## 4 Material Model for Progressive Damage Failure Analysis of Composite Structure  
Composite structures have the potential to exhibit different and simultaneous failure mechanisms when subjected to multiple loading conditions. Tensile and compressive fiber fracture, intralaminar matrix fracture within CFRP plies, delaminations between adjacent plies, adhesive debonding, sandwich core crushing, etc. All these failure mechanisms interact and influence the final failure mode of the evaluated composite sample. No strain-dependent material properties were evaluated in this research activity. Multiple intralaminar material models are available and different numerical methodologies can be selected to reproduce the composite elastic and fracture evolution in LS-DYNA [17]. The presented research focused on the material formulation defied by the MAT58 scheme according to [PERSON] et al. [26] and related to the composite laminate simulation. MAT58 defines a smooth stress-strain relation and implements classic Hashin failure criteria to control the failure initiation inside a specific element. An effective strain parameter (ERODS) is used to control the erosion of FE elements inside this numerical model during a post-failure condition. Post-failure softening values can be defined in the described material model to manage the residual strength of selected material directions (SLIMox). Moreover, a detailed application of several composite material models was reported in [12]. Material and numerical parameters obtained through a calibration process and adopted in the selected composite material formulation are proposed in Table 5.  
The Cohesive Zone Model (CZM) is a mathematical technique that tries to represent the stress-displacement relation that exists around an initiated material crack. The CZM method is based on a nonlinear numerical procedure that consists of reproducing the initiation and evolution of cracks or delaminations in a known a priori propagation path. A characteristic bi-linear cohesive curve is described in terms of local stress versus crack opening displacement Figure 7.  
Fracture surface initiates inside a cohesive zone only upon satisfaction of characteristic failure strength. After failure initiation, the stiffness properties of damaged elements soften with further deformation. The area underneath the CZM constitutive law defines the energy necessary to propagate the fracture, which is commonly associated with fracture toughness (\(G_{\mathrm{C}}\)). *MAT_COHESIVE_MIXED_MODE (MAT38) was selected for the definition of the \begin{table}
\begin{tabular}{l l r} \hline \hline
**Variable** & **Definition** & **Inserted Value** \\ \hline SLIMT1 & Post failure maximum stress limit 1-tension & 0.25 \\ SLIMT2 & Post failure maximum stress limit 2-tension & 0.90 \\ SLIMC1 & Post failure maximum stress limit 1-compression & 0.45 \\ SLIMC2 & Post failure maximum stress limit 2-compression & 0.90 \\ SLIMS & Post failure maximum plane stress shear direction & 0.90 \\ TFAIL & Time step criteria for element deletion & \(1.0\times 10^{-8}\) \\ ERODS & Maximum effective strain for element failure & \(-0.3\) \\ E11C & Strain at longitudinal compressive strength & 0.0065 \\ E11T & Strain at longitudinal tensile strength & 0.012 \\ E22C & Strain at transverse compressive strength & 0.00231 \\ E22T & Strain at transverse tensile strength & 0.0012 \\ GMS & Engineering shear strain at shear strength & 0.03144 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: MAT58 intralaminar numerical parameters. CZM elements in the interlaminar region between adjacent composite plies. Cohesive material properties and cohesive stiffness values obtained with the formulations proposed by [PERSON] et al. [11] are reported in Table 6.  
Crushable closed-cell foam core shows fragile behavior in tension and non-linear evolution in compression. *MAT_CRUSHABLE_FOAM (MAT63) was selected to numerically reconstruct the physical behavior of closed-cell foam Rohacell 51 used in the advanced joint sandwich samples.