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In simulations of forged and stamping processes using the finite element method, load displacement paths and three-dimensional stress and strains states should be well and reliably represented. The simple tension test is a suitable and economical tool to calibrate constitutive equations with finite strains and plasticity for those simulations. A complex three-dimensional stress and strain states are developed when this test is done on rectangular bars and the necking phenomenon appears. In this work, global and local numerical results of the mechanical response of rectangular bars subjected to simple tension test obtained from two different finite element formulations are compared and discussed. To this end, Updated and Total Lagrangian formulations are used in order to get the three-dimensional stress and strain states. Geometric changes together with strain and stress distributions at the cross section where necking occurs are assessed. In particular, a detailed analysis of the effective plastic strain, stress components in axial and transverse directions and pressure, and deviatoric stress components is presented. Specific numerical results are also validated with experimental measurements comparing, in turn, the performance of the two numerical approaches used in this study.

Currently there are many technological applications where computational modeling with the finite element method is extremely useful when, in particular, the problem studied leads to complex triaxial stress states. Problems such as metal forming, impact, and behavior of energy dissipation devices, among others, are typical examples in which it is necessary to consider large strains and elastoplasticity in the finite element models. In all of these cases, different finite element types and formulations are used. However, it is not usual to assess their predicting capabilities of the global and local mechanical responses of the problem.

Calibration of the constitutive equations with experimental results in the large strain range is a requisite for reliable finite element modelling. For this purpose, the simple tension test is one of the most appropriate ways in practice to characterize the mechanical response of metals at large plastic strain ranges and triaxial stress states.

The simple tension test has a first stage where small or moderate strains are observed. In this case, the material behavior is practically linear. In a second stage, strains increase with little changes on the applied load. Finally, after the maximum load is reached, strains are concentrated in a zone where large geometric changes produce the necking in the specimen. In the necking zone, large plastic strains and a triaxial state of stresses can be found.

Bridgman [

Both the Updated and Total Lagrangian formulations (see Bathe’s textbook [

The aim of this paper is to present a detailed study of the global and local mechanical responses of rectangular samples subjected to simple tension conducted in order to obtain strain and stress distributions at the minimum cross section. In particular, effective plastic strain, stress components in axial and transverse directions, and pressure and deviatoric stress components are presented and discussed. To this end, finite element simulations are carried out using the ULF approach with H1/P0 elements [

The large strain kinematics and constitutive model are provided in Sections

The large strain kinematics is defined in terms of deformation gradient tensor

Elastoplastic kinematics of a continuum body.

For the elastoplastic case, the multiplicative decomposition of the deformation gradient tensor

In the original configuration

In the current configuration of Almansi strain tensor

It is convenient to introduce the rate of deformation tensor

It is important to point out that, from the multiplicative decomposition, additive decompositions for

The constitutive law is written in the context of irreversible thermodynamics of solids by using the free energy

For metals under large strains, the elastic strains are negligible, even for the case of large plastic deformations. The elastic component of the gradient tensor

The Cauchy stress tensor can be obtained from (

Plasticity is taken into account by a classical J2 model:

The actual yield stress has a power hardening law given by

Furthermore, the spatial tensor

The finite strain constitutive elastoplastic model presented previously in Section

The global discretized equilibrium equations, for load level at the present quasistatic case, can be written in matrix form for a certain time

For a given load increment this equilibrium equation is solved at time

The displacements and the spatial coordinates are updated to fulfill a convenient convergence criterion.

In order to compute the elastic predictor problem the elastic Finger tensor

The elastic part of the Finger tensor plays the role of an internal variable in this model. Details of the derivation of (

If necessary the plastic corrector is based on a backward Euler integration scheme, leading to

Then, substituting (

Replacing (

To avoid numerical locking due to incompressible plastic deformation, an H1/P0 mixed finite element is used. The H1/P0 element is the extension to the three-dimensional case of the well known Q1/P0 element proposed by Nagtegaal et al. [

The internal force vector, in the current configuration

The tangent stiffness matrix

In the elastic predictor problem the trial elastic Green Lagrange tensor is computed first as

Then the trial Second Piola Kirchhoff stress [

If necessary in the plastic, corrector problem, the plastic multiplier

Finally, the Second Piola Kirchhoff stress tensor results:

Finally, the contributions of

The geometry of the rectangular bar used in the numerical simulation of the simple tension test is shown in Figure

Rectangular bar.

Geometry of the specimen

Finite element mesh

Due to the symmetry of the specimen only one-eighth of it has been modeled. Thus the length, thickness, and width in the finite element model are

To ensure location of necking at the central part of the specimen, a geometric imperfection is imposed as a linear width variation along its length. The width is reduced to 1.376% in the central area of the specimen.

Boundary conditions are imposed to satisfy symmetry conditions. Displacements

To simulate the tensile test, displacements

The material considered in the numerical analysis is SAE 1045 steel with the following mechanical properties: elastic modulus

In what follows the results obtained with the two implementation processes presented in previous section are compared.

The evolution of the tensile force

Tensile force as a function of logarithmic strain in the neck.

An overall good agreement between the two finite element implementations can be observed in these curves. In both cases is obtained a linear behavior for loads of less than 33950 N, which is the initial yield load, presenting deformations very close to zero. On the other hand, a nonlinear relationship between deformation and load is obtained for greater values than 33950 N.

We can state that both implementation processes have a behavior very close to each other for the global parameter behavior of the tensile test given by applied load-true strain ratio.

As characteristic parameter of the necking phenomenon it is possible to consider the evolution of the specimen cross-sectional dimensions in the necking zone. The symmetry plane

Figure

Ratio of current to initial width versus engineering strain.

The results obtained numerically in this work and experimentally by Cabezas and Celentano [

The evolution of the ratio between the current thickness

Ratio of current to initial thickness versus engineering strain.

From comparing Figures

Deformed configurations in the necking zone for a final elongation of 20%.

It can be seen in Figure

The two implementation processes represent very well the evolution of necking in the cross section of the specimen corresponding to the plane of symmetry. It should be noted that the cross section in the necking zone has a greater geometric change in the direction of smaller dimension.

The effective plastic strain contours at the end of the simulation, corresponding to the fracture stage for elongation of 20.0%, are shown in Figures

Comparison of effective plastic strain contours for a final elongation of 20%.

TLF

ULF

The maximum effective plastic strain value occurs in the central point of the cross section of symmetry of the specimen, where maximum necking is developed. The effective plastic strain obtained with TLF is 1.027. It is slightly higher than that obtained with ULF which is 0.906.

Since the maximum effective plastic strains are slightly different it is of interest to analyze the distribution of these in the central section of the necking. This is done through analyzing the strain values on Gauss points of the used finite element mesh. The location of these Gauss points at the

Gauss points in the undeformed configuration.

The effective plastic strain level curves plotted at the deformed configuration are shown in Figures

Level curves of effective plastic strain for TLF and 20% of final elongation.

Level curves of effective plastic strain for ULF and 20% of final elongation.

In order to analyze the effective plastic strain development, values greater than 0.9 are compared in Figures

Two interesting straight lines are shown in Figure

The effective plastic strain in the necking section as a function of the deformed

Effective plastic strain in the neck section along coordinate axis

Although not shown, according to the hardening law given by (

It is also important to analyze the behavior of some of the Cauchy stress tensor components. The more representative ones are the axial component

Figure

Axial stress

Figure

Transverse stress

Once again, smoothed values for the pressure and deviatoric normal stress components are presented below, using the procedure previously described and discussed. As has been suggested by Gabaldón [

Pressure (MPa) in the neck section along coordinate axis

Deviatoric stresses (MPa) in the neck section along coordinate axis

In these figures, the analytical approximations for the pressure and the axial deviatoric stress for cylindrical specimens (see Appendix

The response of the simple tension test of a rectangular specimen of SAE1045 steel into a postnecking behavior has been obtained using two well known large strain finite element implementation processes. An Updated Lagrangian formulation with a H1/P0 element (ULF implementation) and a Total Lagrangian formulation with a B-bar element (TLF implementation) both using hexahedral isoparametric elements have been used for the numerical simulations. Although the plastic constitutive model is the same for both approaches, the implementation in each formulation is quite different, and their performances on the prediction of complex three-dimensional stress and strain states such as those occurring at the necking section of a tensile sample are not usually discussed in literature.

Global and local mechanical results have been obtained and compared using the two finite element implementation processes. Moreover, some specific numerical results at the necking section have been also validated with experimental measurements.

As a global comparison parameter the applied load as a function of the true strain has been used. These curves obtained with both finite element implementation processes are very close to the experimental results.

At the central cross section where the necking is more evident, some local parameters have been evaluated and discussed. Particularly, the deformed shape of the central cross section and local constitutive parameters, such as the effective plastic strain, stress components, pressure, and deviatoric stresses have been compared and analyzed.

Both numerical solutions represent very well the geometric evolution of the necking in the cross section of the specimen corresponding to the plane of symmetry. Very well fitting has been found regarding the changes in thickness and width of the cross section in terms of engineering strain compared with the experimental results.

It should be noted that the cross section of the specimen in the necking zone presents greater geometric change in the direction of smaller dimensions. Both implementation processes predict in excess these deformations compared with the experimental results. Particularly, the TLF results are a bit higher than the ULF.

The effective plastic strain distributions obtained with both implementation processes do not lead to significant differences in the results. Particularly in the cross section of the specimen in the central area of the neck (plane of symmetry

Both axial stress components analyzed, the longitudinal and transverse ones, have quite similar responses, with maximum values in the central area and minimum values in the width of the cross section where necking occurs. Compression values in the width of the specimen are obtained for the transverse stress component.

Pressure and deviatoric stress distributions have quite similar behaviors for both implementation processes. Pressure distributions have larger gradients along the

In summary, the simple tension test in large strain range, in which the necking phenomenon appears, is an appropriate alternative to calibrate the constitutive plasticity J2 models in both ULF and TLF implementation processes. The J2 constitutive model implemented in such formulations was found to provide a realistic response during the tensile test of rectangular bars. Finally, the results for the global and local variables obtained with the two implementation processes exhibit a good agreement.

By means of an improved strain-displacement matrix

In 3D problems the expression of

For 3D case

The pressure in a cylindrical specimen can be defined as

Considering that the J2 yield criterion for metals in the neck section can be expressed as

Finally, if in (

The authors declare that they have no competing interests.

The financial support provided by SeCTyP-Universidad Nacional de Cuyo projects 06B/253, 06B/308, and B008 is gratefully acknowledged. Diego Celentano acknowledges the financial support provided by FONDECYT (Project no. 1130404).