Hardening Law Research Articles

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.

Whip-Bezier: A C1-continuous hardening law for efficient direct and inverse identification

In 3D constitutive models, the hardening law is an essential model ingredient that has a first order effect on the predicted stress–strain response. In most phenomenological plasticity models, the hardening law coincides with the stress–strain curve for monotonic uniaxial loading. When identifying the hardening law for large strains through hybrid experimental–numerical approaches, the numerical properties of the hardening law play an important role. Here, we propose a novel piecewise-defined hardening law based on a composition of quadratic Bezier curve segments. The Whip-Bezier formulation preserves C1 continuity and achieves a parametrization with less parameters than its piecewise-linear counterpart. Both the possibility of identifying the hardening parameters in a decoupled optimization (e.g. from in-plane torsion experiments) and the use of a fully-coupled identification scheme (e.g. from notched tension experiments) are detailed. When contrasted with widely-used analytical expressions, the Whip-Bezier model consistently demonstrates enhanced performance with increasing number of parameters. This improvement comes with minimal statistical variance and reduced dependence on the initial optimization guess. Furthermore, the fitting quality is at least equivalent to that of the Swift-Voce hardening law with the same number of parameters.

Modelling of shakedown, ratcheting and liquefaction of saturated granular soils using kinematic hardening with memory surface

The present paper proposes a kinematic hardening model with a memory surface to more accurately reproduce cyclic hardening and the behaviour of saturated granular soils subjected to cyclic loads. The new cyclic hardening law offers improved accuracy in predicting soil ratcheting in drained conditions and failures due to liquefaction in undrained conditions. Simulating these two aspects has been challenging due to issues such as significant loss of accuracy and numerical instability, particularly under extended (or a large number of) cyclic loads. To deal with the challenge, in addition to the novel theoretical framework, we also developed a robust sub-stepping algorithm based on an explicit stress integration scheme with automatic error control. The proposed model is compared with several existing models and its performance is evaluated by conducting sensitivity analyses and performing validations against a wide range of laboratory test results.

Experimental and Numerical Investigation of Folding Process—Prediction of Folding Force and Springback

The folding process is characterized by the springback phenomenon. Several experimental folding tests are elaborated and illustrated in this paper. The precision and the quality of the folded sheet workpiece are related to the reduction in the springback phenomena. For that, two tools are designed and used for the folding process. An accurate design of the folding tool plays a significant role in contributing to the folding process and reducing potential defects related to springback. An experimental solution is presented to avoid the forming of defaults and compensate the workpiece springback after its removal from the die. Moreover, an accurate numerical modeling enables an efficient prediction of the springback. This allows us to obtain precise parts through the folding process. For that, a modified Johnson–Cook model is implemented on ABAQUS software in order to predict the folding force and the springback in a U-die folding process. In addition to the isotropic hardening law, a nonlinear kinematic hardening rule is used. To ensure the model’s accuracy and reliability, we conducted validation experiments. The model’s predictions are compared with experimental tests to show its capability to simulate the folding process effectively. The developed mechanical model can adequately predict and analyze springback effects and folding force evolution, helping designers compensate for them and achieve the desired final shape.

Predictive Modeling of Spring-Back Behavior in V-Bending of SS400 Steel Sheets under Elevated Temperatures Using Combined Hardening Laws

This research presents an innovative methodology for accurately predicting spring-back tendencies in V-bending of SS400 steel sheets under elevated temperatures. The study leverages extensive tensile test data to determine parameters for pure isotropic and kinematic hardening laws at varying temperatures, crucial inputs for Finite Element Method (FEM) simulations. While using pure isotropic or kinematic hardening laws alone has limitations, especially at elevated temperatures, a hybrid approach is recommended for robust predictive models in ABAQUS 6.13 software. To address this challenge, a novel method is introduced, utilizing flow stress curve ratios between elevated and room temperatures as a function of equivalent strain to derive combined hardening law parameters. Rigorous comparison of simulation and experimental results confirms the model’s effectiveness in predicting spring-back in the V-bending of SS400 steel sheets, particularly under elevated temperatures. This innovative approach enhances understanding of material behavior at high temperatures and improves predictive capabilities for designing and optimizing complex V-bending processes.

A lode-dependent plasticity model for high-strength structural steel

In this study, a new plasticity model was developed to characterize the mechanical behavior of high-strength structural steel under multiaxial stress states. This model utilizes a Lode-dependent yield criterion in conjunction with a non-associated flow rule and isotropic hardening law to determine the evolution of the plastic flow stress. Further, a detailed numerical implementation algorithm for this plasticity model is presented, including the integration algorithm of the constitutive equations using an implicit elastic predictor-return mapping method and formulation of the consistent tangent modulus. To validate the proposed plasticity model, six types of Q690 high-strength steel specimens were tested under monotonic loading. These specimens can be used to obtain the elastoplastic response of the material under uniaxial tension, compression, pure shear, tensile shear, and plane strain state, respectively. Corresponding numerical simulations were performed for each specimen using ABAQUS/Standard with the proposed model being introduced into the analysis via the UMAT user subroutine. Then, the prediction performance of the proposed plasticity model was investigated by comparing the load-displacement curves of each test specimen obtained from both experiments and numerical simulations. It is shown that the proposed plasticity model can accurately predict the mechanical response of the high-strength steel under different stress states and enables to visualize the key stress state parameters and material strength variation at the failure region of each specimen, which confirms the promising prospect of this new model in practical applications.

A standard thermodynamic-based extension of the Modified Cam-Clay soil model and its applications

For mechanical behaviours of granular soils considered here (sands, silts and stiff clays), we propose a thermodynamic-based formulation which relies on the Modified Cam-Clay constitutive model. To this end, and in the framework of Generalized Standard Materials (GSM) framework, we introduce a free energy and a yield criterion (or equivalently a dissipation potential) which account for an isotropic nonlinear elastic behaviour and a volumetric hardening law. Based on available works with coupled hardening laws for von Mises plasticity, we propose for pressure-sensitive materials a new nonlinear deviatoric hardening preserving the well-known critical state existence.Taking advantage of the GSM framework, we establish an incremental variational formulation of the model whose solution obeys to a minimization principle. This has required an approximation of the incremental dissipation. The resulting equations are detailed for the use of an implicit resolution scheme with a standard Newton–Raphson algorithm. The constitutive model parameters are calibrated on a pool of experimental data on isotropic, shearing and triaxial tests on loose sand specimen. The investigations of the model responses allow to better predict the nonlinear regimes of soils specimen on monotonic and cyclic loading paths. Numerical finite element simulations finally underline the predictive capabilities of the model on structures.

Modelling the viscoplastic behaviour of Callovo-Oxfordian claystone with consideration of damage effect

In order to evaluate the performance of deep geological disposal of radioactive waste, an underground research laboratory (URL) was constructed by Andra in the Callovo-Oxfordian (COx) claystone formation at the Meuse/Haute-Marne (MHM). The construction of URL induced the excavation damage of host formations, and the ventilation in the galleries desaturated the host formation close to the gallery wall. Moreover, it is expected that the mechanical behaviour of COx claystone is time-dependent. This study presents a constitutive model developed to describe the viscoplastic behaviour of unsaturated and damaged COx claystone. In this model, the unsaturation effect is considered by adopting the Bishop effective stress and the van Genuchten (VG) water retention model. In terms of the viscoplastic behaviour, the nonstationary flow surface (NSFS) theory for unsaturated soils is used with consideration of the coupled effects of strain rate and suction on the yield stress. A progressive hardening law is adopted. Meanwhile, a non-associated flow rule is used, which is similar to that in Barcelona basic model (BBM). In addition, to describe the damage effect induced by suction change and viscoplastic loading, a damage function is defined based on the crack volume proportion. This damage function contains two variables: unsaturated effective stress and viscoplastic volumetric strain, with the related parameters determined based on the mercury intrusion porosimetry (MIP) tests. For the model validation, different tests on COx claystone under different loading paths are simulated. Comparisons between experimental and simulated results indicated that the present model is able to well describe the viscoplastic behaviour of damaged COx claystone, including swelling/shrinkage, triaxial extension and compression, and triaxial creep.

Ductile fracture of anisotropic QP980 steel sheet by using the isotropic/anisotropic modified Mohr-Coulomb models

The modified isotropic Mohr-Coulomb (iso-MMC) fracture model, which utilizes the Mises yield function and isotropic hardening law, was calibrated through tensile tests on various specimens. The resulting calibrated model was constructed in (η, L, ε¯f) space to represent the isotropic fracture locus of QP980 steel sheet. Subsequently, this calibrated model was utilized to predict the ductile fracture behavior of QP980 steel sheet loaded along rolling direction under different stress states. To comprehensively consider the impact of anisotropic properties on fracture evolution, an anisotropic MMC (aniso-MMC) ductile fracture model was developed in terms of strain space. Under the assumption of plane stress, the Mohr-Coulomb (MC) criterion was converted from stress space to strain space through an anisotropic constitutive model in order to characterize ductile fracture in QP980 steel sheet. The quadratic yield function of Hill's 48 yield function with the isotropic hardening law was employed to characterize the plastic behavior during loading. The anisotropic fracture loci of QP980 steel sheet were obtained in various loading directions. The scanning electron microscope (SEM) was utilized to characterize the microscopic morphology of the fracture surfaces of QP980 steel sheet, investigating the fracture mechanism of anisotropic ductility under different stress states. The results of both tensile experiments and simulations have confirmed that the plastic anisotropy of materials plays a critical role in determining ductile fracture behavior. The aniso-MMC fracture model was ultimately validated through hole expansion tests and equi-biaxial tensile tests, demonstrating its capability to predict ductile fracture behavior under both proportional and non-proportional loading conditions.

Application of ductile fracture model for the prediction of low cycle fatigue in structural steel

The complexity of the loading conditions makes it challenging to accurately predict the damage evolution and resulting ductile fracture in materials and structures. This paper presents an application method to predict fracture initiation for low cycle fatigue of 4-point bending in structural steel SS275. The swift isotropic hardening law was used in the numerical simulation to account for large strain deformation. The damage indicator framework with the Hosford-Coulomb (HC) model was adopted to characterize the damage accumulation and fracture behavior. Uniaxial tension tests on specimens with various fracture modes were conducted to determine the material properties required for calibration of the plasticity model and fracture model param1eters. Non-linear finite element analyses were performed to identify fracture model parameters based on the loading path of fracture initiation. The presented HC model was applied to an extremely low fatigue test (less than 100 cycles) and demonstrated good accuracy in predicting fracture initiation for complex loading scenarios.

Patterns of localized deformation at pre-fracture stage in carbon steel – stainless steel bimetal

The work is devoted to the study of strain localization at macroscale level during parabolic mechanical hardening and pre-fracture under quasi-static loading of a carbon steel – stainless steel bimetal. The problem of estimating the scale of the phenomena that determine plasticity is decisive in the development of any theories of plastic deformation, in particular, dislocation theories. The main difficulty in constructing such theories is the reconciling the dislocation scales, characteristic for most deformation and mechanical hardening mechanisms, with macroscopic parameters of deformation processes. In the framework of the autowave model of localized plastic deformation, this problem can be reduced to the possibility of obtaining parameters from the results of macroscale observations of localized plastic flow development. During the experiments, it was confirmed that in a bimetal at any forming stage, a specific pattern of localization centers distribution is spontaneously generated - a pattern of localized plastic flow. The shape of such patterns is determined by the law of mechanical hardening acting in the material. It is shown that the observed localization patterns can be used as an informative feature in predicting the plasticity margin. In the process of uniaxial tension at the stage of parabolic mechanical hardening of the bimetal, the deformation mode is realized with the formation of several potential fracture centers. It was established that at the pre-fracture stage, during the time evolution of the wave pattern of deformation localization, the zone of active plastic deformation narrows, but the number of centers in it either remains the same with a decrease in the distance between them, or even increases. The result of this process is the formation of a macroscopic neck, and then fracture. At the pre-fracture stage, the collapse point indicates the place of future fracture and signals the need to stop the deformation process in order to avoid the fracture of the bimetallic material. Thus, the well-known manifestation of deformation macroscopic localization – formation of a neck – is preceded by complex phenomena of mutually coordinated motion of localized plasticity centers at the pre-fracture stage.

Steady-state and transient mechanical response analysis of superelastic nitinol lattice structures prior to additive manufacturing: An in-silico study

The additive manufacturing (AM) of nitinol (NiTi) has gained considerable attention due to the capability of this material and processing combination to produce complex structures with desired shape memory or superelastic properties. Finite element analysis models were developed in this work to simulate the load-bearing capability of Laser Powder Bed Fusion (PBF-LB) produced NiTi lattice structures. The Auricchio numerical method, with exponential hardening law, was implemented within the FEA model to take account of the martensite-austenite reorientation upon change of stress state. The strut diameter and the unit cell side length were varied which in turn varied the lattice relative density. The lattice structures were subjected to 7 % compressive strain under isothermal conditions. The highest energy absorption efficiency of 40 % was achieved from the lattice structure with a strut diameter of 2 mm, a 6 mm unit cell side length, and a relative density of 42 %; however, the highest specific peak stress was recorded within the lattice structure having a strut diameter of 2.5 mm, a 4 mm unit cell side length, and a relative density of 62 %. Transient analysis was performed by varying the compression rate from 0.25 mm/s to 1 mm/s. For all loading conditions, the energy absorption increased linearly until a stable stress plateau was reached. Moreover, for these examined compression rates, the maximum stress absorption efficiencies varied within the range of 25 %–33 %.

Low amplitude strain accumulation model for natural soft clays below railways

An improved constitutive model for strain accumulation of natural clays under undrained cyclic loading is presented. The proposed model includes a formulation for the non-linear small-strain stiffness in the overconsolidated regime, along with a modified hardening law for cyclic accumulation to improve the tracking of strain accumulation at small stress amplitudes. To calibrate and validate the proposed model, a series of laboratory tests were conducted to study the cyclic response of natural Swedish clays, the effect of loading amplitude and pre-shearing history. Good agreement between predicted and measured accumulated axial strains and excess pore water pressures was obtained with different loading amplitudes. The findings reveal that the undrained pre-shearing has a substantial impact on the rate of accumulated strain, with pre-sheared samples exhibiting lower resistance values. The proposed and validated model opens up possibilities to study the monotonic and non-monotonic quasi-static response of soft clays below railway embankments over the lifetime of the structure, i.e. including the effects of construction, operation and decommissioning.

A novel elastoplastic model for Yunnan sandstone under poly-axial loading

The lack of understanding of plastic hardening (softening) laws, especially under anisotropic stress conditions, results in inappropriate geotechnical management. Most of the yielding envelopes do not consider the effect of intermediate principal stress and the influence of Lode’s angle. In addition, the application of plastic flow rules regarding yielding surfaces compromises the softening of rock internal friction as well as the influence of Lode’s angle on the plastic potential. Moreover, the ductility to brittleness transition in the intermediate principal stress direction still requires a theoretical foundation. In this study, based on poly-axial testing results of Yunnan sandstone, we adopted a failure criterion with the intermediate principal stress proposed by Menétrey and Willam. The proposed new failure envelope was applied to capture the plastic evolution of rock samples. A plastic hardening-softening model is constructed, based on the framework of the plastic theory. The softening envelope is modified to better present the stress drop and considers the deterioration of rock internal friction in the post-peak stage of poly-axial loading. The differential of plastic potential according to the principal stresses is also modified, considering the rotation of Lode’s angle in the poly-axial loading tests. The model results were compared with laboratory testing results, which showed great consistency across 9 different loading tests (5 under triaxial stress and 4 under poly-axial stress with 22 stress-strain curves in total). The induced brittleness by the intermediate principal stress is also well captured by the proposed model.

Modified plasticity constitutive model for extruded aluminum alloys

Aluminum alloys have been increasingly used in a wide range of construction applications, especially in important public buildings. A plasticity constitutive model that can accurately describe the elastoplastic behavior of aluminum alloys under complicated stress states is of great importance in precisely simulating aluminum alloy structures under complicated working conditions. Also, a precise plasticity model serves as the foundation for studying the ductile fracture of metals, and is therefore crucial for structural analysis involving large deformation, such as progressive collapse analysis. The classic J2 plasticity model based on the von Mises yield criterion is the most commonly utilized model for ductile metal materials. However, it ignores the effect of stress state on metal plasticity behavior, making it inapplicable to aluminum alloy materials under various complicated stress states. This study experimentally demonstrated that stress triaxiality and Lode angle have a noticeable effect on the plasticity behavior of extruded aluminum alloys (6061-T6, 6082-T6, and 6063-T5). Accordingly, a plasticity constitutive model suitable for extruded aluminum alloys under complicated stress states was proposed and validated, including a full-range hardening law and a new yield criterion. In addition, the parameter calibration method for this new model was detailed. Finally, comparisons between the experimental and numerical results proved the high accuracy of the proposed model in predicting the actual response of a structural extruded aluminum alloy under complicated stress states. Compared with previous studies, the proposed model can not only consider the effects of stress triaxiality and Lode angle, but it can also provide a more reasonable description of yield surface under considerably high or low stress triaxiality.

Strain gradient plasticity based on saturating variables

Standard scalar-based strain gradient plasticity models generally incorporate the gradient of the cumulative plastic strain into the constitutive modeling. The following limitations of this approach have been pointed out in the literature: Unlimited enhanced hardening at large strains, unlimited cyclic hardening in bending, strain localization band broadening at large applied strains for softening materials with saturation, and possible vanishing of the yield stress for too high values of the Laplace term in the enhanced hardening function. The present work proposes an alternative scalar-based micromorphic approach that solves most of the listed difficulties. The theory incorporates the effect of the gradient of saturating variables instead of the gradient of ever-increasing cumulative plastic strain. A phenomenological theory is presented that includes an exponentially saturating internal variable and its gradient. The approach is also applied to dislocation density based plasticity where the dislocation density is known to saturate at large strains. The benefits of the new theory are demonstrated by finite element simulations of various boundary value problems: Monotonic and cyclic tension and simple glide of an infinite strip, strain localization in a plate, bending and torsion of bars. Both hardening and softening plasticity laws are addressed illustrating the size-dependent hardening in the former case and the evolution of the finite width of localization bands in the latter. Weak points of the approach, namely the possibility of negative dissipation and of negative yield stress, are also identified.

Soil particle size-dependent constitutive modeling of frozen soil under impact loading

In this study, the soil particle size dependence of the impact dynamic mechanical properties of frozen soil was investigated using a split Hopkinson pressure bar, and the meso-deformation mechanism of frozen soil was analyzed via electron microscopy. Under impact loading, the dynamic stress-strain curves of frozen soil showed strain hardening, strain softening, strain rate-temperature sensitivity, and soil particle size dependence, and the soil particle size dependence of frozen soil was considered to be significantly influenced by non-uniform meso-deformation, which primarily appeared in the form of microcracks surrounding the coarse particles. A constitutive model comprising a multiplicative hardening law was proposed, and soil particle size dependence was introduced through the equivalent strain gradient of the soil cell element. The impact compression process of frozen soil was simulated in ABAQUS/Explicit using the proposed model, and the mechanism of frozen soil deformation was summarized by the finite element analysis: the stress concentration around coarse particles induces the initiation of microcracks, the propagation of microcracks forms a crack network, eventually leading to the softening behavior of frozen soil.

A numerical study on rotational stiffness characteristics of the disk lock joint

Rotational stiffness characteristics of the disk lock joint have a significant effect on the load bearing capacity and stability of disk lock scaffolds. However, experimental researches on scaffold joints are usually expensive and time consuming, especially when different joint connection configurations and loading modes are involved. This paper explores the feasibility and effectiveness of the 3D nonlinear finite element analysis method for studying the rotational stiffness characteristics of the disk lock joint. Detailed geometric models of the disk lock joints are established and the geometric models are meshed with finite elements. The elastoplastic constitutive model based on the isotropic hardening law and the associated flow rule is used to describe the complete deformation process of the material, and the penalty stiffness method is applied to accurately simulate the interaction between different components. Thus the complete finite element models of the disk lock joints are established. Firstly, the loading processes of the one-way disk lock joint are simulated and verified by experiments. To further demonstrate the robustness and effectiveness of the finite element method, the loading processes of disk lock joints under different connection configurations and loading modes are simulated. The influences of connection configurations and loading modes on joint rotational stiffness are also investigated to reveal the load bearing mechanism of the disk lock joint, and thereby contribute to better predictions of the load bearing capacity of the disk lock scaffolds.

Investigation on residual stress of EQ56 high strength steel butt weld

In the welding process of high-strength steel (HSS), the intense temperature change in joints induces the formation of welding residual stress (WRS), which increases the fatigue crack growth rate. Developing an effective finite element (FE) model for accurately predicting the WRS distribution is critical to improve the quality of welded structures. Experimental tests and numerical simulations were carried out to investigate the WRS distribution of EQ56 HSS. The thermal expansion curves at various heating and cooling rates were measured through thermal simulation experiments for achieving the continuous cooling transformation (CCT) diagram of EQ56 HSS. Based on the CCT diagram, the thermo-metallurgical-mechanical (TMM) FE model was developed to consider the influence of solid-state phase transformation (SSPT) on distribution and magnitude of WRS for welded EQ56 HSS joints, as well as material hardening law. Comparison of predicted WRS distribution and experimental data shows that plastic hardening laws of materials significantly affect the precision of WRS distribution, and combined hardening law gives most reliable prediction. The FE model considering SSPT behavior provides more accurate estimation of WRS distribution than those without considering SSPT behavior.

Nonlocal strain regularisation for critical state models with volumetric hardening

The finite element (FE) method may suffer from numerical instability and mesh dependency when modelling the formation of shear bands using strain-softening constitutive models. Nonlocal methods have been shown to be capable of avoiding these numerical issues effectively. This paper proposes a novel generic algorithm for incorporating existing nonlocal methods into the FE formulation of elasto-plastic constitutive models with volumetric hardening laws. Using the example of the modified Cam Clay (MCC) model this paper shows that instabilities are inevitable if the nonlocal plastic volumetric strains are directly used to update the hardening parameter. The paper further proposes a robust regularisation algorithm to overcome this issue. This algorithm has been implemented into a bespoke FE code and its capability in significantly reducing mesh dependency while maintaining numerical stability is verified through a series of numerical analyses of biaxial compression tests for highly overconsolidated clays. The performance of various existing nonlocal methods is also critically assessed.