MATERIALS SCIENCE & ENGINEERING ELSEVIER
Materials Science and Engineering C2 (1994) 113132
12
Biomimicking of animal quills and plant stems: natural cylindrical shells with foam cores G.N. Karam, L.J. Gibson Department of Civil and Environmental Engineering, Massachussetts Institute of Technology, Cambridge, MA 02319, USA Received 20 April 1994; in revised form 2 August 1994
Abstract
Thin walled cylindrical shell structures are widespread in nature: examples include porcupine quills, hedgehog spines and plant stems. All have an outer shell of almost fully dense material supported by a low density, cellular core. In nature, all are loaded in some combination of axial compression and bending: failure is typically by buckling. Natural structures are often optimized. Here we have investigated and characterized the morphology of several natural tubular structures. Mechanical models recently developed to analyze the elastic buckling of a thin cylindrical shell supported by a soft elastic core (G.N. Karam and L.J. Gibson, Elastic buckling of cylindrical shells with elastic cores, I: Analysis, submitted to lnt. J. Solids Structures, 1994., G.N. Karam and L.J. Gibson, Elastic buckling of cylindrical shells with elastic cores, II: Experiments, submitted to lnt. J. Solids Structures, 1994) were used to study the mechanical efficiency of these natural structures. It was found that natural structures are often more mechanically efficient than equivalent weight hollow cylinders. Biomimicking of natural cylindrical shell structures may offer the potential to increase the mechanical efficiency of engineering structures. Keywords: Biomimicking; Quills; Shells
1. Introduction
Plant stems and animal quills and spines all resist both axial loads and bending moments: bending arises from wind loads in plant stems and from eccentrically applied point loads in animal quills and spines. Loading from any direction is resisted equally by their axisymmetric cylindrical cross section. The bending stiffness and overall stability (Euler column buckling resistance) of a cylindrical tube increase, for the same mass, with increasing radius to thickness ratio, a/t, up to the limit of the onset of local buckling, the controlling failure mechanism in plant stems and animal quills. Many plant stems support the outer cylindrical shell with a foamlike layer of parenchyma cells; animal quills and spines, too, have a similar layer of foamlike cells. Here we estimate the contribution of the foamlike core to the resistance of plant stems and animal quills and spines to local elastic buckling using recent theoretical and experimental work on the elastic buckling of cylindrical shells with soft elastic cores [1,2]. The results suggest that biomimicking of such natural structures can lead to engineering structures with improved mechanical efficiency. The microstructures of different plant stems and animal quills and spines were measured from optical
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and scanning electron micrographs. The elastic moduli of the cores were estimated from measurements of their relative densities using models for cellular solids [3]. Data from our own specimens were supplemented by measurements on micrographs available in the literature. The mechanical efficiency of the natural structures was evaluated by calculating the ratios of the axial buckling load and the local buckling moment, to those of a hollow cylinder of equal radius and mass. In addition the core depth was compared with that at which the normal stress in the core is calculated to decay to 95% of the maximum value. Potential biomimetic applications of these structures are described.
2. Materials and methods
Fully grown live plant specimens from the natural wild growth in the New England area were collected and identified at the Herbarium of Harvard University [4]. Stem cross sections were obtained from the lower quarter of the stems by sectioning with a sharp razor blade. Measurements of the microstructure of the larger specimens were performed with an electronic caliper (MaxCal, Japan) and an optical microscope with a precision glass reticle (Edmund Scientific Co, Bar
114
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132
rington, N J). Smaller specimens were gold coated, examined and photographed in the scanning electron microscope (Cambridge Instruments, Model $240). North American porcupine (Erethizon) quills, 5060 mm long, were obtained from live animals at the Franklin Park Zoo in Boston while hedgehog (Erinaceus Europaeus) spines, 1520 mm long, were obtained from recently killed animals at the Zoology Department of Harvard University. We also obtained, from the Museum of Comparative Zoology at Harvard University, conserved quill specimens for two Old World porcupine species (Hystrix Galeata and Hystrix Subcristata) and an Australian echidna (Tachyglossus Aculeatus) of lengths 235,360 and 35 mm respectively. Sections were prepared by freezing in liquid nitrogen and sectionning with a sharp razor blade. Specimens were then gold coated and prepared for examination in the scanning electron microscope. Geometric property measurements were made on cross sections obtained from the central part of the quills where the diameter is more or less constant along the length. The recent botanical and zoological literature was also surveyed for micrographs of plant stems and animal quills from which microstructural measurements could be made [58]. The radius, shell thickness and core depth of each natural tubular structure were measured either by direct optical microscopy, or on scanning electron micrographs. The solid area fractions, or relative densities, of the outer shell and of the inner core were obtained by the classical stereological method of point counting: the relative density was found from the ratio of the number of grid points that fell on solid material to the total number of grid points falling in the observed region [9]. In the case of the optical measurements, the glass reticle grid was used as a reference while in the case of the micrographs a grid was overlain on top of the micrograph.
3. Microstructure and material properties of natural cylindrical shells with compliant cores
3.1. Microstructures The porcupine quills and hedgehog spines all had a dense outer shell with a more compliant inner core; their geometrical measurements are listed in Table 1. Four types of core microstructure were observed. The simplest microstructure, a foamlike core filling the outer shell, was observed in the North American porcupine (Erethizon), (Fig. 1), in the Brazilian tree porcupine (Coendou prehensilis) [10], and in one of the two specimens of the echidna (Tachyglossus Aculeatus) that were investigated (Figs. 2(a)(c)). The other echidna specimen was hollow (Fig. 2(d)). The cell wall
thickness of the foamlike core is uniform over the cross section while the cell size decreases from the center of the cross section to the outer shell, resulting in a radially increasing relative density, (Figs. l(a)(c)). The foam relative density reported in Table 1 is an average across the section. The second core microstructure resembled the first, but with additional thin, solid, longitudinal stiffeners running radially from the outer shell of the quill towards the center. This microstructure was observed in the Old World porcupine quills Hystrix Subcristata (Fig. 3), H. Galeata, (Fig. 4), and H. Indica x Cristata [10]. The stiffeners decrease in thickness as they converge at the central axis. The foam filling the remaining core stabilizes both the outer shell and the thin stiffeners (Figs. 3(c), 3(d)). The volume fraction of the cross section occupied by the stiffeners was reported as the core relative density due to the ribs and it was obtained by dividing the area occupied by the solid ribs by the total area of the section. The three dimensional foam relative density was reported separately (Table 1). In the third microstructure the outer shell is stabilized by closely spaced longitudinal and radial stiffeners; this structure was observed in the spines of the hedgehog (Erinaceus Europaeus) [10] (Fig. 5) and the spiny rat (Hemiechinus spinosus) [10]. The longitudinal stiffeners do not fully extend into the center of the cross section (Figs. 5(a), 5(b)). The radial ring stiffeners span the spaces between the longitudinal stiffeners; at the radius at which the longitudinal stiffeners end, every three to four ring stiffeners converge to form a thin diaphragm or septum that spans across the open central core (Figs. 5(c), 5(d)). This configuration acts as a square honeycomb supporting the inside surface of the shell (Figs. 5(d), 5(e)). The relative density of the honeycomb reported in Table 1 does not include the central septa. The fourth microstructure type was observed only in the tenrec (Setifer) spine which has a filled foam core exclusively made out of thin closely spaced septa [10]. The plant stems surveyed (Table 2) had a microstructure with a foamlike core similar to the first type observed in the porcupine quills. They could however be divided into two main groups depending on the location of the vascular bundles in the stem. In the first group, (Arena, Eleocharis, Elytrigia, Hordeum, and Secale), the outer shell was made of close to cylindrical sclerenchyma and collenchyma cells, aligned along the main axis (Fig. 6). The cells in the core were elongated parenchyma cells that are much shorter, more equiaxed and of less regular geometry than the sclerenchyma and collenchyma cells making up the outer shell [5]. The cells in the outer shell layer have small diameters, very thick walls and virtually no lumen, while those in the core have thinner walls and much larger diameters resulting in a clear density change (Figs. 6(a), 6(b)). The vascular bundles (xylem and phloem) through which
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132
115
Table 1 Section properties of animal quills and spines Animal genus/specie (common name)
Information source
Outer radius (ram)
Shell thickness (ram)
Shell relative density p ( )
Core relative density, Pc ( )
0.48
0.033
1.0
0.2
foam filled core
SEM investigation
0.89
0.048
1.0
0.10.15
foam filled core
SEM investigation
1.34
0.48
1.0
0.11
foam filled core
SEM investigation
0.90
0.33
1.0
0.0
empty thick walled tube
Core depth, c (mm)
Type 1: isotropic three dimensional foam core
C oendou prehensilis
[10]
(Brazilian Porcupine)
Erethizon, (North American porcupine)
Tachyglossus Aculeatus, specimen 1 (echidna)
Tachyglossus Aculeatus, specimen 2 (echidna)
Type 2: longitudinal solid ribs with isotropic three dimensional foam core
Hystrir Galeata
SEM investigation
1.34
0.074
1.0
1.33
0.133
1.0
1.25
0.12
1.0
(porcupine)
Hystrix lndica Cristata
[1O]
0.037 ribs, 0.10 foam 0.03 ribs, 0.15 foam
foam filled core
0.07 ribs, 0.11 foam
foam filled core
foam filled core
(porcupine)
Hystrix Subcristatus
SEM investigation
(porcupine) Type 3: orthogonal longitudinal and circumferential stiffeners in a square honeycomb
Erinaceus Europaeus
SEM investigation
0.36
0.025
1.0
0.1
0.17
[10]
0.52
0.04
1.0
0.1
0.285
[10]
0.53
0.04
1.0
0.1
0.21
2.65
0.53
1.0
0.1
(hedgehog)
Erinaceus Europaeus (hedgehog)
Hemiechinus spinosus (spiny rat)
Type 4: closely spaced thin septa
Setifer (tenrec)
filled core [10]
water and nutrients circulate, and the stiff bundle sheath enclosing them, are part of the core and were included in its estimated density. All of the specimens surveyed in this first group had a central hole. In the second group the outer shell, again made up of elongated cells, contained the vascular bundles while the core was made up of foamlike, roughly equiaxed parenchyma cells. Most specimens in this group had fully filled cores. Examples include Artemisia, Cenchrus Ciliaris, Latuca Biennis and Phytolacca Americana. The
dimensionless geometrical parameters needed for the analysis of mechanical efficiency were calculated from the information in Tables 1 and 2 and are presented in Tables 3 and 4 along with the required material properties. The radius to thickness ratio, a/t, was obtained from the outer radius and shell thickness measurements by subtracting half the thickness from the outer radius and dividing the result by the thickness. The core to shell density ratio, pJp, was obtained as the ratio of the core relative density to that of the
116
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132
(a)
(b)
tel
(d)
Fig. 1. Micrographs showing North American porcupine quill (Erethizon) : (a), (b), cross section; (c), (d) longitudinal section
i
2~
(a)
(b)
Icl
Idl
Fig. 2. Micrographs showing echidna ( Tachyglossus Aculeatus) quills: (a), (b), (d), cross sections; (c) longitudinal section.
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132
(a)
117
(b)
Z
\ (dl
Icl
Fig. 3. Micrographs showing Hystrix Subcristata quill : (a), (b) cross section; (c), (d), longitudinal section.
shell. The core depth to thickness ratio c/t, was calculated as the ratio of the measured values. In the case of a foamlike core filling the shell c/t=a/t1/2.
3.2. Material properties
(a)
/ ! j
(b)
Fig. 4. Micrographs showing Hystrix Galeata quill cross section.
To apply the analysis for the local buckling of a cylindrical shell with a compliant core, the ratio of Young's moduli of the shell to that of the core must be known. Here, we estimated the core moduli using previous models for cellular solids [3]: the ratio of Young's modulus of a cellular solid to that of the solid cell wall material, E3E, can be estimated simply from their density ratio, pJp, as (pjp)n, where the exponent n depends on the geometry of the cellular core. Here we assume, like Vincent and Owers [10], that the outer shell and solid material of the core of the animal quills and spines are identical and have the same mechanical properties. In plant stems the mechanical properties of the solid cell wall vary from one tissue to another, depending on composition and function. Sclenrenchyma cell walls are lignified and have the highest cellulose content, parenchyma cell wails are not lignified and have the lowest cellulose content, and collenchyma cells and vascular tissues have intermediate compositions and properties. In addition, the internal turgor pressure in parenchyma cells increases their apparent stiffness [11]. The exact properties of each cell wall type are difficult
118
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132
(a)
(b)
(c)
Ld)
le)
Fig. 5. Micrographs showing hedgehog (Erinaceus Europaeus) spine: (a), (b), cross section; (c), (d), (e) longitudinal sections.
to measure with only some partial and limited data available [1215]. The "rind" or outer shell of the plant stem is made up of sclerenchyma, vascular bundles and some collenchyma, while the core is made of parenchyma, vascular bundles and some collenchyma. Determination of the exact material properties of the shell and core is made difficult by the limited data available for each type of cell, the variation in the volume fraction of each type of cell in different plant types, and by variations in turgor pressure. A common simplifying assumption is to adopt the same cell wall properties for all tissues and to neglect the effect of turgor pressure [1417]. As a first approximation, we also make this simplifying assumption for the plant stems; a more sophisticated analysis is presented in Appendix B.
Our local buckling analysis [1], assumed that both the shell and the core materials were isotropic. The natural materials of this study do not satisfy this assumption. The outer shells of the quills and spines are made of a fibre reinforced composite laminate which most likely has its keratin microfibrils oriented to resist stresses in the most efficient way, as was observed, for example, in the bessbeetle cuticle [18]. The shells of the plant stems, consisting of longitudinally oriented, thick walled fibres, are also anisotropic [11]. Due to the lack of published information on the anisotropy of the natural materials in this study and for the sake of analytical simplicity, we assume the shell materials to be isotropic. Young's moduli of the core materials were estimated in the longitudinal, circumferential and radial directions
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132
119
Table 2 Section properties of plant stems Plant name (description)
Information source
Outer radius (mm)
Thickness (mm)
Shell relative density p
()
Core depth, c (mm)
Core relative density, Pc
()
Group 1: vascular bundles in core Avena (oat)
E leocharis (sedge grass) E~trigia repens specimen 1 (grass) Elytrigia repens specimen 2 (grass) Hordeum vulgare (barley) Secale (rye)
[5]
3.82
0.064).08
0.951.0
[7]
0.35
0.016
0.9
SEM investigation
0.97
0.036
0.91.0
SEM investigation
1.17
0.05
0.91.0
0.1 without vascular bundles 0.2
0.824
0.1
0.13 (0.2 with vascular bundles) 0.13, (0.2 with
0.285
0.4
v.b.) [6]
1.52.0
0.030.09
0.81.0
0.10.2
0.30.68
[5]
2.90
0.06
0.8
0.075
0.704
1.6
0.381
0.91.0
0.10.3
foam filled
2.55
0.762
1.0
0.2
foam filled
0.83
0.14
0.8
0.2
foam filled
7.5
0.127
1.0
0.070.10
1.54
7.75
0.41
0.95
0.03
5.0
G~oup II: vascular bundles in shell Artemisia specimen 1 Artemisia specimen 2 Cenchrus Ciliaris (buffel grass) Latuca Biennis Phytolacca Americana (wildberry type)
SEM investigation SEM investigation [8]
Optical microscopy Optical microscopy
using the models of Gibson and Ashby [3] for cellular solids. The results are summarized in Fig. 7, which shows the four types of microstructure schematically. Young's moduli of the echidna and porcupine quills with a simple, isotropic, closedcell foamlike core, classified as Type 1 microstructure (Coendou, Erethizon, and Tachyglossus Aculeatus), are estimated to be (Fig. 7(a)):
Here, we have assumed that the bending of the cell edges is the dominant mechanism controlling the moduli; axial stretching of the cell faces has been neglected. If the axial face stretching is accounted for, we find
[31:
where th is the fraction of the solid material in the cell edges. The more conservative, former estimate of the foam stiffness will be used in the following analysis, neglecting the wall stretching contribution. The second type of microstructure, exhibited by the Old World porcupines (Hystrix Galeata, Subcristatus and Indica x Cristata (Fig. 7(b))), has an isotropic foam core with longitudinal rib stiffeners, producing orthotropic properties. Their relative moduli in the longitudinal and radial directions, x and z, can be estimated from a rule of mixtures upper bound, as: 2
+ 1
Pc
Pc
(3)
2
and the relative modulus in the circumferential direction, y, can be conservatively estimated by:
G.N. Karam, L.Z Gibson / Materials Science and Engineering C2 (1994) 113132
120
an isotropic material [11], (Fig. 7(a)). They are treated similarly to the procupine quills of Type 1 microstructure. The relative stiffness of these cores can be conservatively estimated from Eq. 1, neglecting the turgor pressure and wall stretching effects.
4. The mechanical efficiency of natural cylindrical shell strucutures with compliant cores
(a)
(b)
Fig. 6. Micrographs showing grass (Elytrigia repens) stems: (a), (b), cross sections.
P/ribs x P/foam
\ P ] ribs/
x
(4) where (Pe/P)ribs and (PJP)foam represent the relative density of the ribs and foam, obtained by dividing the solid cross sectional area occupied by the ribs and the foam respectively over that of the total cross section (Table 1). In the case the third type of microstructure, typified by the hedgehog spines (Erinaceus and Hemiechinus) (Fig. 7(c)), the square honeycomb core has a radial relative stiffness equal to its relative density, while the longitudinal and circumferential stiffnesses are equal to half that value because only half the core material is oriented in those directions: Ec
E~
E~
p~
The last type of core microstructure observed in the spines is that of the tenrec ($etifer) (Fig. 7(d)). Its circumferential and radial relative stiffnesses are equal to its relative density while its longitudinal stiffness is negligible. The cores of the plant stems consist mainly of parenchyrnatous tissue which can be considered as
Cylindrical shell structures typically fail by local buckling. Here we use the analysis of Karam and Gibson [1,2] for the buckling of a cylindrical shell with a compliant core to estimate the mechanical efficiency of the natural structures described above; the details of the analysis are given in Appendix A. The analysis indicates that under uniaxial load the shell buckles axisymmetrically; under a bending moment the shell buckles when the compressive stress reaches the uniaxial value required to produce axisymmetric buckling. The core acts as an elastic foundation: the stresses in the core are maximum at the shellcore interface and decay radially inward. One measure of the mechanical efficiency of natural structures is the ratio of the failure load of the shell with the compliant core to that of an equivalent hollow shell with no core of equal radius and mass. The ratios of the axial failure load, Pcr/(Po)eq, and local buckling moment, MLb./(MLb.)eq,of the quills and spines to those of equivalent hollow cylinders are shown in Figs. 8(a) and 8(b), respectively. Under axial load, only the hedgehog spines, with the square honeycomblike cores which have a stiffness proportional to their density, lie above the line; their buckling resistance is roughly 50% higher than that of an equivalent hollow cylinder. In bending most of the quills and spines have an equal or better buckling resistance than the equivalent cylinder. The best improvements, over 300%, are achieved by Er/naceus and Hemiechinus, again because of their square honeycomb cores. The Hystrix family yields modest increases in buckling resistance, ranging up to about 40%. Tachyglossus Aculeatus, Erethizon, and Coendou show a decrease in buckling resistance by as much as 30% (Table 5). Note that the model used in the analysis to estimate the stiffness of the foam core for the porcupine quills of Type 1 microstructure (Fig. 7(a)) neglected the wall stretching effect and may have been overconservative. Microscopic observation showed the cell faces of Erethizon's foam core to contain a substantial fraction of the total solid material. A more sophisticated microstructural investigation would have been required in order to correctly account for this stiffening effect. Prior to local buckling, hollow cylindrical shells ovalize, decreasing the moment of inertia and thus their local buckling resistance. The compliant cores in all of the animal quills and spines investigated succeed in virtually
(spiny rat)
NA, not applicable.
(tenrec)
Setifer
4.5
12.75
Hemiechinus spinosus
(hedgehog)
12.5
13.7
10.0
9.5
17.6
2.20
2.3
Erinaceus Europaeus
(hedgehog)
Erinaceus Europaeus
porcupine
Hystrix subcristatus
(porcupine)
Hystrix Indica Cristata
(porcupine)
Hystrix Galeata
specimen 2 (echidna)
Tachyglossus Aculeatus
specimen 1 (echidna)
Tachyglossus Aculeatus
(North American porcupine)
Erethizon
18.0
14.0
Coendou prehensilis
0.1
0.10
0.10
0.10
0.18
0.18
0.137
0.0
0.11
0.125
0.2
( )
( )
(Brazilian porcupine)
Core to shell density ratio pJp
Radius to thickness ratio a/t
Animal genus/specie (common name)
4
5.25
7.1
6.8
9.5
9
17.1
0.0
1.8
17.5
13.5
(  )
Core depth to thickness ratio c/t
Table 3 Dimensionless material and geometrical properties of animal quills and spines
NA
6.75
6.75
6.86
6.77
7.24
8.40
4.08
4.15
10.50
8.25
(  )
for 95% stress decay (c/t)o
c/t ratio
NA
1.36
1.36
1.37
1.35
1.45
1.68
0.0
0.83
2.10
1.65
(  )
Buckling wavelength parameter A~Jt
0.10
0.10
0.10
0.10
0.081
0.052
0.047
0.0
0.012
0.0156
0.04
(  )
Radial core to shell stiffness ratio (EJE)z
(E¢/E)~
0.0
0.05
0.05
0.05
0.081
0.052
0.047
0.0
0.012
0.0156
0.04
()
ratio
Longitudinal core to shell stiffness
~.
3
~ ~
~. ~' ~.
"~
~
"~
.~
(Oat)
(wildberry type)
Latuca Biennis Phytolacca Americana
(buffel grass)
Cenchrus Ciliaris
specimen 2
Artemisia
specimen 1
Artemisia
Group II
(rye)
Secale
(barley)
Hordeum vulgare
specimen 2 (grass)
E~trigia repens
specimen 1 (grass)
Elytrigia repens
(sedge grass)
Eleocharis
0.10 0.032
18.4
0.25
0.20
0.22
0.094
0.17
0.22
0.22
0.22
0.10
Core to shell relative density p~/p ( )
58.6
5.4
2.84
3.7
48
28.7
22.9
26.4
21.4
54
Group I
Arena
Radius to thickness ratio a/t ( )
Plant name (description)
Table 4 Dimensionless material and geometrical properties of plant stems
12.2
12.1
4.9
2.3
3.2
11.7
9
8
7.9
6.35
12
Core depth to thickness ratio c/t ( )
11.7
14.5
5.84
4.53
5.06
14
10.02
8.70
8.78
8.645
14.01
for 95% stress decay (c/t)o ( )
c/t ratio
2.347
2.892
1.169
0.908
1.015
2.80
2.007
1.743
1.76
1.729
2.85
Buckling wave length p a r a m e t e r )t~,/t ( )
0.001
0.01
0.0625
0.04
0.05
0.009
0.03
0.05
0.05
0.05
0.01
Radial core to shell stiffness ratio (EJE)~ ( )
0.001
0.01
0.0625
0.04
0.05
0.009
0.03
0.05
0.05
0.05
0.01
()
Longitudinal core to shell stiffness ratio (E¢/E)~
~a
~. ~ ~'
ga.
"~ ~

"~
to
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132
$
(a) Tvve 1 porcupine quill aq~l plant stem microstructure:
2
'
'
'
'
I
'
'
'
123
'
I
'
'
'
'
I
. . . .
Hemiechinu~D •
(Ec/E)x = (Ec/E)y  (Ec/E)z = (pc/p)
%
•
1.5 A w •
1 x
0.5
Edn~e~
• H. Subcristatus
Tachyglossus Aculeatus
H. G a l e a t a •
• H. I n d i c a x C r i s t a t a
C~Wndou
,
,
• Erethizor
Y ,
0
,
,
,
I
(0)
(Ec/E)x = (Ec/E)z =
,
,
,
I
5
fb) TvDe 2 porcupine quill microstruetutg;
,
,
,
I
10
,
,
,
,
15
20
Radius to thickness ratio, a/t ()
2
(pc /p)ribs +[1(pc /p)ribs](9C /9) foara (Ec ]E)y
given by eqn. 4
4
'
'
'
'
f
'
'
'
'
d
I
'
'
3
(g) Tvoe 3 hedgeho~, spine microstructure:
•~ ~
'
I
.
.
.
.
•
•
Erinaceus
2 FI. S u b c f i s t a t u s •
2(Ec/E)x =2(Ec/E)y = (Ec/E)z = (pc/p) Tachyglossus 0
~
'
Hemiechinus •
"Aculeatus , , ,
0 (b)
H. l n d i c a x C r i s t a t a ,
I
,
,
,
,
I
,
H. G a l e a t a
• Coendou ,
,
,
I
5 10 15 Radius to thickness ratio, a/t ()
• Erethizon ,
,
,
,
20
z
Y
~ i ~
(d) Type 4 tenrec spine microstructure: (Ec/E)y = (Ec/E)z = (pc/p) (Ec/E)x = 0
Fig. 7. Microstructural core types and stiffness models:(a) T y p e 1 quills a n d plant stems; (b)Type 2, quills; (c) T y p e 3, spines; (d) T y p e 4, spines.
eliminating ovalization prior to local buckling (Table
5). The improvements in local buckling resistance under axial load or bending moment are mixed, with most species achieving some improvement in local buckling moment resistance (Fig. 8(b)). The function of the quill or spine may dictate the level of performance. The short spines of the hedgehog (Erinaceus) and the spiny rat (Hemieehlnus) are required to act as shock absorbers as much as armour and protection to discourage predators, hence the high structural efficiency requirement and the need to delay local buckling until the internal stresses have almost reached material failure limits [10]. The longer quills of the porcupines may only be needed to act as a deterrent to predators with less of a mechanical shock absorbing function [10]. The efficiency ranking of the species surveyed varies little from one measure to another in Figs. 8(a) and 8(b) and may be related to the geographic origins of each specie. Both Coendou and Erethizon are New World porcupines, Hystrix is the family of Old World porcupines, Tachyglossus Aculeatus is an echidna native to Australia,
Fig. 8. R a t i o of the failure loads of animal quills a n d spines to those of the no core cylinder with equal radius and mass: (a) axial buckling load; (b) local buckling m o m e n t .
Tasmania and New Guinea. The tenrec (Setifer) is found in Madagascar and the hedgehog (Erinaceus Europaeus) is a European insectivorous mammal. The results for the ratios of axial buckling load and local buckling moment for the plant stems relative to equivalent hollow shells are summarized in Fig. 9. Plant stems have their axial buckling resistance reduced between 10% and 40% (Fig. 9(a)). At a/t above about 20, the local buckling moment resistance increases by 2050% over that of the equivalent hollow shell (Fig. 9(b)). Our initial estimates for the buckling resistance of the plant stems are limited in that they neglect the effect of turgor pressure and they assume that the shell and core have identical cell wall properties. We next investigate these two limitations further. The mechanical performance of stems with mainly pithy parenchymatous cores was estimated assuming that the shell and core had identical cell wall properties and that the effect of turgot pressure on the apparent Young modulus of the core could be neglected. Parenchyma cells are living cells with a protoplast. They fulfill many physiological functions including water storage and act as stacked pressurized containers in the core. The pressurized protoplast stiffens the cell walls against bending and acts as an elastic foundation [11]. The impact of this water pressure (or turgor pressure) on the effective stiffness properties of plant stems and other tissues has been investigated by Steudle et al. [19], Niklas and O'Rourke [20] and Niklas [21]. In the
124
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132
Table 5 Failure load ratios for animal quills and spines Animal genus/specie (common name) Coendou prehensilis (Brazilian porcupine) Erethizon (North American porcupine) Tachyglossus Aculeatus specimen 1 (echidna) Tachyglossus Aculeatus specimen 2 (echidna)
Buckling stress ratio
Axial buckling load ratio
Local buckling moment ratio
o'er/Oo)~q
Pcr/(Po)eq
MJb/(MIb)cq
( )
( )
( )
Ovalisation at local buckling ~lb ( )
0.91
0.51
0.77
0.0074
0.74
0.40
0.73
0.0054
0.91
0.82
0.88
0.134
1.00
1.00
1.00
0.145
1.14
0.73
1.27
0.0055
0.87
0.55
0.96
0.0134
1.14
0.85
1.44
0.0125
2.05
1.70
3.25
0.0107
1.90
1.58
2.98
0.0109
2.06
1.71
3.36
0.0129
Hystrix Galeata (porcupine)
Hystrix IndicaCristata (porcupine)
Hystrix Subcristatus (porcupine) Erinaceus Europaeus (hedgehog) Erinaceus Europaeus (hedgehog) Hemiechinus spinosus (spiny rat)
case of species occupying wet habitats (hydrophytes) mechanical support is mainly provided by hydrostatic tissues [11]. Experimental investigations have shown the turgor pressure to increase the effective stiffness of chive (Alliurn schoenoprasnum ) leaves by twofold [20] and that of parenchyma plugs from potato tubers by four to fivefold [11]; similar results had been reported earlier on giant single cells [19]. Nilsson et al. [22] derived a formula that predicts the apparent elastic modulus of parenchyma for any turgor pressure, by treating the cells as spheres and assuming the solid cell wall contribution to the apparent modulus of the parenchyma tissue arises from compression or tension inside the walls. Gibson and Ashby [3] proposed that Young's modulus of a gas filled, closed cell synthetic foam could be estimated by simply adding the contributions of bending of the cell edges, stretching of the
cell faces and compression of the gas according to the ideal gas law. They did not account for the pretensioning effect of the gas pressure on the stiffness of the cell walls. Both models show the increase in the relative stiffness ratio, EJE, to be proportional to po/E, where Po is the internal cell pressure. The simple staggered cubic model of Gibson and Ashby [3] is extended to account for the pretensioning effect of internal pressure in Appendix B. The pressurization introduces tensile axial forces in the bending elements which apply a restoring moment that counters the bending imposed by external loads (Fig. 10(c)). Eq. (B7) shows the increase in modulus to be dependent on the pressure to cell wall stiffness ratio po/E and the relative density of the cellular material, pc/P. The ratio of the apparent modulus relative to that of the same unpressurized cellular material is plotted in Fig. 11 and
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132 2
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Fig. 9. Ratio of the failure loads of plant stems to those of the n o core cylinder with equal radius and mass: (a) axial buckling load; (b) local buckling m o m e n t .
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Fig. 12. Increase in a p p a r e n t relative m o d u l u s of elasticity over u n p r e s s u r i z e d cellular material (fully restrained bending element).
(a) P
(b) Q
p
Q M
y '1
p
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Fig. 12 for two limiting cases. At equal turgor pressures the lighter foams show the largest increases in apparent modulus, up to about a factor of five at po/E = 0.03. The reported moduli for sclerenchyma cell walls vary between 10 and 35 GPa (35 GPa being that of wood cell walls) while those of parenchyma cell walls fall in the range of 0.14 GPa [1113,15,21]. The moduli of the vascular bundles and collenchyma have intermediate values [11,12,14]. The outer shell is made of sclerenchyma, vascular bundles and collenchyma, while the
core is made of parenchyma and some vascular bundles and collenchyma. From the data collected the ratio of the average cell wall modulus in the shell to the average cell wall modulus in the core can be estimated to be roughly 10. Hence the estimates of the core relative stiffness obtained from Eq. (1) should be reduced by that ratio. On the other hand, the turgor pressure is reported to vary between 0.1 and 2 MPa [11,13,14,19,20,21] giving po/E ranging from 10 3 to more than 10 2. This internal pressurization increases the apparent modulus of the core by a factor of between two and five (Fig. 11) depending on the cells' density and the cell wall modulus. Hence our assumptions of identical cell wall properties for all plant tissue types and neglect of the effect of turgor pressure result in compensating errors which allow an acceptable estimate of the core to shell relative stiffness to be obtained from Eq. (1). A second measure of the mechanical efficiency of the natural structures is to compare the measured core depth with the core thickness at which the core stresses decay to a negligible value (which we take to be 5% of their maximum value). We expect the core to be only as thick as necessary to resist the internal core stresses in an efficient structure. In Fig. 13, the measured core depth to thickness ratio, c/t, is plotted vs. the predicted core depth to thickness ratio at which the
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132
126
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 Quills and Spines,
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core stresses decay to 5% of their maximum value, (c/ t)o , for the quills, spines and plant stems. The data for the plant stems falls consistently along, but slightly below, the c/t=(c/t)o line. It is fitted with a high correlation, r 2 = 0.92, by a line of a slope of 0.95. The data for the quills and spines does not long the c/t = (c/ t)o line. However, it does show a very clear trend of its own, best fitted by a line of slope 2.9, with a correlation factor of r 2 = 0.88. The growth or development of the reinforcing foam cores of the plant stems seems driven by t h e stresses generated in the core at impending local buckling of the shell. The decay of stresses away from the shell reduces the amount of material needed to resist them causing a continuous foam density reduction (Fig. 6(b)); at c/t close to (c/t)o, the foam core can be eliminated: as the stresses tend to zero, no more material is needed. The spines of Erinaceus and Hemiechinus are the only ones in the animal group that have a core only partially spanning the inside of the shell. They do however contain thin widely spaced diaphragms. The data points corresponding to their cores fall at the intersection of the c/t = (c/t)o line and the line describing the trend of the quill and spine data. Though fully filled, the quills of the porcupines (Hystrix, Erethizon, Coendou) all show a marked decrease in the core density towards the center (Figs. l(a), 2(a) and 3(a)) support ress controlled growth hypothesis. The purely theoretical measure of the stresses in the core at local buckling, (c/t)o, derived in the stress decay analysis, is shown to be one, but possibly not the only, major variable controlling the growth of foam cores in quills and spines. Physiological and physical conditions, not investigated in this analysis may well explain the need for fully filled cores or cores with diaphragms. Finally, Fig. 14 compiles the calculated ovalisation at local buckling under bending, ¢'lb, for the quills, spines and stems. It is plotted vs. a/t showing the dramatic decrease in ovalisation from 0.145 for a hollow tube to less than 0.01. This decrease is more important at higher a/t ratios. For a porcupine or a hedgehog the quills and spines may represent a significant fraction of their total weight
4
O
o
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O 00
10
20
.....
30
,...
40
50
60
Radius to thickness ratio, a/t () Fig. 14. O v a l i z a t i o n at l o c a l b u c k l i n g vs. r a d i u s to t h i c k n e s s ratio,
a/t, f o r quills, s p i n e s a n d stems.
and perform a function that is critical for survival. The spines' density was estimated from the hide of one of the sacrificed hedgehogs that furnished the spine specimens, to be as high as 114 spines cm 2. The average spine weight was measured to be around 5.4 mg. Treating the hedgehog as a hemisphere covered with spines, the weight of the spines was estimated to be of the order of 10% of the total weight of the animal. This is a relatively important fraction of the hedgehog weight and it can explain the high mechanical efficiency of its spines. The stem of a plant is also a very important biomass investment, providing structural support and carrying lifelines from the roots to the leaves and reproductive organs. In addition, it may also fulfill other physiological functions such as water and nutrient storage. In a competition for sunlight and exposure, the plant able to minimize its structural weight for the required strength, will maximize its growth rate, attaining greater heights in shorter times, and hence securing its evolutionary niche and insuring its perpetuation.
5. Engineering design and biomimetics The mechanical design problem for natural tubular structures is that they minimize their mass for a given required stiffness and strength. This survey of animal quills and plant stems suggests that at least part of the function of the compliant cores is to increase their local buckling resistance. Here we discuss the observed microstructures with respect to the broader mechanical design problem. In practice, natural organisms also have to satisfy many other requirements imposed by growth and physiological function; we have not attempted to analyze these here. First consider the animal quills and spines. These are modified hairs whose original dimensions, utility and density may have been determined by a minimum heat transfer condition [23]. Structurally they act as a beamcolumn with an eccentrically applied compressive force at the free end [10]. Fig. 15 presents a set of
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132 (a) Steo 1: Global bendin~ stiffness
I,
'O
(b~ Step 2: Brazier ovalization resistance
O® (c) Steo 3: Local bucklin,, resistance
®0 (d) Steo 4: Local buckling resistance improvement
e,
e
(e) Steo 5: Removin~ unstressed material in middle core
•
o
Fig. 15. (a) to (e) E v o l u t i o n a r y d e s i g n p r o c e s s in a n i m a l quills a n d spines.
sketches illustrating the development of increasingly mechanically efficient beamcolumn structures. Starting with a solid hair, or a rod, the first step to improve material efficiency is to go to a thin walled hollow cylinder (Fig. 15(a)). For a given bending stiffness, the mass of the cylinder can be decreased by increasing the ratio of the radius to the wall thickness, a/t. This, however, simultaneously decreases the resistance of the cross section to Brazier ovalization and local buckling. Ovalization can be counteracted by the introduction of a compliant core, providing support against local buckling for the outer shell (Fig. 15(b)). At this stage local buckling of the shell becomes the next problem to solve in order to meet the strength requirement. The problem of local buckling under bending is in fact that of a thin sheet on an elastic foundation under axial compressive stresses. One way to improve the resistance to local buckling is to provide for immediate support of the shell in the most efficient way, which is to align some core material radially. This radial reinforcement can be provided in the form of longitudinal, circumferential or orthogonal stiffeners (Fig. 15(c)). If these stiffeners are massively built they can also resist Brazier ovalization and the isotropic foam core can be dispensed with leaving a honeycomblike core (Fig. 15(d)). Material in the central part of the core, where the stresses have decayed to a negligible value, can be removed (Fig. 15(e)). All of these structures are manifested in the quill and spine microstructures surveyed in this study. The design of plant stems
127
involves additional physiological factors, described by Niklas [11]. In engineering design tubular cylindrical sections represented by Fig. 15(a) are commonly available. Stiffened tubes (Fig. 15(e)) are currently used only in the largest of engineering cylindrical shells, e.g. offshore oil platforms. The recent development of the GASAR process which allows the production of cylindrical metal shells with an integral honeycomblike or foamlike core now allows mimicking of natural tubular structures in smaller scale engineering applications. 6. Conclusions The dense, stiff outer cylindrical shell of many animal quills and plant stems is supported by an inner foamlike or honeycomblike compliant core. Here, we have used measurements of the microstructure of a wide range of animal quills and plant stems to estimate the role of the compliant core in resisting local buckling. The results suggest that at least part of the function of the compliant core is to increase the local buckling resistance of the shell relative to that of a hollow cylindrical shell. The various structures are compared in terms of increasing mechanical efficiency: the most efficient core is the honeycomblike stiffeners of the hedgehog spine (Erinaceus europaeus). The development of new processes for making cylindrical shells with integral compliant cellular cores allows biomimicking of these natural structures in engineering design. Acknowledgements We are grateful to Mr. Michael Lynch of the Franklin Park Zoo, Boston, MA and to Mr. Tomasz Owerkowicz, Ms. M Rutzmaser and Ms.T. McFadden of the Museum of Comparative Zoology, Harvard University, Cambridge, MA for generously supplying hedgehog spines and porcupine quills. We also thank Professor Peter Stevens of the Harvard University Herbarium, who assisted with identification of the plant specimens. Financial support for this project was provided by the National Science Foundation (Grants No.MSS9202202 and EID9023692). Appendix A: Models for the mechanical efficiency of natural structures To model the resistance of natural cylindrical structures with foam cores to local buckling under axial compression and bending Karam and Gibson [1] treated a longitudinal strip of the shell as a beam on dimensional elastic foundation with isotropic properties in the xz plane and of infinite depth. The core resists the move
G.N. Karam, L.Z Gibson / Materials Science and Engineering C2 (1994) 113132
128
ments of the shell by extensioncompression in the z direction and by shearing in the xz plane. The critical uniaxial compression buckling load of the shell, corresponding to an axisymmetric buckling mode, was found to be:
or:,= v~(1  v2)~ji
(A1)
where O'o, the buckling stress of an empty cylindrical shell under an axial load, is given by:
ET fro= ax/3(1 /./2)
(A2)
tube; as in the case of the hollow shell, local buckling always precedes the Brazier moment. The Type 1, 2, and 3 microstructures were analyzed by modifying the results of Karam and Gibson [1] to account for the transversely isotropic properties of the cores and their measured depth as follows. (a) In the elastic foundation calculations of the critical buckling parameter,Acr/t, and the function fl (Eqs. A3A5) the radial stiffness modulus ratio,(EdE)z, was used instead of EdE, and the spring constant of the foundation, kc, was modified to account for the partial depth of the foundation as follows [2].
and sinh
1 alt (Acr/t) 2 f~ = 1 2 ( 1  v2) i (ACr/t) + a'"~ keA, + if (&rlt)(alt)
ke=
cosh
 ~
E c
A (3vc)(l+vc)sinh 2
+(l+vc)
.(c). ~
+4
(A3) (A6)
where ko, the foundation spring constant, is given by: k= =
2EdA (3.
vc)(1 + vc)
(A4)
and. A = Ac,, the half buckling wavelength divided by ~',is found from the solution to:  t3 tA k~A 12(1_v2)A3 + ~ + ~  = 0
(AS)
E, Ec and v, vc are Young's modulus and Poisson ratios of the shell and core materials, respectively. They also analyzed the decay of normal and shear stresses due to axisymmetric buckling with radial distance into the core. At a depth of (c/t)o = 5G Jr, or 1.6 half buckling wavelengths, the stresses decay to about 5% of their maximum values, suggesting that the innermost core resists little load and can be removed without significant reduction in the local buckling resistance. The case of a cylindrical shell with a foam core subject to bending was then analyzed. Hollow cylindrical shells ovalize under a bending moment: as the curvature increases, the crosssection gradually changes from circular to oval, decreasing the moment of inertia of the crosssection and consequently its bending rigidity. The moment resisting capacity of the tube reaches a maximum at the Brazier moment. True local buckling occurs when the maximum compressive stress in the wall of the tube reaches the uniaxial compressionbuckling stress. In hollow cylinders local buckling occurs in the ovalized section at slightly lower moments than the Brazier moment. In cylindrical tubes with a compliant core, the core resists the ovalization of the crosssection and acts as an elastic foundation supporting the shell. Karam and Gibson [1] calculated the Brazier moment and the local buckling moment for a foam filled shell by minimizing the strain energy of the uniformly bent
(b) In the terms correcting for the axial load carried by the core and for the increase in the moment of inertia due to the core the longitudinal stiffness modulus ratio,(EdE)x, was used. (c) The radial stiffness modulus ratio, (EJE)z, was used in the calculations of the ovalisation of the core and the Poisson effect. The effect of the lower circumferential stiffness modulus ratio, (EdE)y,in the case of Type 2 and 3 core microstructures (Figs. 7(b) and (c)) was neglected. The error resulting from this simplification affects only the degree of ovalisation which is negligible for all moduli ratios considered. Applying all the modifications described above to the results of Karam and Gibson [1], the following equations are obtained: crc,
43(1  v 2)
[1+ ~ ~(2
(tro)oq
clt]]f'
Pc, (eo):q
(A7)
altlJ

[ 1+
~
c,, r
2  alti]
"
8 t
3
1+ o , (aft' ( a (M,.~.):q
1 + ~
X
l+
1
1  a/t] ]
1
l
<'b
l
l i t 1i 77P'
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132
/0.31211+~(2
c/t]][ 2 a/t]
(A9)
where the ovalisation at the local buckling moment, ~:,b is obtained from:
W~ 1  ~ ] \ 1  3~:~b] [1+ ~t(1(1
 c/t]*]]'/]2~~] J
~
,
\311/2
fl
(AIO)
1+ g ( 1  ~ , ) ~ ~fl was calculated from Eqs. A3 and A6; Acr/tfrom Eqs. A5 and A6 (also given in Tables 3 and 4); and the parameters a, a', fl, and /3' defined as follows:
129
prevented from rotation at the cell intersections, a simply supported single beam element loaded at its midspan by a force Q and subjected to an axial tensile force P can be used to deduce the apparent modulus of the pressurized cellular material (Fig. 10(c)). The force Q is due to an applied uniform stress and P is the axial pretension introduced by the internal pressurization. Following Timoshenko and Gere [24] the governing equation of the beam is: dZy
P
dx 2
Ei y=
Q (B1)
2EI x
where E is the modulus of elasticity of the beam and I its moment of inertia. The general solution to this differential equation is of the form: y =A sinh(/cc) +B cosh(kx) + ~Qx
(B2)
with kz=P/EI. The constants A and B, evaluated for the left hand side of the beam from the boundary conditions of y ( x = 0 ) = 0 and dy/dx(x=l/2)=O are: A=
O/t~ E  x
a/t] ]
Q
2kP cosh(kl/2) and B = 0
The maximum deflection, 6 is at midspan and is given by:
[3'=0"762(1( l  c/t~4~a/]t]
6=Yxm 2kP~ 2
These failure load ratios are presented in Tables 5 and 6, and Figs. 8 and 9. The analysis is not applicable to type 4 microstructures (Setifer spines) where the foundation cannot provide any shearing resistance due to its longitudinal discontinuity in the x direction. The Setifer spine specimen was therefore dropped from the mechanical efficiency analysis.
Let cra and Po be the applied stress and the internal fluid pressure (greater than atmospheric pressure) in the cellular material, respectively, then Q and P are proportional to:
Appendix B: Elastic modulus of internally pressurized cellular material To model the effect of turgor pressure in stiffening the living tissues of plants the staggered cubic representation of Gibson and Ashby [3] is used. The cellular parenchyma (or any other pressurized cell type) is represented by a series of staggered closed cubic ceils of length 1,with most of their solid material concentrated at the cell edges in the from of small square section span l, and depth t (Fig. 10(a)). Following Gibson and Ashby [3] the relative density of the foam, pJp, can be estimated as (t/l)2. Under an applied vertical stress, the major deformation mechanism is bending of the horizontal beam elements loaded at their midspan by the vertical struts (Fig. 10(b)). If the edges are not
Q c~ tYa12 and
P~pol 2
(B4)
The apparent strain of the pressurized cellular material will be given by ea tx 6/1 and the apparent modulus of elasticity obtained from Ea = tra/e.. Using these relationships and combining B4 with B3 we obtain:
Po
Ea=C
E t 2 3pol 2 0.25 __14__~p0(i) t a n h ( ~ / @ ( t ) )
(B5)
where C is a constant that represents the exact geometric relationships between the loads and the displacements. Noting that (pJp)~ (t/l)2, Eq. B5 can be rewritten: po E._~. = C
E 0.25 
(B6)
tanh
For the case of the unpressurized cellular solid, Gibson and Ashby [3] found that EJE = (pjp)2 in Eq.
G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132
130
Table 6 Failure load ratios for plant stems Plant name (description)
Buckling stress ratio
O'er](O'O)eq
Axial buckling load ratio ecr/(Po)eq
Local buckling moment ratio Mib/(M,b)eq
Ovalisation at local buckling
1.31
0.70
1.36
0.0024
1.396
0.81
1.54
0.0084
1.48
0.80
1.52
0.0068
1.32
0.72
1.35
0.007
1.28
0.69
1.31
0.005
1.19
0.66
1.28
0.0026
0.84
0.66
0.96
0.0667
0.86
0.71
0.91
0.0989
0.83
0.58
0.96
0.0334
1.39
0.74
1.44
0.0024
0.83
0.66
1.15
0.0297
Group I
Arena (oat) Eleocharis (sedge grass) Elytrigia repens specimen 1 (grass) Elytrigia repens specimen 2 (grass) Hordeum vulgare (barley) Secale
(rye) Group II
Artemisia specimen 1 Artemisia specimen 2 Cenchrus Ciliaris (buffel grass) Latuca Biennis Phytolacca Americana (wildberry type)
B6 the apparent modulus ratio, Ea/E, should then converge to. (pjp)z as the turgor pressure decreases or po/E tends to zero. The constant C can be determined from this last condition, it is found to be equal to 1/ 4. Hence Eq. B6 can be simply stated as:
E 1.0
Pc tanh
V3po\P]
dZy
P
dx 2
~Y= E1
M
Q
2EI x
(B8)
with a general solution in the form of:
Po E___~=
simple bending element model (Fig. 10(d)). The governing equation of the beam becomes:
(B7) P
\ V E \Pc/}
The gain in the apparent modulus of the cellular solid due to the internal pressurization can be calculated from the ratio of Ea/E to (pjp)Z. Fig. 11 plots this ratio for different values of po/E at different core relative densities. If the cell wall edges are assumed to be totally restrained from rotations at the cell wall intersections, then a fixed end conditions should be applied to the
y = A sinh(kx) + B cosh(kx) + Q
M e
(B9)
where k is the same as before. The boundary conditions of zero slope and deflection at x = 0 and zero slope at midspan allow us to calculate the constants: Q A= ;B= 2kfl
M P
and
Q (cosh(kl/2)  1 M= 2k~ sinh(kl/2) I
G.N. Karam, L.Z Gibson / Materials Science and Engineering C2 (1994) 113132
The maximum deflection at midspan is found as:
Q_Q{kl 2(1cosh(kl/2))] 6=Yx=1/2= 2kP\ 2 + sinh(kl/2) ]
(BIO)
Following the same dimensional arguments as before for estimating Q and P we find:
Po
a t
Po Ea
E
~ = C
[~]~n[V\2\
1  cosh
(Bll)
I~~['t) )
0.25 + 2
3po 1
. smh
3po 1
by imposing the condition that the apparent modulus ratio, EJE, should converge to (pJp) 2 as the turgor pressure decreases or pJE tends to zero, C is found to be equal to 1/16. By noting that (t/l)Z~ (OJO)2 an equation similar to B7 can be derived for the gain in apparent modulus due to internal pressurization if the bending elements are assumed to be completely restrained at the edges. Fig. 12 plots these results similarily to Fig. 11. The gains calculated in Figs. 11 and 12 are the upper and lower bounds depending on the stiffness of the edge restraint of the bending elements. The modulus of parenchymatous tissues has been measured under different turgor pressures but no analysis was performed for lack of applicable mechanical models [13,19,20]. These preliminary results provide a model for the analysis of turgidity in plant stems and organs.
Notation
A,B,C constants a c
E Ea Ec I k k~ 1 M Mlb
P
Vo P.
radius of cylindrical shell to midplane of thickness core thickness Young's modulus of shell apparent modulus of pressurized foam Young's modulus of core moment of inertia of beam element in pressurized foam model
(P/EI)°'5 spring constant for compliant core beam span in pressurized foam model end moment in beam element of pressurized foam model local buckling moment axial pretension force in beam element of pressurized foam model axial compressive buckling load of hollow shell axial compressive buckling load of shell with compliant core
~a
A ~cr 19
v~ P p~ 0%
O'er
131
internal fluid pressure inside foam core (turgor pressure) force applied at midspan of beam element in pressurized foam model thickness of shell (Appendix A), thickness of cell wall (Appendix B) deflection of beam element in foam model (Appendix B) apparent strain in pressurized foam model degree of ovalization at local buckling buckling wavelength parameter value of A at buckling Poisson's ratio of shell Poisson's ratio of core density of the shell density of the core applied stress on pressurized foam theoretical buckling stress in uniaxial compression of hollow shell axisymmetric buckling stress of shell with compliant core under uniaxial compression
References [1] G.N. Karam and L.J. Gibson, Elastic buckling of cylindrical shells with elastic cores, I: Analysis, submitted to Int. J. Solids Structures, 1994. [2] G.N. Karam and L.J. Gibson, Elastic buckling of cylindrical shells with elastic cores, II: Experiments, submitted to Int. Z Solids Structures, 1994. [3] L.J. Gibson and M.F. Ashby, Cellular Solids Structure and Properties, Pergamon, 1988, New York. [4] Peter Stevens, personal communication, Herbarium, Harvard University, Cambridge, MA, 1993. [5] K. Esau, Anatomy o f Seed Plants. 2nd edn., Wiley, New York, 1977. [6] G.J. Dunn and K.G. Briggs, Variation in culm anatomy among barley cultivars differing in lodging resistance, Can. J. Bot., 67 (1989) 18381843. [7] O. Ue.no, M. Samejima and T. Koyama, Distribution and evolution of CA syndrome in Eleocharis, a sedge group inhaiting wet and aquatic environments, based on culm anatomy and carbon isotope ratios, Anr~ Bot., 64 (1989) 425438. [8] J.R. Wilson, K.L. Anderson and J.B. Hacker, Dry matter digestibility in vitro of leaf and stem of buffel grass (Cenchrus ciliaris ) and related species and its relation to plant morphology and anatomy, Aust. J. Agr. Res., 40 (1989) 281291. [9] E.E. Underwood, Quantitative Stereology, AddisonWesley, Reading, MA, 1970. [10] J.F.V. Vincent and P. Owers, Mechanical design of hedgehog spines and porcupine quills, J. Zool. Lond., A210 (1986) 5575. [11] K.J. Niklas, Plant Biomechanics, an Engineering Approach to Plant Form and Function, University of Chicago, Chicago, IL, 1992. [12] J.F.V. Vincent, The mechanical design of grass, J. Mater. Sci., 17 (1982) 856860. [13] K.J. Niklas, Dependency of the tensile modulus on transverse dimensions, water potential, and cell number of pith parenchyma, Am. J. Bot., 75(9) (1988) 12861292.
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G.N. Karam, L.J. Gibson / Materials Science and Engineering C2 (1994) 113132
[14] K.J. Niklas, Biomechanics of Psilotum Nudum and some early paleozoic vascular sporophytes,Am. J. Bot., 77(5) (1990) 590606. [15] L.J. Gibson, M.F. Ashby and K.E. Easterling, Structure and mechanics of the Iris leaf, Z Mater. Sci., 23 (1988) 30413048. [16] W.K. Silk, L.L. Wang and R.E. Cleland, Mechanical properties of the rice panicle, Plant Physiol., 70 (1982) 460464. [17] A. Kokubo, S. Kuraishi and N. Sakurai, Culm strength of Barley, Plant Physiol., 91 (1989) 876882. [18] S.L. Gunderson, K.E. Gunnison and J.W. Sawvel, Hierarchical structure of a natural composite:insect cuticle, Mater. Res. Soc. Syrup. Proc., Vol. 255, 1992, pp. 159169. [19] E. Steudle, U. Zimmerman and U. Luttge, Effect of turgor pressure and cell size on the wall elasticity of plant cells, Plant Physiol., 59 (1977) 285289.
[20] K.J. Niklas and T.D. O'Rourke, Flexural rigidity of chive and its response to water potential, Am. J. Bot., 74(7) (1987) 10341044. [21]. K.J. Niklas, Mechanical behavior of plant tissues as inferred from the theory of pressurized cellular solids, Am. J. Bot., 766) (1989) 929937. [22] S.B. Nilsson, S.H. Hertz and S. Falk, On the relation between turgor pressure and tissue rigidity 2. Theoretical calculations on model systems, Physiol. Plant., 11 (1958) 818837. [23] A. Bejan, Surfaces covered with hair: Optimal strand diameter and optimal porosity for minimum heat transfer, Biomimetics, 1(1) (1992) 2540. [24] S.P. Timoshenko and J.M. Gere, Theory o f Elastic Stability, 2nd edn., McGrawHill, 1961.