1  

The  Bending  and  Failure  of  Sandwich  Structures  with  

Auxetic  Gradient  Cellular  Cores    

Y. Hou1,2, Y. P. Tai3, C. Lira2,4, F. Scarpa2*, J. R. Yates5, B. Gu1* 1 College of Textiles, Key Laboratory of high-performance fibers & products, Ministry of

Education, Donghua University, Shanghai, China, 201620

2 Advanced Composites Centre for Innovation and Science, University of Bristol, BS8 1TR

Bristol, UK

3 Department of Mechanical Engineering, University of Sheffield, Sheffield S1 3JD, UK

4 National Composites Centre, Feynman Way Central, Bristol & Bath Science Park, Bristol

BS16 7FS

5 MACE, University of Manchester, Manchester M60 1QD, UK

* Corresponding Authors

Abstract    

We describe the bending and failure behaviour of polymorphic honeycomb topologies

consisting of gradient variations of the horizontal rib length and cell internal across the

surface of the cellular structures. The novel cores were used to manufacture sandwich

beams subjected to 3-point bending tests. Full-scale nonlinear Finite Element models

were also developed to simulate the flexural and failure behaviour of the sandwich

structures. Good agreement was observed between the experimental and FE model

results. And the validated numerical model was then used to perform a parametric

analysis on the influence of the gradient core geometry over the mechanical

performance of the structures. It was found that the aspect ratio and the extent of

 

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gradient (i.e. the horizontal rib length growth rate or the internal angle increment) have

a significant influence on the flexural properties of the sandwich panels with angle

gradient cores.

Keywords

A. Sandwich; B. Auxetics; C. Finite Element Modelling (FEM); D. gradient honeycomb.

Nomenclature

A reduced area bds2/c l Core oblique rib length

b Sandwich width L Specimen total length

c Core thickness Pmax Maximum force prior to failure

d Sandwich thickness S Support span length

ds specimen depth between the two

neutral lines of the skins t Cell wall thickness

D Sandwich bending stiffness tf Skin thickness

D* Specific sandwich bending stiffness W Total weight of sandwich panel

E Sandwich bending modulus α Aspect ratio, h/l

E* Specific sandwich bending modulus α0 Aspect ratio of the first row

Fult Core shear ultimate strength β Aspect ratio of the oblique rib

thickness, t/l

Fult* Specific core shear ultimate strength θ Cell internal angle

Gyz Core shear modulus in yz plain △θ Increment of cell internal angle

Gyz* Specific core shear modulus in yz plain δ deflection at maximum load

Gs Shear modulus of the solid material the

core was made of σ Skin stress

h Core horizontal rib length ρc Core density

hg Growth rate of horizontal rib length ρs Density of the core solid

 

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material

1  Introduction  

 

Lightweight cellular honeycombs or foams have been widely used in automotive,

marine and aerospace applications due to superior indentation resistance, acoustic

attenuation, thermal insulation and energy absorption ability (Aktay et al., 2008; Gibson

and Ashby, 1999; Hanssen et al., 2006; Mills et al., 2003; Papka and Kyriakides, 1998;

Sun et al., 2010). The great diversity in terms of topology and material composition in

cellular solids provides the material and structural designer with a large range of

possible multifunctional properties, as well as unusual deformation characteristics. A

typical example of cellular materials with extreme material characteristics is represented

by auxetic (negative Poisson’s ratio) honeycombs and porous solids, manufactured for

the first time by Lakes (Lakes, 1987) in the form of thermoplastic open cell foam. Since

Lake’s seminal paper, negative Poisson’s ratio structures have been extensively

investigated due to their stiffening geometric effects (Evans and Alderson, 2000; Grima

et al., 2012a; Grima et al., 2012b; Grima et al., 2011; Yang et al., 2004), enhanced

in-plane indentation resistance (Evans and Alderson, 2000; Grima et al., 2012a; Scarpa

et al., 2007), shear modulus (Ju and Summers, 2011; Prawoto, 2012), impact absorption

(Ju and Summers, 2011; Scarpa et al., 2003a) and damage tolerance (Liu and Hu, 2010).

Functionally graded cellular structures are characterized by a gradual variation of cell

 

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size, shape and wall thickness over a prescribed volume. The gradient topology leads to

a continuous distribution of curvatures, stiffness and energy absorption capability. In

nature, gradient topologies in plant stems are widely present to provide resistance to

high bending loading. Some typical examples are the honeycomb-like vascular bundles

of Cocus nucifera, and the radial density of the cells distribution in Iriartea gigantea

providing higher bending stiffness compared to a uniform density section (Gibson et al.,

2010)Error! Reference source not found.. Ajdari et al (Ajdari et al., 2011) introduced

a density gradient cellular structure by a gradual change of the cell wall thickness, and

investigated its deformation mechanism under low and high crushing velocities. The

decrease of the honeycomb relative density along the crushing direction significantly

enhanced the energy absorption at early stage. Cui et al (Cui et al., 2009) have explored

in detail the energy absorption capability of functionally graded foam with different

density parameters. Lee et al (Lee et al., 2007) have used a finite element method

approach to understand the fracture behavior in functionally graded foams. Kirugulige

et al (Kirugulige et al., 2005) showed that the introduction of gradients in the core

would considerably reduce the stress intensification of the equivalent sandwich

structure under dynamic loading. While references (Ajdari et al., 2011; Cui et al., 2009;

Kirugulige et al., 2005; Lee et al., 2007) concern foam configurations, Lim (Lim, 2002)

proposed for the first time a gradient cellular structure made of centresymmetric regular

cells by varying the internal cell angle row by row. The gradient cellular core provided

the design of a functionally graded beam with Poisson-curving characteristics. Lira et al

 

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(Lira and Scarpa, 2010) provided a feasible solution to optimize the transverse shear

stiffness of the honeycomb through changing the horizontal rib thickness across the

whole panel. The use of a step-wise honeycomb core was introduced by Lira et al (Lira

et al., 2011) as a filler for aeroengine fan blades, showing a significant decrease in terms

of mass and mass modal displacement compared to existing baseline fan blade

configurations. More recently, Alderson et al (Alderson et al., 2012) have produced

gradient open-cell PU foams with complex geometry moulds to achieve gradient elastic

properties along the length of whole specimen. These gradient cellular solids are also

called polymorphic (Alderson et al., 2012), because they can assume varying shape

changes and Poisson’s ratio values along the principal manufacturing direction. The

gradual Poisson’s ratio change and variation of curvature is indeed one of the main

characteristics of the gradient cellular cores. The topologies shown in Fig. 1 (which will

be the subject of this work) are all characterized by a re-entrant (butterfly) shape. The

configuration with gradually changing aspect ratio will be called topology #1, while the

cellular layout with varying internal cell angle is denominated as topology #2. When

cellular panels made with these configurations are subjected to a distributed bending

moment along their edges, the resulting curvatures assume a complex shape. A FE

model has been developed to simulate this particular loading case. The curvature of

each surface segment along the horizontal rib direction is calculated from the

displacement of the nodes belonging to the horizontal ribs (aligned along the X-axis).

The distribution of the X-curves is displayed in Fig. 2. Bent plates made with positive

 

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Poisson’s ratio materials assume a single anticlastic curvature, while negative Poisson’s

ratio provides a synclastic shape. Topology #1 features cells having re-entrant unit cell,

therefore showing an auxetic effect under uniaxial loading. The curvatures are all

synclastic, but with different magnitudes because of the varying unit cell shape (Fig. 2a).

On the contrary, the panel made with topology #2 (Fig. 2b) has a continuous

distribution of curvatures, ranging from synclastic (where the cells have a re-entrant

shape) to anticlastic corresponding to convex hexagonal units. The honeycomb with

topology #1 is composed by cells all having a re-entrant shape, however the magnitude

of the synclastic curvature is different along the surface of the cellular panel (Fig. 2a).

Multiple curvature cores could be very useful for developing sandwich structures with

complex shape geometry, ranging from radar radome to acoustic liners and ducts. At the

same time, little is understood about the distribution (and variation) of the mechanical

properties due to gradient configurations, in particular when failure within the core is

considered.

The work described in this paper concerns two different angle gradient honeycomb

configurations based on the topology of auxetic unit cell, with a variation in horizontal

rib length and cell internal angle separately. Specimens of the two gradient structures

were produced using rapid prototyping (RP) technique. Sandwich panels were made by

bonding two layers of thin and stiff skin to the gradient core and subjected to three point

bending tests according to ASTM standard. The experimental results were used to

validate the Finite Element model (FEM), and subsequently used to perform a

 

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parametric analysis on the bending properties (linear and nonlinear) of the angle

gradient honeycombs for various cell wall aspect ratios, horizontal rib length growth

rate and increment of the cell internal angle. To the best of the Author’s knowledge, this

is the first time a comprehensive analysis on the bending and failure of man-made

gradient cellular structures has been performed.

2  Manufacturing  and  Experiments  

2.1  Angle  gradient  configurations  

 

As mentioned in the introduction, the fundamental topology of re-entrant unit cell is

characterized by the cell internal angle θ, the cell wall aspect ratio α=h/l and relative

thickness of the oblique rib β=t/l (Fig. 1(a)). The two gradient honeycomb

configurations (#1 and #2) have been designed by varying the horizon rib length h and

internal cell angle θ row by row along the y direction (Fig. 1 (b)). Both configurations

are made by changing the position of the point connecting the three ribs. The position of

the node can only change along the y direction. For the configuration #1, the horizontal

rib length of the (i+1)th row increases by 5% compared to the one belonging to the ith

row (i.e. hg = (hi+1-hi)/hi = 0.05). In this configuration the connecting nodes move along

the same direction, with the angle change provided by the progressing shortening of the

horizontal ribs. For configuration #2, it is the internal cell angle at each row which

increases (by 5°, in our case), progressing along the y axis (i.e. Δθ = (θi+1-θi)/θi = 0.05).

 

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The first row from the bottom of the two configurations (Figure 1b) has different

starting angle (-10º for #1, -25º for #2). In this way, configuration #2 maintains an

integer number of row with the same gradient, with both the lower and upper rows

terminating aligned to the horizontal rib.

The geometric parameters of the first row (labeled by subscript 0) of the two cell

gradient structures are detailed in Table 1.

Fig. 1 Topology of re-entrant unit cell and geometric configurations of angle gradient honeycombs

Table 1 Initial parameters belonging to the first row and size of the specimens.

Specimen Re-entrant Unit cell in the first row Specimen size (mm)

0θ 0l 0α t L b c ft S

#1 -10° 12 1 0.6 154 79 22 1.2 108 #2 -25° 12 1 0.6 138 69 22 1.2 88

Gradient honeycomb samples were manufactured using a Fused Deposition Molding

Rapid Prototyping technique (FDM RP, Stratasys® Dimension® Elite). The FDM

 

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machine is able to deposit minimum thickness of 0.178mm of ABSplus (Stratasys®)

material, including also the support material. The samples of the gradient honeycombs

have been cut into dimensions according to the ASTM C393/C393M-06 standard

(393M-06) (Table 1).

Fig. 2 Gradient curvature displayed by angle gradient honeycombs

2.2  Three-­‐point  bending  tests  

 

Two composite face skins have been bonded to the gradient cores using epoxy super

glue (Hexcel Redux® 810) cold cured for 72 hours (Fig. 3 (a)) at room temperature.

The composite skins (stacking sequence of [±25°]2s) have been obtained from IM7/8552

 

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carbon/epoxy unidirectional prepreg (Hexcel Corporation) with a sheet thickness of

0.15mm. The particular stacking sequence used provides a through-the-thickness

negative Poisson’s ratio effect, which shows strong localization of the damage after

low-kinetic energy impacts typical of damage tolerance problems (Alderson and Coenen,

2008; Hadi Harkati et al., 2007). However, the effect of the stacking sequences over the

global mechanical behavior of the composite sandwich beams is not evaluated within

the context of this work. The three point bending tests were performed using a servo

hydraulic machine Instron 8501 with a calibrated load cell of 50kN (Fig. 3(b)). Three

rigid rollers having 12 mm of diameter have been used as loading and supporting

devices. Rubber pressure pads of 12mm width were placed between the bars and

specimen to prevent local damage to the face skins (Fig. 3 (c)). The tests have been

carried out under displacement-control mode at a constant velocity of 0.1mm/s, with

data acquisition sampled at 3Hz. The deflection of the specimen has been calculated

from the displacements of the skin under the top roller, which were tracked using a 2D

digital image correlation system (Davis 7). All the tests have been conducted at ambient

room temperature and humidity around 50 %. The load-displacement curves of the two

gradient cellular core sandwich panels are shown in Fig. 4, while the corresponding

damage morphologies are represented in Fig. 5. It is possible to observe the presence of

shear cracks in the gradient cellular core, and distinct delamination occurring between

the core and skin. The core shear strength, skin stress is calculated from the

load-displacement curve based on the ASTM C393/C393M-06 standard (Eq. 1 & Eq. 2).

 

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The pure bending stiffness, bending modulus of the sandwich panel and the shear

modulus of the core can also be obtained using simple bending theory (Eq. 3 ~ Eq. 5)

(Fan, 2006). To consider the effect of the mass, the mechanical properties of Eq. 2 ~ Eq.

5 have been normalised with the total weight W of the sandwich configurations. The

normalized (specific) parameters are represented with the superscript *. The comparison

between the specific mechanical properties of the two gradient cellular sandwich beams

is illustrated in Table 2. The angle gradient configuration #1 has a lower relative density

than the one exhibited by topology #2. However, the #2 layout (i.e., the angle-gradient

centre-symmetric configuration) shows the highest specific flexural properties. When

compared against the aspect ratio gradient configuration (#1), samples #2 show higher

average value of core shear strength (38.4%), shear modulus (44%), bending stiffness

(19.7%), flexural modulus (19.7%), and skin strength (26%). The cellular configuration

#2 is characterised by cells with a higher degree of topology change – the unit cells vary

continuously between re-entrant, rectangular and convex hexagonal shapes, while

configuration #1 has a single topology, but with different aspect ratio. The lower

equivalent density of the configuration #1 provides also a smaller contact surface

between the face skins and the core itself, therefore decreasing the amount of transfer

load during the bending deformation of the sandwich beams.

bcdPFult )(max

+=

(1)

bcdtSP

f )(2max

+=σ

(2)

 

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bSPDδ48

3max=

(3)

3

3max

3 412

bdSP

dDE

δ==

(4)

DAS

SPA

G 1241

2

max−=

δ

(5)

Table 2 Flexural properties of the gradient core sandwich panels.

Parameters 1-1 1-2 2-1 2-2

sc ρρ 0.092 0.092 0.119 0.119

∗ultF ( gMPa / ) 2.04×10-2 2.17×10-2 3.43×10-2 3.39×10-2

∗yzG ( gGPa / ) 0.59 0.65 1.11 1.08

∗D ( gmN /⋅ ) 13.05 14.19 17.16 16.73

∗E ( gMPa / ) 10.78 11.72 14.18 13.82

∗σ ( gMPa / ) 0.88 0.94 1.24 1.23

 

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Fig. 3 (a) Sandwich panels with the gradient cores produced; (b) experimental setup for the

3-point bending tests; (c) typical sandwich beam configuration for the ASTM C393/C393M-06 standard.

Fig. 4 Load displacement curves for sandwich panels with gradient cellular core under the

3-point bending tests.

 

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Fig.5 Damage morphologies of the gradient cellular sandwich panels under 3-point bending

loading. (a) Specimen #1; (b) Specimen #2.

3  Full  scale  three-­‐point  bending  model  

 

The Finite Element model representing the sandwich beams with the gradient cellular

core has been developed using the commercial software ANSYS 11.0. The full-scale

 

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geometric model of the sandwich panel is shown in Fig.6 (a). For simplicity, no

interface regions between the skin and the core elements have been modeled, therefore

the model is representative only of the maximum static load, flexural behavior and core

failure, while skin-core debonding observed in the experimental results cannot be

simulated. The composite skin is represented using triangular linear layered structure

shell elements (SHELL99) with eight nodes and six degree freedoms. The triangular

meshing allowed following the complexity of skin-core geometry through a mapped

mesh approach. The core material was represented via quadrilateral eight-node

structural shells (SHELL93). The cell wall thickness (0.6mm) and the parameters of the

carbon/epoxy prepreg (0.15mm×8 layers, [±25°]2s) were specified in the real constants

of the element type. A convergence analysis (mesh size equal to l/2, l/3 and l/4) was

performed to obtain an acceptable compromise between accuracy and CPU time. An

average mesh size equal to l/2 was adopted for both the SHELL99 and SHELL93

elements. The skin material (IM7/8552 carbon/epoxy unidirectional prepreg) was

defined as a transversely isotropic but in-plane special orthotropic material, with

Ex=171.42 GPa, Ey=9.08 GPa, Gxy=5.29 GPa, vxy=0.32, and density ρ=1430 kg/m3. The

core material (FDM ABSplus) properties were determined by performing tensile tests

(ASTM D638-08) on dog-bone shaped specimens. The FDM ABSplus material

presented a transversely isotropic mechanical behaviour, with Ex=2.016 GPa, Ez=1.53

GPa, Gxz=0.626 GPa, vxy=0.43, ρ=1040 kg/m3. The mechanical properties of the core

plastic material are consistent with data used for other types of FDM-produced cellular

 

  16  

configurations. In the full-scale three point bending model, boundary conditions were

applied on three nodal regions (A, B, C) of 12mm width to simulate the experimental

setup. All the nodal degree freedoms of the A and B regions at the bottom surface have

been constrained, whereas the nodes in A region have been subjected to a displacement

along the negative Z direction (Fig. 6(b)). The constant loading velocity used during the

test has been represented using a total time approach. Tsai-Wu failure (Tsai and Wu,

1971) and maximum stress failure criterion have been also used to predict the failure of

the composite skin and core respectively and the strength values are listed in Table 3.

Element birth and death technology was introduced to kill elements meeting the defined

failure criteria during the loading process. The force-displacement results obtained from

the FE model is shown in Fig.7, revealing a general good agreement between the FE

model and experimental data, in particular when considering the initial stage of the

loading and the maximum load sustained by the whole sandwich panel. The

load-displacement relations obtained from the FE model imply an earlier and larger

material softening ascribed to the constitutive relationship of the stress and strain of

FDM ABSplus represented by multilinear isotropic hardening (miso option in ANSYS

11.0). However, multilinear isotropic hardening does not allow negative slope between

the stress and strain, meaning that the FE model could only simulate the softening stage

of the core material and not its failure stage (which is represented through element death

using failure criteria). An example of the numerical failure simulation process is given

in Fig. 8, in which the specimen #2 is considered. The angle gradient core fails due to

 

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shear crack under the bending loading (Fig. 8(c)), and no elements belonging to the

composite skin are eliminated, although the stress levels are significantly higher than

within the ones present in the core.

The FE model validated by the experimental results represents a toll showing a robust

fidelity for what concerns the prediction of the elastic flexural properties, transverse

shear core and onset of failure. From this point onwards it will be used to perform a

parametric analysis and predict the dependence of the mechanical properties of

sandwich beams with auxetic gradient core versus the geometry of gradient topology.

Fig. 6 (a) Geometric model in FE analysis; (b) mesh scheme and boundary conditions.

Table 3 Failure criteria of the skin and core material (unit: MPa) Composite skin (Tsai-Wu) Core

XT YT ZT XC YC ZC XYS Coupling Von Mises

Maximum Stress 2326.2 62.3 62.3 1200.1 199.8 199.8 92.3 -1 35.5

 

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Fig. 7 Comparison in load displacement curves between experimental and FE results.

 

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Fig. 8 Nodal contour plots of Von Mises Stress of angle gradient sandwich panels: (a) overall;

(b) angle gradient core; (c) angle gradient core after elements killing (sideview).

A direct comparison between Fig.5 (a, b) and Fig. 8 (c) shows that the FEM model

can also approximate the damage morphology observed during the experiments. For

configuration #1, the crack first propagates vertically and then horizontally, a

behaviour which is also observed in the FEM model (Fig. 8 (c)). In configuration #2,

the crack propagates simultaneously both vertically and horizontally, as predicted also

by the FEM representation.

4  Parametric  analysis  

 

Full-scale FE models representing the sandwich beams have been parameterized

against the unit cell initial aspect ratio α0, horizontal rib length growth rate hg and cell

 

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internal angle increment Δθ. The gradient auxetic cores have been produced using

rapid prototyping techniques, hence also the use of thermoplastics as core material.

However, for practical manufacturing applications, gradient cellular cores could be

produced using more composite production-oriented techniques, like Kirigami

(Origami plus ply-cut patterns) adopted to build complex cellular shapes using

thermoset fabrics and thermoplastic films (Saito et al., 2011). Hence the core material

within the parametric FE has been replaced by standard Kevlar 49/914 epoxy prepreg

(Hexcel Composites Ltd, UK), according to (Saito et al., 2011). Kevlar 49 plain

woven fabric/914 epoxy prepreg has excellent thermal stability as well as high

specific stiffness and strength, with E=29 GPa, v=0.05, ρ=1380 kg/m3. The skin of the

sandwich panel in the FE models is the same IM7/8552 carbon/epoxy prepreg. The

geometry nondimensional groups used for the parametric analysis range from 0.8 to 2

for the cell wall aspect ratio, from 0 to 0.05 for the horizontal rib length growth rate in

topology #1, and cell internal angle increment varying between 0º and 5° for #2

configuration. The dimensions of the specimens are all based on the ASTM

C393/C393M-06 standard and are detailed in Tables 4 and 5. For specimen of the #1

type, the average length is 156.9 mm (standard deviation of 3.13 mm), and the width

average is 70.4 mm (standard deviation of 7.4 mm). For specimen #2, the average

dimensions are 153.8 mm×71.4 mm (standard deviations being 8.4 mm and 4.1 mm

respectively). The height for all specimens is 25 mm. The images contained in Tables

4 and 5 include also the locations of the failure onset in the cores. The core failure

behaviour will be the subject of a detailed analysis in Paragraph 4.3.

 

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Table 4 Topologies of #1 under different geometric parameters and the according specimen size.

#1 l0=12, t=0.6, θ0= -10° α0=0.8 α0=1.0 α0=1.2 α0=1.6

hg=0

153.9×60.1×25

153.9×79.3×25

159.9×73.9×25

159.9×68.4×25

hg=0.02

153.9×60.1×25

153.9×79.3×25

159.9×73.9×25

159.9×68.4×25

hg=0.05

153.9×60.1×25

153.9×79.3×25

159.9×73.9×25

159.9×68.4×25

 

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Table 5 Topologies of #2 under different geometric parameters and the according specimen size.

#2 l0=12, t=0.6, θ0= -25° α0=1.0 α0=1.2 α0=1.8 α0=2

Δθ=0°

152×69.3×25

152×74.6×25

152×66.1×25

152×75.7×25

Δθ=3°

140×69.3×25

164×74.6×25

164×66.1×25

164×75.7×25

Δθ=5°

138×69.3×25

156×74.6×25

156×66.1×25

156×75.7×25

4.1  Relative  density    

 

The relative density is defined as ρc/ρs, where ρc is the density of the cellular structure

and ρs is that of the solid core material. The relative density plays a crucial role in the

mechanical, thermal and dielectric properties of cellular solids (Gibson and Ashby,

 

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1999). The variation of ρc/ρs versus the gradient core cell aspect ratio and different

parameters (hg for configurations #1, Δθ for topologies #2) is shown in Fig. 9. A

common feature of the two gradient configurations is the decrease of the relative

density for increasing cell wall aspect ratio values. Configuration #1 has also a

general lower relative density than the angle gradient #2. Topology #1 does show

almost no sensitivity versus the hg parameter, with the relative density diminishing by

50 % when α0 changes from 0.8 to 2. On the contrary, angle-gradient configurations

#2 have significantly higher relative densities. The denser honeycombs correspond to

Δθ=0° (i.e., no gradient and fully auxetic configuration), and decrease on average by

15 % for Δθ=3°and Δθ=5°. A direct comparison between the configuration #1 (l=12,

α0=1.0, t=0.6, θ0=-10°, hg=0) and the #2 (l=12, α0=1.0, t=0.6, θ0=-10°, Δθ=0°) leads to

conclusion that relative density of auxetic structures with the same wall thickness

decreases with the increase of the cell internal angle (-90°~0°) when the aspect ratio

equals 1.

 

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Fig.9 Relative density versus the aspect ratio and geometry parameters for the two auxetic gradient core configurations.

4.2  Specific  mechanical  properties    The flexural mechanical properties (Fult

*, Gyz*, D*, E*, σ*) versus the base (first row)

cell wall aspect ratio α0 have been normalised using the total weight of each gradient

core configuration (Figs. 10 and 11). The gradient topology #1 shows a common trend

of decrease versus the base cell wall aspect ratio for all the mechanical properties

considered. The specific critical load decreases by 61 % when α0 varies from 0.8 to

1.6, similarly to the specific transverse shear modulus (53%), and skin stress (56 %).

The sensitivity to the hg parameter is small, and only observable at higher values of

base cell wall aspect ratio for the specific critical load and skin stress. The angle

gradient configuration #2 shows however a more significant dependence over the

angle increment Δθ, maintaining however an overall decreasing trend for increasing

 

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α0 values (Fig. 11). For base cell wall configurations those α0 > 1.2, the specific

critical loads are on average 10 % higher for the non gradient configuration (Δθ = 0º),

while positive angle variation do only provide a negligible contribution (Fig. 11(a)).

The specific transverse shear modulus receives a positive contribution from non-zero

angle increments only for α0 < 1.2, otherwise no effect is observed for the Δθ values

considered in this work (Fig. 11 (b)). More sensitive to the angle increment is the

specific flexural stiffness of the sandwich beams (Fig. 11 (c)). For α0 = 1.2, Δθ = 1º

provides a 14 % increase in flexural stiffness compared to the non gradient

configuration. Increasing Δθ to 3º leads to a 28 % enhancement, this is approximately

maintained also for higher values of base cell wall aspect ratio. The specific skin

stress is only mildly affected by the internal cell angle increments, and for α0 >1.2 the

presence of non-zero angle increments alleviates on average by 9 % the skin stress

compared to the non-gradient configuration.

The gradient configuration #2 shows general improved specific flexural properties

compared to the #1 topology for the same aspect ratio values. At α0 = 1.2, the specific

critical load is on average 45 % higher in the #2 layout. The same behaviour is

observed for the specific shear modulus (40 %), flexural stiffness (31 %) and skin

stress (38 %). At higher aspect ratios (α0 > 1.8), the enhancement is significantly

decreased, and the two configurations do offer similar global mechanical properties.

 

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Fig.10 Specific flexural properties of gradient core sandwich panels (#1) under different

honeycomb geometry. (a) critical load; (b) core shear modulus in yz plane; (c) bending

stiffness; (d) skin stress.

 

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Fig.11 Specific flexural properties of gradient core sandwich panels (#2) for various

honeycomb geometry. (a) critical load; (b) core shear modulus in yz plane; (c) bending

stiffness; (d) skin stress.

Three-point bending allows identifying the transverse shear modulus of the

honeycomb core along the plane loading. In centersymmetric honeycomb

configurations the transverse shear modulus Gyz (i.e., in the plane of the transverse

bending loading) is expressed in closed form as a function of the geometry and core

material shear modulus Gs (see Fig. 12 (a)) (Gibson and Ashby, 1999). A convenient

way to describe the shear modulus for gradient configurations is to express the

relation between the FE-generated results and equivalent shear moduli TopyzG and

BottomyzG related to the unit cell shapes at the top and bottom configuration of the

gradient honeycomb panel as ( )Bottomyz

Topyz

FEMyz GGG −= λ . The dependence of the

non-dimensional transverse shear modulus on the various geometric parameters (α0,

 

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hg, Δθ) can be expressed by nonlinear regression analysis. For the configuration #1, a

suitable expression for the λ parameter is:

)exp(32

010 A

hA

AA g−−⋅+=α

λ (6)

The sensitivity of λ versus the parameter hg is stronger for small base cell wall aspect

ratios, while for higher values of α0, l tends to unity, with negligible effect from the

horizontal rib gradient (Fig. 12(b)). The configuration #2 has otherwise another

expression for the λ parameter:

θαθαθαλ Δ⋅⋅+Δ⋅+⋅+Δ⋅+⋅+= 052

42032010 AAAAAA (7)

The sensitivity of λ versus the cell wall aspect ratio is stronger for small angle

gradient increments above α0 = 1.4. For the various Δθ values, λ increases by a factor

of 3 passing from low to high aspect ratios. The smaller angle gradient configuration

(Δθ = 1º) provides the highest λ values (and therefore the highest shear modulus).

With a small angle gradient value, the configuration #2 is composed by a complete set

of re-entrant cells (i.e., fully auxetic). The fully auxetic configuration provides the

highest values for shear modulus Gyz at constant cell wall aspect ratio (Scarpa et al.,

2003b). The fitting coefficients for each expression are listed in Table 6.

 

  29  

Fig. 12 Relationship between non-dimensional parameter of transverse shear modulus and

aspect ratio and gradient parameters. (a) definition of non-dimensional transverse shear

modulus; (b) specimen #1; (c) specimen #2.

Table 6 Fitting coefficients for the regression formulas between non-dimensional transverse shear modulus and other geometric parameters.

Fitting

coefficients A0 A1 A2 A3 A4 A5 R2

#1 -0.8388 -5421.15 -0.1585 -0.022 -- -- 0.985 #2 16.3664 -12.0687 -3.7773 5.9784 0.5932 -0.9797 0.942

4.3  Onset  of  damage  in  the  core  

A first examination of the results presented in Tables 3 and 4 shows that the majority

of the failure points lies in the proximity of the free lateral surfaces. However, the

 

  30  

gradient core topology provides also an influence on the initiation of the core failure

within the sandwich beams. The crack initiation position for each configuration,

where the first elements were de-activated due to the maximum stress failure criterion,

is shown with a ‘×’ mark for topologies featured in Tables 4 and 5. We use the

nondimensional variable ξ = x / (L/2) to locate the failure onset along the x-direction

of the sandwich panel (Fig.13 (a)). The variation of ξ versus the characteristics

parameters hg and Δθ are visualised in the maps shown in Fig. 13(b, c).

Fig. 13 Initial crack positions for each gradient configuration. (a) definition of nondimensional parameter ξ. (b) specimen #1. (c) specimen #2.

Note: NPR stands for Negative Poisson’s ratio, ZPR for Zero Poisson’s ratio, PPR for Positive Poisson’s ratio.

 

  31  

The gradient topology #1 has only the 32.3 % of the failure locations occurring under

the centrally loaded area of the sandwich beams (Figure 13(b)). The majority of the

failed sections of the core (64.7 %) are concentrated on the top supporting area

(ξ→1). It is worth noticing that aside from the non-gradient configuration (i.e. hg =

0.0), failure does not occur above the lower support end (ξ = -1). Crack initiation on

the core therefore happens in the areas with the cells having the largest dimensions,

and in the oblique ribs. The increase of the gradient parameter hg leads to a

concentration of the failure in the ξ = 1 region, again due to the relative increase with

the gradient of the core cell sizes in that area.

The topology #2 has a different type of behaviour. The majority of the failure occurs

under the centrally loaded area (70.8 %). The support at ξ = 1 features a 16.7 %

localisation of the crack initiation, while 12.5 % of the cracks are concentrated on the

area with ξ = -1 (Figure 13(c)). It is apparent that the central portion of the core tends

to concentrate the overall deformation of the sandwich panel. The core under the

loaded area assumes unit cell shapes with small positive and negative internal cell

angles in rapid succession. Small values of internal cell angles tend to provide large

negative or positive equivalent Poisson’s ratios values νxy (between ~-8 and +7 for α0

= 1) (Smith et al., 2002). At the same time, the Poisson’s ratio νyx tends to be zero, in

the view of the quasi-rectangular local shape of the cell units (Hayes et al., 2004). The

presence of a small concentrated are with equivalent large negative Poisson’s ratio

behaviour gives a strong concentration of the overall deformation of the sandwich,

similarly to what has been observed in composite laminates with

through-the-thickness negative Poisson’s ratio subjected to low kinetic energy impact

(Alderson and Coenen, 2008). A higher portion of failure onset is however observed

 

  32  

in the cells with local equivalent positive Poisson’s ratio (PPR). In general, re-entrant

honeycomb configurations do show a higher out-of-plane shear buckling stress

(Zhang and Ashby, 1992), making therefore the areas with convex hexagonal shape

more prone to fail due to core shear. When no gradient angle is present (i.e., Δθ = 0º),

failure occurs close the loaded and supported areas of the sandwich beam. Higher

values of gradient angle Δθ appear to concentrate further the location of the failure

onset under the centrally loaded region, leaving the area with the highest equivalent

negative Poisson’s ratio effect undamaged.

Conclusions  

Gradient honeycomb cored with variable cell-wall aspect ratios and internal cell

angles have been manufactured and used as fillers for sandwich beams. The gradient

cores show a multiple curvature (polymorphic) behaviour, and different specific

mechanical properties depending on the gradient topology are considered.

Angle-gradient configurations with constant cell-wall aspect ratio provide the highest

specific critical load, shear modulus and bending stiffness of composite sandwich

panels with gradient filler. Gradient core topologies with varying cell horizontal rib

length (and therefore cell wall aspect ratio) are sensitive to the relative size of the cells

versus the overall dimension of the honeycomb panel, in particular with regards of the

location of the failure onset in the core. Angle-gradient configurations tend to localise

strongly the damage around the loaded area, the more the higher the angle increment

used to build the gradient topology. The two gradient configurations show that it is

possible to tune and control the global and local mechanical response of polymorphic

cores based on geometry and selection of the core material.

 

  33  

Acknowledgements  

 

The manufacturing of the sandwich samples and their mechanical testing has been

performed under the auspices of the UK TSB 16093 REACTICS project. YH would

also like to thank the Chinese Scholarship Council for the provision of a bursary to

support her research. Further support for the FE simulations has been provided by the

logistics of the European Project FP7-NMP-2009-LARGE-3M-RECT. The Authors

would like also to thank the anonymous Referees for their useful and insightful

suggestions.

 

  34  

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