ON THE CHARACTERISATION OF CELLULAR MATERIALS USED IN
SANDWICH CONSTRUCTION
R.A.W.Mines,
Impact Research Centre, Department of Engineering,
University of Liverpool, Brownlow Street,
Liverpool, L69 3GH, UK
ABSTRACT
The paper gives an overview of issues relating to the characterisation of the progressive collapse of core cellular materials
used in sandwich construction. The specific structural application addressed is foreign object impact, and in this case the core
cellular material is subject to multi axial stresses, progressive collapse and possible rupture. The paper gives an overview of
various theoretical and modelling issues, which are then related to experimental materials and structural tests for model
development, calibration and validation. Most discussion concerns polymeric foam, metal foam and metallic micro lattice
structures.
Introduction
Sandwich construction is being used for lightweight structures in aerospace, marine and automotive industries. The application
driver for this is good specific stiffness and strength properties. Skin materials can be polymer composite, metal or hybrid
materials, such as fibre metal laminates. Design parameters include skin and core thicknesses, skin and core mechanical
properties. Structural design criteria include skin buckling, core shear failure and core skin debonding. Loadings not only
include one off loads, but also fatigue behaviour. An important design driver is foreign object impact, e.g. dropped tools and
ballistic impact for aerostructures.
Foreign object impact will be the starting point for discussion here. Mines et al [1] give a discussion of low velocity impact of
sandwich panels with syntactic foam type and aluminium honeycomb cores. They experimentally study the localised crush
behaviour of core materials under multi axial stresses, and also the rupture behaviour of cores. For the case of aluminium
honeycomb (low density) , the indenter (diameter 50mm) deforms the upper skin and core, then perforates the upper skin and
crushes the core, and finally perforates the lower skin. For the case of the syntactic foam type of material (high density), the
core plays a greater role in the perforation process as its failure stress is similar to the failure stress of the skin.
Crushable Foams
Crushable foams have been used for many years as core materials in sandwich construction. Possible foam materials are
based on polymers (e.g. Divinycell [2], Rohacell [3]), metals (e.g. Alporas [4]) and graphite. It has been shown that the
mechanical properties of cellular materials, e.g. elastic behaviour, uniaxial crush and densification, are dependent on the cell
geometry, on the cell size (foam density) and on the parent material [5]. Figure 1 shows the microstructures for Rohacell 51
WF and Alporas foams [3,4]. Rohacell 51WF (density = 51 kgm-3) is a PMI foam with a well defined cell geometry and a fine
structure (cell size typically 0.5mm). Alporas (density 200 kgm-3) is a closed cell alumimium foam with a coarse and variable
cell size (2 to 10mm).
A feature of foam behaviour during the progressive collapse of a sandwich beam or the perforation of a sandwich panel is multi
axial crush. In this, not only the initiation of damage is of interest, but also progressive damage and large scale rupture. These
behaviours can be simulated using finite element analysis and continuum models, in which not only deviatoric deformation is
taken into account but also hydrostatic effects.
5MM
Figure 1. Microstructure for (a) Rohacell 51WF [3] and (b) Alporas [4] foams
Deshpande and Fleck [6] summarise a basic model for this, and their equation for the initiation of damage ( σ ) is:
⎛
⎞
⎜
⎟
1
⎜
⎟ 2
σ= ⎜
σ e + α 2σ 2 m
2 ⎟
⎜ 1 + ⎛⎜ α ⎞⎟ ⎟
⎜
⎟
⎝ ⎝3⎠ ⎠
(
)
(1)
Where σe = deviatoric stress, σm = hydrostatic stress and α = curve fitting parameter that defines the shape of the yield
surface. They also describe the progression of failure in terms of a linear tangent modulus. This model has been implemented
in the explicit finite element code DYNA [7]. DYNA has a number of other foam crush simulation models of varying complexity.
Langseth et al [8] compare these for metal foams and for various loading conditions. They conclude that current models do not
accurately simulate all behaviour, and also that a more complex model does not necessarily give more accurate results. Also,
a more complex model usually requires more experimental data as input. Hence, there is a balance to make between
sophistication of material model and its robustness and ease of calibration.
Such models require extensive experimental data as input, e.g. multi axial data, hydrostatic compression and even hydrostatic
tension data [2,3]. The Arcan test can be adapted to test foams under combined compression/shear or tension/shear (see
Figure 2). As the grips are rotated with respect to the uniaxial testing grips, so various combinations of stress can be
achieved. The aim is to construct the failure envelope in principal stress space. In the case of hydrostatic testing, Deshpande
and Fleck [ 9 ] use a complex 3D cruciform geometry, Mines et al use a cylindrical specimen [3] and Peroni et al [10] also use
a cylindrical specimen.
In some cases, cell walls can rupture giving rise to large scale tearing in a structure, e.g. core shear failure in a bending
sandwich beam [11]. Hence, in this case, a multi axial failure criterion has to be developed to integrate with a continuum
model. Langseth et al [12] suggest a two parameter approach for metal foam, which includes maximum plastic volumetric
strain and maximum principal stress. Mines et al [13] suggest a single parameter, i.e. maximum effective strain criterion, for
polymeric foam. Daniel and Gdoutos [14] suggest a point interactive criterion based on Tsai Wu for polymeric foam. These
parameters have been validated for specific foams and specific loading conditions. However, there is a need for more
fundamental investigation of these types of failure criteria for foams. An issue here is the modelling of progression of damage.
If finite element simulations are being used, then element erosion or deletion can be used, however this leads to mesh
sensitivity [11].
Complicating factors in describing the behaviour of foams
One of the disadvantages of the continuum modelling approach for foams is the occurrence of strain softening in some foams,
e.g. Rohacell. This gives rise to strain localisation, which means that strains are discontinuous through a volume being
crushed [15]. This has implications for continuum modelling.
Other issues relating to the characterisation of foams are as follows. Metallic foams (e.g. Alporas) have large cell size that can
make a continuum analysis inappropriate for cores with small relative dimension. Also, such foams have variable cell size by
up to a factor of five, which means that there are large property variations through a core volume [11].
Figure 2. Arcan test for multi axial testing of metallic foams
Microstructural models for foam
Given the wide variety of foams, and other cellular structures, there is interest in looking at the microstructure and to relate the
microstructure to continuum models. This is called homogenisation.
For example, truncated octahedral structures have been proposed to characterise elastic and collapse behaviour of Rohacell
foam [16,17]. Figure 3 gives the geometry in the form of a frame work, which can be used for open cell foams. There is the
possibility to use this geometry as a shell structure, which is applicable to closed cell foams, e.g. Rohacell. Chen and Lakes
[16] quantify such properties as relative stiffness and strength with these models. Mills and Zhu [17] develop this truncated
octahedral approach to model a single cell with edges and faces. They quantify the deformation of cells and faces, and they
include elasto plastic response. It should be noted that cell edges contain 40% of the material in a metallic closed cell foam,
and so a combination of framework and shell models may be required. Additional complexity comes from the variation of cell
size in foams. This issue has been addressed by Bazant et al. [18], and they model the behaviour of Divinycell H100 foam
using a micro plane model. They model each sphere individually, and they assume perfect friction and no slip between
spheres.
Figure 3. Framework model for truncated octahedral geometry
The direct approach in foam modelling
It is perhaps worth noting here the direct approach of taking a computed X ray tomography picture of the full depth of the foam
and inputting the geometry into a finite element analysis [19]. This is a useful approach for small blocks of material but it is still
problematic for the non linear behaviour of large scale structures. Also, the approach does not give transparent understanding
of micro structural effects.
New generation metallic lattice structures
Another driver for micro mechanical models is the development of manufacturing procedures for lattice structures. These
include conventional manufacturing techniques [20] as well as more advanced manufacturing techniques [21]. In the latter, the
rapid prototyping selective laser melting (SLM) technique allows the realisation of metallic open cellular lattice structures at the
micro meter scale [21]. For example, micro lattice structures can be developed with strut diameters of 150 micro meters, and
with aspect ratios from 1 to 100 plus. Figure 4 shows a body centred cubic structure (Stainless Steel 316L) block tested under
compression and a beam after test under three point bend. In the latter, the two ply skin is woven carbon epoxy pre preg.
There has been no attempt as yet to optimise lattice geometry for crush efficiency, as manufacturing issues are non trivial and
are still being sorted out. This SLM manufacturing technique is being used at Liverpool to investigate body centred cubic
[22,23], octet truss [24], tetrahedral [25], and Kagome [26] structures. The preferred core topologies are those of stretch and
compression without bending [24]. Tetrahedral trusses are good for plates [25], Kagome trusses have superior isotropy and
greater resistance to softening modes such as plastic buckling [26] and octet trusses are good for stretching not bending
behaviour but they are complex [24].
Figure 4. bcc block at 6% compression [ 22] and beam after three point bend [ 23]. Cell size 2mm
Given the generality of the SLM manufacturing approach, there is the opportunity to realise lattice structures that have been
formally optimised. This ties in with the Topology Optimisation approach proposed by Sigmund and Bendsoe [27]. However,
these methods are mostly focussed on elastic behaviour, and methodologies are less well developed for the progressive
collapse of lattice structures. However, Pedersen [28], has made a start in the area. Multiple modes of failure (e.g. strut
yielding, elastic buckling, plastic buckling and rupture) make formal optimisation difficult. For example, modes of strut failure
during crush will be dependent of the aspect ratio of struts.
It should be noted that the quantification of behaviour of such lattice structures requires the measuring of strut properties. This
requires tension and compression testing at the micro scale, where failure loads are the order of one Newton. The properties
of the struts will be dependent on their orientation, as in the SLM manufacturing process, lattice structures are built up layer by
layer [21].
Graded cellular structures
Given the ability to tailor cellular structures, there is the opportunity to realise graded structures. For example, in the case of
foreign object impact of sandwich panels, there would be interest in having a finer cellular structure nearer the skin and a more
coarse cellular structure towards the centre of the panel [29]. These stratified structures have been realised for open cell
foams using investment casting manufacturing processes [30]. Figure 5 shows a possible graded geometry for selectively
laser melted bcc,z lattice structures of the type described in [21]. The transition region doubles the cell size in going from
bottom to top, i.e. 4 cells into 1. The picture shows the 4 cells and the transition region. A design aspect is structural continuity
for the struts. The general structural design issue here would be the transition of properties from fine to coarse regions, and
making sure that the transition region does not become a weak point in the structure.
Strain rate effects in cellular solids
It is not the purpose of the current paper to discuss strain rate effects in depth, but fairly obviously these may well be important
for foreign object impact damage. The dynamic response of cellular materials can be thought to be dependent on global strain
rates, although in the case of strain localisation, local strain rates will be higher than global rates.
Figure 5. Model of transition in cell size in a bcc,z micro lattice structure
Impact compression, shear and tension tests are possible using drop weights or dynamic servo hydraulic machines [31]. For
Rohacell 51WF, it has been shown that [32]:
σ dy
⎛ ε& ⎞
A
log⎜⎜ ⎟⎟.....when.....10 −3 ≤ ε& ≤ 100 s −1
= 1+
σy
σy
⎝ ε&r ⎠
σ dy
= 1.21.....when.....10 0 ≤ ε& ≤ 10 2 s −1
σy
(2)
(3)
Where σdy is the dynamic failure stress, σy is the static failure stress, A is a material value, έ is the strain rate and έr is the static
reference strain rate. This foam shows some rate sensitivity at low rates and insensitivity at higher rates. Gibson and Ashby
[5] have identified the physical bases for these behaviours as cell wall inertia and strain localisation. At the very high rates
(strain rates greater than 104 per sec), shock waves form in the material and so there is a maximum speed at which a
disturbance can travel through foam [34].
Summary of issues to be addressed
Manufacturing technology is now developed to such a point that we can now think about tailoring and optimising cellular
structures at the micro scale. Although foams and lattice structures have been the focus of discussion here, manufacturing
technologies are also developing for other cellular cores such as honeycomb and folded cores [34]. In the latter, folded
geometry can been pre defined to give specific sandwich panel properties. The manufacturing process consists of rolled
Nomex paper [33].
To summarise, the study of cellular microstructure and the relating of this to structural behaviour is currently a very active
research field. Some of the current experimental mechanics research issues are:
1.
2.
Deriving parent material properties for structural features with micro meter dimension.
Development of homogenised models which can simulate the complex micro structural behaviour but which can also
be used efficiently to simulate the non linear behaviour of large sandwich structures.
3.
4.
5.
The development of various global materials tests that can be used to develop, calibrate and validate these models.
These include Arcan and hydrostatic tests.
Deriving impact material properties, and distinguishing between parent material strain rate effects and cell wall inertial
effects.
Materials tests and models need to be able to cope with graded structures and to provided bases for the development
of optimum configurations, taking into account the limitations of the specific manufacturing process.
Acknowledgements
The work described in this paper was supported from contracts:
EPSRC/EP/C009525/1, EPSRC/EP/009398/1, and EU FP6 CELPACT.
EPSRC/GR/F10057/01, EU FP4 CRASURV,
References
1. Mines, R.A.W., Worrall, C.M., Gibson, A.G., Int. J. Imp. Engg., 21 (10), 855-879, 1998
2. Mines, R.A.W., Alias, A., Composites Part A, 33, 11-26, 2002
3. Li, Q.M., Mines, R.A.W., Birch, R.S., Int J Sol Struct, 37, 6321-6341, 2000
4. McKown, S., Mines, R.A.W., Journal of Applied Mechanics and Materials, 1-2, 211-216, 2004
nd
5. Gibson, L.J., Ashby, M.F., Cellular Solids (2 Edition), CUP,1997
6. Deshpande, V.S., Fleck, N.A., J. Mech Phys Sol, 48, 1253-1283, 2000
7. Hallquist, J.O., LS-DYNA Theory Manual, LSTC, 2006
8. Hanssen, A.G., Hopperstad, O.S., Langseth, M., Ilstad, H., Int. J. Mech. Sc., 44(2), 359-406, 2002
9. Deshpande, V.S., and Fleck, N.A., Acta. Materiala, 49, 1859-1866, 2001
10. Peroni, L., Avalle, M., Martella, P., Multi axial characterisation of the mechanical behaviour of aluminium foam, In HPSM
2006 (ed. C. A. Brebbia), WIT Press, 249-258, 2006
11. McKown, S., Mines, R.A.W., Impact behaviour of metal foam cored sandwich beams, In ECF16 (Ed. E.E. Gdoutos), Paper
No. 254, 2006
12. Reyes, A., Hopperstad, O.S., Berstad, T., Hanssen, A.G., Langseth, M., Eur. J of Mech. Part A Solids, 22(6), 815-835,
2003
13. Mines, R.A.W., Li, Q.M., Birch, R.S., Rigby, R., Al Khalil, M., Tanner, A., Static behaviour of pre stressed polymer
composite sandwich beams, In Recent Advances in Experimental Mechanics (Ed. E.E. Gdoutos), Kluwer, 671-682, 2002
14. Gdoutos, E.E., Daniel, I.M., Wang, K.A., Composites Part A, 33, 163-176, 2002
15. Li, Q.M., Mines, R.A.W., Strain, 38, 132-140, 2002
16. Chen, C.P., Lakes, R.S., Cellular Polymers, 14 (3), 186-202, 1995
17. Mills, N.J., Zhu, H.X., J. Mech Phys Sol, 47, 669-695, 1999
18. Brocca, M., Bazant, Z.P., Daniel, I.M., Int. J. Sol. Struct., 38, 8111-8132, 2001
19. Youssef, S., Maire, E., Gaertner, R., Acta Materialia, 53, 719-730, 2005
20. Wadley, H.N.G., Fleck, N.A., Evans, A.G., Composites Science and Technology,63(16),2331-2343, 2003
21. Santorinaios, M., Brooks, W., Sutcliffe, C.J., Mines, R.A.W., WIT Transactions on the Built Environment, 85, 481-490, 2006
22. Mines, R.A.W., McKown S., Cantwell W., Tsopanos, S, Brooks W., Sutcliffe C.J., On the progressive collapse of micro
lattice structures, In ICEM13 (Ed. E.E. Gdoutos), Paper No.158, 2007
23. McKown, S., Cantwell, W.J., Brooks, W.K., Mines, R.A.W., Sutcliffe, C.J., High performance sandwich structures with
hierarchical lattice cores, SAMPE Europe, 2006
24. Deshpande, V.S., Fleck, N.A., Ashby, M.F., J. Mech. Phys. Sol.., 49, 1747-1769, 2001
25. Deshpande, V.S., Fleck, N.A., Int. J. Sol. Struct., 39, 6275-6305, 2001
26. Wang, J., Evans, A.G., Dharmasena, K., Wadley, H.N.G., Int. J. Sol. Struct., 40, 6981-6988, 2003
27. Bendsoe, M.P., Sigmund, O., Topology Optimisation, Springer, 2004
28. Pedersen, C.B.W., Topology optimisation of energy absorbing frames, In WCCM V (Eds. H.A.Mang, F.G. Rammerstorfer,
J. Eberhardsteiner), Paper No. 81219, 2002
29. Brothers, A.H., Dunand, D.C., Adv. Engg. Mat, 8 (9), 805-809, 2006
30. Kirugulige, M.S., Kitey, R., Tippur, H.V., Comp. Sc. Techn., 65, 1052-1068, 2005
th
31. Mines, R.A.W., Strain rate effects in crushable structural foams, In 5 BSSM International Conference on Advances in
Experimental Mechanics, (Eds. P.Withers, J Fonseca), 2007
32. Li, Q.M., Mines, R.A.W., Birch, R.S., Combined strain rate and temperature effects on compression strength of Rohacell
rd
51WF structural foam, In 3 Asia Pacific Conference on Shock and Impact Loads on Structures (Eds. T.S. Lok and C.H.
Lim), 221-226, 1999
33. Mines, R.A.W., Comp Struct., 64, 55-62, 2004
34. Nguyen, M.Q., Jacombs, S.S., Thomson, R.S., Hachenberg, D., Scott, M.L., Comp. Struct., 67, 217-227,2005