Mechanics and Design of
Flexible Composite Fish Armor
by
MASSACHUSETTS INSTIt9TE
OF TECHNOLOGY
Ashley Browning
JUN 28 2012
B.S., University of Texas at Austin (2008)
LIBRARIES
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
ARCHNVES
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2012
o Massachusetts
Institute of Technology 2012. All rights reserved.
A uthor ........................... ,...........
Department oMechanical Eriering
May_2 , 2012
Certified by ..........................
Ford Professor of
Mary C. 4 oyce
IE
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Certified by ........................
Christine Ortiz
Professor of Material Science and Engineering
ThesiSpervp
A ccepted by ..........................
David Hardt
Chairman, Department Committee on Graduate Students
Mechanics and Design of
Flexible Composite Fish Armor
by
Ashley Browning
Submitted to the Department of Mechanical Engineering
on May 25, 2012, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
Inspired by the overlapping scales found on teleost fish, a new composite architecture
explores the mechanics of materials to accommodate both flexibility and protection.
These biological structures consist of overlapping mineralized plates embedded in a
compliant tissue to form a natural flexible armor which protects underlying soft tissue
and vital organs. Here, the functional performance of such armors is investigated, in
which the composition, spatial arrangement, and morphometry of the scales provide
locally tailored functionality. Fabricated macroscale prototypes and finite element
based micromechanical models are employed to measure mechanical response to blunt
and penetrating indentation loading. Deformation mechanisms of scale bending, scale
rotation, tissue shear, and tissue constraint were found to govern the ability of the
composite to protect the underlying substrate. These deformation mechanisms, the
resistance to deformation, and the resulting energy absorption can all be tailored by
structural parameters including architectural arrangement (angle of the scales, degree
of scale overlap), composition (volume fraction of the scales), morphometry (aspect
ration of the scales), and material properties (tissue modulus and scale modulus). In
addition, this network of armor serves to distribute the load of a predatory attack
over a large area to mitigate stress concentrations. Mechanical characterization of
such layered, segmented structures is fundamental to developing design principles for
engineered protective systems and composites.
Thesis Supervisor: Mary C. Boyce
Title: Ford Professor of Mechanical Engineering
Thesis Supervisor: Christine Ortiz
Title: Professor of Material Science and Engineering
3
4
Acknowledgments
I would like to thank my advisors, Professor Mary C. Boyce and Professor Christine
Ortiz for their endless support, wisdom, and enthusiasm. They have helped me grow
and develop into the confident woman I am today. They are both excellent role
models and have taught me the importance of professional success in the balance of
life.
Thanks to the all of the Boyce and Ortiz group members for your help and advice.
Special thanks to Meredith Silberstein, Juha Song, and Shawn Chester for showing
me the ropes and to Narges Kaynia, Ting Ting Chen, Swati Varshney, and Matt
Connors for being an absolute pleasure to work with.
I'm grateful for the technical assistance of Dr. Ara Nazarian at Beth Israel Deaconess Medical Center (BIDMC) Orthopedics Biomechanics Laboratory and Dr. Alan
Schwartzman at the MIT Nanolab. I'd also like to acknowledge Mark Belanger at the
MIT Edgerton Shop, Bill Buckley at the Laboratory for Manufacturing and Productivity (LMP) Shop, and Ken Stone at the MIT Hobby Shop for doing things faster,
safer, and more easily than I thought was possible since before I was born.
None of this would have been possible without the support of my friends, family,
and fiance. Thanks to Adam and Jason for visiting me in this arctic tundra and Jaclyn
for being the sister I never had. Thanks Mom and Dad for the weekly reminders to
graduate, in case I somehow forgot. Thanks James for his patience and support of
my career, in spite of the challenges it created. Thanks JetBlue flight 1264 for getting
him here every other week and Beyonce for reminding him to put a ring on it. I'm so
grateful that Joe and Jenna Petrzelka have been my rock and source of coffee since
day one and that Tiffy and Mark spice up my ramen noodle diet with molecular
gastronomy. And thank you, litter, for coming into my life and making everything so
much more enjoyable.
This research was supported by the U.S. Army Research Office through the MIT
Institute for Soldier Nanotechnologies under contract W911NF-07-D-0004.
5
6
Contents
15
1 Introduction
2
1.1
Bioinspired flexible protective systems
. . . . . . . . . . . . . . . . .
15
1.2
Fish scales as a structural biological material . . . . . . . . . . . . . .
17
1.3
O verview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.1
2.2
3
19
Background
Geometry of elasmoid fish scale assembly
. . . . . . . . . . . . . . .
2.1.1
Fish integument overlap and arrangement
. . . . . . . . . . .
19
2.1.2
Ex-vivo characterization with pCT . . . . . . . . . . . . . . .
22
2.1.3
Model formulation
. . . . . . . . . . . . . . . . . . . . . . . .
24
Fish scale and tissue materials properties . . . . . . . . . . . . . . . .
26
29
Mechanical protection under blunt loading
3.1
3.2
Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.1.1
Prototype fabrication and mechanical testing . . . . . . . . . .
29
3.1.2
Finite element micromechanical model
. . . . . . . . . . . . .
33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
Infinitely long (periodic) structures . . . . . . . . . . . . . . . . . . .
48
. . . . . . . .
48
Experimental and simulated finite-sized
structures
3.3
3.4
19
3.3.1
Deformation mechanisms in periodic structures
3.3.2
Energy absorption
. . . . . . . . . . . . . . . . . . . . . . . .
60
3.3.3
Effect of scale morphometry . . . . . . . . . . . . . . . . . . .
64
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
7
4
Mechanical protection under predatory tooth attack
69
4.1
Finite element micromechanical model
. . . . . . . . . . . . . . . . .
70
4.2
Indentation simulations results . . . . . . . . . . . . . . . . . . . . . .
71
4.3
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5 Bending and flexibility
6
79
5.1
Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.2
R esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
Conclusions
87
6.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
6.2
Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
6.3
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
A Materials characterization of the scales of Arapaima gigas
91
A. 1 Elemental analysis with EDX and BSEM . . . . . . . . . . . . . . . .
91
A.2 Mineralization analysis with TGA . . . . . . . . . . . . . . . . . . . .
94
A.3 Nanoindentation
94
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B DIC and load synchronization
99
C Periodic boundary conditions
103
C.1
General mathematical formulation for periodicity in two directions
C.2 Case I: Fixed corner node with periodicity in two directions
. . . . .
103
105
C.3 Case II: Fixed corner node, roller surface, with periodicity in one direction 109
C.4 Abaqus input file . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
C.5 MATLAB code to generate Abaqus input file . . . . . . . . . . . . . .
115
8
List of Figures
. . . .
16
. . . . . . . . . . . .
20
2-2 Scale overlap for the common rudd . . . . . . . . . . . . . . . . . . .
21
2-3 Scale overlap for the family of Cyprinidae fish . . . . . . . . . . . . .
21
. . . . . . . . . . . . . .
23
2-5 pCT slices of salmon integument across body length . . . . . . . . . .
24
2-6
Overview of fish assembly and geometric variables . . . . . . . . . . .
25
2-7
Map of scale volume fraction
0, Ra = 50) . . . . . . . . . . . . .
26
2-8
Materials properties of fish scales and tissue . . . . . . . . . . . . . .
27
3-1
3-D printed prototypes . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3-2
Compression testing of silicone rubber
. . . . . . . . . . . . . . . . .
31
3-3
Tensile testing of ABS dogbones . . . . . . . . . . . . . . . . . . . . .
32
3-4
Selected experimental geometries for testing . . . . . . . . . . . . . .
33
3-5
Experimental setup on Zwick
. . . . . . . . . . . . . . . . . . . . . .
34
3-6
Undeformed finite-sized geometries
. . . . . . . . . . . . . . . . . . .
35
3-7 Undeformed geometries for periodic simulations . . . . . . . . . . . .
36
3-8
Definitions of energy and stiffness. . . . . . . . . . . . . . . . . . . . .
37
3-9
Definitions of deformation mechanisms. . . . . . . . . . . . . . . . . .
38
1-1
Fish armor and its human analogues.
2-1 Histology of fish integument in different species
2-4 Sample prep for pCT of salmon integument
#(Kd,
3-10 Experimental deformation under blunt loading for select geometries.
40
3-11 Experimental deformation under blunt loading for all geometries.
41
3-12 Simulations of select finite-sized geometries under blunt loading.
.
3-13 Experimental cracking and delimitation . . . . . . . . . . . . . . . . .
9
44
45
3-14 Stiffness and back deflection for experiments . . . . . . . . . .
46
3-15 Stiffness and energy absorption for finite-sized simulations
. .
47
3-16 Spring model of layered composite. . . . . . . . . . . . . . . .
48
3-17 Periodic simulations for select geometries under blunt loading
49
3-18 Loading curves for all periodic simulations under blunt loading
50
3-19 Deformation mechanisms, stiffness, and energy absorption
52
3-20 Simulated contours of u2 for blunt loading . . . . . . . . . . . . . . .
53
3-21 Simulated contours of
622
for blunt loading
.
. . . . . . . . . . . .
54
3-22 Simulated contours of
612
for blunt loading
.
. . . . . . . . . . . .
55
3-23 Simulated contours of ol for blunt loading . . . . . . . . . . . . . . .
56
3-24 Simulated contours of o 22 for blunt loading . . . . . . . . . . . . . . .
57
3-25 All deformation mechanisms for blunt loading
. . . . . . . . . . . .
58
3-26 Key deformation mechanisms for blunt loading
. . . . . . . . . . . .
59
3-27 Deformation resistance vs. energy absorption for blunt loading . . . .
62
3-28 Maps of energy absorption for blunt loading . . . . . . . . . . . . . .
63
3-29 Effect of scale morphometry . . . . . . . . . . . . . . . . . . . . . . .
65
3-30 Deformation mechanisms for blunt loading and changing Ra
66
. . . . .
4-1
Predatory attack by a tooth .................
4-2
Indentation loading conditions . . . . . . . . . . . . . . . . . . .
71
4-3
Contours of indentation of fish scale assembly
. . . . . . . . . .
73
4-4
Indentation force and back deflection for R/t, = 0.1 . . . . . . .
74
4-5
Indentation loading curves for varying indenters (R/t = 0.1, 10)
75
4-6
Deformation resistance vs. energy absorption for indentation . .
76
5-1
Experimental setup for bending over die. . . . . . . . . . . . . .
80
5-2
Experimental deformation under bending for all geometries.
81
6-1
Summary of deformation mechanisms under blunt loading.
6-2
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
A-1 Overview and SEM of Arapaima gigas scales . . . . . . . . . . .
92
10
......
69
.
88
A-2 Materials characterization of Arapaima gigas scales by EDX and BSEM 93
A-3 Mineralization of Arapaima gigas scales using TGA . . . . . . . . . .
95
. . . . . . . .
96
. . . . . . . . . . . . . . .
97
A-4 Location for nanoindentation of Arapaima gigas scales
A-5 Nanoindentation of Arapaima gigas scales
C-1 Representative volume element for periodic boundary conditions. . . . 104
C-2 Node sets for periodic boundary conditions.
. . . . . . . . . . . . . . 105
C-3 Periodic boundary conditions used in model. . . . . . . . . . . . . . . 109
11
12
List of Tables
2.1
Control parameters for pCT scan . . . . . . . . . . . . . . . . . . . .
22
2.2
Description of parameters and variables used in study . . . . . . . . .
28
3.1
Volume fractions
studied where Ra = 50. . . . . . . .
30
3.2
Material properties used in compression simulations . . . . . . . . . .
37
3.3
Normalized areal density PA for
0, Ra = 50) where p, ~ pt. . . .
60
3.4
Normalized areal density PA for
#(Kd,
#(Kd,
0, Ra = 50) where p8 = 2 .6pt.
61
3.5
Aspect ratio study parameters.
. . . . . . . . . . . . . . . . . . . . .
65
4.1
Material properties and geometry used in indentation simulations. . .
70
A. 1 Thermal profile for thermogravimetric analysis . . . . . . . . . . . . .
94
# for geometries
13
14
Chapter 1
Introduction
1.1
Bioinspired flexible protective systems
Natural protective systems may hold the key to the design of new structural materials that are flexible and lightweight while still affording mechanical protection.
These biological materials have the potential to inspire rapid advances in the design
and engineering of new structures for a variety of applications, from microsystems to
body armor to vehicle and building architecture. In order to realize these designs, a
full understanding of the structure-function relationships and underlying mechanics
of such systems is essential. Biological exoskeletons have evolved over many millions
of years to maximize survivability by incorporating various structural design principles (Ortiz and Boyce, 2008).
In the case of ancient fish, dermal armor became
thinner, lighter, and subdivided into small plates to accommodate threats without
compromising mobility (Colbert and Morales, 2001; Arciszewski and Cornell, 2006).
Today, most species of bony fish (e.g. zebrafish Danio rerio, Arapaima gigas) possess
a network of flexible yet stiff and strong scales that provide a lightweight defense
against predators (Fig. 1-1). These scales are angled, overlapped, and embedded in
the tissue to provide spatially homogeneous protective coverage and flexibility that
can be tailored across the body. Unlike the articulating exterior scales on creatures
such as snakes, lizards, pangolins, or butterfly wings, the teleost fish scales are mineralized and deeply embedded within the dermis of the integument (Whitear, 1986).
15
Embedding scales fully within the tissue creates a unique robust composite system in
which the interplay between the mineralized scales and the compliant tissue plays a
nontrivial role in tailoring and controlling deformation mechanisms. A careful balance
of mobility and protection is an important goal in animal evolution, just as it is in the
engineering of human armor (Arciszewski and Cornell, 2006), and hence imbricated
fish scales are an appealing biomimetic system.
(a)
(b)
(c)
Figure 1-1: Fish armor and its human analogues. (a) Teleost fish Atlantic salmon
(Salmo salar) is covered in scales. A cross section (below) shows that scales are overlapping and embedded in a tissue. (b) Backplate of an early 16th century brigandine.
Brigandines were used as body armor and composed of small iron plates riveted to a
canvas or linen fabric backing. (c) Dragonskin is modern segmented armor developed
by Pinnacle Armor, consisting of ceramic discs encased in an aramid textile cover.
Images adapted from Ostman, Elisabeth, Iduns kokbok, 1911, Last accessed on May
1,2012. http: //commons. wikimedia. org/wiki/Salmosalar; Evans (1994); Mitsuhiro Arita, Exoroche on Arms and Armor, Last accessed on May 1, 2012. http:
//wiki.ffxiclopedia.org/wiki/Exoroche-onArmsandArmor; Daderot, Brigandine backplate, probably Italy, early 16th century - Higgins Armory Museum,
Last accessed on May 1, 2012. http://commons.wikimedia.org/wiki/Category:
PiecesofarmorintheHigginsArmoryMuseum; Ling Li, April 27, 2012; Pinnacle Armor, Dragon Skin, Last accessed on May 1, 2012. http: //www .pinnaclearmor.
com/body-armor/dragon-skin/; Wikipedia, Dragon Skin, Last accessed on May 1,
2012. http://en.wikipedia.org/wiki/DragonSkin; Neal and Bain (2004).
Modern elasmoid scales evolved from the ganoid and rhomboid scales of primitive
fish and are now nearly ubiquitous in present day. Compared to their ancient counterparts, elasmoid scales are thinner, lighter, and more flexible, having eliminated bone
and reduced hypermineralized components (Sire, 1990; Sire et al., 2009). Fish scales
16
vary in both morphology (Burdak, 1979; Ibafiez et al., 2009) and mechanical properties (Garrano et al., 2012) across the surface of the body to accommodate inherent
body curvature and performance requirements, e.g., increased biomechanical flexibility near the tail for locomotion, and on the main body for increased protection. Such
structure-function relationships have evolutionary implications for swimming mode
(Breder, 1926; Lindsey, 1979), body stiffness and flexural response (Long et al., 1996;
Hebrank and Hebrank, 1986; Hebrank, 1980), hydrodynamics (Whitear, 1986), and
defensive armor (Burdak, 1979; Vermeij, 1983) in which species have adapted their
integument in response to environmental stimuli to maximize survivability.
1.2
Fish scales as a structural biological material
The elasmoid fish scale has recently been investigated as a viable structural biological
material. Material structure and properties vary by species, but elasmoid scales have
been reported to be mineralized 16-59%, slightly anisotropic, and have hydrationdependent material properties (Whitear, 1986; Torres et al., 2008; Zhu et al., 2012;
Garrano et al., 2012). They are composed of a bony mineralized outer layer and a
compliant basal collagen layer (Sire, 1990; Sire et al., 2009). Experimental measurements of mechanical properties have been limited to the mechanics of an individual
scale resisting penetration by a sharp indenter or uniaxial tensile testing. Testing
of scales in a hydrated state found the tensile strength to range 22-65 MPa and
toughness ~1 MPa (Meyers et al., 2011; Zhu et al., 2012; Ikoma et al., 2003; Torres
et al., 2008; Garrano et al., 2012). The multilayered fish scale localizes penetration
by promoting circumferential (Bruet et al., 2008; Song et al., 2011) or cross-like (Zhu
et al., 2012) fracture patterns in the outer layer. On a larger length scale, work by
Vernerey and Barthelat (2010) introduced a two-dimensional mathematical model of
overlapping fish scales in accommodating larger length scale bending and radius of
curvature as a function of scale geometry and material properties. They find a strainstiffening behavior of the assembly with increasing scale overlap provides the ability
to distribute deformation over a large area during bending. However, the mechan17
ics of embedded overlapping scales in response to blunt and predatory penetration
loading remains unexplored. The ability of fish scales to function as a dermal armor
and provide protection via minimizing back deflection and compressive stresses in
underlying tissues remains to be explored. In addition, a composite model is needed
to fully understand the interplay of the mineralized scales with a hyperelastic tissue
as a means of tailoring deformation mechanisms.
1.3
Overview
Here, the architectural structure of the overlapping scales embedded in a soft tissue is
investigated as a key determinant in mechanical protection to advance understanding
of the mechanical design principles of flexible protective biological exoskeletons. The
composite system developed herein considers fish scales fully embedded in a tissue
matrix, more closely emulating physiological conditions. The connections amongst
the composite structure, the deformation mechanisms, and the macroscopic response
to a variety of physiological loading conditions are explored. In particular, the roles
of tissue shear and scale bending and rotation as a function of scale geometry are
identified.
To accomplish these objectives, a finite element based micromechanical model was
developed in which protective response was determined as a function of scale aspect ratio, scale orientation angle, spatial overlap, and volume fraction of scales. Macroscale
synthetic models were constructed using a combination of three-dimensional printing and molding methods; mechanical behavior under blunt compressive loading was
measured using mechanical testing where digital image correlation (DIC) was used to
measure the strain field in the composite material. These experimental and numerical models yield detailed insights into the roles and the tradeoffs of the composite
structure providing constraint, shear, and bending mechanisms to impart protection
and flexibility.
18
Chapter 2
Background
2.1
2.1.1
Geometry of elasmoid fish scale assembly
Fish integument overlap and arrangement
To simplify the complex geometry of a fish, this study focuses only on the integument
(i.e. the epidermis, dermis, and embedded scales).
A superficial examination of
the surface of a teleost fish reveals articulating scales; however, histological sections
(Whitear, 1986; Park and Lee, 1988; Hawkes, 1974; Sire, 1990; United States Fish
and Wildlife Service, 2010) further reveal that the scales are fully surrounded by soft
tissue comprised by the dermis and epidermis and also embedded deeply into the soft
tissue at an angle (Fig. 2-1). The primary features of the structure of mineralized
scales embedded in a soft tissue are captured in a micromechanical model consisting of
(1) an upper layer of relatively high modulus (E,) scale of aspect ratio L8 /t, oriented
at angle 0 and embedded in a low modulus (Et) tissue, and (2) a soft (low modulus
Et) tissue substrate support representing the underlying collagenous layer.
The scales are initially oriented at angle 0 relative to a compliant tissue support.
Scales overlap each other such that a fraction of the scale is embedded beneath an
adjacent scale. The length d is the exposed length of scale, while the remaining
length (L, - d) is embedded beneath another scale. This degree of imbrication, or
overlap, Kd - d/L, is a measure for the spatial overlap defined by Burdak (1979),
19
(a)%%
_
_
-__ME
400
(b)
V
100 pm
(d)
(d)
Figure 2-1: Histology of integument in different species of teleost fish. Scales are
shown to be embedded in a dermal tissue at varying angles 0 and overlap Kd. Adapted
figure showing section of integument of (a) trout (United States Fish and Wildlife
Service, 2010), (b) mangrove killifish (Park and Lee, 1988), (c) unknown (Guinan),
(d) cichlid (Sire, 1990), (e) coho salmon (Hawkes, 1974), and (f) unknown (Smith,
2012).
who measured Kd for 16 species of the Cyprinidae family of fish and found that values
range from 0.24 < Kd
1 depending on species and body location (Fig. 2-2, 2-3).
Fish with highly overlapping, heavily armored skins such as the tench (Tinca tinca),
have Kd ~ 0, whereas lightly scaled fish have Kd ~ 1, as is the case for undulating
eel-like fish. The number of scales per unit length is given by Ki-.
The overlap along
the circumference was determined to be constant for all species studied by Burdak
(1979).
Typical values for the scale thickness in teleost fish are on the order of t, ~ 0.1
20
(a)
(b)
(c)
1(d
Figure 2-2: Scale overlap Kd along the length of the common rudd (Scardinius erythrophthalmus). Image adapted from Burdak (1979).
Kd along length of fish
alburnus alburnus alburnus
rutilus rutilus rutilus
blicca bjoerkna bjoerkna
leuciscus cephalus cephalus
scardinius erythrophthalmus
chondrostoma nasus variabile
carassius auratus gibelio
carassius carassius
cyprinus carpio carpio
leuciscus idus idus
tinca tinca
gobio gobio gobio
misgurnus fossifis
cobitis taenia taenia
nemachilus barbatulus barbatulus
1
0.8
cc 0.6
a)
0
0.4
0.2
increasing fish size
0
3
Kd = d/L
region of fish
Figure 2-3: Scale overlap Kd along the length body for the family of Cyprinidae fish.
Image adapted from Burdak (1979).
21
mm, but reported values range from 44 pum to 1 mm, (Whitenack et al., 2010; Meyers
et al., 2011).
Typical scale lengths, LS, range from 5 to 15 mm, but have been
reported up to 100 mm long (Francis, 1990; Garrano et al., 2012; Zhu et al., 2012;
Meyers et al., 2011; Torres et al., 2008; Jawad and Al-Jufaili, 2007).
Despite the
large range of observed scale lengths and thicknesses, the geometry of the fish scale
preserves an aspect ratio Ra = L8 /t, between 25 to 100 based on available literature
and measurements. The effect of various geometric parameters will be studied (as
discussed later) but the scale will be held at constant aspect ratio of either Ra
=
50
or 100, which were taken as intermediate and maximum values, respectively.
2.1.2
Ex-vivo characterization with
CT
The aspect ratio of scales has been observed to vary not only from species to species,
but also across the length of an individual fish. Micro-computed tomography (iCT)
was employed to obtain full three-dimensional characterization of the integument of
a fresh Atlantic salmon (Salmo salar) obtained from a local fishery (New Deal Fish
Market, Cambridge, MA). Samples were excised from the main body and caudal
peduncle (tail) near the lateral line and immediately scanned to reduce the effects of
dehydration (Fig. 2-4). Scans were performed with a pCT40 (Scanco Medical AG,
Switzerland) using the parameters in Table 2.1 and analyzed following previously
reported methods (Song et al., 2010; Connors et al., 2012).
Mimics (Materialise,
Belgium) and Adobe Photoshop were used to process the DICOM slices and threshold
the partially mineralized scales. The ttCT slices reveal the aspect ratio, overlap, and
angle of the scales of the salmon vary as a function of location along the body shown
in Fig. 2-5.
Table 2.1: Control parameters for puCT scan
E (energy)
I (current)
FOV (tube 0)
voxel (resolution)
45 kVp
177 pA
36.9 mm (largest)
18 Pm (high)
22
(a)
Figure 2-4: Sample prep for pCT of fish integument. (a,b) Scales of Atlantic salmon
(Salmo salar) are - 10 mm long and transparent. (c) After underlying muscle is
removed, resulting tissue and scale composite is flexible. (d,e) Tissue is cut into
~ 25 x 25 mm sections and (f) scanned in puCT tube between layers of foam. Salmon
image adapted from Ostman, Elisabeth, Iduns kokbok, 1911, Last accessed on May
1, 2012. http: //commons. wikimedia.org/wiki/Salmosalar.
23
2 mm
Figure 2-5: (a) Teleost fish Salmo salar,the Atlantic salmon, covered with (b) integument of overlapping stiff scales. (c) Two-dimensional pCT slices of structure (false
coloring), showing scales embedded in a dermal tissue at various geometric configurations. Salmon image adapted from Ostman, Elisabeth, Iduns kokbok, 1911, Last
accessed on May 1, 2012. http://commons.wikimedia.org/wiki/Salmo.salar.
2.1.3
Model formulation
Relevant geometric parameters and structures are generalized in a model geometry
shown in Fig. 2-6. This non-dimensional formulation allows for length-independent
analysis of overlapping composite assemblies across a range of length scales.
All
parameters and variables in model are shown in Table 2.2.
In most teleost fish species, the upper epidermis follows the contours of the scale
pockets (Whitear, 1986), and hence, the irregularity of the surface topology increases
as a function of Kd and 0. A variable of scale volume fraction,
#,
can be calculated
in terms of the thickness of the dermis between scales td, and scale thickness to, as
#
=
ts/(ts + td).
This fraction # represents the scale-to-tissue ratio in the upper,
scaled layer and does not include the tissue support, as this bottom layer serves only
to capture physiologically relevant back deflection in the underlying integument. This
bottom tissue layer is fixed at height Htissue = Ls/2. Here, the scale volume fraction
is reported as
#(Kd,
0, Ra) as shown in Fig. 2-7, whereas previous models, neglecting
a composite system, only consider a linear
#
~ 1/Kd. This study will vary geometric
parameters as governed by the following relationship:
tan2 0 +1
KdRatan0
24
k
I
9
CliJ
Figure 2-6: Overview of fish assembly assembly and variables. This non-dimensional
formulation allows for length-independent analysis of overlapping composite assemblies across a range of length scales.
An approximate biologically relevant range of
#(Kd,
0, Ra) is shown in Fig. 2-
7 based on available histological sections and puCT scans (Whitear, 1986; Park and
Lee,.1988; Hawkes, 1974; Sire, 1990; United States Fish and Wildlife Service, 2010;
Guinan).
To compare relative weights of architectures given tissue density pt and scale
density ps, the approximate areal density PA (mass per unit area) of the composite
prototypes is given by
PA -
ptL2
+ L, sin[p# -+pt(1 - @)]
fixed in study
function of architecture
The areal density of the structure PA is normalized by the areal density of the bottom
layer alone (fixed), highlighting the additional weight of the scale layer.
PA =
1 + 2 sin0[(ps/pt)4 + 1 - $)]
Assuming similarity of the scale and tissue densities (p8 ~ pt), this weight is primarily
25
Volume Fraction <p
40
__
_
_00__
_
__
_
__
_
.0e_
30
OC 20
10
0
0.5
00
1-
Kd (scale overlap)
0
0.2 0.4 0.6 0.8
1
Figure 2-7: Map of scale volume fraction q(Kd, 0, Ra = 50) indicating approximate
biological range based off histological sections and measurements in Figs. 2-1 and
2-5. Corresponding geometry from model shown on right.
a function of scale angle: PA - PA(O), as discussed later in Section 3.3.2.
2.2
Fish scale and tissue materials properties
Fish scales are composed of a cross-ply collagen layer and an outer mineralized hydroxyapatite layer as detailed in Appendix A (Sire, 1990; Sire et al., 2009) and, for
the purpose of this work, can be approximated as an elastic material. Tensile testing
of fish scales reports an effective scale elastic modulus is ~ 1 GPa but varies among
species and along the length of the body of the fish (Fig. 2-8). Testing of scales in
a hydrated state (more closely resembling in vivo conditions) yields a reduced elastic
modulus 0.1-0.8 GPa and tensile strength to range 22-65 MPa (Torres et al., 2008;
Zhu et al., 2012; Garrano et al., 2012; Hebrank and Hebrank, 1986; Meyers et al.,
2011; Ikoma et al., 2003).
The elastic properties of fish scales are in the range of
common engineering polymers (e.g., polycarbonate, acrylonitrile butadiene styrene),
as shown in Fig. 2-8. The density of fish scales (ps) is approximated using results
obtained from thermogravimetric analysis (see Appendix A.2) and the relative densities of hydroxyapatite (mineralized) and collagen (organic). The density of fish scales
26
ranges 1.92-2.43 g/cc based on available literature and measurements (Ikoma et al.,
2003; Torres et al., 2008).
Little work has been done to characterize the tissue in the integument of fish.
Work by Hebrank and Hebrank (1986) reported tensile testing of fish tissue samples
(both with and without scales) to determine an approximate modulus. For comparison, these results are shown along with other biological tissues in Fig. 2-8. Many
biological tissues exhibit hyperelastic, nearly incompressible materials properties similar to rubber. In general, the tissue is orders of magnitude more compliant than the
embedded scales. For reference, the density of fish tissue (pt) is approximated as an
intermediate value between human adipose tissue (p = 0.92 g/cc) and human muscle
(p = 1.06 g/cc) density (Heymsfield et al., 2002). As an upper bound, the scales are
approximately 2.6 x as dense as the surrounding tissue.
(a)
scale elastic
modulus, E
3D printed ABS
extruded ABS
polymer
fish scale1
(wet or
0
1 GlPa
2GPa
0
)
(b)
tissue shear
modulus, G
10 kPa
100 kPa
1 MPa
10OMPa
Hebrank, 1986; Meyers et al., 2011; Ikoma et al., 2003).
27
100OMPa
1GPa
Table 2.2: Description of parameters and variables used in study
Parameter
Description
Ls
ts
scale length
scale thickness
dermal tissue thickness
epidermal tissue thickness
exposed scale length
td
te
d
Htissue
Hscale
total height
scale overlap
scale angle
scale volume fraction
scale modulus
tissue modulus
scale density
tissue density
normalized areal density
PA
= t,(1/
te
-
Property
geometric
geometric
geometric
geometric
geometric
1)
ts
Htissue Ls/2
Hcae ~ L, sin 0
Htotal = Hcae + Htissue
Kd = d/L
scale layer height
Kd
Es
Et
Ps
Pt
td
tissue support height
Htotai
0
Formula
#
PA
28
tan
=1 + 2 sin 9[-Pt#
10-
#)]
geometric
geometric
geometric
spatial
spatial
composition
material
material
material
material
material &
composition
Chapter 3
Mechanical protection under blunt
loading
3.1
3.1.1
Materials and methods
Prototype fabrication and mechanical testing
Macroscale synthetic prototypes were fabricated and tested to characterize a row of
scales over a range of geometric architectures. A computer aided drawing package
(SolidWorks, Dassault Systemes) was used to create models of the scales in a variety
of geometric configurations. Scale aspect ratio was kept constant at Ra - 50 and
scale volume fraction <5 was varied from 0 to 1 by parametrically varying overlap
Kd
(0.1, 0.3, 0.5, 0.7, 0.9) and scale angle 0 (2.50, 50, 10', 200, 40') for a total of 21
arrangements (Table 3.1).
These models were exported as stereolithography (.STL) files for three-dimensional
printing using fused deposition modeling (Dimension BST 1200es, Stratasys). Parts
were built with thermopolymer acrylonitrile butadiene styrene (ABS) with a layer
thickness of 0.010 inches and a breakaway support. The resulting parts were placed
in an acrylic frame and used as a mold for a translucent silicone rubber (Mold Max
10T, Smooth-On). After the rubber cured, the ABS scales were permanently embedded in the silicone and the frame was removed, creating the prototype parts shown in
29
Table 3.1: Volume fractions
#
#
0.1
2.50
50
-
bZ
100
-
d
200
400
0.62
0.41
for geometries studied where Ra = 50.
overlap Kd
0.3
0.5
0.7
0.77
0.39
0.21
0.14
0.92
0.46
0.23
0.12
0.08
0.66
0.33
0.17
0.09
0.06
0.9
0.51
0.26
0.13
0.07
0.05
(d)
Figure 3-1: (a) Fused deposition modeling (FDM) 3-D printing was used to obtain
macroscopic prototypes for a variety of architectures. ABS parts were placed in an
acrylic frame, creating a mold. (b) Silicone rubber was cured in the mold, embedding
the rigid ABS scales in the prototypes, which easily broke away from the PLA support
material. (c-d) Resulting fabricated samples with a U.S. quarter for reference.
Fig. 3-1. The final macroscale models had scale geometry LS = 50 mm, t' = 1 mm,
were 100 x 40 x -50 mm in size, and contained 2-6 repeating scale units depending
30
on angle 0 and overlap Kd. Material properties of the silicone rubber and ABS were
measured Eo,t = 390 kPa and E,
=
880 MPa using compressive and tensile testing,
respectively (Figs. 3-2, 3-3). These properties are in the range of biological materials
for wet elasmoid scales (Lin et al., 2011; Torres et al., 2008; Garrano et al., 2012) and
fish tissue (Hebrank and Hebrank, 1986), as shown in Fig. 2-8. The ABS to silicone
rubber modulus ratio is -2000.
H =25mm
W =40 mm
L = 100 mm
A = L*WF
0.2
expAFE
FF
uniaxial compression
CU
~0
6
Eo =0.39 MPa
H0
plane strain
W
L
0
0
e=6/H
o=F/A
0.1
e from DIC
0.2
Figure 3-2: Elastic modulus of the silicone rubber was evaluated using uniaxial compression testing of rectangular samples. A rigid frame was used (see Fig. 3-5) to
impose experimental plane strain conditions and results were verified with 2-D model
in Abaqus.
Plane strain compression testing of the synthetic prototypes was performed (Zwick
Z2.5kN, Zwick/Roel) at a loading rate of 0.1 mm/s to a maximum force of 400 N.
Each sample was tested three times and repositioned between each test to estimate
measurement errors. Error bars in the experimental data correspond to the standard
deviation of these results. Macroscopic engineering stress o- = F2 /A is calculated from
the measured vertical reaction force, F2 , and the initial total sample cross-sectional
(surface) area, A. To provide plane strain conditions for finite thickness geometry,
a frame was constructed with thick clear acrylic (PMMA) sheets, constraining the
sample in the 3-direction, which was precisely adjusted to account for slightly varying
sample thicknesses (Fig. 3-5). Chalk powder and Teflon sheets were used to reduce
friction on the sides and top/bottom of the rubber, respectively. A speckle pattern
was applied on the samples with an airbrush and India ink for use with optical ex31
(a)
V
(b)
I
(d)
(e)
Zwick Data
80
a-e from DIC
10
8
60
6
40
0-
4
20
0
0
2
0
0.1
0.2
0.3 0.4
gage elongation (mm)
0
'
0.005
0.01
8
0.015
0.02
Figure 3-3: Elastic modulus of the 3-D printed ABS was evaluated using tensile testing
of laser cut dogbone samples. Data from the digital extensometer is synchronized
to mechanical testing data to obtain engineering stress-strain curves as detailed in
Appendix B.
tensometry via digital image correlation (DIC). Images of the testing were taken at a
rate of 1 fps (VicSnap, Correlated Solutions) and DIC (Vic-2D 2009, Correlated Solutions) was used to track global deformation and obtain local contours of logarithmic
(Hencky) strain. Engineering strain E is calculated as the total vertical displacement
from DIC, 6, normalized by the initial total height, Htotai. Shear strain is reported as
engineering shear strain. Data from the digital extensometer was then synchronized
to mechanical testing data to obtain engineering stress-strain curves as detailed in
Appendix B.
32
(a)
(b)
Volume Fraction <
40
30
(c)
C 20
10
0
0
0.5
Kd (scale overlap)
0
0.5
(dl
1
Figure 3-4: (a) Map of scale volume fraction # and selected geometries for Ra = 50.
Macroscale 3-D printed and molded synthetic fish scale inspired assembly of ABS
(yellow) and silicone rubber (translucent): (b) Kd = 0.3, 0 = 200, # = 0.21, (c) Kd
= 0.3, 0 = 5, # = 0.77, (d) Kd = 0.9, 0 = 50, #S 0.26. Selected geometries have
similar scale overlap (b, c), angle (c, d), and volume fraction (b, d) to demonstrate
the functional effect of each structural parameter.
3.1.2
Finite element micromechanical model
A two-dimensional plane strain composite scale-tissue micromechanical model was
simulated using finite element analysis (FEA). The composite structure was discretized and modeled with quadratic continuum elements in ABAQUS/Standard (library types CPE6H and CPE8RH) and a fine mesh with elements of length
-
t,/4
was needed to capture the deformation within the scales. Tissue and scale elements
are modeled as perfectly bonded with properties found in Table 3.2. A blunt loading
event was conducted in the form of a distributed compression of the scale assembly
with a rigid plate. Non-linear, large deformation theory with frictionless contact was
assumed.
Simulations which consider the same finite length geometries and the same loading
conditions as the experimental samples, were conducted and allow for direct comparison of model with experimental results. These geometries correspond to the archi33
Figure 3-5: Experimental setup on Zwick with plane strain acrylic and aluminum
frame, Teflon and chalk powder for frictionless boundaries, and pre-loading (rubber
bands) to secure the sample against the base. The load from the platens is transferred
via a polycarbonate plate the width of the sample. Plastic shims (here, yellow and
brown) were used to precisely adjust the width of the aluminum frame to account for
varying sample thicknesses.
tectures
#(Kd,
0, Ra = 50) found in Table 3.1 as and shown in Fig. 3-6. To simulate
structures with many scales, such as those found on the entire length of a fish, periodic boundary conditions were also imposed on the lateral edges of the scale assembly
while allowing overall expansion in the 1-direction, as detailed in Appendix C. These
geometries are the same architectures
#(Kd,
6, Ra = 50) found in Table 3.1 and have
varying length proportional to LS cos 0 as shown in Fig. 3-7. Note that the representative volume element (RVE) used is 3x as long as necessary, but used to better
visualize the deformations in the structures. Further simulations were carried out
changing the aspect ratio of the scales, Ra = L/t,, from 50 to 100 by increasing the
length of the scales L, while holding scale volume fraction
results in overlap Kd half of that shown in Table 3.1.
34
# and
angle 0 fixed. This
Kd = 0.3
Kd = 0.1
K, - 0.7
Kd = 0.5
Kd = 0.9
0
LO
$
=
0.92,1PA= 1.07 (1.22)
0.66,
A
1.07 (1.18)
$P= 0.51, p5A
= 1.08 (1.16)
a)
0
LO
11
$0.46, j5= 1. 15 (1.30)
4 = 0.33, AA= 1.16 (1.27)
$= 0.23, PA= 1.33 (1.48)
$P 0.17, PA= 1 .3 3 (1.44)
P = 0.13, 5A= 1. 7(1.76)
1.65 (1.91)
$P =0.12,1A= 1.66 (1.82)
$ =0.09, p= 1.67 (1.78)
$=0.07, PA =1.67 (1.76)
2.25 (2.56)
(P ='0.08, PA = 2.26 (2.45)'
CD
$ =0.77, PA= 1.14 (1.39)
0
$= 0.39 PA= 1. 3 1 (1.56)2
I0.26, 5=
1.16 (1.25)
0
I
11
S= 0.62,
= 1.58 (2.37)
$P 0.21,
PA
CID
=O.41, p = 2.15 (0.14,
$= 0.05, PA= 2 .2 7 (2.38)
Figure 3-6: Undeformed finite-sized geometries #(Kd, 0, Ra = 50) used to simulate experiments. Areal density PA is given for
experimental and silicone rubber and ABS and biological tissue and scales in parentheses.
..
.. ..
....
....
I..........
Kd = 0.3
Kd = 0.1
Kd = 0.5
K, = 0.7
Kd =0.9
0
LO
Ci
II
CD
$= 0.21A
1.07 (1."22)
$= 0.66 TA= 1.07 (1.18)
4)=0.51, I,
1.08 (1.16)
4) =046
1.15 (1.30)
4) 0.33, 1& 1.16 (1 27)
P = 0.26,15A
1.16 (1.25)
0
LO
$ = 0.77,
15A
14 (1.39)
11
0
0
4= 0.39,
CD
A=
1. 3 1 (1.56)
t= 0.23,
5A1.33
(1.48)
4) = 0.17, jY
= 1.33 (1.44)
0
0
C\J
CID
$ I=0.6 ,
PTA=
.58 (0.37)
PA
$. =021,
91
(1 *165
4)=0.12, FA*= 1.66 (.82)
*.07,5A= *.7(.6
0
0D
041,
pTA .1 (3.1 2)
$0K.08,15A=2.26
(2 5)
Figure 3-7: Undeformed geometries #(Kd, 0, R, = 50) with periodic boundary conditions used to simulate very long structures.
Areal density PA is given for experimental and silicone rubber and ABS and biological tissue and scales in parentheses. These
structures have varying length proportional to L, cos 0. Note that the representative volume element (RVE) used is 3 x as long
as necessary, but used to better visualize the deformations in the structures.
Table 3.2: Material parameters experimentally determined from synthetic prototype
and employed in simulations.
Scale
Tissue
Material
Behavior
Modulus
3-D printed ABS
Elastic
Es = 880 MPa
Silicone rubber
Neo-Hookean
Eo,t = 0.39 MPa
Poisson's ratio
Density
v, = 0.4
Ps = 0.80 g/cc
vt - 0.49
pt = 1.07 g/cc
Macroscopic engineering stress and strain were calculated in the same manner as
done experimentally (stress o = F2 /A and strain c = 6 /Hota). Local true Cauchy
stress and logarithmic Hencky strain contours were evaluated at different load levels to identify the deformation mechanisms which accommodate the loading for the
different composite geometries. An effective secant stiffness Elo-=.o5
MPa
=
a/E was
determined (Fig. 3-8) and normalized by tissue elastic modulus Et, highlighting the
stiffness amplification effect of the scales. Strain energy density U o0.05
MPa
=
o Ed
was determined and normalized by the energy absorbed by the tissue at a same macroscopic stress, Ut.
E
0*
Figure 3-8: Definitions of energy E and stiffness U taken at a*. For this work, both
E and U were evaluated at a* = 0.05 MPa.
Angular rotation dO of individual scales in response to loading was calculated
37
with respect to initial angle: dO = 9' - 9 (Fig.
3-9).
For small bending within
an individual scale, the local deformed curvature is described by
K
= a2y'/Ox'2 in
a rotated coordinate frame (1', 2') aligned with original scale angle 9. The average
bending curvature along a scale of deformed length s is given by ks =
fLS Kds and
found by extracting the displacements ui and u 2 of nodes along the scale in the mesh.
(a)
LC
0
0
0
0
0
0
0
0
0
11
(b)
1
K(S)
21
Figure 3-9: Definitions of deformation mechanisms under blunt loading with periodic
boundary conditions. (a) Undeformed configuration and boundary conditions for Kd
= 0.5, 9 = 5', # = 0.46, PA = 1.33 and (b) deformed configuration exhibiting local
scale curvature K(s), rotation dO = 0' - 9, and shear strain E12 in tissue.
3.2
Experimental and simulated finite-sized
structures
The geometric arrangement of the scale and tissue structure governs the response to
blunt loading, resulting in a substantial dependence of back deflection (6 tissue, refer to
Fig. 2-6) and stiffness (E) on geometric arrangement of the assembly. Macroscopic
loading curves for the finite-sized geometries are shown in Fig. 3-10a for three selected
geometries (all with scale aspect ratio Ra = 50):
(i) Kd = 0.3, 9 = 200, # = 0.21
(ii) Kd = 0.3, 9 = 50, # = 0.77
38
(iii) Kd= 0.9, 0 = 50,
#
= 0.26
These particular geometries are highlighted because they demonstrate the functional effect of each structural parameter individually by sharing similar scale overlap
Kd
(i, ii), scale angle 0 (ii, iii), or scale volume fraction
#
(i, iii). The experimen-
tal and simulated loading curves for the same finite-length geometries show that all
structures studied have an initial nonlinear response due to contact conditions from
the irregular loading surface. It was also observed that the bilayer structure of the
finite-sized composite allowed the bottom tissue layer of the samples to laterally expand significantly (seen in Fig. 3-12).
This edge effect has a strong influence on
several aspects of the results of the simulations, as discussed later in Section 3.3.1
where the influence of finite-sized specimens are compared to infinitely long periodic
structures.
When evaluated at a fixed macroscopic strain, the effect of scale geometry on the
maximum back deflection in the underlying tissue was found to be modest. As shown
in Fig. 3-10b, increasing Kd from 0.9 to 0.3 (a -3x increase in
#)
offered a <10%
decrease in back deflection tissue at any fixed strain. However, the effective stiffness
of the composites is strongly dependent on the scale arrangement, and hence, at a
fixed macroscopic stress, the same ~3x increase in
# offers
a 30-40% decrease in back
deflection (Fig. 3-10c). Given that predatory attacks are often force-limited, these
results show how geometric scale configurations and compositions observed in nature
can protect underlying tissue by limiting tissue straining as expected. By tailoring
the structural parameters, species are able to control the local protective function of
their armors.
39
(a)
0.1
0.08
(c) 0.12
*
Kd=0.3, 0=5*,
III
* 0.08
$=0.77, PA114
E- 0.06
0
(b) 0.12
6=20*,
P=0.21, P,= 1 .6 5
Kd=0.3,
0.08
K,=0.9, 0=5*,
0.04
P=026, PA=
0.04
0.02
0
0.00
0
0.05
0.1
0
0.05
U2 (mm)
0.00
0
0
-0.15
0.05
a (MPa)
0.1
-12
-22
-2.5
(d)
0. 1
0
-0.15
-0.
15
1
K-=0.3, 0=20*, $)=0.21, PA= .65
(e)
(f)
K =0.9 9=50 <=0_26
5
=1.16
Figure 3-10: Effect of blunt loading on ABS and silicone rubber prototypes under
plane strain compression for three selected geometries with Ra = 50. (a) Corresponding experimental and simulation compression stress-strain curves, and back deflection
as a function of (b) strain and (c) macroscopic stress. Error bars represent standard
deviation of n = 3 experiments and solid lines represent simulations of finite-sized
geometry. (d-i) Selected experimental contours of displacement and strain are shown
at macroscopic stress o = 0.05 MPa and indicate deformation within the composite
structures.
40
undeformed
s=0.05
6=0. 10
e12 at e=0.10
10.15
-0.15m
Figure 3-11: Undeformed geometry and experimental deformation under blunt loading at E= 0.05, 0.1 and contours of shear strain evaluated at c = 0.1.
41
e1
undeformed
e=O.10
s=0.05
-0.15
at
e=O.10
10.15
U
I
I
(continued) Undeformed geometry and experimental deformation under blunt loading
at e = 0.05, 0.1 and contours of shear strain evaluated at c = 0.1.
42
e1 at e=0.10
undeformed
e=0. 10
&=0.05
-0.15 iUKWJO.15
(continued) Undeformed geometry and experimental deformation under blunt loading
at E= 0.05, 0.1 and contours of shear strain evaluated at E= 0.1.
43
u2(mm)
e22
0
-2.5
12
0
-0.15
-0.15
0.15
(a)
(b)
Figure 3-12: Simulated contours of displacement and strain are shown at macroscopic
stress o = 0.05 MPa corresponding to experiments of blunt loading of physical models
in Fig. 3-10. (a) Kd = 0.3, 0 = 200, corresponding to # = 0.21, PA =1.65 (b) Kd =
0.3, 0 = 50, # = 0.77, PA = 1.14 and (c) Kd = 0.9, 0 = 5', 4 = 0.26, PA = 1.16.
Experimental contours of displacement and strain agree well (Figs. 3-10, 3-12)
with simulated results of the same geometry, demonstrating the quantitative capabilities of the model and providing insights into the deformation mechanisms. Depending
on the geometric arrangements and volume fractions of the scales, macroscopic loading
is accommodated by either (1) global tissue shear between scales and scale rotation,
(2) local tissue shear at scale tips and scale bending, or (3) constraint of the tissue
and overall stiffening. These deformation mechanisms are illustrated using the three
selected geometries in Figs. 3-10, 3-12.
Tissue shear and scale rotation. As shown in the contours of Fig. 3-10d for Kd =
0.3, 0 = 20', the anisotropy from the angled scales couples the compressive loading
to a shear strain
612
within the tissue.
For similar architectures with high scale
overlap (low Kd) and initial angle 0, this region of shear strain is broad and evenly
distributed over the length of the scales. This mechanism of shear between scales
promotes uniform scale rotation, as shown in Figs. 3-10d, 3-12a. Here, the scales
deform minimally as they rotate relative to the tissue; deformation is accommodated
through tissue shearing. This effect was largely observed for structures with 0 > 200.
Experimental results capture this effect well with the exception of geometries with
44
#
~ 1, in which the imaging pattern could not resolve the very narrow tissue layer
between scales (Fig. 3-10e).
Scale bending. Upon blunt loading, it can be seen in Fig. 3-10f that the scales in
the arrangement with Kd = 0.9, 0 = 50 have noticeably deformed from their original
configuration (Fig. 3-4). Here, the scales locally bend to accommodate the loading,
resulting in highly localized shear strain in the regions of scale overlap. Such low
overlap results in undesirable localized regions of strain in the tissue substrate, shown
in the contours of
622
and
612.
Due to the small feature size and non-linear deformation
of the scales during bending, digital image correlation (DIC) was not able to fully
capture the motion of the scales. It can also be seen for other architectures with
Kd
> 0.5 that the scales with low overlap underwent significant local bending, often
constrained by three or more points of inflection. At high load (F > 500 N), the
scales in same samples were observed to crack and subsequently delaminate from the
rubber tissue (Fig. 3-13).
delamination
delamination
(a)
(b)
Figure 3-13: Cracking and delimitation observed experimentally under high loads
(F > 500 N) for (a) Kd = 0.7, 0 = 400, # = 0.06, (b) Kd = 0.7, 0 = 20', # = 0.09.
Tissue constraint. The scale layers imposed a significant constraint on the underlying tissue, constraining lateral expansion. This constraint is apparent in the
deformed images of Fig. 3-12, which show the constrained bulging of the underlying
45
tissue in the finite strained samples. This constraint effect will be shown to provide
significant stiffening and hence dramatic reduction in back deflection of the overall
composites when evaluating the periodic structure.
(b) 0.10
(a) 4
experiment
FE (periodic)
0.08
experiment
FE (periodic)
O
0.06
0.04
io
U)
O
O
I
1 I0.02
01$'
1
0.5
scale fraction $
00
0
0.5
scale fraction $p
1
Figure 3-14: Experimental and finite length simulation results for 21 total geometries
with Ra = 50. (a) Stiffness and (b) back deflection in response to blunt loading are
evaluated at a = 0.05 MPa.
Loading curves from experiments on physical prototypes agree well (Fig. 3-10a)
with simulations of the same geometry and display the behavior of the composite
structure. A secant stiffness was extracted from loading curves at a fixed macroscopic
stress of a = 0.05 MPa (corresponding to e = 0.02 - 0.17) and was found to range
from
E
= 1 to 3 across the geometries studied (Fig. 3-14a). This secant stiffness
captures the effective "springs in series"-like stiffness of the tissue support layer and
the scaled layer (Fig. 3-16). Assemblies with a low volume fraction of scales had
E ~ Et; indirectly, these low volume fraction scale assemblies do not limit back
deflection. Increasing
#
from 0 to 0.25 promotes a nearly linear increase in E from
1 to 2 and further increasing
#
gives a plateau of E = 3 at
show the effect of increasing
#
on limiting back deflection and suggest a plateau of
influence at
#=
#
= 0.5. These results
0.5. Effective stiffness for all geometries is shown in Fig. 3-15.
While effective stiffness of the composite is one indicator of the protective response
of the system, it does not explicitly describe the effects experienced in the underly46
Kd = 0.1
Kd
= 0.3
Kd = 0.5
K, = 0.7
K = 0.9
0
LO
C\i
4)= 0.9 2 , pA = 1.07 (1.22)
11
CD
E=2.82
U=0.49
-E/p=2.64-Ulp=0.4a
$=O.66,
=1.07 (1.18)
E=2.74
U=0.65
E./p=5BUp.=1L60.
$=0.51, p =1.08 (1.16)
E=2.46
U=0.70
E4p=22B___U4L=065
= 0.46,
E=2.49
F /p-9&i.
$ = 0.33, Fp= 1.16 (1.27)
U=0.94
E=2.05
U/p=081
$=0.26, A= 1.16 (1.25)
E=2.08
U=O.91
$=0.17, Y =1.33 (1.44)
E=1.66
U=1.11
U/p=0.83
E/p=1.25
$ =0.13, p=
1.34 (1.42)
U=1.33
U/p=0.99
$
1067 76)
U-.
U/ =Q-57
0
4)= 0.77,
E=2.90
CD
F/p=2.54
=1.14 (1.39)
U=0.47
Up=~041
i
= 1.15 (1.30)
U=0.72
UI/p-6AF/p=177
C
a)
$ =0.39, A= 1.31 (1.56)
E-2.52
U=0.65
E/p=1.92
U/p=0.50
$=0.23, PA= 1.33 (1.48)
E=2.10
U=0.78
E/p=1.58
U/p=0.59
1.65 191)
p0.21,>
U=0.78
E/p=123 Ulo-p.47
$=0.12
1
.82)
E=1.'61
= 0 82
FE/p~l 011 PQ65
U/p=0.7
F/p=1.8
E=1.48
E/p=1.11
0
CJ
I
E./p16
E=2.4WA
Elp=1.79
IU=3P
$
=U=0.,59
U/p= .38
L
E=2.02
E=7
E/p=0.73
1.67 1.78)
U=.50
U/p=0.50
.07
E=1.64
E/P=0.64
0
II
CD
$ = 0.41,
A = 2.15(3.12)
E=2.05
E/p=0.95
U=0.84
$= 0.14,
E=1.64
JU/p=3
E/
= 2.25 (2.56)
U=1.08
p0.8
$ = 0.08, jA= 2.26 (2.45)
E=1.39
U=1.21
/p=06 9I
/p=n.
$ = 0.06, p = 2.27 (2.40)
E=0.97
U=0.67
E/p03U/p=0.3i
4)= 0.05,
E=1.08
E/p=0.47
A = 2.27 (2.38)
U=1.60
U/p=0.71
Figure 3-15: Resistance to deformation (E, E/PA) and the resulting energy absorption (U, U/PA) for each geometry under blunt
loading for the finite-sized structure simulated for experimental geometries. Each structure is characterized by #(Kd, 0, Ra = 50)
with areal density PA given for experimental and silicone rubber and ABS and biological tissue and scales in parentheses.
ing tissue. To function as an effective dermal armor, fish scales must minimize the
deflections, strains and/or stresses beneath the scale layer to protect the body. Experimental results show that back deflection in the underlying tissue, 6 tissue, decreases
as scale volume fraction increases for
4
< 0.5 and then remains at 6 tissue ~ 1 mm (ts)
for #$> 0.5 (Fig. 3-14b).
F
F
&scale
Utissue
6total
-
kscale
(variable)
kscaie
kti
ektotal
= E
ktissue (constant)
6scale
6total-6tissue
ktotai = (1/kscaie+1/ktissueY
Figure 3-16: "Springs in series" model of tissue support layer and the scaled layer is
captured by effective secant stiffness E.
3.3
3.3.1
Infinitely long (periodic) structures
Deformation mechanisms in periodic structures
Since the actual structures consist of many repeating scales along the length of the
fish body, simulations with periodic boundary conditions which capture an infinitely
long structure provide a more physiologically relevant assessment of the mechanical
response than the simulations of the finite-sized specimens. Recall the finite-sized
specimens exhibited significant lateral expansion (bulging) of the bottom soft tissue
substrate (Fig. 3-12) due to the unconstrained lateral edges. The relaxed boundary
conditions in these finite-sized specimens causes the structures to exhibit a more
compliant response. In an infinitely periodic structure, the lateral strain of both layers
48
must be identical based on compatibility. The implications of this constraint have
a dramatic impact on tissue deformation for geometries with stiff upper scale-tissue
layers, in which the expansion of the lower layer is constrained by the presence of and
compatibility with the scale layer. To eliminate these boundary effects, a periodic
structure was simulated to more accurately model a larger number of repeating scales
along the length, such as those found along the length of a fish body.
0.1
0.08
'C 0.06
C.
o 0.04
(a)
Kd=0.3, 0=20-,P=0.21, A=1.65
Kd= 0.3, 6=5', $p=0.77, PA11
Kd=0.9, 6=5*, $=0.26, yA= 1 . 1 6
0.02
0'
0
-2.5
U2
0.05
e22
i
0
-0.15
0.1
e
E,2
1 0 -0.15
in
o, (MPa)
0.15 -2.5
a22 (MPa)
2.5
-0.1
0.i
(b)
(c)
(d)
Figure 3-17: Periodic simulations for Ra = 50 yield insights into deformation mechanisms of periodic scale assembly under blunt loading. (a) Loading curves demonstrate
stiffening constraint for structures with high #. Contours are evaluated at o- = 0.05
MPa. (b) Large initial scale 0 promotes rotation of scales via uniform tissue shearing (612). (c) High scale volume fraction directly results in a stiff composite, which
reduces back deflection in the underlying tissue u 2. (d) Geometries with little scale
overlap (high Kd) exhibit excessive scale bending (ou) and localized high stress (o 22 )
and strain in the underlying tissue.
The resulting loading curves and contours for the periodic structure are shown in
49
Figs. 3-17 through 3-24, which demonstrate the large difference the constraint from
periodicity has on the results for geometries with high
#.
Additionally, all simulated
loading curves can be seen on a single plot in Fig. 3-18, where it can be seen that the
non-flat loading surface of several geometries result in a softer overall response than
the tissue (rubber) alone. Deformation mechanisms (Jk, dO), resistance to deformation
(E, E/PA), and the resulting energy absorption (U,
U/PA)
can be shown for each
geometry in Fig. 3-19.
tissue
scales
0.1
irregular loading surface
0.05"void
oid"
0--
0
0.05
0.1
0.15
Figure 3-18: Loading curves for all periodic simulations under blunt loading. The
non-flat loading surface of several geometries with scale volume fraction # ~ 0 (e.g.
bottom right structure) result in a softer overall response than the tissue (rubber)
alone due to "voids". Future work should normalized the response of each structure to
a similar structure compose of tissue (rubber) only to highlight the stiffening response
of the scales.
Imposing periodic boundary conditions has little effect for small volume fraction
(# < 0.2), but as
#
increases, the effective stiffness of the periodic scale structure
increases by an order of magnitude over the finite length structures (Fig. 3-26a),
which had permitted lateral deformation (bulging) of the underlying tissue. The
effective stiffness of the periodic structure follows a nearly linear relationship with
scale fraction
#
(Fig. 3-26a) and shows the direct effect that increasing scale armor
has on limiting back deflection at a given macroscopic stress level (Fig. 3-26b). This
relationship between back deflection and scale volume fraction demonstrates that the
50
volume fraction of scales governs the constraint of the tissue layer and in turn, has a
significant influence on protection. Therefore, effective stiffness and back deflection
can be adjusted by tailoring the scale fraction
#(Kd,
One aspect of the constraint effect of increased
0).
# on effective
stiffness is demon-
strated in the geometry of Fig. 3-17c, which has the same overlap as Fig. 3-17b but a
significantly higher scale volume fraction, which was achieved by reducing the angle
of the scales from 200 to 5', resulting in a stiffer loading curve and far less penetration
in the underlying tissue
(U 2 , 6 tissue)
at a given force. In this case, by reconfiguring
the angle of the same number of scales per unit length (Kd-1), the system can achieve
a 4x increase in penetration resistance. These results show how the geometric arrangement of the scales and the scale volume fraction, together with their influence
on constraining the deformation of the underlying tissue, serve to provide protection.
Furthermore, such scale-imposed tissue constraint could serve as a design principle
for these composite structures to locally tailor stiffness.
The underlying deformation mechanisms for the structures with periodic boundary
conditions were also identified. All mechanisms (scale bending, scale rotation, and
tissue shear) are shown as a function of architecture (Kd, 0) and composition (#)
in Fig. 3-25.
Similar to results from finite-length geometry, scale bending, scale
rotation, and tissue shear were found to be the primary modes of deformation. The
mechanisms of scale bending and rotation are dependent on initial overlap (Kd) and
angle (0) (Fig. 3-26). Architectures with low overlap (high Kd
-
1) are dominated by
scale bending, while those with high initial angle 0 are dominated by tissue shearing
and corresponding scale rotation. This behavior is demonstrated by the high values
of contours of bending stress (oll) found within the minimally-overlapping scales in
Fig. 3-17d for a macroscopic stress of or = 0.05 MPa. Scales in such configurations
at this load have an effective curvature of , > 0.25LS 1, or equivalently, an effective
bending radius of R < 4L,. Conversely, the large initial angle 0 in Fig. 3-17b results
51
0
LO)
$ = 0.92, PA=1.07 (1.22)
U=0.17
E=9.78
K=0.08
dG=0.02* E/p=9.17 U/p=O.1
CD
Kd = 0.9
K, = 0.7
Kd = 0.5
Kd = 0.3
Kd = 0.1
$ =0.66, PA = 1 07 (1 .18)
K=0.17
E=6.60
U=0.28
d8=0.45* E/p=6.15 U/p=0.26
0
LO
4P= 0.46, QA= 1.15(13)
U=0.36
K=0.11
E=4.71
U=0 19
E=8.18
E/p=7.17 U/p- .17 d8=0.25* E/p=4.08 U/p=0.31
(P= 0-77, PA
11
CD
K=0.02
d6=0.0*
1.14 (1.3 )
4= 0.33, pA = 1.16 (1.27)
K=0.20
E=3.36
U=0.54
d=0.840 E/p=2.90 U/p=0.46
4 = 0.51, PA = 1.08 (1.16)
U=0.45
E=4.28
K=0.19
0
dO=-0.03 E/p=3.97 U/p=0.42
=.26, P = 1.16 (1.25)
E=2.51
U=0.81
dO=0.52* E/p=2.16 U/p=0.70
K=0.26
0
0
39
$ = 0. , PA= 1.31 (1.56)
U=0.47
E=3.77
K=0.05
0
dB=0.35 E/p=2.87 U/p=0.36
a)
= 0.62,
o K=0.0
od8=-0.5
I=045
CD
o
t
U/
$p= 0.137, p, = 1 .33 (1.44)
K=0.26
E=2.01
d8=1.62* E/p=1.51
U=0.89
U/p=1.67
1 42
$= 0.13, p= 1.34 ( . )
U=1.37
E=1.36
K=0.31
d=1.87* E/p=1.02 U/p=1.02
1.58 (2.37
3.66
=2.31
$ = 0.07, pA= 1.67 (1. 7 8 )
K=0.05
E=1.09
U=1 .58
$ =10.21, p= 1.6 5 (1 91)
U0.78
E=2.21
U/ =0.2 K=0.1
4
U/ =0.47
d8=1.2)* E/p=1.3
$=0.41,
K=0.1
dO=3.
CD
$= 0.23, P= 1.33 (1.48)
U=0.65
E=2.68
K=0.16
0
8=0.90 E/p=2.03 U/p=0.49
dO=2.450 E/p=0.65 U/p=0.95
.15 (3.12,
=2.10
p=0.97
8 $=0. 4,p =2. 25( 2
K=0.20
__=4.110
E=1.65
E/p=0.74
56
)
U=-1.04
U/ =0.47
0.08,P =2.26(2.4)
U=1. 5
K.3=20 E=1.36
d644.98* E/p=0.60 U/p=0 55
Figure 3-19: Deformation mechanisms ('-, dO), resistance to deformation (E, E/PA), and the resulting energy absorption
(U, U/PA) for each geometry. Each structure is characterized by <(Kd,0, Ra = 50) with areal density PA given for experimental and silicone rubber and ABS and biological tissue and scales in parentheses.
K= = 0.5
Kd = 0.1
LO
C\J
-2.5
U2
Kd = 0.9
0
0n
I0
LO
a)
0
0
CD
'I
a)
Q1
0
a)
3
Kd= 0.
Kd= 0.1
0.1
U
Kd= 0.5
Kd=0. 9
Kd= 0. 7
-=2.5'
- - -0=5*
cd
a .
20
.
-
0U
0
0.05
0.1
'0
0.05
0.1
0.1
0.05
=200
8=400
0.1
Figure 3-20: Loading curves and contours of u2 for blunt loading of architectures with periodic boundary conditions for Ra = 50
evaluated at o = 0.05 MPa.
Kd = 0.9
Kd = 0.5
Kd = 0.1
- 22
0
-0.15
CD
LO
OD
0
0)
0
CDi
Kd= 0. 3
Kd= 0.1
as
IIa)
Kd= 0. 7
Kd= 0.5
Kd=0.9
E=2.5*
- - - 0=50
0=10*
a.
20
--
0 II
0
0.05
C
0.1
0.05
PE
0.1
~0
0.05
C
0.1
0=20*
0=400
0.1
Figure 3-21: Loading curves and contours of E22 for blunt loading of architectures with periodic boundary conditions for Ra = 50
evaluated at o = 0.05 MPa.
-
-
K 0= 0.5
Kd = 0.1
0'
c'O
II
Kd = 0.9
812
-0.15
0.15
0
LO
a)
0
C0
C14
11
CD
C-"
V-I
0
0
Kd= 0.1
Kd= 0.3
Kd= 0. 7
Kd= 0.5
Kd= 0.9
0.1
Figure 3-22: Loading curves and contours of e2 for blunt loading of architectures with periodic boundary conditions for Ra = 50
evaluated at o = 0.05 MPa.
..
......
..
....
....
....
......
..
....
.
............
.... -
K= = 0.5
Kd = 0.1
Kd = 0.9
0
LO
al
-2.5
(MPa)
i
2.5
CD
0
LO
CD
0
C\D
0
.0I
a)
Kd= 0.1
0.1
Kd= 0. 7
Kd= 0.3
-0.1
Kd= 0. 9
9=2.5
- - - 8=5*
-0=10'
-
20 .05
0=20
9=40*
0
0.05
C
0.1
0
0
0.05
C
0.1
0
0.05
C
0.1
0.05
C
0.1
0.05
r-
0.1
Figure 3-23: Loading curves and contours of oa for blunt loading of architectures with periodic boundary conditions for Ra = 50
evaluated at -= 0.05 MPa.
Kd = 0.1
K
LO
CD
-0.1
022
ino
= 0.5
Kd = 0.9
(MPa)
0.1
0
a)
0
C).
I
CD
Q1
0
CL
Kd= 0.1
Kd= 0.
3
Kd= 0.7
Kd= 0.5
0.1
~0
0.05
0.1
0.05
Kd= 0. 9
0.1
Figure 3-24: Loading curves and contours of o 22 for blunt loading of architectures with periodic boundary conditions for Ra = 50
evaluated at -= 0.05 MPa.
.................
in a shear straining
of the tissue, which in turn, leads to the rotation of the
(E12)
scales. Structures with both high Kd and 0 (corresponding to
#<
0.1) exhibit both
significant scale bending and scale rotation, but such configurations are typically not
observed in nature.
0
K
6
0.01
vs overlap Kd:
E
a
C
SHEARING
ROTATION
BENDING
0
0
00
7-.
0.5
o = 2.50
0.6
M
* 4
0
S2
.005
S21
1
'4
6=5
0.4
-'
o=
10'
o
40'
A)
CM
0
W
00
1
1
Kd (scale overlap)
Kd (scale overlap)
Kd (scale overlap)
0.5
0
vs volume
fraction $s:s
E
S6
.2
4
0
0.01
E
.005
?0.4
0
A~ 0.2 okiP;,
C
W
0.
0
0.5
scale fraction (P
0
scale fraction $p
4
0.5
scale fraction $p
06
0.01
vs angle60:
E = 2.5*
0 5
0 = 10
0.6
Kd = 0.1
-c 0.6
E
E
ad
.005
- 0
04
CM
Ne
0
0
40
20
0 (scale angle, *)
0
40
20
0 (scale angle, *)
C
Ka = 0.3
0.4 9
0.2
Kd = 0.5
0-
Kd =0.9
K =
0.
0 (scale angle, *)
Figure 3-25: All deformation mechanisms for uniform compression evaluated at a
macroscopic stress of a = 0.05 MPa for 21 total geometries with Ra = 50. Scale
bending P, is primarily a function of scale overlap Kd, scale rotation dO is primarily a function of scale angle 0, and average inter-tissue shear strain (Ce) is only a
mechanism to promote scale rotation.
58
(a)
10
(b) 0.10
experiment
FE (finite length)
8
FE (periodic)
O
0.081
0
1W
an
a)
0
NC
4
0
0)( O9 0
0.04
II
0.02 F
1
0.5
0
01
10
0.5
1
scale fraction $p
scale fraction $
(c) 0.01
0.008
0
0
2 '
00
0
0
0
0.06
6
U,
experiment
FE (finite length)
FE (periodic)
(d)
6
e=40*
0
E 0.006
C
.4
0
~0.004
*0
2
2.50
0.002 f
01
0
0.5
scale overlap Kd
1
0
40
20
0 (scale angle,
0)
Figure 3-26: Deformation mechanisms for uniform compression evaluated at a macroscopic stress of o- = 0.05 MPa for 21 total geometries with Ra = 50.
(a) For low
scale volume fractions (0 < 0.25) in experiments and finite length simulations, effective loading stiffness E of the composite increases linearly with scale fraction then
plateaus for # > 0.5. Similarly, back deflection in the underlying tissue, tissue, decreases linearly for # < 0.5. Periodic simulations suggest that this plateau for large
# is a boundary effect. For periodic simulations, (c) microscopic scale bending is
primarily a function of overlap Kd, where as (d) scale rotation is dependent on initial
scale 0.
59
3.3.2
Energy absorption
To determine the ideal armor architecture under blunt loading, the deformation resistance of the composite must be evaluated against competing mechanical functionalities. Assuming similarity of the scale and tissue densities (PS ~ Pt) in this study
for the experimental samples of silicone rubber and ABS, this weight is primarily a
function of scale angle: PA
-
PA(0),
as shown in Table 3.3. Thus to minimize the
weight of the armor while stiff maximizing its protection against blunt loading via
increased stiffness, the composite should have both a minimum scale angle 0 and high
scale density
#.
Given that the stiffness of structures (E) and their energy absorption
(U) are inversely related when evaluated at a fixed load, it is problematic to select a
structure that combines both high stiffness and high energy absorption. Thus, if both
properties of high E and U are desired, an intermediate value of
#
must be selected
to balance these two criteria, as shown in Fig. 3-27b and Fig. 3-28a. Furthermore,
it can be seen in Fig. 3-27a that the stiffness (in blue) exponentially decreases with
increasing weight, while the energy absorption (in red) nearly linearly increases with
weight. Hence, it is advantageous to select a structure where PA
<
1.5 (equivalently,
O < 100, after which increasing weight has diminishing returns in terms of stiffness
and energy absorption.
Table 3.3: Normalized areal density PA for
#(Kd,
0, Ra = 50) where p, ~ pt.
overlap Kd
0.7
0.5
0.3
0.9
PA
0.1
2.50
-
-
1.06
1.07
1.08
50
-
1.14
1.15
1.16
1.16
0
100
-
Cd
200
400
1.58
2.17
1.31
1.65
2.24
1.33
1.66
2.26
1.33
1.67
2.27
1.34
1.67
2.27
<
Alternatively, if the density of the mineralized scales is greater than that of the
60
tissue, as often the case in biological systems, the resulting weight is a function of both
scale volume fraction <5 and angle 0. For physiologically relevant biological materials
of tissue and fish scale, an approximate density ratio is p,
= 2 .6pt
where PA is shown
in Table 3.4. Energy absorbed and energy absorbed per unit weight for this density
ratio is shown in Fig. 3-27b and Fig. 3-28b. As the density of the scales increases,
E
the intersection of the
and U curves is increasing on the lower end of the weight
spectrum, suggesting that added weight doesn't necessarily imply added protection.
Table 3.4: Normalized areal density PA for
<
b-
<;(Kd,
0, Ra = 50) where ps = 2.6pt.
overlap Kd
0.7
0.5
0.3
0.9
PA
0.1
2.50
50
-
1.39
1.22
1.30
1.18
1.27
1.16
1.25
10
200
-
2.37
1.56
1.91
1.48
1.82
1.44
1.78
1.42
1.76
400
3.12
2.56
2.45
2.40
2.38
61
2.50
50
100
20*
40*
(a) 2
10
0.9
0
0
E3
'j AK,
y0
0.7
0.5
Kd
0
0
E3
0.1
0.3
D1
-5 'ui
o0O
0.3
0.1
E3
0
0
0
E30
0.7
0.9
00
E3
009
01
0.5
1.5
2
2
-0
PA
2.50 50
(b) 2
100
03
0.9
0
0
0
0
0
0.1
0
1
0.1
0
0.3
0
Kd
0
0
I3 1 F
0.5
400
10
Kd
0.7
20*
1.5
5 ILo
0
0
0.5
0.7
0.9
3
2.5
2
0.3
S'o
3.5
3
j5A
)0
2.50 50 100 20* 40*
(c) 2
0
i
O1
AAK
Kd
0.9
00
0.1
0
0.7 iD
1
0.5
1W
0.5
0.3
3
0
0.7
E30E
0.1
0.9
0
1
1
2
0
3
4
j5A
5
60
6
7
Figure 3-27: Deformation resistance and energy absorption vs.
loading for
0.3
#(Kd, 0, Ra
=
weight for blunt
50), evaluated at a macroscopic stress of o = 0.05 MPa.
Stiffness E and energy absorption U for (a) the experiments and simulations where
Ps ~ pt, (b) biological densities of tissue and scales where p, = 2 .6pt, and (c) for an
hypothetical case where ps = 10pt.
62
(b)
Stiffness E
(a)
Energy Absorption U
40
35
30
25
a) o
0.2
0.4
0.6
Kd(scale overlap)
(c)
40
0.2
0.4
Kd(scale
(d)
40
E/o.
35
35
30
30
25
25
* 20
20
0.6
overlap)
O/S
15
10
5
0.3
0.1
(e)
0.5
0.7
Kd(scale overlap)
0.1
0.9
(f)
ff/PA
40
0.3
0.5
0.7
Kd(scale overlap)
0.9
U/PA,
40
35
35
30
30
25
25
C 20
20
15
15
10
10
5
5
0.2
0.4
0.6
Kd(scale overlap)
0.2
0.4
0.6
Kd(scale overlap)
Figure 3-28: Deformation resistance and energy absorption for blunt loading for
#(Kd, 0, Ra = 50), evaluated at a macroscopic stress of o- = 0.05 MPa. Corresponding
map of scale density # is shown in Fig. 2-7. (a) Stiffness E and (b) energy absorption
U as a function of scale architecture (scale overlap Kd and scale angle 0. E/IPA and
U/pA are normalized by areal density to reflect the stiffness and energy absorption
as a function of weight for (c-d) the experiments and simulations where p, ~ pt and
(e-f) simulations of biological densities of tissue and scales where ps = 2.6pt.
63
3.3.3
Effect of scale morphometry
Increasing the aspect ratio of scales Ra = L/t, from 50 to 100 was performed in
three ways: (a) fixing
fixing
#, Kd
#, 0 and
varying Kd, (b) fixing Kd, 0 and varying
#,
and (c)
and varying 0 as shown in Table 3.5. The resulting geometries, stress-
strain curves, and contours of o22 are shown in Fig. 3-29. To compare the same
composition and contact geometry of varying architectures, scale volume fraction
#
and angle 0 were held constant and evaluated for all geometries as a function of
#(Kd,
0, Ra
=
100). This combination of parameters is reflected in Fig. 3-29b, Fig.
3-29a shows the original structure from the previous discussion.
Here for the same volume fraction of scales
# and
a given fixed 0, the increase of
aspect ratio reveals a tradeoff between dominant deformation mechanisms. Increasing
the aspect ratio via reducing Kd increases the effective stiffness E of the assemblies,
resulting in a reduced back deflection in the underlying tissue (Fig. 3-26). To examine the mechanics that govern this response, scale bending dO and rotation i were
evaluated for the infinitely-periodic case. Bending within the scales was reduced for
higher aspect ratio simulations, which is explained by the corresponding decrease in
scale overlap. However, the trend in Fig. 3-26c suggests that scale with the same
overlap Kd and increased aspect ratio would bend much more than those with the
original aspect ratio, showing the large influence of Ra on the local deflections and
bending curvature of the scales. Despite the same fraction of tissue in the composite,
shearing of the tissue was notably reduced. Hence, while scale bending increased,
scale rotation was found to significantly decrease. This is noteworthy because previous results suggest scale angle 0 (here, fixed) governs tissue shear and scale rotation.
This transition from tissue shear plus scale rotation to scale bending with increasing
aspect ratio served to stiffen the composite, yet resulted in elevated levels of local
bending stress within the scales. These higher bending stresses will lead to early
failure of the scales via yielding or fracture.
64
Table 3.5: Aspect ratio study parameters.
(a)
(b)
(c)
(d)
Ra
Kd
0
$
PA
50
100
100
100
0.3
0.15
0.3
0.3
20
20
20
10
0.21
0.21
0.10
0.19
1.65
2.30
1.67
1.66
0.1
0.08
-
0 06
.
0.04
0.02
00
0.05
0.1
8
(b)-
(a)
(c)
(d)
undeformed
Figure 3-29: Effect of scale morphometry. Increasing scale aspect ratio Ra = Ls/t,
from 50 to 100 as a function of varying Kd, 0, #. Stress-strain curves and contours of
-22 are shown for these combinations of geometric parameters corresponding to Table
3.5. Architecture in (a) is used in Section 3.3.1 and (b) in Section 3.3.3.
Alternatively, the aspect ratio of the scales can be increased by fixing the scale
volume fraction
# and
the scale overlap Kd and varying the scale angle 0. Changing
65
(a)
(b)0.10
15
V FE (Ra= 100, periodic)
O FE (R ,= 50, periodic)
10 1
I1W
0.08
o FE (Ra= 50, finite len.)
I experiment
. 0.06
to
CO
0)
V
C
e 0.04
V V"-
5
0.02
(c)
0.01
0
0
1
0.5
scale fraction p
0.5
1
scale fraction $)
(d)
8 = 2.5*
50
100
200
40*
0.008 -
:r
E 0.006
C
a)
Ra = 100
00
0.004
0
M
R, = 50
0.002
0
00
Kd
0.5
(scale overlap)
1
20
30
0 (scale angle,
0)
Figure 3-30: Deformation mechanisms for uniform compression evaluated at a macroscopic stress of o- = 0.05 MPa for 21 total geometries with Ra = 50 and 100. (a) For
low scale volume fractions (# < 0.25) in experiments and finite length simulations,
effective loading stiffness E of the composite increases linearly with scale fraction
then plateaus for # > 0.5. Similarly, back deflection in the underlying tissue, tissue,
decreases linearly for # < 0.5. Periodic simulations suggest that this plateau for large
# is a boundary effect. For periodic simulations, (c) microscopic scale bending is
primarily a function of overlap Kd, where as (d) scale rotation is dependent on initial
scale 0. Transition from rotation to bending is evident for large Ra (dashed line).
the fixed parameters results in a completely different structural arrangement, as shown
in Fig. 3-29d. This structure with fixed
#, Kd, Ra
results in both more bending and
rotation than the original structure, yet is less stiff. These results suggest that aspect
ratio alone can not govern the deformation mechanisms of the structure and that
66
the response is a function of composition
(#),
structural arrangement (Kd, 0) and
morphometry (Ra).
In general, the morphometry of the scales helps to govern the dominant deformation mechanism of the assembly, but is dependent on the spatial arrangement of the
scales. Typically, thin, high aspect ratio scales result in deformations primarily dominated by bending, whereas thick (low aspect ratio) scales are dominated by tissue
shear plus scale rotation. This tradeoff can be tailored by adjusting individual scale
aspect ratios in addition to other properties, e.g. elastic modulus, internal structure,
and porosity of the scales.
3.4
Conclusions
Scale architecture (Kd, 0), composition (#), and morphometry (Ra) can be used
to locally tailor composite stiffness and back deflection in the underlying tissue in
order to achieve protection under blunt loading events. Overlapping scales distribute
stresses across a large volume of material and provide penetration resistance at a
reduced weight compared to a continuous armor layer. The flexibility of the scales
and tissue permits the scales to rotate and bend under applied loading. Assemblies
of these scaled structures are formed without joints or hinges, forming a simple,
easily adaptable design. The degree of flexibility or protection is easily tailored and
adaptable by simply changing morphometry and the overlap distribution of the scales.
Inclusion of a compliant tissue is key to understanding the true behavior of the
composite, which is dominated by tissue constraint, tissue shearing (and resulting
scale rotation), and scale bending.
The composition primarily governs the effec-
tive response of the composite, which is achieved by a combination of deformation
mechanisms controlled by the microstructural arrangement. Volume fraction
# gov-
erns constraint, effective stiffness, and back deflection; scale overlap Kd governs scale
67
bending, and initial scale angle 0 governs tissue shearing and scale rotation. The
influence of these mechanisms can be tailored by adjusting the scale morphometry:
for scales with a large aspect ratio, the deformation transitions will be dominated
by scale bending. Conversely, scales with small aspect ratio will rotate under blunt
loading, accommodated by tissue shearing.
68
Chapter 4
Mechanical protection under
predatory tooth attack
While blunt loading captures the response of the assembly to large indenters (R >
KdLS), threats are often localized to a small area. This is the case for predatory
biting with a sharp tooth, with common fish predators shown in Fig. 4-1. These
biting attacks are idealized as force-limited penetration with indenter of radius R.
(a)
(b)
(C)
Figure 4-1: Predatory biting attacks by (a) birchir Polypterus senegalus (b) nothern
pike Esox lucius biting salmon and (c) river otter. Images adapted from Song et al.
(2011); Alaska Department of Fish and Game, http://www.adfg.alaska.gov/
index. cfm?adfg=wildlifenews.view_ article&articlesid=50&issueid=13;
Richard
Mousel,
http://www.critterzone. com/animal-pictures-nature/
mammal-otter-river.htm last accessed on May 1, 2012.
69
4.1
Finite element micromechanical model
A two-dimensional plane strain composite scale-tissue micromechanical model was
simulated using finite element analysis (FEA) similar to blunt loading simulations
in Section 3.1.2. Quadratic continuum elements were used in ABAQUS/Standard
(library types CPE6H and CPE8RH) and a fine mesh with elements of length ~ t,/4
was needed to capture the deformation within the scales. Scales and tissue were
modeled with isotropic linear elastic properties from experimental characterization
as shown in Table 4.1. Tissue and scale elements are modeled as perfectly bonded.
Non-linear, large deformation theory with frictionless contact was assumed. Here,
predatory attack via tooth indentation was simulated with rigid indenters of varying
radii (R/ts = 0.1, 1, 10) and a fixed half cone angle (0/2) of ~ 30 .
Table 4.1: Material parameters experimentally determined from synthetic prototype
and employed in indentation simulations.
Scale
Tissue
Behavior
Modulus
Elastic
E, = 880 MPa
Neo-Hookean
Eot = 0.39 MPa
Poisson's ratio
Density
Geometry
v, = 0.4
Ps = 0.80 g/cc
L, = 50 mm, t, = 1 mm
vt
0.49
pt = 1.07 g/cc
Htissue= L,/2 = 25 mm
-
The thin epidermal tissue directly beneath the indentor was removed to increase
computational efficiency, as this thin outer layer of tissue would be pierced in a
physiological setting. In addition, these indentations were performed on a very large
sample (length > L cos0) to neglect edge effects (Fig. 4-2).
All indenters were
restricted to motion in the 2-direction and reaction forces F 1 , F2 , and displacement
6 were computed. Lateral reaction force F1 was negligible compared to normal force
F2 for all indentations, so force reported is F2 .
Local true Cauchy stress and logarithmic Hencky strain contours were evaluated
70
Figure 4-2: Indentation loading conditions, showing very large sample (length
> L cos 0) to neglect edge effects, removed tissue at loading surface to increase computational efficiency, and a constrained (fixed) bottom edge.
at different load levels to identify the deformation mechanisms which accommodate
the loading for the different composite geometries. Indentation force is reported F2
per unit width (plane strain) and indentation depth is normalized by sample height,
6 /Htotai.
Deflections in the underlying tissue 6 tissue were reported at depth t, below the
scales. A normalized indentation force per depth (stiffness) is given by F2H/6tisse
and evaluated at F = F2 = 5 N/mm. Strain energy density per unit thickness
UlF=5N/mm
4.2
OF
ydF was also determined.
Indentation simulations results
Idealizing predatory biting as a sharp tooth of radius R, the corresponding contact
area (-
2R) and the resulting deformation-induced scale-tissue interactions govern
the number of scales that are involved in the deformation. For physiologically relevant
indentation (R < KdL,), the indenter only comes into direct contact with one scale.
Underlapped and adjacent scales are also deformed, as governed by Kd. Under low
penetration loads, the scales are not locally indented, but rather undergo significant
bending with the compliant tissue between the scales shearing and the tissue beneath
the scales also shearing (Fig. 4-3). For the case of a conical rigid indenter tip at low
71
loads (F = 5 N), there is a single point of contact and the resulting loading curves
and underlying tissue deformation were found to be nearly identical for the range of
R/t = 0.1, 1, 10 simulated. Note that this force of F = 5 N is only relevant for the
materials properties and scale size shown in Table 4.1.
The similarity of both the loading curves and the back deflection corresponding
to Kd = 0.3, 0 = 20',
#
common overlap of Kd
=
= 0.21, and Kd = 0.3, 0 = 5',
#
= 0.77 indicate that the
0.3 controls the mechanical response of the composite. In
contrast, the response for Kd = 0.9, 0 = 5',
#
= 0.26 is much more compliant and
permits more back deflection tissue in the underlying tissue because of the minimally
overlapping scales of Kd
=
0.9. Localized indentations of R/t, = 0.1 are primarily
accommodated through the mechanism of scale bending, which is isolated to only
a few scales. Evaluating across all architectures
#(Kd,
0, Ra
=
50), the overlap of
the scales (Kd) governs the ability of the composite to resist penetration (Fig. 44). Normalized indention force (local stiffness) increases with increasing scale overlap
(decreasing Kd).
Similarly, back deflection in the underlying tissue was found to
decrease with overlap such that architectures with highly overlapping scales (Kd
-
0)
are very stiff and had minimal back deflection. Due to the localized, nonuniform
deformation of the assembly under finite indentation, scale rotation is no longer a
dominant deformation mechanism. In these layered composite structures, the overlap,
angle, and volume fraction work together to give interplay between scale bending and
shearing of the tissue to involve more material. These geometries are mechanisms to
absorb more energy by involving greater amount of material laterally as well as to
deeper depths.
Larger indenters (R > KdL, and simulated for R/t, = 10 as shown in Fig. 4-5)
under large penetration loads permit the scale to bend around the indenter. Above
F
=
10 N, these larger radius indenters produce a stiffer behavior as the contact
area between scale and tooth increases.
For indenters whose radii span multiple
72
(a)8
(b) 0.08
-
0
-0.08
-1
Kd=0.3, 0=20*, $)=0.21, PA=1 .65
Kd= 0.3, 0=5, $=0.77, PA 1 1 4
Kd=0.9, 0=50, $=0.26, PA 1.16
-0.16
0
0.15
.0
6/H
u2 (mm)
-2.5
0
-0.2
0
40
tissue path (mm)
822
uj0.2
E12
-0.2
i
o
0.2
-2.5
(MPa)
ui2.5
022
-0.1
(MPa)
inu0.1
(d)
(e)
(f)
Figure 4-3: Indentation on fish scale assembly for radius R/t, - 0.1. (a) Loading curves for three geometries show that indentation force is nearly identical for architectures with similar overlap, despite very different 6, $. (b) Corresponding underlying
back deflection and (d-f) contours of stress, strain taken at F = 5 N.
(b)0.20
(a) 80
00
0
70
0
0
0
0
0.12
z
00
g0
40
00
0
0
0
0
2 50
M
U-
30
0
00
0
0.16
EO
E 60
0
0
0.081
0
OD 00
0
0
0.04'
0
0.5
q)(scale fraction)
0
(c) 80
(d)0.20
70
0.16
E 60
z
V7 7
1050
7
7
v'
v
V
0.12 *vV7
7
~2 5
1
0.5
$ (scale fraction)
1
V7
V7
0.08
40
30
7
0 (scale angle,
0.04 L
0
4
20
0
40
20
0 (scale angle,
0)
0)
(f) 0.20
(e) 80
70
0.16
LI
El
E 60
E
_r
S0.12
z
LI
50
U-
0.08[ El
40
30
0
Kd
0.5
(scale overlap)
0.04'
0
1
0.5
Kd
1
(scale overlap)
Figure 4-4: Indentation force and back deflection evaluated at F = 5 N across all scale
assembly architectures
#(Kd, 0,
Ra = 50) for a sharp indenter with radius R/t, = 0.1.
(a-d) While little relationship existed between scale angle 6 and volume fraction 4,
(e-f) there is a direct trend between overlap Kd and the ability of the composite to
resist penetration.
scales, the load is distributed across the scale assembly which reduces deflections, scale
deformation, and stress concentrations, thereby protecting the fish. This explains
74
why predators benefit from having sharper teeth: if one isolated scale sees too much
bending, the scale can be defeated. By localizing the indentation deformation to a
finite region, predators with a given biting force are able to penetrate the underlying
tissue. While this elastic model does not incorporate failure mechanisms such as
fracture, delamination, or plasticity, the stress distributions within the scales and
tissue give insight to failure modes and locations resulting from excessive deformation.
High shear strain 12 of the tissue promotes failure by delamination of the tissue-scale
interface, which permits scales to dislodge and damage the structure of the composite.
Alternatively, regions of high bending stress o-l of the scale promote yield and fracture
of the scale.
30
radius: R/t = 0.1
.indenter
25
K= 0.3, = 20*, $=0.21
Kd 0.3, 0 = 5-, $=0.77
Kd= 0.9, 0 = 5*, P=0.26
20
E
Z_ 15
u.-
10
indenter radius: R/t = 10
----- Kd =0.3,E=5', $=0.77
----- Kd=0.9,6=5*, p=0.26
5
0
0
0.1
0.2
0.3
0.4
0.5
6/H
Figure 4-5: Indentation loading curves for varying indenter geometry (R/t = 0.1, 10)
show little effect of R on mechanical response due to point-wise interaction for small
loading. At larger loads, the scale deforms around the indenter radius and the reaction
force increases. Contours shown of &.
To determine the ideal armor architecture, the penetration resistance of the com-
posite must be evaluated against competing mechanical functionalities. While segmented scale armor is inherently more flexible than a solid plate of the same thickness,
75
(a) 0. 4 0
j Kd
0.
0.3 5
00
0 .3
0
-4
0
200
100
2.50 50
0
40*
80
0
0
o
0.1
60 _E 0.3
0.9
0. 7
0.
E
0
03
0 .2
0.1
50
130
0.
0.1
o
0
0
5 0.2
1
Kd
70
E0
0
Q
1.6
0.
1.4
5% 8B
1.2
1.8
2.2
2
0.5
U- 0.7
0.9
40
30
2 .4
TA
2.5050 100
(b)
0.
400
200
20*0
40*
O0
0
-
-70
0.315-00
-Kd
3
0.1
0.
0. 9
0.
0.
Kd
E
5
3 D
--
0.
1
0.3
- 60
0 03
E
$ 0.3
LL
E
0.2 5
0 1
SO
- 50
0
0 .2
0.1 5b
1
.
29
1.5
0
40
3
30
3.5
O.
2.5
2
0.5
M.. 0.7
,_ 0.9
5A
Figure 4-6: Deformation resistance and energy absorption vs. weight for indentation
for <(Kd, 0, Ra = 50) and R/t = 0.1, evaluated at indentation load of F=5 N/mm.
Stiffness FH/6 and energy absorption U for (a) the experiments and simulations
where p,
-
pt and (b) biological densities of tissue and scales where p, =
2
.6pt.
ongoing and future work remains to characterize the flexibility of the these systems.
The relative weight of the composites is given by the areal density PA of each architecture, which is primarily a function PA(L sin 0) given the similar density p of the
scale and tissue. It can be shown in Fig. 4-4 that geometries with high overlap (e.g.
Kd
= 0.1) resist penetration best. However, the geometric constraints of the structure
(as shown in Fig. 2-7) restrict that high overlap is correlated with high scale angle 0,
thus increasing the weight PA of the structure. Thus to minimize the weight of the
armor while still maximizing its protection against indentation, the composite should
76
have both a minimum scale angle 0 and high overlap Kd. As these are conflicting
requirements, a median value such as
Kd
~ 0.5 is a compromise (as found in nature),
though other factors such as flexibility should be considered. This tradeoff is shown
in Fig. 4-6.
4.3
Conclusions
Finite indentation of overlapping scaled structures models physiologically relevant
predatory biting attacks. Simulations reveal that as the contact area of the loading
surface is reduced to a finite radius penetration indentation (R = oc -
R ~ 0),
there is a transition from tissue constraint and scale rotation (under blunt loading)
to a deformation dominated by localized scale bending, which is accommodated by
shearing of the tissue interlayers and the underlying tissue substrate and governed by
the overlap of the scales. Architectures with high overlap (low Kd) resist penetration
by distributing loading over a large area. Under penetration, back deflection in the
underlying tissue is nearly proportional to the overlap of the scales (6 tsase oc
Kd).
Deformation of a scale during localized bending could risk failure by plastic deformation or fracture, exposing the underlying tissue and defeating the armor. Here,
highly overlapping scales are beneficial as they provide multiple layers of defense; in
order to penetrate the tissue, the indenter must deform and defeat Kd-1 scales. Alternatively, scale rotation deforms the compliant tissue which could delaminate or tear
gracefully, leaving the stiff scales intact.
77
78
Chapter 5
Bending and flexibility
A segmented armor system, such as the layered, overlapping scale and tissue structure
found on fish is advantageous due to its unique combination of mobility and protection. While segmented scale armor is inherently more flexible than a solid plate of
the same thickness, the flexibility of the these systems must be characterized. This
work presents an experiential introduction to evaluating the flexibility of overlapping,
layered composite structures under microscopic bending.
5.1
Materials and methods
To roughly evaluate the response of the composites under bending, the samples were
manually deformed over a steel die, as shown in Fig. 5-1. The die is of radius R
-
80 mm and was chosen because it was the largest die readily available. Smaller
radius dies were hypothesized to easily break the samples under a small applied load.
The tested samples correspond to those tested under blunt compression loading in
Chapter 3 for <(Kd, 0, Ra = 50) as shown in Table 3.1. They are fabricated of 3-D
printed ABS and silicone rubber and measure 100 mm by 40 mm by -50 mm tall,
depending on scale angle 0.
79
5.2
Results
The resulting deformations are shown in Fig. 5-2. Delimitation was the primary mode
of failure for these samples under bending. Structures with high scale angle 0 delaminated most easily. The scales showed little sign of mechanical deformation (scale
bending) and instead rotated with the underlying tissue substrate to accommodate
the curvature. These results of tissue delamination correspond to results observed experimentally with the true biological fish integument: under extreme curvature (e.g.
folding the integument upon itself) the scales "popped out" of the composite. In contrast, the same samples were found to fracture and crack under blunt and indentation
loading. Hence, to function as a flexible armor, the composite must resist multiple
failure mechanisms.
Future testing should be performed on longer samples of length of at least 3-4
L, to neglect boundary effects. Testing should be done using a fixture and using
instrumented equipment to record the load applied evenly to the samples. In addition, digital image correlation (DIC) should be used to track the deformation of the
samples. Finally, these samples should be in good condition; several of the samples
were damaged from previous compression testing.
Figure 5-1: Experimental setup for bending over die of radius R = 80 mm.
80
LO,
Ci
CD
LO
CD
Figure 5-2: Experimental bending of 3-D printed ABS and silicone rubber structures
over a die of radius R = 80 mm for 0 = 2.5'.
81
LO
a)
'LO
CD
II
(continued~I)
f3DpitdASadslcn
edn
Eprmna
0m
frdu
overa d~~~~~~~i
08
o
'
ubrsrcue
0r
t6
6
edn
(cniue)Eprietl
ovradi
0-1'
frdisR
-80m
I83
o
o
-
ritdAB
n
ilcn
rbe
trcue
0
iI
N
0
Ci
a
*o
I
Ci
CD
silicone rubber structures
(continued) Experimental bending of 3-D printed ABS and
over a die of radius R = 80 mm for 0 = 20'.
84
0
a)
0
0
LO
a)
(cnine)
ovradi
xermntlbedngo
frdisR
0-4'
=80m
3DprnedAS
o
85
n
slcoerubr
tucue
86
Chapter 6
Conclusions
6.1
Conclusions
Fish scale armor provides a model biomimetic composite for the design of structures
to provide global penetration resistance and flexibility. Biological systems have a limited library of materials available, often consisting of rigid mineralized components
and compliant organic tissues. The morphology and architecture of the material organization is one unique way to provide a spectrum of structural functionalities. Overlapping scale units distribute stresses across a large volume of material and provide
penetration resistance at a reduced weight (and subsequent cost of mineralization)
compared to a continuous armor layer. Scale architecture (Kd, 0), composition (#),
and morphometry (Ra) can be used to locally tailor composite stiffness and back
deflection in the underlying tissue in order to achieve biomechanical protection.
Under blunt loading, the composition primarily governs the effective response
of the composite, which is achieved by a combination of deformation mechanisms
controlled by the microstructural arrangement. Inclusion of a compliant tissue is key
to understanding the true behavior which is dominated by tissue constraint, tissue
shearing (and resulting scale rotation), and scale bending. Scale volume fraction
87
#
governs constraint, effective stiffness, and back deflection; scale overlap Kd governs
scale bending, and initial scale angle 0 governs tissue shearing and scale rotation. The
influence of these mechanisms can be tailored by adjusting the scale morphometry:
for scales with a large aspect ratio, the deformation transitions will be dominated by
scale bending. These results are summarized in Fig. 6-1.
high mineralization cost
increasing <b,
tissue constraint
decreasing overlap (increasing Kd)
increasing indentation deflection
CD
CO,
.C
scale bending
E12
high back
deflection
scale rotation,
tissue shear
Figure 6-1: Summary of deformation mechanisms under blunt loading.
As the contact area of the loading surface is reduced to a finite radius penetration indentation, there is a transition from tissue constraint and scale rotation to a
deformation dominated by localized scale bending, which is accommodated by shearing of the tissue interlayers and the underlying tissue substrate and governed by the
overlap of the scales. Architectures with high overlap (low Kd) resist penetration
by distributing loading over a large area. Deformation of a scale during localized
bending could risk failure by plastic deformation or fracture, exposing the underlying
tissue and defeating the armor. Here, highly overlapping scales are beneficial as they
provide multiple layers of defense; in order to penetrate the tissue, the indenter must
deform and defeat K - scales. Alternatively, scale rotation deforms the compliant
88
tissue which could delaminate or tear gracefully, leaving the stiff scales intact. Such
mechanical characterization could have useful applications in developing principles for
protective systems (Neal and Bain, 2004), composite textiles (Hatjasalo and Rinko,
2006), and even electronics (Kim et al., 2012).
6.2
Contributions
This thesis presents three main contributions that have been achieved:
1. A parametric model is developed to describe the fish scale assembly of the integument. Here, the scale architecture (Kd, 0), composition (#), and morphometry (Ra) are systematically varied to determine the effect of each variable.
2. Micromechanical models are used to simulate these various geometries under
blunt and indentation loading to describe the effective response and protective
features of the structures. Deformation mechanisms are characterized, revealing
transitions across scale and indenter geometries.
3. Physical prototypes are fabricated and tested, showcasing the strengths and
limitations of experimental techniques such as 3-D printing and digital image
correlation.
6.3
Future work
Simulations should explore the role of these layered structures as designs to involve
more material energy absorption (Fig. 6-2a). Macroscopic bending of the scale-tissue
structure should be investigated (Fig. 6-2b). It is important to understand the tradeoff between protection and mobility in flexible, segmented armor design. This could
be performed analytically, simulated using a finite element micromechanical model,
89
and compared to preliminary bending experiments performed in Chapter 5. In addition, advanced micromechanical models should incorporate viscoelasticity, plasticity,
failure (delimitation shown in Fig. 6-2c).
Experimental work should focus on validating indentation simulations in Chapter
4. These indentation experiments need to be performed on samples with a larger
length than those used for blunt loading to neglect boundary effects. Simulations
predict that a sample length of at least 3-4 L, is needed for the stress fields to
approach zero given similar loading. Indenters of varying radii have already been
fabricated (Fig. 6-2d).
(a)
(b) B'(x,y) = (R-h)sine, (R-H)(1-cosO)+h
= UR=LKB'
B
L
(d)
Figure 6-2: Future work. (a) Bilayered composite structures that involve greater
material in energy absorption (contours of e12 shown). (b) Macroscopic bending of
composite. (c) Delamination and failure observed experimentally in early prototype
under bending needs to be characterized. (d) Experimental indentation of samples
could be achieved with fabricated plane strain indenters.
90
Appendix A
Materials characterization of the
scales of Arapaima gigas
The primary objective of this research is to study the mechanical behavior of the
elasmoid scales of a variety of fish. Giant fish measuring over two meters in length
and weighing in at over 100 kilograms like the Amazonian fish Arapaima gigas are
fascinating biological systems. These fish possess large scales of varying degrees of
mineralization that serve as a dermal armor to natural predators. These scales are
elasmoid and consist of an elasmodine basal plate of a twisted plywood arrangement
of collagen fibers capped with different hypermineralized tissues (Meunier and Brito,
2004; Sire et al., 2009; Torres et al., 2008). Scales of A. gigas have a partially mineralized base and well mineralized limiting layer of hydroxyapatite. These layers can
be easily seen with optical and scanning electron microscopy (SEM) as shown in Fig.
A-i.
A.1
Elemental analysis with EDX and BSEM
Energy-dispersive X-ray spectroscopy (EDX) and backscattered scanning electron
microscopy (BSEM) were performed of the fish scales of Arapaima gigas to determine
91
dorsal
(a)
anterior
(b)
posterior
ventral
(c)
anterior/
rostral field,
embedded in
dermis
posterior/caudal
field, covered in
epidermis
(exposed)
Figure A-1: (a,b) Arapaima gigas are large fish that possess equally large scales (-
50 mm long and ~1 mm thick) with human hand shown as reference with dry scales.
(c) Off-white portion of scale in embedded in dermis, while dark portion is covered in
the epidermis (exposed). Scanning electron microscopy (SEM) images of embedded
(d,f) and exposed (e,g) section of scale showing external mineralized layer and basal
collagen layer. Scales in (d,e) are sectioned with scissors, while those in (f,g) are
mounted in epoxy and polished.
92
the material composition. A section of the scale was embedded in a room-temperature
curing epoxy resin and cross-sectioned in the transverse and longitudinal directions
To reduce
using a diamond-impregnated annular wafering saw at 800-900 r.p.m.
surface roughness, samples were polished on a polishing wheel with 15 Pm, 6 pm, and
1 pm silica nanoparticles on a soft pad, and again with 50 nm silica nanoparticles
on a cloth pad in distilled water. Surface roughness was verified using atomic force
microscopy (AFM). Sample was sputter coated with gold for conductivity.
Samples were scanned in a JEOL JSM-6700F at 15 kV and 10.0 pA using a 26 mm
sample holder. EDX was used for spatial imaging as shown in Fig. A-2a. Spectral
analysis with BSEM (Fig. A-2b) confirms composition of an internal layer of collagen
and and outer layer of hydroxyapatite. Silicon was also present, likely from polishing
contamination.
phosphorus
oxygen
(a)
(b)
Z
IP
0
1
Z:2
gradient
even distribution
Ca
internal
2
4
3
5
keV
-
collagen
6
7
8
9
10
Figure A-2: Materials characterization of Arapaima gigas scales by (a) energydispersive X-ray spectroscopy (EDX) and (b) backscattered scanning electron microscopy (BSEM), confirming an internal collagen layer and an outer mineralized
hydroxyapatite layer. Agrees with similar results published in Meyers et al. (2011).
93
A.2
Mineralization analysis with TGA
Thermogravimetric analysis (TGA) was used to assess the mineralization of the scales
of Arapaima gigas using a TGA Q50. It was found that a mortar and pestal created
a paste that was not suitable for TGA so scissors were used to create a very fine
powder of dried fish scales. A 10 mg powdered sample was placed in a platinum pan
and into the instrument. After the thermal profile in Table A.1, air was used to cool
the device.
Table A.1: Thermal profile for thermogravimetric analysis
Equilibrate temperature
Initial isothermal time
30 0 C
5 minutes
Ramp rate
10-20 'C/min
Maximum temperature
Final isothermal time
800 0 C
5 minutes
After water evaporated from samples, organic and inorganic composition were
determined by computing the mass that burns off during thermal ramp and the
remaining mass, respectively (Fig. A-3). Density of hydroxyapatite (p = 3.17 x 103
kg m3 ) and collagen (p = 1.33 x 103 kg M3 ) were used to determine volume fraction
of mineralized and organic components and compared against previously published
values (Ikoma et al., 2003; Torres et al., 2008). The results conclude that the scales
of Arapaima gigas are
A.3
-
24% mineralized by volume.
Nanoindentation
Nanoindentation was employed to evaluate the spatially-specific stiffness, hardness,
and deformation mechanisms of the different scales through the thickness of the two
different material layers. Samples were embedded and polished as in Section A.1.
Nanoindentation of the scale samples was performed on a Hysitron Triboindenter
94
(collagen)
-0.3
-0.2
0.2
% inorg 0.150
40
-0.1
40
0.05
Temperature ("C)
experimental
Ikoma et al, 2003
Torres et al 2008
inorganic
organic
24
62
S
2
25
20
43
5
54
Figure A-3: Mineralization of Arapaimagigas scales using TGA. Density of hydroxyapatite and collagen were used to determine volume fraction of mineralized and organic
components. Results for volume % characterization consistent with other elasmoid
fish scale mineralization data (Ikoma et al., 2003; Torres et al., 2008).
using a Berkovich (triagonal pyramid) diamond probe tip following work by Bruet
et al. (2008). Due to the nonideal tip geometry of the indenter and frame compliance,
the system was calibrated prior to each set of experiments using a standard fused
quartz sample. This calibrated indentation also provides the probe geometry (tip end
radius and truncate height) for use in data analysis.
The piezoelectric transducer was allowed to equilibrate for 660 seconds prior to
each indent. The loading profile consisted of constant loading rate of 50 pN/s until the
maximum set peak load is reached, hold period of 3 s at this maximum set peak load,
a constant unloading rate of -50 pN/s until a zero force load is reached. A maximum
set load of 500 MN was used as an initial estimate based on successful penetration
depth. Indentions was performed along the length of the sample approximately every
2 pm to observe the different material layers.
All indentation load-displacement data collected from the experiments were ex95
ported for processing in MATLAB. Material parameters including elastic modulus and
hardness were determined from the indentation unloading curve based on the OliverPharr method (Oliver and Pharr, 1992) and preliminary experiments are shown in
Fig. A-5. The reduced elastic modulus is calculated as
Eo-p = '/
2
S
Amax
where S is the slope of the initial region of the unloading indentation-depth curve, and
Amax is the projected area of contact at the maximum contact depth. This projected
area is derived using the geometry of the probe tip along with contact depth from
the instrumented indentation curves. The hardness is calculated as
H0
P -- Pmax
Amax
where Pmax = 500 pN is the maximum load of the indentation curve. The hardness
obtained represents the load-bearing capacity of the material.
;
ocation o;
Lnanoindentation
Figure A-4: Location for nanoindentation of Arapaima gigas scales.
croscopy of scale embedded in epoxy, ground and polished.
96
Optical mi-
60
C!,
20 .
0-
w
-......
............
V+
0
z
-.
- -
40 .--- .-..
12.
...-.
.. . .. . .... .... .. .. ..
. . . . . . . . . .*. . ..
3
-V-w-
#W+*
.....
-------L_
100
0
200
300
:L
500
400
600
700
Distance (t m)
3r
200
400
600
Indentation Depth (nm)
800
a- 2k
6
I
....
-.
-.-.. ...
0T
0
............
1.0
0.5
-.............
*:'**.~
-~
.
I
100
***4*~*
4.
*
*
I
*.
200
**~
44~4*
Y--Y
70
I
300
*~
400
.
500
600
700
Distance (t m)
IK
Figure A-5: Preliminary data for Oliver-Pharr stiffness and hardness from nanoindentation of external region of Arapaima gigas
scale where Pmax = 500 pN. This work is preliminary and needs refinement to determine appropriate indentation loading profile
and indentation spacing to accurately capture the laminate collagen layers. While tensile testing reports elastic modulus an
order-of-magnitude lower around E ~ 2 GPa (Torres et al., 2008), similar results are published in Meyers et al. (2011).
.........
.
..........
..
98
Appendix B
DIC and load synchronization
The following programs are meant to extract and analyze data from a mechanical
testing machine (e.g., Zwick). The program importZwickCycle .m I reads the . tra file
and produces a N x 1 cell array of data (e.g., "Test time" ;"Cycle number"; "Standard
travel"; "Standard force") or any output specified by the Zwick program file. These
vectors will all be the same length N (and typically quite large due to the data
acquisition rate of the Zwick).
The displacement or strain data from digital image correlation (DIC) extensometry is imported in a similar manner. Due to the finite frame rate of the camera, there
will be far less data points from DIC than from the Zwick (vector of length M < N).
These data sets are conditioned as needed (e.g. noise removed below a threshold,
zeroed) such that the curves all start from the same time corresponding to the start
of the test.
Synchronize the strain data from DIC to load data from Zwick using interpZwCam. m.
This program will linearly interpolate the camera strain over the Zwick data such that
the strain vector is also of length N. This will permit you to plot displacement (or
strain) vs the force (or stress) to extract true loading data and calculate elastic moduli
without concern about frame compliance.
'Code written for MATLAB R2009b.
99
function
[data]
= importZwickCycle(filename)
% written by ashley browning, 2011
X reads force, displacement,
etc
data from a zwick.tra file
X---------------------------------------------------------------fid = fopen(filename,'r');
% read line from file
tline
=
fgetl(fid);
~= -1
while tline
if
or(tline(2:5) ==
'Test',
tline(2:5)
==
'Stan')
% "Test time";"Cycle number";"Standard travel";"Standard
tline
= fgetl(fid); % skip dimensions
data = textscan(fid,'% f u%fu % fu%f','delimiter',';');
else
tline
=
fgetl(fid);
end
end
fclose(fid);
end
100
force"
= interpZwCam(timeCam,
[strainInt]
function
strainCam,
timeZwick)
. ----------------------------------------------------------------
%
written by ashley browning, 2011 to interpolate
Y from DIC to the Load data from Zwick
%/
%
the strain
data
timeCam is time from extensometer (seconds)
timeCam = (frame number) / (frame rate in fps)
% strainCam is from extensometer (unitless)
%/
/0
%
timeZwick
timeZwick
is real time from zwick
can be read from the .tra
fiLe in seconds,
with
the
Loading
rate and displacement
or computed
ideally, timeCam, strainCam, timeZwick have all been zeroed and
thresholded appropriately so they coincide with true start
shortest time (all common time)
so that you interpolate over the same
if timeCam(end) >= timeZwick(end)
% truncate extensometer time
if use
I =
timeCam
<=
time interval
timeZwick(end);
timeCam = timeCam(I);
strainCam = strainCam(I);
else
%
truncate zwick time
display('youucutuoffutheuextensometerutoousoonu:(')
display('timeZwick(end)utimeCam(end)')
[timeZwick(end) timeCam(end)]
end
%
interpolate over finer time abscissa
strainInt = interpl(timeCam, strainCam, timeZwick);
if(length(strainInt) ~= length(timeZwick))
error('fixulengthsuofuinterpolateduvectors')
end
% fix
weird interpolation
dstrain
=
strainInt(end-1)
error at end
-
strainInt(end-2);
strainInt(end) = strainInt(end-1)+dstrain;
end
101
102
Appendix C
Periodic boundary conditions
This methodology was developed by Danielsson, Parks, and Boyce Danielsson et al.
(2002) and based off Nuo Sheng's M.S. thesis with the input of Narges Kaynia.
C.1
General mathematical formulation for periodicity in two directions
Deformation
Often when analyzing complex structures, it is convenient to only model a single repeating unit, or representative volume element (RVE). Using RVEs is computationally
efficient and neglects artificial boundary effects from a larger, finite structure. In order
to simulate the RVE in a finite element package such as Abaqus, periodic boundary
conditions are needed.
To develop equations to implement in Abaqus, consider the 2-D solid with periodic
boundaries in the 1 and 2-directions undergoing deformation in Fig. C.1.
103
TL
2
TR
YY2
Ia
XX2
Xxi
t-1
YY1
BR
BL
4
Li
-
Figure C-1: Representative volume element for periodic boundary conditions. BL=
bottom left node, BR= bottom right node, TL= top left node, TR= top right node,
XX1= left node set, XX2 = right node set, YY1 = bottom node set, YY2 = top node
set.
For a given 2-D solid under plane strain, let the deformation gradient tensor F be
Ox
F
ax
Fn
F 12
F
F 22
21
Let the displacement tensor H be
F 11 - 1
H=F-1=
F
F 12
F 22
21
-1
Hn
H
H
H
21
12
2 2
Periodicity
The following general relationships exist for displacement u, given symmetry in the
1- and 2-direction for a 2-D, element shown in Fig. C.1.
Here, we relate both the vertical displacement (ui) and the horizontal displacement
(U2)
on the two sets of paired edges (top and bottom, left and right). These
relationships are described by the following equations:
UXX2
u1v2
UYY2
_
-
xXi1
HnL1
12L2
YY1
H
U XX2
xX2
UYY2
U2
104
_
-
=
Xx11=i
UYY1
U2
H21L1
H2 2 L2
(C.1)
Depending on boundary conditions and specific periodic constraints used, some
or all corner nodes may or may not be included in the edge sets. Assume here that all
corner nodes are excluded from the edge sets. To fully define periodicity on the RVE,
these corners must also be related. All relationships between the four corners (top
left, bottom left, top right, bottom right) are described by the following equations:
uf"
=HnL1
-uL
TL
u
=
uBR
HL
_ UBL =
UTR -
L =
H21L1
H21L1
(C.2)
uT
TR
BR
-
2
uUTL
1
C.2
u BL
1
BR =H22L2
-
_
TL -
H12
2
BL
H2
Case I: Fixed corner node with periodicity in
two directions
To implement the most general case of the framework developed in Section C.1, let's
consider the geometry shown in Fig. C.2. Here we've fixed the bottom left (BL) node
and redefined our edge sets.
TL
2
TR
L2
t-1
1BI
I1
R
Figure C-2: Node sets for periodic boundary conditions.
When defining our edge sets, we have to be sure to not over-constrain any nodes.
Each degree of freedom (ui, U2 ) must only be declared once. In addition, Abaqus
requires that the equations are in the simplest possible form. To do this, we must
105
ensure that the equations have been reduced as much as possible and that we don't
repeat variables. We've made the following modifications to our edge sets:
1. XX1 is the left side, including BL node, not including TL node
2. XX2 is the right side, including BR node, not including TR node
3. YY1 is the bottom side, including BL node, not including BR node
4. YY2 is the top side, including TL node, not including TR node
Now only the TR node must need corner periodic equations (C.2), as all the other
corners are a part of the edge sets and defined by (C.1):
Boundary conditions
To prevent pure translation and rotation, a single node must be fixed. Here, we choose
BL to be the fixed node, but this is arbitrary. The following boundary conditions
must be specified:
UBL
U1
UBL
2
-
-
o
To respecify the sets to include/exclude corners, we must go back and simplify
our corner periodic equations (C.2):
0
BR -
Ui
=
-[L
TR
HnL1
=HL
uBR -
e=
H 21 L 1
T2 L
uTR
U2
=H21L
1
ui
- UBR
UTL
-K=
UTR
12L2
-
0
BR
0
H12L2
U2L -e=H22L2
Rearranging (and subsequently eliminating any equations with variables
uTL),
uBR
or
this reduces to
UJR = Hn L1 + H 12 L 1
u2'
106
=
H 21 L 1
+ H 2 2L 2
(C.3)
Together, equations (C.1) and (C.3) define our periodic constraints. Double checking with the way we defined our edges in Fig. C.2, we can verify that each degree of
freedom is only declared once.
Loading
An external load is applied on the body by prescribing the displacement gradient
H = F - 1. Let H be defined be creating two arbitrary virtual dummy nodes, VI
and V2, such that the displacements u1 , U2 of these virtual nodes correspond to the
components of H.
H=
Hu
H12
U 71
H 21
H22
U
U2
Hence, to apply uniaxial strain of A
F 22
uV2
1 = U2=
-
U V2
2
'
2
1.1 in the 2-direction, prescribe H 22 =
0.1. Alternatively, to apply simple shear y = 0.1, prescribe H 12 =
0.1.
Implementation
Refer to Section C.4 for implementation in an Abaqus input file.
Virtual nodes V1, V2 are unattached to the meshed part. If included at the *Part
level, their number must be larger than the maximum number of nodes in the mesh.
If included at the *Assembly level, they are not related to the part at all and their
number is arbitrary (can be 1, 2, etc.).
H=
Hu1
H21
H
12
H22
u
1
=
UV2
1
99971
99981
1
U V1
U2
99972 99982
They can be defined at any location, though placing them at the origin simplifies
107
interpreting their displacements:
*Node, nset=FakeDefNodes
9997, 0., 0.
9997, 0., 0.
The previous relationships can be simplified as the following, which can then be
programmed into Abaqus using *Equation (for L1 =10 and L1=20):
xi
- HuL 1 = 0
Yx1
-
YY1
1
- H12L2
urx2 _
x2 _
YY2
U1Y2
U
Y
2
_
Y1 -
U R =
TR
U2
NOTE:
-
H
=H
H21L1=0
XX1,
1, 1, XX2,
XX1, 2,
1, XX2,
1, -1,
9997,
2, -1,
9997, 2,
1, 10.
10.
0
YY1, 1, 1, YY2, 1,
-1,
9998,
1, 20.
H22L= 0
YY1, 2, 1, YY2, 2,
-1,
9998,
2,
1 L1
21 L,
=
_
+ H 1 2 Li
TR, 1, 1, 9997, 1, -10.
+ H22L2
TR, 2, 1, 9997, 2,
9998,
-10.
20.
1, -10.
9998, 2,
-20.
The order of the variables in these equations has a significant effect
on how the stiffness matrix is solved. When defining the node sets, ensure that they
remain in the correct order by declaring unsorted in the node definition. In addition,
assure that the x, y coordinates of these node sets match up identically. For complex
geometries, this can be done by verifying in MATLAB, as shown in Section C.5.
The boundary conditions for true and dummy node sets can be implemented using
*Boundary:
U BL
U
F11 = 1.1
0
uBL = 0
H11 = 0.1
108
BL,
1, 2,
9997,
0.
1, 1, 0.1
C.3
Case II: Fixed corner node, roller surface, with
periodicity in one direction
This is the periodic boundary condition used in the model in Chapter 3.3.1.
Consider the 2-D solid with periodic boundaries in the 1-direction undergoing
deformation in Fig. C.3. This would be used for modeling compression or tension of
a very long structure with finite height.
1. XX1 is the left side, not including BL node
2. XX2 is the right side, not including BR node
3. YY1 is the bottom side, including both BL and BR nodes
TL
TR
YY2
2
L2
Xxi
XX2
1BL
I
;M
R
Li
Figure C-3: Periodic boundary conditions used in model. BL= bottom left node,
BR= bottom right node, TL= top left node, TR= top right node, XX1= left node
set, XX2 = right node set, YY1 = bottom node set, YY2 = top node set.
Boundary conditions
To prevent pure translation and rotation, a single node must be fixed. Here, we choose
BL to be the fixed node. The following boundary conditions must be specified:
u1
BL
B
2uL=
109
0
To allow frictionless free expansion in the 1-direction and to prevent rotation,
impose a roller condition for bottom surface YY1:
? 4 Y1
=o
The periodic equations are as follows. Note the corner equation is simplified based
on our fixed node boundary condition:
XX2
XX2
_
Xxi =
HuLi
Xxi
H21Li
uB=
H1 1 L 1
Loading
For this problem, we will apply a uniform pressure in the 2-direction for plane strain
elements. We want to specify no shear and ensure that the elements are free to
expand in the 1-direction.
Thus, we need to set the shear deformation F2 1 = 0.
Apply traction, displacement, contact, or pressure conditions as normal in Abaqus
CAE, e.g. we can just specify the pressure using *Dsload. You can also impose a
finite indentation, but note that the indentation would be repeating along the length
due to the periodic boundary conditions.
Alternatively, you can impose a uniform compression using the deformation gradient F by declaring a displacement on the F 22 (99982) node.
Implementation
Refer to Section C. 5 for implementation by defining all boundary conditions in MATLAB.
Now the previous relationships can be simplified as the following, which can then
110
be programmed into Abaqus using *Equation (for L 1=10):
XX2* - XX1* - 9997 1 L1 = 0
xx1,
1,
1, XX2,
1, -1,
9997,
XX2* - XX1* - 9997 2 L1
xx1,
2, 1, XX2,
2, -1,
9997, 2,
=
0
BR 1 - 9997 1 L1 = 0
BR,
1,
1, 9997,
1, 10.
10.
1, -10.
The boundary conditions for true and dummy node sets can be implemented using
*Boundary:
BLo
BL,
1
1=
H21
C.4
0
=0
1, 1, 0.
YY1, 2,
2,
9997, 2, 2,
0.
0.
Abaqus input file
The following Abaqus input file implements the above formulation of periodicity in
both x and y by use of virtual "dummy" nodes. In addition, this input file demonstrates the ability to impose a deformation by declaring the deformation gradient F.
By selectively uncommenting the ** lines and changing the components of F, the
load can be changed from uniaxial compression/tension, biaxial compression/tension,
or simple shear.
111
*Heading
**
Job name: Mesh2D Model name: Model-1
**
Generated by: Abaqus/CAE Version 6.8-2
*Preprint, echo=NO, model=NO, history=NO,
**
**
PARTS
**
*Part, name=Part-1
*End Part
**
**
ASSEMBLY
**
*Assembly,
name=Assembly
**
*Instance, name=Part-1-1, part=Part-1
*Node,
nset=Nall
1,
1.,
0.
2,
0.,
0.
3,
1.,
1.
4,
0.,
1.
5,
0.5,
0.
6,
1.,
0.5
7,
0.5,
1.
8,
0.,
0.5
9,
0.5,
0.5
*Element, type=CPE4H, Elset=ElAll
1, 5, 9, 8, 2
2, 8, 9, 7, 4
3, 1, 6, 9, 5
4, 3, 7, 9, 6
*Solid Section, elset=ElAll, material=NH
*End Instance
**
*Nset,
2,
*Nset,
1,
*Nset,
4,
*Nset,
3,
nset=BLNode ,
instance=Part-1-1
nset=BRNode,
instance=Part-1-1
nset=TLNode,
instance=Part-1-1
nset=TRNode,
instance=Part-1-1
**
**
PERIODIC BOUNDARY CONDITIONS:
**
*Node, nset=FakeDefNodes
9997, 0., 0.
112
contact=NO
9998, 0., 0.
** XX1 is the left
side,
including BLnode , not including TLnode
*Nset, nset= XX1,
instance=Part-1-1,
unsorted
2,8,
** XX2 is the right side, including BRnode , not including TRnode
*Nset,
nset= XX2,
instance=Part-1-1,
unsorted
1,6,
** YY1 is the bottom side, including BLnode , not including BRnode
*Nset,
nset= YY1,
instance=Part-1-1,
unsorted
2,5,
** YY2 is the top side, including TLnode , not including TRnode
*Nset, nset= YY2,
instance=Part-1-1,
unsorted
4,7,
*Equation
3
XX2,
1,
XXi, 1,
9997, 1,
3
XX2,
2,
XX1, 2,
9997, 2,
3
YY2,
1,
YY1,
1,
9998, 1,
3
YY2,
2,
YY1, 2,
9998, 2,
3
TRNode,
9997, 1,
9998, 1,
3
TRNode,
9997, 2,
9998, 2,
1
-1
-1.0
1
-1
-1.0
1
-1
-1.0
1
-1
-1.0
1, 1
-1.0
-1.0
2, 1
-1.0
-1.0
**
*End
Assembly
**
*Material,
name=NH
*Hyperelastic,
neo
hooke
50.,0.0004027
*Step, name=Strain, nlgeom=YES
*Static
0.01, 1.,
le-05, 1.
113
*Boundary
BLNode, 1, 2, 0.0
** APPLY LOAD TO DEFORMATION
**
unixial
**9997,
**9998,
**9998,
**
**
compression
2, 2, 0.0
1, 1, 0.0
2, 2, -0.1
biaxial
**9997,
**9997,
**9998,
**9998,
compression
1,
2,
1,
2,
simple
9997,
9997,
9998,
9998,
GRADIENT
1,
2,
1,
2,
1,
2,
1,
2,
-0.1
0.0
0.0
-0.1
shear
1,
2,
1,
2,
0.0
0.0
0.1
0.0
**
*NODE PRINT, NSET=FakeDefNodes
u,tf
*Output, field, variable=PRESELECT
*node output, nset=FakeDefNodes
u
*End
Step
114
C.5
MATLAB code to generate Abaqus input file
The following MATLAB programs take an input file from Abaque CAE, invoke periodic boundary conditions in x (as in Section C.1), and rewrites the original input file
to a new file.
115
maininput m
written by ashley browning 2010 to make a bunch of input files
periodic in x by the program 'nodesortalignrewrite ',
location, part name
location, new file
file
which inputs original
0/
%0
when CAE adds a part to an assembly , it calls 'part' -- > 'part-i'
can be added in loop below
'-1' or any other postscripts
this
'input2red ', 'input3red ',
if you have 'inputired',
simi larly,
add 'red' in the loop below
you can just
clear all;
clc;
% name of input file,
INPUTNAME={'input1',
ie:
inputl.inp
'input2','input3'};
X name of part in associated input file
PARTNAME={'partl', 'part2','part3'};
for i=1:length(INPUTNAME)
display(INPUTNAME(i))
nodesortalignrewrite (...
char(strcat('I:/AbaqusTemp/',INPUTNAME(i),'.inp')),...
char(strcat('E:/Ash/Abaqus/pbc/',INPUTNAME(i),'pbc.inp')),...
char(strcat(INPUTNAME{i},'-1')))
end
116
function [] = nodesortalign_2Drewrite3(fname,fnnewpart)
% Ashley Browning modified from code by Matthew J. Rosario
% do NOT specify any boundary conditions on the part in CAE
Z all of the BCs are written in this program.
% only specify loading BCs (indentation, pressure, etc)
% example:
(original file loc, new file loc, part name)
% nodesortalign_2Drewrite3('I:\AbaqusTemp\frommodel.inp',
'E:\Ash\Abaqus\pbc\input.inp',
'part -1')
% Open file for reading
fid1=fopen(fname,'r');
fid =fopen(fnnew,'wt');
if
(fid1
%
%
input
output
< 0)
error('couldunotuopenufileus',fname);
end
% Find the end of header
head = 0;
key = '*Node';
while
1
readin = fgetl(fidl);
len = length(readin);
if
len~=5
head
=
head + 1;
fprintf(fid,readin); Zcopy this to new file
fprintf(fid,'\n');
Zcopy this to new file
elseif and(readin(len)==key(len),len==5)
head
=
head + 1;
break
elseif head>20
error('headeroidentificationuerror')
end
end
% Store nodes and their positions
data = textscan(fidl,'%du%fuf','delimiter',',');
node=data{1 ,1};
xpos=data{1,2};
ypos=data{1 ,3};
le-4;
rtol = 1/tol;
% Round all data to 4 decimal places
xpos = round(xpos*rtol)/rtol;
ypos = round(ypos*rtol)/rtol;
tol
=
117
% Find maximum dimensions of prism
xmax=max(xpos);xmin=min(xpos);
ymax=max(ypos);ymin=min(ypos);
% The following procedures sort through the node sets
%
to seIect nodes
%
Define the face nodes
that are found on faces,
edges,
and corners
XX1=[];
for k=1:length(xpos)
if
xpos(k)==xmin
XX1=[XX1; double(node(k)) xpos(k) ypos(k)];
end
end
XX1=sortrows(XX1,3);
XX2=[];
for k=1 :length(xpos)
Yif xpos (k)==xmax
if abs(xpos(k)-xmax)<=2*tol
XX2=[XX2; double(node(k)) xpos(k) ypos(k)];
end
end
XX2=sortrows(XX2,3);
YY1=[];
for k=1:length(ypos)
if ypos(k)==ymin
YY1=[YY1; node (k) xpos(k)
end
end
YY1=sortrows(YY1,2);
ypos(k)];
YY2=[];
for k=1:length(ypos)
if ypos(k)==ymax
YY2=[YY2; node (k) xpos(k) ypos(k)];
end
end
YY2=sortrows(YY2,2);
Y make sure the XX1 and XX2 (Left
% have identicaL ypos and rewrite
118
and right
if
not
sides)
%
display this
on the screen so you can determine aliasing
LengthLeft=length(XX1)
LengthRight=length(XX2)
maxDiff = max(abs( ypos(XX1(:,1))-ypos(XX2(:,1))))
% error before alignment: (XX1,
XX2,
difference):
[ypos(XX1(:,1)), ypos(XX2(:,1)), ypos(XX1(:,1))-ypos(XX2(:,1))];
% adjust the ypos of XX2 to match that of XX1
for k=1:length(XX2)
ypos(XX2(k,1))=ypos(XX1(k,1));
end
Y error after alignment: (XX1, XX2, difference):
[ypos(XX1(:,1)), ypos(XX2(:,1)), ypos(XX1(:,1))-ypos(XX2(:,1))];
L1=xpos(YY1(length(YY1),1))-xpos(YY1(1,1));
L2=ypos(XX1(length(XX1),1))-ypos(XX1(1,1));
cornernodes=[XX1(length(XX1),1)
XX2(length(XX2),1);
XX1(1,1) XX2(1,1)1
% Write sorted node numbers to file
% fid = fopen(fnnew,'wt');
Z Write new aligned node assigments to
the same file
fprintf
(fid,'*Node\n');
for k=1:length(node)
fprintf(fid,'\td,\tf,\tf\n',node (k),xpos(k),ypos(k));
end
% find the
key
=
body
while
end of instance
'*EnduInstance';
0;
1
readin = fgetl(fid1);
len = length(readin);
if
len~=13
body = body + 1;
%copy this
fprintf(fid,'\n');
Xcopy this
elseif and(readin(len)== key (len ) ,len
body = body + 1;
fprintf(fid,readin); Xcopy this
fprintf(fid,'\n');
%copy this
break
end
fprintf(fid,readin);
to new file
to new file
==1 3)
to
to
end
fprintf(fid,'*Node,unset=FakeDefNodesu\n');
119
new file
new file
fprintf(fid,'9997,uO.,uO.u\n9998,uO.,uO.u\n');
fprintf(fid,'*Nsetanset=BLNode,uinstance=Xsu\nd,u\n',partXX1(1));
fprintf(fid,'*Nsetunset=BRNodeuinstance=%su\nd,u\n',partXX2(1));
fprintf(fid,'*Nset,unset=XX1,uinstance=%s,uunsortedu\n',part);
for k=2:length(XX1)
Y starts at k=2 to
fprintf(fid,'XAd,\n',XX1(k));
omit BLnode
end
fprintf(fid,'*Nset,anset=XX2,uinstance=%s,uunsortedu\n',part);
for k=2:length(XX2)
Z starts at k=2 to
fprintf(fid,'X/d,\n',XX2(k));
end
omit BRnode
fprintf(fid,'*Nset,anset=YY1,uinstance=%s,uunsortedu\n',part);
for k=1:length(YY1)
fprintf(fid,'Xd,\n',YY1(k));
end
fprintf(fid,,'*Equationu\n');
fprintf(fid,'3u\nXX1,u1,u1a\nXX2,u1,u-1u\n9997,u1,u%du\n',
fprintf(fid,'3u\nXX1,u2,ulu\nXX2,u2,u-1u\n9997,u2,udu\n',
fprintf(fid,'2a\nBRNode,u1,ula\n9997,u1,u-Xdu\n',
Li);
key =
while
Li);
Li);
'*Boundary';
1
readin = fgetl(fidl);
len = length(readin);
if len~=9
fprintf(fid,readin);
fprintf(fid,'\n');
%copy
%copy
this
this
to new file
to new file
elseif and(readin(len)~=key(len),len==9)
fprintf(fid,readin); %copy this to new file
fprintf(fid,'\n');
Xcopy this to new file
elseif and(readin(len)==key(len),len==9)
break
end
end
%
No Deformation BCs
fprintf(fid,'*Boundaryu\n');
fprintf(fid,'BLNode,u1,u1u\n');
fprintf(fid,'YY1,u2,u2u\n');
fprintf(fid,'9997,u2,u2u\n');
fprintf(fid,'*Boundaryu\n');
% Copy other boundary conditions
while
1
readin = fgetl(fidl);
len = length(readin);
120
and
the
rest
of
the
input
file
if
len>1
fprintf(fid,readin);
fprintf(fid, '\n');
%copy
7.copy
else
break
end
end
fclose(fidl);
fclose(fid);
disp('complete')
end
121
this
this
to
to
new file
new file
122
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