# Inverse identification of anisotropic plasticity model parameters using FEMU and a heterogeneous test

Mafalda Gonçalves, Sandrine Thuillier, António Andrade-Campos

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Mafalda Gonçalves, Sandrine Thuillier, António Andrade-Campos. Inverse identification of anisotropic plasticity model parameters using FEMU and a heterogeneous test. ESAFORM 2025, May 2025, Paestum, Italy. pp.1548-1557, 10.21741/9781644903599-167. hal-05062346

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# Inverse identification of anisotropic plasticity model parameters using FEMU and a heterogeneous test

Mafalda Gonçalves¹,ª*, Sandrine Thuillier²,ᵇ and António Andrade-Campos¹,ᶜ

¹Centre for Mechanical Technology and Automation (TEMA), LASI - Intelligent Systems
Associate Laboratory, Department of Mechanical Engineering, University of Aveiro, Portugal

²Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France

ªmafalda.goncalves@ua.pt, ᵇsandrine.thuillier@univ-ubs.fr, ᶜgilac@ua.pt

**Keywords:** Heterogeneous Test, Anisotropic Plasticity, Inverse Identification

**Abstract.** Accurate characterization of the anisotropic plastic behavior of sheet metals is critical for predicting their performance in forming operations. Traditional testing methods often struggle to capture complex material responses under varied loading conditions. This study focuses on combining a heterogeneous mechanical test with the Finite Element Model Updating (FEMU) technique to inversely identify the parameters from an anisotropic plasticity model. The experiments employed a specimen geometry designed through topology optimization – TopOpt, from which full-field data was extracted using Digital Image Correlation (DIC) from both surfaces. Numerical simulation was conducted using a finite element model reproducing the experimental boundary conditions, with material parameters calibrated through quasi-homogeneous tests assuming isotropic behavior. The FEMU procedure was then applied to calibrate the Swift-Voce hardening law and the Yld2004-18p yield criterion, using experimental data from both surfaces of the TopOpt specimen. This work allowed for the evaluation of the TopOpt specimen's capability to provide sufficient data for parameter identification with the results highlighting the complexity of calibrating such an advanced material model with a single experiment.

**Introduction**

Accurately characterizing the material behavior of sheet metals is critical for predicting their performance in forming operations, particularly in industries such as automotive and aerospace. These processes often involve complex material behaviors, including strain hardening and plastic anisotropy, which require advanced constitutive models capable of capturing these phenomena. Traditional approaches to characterize anisotropic plasticity rely on standard mechanical tests, such as uniaxial tension or simple shear tests, often conducted in multiple orientations. While effective, these methods can be time-consuming and resource-intensive, requiring a series of experiments to fully capture the material's anisotropic behavior.

Heterogeneous mechanical tests have emerged as a promising alternative to traditional methods, offering the ability to capture a broad range of strain and stress states in a single experiment. These tests reduce the number of experiments required and provide a more comprehensive representation of material behavior under complex loading conditions. Unlike quasi-homogeneous tests, where strain distributions are relatively uniform and directly linked to applied forces, heterogeneous tests exhibit non-uniform stress and strain distributions [1]. This complexity excludes direct analytical solutions, necessitating the use of advanced techniques such as Digital Image Correlation (DIC) [2] to extract full-field measurements and inverse methods for material parameter identification.

Among inverse identification techniques, the Finite Element Model Updating (FEMU) technique [3] has gained significant attention. FEMU iteratively updates material parameters until obtaining the minimum discrepancy between experimental data and numerical simulation predictions, enabling an accurate identification of constitutive model parameters. Previous studies

<figure><img src="image_2.png" /></figure>
have applied FEMU to virtual data with success [4-6], but fewer investigations have focused on using experimental data, particularly from heterogeneous tests, to characterize anisotropic plasticity [7,8]. Moreover, challenges remain in identifying all model parameters to characterize the anisotropic plasticity behavior using data from a single heterogeneous experiment.

This study focuses on the TopOpt test, an advanced heterogeneous test geometry designed by a topology-based optimization approach to achieve the highest heterogeneity of strain states [9]. This test exhibits a characteristic behavior due to its out-of-plane deformation, resulting in distinct mechanical responses on each surface [10]. A finite element model of the test is implemented using the experimental boundary conditions that were retrieved from the full-field data captured by digital image correlation [11]. A numerical simulation is conducted using the parameters calibrated from quasi-homogeneous mechanical tests, assuming isotropic behavior. The potential of the TopOpt test to provide sufficient information for characterizing the anisotropic plasticity of sheet metals is then evaluated using FEMU, based on the previous numerical model. The model parameters from the Swift-Voce hardening law and the Yld2004-18p yield criteria are identified simultaneously.

## Experimental procedure and numerical model

Specimen geometry and material model. The TopOpt test specimen, designed using a topology optimization approach [9], is presented in Figure 1 along with its general dimensions. A rectangular section of 55 mm in length was added to both ends of the specimen. The final geometry was fabricated using water jet cutting.

<figure><img src="image_3.png" /><figcaption>Figure 1 – Specimen geometry and general dimensions.</figcaption></figure>

The material investigated in this work is a dual-phase DP600 steel, provided in sheets of thickness 0.8 mm. The mechanical properties of this material were determined through tensile tests conducted at various orientations relative to the rolling direction, as summarized in Table 1.

<table><caption>Table 1 – Mechanical properties of the DP600 steel used in this work.</caption><thead><tr><th colspan="4">DP600 steel</th></tr><tr><th><i>R</i><sub>m</sub> [MPa]</th><th><i>r&#773;</i> [-]</th><th>Δ<i>r</i> [-]</th><th><i>t</i> [mm]</th></tr></thead><tbody><tr><td>661.9</td><td>0.999</td><td>0.085</td><td>0.8</td></tr></tbody></table>

Given the determined mechanical properties of DP600, appropriate constitutive models were selected to accurately reproduce virtually its behavior. The isotropic linear elastic behavior is reproduced by Hooke's law while the anisotropic plastic behavior, assuming isotropic hardening, can be expressed by the yield function given as:

$$f(\boldsymbol{\sigma}, \boldsymbol{\varepsilon}^{\text{P}}) = \bar{\sigma}(\boldsymbol{\sigma}) - \sigma_y(\bar{\boldsymbol{\varepsilon}}^{\text{P}}), \quad (1)$$

where $\bar{\sigma}(\boldsymbol{\sigma})$ stands for the equivalent stress defined based on the Cauchy stress tensor $\boldsymbol{\sigma}$. The evolution of the flow stress, $\sigma_y$, defined as a function of the equivalent plastic strain, $\bar{\boldsymbol{\varepsilon}}^{\text{P}}$, is described according to the Swift-Voce hardening law as
$$ \sigma_y(\bar{\varepsilon}^p) = (1 - \alpha)(K(\varepsilon_0 + \bar{\varepsilon}^p)^n) + \alpha(\sigma_0 + Q(1 - \exp(-b\bar{\varepsilon}^p))), \quad (2) $$

where the material parameters $\alpha$, $K$, $\varepsilon_0$, $n$, $\sigma_0$, $Q$, and $b$ need to be identified according to the material under study. This material model consists of a linear combination of two hardening models to capture both the rapid initial increase in stress from the Swift (first term) and the asymptotic behavior of the Voce (second term) laws. Combining the advantages of both models, the parameter $\alpha$ represents the weight coefficient, assuming values between 0 and 1.

The Yld2004-18p yield criterion, proposed by Barlat et al. [12], was used in this work. It is formulated for a 3D case with 18 anisotropic coefficients as follows

$$ 4\bar{\sigma}^a = |s'_1 - s''_1|^a + |s'_1 - s''_2|^a + |s'_1 - s''_3|^a + |s'_2 - s''_1|^a + |s'_2 - s''_2|^a + |s'_2 - s''_3|^a + |s'_3 - s''_1|^a + |s'_3 - s''_2|^a + |s'_3 - s''_3|^a, \quad (3) $$

where $s'_i$ and $s''_i$ are the eigenvalues of the tensors $\mathbf{S}'$ and $\mathbf{S}''$. These tensors are computed based on the actual stress tensor, $\sigma$, after two linear transformations

$$ \mathbf{S}' = \mathbf{C}'\mathbf{T}\sigma \quad \text{and} \quad \mathbf{S}'' = \mathbf{C}''\mathbf{T}\sigma, \quad (4) \quad (5) $$

where $\mathbf{T}$ is a standard linear transformation. $\mathbf{C}'$ and $\mathbf{C}''$ contain 18 anisotropic model parameters and are given as

$$ \mathbf{C}' = \begin{bmatrix} 0 & -c'_{12} & -c'_{13} & 0 & 0 & 0 \\ -c'_{21} & 0 & -c'_{23} & 0 & 0 & 0 \\ -c'_{31} & -c'_{32} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c'_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & c'_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & c'_{66} \end{bmatrix} \quad (6) $$

and

$$ \mathbf{C}'' = \begin{bmatrix} 0 & -c''_{12} & -c''_{13} & 0 & 0 & 0 \\ -c''_{21} & 0 & -c''_{23} & 0 & 0 & 0 \\ -c''_{31} & -c''_{32} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c''_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & c''_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & c''_{66} \end{bmatrix} \quad (7) $$

where $c'_{ij}$ and $c''_{ij}$ with $i, j = 1, ..., 6$ are the constitutive model parameters that define the Yld2004-18p yield criterion. Since through-thickness shear measurements do not exist, the value of 1.0 has been assigned to the parameters $c''_{44}$, $c''_{55}$, $c''_{44}$ and $c''_{55}$, assuming isotropic behavior through the thickness.

Experimental testing. A uniaxial tensile loading test was carried out, in which a uniaxial external load was applied along the Y-axis of specimen, oriented at 45° to the rolling direction (cf. Fig. 1). Experimental tests were conducted on an Instron 5969 testing machine with a 50 kN load cell under quasi-static conditions. A displacement speed of 3 mm/min was used to ensure a consistent strain evolution during testing.

Full-field strain measurements were obtained using two stereo digital image correlation systems, each one composed of two 5-megapixel Basler acA2440-35µm cameras (2048 x 2448
pixels resolution) capturing images at 1 Hz. The specimen was prepared with a random speckle pattern applied on both surfaces using a white base and black speckles for optimal contrast. Cameras were vertically positioned and calibrated for an accurate capture of the out-of-plane displacement, a characteristic feature of the TopOpt test. This behavior is characterized by an out-of-plane movement of the specimen throughout the test and Figure 2 illustrates its evolution in the transverse direction at different time steps until rupture. Two independent DIC systems monitored both the front and back surfaces of the TopOpt specimen, which exhibited distinct mechanical behaviors due to out-of-plane deformation of the specimen during the test.

Experimental data was collected using VIC-3D software, and post-processing was performed with MatchID software. Optimal DIC settings, including correlation algorithm, subset size, and step size, were determined through a performance analysis to balance noise and signal reconstruction. Further details on testing conditions and the DIC setups are available in [11].

<figure><img src="image_4.png" /><figcaption>Figure 2 – Evolution of the out-of-plane displacement across the transversal direction through the test time, from the grey (no grips' displacement) to the black (end of the test) lines.</figcaption></figure>

FE modeling. A finite element model of the test was developed to replicate the behavior of the TopOpt specimen subjected to uniaxial tensile loading (cf. Fig. 1). The model does not include the section for accommodating the grips, focusing instead on the active test region. Material behavior was implemented using a User-defined Material subroutine (UMAT) in Abaqus/Standard.

Full-field displacement maps of the specimen's surfaces were obtained experimentally due to DIC. To reduce computational costs, a subset of equally spaced timesteps was selected, with corresponding times extracted to align numerical simulations with experimental data. The DIC frame was aligned with the finite element model frame using MatchID software to ensure proper transformation after post-processing.

Boundary conditions were derived directly from the experimental x-, y-, and z-displacement maps. Displacement values at the DIC points along the top and bottom edges of the specimen were extracted at the last frame. Due to field of view limitations, displacements from noise-free regions near the edges were extracted and interpolated onto finite element mesh nodes using a nearest-neighbor method, with extrapolation for regions outside the field of view. A smoothing procedure was applied to avoid abrupt peaks in the applied displacements.

Displacement-driven simulations were conducted, applying the interpolated x-, y-, and z-displacements to the nodes along the top and bottom edges of the finite element model, corresponding to the values of the last frame of the experimental test. This approach ensured that the numerical boundary conditions closely matched the experimental deformations, providing an accurate representation of the physical behavior in the simulations.
The TopOpt finite element model employed eight-node linear brick elements with reduced integration, ensuring appropriate element types for the complex geometry and loading conditions. The model consisted of 57 086 elements distributed across three layers to capture the out-of-plane behavior accurately. Although experimental full-field data was available on both surfaces of the TopOpt specimen, a single finite element model was used, with numerical information extracted from either the front or back surfaces depending on the surface being analyzed.

<figure><img src="image_5.png" /><figcaption>Figure 3 – Experimental boundary conditions applied to the finite element model: (a) finite element mesh and (b) vertical displacement at the last frame extracted from experimental data and interpolated to the FE nodes at the top edge.</figcaption></figure>

## Inverse identification process

The finite element model updating is a powerful technique employed to identify the material model parameters that minimize the discrepancies between numerical predictions and experimental data. The process typically involves setting up an initial finite element model with the reference parameters and comparing the obtained data from the numerical simulation to experimental measurements. The differences, or residuals, between the simulated and experimental data are used to compute the objective function given as

$$ \Phi(\chi) = \frac{1}{n_t} \sum_{i=1}^{n_t} \left\{ \frac{1}{3n_p} \sum_{j=1}^{n_p} \left[ \left( \frac{\varepsilon_{xx}^{\text{num}}(\chi) - \varepsilon_{xx}^{\text{exp}}}{\varepsilon_{\text{max}}^{\text{exp}}} \right)^2 + \left( \frac{\varepsilon_{yy}^{\text{num}}(\chi) - \varepsilon_{yy}^{\text{exp}}}{\varepsilon_{\text{max}}^{\text{exp}}} \right)^2 + \left( \frac{\varepsilon_{xy}^{\text{num}}(\chi) - \varepsilon_{xy}^{\text{exp}}}{\varepsilon_{\text{max}}^{\text{exp}}} \right)^2 \right]_j + \left( \frac{F^{\text{num}}(\chi) - F^{\text{exp}}}{F_{\text{max}}^{\text{exp}}} \right)^2 \right\}_i \quad (8) $$

which the optimization algorithm seeks to minimize. The cost function quantifies the difference between experimental and simulated data using the $\chi$ vector of the material parameters. $F$ corresponds to the grips' reaction force that was recorded using a load cell. Regarding the strain components $\varepsilon_{xx}$, $\varepsilon_{yy}$, and $\varepsilon_{xy}$, these are computed based on the displacement fields, both numerical and experimental, by the identification software. To enable a straightforward comparison of these quantities, the experimental displacements at the DIC points were interpolated to the FE nodes. The superscripts num and exp refer to the data iteratively generated along the numerical optimization process and to the experimental data, respectively. $n_t$ is the number of time instances decided to be used by the authors and $n_p$ is the number of measured points. $F_{\text{max}}^{\text{exp}}$ refers to the maximum load value achieved by the test and $\varepsilon_{\text{max}}^{\text{exp}}$ to the maximum strain value of all the strain components.
## Results and discussion

Numerical predictions. Numerical simulation was carried out with the finite element model presented previously and with a set of Swift-Voce hardening law parameters calibrated using quasi-homogeneous tests, which are presented in Table 2 [13]. The test was carried out under the assumption of isotropic material behavior, since the used material presents a weak in-plane anisotropy, as can be noted in Table 1. The objective was to understand if the overall behavior of the test could be reproduced numerically since the experimental boundary conditions are applied in the numerical model.

<table><caption>Table 2 – Elastic properties and Swift-Voce hardening law parameters for the DP600 steel calibrated using quasi-homogeneous tests.</caption><thead><tr><th colspan="3">Elastic properties</th><th colspan="5">Swift-Voce hardening law parameters</th></tr><tr><th><i>E</i> [MPa]</th><th><i>ν</i></th><th>α</th><th>σ<sub>0</sub>[MPa]</th><th><i>Q</i></th><th><i>b</i></th><th><i>K</i>[MPa]</th><th>ε<sub>0</sub></th><th><i>n</i></th></tr></thead><tbody><tr><td>209990</td><td>0.3</td><td>0.38</td><td>583.01</td><td>467.33</td><td>13.06</td><td>1115.15</td><td>0.0013</td><td>0.31</td></tr></tbody></table>

Using these parameters, a simulation was performed to replicate the test behavior under conditions analogous to the experiments. Figure 4 depicts the comparison between the experimental load and the reaction force extracted from the numerical simulation throughout the test. The average displacement for both setups was calculated as the mean displacement recorded by three virtual extensometers tracking the specimen's deformation. The numerical fields (second row), titled as Reference, are plotted against the experimental ones (first row) in Figure 6. The results show a close correlation between experimental data and numerical predictions, particularly on the load evolution. There are some differences on the magnitude of the strain components that point out the importance of an accurate parameter identification for achieving realistic predictions.

<figure><img src="image_6.png" /><figcaption>Figure 4 – Comparison between the experimental load and the reaction force extracted from the simulation throughout the test.</figcaption></figure>

Parameter identification. The results of the parameter identification procedure with the TopOpt test are presented in Table 3 with the obtained hardening law curve and yield criterion plotted in Figure 5. The hardening law calibrated using quasi-homogeneous tests and the von Mises yield criterion are presented for comparison purposes since this behavior was used as starting point for the optimization.
<table><caption>Table 3 – Identified constitutive parameters from the Swift-Voce hardening law and Yld2004-18p yield criterion using the TopOpt test.</caption><thead><tr><th colspan="11">TopOpt test – Identified parameters</th></tr><tr><th>c'<sub>12</sub></th><th>c'<sub>13</sub></th><th>c'<sub>21</sub></th><th>c'<sub>23</sub></th><th>c'<sub>31</sub></th><th>c'<sub>32</sub></th><th>c'<sub>66</sub></th><th>c''<sub>12</sub></th><th>c''<sub>13</sub></th><th>c''<sub>21</sub></th><th>c''<sub>23</sub></th></tr></thead><tbody><tr><td>0.800</td><td>0.800</td><td>1.200</td><td>1.191</td><td>1.199</td><td>1.199</td><td>1.086</td><td>1.200</td><td>1.199</td><td>0.800</td><td>0.802</td></tr><tr><th>c''<sub>31</sub></th><th>c''<sub>32</sub></th><th>c''<sub>66</sub></th><th>α</th><th>σ<sub>0</sub></th><th>Q</th><th>b</th><th>K</th><th>ε<sub>0</sub></th><th>n</th><th>φ</th></tr><tr><td>0.800</td><td>0.800</td><td>0.850</td><td>0.31</td><td>677.87</td><td>413.08</td><td>12.69</td><td>1050.68</td><td>0.0013</td><td>0.372</td><td>2.5x10<sup>-2</sup></td></tr></tbody></table>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Figure 5 – Identified behavior with the TopOpt test: (a) Swift-Voce hardening law and (b) Yld2004-18p yield criterion. The hardening law calibrated from quasi-homogeneous tests and the von Mises yield criterion are presented as a reference solution.</figcaption></figure>

Figure 7 presents the experimental load evolution during the TopOpt test compared to the one extracted from the numerical simulation with the identified parameters. The calibrated model closely follows the experimental load-displacement curve throughout the test, with some differences in the plastic region. The predicted numerical fields (strain components and out-of-plane displacement) are plotted at the last row of Figure 6.
<figure><img src="image_9.png" /><figcaption>Figure 6 – Comparison between experimental and numerical data: (a) $\varepsilon_{xx}$, (b) $\varepsilon_{yy}$, (c) $\varepsilon_{xy}$ and (d) W. The Reference and Identified maps result from the numerical simulation with the parameters on Table 2 and Table 3, respectively.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 7 – Comparison of the experimental load and the rection force extracted from the simulation with the identified parameters.</figcaption></figure>

The numerical model with the identified behavior was able to reproduce the overall behavior of the test, despite its complex behavior. However, some differences in the magnitude values of the strain components and out-of-plane displacement exist and several factors may contribute to this discrepancy. First, the FEMU cost function does not incorporate out-of-plane components, such as strain components or displacement, limiting its accuracy in capturing the three-dimensional material response. Second, the strain computations from both experimental and numerical displacements assume plane stress conditions, which may oversimplify the complex stress state in the TopOpt test. These limitations highlight areas for methodological improvement in future studies.

**Concluding remarks**

This study investigated the TopOpt test, an innovative heterogeneous mechanical test designed through topology optimization, and its application in identifying parameters for advanced anisotropic plasticity models. By gathering digital image correlation and the finite element model updating technique, the framework aimed to calibrate the Swift-Voce hardening law and the Yld2004-18p yield criterion from experimental data on the TopOpt test. Initial numerical simulations used material model parameters calibrated from quasi-homogeneous tests under isotropic assumptions, serving as a baseline for comparison. The results demonstrated that the overall behavior of the test is able to be reproduced numerically. However, discrepancies between the experimental and numerical results in terms of load evolution and strain magnitude pointed to methodological shortcomings.

**Acknowledgements**

This project has received funding from the Research Fund for Coal and Steel under grant agreement No 888153. The authors also acknowledge the financial support under the projects UIDB/00481/2020 and UIDP/00481/2020 – FCT – Fundação para a Ciência e Tecnologia; and CENTRO-01-0145-FEDER-022083 – Centro Portugal Regional Operational Programme (Centro2020), under the PORTUGAL 2020 Partnership Agreement through the European Regional Development Fund. M. Gonçalves is grateful to the FCT for the Ph.D. grant Ref. UI/BD/151257/2021 and ESAFORM association for the mobility grant.

**Disclaimer**

The results reflect only the authors' view, and the European Commission is not responsible for any use that may be made of the information it contains.
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