Hardening Law Research Articles

The role of accumulated plasticity on yield surface evolution in pearlitic steel

In this paper, we analyze one specific assumption used in many cyclic plasticity models: The dependence of the yield surface evolution on the accumulated plasticity, i.e., the time-integrated plastic multiplier. We measure this scalar variable in carefully designed experiments and track the yield surface evolution. Our results show that this variable is insufficient to describe the isotropic or distortional hardening of the investigated steel, invalidating the commonly used assumption. The results highlight requirements for developing and verifying new isotropic and distortional hardening laws.

Constitutive relationship and characterization of fracture behavior for WE43 alloy under various stress states

The plasticity and fracture behavior of the WE43 alloy were investigated under various stress states. Mechanical experiments were conducted with special designed specimens for tension, compression and shear. The testing process was recorded and handled by digital image correlation technology. Experimental results show that WE43 alloy possesses the low tension−compression asymmetry and the fracture mechanism belongs to the ductile fracture. The plastic deformation behavior under uniaxial tension and compression was simulated with different hardening laws based on the Drucker yield function. The stress state and fracture strain were obtained by the numerical simulation. The ductile fracture behavior was numerically predicted by Brozzo, Oh, Ko-Huh and DF2016 criteria to compare with the experimental results. The results suggest that the plastic deformation can be reasonably modeled by the Swift−Voce hardening law and Drucker yield function. It is also demonstrated that the DF2016 criterion can accurately predict the fracture behavior of the alloy under various stress states.

A novel integrated framework for reproducible formability predictions using virtual materials testing

Background: Formed aluminium alloy sheet materials are increasingly adopted in production processes such as vehicle manufacturing, due to the potential for weight-saving and improved recyclability when compared to more traditional steel alloys. To maximise these benefits whilst maintaining sufficient mechanical properties, the link between formability and microstructure must be better understood. Virtual materials testing is a cost-effective strategy for generating microstructure-informed formability predictions. Methods: We developed an open-source hybrid framework, combining experimental and computational tasks, for generating reproducible formability predictions. Starting with experimental texture measurements and stress-strain curves, we calibrated crystal plasticity (CP) model parameters. The framework used these parameters to perform a large set of multiaxial full-field CP simulations, from which various anisotropic yield functions were fitted. With these anisotropy parameters, we then employed a Marciniak-Kuczyński finite-element model to predict forming limit curves, which we compared with those from experimental Nakazima tests. Results: We executed the workflow with the aluminium alloy Surfalex HF (AA6016A) as a case study material. The 18-parameter Barlat yield function provided the best fit, compared to six-parameter functions. Predicted forming limits depended strongly on the chosen hardening law, and good agreement with the experimental forming limit curve was found. All of the generated data have been uploaded to the Zenodo repository. A set of Jupyter notebooks to allow interactive inspection of our methods and data are also available. Conclusions: We demonstrated a robust methodology for replicable virtual materials testing, which enables cheaper and faster formability analyses. This complete workflow is encoded within a simple yet highly customisable computational pipeline that can be applied to any material. To maximise reproducibility, our approach takes care to ensure our methods and data — and the ways in which that data is processed — are unambiguously defined during all steps of the workflow.

Theoretical and Numerical Investigation of the Limit Strain of a 5754-O Aluminum Alloy Sheet Considering the Influence of the Hardening Law

Theoretical and Numerical Investigation of the Limit Strain of a 5754-O Aluminum Alloy Sheet Considering the Influence of the Hardening Law

A Cohesive Traction Embedded Constitutive Law Combined with Shear-Induced Damage and Kinematic Hardening Law

A Cohesive Traction Embedded Constitutive Law Combined with Shear-Induced Damage and Kinematic Hardening Law

Extending modified Cam-Clay to unsaturated soils by incorporating a scaled stress variable

This paper extends the modified Cam-Clay model to describe the behaviour of unsaturated soils by replacing the effective stress with the scaled stress. The scaled stress accounts for the mechanical effect of capillarity by factoring Bishop’s stress with a power function of degree of saturation. The extendedmodified Cam-Clay model postulates the existence of a yield curve in the mean scaled stress 𝑝̅– deviatoric scaled stress 𝑞̅ plane. Coherently with the modified Cam-Clay model, the stress-strain behaviour is predicted by means of an elastic law inside the yield curve. As the scaled stress state evolves onto the yield curve, the behaviour becomes elasto-plastic and this is modelled by a plastic flow rule combined with a volumetric hardening law. The extended modified Cam-Clay model is then validated against experimental data published in the literature. Results show that the proposed approach is capable of well predicting the mechanical behaviour of unsaturated soils during triaxial loading. Future work will be directed at further validating the present formulation along different stress paths as well as accounting for the combined effect of partial saturation and cementation within a critical state framework based on bounding surface plasticity.

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.

Implementation of the Lemaitre damage model with kinematic hardening in the ANSYS finite element complex

Currently, one of the most popular in fracture mechanics is taking into account damage and its effect on the stress-strain state of structural elements. In this work, the Lemaitre damage model was integrated taking, into account the combined hardening law, which combines the Armstrong - Frederick kinematic hardening law and the Voce isotropic hardening law, into the ANSYS finite element software. The model is implemented in the form of a dynamically linked library of user material for three-dimensional objects, which is tested on a cylindrical specimen with an external annular notch, both in an axisymmetric setting and in a three-dimensional one. The article presents model representations of the above-listed standard systems. This work demonstrates only one of the two stages of verification of the created program - comparison of damage fields under monotonic loading with data known in the literature - and doesn’t take into account the verification of cycle-by-cycle kinetics of plastic deformation accumulation with experimental data for low-cycle fatigue. The result of verification, consistent with similar experiments known in the literature, is confirmed in accordance with similar experiments. In addition, an analogy was found using the TSL law of the cohesive fracture mechanics approach. Despite the fact that two different types of constitutive equations are used in the cohesive model and Lemaitre, the physical meaning of these equations consists in one thing - visualization and identification of mechanisms and coordinates of damage.

Dynamic numerical prediction of plasticity and damage in a turbofan blade containment test

For the light designs of new aircraft engines, constitutive plasticity and fracture models sensitive to strain rate and temperature are essential for the accurate prediction of deformations and internal stresses of the components during simulations of impact and explosion events. The work described in this article consists of the development and numerical analysis by finite elements of the blade containment test of a commercial aircraft turbofan engine, conducted to evaluate the structural integrity of the casing after being impacted by a detached fan blade. Two simulation models of the test are proposed, in which the resistance behavior of the strain rate-dependent material is described by isotropic laws of strain hardening and Johnson–Cook damage. The strength analysis is based on the numerical field results of equivalent stresses and deformations, along with the internal damage rates of the casing. The first simplified model considers half of the casing impacted by a blade at different speeds and angles of impact. The second model consists of a complete discretization of casing and rotating turbofan, with the initial detached condition of a blade simulated at different rotating speeds of the fan. The material used in this study is the Ti-6Al-4V alloy. The results analysis and advances obtained make it possible to approach an efficient computational tool with more accurate calculations to study a casing redesign with a safe reduction in mass and that fulfills the certification requirements using the blade containment test.

Machine learning-based multi-objective optimization for efficient identification of crystal plasticity model parameters

A set of constitutive model parameters along with crystallography governs the activation of deformation mechanisms in crystal plasticity. The constitutive parameters are typically established by fitting of mechanical data, while microstructural data is used for verification. This paper develops a Pareto-based multi-objective machine learning methodology for efficient identification of crystal plasticity constitutive parameters. Specifically, the methodology relays on a Gaussian processes-based surrogate model to limit the number of calls to a given crystal plasticity model, and, consequently, to increase the computational efficiency. The constitutive parameters pertaining to an Elasto-Plastic Self-Consistent (EPSC) crystal plasticity model including a dislocation density-based hardening law, a backstress law, and a phase transformations law are identified for two materials, a dual phase (DP) steel, DP780, subjected to load reversals and a stainless steel (SS), 316L, subjected to strain rate and temperature sensitive deformation. The latter material undergoes plasticity-induced martensitic phase transformations. The optimization objectives were the quasi static flow stress data for the DP steel case study, while a set of strain-rate and temperature sensitive flow stress and phase volume fraction data for the SS case study. The procedure and results for the two case studies are presented and discussed illustrating advantages and versatility of the developed methodology. In particular, the efficiency of the developed methodology over an existing genetic algorithm methodology is discussed. Additionally, the parameters identified for the SS case study were utilized to simulate three biaxial tensile loading paths using a finite element implementation of EPSC for further verification.

Investigation of crossed-twin structure formation in magnesium and magnesium alloys

In this work, the effect of alloying addition on the propensity for twin-twin interactions to transform into crossed-twin structures in magnesium alloys is investigated. A full-field elasto-viscoplastic fast Fourier transform (EVP-FFT) framework combined with a discrete twin model and dislocation density-based hardening law for slip strengths is used to calculate the micromechanical fields in the crystalline matrix around the interacting twins. AZ31 and MgLi alloys are selected along with pure Mg to study the influence of plastic anisotropy in connection with alloying elements. These alloys were selected since their plastic anisotropy measure, which is defined as the ratio between the critical resolved shear stress for pyramidal 〈c+a〉 and basal 〈a〉 slip modes, spanned a wide range. To quantify the role of twin thicknesses, we probe a range of impinging twin thicknesses while fixing the recipient twin thickness. The analysis reveals that: (i) the local driving stress for crossed twin structure formation generated from the interaction of the two twins is lower in a low plastically anisotropic material, like a MgLi alloy, than a high plastically anisotropic material like pure Mg and (ii) the critical impinging twin thickness needed to form the crossed twin structure in pure Mg, AZ31 and MgLi alloys is ∼0.5, ∼0.75 and ∼1.5 times the recipient twin thickness. We propose a relationship between the tendency for crossed twin structure formation and the experimentally observed higher ductility in MgLi alloys compared to pure Mg. One key implication of the findings is that crossed-twin structure formation can be hindered and the ductility of magnesium alloy thereby improved by properly choosing alloying elements that lower the slip strength for pyramidal c+a slip.

A unified amorphous–crystalline viscoplastic hardening law for non-isothermal modelling of thermoplastics and thermosets

The constitutive modelling of semi-crystalline polymers (SCP) has to consider several aspects as rate- and temperature-dependence, self-heating, and in particular, the double yield (DY) phenomenon. A full characterization of all these complex features involves prominent efforts in terms of material testing and parameter identification (PI). The contribution of the crystalline phase plays an important role in the evolution of the plastic yield in the SCPs. In this work, a constitutive model, named Unified SCP (USCP), is proposed by modifying the physically-based Boyce–Parks–Argon (BPA) glassy model. The contribution of the crystalline phase is introduced in the strain softening/hardening evolution law of the strength, providing an alternative interpretation of the underlying morphological changes caused by the crystalline phase embedded in the amorphous phase. The proposed formulation extends the BPA model with a new contribution to capture the crystalline phase. A full thermo-mechanical coupled numerical framework is developed for the USCP model validation. The DY phenomenon at different strain rates with self-heating and thermal softening effects is investigated and predicted. The proposed model extension needs three material constants with clear physical meaning. To identify them, a fast in-house optimization process based on Nelder–Mead is used, in which only a single element test is required. The model accurately predicts the experimental results for both thermosets and thermoplastics such as epoxy, nylon 101, PA6 and LDPE under monotonic loadings reported by different authors.

Dynamical Models of Plasticity with Nonmonotonic Deformation Curves for Nanomaterials

Nanomaterials are widely used in different fields, such as microelectronics, industry, and nanocomposites, and they can exhibit unstable deformation behaviour depending on the strain rates. Under strain rates of 10−4–10−1 s−1, the deformation of nanomaterials, unlike the quasi-static deformation of micromaterials, is characterized by the presence of the rate sensitivity as a possible scale phenomenon in dynamic plasticity. In this paper, the relaxation model of plasticity for the prediction of deformation curves at different strain rates is used. It allows us to comprehensively study the effects of strain hardening in a wide range of deformation conditions for coarse-grained materials and nanomaterials. Considering the plastic deformation of the nanosized samples in the early stages, dynamical softening, associated with a generation of new defects, and dynamic hardening, are crucial. The proposed model, using one parameter or the classical hardening law as an example of nanosized gold whisker crystals, tungsten single-crystal pillars, and single-crystalline Au-Ag alloy nanowires, is verified. Calculated sets of parameters of characteristic time, as a parameter of rate sensitivity of a material, and hardening parameters for different nanomaterials are compared. It is shown that the characteristic relaxation times for the single-crystal nanomaterials (100–103 s) are greater than for the nanostructured materials (10−6–10−4 s). Despite the manifestation of dynamics at different strain rates of nanomaterials, single crystal and nanostructured materials, the proposed model can be successfully applied to materials with different degrees of hardening or softening.

Two‐invariant modular implicit plasticity solver framework for geomechanical simulators

AbstractA modular, computationally efficient integration framework for two‐invariant‐based elastoplastic constitutive models is proposed. It is designed to allow a range of constitutive models to be incorporated with minimal programming effort. The development is intended for efficient numerical schemes in large‐scale geomechanical simulations. The modular format consists of four components: isotropic elastic relation, isotropic yield criterion, two‐invariant smooth flow potential plastic flow rule and isotropic hardening law. Linear and non‐linear bulk elasticity are considered. For the non‐linear case, a numerical assessment of the accuracy of bulk elasticity integration schemes is presented, where pure Euler and semi‐analytical integrators are compared. The deviatoric component of the elastic law considers both the constant shear modulus and constant Poisson's ratio models. The combination of the classic Terzaghi compaction law with different bulk elasticity models may lead to a physical inconsistency, which is removed by a straightforward modification of the hardening law. When used in conjunction with the proposed bulk elasticity models, it results in a family of compaction laws that significantly reshape the normal compaction line – a fact that seems to have been overlooked in the literature. The main contributions concerning the plasticity integrator are: a novel integration scheme that considers the variation of specific volume within each step; and a condensed system comprising two scalar return‐mapping equations. A Modified Cam‐Clay (MCC) model is used in the assessments. A comparison with an existing, well‐known algorithm is also provided and, in spite of its modularity/generality, the proposed scheme is shown to yield significant computational gains.

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

Large strain flow curve identification for sheet metal

The large strain flow curve is an essential input for the calibration of a myriad of plasticity models required for accurately simulating sheet metal forming processes and the mechanical performance of formed components. Studies on large strain flow curve identification can be divided in three main categories. The first class of experiments generate statically determinate stress states which unambiguously correlate analytically to the external load and measured strains, hence enable a direct measurement of the large strain flow curve. The second category relies on the intermittently measured flow stress using pre-strained samples to construct the large strain flow curve. The third category consists of methods that minimize the discrepancy between the experimentally acquired and the numerically computed measurand to inversely identify a chosen hardening law representing the large strain flow curve. The paper presents an overview of the state-of-the-art methods to acquire the large strain flow curve of sheet metal. A selected number of methods are reviewed and recent developments are discussed in detail. Finally, an objective assessment of the selected methods is pursued by applying them to a DP600 steel sheet.

Some Issues with Statistical Crystal Plasticity Models: Description of the Effects Triggered in FCC Crystals by Loading with Strain-Path Changes.

The justification of the applicability of constitutive models to exploring technological processes requires a detailed analysis of their performance when they are used to describe loadings including the complex loading mode that is characteristic of these processes. This paper considers the effect of equivalent stress overshooting after the strain-path changes known to occur in metals and alloys. The macrophenomenological and multilevel models, which are based on crystal plasticity, account for this effect by applying anisotropic yield criteria at the macro- and mesolevels, respectively. We introduce a two-level constitutive statistical inelastic deformation model (identified for aluminum) that incorporates the popular simple phenomenological anisotropic hardening law for describing the behavior of FCC polycrystals. The results of the numerical simulation are in satisfactory agreement with existing experimental data. Statistical analysis of the motion of a mesostress in the stress space on the crystallite yield surface is performed. The obtained data are compared with the results found using the isotropic hardening law. The results clarify the simulation details of statistical crystal plasticity models under loading with strain-path changes in materials and demonstrate their suitability for describing the processes under consideration.

Dynamic Hardening Behavior and Ductile Fracture of High-Strength Steel at Intermediate Strain Rates

The mechanical properties of constructional steels at different strain rates were the basis for the dynamic analysis of the progressive collapse of steel structures. The effect of strain rate on the hardening behavior and ductile fracture for the high-strength steel Q690 is investigated in this paper. Quasistatic and intermediate strain-rate experiments were performed. A smooth sheet and a V-shaped notched specimen were carefully designed to cover intermediate and high-stress triaxialities. The experimental results revealed that the effect of strain rate on the hardening behavior is sensitive to stress state. A novel dynamic isotropic hardening law was developed, taking advantage of the Cowper–Symonds and Johnson–Cook hardening models. Also, negative and positive strain-rate effects on the ductility of Q690 were observed in the smooth and notched specimens, respectively. An attempt was made to introduce a new strain-rate term into an uncoupled ductile fracture model to describe the differences of the strain-rate effects at various stress states. The new hardening law and the strain-rate-dependent ductile fracture model were validated with satisfactory accuracy.

Efficient and robust stress integration algorithm for anisotropic distortional hardening law under cross-loading with latent hardening

A fast and robust stress-update algorithm based on the general cutting-plane method (GCPM) was developed for a distortional hardening model, known as the HAH-DPS model. It captures the anisotropic hardening behaviors such as the Bauschinger effect, transient hardening, differential permanent softening, and cross-loading effects. The lower computational efficiency of the direct application of GCPM was rectified by considering the all-evolutionary plastic state variables during iterations. The newly proposed algorithm was formulated on the dependence of the equivalent plastic strain and the other state variables defined in the distortional hardening model. And it was implemented in a commercial finite element software using a user-defined material subroutine (UMAT). Finite element simulations under strain-path change were carried out to demonstrate the performance of the new numerical algorithm in terms of the convergence behavior locally as well as globally.

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.