## 2 Materials and Methods
### Scots Pine Beam and Its Thermodynamic Equilibrium
#### 2.2.2 Thermo-Dynamic Equilibrium of Moisture Content  
In this study, the equilibrium moisture content (EMC) was used as the moisture gradient parameter since it only depends on the environmental conditions surrounding the beam and it is not experimental derived. The EMC was determined by calculation in each slice along the longitudinal and radial directions, because generally wood has an increase in EMC from the edge of the beam to the path [22]. The tested Scots pine beam was left outdoors for more than two years (January 2016-August 2018) at the campus of the Norwegian University of Science and Technology (NTNU, Trondheim, Norway) [23], where there was an average temperature (T) of 5 \({}^{\circ}\)C and relative humidity (RH) of 85%. In September 2018, the beam was moved inside the Fatigue, Fracture and Mechanical Characterization Laboratory (NTNU campus), under new conditions of T = 21 \({}^{\circ}\)C and RH = 30%. It was assumed that, from 1 September 2018 until 9 February 2019 (i.e., 160 days), the beam--from its external surface to a certain depth towards the path--reached an hygothermal equilibrium state with the indoor conditions, the so-called equilibrium moisture content (EMC), which is expressed as the percentage of the mass of water per unit mass of anhydrous material [24]. The response of the beam to the variations in the indoor climate occurred differently along longitudinal and radial \begin{table}
\begin{tabular}{c c c c c} \hline
**ID Number** & **Group** & **Directionality** & \multicolumn{2}{c}{**Grain Angle**} \\ \hline  & & & 50 th percentile & 90 th percentile \\ \hline S081, S091, S101, S111, & & & \\ S123, S142, S153, S173 & Group 1 & Parallel to grain & 4\({}^{\circ}\) & 9\({}^{\circ}\) \\ S093, S122, S143, S152, S163, S172 & Group 2 & Not parallel to grain & 5\({}^{\circ}\) & 10\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 1: ID number of the samplings, their classification into the group (1 or 2) based on the directionality in the beam slices. Figure 3: (**a**) Samples carved from the slice with a grain angle of 0\({}^{\circ}\), i.e., fracture follows the direction of the growth rings; (**b**) samples carved with a grain angle different from 0\({}^{\circ}\) with respect to the growth ring direction. directions, determining peculiar moisture gradients inside the wood. Since the beam was naturally dried, it was assumed that, after 160 days, only the outermost layers were in complete equilibrium with the indoor climate conditions, while the innermost layers (i.e., closer to the pith and in the middle of the beam along the longitudinal direction) were still--in some part--in equilibrium with the previously experienced outdoor climate conditions. In this way, it was possible to determine the EMC at each point of the beam using RHresponse. Indeed, at a constant temperature, T (=21 \({}^{\circ}\)C), RHresponse is the response of a wooden material at a certain depth of the specimen with respect to adjacent material layers and environmental relative humidity. This parameter is determined using Equation (2) in [25]. \[\text{RH}_{\text{response,i}}=\frac{\text{RH}_{\text{response, i-1}}+\frac{\text{RH}_{\text{i}}}{\text{n/3}}}{1+\frac{1}{\text{n/3}}} \tag{2}\]  
where RH is the relative humidity (%); RHresponse is the material response in RH (%); n is the number of data points over the response time expressed in days and multiplied by 24 h; and i is the ith day, and therefore it refers to the time expressed in days occurred since the environmental change (from day 1 to 160). Thus, RHi was set equal to 85% in the first few days of drying (i.e., corresponding to the averaged outdoor conditions) up to 30% going towards the 160 th day (i.e., approx. equal to the stable laboratory room conditions). RHresponse depends on the response time, defined as the time needed for an object to reach 95% of the end value of a step change in RH [24]. The response time depends on the geometry and the depth of the object.