FAILURE BEHAVIOUR INVESTIGATION OF METALLIC OPEN LATTICE
CELLULAR STRUCTURES
G.N. Labeas, M.M. Sunaric
Department of Mechanical Engineering and Aeronautics
University of Patras
Laboratory of Technology and Strength of Materials (LTSM)
Panepistimioupolis
Rion 26500
Greece
labeas@mech.upatras.gr, msunaric@mech.upatras.gr
ABSTRACT
The cellular core of sandwich cellular structures is investigated in the present paper. The elastic moduli and the critical
buckling loads are predicted for the metallic open lattice cellular structures by means of non-linear static numerical analysis.
Predicted values are compared to experimental data involving compressive loading of cellular structures. Numerical studies
are performed to determine the influence of the strut radius, basic cell geometry and size on the behaviour of the structure.
The maximum deformation of the structure, which characterizes its energy absorbing capabilities, is approximated using a
simple engineering approach.
Introduction
Cellular structures are sandwich structures characterized by the construction of their core, which is made of an interconnected
network of solid struts or shell-type shapes. Sandwich structures made of metallic or plastic core and composite skins have
attracted the attention of the engineering industry since many years. The applications of cellular structures are widespread.
Thermal insulation, packaging, structural, buoyancy and many other engineering sectors are the most common fields for the
utilization of cellular structures [1]. The usual design parameters of cellular sandwich structures are the relative density, the
elastic specific stiffness and strength [2]. The mechanical behaviour of sandwich structures is dependent on the mechanical
properties of both core and composite skin. The cellular core behaviour is dependent on the cell geometry and cell size.
The present work focuses on the failure behaviour of metallic open lattice cellular cores. This behaviour is considered to be
one of the keys to the successful development of improved sandwich structures with tailored properties. The failure behaviour
is analysed by means of numerical simulation via Finite Element modelling of cellular cores. Due to the high difference of size
scale between the cell structure (which is usually of the order of a few millimetres) and the entire sandwich cellular structure,
the ordinary numerical analysis methodologies lead to very large models, requiring high computing power for their solution. In
contrary, alternative and multiscale modelling strategies are preferable. They are based on very detailed unit cell models using
material properties, cell geometry and cell size for determining the cellular metallic core properties.
In the present paper, three types of cellular cores, based on different unit cell configurations, namely, ‘bcc’, ‘bcc,z’ and ‘f2fcc,z’
are investigated. The compression behaviour of the three types of cellular structures is simulated by adequate beam-type
Finite Element models. The numerical analysis comprises a linear eigenvalue buckling analysis followed by a non-linear
elastoplastic static analysis. The buckling analysis is used to identify the buckling load and buckling modes of the structures,
as well as, to calculate the artificial imperfections to be introduced in the non-linear elastoplastic analysis. The latter is used to
predict the failure load and mode of the structures. A strong effect of the radius of the unit cell struts and the unit cell size on
the computed elasticity modulus and failure load is observed. Therefore, numerical results for varying strut radius and unit cell
size are presented. It may be concluded that, if the detailed geometrical description of the unit cell structure is known and
material properties at the size scale of the struts are provided, then the sufficiently accurate prediction of the cellular structure
response is possible.
Numerical model description
The numerical simulations of the cellular structure deal with predicting the elasticity modulus and failure load of the structure in
uniaxial compression. The numerical study is performed to analyze the effect of different arrangement of strut members inside
the cell, as well as, the cell size and strut radius on the structure’s elasticity modulus.
The cellular structure modelling and simulations refer to the cellular structures developed and tested in [3]. The cubic
structures of dimensions 25x25x25 mm3 involve three different cell geometries, as presented in Figure 1, namely the ‘bcc’ cell
3
3
with density of 994kg/m , the ‘bcc,z’ cell having a higher density of 1258kg/m , and the ‘f2fcc,z’ geometry with density of
983kg/m3. The size of the examined cells varies from 1.5mm to 5mm. The unit cell structure, which consists of vertical
members with diagonal cross members only, is constructed using the selective laser melting technique with Stainless steel
316L as powder material, [4].
bcc
Cell size:
5 mm
bcc, z
Cell size:
2.5 mm
f2fcc, z
Cell size:
1.5 mm
a)
b)
Figure 1: a) The three investigated unit cells types and
3
b) Cellular core structures of 25x25x25 mm with cell sizes of 5mm, 2.5mm and 1.25mm
The bcc and the bcc,z geometries have bulk-centred geometries, i.e. the members cross at the centroid of a cubic cell. The
bcc cell is spatially symmetric, having members along the spatial diagonals of the unit cell, thus showing the same stiffness
when loaded in x, y or z direction. The bcc,z cell is a modification of the bcc cell, with added members on the sides in vertical
direction. This implies larger stiffness of this cell type when loaded in z direction, but also affects the moduli in directions x and
y. The f2fcc,z geometry differs from the previous two in the way that it is facet-centred, i.e. the members cross as diagonals on
the faces of the cubic cell. As in the bcc,z geometry, the additional members are present on the vertical edges of the cube.
The examined cell side lengths are 1.5mm, 2.5mm and 5mm.
The beam element type BEAM188 is used in ANSYS finite element code to model the unit cell. It is a three-dimensional beam
element with six degrees of freedom per node, ideally applicable to problems involving geometrical non-linearity and plasticity.
The FE refinement level in the present case studies varies from 1 to 4 elements per strut-member. Higher refinement level
contributes to an improved modelling of buckling phenomena in members under axial compression; however, it has a major
3
influence on the required computation time, even for models of 25x25x25 mm size presently analysed. By multiplying the unit
cell in three dimensions the cubic cellular structure is generated. Two surfaces of the structure that are perpendicular to the
loading direction are selected for applying the compressive load and boundary conditions, as shown in Figure 2. The load is
applied as axial force on the nodes of the loading surface. The rigid movement of the loaded surface is achieved by applying
coupled displacement in loading direction on all nodes of the loaded surface. On the opposite surface one corner node is
selected to be constrained for translation in all directions, while for the remaining nodes only the translation in the loading
direction is constrained, thus enabling the lateral displacement of the structure. A linear static numerical analysis is initially
executed for predicting the elasticity modulus of the cellular structure. Consequently, a non-linear static analysis including
geometrical and material non-linearity is performed to predict the failure load of the analyzed component; where required
artificial imperfections are introduced by means of an eigenvalue buckling analysis. The material used for the strut construction
3
is stainless steel 316L, with density of 8000 kg/m and elasticity modulus of 23 GPa. The material non-linear data of the 316L
stainless steel material, used in the present analysis, is part of a testing campaign performed by the University of Liverpool.
Figure 2: Compressive loading and boundary conditions of the investigated cellular structures
Results and discussion
I. Prediction of elasticity modulus of the cellular structure
The elasticity modulus of a cellular structure depends directly on the unit cell geometry and size, strut radius and material
properties. The current study focuses on the effect of the cell geometry on the elasticity modulus of the structure. For simple
two-dimensional or three-dimensional geometries it is possible to calculate theoretically the elasticity modulus of the unit cell or
the cellular structure by using the elementary mechanics of materials theory [1]. However, if the cell geometry becomes more
complex, the theoretical approach may only roughly approximate the real elasticity modulus of the structure; in such cases, it is
usually determined by experimental measurement [3]. To minimize the rather expensive testing, especially in cases of novel
cell-type development and optimization, which require the identification of modulus variation with respect to shape, size and
material, a numerical analysis is preferable.
The prediction of elasticity modulus of cellular structures is currently performed for the previously described cell geometries
bcc, bcc,z and f2fcc,z, with cell sizes of L=1,5 mm and L=2,5 mm. The average strut member cross-section radius has been
experimentally measured in [3] to be r=0,16mm; however, as it is very difficult to produce struts with constant radius
throughout the entire structure, the measured value of radius introduces some uncertainty into the numerical analysis. For the
present simulations the average strut radius is estimated for each structure separately by comparing the predicted values of
the elasticity modulus with the respective experimental ones.
Applying the compressive axial load and assuming linear elastic static analysis, the elasticity modulus is estimated as the ratio
of computed ‘global’ axial stress over the ‘global’ axial strain. The ‘global’ axial stress is calculated as the overall axial load
over the loading surface (Figure 3a). The ‘global’ axial strain is calculated as the axial displacement of the loaded surface over
the initial height of the structure (Figure 3b).
F
a)
b)
∆Η
Η
L
D
σ = F / (L·D)
ε = ∆Η / Η
Figure 3: a) Calculation of the ‘global’ axial stress; b) Calculation of the ‘global’ axial strain
The calculated elasticity moduli are compared to the measured values from [3] in Table 1. It can be observed from both the
experimental measurements and calculated values that a variation of the unit cell geometry or size results to a radical change
in the structure’s elasticity modulus. As observed from Table 1, the elastic moduli of the structures with bcc cell type are lower
compared to that of structures with bcc,z and f2fcc,z cell type; this can be explained by the existence of additional axial
members in the direction of loading in the bcc,z and f2fcc,z cells. Furthermore, an increase in the unit cell size decreases the
structure stiffness and the respective elasticity modulus, which can be explained by observing that, as the cell size decreases,
the effective loaded area tends to the value LxD (see figure 3.a) and the elasticity modulus of the structure tends to approach
the bulk material elasticity modulus.
Table 1. Measured and predicted elasticity moduli for different cell geometries and sizes
Measured
Estimated
Elasticity modulus
average
cross-section
Cell size
Cell type
experimental (GPa)
cross-section
radius
radius
r=0,22mm
1,5 mm
0,44
bcc
r=0,21mm
2,5 mm
0,05
bcc,z
f2fcc,z
r=0,16mm
Elasticity modulus
numerical (GPa)
0,443
0,047
r=0,24mm
1,5 mm
2,7
2,696
r=0,18mm
2,5 mm
0,45
0,479
r=0,2mm
1,5 mm
1,9
2,18
r=0,23mm
2,5 mm
1
0,923
A comparison between the two last columns of Table 1, indicates that the elastic moduli predicted by the numerical simulations
can match the experimentally measured values, if an appropriate adjustment of the strut radius for the numerical analysis is
performed. This is done by varying the beam cross-section radius in the analysis, such that the calculated elasticity modulus
correlates well to the measured values. This numerical adjustment can be considered as reasonable, taking into account that
the real strut cross-section radius is not constant, as the strut is thicker close to the region of connection with other struts; in
addition there are cross-section differences between the individual struts. In Figure 4 the influence of strut cross-section radius
on the structure elasticity modulus for different cell geometries is depicted, which underlines the importance of the accurate
knowledge of the strut cross-section radius for the characterization of the cellular structure. As shown in Figure 4, in certain
cases, even a small increase of the strut cross-section radius can cause a high modulus increase. A comparison between the
behaviour of bcc and bcc,z geometries indicates that, while the modulus of the bcc cell structure changes exponentially with
respect to the strut radius, the modulus of the bcc,z geometry changes rather linearly.
2,5mm bcc geometry
2,5mm bcc,z geometry
1400
140
Elasticity modulus, MPa
Elasticity modulus, MPa
simulated
120
experimental
100
80
60
40
20
0
0.09
0.12
0.15
0.18
0.21
0.24
0.27
1200
simulated
experimental
1000
800
600
400
200
0
0.09
0.30
0.12
Cross-section radius, mm
0.15
0.18
0.21
0.24
0.27
0.30
Cross-section radius, mm
Figure 4: Dependence of the structure elasticity modulus on the member strut cross-section radius
for the bcc and the bcc,z cell geometries
II. Failure of cellular structures in axial compression loading
In order to determine the dominant failure mode for the metallic open lattice cellular structures, non-linear analyses with
different options have been performed. The different options comprise non-linear analysis with and without material plasticity,
as well as, with and without geometrical non-linearity. The results of the three analysis types for the case of bcc cellular
structure with 5 mm cell size are presented in Figure 5. It can be concluded that both the geometrical and the material nonlinearity, i.e. buckling and plasticity, contribute to the failure of the structure, which implies that the basic failure mechanism of
the structure is plastic buckling instability.
180
160
140
Stress, kPa
120
100
80
Analysis type A: geometric linearity and
plastic material properties
60
40
Analysis type B: geometric non- linearity
and elastic material properties
20
Analysis type C: geometric non- linearity
and plastic material properties
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Strain
Figure 5. Numerical ‘global’ stress-strain curves for different analysis types
The static compressive loading of the cubic cellular structures is numerically simulated and the results in the form of ‘global’
engineering stress-strain data are compared to the experimental data of Ref. [4]. The comparison of the experimental and
numerical results referring to bcc and bcc,z geometries calculated with r=0,21mm and r=0,18mm, respectively, is presented in
Figure 6. In Figure 7 the deformation sequence of the cellular structure with bcc geometry and cell size 2,5mm is presented.
6
bcc experimental
bcc,z experimental
5
bcc numerical
bcc,z numerical
Stress, MPa
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Strain
Figure 6: Comparison between experimental and numerical ‘global’ stress-strain curves
for bcc and bcc,z structures (cell size 2,5mm)
Figure 7: Deformation sequence of bcc structure with cell size 2.5mm
As can be observed in the Figure 6, the numerically predicted critical loads for both structures are slightly overestimated. For
the bcc geometry, the plateau after the initial deviation of the linear regime is calculated to be at higher stress level compared
to the experiment. For the bcc,z geometry, the numerical stress-strain curve indicated a stiffer structure, with a slightly
overestimated critical buckling load, after which the numerical model becomes unstable and does not converge. At this loading
level the plastic hinging of many strut connecting nodes may be observed. The modelling of plastic hinges is not considered in
this paper.
In order to overcome the problem of the critical load overestimation, a study is performed on the influence of the strut crosssection radius on the critical load of the structure. The results of this study for the bcc and bcc,z geometry structures with cell
size of 2,5mm are presented in Figure 8a and 8b respectively from which a strong dependence of the strut cross-section
radius on the critical loading may be noticed. As may be observed from Figure 8, in both cases the critical load correlates
much better to the experimental one, when the cross-section radius r=0,16mm is used for both the bcc and the bcc,z
geometry.
6
5
bcc experimental
4
bcc,z r=0,18mm
bcc,z r=0,16mm
Stress, MPa
bcc r=0,16mm
Stress, MPa
bcc,z experimental
5
bcc r=0,20mml
3
2
1
bcc,z r=0,10mm
4
3
2
1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0
Strain
0.05
0.1
Strain
0.15
0.2
Figure 8: Effect of strut cross-section radius on the critical load of the cellular structure:
a) bcc cellular structure, b) bcc,z cellular structure
The numerical difficulties of the current implicit finite element simulation to predict the structural deformation deep in the postbuckling regime, limit the numerical analysis to the prediction of initial critical plastic buckling load, as well as to the first part of
the ‘plateau’ of the load-displacement curve. However, in order to estimate the entire energy absorption capability, prediction
of the maximum deformation is required. To this scope, the following simple engineering approach is currently used; it is
assumed that, as the structure progressively collapses due to local plastic buckling of its members, the deformation reaches a
stage where the beams are stacked over each other stuffing the area of the structure; at this stage, which corresponds to the
end of loading ‘plateau’, the load starts to increase again, indicating that load is actually transferred only by the contact
between the touching beams. The current numerical model is not capable of predicting the final stress build-up and the
maximum deformation, since including contact analysis would lead to enormous computational effort. This deformation point is
estimated by the fraction of the total volume of the beams to the total volume of the structure, assuming that the “vertical” view
of the structure remains unchanged during its deformation. For the bcc, bcc-z and f2fcc,z cell types, the maximum deformation
is estimated to the values of 85%, 75% and 65%, respectively.
Conclusions
A numerical simulation model, for the prediction of elasticity modulus, critical plastic buckling load and maximum strain of
compressively loaded cellular structures is presented in this paper. A non-linear elastoplastic analysis including initial
imperfections and geometrical non-linearity is required for the proper simulation of the cellular structure behaviour, as the basic
failure mode is elastoplastic buckling. The numerical model is capable of predicting the critical buckling load of the cellular
structure. Numerical results have shown that the structural response is strongly influenced by the unit cell strut cross-section
radius and unit cell size and shape. The bcc,z and the f2fcc,z geometries, having additional vertical struts, exhibit much higher
stiffness and critical buckling load compared to the bcc structure. Prediction of the ‘global’ stress strain curve at the large
deformation region is not feasible by the present approach, as it requires contacts analysis of the collapsed structure. In order
to overcome this difficulty, the maximum deformation of the structure is approximated by a simple engineering method.
Acknowledgments
The authors wish to acknowledge the European Union for their financial support to this research. The results presented in this
paper are partially obtained in the frame of the European Union funded research program “CELPACT – Cellular Structures for
Impact Performance”.
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