The Finnie Model of Erosive Wear

The following model is what I use for evaluating wear in DEM. It’s based on the work of Iain Finnie.

Finnie I., Erosion of Surfaces by Solid Particles. Wear, 3(2):87-103, 1960.

The wear model I use for granular contact is based on two parts. The first part is the calculation of the number, direction (angle of impact) and velocity of the particles striking the ductile surface.  Brittle materials can see fractures upon impact and require a different approach to the mathematical model that is solely based on the material properties and will not be discussed here.

The model assumes ductile material so the model will calculate the amount of surface material removed based on the particle trajectories. Note: the model though it calculates the volume of material removed, the particle velocities are dependent on the material size which is an estimate to begin with so the effect of particle size on erosion is relatively uncertain and is a qualitative approach to wear.

Materials such as steel undergo wear by a process of plastic deformation in which the material is removed by the displacement or cutting action of the eroding particle. The assumptions behind the model are that there is a ratio between the normal force component and the shear component of constant value K.  This is true if the particle rotation during the cutting/impact is fairly small. In addition, it is assumed that there is a ratio between the depth of contact l and the depth of cut y_t as seen in the figure below of value \psi.

finnie

Wear, 3(1960) 93: Idealized picture of abrasive grain striking a surface and removing material. Initially the velocity of the particle’s center of gravity makes an angle \alpha with the surface.

 

Further assumptions require that a constant plastic flow stress p  is reached immediately upon impact.  This is for the traction analysis so that the particle cutting face is of uniform width which is large compared to the depth of cut. The volume of material removed by the particle is then taken as the product of the area swept out by the particle tip and the width of the cutting face.

The cases under which material is removed are as follows:

  1. The particle comes down as a low angle, cuts out part of the surface and then leaves again which means the depth of cut goes to zero as the particle departs.
  2. At the higher angles the horizontal motion of the particle ceases before it leaves the surface so the cutting stops when the velocity goes to zero.

For (1) think of a particle rubbing along the surface when impacting at low angles and the particle striking the surface head on and creating craters for high angle impacts (2).

Integrating over the duration of impact or cutting period provides the following expressions for the volume removed:

Q = \frac{mV^2}{p \psi K}(\sin 2 \alpha - \frac{6}{K} \sin^2 \alpha)          if \tan \alpha \leq \frac{K}{6}

Q = \frac{mV^2}{p \psi K}(\frac{K \cos^2 \alpha}{6})          if \tan \alpha \geq \frac{K}{6}

The first equation is for low angle impact, the second for high angle impact. The variables are defined as follows:

Q = the volume of material removed

m = the mass of the particle or effective mass of the particles impacting the surface

V = the velocity of impact

p = constant of plastic flow stress

\psi = ratio of depth of contact to depth of cut

\alpha = angle of impact

K = ratio between the normal force and the shear force

Down side to this model is that the formulation assumes the particle impacts a smooth surface every time when in reality the surface wastes away by previous impacts and becomes rough.  Since the surface roughness increases throughout the duration of the impact period there is a correction made to the model. This correction is made under the observation that the surface roughness increases with each impact.  Therefore, in an impact with a rough surface more material is removed than from a smooth surface.  The second observation is that not all particles that impact the surface will remove material in an idealized manner and can sometimes not remove material at all.  Thus by inspection of erosion craters due to a known number of abrasive grain cuts, the volume removed is assumed to be 50% of the predicted erosion.

In addition,  \psi , is assumed to be 2 from metal cutting experiments, according to Finnie.  This leaves one variable left for the user to define (K).  If using K = 2 as approximated from angular abrasive grain erosion tests, the corrected volume removed is approximately defined as follows:

Q \approx \frac{mV^2}{8p}(\sin 2 \alpha -3 \sin^2 \alpha)          \alpha \leq 18.5 ^{\circ}

Q \approx \frac{mV^2}{24p} \cos^2 \alpha          \alpha \geq 18.5^{\circ}

Maximum erosion has been observed in the impact angle range of 15-20° so the estimated maximum volume removal is given by

Q \approx 0.075 (\frac{MV^2}{2}) \frac{I}{p}

which is 7.5% of the particle’s kinetic energy divided by the flow pressure of the material.

What the I use as an input to the DEM engine is the K variable relating the normal force to the shear force.  This is also a measure of the shape of the material impacting the surface.  If the material is near spherical the K value will increase, for material that is abrasive (has a rough surface) the K value will be small.

Most abrasive material has a K value in the range of 1.5 to 2.5.  Perfectly spherical material would have a K value of 5.0.  Studies show that K is approximately 2 for angular abrasive grain. As most material modeled with spherical particles is rough, the K value should be on the lower end of the K value range.

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