Finite element modelling of the mechanics of solid foam...

Finite Element Modelling of the Mechanics

of Solid Foam Materials

STEVEN RIBEIRO-AYEH

Doctoral Thesis

Stockholm, Sweden 2005

TRITA AVE-2005:05ISSN 1651-7660 KTH Farkost och flygISRN KTH/AVE/DA–05:05–SE SE-100 44 StockholmISBN 91-7178-002-5 SWEDEN

Akademisk avhandling som med tillstand av Kungl Tekniska hogskolan framlaggestill offentlig granskning for avlaggande av teknologie doktorsexamen onsdagen den 30mars 2005 klockan 10.00 i Kollegiesalen, Administrationsbyggnaden, Kungl Tekniskahogskolan, Valhallavagen 79, Stockholm. Fakuletsopponent ar Dr. Andrew M.Kraynik, Sandia National Laboratories, Albuquerque, New Mexico, USA.

c© Steven Ribeiro-Ayeh, mars 2005

Tryck: Universitetsservice US AB

Veniet tempus quo posteri nostri tam aperta nos nescisse mirentur.

(The time will come, when our ancestors will wonder why we did notknow such obvious things.)

Lucius Annaeus SenecaRoman Statesman, philosopher

FE Modelling of the Mechanics of Solid Foam Materials i

Preface

The work presented in this doctoral thesis has been carried out at the Departmentof Aeronautical and Vehicle Engineering, School of Engineering Sciences, KungligaTekniska Hogskolan (KTH). Financial support was kindly granted by the SwedishResearch Council (Vetenskapsradet) and in part by the European Commission,within the Fifth RTD Framework Programme.

I would like to thank my supervisor and co-author Dr. Stefan Hallstrom forhis guidance, support and for many encouraging discussions. I wish to express mygratitude to Professor Dr. Dan Zenkert for accepting me as a Ph.D. student andfor giving me the opportunity to work at this Department. Dan and Stefan did anexcellent job in introducing me to the sometimes quite challenging field that shouldfinally evolve into this thesis.

Stefan must be given credit for encouraging me whenever I came from the testinglab, face hanging and claiming that all tests failed, since the specimens “just broke”.I admired his endurance and his never failing attempts to read my reports and tofind the actual meaning in my words when I again had handed him a few pagesfilled with some seven mile-long, hard to understand sentences, being a combinationof several thoughts in multiple subordinate clauses, assuming that it was the mostconcise and best to understand form of explaining that complicated things couldbe grasped instantly if neatly wrapped in a cascade of words, some commas and atrailing period. I appreciate his work which certainly helped to improve the qualityof my papers.

Especially cordial thanks go to Hans Kolsters, whom I shared the office with formany years. I will certainly miss the days we spent together, our long discussions,having a second serving of lunch, and listening to you speaking dutch on the phone.

I want to thank Andrey Shipsha, Gordon Kelly and Lance McGarva, all of whomI spend many happy hours with, both at work and in other fun places. Thanks toDavid Eller, who sometimes shook his head in disbelief but kindly helped me gettingmy ideas coded.

And thanks you all, my colleagues in the Division of Lightweight Structures andthe rest of the department for all the various kinds of support to my work. I reallyenjoyed the times I spent here, working in a friendly, comfortable and just enjoyableatmosphere.

Finally, words of thank go to my dear Susanne who supported me with all herlove and understanding, to my parents, and last not least to the Osquars.

Stockholm, March 2005

FE Modelling of the Mechanics of Solid Foam Materials iii

Abstract

Failure of bi-material interfaces is studied with the aim to quantify the influence ofthe induced stress concentrations on the strength of the interfaces. A simple point-stress criterion, used in conjunction with finite element calculations, is evaluated toprovide strength predictions for bi-material bonded joints and inserts in polymerfoam. The influence of local stress concentrations on the initiation of fracture atopen and closed wedge bi-material interfaces is investigated. The joint combinationsare analysed numerically and the strength predictions obtained from the point-stresscriterion are verified in experiments.

The predictions are made using a simple point-stress criterion in combinationwith highly accurate finite element calculations. The point-stress criterion wasknown from earlier work to give accurate predictions of failure at cracks and notchesbut had to be slightly modified to become applicable for the studied configurations.The criterion shows to be generally applicable to the bi-material interfaces studiedherein. Sensible predictions for the tendentious strength behaviour are made withreasonable accuracy, including the prediction of crossover from local, joint-inducedfailure to global failure.

To study the micromechanical properties of a cellular solid with arbitrary topo-logy, various models of a closed-cell foam are created on the basis of random Voronoitessellations. The foam models are analysed using the finite element method andthe effective elastic properties of the model cellular solids are determined. The cal-culated moduli are compared to the properties of a real reference foam and thenumerical results show to be in very good agreement.

The mechanical properties of closed-cell, low-density cellular solids are governedby the stiffnesses of the cell edges and the cell faces. Idealised foam models withplanar cell faces, cannot account for the curved faces found in some metal andpolymer foams. Finite element models of closed-cell foams are created to analysethe influence of cell face curvature on the stiffness of the foam. By determining theelastic modulus for foams with non-planar cell faces, the effect of cell face curvaturecan be analysed as a function of the relative density and the distribution of solidmaterial between cell edges and faces.

Foam models are generated from disturbed point distribution lattices and com-pared to models obtained from random distributions. The aim is to analyse if andhow the geometry of the cells and their spatial arrangement influences the mech-anical properties of a foam. The results suggest that the spatial arrangement andthe geometry of the cells have significant influence on the properties of a foam. Theelastic properties calculated for models from disturbed foam structures underestim-ate the elastic moduli of the foam, whereas models from random structures provideresults which were in very good agreement with a reference foam.

FE Modelling of the Mechanics of Solid Foam Materials v

Dissertation

This thesis consists of a brief introduction to the area of research and the followingappended papers:

Paper A

S. Ribeiro-Ayeh and S. Hallstrom: Strength Prediction of Beams with Bi-MaterialButt-Joints, 2002.Published in in Engineering Fracture Mechanics Volume 70, Issue 12 , August 2003,Pages 1491-1507

Paper B

S. Ribeiro-Ayeh and S. Hallstrom: Strength Prediction of Bi-Material InterfaceCorners on Inserts in Polymer Foam, 2002.submitted for publication

Paper C

S. Ribeiro-Ayeh and S. Hallstrom: Stochastic Finite Element Models for the Char-acterisation of Effective Mechanical Properties of Cellular Solids, 2005.submitted for publication

Paper D

S. Ribeiro-Ayeh and S. Hallstrom: The Influence of Cell Face Curvature on theStiffness of Cellular Solids, 2005.submitted for publication

Paper E

S. Ribeiro-Ayeh: The Effect of Topological Disorder on the Mechanical Propertiesof Cellular Solids, 2005.submitted for publication

Papers A and B were in part presented in: S. Ribeiro-Ayeh: On the Bi-MaterialInterface Strength of Inserts in Polymer Foam in Proceedings of the Sixth InternationalConference on Sandwich Structures, Ft.Lauderdale, Florida, USA, 2003

Paper C was in part presented in: S. Ribeiro-Ayeh: Development of A MicrostructuralFinite Element Model for Foam Structures in Proceedings of the 17th Nordic Seminar onComputational Mechanics, Stockholm, Sweden, 2004

FE Modelling of the Mechanics of Solid Foam Materials vii

Division of work between the authors

Paper A

Ribeiro-Ayeh performed the experiments, the finite element analyses and wrote thepaper. Hallstrom initiated and guided the work and contributed to the paper withvaluable comments and revisions.

Paper B

Ribeiro-Ayeh performed the experiments, the finite element analyses and wrote thepaper. Hallstrom initiated and guided the work and supplied valuable commentsand revisions to the paper.

Paper C

Ribeiro-Ayeh developed the foam models, performed the finite element analyses andwrote the paper. Hallstrom initiated and supported the work and supplied valuablecomments and revisions to the paper.

Paper D

Ribeiro-Ayeh developed the foam models, performed the finite element analyses andwrote the paper. Hallstrom supplied valuable contributions to the interpretation ofthe results, and the content and structure of the paper.

FE Modelling of the Mechanics of Solid Foam Materials ix

Contents

Preface i

Abstract iii

Dissertation v

Division of work between the authors vii

1. Introduction 1

2. Strength analysis 22.1. Failure assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2. Analysis of bi-material interfaces . . . . . . . . . . . . . . . . . . . . 4

3. Stiffness analysis 63.1. Previous analytical and numerical models . . . . . . . . . . . . . . . . 73.2. Cell growth - Voronoi structure . . . . . . . . . . . . . . . . . . . . . 83.3. Finite element modelling . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3.1 Periodic boundary conditions . . . . . . . . . . . . . . . . . . 103.3.2 Evaluation of elastic moduli . . . . . . . . . . . . . . . . . . . 11

3.4. Variations of the cell morphology . . . . . . . . . . . . . . . . . . . . 123.4.1 Topological disorder . . . . . . . . . . . . . . . . . . . . . . . 123.4.2 Non-planar cell faces . . . . . . . . . . . . . . . . . . . . . . . 133.4.3 Distorted cells . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4. Future work 14

Bibliography 19

Paper A A1–A17

Paper B B1–B18

Paper C C1–C20

Paper D D1–D23

Paper E E1–E24

FE Modelling of the Mechanics of Solid Foam Materials 1

1. Introduction

The sandwich concept is a widespread technique for achieving stiff and strong struc-tures while maintaining a low structure weight. It is utilised in numerous applica-tions for example in the vehicle, transportation and building sectors.

In commercial aircraft, sandwich design is well established mainly in cabin in-teriors, hatches, doors and various control surfaces, but even large structures suchas cabin floor panels or fuselage parts may be produced in sandwich design (Fig. 1).Carbon fibre sandwich panels offer high structural stiffness and superior thermalproperties and are commonly used in aerospace launch vehicles and satellite struc-tures, both in interior and exterior structures (Fig. 2).

Figure 1: Airbus A380 aircraft (Airbus) Figure 2: Ariane 5 heavy-lift launcher(European Space Agency)

For railway vehicles, interior modules such as partition walls, ceiling, luggageracks, floor panels or the whole of the car body may be built in sandwich design(Fig. 3). The construction industry makes use of sandwich panels for the lightweightdesign of large structures, such as the Stockholm Globe Arena, shown in figure 4.

Figure 3: Gardermoen high-speed train(Adtranz ABB Strommens)

Figure 4: Stockholm Globe Arena(Stockholm Globe Arenas)

Ongoing active development continues on the field of sandwich structures andthe question of load introduction is a remaining and recurring challenge for designersand engineers. As of today, the difficulties encountered with load introductions are

2 Steven Ribeiro-Ayeh

frequently approached as isolated problems and not integrated in the overall designprocess at an as early stage as one could wish. The premiere reason for this is thelack of reference cases and documented knowledge in terms of systematic, analyticalmethods for the design of these load introductions. In consequence, considerableweight may be added to structures as a result of inferior design; both through extramaterial (such as denser cores, thicker faces, larger inserts and more adhesives) butalso due to low allowable design loads and cautiously high safety factors to com-pensate for poor efficiency and reliability. Thus, the design of inserts is oftentimesless than optimal and stands in sharp contrast to the substantial weight savings thatcan be achieved by means of advanced sandwich design and improved manufacturingmethods.

2. Strength analysis

A sandwich construction is an inhomogeneous compound of very specialised, overallhigh but locally low performance materials. Sandwich plates with commonly usedtransversely flexible core materials, such as cellular foams or honeycombs lack thenecessary stiffness and strength to sustain localised loads. Therefore, load introduc-tion into such a sandwich structure relies on the integration of adequately designedtransition media, commonly referred to as inserts (Fig. 5). The purpose of an insertis to distribute localised loads in the panel in an appropriate manner [1]. The core

Figure 5: Aluminium insert in a sandwich beam [2]

is often replaced locally by a more high-performing material, such as foam of higherdensity or by composite or metal inserts which provide sufficient local strength andcan accommodate mechanical fasteners for the transfer of external loads into thesandwich structure.

2.1. Failure assessment

The material properties, the mechanical sandwich properties and the insert geo-metry are the foremost factors determining the load bearing capabilities of an in-sert. Comprehensive formulas for the design and dimensioning of load introductionsand inserts have yet to be developed. For specific design cases, such as some spacevehicles, handbooks presenting design charts and formulae for inserts exist. Here thecapabilities of inserts are determined by utilisation of measured material properties


in combination with experimentally verified analytical models. These may be usedfor preliminary design but they may then need to be substantiated by testing of theactual configuration [3, 4].

(a) internal metal doubler [5] (b) Hour-glass shaped insert [6]

Figure 6: Examples of failure near inserts

The use of local reinforcements creates an assembly of materials of differentstiffness, with differences in elastic constants and mode of deformation. Structuralfailure is oftentimes observed near or at an insert (Fig. 6). The existence of multi-material corners of dissimilar material and challenging local geometries are believedto determine the stress field in the region closest to the insert. Due to geomet-ric and/or material mismatch, areas of high local stress may be observed at theselocations [7]. The local stress fields could in part be described by use of fracturemechanics terms, yet they are set aside in many analyses [8, 9, 10] and only a lim-ited number of attempts to actually characterise the stress fields in the immediatevicinity of an insert exists.

The local stress fields are believed to contribute substantially to the initiationof fracture and material failure and should therefore be given special considera-tion when estimating the load bearing capabilities of a structure [11]. Analyticalapproaches [12, 13] and numerical solution routines [14, 15] have been derived forspecific material combinations and geometries, but the analysis of practical prob-lems often requires a combined approach of both analytical and numerical descrip-tions [16].

Failure criteria

A more general approach which may be used within the scope of a detailed finiteelement analysis is desirable for the analysis of stress fields in the development ofload introductions and a stress-based failure criterion is suggested herein.The point-stress criterion, originally proposed by Whitney and Nuismer [17] mayprovide a tool for the determination of the strength of a brittle material when themaximum stress is highly localised. Independent of the local geometry, the point-stress criterion relates relates general stress level at a certain distance from a stressconcentration to the un-notched strength and the fracture toughness of the material.


The point-stress criterion has been successfully applied for the prediction of fail-ure due to geometric discontinuities in structural PVC foam [18, 19], as well as forfailure prediction of specific insert configurations in sandwich plates with materialand geometric discontinuities [5, 6].

As it had been utilised in earlier work [18, 19, 20], the point-stress criterion wasonly applicable locally at stress concentrations, operating on the maximum tensiletangential stress σϕϕ at a characteristic distance rch from a notch tip (Fig. 7). Ageneralisation of the criterion by formulating it to work on the first principal stressσ11, allowed for the use of the criterion beyond the immediate vicinity of a stressconcentration [21]. Then the strength of bi-material interfaces could be analysed,taking material properties and the local geometry into consideration. By establishinga relationship between the failure strength and the geometry of the bi-materialinterface (Fig. 8) a tool was obtained, which in the future could permit the use ofthe point-stress criterion for the selection of suitable material pairings and for thedesign of optimised interface geometries.

σσ

σϕϕ

σϕϕ

r

ϕ

Figure 7: Local geometry at an arbit-rary wedge notch [19].

E2, ν2

E1, ν1

ϕφ1

rφ2

Figure 8: Bi-material interface corner

2.2. Analysis of bi-material interfaces

The proposed generalisation of the point-stress criterion allowed for its applicationbeyond the immediate vicinity of a stress concentration (Fig. 9). Evaluated within

E2 E1

rch

rch

ν2 ν1

Figure 9: Evaluation of the point-stress criterion (hatchedarea) Figure 10: Fracture near an insert [22]

the whole of the test specimens, the criterion could be used to identify failure critical


areas (Fig. 10) and to estimate the corresponding failure load.

In the case of bi-material beams, it showed to be possible to obtain numericalestimates of the failure strength which were in very good agreement with experi-mental data. Furthermore, a transition between different modes of failure could beidentified, which could confirm the usability of the point-stress criterion as a localcriterion in the vicinity of a stress concentration as well as globally in the whole ofthe test specimens.In the analysis of aluminium inserts embedded in Rohacell r© foam [23], where mul-tiple stress concentrations were present, the point-stress criterion could be used tosuccessfully identify the failure critical areas. Estimates of the failure strength wereconservative and in reasonable agreement with experimental results.

(a) Insert (lower corner is not critical) (b) Bi-material beam

Figure 11: Failure locations distant from a stress concentration

Figures 11 and 12 show the experimental analysis of inserts and of bi-materialbeams respectively. The fracture locations clearly differ, depending on the existence

(a) Insert (lower corner is critical) (b) Bi-material beam

Figure 12: Failure near a critical corner

and location of a failure critical stress concentration. The large fracture surfacewhich does not reveal the lower corner of the insert (Fig. 11(a)) and the fracture


location distant from the bi-material interface (Fig. 11(b)) suggest, that for thesegeometries failure was probably not due to local stress concentrations. Fracture nearthe critical lower corner of the insert (Fig. 12(a)) and near the bi-material interface(Fig. 12(b)) on the other hand, may lead to the conclusion that the existence of highlocal stresses may have played a role.

In correspondence with the experiments, the numerical predictions captured therelationship between interface angle or insert geometry and failure load.

3. Stiffness analysis

A foam consists of gas cells or bubbles in a liquid medium, separated by thin filmsthe cell walls or cell faces. Depending on their size and gas pressure, the cells maybe approximately spherical (wet foam or soap froth) or form polyhedra (dry foamor cellular solid). In the current analysis, the latter kind of foams are considered

(a) Closed-cell polystyrene foam(Ohio State University)

(b) Open-cell ceramic foam (Penn State Uni-

versity)

(c) Closed-cell PVC foam (d) Stabilised aluminium foam

Figure 13: Examples for the microstructure of different foam materials

as they can be found for example in the aforementioned engineering applications.


Since the technical properties of these dry foams are comparable to the propertiesof a solid material with voids, the foams are also referred to as cellular solids.

In the stiffness analysis of cellular solids, the description of the material is shiftedfrom a macroscopical point of view, where the material was treated as a continuumsolid to a microscopical approach, taking into account the structural complexity ofthe foam. In figure 13 examples are given of the variety of microstructures foundin different cellular solids. It is evident that the foams are irregular structures,consisting of a network of interconnected cells of individual sizes and geometries.

The mechanical properties, such as strength and modulus, of cellular solids arelargely determined by the geometrical features of the material and by the propertiesof the solid material from which the foam is made [24].

3.1. Previous analytical and numerical models

The mechanical properties of cellular solids may be determined experimentally onthe microscale level [25], but advanced specimen preparation is required [26] andthe results may be influenced by local inhomogeneities of the material.

Gibson and Ashby [24] developed analytical foam models based on an idealisedsingle cell geometry, well suitable for the description of open- and closed-cell foams.Despite their simplicity, they offer a good representation of the cell mechanics andwere successfully used for the derivation of various scaling laws for the properties ofcellular solids [24].

Figure 14: Foam model of monodispersetetrakaidecahedral cells

Figure 15: Beam model of a randomopen-cell foam

Micro-mechanical models of open- and closed-cell foam may be created on thebasis of originally ordered lattice structures. Examples of such are periodic Kelvinfoam structures (Fig. 14) which were examined in ordered and in randomised versionsby various authors [27, 28]. Finite element simulations of various types of open-cellfoams (Fig. 15) may be made by modelling the cell edges as beams [27, 29, 30]. For


periodic, repeating structures of geometric regularity, the analysis may be simplifiedby using partial models of foam cells [27, 31, 32].

Other modelling techniques in which a representation is sought which closelyresembles the microstructure of a real cellular solid may be the use of tomographicimages, digitised images [33, 34], or x-ray scans [35, 36] of a foam. While being aclose digital reproduction of the structure of a real foam, these models are unique foreach individual foam sample and may not necessarily be representative. Generally,digitised models are not periodic, which may hamper the application of boundaryconditions in finite element models.The mechanical properties of the model foam may depend on the digitising processand the obtained resolution of the model [37]. A coarse resolution may introducenumerical errors to the FE analysis which could be avoided by a including a highlevel of detail in the digitising but then it may not always be possible to create afinite element model from the data as the model may be very large [38].

3.2. Cell growth - Voronoi structure

Models of cellular solids may be generated by means of Voronoi tessellation of dis-

(a) nucleation (b) cell growth (c) cell growth

(d) cell growth (e) consolidation (f) Voronoi tessellation

Figure 16: Two-dimensional cell growth simulation


tributions of seed points in space. In a Voronoi tessellation, a cell is defined by thespace that is closer to a specific seed point than to any other. Mathematically, theVoronoi tessellation is obtained by allowing spherical bubbles to grow with uniformvelocity from each of the seed points (Fig. 16). Wherever the bubbles touch, growthis halted at the contact surface, but allowed to continue elsewhere (Figs. 16(b)–16(e)). In this respect the tessellation is similar to the actual process of liquid foamformation [39].

The amount of structure or disorder in the Voronoi tessellation depends on thespatial distribution of the seed points. If they are arranged in regular arrays or grids,models of ordered foams may result (Fig. 14). Seed points in a Poisson distribution,randomised lattice arrangements or seeds deployed by random sequential adsorption(RSA) algorithm [40, 41] may result in unstructured, random Voronoi tessellations.The seed distribution influences the cell morphology, the spatial disposition of thecells, and the cell size distribution (Fig. 17).

3.3. Finite element modelling

The micro-mechanical foam model was prepared as a spatially periodic model builtfrom a cubic unit element of foam, a so called “representative volume element”(RVE) or “statistically homogeneous specimen” [42]. Countless identical copies ofthe RVE fit together to fill space, forming a space-tiling periodic domain. Eachunit element must be large enough to be statistically representative and it may thuscontain numerous foam cells of different size and shape (Fig. 18).

Figure 17: Cells in a random Voro-noi tessellation

Figure 18: Representative volumeelement (RVE)

Finite element (FE) models of open-cell cellular solids may be created by mesh-ing the cell edges with beam elements (Fig. 15). For the analysis of closed-cellfoams, the cell faces can be meshed with either shells or membrane elements andreinforcing beam elements must be used along the cell edges. Besides the relative


density, the distribution of solid between cell faces and edges must be taken intoaccount to obtain reasonable stiffness estimates. Models made of shell elements,where all material is located the cell faces may overestimate the properties of poly-mer foams [27, 30]. The geometry of the beam elements may be simplified to acircular cross section instead of the triangular Plateau border shape. This may leadto a reduction of the principal second moment of area and thus a somewhat lowerbending stiffness of the edges with respect to the real material [27]. The stiffnessreduction may be negligible in the modelling of closed-cell foams, where the cellfaces predominantly contribute to the foam stiffness [43].

3.3.1. Periodic boundary conditions

The use of a parallelepipedal RVE facilitates the application of periodic boundaryconditions to the finite element model and can allow for multi-axial loading of themodel. The use of periodic boundary conditions implies that the RVE is surroundedby images of itself ad infinitum, meaning that nodes on any pair of boundary surfaceson opposite sides of the RVE have identical opposite displacement components [44,45].

A B

C

D

Figure 19: Schematic periodicmodel Figure 20: Finite element mesh

When denoting a pair of opposite boundary surfaces A and B (Fig. 19), theboundary condition may be stated as uA

i = −uBi for the displacement vectors ui of

a node pair. The displacement conditions on the boundaries of the RVE may thenbe written by specifying the average strain over the volume of the RVE, εij, as [46]:

εij =δu

δx=

uBi − uA

i

xBj − xA

j

(1)

For the implementation of the periodic boundary conditions in the FE model(Fig. 20), the mesh on any opposite sides A and B needs to match exactly, whichmay be achieved by using a mesh generator such as the Cubit [47] where such acondition can be prescribed.


3.3.2. Evaluation of elastic moduli

Series of FE analyses (Fig. 21) allow for the population of the compliance matrix Sof each RVE. Subsequently, the elastic properties may be determined by identifying

(a) Uniaxial tension (b) Shear

Figure 21: Nodal displacement plots [48]

the elements sij of the compliance matrix. The directional material properties maythen be averaged by calculating the effective isotropic properties [49].

In figure 22 the elastic properties of a low-density polymer foam obtained from aseries of different FE models are compared to the data for an existing foam, RohmRohacell r© 51WF. The material properties of the Rohacell r© solid material [50] were

100 200 300 400 500 600

60

65

70

75

80

85

90

Number of cells

Eff

ectiv

e Y

oung

’s m

odul

us E

eff [M

Pa]

Numerical simulationSeries averageExperimental dataManufacturer data

(a) Young’s modulus

100 200 300 400 500 600

16

18

20

22

24

26

28

30

32

34

Number of cells

Eff

ectiv

e s

hear

mod

ulus

Gef

f [MP

a]

Numerical simulationSeries averageExperimental dataManufacturer data

(b) Shear modulus

Figure 22: Elastic moduli of Rohacell r© foam models [48]


used in the finite element models and the results were in very good agreement withexperimental data [50, 51] and the specifications of the real foam [23].

3.4. Variations of the cell morphology

3.4.1. Topological disorder

The influence of the seed distribution on the cell morphology and the elastic moduliof a model foam may be studied by evaluating and comparing the results for differentordered and disordered seed distributions [52].

(a) Ordered FCC lattice (b) Disturbed cubic lat-tice

(c) Random seed distri-bution (RSA)

Figure 23: Examples of more or less disturbed Voronoi foam modelscreated from different seed point distributions

To represent the structural disorder in a real foam (Fig. 13), Voronoi tessella-tions for the modelling of cellular solids may be created from random seed pointdistributions [53] or from randomised perturbations of initially structured seed lat-tices [27, 28, 29].

Models created from ordered seed distributions (Figs. 23(a), 14) may be usedto mimic the structural disorder of real foams as the randomness in the seed dis-tribution is gradually increased (Fig. 23(b)). The degree of randomness and thecorresponding geometric disorder may be described through the definition of a “to-pological disorder” [53] or “topological energy” [54] which measures the deviationof the individual cell shapes from geometrically symmetric polyhedra. The higherthe topological disorder, the less uniform the cells.

In the analysis, Voronoi tessellations based on ordered BCC and FCC seed struc-tures were randomised and compared to foam models created from random RSA seedstructures. It was found, that the spatial distribution of the seed points significantlyinfluenced the cell shapes and subsequently affected the mechanical properties of thefoam. The calculated elastic moduli of the ordered structures showed a considerabledegree of anisotropy and they underestimated both Young’s modulus and shear mod-ulus of the real reference foam [52]. Increased randomness in the initially orderedlattices resulted in more isotropic properties and a better estimate of the elastic


moduli, but it was accompanied by a significant increase in cell face area. Denselypacked RSA point distributions on the other hand provided models of random cellstructures, which appeared to be the most suitable representation of a cellular solid(Fig. 23(c)).

3.4.2. Non-planar cell faces

The gas pressure between adjacent cells and the production process may affect theshape of the cell faces. In an ideal, dry foam the cell faces are planar, which maynot be the case for a real foam (Fig. 13). Especially in the case of aluminium foams(Fig. 13(d)) it is believed that the stiffness of the foam can be reduced by curved orcorrugated cell faces [32, 55, 56].

Figure 24: Foam with non-planar cell faces [57]

In order to investigate the influence on the elastic moduli of the foam, an analysiswas performed where the cell faces were given different curvatures (Fig. 24) by meansof Schwarz-Christoffel mapping [58], allowing for an arbitrary definition of an out-of-plane shape of the cell faces.

The results showed, that the stiffness of the foam may be significantly influencedby the shape of the cell faces [57], which was in accordance with related studies onopen-cell foams [59]. It was found, that the elastic moduli were considerably lowerwhen the cell faces were non-planar. The lower the distribution of solid and themore material in the cell faces, the greater was the reducing effect of the cell facecurvature. While thickness variations of the cell faces alone may cause a decrease inshear and bulk moduli [60], it was also specifically shown that the observed reductionin stiffness was due to the face curvature and not only due to the reduction of facethickness associated with the imposed curvature.


3.4.3. Distorted cells

In some foams, the cell shapes may exhibit a substantial deviation from idealisedspherical bubbles (Fig. 13(c)). This distortion of the cells, which may be causedby the foam production process can cause unwanted anisotropic properties of thefoam [61].

Different variations of the cell geometry may be analysed by creating foam modelsof differently distorted cells. The seed point distribution within a RVE may be

(a) elongated cells (b) disc shaped cells

Figure 25: Foam models with differently distorted cells

influenced such that the cells of the foam are of different shape, creating a foamwith controlled anisotropic properties. Figure 25(a) shows a models with cells whichwere elongated to an aspect ratio of [8:1:1] giving them almost cylindrical shapeswith a height of eight times the diameter. In the same manner cells may be givena disc-like appearance such as in the model in figure 25(b) where the cells have anaspect ratio of [1:5:5], meaning the diameter is five times the cell height.

Ongoing research suggests, that the cell shapes might have a profound effect onthe elastic moduli of the foam [62].

4. Future work

Possible further research could include a revision of the point-stress criterion and toimprove it to better match experimental results. This might be accompanied by amodification of the criterion to the analysis of multi-axial stress states.

For the analysis of inserts in sandwich structures it would be of great interest toextend the point-stress criterion and to investigate tri-material interfaces, such asthe junction between sandwich face, core and insert.


In the area of micromechanical modelling of cellular solids, it would be a fascin-ating challenge to use the models for simulations in the field of fracture mechanics(Fig. 26). This could involve the investigation of crack growth and the analysis offracture paths in the foam. Of interest could be the investigation of the shape ofthe crack front and the progress of fracture through the material, which could beeither through the cell faces or along the cell edges.Considering the use of the point-stress criterion, it would be very interesting topossibly establish a link between fracture on the cell-level and failure of the cellularsolid on a macroscopic scale.

Figure 26: Fracture mechanics analysis Figure 27: Foam crushing simulation

Considering the mechanics of cellular solids and their use in and as crash ab-sorbing structures it would be appealing to simulate the crushing of a foam on amicromechanical level (Fig. 27). The transition from elastic deformation to the onsetof failure by cell wall buckling, the plateau phase and the progressing densificationof the foam could be simulated.

The analysis of local defects in the foam might be of interest for practical applica-tions and for the production of foams. One could for example analyse the mechanicsof partially closed-cell foams by including the effects of missing cell walls. The in-fluence of unwanted large gas bubbles, as they might occur in the foam productionprocess could be investigated.

Lastly, the presented micromechanical models might very well be used to tailornew foams by demand. Thus, one could determine the required properties, such ascell size and solid material according to the desired elastic properties of the foam.


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