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

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

Introduction
Framework
Introduction
Framework
SHEET METAL FORMING

Industry
requirements
• Costs
• Time
• Material waste
Virtual
manufacturing
processes
Numerical
simulation
Accurate material
behavior
reproduction
Introduction
Framework

Experimental data
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?
Introduction
Main challenge

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]
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
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.”
Design by topology optimization
Main idea

A compliant mechanism is known for transforming the displacement through the
deformation of its flexible members [7].
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.

Design methodology
Framework
Topological design of
compliant
mechanisms
Enrichment of the
strain field
Induce specific
mechanical states
Heterogeneous
displacement map

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
Reproduction of a uniaxial tensile loading test
Fin

Design methodology
Fin

Design methodology
uin
uout
Fin

Design methodology
Problem formulation
𝑇(𝐗) =
𝑢out
𝑢in
maximize
subject to
Design variables bounds
uin
uout
Fin
Structural equilibrium
Volume constraint

Design methodology
Test conditions
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

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

Results and analysis
Design optimization evolution

Specimen geometry

Stress states heterogeneity
Linear analysis
id = 0.0359
Nonlinear geometric
analysis
id = 0.0190
Nonlinear material and
geometric analysis
id = 0.0215
[6] [8]
[8] Gonçalves M, Andrade-Campos A, Thuillier S. ESAFORM 2022 - 25th International Conference on Material Forming, 27-29 April, Braga, Portugal

Tensile test reproduction

Stress and strain fields
Linear
Geometric
nonlinearities
Material and geometric
nonlinearities

Principal strains diagram
▪ Stress states heterogeneity
▪ Uniaxial tension
▪ Uniaxial compression
▪ Pure shear
▪ 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;

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

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

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

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