New modelling approaches to predict wood properties from its cellular structure: imagebased representation and meshless methods
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Abstract
Key message
The real tissue structure, including local anisotropy directions, is defined from anatomical images of wood. Using this digital representation, thermal/mass diffusivity and mechanical properties (stiffness, large deformation, rupture) are successfully predicted for any anatomical pattern using suitable meshless methods.
Introduction
Wood, an engineering material of biological origin, presents a huge variability among and within species. Understanding structure/property relationships in wood would allow engineers to control and benefit from this variability. Several decades of studies in this domain have emphasised the need to account simultaneously for the phase properties and the phase morphology in order to be able to predict wood properties from its anatomical features. This work is focused on the possibilities offered by meshless computational methods to perform upscaling in wood using actual tissue morphologies obtained by microscopic images.
Methods
After a section devoted to the representation step, the digital representation of wood anatomy by image processing and grid generation, the papers focuses on three meshless methods applied to predict different macroscopic properties in the transverse plane of wood (spruce earlywood, spruce latewood and poplar): Lattice Boltzmann Method (LBM) allows thermal conductivity and mass diffusivity to be predicted, Material Point Method (MPM) deals with rigidity and compression at large deformations and peridynamic method is used to predict the fracture pathway in the cellular arrangement.
Results
This work proves that the macroscopic properties can be predicted with quite good accuracy using only the cellular structure and published data regarding the cell wall properties. A whole set of results is presented and commented, including the anisotropic ratios between radial and tangential directions.
Keywords
Lattice Boltzmann Method Material Point Method Peridynamic Real anatomical structure Mechanical properties Stiffness Fracture Diffusion Thermal conduction Collapse1 A short review of structure/properties relationships in wood

The huge variability of wood properties, within species or between species

The intuition, provided by studying wood anatomy, that the anatomical pattern is able to explain, at least partly, this variability
When wood is used as a structural material, two properties are of primary matter: longitudinal stiffness and transversal shrinkage. This is why most research efforts have focused on these properties. In this case, the term “property” refers to the socalled macroscopic property, as defined using solid wood samples, with typical sizes of several centimeters in each direction, possibly some tens of centimetres in the longitudinal direction. Studies showed that density, which represents the quantity of lignocellulosic matter embedded in the wood, is highly variable (ranging from 100 to 1200 kg m^{−3} among species) and is likely to account for most of the variability in the properties of wood. Therefore, it is not surprising that the first attempts to predict wood properties were in the form of linear or nonlinear correlations dependent on density. For example, this strategy works nicely for longitudinal stiffness and hardness (Kollmann and Côté 1968; Bosshard 1984). The hidden, and coarse, assumption made in this simple approach is that all phases of wood are in parallel and aligned along the longitudinal direction. Using this simple upscaling strategy, the macroscopic property is simply a weighted average of the microscopic property over all phases of the heterogeneous medium. This explains why poor correlations are obtained for certain properties, such as transverse shrinkage. In addition, even if a rather good correlation is obtained, the residual variability is too large to accurately predict wood properties. This means that, for a given sample, the deviation from the general correlation might be large in terms of relative error.
To improve the knowledge of wood properties and, more specifically, to elaborate models to explain the dramatic variability observed in its properties, microscopic features (at the anatomical and ultrastructural levels) have to be considered. The task is not easy, as many spatial scales contribute to macroscopic behaviours. It is now well established that three spatial scales are particularly relevant: the cell wall level, namely through the microfibril angle (MFA); the cellular structure, which explains the tissue properties; and the anatomical pattern, in which all the anatomical tissues are organised in proportion and in space. A multiscale approach would be ideal to account for all these wood features. In practice, key factors were gradually introduced to explain the deviations observed from the statistical correlations.
For example, regarding the stiffness or shrinkage values in the longitudinal direction, the MFA in the secondary cell wall was proposed as an explanatory parameter decades ago (Harris and Meylan 1965; Meylan and Probine 1969). These findings were a major improvement in the understanding of the longitudinal behaviour of wood. The determination of the MFA by Xray diffraction (Cave 1966) played an important part in this progress. It is noteworthy to mention that such a clear influence of the MFA on the longitudinal properties of wood is a rare example where a factor at a low spatial scale (nanoscale) has a straightforward effect on a macroscopic property: scaling in material sciences is usually more complex and involves successive upscaling steps. This relative simplicity, due to the fact that all solid components act in series in the longitudinal directions, allowed analytical models to be proposed in the same period (Barber 1968).
The understanding and prediction of structureproperty relationships in the transverse plane of wood (radialtangential) are more complicated. In this plane, solid components act both in parallel and in series at different spatial scales (multilayered cell wall, cellular morphology, anatomical pattern).
Again, observations and measurements came before modelling. Several scientists tried to use anatomical features as input parameters in statistical explanations. For example, the occurrence of ray cells (Barkas 1941; Boutelje 1962; Kelsey 1963; Keller and Thiercelin 1975; Guitard and El Amri 1987) and the shape of cells (Mariaux and Narboni 1978; Masseran and Mariaux 1985) were tested as possible explanations of shrinkage variability. In 1989, Mariaux observed that the transverse anisotropy of tissues depends on the mean elongation of the cell, but that shrinkage was not isotropic for “isotropic” cells (same mean diameter in both the radial and tangential directions).
In the meantime, theoretical works were proposed to explain transverse properties from the cellular structure (Barber and Meylan 1964; Gillis 1972; Koponen et al. 1991; Gibson and Ashby 1988). These works are based on analytical models and assume that the cellular structure is represented by a unique tracheid.
Indeed, earlier works pointed out the need to account for the real morphology of the cellular structure for a successful prediction of transverse properties (Farruggia 1998; Holmberg et al. 1999; Perré 2001; Nairn 2006; Abbasi 2013). To do this, computational approaches, which take advantage of advances made in applied mathematics and mechanics regarding scaling approaches (SanchezPalencia 1980; Suquet 1985), need to be applied to wood science.
 1.
Representation: choice of the representative elementary volume (REV), also called the Unit Cell. This REV should be defined in a suitable way for subsequent calculation (finite element mesh, collection of material points…).
 2.
Characterisation of the properties for each phase of the unit cell.
 3.
Solution: the theoretical formulation (i.e. the homogenisation of periodic media) has to be solved using a suitable computational method.
 4.
Validation: the predicted macroscopic properties should be tested against experimental data.
 5.
Localisation: this step is not mandatory, but it allows the local (microscopic) fields (shrinkage, strain, stress, temperature, etc.) to be computed inside the REV under the macroscopic conditions applied to the product.
Figure 1 indicates that a macroscopic property depends on both the local properties of the different phases of the REV (2) and their spatial organisation (morphology) (1). For example, in the transverse plane, the anisotropy of tissue stiffness is mostly explained by the cellular structure (morphology) (Farruggia 1998; Perré 2001), while shrinkage, namely the difference between normal wood and reaction wood, strongly involves the cell wall behaviour in the transverse plane (local property) (Watanabe and Norimoto 1996; Perré and Huber 2007) and, eventually, by the alternation of earlywood and latewood (Lanvermann 2014; Perré and Turner 2002, 2008). In the case of a biological product such as wood or lignocellulosic materials, steps 1 and 2 are particularly difficult. In addition, they are strongly entwined, yet the quality of the prediction depends mostly on the qualities of these two first steps. Whatever the target scale of the scaling approach (the macroscopic scale), the different phases of the unit cell have to be well defined, both in shape and in values, at a smaller scale. For example, if the goal is to obtain the properties of the cellular structure, the representation step consists of defining the size of the REV and the cell morphology inside this volume. Then, computing the macroscopic properties requires the cell wall behaviour to be used as input data (step 2). Regarding this step, products of biological origin are different from other materials in the sense that the constituents of the unit cell do not exist alone and are, therefore, very difficult to characterise. Indeed, two strategies coexist. The first one consists of direct characterisation. In this case, the size of the sample has to be sufficiently reduced so that it is representative of the local scale. A number of researchers have proposed this approach for mechanical properties (Bergander and Salmén 2000; SedighiGilani and Navi 2007; Farruggia and Perré 2000; Perré et al. 2013) and for shrinkage (Perré and Huber 2007; Perré 2007; Almeida et al. 2009; Almeida et al. 2014). In the second strategy, the local properties are deduced from previous scaling approaches (Holmberg et al. 1999; Hofstetter et al. 2005; Neagu and Gamstedt 2007). If the phase morphology is correctly represented, a third possibility exists: a complete scaling approach is performed, but, instead of using this approach to predict the macroscopic properties, an inverse analysis allows local properties to be deduced from macroscopic measurements (arrow 6 of Fig. 1). This approach was applied successfully to demonstrate an equivalent stiffness value of the cell wall in the transverse plane (Farruggia 1998).
As this paper is devoted to the behaviour of wood tissues, the local scale is, therefore, the cell wall, whereas the macroscopic scale will be that of the tissue (a representative subset of cells). It is well known that the cell wall is multilayered and presents a specific microfibril orientation and macromolecular organisation, which explains its heterogeneous and anisotropic properties (Salmen 2004; Gierlinger and Schwanninger 2006).
Several options exist regarding the modelling approach. The simplest one would assume the cell wall to be homogeneous and isotropic. At this level, one has to be aware that the assumptions made regarding the solid phase of the unit cell have dramatic effects on steps 1 and 2. The representation step should provide a geometrical description relevant to the assumptions. If needed, heterogeneities and material directions must be generated, ideally using real anatomical structures, which makes image processing difficult. In this sense, modern imaging tools, such as environmental electron scanning microscopy (ESEM), confocal scanning laser microscopy (CLSM), confocal Raman microscopy and computed Xray μtomography, are of great interest (Perré 2011). It is now quite common to acquire chemical images of wood sections (Gierlinger and Schwanninger 2006; Perré 2011; Cabrolier 2012).

The Lattice Boltzmann Method (LBM) was chosen for thermal and mass diffusion. This is an elegant method in which both the value and the flux are known at each grid point (Succi 2001; Mohamad 2011).

The Material Point Method (MPM) was chosen for the mechanical behaviour. This method predicts stiffness, but also analyses the behaviour of the cellular structure in large deformations (Sulsky et al. 1994; Sulsky and Schreyer 2004; Bardenhagen and Kober 2004).

Finally, fracture in the cellular structure was simulated using the peridynamic method (Silling 2000; Silling et al. 2007; Silling and Askari 2005).
The next section of this paper is devoted to the imagebased representation of cellular structures. Three tissues will serve as input files in Section 4: spruce (Picea abies) earlywood, spruce latewood and poplar (Populus tremula x alba). Poplar is proposed here as one example of a dualporosity organisation (vessel lumens and fibre lumens), which requires ca. 200,000 points to accurately represent the actual morphology (the same number of points allows the sublayers of the cell wall to be represented in singleporosity tissues). Each method is then briefly presented, and the last section presents results and discussion.
2 Imagebased representation
3 The meshless methods used in this work
3.1 Lattice Boltzmann Method
The LBM was first developed to solve the macroscopic momentum equation for viscous flows (Succi 2001). Based on velocity distributions on a regular lattice, this method simulates the macroscopic behaviour as an emerging property of the discrete movement of particles (propagation and collision). Indeed, an asymptotic development of the lattice rules allows the macroscopic set of equations to be theoretically derived from the discrete rules. Following this strategy, LBM progressively became a general numerical method to solve any kind of partial differential equations. It belongs to the family of socalled meshless methods (Belytschko et al. 1996; Frank and Perré 2010) and, therefore, has interesting properties, such as simplicity of development, suitability for parallel computing and flexibility in geometrical shape (Frank et al. 2010).
In LBM formalism, Eqs. (1) and (2) consist of two steps, collision and streaming. Then, boundary conditions must be applied to complete one time step (Fig. 6). On the domain boundaries, the components of the distribution functions for velocities going outwards are known from the streaming process. On the contrary, they are unknown for velocity components getting towards the domain. For example, the distribution functions (f_{1}, f_{5}, f_{8}) and (f_{3}, f_{6}, f_{7}) associated to velocities (c_{1}, c_{5}, c_{8}) and (c_{3}, c_{6}, c_{7}) are known at the lattice faces x = x_{1} and x = x_{n}, respectively. On the contrary, (f_{3}, f_{6}, f_{7}) at x = x_{1} and (f_{1}, f_{5}, f_{8}) at x = x_{n} are unknown. For more details, see Bao et al. (2008) and references therein.
As our goal is to compute the equivalent macroscopic properties of the REV, the method has to be applied to heterogeneous domains. To do this, the LBM equation (Eq. 1), including the BGK approximation (Eq. 2), can be used by distinguishing two relaxation times, τ_{1} and τ_{2}. The values of these two relaxation times are related to the two diffusion coefficients of the respective phase, as defined in Eq. (7). The discrete particle velocities are represented in a computational domain, such as earlywood of spruce from Fig. 2. The rectangular computational domain [x_{1}, x_{n}] × [y_{1}, y_{m}] is, therefore, divided into two subsets: the cell lumens, Σ_{1}, and the solid phase, Σ_{2}, which are related to the macroscopic diffusion coefficients D_{1} and D_{2}, respectively.
 1.
Calculate the equilibrium density function and perform the collision procedure at each node.
 2.
Stream the density distribution populations.
 3.
Apply boundary conditions for the density distribution function.
A Fortran95 LBM code was developed at LGPM (CentraleSupelec) to model the thermal and mass diffusion in heterogeneous medium. This code is flexible and can take account for different morphologies. To compute the equivalent property of the REV, this set of steps is iterated until equilibrium is reached. In general, a large number of iterations (10^{5}–10^{6}) is required to obtain the steadystate regime. The equivalent diffusivity is then computed as the average flux divided by the macroscopic gradient imposed by the Dirichlet boundary conditions.
3.2 Material Point Method
The Material Point Method (MPM) is derived from the Particle in Cell (PIC) method originally developed in the 1960s by Harlow in computational fluid mechanics. The MPM has been successfully applied to computational solid mechanics problems (Sulsky et al. 1994; Sulsky and Schreyer 2004; Bardenhagen and Kober 2004; Guilkey et al. 2006). One of the most important features of the MPM is that it allows a straightforward discretisation of complex material shapes, including direct discretisation from 2D or 3D images, as well as an efficient and robust handling of contacts between material surfaces.
In Eq. (16), \( {\overline{\overline{\sigma}}}^S \) is the specific stress tensor at point p. The external forces include the boundary conditions, which can be applied either to points or to nodes.
The stress tensor at each material point is then simply updated by applying the constitutive equations. The updated stress tensor, point position and velocity are now available to proceed further in time. The grid is usually reset at its original shape before proceeding to the next time step (step 6 of Fig. 7). MPM simulations have been proposed for wood, elastic and plastic constitutive equations (Nairn 2006), but with a quite poor morphological description (no local anisotropy nor a refined cell wall description) compared with the present work.
The MPM code developed at LGPM (CentraleSupelec), MPM_Pore accounts for large displacements and for anisotropic materials. MPM_Pore is written in Fortran 95 and parallelised by domain decomposition using Message Passing Interface (MPI) instructions.
3.3 Peridynamic approach
The growth of cracks in heterogeneous materials is of crucial interest in many fundamental and industrial domains, and has been extensively studied using various numerical approaches, such as the finite element method (Zavattieri et al. 2001; De la Osa et al. 2009; Sancho et al. 2007; Ruiz et al. 2000; Itakura et al. 2005), deformable lattice methods (Kitsunezaki 2013) and lattice element methods (Topin et al. 2007; Affes et al. 2012), to cite a few. Recently, the peridynamic approach emerged as an alternative method, based on integral equations, rather than partial differential equations (Silling 2000).
To take heterogeneous mechanical properties into account, both k and s_{0} depend upon the phase (i.e. cell wall or middle lamella) of points \( {\overrightarrow{r}}_i \) and \( {\overrightarrow{r}}_j \). Concretely, if \( \varphi \left({\overrightarrow{r}}_i\right) \) is a phase index, k becomes k_{ij} = k(φ_{i}, φ_{j}) in Eq. (29), where \( {\varphi}_i=\varphi \left({\overrightarrow{r}}_i\right) \) and \( {\varphi}_j=\varphi \left({\overrightarrow{r}}_j\right) \). Critical elongation s_{0} becomes s_{oij} = s_{0}(φ_{i}, φ_{j}) in the same way. As a consequence, a stiffness k_{αβ} and a critical elongation s_{0αβ} can be attributed to each {α, β} phasephase couple. Bulk phase elastic moduli E_{α} can be deduced from k_{αα} values following Eq. (27), bulk phase fracture energies G_{α} can be deduced from s_{0αα} following Eq. (28) and interface fracture energies G_{αβ} can be deduced from s_{0αβ} when a ≠ β.
4 Results
4.1 Thermal and mass diffusivity
The LBM was used to compute the equivalent thermal conductivity and the mass diffusivity of the cellular structure of spruce, which was obtained in Section 2. The simulation was performed in the two transverse directions of wood (radial and tangential), which means a flux along the horizontal and vertical directions, respectively, in the images depicted in Fig. 5.
The solid fraction ε_{s} was equal to 0.40 and 0.78 for earlywood and latewood, respectively. The thermal conductivities of the solid and the gas were set to λ_{s} = 1 W m^{− 1} K^{− 1} and λ_{air} = 0.023 W m^{− 1} K^{− 1}, respectively. For mass diffusion, we used the dimensionless mass diffusivity f, which accounts for the diffusion resistance relative to binary diffusion in air (Perré and Turner 2001b). Therefore, this value ranges from 0 to 1. The values of f used for the solid and gaseous phases were f_{s} = 0.004 and f_{air} = 1, respectively. These values are representative of bound water diffusion in wood at a moisture content value of ca. 12 % (Siau 1984). A full LBM computational run on a grid of about 200 × 200 points requires a couple of hours (2 to 3) using an Intel I7 processor at 3.4 GHz. Note that 3–10 h were required for simulating mass diffusion because the connected phase (the solid) phase has a low diffusivity in this case.
4.2 Mechanical behaviour
The mechanical behaviour of the three tissues (Fig. 5) was simulated using MPM in both the radial and tangential directions. One run basically consists of a compression test with large deformations. This method was implemented in a custom code written in FORTRAN95 and parallelised using Message Passing Interface (MPI) routines. A full computational run on a grid of about 200,000 points requires a couple of hours (2 to 3) using 4 cores on an Intel I7 processor at 3 GHz. To allow the REV to pave the plane, the lateral faces of the domain are forced to stay straight, and an iterative algorithm was derived to keep the average lateral force at zero. This strategy assumes that the REV has two planes of symmetry (Farruggia 1998). The local cell wall properties were deduced from the paper by Neagu and Gamstedt (2007). A blocked rotation was applied to the values of this work to account for an AMF of ca. 20°. We finally used 8 GPa along the tangential direction (parallel to the lumen contour) and 6.4 GPa across the cell wall.
Summary of the stiffness computed by MPM modelling for the three tissues selected in the present work
Tissue  Solid fraction (%)  Stiffness (MPa)  

Tangential  Radial  
Spruce, earlywood  40  416  1090 
Spruce, latewood  78  3470  2430 
Poplar  38  364  811 
However, one should emphasise that the stress levels and the domain of elasticity are higher than expected. Several effects are likely to explain these differences, including boundary conditions, cell wall behaviour (assumed to be elastic here) and refinement of the background grid. All these effects will be presented in detail in a future paper that will focus on MPM.
4.3 Fragmentation
This algorithm was implemented in parallel in a custom code written in FORTRAN 95 and using Message Passing Interface (MPI) routines. A typical computing time is 8 h on a 16core server.
In the present application, we used a horizon, δ = 4Δx, where Δx is the grid step, which is a good compromise between precision and numerical weight. Two phases have to be distinguished: the cell wall phase and middle lamella. The tenacity, \( K=\sqrt{EG} \), is defined for each phase: K_{cw} for the cell wall and K_{ml} for lamella. Here, we fixed the ratio between tenacities as K_{cw}/K_{ml} = 4. A prenotch was performed on one side of the spruce latewood sample and the tensile fracture test was performed. The same test was performed for the earlywood and latewood of spruce.
Such a test clearly shows that the peridynamic approach is a convenient framework to simulate crack growth in wood, despite the complex structure and mechanical properties inside this material.
5 Conclusion
This paper presents a comprehensive strategy to predict different wood properties from anatomical images. The starting point is an imagebased representation that is able to benefit from any present and future imaging tools, either in 2D or in 3D. This representation step accounts for the local anisotropy and heterogeneity of the cell wall. Then, several meshless methods are proposed to compute the properties of wood tissues from its cellular structure. The emphasis here was on meshless methods, including the Lattice Boltzmann Method, Material Point Method and peridynamic method, which are able to account quite easily for any complex geometries and to predict thermal and mass diffusivities, stiffness and fracture, respectively.
A selected set of computational results proved the predictive ability of this modelling strategy and its potential to predict properties that would be difficult, if not impossible, to measure.
Further studies are in progress to extend this modelling approach in 3D and to extract general trends by comparing various anatomical patterns. The extension from 2D to 3D is today perfectly possible, namely thanks to μtomography available at synchrotron facilities. However, it requires much higher computational resources for three major steps: image processing, the computational part itself and postprocessing.
Notes
Acknowledgments
The authors would like to thank the French Agence Nationale de la Recherche (ANR, project ANR 10 – HABISOL00503) for its participation in founding this research work.
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