DEGREE PROJECT IN SOLID MECHANICS,,SECOND CYCLE, 30 CREDITSSTOCKHOLM, SWEDEN 2019

Methods for modelling latticestructures

MONA KOUACH

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

Figure on cover: Lattice structure of size 2x2x2.Copyright c© 2019 Mona KouachAll rights reserved

ROYAL INSTITUTE OF TECHNOLOGY

MASTER’S THESIS

Methods for Modelling Lattice Structures

Author:Mona Kouach

A thesis submitted in fulfillment of the requirementsfor the degree of Master in engineering

in the

Department of Solid Mechanics

September 8, 2019

i

AbstractThe application of lattice structures have become increasingly popular as additivemanufacturing (AM) opens up the possibility to manufacture complex configura-tions. However, modelling such structures can be computationally expensive. Thefollowing thesis has been conducted in order for the department of Structural Analy-sis, at SAAB in Järfälla, to converge with the future use of AM and lattice structures.An approach to model lattice structures using homogenization is presented wherethree similar methods involving representative volume element (RVE) have beendeveloped and evaluated. The stiffness matrices, of the RVEs, for different sizesof lattice structures, comprising of BCC strut-based units, have been obtained. Thestiffness matrices were compared and analysed on a larger solid structure in order tosee the deformational predictability of a lattice-based structure of the same size. Theresults showed that all methods were good approximations with slight differencesin terms of boundary conditions (BCs) at the outer edge. The comparative analysesshowed that two of the three methods matches the deformational predictability. TheBCs in all methods have different influences which makes it pivotal to establish theBCs of the structure before using the approach presented in this thesis.

Keywords: Lattice structures; Finite element method; Homogenization; Representa-tive volume element

ii

SammanfattningÖkad implementering av gitterstrukturer i komponenter är ett resultat av utvecklin-gen inom additiv tillverkning. Metoden öppnar upp för tillverkning av komplexastrukturer med färre delmoment. Dock så uppkommer det svårigheter vid simuler-ing av dessa komplexa strukturer då beräkningar snabbt tyngs ner med ökad kom-plexitet. Följande examensarbete har utförts hos avdelningen Strukturanalys, påSAAB i Järfälla, för att de ska kunna möta upp det framtida behovet av beräkningarpå additivt tillverkade gitterstrukturer. I det här arbetet presenteras ett tillvägagångsättför modellering av gitterstrukturer med hjälp av represantiva volymselement. Styvhets-matriser har räknats fram, för en vald gitterkonfiguration, som sedan viktats mottre snarlika representativa volymselement. En jämförelseanalys mellan de olikastyvhetsmatriserna har sedan gjorts på en större och solid modell för att se hurväl metoderna förutsett deformationen av en gitterstruktur i samma storlek. Re-sultaten har visat att samtliga metoder är bra approximationer med tämligen småskillnader från randeffekterna. Vid jämförelseanalysen simulerades gitterstrukturenbäst med två av de tre metoder. En av slutsatserna är att det är viktigt att förståinverkan av randvillkoren hos gitterstrukturer innan implementering görs med dettillvägagångssätt som presenterats i det här examensarbetet.

iii

Acknowledgements

During the spring of 2019, this thesis project was carried out at the department ofStructural Analysis, at SAAB Surveillance in Järfälla.

I would like to give my sincerest gratitude to my supervisor at SAAB, M.Sc. AndersMolin for the guidance, knowledge and support. It has been truly wonderful to workwith you again. I would also like to extend my warmest gratitude to my supervisorat KTH, Dr. Carl Dahlberg for the expertise, ideas and supervision.

And to all the wonderful colleagues at SAAB Surveillance in Järfälla, I thank youall for making my time at the office very welcoming and delightful. It has been aninvaluable experience that I will bring with me further on.

Last but not least, I am indebted to M.Sc. Marcus Nilsson at SAAB, for my extensiveexperience within SpaceClaim and ANSYS, which has helped me a lot during theproject course.

Mona KouachStockholm, SwedenJune 2019

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Table of Contents

List of Figures v

List of Tables vi

List of Acronyms vii

1 Introduction 11.1 Cellular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Lattice structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Evaluation methods . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Homogenization with RVE . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Methodology 102.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 RVE Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Results and Discussion 18

4 Conclusions and Further Work 28

References 29

v

List of Figures

1.1 Examples of cellular materials . . . . . . . . . . . . . . . . . . . . . . . . 21.2 General overview of additive manufacturing with selective laser melt-

ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Example of a lattice configuration. . . . . . . . . . . . . . . . . . . . . . 61.4 The outer boundary surfaces of the example lattice configuration. . . . 7

2.1 Flow chart of the approach. . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Overview of the constructed lattice unit. . . . . . . . . . . . . . . . . . . 122.3 Different expanded sizes of lattice structures. . . . . . . . . . . . . . . . 122.4 Overview of the RVEs for each method. . . . . . . . . . . . . . . . . . . 132.5 Overview of benchmark lattice geometry. . . . . . . . . . . . . . . . . . 152.6 Overview of the solid benchmark geometry. . . . . . . . . . . . . . . . . 162.7 The boundary conditions of the benchmark lattice structure. . . . . . . 17

3.1 One of the resulting stiffness matrix for lattice size N = 7, evaluatedwith M1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 C11 stiffness values against lattice size N. . . . . . . . . . . . . . . . . . 193.3 C22 stiffness values against lattice size N. . . . . . . . . . . . . . . . . . 193.4 C12 stiffness values against lattice size N. . . . . . . . . . . . . . . . . . 203.5 C13 stiffness values against lattice size N. . . . . . . . . . . . . . . . . . 203.6 Illustration of the lateral contraction on the lattice unit. . . . . . . . . . 213.7 Illustration of shear strain on the lattice unit. . . . . . . . . . . . . . . . 213.8 C44 stiffness values against lattice size N. . . . . . . . . . . . . . . . . . 223.9 C66 stiffness values against lattice size N. . . . . . . . . . . . . . . . . . 223.10 Comparison between the three methods in terms of all the stiffness

matrix components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.11 Comparison between the three methods in terms of the stiffness ma-

trix components C31, C13 & C66. . . . . . . . . . . . . . . . . . . . . . . . 243.12 Benchmark analysis: Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . 253.13 Benchmark analysis: Method 1. . . . . . . . . . . . . . . . . . . . . . . . 253.14 Benchmark analysis: Method 2. . . . . . . . . . . . . . . . . . . . . . . . 263.15 Benchmark analysis: Method 3. . . . . . . . . . . . . . . . . . . . . . . . 26

vi

List of Tables

1.1 Displacement input depending on the strain state. . . . . . . . . . . . . 8

2.1 Material parameters for Aluminium Alloy. . . . . . . . . . . . . . . . . 14

3.1 Results of directional deformation for all benchmark analyses. . . . . . 243.2 Number of elements, nodes and computational time in the benchmark

analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

vii

List of Acronyms

AM additive manufacturing.

BCs boundary conditions.

CS coordinate system.

FE finite element.

FEA finite element analysis.

FEM finite element method.

M1 Method 1.

M2 Method 2.

M3 Method 3.

PBCs periodic boundary conditions.

RVE representative volume element.

1

Chapter 1

Introduction

In the aerospace industry, lightweight structures that uphold robust properties areof high interest. Due to traditional manufacturing constraints, parts can tend to beover designed and thus have more weight than needed. However, additive man-ufacturing (AM) opens up for structural designs that previously were not possibleto manufacture such as lattice structures. With the extensive research published [1],one can see it implemented in a number of industries and companies, includingSAAB.

With implementation of lattice structures in a design, the same structural benefits ase.g. a sandwich composite can be achieved [2]. Lattice structures can be engineeredto have the required mechanical properties depending on the surrounding loads.

A lattice structure is often detailed and fine in its design, which easily gets costly orimpossible to simulate with today’s engineering simulation software, in an efficientway. In order for the department of Structural Analysis, at SAAB in Järfälla, to meetwith the future use of AM, some of these constraints need to be cleared up.

The aim for this master thesis is to increase the knowledge in the different meth-ods to simulate lattice structures with finite element (FE)-software and to present anapproach for such simulations.

This will be achieved by evaluating different methods of homogenization on a lat-tice configuration unit, analysed with 3D elements. The methods will then be com-pared with each other to see how well their deformational predictability match witha larger and more complex structure consisting of only the chosen lattice units.

This report is limited by the following assumptions:

• Open cell structures

• Isotropic materials

• Linear elastic strains

• Homogenization using cuboid representative volume element (RVE)

Chapter 1. Introduction 2

1.1 Cellular Materials

Cellular materials, also denoted as cellular solids or foams, are space-filling unit cellsthat can be tessellated along any axis with no gaps between the cells. A lightweightand robust design of a motorised system has an advantage due to its lower fuel con-sumption during operations. The unique traits of heat dissipation, weight-ratio andenergy absorption in cellular materials [3] are what makes the use of cellular materi-als in lightweight designs popular. As mentioned by L.J. Gibson and M.F. Ashby [3],"The subject is important to us here because the properties of cellular solids dependdirectly on the shape and structure of the cells."

There are two major groups that cellular materials can be divided into, the first iscalled stochastic, which include materials with voids that have been randomly dis-tributed during the manufacturing process. Examples of stochastic cellular materialsare natural materials such as sponges, cork and wood. [3] The second group is calledregular and comprise of cellular materials which are ordered. Examples of cellularmaterials can be seen in Figure 1.1.

FIGURE 1.1: Examples of cellular materials (a) An open cell foam. (b)A closed cell foam. (c) A multi-layer lattice truss. (d) A prismatic

corrugated core. [4]

Man-made cellular materials, shown in Figures 1.1a-1.1b, can be manufactured byeither liquid or solid state processing. Polymer foams can be foamed by adding inertgas bubbles into hot polymer or liquid monomers, which then grows and solidifiesinto stochastic foams. Open-celled metal foams can be made by casting metallicpowder together with space-holding particles. [5]

The manufacturing process of closed cell metal foams are a bit similar to polymericfoams, as inert gas is injected into molten metal. This process can generate both astochastic and regular foam depending on the sizes of the space-holding particlesand how they are inserted.

Chapter 1. Introduction 3

Until recently, the manufacturing of complex geometries with cellular materials wasa challenge. With the advancements in AM, creating cellular geometries in variousdifferent materials have become more accessible. The general definition of AM in-cludes all methods that repeatedly adds material together, where the more commonname is 3D-printing. A sandwich beam, for instance, can be made in one step insteadof several, which makes it an interesting process when looking at the manufacturingprocesses and lead time. Several components can be made simultaneously when as-suming a big enough building plate.

FIGURE 1.2: General overview of Additive Manufacturing with se-lective laser melting. [6]

Regular cellular materials can have either a two-dimensional cell configuration or athree-dimensional configuration.

Two-dimensional arrays with assemblies of triangles, squares or hexagons are calledhoneycombs, and there exists both man-made and natural cellular structures. How-ever, naturally occurring cellular structures include frequent deviations from its reg-ular form due to the individual cells ability to nucleate and grow. [3].

Three-dimensional cell configurations, also called lattice structures, are made up ofan array of one specific cell unit, e.g. the cell unit seen in Figure 1.1c. The majority ofthree-dimensional cellular materials are anisotropic, which often is a desired quality,as one direction can be designed to be more stiff than the others.

1.1.1 Lattice structures

There are a few different ways of achieving a design containing lattice structures,depending on the lattice type. The most common way is by newly added featuresin many popular CAD-software. It is also possible to use optimisation methodsto generate different types of lattice structures depending on the loading and thepreset conditions. The optimisation tools are often found in FE-software providedby companies such as ANSYS and Altair.

As always, it is important for the mechanical design engineer to know beforehandabout the manufacturing methods and their attributes. One essential attribute toconsider is that lattice structures need to be seen as materials engineered dependingon its’ use rather than general materials to use in a design. The rule of thumb with

Chapter 1. Introduction 4

AM is that if one can avoid using this method, then one should do so. Especially ifthe manufacturing infrastructure already exists. [7]

One of the disadvantages with AM is with the thermal distortions that can arisein a part that is being manufactured. This is due to high temperature differencesduring the printing process and it could have a large negative effect on the part.This can be managed by foreseeing the distortions, early in the process, and reducethe possible trial-and-error experiments. In those cases, ANSYS Additive PrintTM

can be used to simulate how 3D-printed parts distort during manufacturing with ahigh predictability.1

1.1.2 Evaluation methods

The most regular way of evaluating the properties of lattice structures is with exper-imental methods. It is also common to combine experimental data with FE-models.The two primary methods of simulating lattice structures are with either finite ele-ment analysis (FEA), also referred as finite element method (FEM), or with homog-enization, where the latter is a simplified FEA and will be further described in thisreport.

Al-Ketan et al. [8] investigated numerous of cellular structures to find the topology-property relationship of the different cellular classes. This was done by conductingcompressive testing to evaluate the mechanical properties and the results showedthat the sheet-Diamond cellular structure had the best mechanical performance.

With FEA, the lattice structures can be modelled by using 3D-elements or beamelements depending on the analysis type. Beam elements are cheaper and moreefficient in terms of computational time. Timoshenko beam elements, which is ofhigher order, has been shown to be a very suitable element type for modelling thenonlinear deformation of open cell structures. Luxner et al. [9] compared the simu-lations on various lattice structures by using continuum elements and Timoshenkobeam elements in order to conclude the adaptation of beam elements. The con-clusions showed that the application of beam elements represented the mechanicalbehaviour very well. The results of the simulations were also compared with ex-periments with printed lattice structures, where uniaxial compression was placedon one of the principal directions. Smith et al. [10] also compared both FE-modelsusing 3D and beam elements, of two variations of body centred cubic (BCC) latticestructures, with experimental data. The analyses were set with compressive loadsand the results showed that the FE-models were in agreement with the experiments.

To achieve an accurate model that matches the reality, periodic boundary conditions(PBCs), which forces two parallel faces to deform in the same way can be applied[11]. These are a set of prescribed BCs used on a unit cell in order to approximatea much larger structure. When using PBCs with FEM, the nodal placements needsto be controlled and can be done easily with most simple geometries. However, asthe complexity increases with lattice geometries, so does the difficulty with PBCs.According to Nanakorn and Theerakittayakorn [12], the use of PBCs is uncompli-cated with solid elements but not with beam elements. In this study, the distinctionsbetween solid elements and Euler beam elements are discussed. Tollenaere and Cail-lerie [13] suggests a method for continuous modelling of quasi-periodic lattice struc-tures by using discrete homogenization and PBCs.

1https://www.ansys.com/products/structures/ansys-additive-print

Chapter 1. Introduction 5

Titanium foams, which have been manufactured using the space holder method,was by Simoneau et al. [14] compared with numerically-generated foams in orderto validate their algorithm. The smallest numerically converging representative vol-ume element (RVE) in a foam, could be found with the algorithm which is based ona sampling method called stereology.

In order to evaluate which of the different types of BCs are suitable on lattice struc-tures, Terriault and Brailovski [11] chose an approach with homogenization. Adiamond-type lattice unit structure was simulated under compression, with solid el-ements, and the concluding remarks were that PBCs are suitable for free-deformingstructures and frictionless BCs are applicable for structures that are rigidly enclosed.The structural convergence, described in [11], is similar to the methodology de-scribed in this report.

Chapter 1. Introduction 6

1.2 Homogenization with RVE

With anisotropic elastic properties, the stiffness differs in the different directions ofthe material. Most metallic materials are usually considered to be isotropic, butdepending on the symmetry of the lattice configuration, the behaviour might beanisotropic.

The methodology described in this report is applicable on any lattice structure, re-gardless of geometrical symmetries and materials as long as the analyses are consid-ered to be elastic. Note however that full anisotropy cannot be represented with thismethod.

The method that is used in this project is called homogenization, which is a processfor extracting macro-mechanical behaviour from micro-mechanical properties [15,16]. The general idea is that the micro-mechanical properties are averaged over achosen representative volume called representative volume element (RVE). An RVEis considered to be the smallest constituent volume sample of a material which stillrepresents the whole material. Homogenization is commonly used with compositematerials, but also with voids and defects in materials e.g. with steels. It is nonethe-less important to be aware of the relationship between the micro-mechanical prop-erties and the macro-mechanical in all materials.The general weighted formula of homogenization looks like,

⟨f⟩=

1V

∫V

f (x) dV, (1.1)

where the function f, in this report, represents the properties of the lattice,⟨f⟩

is theeffective function value and V is the volume of the RVE.

Consider the lattice unit shown in Figure 1.3 with the coordinate system (CS) as 123.

FIGURE 1.3: Example of a lattice configuration.

In each point there will be a stress σ and the macroscopic stress 〈σ〉 is searched for thewhole RVE. In particular, the relationship between a stress 〈σ〉 and its correspondingstrain 〈ε〉 which is described with the stiffness matrix C, see Equation 1.2.

Chapter 1. Introduction 7

〈σ1〉〈σ2〉〈σ3〉〈σ4〉〈σ5〉〈σ6〉

︸ ︷︷ ︸〈σ〉

=

C11 C12 C13 C14 C15 C16

C21 C22 C23 C24 C25 C26

C31 C32 C33 C34 C35 C36

C41 C42 C43 C44 C45 C46

C51 C52 C53 C54 C55 C56

C61 C62 C63 C64 C65 C66

︸ ︷︷ ︸

C

〈ε1〉〈ε2〉〈ε3〉〈ε4〉〈ε5〉〈ε6〉

︸ ︷︷ ︸〈ε〉

(1.2)

where C44 = C55 = C66 = 2 G and G is the shear modulus. With the symmetry ofthe stiffness matrix, the tensor notation becomes,

Cij = Cji (1.3)

The strains are described through nodal displacements which lie on the outer bound-ary surfaces on the lattice unit, can be seen as the orange surfaces on Figure 1.4.

FIGURE 1.4: The outer boundary surfaces where the prescribed dis-placements are applied.

The first three components in 〈ε〉 represents tensile strain states and the three latterrepresents shear strain states. For instance, the first strain state is prescribed dis-placements on the 1-direction, then the macroscopic strain would look like,

〈ε〉T =[〈ε1〉 0 0 0 0 0

](1.4)

from which the inserted nodal displacements on the outer boundary surfaces be-come,

Chapter 1. Introduction 8

u1 = 〈ε1〉 x1

u2 = 0u3 = 0

(1.5)

where ui is the displacement in the i-direction, 〈ε1〉 is the strain input and x1 de-scribes the nodal coordinates.

In the case of the fourth strain state, shear strain in e.g. 12-plane, the macroscopicstrain would look like,

〈ε〉T =[0 0 0 〈ε4〉 0 0

](1.6)

where the inserted nodal displacements become,

u1 =12〈ε4〉 x2

u2 =12〈ε4〉 x1

u3 = 0

(1.7)

This is then solved with FEM. For all of the six strain states, non-zero 〈ε〉 yields sixdifferent solutions, see Table 1.1.

〈ε1〉 〈ε2〉 〈ε3〉 〈ε4〉 〈ε5〉 〈ε6〉

u1 〈ε1〉 x1 0 0 12 〈ε4〉 x2 0 1

2 〈ε6〉 x3

u2 0 〈ε2〉 x2 0 12 〈ε4〉 x1

12 〈ε5〉 x3 0

u3 0 0 〈ε3〉 x3 0 12 〈ε5〉 x2

12 〈ε6〉 x1

TABLE 1.1: Displacement input depending on the strain state.

For each FEM solution, a stress field is given in each point. With that, a mean stressfield can be evaluated for the whole RVE.

〈σij〉 =1

VRVE

∫Vtot

σij dV (1.8)

where VRVE is the volume of the chosen RVE and Vtot is the volume of all N elementsof the lattice unit. An approximation can be applied as,

〈σij〉 =1

VRVE

N

∑k=1

σkij Vk, (1.9)

where k is the element number, σkij is the element average stress value and Vk is the

volume of the element.

Every strain state solution yields a column in the stiffness matrix. In the case of thefirst strain state, the expression would look like,

Chapter 1. Introduction 9

〈σ1〉〈σ2〉〈σ3〉〈σ4〉〈σ5〉〈σ6〉

=

C11 C12 C13 C14 C15 C16

C21 C22 C23 C24 C25 C26

C31 C32 C33 C34 C35 C36

C41 C42 C43 C44 C45 C46

C51 C52 C53 C54 C55 C56

C61 C62 C63 C64 C65 C66

〈ε1〉00000

(1.10)

and the first column of C rewritten,

⇒

〈σ1〉〈ε1〉〈σ2〉〈ε1〉〈σ3〉〈ε1〉〈σ4〉〈ε1〉〈σ5〉〈ε1〉〈σ6〉〈ε1〉

=

C11

C21

C31

C41

C51

C61

(1.11)

And like so, the whole stiffness matrix of the RVE can be constructed.

10

Chapter 2

Methodology

In this section, the method of this project is presented. The developed and resultingapproach has been translated into a flow chart, which can be seen in Figure 2.1 wherethe three headlines, Design, FEM and Evaluation, will be further detailed in theircorresponding sections.

The CS shown in the figures which are in the form of XYZ-system are equivalent tothe 123-system used in Section 1.2. The FE-software, used during in the project, hasbeen ANSYS Workbench and both Excel and MATLAB have been utilised duringthe evaluation step.

Chapter 2. Methodology 11

FIGURE 2.1: Flow chart of the approach.

Chapter 2. Methodology 12

2.1 Design

An open cell lattice core design was constructed with equal sizes of height, depthand width, see Figures 2.2a-2.2b, and its’ configuration is called BCC strut-based lat-tice. The struts of the lattice are solid and have an angle of 45◦ which is the maximumpossible angle to print with AM, without using support structures.

In order to evaluate whether components in the stiffness matrices converges to spe-cific values, the lattice structure was further expanded. By adding a repetition of theconstructed lattice configuration unit at each direction of the CS, see Figures 2.3a-2.3f.

(A) Height of the lattice unit. (B) Depth, width and strut diameter of the lattice unit.

FIGURE 2.2: Overview of the constructed lattice unit.

(A) N=2 (B) N=3 (C) N=4

(D) N=5 (E) N=6 (F) N=7

FIGURE 2.3: Lattice structures with sizes referring to NxNxN.

Chapter 2. Methodology 13

2.1.1 RVE Methods

The three RVE methods that have been developed are all based on the theory de-scribed in Section 1.2 and are mentioned as Method 1 (M1), Method 2 (M2) andMethod 3 (M3).

M1 can be applied at any lattice structure size and its’ RVE is of the whole structure,see Figures 2.4a & 2.4d. Starting from a lattice structure size of N ≥ 3, M2 can also beutilised. The RVE of this method can be seen in Figures 2.4b & 2.4e, where the RVEis one layer less, of the repeated lattice structure, at each direction of the structure.M3 can too be utilised from a size of N = 3 but with only odd numbered sizes asthe RVE encases the most centred lattice unit in the structure, see Figures 2.4c & 2.4f.This method resembles the structural convergence method described in [11].

(A) M1 (B) M2 (C) M3

(D) M1 (E) M2 (F) M3

FIGURE 2.4: Overview of the RVEs for each method, for lattice size ofN=7.

2.2 FEM

Aluminium alloy has been used in the analyses, see material properties in Table 2.1,as it is one of the most commonly used materials with AM.

As the design is imported into the FE-software, the yield criteria of the isotropicmaterial is removed as the analyses are only elastic. By leaving the yield criteria, theiterations can be prolonged as the FE-software always needs to check whether thecriteria is achieved or not.

The type of mesh elements used are tetrahedral elements which are a higher ordertype of 3D-elements with quadratic deformations behaviour. [17] During the project,the same element size, in the mesh, was used in order for the results to be compara-ble.

Chapter 2. Methodology 14

TABLE 2.1: Material parameters for Aluminium Alloy.

Variable Value Unit DescriptionE 71 GPa Young’s modulusG 26.7 GPa Shear modulusν 0.33 1 Poisson’s ratio

With the use of command inputs in the FE-software, the nodes on the outer bound-ary surfaces are grouped. The command inputs also calculates the volume of thecorresponding RVE, VRVE, depending on the chosen method and the size of the lat-tice structure, see Section 2.1.1.

During each of the six strain states, the boundary conditions (BCs), for each coordi-nate direction, are placed onto each node in the group, the strain states can be seenin Table 1.1.

As the solver is finished, the command input sums all the stress components in eachelement that reside inside of the chosen RVE. The summed components are thendivided with the calculated VRVE, see Equation 1.9, to which the results are saved.

2.3 Evaluation

The stiffness matrix, for the lattice size and chosen method, is evaluated using Equa-tion 1.2.

The highest level of anisotropy that can be captured with this method is a cubicanisotropy. For simplicity however, the behaviour of the chosen lattice structureis described as transversely isotropic, due to the geometrical symmetry of the XZ-plane, see Figure 2.2b. With that in mind, the stiffness matrix in Equation 1.2 wasexpected to look like,

C11 C12 C13 0 0 0C22 C12 0 0 0

C11 0 0 0C44 0 0

C44 0

sym. C11−C132

(2.1)

which yields five independent stiffness components. However, by rewriting C11−C132 =

C66, six indifferent non-zero coefficients are expected to define the stiffness matricesfrom the three methods,

Chapter 2. Methodology 15

C11 C12 C13 0 0 0C22 C12 0 0 0

C11 0 0 0C44 0 0

C44 0sym. C66

(2.2)

2.4 Benchmarking

In order to compare all three methods and identify which methods that predictedthe behaviour of a much larger structure, a benchmark structure was created, seeFigures 2.5a-2.5b.

(A) Side view

(B) Top view

FIGURE 2.5: Overview of benchmark lattice geometry.

Chapter 2. Methodology 16

The limitations of the design space was set by the remote computer, used for FEAat SAAB Surveillance. With the known RAM and the memory of a single meshedlattice unit, the design space was spanned.

The aim of the benchmark analyses was to activate as many deformation modes inthe stiffness matrix during loading. By including a corner, tensile, shear, bendingand torsional deformation was expected to be achieved. The angular cut, seen inFigure 2.5a, was added to increase the complexity of the lattice geometry.

(A) Side view

(B) Top view

FIGURE 2.6: Overview of the solid benchmark geometry.

A similar but completely solid structure was made for the evaluated stiffness matri-ces in the three methods, see Figures 2.6a & 2.6b.

From the resulting stiffness matrices of lattice size N = 7, the equivalent compo-nents, shown in Equation 2.2, were averaged with each other. These averaged stiff-ness matrices were then placed onto the solid structure, creating three comparablebenchmark analyses. It should be noted that the CS, of the obtained stiffness ma-trices on the solid geometries, were checked to match the orientation of the latticeconfiguration, seen in Figure 3.12.

Chapter 2. Methodology 17

The benchmark lattice geometry was meshed with the same element size as in previ-ous analyses whilst the mesh was much coarser for the solid structures. The elementsizes was set to be the same size as one lattice unit.

A fixed support was placed on one of the sides of the benchmark geometry, markedblue in Figure 2.7. With the aim to induce tensile, shear, bending and torsional de-formation all at once, the red part of Figure 2.7 was loaded upward with a force of100N. The magnitude of the force was set in order to only yield elastic deformationsin the structure. The BCs were also set to the solid structures and the maximum de-formation, in the loading direction, of all analyses were compared with each other,see Section 3.

FIGURE 2.7: The boundary conditions of the benchmark lattice struc-ture.

18

Chapter 3

Results and Discussion

The results have been normalised with the elastic modulus, making the results com-patible with any isotropic material which have the same Poisson’s ratio, listed inTable 2.1.

The list down below shows a summary of the methods which are referenced in theplots.

• Method 1 (M1): RVE is of the whole lattice structure.

• Method 2 (M2): RVE is of one layer of lattice units less than the whole structure.

• Method 3 (M3): RVE encases the centred lattice unit.

The example from Figure 3.1 is a resulting stiffness matrix evaluated at lattice sizeN = 7 with M1, where the values represent percentages of Young’s modulus. Theyellow marked cells are values that are less than 0.0005% and the green marked cellsare values above. During all analyses, numerical values were also achieved in theplace of the yellow marked areas. These were set to zero as they are relatively closeto zero, compared with other components.

FIGURE 3.1: Resulting stiffness matrix for lattice size N = 7, evalu-ated with M1. Values are shown in terms of percentage of Young’s

modulus of Aluminium alloy.

As mentioned in Section 2.3, the resulting stiffness matrices is described to behavetransversely isotropic due to symmetry of the chosen lattice configuration. Figure3.1 shows the behaviour very well, where the number of indifferent and non-zerostiffness coefficients are six, see Equation 2.2. The resulting six indifferent stiffnesscoefficients, from each of the three methods, were plotted over the N size of thelattice structures, see Figures 3.2-3.9.

The variations of M2 and M3 are significantly less compared with M1, which can beseen in Figures 3.2, 3.5 & 3.9. The reason for the decreasing variations in M1 couldbe in that the rigidly placed BCs are evened out more with a larger lattice structure.As the size increases, so does the number of lattice units that do not come in direct

Chapter 3. Results and Discussion 19

contact with the rigid BCs. Which is also the main motive for the development ofM2 and M3. These methods were intended to exclude all rigidly set lattice units.However, the methods are not without its’ errors, as the strains placed on the exteriorof the whole lattice structure may not be the actual strains in the elements that aretaken into consideration during evaluation. If the errors are too large then furtherevaluations could be done by extracting all the strain components in each consideredelement and use those values in Equation 1.2.

1 2 3 4 5 6 7

N

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

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%]

C11

Method 1

C11

Method 2

C11

Method 3

FIGURE 3.2: C11 stiffness values against lattice size N.

1 2 3 4 5 6 7

N

0

1

2

3

4

5

6

7

Pe

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%]

C22

Method 1

C22

Method 2

C22

Method 3

FIGURE 3.3: C22 stiffness values against lattice size N.

Chapter 3. Results and Discussion 20

The results are, on the contrary, almost inverted for stiffness component C12, seeFigure 3.4, for M1 where the results seem to be constant regardless of the latticesize. For component C12, all methods are very close to each other, where it canbe concluded that there is little influence of the rigidly set BCs for this particularcomponent.

1 2 3 4 5 6 7

N

0

0.5

1

1.5

2

2.5

3

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%]

C12

Method 1

C12

Method 2

C12

Method 3

FIGURE 3.4: C12 stiffness values against lattice size N.

1 2 3 4 5 6 7

N

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Pe

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%]

C13

Method 1

C13

Method 2

C13

Method 3

FIGURE 3.5: C13 stiffness values against lattice size N.

It should be noted that for the stiffness components C13 and C66, the magnitudes

Chapter 3. Results and Discussion 21

are far smaller compared to the other components. The reason could be in the ge-ometry of the chosen lattice unit, where C13 represents the lateral contraction of thegeometry. The geometry can be seen as weak when wanting to resist expandingin the 3-direction whilst being compressed in the 1-direction, see Figure 3.6 for anillustration.

FIGURE 3.6: Illustration of lateral contraction on the lattice unit forstiffness component C13.

The same argument goes for stiffness component C66, which has a weak shear plane,with few members to resist the deformation, see Figure 3.7.

FIGURE 3.7: Illustration of shear strain on the lattice unit for stiffnesscomponent C66.

Chapter 3. Results and Discussion 22

1 2 3 4 5 6 7

N

0

0.5

1

1.5

2

2.5

3

Pe

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%]

C44

Method 1

C44

Method 2

C44

Method 3

FIGURE 3.8: C44 stiffness values against lattice size N.

1 2 3 4 5 6 7

N

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Pe

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%]

C66

Method 1

C66

Method 2

C66

Method 3

FIGURE 3.9: C66 stiffness values against lattice size N.

The results for M1 in Figures 3.5 & 3.9, seem to not converge as easily as with M2and M3, with increasing lattice size. It would therefore be interesting to see howmuch more the lattice size needs to be increased, in order to get a more convergingplot.

A comparison between the methods can be seen in Figure 3.10, where the values,from the lattice size of N = 7, are plotted as bars.

Chapter 3. Results and Discussion 23

C11 C33 C22 C21 C12 C32 C23 C31 C13 C44 C55 C66

Stiffness matrix component

0

1

2

3

4

5

6

7

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Method 1

Method 2

Method 3

FIGURE 3.10: Comparison between the three methods in terms of allthe stiffness matrix components, values taken from lattice size N = 7.

As discussed previously, the number of indifferent and non-zero stiffness compo-nents can also be confirmed with Figure 3.10, which is six.

It should also be noted that all methods match each other well, according to Figure3.10. One interesting detail nonetheless, is that M3 seem to be a bit stiffer than M1.As the concept behind M3 was to exclude the rigidly set BCs, the expectation wasthat M1 would be the stiffest method. However, as mentioned earlier in this chapter,the strains on the elements in M3 are not the same as the strains applied on the outerboundary elements. The same can be concluded for M2.

The stiffness component C22 has the highest overall stiffness, which is verified by thechosen lattice geometry, see Figure 2.2a. By comparing the components C11, C33 withC22, there arises a factor of two in difference. This could be explained by the numberof struts, in the lattice unit, that acts on loading. There are two struts operating withcomponents C11 and C33 whilst all four struts act with component C22.

The stiffness components C31, C13 & C66 are re-plotted in Figure 3.11 for a bettervisual. It can be noted here that M1 is the stiffest method of all three methods andcomponents.

Chapter 3. Results and Discussion 24

C31 C13 C66

Stiffness matrix component

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

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Method 1

Method 2

Method 3

FIGURE 3.11: Comparison between the three methods in terms of thestiffness matrix components C31, C13 & C66.

In Figures 3.12-3.15, deformation plots, in the y-direction, of the benchmark analysescan be seen. Figures 3.13-3.15 have been scaled according to the results from thelattice benchmark analysis.

The maximum deformation values, in the y-direction, of all analyses have been tab-ulated in Table 3.1.

TABLE 3.1: Results of directional deformation for all benchmark anal-yses.

Analysis Target uy,max reached [%]Lattice -

M1 61%M2 93%M3 97%

It can be noted that M3 matched the directional deformation, in the lattice bench-mark analysis, the best of all methods with 97%. M1 resulted as the stiffest of all themethods by only reaching the target value with 62%, an explanation could here bean effect of the rigidly set BCs. It could also be that the very weak stiffness compo-nents C31, C13 & C66, influenced the structure by being more compliant in M2 andM3. In Figure 3.11, M1 is almost twice as stiff in the stiffness components comparedwith the other methods.

A check was made to see where the maximum deformations occurred and they wereall at the same edge, colored as the maximum magnitude in the Figures 3.12-3.15.However, for the lattice structure, the maximum deformation was found at the bot-tom element whilst in the others, they were found to be at the top.

Chapter 3. Results and Discussion 25

FIGURE 3.12: Benchmark analysis: Lattice.

FIGURE 3.13: Benchmark analysis: Method 1.

Chapter 3. Results and Discussion 26

FIGURE 3.14: Benchmark analysis: Method 2.

FIGURE 3.15: Benchmark analysis: Method 3.

As the number of elements, see Table 3.2, being far less with the solid structurescompared with the lattice structure, the computational time decreased drastically.The CPU-time of the benchmarked solid structures was slightly over 400 times faster,which makes the approach with homogenization very promising.

Chapter 3. Results and Discussion 27

TABLE 3.2: Number of elements, nodes and computational time inthe benchmark analyses.

Analysis # of elements # of nodes CPU time [s]Lattice 4.8× 106 7.7× 106 1250

M1 2.3× 103 4× 102 3M2 2.3× 103 4× 102 3M3 2.3× 103 4× 102 3

28

Chapter 4

Conclusions and Further Work

A homogenization approach to model lattice structures is presented where threesimilarly developed RVE methods are used to evaluate the stiffness matrices of adesigned lattice configuration. The RVE methods are placed on a larger solid struc-ture to compare the predictability in deformation of a larger benchmarked structurecontaining only the chosen lattice configuration. The methods are found to alreadyconverge at a lattice size of N = 3 for a majority of the stiffness matrix components,for the chosen lattice configuration. It is also found that all the RVE methods gen-erate similar values and are a good approximation for modelling the selected latticestructure. The results of the benchmark analyses shows that the third RVE methodM3, which extracts the results of the middle lattice unit, predicts the benchmarkedstructure the best of all methods, with the results from M2 nearby. The first RVEmethod M1 is shown to be too stiff for the constructed benchmark structure as a re-sult of rigidly set BCs. However, the application of the methods depends much onthe boundaries of the lattice structure that is to be modelled.

Recommendations for further work is to verify the results with experimental dataand to test the approach with other lattice configurations.

The concluding remarks from previous paragraphs have been listed below;

• All methods are good approximation of the constructed lattice configuration.

• Properties vary depending on the surroundings of the lattice structure.

• Method 3 and Method 2 predicted the benchmarked lattice structure the best.

• Further analyses are needed to establish a reliable approach in terms of otherlattice configurations.

• In order to fully capture the behaviour of lattice structures, it is recommendedthat further studies should be conducted with periodic boundary conditions(PBCs).

29

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