Computational modeling of the dynamics and interference effects of an erosive granular jet impacting a porous, compliant surface

Abstract

The general problem of a loosely flowing erosive granular jet undergoing impact with a compliant surface is common in many manufacturing processes, and also in the operating environment of a variety of machine parts. This paper presents a three-dimensional, collision-driven discrete particle simulation framework for investigating the dynamics of a jet of erosive particles impacting a surface with a specified porosity and compliance. The framework is capable of handling repeated collisions between incoming particles and rebounding particles, and between particles and surfaces. It is also capable of performing a coupled simultaneous calculation of sub-surface stresses in the material, assuming a certain porosity. Well illustrated numerical examples are presented with detailed analysis for investigations on the mechanics and energetics of the interfering collisions in eroding jets close to the target surface, on the effect of such interference on the material erosion, and on the evolving stress levels and potential damage zones under the action of impact. Particularly, the assumption of considering first-order collisions between oncoming and rebounding jet particles is re-examined. The influence of repeated collisions on energy transferred to the surface was found to be significant under conditions which involves high particle numbers or fluxes, and also high degrees of inelasticity. The overall trends for parametric variations were found to be in accordance with reported trends in the literature.

Appendix: Stresses due to point loadings

For completeness of presentation, we outline here the expressions for the stresses due to point normal and point tangential loading which have been employed for the sub-surface stress calculations for the particle jet impacting the porous material layer. In the following, it is assumed that the loading acts on a surface located along the $x$–$y$ plane—at $z=0$. The point where the load is applied is assumed to be the origin. For any other loading point, simple coordinate transformations need to be employed. Assuming that a normal loading of magnitude $F_N$ is applied along the $z$ direction, the stresses on the surface, and sub-surface is given in cylindrical polar coordinates as follows:

$$\begin{aligned} \sigma _{rr}= & {} \frac{F_N}{2\pi } \left[ (1 - 2\nu ) \left( \frac{1}{r^2} - \frac{z}{\rho r^2} \right) - \frac{3 z r^2}{\rho ^5} \right] \end{aligned}$$

(49)

$$\begin{aligned} \sigma _{\theta \theta }= & {} - \frac{F_N}{2\pi } \left( 1 - 2\nu \right) \left[ \frac{1}{r^2} - \frac{z}{\rho r^2} - \frac{z}{\rho ^3} \right] \end{aligned}$$

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$$\begin{aligned} \sigma _{zz}= & {} - \frac{3 F_N}{2\pi } \frac{z^3}{\rho ^5} \end{aligned}$$

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$$\begin{aligned} \sigma _{rz}= & {} - \frac{3 F_N}{2\pi } \frac{r z^2}{\rho ^5}, \sigma _{\theta z} = \sigma _{\theta r} = 0 \end{aligned}$$

(52)

where $z \ge 0$, $r^2 = x^2 + y^2$, and $\rho ^2 = x^2 + y^2 + z^2$. The expressions are radially symmetric for a single point loading along the loading axis. Corresponding expressions for stresses due to a tangential loading of magnitude $F_X$ applied along the positive $x$ direction can be written as follows:

$$\begin{aligned} \sigma _{xx}&= \frac{F_X}{2\pi } \Biggl [ -\frac{3 x^3}{\rho ^5} + (1 - 2\nu ) \Biggl \{ \frac{x}{\rho ^3} - \frac{3x}{\rho ( \rho + z )^2} \nonumber \\&\quad + \frac{x^3}{\rho ^3 ( \rho + z )^2} + \frac{2x^3}{\rho ^2 ( \rho + z )^3} \Biggr \} \Biggr ] \end{aligned}$$

(53)

$$\begin{aligned} \sigma _{yy}&= \frac{F_X}{2\pi } \Biggl [ -\frac{3 x y^2}{\rho ^5} + (1 - 2\nu ) \Biggl \{ \frac{x}{\rho ^3} - \frac{x}{\rho ( \rho + z )^2} \nonumber \\&\quad + \frac{x y^2}{\rho ^3 ( \rho + z )^2} + \frac{2x y^2}{\rho ^2 ( \rho + z )^3} \Biggr \} \Biggr ] \end{aligned}$$

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$$\begin{aligned} \sigma _{xy}&= \frac{F_X}{2\pi } \Biggl [ -\frac{3 x^2 y}{\rho ^5} + ( 1 - 2\nu ) \Biggl \{ -\frac{y}{\rho ( \rho + z )^2}\nonumber \\&\quad + \frac{x^2 y}{\rho ^3 ( \rho + z )^2} + \frac{2 x^2 y}{\rho ^2 ( \rho + z )^3} \Biggr \} \Biggr ] \end{aligned}$$

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$$\begin{aligned} \sigma _{zz}&= \frac{F_X}{2 \pi }\left[ -\frac{3 x z^2}{\rho ^5} \right] \end{aligned}$$

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$$\begin{aligned} \sigma _{yz}&= \frac{F_X}{2 \pi }\left[ -\frac{3 x y z}{\rho ^5} \right] \end{aligned}$$

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$$\begin{aligned} \sigma _{zx}&= \frac{F_X}{2 \pi }\left[ -\frac{3 x^2 z}{\rho ^5} \right] \end{aligned}$$

(58)

The expressions for point loading along $y$ direction can be obtained from the above by simply interchanging $x$ and $y$.