Abstract

Evidence from cyclic tests on metals, elastomers, and sandy soils reveals that damping forces are nearly rate-independent and structural (hysteretic or rate-independent) damping was widely adopted since the 1940s. While there is no time-domain constitutive equation for a linear spring connected in parallel with a rate-independent dashpot, the dynamic stiffness (transfer function) of this mechanical network can be constructed in the frequency domain; and it was known since the early 1960s that this mechanical network exhibits a non-causal response. In view of its simplicity in association with the wide practical need to model rate-independent dissipation, this mechanical network was also implemented in time-domain formulations with the label complex stiffness where the force output, P(t) is related in the time domain to the displacement input, u(t), with P(t) = k(1 + iη)u(t). This paper shows that the complex stiffness, as expressed in the time domain by various scholars, is a fundamentally flawed construct since in addition to causality it violates equilibrium.

References

1.
Harris
,
C. M.
, and
Crede
,
C. E.
,
1976
,
Shock and Vibration Handbook
,
McGraw-Hill
,
New York
.
2.
Ferry
,
J. D.
,
1980
,
Viscoelastic Properties of Polymers
,
John Wiley and Sons
,
New York
.
3.
Koeller
,
R.
,
1984
, “
Applications of Fractional Calculus to the Theory of Viscoelasticity
,”
ASME J. Appl. Mech.
,
51
(
2
), pp.
299
307
.
4.
Bird
,
R. B.
,
Armstrong
,
R. C.
, and
Hassager
,
O.
,
1987
,
Dynamics of Polymeric Liquids. Vol. 1: Fluid Mechanics
,
Wiley
,
New York
.
5.
Tschoegl
,
N. W.
,
2012
,
The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction
,
Springer
,
Berlin, Germany
.
6.
Giesekus
,
H.
,
1995
, “
An Alternative Approach to the Linear Theory of Viscoelasticity and Some Characteristic Effects Being Distinctive of the Type of Material
,”
Rheol. Acta
,
34
(
1
), pp.
2
11
.
7.
Schiessel
,
H.
,
Metzler
,
R.
,
Blumen
,
A.
, and
Nonnenmacher
,
T.
,
1995
, “
Generalized Viscoelastic Models: Their Fractional Equations With Solutions
,”
J. Phys. A Math. Gen.
,
28
(
23
), p.
6567
.
8.
Makris
,
N.
, and
Kampas
,
G.
,
2009
, “
Analyticity and Causality of the Three-Parameter Rheological Models
,”
Rheol. Acta
,
48
(
7
), pp.
815
825
.
9.
Makris
,
N.
, and
Efthymiou
,
E.
,
2020
, “
Time-Response Functions of Fractional Derivative Rheological Models
,”
Rheol. Acta
,
59
(
12
), pp.
849
873
.
10.
Lighthill
,
M. J.
,
1958
,
An Introduction to Fourier Analysis and Generalised Functions
,
Cambridge University Press
,
Cambridge, UK
.
11.
Makris
,
N.
,
1997
, “
Stiffness, Flexibility, Impedance, Mobility, and Hidden Delta Function
,”
J. Eng. Mech. ASCE
,
123
(
11
), pp.
1202
1208
.
12.
Makris
,
N.
,
2019
, “
The Frequency Response Function of the Creep Compliance
,”
Meccanica
,
54
(
1
), pp.
19
31
.
13.
Clough
,
R.
, and
Penzien
,
J.
,
1997
,
“Dynamics of Structures
,
McGraw-Hill
,
New York
.
14.
Kelly
,
J. M.
,
1989
,
Earthquake-Resistant Design With Rubber
,
Springer-Verlag
,
London
.
15.
Hardin
,
B. O.
, and
Drnevich
,
V. P.
,
1972
, “
Shear Modulus and Damping in Soils: Design Equations and Curves
,”
J. Soil Mech. Found. Div.
,
98
(
7
), pp.
667
692
.
16.
Hardin
,
B. O.
, and
Drnevich
,
V. P.
,
1972
, “
Shear Modulus and Damping in Soils: Measurement and Parameter Effects
,”
J. Soil Mech. Found. Div.
,
98
(
6
), pp.
603
624
.
17.
Tatsuoka
,
F.
,
Iwasaki
,
T.
, and
Takagi
,
Y.
,
1978
, “
Hysteretic Damping of Sands Under Cyclic Loading and Its Relation to Shear Modulus
,”
Soils. Found.
,
18
(
2
), pp.
25
40
.
18.
Theodorsen
,
T.
, and
Garrick
,
I. E.
,
1940
,
Mechanism of Flutter a Theoretical and Experimental Investigation of the Flutter Problem
, Report No. 685,
National Aeronautics and Space Administration
,
Washington, DC
.
19.
Myklestad
,
N. O.
,
1952
, “
The Concept of Complex Damping
,”
ASME J. Appl. Mech.
,
19
(
3
), pp.
284
286
.
20.
Bishop
,
R.
,
1955
, “
The Treatment of Damping Forces in Vibration Theory
,”
Aeronaut. J.
,
59
(
539
), pp.
738
742
.
21.
Neumark
,
S.
,
1957
,
Concept of Complex Stiffness Applied to Problems of Oscillations With Viscous and Hysteretic Damping
,
Aeronautical Research Council Reports and Memoranda, Ministry of Aviation
,
London, UK
.
22.
Chopra
,
A.
,
2000
,
Dynamics of Structures: Theory and Applications to Earthquake Engineering
,
Pearson Education, Inc.
,
Hoboken, NJ
.
23.
Crandall
,
S. H.
,
1961
, “Dynamic Response of Systems With Structural Damping,”
Air, Space and Instruments, Draper Anniversary Volume
,
S.
Lees
, ed.,
McGraw-Hill
,
New York
.
24.
Crandall
,
S. H.
,
1970
, “
The Role of Damping in Vibration Theory
,”
JSV
,
11
(
1
), pp.
3
18
.
25.
Crandall
,
S. H.
,
1991
, “
The Hysteretic Damping Model in Vibration Theory
,”
Proc. Inst. Mech. Eng. C: Mech. Eng. Sci.
,
205
(
1
), pp.
23
28
.
26.
Caughey
,
T. K.
,
1962
, “
Vibration of Dynamic System With Linear Hysteretic Damping (Linear Theory)
,”
Proceedings of Fourth US National Congress of Applied Mechanics
,
Berkeley, CA
,
June 18–21
,
Vol. 1
, pp.
87
97
.
27.
Makris
,
N.
,
1997
, “
Causal Hysteretic Element
,”
J. Eng. Mech. ASCE
,
123
(
11
), pp.
1209
1214
.
28.
Luo
,
H.
, and
Ikago
,
K.
,
2021
, “
Unifying Causal Model of Rate-Independent Linear Damping for Effectively Reducing Seismic Response in Low-Frequency Structures
,”
Earthq. Eng. Struct. Dyn.
,
50
(
9
), pp.
2355
2378
.
29.
Liu
,
W.
, and
Ikago
,
K.
,
2021
, “
Feasibility Study of the Physical Implementation of Rate-Independent Linear Damping for the Protection of Low-Frequency Structures
,”
J. Build. Eng.
,
44
(
1
), p.
103319
.
30.
Daboul
,
J.
, and
Delbourgo
,
R.
,
1999
, “
Matrix Representation of Octonions and Generalizations
,”
J. Math. Phys.
,
40
(
8
), pp.
4134
4150
.
31.
Erdelyi
,
A.
,
1954
,
Bateman Manuscript Project, Tables of Integral Transforms
,
McGraw Hill
,
New York
.
32.
Makris
,
N.
,
1994
, “
The Imaginary Counterpart of Recorded Motions
,”
Earthq. Eng. Struct. Dyn.
,
23
(
3
), pp.
265
273
.
33.
Makris
,
N.
,
2020
, “
On the Physical Meaning of Time-Domain Constitutive Models With Complex Parameters
,”
Meccanica
,
55
(
3
), pp.
453
467
.
34.
Papoulis
,
A.
,
1962
,
The Fourier Integral and Its Applications
,
McCraw-Hill
,
New York
.
35.
Bracewell
,
R.
,
1965
,
The Fourier Transform and Its Applications
,
McGraw-Hill
,
New York
.
36.
Morse
,
P. M.
, and
Feshbach
,
H.
,
1953
,
Methods of Theoretical Physics
,
McGraw-Hill
,
New York
.
You do not currently have access to this content.