On the design of a heterogeneous mechanical test using a nonlinear topology optimization approach

M. Gonçalves, A. Andrade-Campos and S. Thuillier
On the design of a heterogeneous mechanical test
using a nonlinear topology optimization
approach
25th International Conference on Material Forming
27-29th April 2022
Braga, Portugal
Centre for Mechanical Tecnology and Automation, Department of Mechanical Engineering, University of Aveiro, Portugal
Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France
1
2
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1

Outline
01
02
03
04
Motivation
Design methodology
Results and Discussion
Conclusions
2
M. Gonçalves, A. Andrade-Campos and S. Thuillier – 25th International Conference on Material Forming, 27-29th April, Braga, Portugal

Full-field
measurements
Material
characterization
Heterogeneous
tests
Sheet metal
forming
processes
Innovative
design
approaches
Constitutive
models
calibration
Motivation
01
3

Full-field
measurements
Material
characterization
Heterogeneous
tests
Sheet metal
forming
processes
Innovative
design
approaches
Constitutive
models
calibration
Motivation
01
▪ Replace a whole range of standard mechanical tests
▪ Heterogeneous displacement and strain fields
▪ Larger quantity and quality of information about the
material behavior
4
[1] E. M. C. Jones et al., Comput. Mater. Sci., vol. 152, pp. 268–290, 2018.
[2] J. M. P. Martins et al., Int. J. Solids Struct., vol. 172-173, pp. 21-37, 2019.
[3] T. Pottier et al., Exp. Mech., vol. 52, pp. 951-963, 2012.
[4] N. Souto et al., Int. J. Mater. Form., vol. 10, pp. 353-367, 2017.
[5] M. Gonçalves et al., submitted.
[1] [2] [3] [4] [5]

Design of a heterogeneous mechanical test that presents a high diversity of mechanical states
▪ Nonlinear behavior
▪ Topology optimization approach
▪ Mechanism design theory
Motivation
01
5

Design methodology
02
Topological design
of compliant
mechanisms
Enrichment of the
strain field
Induce specific
mechanical states
Heterogeneous
displacement map
Framework
6

Design methodology
02
u1
u2
u1 u2
(a) (b) (c)
u1 u2
The theory of compliant mechanisms
This figure shows how two displacements applied in predefined locations can induce a specific
mechanical state: (a) shear, (b) tension, and (c) compression.
7
A compliant mechanism is known for transforming the displacement through the deformation
of its flexible members [6].
The goal is to introduce heterogeneity through the displacement field by applying
displacements in specific locations of the specimen.
[6] B. Zhu et al., Struct Multidiscip Optim. , vol. 30, pp. 125-132, 2021.

Design methodology
02
Initial design domain
uin
Kout
uout Kin
Fin
y
x
A uniaxial tensile loading test is reproduced:
▪ 𝐅in, the load applied by the grips
▪ 𝐮out and 𝐮in, the applied displacements in output and
input locations
▪ 𝐾in and 𝐾out, artificial stiffnesses to avoid numerical
issues
8

Design methodology
02
Design by Topology Optimization
9
Void, 𝜌min
Material, 1
𝑋𝑒 0 < 𝜌min ≤ 𝑋𝑒≤ 1
The design domain is defined by a discretized material
distribution of solid and void elements.
Each one is described by a design variable, its relative
density:
At each iteration, the material layout is updated, and
material is removed or added to each element until the
optimal solution is reached.

Design methodology
02
Problem formulation
𝑇(𝐗) =
𝑢out
𝑢in
maximize
subject to 𝐑 = 𝟎
σ𝑒=1
𝑀
𝑋𝑒𝑉
𝑒
σ𝑒=1
𝑀
𝑉
𝑒
− V∗
≤ 0
0 ≤ 𝜌min ≤ Xe ≤ 1, 𝑒 = 1, 2, . . , 𝑀.
9

Design methodology
02
Nonlinear finite element analysis
Non-linear FEA
Element stress update
▪ Geometric nonlinearity to account for large deformations
𝐄 = 𝐄L + 𝐄N = 𝐁L𝐔 + 𝐁N𝐔
▪ Material behavior remains linear elastic
(linear relation between strains and stresses)
10

Design methodology
02
Non-linear FEA
Element internal forces
and stiffness
Stiffness matrix assembly
▪ Stiffness matrix
(small and large displacements and initial stress state, respectively)
𝐊T = 𝐊L + 𝐊N + 𝐊S
11

Design methodology
02
Non-linear FEA
Element internal forces
and stiffness
Stiffness matrix assembly
Converged
Compute residual
𝐔k = 𝐊t
−1
𝐑
𝐔 = 𝐔 + 𝐔k
Incremental load steps
Yes
No
𝐑 = න𝐁T
𝐒 𝑑𝑉 − 𝐅ext
= 𝐅int
− 𝐅ext
▪ Balance between internal and external loads
(iterative procedure until residual reaches zero - equilibrium)
13
▪ Stiffness matrix
(small and large displacements and initial stress state, respectively)
𝐊T = 𝐊L + 𝐊N + 𝐊S

Design methodology
02
Sensitivity analysis
𝑑𝑇
𝑑𝑋
=
𝑑𝑢out 𝑢in − 𝑑uin 𝑢out
𝑢𝑖𝑛
2
𝑑𝑢in = −𝑝 1 − 𝑋𝑒 𝜌𝑚𝑖𝑛
𝑝−1
+ 𝑋𝑒 𝛌T
𝐅int
𝑑𝑢out = −𝑝 1 − 𝑋𝑒 𝜌𝑚𝑖𝑛
𝑝−1
+ 𝑋𝑒 𝛄T
𝐅int
14
▪ Objective-function
▪ Element sensitivity
𝐊T𝜸 = 𝐋out
𝑇(𝐗) =
𝑢out
𝑢in
𝐊T𝝀 = 𝐋in

Design methodology
02
Solution evaluation
𝑖𝑑 = ෑ
𝑠=1
3
3
σ𝑒=1
𝑛
𝑋𝑒
෍
𝑒=1
𝑛
𝑠
𝛿𝑒𝑍𝑒𝑋𝑒
▪ Scalar indicator to evaluate the performance of the specimen in inducing a diversity of
stress states [7]
▪ The number of points in each stress state (tension, compression and shear)
▪ Unstress material and stress concentrations are penalized
𝑍𝑒 =
1
1+ 𝑏𝜎𝑒
∗ 2 with 𝜎𝑒
∗
=
𝜎𝑒
VM− 𝜎
VM
𝜎
VM
15
𝑠
𝛿𝑒- assumes the value of 1 if the element is in the stress state s
[7] B. Barroqueiro et al., Int. J. Mech. Sci., vol. 181, p. 105764, 2020.

Results and discussion
03
Specimen geometries and stress states distribution
Obtained specimen geometries with (a) linear and (b) nonlinear analyses.
(a) 𝑖𝑑 = 0.0367 (b) 𝑖𝑑 = 0.0120
16
▪ Material similarly placed at the locations
where the displacements and load are applied
▪ Different material layouts due to the
consideration of geometrical nonlinearities in
FEA
▪ Heterogeneous stress field with material
subjected to tension, compression and shear

Results and discussion
03
Strain and stress fields
17
▪ FEA conducted in Abaqus/Standard to evaluate the performance of
the specimens
▪ Hooke’s and Swift’s isotropic hardening laws for DP600
▪ FLD criterion to predict rupture
▪ Linear solution
▪ Early rupture, presenting the maximum von Mises stress at
specific locations.
▪ Plastic strains only appear at those locations.
▪ Low values of von Mises stress
▪ Nonlinear solution
▪ More homogeneous distribution of the von Mises stress
▪ Larger area in the plastic regime
▪ Higher duration of the test, leading to a higher quantity of
information about the material behavior.
Obtained results from FEA at the moment just before rupture: the
equivalent plastic strain, 𝜀𝑃, and the von Mises stress, σVM, of (a) linear
and (b) nonlinear solutions.
(a)
(b)

Concluding remarks
04
▪ A nonlinear topology-based optimization methodology to design heterogeneous mechanical tests is
proposed;
▪ Geometric nonlinearity is introduced in FEA to account for the large deformations;
▪ The use of a nonlinear analysis is proved to be required for a more correct reproduction of the
specimen behavior;
▪ The introduction of material nonlinearity is of major relevance for an accurate test design.
20

Concluding remarks
04
17
At IDDRG (International Deep Drawing Research Group) Conference 2022:
▪ The nonlinear elastoplastic material behavior is introduced in the test design procedure;
▪ Both material and geometric nonlinearities are taken into account.

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.
Thank you!
Any questions?
Acknowledgments

On the design of a heterogeneous mechanical test using a nonlinear topology optimization approach

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