Rheologically complex materials are described by function-valued properties with dependence on a timescale (linear viscoelasticity), input amplitude (nonlinear material behavior), or more generally both (nonlinear viscoelasticity). This complexity presents a difficulty when trying to utilize these material systems in engineering designs. Here, we focus on linear viscoelasticity and a methodology to identify the desired viscoelastic behavior. This is an early-stage design step to optimize target (function-valued) properties before choosing or synthesizing a real material. In linear viscoelasticity, it is not obvious which properties can be treated as independent design variables. Thus, it is nontrivial to select the most design-appropriate constitutive model, to be as general as possible, but not violate fundamental restrictions. We use the Kramers–Kronig constraint to show that frequency-dependent moduli (e.g., shear moduli G(ω) and G(ω)) cannot be treated as two independent design variables. Rather, a single function such as the relaxation modulus (e.g., K(t) for force-relaxation or G(t) for stress relaxation) is an appropriate function-valued design variable. A simple case study is used to demonstrate the framework in which we identify target properties for a vibration isolation system. Viscoelasticity improves performance. Different parameterizations of the kernel function are optimized and compared for performance. While parameterization may limit the generality of the kernel function, we do include a nonobvious representation (power law) that is found in real viscoelastic material systems and in the spring-dashpot paradigm would require an infinite number of components. Our methodology provides a means to answer the question, “What viscoelastic properties are desirable?” This ability to identify targeted behavior will be useful for subsequent stages of the design process including the selection or synthesis of real materials.

References

References
1.
Autumn
,
K.
,
Liang
,
Y. A.
,
Hsieh
,
S. T.
,
Zesch
,
W.
,
Chan
,
W. P.
,
Kenny
,
T. W.
,
Fearing
,
R.
, and
Full
,
R. J.
,
2000
, “
Adhesive Force of a Single Gecko Foot-Hair
,”
Nature
,
405
(
6787
), pp.
681
685
.
2.
Shadwick
,
R. E.
,
1999
, “
Mechanical Design in Arteries
,”
J. Exp. Biol.
,
202
, pp.
3305
3313
.
3.
Denny
,
M.
,
1980
, “
The Role of Gastropod Pedal Mucus in Locomotion
,”
Nature
,
285
(
5761
), pp.
160
161
.
4.
Caldwell
,
1972
, “
Vibration Damping in World Trade Center Using Viscoelastic Material
,” 3M.
5.
Mahmoodi
,
P.
,
Robertson
,
L. E.
,
Yontar
,
M.
,
Moy
,
C.
, and
Field
,
I.
,
1987
, “
Performance of Viscoelastic Dampers in World Trade Center Towers
,” 3M.
6.
de Gennes
,
P. G.
,
1996
, “
Soft Adhesives
,”
Langmuir
,
12
(
19
), pp.
4497
4500
.
7.
Ewoldt
,
R. H.
,
2014
, “
Extremely Soft: Design With Rheologically Complex Fluids
,”
Soft Rob.
,
1
(
1
), pp.
12
20
.
8.
Yang
,
H.
, and
Chang
,
E.
,
1997
, “
The Role of Viscoelastic Properties in the Design of Pressure-Sensitive Adhesives
,”
Trends Polym. Sci.
,
5
(
11
), pp.
380
384
.
9.
Nachbar
,
W.
, and
Schipmolder
,
J. B.
,
1969
, “
Optimization of a Viscoelastic Structure: The Seat-Belt Problem
,”
ASME J. Appl. Mech.
,
36
(
3
), p.
565
.
10.
Fischer
,
C.
,
Braun
,
S. A.
,
Bourban
,
P.-E.
,
Michaud
,
V.
,
Plummer
,
C. J. G.
, and
Månson
,
J.-A. E.
,
2006
, “
Dynamic Properties of Sandwich Structures With Integrated Shear-Thickening Fluids
,”
Smart Mater. Struct.
,
15
(
5
), pp.
1467
1475
.
11.
Lee
,
Y. S.
,
Wetzel
,
E. D.
, and
Wagner
,
N. J.
,
2003
, “
The Ballistic Impact Characteristics of Kevlar Woven Fabrics Impregnated With a Colloidal Shear Thickening Fluid
,”
J. Mater. Sci.
,
38
(
13
), pp.
2825
2833
.
12.
Nachbar
,
W.
, and
Schipmolder
,
J. B.
,
1971
, “
Optimization of a Standard Linear Viscoelastic Material for Use in a Seat Belt
,”
ASME J. Appl. Mech.
,
38
(
2
), p.
532
.
13.
Ferry
,
J. D.
,
1980
,
Viscoelastic Properties of Polymers
,
3rd ed.
,
Wiley
,
New York
.
14.
Xu
,
H.
,
Li
,
Y.
,
Brinson
,
C.
, and
Chen
,
W.
,
2014
, “
A Descriptor-Based Design Methodology for Developing Heterogeneous Microstructural Materials System
,”
ASME J. Mech. Des.
,
136
(
5
), p.
051007
.
15.
Asbeck
,
A.
,
De Rossi
,
S.
,
Galiana
,
I.
,
Ding
,
Y.
, and
Walsh
,
C.
,
2014
, “
Stronger, Smarter, Softer: Next-Generation Wearable Robots
,”
Rob. Autom. Mag.
,
21
(
4
), pp.
22
33
.
16.
Shepherd
,
R. F.
,
Ilievski
,
F.
,
Choi
,
W.
,
Morin
,
S. A.
,
Stokes
,
A. A.
,
Mazzeo
,
A. D.
,
Chen
,
X.
,
Wang
,
M.
, and
Whitesides
,
G. M.
,
2011
, “
Multigait Soft Robot
,”
Proc. Natl. Acad. Sci. U. S. A.
,
108
(
51
), pp.
20400
20403
.
17.
Mahdavi
,
A.
,
Ferreira
,
L.
,
Sundback
,
C.
,
Nichol
,
J. W.
,
Chan
,
E. P.
,
Carter
,
D. J. D.
,
Bettinger
,
C. J.
,
Patanavanich
,
S.
,
Chignozha
,
L.
,
Ben-Joseph
,
E.
,
Galakatos
,
A.
,
Pryor
,
H.
,
Pomerantseva
,
I.
,
Masiakos
,
P. T.
,
Faquin
,
W.
,
Zumbuehl
,
A.
,
Hong
,
S.
,
Borenstein
,
J.
,
Vacanti
,
J.
,
Langer
,
R.
, and
Karp
,
J. M.
,
2008
, “
A Biodegradable and Biocompatible Gecko-Inspired Tissue Adhesive
,”
Proc. Natl. Acad. Sci. U. S. A.
,
105
(
7
), pp.
2307
2312
.
18.
Laulicht
,
B.
,
Langer
,
R.
, and
Karp
,
J. M.
,
2012
, “
Quick-Release Medical Tape
,”
Proc. Natl. Acad. Sci. U. S. A.
,
109
(
46
), pp.
18803
18808
.
19.
Kim
,
D.-H.
, and
Rogers
,
J. A.
,
2008
, “
Stretchable Electronics: Materials Strategies and Devices
,”
Adv. Mater.
,
20
(
24
), pp.
4887
4892
.
20.
Kim
,
D.-H.
,
Xiao
,
J.
,
Song
,
J.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2010
, “
Stretchable, Curvilinear Electronics Based on Inorganic Materials
,”
Adv. Mater. (Deerfield Beach, Fla.)
,
22
(
19
), pp.
2108
2124
.
21.
Ashby
,
M. F.
,
1999
,
Material Selection in Mechanical Design
,
Butterworth-Heinemann
,
Boston, MA
.
22.
Dieter
,
G.
,
2009
,
Engineering Design: A Materials and Processing Approach
,
4th ed.
,
McGraw-Hill Higher Education
,
New York
.
23.
Olson
,
G. B.
,
2000
, “
Designing a New Material World
,”
Science
,
288
(
5468
), pp.
993
998
.
24.
Cussler
,
E. L.
, and
Moggridge
,
G. D.
,
2012
,
Chemical Product Design
,
2nd ed.
,
Cambridge University Press
,
Cambridge, UK
.
25.
Nelson
,
A.
, and
Ewoldt
,
R. H.
,
2015
, “
Design of Yield-Stress Fluids: A Rheology to Structure Inverse Problem
,” (in press).
26.
Nelson
,
A.
,
2015
, “
Expanding Yield-Stress Fluid Paradigms
,” M.S. thesis, University of Illinois, Champaign, IL.
27.
Grindy
,
S. C.
,
Learsch
,
R.
,
Mozhdehi
,
D.
,
Cheng
,
J.
,
Barrett
,
D. G.
,
Guan
,
Z.
,
Messersmith
,
P. B.
, and
Holten-Andersen
,
N.
,
2015
, “
Control of Hierarchical Polymer Mechanics With Bioinspired Metal-Coordination Dynamics
,”
Nat. Mater.
,
14
(
12
), pp.
1210
1216
.
28.
Chen
,
P.
,
Li
,
Q.
,
Grindy
,
S.
, and
Holten-Andersen
,
N.
,
2015
, “
White-Light-Emitting Lanthanide Metallogels With Tunable Luminescence and Reversible Stimuli-Responsive Properties
,”
J. Am. Chem. Soc.
,
137
(
36
), pp.
11590
11593
.
29.
Kim
,
H. M.
,
Michelena
,
N. F.
,
Papalambros
,
P. Y.
, and
Jiang
,
T.
,
2003
, “
Target Cascading in Optimal System Design
,”
ASME J. Mech. Des.
,
125
(
3
), p.
474
.
30.
Kim
,
H. M.
,
Rideout
,
D. G.
,
Papalambros
,
P. Y.
, and
Stein
,
J. L.
,
2003
, “
Analytical Target Cascading in Automotive Vehicle Design
,”
ASME J. Mech. Des.
,
125
(
3
), p.
481
.
31.
Choi
,
H.
,
McDowell
,
D. L.
,
Allen
,
J. K.
,
Rosen
,
D.
, and
Mistree
,
F.
,
2008
, “
An Inductive Design Exploration Method for Robust Multiscale Materials Design
,”
ASME J. Mech. Des.
,
130
(
3
), p.
031402
.
32.
Linsey
,
J. S.
,
Tseng
,
I.
,
Fu
,
K.
,
Cagan
,
J.
,
Wood
,
K. L.
, and
Schunn
,
C.
,
2010
, “
A Study of Design Fixation, Its Mitigation and Perception in Engineering Design Faculty
,”
ASME J. Mech. Des.
,
132
(
4
), p.
041003
.
33.
Hilton
,
H. H.
, and
Yi
,
S.
,
1992
, “
Analytical Formulation of Optimum Material Properties for Viscoelastic Damping
,”
Smart Mater. Struct.
,
1
(
2
), p.
113
.
34.
Hilton
,
H. H.
,
2005
, “
Optimum Linear and Nonlinear Viscoelastic Designer Functionally Graded Materials-Characterizations and Analysis
,”
Composites, Part A
,
36
(
10
), pp.
1329
1334
.
35.
Prasad
,
J.
, and
Diaz
,
A. R.
,
2008
, “
A Concept for a Material That Softens With Frequency
,”
ASME J. Mech. Des.
,
130
(
9
), p.
091703
.
36.
Bharadwaj
,
N. A.
,
Allison
,
J. T.
, and
Ewoldt
,
R. H.
,
2013
, “
Early-Stage Design of Rheologically Complex Materials Via Material Function Design Targets
,”
ASME
Paper No. DETC2013-13462.
37.
Dealy
,
J. M.
,
1995
, “
Official Nomenclature for Material Functions Describing the Response of a Viscoelastic Fluid to Various Shearing and Extensional Deformations
,”
J. Rheol.
,
39
(
1
), pp.
253
265
.
38.
Bird
,
R. B.
,
Armstrong
,
R.
, and
Hassger
,
O.
,
1987
,
Dynamics of Polymeric Liquids: Volume 1 Fluid Mechanics
,
2nd ed.
,
Wiley
,
New York
.
39.
Johnson
,
C. D.
,
1995
, “
Design of Passive Damping Systems
,”
ASME J. Mech. Des.
,
117
(
B
), p.
171
.
40.
Hilton
,
H. H.
,
Lee
,
D. H.
, and
El Fouly
,
A. R. A.
,
2008
, “
Generalized Viscoelastic Designer Functionally Graded Auxetic Materials Engineered/Tailored for Specific Task Performances
,”
Mech. Time-Depend. Mater.
,
12
(
2
), pp.
151
178
.
41.
Breuer
,
S.
, and
Onat
,
E. T.
,
1962
, “
On Uniqueness in Linear Elasticity
,”
Q. Appl. Math.
,
19
(
4
), pp.
355
359
.
42.
Rabotnov
,
Y. N.
,
1977
,
Elements of Hereditary Solid Mechanics (English Translation)
,
Mir
,
Moscow, Russia
.
43.
Chambon
,
F.
, and
Winter
,
H. H.
,
1987
, “
Linear Viscoelasticity at the Gel Point of a Cross-Linking PDMS With Imbalanced Stoichiometry
,”
J. Rheol.
,
31
(
8
), pp.
683
697
.
44.
Winter
,
H. H.
, and
Chambon
,
F.
,
1986
, “
Analysis of Linear Viscoelasticity of a Cross-Linking Polymer at the Gel Point
,”
J. Rheol.
,
30
(
2
), pp.
367
382
.
45.
Koike
,
A.
,
Nemoto
,
N.
,
Inoue
,
T.
, and
Osaki
,
K.
,
1995
, “
Dynamic Light Scattering and Dynamic Viscoelasticity of Poly(Vinyl Alcohol) in Aqueous Borax Solutions. 1. Concentration Effect
,”
Macromolecules
,
28
(
7
), pp.
2339
2344
.
46.
Corman
,
R.
,
2015
, “
Enabling Design With Rheological Complexity: Intuition and Optimization of Viscoelastic Materials
,”
M.S. thesis
, University of Illinois at Urbana-Champaign, Champaign, IL.
47.
Tsiklauri
,
D.
, and
Beresnev
,
I.
,
2001
, “
Non-Newtonian Effects in the Peristaltic Flow of a Maxwell Fluid
,”
Phys. Rev. E
,
64
(
3
), p.
036303
.
48.
Cordobes
,
F.
,
Partal
,
P.
, and
Guerrero
,
A.
,
2004
, “
Rheology and Microstructure of Heat-Induced Egg Yolk Gels
,”
Rheol. Acta
,
43
(
2
), pp.
184
195
.
49.
Trepat
,
X.
,
Deng
,
L.
,
An
,
S. S.
,
Navajas
,
D.
,
Tschumperlin
,
D. J.
,
Gerthoffer
,
W. T.
,
Butler
,
J. P.
, and
Fredberg
,
J. J.
,
2007
, “
Universal Physical Responses to Stretch in the Living Cell
,”
Nature
,
447
(
7144
), pp.
592
595
.
50.
Winter
,
H. H.
, and
Mours
,
M.
,
1997
, “
Rheology of Polymers Near Liquid-Solid Transitions
,”
Neutron Spin Echo Spectroscopy, Viscoelasticity, Rheology (Advances in Polymer Science)
, Vol.
134
,
Springer
,
Berlin
, pp.
165
234
.
51.
Kollmannsberger
,
P.
, and
Fabry
,
B.
,
2011
, “
Linear and Nonlinear Rheology of Living Cells
,”
Annu. Rev. Mater. Res.
,
41
(
1
), pp.
75
97
.
52.
Athans
,
M.
, and
Falb
,
P. L.
,
2006
,
Optimal Control: An Introduction to the Theory and Its Applications
,
Dover Publications/McGraw-Hill
,
New York
.
53.
Betts
,
J. T.
,
2010
,
Practical Methods for Optimal Control and Estimation Using Nonlinear Programming
,
SIAM
,
Philadelphia, PA
.
54.
Allison
,
J. T.
,
Guo
,
T.
, and
Han
,
Z.
,
2014
, “
Co-Design of an Active Suspension Using Simultaneous Dynamic Optimization
,”
ASME J. Mech. Des.
,
136
(
8
), p.
081003
.
55.
Karnopp
,
D.
,
1995
, “
Active and Semi-Active Vibration Isolation
,”
ASME J. Mech. Des.
,
117
(
B
), p.
177
.
56.
Inman
,
D. J.
,
2013
,
Engineering Vibration
,
4th ed.
,
Prentice-Hall
,
New York
.
57.
Corman
,
R. E.
, and
Rao
,
L. G.
,
2015
, Linear Viscoelastic Design Optimization-File Exchange-MATLAB Central.
58.
Donelan
,
M. A.
,
Hamilton
,
J.
, and
Hui
,
W. H.
,
1985
, “
Directional Spectra of Wind-Generated Waves
,”
Philos. Trans. R. Soc., A
,
315
(
1534
), pp.
509
562
.
59.
Latheef
,
M.
, and
Swan
,
C.
,
2013
, “
A Laboratory Study of Wave Crest Statistics and the Role of Directional Spreading
,”
Proc. R. Soc. A
,
469
(
2152
), p.
20120696
.
60.
Larson
,
R. G.
,
1999
,
The Structure and Rheology of Complex Fluids
,
Oxford University Press
,
Oxford, UK
.
61.
Rubinstein
,
M.
, and
Colby
,
R. H.
,
2003
,
Polymer Physics
,
Oxford University Press
,
Oxford, UK
.
62.
Mewis
,
J.
, and
Wagner
,
N. J.
,
2012
,
Colloidal Suspension Rheology
,
Cambridge University Press
,
Cambridge, UK
.
63.
Rao
,
L. G.
, and
Allison
,
J. T.
,
2015
, “
Generalized Viscoelastic Material Design With Integro-Differential Equations and Direct Optimal Control
,”
ASME
Paper No. DETC2015-46768.
You do not currently have access to this content.