Analysis of drying stresses in green-glued plywood of Bete (Mansonia Altissima) specie

The wood veneer dried artificially before gluing for the manufacture of plywood generally suffers several damages (cracks, deformations), decreasing the competitiveness of the process and limiting the use of nervous species of wood. The proposed solution is to glue them at the green state. However, during the drying process, superficial and internal cracks are observed. These cracks appear when stresses due to the anisotropic shrinkage of the plywood exceed the mechanical strength of the wood material. In the present study, the response of a 3-ply plywood of Bete (Mansonia altissima) glued at the green state is simulated in order to evaluate these stresses and deformations during drying using a finite element method program. The numerical results and associated experimental data make it possible to better analyze and understand the mechanical behavior of the plywood during drying in order to look for reliable plywood through new process.


Introduction
Plywood factories have been using wood veneer obtained by unrolling some species for a long time. The artificial drying of these wood veneers, the only industrial process, leads to a certain number of incidents that may occur during the process. Some defects are visible (superficial cracks and deformations) while others may be non-visible (internal cracks). Nevertheless, they are susceptible to induce future failure [1]. To provide solutions, several works have developed products glued at the green state [2,3], and in particular plywood panels using different technologies [4][5][6][7]. At the industrial level, this technique may allow some energy savings and reduce the degradation of the material integrity of veneers during handling. Lavalette [8] has shown that the product from the vacuum drying of green-glued plywood processing, meets the standard requirements for shear stress according to EN 314-1 [9]. But, she showed some superficial cracks probably due to the stresses induced by the gradient of moisture and anisotropic shrinkage.
To forecast these internal stresses and strains, which is a prerequisite to design the products, it is difficult to build an experimental design without the use of numerical simulations. The numerical model of the hydro-mechanical behavior during the drying of solid wood products has been the subject of several studies on sawn timber products [10,11] and crossed lamellae panels [12]. These questions are still to be answered for plywood, especially as the thickness of the wood veneer and the method of their production are different from those of the products mentioned above. Thus, finite element simulations can be performed to study how the internal structure and properties of the material affect the plywood shape stability.
The objective of the study is to determine the drying stresses in green-glued plywood by FEM analysis (Cast3M Finite Element Software). To perform an accurate simulation, it is essential to have a sufficiently detailed description of the properties of the wood. The mechano-sorption, the elastic, visco-elastic behaviors are implemented in Cast3M finite element software during drying. For this purpose, experimental drying tests of plywood consisting of three plies of size 100 × 100 × 6 mm 3 (Fig. 1) permit to follow their deformation. The hydro-mechanical test setup for the veneers have been carried out and the results used in the numerical drying tests. They have made it possible to reproduce the experimental behavior (deformations, drying kinetics, etc.) and to provide estimates of the internal stresses in plywood.

Numerical model
Several models of wood behaviors were offered during those last years. The models of stress/deformation were developed, and most of them assumed that mechanical properties change during the drying. They provided a better knowledge of the mechanical behavior and may aid to improve the quality of the wood during drying [13,14]. The rheological model implemented in the numerical approach is briefly described below. The constitutive equation given by Eq. (1) expresses the total strain �ε as a sum of three separate strains, describing the fundamental behaviors of wood. It is illustrated by the Maxwell figure of r branches (Fig. 2).
where �ε w is the free shrinkage/swelling that occurs when the water content is below the fiber saturation point (W fsp ). It is defined as: with α as the shrinkage/swelling coefficient (independent of moisture) and w is the variation of moisture content; �ε eve is the elastic and visco-elastic strain defined by using the chain model of Maxwell as follows: for 1 − e − α µ �t σ µ (t) , and ∼ K is the fictious rigidity. It depends on the length of the time step, the parameters of Maxwell; E 0 (w) is the elastic modulus at the beginning of the increment. It depends on moisture content w; E µ andη µ are, respectively, the elastic modulus and the viscosity of the branch µ . σ his (t) is the term of the history. Its depends on the length of the step time Δt, the state of the deformation ε(t) and the recent values gained by the internal stress σ µ (t) at the beginning of the increment. �ε ms is the mechano-sorptive deformation, and depends linearly on the stress and the variation of the water content. This definition leads to the formulation of the following expression of deformation which does not depend on the previous ones [10]: where m is the compliance of mechano-sorptive creep.
From Eq. (1), the total formulation then can be written:

Identification and experimental determination of input parameters of the model
In order to validate the numerical simulation by experiments, we choose the Bete specie as a reference. It permits to describe and identify the input parameters of the model like the isothermal desorption, coefficients of diffusion and exchange, the test boundary condition and material data.

Description of Bete specie
Bete (Mansonia altissima) is one of the most abundant hardwood species from the Congo Basin forest [15]. The color of the wood is brown and the grain is straight. The wood can be used in frame, parquet and paneling. According to the CIRAD (Agricultural Research Centre for International Development, in French) technological database [16], the density at 12% moisture content varies from 0.59 to 0.72. The average elastic modulus in the longitudinal direction (MOE) is 13,600 MPa, with a standard deviation of 1124 MPa. The mean value of the modulus of rupture in bending (MOR) is 110 MPa, with a standard deviation of 10 MPa. The compressive rupture stress is 60 MPa, with a variation coefficient of 10%. The average W fsp value is 28% [15]. The tangential and radial shrinkage coefficients vary, respectively, from 0.241% to 0.286% and from 0.15% to 0.178%.

Determination of the isothermal desorption of Bete specie
Ten samples in green state (saturated) of size 20 × 20 × 2 mm 3 were placed in a climatic chamber at constant temperature of 40 °C. Then the relative humidity (RH) inside the device was varied from 95 to 10%. For each RH value, the samples remained in the oven until their masses stabilized. A given mass was considered as stabilized in this study when its relative variation was equal or less than 0,001. Knowing the saturated and anhydrous mass values (obtained at 105 °C), we deduced the water content W c , and the couple (RH, W c ) provided each point of the isotherm desorption curve.

Determination of the shrinkage coefficient of Bete specie
The full shrinkage radial and tangential of Bete specie were determined experimentally by using 5 samples in green state (saturated) of size 20 × 20 × 2 mm 3 . We assumed that the coefficients are constants. They were placed in an oven at constant temperature of 40 °C and humidity of 90% until their mass stabilized in order to have the W fsp value. The tangential and radial dimensions were measured. The same sizes were also measured for the same specimens in anhydrous state. From those dimensions, we deduced the radial and tangential shrinkage coefficients.

Determination of the density of the Bete species
To determine the density, the same specimens for the determination of shrinkage coefficients were used. We assume that the longitudinal shrinkage of the specimen is neglected. Knowing the dimension and the mass, we deduced the density ρ w at 0% of W c and by the relation ρ w (W%) = ρ w (0%) (1 + W/100) [22], the density versus the W c .

Determination of elastic properties of the Bete species
For the determination of elastic properties of Bete plywood, the tensile tests were carried out on specimen obtained after unrolling and conditioning at a temperature of 40 °C and a relative humidity of 70% (which corresponds to an average moisture content of about 12% at the hygroscopic equilibrium). The tensile tests are inspired by the standard NF EN 326-1 [17]. However, a different specimen geometry was used because preliminary tests showed that the rupture occurred in the jaws of the standard specimen. More precisely, we used a "dumbbell" specimen [8], reinforced by wooden heels at the jaws, in order to ensure the rupture in the useful part of the sample. It was cut, respectively, in the parallel and perpendicular directions of the fibers in order to obtain the elasticity modulus in those two principal directions.

Fabrication of 3-ply plywood
About 2-mm-thick Mansonia altissima rotary cut veneer (from the production forest of Cameroon) was used for plywood manufacture. Wood plies were selected defect free, with a regular slope of grain in order to avoid their effect on test results. They were cut to the panel dimensions (600 × 600 × 2 mm 3 ) and initially stored in a conditioning chamber at 4 °C to keep their moisture content beyond the W fsp . Then the wood plies were taken off the chamber and glued. The adhesive used was a one-component polyurethane (ref: Collano RP 2554) with a viscosity of 1000 mPa/s at 20 °C, developed from the adhesive patented for green plywood gluing [18]. It was spread on the plies by using a notched squeegee so that the glue was evenly distributed. 3-ply plywood panels were manufactured by using the vacuum process technique [7,8]. The plies were oriented according to two arrangements options. The first one was antisymmetric ( − 15°/0°/ + 15°). It was considered in order to validate the numerical model of drying plywood. The second one was symmetric (0°/90°/0°) which represents the conventional plywood. The panels were placed in a vacuum dryer set to 150 mbar to ensure the bonding and equipped with a device (microprocessor) indicating their average moisture content (about 55%). The average moisture content of the panels when they were removed from the drier was 35%. The panels were cut according to the dimensions of the samples (section of 100 × 100 mm 2 ) and placed in an oven where the relative humidity and the temperature were, respectively, 95% and 10 °C. Thus, their moisture content could be close to the W fsp . We remind that the W fsp average value, according to the literature is 28%.

Determination of coefficients of diffusion and exchange
The relationship between the diffusion coefficient K D and the water content in the wood is far from being definitively established. The K D values found in the literature depend on wood species and widely vary. The transfer of water in the wood is hindered by two resistances: an internal one which can be described by the diffusion coefficient K D, and a second one which is developed at the interface between the specimen and the external environment, which can be described by the surface exchange coefficient K C [19][20][21]. The boundary conditions associated with the mass diffusion equation are literally written in the unidirectional framework by the following system of equations as (Eq. 6): with K D as the coefficient of diffusion of the material; W, the moisture content in the material; W surf , the moisture content on the surface of the material; W ini , the initial moisture content in the material (at the beginning of drying); W eq , the equivalent moisture content to the relative humidity of the study environment; K C , the exchange coefficient between the material and the ambient environment; a, the specimen thickness; t, the time; and Q m , the flux of surface water content. For experimental determination, 10 green-glued plywood specimens were placed in a climatic chamber in which the air relative humidity and the temperature were, respectively, 50% and 40 °C. In such conditions, the moisture content at the hygroscopic equilibrium was 10%. These parameters correspond to a desorption of the specimens. The four lateral faces (RT and RL planes) of each sample were insulated with an EPI (emulsion polymer isocyanate) adhesive of a Kleiberit brand in order to impose the diffusion in the radial direction of the sample (Fig. 3).
At each hour, the samples were weighed. The anhydrous masses were obtained after setting the samples in the oven at 105° C till their mass stopped varying. Then, we deduced the evolution of the moisture content with the time in order to determine the diffusion and exchange coefficients.

Validation of the numerical model
In order to validate the numerical results, the configuration of plywood chosen was arranged with asymmetrical ring orientations (longitudinal direction). The first and third plies are inclined by ± 15° about the direction of the fiber as indicated in Fig. 4. The process was conducted by following to steps.
In the first step, five green-glued plywood specimens with an initial average moisture content of approximately 32% were placed in a drying oven with the diffusion and theoretical mechanical boundary conditions mentioned in Fig. 5. These mechanical boundary conditions corresponded to simple supports experimentally. During the experiment, the specimens were placed on a grilling of the dryer, without holding, allowing them to perform free flexural deformation and to avoid any generation of external stress. The displacements U z of the point P 1 of the plywood (Fig. 5) of each sample were measured each hour using a digital caliper.
In the second step, the fitting between experimental and numerical values was realized by minimizing the quadratic gap between the numerical and experimental displacements of the point P 1 according to two scenarios. The first one did not take into account the visco-elastic behavior of the plywood. In the second one, that behavior was considered.
The material parameters used for the numerical simulation are listed in Table 1 and are representative of the wood veneers of the Bete species.  The following assumptions were considered: a-The influence of the moisture content W (%) on the elastic properties specified in Table 1 and the density of the woods was taken into account by using linear corrections given by Guitard [22] on the modulus of elasticity (Eq. 7) and density (Eq. 8): with C ij constant coefficients and S −1 ij (W) the elastic properties. The validated numerical model permits to quantify and predict the fields of deformations and stresses according to two configurations: the first one was asymmetrical and the second was conventional (Fig. 6). The main interest of the asymmetrical configuration was the comparison of experimental and numerical results. In the second configuration, veneers are arranged orthogonally (Fig. 7) and the main purpose was the comparison of drying stresses and the ultimate strength of the veneers. Figure 7 presents the plot of the equilibrium moisture content versus the relative humidity of the Bete species. Experimental results were fitted according to the Henderson model [24] (Eq. 10):

Isothermal desorption of the Bete
The estimated values of A, B and C at the end of the fitting process were, respectively, 0.14, 100 and 1.9. T is the temperature in Kelvin degree.
The W fsp value, corresponding to the limit of nonhygroscopic domain, was 32% (Fig. 7).

Shrinkage coefficient of Bete specie
The experimental result of tangential and radial shrinkage is given in Table 2. We notice that the value of the tangential shrinkage is bigger than the one found in the literature. On the other hand, we have almost the same result of radial shrinkage as in the literature.

Volumic mass and density of specie of Bete
The average density value obtained experimentally in anhydrous conditions is 0.59, with a standard deviation of 0.05. By using Eq. 8, the value at 12% practically showed no variation (from 0.655 to 0.666). Such results correspond to the literature mean value of 0.66 (volumic mass of 660 kg/m 3 ).

Elastic properties of plywood
The results of tensile tests of the Bete plies are shown in Table 3.
The results show that the elastic modulus and the failure strength of the samples in the longitudinal direction  are close to those of the solid wood obtained from the reference [16]. In the direction perpendicular to the fibers, the average elastic modulus is much lower than the module found in the literature. This can be explained by the presence of peeling slots. Figure 8 shows an exponential decrease of the moisture content with the drying time. It was therefore possible to determine the expression of diffusion coefficient K D as an exponential function of the moisture content. Similar trends are available in the literature [19][20][21]. By fitting the experimental curve of the evolution of the moisture content (Fig. 8), we found the analytical expression of the diffusion coefficient (Eq. 11). We remind that the relative humidity and the temperature conditions are, respectively, 50% and 40 °C:

Diffusion and exchange coefficients
with K 0 , as the constant diffusion (5,84 10 -10 m 2 /s); K 0w , as the constant coefficient (1,97), and w, the moisture content (%) ( Table 4). The value of the coefficient K C and the expression of K D are presented in Table 4.
In the literature, several authors conducted investigations concerning the mass transfer properties of wood (11) K D = K 0 exp(K 0w w), [25,26]. Concerning central Africa tropical woods, some results on species like sapelli (Entandrophragma cylindricum) and sipo (Entandrophragma utile) are available [27][28][29]. Such timber species present similar technological properties with Bete [30]. For instance, their mean density varies from 0.62 to 0.66. Explicit results concerning the diffusion coefficients were determined by Nsouandelé [29]. At a temperature of 40 °C and a RH of Obtained in the literature [13,23] m i = compliance in the direction i m ij = compliance in the plan ij with i, jǫ{L, T, R} : direction of wood These parameters were obtained by fitting the models and numerical curves to that resulting from the experiment With ǫ{1, 2, 3} : number of branches and iǫ{L, T , R} : direction of wood Therefore, significant comparisons between the diffusion coefficients of green-glued plywood and solid wood may not be easily established because the green-glued plywood is a highly heterogeneous material, compared to the solid wood. Therefore, diffusion coefficients determined in this study should be considered as reference values for further studies concerning plywood.

Experimental results and numerical validation of the model
Concerning the experimental results, Fig. 9 presents the shape of the asymmetrical green-glued plywood after 50 h of drying. Visible and considerable displacements of the plywood shape were observed.
The displacement U z of the point P 1 of the plywood (Fig. 9) of each sample was measured at each hour using a digital caliper and is shown in Fig. 10.
Concerning the numerical validation of these results, we remind that two scenarios were considered. The first one, we considered just the elastic and the mechanosorptive model without taking into account the viscoelastic behavior of the plywood. In the second one, that behavior was considered.

Scenario 1: the visco-elastic behavior of the plywood is not considered
Simulations were carried out. Only the change in tangential shrinkage coefficient greatly influences the displacement of the plywood sample. Variation of the tangential shrinkage coefficient (α T ), starting from the data obtained in the literature for the specie Bete [15] and the experimental result, provides the evolution of the displacement at the point P1 of the plywood presented in Fig. 11 for four cases of tangential shrinkage coefficient values retained (α T1 = 0.246, α T2 = 0.339, α T3 = 0.634, α T4 = 0.780).
During the first 5 h of drying, the displacement as a function of time is practically linear and in this range, all the coefficients of tangential shrinkage used predict well the elastic behavior of drying of the plywood. The likelihood of this scenario is limited by its ability to predict displacements in the post-elastic phase.
The value of the tangential shrinkage coefficient of the veneer is retained by minimizing the quadratic criterion between the experimental and numerical results of the first 10 h, assuming that at this time the visco-elastic effect is negligible. This scenario's value (α T3 = 0.634) differs considerably from that of solid wood. This can be justified by the presence of cracks resulting from the unwinding operation. It is therefore chosen for the rest of the numerical model.

Scenario 2: the visco-elastic behavior of the plywood is considered
The three-branched Maxwell model was considered and the parameters of viscosity γ i 1 and α i 1 −1 case studied and retained are presented in Table 5.
The displacements illustrating the evolution of the plywood at the point P 1 according to the variation of the viscosity parameters are shown in Fig. 12. We remind that the various displacements at the point P 1 were measured on asymmetrical plywood samples.
The visco-elastic model parameters chosen in this scenario are those corresponding to the curve U z _ Numerical_α T3 _v 2 obtained by minimizing the quadratic criterion between the experimental results to those obtained numerically. The numerical displacement field on asymmetrical samples after 50 h of drying is shown in Fig. 13.
Since the scenario 2 presented a more interesting ability to predict displacements in the post-elastic phase, it has been considered in the next parts of the study. The validation of the model attached to that scenario will subsequently permit to determine the deformations and the stress in the conventional plywood configuration during drying.

Experimental and numerical results of the conventional configuration
The conventional configuration corresponded to a plies' arrangement of 0°/90°/0°.

Experimental result
The experimental results in the same condition of drying shows that there was some surface cracking at the both sides of the plywood (Fig. 14).

Numerical results of stress fields and evolution of the stress at a plywood surface point
The numerical results of the stress fields presented in Fig. 15 show that: • along the X direction (aligned with the longitudinal direction of wood on the outer plies), the maximum surface stress of the plywood is in compression, corresponding to a value of -4.3 MPa; the inner part of the plywood is in tension with a maximum value of 6 MPa. • along the Y direction (the transverse direction of the outer plies), the surface in under tension with a maximum value of 4.4 MPa; the inner part of the plywood is under compression with a maximum stress value of -8.6 MPa. Tensile stress in Y direction which is due to the orientation of the central ply (longitudinal in Y direction) that prevents the deformation in y direction (transversal) of the external plies, and put them in tension.
In order to understand the behavior of the stress at the surface of the plywood, the stress evolution curve of a point at the center of the plywood is plotted in the longitudinal and tangential directions. The results are illustrated by Fig. 16. The figure shows that, after five hours of drying, any stress at the surface of the plywood almost reaches its maximum value.
The numerical model provides valuable information concerning the importance of an accurate description of the green-glued plywood during drying, regarding stress, deformation and the drying kinetics.
In the conventional configuration where the plies are orthogonally arranged, the numerical results show that the stresses of the plywood, in the direction tangential to the fibers at the surface at 12% moisture content (4.4 MPa) are greater than the average rupture stress obtained experimentally (4.05 MPa). The accuracy of these results should make possible for plywood manufacturers to correctly plan drying programs before they are actually put into practice. In other words, the model can aid in predicting the quality (levels of shape stability and material integrity) of green-glued plywood before starting a drying program.
Concerning the drying kinetics (the evolution of the stress during drying in Fig. 16), we noted that the threshold of the maximum stress is practically reached after the first 5 h. In other words, from 32 to 25% of water content, the stresses in the plywood are practically at their maximum. As a result, it is therefore possible for plywood manufacturers to quantify correctly the energy savings for two drying situations. In the first one, the automatic drying program is managed until one reaches the desired moisture content. In the second one, the automatic drying program is interrupted and substituted by a natural drying of the products. In addition, some early stage measures concerning the drying program could therefore be taken if one wants to reduce the drying stress.

Conclusion
In this paper, the behavior of green-glued plywood during the drying process was simulated. A constant relative humidity (50%) and temperature (40 °C) were considered during drying. Small plywood specimens, manufactured by using Bete (Mansonia altissima), an abundant species from the Congo Basin, were used.  The first step of the study focused on the experimental determination of some important material properties of the plywood and veneers, namely diffusion and exchange coefficients, tensile strength and modulus of elasticity, shrinkage coefficients and density. In the second step, the drying behavior of the plywood was quantified thanks to FEM model in which elastic, visco-elastic and mechano-sorptive aspects of the wood veneer were considered. The surface cracks in the plywood during drying are justified by the fact that the drying stress is higher than the tensile ultimate strength of the veneer. In addition, the stress at the surface of plywood reaches practically its maximum value during the first 5 h of drying. The simulation yields information about unfavorable deformations and stresses during the drying process. Such information presents an interesting potential to be valuable in plywood factories,   Table 4 The data used in the diffusion model Exchange coefficient (m/s) K C = 2.34 10 -8 Table 5 Evolution and selection of the visco-elastic parameters especially the quality control and the planning program of the green-glued plywood. Further investigations will be needed in order to improve the FEM model, in the perspective to be applied on practically sized plywood according to the length and width. They concern the influence of specimen's dimensions and certain wood veneer defects on the drying behavior of the plywood and also to look for solutions at the level of the manufacturing process to improve the quality of plywood during the vacuum drying process, in particular, to reduce the cracks observed previously by relaxing the induced stress.