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 \(\Delta \varepsilon\) 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).
$$\Delta \varepsilon = \Delta {\varepsilon _{eve}} + \Delta {\varepsilon _{ms}} + \Delta {\varepsilon _w},$$
(1)
where \({\Delta \varepsilon}_{\text{w}}\) is the free shrinkage/swelling that occurs when the water content is below the fiber saturation point (Wfsp). It is defined as:
$$\Delta {\varepsilon _w} = \alpha \Delta w,$$
(2)
with α as the shrinkage/swelling coefficient (independent of moisture) and \(\Delta w\) is the variation of moisture content; \({\Delta \varepsilon}_{\text{eve}}\) is the elastic and visco-elastic strain defined by using the chain model of Maxwell as follows: for
$$\Delta t,\Delta {\varepsilon _{eve}} = {\rm{ }}{\left[ {\tilde K} \right]^{ - 1}}\Delta \sigma - {\left[ {\tilde K} \right]^{ - 1}}{\sigma^{his}}\left( t \right),$$
(3)
with \(\tilde K = {E^0}({\rm{ }}w)\left[ {1 + \sum\limits_{\mu = 1}^r {{\gamma _\mu }} {\mkern 1mu} \left( {\frac{{1 - {e^{ - {\alpha _\mu }\Delta t}}}}{{{\alpha _\mu }\Delta t}}} \right)} \right],\) where \({\gamma _\mu } = \frac{{{E^0}(w)}}{{{E^\mu }}}{\rm{ }}and{\rm{ }}{\alpha _\mu } = \frac{{{E^\mu }}}{{{\eta ^\mu }}}(unit {\rm{ }} in {\rm{ }}{s^{ - 1}})\), and \({\sigma ^{his}}\left( t \right) = \sum\limits_{\mu = 1}^r {\left( {1 - {e^{ - {\kern 1pt} {\alpha _\mu }\Delta t}}} \right)} {\mkern 1mu} {\sigma ^\mu }(t)\), and \(\stackrel{\sim }{K}\) 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; \({E}^{\mu } and {\eta}^{\mu }\) are, respectively, the elastic modulus and the viscosity of the branch \({\mu }\). \({\sigma }^{his}\left(t\right)\) 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 \({\sigma }^{\mu }(t)\) at the beginning of the increment. \({\Delta \varepsilon}_{\text{ms}}\) 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]:
$$\Delta {\varepsilon_{ms}} = m{\rm{ }}\sigma{\mkern 1mu}\Delta w$$
(4)
where m is the compliance of mechano-sorptive creep.
From Eq. (1), the total formulation then can be written:
$$\Delta \varepsilon = \alpha \Delta w + {[\tilde K]^{ - 1}}\Delta \sigma - {[\tilde K]^{ - 1}}{\sigma ^{his}}\left( t \right){\rm{ }} + m{\rm{ }}\sigma {\mkern 1mu} \Delta w$$
(5)
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 Wfsp 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 mm3 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 Wc, and the couple (RH, Wc) 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 mm3. 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 Wfsp 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 ρw at 0% of Wc and by the relation ρw(W%) = ρw(0%) (1 + W/100) [22], the density versus the Wc.
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 mm3) and initially stored in a conditioning chamber at 4 °C to keep their moisture content beyond the Wfsp. 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 mm2) 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 Wfsp. We remind that the Wfsp average value, according to the literature is 28%.
Determination of coefficients of diffusion and exchange
The relationship between the diffusion coefficient KD and the water content in the wood is far from being definitively established. The KD 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 KD, 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 KC [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):
$$\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial w}}{{\partial t}} = \frac{{\partial w}}{{\partial z}}\left( {{K_D}(w)\frac{{\partial w}}{{\partial z}}} \right)}&{with\;0 < z < a}\\ {w = {w_{ini}}}&{for\;z \in [0,{\rm{ }}a]{\rm{ }}\;at\;t = 0}\\ {{K_D}(w)\frac{{\partial w}}{{\partial z}} = {K_C}\left( {{w_{surf}} - {w_{eq}}} \right) = {Q_m};}&{\left( {z = a,\;t > 0} \right)} \end{array}} \right.$$
(6)
with KD as the coefficient of diffusion of the material; W, the moisture content in the material; Wsurf, the moisture content on the surface of the material; Wini, the initial moisture content in the material (at the beginning of drying); Weq, the equivalent moisture content to the relative humidity of the study environment; KC, the exchange coefficient between the material and the ambient environment; a, the specimen thickness; t, the time; and Qm, 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 Uz of the point P1 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 P1 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):
$${\text{S}}_{\text{ij}}^{-1}\left({\text{W}}\right)\text{=}{ \, {\text{S}}}_{\text{ij}}^{-1}\left(\text{12\%}\right)\left[{1}- \text{ } {\text{C}}_{\text{ij}}\text{(W-}{12}\text{)}\right]$$
(7)
with Cij constant coefficients and \({\text{S}}_{\text{ij}}^{-1}\left({\text{W}}\right)\) the elastic properties.
$${\rho }_{\mathrm{W }\left(\mathrm{W \%}\right)}={\rho }_{\mathrm{W }\left(0\mathrm{ \%}\right)}\left(1+W/100\right)$$
(8)
\({\text{S}}_{\text{ij}}^{-1}\left({\text{W}}\right)\) are defined as follows (Eq. 9):
$$S_{11}^{ - 1} = {E_R};{\rm{ }}S_{22}^{ - 1} = {E_T};{\rm{ }}S_{33}^{ - 1} ={E_L};{\rm{ }}S_{44}^{ - 1} = {G_{LT}};{\rm{ }}S_{55}^{ - 1} = {G_{LR}};{\rm{ }}S_{66}^{ - 1} = {G_{RT}}$$
(9)
-
b-
The shrinkage coefficients were constant during drying.
-
c-
We assumed that the viscosity parameters of the Maxwell model with three branches (n = 3) are the same for the direction of the plywood.
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.