## 2 Material and Modelling
### Material Model: CODAM2  
To describe the mechanical behaviour of the \([90/45/0/-45]_{s}\) beech veneer laminates, the strain-based COMposite DAMage Model (CODAM2) was applied, which is incorporated as MAT219 in the commercial FE software LS-DYNA [26]. In the following, the governing equations of this material model are presented. More details can be found in related literature [27; 28; 29].  
Figure 1b illustrates a general stress-strain curve to describe progressive damage in the two principal axes in the grain direction (\(\alpha=1\)) and perpendicular (\(\alpha=2\)) to it. Strains are evaluated in each ply of the laminate. Hence, ply-based damage is formulated in terms of equivalent strain functions. The longitudinal (associated with the fibre or grain) equivalent strain \(\varepsilon_{1}^{\text{eq}}\) is taken to be equal to the magnitude of the longitudinal normal strain \(\varepsilon_{11}\)  
\[\varepsilon_{1}^{\text{eq}}=|\varepsilon_{11}|. \tag{1}\]  
Figure 1: (a) Illustration of considered quasi-isotropic \([90/45/0/-45]_{s}\) laminates and (b) ply-based stress-strain relationship as input into CODAM2 finite element model to simulate progressive damage in wood veneer laminates.  
The transverse (matrix) equivalent strain \(\varepsilon_{2}^{\text{eq}}\) considers transverse tensile \(\varepsilon_{22}\) and shear strains \(\gamma_{12}\) such that  
\[\varepsilon_{2}^{\text{eq}}=\text{sign}(\varepsilon_{22})\sqrt{\left( \varepsilon_{22}\right)^{2}+\left(\frac{\gamma_{12}}{2}\right)^{2}}. \tag{2}\]  
The sign of the transverse normal strain \(\varepsilon_{22}\) is used to determine compressive (negative) or tensile (positive) loadings in the transverse direction. Here, only tensile load cases are considered.  
The damage variables \(\omega_{\alpha}\) in the longitudinal \((\alpha=1)\) and transverse \((\alpha=2)\) directions are defined in terms of damage initiation strain \(\varepsilon_{\alpha}^{i}\) and damage saturation strain \(\varepsilon_{\alpha}^{s}\) such that  
\[\omega_{\alpha}=\left(\frac{\left|\varepsilon_{\alpha}^{\text{eq}}\right|- \varepsilon_{\alpha}^{i}}{\varepsilon_{\alpha}^{s}-\varepsilon_{\alpha}^{i}} \right)\left(\frac{\varepsilon_{\alpha}^{s}}{\varepsilon_{\alpha}^{\text{eq} }}\right)\quad\text{for }\left(\left|\varepsilon_{\alpha}^{\text{eq}}\right|- \varepsilon_{\alpha}^{i}\right)>0. \tag{3}\]  
A damage variable \(0<\omega_{\alpha}<1\) indicates partial damage while \(\omega_{\alpha}=1\) denotes fully saturated damage.  
The parameters for the damage initiation and saturation strains, \(\varepsilon_{\alpha}^{i}\) and \(\varepsilon_{\alpha}^{s}\), are crucial in modelling progressive fracture. The fracture energy of each ply \(G_{\alpha}^{f}\) can be related to the damage saturation strains \(\varepsilon_{\alpha}^{s}\) and ply strength \(X_{\alpha}\) by  
\[\varepsilon_{\alpha}^{s}=\frac{2G_{\alpha}^{f}}{X_{\alpha}l^{*}}. \tag{4}\]  
Note that the fracture energy \(G_{\alpha}^{f}\) in CDM is typically distributed over the volume of a finite element by considering a characteristic element length \(l^{*}\) according to [PERSON]'s crack band scaling law [30]. This characteristic length plays an important role when different element sizes are considered.  
CODAM2 is capable of describing laminate behaviour using only one through-thickness integration point, which is particularly advantageous when dealing with \([90/45/0/-45]_{s}\) been laminates. This feature makes CODAM2 a much more efficient material model compared to other CDM-based models that require assigning ply-based material properties to multiple integration points through the thickness to describe laminates with various ply angles. The enhanced efficiency of CODAM2 also allows for its integration with data-driven calibration methods, such as genetic algorithms, as discussed in Section 3.