Transverse Anisotropy in Softwoods

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1 Transverse Anisotropy in Softwoods Modeing and Experiments CARL MODÉN Licenciate Thesis Stockhom, Sweden 2006

2 TRITA-AVE 2006:30 ISSN ISBN KTH Engineering Sciences Department of Aeronautica and Vehice Engineering Teknikringen 8 SE Stockhom SWEDEN Akademisk avhanding som med tistånd av Kung Tekniska högskoan framägges ti offentig granskning för aväggande av teknoogie icenciatexamen i ättkonstruktioner torsdagen den 8 juni 2006 kockan i S40, Teknikringen 8, Kung Tekniska Högskoan, Stockhom. Car Modén, june 2006 Tryck: Universitetsservice US AB

3 iii Abstract Transverse anisotropy is an important phenomenon of practica and scientific interest. Athough the presence of ray tissue expains the high radia moduus in many hardwoods, experimenta data in the iterature shows that this is not the case for pine. It is possibe that anisotropy in softwoods may be expained by the ceuar structure and associated deformation mechanisms. An experimenta approach was deveoped by which oca radia moduus in spruce was determined at sub-annua ring scae. Digita specke photography (DSP) was used, and the density distribution was carefuy characterized using x-ray densitometry and the SiviScan apparatus. A unique set of data was generated for radia moduus versus a wide range of densities. This was possibe since earywood density shows arge density variations in spruce. Quaitative comparison was made between data and predictions from stretching and bending honeycomb modes. The hypothesis for presence of ce wa stretching was supported by data. A mode for wood was therefore deveoped where both ce wa bending and stretching are incuded. The purpose was a mode for predictions of softwood modui over a wide range of densities. The reative importance of the deformation mechanisms was investigated in a parametric study. A two-phase mode was deveoped and radia and tangentia modui were predicted. Comparison with experimenta data showed good agreement considering the nature of the mode (density is the ony input parameter). Agreement is much better than for a reguar honeycomb mode. According to the mode, ce wa bending dominates at both ow and high densities during tangentia oading. In radia oading, ce wa stretching dominates at higher densities.

5 Preface This work is part of the WoodComp B project in the Biofibre Materias Centre (BiMaC) program funded by KTH and Skogsindustrierna. First I woud ike to thank my supervisor Prof. Lars Bergund. You have not ony introduced me to the fascinating word of wood mechanics, but you have aso taught me how to start thinking ike a materias scientist. Your devoted supervision has inspired me deepy. I woud aso ike to thank everyone in the Biocomposites research group, Andrey, Jonas, Mariee, Main, Saïd and Anna. Our Monday morning meetings make it much easier to start the week and I think it is great to work with peope with such different backgrounds and interests, but one common goa. I am aso part of the division of Lightweight Structures at the Aeronautica and Vehice Engineering department. I ike the discussions during unches and in the corridors as they have heped me to see things a ot cearer. Finay much kudos to my parents. It is your everasting beief in me and your support, both with encouragement and with practica things that has made me come this far. Thank you for aways being there. v

6 Contents Preface Papers v vii 1 Introduction Mechanics of honeycombs Appication of honeycombs to wood Thesis objective Bibiography 9 Paper 1 Eastic deformation mechanisms of softwoos in radia tension ce wa bending or stretching? Paper 2 Modeing transverse anisotropy in Softwoods Anaysis of deformation vi

7 Papers 1. Modén, Car S, and Bergund, Lars A., Eastic deformation mechanisms of softwoods in radia tension ce wa bending or stretching?, to be submitted to Hozforschung, Modén, Car S, and Bergund, Lars A., Modeing transverse aniostropy in Softwoods anaysis of deformation, manuscript, vii

9 Chapter 1 Introduction Wood is a heterogeneous materia. This apparent from on the cross-section of coniferous wood. The annua rings form ight and dark patterns. If the same surface is viewed in arger magnification, the previousy homogeneous rings become heterogeneous where the tracheids form a ceuar materia. The ceuar structure and annua ring structure are two important microstructura scaes controing wood properties. Transverse anisotropy is an important phenomenon in wood science of both scientific and industria interest. Simpe hexagona honeycomb modes for transverse modui of softwoods may offer improved understanding through identification of critica materia parameters and their infuence on transverse modui. 1.1 Mechanics of honeycombs Gibson and Ashby [4] cacuated the mechanica properties of structures having the same hexagona geometry as the honeycombs in bee hives. This kind of structure is found in many paces in nature and in man-made products. The foowing discussion essentiay foows Gibson and Ashby [4] The ideaised geometry of a honeycomb ce is presented in Figure 1.1. A genera honeycomb has four geometrica parameters. The four parameters are two ce wa engths, h and, controing the dimensions of the ce, the ce shape ange θ controing the shape of the honeycomb and finay ce wa thickness t. A honeycomb where h = and θ = 30 is termed a reguar honeycomb. Such a honeycomb has an interesting symmetry, since it wi ook exacty the same if rotated 60. Figure 1.1 aso contains a coordinate system. The axes R and T correspond to the radia and tangentia directions found in wood. The R and T -directions correspond to the x 1 and x 2 -directions in the notation used by Gibson and Ashby. There is aso a thickness dimension of the honeycomb out of the pane. Throughout this text, this thickness is assumed to be unity. 1

10 2 CHAPTER 1. INTRODUCTION t R T θ h Figure 1.1: Geometry of a honeycomb ce. Figure 1.1 shows one ce of the honeycomb. This geometry is usefu in order to understand the structure of the honeycomb. There is however an even simper geometrica representation of the honeycomb, that is usefu when deriving properties of the honeycomb. Figure 1.2 shows this representation of the smaest repeat unit of a honeycomb structure. This unit consists of ony three ce was. The dimensions of the repeat unit are h + sin θ and 2 cos θ. h+ sin θ 2 cos θ R T θ h Figure 1.2: Geometry of the smaest repeat unit in a hexagona honeycomb. A very important property of a honeycomb is the reative density. This is cacuated as the density of the ceuar structure ρ divided by the density of the soid ce wa ρ s. The reative density is thus given as the voume fraction of materia in the repeat unit. From Figure 1.2 it cear that the reative density can be expressed as ρ ρ s = 2t + th 2 cos θ(h + sinθ) = t h/ +2 2 cos θ(h/ + sin θ) (1.1) In the second step, the aspect ratio of the ce wa t/ is separated from the rest of the geometry. As expected, the reative density is directy proportiona to the

11 1.1. MECHANICS OF HONEYCOMBS 3 aspect ratio of the ce wa. In the case of reguar honeycombs, the density simpifies to ρ = t 2 (1.2) ρ s 3 Cassica beam theory can be used to derive the mechanica properties of honeycombs in the pane. This means that the members wi not stretch and cross-sections of the was wi be pane during beam deformation. The corners of the honeycomb are assumed to be rigid, i.e. there is no hinging in the corners. The corners can transate, but rotation of the corners are imited, because of the repeated structure, so that a corners must rotate equay. Due to symmetry, no rotation of the corners is possibe at a if the honeycomb is oaded parae to any of the coordinate axes. Symmetry aso prevents deformation of the ce was parae to the T -axis for the cases when oad is parae to any of the two coordinate axes. Figure 1.3 a) shows a honeycomb oaded in the R-direction. The reationship between the goba stress σ R and the force P R in one ce wa is σ R = P R h + sin θ (1.3) The denominator, h+ sin θ, is the size of the smaest repeat unit in the T -direction. The component of the oad perpendicuar to the ce wa, P sin θ wi cause the wa to bend. According to beam theory, the defection of the wa wi be δ = P R sin θ 3 12E s I (1.4) Where E s is the eastic moduus of the ce wa and I is the second moment of inertia of the ce wa. For a rectanguar cross section with thickness t and unity width this is I = t 3 /12. Strain is dispacement divided by the ength over which the dispacement is distributed. In the case of deformation of a honeycomb in the R-direction, the dispacement is the component of beam defection in the R-direction, δ cos θ. This is the tota deformation of a unit ce and hence strain in the R-direction is reated to defection δ as ε R = δ sin θ cos θ = P R sin 2 θ 2 E s cos θt 3 (1.5) Hookes aw, σ = Eε, tes us the reationship between stress and strain for a inear eastic materia. Since we have derived reations for σ R and ε R, it is easy to derive the Young s moduus in the R-direction. E R = σ R E s t 3 cos θ = ε R (h + sin θ) sin 2 (1.6) θ 2 which can be rewritten to express the reative moduus E R E s = ( t ) 3 cos θ (h/ + sin θ) sin 2 θ (1.7)

12 4 CHAPTER 1. INTRODUCTION σ R a) b) δ θ P R h R T σ R Figure 1.3: a) Honeycomb oaded in the R-direction. b) The deformation of a ce wa oaded in the R-direction. The same arguments can be used for deriving expressions for stresses and strains when the honeycomb is oaded in the T -direction, see Figure 1.4. The stress in the T -direction is σ T = P T (1.8) cos θ a) b) δ P T θ σ T h σ T R T Figure 1.4: a) Honeycomb oaded in the T -direction. b) The deformation of a ce wa oaded in the T -direction. Reation between beam defections and force P T on the beam is δ = P T cos θ 3 12E s I (1.9)

13 1.2. APPLICATION OF HONEYCOMBS TO WOOD 5 The strains wi be ε T = Moduus in the T -direction is δ cos θ h + sin θ = P T cos 2 θ 3 E s (h + sin θ)t 3 (1.10) E T = σ T = E s(h + sin θ)t 3 ε T 2 cos 3 θ (1.11) Again this expression can be rewritten to express the reative moduus of the honeycomb in the T -direction. ( ) 3 E T t h/ + sinθ = E s cos 3 (1.12) θ Two interesting specia cases are a honeycomb with reguar hexagons and a honeycomb with θ =0. In the case of a reguar honeycomb E R = E ( ) 3 T t 4 = 3 (1.13) E s E s Since E R = E T and because of symmetry for rotations of 60, the reguar honeycomb is isotropic. In order to study the other interesting specia case, we study the imit when θ approaches 0. E R im = im θ 0 E s θ 0 E R im = im θ 0 E s θ 0 ( ) 3 t cos θ (h/ + sin θ) sin 2 θ = (1.14) ( ) 3 ( ) 3 t h/ + sin θ t h cos 3 = (1.15) θ The reative moduus E R /E s is infinite when ce was are parae to R-direction and this consequence is not discussed by Gibson and Ashby. The infinite moduus does ceary not exist in a rea materia, but is a imitation caused by the assumption that ce was ony deform through bending. We wi return to this probem when we discuss the appications of honeycomb theories to wood. 1.2 Appication of honeycombs to wood Honeycomb modes have been appied to wood. Eastering et a.[3] studied the mechanics of basa using reguar honeycombs. They proposed that the ray ces woud act as stiffeners in the radia direction, thus expaining the anisotropy in basa. Gibson and Ashby [4] generaised the mode and appied it to a species of wood. They aso assume a honeycomb with reguar hexagons and use data from the iterature and fit a ine foowing the equation ( ) 3 E T ρ = C 1 (1.16) E s ρ s

14 6 CHAPTER 1. INTRODUCTION They use axia data from Cave [2] for the ce wa moduus, E s = 35 GPa. The constant C 1, fitted to data wi incude the ce wa anisotropy and effects from ceuar arrangements, and is reported as to The figure from Gibson and Ashby is reproduced in Figure 1.5. Figure 1.5: Wood moduus as a function of density. After Gibson and Ashby [4]. The points are experimenta data in three directions from various species. The fitted ines are proportiona to density for the axia data, and proportiona to the cube of radia and tangentia data. Note the discrepancy between radia data (fied circes) and the theory Since Gibson and Ashby assume reguar hexagons, the mode behaves isotropic. The higher moduus in the radia direction is expained by the existence of rays. Gibson and Ashby argue that the radia moduus is E R = V R R 3 E T + (1 V R )E T (1.17)

15 1.3. THESIS OBJECTIVE 7 where V R is the voume fraction and R is the density ratio of rays and tracheids. A typica vaue for the radia moduus is E R =1.5E T. A probem with the argument used by Gibson and Ashby is that the rays might not act as stiffeners, at east not in softwoods. Bouteje [1] measured the anisotropy in sweing during water sorption of sma pine specimen of varying density. Some of the specimens had the rays removed whereas the rays where sti present in other sampes. Bouteje showed that anisotropy is ceary dependent on density, but not dependent on whether the rays were removed or not. Bouteje s [1] observations ead to the concusion that Gibson and Ashby s [4] assumptions of rays, that increase the radia moduus, is wrong for pine, and possiby most softwoods. It is necessary to find another anaytica method to mode the anisotropy of transverse moduus. One way of doing this is to remove the restriction of a reguar honeycomb by changing the ce shape ange θ. Kahe and Woodhouse [5] impemented the mode by Gibson and Ashby in a numerica approach, and adjusted θ according to experimenta data. Watanabe et a. [7, 6] determined ce geometries in earywood for seven species of softwood. They used a honeycomb where bending and stretching of the ce was was incuded. The agreement between predictions and data was good. 1.3 Thesis objective The objective of the present study is to improve the understanding of transverse anisotropy in softwoods through the combination of experimenta characterisation and theoretica modeing. In particuar, the hypothesis of ce wa stretching as a contributing deformation mechanism is investigated. This mechanism has not received much attention previousy.

17 Bibiography [1] J. B. Bouteje. The reationship of structure to transverse anisotropy in wood with reference to shrinkage and easticity. Hozforschung, 16:33 46, [2] I.D. Cave. The ongitudina young s moduus of Pinus Radiata. Wood Science and Technoogy, 3:40 48, [3] K.A. Eastering, R. Harryson, and L.J. Gibson. The mechanics of twodimensiona ceuar materias. Proceedings of the Roya Society of London, A382:31 41, [4] L.J. Gibson and M.F. Ashby. Ceuar Soids: Structure and Properties. Pergamon Press, Oxford, [5] E.E. Kahe and J.E. Woodhouse. The infuence of ce geometry on the easticity of softwood. Journa of Materias Science, 29: , [6] U. Watanabe, M. Fujita, and M. Norimoto. Transverse Young s Modui and Ce Shapes in Coniferous Eary Wood. Hozforschung, 56:1 6, [7] U. Watanabe, M. Norimoto, T. Ohgama, and M. Fujita. Tangentia Young s moduus of coniferous eary wood investigated using ce modes. Hozforschung, 53: ,

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