A New in-Plane Bending Test to Determine Flow Curves for Materials with Low Uniform Elongation

The maximum strain in sheet metal forming processes is often higher than the uniform strain that can be achieved in a uniaxial tensile test. For simulation, the flow stress from a uniaxial tensile test is often extrapolated by fitting one of the available hardening laws. This introduces significant uncertainty in the simulation results. Especially, the prediction of stability of deformation is highly affected by the actual hardening rate and therefore accurate flow stress data is required for the full strain range. Alternative sheet forming tests such as a bending test, a shear test and a bulge forming test are comparatively more stable than a tensile test and higher strains can be reached. There are two main challenges with these alternative tests. Firstly, for accurate and robust studies of material behavior it is necessary that the pure bending, simple shear and biaxial conditions are reliably and sufficiently homogeneously imposed during the experiments. Secondly, the output from the test must be measured accurately and then converted into the stress–strain relation of the material.

Conventional simple three point bending and four point bending tests can be easily performed in a universal tensile testing machine. The issue with these tests is the involvement of axial and transverse forces in the bending deformation. Additionally friction and local deformation under the contact points can be sources of errors. Bending setups have been developed where loads are applied without contact points which are often referred to in the underneath mentioned literature as pure bending, near pure bending or free bending setups. In this article, these setups will be referred to as pure bending setups. Compared to a three point or four point bending setup, these setups are complex and carry their own actuation and output measurement mechanisms.

Earliest pure bending setups were developed by [1, 2]. In [1], Perduijn and Hoogenboom developed a specially designed out of plane pure bending setup to study sheet material. Ben Zineb et al. [2] developed an out of plane pure bending setup for fatigue study of composite materials by bending to 2% strain. In their setup, by using sliding and pivot interfaces, a clamping system can be rotated by loading in a tensile machine. The rotation of the clamps is used to produce pure bending in the clamped beam sample. Weiss et al. [3] introduced a mechanism that can be used in a standard tensile test machine to perform out of plane pure bending. They showed good agreement of the moment–curvature results with a plane stress bending model for low curvature bending. The setup of [3] was used by Badr et al. in [4] to perform cyclic bending on Titanium alloy. The moment–curvature data along with uniaxial tension/compression experimental data and a finite element model were used to fit the parameters of the YLD2000-2D yield function incorporated with the homogenous anisotropic hardening (HAH) model. The above pure bending mechanism was limited to 1% strain in the outer most fibers [4]. Maeda et al. [5] developed an out of plane bending setup to investigate tension-compression asymmetry (TCA) for dual phase steel DP980. The experimental moment–curvature results from the bending setup were validated with the moment–curvature results iteratively calculated with pre-determined tension and compression data from a uniaxial test. They showed that if TCA is not taken into account the calculated moment–curvature results deviate from the measured one. Boers et al. [6] presented an out of plane bending setup to investigate the Bauschinger effect. The measured force and bend angle results were used to perform such an investigation. Kim et al. [7] developed an out of plane bending setup to study TCA in shape memory alloys. They measured the strain and moment–curvature relation. Cyclic bending tests were performed under ~4% maximum deformation. A similar setup was presented by Sanchez et al. [8] to investigate Bauschinger effects in dual phase steels and aluminum alloys. The moment–strain relations were validated with those of numerically calculated moment–strain relations for the bending process with pre-determined tensile test data. In [9] Denk et al. proposed an in-plane bending setup for cyclic fatigue testing. The strains were measured using digital image correlation (DIC). From the forces the maximum linear stress is calculated using the flexure relation, which is used to generate S-N curves. A miniaturized version of the pure bending setup was developed by Hoefnagels et al. [10]. The size of this setup allows for performing pure bending tests under a Scanning electron microscope (SEM).

Other than for sheet metals, pure bending devices have been developed and used for various materials, such as for composite materials by Ben Zineb et al. [2], weldments by Bu & Gardner [11], flexible electronics by Hoefnagels et al. [12], micro electro-mechanical systems (MEMS) by Elhebeary & Saif [13], micro thin sheet by Stölken & Evans [14] and metal tubes by Guo et al. [15]. Alternative testing methods such as shear tests [16‐18] and bulge tests [19‐21] can also be used to obtain stress–strain data at high strains. These tests are in general complex in design. The shear test in particular requires efforts to perform the test without wrinkling of the specimen. Peirs et al. [22] developed a shear sample design which doesn’t require an anti-buckling mechanism to perform a shear test. This sample design was used by Rahmaan et al. [23] to extract stress–strain data for DP600 steel and aluminium AA5182-O. An equivalent plastic work methodology is used to obtain the work hardening response to large strain levels using shear and tensile tests together. Another test procedure is the in-plane torsion test, which can be used to extract the material behavior, as in the work by Yin et al. [24].

To convert the measured output from these tests into stress–strain response of the material, inverse fitting methods are commonly used in literature [4, 7, 25‐27]. These inverse fitting methods require accurate modeling of the processes with a predetermined hardening model and costly optimization loops. Another way to convert the moment–curvature relation directly into a stress–strain relationship is by using the analytical derivation of Nadai [28]. The derivation is explained in detail in Section 2. To the authors’ knowledge this formulation has hardly been used in the sheet metal forming field. Two notable uses of such a derivation can be found in [29, 30]. The aim of these two works was to measure the TCA in stress–strain behavior using a 4 point bending test. Both studies used rather low ductility material with TCA and strains up to 1%, with an exception in [29] where copper was used also to investigate the pre-straining effect up to a strain of 5%. In the overview above, most tests are out of plane bending tests up to a few percent of maximum strain. Some in-plane bending tests exist, but they are not pure bending tests and the transverse forces directly influence the deformation in the gauge area.

The aim of this paper is to demonstrate a simple and novel in-plane pure bending test setup to investigate large plastic deformation in metals. The proposed bending device is simple in design and can be used in a tensile test machine. Apart from that, the proposed bending setup has four advantages over the previously mentioned devices.

These advantages can be used to achieve high strains, which is a limitation in previously mentioned bending setups. The applied moment is calculated from the forces measured by the tensile test machine and the curvature is obtained from image processing of the specimen deformation respectively. The moment–curvature relation is directly converted into a stress–strain relationship by using the analytical derivation of Nadai [28].

With the new bending setup, high curvatures can be reached at which the validity of the analytical reverse calculation method becomes questionable. It is therefore necessary to perform a detailed validation to determine the range of applicability. Both numerical and experimental validations are performed. The numerical validation is performed with a 3D FEM model of the bending test. The stress–strain curve evaluated from the moment–curvature relation of the model is compared with the exactly known input flow curves of the simulation. In this way, the validity of the analytical equations can be checked for a ‘perfect experiment’. The experimental validation is performed with a highly ductile mild steel to assess the practical applicability and limitations of the test for a material for which flow curves up to high strains are available.

To demonstrate the relevance of the test procedure, the proposed bending stage is used to evaluate the stress–strain curve for a very low ductility and high strength Martensitic Steel. In this class of steels the microstructure is almost fully martensitic with small islands of ferrite and bainite [31]. The application of these steels include rocker outer, side intrusion beams, bumper beams, and structural reinforcement components of the automotive body. In a uniaxial tensile test, the uniform strain is limited to 3%–5%, while much higher strains are applied in forming operations, demonstrating the necessity for high-strain flow curves.

The content of this paper is described in sections as follows. In Section 2 the formulation for the conversion of moment–curvature into stress–strain relation is revisited. In Section 3 the design and development of the proposed bending setup is described. This is followed by the description of test method and evaluation of the output in Section 3. The validation and limitation of the testing and evaluation procedure is investigated using numerical modeling in Section 4. In Section 5 various experimental results from the test are discussed, including experimental validation and application results. The discussion of the various sections is concluded in Section 6.

For bending of a beam with a monotonically increasing moment, the stress and strain in the cross-section can be obtained from the moment–curvature diagram, under certain assumptions. The expression for the stress in the outer fibre from a known moment and curvature was described already in 1931 by Nadai [28, Chapter 23, p. 164]. The derivation by Nadai starts with assuming that the material has asymmetric tension–compression flow stress. The equations are then derived for the tension and compression stress separately. Subsequently, the equations are simplified for symmetry in tension and compression. In order to derive an analytical equation, it is assumed that the cross section remains constant, the engineering strain follows a linear relation over the height, the neutral line is always in the middle of the cross section (Bernoulli hypothesis) and that the stress state is uniaxial. The implications of these assumptions are described below.

The implications described under 2 have negligible influence on the results for small curvatures, but the accuracy will decrease for higher curvatures. In Section 4, the influence of these simplifying assumptions will be investigated by a numerical example. For simplicity, the implication of point 1 is accepted as a limitation and only materials with a symmetric tension/compression behaviour are considered in this paper.

The derivation below follows a slightly different path than in [28]. Consider a beam with a constant rectangular cross section of height h and width b under pure bending such that the neutral line stays at y = 0. The moment M depends on the stress distribution σ(ϵ(κ, y)) following:

The engineering strain ϵ(κ, y) at any point y in the cross section and curvature κ are related by:

In Fig. 1, a typical stress and strain distribution over the height of the beam at a certain curvature κ is shown.

It is important to note from Fig. 1, that σ is only a function of ϵ and ϵ is function of κ and y. We can take the derivative of Eq. (1) on both sides with respect to κ.

The integral limits on the right hand side are not a function of κ, then according to Leibniz’s rule:

Using the chain rule and realizing that κ and y are independent yields:

From Eq. (2) follows \( \frac{\partial \epsilon }{\partial \kappa }{\vert}_y=\kern0.5em y \), thus simplifying Eq. (5) to:

For the bending case under consideration, according to Eq. (2), the derivative of stress with respect to strain can also be derived from the partial derivative of stress with respect to coordinate y at constant curvature κ:

\( \frac{\partial y}{\partial \epsilon }{\vert}_{\kappa }=\frac{1}{\kappa } \) then leads to:

Eq. (8) can be integrated using Integration by parts:and with \( {\int}_{-h/2}^{h/2}\ y\sigma\ dy=\frac{M}{b}\kern0.75em \)

Based on the assumption that the material behaves symmetric in tension and compression σ(−h/2) =  − σ(h/2).such that:

The engineering strain ϵ_l at the outer most fibre (h/2) can be obtained from the curvature using:

The engineering strain can be converted to true strain in tension and compression ε_T and ε_c, respectively, using:

It is now shown that the engineering strain is a first order approximation of the average of the ε_T and ε_C. The average true strain can be written as:

Substituting Eq. (14) in Eq. (15):

The term \( \ln \left(\frac{1+\epsilon \left(h/2\right)}{1-\epsilon \left(h/2\right)}\right) \) can be approximated by a Taylor series expansion, resulting in:

Putting the Taylor expansion results in Eq. (16), and neglecting the third and higher order terms gives:

Thus, the average true strain from Eq. (15) is coupled with the stress from Eq. (12) for comparison and validation purpose. Although several assumptions are made in the derivation of Eq. (12) and Eq. (15), in Section 4 and Section 5 it will be shown to give good results even for relatively high strains.

A new test design is proposed that is able to perform pure bending in the plane of the sheet. This bending setup can be mounted in a tensile test machine. A special sample design is suggested for this process as illustrated in Fig. 2a). The sample has a reduced cross section in the middle that acts as a beam between two rectangular support sections. The support sections have holes for pins through which loads can be applied vertically. The pin-hole interface offers free rotation and are loaded as shown in Fig. 2. The applied boundary conditions will result in rigid rotation of the rectangular support sections causing pure bending in the beam as shown in Fig. 2 b). This in-plane bending of the sheet allows for a high curvature to height ratio, resulting in a high strain at the outer fiber.

In this section, the equations for the reverse calculation of the stress–strain relation from the moment–curvature relation are validated by using a numerical model with exactly known hardening curve. The difference between the input hardening curve and the result from the analytical reverse calculation shows up to what level the assumptions that were made in the derivation are valid. The numerical simulation is performed with MSC.Marc 2017, where the beam section is modelled using linear 3D brick elements (element code 7) with enhanced assumed strain formulation to improve the accuracy in bending. Specifications of the specimen and mesh sizes are presented in Fig. 9 and Table 1 with beam length 10 mm beam height of 4 mm and sheet thickness of 2 mm. No symmetry is used in the model for this initial analysis. Frictionless contact is used for the pin-hole interface. The simulation is performed with a von Mises yield function. Isotropic hardening behaviour is applied, using the Swift hardening law:

With K, ε_o and n material parameters, equal to 550.8 MPa, 0.01024 and 0.2163 respectively. These are representative values for mild steel.

Curvature and moment are calculated by processing the nodal data of the simulation output files in MATLAB and then using the procedure described in Sections 3.3.1 and 3.3.2. The calculated moment and curvature are used in Eq. (12) to determine the stress. Eq. (14) and (15) are used to determine the average true strain. The stress–strain curve achieved by using the reverse analytical calculation is then compared to the input flow stress curve. The comparison is shown here for a maximum curvature of 0.11 mm⁻¹ with true strains up to ~23% in the outermost fibres Fig. 10. The axial component of stress and strain is also separately evaluated at the nodal position in the outermost tension and outermost compression fibre of the beam section. The comparison shows that the reverse calculation is in good agreement until ~12% strain. The deviation becomes significant after 15% strain. Thus, the assumptions that were made in Section 2, leading to the reverse analytical calculation are acceptable for strains up to 12%.

The equivalent plastic strain, axial strain and radial stress σ₂₂ are plotted in Fig. 11 at 12% and 42% maximum strain in the beam. It can be seen that the deviation between the linear engineering strain and nonlinear true strain is small for the case of 12% maximum strain. Also at 12% of strain, the maximum value of the stress σ₂₂ is small compared to the maximum value of axial stress σ₁₁ in Fig. 10. At a maximum strain of 42%, σ₂₂ becomes significant. Therefore, at high strain the stress–strain curve derived from the reverse analytical calculation deviates from the input flow curve for strains above 12%. It should be noted that the limitation for accurate evaluation of stress–strain relations to 12% strain is due to the assumptions made in the analytical conversion. With a numerical evaluation framework where all phenomena of the pure bending process are properly taken into account, the stress–strain relation can be obtained up to higher strains, still without assuming a pre-defined hardening formula. It should be noted that even for material with symmetric tension and compression behaviour, the engineering strains are only symmetric for low curvatures. At higher curvature due to shift of the zero strain position, the engineering strains become non-symmetric. If the material behaviour has TCA, the engineering strains are asymmetric in tension and compression from the beginning. In these cases, it is necessary to measure the strains in the outermost fibres of the beam or measure curvature and one of the outermost fibre strain. For the case of asymmetric tension and compression strains the curvature is given by.where ϵ_T and ϵ_C are the engineering strains of the outermost fibres in tension and compression, respectively. The equation for reverse calculation of stress for the asymmetric stress–strain material can be found in [28]. The application of these equations has already been demonstrated by [29, 30] for four point bending, but only for strains in a range of 1–5%.

Experiments have been performed with the new bending setup according to the methodology described in Section 3. Experiments are divided into two categories: validation and application. In the experimental validation, a very ductile mild steel is used for which the stress–strain curve can also be determined up to high strain in a tensile test. The curve from the reverse analytical calculation is compared with the direct tensile experiment. The difference with the numerical validation is that now also measurement uncertainties are included. In the experimental application a low-ductility material is used with less than 3% uniform strain in a tensile test.

After the numerical and experimental validation and the first application to a low ductility material, the following conclusions can be drawn.

Overall, the bending setup offers a simple and unique solution for in-plane pure bending of sheet material. The moment–curvature measurement from this setup can be used to study hardening behaviour up to relatively high strain.