Methods for Modelling Lattice Structures 1355716/ ¢  A lattice structure is often detailed

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  • DEGREE PROJECT IN SOLID MECHANICS,, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2019

    Methods for modelling lattice structures

    MONA KOUACH

    KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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

  • ROYAL INSTITUTE OF TECHNOLOGY

    MASTER’S THESIS

    Methods for Modelling Lattice Structures

    Author: Mona Kouach

    A thesis submitted in fulfillment of the requirements for the degree of Master in engineering

    in the

    Department of Solid Mechanics

    September 8, 2019

  • i

    Abstract The application of lattice structures have become increasingly popular as additive manufacturing (AM) opens up the possibility to manufacture complex configura- tions. However, modelling such structures can be computationally expensive. The following 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 where three similar methods involving representative volume element (RVE) have been developed and evaluated. The stiffness matrices, of the RVEs, for different sizes of lattice structures, comprising of BCC strut-based units, have been obtained. The stiffness matrices were compared and analysed on a larger solid structure in order to see the deformational predictability of a lattice-based structure of the same size. The results showed that all methods were good approximations with slight differences in terms of boundary conditions (BCs) at the outer edge. The comparative analyses showed that two of the three methods matches the deformational predictability. The BCs in all methods have different influences which makes it pivotal to establish the BCs of the structure before using the approach presented in this thesis.

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

  • ii

    Sammanfattning Ökad implementering av gitterstrukturer i komponenter är ett resultat av utvecklin- gen inom additiv tillverkning. Metoden öppnar upp för tillverkning av komplexa strukturer 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, på SAAB i Järfälla, för att de ska kunna möta upp det framtida behovet av beräkningar på additivt tillverkade gitterstrukturer. I det här arbetet presenteras ett tillvägagångsätt för modellering av gitterstrukturer med hjälp av represantiva volymselement. Styvhets- matriser har räknats fram, för en vald gitterkonfiguration, som sedan viktats mot tre snarlika representativa volymselement. En jämförelseanalys mellan de olika styvhetsmatriserna har sedan gjorts på en större och solid modell för att se hur väl metoderna förutsett deformationen av en gitterstruktur i samma storlek. Re- sultaten har visat att samtliga metoder är bra approximationer med tämligen små skillnader från randeffekterna. Vid jämförelseanalysen simulerades gitterstrukturen bäst med två av de tre metoder. En av slutsatserna är att det är viktigt att förstå inverkan av randvillkoren hos gitterstrukturer innan implementering görs med det tillvägagångssätt som presenterats i det här examensarbetet.

  • iii

    Acknowledgements

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

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

    And to all the wonderful colleagues at SAAB Surveillance in Järfälla, I thank you all for making my time at the office very welcoming and delightful. It has been an invaluable 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 extensive experience within SpaceClaim and ANSYS, which has helped me a lot during the project course.

    Mona Kouach Stockholm, Sweden June 2019

  • iv

    Table of Contents

    List of Figures v

    List of Tables vi

    List of Acronyms vii

    1 Introduction 1 1.1 Cellular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.1.1 Lattice structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Evaluation methods . . . . . . . . . . . . . . . . . . . . . . . . . 4

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

    2 Methodology 10 2.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    2.1.1 RVE Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 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 . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 General overview of additive manufacturing with selective laser melt-

    ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Example of a lattice configuration. . . . . . . . . . . . . . . . . . . . . . 6 1.4 The outer boundary surfaces of the example lattice configuration. . . . 7

    2.1 Flow chart of the approach. . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Overview of the constructed lattice unit. . . . . . . . . . . . . . . . . . . 12 2.3 Different expanded sizes of lattice structures. . . . . . . . . . . . . . . . 12 2.4 Overview of the RVEs for each method. . . . . . . . . . . . . . . . . . . 13 2.5 Overview of benchmark lattice geometry. . . . . . . . . . . . . . . . . . 15 2.6 Overview of the solid benchmark geometry. . . . . . . . . . . . . . . . . 16 2.7 The boundary conditions of the benchmark lattice structure. . . . . . . 17

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

    3.2 C11 stiffness values against lattice size N. . . . . . . . . . . . . . . . . . 19 3.3 C22 stiffness values against lattice size N. . . . . . . . . . . . . . . . . . 19 3.4 C12 stiffness values against lattice size N. . . . . . . . . . . . . . . . . . 20 3.5 C13 stiffness values against lattice size N. . . . . . . . . . . . . . . . . . 20 3.6 Illustration of the lateral contraction on the lattice unit. . . . . . . . . . 21 3.7 Illustration of shear strain on the lattice unit. . . . . . . . . . . . . . . . 21 3.8 C44 stiffness values against lattice size N. . . . . . . . . . . . . . . . . . 22 3.9 C66 stiffness values against lattice size N. . . . . . . . . . . . . . . . . . 22 3.10 Comparison between the three methods in terms of all the stiffness

    matrix components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.11 Comparison between the three methods in terms of the stiffness ma-

    trix components C31, C13 & C66. . . . . . . . . . . . . . . . . . . . . . . . 24 3.12 Benchmark analysis: Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.13 Benchmark analysis: Method 1. . . . . . . . . . . . . . . . . . . . . . . . 25 3.14 Benchmark analysis: Method 2. . . . . . . . . . . . . . . . . . . . . . . . 26 3.15 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. . . . . . 24 3.2 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 are of high interest. Due to traditional manufacturing constraints, parts can tend to be over designed and thus have more weight than needed. However, additive man- ufacturing (AM) opens up for structural designs that previously were not possible to manufacture such as lattice structures. With the extensive research published [1], one can see it implemented in a number of industries and companies, including SAAB.

    With implementation of lattice structures in a design, the same structural benefits as e.g. a sandwich composite can be achieved [2]. Lattice structures can be engineered to have the required mechanical properties depending on the