Introduction
When a wood board is exposed to a change in relative humidity on only one of its surfaces, e.g. in case of flooring or a panel painting, the resulting asymmetric moisture content profile induces differential expansion over the thickness. The basic idea in this study, shown in Fig. 1, is analogous. Here we will consider a wood board with a thickness much smaller than its length and width, clamped at one end. The board has an impermeable layer on five surfaces, leaving one surface exposed to the atmosphere surrounding it. A semi one-dimensional moisture transport experiment is thus created. Here we will assume that the impermeable layer has no influence on the mechanical behavior of the board. The board itself is initially in equilibrium with the surrounding air, and therefore has a flat moisture content profile (Fig. 1a) when suddenly the relative humidity of the surrounding air is changed. The moisture content at the exposed surface consequently rises, and moisture is transported into the material resulting in a moisture content gradient. The consequent asymmetric moisture profile causes unequal expansion over the thickness; the expansion near the exposed surface is the highest and decreases towards the coated surface. These differences in expansion result in a constant moment, and as a consequence the board bends (Fig. 1b).



Fig 1 Schematic diagram of the bending experiment. A board of wood is exposed to a step change in the ambient relative humidity (RH) at one side. The board is sealed on all sides except the one exposed to the environment resulting in a one-dimensional moisture transport.
(a)  The sample is in equilibrium and the moisture content (MC) is constant throughout the sample.
(b)  A increase in the relative humidity causes a moisture content gradient in the sample resulting in a bending.
 (c)  In the final situation the moisture has equilibrated throughout the sample and the board is straight again.

    Cycling
The aim of this study is the analysis of the dynamic moisture-induced bending of a wooden board. A theoretical model is introduced by coupling moisture transport to a linear elastic model for the moisture induced bending. Experimental verification is provided with boards with different thicknesses. We have looked at the frequency response and the experimental verification with sinusoidal relative humidity fluctuations. In fig 2 we have shown a movie of an experiment where the oak is subjected to a sinusoidal changing relative humidity. As can be seen, the oak mechanical reaction is also a sinusoidal bending

    Fig 2. A movie of the mechanical response of a wooden board to sinusoidal relative humidity fluctuations

 The resulting Bode amplitude plots of the deflection of the board in x- and y-direction are shown as a function of the frequency in Fig. 3. Three regimes can be distinguished. At low frequencies, the board is seen to deflect to a smaller extent in the x-direction, but its amplitude in the y-direction reaches a constant value. During these slow fluctuations, the asymmetry in the moisture content distribution is minor, resulting in small x-deflections. The moisture penetration is high, resulting in a high amplitude in the y-deflection. In an intermediate frequency interval, the board reacts heavily to the imposed fluctuations in relative humidity. For increasing frequencies, the amplitude in both x- and y-deflection is seen to decline; the variations in relative humidity are too fast to follow, i.e., the moisture does not penetrate far enough into the material to cause deflections. 



Fig 3: Bode amplitude plots for the x- and y-deflection of the free end of a 2 mm thick board, which results from application
50–90%.  For the experimental values, the slowest variation is134 h, the fastest 2 min



    Damage
We have used the derived frequency bending behavior to determine maximum allowable fluctuations in RH for failure in the finishing layer of an unrestrained panel. It was assumed that the stresses in the wood itself were small due to the absence of mechanical restrictions. In case the board is not free to deform, stresses build up inside the wood when exposed to a change in RH. Here, we make a first estimation of the consequences for damage in the wood in case the panel is mechanically retrained. Figure 4 shows a frequency versus thickness plot, indicating three different regions for two different amplitudes in RH (30% and 15%). For boards thicker than 12 mm, a daily fluctuation with an amplitude of 30% is in the elastic deformation region. The same fluctuation results in plastic deformation for a board with a thickness between 5 and 12 mm, and in damage for a thickness smaller than 5 mm. For a daily fluctuation with an amplitude of 15% , boards thicker than 5 mm are in the safe elastic region. Boards with a thickness smaller than 5 mm are subjected to plastic deformation, whereas damage does not occur at this amplitude in RH. If we consider a yearly fluctuation with an amplitude of 30%, panels with all thicknesses considered here (up to 40 mm thickness) are in the damage region. On the other hand, all panel thicknesses considered here result in plastic deformation for a fluctuation of 15%.



Fig 4: Frequency versus panel thickness plot for a restrained oak board, indiicating three different regimes for two diffenrent RH amplitudes
a) 30% and b) 15%



     

T. Arends . L. Pel . H. P. Huinink, Hygromorphic response dynamics of oak: towards accelerated material characterization, Materials and Structures 50 181 (2017)

T.Arends, L. Pel, H.P.  Huinink, H.L Schellen, D.M.J. Smeulders, Dynamic Bending of an Oak Board Due to a Moisture Content Gradient, Poromechanics 2017 - Proceedings of the 6th Biot Conference on Poromechanics 2017, p 386-394,  6th Biot Conference on Poromechanics, Poromechanics,  Paris France; 9-13 July 2017

Thomas Arends, Leo Pel, David Smeulders, Moisture penetration in oak during sinusoidal humidity fluctuations studied by NMR, Construction and Building Materials 166, 196–203 (2018)

T. Arends, L. Pel & D. M. J. Smeulders, The frequency dependence of hygro-expansive scaling of oak  Wood materials science & engineering 2018 https://doi.org/10.1080/17480272.2018.1472141 (open access)

Thomas Arends, Leo Pel, Henk L. Schellen and David M. J. Smeulders, Relating relative humidity fluctuations to damage in oak panel paintings by a simple experiment, Studies in Conservation 64, 101-114 (2019), https://doi.org/10.1080/00393630.2018.1481351 (open access)