# Equivalent Analysis of Thermo-Dynamic Blow-Off Impulse under X-ray Irradiation
## 6 Effect of X-ray Angle of Incidence of Blow-Off Impulse  
When a parallel X-ray irradiates the structure or material surface, there will be different angles at different directions for the normal structure surface and radiated direction of X-ray. It directly affects the distribution of X-ray energy fluxes on the surface, and the distribution law of blow-off impulse, and the load of different distribution directly affects the structural response law. Accordingly, it is of great significance to study the effect of X-ray angle of incidence of blow-off impulse. No definite conclusion has been drawn about the distribution of X-ray blow-off impulse with the angle. When [PERSON] [1,4] calculated the impulse of cylindrical shell by X-ray, the X-ray on the surface of cylindrical shell is in fact cosine distribution with the angle of incidence. Thus, it was intuitively assumed that the blow-off impulse is cosine distribution, whereas there is no corresponding proof and applicable conditions. Subsequently, researchers in this field have been exploiting this hypothesis to the load design of structural response laboratory simulation [2,3,5,6]. However, the relation of blow-off impulse with  
Figure 8: Equal energy flux curve on impulse plane. Figure 9: Equal impulse curve on energy flux plane. the angle of incidence has been explored. A clear conclusion from analytical deduction and derivation is difficult to draw since formula is overly sophisticated about X-ray energy deposition of blackbody spectrum. [PERSON] conducted an analytical deduction of the energy deposition on the surface of cylindrical shells irradiated by X-ray in different directions [4], and the conclusion was also unsatisfactory. He also calculated and analyzed the blow-off impulse of cylindrical shell by X-ray, whereas the numerical results cannot account for the problem since the relation is not the same between the different direction of two-dimensional mesh and the incidence direction of X-ray. Thus, there has been no effective progress in the study on the relation between the pulsed X-ray blow-off impulse and the angle of incidence over the past few years. Since the problem originates from the pulsed X-ray effect of two-dimensional cylindrical shells, the method was also illustrated in the two-dimensional framework, thereby to some extent limiting the solution. In this study, the oblique incidence of X-ray was discussed in a two-dimensional frame, and the relation between the pulsed X-ray blow-off impulse and the angle of incidence was obtained by numerical calculation. Figure 10 presents the oblique incidence model of X-ray; the dotted line is in perpendicular incident direction, while the solid line is in oblique incident direction. From the geometry in the figure, the energy deposition thickness of X-ray in the identical size and location of the grid can be concluded. The formula is written as:  
\[\Delta R=\frac{\Delta r}{\cos\theta} \tag{47}\]  
where \(\Delta R\) denotes the energy deposition thickness of oblique incident X-ray in the grid, \(\Delta r\) is the energy deposition thickness of vertical incident X-ray and \(\theta\) represents the angle between the incident X-ray direction and the normal direction of the target surface. For mono-energetic X-ray irradiation, the energy deposition at \(r\) can be deduced as:  
\[E=\Phi_{0}\mu e^{\frac{-\mu\mu\
u}{\cos\theta}} \tag{48}\]  
Since material melting is an essential condition to blow-off impulse. \(E_{\
u}\) is the melting energy of material, so the thickness of melting layer can be computed from the above formula. The formula is written as:  
\[r_{\
u}=\frac{\cos\theta}{\mu\rho}\ln\frac{\Phi_{0}\mu}{E_{\
u}} \tag{49}\]  
Figure 10: Geometric relation between oblique incidence of X-ray and computational mesh. The melt thickness of the material is basically cosine distribution with angle of incidence for mono-energetic X-ray. Blow-off impulse is expressed as:  
\[I=\int_{0}^{r_{p}}u\rho dr \tag{50}\]  
where \(u\) denotes the particle velocity.