while the anisotropy of the weld zone was ignored for simplicity. For numerical simulations, mechanical properties previously characterized [1] were applied. For validation purposes, three springback tests including the unconstrained cylindrical bending, 2-D draw bending and OSU draw-bend tests were carried out. The numerical method performed reasonably well in analyzing all verification tests and it was confirmed that the springback of TWB as well as of base samples is significantly affected by the ratio of the yield stress with respect to Young’s modulus and thickness.

Macro-performance evaluation of friction stir welded automotive tailor-welded blank sheets: Part I – Material properties

In order to evaluate the macroscopic performance of friction stir welded automotive tailor-welded blank (TWB) sheets, the hardening behavior, anisotropic yielding properties and forming limit diagram were characterized both for base (material) and weld zones. In order to describe the Bauschinger and transient hardening behaviors as well as permanent softening during reverse loading, the modified Chaboche type combined isotropic–kinematic hardening law was applied. As for anisotropic yielding, the non-quadratic anisotropic yield function, Yld2000-2d, was utilized for base material zones, while isotropy was assumed for weld zones for simplicity. As for weld zones, hardening properties were obtained using the rule of mixture and selectively by direct measurement using sub-sized specimens. Forming limit diagrams were measured for base materials but calculated for weld zones based on Hill’s bifurcation and M–K theories. In this work, four automotive sheets were considered: aluminum alloy 6111-T4, 5083-H18, 5083-O and dual-phase steel DP590 sheets, each having one or two thicknesses. Base sheets with the same and different thicknesses were friction-stir welded for tailor-welded blank (TWB) samples.

시간 의존성 구성방정식을 이용한 AA6022-T4 판재의 탄성 복원 예측

The time-dependent constitutive law was utilized based on viscoelastic-plasticity to predict the time-dependent spring-back behavior of aluminum alloy 6022-T4 sheets. Besides nonlinear viscoelasticity, non-quadratic anisotropic yield function, Yld2000-2d, was used to account for the anisotropic yield behavior, while the combined isotropic-kinematic hardening law was used to represent the Bauschinger effect and transient hardening. For verification purposes, finite element simulations were performed for the draw-bending and the results were compared with experimental results.

Springback prediction of TWIP automotive sheets

In an effort to reduce the weight of vehicles, automotive companies are replacing conventional steel parts with light weight alloys and/or with advanced high strength steels (AHSS) such as dual-phase (DP), twinning induced plasticity (TWIP), and transformation induced plasticity (TRIP) steels. The main objective of this work is to experimentally and numerically evaluate the macro-performance of the automotive TWIP sheet in conjunction with springback. In order to characterize the mechanical properties, simple tension and tension-compression tests were performed to determine anisotropic properties, as well as the Bauschinger, transient, and permanent softening behaviors during reverse loading. For numerical simulations, the anisotropic yield function Yld2000-2d was utilized along with the combined isotropic-kinematic hardening law based on the modified Chaboche model. Springback verification was performed for the unconstrained cylindrical bending and 2D draw bending tests.