Mechanical properties of cellular materials

Transcription

1 Mechanical properties of cellular materials Author: Mateja Erjavec Advisor: doc. dr. Primož Ziherl March 2011 Abstract Materials with cellular structure occur widely in nature. This seminar discusses the mechanical models of two and three dimensional cellular solids. We introduce the honeycomb-like structure of wood and the foam-like structure of the trabecular bone. The third example of cellular material, glass sponge Euplectella sp., is presented as well. 1

2 Contents 1 Introduction 3 2 Cellular solid Mechanics of honeycombs Mechanics of foams Wood 10 4 Trabecular bone 11 5 Glass sponge Euplectella sp Conclusion 14 2

3 1 Introduction When modern man builds large load-bearing structures, he uses dense solids: steel, concrete, glass. When nature does the same, she generally uses cellular materials: wood, bone, coral. There must be good reasons for it [1]. M. F. Ashby The use of cellular structure allows a material to have good mechanical properties at low weight. Materials with cellular structure widely occur in nature and have many potential engineering applications [2]. Examples of cellular structures in nature are wood, bone, cork, plant stems, glass sponges and bird beaks [2]. Man has made many artificial cellular structures, such as honeycomb-like materials used for lightweight aerospace components. Cellular solids can be divided according to shape and size of the cells and the way the cells are distributed [3]. The most important feature of a cellular structure is the relative density, defined as the density of cellular solid (ρ ) divided by the density of the solid it is made from ( ) relative density = ρ (1) As the relative density increases, the cell walls become thicker [3]. Relative density is equivalent to the volume fraction of a solid [4]. Typical relative densities of some cellular material are shown in Table 1. material relative density special ultra-low-density foams polymeric foams (packaging and insulation) cork 0.14 softwoods porous solids > 0.3 Table 1: Typical relative densities of some artificial and natural cellular materials [3]. There are three regimes in the stress-strain curve for a cellular material upon compression (Figure 1a). The initial linear elastic response, which corresponds to cell edge bending or face stretching, is followed by a stress plateau corresponding to progressive cell collapse by elastic buckling, plastic yielding or brittle crushing [3]. The stress plateau depends on the nature of the solid from which the material is made. When the opposite cell walls come into contact, cell collapse ends. This is the third regime called densification which 3

4 denotes a collapse of the cells throughout the material. Upon compression, the stiffness of the cellular material increases and converges towards the stiffness of the base material. Many cellular solids have low relative densities and they can be deformed up to large strains before densification occurs. At small tensile strains, the linear elastic response is the same as during compression (Figure 1c). As the strain increases, the cells become more aligned along the loading direction, increasing the stiffness of the material until tensile failure occurs [4]. Figure 1: Uniaxial stress-strain curve for (a) elastomeric foam upon compression, (b) elastic-plastic foam upon compression, (c) elastomeric foam upon tension and (d) elastic-plastic foam upon tension [4]. 2 Cellular solid A cellular solid consists of an interconnected network of solid struts or plates forming edges and faces of cells[5]. Honeycomb structures are two-dimensional cellular solids and are less complex than the three-dimensional structures like 4

5 foams. Foams are either open, meaning that they have solid only at the edges of the polyhedral, or closed, indicating a solid membrane over the faces of polyhedral cells [4]. 2.1 Mechanics of honeycombs Honeycombs are the commonest two-dimensional structures (Figure 2a). It is important to understand the mechanical behavior of these structures, particularly since they are mainly used in load-bearing structures. Figure 2: Hexagonal cell (a) undeformed, (b) upon in-plane compression initially deforms by bending, (c) upon in-plane compression at a sufficiently high load collapse by elastic buckling [4]. 5

6 In this section, we discuss the in-plane and the out-of-plane deformation of honeycombs and the corresponding elastic moduli. The mechanical properties of honeycomb depend on the direction of loading. The in-plane properties depend on the relative density and cell geometry, whereas the out-of-plane properties depend only on relative density. Properties in the x 1 x 2 plane are defined as in-plane properties [3]. If the cell walls of honeycomb are compressed in-plane (Figure 2b), they initially deform by bending [4], causing a linear elastic deformation [5]. This is why the in-plane stiffnesses and strength are the lowest. If the compression increases beyond a critical strain, the cells collapse by elastic buckling in elastomeric materials, by plastic yielding in materials with a plastic yield point, by creep or brittle fracture in brittle materials [3]. The last two modes of collapse are irreversible [5]. Ifthecellwallsofhoneycombareexposedtotensiontheybend, butnotto the point of elastic buckling. Instead they show plasticity, wherein a fracture can be observed if the cells are brittle [3]. If the hexagonal honeycomb is regular, which means that all the angles measure 30, wall lengths are all equal and all wall thicknesses t are also equal, then the in-plane properties are isotropic [3]. This means that we can describe the in-plane properties of the honeycomb by Young and shear modulus E and G, respectively [3]. The notation * denotes the effective value. Conversely, if the honeycomb is not regular, the in-plane properties are anisotropic,andtheircompletedescriptionrequires4moduli(e 1,E 2,G 12,ν 12 where ν 12 is a Poisson ratio) and two values for the plateau stress σ 1 and σ 2 [5]. If we assume that the honeycomb has a low relative density, so that wall thickness t is small compared to wall length l [5]. Simple geometry gives [5] ρ = t h/l+2 l2cosθ(h/l+sinθ), (2) where h is the length of the side of the cell wall (Figure 2a). The relative density is proportional to t/l. A load P on the end of the sloping cell wall P = σ 1 (h+lsinθ)b, where σ 1 is stress acting in the x 1 direction, b is depth of the cell wall, h/l and θ are related to the cell geometry. The cell wall is deflected by δ = Pl3 sinθ 12E 0 I, (3) where E 0 is the Young modulus of the solid cell wall material and I = bt 3 /12 is the moment of inertia of the cell wall [4]. The x 1 -component of the 6

7 deflection equals δsinθ and gives the strain ǫ 1 = δsinθ lcosθ = σ 1(h+lsinθ)bl 2 sin 2 θ. (4) 12E 0 Icosθ The Young modulus parallel to x 1 is E 1 = σ 1 /ǫ 1, giving E1 ( ) t 3 ( ) cosθ t 3 = E 0 l (h/l+sinθ)sin 2 θ. (5) l Young modulus for in-plane loading depends on relative density E 1 E 0 ( ) ρ 3. (6) Travertine and tufa are both a variety of limestone, but tufa is refered as porous traventine. The compressive strenght is N/mm 2 for traventine and up to 100 N/mm 2 for tufa. In a similar way we can find the Young modulus in the x 2 direction and the shear modulus in the x 1 x 2 plane. It depends on the relative density, a factor related to the cell geometry and on E 0 [5]. At a sufficiently high load, the cells will collapse by elastic buckling, plastic yielding, creep or brittle crushing [3]. The elastic buckling collapse stress is proportional to the Euler buckling load σ el = P crit 2lbcosθ, (7) where Euler load is P crit = n 2 π 2 E 0 I/h 2 and n describes the rotational stiffness of the node where three cell walls meet (Figure 2c). The rotational stiffness depends on the degree of constraint to rotation at the node B caused by the walls AB and BC. If rotation is freely allowed, n = 1, for both ends fixed n = 2 [5]. The plastic collapse stress and the brittle crushing stress are calculated knowing the modulus of rupture required to fracture the walls and plastic hinges in the cell walls [4]. When all cells collapse, the cell walls press against one another, the stress rise sharply, which we refer to as densification [4]. Loading in the out-of-plane direction means loading along the prism axis, i.e along the x 3 -direction. The function of a honeycomb core is to carry the shear and normal loads in the x 3 direction [3]. The cell walls of honeycomb initially compress axially [4] and therefore the moduli, collapse stresses, as well as strength will be much larger than the in-plane ones [3]. A total of 9 moduli are needed to describe the out-of-plane deformation [3, 5]. 7

8 Upon loading in the out-of-plane direction, the cell walls of honeycomb initially compress axially, so the Young modulus E 3 is proportional to the volume fraction of solid or to the relative density: E 3 E 0 = ρ. (8) Forloadinginthex 3 directiontheyieldstrengthandbrittlecrushingstrength both vary linearly with the relative density [4]. 2.2 Mechanics of foams The mechanical properties of foams, three-dimensional cellular solids, are related to the properties of the cell wall and to the cell geometry. Man-made foams are used for absorption of the energy of impacts and in lightweight structures [5]. We can estimate their mechanical properties with dimensional arguments, based on the failure in foams and their mechanism. This argument assumes that the cells in foams are geometrically similar [4]. We will use a cubic cell, but the same result is obtained for any cell geometry, as long as the mode of deformation or failure is the same [4]. In the first regime of the stress-strain curve, which is the linear elastic regime, under uniaxial stress, open-cell foams deformation is caused primarily by bending of the cell edges [4]. We can estimate E. The bending deflection under a transverse load, F, is like point load on a rod (Figure 3) δ Fl3 E 0 t4, (9) where l is length of a strut, E 0 is the Young modulus of the solid, and cross-sectional area is proportional to t 2. The strain acting on the cell is proportional to δ/l [Eq. (4)] and the stress is proportional to F/l 2. For any open-cell foam the relative density is, using dimensional arguments, proportional to (t/l) 2 [4], giving E E 0 = C 1 ( ) ρ 2. (10) The Young modulus depends on the square of the relative density. All constants of proportionality related to the cell geometry are combined in one constant C 1. By fitting Eq. (10) to data, we find that C 1 1. Poisson ratio depends on cell geometry, and its typical values lie around 1/3 [4]. 8

9 Figure 3: An open-cell foam under a transverse load [4]. In (10) is assumed that all of the material of the foam is found in the struts, that define the cells. In a closed-cell foam, some fraction of the polymer resides in the cell walls or faces and we have to take that into consideration. The elastic collapse stress is proportional to the Euler buckling load divided by l 2 : σel P crit /l 2 E 0 t 4 /l 4. Thus the elastic collapse stress depends on the square of the relative density σ el E 0 = C 2 ( ) ρ 2. (11) Fitting Eq. (11) to data gives C The applied moment on a strut fromatransverseforcef ism Fl σ pll 3 andtheplasticmoment, required to form plastic hinges is M p σ ys l 3, where σ ys is the yield stress of the solid cell wall material [4]. The plastic moment M p is the moment at which the entire cross section has reached its yield stress. If we equate both moments we find the plastic collapse stress σ pl σpl ( ) ( ) t 3 ρ 3/2 = C 3. (12) σ ys l C 3 is approximately 0.3 [4]. The brittle crushing strength is established in a similar way [4]. Mechanical properties can also depend on the fluid within 9

10 the cells. In open-cell foams, there is a viscous dissipation of fluid moving through the cells, which depends on the fluid viscosity, the strain rate, the sample size and the cell size [4]. 3 Wood Wood is an almost pure polymeric composite [2] and it primarily comprises parallelcellsthatarearrangedalongthetrunkofatree[6]. Itcanberegarded as a cellular material at the scale of a hundred micrometers to centimeters [2]. The cellular structure of wood is similar to that of the honeycomb. Loading along the grain, which is analogous to the out-of-plane direction in the honeycomb, compresses the cell walls axially. When loaded across the grain, analogous to the in-plane direction in the honeycombs, cell walls bend [4]. The stiffness and strength of wood depend on its relative density [4]. The compressive strength of the wood is determined by uniaxial yielding for loading along the grain and varies linearly with relative density. The strength across the grain varies with density square and is determined by the formation of plastic hinges while bending [4]. The value of the strength of the cell walls depends on the loading direction. The mechanical properties of wood along the cells, are much more prominent than in perpendicular direction [2]. If the density of the wood varies, we can describe wood s compressive strength using the honeycomb model which explains the dependence of the mechanical properties on the density and the anisotropy of wood [2, 4]. Young modulus and strength along the grain vary linearly with relative density. Because of this wood has remarkable efficiency, which is most apparent in resisting bending and also buckling loads. For a beam of a given stiffness or a column of an elastic buckling load, the material that minimizes themassofthebeamorcolumnmaximizestheperformanceindexe 1/2 /ρ [4]. Rearranging Eq. (8), gives E 1/2 = E1/2 0 ρ ( ρ0 ρ ) 1/2. (13) This result describes loading along the grain [4]. Wood is more efficient than thesolidfromwhichisitmade[4]. Itisbetterbyafactorof( /ρ ) 1/2, which is about 2 for typical softwoods with a relative density of 0.3 [4]. Spruce has a value of 7.1 kpa 1/2 (gm 3 ) 1 and a unidirectional carbon fibre composite has a value of about 9 kpa 1/2 (gm 3 ) 1. 10

11 The performance index for the strength of the beam in bending is defined by σ 2/3 /ρ. The performance of wood is better than that of the solid from which is it made [4] σ 2/3 ( ) 1/3 = σ2/3 ys ρ0. (14) ρ ρ Spruce has a value of 92 Pa 2/3 (gm 3 ) 1 and a carbon fiber composite has a value of 75 Pa 2/3 (gm 3 ) 1 [4]. It is the bending and buckling performance of wood that is critical in trees. Branches bend because of their weight and the wind [4]. The trunk bends because of the wind, but it also supports the weight of the entire tree above any given height [4]. Dimensional analysis of elastic buckling indicates thattreediametershouldincreasewiththe3/2poweroftheheighth[4]. The EulerbucklingloadisP crit EI/h 2 andtheweightofthetreeisw ρgd 2 h. Equating the weight to the buckling load and rearranging gives ( ) Cgρ 1/2 d = h 3/2, (15) E where E is the Young modulus of the wood, g is the acceleration of gravity, ρ is the density of the wood and C is a numerical constant. The absolute height of the tallest trees is limited by water transport constraints [4]. 4 Trabecular bone The trabecular bone is a hierarchically structured material with remarkable mechanical properties. The structure itself is extremely complex and variable, but the mineralized collagen fibril, its basic building block, is rather universal. Bones consist of roughly half polymer and half mineral components interconnected at the nanoscale. Their mechanical performance depends on bone s architecture at all levels of hierarchy [7]. Trabecular bone can be considered as a foam-like network of bone trabeculae [2]. It can be found at the ends of long bones, proximal to joints and within the interior of vertebrae, and also in the core of shell-like bones such as the skull [4]. The porous space left free by the bone material is filled with marrow and living cells. The thickness of the strut, a trabeculae, in the trabecular bone is fairly constant between one and three hundred micrometers [2]. The typical relative density of trabecular bone is between 0.05 and 0.3, but it generally depends on the magnitude of the loads. The orientation is determined by the load distribution in the bone [2]. Trabecular bone grows in response to load. Trabecular bone with low-density resembles an 11

12 open-cell foam, while one with high-density denotes a more plate-like structure. Strength depends on the interplay between different structural levels, whereas stability depends on trabecular architecture [7]. Fracture toughness and elastic constants of bone are anisotropic due to the lamellar structure of the bone matrix [7]. Stiffness and fracture strain depend on amount of minerals in the collagen matrix. An increase of stiffness was observed as a result of an increase of mineral density [7]. Young modulus for open-cell foams varies with the square of the relative density, as we have seen in Eq. (10). In Figure 4a, Young modulus is plotted against relative density. The data in the log-log plot are well described by a line of slope 2. Bending is the dominant mode of linear elastic deformation in trabecular bone [4]. Scatter in the data is caused by variations in the trabecular architecture [4]. In Eq. (10), we have constant C 1 1, but the variation in the architecture of trabecular bone amounts to the variation of C 1 [4]. Figure 4: (a) Young modulus and (b) the compressive strength of trabecular bone plotted against density [4]. 12

13 In Figure 4b the compressive strength is plotted against relative density. The data lie roughly on the line slope of 2. This corresponds to failure by elastic crushing [Eq. (11)] [4]. The density dependences of the Young modulus and compressive strength suggest that the failure strain is a constant and that it is independent of density [4]. Understanding the properties of trabecular bone is important for the treatment and correct assessment of diseases such as osteoporosis. An osteoporotic bone has less strength and less stiffness than a normal bone [8]. In patients whit osteoporosis, the bone mass is lost through uniform thinning of the trabeculae [2] and the osteoporotic bone is not able to absorb as much energy before fracture as the normal bone [8]. After reaching a certain critical thickness of the trabucelae, bone mass is also lost through resorption of individual struts [2]. A random removal of trabeculae is more damaging to bone strength than uniform thinning of the trabeculae [2]. A subsequent increase of bone mass according to the original value by thickening the remaining struts does not restore its mechanical properties. It can be inferred that the preservation of trabecular connectivity should be the aim of future drug therapy pursuits [2]. 5 Glass sponge Euplectella sp. Glass sponge Euplectella sp. is a deep-sea, sediment dwelling sponge from WesternPacific[9]. Theskeletonofaglassspongeiscomposedofalmostpure silica mineral [2]. Spicules in siliceous sponges exhibit exceptional flexibility and toughness compared with brittle synthetic glass rods of similar length scales [2]. They have optical properties which can be compared to manmade optical fibers and are structurally resistant. The individual spicules are just one structural level in an almost purely mineral skeleton of Euplectella. The cylindrical glass cage (20-25 cm long, 2-4 cm in diameter) with lateral openings is exposed to ocean currents. Its function is to support the living portion of the sponge, responsible for filtering and metabolite trapping. At the macroscopic scale, the cylindrical structure is reinforced by ridges that extend perpendicularly to the surface of the cylinder and spiral the cage. The surface of the cylinder consists of a regular square lattice, composed of a series of cemented vertical and horizontal struts each consisting of bundled spicules parallel to one another [9]. Despite its fragility, glass is widely used as a building material in the biological world. Its strength is compromised by surface defects where there is a concentration of applied stresses. A scratch in the surface is likely to induce fracture [9]. The surface defects in the silica may be induced by 13

14 external point loads, meaning that a scratched plain glass spicule would fracture when subsequently loaded by tension or bending.let us denote the sizeofthelargestdefectbyh. Forh h,theyieldstrengthcanbecalculated from h σ f σf th h, (16) where σf th is the theoretical strength of the defect-free material and h is a characteristic length. For typical ceramic materials, this length is on the order of 10 to 30 nm [9]. If h h the strength of the material is equal to its defect-free value σ f = σf th [9]. The low strength of the glass is balanced at the next structural level. A large number of individual glass layers protect the spicule efficiently. Thin organic interlayers seem to be important in preventing cracks from extending to inner layers. Thicker inner layers help to improve rigidity of the spicule. At a higher level of hierarchy, spicules join to form parallel bundles, with larger defect tolerance than that of the individual fibers. If a fiber fails, the neighboring ones still hold, and the crack in the first fiber will be deflected at interface to its neighbors. A weak lateral bonding between fibers is crucial for this toughening mechanism to work [9]. The rigid structures ensure that the cylinder is stable with a number of measures to reduce the brittleness of glass [9]. 6 Conclusion In the seminar, three cellular structured materials of entirely different chemical compositions have been described. The wood cell wall is an almost pure polymeric composite, whereas the bone is an organic-inorganic composite and the skeleton of a glass sponge is an almost pure silica mineral composite. It was shown that the most important feature in describing mechanical properties of cellular structure is the relative density. The elastic properties of cellular materials depend on the solid of which they are made from and on cell geometry. The cellular structure of wood is similar to that of honeycomb and the mechanical properties of wood linearly depend on the relative density. The stiffness and strength are higher when loaded along the grain than across it. The performance of wood in bending and buckling is better than that of the solid from which it is made. The cellular structure of the trabecular bone resembles that of a foam. Trabecular bone grows in response to the load and its direction. The Young 14

15 modulus and compressive strength vary with the square of the relative density. The structural hierarchy of the skeleton of glass sponge Euplectella sp. is very complex and it serves as an example of nature s ability to improve deficient building materials. References [1] C. Tekoglu, Size effects in cellular solids (PhD thesis, University of Groningen, Groningen, 2007); available at ( ). [2] P. Fratzl and R. Weinkamer, Prog. Mater. Sci. 52, 1263 (2007). [3] ( ). [4] L. J. Gibson, J. Biomech. 38, 377 (2005). [5] L. J. Gibson and M. F. Ashby, Celullar solids (Cambridge University Press, Cambridge, 1997). [6] R. B. Miller, Structure of Wood (Forest Products Laboratory, Madison, 1999). [7] I. Jager and P. Fratzl, Biophys. J. 79, 1737 (2000). [8] R. P. Dickenson, W. C. Hutton, J. R. R. Stotf, J. Bone Jt. Surg. 63, 233 (1981). [9] J.Aizenberg, J.C.Weaver, M.S.Thanawala, V.C.Sundar, D.E.Morse and P. Fratzl, Science 309, 275 (2005). [10] ( ). 15