# Determination of Stress Intensity Factors under Shock Loading Using a Diffraction-Based Technique
## 2 Materials and Methods
### Dynamic Stress Intensity Factors Evaluation-Experiment vs. Finite Element Method (FEM)  
When the task to assess the time development of stress intensity factor and its peak value determination under impact loading of the structure is assigned, the value of measured DSIF can be computed directly from the asymptotic displacement field around the crack tip [16]:  
\[K_{i}(t)=\lim_{r\to 0}\frac{\text{E}\Delta_{xx}(\eta_{r},t)}{8(1-\
u^{2})} \sqrt{\frac{2\pi}{\text{r}}} \tag{4}\]  
where  
\(i\) considered mode I, II,  
\(K_{i}(t)\) the stress intensity factor of \(i\)th mode in time \(t\),  
\(\Delta_{xx}\) time-dependent displacements difference in the local coordinate system; ax = x for K\({}_{\text{II}}\),  
y for K\({}_{\text{I}}\),  
\(\eta_{r}\) point on the notch face in distance \(r\) from notch tip,  
E Young modulus of elasticity,  
v Poisson ratio,  
r distance from notch tip. Nevertheless, the Equation (4) presents only the first term in the displacement field series of asymptotic analysis around the crack tip. Since the notch face movements were measured only in one point on notch faces, direct evaluation of the stress intensity factor would be affected by an error. In experimental measurements, the laser beam illuminated the prearranged slit in a chosen position with polar coordinate r = 6 mm from a notch tip. When using only the first term of analytical displacement field solution, an obtained error calculating the DSIF directly from notch face movements in the located region was around 30%. For a more accurate evaluation of the stress intensity factor from the experimental data, it was needed to measure the movements of the set of points lying on the traction-free  
Figure 4: Measured impulse forces and their comparison to the first notch opening displacement period: (**a**) beam specimen D = 17.5 mm; (**b**) angled specimen D = 9.5 mm where D is a diameter of a steel ball. The grey color represents the experiment range. crack surface. This leads to several lasers and detectors that are recording notch face movements simultaneously. Due to this, it is practical to assign the stress intensity factor to measured notch face movements numerically. Numerical modeling and the calculation of the dynamic stress intensity factors have been performed by a finite element method using ANSYS software (Canonsburg, PA, USA). A 2D finite element with quadratic displacement behavior and two degrees of freedom for each node was selected. The chosen type of element consisted of eight nodes in a rectangular shape and could be degenerated into the six-nodes triangular shape element. The task was solved as a plane strain state problem. A linearly isotropic material model was considered. For modal analysis, the block Lanczos mode extraction method was used and, for transient analysis, the Distributed sparse matrix direct solver was applied. Using numerical analysis, the effect of the notch geometry as the simulated crack has been evaluated. The geometry of the notch is shown in Figure 2, and a sharp crack is shown in Figure 5. It has been found that this has a minor effect on the observed results of notch face movements in a given position from a notch/crack tip \(\pm\)3.27% between obtained local extremes of the time-dependent notch and crack face movements function. Therefore, numerical results of the real notch geometry from Figure 2 were used for notch face movements comparison with experimental results, and crack-like geometry was used to assign the DSIF values to achieved notch face movements. The numerical model of the beam specimen consisted of 6227 elements/19,188 nodes, and 6589 elements/20,254 nodes for notch and crack-like geometry was used, respectively. The L-shaped specimen consisted of 9632 elements/29,721 nodes, and 10,456 elements/32,197 nodes for notch and crack-like geometry, respectively. Global finite element edge length in both cases of notch/crack geometry was set as one millimeter to capture the propagation of stress waves in the material.