ESAFORM 2023 M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos Centre for Mechanical Technology and Automation (TEMA), Department of Mechanical Engineering University of Aveiro, Portugal Univ. Bretagne Sud, UMR CNRS 6027, IRDL F-56100 Lorient, France

1. On the comparison of heterogeneous
mechanical tests for sheet metal
characterization
M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
1 2 1
1,2
Centre for Mechanical Technology and Automation (TEMA),
Department of Mechanical Engineering
University of Aveiro, Portugal
Univ. Bretagne Sud, UMR CNRS 6027, IRDL
F-56100 Lorient, France
1
2
International ESAFORM Conference on Material Forming, 19-21 April 2023, Krakow, Poland

2. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Framework
Sheet metal forming
Sheet metal forming
processes
Crucial for part manufacturing in several
industries
Process virtualization
Industry requirement for reducing time,
costs and material waste
Realistic simulations & material
behavior reproduction
Need for an accurate characterization
of complex material behaviors
2

3. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Framework
Characterizing sheet metal behavior
Material Testing 2.0 [Pierron et al., 2021]
Use optimized test geometries that are designed to make model calibration procedures more cost-
efficient using full-field measurements and inverse identification techniques.
Replace classical mechanical tests and the use of strain gauges and extensometers.
3

4. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Framework
Why optimized tests?
Also called Heterogeneous mechanical tests.
These are known for presenting complex boundary conditions or geometries and, therefore,
presenting a large diversity of mechanical phenomena with just a single experiment.
The choice of the best test design to characterize a chosen material behavior is not straightforward.
4
[Bertin et al., 2016] [Jones et al., 2018] [Pottier et al., 2012] [Souto et al., 2016] [Barroqueiro et al., 2020]

5. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Framework
Objectives
This work aims at proposing Key Performance Indicators (KPIs) for ranking mechanical tests to improve
material behavior characterization and model calibration procedures.
▪ Strain field richness
▪ Strain states heterogeneity
▪ Test sensitivity to anisotropy
Three advanced mechanical tests are chosen to be analyzed and compared considering the quantity and
quality of information about the material behavior these can provide.
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6. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Key Performance Indicators
Strain field richness
Table 1 – Absolute values and relative weights for the adjustment and
normalization of the 𝐼t indicator terms.
The mechanical indicator 𝐼t evaluates the strain field richness based on the (i) strain state range and
heterogeneity and (ii) plastic strain level.
𝜀2/𝜀1 - Principal strains’ ratio
ҧ
𝜀p
, ҧ
𝜀max
p
- Equivalent plastic strain distribution and maximum value
6
𝐼t = 𝑤r1
Std( Τ
𝜀2 𝜀1)
𝑤a1
+ 𝑤r2
Τ
𝜀2 𝜀1 R
𝑤a2
+ 𝑤r3
Std(ത
𝜀p)
𝑤a3
+ 𝑤r4
ത
𝜀max
p
𝑤a4
+ 𝑤r5
Av(ത
𝜀p)
𝑤a5

7. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Key Performance Indicators
Strain states heterogeneity
The mechanical indicator 𝐼b evaluates the strain states heterogeneity (tension, compression and shear).
Based on the equivalent von Mises stress distribution, it penalizes the existence of stress concentrations.
7
𝐼b = ෑ
𝑠=1
3
3
σ𝑒=1
𝑛
𝑋𝑒
෍
𝑒=1
𝑛
( 𝑠
𝛿𝑒𝑍𝑒𝑋𝑒)
𝑠
𝛿𝑒 - 1 or 0 if the material point is at the strain state in evaluation
[Barroqueiro et al., 2020]

8. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Key Performance Indicators
Strain states heterogeneity
The mechanical indicator 𝐼b evaluates the strain states heterogeneity (tension, compression and shear).
Based on the equivalent von Mises stress distribution, it penalizes the existence of stress concentrations.
𝑠
𝛿𝑒 - 1 or 0 if the material point is at the strain state in evaluation
𝑋𝑒 - Element volume
8
𝐼b = ෑ
𝑠=1
3
3
σ𝑒=1
𝑛
𝑋𝑒
෍
𝑒=1
𝑛
( 𝑠
𝛿𝑒𝑍𝑒𝑋𝑒)
[Barroqueiro et al., 2020]

9. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Key Performance Indicators
Strain states heterogeneity
The mechanical indicator 𝐼b evaluates the strain states heterogeneity (tension, compression and shear).
Based on the equivalent von Mises stress distribution, it penalizes the existence of stress concentrations.
𝑠
𝛿𝑒 - 1 or 0 if the material point is at the strain state in evaluation
𝑋𝑒 - Element volume
9
𝐼b = ෑ
𝑠=1
3
3
σ𝑒=1
𝑛
𝑋𝑒
෍
𝑒=1
𝑛
( 𝑠
𝛿𝑒𝑍𝑒𝑋𝑒)
𝑍𝑒 =
1
1 + 𝑏𝜎𝑒
∗ 2 𝜎𝑒
∗
=
𝜎VM,𝑒 − ത
𝜎VM
ത
𝜎VM
[Barroqueiro et al., 2020]
ത
𝜎VM, 𝜎𝑉𝑀,𝑒 - Mean and element equivalent von Mises stress
𝑏 - “aggressiveness” parameter

10. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Key Performance Indicators
Test sensitivity to anisotropy
The rotation angle γ allow to assess the sensitivity of the test to anisotropy.
It is based on the principal angle β and covers all the stress states in the Mohr circle in two conditions.
It ranges from 0 to 90.
𝜎1, 𝜎2 - Major and minor principal stresses
𝜎11, 𝜎22, 𝜎12- Stress components in the material frame
10
𝑞 =
𝜎11 − 𝜎22
|𝜎11 − 𝜎22|
𝜎1 − 𝜎2
| 𝜎1 − 𝜎2 |
𝛾 = ቊ
45
45 1 − 𝑞 + 𝑞 𝛽
if 𝜎𝑥𝑥 = 𝜎𝑦𝑦 and 𝜎𝑥𝑦 ≠ 0
otherwise
[Oliveira et al., 2022]

11. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Key Performance Indicators
Test sensitivity to anisotropy
The rotation angle values are usually represented in a histogram in order to analyze its distribution over
the range.
The wider the distribution, the higher sensitivity to anisotropy.
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12. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Key Performance Indicators
Test sensitivity to anisotropy
An optimal distribution would be characterized by a uniform dispersion over the whole range since it
means a higher sensitivity to anisotropy. In this case, each bin would have the same density.
This can be defined as reference (or ideal) distribution.
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13. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Key Performance Indicators
Test sensitivity to anisotropy
SR = 1 −
σ𝑗=1
𝑏
|dref,𝑗 − dR,𝑗|
2
𝑑ref,𝑗 - density of a reference bin
𝑑R,𝑗 - density of a real distribution bin
𝑏 – number of bins
It is proposed an indicator that quantifies the information given by each rotation angle distribution.
The indicator computes the difference between the reference and real distributions and achieves a higher
value if both distributions are similar.
dref,𝑗
dR,𝑗
dR,𝑗
dref,𝑗
13

14. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Analysis of advanced tests
Specimen designs
(a) Notched (b) D (c) TopOpt
[Rossi et al., 2022] [Jones et al., 2022] [Goncalves et al., 2023]
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15. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Analysis of advanced tests
Material behavior
Experiment after uniaxial tensile test Numerical simulation
A large out-of-plane movement was observed due to the occurrence of buckling.
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16. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Analysis of advanced tests
Material behavior
Notched D TopOpt
Specimen
design
Material DP600 steel, 0.8 mm thickness
Elastic
behavior
Isotropic, Hooke’s law
Hardening Isotropic hardening, Swift’s law
Yield
criterion
Yld2000-2d Yld2004-18p
Table 2 - Elastic and constitutive model parameters for Swift law, Yld200-2d
and Yld2004-18p of DP600 steel.
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17. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Analysis of advanced tests
Numerical simulation
Notched D TopOpt
Specimen design
Test type Uniaxial tensile loading
Assumptions 2D plane stress conditions 3D
Element type
Four-node shell elements with reduced integration
and hourglass control
Eight-node brick
elements
Element size 0.5 mm
Boundary conditions
x- and z- constrained
Applied displacement in y-direction at the top edge
All degrees of freedom constrained at the bottom edge
Stopping criterion Forming Limit Diagram (FLD) to predict localized necking
x
y
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18. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Results and discussion
Principal stress and strain diagrams
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19. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Results and discussion
Principal stress and strain diagrams
Principal strains diagram
▪ D and TopOpt from plane strain
tension to plane strain compression
▪ TopOpt reaches more compressive
states
19

20. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Results and discussion
Principal stress and strain diagrams
Principal stresses diagram
▪ Notched mainly uniaxial tension
▪ Equibiaxial tension to equibiaxial
compression for the others
▪ Largest range associated with the
TopOpt
20

21. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Results and discussion
Principal stress and strain diagrams
PEEQ distribution
▪ Notched with higher values localized in
the center
▪ D with a more spread distribution
▪ TopOpt with the largest area but higher
values very localized
21

22. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Results and discussion
Strain field richness
This indicator evaluates the strain heterogeneity and level in the specimens, being the D and the TopOpt
specimens the ones with the highest values.
The TopOpt specimen presents material points in every strain state considered. The bounds of the strain
state range are similar in the D and TopOpt specimens.
Compressive states are only achieved by the TopOpt specimen.
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23. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Results and discussion
Strain states heterogeneity
This indicator evaluates the diversity of strain states induced in the specimens, being the highest value
associated with the TopOpt specimen.
All specimens present a higher fraction of points in tension.
The Notched and D specimens present a higher partial value related to shear. In the TopOpt specimen, the
material points subjected to shear are in a stress concentration area which is penalized by this indicator.
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24. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Results and discussion
Test sensitivity to anisotropy
(a) Notched (b) D (c) TopOpt
The more dispersed the rotation angle distribution, the higher the sensitivity to the anisotropic behavior.
The Notched and D specimens present distributions with similar range of values. The Notched has a higher
fraction of points in elasticity concentrated at 45°.
The TopOpt presents the best distribution with material points in the plastic regime covering the whole
range.
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25. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Results and discussion
Test sensitivity to anisotropy
This indicator allows to quantify the information on the rotation angle distribution.
The Notched and D specimens present a similar range of rotation angle values and, therefore, similar
indicator values.
The TopOpt specimen presents the highest indicator value due to the highest range of rotation angle values.
Highest sensitivity to the anisotropic behavior.
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26. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Concluding remarks
Final ranking
1
2
3
It can be concluded that the TopOpt specimen presents the highest strain field richness, followed closely by
the D specimen. The highest strain state range as well as an interesting plastic strain distribution.
The highest heterogeneity of stress states was also noticed in the TopOpt.
The D and the TopOpt specimens were the ones with the largest range of rotation angle values, presenting
the higher sensitivity to the anisotropic behavior.
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27. M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
Concluding remarks
Future considerations
This work is a step closer to a more straightforward approach to choose the most suitable test for material
behavior characterization and model calibration procedures.
The proposed KPIs evaluate the diversity of mechanical phenomena presented by each specimen.
There is still a need for metrics that take into account the inverse identification quality and the extraction
quality by full-field measurement techniques.
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28. mafalda.goncalves@ua.pt
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.
Acknowledgments
Thank you!
Any questions?

29. On the comparison of heterogeneous
mechanical tests for sheet metal
characterization
M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos
1 2 1
1,2
Centre for Mechanical Technology and Automation (TEMA),
Department of Mechanical Engineering
University of Aveiro, Portugal
Univ. Bretagne Sud, UMR CNRS 6027, IRDL
F-56100 Lorient, France
1
2
International ESAFORM Conference on Material Forming, 19-21 April 2023, Krakow, Poland