## 3 Methodology
### Inverse Characterization of Elastic Constants  
This work considers a self-sufficient methodology based on DIC strain measurements and finite element model updating (FEMU) allowing simultaneous measurement of the material interlaminar constitutive properties (\(E_{11}\), \(E_{33}\), \(v_{13}\), \(G_{13}\)) along with the ILT strength. The idea is to take advantage of the full heterogeneous strain field measured on the surface of the OHC specimens, as illustrated in Figure 5, and evaluate if such data can be used in inverse characterization. At the end of the FEMU procedure, optimized material parameters can be used in an additional FE analysis at failure load for assessment of the ILT strength properties. Such advancement would eliminate reliance on additional tests to determine material constitutive properties, which could be unknown prior to the ILT testing. This DIC data-driven OHC ILT test method does not require prior knowledge of any other additional material properties. The DIC data-driven FEMU method is based on the nonlinear minimization of the least square error between DIC-measured strain field and FEM-predicted strains by fine-tuning the parameters of  
Figure 6: 2D plane stress FE mesh used for stress analysis and finite element model updating (FEMU) inverse characterization using Abaqus FE software. Figure 7: ILT stress concentration in a typical unidirectional OHC specimen using FE analysis for (**a**) the carbon/epoxy tape and (**b**) the carbon/epoxy fabric materials considered in this work. the material constitutive model. The concept of the FEMU algorithm using DIC data from unidirectional OHC specimens is shown in Figure 8.  
The FEMU methodology uses a weighted Levenberg-Marquardt optimization algorithm [32] where the cost function is defined as the weighted root mean square (RMS) error between FEM-computed and DIC-measured surface strains. The reader is referred to Reference [15] for more details on the optimization methodology, including iterative update of the material parameters using the Levenberg-Marquardt algorithm with smooth damping parameter variation [33]. Strain field optimization is carried out for about 20 DIC images taken throughout the loading history and up to ultimate failure. The two normal strain components \(\varepsilon_{11}\) and \(\varepsilon_{33}\) and the interlaminar shear strain component \(\gamma_{13}\) are considered in the optimization algorithm. The weights used in the cost function are defined at each node as a function of the local and maximum strain quantities:  
\[w_{ii}{}^{j}=\left\{ \begin{array}{cc}\left(\frac{\varepsilon_{ii}{}^{j}}{max_ {j}[\{\varepsilon_{ii}\}]}\right)^{2}&j\text{ node index}\\ 0\text{ \emph{if }}ii=11\text{ \emph{and }}|\varepsilon_{ii}|^{j}<0.25 max_{j} \Big{\{}|\varepsilon_{ii}|^{j}\Big{\}}&ii=\{11,33\text{ or }13\}\text{ strain component index}\end{array} \right. \tag{2}\]  
As shown in Equation (2), the weights are selected, such as more weight is given to data points with large strain values. For longitudinal fiber-direction strains, which are typically very small due to anisotropy of the composite material, a minimum threshold of 25% of the absolute maximum strain value is also used. This definition of weighted residuals is one way of reducing the sensitivity of the optimization algorithm to measurement noise. Four optimization parameters (\(E_{11}\), \(E_{33}\), \(\
u_{13}\), \(G_{13}\)) are considered, which are the longitudinal and interlaminar Young's Moduli, the Poisson's ratio, and the shear modulus, respectively. A finite difference method is used to compute the sensitivity matrix used in the optimization algorithm. It is worth noting that using finite difference provides an accurate approximation of the sensitivity matrix, which can be also updated at each FEMU iterations, and results in optimum convergence. However, the finite difference approach can also be very costly in terms of computation times since multiple FE analyses are required at each iterations to compute the coefficients of the sensitivity matrix.