Isotropic-kinematic Hardening Research Articles

Mastering the art: Proficient finite element implementation and robust evaluation of a strain-hardening porous ductile material crack growth prediction model at finite strain

Accurate prediction of crack extension is crucial for ensuring the structural integrity of metal components exposed to diverse loads. While the Gurson model and its extensions are widely accepted for delineating ductile fracture stages, especially in porous materials with a rigid-perfectly plastic matrix, their efficacy diminishes in the presence of strain-hardening matrices. To address this limitation, our study introduces a robust numerical implementation and assessment of Perrin's model. In contrast to Gurson's approach, Perrin's model incorporates two distinct strain-hardening parameters derived from an intricate analysis of a strain-hardening hollow sphere subjected to axisymmetric loading.Moving beyond theoretical discussions, this paper actively engages in the rigorous evaluation of Perrin's model under various scenarios, encompassing isotropic, kinematic, and mixed isotropic-kinematic hardening conditions. The model's effectiveness is convincingly demonstrated through meticulous comparisons between numerical simulations and real-world experimental observations conducted on pre-cracked specimens.Facing challenges related to achieving global elastic-plastic convergence in large-scale simulations, our research introduces a sophisticated approach. Leveraging stiffness tangent moduli ensure the maintenance of quadratic convergence in global Newton method iterations. This not only enhances the reliability of our simulations but also underscores the practical applicability of our findings.The study also focuses on the necking and crack propagation of a cylindrical specimen through meticulous computational modeling. Our mesh captures intricate details and addresses stress gradients, revealing coplanar crack propagation contrary to prior beliefs. The confirmation of the cup-cone fracture effect challenges previous interpretations, emphasizing the critical role of void nucleation. This reinforces the credibility and applicability of our approach in advancing the understanding of material fracture behavior.In summary, this study significantly contributes to existing knowledge in fracture prediction models, offering a robust framework for predicting ductile fracture under substantial deformations. The presented findings pave the way for advancements in structural engineering, providing invaluable insights for optimizing the performance and resilience of metal structures in real-world applications.

A Constitutive Model for Asymmetric Cyclic Hysteresis of Wrought Magnesium Alloys under Variable Amplitude Loading

A cyclic plasticity constitutive model was developed for materials with asymmetric cyclic behavior to explain the stabilized stress–strain response under variable amplitude loading. The proposed constitutive model incorporated the von Mises yield function with an adjustment to accommodate asymmetric yielding under tension and compression. A combined isotropic–kinematic hardening model was proposed to describe the evolution of the yield surface in the reference uniaxial frame and the actual frame. The history of plastic deformation is memorized by introducing internal variables, accumulated slip, and residual twins, which govern the cyclic flow behavior in the subsequent reversal. The additional conditions required to predict the stabilized hysteresis response of a material under variable amplitude loading were set out and incorporated into the constitutive model. The model was numerically implemented and programmed into a user material (UMAT) subroutine to run with the commercial finite element program, Abaqus/Standard 2019. The model was calibrated using the stabilized hysteresis response of ZEK100 and AZ31B sheets under constant amplitude strain-controlled cyclic loading for different strain amplitudes. To verify the model, constant amplitude and four different variable amplitude load spectra tests were performed and the stabilized stress–strain hysteresis response predicted by the model was compared with test results. It was demonstrated that the results are in very good agreement.

Explicit and Implicit Integration of Constitutive Equations of Chaboche Isotropic-Kinematic Hardening Material Model

The proper selection of the material model and its parameters in numerical simulations of the material response under loading is a very important task. In the cyclic load one should assume different phenomena, e.g. ratchetting effect, the mean stress relaxation, creep, etc. The Chaboche kinematic hardening model is mainly applied in order to capture the Bauschinger effect due to its good description of the material behaviour under the cyclic loading. The modified Chaboche isotropic-kinematic hardening model is considered in this paper in order to simulate the strain-controlled and stress-controlled uniaxial cyclic tension-compression test. The implicit and explicit integration procedures are tested here, showing advantages and disadvantages of both methods. It is shown that both approaches provide similar results. The implicit integration method of constitutive equations is unconditionally stable and thus less calculation steps are required in comparison to the explicit procedure. The implicit and explicit integration of the Chaboche model including isotropic and kinematic hardening are then implemented for the 3D case in commercial ABAQUS program in the form of the user material procedure. The correctness of results obtained for the user material model is verified by comparison to results for commercially implemented Chaboche model.

Modeling strain-induced martensitic transformation in austenitic stainless steels subjected to cryogenic temperatures using incremental mean-field homogenization schemes

Austenitic stainless steels are commonly used as structural materials in high-field superconducting magnet systems because they retain high strength, ductility, and toughness at very low temperatures, and they are paramagnetic or antiferromagnetic under the Néel temperature in their fully austenitic state. However, they are susceptible to strain-induced martensitic transformation, especially at cryogenic temperatures, which modifies the material properties, induces volume changes and additional strain hardening, and leads to ferromagnetic behavior. Thus, accurate predictions of the structural performance of these materials at very low temperatures are of great interest for the conception and design of these cryo-magnetic systems. In this paper, we propose an adequate constitutive model for the evolving bi-phase material—austenite and martensite—based on a Hill-type incremental formulation. Two different versions of the model are proposed based on the linear mean-field homogenization scheme: Mori-Tanaka and Self-Consistent. Moreover, a rate-independent nonlinear mixed kinematic-isotropic hardening law is used for each phase, and the martensitic transformation is described by the nonlinear kinetic law proposed by Olson and Cohen (1975). The constitutive model is implemented in ABAQUS/Standard through a UMAT user subroutine, for which a return mapping algorithm based on the implicit backward Euler integration scheme is used and a closed-form expression of the consistent Jacobian tensor is provided. The Mori-Tanaka and Self-Consistent approaches are evaluated in terms of their ability to describe the mechanical behavior of the bi-phase aggregate by comparing the predictions of the homogenization schemes with unit-cell finite element calculations with an explicit description of the martensite inclusions and the austenite matrix. The comparison is carried out for different stress states with controlled triaxiality and Lode parameter under monotonic and cycling loading, paying special attention to the evolution of the mechanical fields in each phase. The unit-cell calculations are performed for both constant and evolving martensite volume fractions. In addition, numerical simulations of tensile tests on samples subjected to different initial temperatures are carried out for the transforming bi-phase material and the results are compared to experimental data for AISI 304L and AISI 316LN steels.

Ratcheting–Fatigue Damage Assessment of Additively Manufactured SS304L and AlSi10Mg Samples under Asymmetric Stress Cycles

The present study aims to investigate the interaction of ratcheting and fatigue phenomena for additively manufactured (AM) samples of SS304L and AlSi10Mg undergoing uniaxial asymmetric stress cycles. Overall damage was accumulated through fatigue and ratcheting on AM samples prepared from three-dimensional-printed plates along vertical and horizontal directions. Fatigue damage was evaluated based on the strain energy density fatigue approach and ratcheting damage was calculated through use of an isotropic–kinematic hardening framework. The isotropic description through the Lee–Zavrel (L–Z) model formed the initial and concentric expansion of yield surfaces while the Ahmadzadeh–Varvani (A–V) kinematic hardening rule translated yield surfaces into the deviatoric stress space. Ratcheting of AM samples was simulated using finite element analysis through use of triangular and quadrilateral elements. Ratcheting values of the AM samples were simulated on the basis of Chaboche’s materials model. The predicted and simulated ratcheting damage curves placed above the experimental fatigue–ratcheting experimental data while predicted fatigue damage curves collapsed below the measured values. The overall damage was formulated to partition damage weights due to fatigue and ratcheting phenomena.

Monotonic and Cyclic Modelling of Structural Steel for Finite Element Analysis

Nowadays, as a response to the promoted concept of performance-based seismic design, the finite element method (FEM) has become an omnipresent practice that supports the simulation of structural elements. This approach requires constitutive numerical models, which could reproduce both elastic and plastic behaviour of steel. Looking forward to a reliable FEM simulation, the modelling and calibration of steel undergoing monotonic and cyclic loading tests is investigated in this paper. With respect to monotonic tensile test, a sensitivity analysis regarding mesh refinement and finite element type is developed. Another inspected issue, such as high-strength steel modelling, taking into consideration low ductility and absence of yield plateau, is also presented. And finally, a model for structural steel, using the combined isotropic-kinematic hardening model, is introduced to simulate metal non-linear behaviour under ultra-low-cycle fatigue regime. All these numerical simulations are processed via Abaqus software package, and are validated by a set of experimental tests.

Effect of the yield surface evolution on the earing defect prediction

Although the prediction of earing in the cup drawing process is considerably related to the yield surface shape, the yield surface evolution is also essential for the final ear form. The bending-unbending issue is a fundamental subject occurring on the die and punch shoulders. Since the yield stress is loading path dependent in reversal loadings, the conventional hardening models used in the monotonic loading conditions bring about inaccurate outcomes for predicting the ultimate earing profile, and a kinematic hardening model should be incorporated into the constitutive equations. This study elucidates the yield surface evolution effect involving expansion and translation simultaneously on the ear formation. A sixth-order polynomial yield function was employed to precisely characterize the yield surface shape, while a combined isotropic-kinematic hardening model was implemented to represent the evolution of the yield surface. The translation of the yield surface position was defined by the Armstrong-Frederic hardening model. Punch force-stroke responses and the ear form profiles were predicted by the implemented plasticity model in Marc using the Hypela2 user subroutine and compared with the experimental results. The combined hardening assumption yielded an increase in the mean cup height when compared to the isotropic hardening assumption. Moreover, The HomPol6 coupled with the combined hardening showed a better agreement with the experimental results.

Failure of high-speed bearing at cyclic impact-sliding contacts: Numerical and experimental analysis

The thermal-induced failure mechanism of the bearing outer-ring guiding-surface is investigated within this work when subjected to cyclic impact and sliding actions. The paper combines numerical simulations and experimental analysis. A high-speed bearing oil interruption experiment is carried out for testing the severe damage of the bearing steel at high-speed impact-sliding contacts. A coupled thermo-elasto-plastic phase-field model is established and validated by experimental results. It then allows, by simulating the multi-physics problem, the predictions of damage propagation and failure for ductile materials at cyclic impact-sliding contacts. To this end, a temperature-dependent isotropic-kinematic hardening model combined with thermal softening, cyclic strain hardening, and damage degradation is employed. The results show that under high-speed cyclic impact-sliding conditions, the damage initiated and accumulated at the contact near-surface is accompanied by instantaneous high temperature and plastic deformation. The failure of bearing is induced by a strong thermal softening effect at high-speed sliding and rapidly propagated under cyclic impact loading. In addition, the impact velocity, impact frequency, and friction coefficient have significant effects on damage initiation and accumulation.

Experimental and numerical analysis of mechanical fields on cross-shaped specimens for stress corrosion cracking of cold-worked austenitic stainless steels exposed to primary environment

A stress corrosion cracking (SCC) test was performed on a cold-worked austenitic stainless steel in simulated primary water, using a cross-shaped specimen permitting sequential loading. Crack density and location were investigated by scanning electron microscopy after the SCC test. To analyse the mechanical fields in the cracking areas, finite element simulations of the whole mechanical loading were conducted, involving both strain-path and temperature changes. Combined isotropic-kinematic hardening was used as constitutive equation and identified with tensile tests performed at room temperature and at 340 °C. Partial validation of the model was obtained by comparison of numerical strain fields with experimental measurements obtained by digital image correlation performed on a representative “ex-situ” test in air. Variations of the strain and stress fields during this test were discussed in relation with the cracking network observed at the end of the SCC test.

Analyses of the model parameters of kinematic-isotropic hardening rule using genetic algorithm approach for predicting cyclic-plasticity of metallic structural materials

Modeling of engineering components under cyclic loading is complex because cyclic-plastic phenomena like the Bauschinger effect, ratcheting, shakedown, and mean stress relaxation are to be considered in the constitutive modeling. This has led to several investigations on modeling the cyclic-plastic behaviour of materials under stress-controlled and strain-controlled testing; but amongst these, Chaboche's isotropic-kinematic hardening (CIKH) model is quite popular and a frequently used one. A new methodology has been recently proposed by some of the present authors to use genetic algorithm towards optimization of the parameters of CIKH modeling with a demonstration of its applicability for a few materials. This investigation aims to elucidate that this approach is applicable to both cyclically stable materials as well as cyclic softening or hardening materials using several examples on structural materials. The considered metallic materials include several aluminum (like AA7075-T6 alloy), iron (like CS-1026and Sa333 C-Mn steel), titanium (like TA16 alloy), zirconium (like Zr-4 alloy)-base alloys or superalloys (like INCONEL718). The merit of the analysis of cyclic plastic deformation behaviour of materials with the current approach is inherent in its achieving higher accuracy of fitting to the experimental data with a single set of material parameters under both strain- and stress-controlled cycling. This has been established with a comparative analysis of the accuracy of fitting by the present approach with the ones available from the existing reports. The parameters obtained using the approach for the different materials are compared to get insights into the mechanism of plastic deformation.

A computationally fast and accurate procedure for the identification of the Chaboche isotropic-kinematic hardening model parameters based on strain-controlled cycles and asymptotic ratcheting rate

The Chaboche isotropic-kinematic hardening (CIKH) model provides a versatile and realistic description of the material stress–strain behavior under generic multiaxial cyclic loadings. However, identifying the backstress parameters is challenging, and can be formulated as an optimization problem using different approaches. Instead of a computationally expensive pointwise search, in this paper the global properties of the cyclic curves are fitted to the experimental data. The conditions introduced are the hysteresis areas, peak stress values and tangent moduli at the extreme points, however the framework can be easily adapted to other target quantities. One linear and two non-linear backstress components of the kinematic hardening model are introduced, although the analytical equations developed can be used to refine the model further, with more components. Two stabilized cycles are required to identify the main kinematic parameters. New analytical expressions for asymptotic ratcheting rates in uniaxial tests are developed and then used to tune the dynamics of the slightly non-linear (hence, slowest) backstress component. After obtaining the kinematic parameters, isotropic hardening laws can also be identified, by considering the evolution of the extreme points of the strain-controlled cycles before stabilization. Practical demonstrations of the procedure are provided by experimental tests carried out on a 7075-T6 aluminum alloy, 42CrMo4+QT steel, and a high-silicon ferritic ductile cast iron. An accurate reproduction of the material behavior is achieved, at a negligible computational cost.

Ratcheting evaluation of additively manufactured 4043 aluminum samples through a combined isotropic–kinematic hardening framework

Ratcheting evaluation of additively manufactured 4043 aluminum samples through a combined isotropic–kinematic hardening framework

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.

Parameter Identification and Modeling of Hardening to Preserve Identical Predictions under Monotonic Loading

Advanced hardening models accurately describe the transient plastic behavior (reyielding, stagnation, resumption...) after various strain-path changes (reverse, orthogonal...). However, a common drawback of these models is that they usually predict monotonic loading with lower accuracy than the regular isotropic hardening models. Consequently, the finite element predictions using these models may sometimes lose in accuracy, in spite of their tremendous theoretical superiority. This drawback has been eliminated in the literature for the Chaboche isotropic-kinematic hardening model. In this work, a generic approach is proposed for advanced hardening models. Arbitrary models could be successfully compensated to preserve rigorously identical predictions under monotonic loading. A physically-based model involving a 4th order tensor and two 2nd order tensors was used for the demonstration. The parameter identification procedure was greatly simplified by rigorously decoupling the identification of isotropic hardening parameters from the other parameters.

Geometric and Material Nonlinear Analyses of Trusses Subjected to Thermomechanical Loads

In this article, geometric and material nonlinear analyses of planar and space trusses subjected to thermo-mechanical loads are performed using the Total Lagrangian formulation. The material properties are Thermally InDependent (TID). By obtaining the equilibrium path of these structures, their buckling and post-buckling behaviors are predicted under thermal and mechanical loads. To consider material nonlinearity, a mixed isotropic–kinematic hardening criterion is developed in this formulation. The behavior of material is assumed to be bilinear. To solve nonlinear equations iteratively, the Generalized Displacement Control Method (GDCM) is used. The authors’ scheme is general and can be employed for different types of planar and spatial truss structures having complicated geometries and behaviors. Several popular benchmarks are considered to show the accuracy and correctness of the proposed formulation. Corresponding equilibrium paths and stress–strain curves are reported for each problem.

Evolution of Lüders banding under axial loading and reverse loading

Low slenderness steel rods that exhibit Lüders banding are used to examine the evolution of localized deformation when a specimen is unloaded and reverse loaded. Special attention is given to how a specimen behaves when the initial loading is interrupted with part of it Lüders deformed and the rest still elastic. Digital image correlation revealed that both parts unload elastically, but on reverse loading the previously elastic zone yields and develops Lüders strain of the opposite sign. By contrast, the plastically deformed zone follows the usual Bauschinger rounding. Thus, the specimen behaves as a structure with its overall response being a combination of the two distinctly different stress-strain histories, and depends on the length of each zone. The experiments are simulated using finite element analysis coupled to a custom combined isotropic-kinematic hardening plasticity model with a partially softening branch spanning the Lüders stress plateau. The analysis reproduces the measured responses and evolution of localized deformation quite accurately demonstrating the fidelity of the constitutive model adopted. The results also demonstrate that the propagation of the Lüders zone has a more complex three-dimensional profile than that reported for thin strips. Overall, the study provides a guide on how Lüders banding can be treated in structures that undergo loading and reverse loading.

A Microstructure Based Elasto-Plastic Polygonal FEM and CDM Approach to Evaluate LCF Life in Titanium Alloys

This work presents the influence of microstructure on the LCF life of two-phase titanium alloys. An elasto-plastic polygonal FEM and continuum damage mechanics approach has been used in this work. Damage coupled Chaboche-Lemaitre combined isotropic-kinematic hardening model is implemented to accurately capture the effects of deformation inhomogeneity and material plasticity. A new strain-based damage evolution law is proposed to predict the effect of mean grain size on the LCF life. The effect of different microstructural parameters i.e., topological variation, elastic anisotropy, mean grain size and shape, volume fraction of phases and initial voids is statistically analyzed on the LCF life.

Load step reduction for adjoint sensitivity analysis of finite strain elastoplasticity

In this paper, load step reduction techniques are investigated for adjoint sensitivity analysis of path-dependent nonlinear finite element systems. In particular, the focus is on finite strain elastoplasticity with typical hardening models. The aim is to reduce the computational cost in the adjoint sensitivity implementation. The adjoint sensitivity formulation is derived with the multiplicative decomposition of deformation gradient, which is applicable to finite strain elastoplasticity. Two properties of adjoint variables are investigated and theoretically proved under certain prerequisites. Based on these properties, load step reduction rules in the sensitivity analysis are discussed. The efficiency of the load step reduction and the applicability to isotropic hardening and kinematic hardening models are numerically demonstrated. Examples include a small-scale cantilever beam structure and a large-scale conrod structure under huge plastic deformations.

Notch fatigue analysis and crack initiation life estimation of maraging steel fabricated by laser beam powder bed fusion under multiaxial loading

This paper deals with the notch fatigue behaviour and crack initiation life estimation in maraging steel fabricated by laser beam powder bed fusion under multiaxial loading. Tests are conducted in tubular geometries with lateral holes considering different normal stress to shear stress ratios and multiaxial loading levels. The cyclic stress–strain response at the notch-controlled process zone is simulated numerically using two alternative approaches: a generalised isotropic plasticity model with mixed isotropic-kinematic hardening, and a linear-elastic model. Both approaches demonstrated to be suitable for predicting the crack initiation sites, the directions of crack growth, and the fatigue life. Fatigue life was calculated from a SWT-based model combined with the Theory of Critical Distances. Elastic-plastic predictions led to smaller errors but slightly shifted to the non-conservative side.