Non-proportional Strain Paths Research Articles

Compression and Fatigue Testing of High-Strength Thin Metal Sheets by Using an Anti-Buckling Device

BackgroundThe modelling of the sheet metal forming operations requires accurate and precise data of the material plastic behaviour along non-proportional strain paths. However, the buckling phenomenon severely limits the compressive strain range that could be used to deform thin metal sheets.ObjectiveThe main aim of this paper was to propose an effective device, that enables to determine of accurate stress-strain characteristics of thin metal sheet specimens subjected to axial deformation without buckling and with a special emphasis on friction correction.MethodsIn this paper, an anti-buckling fixture was proposed to assess the deformation characteristics of X10CrMoVNb9-1 (P91) power engineering steel, and DP500 and DP980 dual-phase steels, under compression loading. The fixture enables monitoring of the friction between the specimen and supporting blocks during the test, and thus the precise stress response of the material could be determined.ResultsThe effectiveness of the fixture was evaluated under tension–compression cyclic loading and during the compression tests in which high-strength thin metal sheets were successfully deformed up to 10% without specimen buckling. Furthermore, the successful determination of a friction force variation between supporting blocks and the specimen during tests enabled to determine an actual force acting on the specimen.ConclusionsThe proposed testing fixture was successfully assessed during the compression and cyclic tension–compression of high-strength thin metal sheets as no buckling was observed. Its advantage lies in adapting to change its length with specimen elongation or shrinkage during a test. The friction force generated from a movement of both parts of the device could be effectively monitored by the special strain gauge system during testing and thus its impact on the stress-strain characteristics could be successfully eliminated.

An improved strain path dependent model under multiaxial cyclic loading for simulating material response of low C-Mn steel

In this work, a prominent cyclic plasticity model based on modified AbdelKarim-Ohno kinematic hardening rule and Calloch isotropic hardening is implemented and validated w.r.t. test results of low C-Mn steel. In reference to the shortcomings of the model, an improved model based on modified Ohno-Wang and Tanaka non-proportionality parameter is introduced. Since the precise assessment of stress–strain response is an essential precondition for carrying out fatigue life analysis, therefore hysteresis loop responses under various multiaxial LCF loading having different strain paths are compared. The simulation result with the proposed model resulted in improved assessment for non-proportional strain paths.

Experimental and Numerical Analysis of Prestrain on the Formability of Zn-Cu-Ti Alloy Zinc Sheet

The forming limit diagrams (FLDs) characterizing the formability of sheet metals are usually obtained by applying proportional loadings. Nevertheless, the industrial processes involve strain path changes that can modify the limit-strain values. In addition, for strongly anisotropic sheet metals such as the Zn-Cu-Ti zinc alloy, large differences in forming limit curves (FLCs) with respect to the sheet rolling direction are observed. In the present work, the analysis of the effect of bilinear strain paths on the FLC is addressed by both experimental measurements and numerical simulations. For this purpose, a miniature testing device was used that allows evaluation of the influence of strain path changes on the limit strain on samples at 0°, 45° and 90° with respect to the sheet rolling direction cut from non-standard large samples previously subjected to a prestrain along the RD up to an early deformation of ~0.12. Numerical simulations were carried out using the well-known Marciniak and Kuczynski (MK) theory in conjunction with the viscoplastic self-consistent (VPSC) crystal plasticity model. In order to account for the grain fragmentation process due to the continuous dynamic recrystallization (CDRX) mechanism, an ad hoc short-range interaction effect (SRE) model was included in the simulations. Additionally, the measured and simulated texture evolution of Zn-Cu-Ti alloy sheets at the different stages of the deformations were shown. The capacity of the MK-VPSC-SRE model was validated, and the limitations to simulating the texture development, flow stress and forming limit curves, including a non-proportional strain path, were discussed.

A new method for the toughness assessment of mobile crane components based on damage mechanics

The characterization of toughness properties in standard Charpy or fracture mechanics tests calls for thickness requirements to be met. Therefore, the characterization of toughness properties is a problem for thin-walled structures. Replacing Charpy impact toughness testing by impact notch tensile testing can solve this problem. However, the toughness requirements are still expressed in terms of standard test results. Therefore, a framework is proposed here for translating these standard test requirements into impact notch tensile test requirements. The proposed framework relies on numerical simulations with a phenomenological damage mechanics model, which uses state-of-stress-dependent, strain-based criteria for the prediction of local damage and global fracture. This model takes the effects of non-proportional strain paths into account and applies different criteria for cleavage and ductile fracture in order to predict correctly the activation of cleavage and ductile fracture mechanisms in the corresponding numerical simulations.

Investigation of non-proportional load paths by using a cruciform specimen in a conventional Nakajima test

The prediction of necking in sheet-metal forming is one of the most important tasks in the simulation of forming processes. The most common way to predict the necking for linear load paths is the Forming Limit Diagram (FLD) according to ISO 12004-2. This criterion is valid only for linear load paths. For non-proportional loading paths, Volk introduced a Generalized Forming Limit Concept (GFLC). This model enables the prediction of the formability for any non-proportional loading paths. The experimental investigation of those loading paths is still very challenging and complex. It involves a wide range of testing equipment for the pre- and post-strain of specimens. Jocham introduced a new method that allows the investigation of various non-proportional loading paths by using a cruciform specimen and a draw bead tool in a conventional Nakajima test setup. The height of the draw beads can be varied to create different load paths in the specimen centre. Nevertheless it is necessary for the thinnest part of the specimen to be in the middle of the specimen. This is achieved by milling an indentation into the centre. This mechanical procedure affects the formability of the material. In this paper, the cruciform specimens are laser weld from three single metal sheets in order to minimize the influence of the manufacturing process on the specimen. With this procedure, it is possible to test unmanufactured sheets. However, the crack still occurs in the centre of the specimen. With the new cruciform specimen, we are able to create arbitrary non-proportional strain paths.

Loading path dependent distortional hardening of Mg alloys: Experimental investigation and constitutive modeling on cruciform specimens

The forming process of Mg alloys is strongly affected by the anisotropic mechanical behavior due to crystallographic texture. Traditionally, the anisotropic behavior is captured by uni-directional loadings along different directions (e.g. Rolling Direction, RD, Transverse Direction, TD and 45∘ from RD directions). However, two more important factors have been ignored. Firstly, the subsequent deformation behavior and plastic flow of metals are determined by the whole evolution of mechanical behavior along all dimensions (i.e. σxx, σyy, σzz, σxy, σyz and σzx) instead of uni-directional mechanical property. Secondly, since the metal forming process is under multi-axial instead of uni-directional loading, the anisotropic plastic flow behavior of metals under multi-axial loading are crucial to the forming technology. Clearly both factors cannot be clarified by traditional uni-directional loading tests. In line with the state of the art progress in Elasticity and Plasticity theory at finite strain, the aforementioned factors can be clarified by the anisotropic evolution of yield surfaces, i.e., distortional hardening. For wrought Mg alloys, the anisotropic evolution of yield surfaces in σxx and σxy space for extruded bar of Mg alloys has been clarified in [1]. As a series work, the evolution of yield surfaces of AZ31 rolled sheet in σxx and σyy space under uniaxial tension was probed experimentally by means of compressive and biaxial tensile tests with specifically designed non-proportional strain paths. The anisotropic evolution of yield surface is observed under uniaxial tension, and the underlying deformation mechanism is discussed based on microstructure and texture analysis. It is found that the yield surface evolution is ascribed to dislocation hardening and texture rotation induced by basal slip with biaxial in-plane tension. A thermodynamically consistent constitutive model at finite strain is employed to capture the anisotropic behavior. The anisotropic evolution of yield surfaces of AZ31 rolled sheet under uniaxial loading along RD and TD directions are predicted and validated after model parameters identification.

Springback Prediction of Aluminum Alloy Sheet under Changing Loading Paths with Consideration of the Influence of Kinematic Hardening and Ductile Damage

Springback prediction of sheet metal forming is always an important issue in the industry, because it greatly affects the final shape of the product. The accuracy of simulation prediction depends on not only the forming condition but also the chosen material model, which determines the stress and strain redistributions in the formed parts. In this paper, a newly proposed elastoplastic constitutive model is used, in which the initial and induced anisotropies, combined nonlinear isotropic and kinematic hardenings, as well as isotropic ductile damage, are taken into account. The aluminum alloy sheet metal AA7055 was chosen as the studied material. In order to investigate springback under non-proportional strain paths, three-point bending tests were conducted with pre-strained specimens, and five different pre-straining states were considered. The comparisons between numerical and experimental results highlighted the hard effect of both kinematic hardening and ductile damage on the springback prediction, especially for a changed loading path case.

A Multiscale Model to Incorporate Texture Evolution into Phenomenological Plasticity Models

Crystal plasticity is a micromechanics-based model that is regularly used to simulate plastic spin during large deformation. Although crystal plasticity can provide an accurate description of local deformation behaviour, it is often computationally expensive and usually replaced by flow rule-based phenomenological models that do not capture this phenomenon. This work presents a phenomenological-based texture evolution (PBTE) model that allows for the enhancement of flow rule-based models to capture microstructural spin in a phenomenological manner. A numerical framework is presented for generating and calibrating the microstructural evolution for the PBTE model using crystal plasticity. The PBTE Model is calibrated and employed to predict the macroscopic mechanical response and the generated microstructural spin for single crystal FCC cube during non-proportional strain paths.

Modeling of stable cyclic stress‐strain responses under non‐proportional loading

AbstractThis paper proposes a simple two‐surface plasticity model for strain controlled conditions which includes only five material constants to describe the stable stress‐strain relationship under extensive non‐proportional loadings. In this model, the Mróz kinematic hardening rule is adopted to describe the instantaneous translation of the yield surface, and a larger radius of the limit surface is selected to avoid overlapping of the one with the limit surface. A non‐proportionality measure of strain path on the basis of the minimum normal strain range is proposed. Procedures to determine the minimum normal strain range are also presented for general multiaxial loadings. The proposed two‐surface model is relatively convenient to be used since it is not overly sensitive to the modeling parameters. Four kinds of material (type 304 stainless steel, S460N steel, 30CrNiMo8HH steel, and 1%CrMoV steel) tested under strain controlled conditions are used to validate the reliability of the proposed model. Comparisons between test results and model predictions under 19 types of nonproportional strain paths show that the proposed model predicts relatively accurate stress responses under non‐proportional loading paths by introducing non‐proportional cyclic hardening coefficient and non‐proportionality factor into the developed evolution equations.

Applying a new constitutive model to analyse the springback behaviour of titanium in bending and roll forming

This paper is an extension to previous work that dealt with the development of a strain path dependent constitutive model to describe the inelastic behaviour of Ti-6Al-4V at room temperature based on the homogenous yield function combined with the anisotropic hardening characteristics, the so-called HAH model. The present work is to apply and verify the accuracy of the proposed model in the finite element (FE) analysis of springback in bending dominated forming processes such as the V-die bending and the roll forming process. In addition, the model is applied to develop a greater insight into the nature of springback in the roll forming process where springback is generally lower compared to that found for simple bending. For this the hardening characteristics of Ti-6Al-4V were identified applying an inverse analysis approach in Abaqus Standard and the model used to describe the evolution of the anisotropic yield surface during non-proportional strain path deformation; this included the cyclic pure bending and cyclic tension–compression tests to generate experimental target curves. The constitutive model parameters were optimised to capture the cyclic behaviour of Ti-6Al-4V in both the pure bending and tension—compression and incorporated into the numerical models of a V-die bending test and a roll forming procedure to analyse springback.The model achieved good agreement with experimental results and reproduced the lower springback observed in roll forming compared to simple bending. In contrast to this a conventional isotropic hardening model significantly overestimated springback for the roll forming process. This suggests that the lower level of springback observed in roll forming compared to V-die bending may be due to kinematic hardening effects. In addition, a lower level of accumulated effective stress and the presence of redundant shear strain was numerically observed in the bending regions of the roll formed section which also may have contributed to the reduction of springback. The results of this study suggest that for the accurate numerical prediction of springback in roll forming kinematic hardening effects need to be accounted for. For this the study presents an effective numerical approach and prove its applicability by numerical roll forming and V-die bending studies performed on high strength Ti-6Al-4V at room temperature and verified with experimental results.

Distortional hardening concept for modeling anisotropic/asymmetric plastic behavior of AZ31B magnesium alloy sheets

A continuum-based approach for describing the plastic hardening behavior of magnesium alloy sheets subjected to non-proportional strain path changes at room temperature is presented. The constitutive model is based on an anisotropic distortional yield function that combines a stable component and a fluctuating component. The stable component initiates the yield criterion that characterizes the typical strength differential (SD) effect between tension and compression in magnesium alloys at room temperature. The evolution of the fluctuating component is reformulated based on its cubic metal counterpart to represent the deformation behavior of magnesium alloys, which consists of slip- and twinning-dominant deformation modes. To validate the proposed model, a predictor–corrector numerical algorithm was implemented in a finite element (FE) program by using a user material subroutine, and its numerical accuracy was evaluated by iso-error map analysis. Comparison of the experimental and simulated results obtained for various loading path changes showed that, although the model was not formulated using the kinematic hardening concept, the model could reproduce complex features of the stress–strain responses of an AZ31B magnesium alloy sheet under non-proportional loading paths, i.e., asymmetric hardening behavior under tension and compression, the sigmoidal nature of the hardening curve during monotonic compression and compression followed by tension.

Creep-fatigue life evaluation of high chromium ferritic steel under non-proportional loading

T Multiaxial creep-fatigue tests under non-proportional loading conditions with various strain rates were carried out using a hollow cylinder specimen of a high chromium ferritic steel at 823K in air to discuss the influence of non-proportional loading on failure life. Strain paths employed were a push-pull loading and a circle loading. The push-pull loading test is proportional strain path test. The circle loading test is non-proportional strain path test in which sinusoidal waveforms of axial and shear strains have 90 degree phase difference. The failure life is affected largely by the strain rate and the non-proportional loading. This paper presents a modified strain range for life evaluation considering the strain rate based on a non-proportional strain parameter proposed by authors. The strain range is a suitable parameter for life evaluation of tested material under non-proportional loading at high temperature. KEYWORDS. Life evaluation; Creep-fatigue; Multiaxial strain; Nonproportional loading; High chromium ferritic steel.

Constitutive modelling of magnesium alloy sheets under strain path changes

In this study, a continuum-based approach for the description of the plastic hardening behavior of magnesium alloy sheets subjected to non-proportional strain path changes is discussed. The constitutive model is based on an anisotropic distortional yield function combining a stable component and a fluctuating component. The stable component initiates the yield criterion that characterizes the typical strength differential between tension and compression in magnesium alloys at room temperature. The evolution of the fluctuating component is reformulated based on its cubic metal counterpart to represent the deformation nature of magnesium alloys that consist of slip and twin dominant modes. The model is not formulated with a kinematic hardening rule, but it reasonably reproduces complex features of the stress-strain responses under the load reversal in magnesium alloy sheet: i.e., asymmetric hardening behavior under tension and compression, sigmoidal nature of hardening curve during monotonic compression and compression followed by tension, strong anisotropy etc.

Non-proportional hardening models for predicting mean and peak stress evolution in multiaxial fatigue using Tanaka’s incremental plasticity concepts

Non-proportional (NP) hardening can have a significant effect on multiaxial fatigue lives due to the increase in mean and peak stresses it can cause when compared to proportional loads of similar range. Its effect depends both on the material and on the load history path, quantified respectively by the additional NP hardening coefficient αNP and the NP factor FNP. However, since FNP depends on the load path, it is not easy to evaluate under complex multiaxial loading conditions. So, several estimates for the steady-state value of FNP have been proposed to deal with such cases, based on enclosing ellipses or on integrals calculated along the stress, strain, or plastic strain path. Tanaka’s incremental plasticity model, on the other hand, is able to predict the NP hardening evolution as a function of the accumulated plastic strain p along any load history, based on the concept of a polarization tensor [PT] that can be correlated to an internal dislocation structure. In this work, it is shown that the eigenvectors of [PT] represent principal directions along which dislocation structures may form, whose intensity is quantified by the respective eigenvalues. Two new integral estimates for the steady-state FNP of periodic load histories are proposed, one that exactly reproduces the incremental plasticity predictions from Tanaka’s model, and a simpler version calculated solely as a function of the eigenvalues of [PT]. Both can be readily used in fatigue design to correctly account for the additional mean or peak stress effects induced by NP periodic histories. For general non-periodic multiaxial histories, it is also shown that Tanaka’s model can be expressed as an evolution equation for FNP(p). Tension–torsion experiments with 316L steel tubular specimens under especially selected discriminating square NP strain paths are conducted to validate the proposed models.

Evaluation and visualization of multiaxial stress and strain states under non-proportional loading

This paper presents a simple method of determining stresses and strains and a severity of loading history under non-proportional loadings with defining the rotation angles of the maximum principal stress and strain in a three dimensional stress and strain space. Based on the method, non-proportional stress and strain ranges are derived and applicability of the range to the life evaluation for low carbon steel under non-proportional random strain paths are discussed. The strain range taking account of intensity of loading path reducing life can be suitable parameter for multiaxial fatigue life evaluation under multiaxial non-proportional loading. This paper also shows a method of visually presenting the principal stress and strain that assists designers to understand the loading mode whether proportional or non-proportional under 3 dimensional (6 stress/strain components) multiaxial loading.

Characterization of the Shear Stress State Under Non-Proportional Strain Paths Realized by Biaxial Stretching in the Marciniak Test

Complex parts with sharp corners are on the rise to satisfy increasing customer demands on the design. To realize these parts in single or multistage forming processes the information of the material behavior under linear and non-linear strain paths is essential. As Bauschinger already showed for the material behavior under cyclic loading, a reduction of the subsequent beginning of plastic yielding can be observed. Different research works studied the Bauschinger effect under cyclic proportional loading or non-proportional strain paths mostly with uniaxial pre-straining of the material. In this paper the material behavior under shearing after previous biaxial stretching in the Marciniak test is analyzed for the aluminum alloy AA5182 and the advanced high strength steel DP600. The Marciniak test is suited for biaxial pre-straining due to an almost almost equibiaxial strain state and a homogeneous strain distribution in-plane on the bottom of the specimen up to higher strains than in a biaxial tensile test. The subsequent shear test is realized with a modified shear test specimen according to the ASTM standard. Finally, the influence of biaxal pre-straining on the beginning of plastic yielding under shear condition, the hardening behavior and the maximum reachable shear strain is investigated for a better understanding of the material behavior under non-proportional loading paths.

Compression deformation behaviors of sheet metals at various clearances and side forces

Modeling sheet metal forming operations requires understanding of plastic behaviors of sheet metals along non-proportional strain paths. The plastic behavior under reversed uniaxial loading is of particular interest because of its simplicity of interpretation and its application to material elements drawn over a die radius and underwent repeated bending. However, the attainable strain is limited by failures, such as buckling and in-plane deformation, dependent on clearances and side forces. In this study, a finite element (FE) model was established for the compression process of sheet specimens, to probe the deformation behavior. The results show that: With the decrease of the clearance from a very large value to a very small value, four defects modes, including plastic t-buckling, micro-bending, w-buckling, and in-plane compression deformation will occur. With the increase of the side force from a very small value to a very large value, plastic t-buckling, w-buckling, uniform deformation, and in-plane compression will occur. The difference in deformation behaviors under these two parameters indicates that the successful compression process without failures for sheet specimens only can be carried out under a reasonable side force.

Cyclic Test of DP600 Steel under Tension-Compression Load for Different Pre-Strain Levels

Modeling sheet metal forming operations requires understanding of the plastic behavior of the sheet metal along the non-proportional strain paths. Measurement of hardening under reversed uniaxial loading is because of its simplicity very effective mechanical test to achieve several important features of material behavior. With the reversed uniaxial loading can be examined features as Bauschinger effect, work hardening stagnation, permanent softening etc., which are necessary to defined kinematic hardening in the numerical simulations. The biggest problem of uniaxial reversed loading is buckling of the sheet metal during the compression phase. In this article is described development of the simple fixture, used for the reversed uniaxial loading and results of tension-compression test for the steel DP 600 in various pre-strain levels are specified.

High temperature nonproportional low cycle fatigue using fifteen loading paths

This paper discusses effects of nonproportional loading on low cycle fatigue lives for Type 304 stainless steel hollow cylinder specimens at 923K. Strain controlled axial–torsion low cycle fatigue tests were performed using fifteen proportional and nonproportional strain paths at a strain rate of 0.1%/s. Nonproportional straining significantly reduced fatigue lives of the steel at 923K. Reduction in fatigue life due to nonproportional straining reached about 80% compared with the proportional straining in specific strain paths. The nonproportional strains, ΔɛNP and ΔεNP∗, and energy parameter, ΔσIΔɛI, successfully correlated proportional and nonproportional fatigue lives but ASME equivalent strain gave significantly unconservative estimates.

Strain hardening response and modeling of EDDQ and DP780 steel sheet under non-linear strain path

The anisotropic hardening behaviors of dual phase (DP780) and extra deep drawing quality (EDDQ) steel sheets under non-proportional strain paths were investigated. Two-step uniaxial tension tests, which consisted of the first loading in the rolling (RD) or transverse (TD) and the second loading in every 15° from the first loading axis, were conducted. For DP780 steel, a significant Bauschinger effect accompanied by a transient hardening behavior after reverse loading was the prominent phenomenon. In contrast, EDDQ exhibited stress overshooting followed by strain hardening stagnation with respect to the monotonic flow curve near cross-loading conditions. The extended HAH model combined with the Yld2000-2d yield function were used to reproduced the anisotropic hardening behavior of the two materials. For DP780, the extended HAH model could capture the Bauschinger effect and transient hardening behavior well for tension reloading at 0°, 15°, 75° and 90° from the RD or TD prestraining direction. However, the predictions at 30°, 45° and 60° were slightly different from the experiments. For EDDQ, this approach reproduced the strain hardening anisotropy well including flow stress overshooting followed by a stage of strain hardening stagnation.