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## Virtual fields definition and identification procedure

Defining a set of virtual fields is one of the main tasks before using VFM. In the framework of plasticity, three main strategies are currently available to select a suitable set of virtual fields ( Marek et al., 2017 ). The most commonly used strategy is manually defined virtual fields. In this strategy, the user has to develop a continuous function adapted to the problem in hands (that depends on the experimental geometry and boundary conditions), that generates the value of the virtual field over the considered domain. Usually, the functions are of polynomial or periodical basis. This approach has the disadvantage of relying on the experience of the user to develop the function. Nevertheless, this approach is easy to implement. The other two strategies were developed to overcome the disadvantage of the previous one, allowing to obtain the virtual field set automatically with some user inputs ( Pierron et al., 2010; Marek et al., 2017 ). Nevertheless, these two automatic strategies have the disadvantage of being more difficult to implement than the manual strategy and require a higher computational effort. Theref ore, in the present study, the manual approach is chosen.

### YLD20 0 0-2d identification

In this section, the combination of the VFM with the cruciform specimen is tested in a model with a more complex yield criterion, the YLD20 0 0-2d. This yield criterion is combined with Swift’s hardening law, resulting in a total of 11 parameters to be identified. Once again, the identification of the hardening and yield criterion parameters is performed simultaneously. The reference material parameters, which correspond to an aluminium alloy AA6016 in T4 state ( Yoon et al., 2004 ), are presented in Table 2 . Cr3 provided the best results for the Hill’48 yield criterion, hence it is chosen for this analysis. The same FE model of Fig. 3 c is used here. The simulation is displacement-driven, and 1 mm of displacement is applied to each arm of the specimen. This value is reduced in the present case to avoid exceeding the maximum force value. By means of a three-dimensional model, it has been confirmed that after the maximum force value, the normal stress through thickness exhibits a sudden increase, which invalidates the plane stress condition and consequently, limits the application of the VFM beyond this point. The number of elements is increased up to 3226 to better describe the strain gradients developed in the arms of the specimen. The methodology adopted in the VFM procedure uses shape functions and the displacement fields extracted from the FE model at the nodes, to compute the strain tensor at the centroid of the element (as explained in Section 2.1 ), which leads to an average strain tensor over the element. Thus, an increase in the number of elements mitigates the loss of information on the gradients present in the elements. In this case, an even finer mesh could be adopted, but no significant improvements in the results are noticed, whereas the computational time increases. Therefore, to have a computationally efficient mesh it is necessary to find a balance between identification results and computational time. In this case, the integration of the constitutive model .

#### FEMU-based approach and optimisation method

The Finite Element Model Updating (FEMU) relies on the simple idea of adjusting the unknown material parameters of a Finite Element (FE) model to minimise the difference between experimental and numerical results. Due to this straightforward idea, combined with the ease of implementation and flexibility in terms of usable data, this method has widespread adoption in many different applications.

The objective function that represents the idea behind FEMU can be built with different data, namely strain/displacements fields or load signals, or even a combination of both types of data. This flexibility in terms of data has contributed to increase the number of formulations presented in the literature. Nevertheless, it should be highlighted that the most recent ones have in common taking advantage of full-field measurements. In the present work, the adopted objective function relies on this common point and can be written as χ is the vector of unknown material parameters and F is the load signal. The superscripts “num” and “exp” define the numerical and experimental data, respectively. tests n , ti n and p n are the number of tests, time instants and number of inplane measurement points, respectively. exp max F is the maximum load value for each test and exp max is the maximum strain value of all in-plane components for each test. The Levenberg-Marquardt optimisation method is selected to minimise the objective function (Eq. 2). This method is a gradient-based method which resorts to the information of the approximated Hessian and Jacobian matrixes. The Levenberg- Marquardt method has the following form diag LM T .

**Table of contents :**

**1 Introduction **

1.1 Motivation and background

1.2 Objectives and achievements

1.3 List of publications and developed tools

1.4 Structure of the thesis

**2 State-of-the-art on inverse analysis methods **

2.1 Comparison of inverse identication strategies for constitutive mechanical models using full-eld measurements

2.2 Final Remarks

**3 VFM – Single test calibration methodology **

3.1 Identication of Material Parameters for Plasticity Models: A Comparative Study on the Finite Element Model Updating and the Virtual Fields Method

3.2 Calibration of anisotropic plasticity models using a biaxial test and the Virtual Fields Method

3.3 Calibration of anisotropic plasticity models with an optimized heterogeneous test and the Virtual Fields Method

3.4 Final Remarks

**4 Contributions to the calibration of thermo-mechanical constitutive models **

4.1 Calibration of Johnson-Cook Model Using Heterogeneous Thermo-Mechanical Tests .

4.2 Calibration of a thermo-mechanical constitutive model using the VFM and a heterogeneous test

4.3 Final Remarks

**5 Conclusions and Perspectives **

5.1 Conclusions

5.2 Perspectives for future work

**A Copyright agreements **