Acta Polytechnica CTU Proceedings doi:10.14311/APP.2018.15.0051 Acta Polytechnica CTU Proceedings 15:51–56, 2018 © Czech Technical University in Prague, 2018 available online at http://ojs.cvut.cz/ojs/index.php/app DETERMINATION OF HYGROEXPANSION COEFFICIENTS OF WOOD USING ANALYTICAL AND NUMERICAL HOMOGENIZATION Lucie Kucíková, Michal Šejnoha∗, Jan Vorel Czech Technical University in Prague, Faculty of Civil Engineering, Thákurova 7, 166 29 Prague 6, Czech Republic ∗ corresponding author: sejnom@fsv.cvut.cz Abstract. This paper is concerned with the determination of hygroexpansion coefficients of wood. Wood is a naturally hygroscopic material, attracting moisture from its surroundings. Its dimensions change depending on the actual moisture content. This dependence is characterized by the coefficient of hygroexpansion. For complex microstructures, this coefficient can be determined, similarly to the coefficient of thermal expansion, either from analytical or numerical homogenization. The homogenization procedure, used in this paper, comprises several steps corresponding to the wood structural composition. Accuracy of this approach is governed by correctness of the chosen input parameters. One of the most important parameters are the volume fractions of the earlywood and latewood. Ascertained dependence of the earlywood and latewood volume fractions on dry wood density is also presented. Keywords: Wood, homogenization, hygroexpansion coefficient. 1. Introduction Environmentally friendly materials are getting to the forefront of global interest and one of them is also wood. The wood is a natural cellular material and as such its properties change depending on growth site and species. Spruce wood, examined in this paper, belongs to conifers and therefore consists mainly of tracheids [1], long closed tube-like cells with cavity in- side (called lumen). There are two types of tracheids. The first one is characterized by almost square cross- section and thin cell wall, and it forms the earlywood part of the growth ring. The second one forms the latewood part of the growth ring. Compared to the previous one, its cross-section is almost half in width in the radial direction and the cell wall is much thicker. Generally, a multilayered cell wall structure is sim- plified to three layers [2]: M layer (middle lamella, primary wall and outer layer of secondary wall S1), S2 layer (middle layer of secondary wall), and S3 layer (inner layer of secondary wall). Each layer is composed of four basic elements: lignin, hemicellulose, cellulose (crystalline and amorphous) and extractives. Due to their low volume fraction, the extractives are often ne- glected, as was also the case in the present study. The cellulose and hemicellulose are in the form of microfib- rils, bonded together by lignin [3]. The microfibrils wound helically along the cell and their deviation from a longitudinal axis is called a microfibril angle (MFA). The microfibril angle generally changes within the cell wall, but for simplicity it is assumed constant in individual layers. Clearly, it is one of the main factors influencing the wood properties, see e.g. [4] Wood is a naturally hygroscopic material, i.e. it attracts moisture from surrounding atmosphere. The water is present in wood in two forms: free water (i.e. liquid water or water vapor in pores) and bound water (i.e. liquid water in the cell wall). The amount of water present in wood is characterized by the mois- ture content (MC), defined as the ratio of the mass of water to the mass of oven-dried material. Moisture content at which the cell walls are fully saturated and with no free water in the lumens is called a fiber satu- ration point (FSP). The average value of FSP is about 30% [5]. Interestingly, several definitions of fiber sat- uration point can be found in literature [6]. Below this point, the major part of dimensional changes oc- curs in the form of shrinkage (contraction) or swelling (expansion). The resulting moisture strain is linked to the change in moisture content through the hygroexpansion coef- ficient αh [-]. Estimating this coefficient at the level of wood is the principal objective of this paper. To that end, several homogenization techniques combin- ing classical micromechanics and unit cell simulations will be adopted. These are briefly outlined in Section 3 preceded in Section 2 by microstructure quantification, both in terms of material composition and geometry. The resulting predictions are presented in Section 4 followed by the summary of essential observations in Section 5. 2. Materials Special attention should be paid to the selection of input parameters used in homogenization. Cell wall constituents are very small particles, which are diffi- cult to measure. In this paper, properties presented 51 http://dx.doi.org/10.14311/APP.2018.15.0051 http://ojs.cvut.cz/ojs/index.php/app L. Kucíková, M. Šejnoha, J. Vorel Acta Polytechnica CTU Proceedings E11 E22 = E33 G12 = G13 ν21 = ν31 ν22 = ν33 αh,11 αh,22 = αh,33 [GPa] [GPa] [GPa] [-] [-] [-] [-] Cellulose 150.00 17.50 4.50 0.01 0.50 0.000 0.000 Hemicellulose 16.00 3.50 1.50 0.10 0.40 0.000 1.368 Lignin 2.75 2.75 1.03 0.33 0.33 0.351 0.351 Table 1. Phase properties at cell wall level [2]. in [2] were taken as the most relevant and used for computations. Properties of individual cell wall con- stituents are summarized in Table 11. Besides the properties, it is necessary to know the volume frac- tions of individual constituents within each cell wall layer. These are stated in Table 2. In each layer the hemicellulose and cellulose are deviated by a differ- ent microfibril angle. The values of MFA in M and S3 layers were taken from [2]. The MFA of S2 layer was computed by comparing the values found from homogenization and nanoindentation measurements, see [7]. Layer M S2 S3 fcellulose 0.20 0.49 0.49 fhemicellulose 0.15 0.27 0.27 flignin 0.65 0.24 0.24 MFA [°] 45.00 32.53 75.00 Table 2. Volume fraction of cell wall constituents and MFA of individual cell wall layers [2]. Nanoindentation is a nano-/micro-scale measure- ment technique based on contact mechanics, usually used to determine the material hardness and elastic modulus. The method grounds on pushing a small hard tip with known geometry and properties to the tested material while simultaneously recording the applied load and the measured displacement. Herein, only the S2 layer, as the principal carrier of the cell wall stiffness, was indented using the Berkovich inden- ter. A certain hold period at the level of maximum force was considered within each loading cycle to elim- inate the viscoelastic response of the material. The indentation moduli were then estimated from the un- loading part of the loading diagram, see e.g. [8], [9] for more details. According to the theory of anisotropic indentation presented in [10], the indentation modulus can also be defined in terms of the homogenized stiffness matrix pertinent to the S2 layer while taking into account the microfibril angle. Note that the effective stiffness ma- trix in the local coordinate system can be acquired for example from Eq. (5) adopting the material data from Table 1 and the volume fractions from Table 2. The stiffness matrix entering the theoretical expression of indentation modulus further calls for transformation 1Axis-1 is oriented along longitudinal (microfibril) direction, axes-22,33 along transverse direction. into the MFA coordinate system. The actual MFA is determined by matching the measured and theoreti- cally calculated indentation moduli as presented, e.g. in [11, 12]. The resulting MFA for S2 layer is listed in Table 2. The distribution of indentation modulus as a function of MFA and the probability density function of MFA derived from a large set of indents are plotted in Fig. 1 for illustration. (a) (b) Figure 1. a) Variation of indentation modulus as a function of MFA, b) Probability density function of MFA However, a word of caution is in place when adopt- ing the results from nanoindentation measurements directly without any adjustment for potential exper- imental error. As demonstrated in [13] such a step may significantly underestimate the resulting effective stiffness when compared, e.g. to macroscopic measure- ments of the effective longitudinal elastic modulus. Consequently, the predicted coefficients of hygroex- pansion may also be highly inaccurate. To address this issue we provide, apart from the measured S2 layer MFA=32.53◦, the results for MFA=10◦, 15◦, 20◦ for the sake of illustration. The numerical part of homogenization requires the formulation of a periodic unit cell (PUC) representing the material microstructure. Altogether, three types of PUCs were developed to represent the earlywood, latewood, and equivalent wood, while accounting for real microstructural dimensions. The equivalent wood is a combination of both the earlywood and latewood and was created such as to reflect the volume fraction of lumens of the two types of wood within a single growth ring. Apart from that, the volume fractions of individual layers building the cell wall are needed. Herein, we adopt the particular values from [2]. To get the volume fraction of lumens correspond- ing to an equivalent wood requires the knowledge of the volume fractions of lumens of the earlywood and latewood as well as the volume fraction of earlywood and latewood at the level of growth rings. A potential 52 vol. 15/2018 Determination of hygroexpansion coefficients method of attack to derive these quantities is provided by image analysis. Color and binary images of the earlywood of spruce are presented in Figs. 2(a,b) for illustration. (a) (b) (c) Figure 2. a) Original gray-scale image of earlywood, b) Transformed binary image of earlywood, c) Images of growth rings. These images were taken at several locations within several growth rings to increase reliability of the esti- mated volume fractions, at least on the average sense. The variability of microstructure over the length of the growth ring as well as the transition zone (a gradual change between earlywood and latewood) were, there- fore, neglected. Fig. 2(c) then offers similar images at the level of growth rings. Considering a constant width of the latewood part equal to 0.2 mm, see also [2], these images allow for obtaining the dependence of the earlywood volume fraction on dry wood density plotted in Fig. 3. Figure 3. Dependence of the earlywood volume frac- tion on dry wood density [kg/m3]. Thus, given the dry wood density ρdry [kg/m3] we can estimate the volume fraction of earlywood fEW directly from the fitting equation2 fEW = 5 · 10−6 · ρ2 dry − 5 · 10−3 · ρdry + 2.093, (1) without passing through a tedious measurement pro- cess. The volume fraction of latewood is then simply given by fLW = 1− fEW . (2) It is now possible to define the equivalent wood as basic weighted average of both the earlywood (EW) and latewood (LW). The volume fraction of lumens of equivalent wood flum is therefore flum = fEW · flum,EW + fLW · flum,LW , (3) where flum,EW , flum,LW are the volume fractions of lumens of earlywood and latewood, respectively. It is now easy to get the desired dimensions of the wood cells. Figure 4. Periodic unit cells of earlywood, latewood, and equivalent wood. The resulting periodic unit cells are shown in Fig. 4. When constructing these unit cells we adopted the inner dimensions of lumens, i.e. the dimensions of the inner rectangle of the earlywood and latewood PUCs, from [1]. Considering the volume fractions of lumens as flum,EW = 0.69 for the earlywood, flum,LW = 0.12 for the latewood [7], and dry wood density ρdry=390 kg/m3 allowed us to calculate the cell wall thickness. The volume fractions of individual layers in the cell wall rendered its final subdivision distinguishing again M layer (blue, b), S2 layer (red, r), and S3 layer (green, g). The actual thicknesses are provided in the left bottom corner of the picture for the sake of completeness. The dimensions of the equivalent wood (EqW) were derived in analogy with Eq. (3) as the weighted average of the dimensions of earlywood and latewood, respectively. The actual arrangement of the unit cells in Fig. 4 may range from perfectly regular to hexagonal array. With reference to Fig. 2 the latter one is considered in the present study. The discretized transverse cross- section of the 3D unit cells are plotted in Fig. 5. Some improvements in the prediction of effective properties 2Point out that this simple quadratic equation is only applica- ble for the plotted range of dry wood density (ρdry ∈ (350−500) [kg/m3]). 53 L. Kucíková, M. Šejnoha, J. Vorel Acta Polytechnica CTU Proceedings can be expected if addressing the microstructure vari- ability more closely [2]. In the light of the present approach this would call for so called statistically equivalent periodic unit cell, which is more refined and matches the actual microstructure in view of two- point correlation function [14]. (a) (b) Figure 5. Geometry and plane view of discretized PUCs: a) earlywood, b) latewood. 3. Methods Wood is often treated as a natural composite. Its structural composition leads to application of hierar- chical homogenization, starting from the level of cell wall constituents up to the level of the solid wood. In this case, a combination of analytical and numerical homogenization is used to determine the effective hy- groexpansion coefficients. However, according to the formula for the overall strain ε = εe + εh, εh = αhu, (4) where εe denotes the elastic strain, εh the hygroex- pansion strain, αh the vector of hygroexpansion co- efficients3 and u the change of moisture content, the effective hygroexpansion coefficients have to be com- puted simultaneously with the effective mechanical properties. The computation comprises several steps. The first step corresponds to the level of cell wall layers. Each layer (M, S2, and S3) is composed of three phases, where lignin is considered as a matrix phase and hemi- cellulose and cellulose as circular cylindrical inclusions. Properties of individual phases are stated in Table 1 and their volume fractions within each layer appear in Table 2. Using the Mori-Tanaka method, one of the most popular analytical methods, we arrive at the estimates of the effective stiffness matrix LMT in the form LMT = L0+ ( N−1∑ r=1 fr(Lr − L0)Adill r ) ( f0I + N−1∑ r=1 frAdill r )−1 , (5) 3Note that αh is assumed constant independent of moisture. Experimental measurements are currently under way to support this assumption. where L0 and Lr are the stiffness matrices of the matrix phase (lignin) and inclusions (hemicellulose, cellulose), I is the second order identity tensor, f0 and fr are the volume fractions of corresponding phases, and Adill r is the partial strain concentration factor defined as Adill r = ( I + SrL−1 0 (Lr − L0) )−1 , (6) where Sr is the Eshelby tensor depending on the shape of the inclusion. The effective hygroexpansion coefficients are com- puted using the Levin formula as αhom = N−1∑ r=0 fr(BMT r )Tαr, (7) where αr denotes the vector of hygroexpansion coef- ficients of individual phases, BMT r is the stress con- centration factor derived again from the Mori-Tanaka method. For individual phases we get BMT 0 = [f0I + N−1∑ r=1 frBdill r ]−1, BMT r = Bdill r BMT 0 , r = 1, . . . , N − 1, Bdill r = [I + L0(I− Sr)(L−1 r − L−1 0 )]−1. (8) Although analytical homogenization may still be exploited even above the cell wall level, we abandon this approach and turn our attention to the periodic unit cell representation of cellular microstructure, re- call Fig. 5. This approach builds upon the application of the Hill lemma 〈δε(x)Tσ(x)〉 = δET Σ, (9) within the framework of the 1st order homogenization theory. The angle brackets in Eq. (9) denote the volume averaging. Bearing in mind that the evaluation of effective hygroexpansion coefficients is identical to that of thermal expansion, we follow [14] and consider the stress based formulation. We begin with the local constitutive law written as σ(x) = L(x)(ε(x)−αh(x)u), (10) where L is the local stiffness matrix, αh is the vector comprising hygroexpansion coefficients and u denotes the moisture content change. Within the 1st order homogenization the local strain ε(x) is split into two parts ε(x) = E + ε∗(x), (11) where E is a macroscopic part constant over the PUC and ε∗(x) is a fluctuation part, which has to disap- pear upon volume averaging. This can be achieved, e.g. through the application of periodic boundary con- ditions [14] in the solution of the discretized system of governing equations (13). The FEM discretization provides the local strain in the form ε(x) = E + B(x)r, (12) 54 vol. 15/2018 Determination of hygroexpansion coefficients E11 E22 E33 G23 G13 G12 ν21 ν31 ν32 α11 α22 α33 [GPa] [GPa] [GPa] [GPa] [GPa] [GPa] [-] [-] [-] [1/K] [1/K] [1/K] M layer 34.24 4.62 3.96 1.32 1.32 1.60 0.03 0.04 0.41 0.0246 0.3920 0.5410 S2,3 layer 78.72 7.15 6.01 2.05 2.05 2.46 0.02 0.02 0.41 0.0078 0.3247 0.5089 Table 3. Effective properties at cell wall level from analytical homogenization. where r represents the nodal displacements of the fluc- tuation displacement field u∗. Substituting Eq. (12), combined with Eq. (10), back into Eq. (9) and col- lecting terms associated with δr and δE finally gives the aforementioned discretized system of algebraic equations in the form[ ∫ Ω L dΩ ∫ Ω L B dΩ∫ Ω BT L dΩ ∫ Ω BT L B dΩ ]{ E r } ={ ΩΣ + ∫ Ω LαhudΩ∫ Ω BT LαhudΩ } , (13) with the macroscopic strain E and the nodal fluctua- tion displacements r as unknown quantities. Solving the system (13), while setting the macroscopic stress Σ = 0 and the moisture change u = 1, provides the effective hygroexpansion coefficients as components of the macroscopic strainE. This step takes into account the microfibril angle as a deviation of inclusions from the longitudinal axis. Thus for each segment along the perimeter of the cell wall a proper transformation of coordinates from local to global coordinate system is needed [2]. If using the PUC corresponding to equivalent wood we get the effective coefficients of hygroexpansion di- rectly at the level of the solid wood. On the contrary, employing the unit cells pertinent to the earlywood and latewood requires an additional homogenization step, where the corresponding effective stiffness matri- ces are also needed. This can be achieved by setting u = 0 in Eq. (13) and load the unit cell in turn by each of the six components of Σ set equal to one to get individual columns of the 6× 6 compliance matrix Mhom as corresponding strain volume averages. The required effective stiffness matrix Lhom then follows from simple matrix inversion. Once having the effective stiffnesses of the early- wood and latewood we may proceed with the deriva- tion of αhom at the level of solid wood. In such a case one may accept the similarity between the structure of growth rings and a laminate and adopt classical lamination theory, see e.g. [15]. Nevertheless, we pro- ceeded in the footsteps of the previous calculations and performed FEM homogenization of a two layer PUC where individual layers represent the earlywood and latewood, respectively. 4. Results Evaluation of effective coefficients of hygroexpansion of wood required several computational steps on in- dividual scales. The homogenized elastic stiffnesses and coefficients of hygroexpansion of individual lay- ers of the cell wall were determined first using the Mori-Tanaka method, particularly Eqs. (5) and (7). The resulting effective properties assuming MFA=0 in all layers are stored in Table 34. As these refer to the level of cell wall, they are the same for earlywood, latewood and equivalent wood. 0 10 20 30 40 50 MFA [deg] 0 5 10 15 20 E la s ti c m o d u li [G P a ] E L E T E RE L = 14 GPa E L = 6 GPa 15 32.53 (a) 0 10 20 30 40 50 MFA [deg] 0 0,1 0,2 0,3 0,4 α [ − ] α h,L α h,T α h,R (b) Figure 6. Variation of selected homogenized prop- erties as a function of MFA: a) elastic moduli, b) coefficients of hygroexpansion. The effective coefficients of hygroexpansion of early- wood, latewood and equivalent wood followed directly from the application of Eq. (13) at the level of lumens. Therein, the properties from Table 3 were properly adjusted to account for the variation of MFA in indi- vidual cell wall layers, recall Table 2. When adopting the earlywood and latewood PUCs in Fig. 5 we exer- cised Eq. (13) one more time at the level of growth rings on a laminate-like PUC. For illustration we plot the variation of the selected homogenized moduli and 4Axis-1 is oriented in longitudinal direction, axis-2 in cir- cumferential direction, axis-3 in cell wall thickness direction. 55 L. Kucíková, M. Šejnoha, J. Vorel Acta Polytechnica CTU Proceedings coefficients of hygroexpansion as a function of MFA5. More specifically, the MFA in S2 layer was varied from 0◦ to 50◦ while keeping the MFA values from Table 2 for the M and S3 layers. Based on the results from simple tensile tests the ex- pected MFA would range from 10◦ to 20◦. As seen in Fig. 6(a) the value of MFA=32.53◦ would greatly un- derestimate the wood stiffness. Thus to compare the resulting predictions of the homogenized coefficients of hygroexpansion provided by the 3 step (cell wall level, level of lumens, level of annual rings) and 2 step (cell wall level, level of lumens assuming equivalent wood) homogenization we set MFA=15◦. The corresponding results are available in Table 4. Comparing the last two rows it can be seen that the equivalent wood PUC can safely be used in homogenization analyses thus skipping the last homogenization step at the level of annual rings. αh,L αh,T αh,R Earlywood 0.002 0.34 0.21 Latewood -0.003 0.32 0.39 Solid Wood 0.001 0.33 0.23 Equivalent wood 0.001 0.34 0.23 Table 4. Effective coefficients of hygroexpansion of wood from numerical homogenization (MFA=15◦). 5. Conclusions In this paper the Mori-Tanaka micromechanical scheme was effectively combined with numerical simu- lations based on the 1st order homogenization theory. While the Mori-Tanaka method provided the effective properties of individual cell wall layers, numerical ho- mogenization was adopted at the level of tracheids and annual rings. In addition, an equivalent wood was introduced in order to avoid the last homogeniza- tion step at the level of annual rings, which is needed when the earlywood and latewood are considered in- dependently. This will prove useful when modeling the evaluation of moisture content in wood through the application of multi-Fickian moisture transport model [16, 17]. As seen in Table 4, the results for the solid wood and equivalent wood are very similar. This clearly promotes the applicability of equivalent wood and thus the derivation of effective properties through a single homogenization step above the cell wall level. Inadequacy of using MFA obtained directly from nanoindentation was also mentioned. 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L. Frandsen, L. Damkilde, S. Svensson. A revised multi-Fickian moisture transport model to describe non-Fickian effects in wood. Holzforschung 61(5):563–572, 2007. 56 http://dx.doi.org/10.1017/CBO9780511613357 Acta Polytechnica CTU Proceedings 15:51–56, 2018 1 Introduction 2 Materials 3 Methods 4 Results 5 Conclusions Acknowledgements References