In-plane Torsion Research Articles

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

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

Reaching Large Strains During Simple Shear Experiments Thanks to Sequential Re-Machining of the Free Edges

BackgroundThe simple shear experiment is widely used for the calibration of plasticity models due to straightforward post processing. The specimen can be as simple as a rectangular strip of sheet metal, but the maximum strain is limited by early initiation of fractures from the free edges. Avoiding this drawback has been a major motivation for the development of new specimens with optimized edge geometries or the in-plane torsion test, but at the cost of a more complex analysis of the test and often a reduction of the gauge section.ObjectiveThe objective of the present work is to overcome the initiation of fracture from the free edges during simple shear experiments. Our goal is to double the achievable maximum strain, while keeping the size of the specimen and the post processing simplicity of a standard simple shear test.MethodsA sequential single shear test is proposed, consisting of several two steps sequences on a notched geometry. First, an interrupted shear test is performed up to a specified displacement value. Then, the damaged free edges of the specimen are removed through milling. The specimen is then ready for the following sequence of shear and re-machining.ResultsExperiments are performed on three engineering materials, with up to five loading-machining sequences. The maximum attained effective strain is up to two times the one reached during a monotonic experiment. Numerical simulations are used to validate the shear stress and strain calculations from experimental measurements. Practical recommendations are derived for the choice of the displacement step size and Digital Image Correlation analysis.ConclusionIt is found that the maximum strain attained before the undesired failure of the specimen during simple shear test can be substantially extended through repeated re-machining of the specimen boundaries, enabling behavior identification at larger strains.

Non-associated and Non-quadratic Characteristics in Plastic Anisotropy of Automotive Lightweight Sheet Metals

AbstractLightweight sheet metals are highly desirable for automotive applications due to their exceptional strength-to-density ratio. An accurate description of the pronounced plastic anisotropy exhibited by these materials in finite element analysis requires advanced plasticity models. In recent years, significant efforts have been devoted to developing plasticity models and numerical analysis methods based on the non-associated flow rule (non-AFR). In this work, a newly proposed coupled quadratic and non-quadratic model under non-AFR is utilized to comprehensively investigate the non-associated and non-quadratic characteristics during the yielding of three lightweight sheet metals, i.e., dual-phase steel DP980, TRIP-assisted steel QP980, and aluminum alloy AA5754-O. These materials are subjected to various proportional loading paths, including uniaxial tensile tests with a 15° increment, uniaxial compressive tests with a 45° increment, in-plane torsion tests, and biaxial tensile tests using laser-deposited arm-strengthened cruciform specimens. Results show that the non-AFR approach provides an effective means for accurately modeling the yield behavior, including yield stresses and the direction of plastic strain rates, simultaneously, utilizing two separate functions and a simple calibration procedure. The introduction of the non-quadratic plastic potential reduces the average errors in angle when predicting plastic strain directions by the quadratic plastic potential function. Specifically, for DP980, the average error is reduced from 3.1° to 0.9°, for QP980 it is reduced from 6.1° to 3.9°, and for AA5754-O it is reduced from 7.0° to 0.2°. This highlights the importance of considering the non-quadratic characteristic in plasticity modeling, especially for aluminum alloys such as AA5754-O.

Influence of mechanical characterization on the prediction of necking issues during sheet flow forming process

Flow forming is an incremental sheet forming (ISF) process during which a sheet metal is compressed and stretched multiple times by means of one or multiple rotating roller tools. The local tool-workpiece contact zone evolves during the entire process. The necking phenomenon, which corresponds to an uncontrolled thinning of the part wall, is introduced. This phenomenon represents a major issue for ISF processes. A review of the state-of-the-art about ISF processes shows that most studies do not consider the loading path complexity when choosing the mechanical characterization test and its associated constitutive model. Besides, the prediction of necking occurring during sheet flow forming is poorly studied in the literature. In this paper, a finite element analysis (FEA) using the FORGE® software enables a detailed understanding of the loading path (strain and stress states) prevailing during the flow forming operation. Based on the peculiarities of this loading path, different mechanical tests associated with adequate constitutive models are chosen to characterize the material behavior. The ability of each constitutive model used within the FE approach to predict necking is then assessed. Results show that the best prediction of a geometry exhibiting necking issues is obtained with the cyclic in-plane torsion test (ITT) associated with its calibrated isotropic – kinematic hardening model. These results suggest that the behavior characterization under cyclic shear loadings is relevant. Using a simple tensile test with associated power-law provides a faster and conservative necking prediction.

Shear property measurement of additively manufactured continuous fibre reinforced plastics by in-plane torsion testing

In-plane shear properties are essential for calculating the plane stress state of composites. Test results for in-plane shear properties of fibre-reinforced polymers (FRPs) are generally obtained with methods that generate multiaxial stress states, and no simple shear response is measured. In this paper, the in plane torsion test method is applied to additively manufactured (AM) continuous FRPs. With this method, planar simple shear loading is achieved for a dedicated, spiral lay-up of the AM specimen. Using digital image correlation, surface strains were obtained for an observed section of the specimen, and shear stress–strain curves were extracted for fibre bundles, co-extruded thermoplastic material, and the composite of both. Shear stiffness and shear strength values are yielded for representative surface areas. Furthermore, the fracture behaviour in shear is discussed. The co-extruded thermoplastic filler material is shown to be the phase where cracks progress is most likely, however as some fibre bundles are split, these can be bridged by the crack. In conclusion, the method is well suited for determination of shear properties of AM continuous FRP material, giving consistent results for the test sample.

Analytical model of the in-plane torsion test

In research and industry, the in-plane torsion test is applied to investigate the material behaviour at large plastic strains: a sheet is clamped in two concentric circles, the boundaries are twisted against each other applying a torque, and simple shear of the material arises. This deformation is analysed within the scope of finite elasto-plasticity. An additive decomposition of the Almansi strain tensor is derived, valid as an approximation for arbitrary large plastic strains and sufficiently small elastic strains and rotations. Constitutive assumptions are the von Mises yield criterion, an associative flow rule, isotropic hardening, and a physically linear elasticity relation. The incremental formulation of the elasticity relation applies covariant Oldroyd derivatives of the stress and the strain tensors. The assumptions combined with equilibrium conditions lead to evolution equations for the distribution of stresses and accumulated plastic strain. The nonzero circumferential stress must be determined from the equilibrium condition because no deformation is present in tangential direction. As a result, a differential-algebraic-equation (DAE) system is derived, consisting of three ordinary differential equations combined with one algebraic side condition. As an example material, properties of a dual phase steel DP600 are analysed numerically at an accumulated plastic strain of 3.0. Radial normal stresses of 3.1% and tangential normal stresses of 1.0% of the shear stresses are determined. The influence of the additional normal stresses on the determination of the flow curve is 0.024%, which is negligibly small in comparison with other experimental influences and measurement accuracies affecting the experimental flow curve determination.

Design of in-plane torsion experiment to characterize anisotropic plasticity and fracture under simple shear

A novel experimental set-up is proposed to perform in-plane torsion experiments on grooved disk specimens. As compared to previously proposed systems, an enhanced clamping system is introduced to enable the torsion testing of high strength materials such as dual phase steels. The clamping is achieved without applying the clamping pressure through an axial actuator. Instead a custom-made washer-nut system is used which allows for the monitoring of the entire specimen front surface with cameras. Experiments are performed on three different steels (DC01, DP980, AISI301) to demonstrate the validity of the technique for measuring the stress–strain curve for strains of up to 2, for strain rate jump testing, for cyclic loading and for fracture testing. In particular, the demonstration experiments elucidate the importance of full field strain measurements during in-plane torsion. The strain fields vary along the gage section circumference in a periodic manner that is related to the anisotropy of the tested material. This important observation is confirmed through a finite element study covering a wide range of anisotropic Hill’48 materials with non-associated plastic flow.

Deformation mechanism and microstructure evolution of medium-Mn AHSS under various loading conditions

Third-generation advanced high-strength steels possess an exceptional combination of high strength and ductility owing to their high work-hardening rate during deformation. In this study, the mechanical properties and microstructure/texture evolution of medium-Mn steel under various loading conditions (uniaxial tension, in-plane torsion, and equibiaxial tension) were investigated. Interrupted electron back-scattered diffraction and X-ray diffraction characterizations were carried out to track the microstructure and texture evolution of individual phases. Digital image correlation was employed for the in-situ measurement of formation and expansion of deformation bands, i.e., the Lüders and the Portevin-Le Châtelier (PLC) bands. The dynamic strain aging effect in terms of the formation of PLC bands and stress serrations was significantly affected by the loading conditions and stress level. Multiple types of stress serrations were observed in all tested specimens at different deformation stages. The texture evolution of the steel under various deformation modes was further tracked. Ferrite and fresh martensite exhibited the plain texture evolution like most body-centered cubic metals, whereas austenite showed a more complex texture evolution. The martensitic transformation notably decreased the RD||<001> intensity of the austenite; austenite debris was rotated with ferrite grains, while some intact austenite grains self-rotated to a stable orientation, which is unfavorable for martensitic transformation. The obvious intragranular orientation gradient observed in the austenite intact grains implies that considerable deformation occurred in the austenite. Finally, the experimental results indicate that equibiaxial tension primarily promoted the martensitic transformation, and the classical Olson–Cohen model was calibrated to capture the volume fraction evolution of austenite under various stress states.

Characterization of Flow Curves for Ultra-Thin Steel Sheets With the In-Plane Torsion Test

AbstractThe in-plane torsion test (IPT) is a shear test that has already been successfully used to determine flow curves up to high strains for thin sheets with thicknesses between 0.5 mm and 3.0 mm. In the same way as with other shear tests, the formation of wrinkles is a major challenge in determining flow curves with the IPT, especially when testing ultra-thin sheets with a thickness between 0.1 mm and 0.5 mm. A new method for suppressing wrinkling is introduced, in which the formation of wrinkles is avoided by arranging and gluing single sheets to multi-layered specimens. The influence of the used adhesive on the determination of flow curves is negligible. The proposed method is used to identify flow curves for two materials, the high-strength steel TH620 and the soft steel TS230, used in the packaging industry. The materials are tested in sheet thicknesses between 0.17 mm and 0.6 mm. The determined equivalent plastic strains for the TH620 with a sheet thickness of 0.20 mm could be increased from 0.38 (bulge-test) to over 0.8 with the new method using four-layered specimens.

Two new in-plane torsion tests for the investigation of self-heating under pure shear cyclic loading

The fatigue limit under a pure shear loading is very complex to determine for thin sheet materials. In this paper, two new specimen geometries are proposed to assess the fatigue limit under this condition using self-heating measurements for isotropic material. Both new geometries are disc-shaped with particular designs to promote a homogeneous stress field. Strain gauge measurements confirmed that the applied strain fields correspond to a pure shear condition. The thermal fields were imaged by an infrared camera, demonstrating the validity of using a 0D thermal approach. In this way, fatigue limits under pure shear loading could be determined.

Robust characterization of anisotropic shear fracture strains with constant triaxiality using shape optimization of torsional twin bridge specimen

In-plane torsion test has been drawing attention as an effective strategy to obtain good shear properties. Herein, a framework of shape optimization for a twin bridge specimen is presented to improve the performance while ensuring better machinability. The framework includes the consideration of material anisotropy and the determination of design variables. In this study, TRIP1180 sheet was employed as a target material. The optimized design was verified by an actual in-plane torsion test. The experimental results demonstrate that the proposed framework can optimize the shape of the twin bridge for maintaining zero triaxiality up to fracture occurrence.

Large strain flow curves of sheet metals by sheet extrusion

Metal sheets are forward extruded at large plastic strains up to 1.6. The sheet specimens are placed between two half-cylindrical billets and cold-extruded collectively. While extruding the sheets, their central zone is plastically deformed nearly homogeneously under a deviatoric stress state equivalent to simple tension. Tensile test specimens are extracted from the extruded sheets at various extrusion strains delivering flow stresses at discrete large plastic strains of the flow curve. Sheet thicknesses as thin as 0.2 mm could be tested successfully. Steel and aluminum alloys with different strengths were investigated. Results were compared with in-plane torsion test measurements.

Strain hardening under large deformation for AA5182

In this study, an aluminium alloy of AA5182 is taken as the research object to study strain hardening under large plastic deformation. Tensile tests are done for four specimens, including dog-bone specimens, notched specimens, specimens with a central hole and in-plane shear specimens. Bulging tests are also conducted to measure strain hardening under balanced biaxial tension. In addition, an experimental method called in-plane torsion test is also used for shear loading. At least three experiments are completed for each type of specimens along the rolling direction (RD), diagonal direction (TD), and transverse direction (DD). The stroke of each tests is measured by a digital image correlation (DIC) system, and the load-stoke curves were obtained for the tests. Combined with an inverse engineering method, the strain hardening properties are calibrated for the alloy under different loading conditions of shear, uniaxial tension, plane strain tension, and balanced biaxial tension. The strain hardening under various loading conditions is compared and modelled by various yield functions to evaluate their performance. It is concluded that inverse engineering approach is a simple but powerful method to obtain the stress-strain curve up to large plastic deformation. It is also observed that it needs to develop yield functions to model yielding behaviour under complex loading conditions.

Enhanced characterization of the yield behavior of sheet metal at torsional load using digital image correlation methods

This work describes an approach for strain determination at the “in-plane” torsional test using digital image correlation (DIC) without brushing a statistical pattern on the specimen. It is well known that the in-plane torsion test represents a appropriate test method for material characterization of sheet metal in terms of yielding and kinematic hardening (Wagner et al. in Application of the in-plane torsion test in an industrial environment—recent advances and remaining challenges. In: Conference Paper of the 12th Forming Technology Forum, Herrsching, Germany, 2019). However, the gained measurement data do not allow the exact reading of material models for direct use in numerical analysis. The determination of shear values represents thereby the main challenge. Besides well-established methods for stress–strain analysis mentioned in this contribution, the applicability of DIC methods for the direct reading of distortion on the specimen has been investigated. Therefore, a torsion test rig has been developed and torsion tests using specimen out of mild- and high-strength steel have been conducted. The in-plane situation allows the distortion tracking of single points on the specimen by using a 2D image correlation software. The needed pictures were taken by using a common digital lens reflex camera. It has been shown that the resolution of the inherent specimen pattern is sufficient for the use of common 2D image correlation software. Torsional test results in the form of computed yield curves of mild- and advanced high-strength steels are aimed to be presented in this contribution.

Experimental assessment of stiffness and energy dissipation properties of disk-shaped polymer-based composite specimens by in-plane torsion testing

Torsion testing machines are widely used either to measure the strength, stiffness and stress-strain properties of materials or to replicate real-life service conditions. In this paper, a novel experimental method is presented, based on the development of a dedicated steel structure to be used in conjunction with a universal testing machine. This equipment allows applying cyclic in-plane torsion loads on disk-shaped components. The proposed approach aims to enable the assessment of stiffness and damping properties on specimens enabling the application of higher loads in comparison with the traditional machines.Specifically, dynamic trials were performed by attaching the composite specimens and the steel structure to the testing machine, such that the uniaxial controlled displacements can be used to exert the desired cyclic loads onto the specimen. Both torsional stiffness and energy loss were measured from the steady-state load-displacement hysteresis cycles. Amplitudes of sine signals, from 0.05 to 0.2 mm, and a frequency ranging between 1 and 20 Hz, were used in the experiments. The results are presented comparing the behaviour of two polymer matrix composites, with the same number of identical laminae, but characterized by different stacking sequences, namely unidirectional and quasi-isotropic configurations.

Microstructure and Crystallographic Texture of Silicon Iron Modified by Torsion Under Quasihydrostatic Pressure

The electron backscatter diffraction, X-ray diffraction analysis, electromotive force instantaneous measurement, microhardness and coercive force measurement techniques are used to explore the development of the microstructure, crystallographic texture and physico-mechanical properties of silicon iron (Fe-3% Si) alloy under quasi-hydrostatic pressure in a Bridgman anvil. It is found that the alloy deformation is accompanied by its significant hardening. In-plane torsion test shows that the average grain size rapidly decreases down to 185 nm after a quater turn of the movable anvil. With the number of turns increased up to six the grain size gradually decreases down to 150 nm. At the same time, the average size of subgrains is less dependent on the deformation degree. The crystallographic texture evolution involves the formation of the strong {001} texture component after comparatively small megaplastic deformation followed by the simple shear texture formation. Such a behavior can be correctly described in terms of non-equilibrium thermodynamics.

Methods for measuring large shear strains in in-plane torsion tests

The in-plane torsion test achieves true strains far beyond 1.0 for sheet metals, especially using specimens with circular grooves. The accurate measurement of these high strains is a challenge for the conventional digital image correlation (DIC). Thus, the determination of flow curves is limited and fracture strains for very ductile materials cannot be measured. A new grooved specimen is introduced to avoid strain localization. Shear stress and shear strain along a defined area in the groove are constant so that strains can be measured independently of the DIC system setting without error due to strain localization. Furthermore, three methods for the measurement of very high shear strains in the in-plane torsion test are presented: Firstly, the limit of the optical strain measurement is extended by multiple renewal of the digital image correlation (DIC) pattern on the samples. Secondly, the shear strain for the planar specimen is calculated exactly from the rotation angle curve. Lastly, a new incremental method is presented. This method enables to determine shear strains for plane and grooved specimen exactly by only measuring the torque and the angle of rotation. All methods were applied for three steel sheet materials namely DP1000, CP1000 and DC04. The equivalent strain in a grooved in-plane torsion test of sheet steel DC04 was determined as 3.3 with the new incremental method. Such high strains far exceed conventional methods for determining the flow curve.

Characterizing plasticity and fracture of sheet metal through a novel in-plane torsion experiment

In-plane shear specimens typically feature free gage section boundaries along which the state of stress deviates from that of pure or simple shear. As a consequence, the geometry of in-plane shear specimens needs to be carefully chosen to avoid any early fracture initiation from the free boundaries, before the actual failure strain for shear is reached at the specimen center. From this perspective, disc specimens for in-plane torsion experiments offer a significant advantage: they do not feature any free boundaries. However, detailed analysis suggests that circular groove need to be introduced (local thickness reduction) to ensure a strain localization away from the clamped specimen shoulders. In most existing in-plane torsion tests, the specimen is clamped on the inner diameter by applying out-of-plane compression to avoid any slipping. In such configurations, it is impossible to monitor the entire sheared circumference with cameras for digital image correlation. It is the goal of the present work to develop in-plane torsion test using grooved specimens with full optical access to the specimen for 2D or 3D DIC measurements. Furthermore, the experimental set-up will be designed for plasticity and fracture characterization at strain rates of up a few 100/s. Its main feature is a new clamping technique. After identifying a suitable specimen geometry through finite element simulations, experiments are performed on specimens extracted from aluminum alloys and steels sheets. The experimental campaign includes proportional loading, reversed loading and strain rate jumps. The full optical access to the sheared gage section area also enables the discussion of the effects of plastic anisotropy on the strain fields in in-plane torsion experiments. The results from the in-plane torsion experiments are also compared with the fracture strain measurements from in-plane shear experiments performed in a conventional uniaxial loading frame.

Influence of the preheating strategy on the deep drawing of extruded magnesium alloy ME20 sheets

Due to the hexagonal crystal structure of magnesium alloys, a high forming limit can only be achieved at elevated temperatures. For the material characterization of extruded magnesium alloy ME20 sheets, at elevated temperatures, the in-plane torsion test and a multi-layer upsetting test were conducted. Also, FLCs were determined at elevated temperatures. For the deep drawing, two different heating strategies are investigated. In the first method, specimens are placed in an oven at 400 °C for around 10 minutes and then rapidly transferred to the tool. In the second method the specimens are directly heated in the deep drawing tool. In both methods the specimens are painted with Bornitrid lubricant. The effect of the preheating on the coefficient of friction is investigated by using strip tensile tests. FEM simulations for the deep drawing comparing two different material models (Barlat 2000, CPB06) are executed. The results show that specimens heated in the tool show a better formability than oven-heated specimens. The numerical results present that there is no significant difference between Barlat 2000 and CPB06 in an isothermal deep drawing condition. The numerical results are in good agreement with the deep drawing experiments which also indicates that the warm FLCs allow for a good failure prediction.

Investigation of evolving yield surfaces of dual-phase steels

Abstract The aim of this paper is to describe the evolving yield behavior of dual-phase steels during plastic deformation characterized for ten loading paths using a series of mechanical tests including uniaxial tension, uniaxial compression, in-plane torsion and cruciform biaxial tension with the aid of digital image correlation techniques for strain measurement. Large plastic strains in the gauge area of cruciform specimens tested were enabled by a laser deposition process to strengthen the arms in order to measure deformation behavior of the sheet without arbitrarily thinning the gauge section. Experimental yield loci were determined for three dual phase steels with different strength levels up to equivalent plastic strains of ˜0.11 for DP590, ˜0.07 for DP780, ˜0.05 for DP980, respectively. Several existing anisotropic yield criteria under both associated flow rule (AFR) and non-associated flow rule (non-AFR) were applied to describe the anisotropic yield behavior of these DP steels. A comparative study was preformed to validate prediction accuracy of yield criteria with experimental measurements including yield loci, yield stresses and r φ -values under uniaxial tension in seven orientations as well as yield stresses and r b -value under equi-biaxial tension. The results show that non-AFR significantly improved prediction accuracy of both stresses and r-values simultaneously. Under non-AFR, an order of two in the yield stress function is sufficient to accurately predict flow stresses. The evolution of both yield stress and plastic potential surfaces of DP steels were illustrated by changing parameters in the yield criterion as functions of equivalent compliance λ ¯ .