Modeling of Rigid Foams ¨ JOONAS KOLL Doctoral Thesis Stockholm, Sweden 2015 TRITA-AVE 2015-21 ISSN 1651-7660 ISBN 978-91-7595-562-9 Kungliga Tekniska H¨ogskolan (KTH) Department of Aeronautical and Vehicle Engineering SE-100 44 Stockholm, Sweden Akademisk avhandling som med tillst˚ and av Kungl Tekniska H¨ogskolan framl¨agges till offentlig granskning f¨or avl¨aggande av teknologie doktorsexamen i L¨attkonstruktioner fredagen den 22 maj kl 9.15 i D3, Lindstedtsv¨agen 5, Kungliga Tekniska H¨ogskolan, Stockholm. © Joonas K¨ oll, 2015 Tryck: E-Print AB iii Acknowledgments The work presented in this thesis has been carried out at the Department of Aeronautical and Vehicle Engineering at KTH Royal Institute of Technology. Financial support was kindly granted by the Swedish Research Council. I thank my supervisor Dr. Stefan Hallstr¨om for introducing me to the amazing world of foams and for giving me the opportunity to write this thesis. Your guidance and support is gratefully acknowledged. I want to express my gratitude to all of my present and former colleagues at the department. In particular I want to thank my two office mates during my time as a PhD student at KTH, Sohrab and Christof. I appreciate your friendship, support, and all the interesting discussions we had. A special thanks goes to Ylva for your endless care and support. Last but not least I want to thank my family for all your love and support. Joonas K¨ oll Stockholm in 2015 v Abstract The morphology and elastic properties of foams are investigated using models accounting for the cellular structure. Different modeling approaches are evaluated and compared to the cellular structures of real foams. The aim is to find an approach that produces realistic models that can be adjusted to match morphology measures taken form real foams. The elastic properties are computed using finite element analysis and the results are compared with both experimental results from mechanical testing and existing analytical models. A generalization of the existing analytical models is suggested for better agreement with the modeling results. In Paper A Voronoi and equilibrium foam models are generated and investigated. The Voronoi models are found to have a high content of short edges and small faces while conversion into equilibrium foams eliminates such small geometrical features. It is also seen that different seed point distribution algorithms generally result in different model topologies. In Paper B the relations between the sphere packing fraction and the resulting degree of volumetric polydispersity are studied as well as the relations between polydispersity and a number of morphology parameters. Both Voronoi and equilibrium foams are investigated and compared with data from real foams. Such comparisons indicate that the used method is somewhat limited in terms of polydispersity but provides a controlled way of varying the foam morphology. In Paper C equilibrium models are used to study the influence of polydispersity, relative density and distribution of solid on the elastic properties of foams. It is found that the elastic moduli are very weakly coupled to the polydispersity while the influence from the relative density and distribution of solid is considerably stronger. Existing analytical models from the literature are fitted to the results and generalizations are suggested for better agreement over the investigated range of relative density and distribution of solid. In Paper D equilibrium foam models with prescribed polydispersity are generated, analyzed and compared to real foams. The polydispersity of real foams is measured in micro-CT images and the results are applied to the models. The elastic properties are computed and the results are compared with experimental results and previous models with significantly lower polydispersity. No clear relation is found between the elastic properties and the polydispersity of equilibrium foams. vii Sammanfattning Morfologi och elastiska egenskaper f¨ or skum unders¨ oks h¨ ar med hj¨ alp av modeller som tar h¨ ansyn till materialens cellstruktur. Olika modelleringsalternativ utv¨ arderas och j¨ amf¨ ors med cellstrukturen hos riktiga skum. M˚ als¨ attningen ¨ ar att hitta en metod som ger realistiska modeller vars geometri kan anpassas till olika typer av riktiga skum. De elastiska egenskaperna ber¨ aknas med hj¨ alp av finita element analys och resultaten j¨ amf¨ ors dels med experimentella resultat och dels med befintliga analytiska modeller. F¨ or en b¨ attre overensst¨ ammelse med modelleringsresultaten f¨ oresl˚ as en generalisering av de befintliga ¨ analytiska modellerna. I Papper A genereras och unders¨ oks Voronoi och j¨ amviktsmodeller av skum. Voronoimodellerna konstateras ha en onormalt h¨ og andel korta kanter och sm˚ a ytor som elimineras n¨ ar modellerna konverteras till j¨ amviktsmodeller. Det konstateras ocks˚ a att olika algoritmer f¨ or sf¨ arpackning resulterar i olika modelltopologier. I Papper B unders¨ oks sambanden mellan packningsgraden i sf¨ ardistributioner och den resulterande cellvolymspridningen i modellerna, samt dess relation till ett antal morfologiparametrar. B˚ ade Voronoi och j¨ amviktsskum analyseras och j¨ amf¨ ors med data fr˚ an riktiga skum. Resultaten visar att endast en berg¨ ansad cellvolymsspridning kan uppn˚ as med den valda metodiken men den tillhandah˚ aller ¨ and˚ a ett s¨ att att kontrollerat variera morfologin. I Papper C unders¨ oks ett antal parametrars inflytande p˚ a dom elastiska egenskaperna. Dessa parametrar utg¨ ors av cellvolymsspridning, relativa densitet och f¨ ordelning av bulkmaterial mellan cellv¨ aggar och kanter. Kopplingen mellan cellvolymsspridningen och de elastiska egenskaperna visar sig vara mycket svag. Den relativa densiteten och f¨ ordelningen av bulkmaterial har d¨ aremot ett mycket starkare inflytande. Analytiska modeller fr˚ an litteraturen anpassas till resultaten och generaliseringar av dessa f¨ oresl˚ as f¨ or en b¨ attre overensst¨ ammelse ¨ over hela spannet av relativ densitet och bulkmaterialf¨ ordelning. ¨ I Papper D genereras och analyseras j¨ amviktsmodeller med p˚ atvingad cellvolymsspridning och modeller j¨ amf¨ ors med riktiga skum. Cellvolymsspridningen i riktiga skum m¨ ats i datortomografibilder och resultaten anv¨ ands i modellerna. De elastiska egenskaperna ber¨ aknas och resultaten j¨ amf¨ ors dels med experimentella resultat och dels med tidigare modeller med v¨ asentligt l¨ agre cellvolymsspridning. Inga tydliga relationer kan ses mellan de elastiska egenskaperna och cellvolymsspridningen i modellerna av j¨ amviktsskum. Dissertation This doctoral thesis is based on an introduction to the area of research and the following appended papers: Paper A J. K¨ oll and S. Hallstr¨ om. Generation of stochastic foam models for numerical analysis. Journal of Cellular Plastics, 50(1) 37-54, 2014. Paper B J. K¨ oll and S. Hallstr¨ om. Influence of sphere packing fraction on polydispersity and morphology of Voronoi-partitioned foam models. Manuscript submitted for publication. Paper C J. K¨ oll and S. Hallstr¨ om. Elastic properties of equilibrium foams. Manuscript submitted for publication. Paper D J. K¨oll and S. Hallstr¨om. Elastic properties from equilibrium foam models with statistically representative polydispersity. Manuscript to be submitted for publication. ix x Parts of this thesis have also been presented as follows: J. K¨oll and S Hallstr¨om. Morphology effects on constitutive properties of foams. In Proceedings of the 18th International Conference on Composite Materials (ICCM-18), Jeju Island (Korea), 2011. Division of work between authors Paper A K¨ oll implemented the modeling principles through development of software tools, performed the simulations and wrote the paper. Hallstr¨om initiated and guided the work and contributed to the paper with valuable comments and revisions. Paper B K¨oll performed the simulations and wrote the paper. Hallstr¨om initiated and guided the work and contributed to the paper with valuable comments and revisions. Paper C K¨ oll developed the preprocessing software, performed the simulations, suggested the generalized scaling laws and wrote the paper. Hallstr¨ om initiated and guided the work and contributed to the paper with valuable comments and revisions. Paper D K¨oll performed the image processing and the simulations, analyzed the results from the mechanical testing and wrote the paper. Hallstr¨ om initiated and guided the work and contributed to the paper with valuable comments and revisions. xi Contents I Introduction 1 1 Background 1.1 Applications for rigid foam materials . . . . . . . . . . . . . . . . . . 3 5 2 Cellular geometry 2.1 Characterization of the cellular structure . . . . . . . . . . . . . . . . 2.2 Generation of model geometries . . . . . . . . . . . . . . . . . . . . . 7 7 10 3 Elastic properties 17 3.1 FE analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Experimental verification . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 Summary of appended papers 23 5 Future work 25 Bibliography 27 II Appended papers 29 xiii Part I Introduction 1 1 Background A range of different models have been used to study the properties of foams, from relatively simple single cell approximations to very advanced geometrical representations consisting of large clusters of cells. Gibson & Ashby developed analytical models describing the mechanics of foams [1]. These models were based on the idealized single-cell geometries illustrated in Figure 1.1 and offer relations that describe how the elastic properties and strength scale with relative density for open- and closed-cell foams. They also introduced the concept of distribution of solid between beams and plates to account for possible bulk material concentrations along the cell edges. Figure 1.1: Illustration of the single-cell geometries used by Gibson & Ashby to develop analytical models describing the mechanics of open- and closed-cell foams. Several single cell polyhedral models exist that pack to fill space, i.e. are geometrically periodic. For a dry foam the edge-connectivity is four and the faceconnectivity is three [2] and that is also the connectivity usually found in other real foams. Most of the periodic polyhedral cells on the other hand have other edgeand/or face-connectivities, but not the tetrakaidecahedron illustrated in Figure 1.2a, also known as the Kelvin cell. It is the product of Kelvins search for the one ideal cell that partitions space with the minimum use of surface area [3]. The mechanical properties of Kelvin cells have been studied [4–9] but there is always the questions 3 4 J. K¨oll (a) (b) Figure 1.2: Illustration of the Kelvin cell (a) and the Weaire-Phelan model (b). whether or not it is representative of real foam materials with their broad range of cell shapes and sizes. The botanist Matzke undertook historical experimens where he characterized and documented the shape of individual bubbles in monodisperse liquid foams [10, 11]. He presented a range of different cell architectures from his experiments but his findings did not include any Kelvin cells. The Weaire-Phelan model illustrated in Figure 1.2b is the most surface-area effective way known today of partitioning space into cells with equal volumes [12]. Having slightly less surface area per unit volume than the Kelvin cell it shows that a combination of several cells can be more effective from a surface energy perspective. The mechanical properties of the Weaire-Phelan model have also been (a) (b) Figure 1.3: Illustration of Voronoi (a) and equilibrium foam (b). Introduction 5 studied [4, 9] but the eight cells in the model are still limited to only two different cell architectures. Recent work has focused on stochastic models based on Voronoi [13] and Laguerre (weighted Voronoi) partitioning as well as equilibrium foams [8, 9, 14–23]. The aim with such approaches is to develop geometrically realistic models of rigid polydisperse foam materials. The work presented in this thesis is focused on Voronoi and equilibrium foams illustrated in Figure 1.3. 1.1 Applications for rigid foam materials Today rigid foam materials can be found almost anywhere. They are widely used in buildings as insulation underneath concrete floors and they are on of the dominating types of chock-absorbing packing-material used for protecting electronic devices and home appliances during storage and transportation. On a more advanced level rigid foams are used as load carrying core members in sandwich structures typically found in sports equipment as for instance skis and boards, in wind turbine blades, boat hulls, various floor panels, and in the cargo area body panels of box trucks, to name a few examples. The common objective for sandwich applications is the need for high bending stiffness in combination with low weight. Lately there has been an increasing demand for high performance polymer foam core material from the wind turbine industry. The increase is related to the global efforts of switching to renewable energy sources. The automotive industry is in need of developing lighter vehicles due to increasing restrictions on emissions. There is already an increase in the use of composite materials for replacement of metal, especially in ancillary components. It is probable that emission regulations will eventually force the automotive industry to a change of technology, even for main structural components. Foam core sandwich construction can be an alternative for any component where bending stiffness is critical. It is therefore possible that there will be an increasing demand for foam core material also from the automotive industry. 2 Cellular geometry Foam materials are stochastic structures that consist of a space filling cluster of cells. Such clusters are built up from an interconnected network of walls creating a closed-cell foam or an interconnected network of struts creating an open-cell foam. The cells vary in size, shape and orientation creating the irregular structure that is typical for foams. 2.1 Characterization of the cellular structure Microscopy Traditional optical microscopy has been available for a long time and gives some possibility to study the cellular structure of foams. It provides a 2D view of a cross-section of the material as illustrated in Figure 2.1a. With a Scanning Electron Microscope (SEM) hight resolution images can be produced that also show the topography of the surface. That gives depth to the image as illustrated in Figure 2.1b. SEM provides more information about the cellular structure than optical microscopy but is still limited to what can be observed from the surface. It is however useful for preliminary estimates of for instance cell sizes and wall thicknesses. (a) (b) Figure 2.1: Optical microscope image (a) and SEM image (b) of Rohacell IG-F. 7 8 J. K¨oll Computed tomography The internal 3D structure of foams can be studied using X-ray computed tomography (CT) [7, 15, 24–26]. The principle is illustrated in Figure 2.2. It is a technology that uses multiple radiographic images (Figure 2.3a), taken around a single axis of rotation, to reconstruct the internal 3D structure of the scanned object. A stack of tomography images is created where each image is representing a one pixel thick slice of the object (Figure 2.3b). The slices are finally combined to a 3D voxel image representing the structure of the scanned object (Figure 2.3c). Detector X-ray source Specimen Figure 2.2: The principle of computed tomography. Image processing Scientific image processing and analysis tools are needed to extract morphology parameters from reconstructed CT images [26, 27]. First the system of cell walls and/or edges needs to be separated from the enclosed empty space (voids) through thresholding. Then the 3D image needs to be segmented into individual cells through what is called water shedding. After the segmentation is done the cells can be analyzed individually. Thresholding is a process that determines what is void and what is structure, so that the two can be separated. Using simple thresholding all voxels with an intensity value below a specified single threshold are considered to be background and are thus removed while all voxels with an intensity value above the threshold are retained. The problem with simple thresholding is that stray voxels with low intensity values are removed even if they obviously belong to the structure. Inversely, stray background voxels with high intensity values might be retained. In hysteresis Introduction (a) 9 (b) (c) Figure 2.3: Radiograph image (a), reconstructed slice (b) and reconstructed CT image (c) of Rohacell IG-F. 10 J. K¨oll thresholding two threshold values are used, one lower and one upper. Voxels with an intensity below the lower threshold are always considered to be background and voxels with an intensity above the upper threshold are always considered to be part of the structure. The voxels that fall between the two thresholds are retained only if connected to an other voxel with intensity above the upper threshold. This gives a smoother and more realistic separation between background and structure. Figure 2.4a shows an example of a micro-CT image that has been thresholded using hysteresis thresholding. Water shedding is a process that segments the structure into individual cells. Starting from seeds approximately centered inside the individual cells the algorithm simultaneously builds particles by adding voxels to each seed until the it reaches the cell walls. If walls or parts of walls are missing the algorithm restores walls where the growing particles meet. In order to do that the algorithm needs additional information i.e. the cell centers and in which directions to shed the water. The first step is therefore to calculate an euclidian distance map that is exemplified in Figure 2.4b. It is computed from the thresholded image so that every voxel is assigned an intensity value corresponding to the distance to the nearest cell wall. In the ideal case this would give one maxima inside every cell with the intensity values strictly declining towards the cell walls that have zero intensity. In reality many cells will contain several local maxima. The surplus maxima can be removed using the H-maxima transform [28]. The transform reduces local maxima until they reach the intensity level of the surrounding or the maximum suppression distance h. By selecting the appropriate value for h, an image can be generated with one approximately centered local maximum within every cell. This image can then be used for the water shedding to segment the individual cells as illustrated in Figure 2.4c. The segmented image can in turn be used for extracting morphology parameters as for instance individual cell volumes. 2.2 Generation of model geometries Voronoi structures For 3D Voronoi structures the partitioning is based on a set of seed points distributed in a model space where each cell is defined by all points that are closer to one particular seed point than to any other. In order to have the seeds relatively evenly distributed they can be obtained from centra of distributions of densely packed hard spheres, as illustrated in Figure 2.5. Inherently all Voronoi cells are strictly convex polyhedra with flat faces as can be seen in Figure 1.3a. Voronoi partitions can be created with for instance the QHull software [29]. A mathematical definition of the partitioning is given in Equation (1) in Paper A and more examples of such models are shown in Figure 3 in the same paper. Even though Voronoi foams have some resemblance with real foam materials they are also associated with a geometrical problem since they inherently contain small geometrical features in form of short cell edges and small cell faces. The actual Introduction (a) (b) (c) Figure 2.4: Thresholded image (a), EDT image (b) and segmented image (c) of Rohacell IG-F. 11 12 J. K¨oll Figure 2.5: Illustration of randomly packed hard spheres. size can be down to the limit formed by the numerical precision in the geometry description. Such small features in the geometry description will create problems later when the mechanical properties are to be computed. For the FE analysis the geometry needs to be meshed with finite elements and for accurate and successful analysis the mesh needs to be smooth and fulfill some requirements on internal angles, aspect ratios and the variation of element size. Such smooth meshing is in practice very hard to achieve for irregular Voronoi foams. The small geometrical features origin from the seed point locations. A Voronoi Figure 2.6: Illustration of the cause of small geometrical features in Voronoi foams. Introduction 13 vertex has an exactly equal distance d to four neigboring seed points in 3D and three different neighbors in 2D. For practical reasons the source of the problem will here be illustrated in 2D but the principles are the same in 3D. As illustrated in Figure 2.6, a short edge emerges when a fourth seed point happens to be located at approximately the same distance r + d to the Voronoi vertex as the true neighbors so that d << r. The size depends on d and the corresponding event in 3D happens when a fifth seed point is located at approximately the same distance. Proportionally increasing triangular constellations of seed points in 2D and tetrahedral constellations in 3D reduces the problem. It can be achieved by using densely packed disks or spheres as source for the seeds. The share of tetrahedrally organized spheres increases with densification when a sphere packing approaches the random close packing limit and peaks at a fraction of about 0.3 [30]. That is the reason why random close packing reduces the problem but does not eliminate it. Equilibrium foams Equilibrium foams resemble liquid foams with a low liquid content, i.e. dry foams. They are at equilibrium from a cell wall surface energy point of view. Corresponding models can be generated with the Surface Evolver software [31]. A combination of surface area minimization and topology transitions is used to find the equilibrium structure. An initial geometry needs to be provided and all equilibrium foams presented in this thesis were generated from Voronoi partitions. When run to convergence the process also entirely eliminates problem with small geometrical features found in the Voronoi foams, which is confirmed by the results presented in Figures 11 and 12 in Paper A. Figure 2.7 illustrates the geometry in a few different stages during the conversion and more examples of equilibrium foams are shown in Figure 7 in Paper A. Converting Voronoi foams into equilibrium foams with Surface Evolver is a delicate process associated with a risk of having both some robustness and repeatability issues. The nature and consequences of these issues is discussed in Paper A but one consequence is that it is very difficult to automate the process. For every single model the conversion into equilibrium state is a time consuming hands on process. Geometry variation The polydispersity of Voronoi foams can to a certain extent be controlled through varying the packing fraction of the sphere distributions from which the seed points are extracted. The polydispersity is in turn related to several key morphology parameters as for instance the number of face per cell and the spread in edge length and face area, while other parameters, as for instance average edge length and face area, are unaffected. Such relations are shown and discussed in Paper B. The achievable polydispersity is however limited. As shown in Paper B a relative standard deviation of cell volume up to 0.5 can be achieved. 14 J. K¨oll (a) (b) (c) (d ) Figure 2.7: Illustration of different stages during the conversion from Voronoi to equilibrium foam, chronologically ordered from (a) to (d ). The initial Voronoi state is shown in (a). In (c) an ongoing topology transition can be seen while (b) and (d ) show the structure before and after that transition. Converting the Voronoi foams to equilibrium foams strongly reduces the influence of polydispersity and brings the morphology parameters to a virtually constant level. The results presented in Paper B also indicate that the equilibrium models are in better agreement with real foams while the conversion to equilibrium foams was found to be difficult for models with a relative standard deviation in cell volume above 0.25. As shown in Paper D higher spread in cell volume can be achieved for equilibrium foams by applying direct cell volume constraints during the conversion from Voronoi foams. The benefit with this approach is that predefined cell volume distributions can be used. As shown in the paper the individual cell volumes as well as their Introduction 15 statistical distribution in real foams can be determined using micro-CT and image processing. That way models with truly or statistically representative cell volumes can be generated. 3 Elastic properties The elastic properties of foams can be computed with FE-models using foamstructure geometries based on the principles explained in section 2. The mechanical behavior of the models can be verified by comparing with results from mechanical testing. From the FE results scaling laws can be derived that describe how the elastic properties scale with relative density and distribution of solid. 3.1 FE analysis The mechanical properties of closed-cell foams can be computed with FE using plate models where shell elements are assigned to the cell walls. Material concentrations along the cell edges can be modeled by relocating material from the shell elements to additional beam elements placed along the edges. Placing all material in the beam elements creates models of open-cell foam. The geometry descriptions provided by the softwares QHull and Surface Evolver cannot be directly imported by commercial FE software nor is there any other software available for converting them to importable formats. The amount of data in the geometry descriptions is too large for manual conversion to be an alternative why a specialized preprocessing software package was developed to handle the necessary file formats and geometry descriptions. The preprocessor also handles the application of periodic boundary conditions and loads. After preprocessing all FE-models were analyzed with the Abaqus standard solver. The principles described here can however be applied with other FE solvers as well. Preprocessing All Voronoi foams presented in this thesis were created with the QHull software that can produce output in the off-file format. An algorithm that converts from the off-file format to a Surface Evolver input file (fe-file) was developed as part of the preprocessing package. In off-file format polygonal surfaces are defined by vertices and bodies are defined by a set of polygonal surfaces while in the fe-file lines are defined by vertices, triangular surfaces are defined by lines and bodies are defined by a set of triangular surfaces. The output from QHull is defined in 17 18 J. K¨oll Euclidian space while a 3-torus space is used for the surface Evolver input file to handle the periodicity. In Euclidian space there is one way to connect two points with a straight line while in a 3-torus space there are 27 different possibilities. When placing a geometry from Euclidian space in 3-torus space parts of the geometry will typically at first be located outside the 3-torus. All vertices need to be moved inside the 3-torus and torus wrapping definitions need to be added to all line definitions. The torus wrapping definitions state which torus borders are crossed by a line and in which direction they are crossed. After the conversion to equilibrium foam Surface Evolver produces an output file (dmp-file) that in principle follows the same geometry definitions as the fe-file. Another algorithm was developed that converts the geometry back to Euclidian space. The same algorithm also changes geometry description from a cell-by-cell definition to a single interconnected structure suitable for FE-analysis. It identifies all periodic node pairs in the model and defines periodic equation constraints for them. Six different sets of constraints are defined for six principal load-cases, axial loading in three orthogonal directions as well as shear loading in three orthogonal planes. The periodic boundary conditions are in the following illustrated in 2D but the principles are the same when expanded to 3D. For a periodic geometry with the periodicity L, exemplified in Figure 3.1a, the nodes located on the boundaries can be grouped depending on which side of the geometry they belong to as a = [1, 2, 3, 4] (3.1) b = [10, 9, 8, 7] (3.2) c = [1, 12, 11, 10] (3.3) d = [4, 5, 6, 7] . (3.4) R1 and R2 are reference nodes used for homogenizing the global displacements between different periodic node-pairs. For the case of axial loading the periodic boundary conditions can be defined as ua(i) − ub(i) = 0 (3.5) vc(i) − vd(i) = 0 (3.6) θa(i) − θb(i) = 0 (3.7) θc(i) − θd(i) = 0 (3.8) for the x- and y-displacements and the nodal rotations respectively. The homogenized global displacements u and v are controlled by R1 and R2 through the relations uc(i) − ud(i) + u = 0 (3.9) vb(i) − va(i) + v 0. (3.10) = When a homogenized strain u/L is applied in the x-direction, as illustrated in Figure 3.1b, it results in a homogenized lateral strain v/L in the y-direction and the Poisson’s ratio is given by Introduction 19 2 4 1 3 L 12 5 L 11 6 9 7 10 8 y R1 R2 x (a) 2 4 1 3 L+u 12 5 L+v 11 6 9 7 10 8 u y R1 R2 v x (b) u 2 4 1 u 3 5 12 L+v 11 6 L+v 9 7 10 8 u y (c) R1 R2 v x Figure 3.1: Illustration of a periodic geometry (a) under axial loading (b) and shear loading (c). 20 J. K¨oll u νxy = − . v (3.11) For the case of shear loading the nodal rotations are controlled as in Equations 3.7 and 3.8 The shear loading is applied by the displacement u through the relations ua(i) − ub(i) + u = 0 (3.12) vc(i) − vd(i) + u = 0 (3.13) and the contractions are homogenized by the displacement v through the relations va(i) − vb(i) + v = 0 (3.14) uc(i) − ud(i) + v = 0. (3.15) The preprocessor also computes the parameters needed for defining the plate and beam cross-sections illustrated in Figure 3.2. Using the given distribution of solid and the total edge length and wall area in the model, the cross-section areas and the area moments of inertia are determined. Adding also material properties the preprocessor finally creates a complete Abaqus input-file. An example of a finished model under axial loading is illustrated in Figure 3.3. After the analysis, homogenized forces and displacements can easily be extracted from the two reference nodes R1 and R2. Figure 3.2: Illustrations of the cross-sections perpendicular to a cell edge. 3.2 Experimental verification The models were compared with experimental results from mechanical testing. Both block shear tests and tensile test were conducted and analyzed. The experimental verification of the models is discussed in Paper D. Introduction 21 Figure 3.3: Illustrations of a FE model showing displacements under axial loading. 3.3 Analytical model The elastic properties depend on the relative density and the distribution of solid. Analytical expressions can be derived to describe such relations so that further simulations don not have to be performed. In Paper C a generalization of Gibson & Ashby’s analytical model is suggested based on the FE results, E∗ = f1 (φ) Es G∗ = f2 (φ) Es ρ∗ ρs 2 ρ∗ ρs 2 + g1 (φ) ρ∗ ρs (3.16) + g2 (φ) ρ∗ , ρs (3.17) where fi is a second order polynomial, gi is a first order polynomial, φ is the distribution of solid, ρ is the density, E is the Young’s modulus, G is the shear modulus and the indices s and ∗ denote solid material properties and foam properties respectively. The determination of fi and gi is discussed in Papers C and D. 4 Summary of appended papers Paper A Voronoi and equilibrium foam models are generated and investigated for their potential use in numerical analysis using finite element methods. A drawback with the Voronoi models is found to be the high content of short edges and small faces while conversion into equilibrium foams is found to eliminate such small geometrical features. It is also seen that different seed point distribution algorithms generally result in different model topologies. Paper B The relations between the sphere packing fraction and the resulting degree of volumetric polydispersity is investigated for Voronoi foams. Further, the relations between polydispersity and a number of morphology parameters are investigated for both Voronoi and equilibrium foams. Comparisons with data form real foams indicate that the used method is somewhat limited in terms of polydispersity but it provides a controlled way of varying the foam morphology. Paper C Equilibrium finite element models are used to study the influence of polydispersity, relative density and distribution of solid on the elastic properties of foam materials. It is found that the elastic moduli are very weakly coupled to polydispersity. The influence from relative density and distribution of solid is considerably stronger. Analytical models from the literature are fitted to the finite element results. A generalization of the analytical models is suggested for better agreement over the investigated range of relative density and distribution of solid. Paper D Polydisperse finite element equilibrium foam models are generated, analyzed and compared to real foams. The polydispersity of real foams is measured from micro-CT 23 24 J. K¨oll images and applied to the models. Finite element analysis is used to study the elastic properties and the results are compared to experimental results from mechanical testing. The results are also compared to previous models with significantly lower polydispersity. No pronounced relation is found between the elastic properties and the polydispersity of equilibrium foams. 5 Future work A reoccurring issue when comparing the models to polymer foams is that accurate bulk material properties are often unknown, even in the linear elastic regime. The properties can change during the foaming process and accurately measuring them in the finished foam is challenging due to the small dimension of cell walls and struts. Finding methods for determining the full non-linear bulk material properties is one of the future challenges in order to reach reliable predictions for large deformations and the strength of foams. The FE-models can also be further improved by adding the contribution from internal gas pressure. Another future subject is to determine the reasons why Laguerre models predict a relation between the polydispersity and the stiffness of the foam and equilibrium models do not. One possible path when it comes to the robustness issues when converting from Voronoi to equilibrium foams is to rearrange seed point distributions through relaxation of Delauny tetrahedra in Surface Evolver. No topology transitions are likely to take place and only a corse mesh is needed. It could dramatically reduce the number of topology transitions needed for the subsequent conversion of the associated Voronoi foam to an equilibrium foam and make the process much more stable. 25 Bibliography [1] L.J. Gibson and M.F. Ashby. Cellular solids. Cambridge University Press, 2:nd edition, 1997. [2] D. Weaire and S. Hutzler. The Physics of Foams. Oxford University Press, 1999. [3] W. Thomson. On the division of space with minimum partitional area. Acta mathematica, 11:121–134, 1887. [4] A.M. Kraynik and D.A. Reinelt. Linear elastic behavior of dry soap foams. Journal of Colloid and Interface Science, 181:511–520, 1996. [5] A.M. Kraynik and D.A. Reinelt. Elastic-plastic behavior of a kelvin foam. Forma, 11:255–270, 1996. [6] W.E. Warren and A.M. Kraynik. Linear elastic behaviour of a low-density kelvin foam with open cells. Journal of Applied Mechanics, 64:787–794, 1997. [7] F. Fischer, G.T. Lim, U.A. Handge, and V. Altst¨ adt. Numerical simulation of mechanical properties of cellular materials using cumputed tomography analysis. Journal of Cellular Plastics, 45:441–460, 2009. [8] W. Jang, S. Kyriakides, and A.M. Kraynik. On the compressive strengt of open-cell metal foms with kelvin and random cell structures. International Journal of Solids and Structures, 47:2872–2883, 2010. [9] Y. Chen, R. Das, and M. Battley. Effects of cell size and cell wall thickness variations of the stiffness of closed-cell foams. International Journal of Solids and Structures, 52:150–164, 2015. [10] E.B. Matzke. The three-dimensional shapes of bubbles in foams. American Journal of Botany, 31:281–289, 1945. [11] E.B. Matzke. The three-dimensional shape of bubbles in foam - an analysis of the role of surface forces in three-dimensional cell shape determination. American Journal of Botany, 33:58–80, 1946. [12] D. Weaire and R. Phelan. A counter-example to kelvin’s conjecture on minimal surfaces. Philosophical Magazine Letters, 69(2):107–110, 1994. [13] G. Voronoi. Nouvelles applications des param`etres continus a ` la th´eorie des formes quadratiques. deuxi`eme m´emoire. recherches sur les parall´ello`edres primitifs. Journal f¨ ur die reine und angewandte Mathematik, 134:198–287, 1908. 27 28 J. K¨oll [14] S. Ribeiro-Ayeh. Finite Element Modelling of the Mechanics of Solid Foam Materials. PhD thesis, KTH Royal Institute of Technology, Stockholm, Sweden, 2005. auble. Prediction of material behaviour of closed [15] R. Schlimper, M. Rinker, and R. Sch¨ cell rigid foams via mesoscopic modelling. ICCM-17, 2009. [16] C. Redenbach. Microstructure models for cellular materials. Computational Material Science, 44:1397–1408, 2009. oll and S. Hallstr¨ om. Morphology effects on constitutive properties of foams. [17] J. K¨ ICCM-18, 2011. [18] S. Gaitanaros, S. Kyriakides, and A.M. Kraynik. On the crushing response of random open-cell foams. International Journal of Solids and Structures, 49:2733–2743, 2012. [19] J. K¨ oll and S. Hallstr¨ om. Generation of periodic stochastic foam models for numerical analysis. Journal of Cellular Plastics, 50(1):37–54, 2014. [20] I. Vecchio, C. Redenbach, and K. Schalditz. Angles in laguerre tesselation models for solid foams. Computational Material Science, 83:171–184, 2014. oll and S. Hallstr¨ om. Influence of sphere packing fraction on polydispercity and [21] J. K¨ morphology of voronoi-partitioned foam models. Submitted for publication, 2015. oll and S. Hallstr¨ om. Elastic properties of equilibrium foams. Submitted for [22] J. K¨ publication, 2015. [23] J. K¨ oll and S. Hallstr¨ om. Elastic properties from equilibrium foam models with statistically representative polydispersity. Manuscript, 2015. [24] M.D. Montminy, A.R. Tannenbaum, and C.W. Macosko. The 3d structure of real polymer foams. Journal of Colloid and Interface Science, 280:202–211, 2004. [25] S. Youssef, E. Maire, and R. Gaertner. Finite element modelling of the actual structure of cellular materials determined by x-ray tomography. Acta Materialia, 53:719–730, 2005. [26] C. Lautensack and T. Sych. 3d image analysis of open foams using random tessellations. Image Anal Stereol, 25:87–93, 2006. [27] J. Ollion, J. Cochennec, F. Loll, C. Escud´e, and T. Boudier. Tango: A generic tool for high-throughput 3d image analysis for studying nuclear organization. Bioinformatics, 29(14):1840–1841, 2013. [28] P. Soille. Morphological Image Analysis: Principles and Applications. Springer, 1999. [29] C.B. Barber, D.P. Dobkin, and H.T. Huhdanpaa. The quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software, 22(4):469–483, 1996. [30] A.V. Anikeenko and N.N. Medvedev. Structural and entropic insights into the nature of the random-close-packing limit. Physical Review E, 77:031101, 2008. [31] K. Brakke. The surface evolver. Experimental Mathematics, 1(2):141–165, 1992. Part II Appended papers 29
© Copyright 2024