Staggered architectures widely seen in load-bearing biological materials provide not only excellent supporting functions resisting static loading but also brilliant protecting functions attenuating the dynamic impact. However, there are very few efforts to unveil the relationship between staggered architectures and damping properties within load-bearing biological and bioinspired materials, while its static counterpart has been intensively studied over the past decades. Here, based on the Floquet theory, we developed a new generic method to evaluate the dynamic modulus of the composites with various staggered architectures. Comparisons with the finite element method results showed that the new method can give more accurate predictions than previous methods based on the tension-shear chain model. Moreover, the new method is more generic and applicable for two- and three-dimensional arbitrarily staggered architectures. This method provides a useful tool to understand the relationship between micro-architecture and damping property in natural load-bearing biological materials and to facilitate the architectural design of high-damping bioinspired composites.

References

1.
Ashby
,
M. F.
,
Gibson
,
L. J.
,
Wegst
,
U.
, and
Olive
,
R.
,
1995
, “
The Mechanical-Properties of Natural Materials. 1. Material Property Charts
,”
Proc. R. Soc. Math. Phys. Eng. Sci.,
450
(
1938
), pp.
123
140
.
2.
Gao
,
H. J.
,
Ji
,
B. H.
,
Jager
,
I. L.
,
Arzt
,
E.
, and
Fratzl
,
P.
,
2003
, “
Materials Become Insensitive to Flaws at Nanoscale: Lessons From Nature
,”
Proc.Natl. Acad. Sci. U.S.A.
,
100
(
10
), pp.
5597
5600
.
3.
Ji
,
B. H.
, and
Gao
,
H. J.
,
2004
, “
Mechanical Properties of Nanostructure of Biological Materials
,”
J. Mech. Phys. Solids
,
52
(
9
), pp.
1963
1990
.
4.
Zhang
,
Z. Q.
,
Liu
,
B.
,
Huang
,
Y.
,
Hwang
,
K. C.
, and
Gao
,
H.
,
2010
, “
Mechanical Properties of Unidirectional Nanocomposites With Non-Uniformly or Randomly Staggered Platelet Distribution
,”
J. Mech. Phys. Solids
,
58
(
10
), pp.
1646
1660
.
5.
Zhang
,
Z. Q.
,
Zhang
,
Y. W.
, and
Gao
,
H. J.
,
2011
, “
On Optimal Hierarchy of Load-Bearing Biological Materials
,”
Proc. R. Soc. Lond. B Biol. Sci.
,
278
(
1705
), pp.
519
525
.
6.
Ritchie
,
R. O.
,
2011
, “
The Conflicts Between Strength and Toughness
,”
Nat. Mater.
,
10
(
11
), pp.
817
822
.
7.
Jackson
,
A. P.
,
Vincent
,
J. F. V.
, and
Turner
,
R. M.
,
1988
, “
The Mechanical Design of Nacre
,”
Proc. R. Soc. Lond. B Biol. Sci.,
234
(
1277
), pp.
415
440
.
8.
Kamat
,
S.
,
Su
,
X.
,
Ballarini
,
R.
, and
Heuer
,
A. H.
,
2000
, “
Structural Basis for the Fracture Toughness of the Shell of the Conch Strombus Gigas
,”
Nature
,
405
(
6790
), pp.
1036
1040
.
9.
Menig
,
R.
,
Meyers
,
M. H.
,
Meyers
,
M. A.
, and
Vecchio
,
K. S.
,
2000
, “
Quasi-Static and Dynamic Mechanical Response of Haliotis Rufescens (Abalone) Shells
,”
Acta Mater.
,
48
(
9
), pp.
2383
2398
.
10.
Khayer Dastjerdi
,
A.
,
Rabiei
,
R.
, and
Barthelat
,
F.
,
2013
, “
The Weak Interfaces Within Tough Natural Composites: Experiments on Three Types of Nacre
,”
J. Mech. Behav. Biomed. Mater.
,
19
, pp.
50
60
.
11.
Meyers
,
M. A.
,
Chen
,
P. Y.
,
Lin
,
A. Y. M.
, and
Seki
,
Y.
,
2008
, “
Biological Materials: Structure and Mechanical Properties
,”
Prog. Mater. Sci.
,
53
(
1
), pp.
1
206
.
12.
Currey
,
J. D.
,
1977
, “
Mechanical Properties of Mother of Pearl in Tension
,”
Proc. R. Soc. Lond. B Biol. Sci.
,
196
(
1125
), pp.
443
463
.
13.
Landis
,
W.
,
1995
, “
The Strength of a Calcified Tissue Depends in Part on the Molecular Structure and Organization of Its Constituent Mineral Crystals in Their Organic Matrix
,”
Bone
,
16
(
5
), pp.
533
544
.
14.
Landis
,
W. J.
,
Hodgens
,
K. J.
,
Song
,
M. J.
,
Arena
,
J.
,
Kiyonaga
,
S.
,
Marko
,
M.
,
Owen
,
C.
, and
McEwen
,
B. F.
,
1996
, “
Mineralization of Collagen May Occur on Fibril Surfaces: Evidence From Conventional and High-Voltage Electron Microscopy and Three-Dimensional Imaging
,”
J. Struct. Biol.
,
117
(
1
), pp.
24
35
.
15.
Roschger
,
P.
,
Grabner
,
B.
,
Rinnerthaler
,
S.
,
Tesch
,
W.
,
Kneissel
,
M.
,
Berzlanovich
,
A.
,
Klaushofer
,
K.
, and
Fratzl
,
P.
,
2001
, “
Structural Development of the Mineralized Tissue in the Human L4 Vertebral Body
,”
J. Struct. Biol.
,
136
(
2
), pp.
126
136
.
16.
Tesch
,
W.
,
Eidelman
,
N.
,
Roschger
,
P.
,
Goldenberg
,
F.
,
Klaushofer
,
K.
, and
Fratzl
,
P.
,
2001
, “
Graded Microstructure and Mechanical Properties of Human Crown Dentin
,”
Calcif. Tissue Int.
,
69
(
3
), pp.
147
157
.
17.
Warshawsky
,
H.
,
1989
, “
Organization of Crystals in Enamel
,”
Anat. Rec.
,
224
(
2
), pp.
242
262
.
18.
Lei
,
H. F.
,
Zhang
,
Z. Q.
, and
Liu
,
B.
,
2012
, “
Effect of Fiber Arrangement on Mechanical Properties of Short Fiber Reinforced Composites
,”
Compos. Sci. Technol.
,
72
(
4
), pp.
506
514
.
19.
Lei
,
H. J.
,
Zhang
,
Z. Q.
,
Han
,
F.
,
Liu
,
B.
,
Zhang
,
Y. W.
, and
Gao
,
H. J.
,
2013
, “
Elastic Bounds of Bioinspired Nanocomposites
,”
ASME J. Appl. Mech.,
80
(
6
), p.
061017
.
20.
Chen
,
B.
,
Wu
,
P. D.
, and
Gao
,
H.
,
2009
, “
A Characteristic Length for Stress Transfer in the Nanostructure of Biological Composites
,”
Compos. Sci. Technol.
,
69
(
7–8
), pp.
1160
1164
.
21.
Zhang
,
P.
, and
To
,
A. C.
,
2014
, “
Highly Enhanced Damping Figure of Merit in Biomimetic Hierarchical Staggered Composites
,”
ASME J. Appl. Mech.
,
81
(
5
), p.
051015
.
22.
Zuo
,
S. C.
, and
Wei
,
Y. G.
,
2007
, “
Effective Elastic Modulus of Bone-Like Hierarchical Materials
,”
Acta Mech. Solida Sin.
,
20
(
3
), pp.
198
205
.
23.
Kotha
,
S. P.
,
Li
,
Y.
, and
Guzelsu
,
N.
,
2001
, “
Micromechanical Model of Nacre Tested in Tension
,”
J. Mater. Sci.
,
36
(
8
), pp.
2001
2007
.
24.
Tang
,
Z.
,
Kotov
,
N. A.
,
Magonov
,
S.
, and
Ozturk
,
B.
,
2003
, “
Nanostructured Artificial Nacre
,”
Nat. Mater.
,
2
(
6
), pp.
413
418
.
25.
Munch
,
E.
,
Launey
,
M. E.
,
Alsem
,
D. H.
,
Saiz
,
E.
,
Tomsia
,
A. P.
, and
Ritchie
,
R. O.
,
2008
, “
Tough, Bio-Inspired Hybrid Materials
,”
Science
,
322
(
5907
), pp.
1516
1520
.
26.
Dimas
,
L. S.
,
Bratzel
,
G. H.
,
Eylon
,
I.
, and
Buehler
,
M. J.
,
2013
, “
Tough Composites Inspired by Mineralized Natural Materials: Computation, 3D Printing, and Testing
,”
Adv. Funct. Mater.
,
23
(
36
), pp.
4629
4638
.
27.
Espinosa
,
H. D.
,
Juster
,
A. L.
,
Latourte
,
F. J.
,
Loh
,
O. Y.
,
Gregoire
,
D.
, and
Zavattieri
,
P. D.
,
2011
, “
Tablet-Level Origin of Toughening in Abalone Shells and Translation to Synthetic Composite Materials
,”
Nat. Commun.
,
2
, p.
173
.
28.
Lakes
,
R.
,
2002
, “
High Damping Composite Materials: Effect of Structural Hierarchy
,”
J. Compos. Mater.
,
36
(
3
), pp.
287
297
.
29.
Taber
,
K. H.
,
Warden
,
D. L.
, and
Hurley
,
R. A.
,
2006
, “
Blast-Related Traumatic Brain Injury: What Is Known?
,”
J. Neuropsychiatry Clin. Neurosci.
,
18
(
2
), pp.
141
145
.
30.
Viano
,
D. C.
,
Pellman
,
E. J.
,
Withnall
,
C.
, and
Shewchenko
,
N.
,
2006
, “
Concussion in Professional Football: Performance of Newer Helmets in Reconstructed Game Impacts—Part 13
,”
Neurosurgery
,
59
(
3
), pp.
591
606
.
31.
Zhang
,
L.
,
Yang
,
K. H.
, and
King
,
A. I.
,
2004
, “
A Proposed Injury Threshold for Mild Traumatic Brain Injury
,”
ASME J. Biomech. Eng.
,
126
(
2
), pp.
226
236
.
32.
Puxkandl
,
R.
,
Zizak
,
I.
,
Paris
,
O.
,
Keckes
,
J.
,
Tesch
,
W.
,
Bernstorff
,
S.
,
Purslow
,
P.
, and
Fratzl
,
P.
,
2002
, “
Viscoelastic Properties of Collagen: Synchrotron Radiation Investigations and Structural Model
,”
Philos. Trans. R. Soc. B
,
357
(
1418
), pp.
191
197
.
33.
Sanjeevi
,
R.
,
Somanathan
,
N.
, and
Ramaswamy
,
D.
,
1982
, “
A Viscoelastic Model for Collagen Fibres
,”
J. Biomech.
,
15
(
3
), pp.
181
183
.
34.
Shen
,
Z. L.
,
Kahn
,
H.
,
Ballarini
,
R.
, and
Eppell
,
S. J.
,
2011
, “
Viscoelastic Properties of Isolated Collagen Fibrils
,”
Biophys. J.
,
100
(
12
), pp.
3008
3015
.
35.
Yi
,
Y.-M.
,
Park
,
S.-H.
, and
Youn
,
S.-K.
,
1998
, “
Asymptotic Homogenization of Viscoelastic Composites With Periodic Microstructures
,”
Int. J. Solids Struct.
,
35
(
17
), pp.
2039
2055
.
36.
Hu
,
R.
, and
Oskay
,
C.
,
2017
, “
Nonlocal Homogenization Model for Wave Dispersion and Attenuation in Elastic and Viscoelastic Periodic Layered Media
,”
ASME J. Appl. Mech.
,
84
(
3
), p.
031003
.
37.
Hu
,
R.
, and
Oskay
,
C.
,
2018
, “
Spatial–Temporal Nonlocal Homogenization Model for Transient Anti-Plane Shear Wave Propagation in Periodic Viscoelastic Composites
,”
Comput. Methods Appl. Mech. Eng.
,
342
, pp.
1
31
.
38.
Tran
,
A.
,
Yvonnet
,
J.
,
He
,
Q.-C.
,
Toulemonde
,
C.
, and
Sanahuja
,
J.
,
2011
, “
A Simple Computational Homogenization Method for Structures Made of Linear Heterogeneous Viscoelastic Materials
,”
Comput. Methods Appl. Mech. Eng.
,
200
(
45–46
), pp.
2956
2970
.
39.
Zhang
,
P.
,
Heyne
,
M. A.
, and
To
,
A. C.
,
2015
, “
Biomimetic Staggered Composites With Highly Enhanced Energy Dissipation: Modeling, 3D Printing, and Testing
,”
J. Mech. Phys. Solids
,
83
, pp.
285
300
.
40.
Hashin
,
Z.
,
1970
, “
Complex Moduli of Viscoelastic Composites—I. General Theory and Application to Particulate Composites
,”
Int. J. Solids Struct.
,
6
(
5
), pp.
539
552
.
41.
Qwamizadeh
,
M.
,
Zhang
,
Z.
,
Zhou
,
K.
, and
Zhang
,
Y. W.
,
2015
, “
On the Relationship Between the Dynamic Behavior and Nanoscale Staggered Structure of the Bone
,”
J. Mech. Phys. Solids
,
78
, pp.
17
31
.
42.
Qwamizadeh
,
M.
,
Zhang
,
Z.
,
Zhou
,
K.
, and
Zhang
,
Y. W.
,
2016
, “
Protein Viscosity, Mineral Fraction and Staggered Architecture Cooperatively Enable the Fastest Stress Wave Decay in Load-Bearing Biological Materials
,”
J. Mech. Behav. Biomed. Mater.
,
60
, pp.
339
355
.
43.
Qwamizadeh
,
M.
,
Lin
,
M.
,
Zhang
,
Z. Q.
,
Zhou
,
K.
, and
Zhang
,
Y. W.
,
2017
, “
Bounds for the Dynamic Modulus of Unidirectional Composites With Bioinspired Staggered Distributions of Platelets
,”
Compos. Struct.
,
167
, pp.
152
165
.
44.
Qwamizadeh
,
M.
,
Liu
,
P.
,
Zhang
,
Z. Q.
,
Zhou
,
K.
, and
Zhang
,
Y. W.
,
2016
, “
Hierarchical Structure Enhances and Tunes the Damping Behavior of Load-Bearing Biological Materials
,”
ASME J. Appl. Mech.
,
83
(
5
), p.
051009
.
45.
Liu
,
J. J.
,
Hai
,
X. S.
,
Zhu
,
W. Q.
, and
Wei
,
X. D.
,
2018
, “
Optimization of Damping Properties of Staggered Composites Through Microstructure Design
,”
ASME J. Appl. Mech.,
85
(
10
), p.
101002
.
46.
Liu
,
J. J.
,
Zhu
,
W. Q.
,
Yu
,
Z. L.
, and
Wei
,
X. D.
,
2018
, “
Dynamic Shear-Lag Model for Understanding the Role of Matrix in Energy Dissipation in Fiber-Reinforced Composites
,”
Acta Biomater.
,
74
, pp.
270
279
.
47.
Manca
,
F.
,
Palla
,
P. L.
,
Cleri
,
F.
, and
Giordano
,
S.
,
2016
, “
Characteristic Lengths in Natural Bundle Assemblies Arising From Fiber-Matrix Energy Competition: A Floquet-Based Homogenization Theory
,”
Eur. J. Mech. A Solid
,
60
, pp.
145
165
.
48.
Pontryagin
,
L. S.
,
1962
,
Ordinary Differential Equations
,
Addison-Wesley
,
New York
.
49.
Cox
,
H. L.
,
1952
, “
The Elasticity and Strength of Paper and Other Fibrous Materials
,”
Br. J. Appl. Phys.
,
3
(
3
), pp.
72
79
.
You do not currently have access to this content.