Abstract

This new machine-learned (ML) constitutive model for elastomers has been developed to capture the dependence of elastomer behavior on loading conditions such as strain rate and temperature, as well as compound morphology factors such as filler percentage and crosslink density. It is based on our recent new generation of machine-learning algorithms known as conditional neural networks (CondNNs) Ghaderi et al. (2020, “A Physics-Informed Assembly of Feed-Forward Neural Network Engines to Predict Inelasticity in Cross-Linked Polymers,” Polymers, 12(11), p. 2628), and uses data-infused knowledge-driven machine-learned surrogate functions to describe the quasi-static response of polymer batches in cross-linked elastomers. The model reduces the 3D stress-strain mapping space into a 1D space, and this order reduction significantly reduces the training cost by minimizing the search space. It is capable of considering the effects of loading conditions such as strain rate, temperature, and filler percentage in different deformation states, as well as enjoying a high training speed and accuracy even in complicated loading scenarios. It can be used for advanced implementations in finite element programs due to its computing efficiency, simplicity, correctness, and interpretability. It is applicable to a variety of soft materials, including soft robotics, soft digital materials (DMs), hydrogels, and adhesives. This model has a distinct advantage over existing phenomenological models as it can capture strain rate and temperature dependency in a much more comprehensive way.

References

1.
Mao
,
G.
,
Huang
,
X.
,
Liu
,
J.
,
Li
,
T.
,
Qu
,
S.
, and
Yang
,
W.
,
2015
, “
Dielectric Elastomer Peristaltic Pump Module With Finite Deformation
,”
Smart Mater. Struct.
,
24
(
7
), p.
075026
.
2.
Li
,
T.
,
Zou
,
Z.
,
Mao
,
G.
,
Yang
,
X.
,
Liang
,
Y.
,
Li
,
C.
,
Qu
,
S.
,
Suo
,
Z.
, and
Yang
,
W.
,
2019
, “
Agile and Resilient Insect-Scale Robot
,”
Soft Robot.
,
6
(
1
), pp.
133
141
.
3.
Yin
,
T.
,
Wu
,
T.
,
Zhong
,
D.
,
Liu
,
J.
,
Liu
,
X.
,
Han
,
Z.
,
Yu
,
H.
, and
Qu
,
S.
,
2018
, “
Soft Display Using Photonic Crystals on Dielectric Elastomers
,”
ACS Appl. Mater. Interfaces
,
10
(
29
), pp.
24758
24766
.
4.
Dal
,
H.
,
Zopf
,
C.
, and
Kaliske
,
M.
,
2018
, “
Micro-Sphere Based Viscoplastic Constitutive Model for Uncured Green Rubber
,”
Int. J. Solids Struct.
,
132
, pp.
201
217
.
5.
Kaliske
,
M.
, and
Rothert
,
H.
,
1997
, “
Formulation and Implementation of Three-Dimensional Viscoelasticity at Small and Finite Strains
,”
Comput. Mech.
,
19
(
3
), pp.
228
239
.
6.
Volokh
,
K.
,
2010
, “
On Modeling Failure of Rubber-Like Materials
,”
Mech. Res. Commun.
,
37
(
8
), pp.
684
689
.
7.
Dal
,
H.
,
Açıkgöz
,
K.
, and
Badienia
,
Y.
,
2021
, “
On the Performance of Isotropic Hyperelastic Constitutive Models for Rubber-Like Materials: a State of the Art Review
,”
ASME Appl. Mech. Rev.
,
73
(
2
), p.
020802
.
8.
Karapiperis
,
K.
,
Stainier
,
L.
,
Ortiz
,
M.
, and
Andrade
,
J.
,
2021
, “
Data-Driven Multiscale Modeling in Mechanics
,”
J. Mech. Phys. Solids
,
147
, p.
104239
.
9.
Frankel
,
A.
,
Hamel
,
C. M.
,
Bolintineanu
,
D.
,
Long
,
K.
, and
Kramer
,
S.
,
2022
, “
Machine Learning Constitutive Models of Elastomeric Foams
,”
Comput. Meth. Appl. Mech. Eng.
,
391
, p.
114492
.
10.
Ayoub
,
G.
,
Zaïri
,
F.
,
Naït-Abdelaziz
,
M.
,
Gloaguen
,
J. M.
, and
Kridli
,
G.
,
2014
, “
A Visco-Hyperelastic Damage Model for Cyclic Stress-Softening, Hysteresis and Permanent Set in Rubber Using the Network Alteration Theory
,”
Int. J. Plast.
,
54
, pp.
19
33
.
11.
Guo
,
Q.
,
Zäıri
,
F.
, and
Guo
,
X.
,
2018
, “
A Thermo-Viscoelastic-Damage Constitutive Model for Cyclically Loaded Rubbers. Part II: Experimental Studies and Parameter Identification
,”
Int. J. Plast.
,
101
, pp.
58
73
.
12.
Diani
,
J.
,
Brieu
,
M.
, and
Vacherand
,
J.
,
2006
, “
A Damage Directional Constitutive Model for Mullins Effect With Permanent Set and Induced Anisotropy
,”
Eur. J. Mech. A Solids
,
25
(
3
), pp.
483
496
.
13.
R. I. Annually
,
1995
, ASTM Standards.
14.
Rodas
,
C. O.
,
Zaïri
,
F.
,
Naït-Abdelaziz
,
M.
, and
Charrier
,
P.
,
2015
, “
Temperature and Filler Effects on the Relaxed Response of Filled Rubbers: Experimental Observations on a Carbon-Filled SBR and Constitutive Modeling
,”
Int. J. Solids Struct.
,
58
, pp.
309
321
.
15.
Tang
,
S.
,
Li
,
Y.
,
Qiu
,
H.
,
Yang
,
H.
,
Saha
,
S.
,
Mojumder
,
S.
,
Liu
,
W. K.
, and
Guo
,
X.
,
2020
, “
Map123-EP: A Mechanistic-Based Data-Driven Approach for Numerical Elastoplastic Analysis
,”
Comput. Meth. Appl. Mech. Eng.
,
364
, p.
112955
.
16.
Liu
,
X.
,
Tian
,
S.
,
Tao
,
F.
, and
Yu
,
W.
,
2021
, “
A Review of Artificial Neural Networks in the Constitutive Modeling of Composite Materials
,”
Compos. B. Eng.
,
224
, p.
109152
.
17.
Tang
,
J.
,
Chen
,
X.
,
Pei
,
Y.
, and
Fang
,
D.
,
2016
, “
Pseudoelasticity and Nonideal Mullins Effect of Nanocomposite Hydrogels
,”
ASME J. Appl. Mech.
,
83
(
11
), p.
111010
.
18.
Shen
,
Y.
,
Chandrashekhara
,
K.
,
Breig
,
W.
, and
Oliver
,
L.
,
2004
, “
Neural Network Based Constitutive Model for Rubber Material
,”
Rubber Chem. Technol.
,
77
(
2
), pp.
257
277
.
19.
Liang
,
G.
, and
Chandrashekhara
,
K.
,
2008
, “
Neural Network Based Constitutive Model for Elastomeric Foams
,”
Eng. Struct.
,
30
(
7
), pp.
2002
2011
.
20.
Song
,
G.
,
Chandrashekhara
,
K.
,
Breig
,
W.
,
Klein
,
D.
, and
Oliver
,
L.
,
2005
, “
J-Integral Analysis of Cord-Rubber Serpentine Belt Using Neural-Network-Based Material Modelling
,”
Fatigue Fract. Eng. Mater. Struct.
,
28
(
10
), pp.
847
860
.
21.
Raissi
,
M.
,
Perdikaris
,
P.
, and
Karniadakis
,
G. E.
,
2019
, “
Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations
,”
J. Comput. Phys.
,
378
, pp.
686
707
.
22.
Raissi
,
M.
, and
Karniadakis
,
G. E.
,
2018
, “
Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
,”
J. Comput. Phys.
,
357
, pp.
125
141
.
23.
Kirchdoerfer
,
T.
, and
Ortiz
,
M.
,
2016
, “
Data-Driven Computational Mechanics
,”
Comput. Methods Appl. Mech. Eng.
,
304
, pp.
81
101
.
24.
Nguyen
,
L. T. K.
, and
Keip
,
M.-A.
,
2018
, “
A Data-Driven Approach to Nonlinear Elasticity
,”
Comput. Struct.
,
194
, pp.
97
115
.
25.
Amores
,
V. J.
,
Benítez
,
J. M.
, and
Montáns
,
F. J.
,
2019
, “
Average-Chain Behavior of Isotropic Incompressible Polymers Obtained From Macroscopic Experimental Data. A Simple Structure-Based WYPiWYG Model in Julia Language
,”
Adv. Eng. Softw.
,
130
, pp.
41
57
.
26.
Ibanez
,
R.
,
Abisset-Chavanne
,
E.
,
Aguado
,
J. V.
,
Gonzalez
,
D.
,
Cueto
,
E.
, and
Chinesta
,
F.
,
2018
, “
A Manifold Learning Approach to Data-Driven Computational Elasticity and Inelasticity
,”
Arch. Comput. Meth. Eng.
,
25
(
1
), pp.
47
57
.
27.
Reimann
,
D.
,
Nidadavolu
,
K.
,
ul Hassan
,
H.
,
Vajraguptaul Hassan
,
H.
,
Glasmachers
,
N.
,
Junker
,
T.
, and
Hartmaier
,
P.
,
2019
, “
Modeling Macroscopic Material Behavior With Machine Learning Algorithms Trained by Micromechanical Simulations
,”
Front. Mater.
,
6
, pp.
181
.
28.
Zopf
,
C.
, and
Kaliske
,
M.
,
2017
, “
Numerical Characterisation of Uncured Elastomers by a Neural Network Based Approach
,”
Comput. Struct.
,
182
, pp.
504
525
.
29.
Ghaderi
,
A.
,
Morovati
,
V.
, and
Dargazany
,
R.
,
2020
, “
A Physics-Informed Assembly of Feed-Forward Neural Network Engines to Predict Inelasticity in Cross-Linked Polymers
,”
Polymers
,
12
(
11
), p.
2628
.
30.
Glorot
,
X.
, and
Bengio
,
Y.
,
2010
, “
Understanding the Difficulty of Training Deep Feedforward Neural Networks
,”
Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, JMLR Workshop and Conference Proceedings
,
Chia Laguna Resort, Sardinia, Italy
,
May 13–15
.
31.
He
,
K.
,
Zhang
,
X.
,
Ren
,
S.
, and
Sun
,
J.
,
2015
, “
Delving Deep Into Rectifiers: Surpassing Human-Level Performance on Imagenet Classification
,”
Proceedings of the IEEE International Conference on Computer Vision
,
Santiago, Chile
,
Dec. 7–13
.
32.
Wang
,
E.
,
Kosson
,
A.
, and
Mu
,
T.
,
2017
, Deep Action Conditional Neural Network for Frame Prediction in Atari Games, Technical Report, Stanford University.
33.
Ioannou
,
Y.
,
Robertson
,
D.
,
Zikic
,
D.
,
Kontschieder
,
P.
,
Shotton
,
J.
,
Brown
,
M.
, and
Criminisi
,
A.
,
2016
, “
Decision Forests, Convolutional Networks and the Models In-Between
,”
arXiv preprint
.
34.
Ehret
,
A.
,
Itskov
,
M.
, and
Schmid
,
H.
,
2010
, “
Numerical Integration on the Sphere and Its Effect on the Material Symmetry of Constitutive Equations—A Comparative Study
,”
Int. J. Numer. Meth Eng.
,
81
(
2
), pp.
189
206
.
35.
Lambert-Diani
,
J.
, and
Rey
,
C.
,
1999
, “
New Phenomenological Behavior Laws for Rubbers and Thermoplastic Elastomers
,”
Eur. J. Mech A Solids
,
18
(
6
), pp.
1027
1043
.
36.
Amin
,
A.
,
Lion
,
A.
,
Sekita
,
S.
, and
Okui
,
Y.
,
2006
, “
Nonlinear Dependence of Viscosity in Modeling the Rate-Dependent Response of Natural and High Damping Rubbers in Compression and Shear: Experimental Identification and Numerical Verification
,”
Int. J. Plast.
,
22
(
9
), pp.
1610
1657
.
37.
Lion
,
A.
,
1996
, “
A Constitutive Model for Carbon Black Filled Rubber: Experimental Investigations and Mathematical Representation
,”
Contin. Mech. Thermodyn.
,
8
(
3
), pp.
153
169
.
38.
Roland
,
C.
,
Twigg
,
J.
,
Vu
,
Y.
, and
Mott
,
P.
,
2007
, “
High Strain Rate Mechanical Behavior of Polyurea
,”
Polymer
,
48
(
2
), pp.
574
578
.
39.
Fu
,
X.
,
Wang
,
Z.
,
Ma
,
L.
,
Zou
,
Z.
,
Zhang
,
Q.
, and
Guan
,
Y.
,
2020
, “
Temperature-Dependence of Rubber Hyperelasticity Based on the Eight-Chain Model
,”
Polymers
,
12
(
4
), p.
932
.
40.
Diani
,
J.
, and
Le Tallec
,
P.
,
2019
, “
A Fully Equilibrated Microsphere Model With Damage for Rubberlike Materials
,”
J. Mech. Phys. Solids
,
124
(
4
), pp.
702
713
.
41.
Mayumi
,
K.
,
Marcellan
,
A.
,
Ducouret
,
G.
,
Creton
,
C.
, and
Narita
,
T.
,
2013
, “
Stress–Strain Relationship of Highly Stretchable Dual Cross-Link Gels: Separability of Strain and Time Effect
,”
ACS Macro Lett.
,
2
(
12
), pp.
1065
1068
.
42.
Liu
,
M.
,
Guo
,
J.
,
Hui
,
C.-Y.
,
Creton
,
C.
,
Narita
,
T.
, and
Zehnder
,
A.
,
2018
, “
Time-Temperature Equivalence in a PVA Dual Cross-Link Self-Healing Hydrogel
,”
J. Rheol.
,
62
(
4
), pp.
991
1000
.
43.
Xiang
,
Y.
,
Zhong
,
D.
,
Wang
,
P.
,
Yin
,
T.
,
Zhou
,
H.
,
Yu
,
H.
,
Baliga
,
C.
,
Qu
,
S.
, and
Yang
,
W.
,
2019
, “
A Physically Based Visco-Hyperelastic Constitutive Model for Soft Materials
,”
J. Mech. Phys. Solids
,
128
, pp.
208
218
.
You do not currently have access to this content.