Strength Differential Effect Research Articles

Low temperature creep in a high strength roller bearing steel

Noticeable low temperature creep was established for a bainitic and a martensitic microstructure of the 100CrMnMo8 high strength roller bearing steel. The response revealed primary creep that differed between the microstructures, following a power law for martensite and the logarithmic description for bainite. The detected creep was pressure sensitive, higher in tension than in compression for the same stress level, following the strength differential effect (SDE) at material yielding. Two models were proposed where the stress variable for the pressure effect was based on the Drucker–Prager yield function and deviatoric creep strains were derived from a non-associated von Mises potential. Model parameters were determined from experimental series on the respective microstructure. When the models were evaluated against the experiments the accuracy was of the same order as the effects of different heat treatment batches and different load application rates. The importance of different material parameters in the descriptions was discussed.

Modeling of tension–compression asymmetry and orthotropy on metallic materials: Numerical implementation and validation

The details concerning the implementation of the yield criterion developed by Cazacu et al. 2006 (CPB06) [1], which accounts for both tension–compression asymmetry and orthotropy of the plastic flow, in the fully implicit FE solver DD3IMP (contraction of ‘Deep Drawing 3-D IMPlicit') are presented in this work. The implemented constitutive model is extensively described, including the analytical first and second order derivatives required to the stress update algorithm. A set of anisotropy parameters describing the mechanical behavior of two metallic materials at room temperature, namely Zirconium and AZ31-Mg alloy, are identified with the DD3MAT (contraction for ‘Deep Drawing 3-D MATerial’) in-house code (Alves, 2004) [2]. The anisotropy parameters are identified for both the CPB06 and the Cazacu and Barlat (2001) (CB2001) [3] yield criteria, in order to emphasize the importance and role of the strength differential effect. The results clearly show that the CPB06 yield criterion is able to accurately describe both the in-plane anisotropy and tension–compression asymmetry, as well a different anisotropic behavior in uniaxial tension and uniaxial compression. The numerical simulation of a four-point bending test is performed, considering different orientations of the beam, i.e. of the hard/soft to deform direction relatively to the load direction, allowing to validate the implementation. The results obtained with the CPB06 show its ability to describe with accuracy the strain fields in the beam's central cross-section, the distribution of the tensile and compressive layers and, consequently, the shift of the neutral layer. The comparison with the results obtained with CB2001 indicates that the strength differential effect affects the final deformed shape of the beam, particularly for materials exhibiting strong tension–compression asymmetry.

Constraints on the applicability range of pressure-sensitive yield/failure criteria: strong orthotropy or transverse isotropy

Yield/failure initiation criteria discussed in this paper account for the three following effects: the hydrostatic pressure dependence, the tension/compression asymmetry, and isotropic or anisotropic material response. For isotropic materials, criteria accounting for pressure/compression asymmetry (strength differential effect) have to include all three stress invariants (Iyer and Lissenden in Int J Plast 19:2055–2081, 2003; Gao et al. in Int J Plast 27:217–231, 2011; Yoon et al. in Int J Plast 56:184–202, 2014; Coulomb–Mohr’s, cf. Chen and Han in Plasticity for structural engineers. Springer, Berlin, 1995 criteria). In narrower case when only pressure sensitivity is accounted for, rotationally symmetric surfaces independent of the third invariant are considered and broadly discussed (Burzyński in study on strength hypotheses (in Polish). Akad Nauk Tech Lwów, 1928; Drucker and Prager in Q Appl Math 10:157–165, 1952 criteria). For anisotropic materials, the explicit formulation based on either all three common invariants (Goldenblat and Kopnov in Stroit Mekh 307–319, 1966; Kowalsky et al. in Comput Mater Sci 16:81–88, 1999) or first and second common invariants (extended von Mises–type Tsai–Wu’s criterion in Int J NumerMethods Eng 38:2083–2088, 1971) is addressed especially for the case of transverse isotropy, when difference between tetragonal versus hexagonal symmetry is highlighted. The classical Tsai and Wu criterion involves Hill’s type fourth-rank tensor inheriting a possibility of convexity loss in case of strong orthotropy, as discussed by Ganczarski and Skrzypek (Acta Mech 225:2563–2582, 2014). In order to overcome this defect, in the present paper the new Mises-based Tsai–Wu’s criterion is proposed and exemplary implemented for the columnar ice. A mixed way to formulate pressure-sensitive tension/compression asymmetric failure criteria-capable of describing fully distorted limit surfaces, which are based on both all stress invariants and the second common invariant (Khan and Liu in Int J Plast 38:14–26, 2012; Yoon et al. in Int J Plast 56:184–202, 2014), is revised and addressed to orthotropic materials for which the fourth-order linear transformation tensors are used to achieve extension of the isotropic criterion.

An advanced criterion based on non-AFR for anisotropic sheet metals

In the current research an advanced criterion with non-associated flow rule (non-AFR) for depicting the behavior of anisotropic sheet metals is presented to consider the strength differential effects (SDEs) for these materials. Owing to the fact that Lou et al. (2013) yield function is dependent on structure of an anisotropic material (BCC, FCC and HCP), an advanced yield function with inspiring of Yoon et al. (2014) yield function is proposed which is dependent upon anisotropic structures. Furthermore, to compute Lankford coefficients, a new pressure sensitive plastic potential function which would be dependent to anisotropic structure is presented and coupled with the proposed yield function with employing a non-AFR in a novel criterion which is called here \'advanced criterion\'. Totally eighteen experimental data are required to calibrate the criterion contained of directional tensile and compressive yield stresses for the yield function and directional Lankford coefficients for the plastic potential function. To verify the criterion, three anisotropic sheet metals with different structures are taken as case studies such as Al 2008-T4 (a BCC material), Al 2090-T3 (a FCC material) and AZ31 (a HCP material).

Numerical analysis of damage evolution for materials with tension-compression asymmetry

In recent decades, with the increasing attention to the carbon emission problem and shortage of energy, the development of lightweight metallic materials with Hexagonal closed packed (HCP) crystal structures, such as magnesium and titanium alloys, has become an important topic of research. The study and prediction of their mechanical behavior has become increasingly more important and has attracted growing interest in both academic and industrial communities. Metallic materials with a Hexagonal closed packed (HCP) crystallographic microstructure have an unconventional mechanical behavior including an anisotropic plastic response and a strength differential effect (SD) in tension and compression. This behavior poses considerable challenges that are intensified in the presence of microstructural damage processes.In this contribution, a fully coupled continuum damage model with elasto-plastic Cazacu’s orthotropic plasticity criterion has been implemented. The coupling between damaging and material behaviour is accounted for within the framework of Continuum Damage Mechanics (CDM). The closest point projection methods (CPPM) are used to implement the continuum damage constitutive model in an implicit quasi-static finite element environment to update stress and state variables. Finite element simulation of damage evolution and fracture initiation in ductile solids is investigated. The results obtained are compared against both numerical and experimental results available in the literature and good agreement is found between them.

Prediction of Forming Limit Diagrams for Materials with HCP Structure

The forming limit diagram (FLD) is an important tool to be used when characterizing the formability of metallic sheets used in metal forming processes. Experimental measurement and determination of the FLD is time-consuming and therefore the analytical prediction based on theory of plasticity and instability criteria allows a direct and efficient methodology to obtain critical values at different loading paths, thus carrying significant practical importance. However, the accuracy of the plastic instability prediction is strongly dependent on the choice of the material constitutive model [1–3]. Particularly for materials with hexagonal close packed (HCP) crystallographic structure, they have a very limited number of active slip systems at room temperature and demonstrate a strong asymmetry between yielding in tension and compression [4, 5]. Not only the magnitude of the yield locus changes, but also the shape of the yield surface is evolving during the plastic deformation [4]. Conventional phenomenological constitutive models of plasticity fail to capture this unconventional mechanical behavior [4, 6]. Cazacu and Plunkett [6] have proposed generic yield criteria, by using the transformed principal stress, to account for the initial plastic anisotropy and strength differential (SD) effect simultaneously. In this contribution, a generic FLD MATLAB script was developed based on Marciniak–Kuczynski analytical theory and applied to predict the localized necking. The influence of asymmetrical effect on the FLD was evaluated. Several yield functions such as von Mises, Hill, Barlat89, and Cazacu06 were incorporated into analysis. The paper also presents and discusses the influence of different hardening laws on the formability of materials with HCP crystal structures. The findings indicate that the plastic instability theory coupled with Cazacu model can adequately predict the onset of localized necking for HCP materials under different strain paths.

New high collapse model to calculate collapse strength for casing

With the increase of the ultra-deep well for oil and gas exploration and development, the conventional America Petroleum Institute (API) casing strength cannot meet the strength criteria of deep and ultra-deep wells. However, the high collapse casing (HCC) shows a higher resistance to collapse pressure so that it has obviously improved collapse properties in excess of API ratings. For limitation of the API model, the ISO collapse strength model with manufacturing imperfections has been proposed, which improves the casing strength calculation accuracy and increases the benefits for casing strength design, rather than just using API model. But the study on the ISO collapse model shows that it is inappropriate to predict the collapse strength of HCC of all sizes. Hence, a new high collapse model with manufacturing imperfections, strength differential (SD) effect, yield-to-tensile strength ratio, material hardening and intermediate principal stress effect, which can predict the collapse strength of HCC, has been presented based on unified strength theory. Numerical and experimental comparisons show that the calculation results of new high collapse model are much closer to the test data than that of API and ISO models, which will make great improvements in the casing design of deep and ultra-deep wells on the basis of HCC material safety.

Assessment of Damage and Anisotropic Plasticity Models to Predict Ti-6Al-4V Behavior

The plastic behavior of the Ti-6Al-4V alloy includes several features as strength differential effect, anisotropy and yield strength sensitivity to temperature and strain rate. Monotonic tensions in the three orthogonal directions of the material are performed to identify the Hill '48 yield criterion. Monotonic compression and plane strain tensile tests are also included in the experimental campaign to identify the orthotropic yield criterion of CPB06. An assessment of the two models is done by comparing the yield loci and the experimental data points for different levels of plastic work. A first approach of the damage modelling of the Ti-6AL-4V alloy is investigated with an extended Gurson-Tvergaard-Needleman damage model based on Hill '48 yield criterion. Finite element simulations of the experiments are performed and numerical results allows checking force-displacement curves until rupture and local information like displacement and strain fields. The prediction ability of the Hill '48, CPB and extended Gurson models are assessed on simple shear and notched tensile tests until fracture.

Prediction of Crashworthiness for Extruded Magnesium Materials

To assess the crashworthiness of simple wrought magnesium structures, the axial deformation behaviour of different square tubes produced from magnesium alloys AZ31 and ZE10 were numerically investigated under quasi-static compressive loading conditions. Finite-element simulations were conducted to predict and assess the plastic buckling and crush behaviour. The necessary data to determine parameters for the plastic potential were taken from compression tests conducted along different orientations. The yield function Hill48 was selected, despite its inability to capture the strength differential effect. The modelling approach pursued is justified by considering the mechanical loading conditions, the fabrication process of the profiles and its implication on strain anisotropy, balancing achievable accuracy and computational efforts. The simulation results revealed that the material work hardening rates evidenced in uniaxial compression tests influenced the buckling modes as well as the energy dissipation.

In-plane biaxial compression and tension testing of thin sheet materials

Detailed experimental data on the behavior of textured sheet metals under compressive loading is important to describe their tension–compression asymmetry. This is particularly needed for materials that exhibit a strength-differential effect, or in cases where the Bauschinger effect occurs. So far, there is no systematic work describing the third quadrant in the 2D stress space under biaxial compressive loading. This paper presents a new device for biaxial, compressive in-plane testing of thin sheets. Biaxial and uniaxial compression experiments are carried out in the strain controlled device, analyzing the behavior of deep drawing steel sheets with and without skin-pass treatment. Moreover, in order to allow for the experimental description of the yield surfaces, biaxial tensile tests are performed. Detailed numerical validations and experimental strain analysis both for the new specimen for biaxial compressive testing and for the cruciform specimen for biaxial tensile testing show that reasonably homogeneous strain distributions can be achieved. The combined experimental and numerical method presented here allows to evaluate the tension–compression asymmetry of thin sheet materials. The results for the skin-passed condition clearly exhibit a tension–compression asymmetry, which highlights the necessity of biaxial compression tests already in the as-received material condition. The biaxial compression test opens a pathway to a more detailed analysis of the flow behavior of thin sheets under biaxial compression loading.

An improved implicit numerical integration of a non‐associated, three‐invariant cap plasticity model with mixed isotropic–kinematic hardening for geomaterials

SummaryThe Sandia GeoModel is a continuum elastoplastic constitutive model that captures many features of the mechanical response for geological materials over a wide range of porosities and strain rates. Among the specific features incorporated into the formulation are a smooth compression cap, isotropic/kinematic hardening, nonlinear pressure dependence, strength differential effect, and rate sensitivity. This study attempts to provide enhancements regarding computational tractability, domain of applicability, and robustness of the model. A new functional form is presented for the yield and plastic potential functions. This reformulation renders a more accurate, robust, and efficient model as it eliminates spurious solutions attributed to the original form. In addition, we achieve a high‐performance implementation, because the local iterative method is allowed to recast residual vectors with a uniform dimensionality. The model is also furnished with a smooth, elliptical tension cap to account for the tensile yielding. Moreover, an efficient algorithm is introduced, which decreases the computational cost by differentiating the updated shear yield surface from the cap surface based on the trial relative stress state. Finally, various numerical examples including a large‐scale boundary value problem are presented to demonstrate the fidelity of the modified model and to analyze its numerical performance. Copyright © 2015 John Wiley & Sons, Ltd.

Correlation between strength differential effects in the plastic flow of the matrix and the rate of damage growth in porous polycrystals

In this paper, we show that there is a strong correlation between the strength differential (SD) effects in the plastic flow of the matrix, which arise from its dependence on the third stress invariant, void evolution, and ultimately the ductility of porous metallic polycrystals. For this purpose, detailed micromechanical finite-element analyses of three-dimensional unit cells are carried out. The plastic flow of the matrix is described by a criterion that accounts for strength-differential effects induced by shear deformation mechanisms of the constituent grains through a macroscopic parameter, k; only if there is no SD, k is zero, and the von Mises criterion is recovered. Numerical analyses are conducted for macroscopic proportional tensile loadings corresponding to fixed values of the stress triaxiality (ratio of the mean stress to the second stress invariant). It is shown that for the same macroscopic loading, the local plastic strains and the local stress distribution are strongly dependent on the sign of the parameter k. This in turn has a huge impact on damage accumulation, and ultimately affects the ductility of the porous polycrystals. Specifically, for axisymmetric loadings at third stress invariant positive, the rate of void growth is the slowest in the material with k negative, while the reverse holds true for equibiaxial tension (third stress invariant negative). Consequently, the ductility in axisymmetric tension at third-stress invariant positive is also markedly different from that in equibiaxial tension (third-stress invariant negative).

Micromechanical study of the dilatational response of porous solids with pressure-insensitive matrix displaying tension-compression asymmetry

In this paper, the dilatational response of porous solids with pressure-insensitive matrix displaying strength differential (SD) effects is investigated. To this end, micromechanical finite-element analyses of three-dimensional unit cells are carried out. The matrix behavior is governed by the isotropic form of Cazacu et al. (2006) criterion that accounts for SD effects through a parameter k. Simulation results are presented for axisymmetric tensile loadings corresponding to fixed values of the stress triaxiality for the two possible values of the Lode parameter, LP. For moderate and high stress triaxialities, it is shown that for materials for which the matrix tensile strength is larger than its compressive strength (k > 0), under tensile loadings corresponding at LP=1 the void growth rate is much faster than in the case of tensile loadings at LP=-1. The opposite holds true for materials with matrix tensile strength lower than its compressive strength (k< 0). This drastic difference in porosity evolution is explained by the distribution of the local plastic strain and stresses, which are markedly different than in a von Mises material (i.e. no SD effects of the matrix).

The strength differential effect in different heat treatment conditions of the steels 42CrMoS4 and 100Cr6

It is well known that a number of metals show different mechanical properties under tensile and compressive loading. In case of steel this asymmetry is called “Strength Differential Effect” (SD-effect). In this work, the steels 42CrMoS4 (1.7227, AISI 4140) and 100Cr6 (1.3505, AISI 52100) are investigated in different heat treatment conditions with high and low strengths as well as different microstructures. Both, tensile and compressive stress–strain curves are compared and evaluated. It was found that the SD-effect mainly occurs in high-strength quenched and tempered conditions. A bainite condition almost shows the absence of a SD-effect. At low-strength normalized conditions a SD-effect can be observed, too. In these cases, the different length of the Lüders strain is assumed to be the reason of the SD-effect.

Twinnability of hcp metals at the nanoscale

Twinning is generally considered to be the primary deformation mechanism for hexagonal close-packed (hcp) metals due to their limited slip systems. Recent microcompression experiments point to strong size effects indicating that pyramidal slips can dominate in deformation under compression. We present analysis on the twinnability of an ideal hcp single crystal at the nanoscale. A criterion for deformation twinning is derived by considering the elastic lattice-rotation strain, and the result tested against molecular dynamics simulations of magnesium and titanium single crystals. We find ⟨c + a⟩ pyramidal slip dominates the compression deformation at the nanoscale, which is consistent with experimental observations on microcompression. This analysis gives an interpretation of size effects in deformation twinning, at the same time it provides an explanation for the so-called strength differential effect.

Mechanical Testing of Thin Sheet Magnesium Alloys in Biaxial Tension and Uniaxial Compression

Tension and compression experiments on magnesium rolled sheets and extruded products of AZ31 (Mg + 3%Al + 1%Zn) and ZE10 (Mg + 1%Zn + 0.3%Ce based misc metal) were performed at room temperature. The tests were conducted along the longitudinal and the transverse direction to quantify the in-plane anisotropy. Samples built from adhesively-bonded layers of sheets were used for in-plane as well as through-thickness compression testing. It was verified that this simple testing method leads to identical results as using comb-like dies and equi-biaxial bulge testing, respectively. In the case of uniaxial loading, the longitudinal and transverse strain components were measured using independent extensometers. R-values were calculated from these signals. The mechanical responses were correlated to the microstructure and the texture. The recorded differences between tensile and compressive response reveal the strength differential effect of the materials. The distortional character of the plastic behaviour is evidenced through their responses to equi-biaxial tensile loading. Significant differences in the compressive responses of the two alloys were identified by comparing the respective hardening rates.

Modeling the tension–compression asymmetric yield behavior of β-treated Zircaloy-4

Zirconium alloys such as Zircaloy-4 are used in nuclear applications due to adequate strength, ductility and resistance to radiation damage. Recent modeling efforts have focused on improvements to the predicted elastic–plastic response, complicated by the strong strength-differential (S-D) effects in HCP materials. This study develops a pressure-insensitive, continuum plasticity model, dependent on the second and third invariants of the stress deviator (J2 and J3), with an internal variable related to the plastic strain to describe the tension–compression asymmetry of a β-treated Zircaloy-4. Plastic deformation drives isotropic and distortional hardening of the non-Mises yield surface. The proposed plasticity model has been calibrated and validated using measured results from an experimental test program. Results show that the proposed model captures the complex elastic–plastic response observed in measured load–displacement and torque–rotation curves over a range of triaxiality and Lode parameter values.

Plastic deformation of high-purity α-titanium: Model development and validation using the Taylor cylinder impact test

In this paper, results of an experimental study on the quasi-static and high-rate plastic deformation due to impact of a high-purity, polycrystalline, α-titanium material are presented. It was found that the material is transversely isotropic and displays strong strength differential effects. Split Hopkinson Pressure Bar tests in tension and compression and Taylor impact tests were conducted. For an impact velocity of 196m/s, plastic deformation extended to 64% of the length of the deformed specimen, with little radial spreading. A three-dimensional constitutive model was developed. Key in the formulation was the use of a macroscopic yield function that incorporates the specificities of the plastic flow, namely the combined effects of anisotropy and tension–compression asymmetry. Comparison between model predictions and data show the capabilities of the model to describe with accuracy the plastic behavior of the α-titanium material for both quasi-static and high-rate loadings. In particular, the three-dimensional simulations of the Taylor impact test show a very good agreement with data, both the post-test major and minor side profiles and impact interface footprints are very well described.

Impact of anisotropy and viscosity to model the mechanical behavior of Ti–6Al–4V alloy

This paper compares the predictions of an isotropic thermo-elasto-viscoplastic approach and of an anisotropic thermo-elastoplastic one with experimental results representative of the mechanical behavior of Ti–6Al–4V at moderate temperatures and low strain rates. The first model is the well known Norton–Hoff viscoplastic constitutive law with isotropic von Mises yield locus identified by using monotonic tension tests performed at strain rates from 10−3s−1 to 10−1s−1 and at temperatures up to 400°C. The second model is a thermo-elasto-plastic one defined by the orthotropic yield criterion CPB06. It takes into account the anisotropy and the strength differential (SD) effect in tension–compression of Ti–6Al–4V at RT, 150°C and 400°C. The identification of the SD effect is done by using tension and compression tests and the anisotropy behavior is identified by using shear, plane strain, tension and compression tests performed in three orthogonal material directions. The accuracy of the load and displacement predictions of the two macroscopic constitutive models are compared to experimental results obtained from tests performed on specimens with multiaxial loadings and large strain at several temperatures.

Asymmetric yield function based on the stress invariants for pressure sensitive metals

A general asymmetric yield function is proposed with dependence on the stress invariants for pressure sensitive metals. The pressure sensitivity of the proposed yield function is consistent with the experimental result of Spitzig and Richmond (1984) for steel and aluminum alloys while the asymmetry of the third invariant is preserved to model strength differential (SD) effect of pressure insensitive materials. The proposed yield function is transformed in the space of the stress triaxaility, the von Mises stress and the normalized invariant to theoretically investigate the possible reason of the SD effect. The proposed plasticity model is further extended to characterize the anisotropic behavior of metals both in tension and compression. The extension of the yield function is realized by introducing two distinct fourth-order linear transformation tensors of the stress tensor for the second and third invariants, respectively. The extended yield function reasonably models the evolution of yield surfaces for a zirconium clock-rolled plate during in-plane and through-thickness compression reported by Plunkett et al. (2007). The extended yield function is also applied to describe the orthotropic behavior of a face-centered cubic metal of AA 2008-T4 and two hexagonal close-packed metals of high-purity α-titanium and AZ31 magnesium alloy. The orthotropic behavior predicted by the generalized model is compared with experimental results of these metals. The comparison validates that the proposed yield function provides sufficient predictability on SD effect and anisotropic behavior both in tension and compression. When it is necessary to consider r-value anisotropy, the proposed function is efficient to be used with non-associated flow plasticity by introducing a separate plastic potential for the consideration of r-values as shown in Stoughton and Yoon (2004, 2009).