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On the topology design of a mechanical
heterogeneous specimen using geometrical and
material nonlinearities
M. Gonçalves, 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
2
1 2 1
41st International Deep Drawing Research Group Conference, 6 - 10 June 2022, Lorient, France
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Introduction
Framework
Introduction
Framework
SHEET METAL FORMING
Industry
requirements
• Costs
• Time
• Material waste
Virtual
manufacturing
processes
Numerical
simulation
Accurate material
behavior
reproduction
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Introduction
Framework
Experimental data
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Introduction
Framework
Material constitutive models
To calibrate:
Accurate material
behavior
reproduction
▪ Yield criteria
▪ Hardening laws
▪ …
Describe numerically the material behavior
Introduction
Classical mechanical tests
▪ Specific strain/stress state
▪ Homogeneous strain fields
▪ Limited data can be extracted
▪ Large number of tests required
Time-consuming and expensive
[1] Oliveira M G, Thuillier S, Andrade-Campos A. Procedia Manuf. 2020;47:831–8.
[1]
How can we make this process more
efficient?
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Introduction
Main challenge
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Introduction
Heterogeneous mechanical tests
[2] E. M. C. Jones et al., Comput. Mater. Sci., vol. 152, pp. 268–290, 2018.
[3] J. M. P. Martins et al., Int. J. Solids Struct., vol. 172-173, pp. 21-37, 2019.
[4] T. Pottier et al., Exp. Mech., vol. 52, pp. 951-963, 2012.
[5] N. Souto et al., Int. J. Mater. Form., vol. 10, pp. 353-367, 2017.
[6] M. Gonçalves et al., submitted.
▪ Inefficient design methodologies (trial and error approaches)
[2] [3] [4] [5] [6]
▪ Solutions dependent on the knowledge of the authors
▪ Nonrealistic assumptions for the test behavior
[2] [3] [4] [5] [6]
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Introduction
Heterogeneous mechanical tests
There is still a need for a mechanical test rich enough to characterize
accurately a complex material model
Nonlinear behavior
Designed by optimization
Topology optimization
Compliant mechanisms approach
Heterogeneous stress fields
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Introduction
Proposed solution
Design an innovative mechanical test
Geometrical nonlinearities
Elastoplastic material behavior
“Topology optimization is a mathematical method which spatially optimizes the distribution of
material within a defined domain, by fulfilling given constraints previously established and
minimizing a predefined cost function.”
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Design by topology optimization
Main idea
A compliant mechanism is known for transforming the displacement through the
deformation of its flexible members [7].
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Compliant mechanisms
How do they work?
[7] B. Zhu, X. Zhang, H. Zhang, J. Liang, H. Zang, H. Li, et al. Mech Mach Theory. 2020;143.
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Design methodology
Framework
Topological design of
compliant
mechanisms
Enrichment of the
strain field
Induce specific
mechanical states
Heterogeneous
displacement map
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Design methodology
Framework
u1
u2
u1 u2
(a) (b) (c)
u1 u2
In the design of compliant mechanisms, two displacements applied in predefined directions can lead to
a specific mechanical state:
(a) shear
(b) tension
(c) compression
Design methodology
Initial design geometry
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Reproduction of a uniaxial tensile loading test
Fin
Design methodology
Initial design geometry
Fin
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Design methodology
Initial design geometry
uin
uout
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Fin
Design methodology
Problem formulation
𝑇(𝐗) =
𝑢out
𝑢in
maximize
subject to
Design variables bounds
M. Gonçalves, S. Thuillier, A. Andrade-Campos
uin
uout
Fin
Structural equilibrium
Volume constraint
Design methodology
Test conditions
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Material behavior
Linear elastic behavior: E = 210 GPa and ν = 0.3
Isotropic von Mises yield criteria
Isotropic hardening - Swift law
Geometrical nonlinearities
Nonlinear relationship between displacements and strains
𝜎𝑦 = 𝐾 𝜀0 + ҧ
𝜀𝑝 n
ε0 =
σ0
K
ൗ
1
n
with
Plane stress conditions
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Solution evaluation
Heterogeneity indicator
▪ Uniaxial tension
▪ Uniaxial compression
▪ Pure shear
▪ Based on the principal strains’ relationship:
▪ Stress concentrations penalized
▪ Heterogeneity of stress states
[6]
Implementation
Overview
Nonlinear FEA
Objective
function
Sensitivity
analysis
Filtering
operations
Material layout
update
Initialize
domain
Optimal
specimen
geometry
convergence?
Heterogeneity
indicator
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Results and analysis
Design optimization evolution
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Results and analysis
Specimen geometry
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Results and analysis
Stress states heterogeneity
Linear analysis
id = 0.0359
Nonlinear geometric
analysis
id = 0.0190
Nonlinear material and
geometric analysis
id = 0.0215
M. Gonçalves, S. Thuillier, A. Andrade-Campos
[6] [8]
[8] Gonçalves M, Andrade-Campos A, Thuillier S. ESAFORM 2022 - 25th International Conference on Material Forming, 27-29 April, Braga, Portugal
Results and analysis
Tensile test reproduction
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Results and analysis
Stress and strain fields
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Linear
Geometric
nonlinearities
Material and geometric
nonlinearities
Results and analysis
Principal strains diagram
▪ Stress states heterogeneity
▪ Uniaxial tension
▪ Uniaxial compression
▪ Pure shear
M. Gonçalves, S. Thuillier, A. Andrade-Campos
▪ Large area in the plastic regime
▪ Interesting magnitude of the equivalent
plastic strain
Concluding remarks
▪ A nonlinear topology-based optimization methodology to design
heterogeneous mechanical tests is proposed;
▪ An optimal test configuration was obtained;
▪ The elastoplastic material behavior was taken into account for the test design;
▪ The use of a nonlinear analysis is proved to be required for a more correct
reproduction of the specimen behavior;
M. Gonçalves, S. Thuillier, A. Andrade-Campos
Thank you!
Any questions?
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 of the Portuguese
Foundation for Science and Technology (FCT) under the project PTDC/EME-APL/29713/2017 by
UE/FEDER through the programs CENTRO 2020 and COMPETE 2020, and UID/EMS/00481/2013-
FCT under CENTRO-01-0145-FEDER-022083.
M. Gonçalves is grateful to the FCT for the Ph.D. grant Ref. UI/BD/151257/2021.
Acknowledgments
On the topology design of a mechanical
heterogeneous specimen using geometrical and
material nonlinearities
M. Gonçalves, 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
2
1 2 1
41st International Deep Drawing Research Group Conference, 6 - 10 June 2022, Lorient, France
Sensitivity BLF’s
Linearized
buckling analysis
Nonlinear FEA
Objective
function
Sensitivity
analysis
Filtering
operations
Material layout
update
Initialize
domain
Optimal
specimen
geometry
convergence?
Heterogeneity
indicator

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On the topology design of a mechanical heterogeneous specimen using geometrical and material nonlinearities

  • 1. On the topology design of a mechanical heterogeneous specimen using geometrical and material nonlinearities M. Gonçalves, 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 2 1 2 1 41st International Deep Drawing Research Group Conference, 6 - 10 June 2022, Lorient, France
  • 2. M. Gonçalves, S. Thuillier, A. Andrade-Campos Introduction Framework Introduction Framework SHEET METAL FORMING
  • 3. Industry requirements • Costs • Time • Material waste Virtual manufacturing processes Numerical simulation Accurate material behavior reproduction M. Gonçalves, S. Thuillier, A. Andrade-Campos Introduction Framework
  • 4. Experimental data M. Gonçalves, S. Thuillier, A. Andrade-Campos Introduction Framework Material constitutive models To calibrate: Accurate material behavior reproduction ▪ Yield criteria ▪ Hardening laws ▪ … Describe numerically the material behavior
  • 5. Introduction Classical mechanical tests ▪ Specific strain/stress state ▪ Homogeneous strain fields ▪ Limited data can be extracted ▪ Large number of tests required Time-consuming and expensive [1] Oliveira M G, Thuillier S, Andrade-Campos A. Procedia Manuf. 2020;47:831–8. [1]
  • 6. How can we make this process more efficient? M. Gonçalves, S. Thuillier, A. Andrade-Campos Introduction Main challenge
  • 7. M. Gonçalves, S. Thuillier, A. Andrade-Campos Introduction Heterogeneous mechanical tests [2] E. M. C. Jones et al., Comput. Mater. Sci., vol. 152, pp. 268–290, 2018. [3] J. M. P. Martins et al., Int. J. Solids Struct., vol. 172-173, pp. 21-37, 2019. [4] T. Pottier et al., Exp. Mech., vol. 52, pp. 951-963, 2012. [5] N. Souto et al., Int. J. Mater. Form., vol. 10, pp. 353-367, 2017. [6] M. Gonçalves et al., submitted. ▪ Inefficient design methodologies (trial and error approaches) [2] [3] [4] [5] [6] ▪ Solutions dependent on the knowledge of the authors ▪ Nonrealistic assumptions for the test behavior
  • 8. [2] [3] [4] [5] [6] M. Gonçalves, S. Thuillier, A. Andrade-Campos Introduction Heterogeneous mechanical tests There is still a need for a mechanical test rich enough to characterize accurately a complex material model
  • 9. Nonlinear behavior Designed by optimization Topology optimization Compliant mechanisms approach Heterogeneous stress fields M. Gonçalves, S. Thuillier, A. Andrade-Campos Introduction Proposed solution Design an innovative mechanical test Geometrical nonlinearities Elastoplastic material behavior
  • 10. “Topology optimization is a mathematical method which spatially optimizes the distribution of material within a defined domain, by fulfilling given constraints previously established and minimizing a predefined cost function.” M. Gonçalves, S. Thuillier, A. Andrade-Campos Design by topology optimization Main idea
  • 11. A compliant mechanism is known for transforming the displacement through the deformation of its flexible members [7]. M. Gonçalves, S. Thuillier, A. Andrade-Campos Compliant mechanisms How do they work? [7] B. Zhu, X. Zhang, H. Zhang, J. Liang, H. Zang, H. Li, et al. Mech Mach Theory. 2020;143.
  • 12. M. Gonçalves, S. Thuillier, A. Andrade-Campos Design methodology Framework Topological design of compliant mechanisms Enrichment of the strain field Induce specific mechanical states Heterogeneous displacement map
  • 13. M. Gonçalves, S. Thuillier, A. Andrade-Campos Design methodology Framework u1 u2 u1 u2 (a) (b) (c) u1 u2 In the design of compliant mechanisms, two displacements applied in predefined directions can lead to a specific mechanical state: (a) shear (b) tension (c) compression
  • 14. Design methodology Initial design geometry M. Gonçalves, S. Thuillier, A. Andrade-Campos Reproduction of a uniaxial tensile loading test Fin
  • 15. Design methodology Initial design geometry Fin M. Gonçalves, S. Thuillier, A. Andrade-Campos
  • 16. Design methodology Initial design geometry uin uout M. Gonçalves, S. Thuillier, A. Andrade-Campos Fin
  • 17. Design methodology Problem formulation 𝑇(𝐗) = 𝑢out 𝑢in maximize subject to Design variables bounds M. Gonçalves, S. Thuillier, A. Andrade-Campos uin uout Fin Structural equilibrium Volume constraint
  • 18. Design methodology Test conditions M. Gonçalves, S. Thuillier, A. Andrade-Campos Material behavior Linear elastic behavior: E = 210 GPa and ν = 0.3 Isotropic von Mises yield criteria Isotropic hardening - Swift law Geometrical nonlinearities Nonlinear relationship between displacements and strains 𝜎𝑦 = 𝐾 𝜀0 + ҧ 𝜀𝑝 n ε0 = σ0 K ൗ 1 n with Plane stress conditions
  • 19. M. Gonçalves, S. Thuillier, A. Andrade-Campos Solution evaluation Heterogeneity indicator ▪ Uniaxial tension ▪ Uniaxial compression ▪ Pure shear ▪ Based on the principal strains’ relationship: ▪ Stress concentrations penalized ▪ Heterogeneity of stress states [6]
  • 21. Results and analysis Design optimization evolution M. Gonçalves, S. Thuillier, A. Andrade-Campos
  • 22. Results and analysis Specimen geometry M. Gonçalves, S. Thuillier, A. Andrade-Campos
  • 23. Results and analysis Stress states heterogeneity Linear analysis id = 0.0359 Nonlinear geometric analysis id = 0.0190 Nonlinear material and geometric analysis id = 0.0215 M. Gonçalves, S. Thuillier, A. Andrade-Campos [6] [8] [8] Gonçalves M, Andrade-Campos A, Thuillier S. ESAFORM 2022 - 25th International Conference on Material Forming, 27-29 April, Braga, Portugal
  • 24. Results and analysis Tensile test reproduction M. Gonçalves, S. Thuillier, A. Andrade-Campos
  • 25. Results and analysis Stress and strain fields M. Gonçalves, S. Thuillier, A. Andrade-Campos Linear Geometric nonlinearities Material and geometric nonlinearities
  • 26. Results and analysis Principal strains diagram ▪ Stress states heterogeneity ▪ Uniaxial tension ▪ Uniaxial compression ▪ Pure shear M. Gonçalves, S. Thuillier, A. Andrade-Campos ▪ Large area in the plastic regime ▪ Interesting magnitude of the equivalent plastic strain
  • 27. Concluding remarks ▪ A nonlinear topology-based optimization methodology to design heterogeneous mechanical tests is proposed; ▪ An optimal test configuration was obtained; ▪ The elastoplastic material behavior was taken into account for the test design; ▪ The use of a nonlinear analysis is proved to be required for a more correct reproduction of the specimen behavior; M. Gonçalves, S. Thuillier, A. Andrade-Campos
  • 28. Thank you! Any questions? 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 of the Portuguese Foundation for Science and Technology (FCT) under the project PTDC/EME-APL/29713/2017 by UE/FEDER through the programs CENTRO 2020 and COMPETE 2020, and UID/EMS/00481/2013- FCT under CENTRO-01-0145-FEDER-022083. M. Gonçalves is grateful to the FCT for the Ph.D. grant Ref. UI/BD/151257/2021. Acknowledgments
  • 29. On the topology design of a mechanical heterogeneous specimen using geometrical and material nonlinearities M. Gonçalves, 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 2 1 2 1 41st International Deep Drawing Research Group Conference, 6 - 10 June 2022, Lorient, France
  • 30. Sensitivity BLF’s Linearized buckling analysis Nonlinear FEA Objective function Sensitivity analysis Filtering operations Material layout update Initialize domain Optimal specimen geometry convergence? Heterogeneity indicator