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## Inverse methodologies for material parameters identification

In the past, the accurate identification of the material parameters was a generalized problem for the scientific community, mainly due to the increasing complexity of the constitutive models (Kleinermann, 2000; Andrade-Campos et al., 2007). At the beginning of the 1990s, the material parameters determination was simply achieved through the use of curve fitting techniques, considering simple constitutive models and experiments with the assumption of uniform distribution of the resulting strains and stresses within the whole volume of the specimen (Ponthot and Kleinermann, 2006). However, curve fitting techniques were not valid for complex constitutive models and non-homogeneous experiments. Then, Chaboche et al. (1991) proposed a trial-and-error procedure for the identification of material parameters. Withal, automatic methodologies based on the coupling of FE codes with optimization methods were also developed (Schnur and Zabaras, 1992; Cailletaud and Pilvin 1993, 1994; Gelin and Ghouati, 1996) and, nowadays, constitute the principal techniques adopted for the material parameters identification. This kind of techniques are commonly designated as Finite Element Model Updating (FEMU) and consists of an inverse problem where the aim is to determine the input data that lead to a final desired result that is previously known. This inverse problem can be solved using an optimization methodology whose main goal is to seek for a set of material parameters that leads to the smallest difference between the experimental observations and the numerical simulation results. In order to do this, the parameters identification framework combines an optimization algorithm, responsible for updating the material parameters, with an objective function, responsible for defining the gap between experimental and numerical data and that will control the all optimization process. During the identification procedure, the material parameters set is iteratively updated with the purpose of minimizing the objective function value. Figure 2.3 depicts a general FEMU strategy applied in material parameters identification.

An evolution relatively to the experimental data considered in the identification strategies was mainly due to the development of FFM methods. Note that FFM methods, that have emerged in the last years (c.f. an overview in Grédiac (2004)), directly provide displacement or strain field data on the overall surface of the specimen. As a result, it becomes possible to analyze complex mechanical tests developing heterogeneous strain fields. Within FFM methods, Digital Image Correlation (DIC) technique is widely used due to its simplicity and versatility (Périé et al., 2009). Several works made use of DIC technique for different purposes, such as damage analysis (Tarigopula et al., 2008; Périé et al., 2009; Abbassi et al., 2013), experimental validation of models (Teaca et al., 2010; Souto et al., 2015) or identification of material parameters (Belhabib et al., 2008; Cooreman et al., 2008; Pottier et al., 2011a; Haddadi and Belhabib, 2012; Tardif and Kyriakides, 2012; Zhang et al., 2014).

### Plane strain tensile test

The plane strain tension characterizes the material behavior between uniaxial and equibiaxial tension, as pointed out in Figure 2.2, and consists of a typical deformation path encountered in sheet metal forming. Bending of wide sheets or tension in a cylindrical cup wall are examples involving this kind of deformation. Furthermore, it was estimated that 85% of stretching failures in automotive stamping occurs near the plane strain state (Ayres et al., 1978), which reveals the importance of this strain state in sheet forming operations. In plane strain tension, the Cauchy stress and logarithmic strain tensors in the frame ( Ιx , Ιy , Ιz ) can be defined by Wagoner (1981) designed both specimen geometries shown in Figure 2.13 a) with the aim of characterizing plane strain tension. These specimens were subjected to tensile loading using a universal testing machine and near-plane strain tension was obtained in the center of the samples.

Kuwabara and Ikeda proposed an adapted cruciform specimen for characterizing plane strain tension (Kuwabara, 2007), as depicted in Figure 2.13 b). The tensile direction is parallel to the shorter arms of the specimen and the strain component of the longer arms of the specimen is maintained at zero using a closed-loop hydraulic system (Kuwabara, 2007).

#### Hydraulic bulge test

Biaxial tests are essential to provide a reliable mechanical characterization of the material since multi-directional deformation is verified in sheet metal forming. Uniaxial tests only characterize the mechanical properties in a single direction and, usually, such experimental data is not enough to accurately describe complex sheet forming processes, like deep drawing. Biaxial tests, such as the hydraulic bulge test, are used to quantify and clarify the effects of stress biaxiality on monotonic stress-strain curves and forming limit diagrams (Makinde et al., 1992).

The hydraulic bulge test is a testing method where a circular or elliptical blank is clamped around the edges and subjected to an increasing fluid pressure, as illustrated in Figure 2.14. This experiment was initially introduced by Prater and Read (Prater and Read, 1949; Dudderar et al., 1977) and several improvements have since been performed, namely, a biaxial extensometer developed by Johnson and Duncan (Johnson and Duncan, 1965; Banabic et al., 2010) and automated hydraulic bulge testers such as the ones developed by Golongrac (Golongrac, 1975; Kuwabara, 2007), Young et al. (1981) and Koh (2008). For example, the bulge testing device proposed by Young et al. (1981) allows to acquire true stress-strain data by controlling the strain rate deformation, while the bulge test apparatus developed by Koh (2008) allows to accommodate material strength up to 1000 MPa with plate thickness between 1.0 and 1.8 mm. Concerning the bulge experiment, the Cauchy stress and logarithmic strain tensors in the frame ( Ιx , Ιy , Ιz ) can be written by The hydraulic bulge test has been applied to provide flow stress properties at large strain levels as well as to determine the equibiaxial yield stress (σb). This test can give equivalent plastic strain levels up to 0.7 whereas the maximum achievable strain in homogeneous tensile tests is reported to be limited about of 0.3 (Gutscher et al., 2004; Koç et al., 2011). In addition, the experimental data are very reproducible, in particular, the pressure and local strain relation. However, for an accurate measure of biaxial stress-strain curve, the geometry of the bulge must be taken into account (Ranta-Eskola, 1979).

**Table of contents :**

**I – Outline **

**1 General frame **

1.1 Motivation

1.2 Aims of this thesis

1.3 Reading guide

**II – Literature survey **

**2 Parameters identification strategies **

2.1 Introduction

2.2 Inverse methodologies for material parameters identification

2.2.1 Homogeneous approach

2.2.1.1 Tensile test

2.2.1.2 Shear test

2.2.1.3 Plane strain tensile test

2.2.1.4 Hydraulic bulge test

2.2.1.5 Disk compression test

2.2.2 Non-homogeneous approach

2.2.2.1 FEMU identification strategy

2.2.2.2 Other identification strategies

2.3 Conclusion

**3 Parameter and shape optimization **

3.1 Introduction

3.2 Optimization problem

3.2.1 Optimization methods

3.2.1.1 Levenberg-Marquardt algorithm

3.2.1.2 Nelder-Mead algorithm

3.3 Conclusion

**4 Mechanical model up to rupture **

4.1 Introduction

4.2 Continuum mechanics

4.3 Anisotropy and hardening modeling

4.4 Macroscopic rupture criteria

4.5 Conclusion

**III – Mechanical characterization of mild steel **

**5 Material parameters identification and rupture **

5.1 Introduction

5.2 Parameters identification process

5.2.1 General structure

5.2.2 Objective function

5.2.3 Numerical simulation of the tests

5.2.4 Process parameters

5.3 Numerical results

5.4 Macroscopic rupture calibration

5.4.1 Experimental data

5.4.2 Numerical modeling

5.4.3 Critical value determination

5.5 Conclusion

**6 Validation in deep drawing **

6.1 Introduction

6.2 Deep drawing experiment

6.3 Numerical modeling

6.4 Results

6.5 Conclusion

**IV – Computational design of mechanical tests **

**7 Development of an indicator**

7.1 Introduction

7.2 Features of the indicator

7.2.1 Strain state range and heterogeneity

7.2.2 Strain level

7.3 Mathematical formulation

7.4 Numerical tests evaluation

7.5 Results

7.5.1 Main characteristics of the tests

7.5.2 Calculation of the indicator

7.5.3 Robustness of the indicator

7.6 Material parameters sensitivity

7.7 Conclusion

**8 Design of mechanical tests **

8.1 Introduction

8.2 Strain field optimization

8.2.1 Test design using rigid tools

8.2.1.1 Automatic model generation

8.2.1.2 Optimization framework

8.2.1.3 Cost function definition

8.2.1.4 Optimization conditions

8.2.1.5 Results

8.2.2 Test design based on local displacements

8.2.2.1 Automatic model generation

8.2.2.2 Optimization framework

8.2.2.3 Cost function definition

8.2.2.4 Optimization conditions

8.2.2.5 Results

8.3 Parameters identification using a heterogeneous test

8.3.1 Numerical model of the butterfly test

8.3.2 Process parameters

8.3.3 Results

8.3.4 Numerical reproduction of the experimental data

8.4 Conclusion

**V – Final remarks **

**9 Conclusions and future works**

9.1 General conclusions

9.2 Future works

**A. Kinematic hardening implementation in a UMAT subroutine **

A.1 General equations

A.2 Validation of kinematic hardening model

**B. Bulge test analysis **

B.1 Numerical model for the bulge test

B.2 Validation of the stress calculation

**C. Shape optimization of the rectangular shear specimen **

C.1 Material and numerical validation

C.2 Numerical modeling and rupture criterion

C.3 Shape optimization process

C.4 Results and discussion

C.5 Conclusion

**D. Cubic splines definition **

**References **