Strength Differential Research Articles

Damage and fracture behavior under non-proportional biaxial reverse loading in ductile metals: Experiments and material modeling

This paper deals with a modified anisotropic stress-state-dependent plastic-damage continuum model incorporating combined hardening and softening rules to predict the plastic, damage, and fracture behavior of ductile metals under monotonic and reverse loading conditions. In the experimental part, a newly designed biaxially loaded cruciform flat HC-specimen of aluminum alloy EN AW 6082-T6 was subjected to biaxial non-proportional loading to generate a wide range of stress triaxialities during experiments. Thus, it is enabled to perform shear monotonic and reverse loading superimposed by various preloads without unloading processes. The experimental data reveal the Bauschinger effect, the strength-differential (SD) effect, and the change in the hardening ratio after shear reverse loading. This is captured by a Drucker–Prager-type yield criterion with an extended Voce isotropic hardening and modified Chaboche’s kinematic hardening rule. In addition, the damage surface is assumed to be translated in the direction of the rate of the damage strain. The numerically predicted global force–displacement curves and local strain fields are verified in terms of the digital image correlation (DIC) technique. Moreover, the scanning electron microscopy (SEM) is used to examine the fracture surfaces and to validate the proposed damage constitutive model.

Appropriateness of hydrostatic pressure-based modeling for strength differential effect in advanced high strength steel

Although numerous models have been proposed to describe the strength differential (SD) effect, their appropriateness and efficiency in the case of advanced high strength steel (AHSSs) have not been investigated. To this end, this study investigates the SD effect in AHSS using a hydrostatic-pressure-dependent plasticity theory. Although pressure model can effectively predict the SD effect in AHSSs with only one coefficient, the original formulation is experimentally inconsistent in the first quadrant of a yield surface. Hence, this study developed a new formulation to remove the inconsistency in a simple manner and validated the pressure model using new experimental data reported for two recently developed AHSSs: DP980 and MART1500Y. The SD effect is characterized based on monotonic compression and tension tests in three different directions with respect to the rolling direction to obtain the stress–strain curves and r-values. Next, the pressure model is used to predict the plastic behavior of both steel sheets. The parallel transition of flow stress and similarity in r-values between the tension and compression states supports the appropriateness of the proposed pressure-based modeling. Furthermore, these insights substantiate the fact that deformation mechanisms cannot explain the SD effect in both steel sheets. Therefore, complex yield functions that describe the SD effect in general materials are not necessary in the case of AHSS. Additionally, the pressure model is incorporated into a distortional plasticity model for applying it to springback in the bending process. The distortional plasticity model with and without the pressure model is calibrated using tension–compression curves, thereby revealing that the SD significantly affects the calibration. Finally, a U-draw bending experiment and the corresponding finite element simulation are conducted to compare the springback; the results show that considering the SD effect improves the springback simulation accuracy.

Constitutive modeling of temperature and strain rate effects on anisotropy and strength differential properties of metallic materials

The thermal effects on tension compression asymmetry and anisotropy of several metallic materials with a hexagonal close-packed (HCP) crystal structure have been discussed in this study. The Yoon2014 asymmetric yield criterion is combined with a simple Arrhenius type temperature function and a modified Johnson-Cook strain rate model to formulate a thermal dependent plasticity model with analytical solutions for parameter calibration. The temperature and strain rate effects on the strength differential (SD) effects of both isotropic and anisotropic materials have been discussed. Both anisotropy and SD affect the shape of yield locus. In order to quantitatively distinguish the thermal effects on the strength differential and anisotropic effects of metallic materials, the evolution of yield locus and distribution of yield stress under two extreme scenarios has been discussed based on virtual experimental results. For the general application of the thermal dependent asymmetric plasticity model at different temperatures and strain rates, the calibration and validation of the model using uniaxial tensile/compressive results along different directions and biaxial tensile/compressive results have been elaborated for various metallic alloys.

Evaluation of loading-path-dependent constitutive models for springback prediction in martensitic steel forming

In this study, the performance of conventional and advanced constitutive models to the prediction of springback for ultra-high strength steel was assessed. The elasto-plastic behavior of a martensitic steel sheet sample was characterized to assess the conventional stress-strain behavior in uniaxial tension. More advanced tests such as uniaxial compression and tension-compression experiments were conducted to assess the strength-differential (SD) effect and hardening fluctuations during non-linear loading paths (NLLP effect). Three constitutive plasticity models, namely, the conventional isotropic hardening, Yoshida-Uemori (YU) two surface kinematic hardening and pure distortional hardening (HAH20) models were calibrated using an optimization procedure. Then, these models were employed for finite element predictions of springback after forming. The isotropic hardening model does not consider both SD and NLLP effects and the YU model captures only the influence of the NLLP effect. On the other hand, the HAH20 model describes both effects simultaneously. The performance of these models with respect to springback prediction was evaluated using three forming applications: wiper bending, C-rail drawing and Roof-Side-Rail crush forming. This work indicates that the HAH20 model is the best candidate for the prediction of forming and springback for such martensitic steel because of its ability to consider both SD and NLLP effects. However, the conventional isotropic hardening model led to results in good agreement with those obtained using the HAH20 model in specific cases. The reason, discussed in detail in this article, is likely due to the compensation of the permanent softening due to load reversal by the higher flow stress due to the SD effect.

On the Effect of the Tension-Compression Ratio on Forming of Isotropic Materials

It is generally believed that the choice of the yield criterion used to describe the plastic behaviour of isotropic metallic materials does not affect much the accuracy of the predictions of forming operations. For this reason, the von Mises yield criterion is widely used for modelling their plastic behaviour. However, according to the von Mises yield criterion, the ratio between the yield stresses in tension and compression is always 1.0. Nevertheless, materials can present same asymmetry, i.e. a strength differential (SD) effect. In this work, the yield criterion proposed by Cazacu et al. [1], is adopted to describe the mechanical behaviour of isotropic materials, with different levels of tension-compression asymmetry. Numerical simulations of the deep drawing of a cylindrical cup were performed considering these different virtual isotropic materials, in order to evaluate its impact on the thickness distribution and cup height. The analysis of the stress and strain paths, for material points initially located in the flange is performed, enabling the correlation between the thickness distributions predicted and the plastic strain ratios obtained analytically, using the yield function and the normality rule. The results show that even small changes in the tension-compression stress ratio can influence the strains distribution, in isotropic materials.

Analysis of ductile damage and fracture under reverse loading

This paper investigates the influence of reverse and cyclic loading on the damage and fracture behavior of the aluminum alloy EN-AW 6082-T6. A series of uniaxial tension–compression and shear tests with newly designed shear specimen has been performed to analyze the development of plastic deformations and damage under different stress conditions. In addition, several monotonic uniaxial tension and compression tests are conducted to detect the strength-differential (SD) effect. Hence, combined hardening law is incorporated into the hydrostatic-stress-sensitive yield function to predict irreversible deformations under cyclic loading. A newly defined strain triaxiality is proposed to describe the strain state and to reveal the damage evolution processes. To accurately model cyclic plasticity, strain triaxiality is incorporated into the extended Voce strain hardening, and the angle parameter between effective back stress and reduced stress is added to Chaboche’s kinematic hardening. Furthermore, a modified anisotropic stress-state-dependent continuum damage model is considered to evaluate the damage and fracture evolution processes. Additionally, the digital image correlation (DIC) technique is applied to measure the strain fields during the experiment. Numerical simulations are used to predict the distribution of stresses and strains. Finally, the distribution and amount of damage are used to explain the fracture behavior detected by scanning electron microscopy (SEM) pictures taken in the critical parts of the specimens. SEM images and numerically predicted damage stains confirm that the different failure modes depend on the loading history.

Finite element implementation of hydrostatic pressure-sensitive plasticity and its application to distortional hardening model and sheet metal forming simulations

• The strength-differential (SD) effect of AHSS with ultimate tensile strength exceeding 1 GPa is modeled by a hydrostatic pressure-dependent distortional plasticity theory. • The finite element implementation of the pressure-dependent plasticity theory is proposed in the manuscript. The implemented theory is validated with the comparison study with the associated stand-alone code and experimental measurements. • Both the SD- and strain patch change effects of TIRP1180 are reproduced in finite element simulation by embedding the distortional plasticity model, HAH 20 , in the pressure-dependent yield condition. The effectiveness and applicability of the FE-implemented pressure-dependent distortional plasticity model in sheet metal forming simulations is validated through the springback prediction studies. In this work, the hydrostatic pressure-dependency of the distortional plasticity model HAH 20 was implemented using a finite element (FE) code to account for the strength-differential (SD) effect that has been observed in advanced high-strength steel sheets. To this end, a fully-implicit stress update algorithm formulation was introduced for the pressure-dependent plasticity theory. The implementation was validated by comparing the FE prediction of the material behavior during a number of tests with those of a stand-alone code of the constitutive description as well as with experimental data. In order to investigate the SD effect on the springback simulation results, the U-draw bending test was analyzed within this FE framework. Furthermore, in order to assess the effectiveness and stability of the formulation for a large-scale example, forming simulations of an automotive structural part were conducted. In addition to the SD effect, strain path changes and geometrical aspects were also investigated in this example. It was shown that the distortional plasticity-based pressure-dependent yield criterion well describes the asymmetric SD behavior in sheet metal forming simulations.

Phase transformation and plastic behavior of QP steel sheets: Transformation kinetics-informed modeling and forming limit prediction

A previous study on quenching and partitioning (QP) steel showed that the transformation-induced plasticity (TRIP) effect strongly depends on the stress triaxiality and Lode angle parameters during the plastic deformation. However, only the impact of stress states on the linear hardening response (first-order effect) was investigated, assuming its yield and plastic flow evolution (second-order effect) follow isotropic behavior for both tension and compression. The high-order effects of phase transformation on plasticity are still elusive and difficult to incorporate owing to the inconsistent usage of the equivalent strain in kinetics law and constitutive modeling. In this study, the hardening law, considering martensitic transformation, was utilized in conjunction with a strength differential (SD) effect-induced asymmetric yield function based on the proposed work-conjugate framework. The contradiction where the previous model correlates the martensitic transformation law based on the von Mises strain and the mechanical responses based on the normalized plastic work-per-unit material (or the work-conjugate equivalent strain) was eliminated. This elimination imposed the phase transformation influence into size evolution of the yield surface for QP980 because the effective stress directly correlated with the martensitic transformation using the work-conjugate equivalent strain. One can use any yield function in conjunction with this work-conjugate framework to correlate the martensitic transformation and its impact on the mechanical behavior for both first- and second-order effects. Thus, the yield surface of QP980 steel was stably normalized through asymmetric function under different plastic strains and deformation modes. The SD effect of QP980 steel has proven to be anisotropic and the martensitic transformation has negligible impact on the shape evolution of its yield surface. The pressure-dependent phenomenon in the plastic flow behavior of the employed QP steel suggested that its volume change during the plastic deformation can be neglected. The correlation between the microstructure volume fraction, evolving yield surface, and the stress–strain response under nonlinear loadings (Bauschinger effect) was predicted accurately. The forming limit imperfection analysis served as a typical application, where the phase transformation was further correlated with the localized necking phenomenon. The framework to understand the impact of microstructure evolution in QP steel sheets on its final forming limits was provided, suggesting that an optimized strain path design will greatly enhance the accumulated plastic limit before the localized necking.

A new asymmetric yield criterion based on Yld 2000-2d under both associated and non-associated flow rules: Modeling and validation

A new asymmetric yield criterion is developed by introducing asymmetrical parameters into Yld 2000-2d. It has been proved that the convexity of the proposed yield criterion is guaranteed. A ‘sum’ form of yield functions is used to increase the flexibility of the yield criterion. One of the advantages of the developed yield criterion is that the convexity of yield criterion is independent of the anisotropic parameters. Besides, the proposed yield criterion not only possesses the inherent characteristics of Yld 2000-2d under tensile stress states but also can be used to predict material's strength differential (SD) effect. By applying the constructed yield criterion to FCC and HCP materials, the accuracy and flexibility of the proposed yield criterion are verified. The results show that both the 11 and 16 parameters yield criteria for non-associated and associated flows, respectively, can accurately capture the SD effects of these materials. In addition, the proposed yield criterion not only can predict the tension-compression asymmetry along the rolling and transverse directions but also can accurately predict the SD effect along the diagonal direction.

Analytical description of an asymmetric yield function (Yoon2014) by considering anisotropic hardening under non-associated flow rule

A simple analytical form for yield surface distortion under non-associated flow rule is developed from the expression of the transformed second and third deviatoric stress invariants in Yoon et al. (2014). The developed analytical yield criterion has the same ability as the original yield criterion to describe material's strength differential (SD) effect. Compared with Yoon et al. (2014) criterion, the material parameters in the analytical form presented in this study can be directly calculated from the tension and compression stresses along 0°, 45°, 90° and equi-biaxial directions, which doesn't need any optimization process and interpolation to describe the evolution of asymmetric yield surface under the proportional loadings. The proposed yield criterion can accurately predict the anisotropic hardening along 0°, 45°, 90° and equi-biaxial directions under tension and compression, which will make the significant improvements in the stress predictions including 15°, 30°, 60° and 75°. To guarantee the convexity of yield surface during loading history, the minimum order of principal minor determinant is checked. The proposed yield criterion has been successfully applied for HCP materials to validate its general applicability for describing anisotropy/asymmetry-induced distorted yield surface during deformation. In addition, an analytical asymmetric plastic potential function is developed based on Poly4 yield function. The accuracy of the proposed plastic potential function for r-value evolution under tension and compression has been verified by applying it to different materials.

Development of In-plane Biaxial Compression Test Method for Thin Sheet and Investigation of SD Effect under Biaxial Stress Condition

The accuracy of the finite element simulation (FE simulation) of press forming has improved as a result of sophisticated material modelling using data of the various experiments such as uniaxial compression, biaxial tension, and in-plane cyclic loading tests. However, the biaxial compression test of sheet material has never been reported lowing to the difficulty in buckling suppression, and differences between material behavior under biaxial tension and that under biaxial compression have not yet been discussed. In this paper, the biaxial compression test of sheet material using special tools to suppress buckling and a specimen shape that leads to uniform deformation in the measurement area is proposed mainly on the basis of FE simulation. The special tool has a urethane plate where the tool come into contact with the measurement area of the specimen to prevent damage of the strain gauges. Absolute values of compressive stresses during plastic work measured by the biaxial compression test were larger than tensile stresses during the same plastic work measured by the biaxial tension test. Similar results observed in uniaxial tension and compression tests are known as the strength differential (SD) effect.

On mechanical response of Zircaloy-4 under a wider range of stress states: From uniaxial tension to uniaxial compression

The present work summarizes the experimental campaign and computational effort performed on Zircaloy-4 material commonly used in nuclear applications. The Zircaloy-4 exhibits a mild strength differential response under compression as compared with tension. Because of the well-known twinning-slip mechanism usually activated in compression after yielding in HCP metals, distortional hardening is required to define plastic flow throughout fracture. A general asymmetric yield function with the first invariant to the Cauchy stress tensor and the third invariant to the deviator stress tensor are considered to model the pressure-sensitive response and the Strength Differential (SD) effect in HCP polycrystal materials. The proposed macroscopic yield function provides results that are in good agreement with experiments. The onset of fracture and subsequent crack propagation in upsetting tests is also investigated. The results show that failure is not only controlled by material properties but also by friction between contact surfaces of specimens and experimental setup. In addition, the concept of a fixed cutoff value for negative stress triaxiality (η=-1/3) condition is analyzed. It turns out that the specimen fails differently of what is observed experimentally when a constant cutoff value is set. However, a cutoff stress triaxiality function, yields a more realistic fracture pattern consistent with tests.

Influence of SD effect on Johnson–Cook hardening constitutive material model: Numerical and experimental investigation for armor steel

The influence of strength–differential (SD) effect has been investigated on Johnson–Cook (JC) material model parameters for armor grade RHA steel. Two set of JC model parameters has been developed with the help of compressive and tensile flow curve. Further, a finite element analysis (FEA) of static (at 10−3 s−1) and dynamic indentation (at 104 s−1) has been performed to validate the effect of tensile and compressive JC material model parameters. In addition, static and dynamic indentation FEA results has been compared with the expanding cavity model (ECM) and experimental Split–Hopkinson pressure bar results, respectively. FEA simulation with compressive data has been shown good agreement with experimental results and analytical models. %SD effect is increased with an increase in plastic strain in armor grade RHA steel.

An enhanced distortional-hardening-based constitutive model for hexagonal close-packed metals: Application to AZ31B magnesium alloy sheets at elevated temperatures

In this study, a new material model for hexagonal close-packed (HCP) metals for predicting their flow responses under monotonic/cyclic loadings at various temperatures is proposed. The temperature-dependent constitutive model is re-formulated using a continuum-based distortional hardening law, which affects the shape of the yield surface depending on the loading direction, and also uses the concept of the dominant deformation modes in magnesium alloys: twin, untwin, and slip-dominate modes, considering the role of the twins in plastic deformation. In terms of the material asymmetry in yield strength, the Cazacu–Barlat–Plunkett (CPB) ’06 yield criterion is extended to include thermal effects. A numerical formulation of the material model has also been developed to allow its implementation into finite element analysis software via a user-defined material subroutine (UMAT). The predicted results of stress–strain response for monotonic and cyclic loadings are excellent agreement with the experimental data. Moreover, the simulated results show the strength differential (SD) effect and unusual flow responses of AZ31B magnesium alloy sheets at room temperature, as well as the variation in the SD effect and anisotropic hardening behavior at elevated temperatures.

Ductile fracture prediction using enhanced CDM model with Lode angle-dependency for titanium alloy Ti-6Al-4V at room temperature

Abstract Fully coupled ductile damage model considering strength-differential (SD) effect is proposed for simulation of metal forming processes for Ti-6Al-4V. The developed constitutive equations are based on non-associative plasticity theory under thermodynamically-consistent framework. The third stress invariant is added to Hill48 criterion to capture the SD effect. Lode angle-dependent enhanced fully coupled damage model is proposed to account for the loading path sensitivity on ductile damage initiation and growth. A detailed material parameters identification procedure is presented. The proposed model is implemented into the FE code Abaqus/Explicit through user defined subroutine VUMAT. The numerically predicted results obtained with the proposed model are compared to experimental data of Ti-6Al-4V and good agreement is observed.

Indentation Size Effect in Pressure-Sensitive Polymer Based on A Criterion for Description of Yield Differential Effects and Shear Transformation-Mediated Plasticity.

Indentation size effects in poly(methyl methacrylate) (PMMA) were studied through nanoindentation. Two factors of indentation size effects in PMMA, namely yield criterion and shear transformation-mediated plasticity, were analysed in detail. The yield criterion that considers strength differential (SD) effects and pressure sensitivity was constructed by performing the combined shear-compression experiments. The relationship between hardness and normal stress can then be obtained based on Tabot’s relation. Shear transformation-mediated plasticity was also applied to model the measured hardness as a function of the indentation depth at different strain rates. Results show that the yield criterion contains the terms of SD effects and pressure sensitivity gives the best description of the yielding of PMMA. Additionally, the volume of single shear transformation zone calculated through the presented criterion agrees well with simulation and exhibits increases with increasing strain rate. Indentation size effects in PMMA under different strain rates were discussed and an appropriate indentation depth range was suggested for calculating the hardness and modulus.

Validation of recent analytical dilatational models for porous polycrystals using crystal plasticity finite element models with Schmid and non-Schmid activation laws

Recent analytic criteria for isotropic porous materials developed by Cazacu et al. (2013) revealed the importance of considering specificities of plastic behavior in the matrix. On one hand it was shown that if the matrix material is governed by the von Mises criterion, the yield surface of the porous material should be centrosymmetric and, with the exception of hydrostatic and purely deviatoric loadings, there are combined effects of the mean stress and third-invariant of the stress deviator on void growth or collapse; but on the other hand if the matrix plastic deformation displays strength differential (SD) effects, the response is also sensitive to third-invariant and there is a lack of symmetry of the yield surface of the porous material. In this paper, we use a unit cell modeling approach in conjunction with a crystal plasticity finite element model to verify these theoretical predictions. It is assumed that each porous polycrystal contains a regular array of initially spherical voids and a random initial texture. At the grain-level, we consider that plastic deformation is governed by Schmid law and a recent non-Schmid formulation by Savage et al. (2017a) that intrinsically accounts for tension–compression asymmetry in a physical sense. Unit cell FE calculations are performed for axisymmetric tensile and compressive loadings corresponding to a fixed value of the stress triaxiality and the two possible values of the Lode parameter. The resulting numerical points representing the homogenized yield surfaces and void growth/collapse curves are found to be in agreement with the analytical model's predictions.

Thermodynamically-consistent constitutive modeling of hardening asymmetry including isotropic ductile damage for Mg alloys

It is well known that the Mg alloys exhibit strong initial anisotropy and hardening asymmetry during plastic flow. These behaviors are mainly caused by their Hexagonal Close-Packed (HCP) lattice structure and the activation of deformation twinning. In this paper, a thermodynamically-consistent plasticity model is proposed to capture the strong anisotropic/asymmetric behavior of Mg alloys with considering the effect of isotropic ductile damage. A detailed calibration method of the proposed model is provided using shear, tension and compression tests along different orientations of the rolled sheets. The material parameters related to initial anisotropy and strength-differential (SD) effect is treated separately from the hardening and damage parameters. The identification procedure is based on minimizing the difference between experimental measurements and FE simulation results over the whole deformation process until final failure. The parametric study of the newly proposed model is conducted to demonstrate its capability in capturing the hardening asymmetry. The predictive capabilities of the proposed model are illustrated through application to two Mg alloys (AZ31B and ZEK100), both the initial tension-compression asymmetry and unusual stress-strain response under monotonic compression are accurately predicted.

High accuracy springback simulation by using material model considering the SD effect

In order to enhance the accuracy of springback simulation, the strength differential (SD) effect, i.e., the difference in flow stress between tension and compression, of a high strength cold-rolled steel sheet with a tensile strength of 980 MPa is measured by means of in-plane tension-compression test apparatus. Bi-axial stress tests are also performed to measure the contour of plastic work of the test material. From those experimental results, the material model which can consider the SD effect is determined. Furthermore, this material model is implemented into commercial FEM code by using user-subroutine function. To check the validity of this model and established FEM analysis system, curvature-hat crush forming experiment is performed. By comparing the experimental result and forming simulation result, the accuracy of the material model which can consider the SD effect is validated. Consequently it is concluded that the use of material model which is capable of reproducing the SD effect is a must to enhance the accuracy of springback simulation.

Study on the influence of orthotropy and tension–compression asymmetry of metal sheets in springback and formability predictions

Most yield criteria possess a point-symmetry with respect to its center, meaning that a stress state and its reverse state have the same absolute value. However, this can be an unrealistic approximation, even for cubic metals (both face-centered cubic and body center cubic)), which can present a small asymmetry between the yield stress in tension and compression, i.e. a strength differential (SD) effect. This work analyzes the influence of taking or not into account the SD effect in the modeling of the sheet orthotropic behavior in the numerical simulation of a cylindrical cup drawing process. The yield criterion adopted is the CPB06 [1], including its version with two linear transformations [2], allowing a better fitting of the experimental data available. The material analyzed presents a quite small tension-compression ratio of 0.963. However, this small SD effect leads to a slightly higher punch force during the bending dominated stage, resulting in a very small influence on the springback prediction. The influence on the thickness evolution during the process is negligible.