Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 1729638, 12 pages

http://dx.doi.org/10.1155/2016/1729638

## Global and Local Mechanical Responses for Necking of Rectangular Bars Using Updated and Total Lagrangian Finite Element Formulations

^{1}Facultad de Ingeniería e ITIC, Universidad Nacional de Cuyo, Centro Universitario, Parque Gral. San Martín, 5500 Mendoza, Argentina^{2}Departamento de Ingeniería Mecánica y Metalúrgica, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, 7820436 Santiago, Chile

Received 24 January 2016; Revised 20 June 2016; Accepted 21 June 2016

Academic Editor: John D. Clayton

Copyright © 2016 Claudio A. Careglio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1] and Davidenkov and Spiridonova [2] proposed analytical expressions in order to describe the relationships between stress components and yield stress at the necking section. Early numerical simulations of the simple tension test are due to Wilkins [3], Chen [4], and Needleman [5], among others. Norris et al. [6] and Goicolea [7] compared numerical and experimental results for steel and aluminum specimens, respectively. Later, Hallquist [8], Simó [9], Ponthot [10], Cabezas and Celentano [11], and García Garino et al. [12] discussed this problem as well. From an application’s point of view, Safari et al. [13], Paul [14], and Kamaya and Kawakubo [15] studied the flow of equivalent stress in terms of effective plastic strains using the simple tension test and different experimental procedures. Moreover, the works by Wang et al. [16], Coppieters and Kuwabara [17], Kim et al. [18], and Yazzie et al. [19] are dedicated to characterizing the global postnecking behavior in rectangular bars. It is important to remark that all of these works are mainly devoted to predicting the global response of the material during the tensile test. Therefore, the analysis of the local response is a relevant aspect that needs further research.

Both the Updated and Total Lagrangian formulations (see Bathe’s textbook [20] and hereafter, resp., denoted as ULF and TLF) have been used for necking simulation in the tensile test. In practice, ULF has been preferred because the constitutive laws obtained from experimental results are usually written in terms of spatial variables [9, 10, 12]. However, Cabezas and Celentano [11] successfully simulated the necking problem using a TLF based approach. Although there is a lot of information in the literature regarding ULF and TLF, it should be noticed that, up to the authors’ knowledge, a comprehensive assessment of both formulations in problems involving metals subjected to large plastic strains has not been previously addressed.

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 [21, 22] and the TLF with B-bar elements [11] considering in both cases the same constitutive model. Thus, the detailed analysis of the strain and stress states at the necking section of rectangular bars using these two numerical approaches is the main original contribution of the present work.

The large strain kinematics and constitutive model are provided in Sections 2 and 3, respectively. The finite element implementation of the ULF and the TLF is presented in Section 4. In Section 5, the numerical simulation of the simple tension test using a SAE1045 steel rectangular sample is studied in order to compare the results obtained with both formulations. Global and local characteristic variables (such as applied load in terms of true strain and geometric changes at the necking section) obtained with such approaches considering the same finite element mesh are analyzed in detail. Furthermore, distributions of effective plastic strain, stresses in the longitudinal and transverse directions, and pressure and deviatoric stresses are discussed at the loading stages showing large plastic strains and a triaxial stress state. Finally, Section 6 summarizes the main conclusions of this work.

#### 2. Large Strain Kinematics

The large strain kinematics is defined in terms of deformation gradient tensor , where and are the coordinates of the material configuration and the current configuration , respectively, in a Cartesian coordinates system, as can be seen in Figure 1.