mesh_module wear

Purpose

Command for enabling wear of the mesh wall by particle impingement.

Syntax

mesh_module wear keyword value

Keywords:

Keywords	Description
id	obligatory, user-defined name for the mesh module
model	available models: finnie, finnie_unresolved, deformation, archard, combined, off
sum_wear_output	specifies the output for total wear (`mass`, `volume`) default: `volume`

Associated material properties

Material properties

hardness: material hardness used for the archard wear model [Pa]

Material interaction properties

k_finnie: coefficient $k_\mathrm{finnie}$ for the finnie and finnie_unresolved wear models [m s²/kg]
k_deformation: coefficient $k_\mathrm{def}$ for the deformation wear model [m s²/kg]
k_archard: coefficient $k_\mathrm{archard}$ for the archard wear model [-]

Examples

mesh_module wear id myWear model finnie
mesh_module wear id myWear model finnie_unresolved
mesh_module wear id myWear model archard
mesh_module wear id myWear model deformation
mesh_module wear id myWear model combined

After defining a mesh such as in

mesh id myMesh material steel file meshes/wall.stl mesh_modules { myWear }

The total eroded mass [Kg] and volume [m³] can be obtained via

id_myMesh.eroded_mass
id_myMesh.eroded_volume

With the combined model it is also possible to obtain the contribution of each individual wear mechanism to the total wear mass:

id_myMesh.eroded_mass.archard
id_myMesh.eroded_mass.deformation
id_myMesh.eroded_mass.finnie_unresolved

and similarly for the eroded volume.

The local wear height [m], at each mesh element and at each output time step, is written to output files for further post processing e.g. in Paraview via

output_settings mesh_properties { wear }

The contribution of each wear mechanism to the total wear height will appear e.g. in the components tab of Paraview, together with the total wear.

Description

The mesh module wear is used to calculate the wear that a mesh experiences due to particle collisions. In order to reproduce different wear mechanisms, four wear models are available: archard, finnie, finnie_unresolved, deformation and combined wear. They reproduce the wear that results from sliding interactions, and the cutting and deformation produced in an impact. The models are described in detail below.

This mesh module is a prerequisite for the mesh module_deform.

Finnie Model

The model keyword with values finnie or finnie_unresolved activates two different modes of the wear model developed by Finnie [1]. The former is based on the soft sphere framework and increments the wear at every time step involved in a contact (see below), whereas the latter disconsiders the remaining contacts after the first impact i.e. it is based on the hard sphere model.

This model for erosive wear relates the rate of wear to the rate of kinetic energy of particle impact on a surface. For finnie_unresolved, it reads

$E = 0.5 \ k_{\mathrm{finnie}} v_p^2 f(\gamma),$

where $E$ [m³/kg] is the worn volume ( $=V$ [m³]) resulting from a single particle colision normalized by the particle mass ( $=m_p$ [kg]), i.e. $E=V/m_p$ , $v_p$ [m/s] is the magnitude of the particle impact velocity, $k_\mathrm{finnie}$ [m s²/kg] is a model parameter and $f(\gamma)$ is a dimensionless function of the impact angle between the approaching track and the surface.

The model coefficient $k_\mathrm{finnie}$ has to be specified as in the following example:

material_properties steel density 8000 k_finnie 0.1 keywords values

where keyword/value denotes additional material properties (e.g., poissonsRatio, coefficientRestitution, etc.).

The function $f$ is defined as

$f(\gamma) = \frac{1}{3} \cos^2(\gamma) \quad \mbox{if}\;\tan(\gamma) > \frac{1}{3} f(\gamma) = \sin^2(2\gamma) - 3 \sin^2(\gamma) \quad \mbox{if}\;\tan(\gamma) < \frac{1}{3}$

This model (finnie_unresolved) was formulated originaly in the framework of the hard sphere model. Here we also provide an alternative mode (finnie), adapted to the soft sphere model (see below).

When used in conjunction with a Lagrangian particle tracks representing a constant mass ﬂow rate of particles of $\dot{m}_p$ , the eroded mass $EM$ in kg during one time-step $\Delta t$ becomes

$EM_{HS}(t) = \rho E \dot{m}_p \Delta t,$

where $\rho$ [kg/ m³] is the density of the worn material and the subscript HS denotes the hard sphere model. In contrast, the soft sphere model resolves a particle-surface contact with multiple time-steps, instead of one step, and therefore the corresponding formulation is adapted for that case.

For model finnie the normalized worn volume is defined as

$E = k_{\mathrm{finnie}} v_p^2 f(\gamma),$

i.e. the definition of $E$ [m³/kg] is different than the finnie_unresolved by a factor of 0.5. Considering a one term Taylor series expansion for the wear volume over small time intervals the variation of the wear volume is obtained. A subsequent integration over one collision time leads to the following relation for the eroded mass caused by one particle during the particle - surface contact:

$EM_{SS}(t) = \rho \int_0^{tc} e\,dt\,m_p,$

where $tc$ [s] is the contact time, the subscript SS denotes the soft sphere model, and

$e = \frac{dE}{dt} = \frac{\partial E}{\partial v_p} \frac{\partial v_p}{\partial t} + \frac{\partial E}{\partial \gamma} \frac{\partial \gamma}{\partial t}.$

Since $\partial v_p/\partial t$ is the dominant part of the time derivative and tracking $\partial \gamma/\partial t$ would be computationally tedious, the latter is neglected. Hence one obtains

$e \approx \frac{\partial E}{\partial v_p} \frac{\partial v_p}{\partial t} = \frac{2E}{v_p} \frac{f_c}{m_p},$

with $f_c$ being the particle-surface contact force. Additionally, $e=0$ for $v \cdot c < 0$ , where $v$ is the particle impact velocity vector, and $c$ is the surface normal vector (pointing into the surface) at the contact point. This follows from the assumption that wear is caused only during the impact phase of the contact and not during the repulsion phase. Thus,

$EM_{SS}(t) = 2 k_\mathrm{finnie} \rho \int_0^{tc} H(v \cdot c) v_p f(\gamma) f_c dt,$

where $H$ is the Heaviside function. For materials with a uniform density, the worn volume is easily obtained from the eroded mass:

$V(t) = EM_{SS} /\rho.$

The total worn volume in m³ and the total eroded mass in kg are simply the summation of $V$ and $EM_{SS}$ over all mesh elements defining the solid object, respectively. As mentioned earlier, the total worn volume or eroded mass are written in the output file simulation_data_aspherix.csv. The sum_wear_output keyword with values volume or mass defines which option is output to this file. The default value is volume.

Since mass and volume are extensive thermodynamic properties, they can not be used to analyse the local wear at each point of the solid object. For a more refined, local analysis of the wear, we also provide the local wear height $h$ [m], which is approximated by the worn volume $V$ [m³] within the region demarcated by each mesh element normalized by the surface area of the corresponding mesh element A [m²]:

$h(\mathbf{x}, t) = V/A.$

The wear height $h$ [m] is output to the post folder and can be used to generate contour plots in Paraview for example.

In principle, the hard and soft sphere frameworks should give equivalent results when the Young’s modulus is made sufficiently large. However, since the finnie and finnie_unresolved models use different definitions of $E$ [m³/kg], and therefore k_finnie, by a factor of 0.5, the k_finnie would have to be adjusted by the same factor if the two models are compared in this limit condition.

Deformation Model

The impact deformation wear model is similar to the finnie_unresolved model but the corresponding equation is

$E = 0.5 \ k_\mathrm{def} v_n^2,$

where $v_n$ [m/s] is the velocity component normal to the wall at the impact point. Other variables have the same meaning as in the impact cutting model. The value of the model coefficient $k_\mathrm{def}$ [m s²/kg] is set by the material property k_deformation.

Archard Model

The model keyword with value archard activates the model proposed by Archard [2]. The model calculates the total worn volume $V$ [m³] according to the following equation:

$V(t) = \frac{k_\mathrm{archard} W L}{H},$

where $k_\mathrm{archard}$ is a non-dimensional constant, $W$ [N] is the total normal load, $L$ [m] is the sliding distance (integrated tangential velocity) and $H$ [Pa] the hardness of the softest contact surface. The model coefficient $k$ and the hardness $H$ have to be specified for each material pair and each material, respectively. A corresponding section in the input script could look like below:

material_properties steel density 8000 k_archard 1.0 hardness 2000.0 keywords values
material_properties plastic density 1200 k_archard 0.9 hardness 1000.0 keywords values
material_interaction_properties plastic steel k_archard 0.9 keywords values

where keywords/values denote additional material properties (e.g., poissonsRatio, coefficientRestitution, etc.).

The eroded mass and local wear height are easily obtained from the worn volume:

$EM_{SS}(t) = \rho V,$

and

$h(\mathbf{x}, t) = V/A,$

respectively.

Combined Model

The model keyword with value combined activates the combined wear model developed by Roessler and Katterfeld [3]. In this case, the total wear is the result of the sum of the wear obtained from three different wear mechanism: impact cutting (finnie_unresolved), impact deformation (deformation), and sliding (archard). This model is able to predict the wear in a variety of different conditions, and it is shown by Roessler and Katterfeld to be the most robust combination of different models. The equation for the worn volume per unit of particle mass is

$E = 0.5 \ \tilde{k}_{\mathrm{finnie}} v_p^2 f(\gamma) + 0.5 \ k_\mathrm{def} v_n^2 + \frac{\tilde{k}_\mathrm{archard} W L}{m_p H},$

where the coefficients $\tilde{k}_{\mathrm{finnie}}$ and $\tilde{k}_{\mathrm{archard}}$ are given by

$\tilde{k}_{\mathrm{finnie}} = k_{\mathrm{finnie}} \text{ and } \tilde{k}_{\mathrm{archard}} = 0 \text{ for } \gamma > 1.0^{\circ}, \\ \tilde{k}_{\mathrm{finnie}} = 0 \text{ and } \tilde{k}_{\mathrm{archard}} = k_{\mathrm{archard}} \text{ for } \gamma \leq 1.0^{\circ}.$

Additional information

This mesh module stores a global scalar for access by various output commands. The output value is accessed via the property name, which is wear, e.g. id_myMesh.wear. This scalar contains the local wear height at each mesh element.

Alternatively for backward compatibility the access via data position in square brackets is also possible. The position of the data in the output depends on the additional mesh modules that are used (in case of mesh/surface/stress/6dof there are 9 components for mesh/surface/stress, the output for the wear module comes afterwards). Therefore, for a mesh with id myMesh, the wear value can be accessed via id_myMesh[10].

See the table below for an overview of the available properties and how to access them.

Mesh module property	property name (dot access)	probable array position
wear volume or mass	wear	10
eroded volume	eroded_volume	11
eroded mass	eroded_mass	12

For the combined wear model the contributions of each wear mechanism are also added to this vector.

Be aware that using the keyword calculate sum with mesh_properties {wear} will calculate the sum of $h$ [m] (and not $V$ or $EM_{SS}$ ) over all mesh elements defining the solid object. By definition of $h$ , this summation grows without bounds in the limit of mesh refinement and therefore its use is not recommended. If one wishes to analyse the total worn volume or eroded mass, one can find these quantities in the simulation_data_aspherix.csv file, as explained earlier. Alternatively, it is also possible to define a variable storing $V$ or $EM_{SS}$ and output its values to a user defined file, which is illustrated below for $V$ .

calculate sum id totalWearVolume meshes {mesh_user} mesh_properties {wear} property_weights {area}
write_to_file string 'id_time id_totalWearVolume[1]' file post/wear_data.txt title 'time total_wear_volume'

Using the modify_command, the wear can be reset via the wear/reset_wear option (old_style must be set to ‘yes’).

References

[1] Finnie, Iain. Erosion of surfaces by solid particles. Wear 3.2 (1960): 87-103.

[2] Archard, JeFoa. Contact and rubbing of flat surfaces. Journal of applied physics (1953): 24(8), 981-988.

[3] Roessler, T. and Katterfeld, A. Calibrated and Validated Wear Prediction for Bulk Material Handling Equipment using DEM Simulations. ICBMH2023 - The 14th International Conference on Bulk Materials Storage, Handling and Transportation 11-13th July, 2023, Wollongong, New South Wales, Australia.