# Prediction of Mechanical Performance of Acetylated MDF at Different Humid Conditions
## 2 Materials and Methods
### Experimental
#### 2.2.7 Finite Element Analysis  
Geometries of finite element (FE) models were identical to the samples manufactured for three-point bending tests and solved as a 3D problem; thus, full models were considered. Specifically, the beam's part was 300 mm in length, 36 mm in width with a thickness of 18 mm, and the cylindrical pins' parts were 36 mm long with a diameter of 30 mm and placed 26 mm from both ends on a lower side of the beam part and in the middle of the beam's span on an upper side of the beam part. The beam part was created as a deformable body and pins as discrete rigid bodies. The boundary conditions (BCs) and load were applied to the reference points (RP) created on pins' circular areas. Six degrees of freedom (df) were fixed in both bottom pins' reference points for applying BCs and five df were set on the upper pin RP that also serves for applying a load. Namely, the following BCs were assigned: all six df are fastened, thus 1, 2, 3, 4, 5, 6 = 0 for pins at the ends on the lower side of the beam part; for the pin in the middle of the span on the upper side of the beam part movement in the \(z\)-axis direction was enabled, only five df are fastened allowing movement in direction 3, thus 1, 2, 4, 5, 6 = 0. The global coordinate system and df notations and plus directions are shown in Figure 1. The load applied to the beam was a concentrated nodal force of value 1200 N, thus, larger than a mean value of the breaking load F\({}_{\text{max}}\) measured in the experiment (Figure 1).  
3D rigid elements _R3D4_ (4-node bilinear quadrilateral) and _R3D3_ (3-node triangular facet) were assigned to mesh pins. To create a mesh for the beam, _C3D8R_ (8-node linear brick) solid elements with reduced integration were defined.  
Normal hard-contact, frictionless tangential behavior, and a general standard surface-to-surface contact, were prescribed to characterize the interaction between pin parts and the beam part. The mechanical properties of the elastic-plastic isotropic material of the studied MDFs had been taken from the experimental data considering the RH of 65% for the standard board perpendicular to bending: thus. [PERSON]'s modulus of 2574 MPa, [PERSON]'s ratio 0.25, and density of 746.2 kg/m\({}^{3}\).  
Additionally, to specify a damage initiation, damage criteria for the XFEM model were prescribed to define the constitutive response for cohesive elements: namely, the maximum principal stress damage criterion (MAXPS) of value 34.47 MPa, taken from experiment measures. This means the damage criterion is using traction separation laws and is established as:  
\[\text{f}=\{\langle\sigma_{\text{max}}\rangle/\sigma^{0}_{\text{max}}\} \tag{9}\]  
where \(\sigma^{0}_{\text{max}}\) represents the maximum allowable principal stress, and the symbol \(\langle\ \rangle\) represents the Macaulay bracket with the usual interpretation. The Macaulay brackets are here to represent that a purely compressive stress does not initiate damage; rather, damage is initiated when the maximum nominal stress ratio reaches a value of one (f = 1) [28]. Additionally, there was no initial crack assigned to the model. Instead, the XFEM crack domain was designated on the whole beam part. For describing a crack geometry and the crack's growth motion in 3D space, two level sets for a crack is assumed. First set is the crack surface \(\Phi\) while second, \(\Psi\), is made so the intersection of two level sets that gives the crack front. The nodal value of the function \(\Phi\) is the signed distance of the node from the crack face and of the function \(\Psi\) is the signed distance of the node from an almost-orthogonal surface passing through the crack front [28].  
Figure 1: Illustration of parts and boundary conditions assigned in the finite element (FE) model.