Advances in Materials Science and Engineering

Volume 2016, Article ID 4152963, 18 pages

http://dx.doi.org/10.1155/2016/4152963

## Inverse Strategies for Identifying the Parameters of Constitutive Laws of Metal Sheets

CEMUC, Department of Mechanical Engineering, University of Coimbra, Rua Luís Reis Santos, Pinhal de Marrocos, 3030-788 Coimbra, Portugal

Received 1 July 2016; Accepted 16 August 2016

Academic Editor: Sutasn Thipprakmas

Copyright © 2016 P. A. Prates 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

This article is a review regarding recently developed inverse strategies coupled with finite element simulations for the identification of the parameters of constitutive laws that describe the plastic behaviour of metal sheets. It highlights that the identification procedure is dictated by the loading conditions, the geometry of the sample, the type of experimental results selected for the analysis, the cost function, and optimization algorithm used. Also, the type of constitutive law (isotropic and/or kinematic hardening laws and/or anisotropic yield criterion), whose parameters are intended to be identified, affects the whole identification procedure.

#### 1. Introduction

Finite Element Analysis (FEA) is now a well-established computational tool in industry for the optimization of sheet metal forming processes. The accurate modelling of these processes is a complex task due to the nonlinearities involved, such as those associated with (i) the kinematics of large deformations, (ii) the contact between the sheet and the tools, and (iii) the plastic behaviour of the metal sheet.

The description of the plastic behaviour of metal sheets is usually performed using phenomenological constitutive models. In this context, the emergence of new steels and aluminium, magnesium, and other alloys, as well as their increasingly widespread use in the automotive and aeronautical industries, has encouraged the development of more reliable models, with increasing flexibility associated with a larger number of parameters to identify [1–14]. In fact, the accuracy of the numerical simulation results of sheet metal forming processes depends on the flexibility of a constitutive material model but also on the procedure adopted to identify its parameters. The complex nature of the plastic behaviour of metal sheets makes their characterization dependent upon factors such as (i) the constitutive model; (ii) the experimental tests performed, comprising the sample geometry, the testing conditions, and the analysis methodologies; (iii) the strategy for identifying the constitutive parameters.

The strategy for identifying the model parameters is generally seen as an optimization problem, where the purpose is to minimise the difference between computed and experimental results of one or more experiments. Two main types of strategies for the identification of the constitutive parameters can be recognised in literature: classical and inverse strategies. The classical identification strategies for the constitutive parameters make use of a large number of standardised mechanical tests, with well-defined geometry and loading conditions, such that homogeneous stress and strain distribution develop in the region of interest (e.g., [15, 16]); nonstandard mechanical tests can also be performed to properly describe other biaxial stress states in the sheet plane (e.g., [17, 18]). However, sheet metal forming processes are carried out under strongly nonhomogeneous stresses and strains fields. Therefore, limiting the characterization of the mechanical behaviour of metal sheets to a restricted number of tests with linear strain paths and homogeneous deformation can lead to a somewhat incomplete characterization of the overall plastic behaviour of the material [19].

Recent developments and accessibility of optical full-field measurement techniques, such as digital image correlation (DIC) technique coupled with FEA, make the inverse identification strategies a common current place. The full-field measurements allow the acquisition of enriched information from mechanical tests, such as displacement and strain fields; an overview on this topic can be found in [20]. This allows attenuating the constraints on the geometry and loading conditions of the mechanical tests used for the identification of materials parameters, so that nonhomogeneous stress and strain distributions can be developed in the region of interest (e.g., [21–29]). In this sense, the identification of constitutive parameters from nonhomogeneous strain fields and complex loading conditions provides a more reliable description of the material behaviour during real sheet metal forming processes [21]. In such complex mechanical tests, it is no longer possible to identify the constitutive parameters based on simple assumptions on the stress and/or strain states, as in the classical identification strategies. Instead, a finite element model of the mechanical test is established and cost functions are defined to minimise the gap between numerical and experimental results of the mechanical test, which demands efficient optimization algorithms. However, the efficiency of any inverse identification strategy directly depends on the information contained by the objective function. In the context of constitutive parameters identification, this is related to the type of experimental results included (e.g., loads, displacements, and strains) but also to the strain paths and levels of deformation attained by the experimental test. It turns out that there is no consensus about the experimental tests (sample geometry and loading conditions), the cost functions, and the optimization procedure that will lead to accurate constitutive parameters identification. Also, a major obstacle to the widespread use of advanced constitutive models in industrial simulations seems to result from the lack of an efficient strategy for parameters identification. In this sense, the developed strategy must be simple, from an experimental point of view, and allow evaluating to what extent the selected constitutive model allows perfectly describing the behaviour of a given material.

The present paper describes recent inverse strategies coupled with FE simulations for the identification of the parameters of constitutive laws that describe the plastic behaviour of metal sheets, resorting to mechanical tests leading to nonuniform strain and stress states. Following this introduction, the paper addresses general concepts for the constitutive modelling and the optimization problem. Afterwards, an overview of inverse identification strategies for the constitutive parameters is presented, with emphasis on inverse identification strategies resorting to FE simulations.

#### 2. Constitutive Modelling

Constitutive models have been developed to predict the onset and evolution of the plastic deformation of a deformable body undergoing a general state of stress. A phenomenological constitutive model is typically a combination of the following components:(i)Yield criterion that describes the yield surface of the material in a multidimensional stress space: The metal sheets are usually assumed to be orthotropic, with invariant anisotropy during plastic deformation. With high incidence in the last decades, the emergence of anisotropic yield criteria with an increasing number of material parameters has been witnessed. They provide the flexibility required for accurately modelling the plastic behaviour of advanced metallic alloys, which are frequently used in automotive and aeronautical industries. Several approaches have been used for deriving yield criteria, based on(1)high-order polynomial functions (e.g., [1, 2]);(2)the generalization to anisotropy of the second and third invariants of the deviatoric stress tensor, and , respectively (e.g., [3]);(3)one or more isotropic yield functions, using the linear Isotropic Plasticity Equivalent (IPE) stress space concept (e.g., [3–9]);(4)the construction of weighted sums of anisotropic yield criteria (e.g., [7]);(5)the capability to model the tension-compression asymmetry, particularly devoted to specific magnesium and titanium alloys (e.g., [3, 6, 10]);(6)the capability to model kinematic hardening [11];(7)the interpolation of second-order Bézier curves [12].(ii)Hardening laws that express the evolution of the yield surface during plastic deformation, as schematized in Figure 1: The isotropic hardening law refers to the homothetic expansion of the yield surface (see Figure 1(a)) while the kinematic hardening law describes its translation in the stress space (see Figure 1(b)). Kinematic hardening laws are recommended for describing plastic deformation under strain path changes, mainly strain path reversal, in materials that exhibit Bauschinger effect (e.g., [14]). The combination of isotropic and kinematic hardening laws provides a flexible model, for simultaneously describing the change in size and the position of the centre of the yield surface, during plastic deformation. Isotropic hardening laws described by power laws (e.g., [32–37]), saturation laws (e.g., [38, 39]), and weighted combinations of isotropic hardening laws (e.g., [40, 41]) have been proposed. Linear (e.g., [42, 43]) and nonlinear (e.g., [13, 14, 44–47]) kinematic hardening laws were proposed, with the latter being more appropriate to describe the Bauschinger effect.(iii)Flow rule, to establish a relationship between the stress state and the plastic strain increment: Typically, an associated flow rule is adopted, that is, using the yield function as plastic potential, although some exceptions can be found in literature (see, e.g., [48]).The general representation of a constitutive model can be described through a function :where is the equivalent stress defined by a given yield criterion and is the hardening law that represents the evolution of the yield stress during the deformation. The equivalent stress, , is a function of the effective stress tensor, , that includes the parameters of the yield criterion, , for describing the anisotropy ( and are the deviatoric Cauchy stress and the deviatoric backstress tensors, resp.) and is a function of the equivalent plastic strain, , in which the parameters are represented by . The yielding is defined based on the function of (1) and can be written as follows:If , the stress state of the material remains inside the yield surface and only elastic deformation occurs. When plastic deformation occurs, the associated flow rule states that the increment of the plastic strain tensor is normal to the yield surface, for a stress state such that . The normality condition, defined by the associated flow rule, assumes that the increment of the plastic strain tensor is normal to the yield surface and is expressed bywhere is the increment of the plastic strain tensor, is a scalar multiplier, and is the equivalent stress function, representing the plastic potential.