Accurate stress computation in plane strain tensile tests for sheet metal using experimental data

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Abstract

The advances achieved in phenomenological constitutive laws and their implementation in finite element codes for predicting material behavior during forming processes have motivated the research on material identification parameters in order to ensure prediction accuracy. New models require experimental points describing a bi-axial stress state for proper calibration, and the features of the plane strain tensile test have made it one of the most used. The test's principal inconvenience is the influence of the free edges on strain field homogeneity and stress computation.

Experimental measurements of the strain field over the gauge zone on a plane strain tensile test specimen during deformation reveals the evolution of the size of the specimen area that represents a plane strain state.

This article proposes a methodology, based on a numerical analysis of a plane strain tensile test for different materials and specimen geometry, to experimentally identify the evolution of the homogeneous strain field zone during deformation. This research defines an expression for computation of the actual stress in the specimen's plane strain state zone along the loading direction using experimental data and including the edge effect evolution in terms of plastic strain.

The influence of the specimen's geometry and material anisotropy over the stress computation error is discussed and quantified. The stress computation expression proposed here can be adapted to other specimen geometries.

Introduction

The development of new materials with complex mechanical behavior, in order to increase the strength-to-weight ratio, has motivated the development of new constitutive laws able to predict the material behavior during forming. Vegter and van den Boogaard (2006) proposed the representation of the yield function with the help of Bezier's interpolation, Haddadi et al. (2006) explain the Teodosiu–Hu model which accurately describes the macroscopic anisotropic behavior, Aretz (2005) introduced a non-quadratic plane stress yield function for orthotropic sheet metals with a very simple mathematical form and Plunkett et al. (2008) introduced the anisotropy to the isotropic yield function using linear transformations of the stress deviator to describe the anisotropic behavior of textured metals. These constitutive laws require a calibration of material parameters, which depend on mechanical tests.

In sheet metal forming, a classical characterization technique for material identification of complex constitutive laws describing the material's plastic behavior under large strains is the inverse method, which consists in finding the model data set to minimize the error of numerical prediction compared with the experimental measurement in a mechanical test designed to present a heterogeneous stress–strain field. For example, Khalfallah et al. (2002) proposed an inverse algorithm composed on finite element calculation combined with an optimization procedure to identify the anisotropic parameter from plane tensile test. Another calibration technique is based on the stress–strain data obtained from mechanical tests presenting homogeneous behavior. For example, Kuwabara et al. (1998) used cruciform specimens on cold-rolled steel in biaxial tensile test in order to obtain the contours of plastic work and to fit with constitutive laws. Aretz et al. (2007) used uniaxial and plane strain tensile tests for yield function calibration and Flores et al., 2007a, Flores et al., 2007b investigates the material behavior with the help of classical tensile tests, Bauschinger shear tests and successive or simultaneous simple shear tests and plane strain tests, performed in a biaxial machine developed by Flores et al., 2005a, Flores et al., 2005b in order to identify the yield locus and the hardening model.

The first technique requires few mechanical tests with complex geometry specimens and high computational demands, while the second one requires a larger number of quasi-homogeneous mechanical behavior mechanical tests.

The accuracy of material identification in this last technique relies on the mechanical configuration's capability to obtain homogeneous stress–strain behavior throughout the specimen.

Kuwabara (2007) presents a large number of mechanical tests, specimen geometries, and mechanical devices for sheet metal characterization, such as bulge test, biaxial compression test, biaxial tension test using cruciform specimens and plane strain tensile test. In particular, the plane strain tensile test provides useful information of the material's mechanical behavior, such as yield function shape and work hardening identification parameters. In addition, this test can be used in strain-path dependency studies and forming limit diagram characterization. One of this test's advantages is that it can be performed on a classical uni-axial tensile test machine, although its ability to obtain a homogeneous or quasi-homogeneous stress–strain field depends on the specimen's geometry. In this test, the edge effect of the specimen perturbs the stress computation, requiring the implementation of correction techniques.

Flores et al., 2007a, Flores et al., 2007b demonstrated a useful technique consists in maximizing the width to a high ratio, although it does not eliminate the edge effect's influence on the stress computation. The advantage is that the stress computation is obtained by the ratio between the force provided by the machine load cell and the specimen cross section calculated using initial specimen width and the specimen's actual thickness at a central point with little loss in accuracy. The principal inconvenience is that the width size has a direct influence on the required machine strength for plastic deformation and grip design in order to avoid specimen sliding.

A second technique developed by Pijlman (2002) is based on the identification of an error factor calibrated by finite element simulations. The problem is that finite element simulation input data must be previously calibrated; hence, the effect of anisotropy and work hardening over the material during plastic deformation will be contained in this error factor.

A third technique, proposed by An et al. (2004), involves the experimental identification of the edge effect's influence on stress computation by plane strain tests using different higher width specimen geometry, although it does not consider the edge effect evolution's in terms of plastic deformation nor actual thickness.

The present article focuses on the determination of the actual stress in a plane strain tensile specimen based on experimental data taking into account the edge effect's evolution.

The next section describes experimental evidence on specimen homogeneous zone evolution in terms of plastic deformation and proposes a criterion to define the homogeneous zone.

The third section is focused on the stress computation in terms of experimental data, i.e. machine load cell data and the two-dimensional strain field over the specimen gauge zone measurements. The study is based on a numerical analysis of the stress–strain field behavior over plane strain tensile test specimens varying geometry and material properties (initial yield surface and hardening behavior).

The fourth section analyses and discusses the results obtained, an expression to compute the actual stress in the plane strain zone in the loading direction.

In Section 5, some conclusions are established and research perspectives proposed.

Section snippets

Experimental equipment and material description

The plane strain tensile test is performed in a bi-axial test machine developed at the ArGenCo department of the University of Liege, based on the design proposed by Pijlman (2002). The machine can carry plane strain and simple shear tests, separately or simultaneously, on flat specimens. Fig. 1 shows the mechanical configuration together with deformed specimens for both types of tests. Former studies and machine development can be found in Flores et al., 2005a, Flores et al., 2005b and Flores

Finite element strategy

The quasi-static room temperature mechanical behavior of a plane strain test is simulated by finite element analysis. The code used for simulation is SAMCEF Mecano developed by Samtech (1991), with Hill (1948) yield criterion developed by Hill (1948) (Eq. (14), written in terms of the Lankford coefficients r) and a Swift type law (Eq. (15)).Φ(σ)=rTD(rRD+1)σ112+rRD(rTD+1)σ2222rRDrTDσ11σ22+(rRD+rTD)(2r45º+1)σ122rTD(rRD+1)σ¯=0,σ¯=Kε0+ε¯pn.

Here, σ¯, ε¯p are respectively the current yield stress

Results and discussion

The evolution of the homogeneous zone during plastic deformation is strongly dependent on the material and the specimen geometry as shown in Fig. 17. Hence, WH has to be experimentally determined for each case using the strain field measurements (Section 2.3).

The stress in the loading direction can be computed in terms of experimental measurements and a constant containing the geometry and material effect. In this case, the α parameter is defined in Eq. (18).

In Table 3, the geometry dependence

Conclusions and perspectives

In this work, a procedure to determine the homogeneous zone evolution during a plane strain tensile test is proposed. The information is used to develop an expression to determine the actual stress in a plane strain tensile test ensuring the error boundaries exclusively using experimental data.

A correction factor, referred to as α parameter, takes into account the edge effect for a particular geometry. The paper results show that this parameter depends both on the material and the specimen

Acknowledgments

The authors acknowledge the Interuniversity Attraction Poles Programme – Belgian State – Belgian Scientific Policy (Contract P6/24). A.M.H. and L.D., the Belgian Fund for Scientific Research FRS-FNRS and the Walloon Region for their support.

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