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A new mechanistic model for abrasive erosion using discrete element method
Journal Pre-proof A new mechanistic model for abrasive erosion using discrete element method
Yunshan Dong, Fengqi Si, Wei Jin, Yue Cao, Shaojun Ren PII:
S0032-5910(20)31070-6
DOI:
https://doi.org/10.1016/j.powtec.2020.11.017
Reference:
PTEC 15994
To appear in:
Powder Technology
Received date:
17 April 2020
Revised date:
29 September 2020
Accepted date:
11 November 2020
Please cite this article as: Y. Dong, F. Si, W. Jin, et al., A new mechanistic model for abrasive erosion using discrete element method, Powder Technology (2020), https://doi.org/10.1016/j.powtec.2020.11.017
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© 2020 Published by Elsevier.
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A new mechanistic model for abrasive erosion using discrete element method Yunshan Dong, Fengqi Si*, Wei Jin, Yue Cao, Shaojun Ren Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of
Corresponding author
School of Energy & Environment, Southeast University, No.2 Sipailou
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Address
Prof. Fengqi Si
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*
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Energy and Environment, Southeast University, Nanjing 210096, China.
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Road, Nanjing, 210096, China +86 25 83794521
E-mail address
[email protected]
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Tel number
Declarations of interest
None
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Abstract
In this investigation an approach to abrasion erosion prediction is developed with consideration of cutting and deformation mechanisms using the discrete element method (DEM). Erosion rate is divided into two parts, cutting and deformation, which are related to the indentation sizes that are
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predicted by DEM. Empirical coefficients, of course, are introduced to establish the relevance between erosion rate and indentation volume. The proposed model is validated against experiments
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and results show the model accurately predicts abrasive erosion, especially for the maximum
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erosion angle. Furthermore, empirical coefficients are deemed to have more associations with ~
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hardness ratio H , and their relevance and influence on erosion are discussed. Cutting removal has
~
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a 2.0~2.7 power-law relation with particle velocity and deformation damage removal has a 2.8 ~
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power-law relation for H <0.15, decreasing for H >0.15. Finally, the universal correlation with
application.
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empirical coefficients is integrated to the proposed model, which makes advances in the industrial
Keyword: Abrasive erosion; Discrete element method; Mechanistic model; Particle impact
1. Introduction
Abrasive erosion due to repetitive impacts of solid particles has been recognized as one form of material degradation in many industries, such as power plant, petroleum and aerospace [1–4]. The erosion damage inevitably reduces the service life of industrial equipments, increases the risk of equipment failure, and may even result in great loss to the industries. So, it is paramount to quantify the erosion and distinguish the location which is most at risk. Prediction of erosion with
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high accuracy is vital to prevent production problems and maintain the operational security, even help in designing the equipment with minimum erosion damage. In order to predict the volume of material removal, efforts of developing analytical erosion models go half a century back and many erosion equations were derived under different assumptions and methodologies. The earliest work of this research direction is from Finnie [5]. In
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his pioneering work, the treatment of erosion is valid only at low incidence angles and fails in predicting the erosion at normal incidence by treating as deformation erosion, which is the primary
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erosion mechanism at this angle, as a completely different physical phenomenon from cutting
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erosion. Bitter [6,7] took deformation erosion to be significant at high angles. However,
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experimental validations for two distinct types of erosion were poor and Bitter model presented
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little justification for his assumptions. Later, Hutchings [8] introduced the mechanism of low-cycle
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fatigue to determine the effect on the erosion mechanism during oblique impact. Many later erosion models, such as Huang et al. [9], Arabnejad et al. [10], Ben-Ami et al. [11] and Uzi et al. [12,13]
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adopted these two forms of erosion mechanisms. Due to the immature mechanism, almost all of the erosion models incorporate empirical coefficients which need to be calibrated in direct measurement. Huang et al. [9] derived the cutting and deformation erosion with empirical coefficients, which are related to hardness and ductility, into the tangential and normal velocity effect. However, its incapability is to predict a change in the maximum erosion angle [11]. Arabnejad et al. [10] combined Finnie's cutting model and Bitter's deformation model together with empirical coefficients and find the relation of these coefficients to material hardness. Ben-Ami et al. [11] found the empirical coefficients are the function of target material hardness from the impact experiment of same particle. The aforementioned models neglect the particle hardness, not giving
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the more general regularity of coefficients. In fact, the coefficients are related to many other parameters except the target hardness and cause the inapplicability for some cases. For instance, many pre-existing analytical erosion models are inappropriate for the erosion of wide screening particles, which may cause the difference between the prediction and the experiment in Ref, [14]. Uzi et al. [12,13] investigated the energy dissipation model in particle impact and found the
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material removal is related with the tangential kinetic energy loss. In their study, the empirical coefficients changed with the particle sizes, which was appropriate for simulation of multi-scale
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particles. Besides, Zhao et al. [15] compared the shear impact energy to the erosion energy, found
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that the ratio of the shear impact energy to the erosion energy changed with the incident angle.
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Ismail et al. [16] applied the finite element to calculate the deformation erosion and found the
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removal.
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empirical coefficient varies with particle when studying the effect of particle size on the material
As mentioned above, the empirical coefficients are associated with many parameters, such as
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the particle size, the target hardness and even more unknown parameters in the pre-existing analytical erosion models. It leads to the inapplicability of the analytical models when changing some operation boundaries. A fully-empirical model [17–19] can be applied to some erosion cases, but the number of empirical coefficients is much more than the analytical models and it is hard to find the patterns of empirical coefficients. Thus, a comprehensive erosion model should guarantee that the empirical coefficients are related with less material properties and operation condition. It can ensure that the proposed model is applicative for more situations, such as the particle size changing, and even different impact case. DEM is a way to solve the problem. In the work of Zhao et al. [15] DEM is applied to simulate the erosion energy, but the varied coefficients show that the
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energy dissipation cannot ensure the commonality of the erosion model for the different angle. Then, the volume method may be another way to eliminate this difficulty like Ref. [9]. In using DEM, the volume method integrates more influential factors into the indentation sizes, rather than the empirical coefficients, ensuring the adaptability of erosion model. In this paper, we focus on developing a new mechanistic model for abrasive erosion using
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DEM. The newly developed model investigates the mechanisms of cutting removal and deformation damage removal. Both two removals are concerned with unknown indentation sizes
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that are calculated by solving the simultaneous equations of micro-particle impact. Furthermore,
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empirical coefficients are still introduced to establish the relevance between the erosion rate and
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indentation volume. The relevance is validated through sufficient experimental data. Finally, these
adaptability.
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2. Mechanistic model
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empirical coefficients are discussed, and developed to be the universal correlations and verify their
As mentioned before, it is common to attribute the abrasive erosion to two mechanisms. Fig.1 shows the indentation geometry for a single impact. Generally, cutting removal is the main mechanism of erosion at low impingement angles, as the material can be squeezed ahead and to the sides to form ridges at low particle velocities or directly cut at high particle velocities [20]. The ridges can be further chipped away from the surface after subsequent impacts. Deformation damage removal is the main mechanism of erosion at high impingement angles. This mechanism attributes to a brittle behavior, whereas more vulnerability at low impact angles is conventionally referred as ductile behavior. Numerous experiments [21,22] show that brittle materials can neglect the cutting
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removal, whereas ductile materials need consider these two types of removals. Therefore, the total removal of the ductile material is as follows: Q QC QD
(1)
2.1 Cutting removal
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Generally, the cutting material removal is proportional to the volume swept by the particle, as follows:
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QC wmax y max Leff
(2)
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The effective length Leff, rather than the maximum length, is introduced to the effective volume
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of indentation. It is noted that because of the elastic recovery, the indentation swept by the particle
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is equivalent to the crater volume. Owing to these complexities, only a part of the volume is taken
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away from the target surface. For the harder target eroded by the same particle, the effective length of the indentation is larger. The formalization of effective length in Ref. [9] is applied to Eq.(2), as
Leff
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follows: m pU x20 n n wmax y max
(3)
In Eq.(2), QC represents the effective volume of cutting. The mechanism of effective length is so complicated that the empirical correlation of Leff is presented. In fact, the exponent should be 0.5~1.0, which will be examined below. At low impingement angles, it is believed that the particle rotation cuts down the cutting removal [11]. Hence, a modified function is integrated to Eq.(2), as follows: f 1 - e -200
2
(4)
Integrating Eqs.(2) ~ (4), the final cutting removal yields: 6
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(5)
In Eq.(5), the coefficient cC is associated with the material properties.
2.2 Deformation damage removal
At low (<~10 m/s) and high impact velocities (~100 m/s), the erosion is due to the plastic
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deformation. While at even higher impact velocity (~1000 m/s), the erosion comes from melt-driven erosion [23]. The field of present study lies in the low and higher impact velocities, and the low
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cycle fatigue model is taken as a comprehensive method to solve the deformation damage removal.
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The low cycle fatigue model regards that the crack propagation is from the multiple impacts, and is
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the primary cause for the deformation damage removal, as shown in Fig. 2. Since the erosion
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process involves an accumulation of plastic deformation in the surface layers of the target, it is
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proposed that a suitable criterion may be a function of the fatigue strength. The fatigue strength is a property of the material and may be thought of as a measure of its ductility under erosion conditions.
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Coffin-Manson failure criterion is adopted to determine the removal of surface material. Coffin-Manson criterion is expressed as [8]: c 2 p N bf
(6)
where b is generally in range of 0.5~0.7, and assumed to be 0.5 in this paper. Besides, the strain magnitude is a function of the maximum depth of the indentation and the effective particle size, as follows: p y max d eff
(7)
where deff is concerned with the particle shape, e.g. deff=βdp. Substituting Eq.(7) into Eq.(6), Nf is defined as follows:
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b2
(8)
According to the failure criterion, an elementary removed volume for Nf impacts equals the indentation volume for a single normal impact, which is equal to: QE
ymax
0
(9)
Ay dy
where Ay is defined as: Ay d eff y
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(10)
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Substituting Eq.(10) into Eq.(9), the elementary volume is as follows: (11)
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2 QE d eff ymax
Therefore, the amount of deformation damage material removal at a single impact is: QE
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QD
Nf
(12)
1b 2 2b 2 d eff y max c1 b
(13)
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Q D c D
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Substituting Eq.(8) and Eq.(11) into Eq.(12), QD is expressed as follows:
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The deformation erosion factor cD is parallel to the cutting erosion factor cC. It is associated with the material properties. By neglecting mutual effect of two mechanisms of erosion, the total erosion rate consists of two parts: - n 1- n E r c~C f w1max y maxU x20 c~D
where
E r Q m p
1b 2 2b 2 d eff y max c1 b
(14)
, c~C c C , and c~D c D m p c1 b . The erosion rate is expressed in terms of volume
loss to the mass of erodent particle. It means that the volume loss is identical under the same condition of particle impact.
2.3 micro-particle impact
Section 2.1 and Section 2.2 have shown the mechanisms of the cutting removal and the 8
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deformation damage removal in the abrasive erosion. However, the indentation sizes wmax and ymax in Eq.(14) are unknown. These two sizes are concerned with the model of particle impact. Therefore, a model of micro-particle impact is introduced to discretely solve wmax and ymax by DEM simulations. The adhesive and elastoplastic behaviors have been integrated into the impact process. The loading phase has been divided into perfectly-elastic region, elastoplastic region and fully plastic region. In
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this paper, the fully plastic region is incorporated into elastoplastic region. The unloading phase is regarded as the release of elasticity. The model has demonstrated a high degree of accuracy than
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This gives force-displacement relationship as follows:
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other model in Ref. [24] . The process of DEM simulation has been referred in other papers[11,25].
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3 4E w 3 12 if n y in loading or unloading under el astic loading 3R 8E w n y 4E 3 w sech e p , full y 3R n y 12 w nM p 0 Fn 1 - sech 8E w 3 if n y in loading nM 2 p , full y w p , full nM m y 4E 3 1 - sech m y 2 p 0 wm w sech e p , full y w nM - 2 p , full y 3R p , full 2 w w w 12 8E w 3 sin 1 1 if unloading under el astoplastic loading w m wm wm
(15)
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where E* = EpEtar/(Ep+Etar), R* = 0.5deff for particle-wall impact. So as it can be imagine, the angular particles bring about more material removal, whereas the smooth particles cause the less removal. In addition, Ref. [24] gives the value of the empirical coefficient of Eq. (15). In Eq.(15), the corresponding contact radius is defined as: we if y in loading or unloading under el astic loading n y y w 1 - sech n w w sech if y in loading p y e p y p m y y we 1 - sech m w if unloading under el astoplastic loading sech p y p y p
9
(16)
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where we and wp satisfy the following relationships: we2 2we R E
n
2 n
12
(17)
2we R E
12
w 2p
(18)
In Eq. (16), both critical parameters δy and δp are unknown, which have to find the quantity by the corresponding radius. The corresponding radius can be obtained as follows:
R
w p, full
2E w y
12
(19)
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2E wy
3 R p0 21 2 E
(20)
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py
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where both py and p0 are the physical property parameter. Substituting wy into Eq. (17), δy is
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obtained. Substituting wp,full into Eq. (19), δp is obtained.
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Equivalent yield strength Py is an important parameter to measure the change of contact process from elastic contact to elastic-plastic contact, which satisfies the relation as follows:
(21)
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p y min p y，p , p y，tar
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The contact pressure P0 for fully plastic deformation can be obtained from the hardness of the two contact objects, as follows: -1
2 2 p0 g 10 6 H p H tar
(22)
The advantage to adopt DEM simulations is adding more influential factors to the indentation sizes. On the other hand, the empirical coefficients n, c~C and c~D are independent of other influential factors, but correlated with individual material properties. For example, wmax and ymax integrate the information of adhesion in Eq.(14), elastoplasticity, and even particle shape. Therefore, the empirical coefficients may be independent of these three influential factors. The present study hopes these empirical coefficients are concerned with only one influential factor. That is, for the
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given particle and target, the empirical coefficients remain const for any impacts. To extend simulations to the particle dynamics, we use the discrete-element method, implemented here in software MATLAB, which have the high efficiency. The particle impact process satisfies Newton's second law of motion, so the corresponding solutions are as follows: vi 1 vi
Fn t mp
Fn t 2 mp
(24)
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of
i 1 i vi t
(23)
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3. Relevance validations
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The complication associated with measurements compels the validations of the present model
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to be done, where sufficient experimental data is available for the erosion rate. The common
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experimental set refers to Fig.3. Compressed air carries the solid particle to strike the target surface. The particle velocity is measured through CCD camera. Various experimental results with different
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materials are collected from the literature and the relevant material properties are provided for the particles in Table 1 and for the target materials in Table 2. Note that these properties are also the ones that are used in sections of validations and results. Besides, the relevance validation focus on the applicability of Eq.(14), giving the relevance between the erosion rate and indentation volume rather than a complete expression including the empirical coefficients. In this section, the empirical coefficients n, c~C and c~D are obtained by fitting the experimental data. The fitting results are demonstrated in Table 3.
3.1 Indentation geometry
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Before verifying the relevance between the erosion rate and indentation volume, it need to validate the accuracy of indentation geometry from Eqs.(15)~(18). Since there are some tests measuring the indentation profile for the abrasive mass flow rate [29], no experiments measure the indentation geometry for a single impact. Whether for the static experiment or for the dynamic process, the indentation geometry depends on Eq.(16). Therefore, a static experiment could be used
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to substitute the dynamic impacting process. It doesn't affect the conclusion. The experiment studies the flattening of an aluminum sphere by an approximately rigid flat silicon carbide surface in Fig. 4.
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An optical interference microscope is used to measure the width of the indentation. Due to the
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limitations of the experiment, only indentation width is measurable as a comparison. In Fig. 5, the
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width is regarded as a function of the normal load. In the case, the present model of micro-particle
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impact is shown to agree strongly with the experiment data. As the data is compared to the other
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models, a different trend is observed: a model that shows poor agreement with the measurements is Du et al. [30] showing the most discrepancy between the predictions and the measurements. The
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remaining models by Chang et al. [31] and Jamari et al. [26] show agreement with the experiment, but show significant discrepancies for the relationship between the width and the normal contact force. This step ensures the indentation sizes are accurate and further applied to the solution of material removal reasonably.
3.2 Erosion rate
As mentioned before, the existing models predict a discrepancy in the maximum erosion angle. Fig. 6 shows the volume loss with Finnie’s experiment data [7] and corresponding values from the present model. The experiments were conducted using 250 μm angular silicon carbide particles
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impacting aluminum, copper and SAE-1055 stainless steel target materials. The impact velocity of abrasive particles is set as 107 m/s. The average error is below 5%, and the fair agreement is observed between the model predictions and experiments. In these three cases, the advantage is presented by introducing the exponent n, which governs the maximum erosion angle. It can be seen that the smaller the exponent n is, the larger the maximum erosion angle will be. It is noted that the
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hardest material is SAE-1055 stainless steel, and the softest material is aluminum. It seems that the exponent n is related to the target material hardness.
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At small impingement angles, abrasive erosion is dominated by the cutting removal. However
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at the large impingement angle, abrasive erosion is dominated by the deformation damage removal.
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In Fig. 5, the open circle represents that the cutting removal equals the deformation damage
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removal. It is believed that the smaller the corresponding angle in the open circle is, the earlier the
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impact goes through the dominated region of the deformation damage removal. The corresponding angles in the open circle are 68 [deg], 64 [deg] and 62 [deg] for aluminum, copper and SAE-1055
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stainless steel respectively. It seems that the trend is also related to the target material hardness. For the larger hardness, the erosion is more likely to be dominated by the deformation damage removal. Besides, it is generally believed that the brittle materials have a larger hardness. This explains why the brittle materials can neglect the cutting removal. In another comparison, model predictions are compared to experimental data of lead, aluminum, iron and grey cast iron eroded with 326 μm semi-rounded quartz particles at the impact velocity of 100 m/s [27]. Fig. 7 presents the erosion rate as a function of impingement angle. The average error is below 6%. Both the experimental data and the model predictions show very good agreement. It is important to note that lead has a smallest hardness, and its maximum erosion angle
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is the minimum. While grey cast iron has a largest hardness, and its maximum erosion angle is maximum. Similarly, this proved the relationship between the maximum erosion angle and the target material hardness. In Fig. 7, the dashed line is the model prediction of lead without small angle effect. The results have an average 40% error than measured at low impingement angle. When considering the effect of the low impingement angle, these discrepancies will be eliminated. These
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two cases show the present model has good relevance between the erosion rate and indentation
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volume.
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4. Results and discussion
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Various experimental results with different materials are collected from the literature and the
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corresponding coefficients of the erosion model are provided in Table 3. As stated before, the
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exponent n is related with the target material hardness and each of previous experiments has the same abrasive particles individually in Fig. 6 and Fig. 7. However, almost experiments have
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different particles impacting different target materials. Therefore, the hardness ratio is introduced to discuss its effect on the exponent n below, and the parameter contains the information of particle hardness and target material hardness. The hardness ratio is defined as: H ~ H tar Hp
(25)
Generally, when the hardness ratio stays below a certain critical value, pure abrasive erosion takes place. The critical value of hardness ratio will be discussed below. In addition, the relationships between other empirical coefficients c~C and c~D and the hardness ratio are also discussed below.
4.1 Impact velocity effect 14
Journal Pre-proof In order to validate that the coefficients n, c~C and c~D are only concerned with the material properties, a set of experiments are conducted by varying the impact velocity and the impingement angle. It means that c~C and c~D should be verified to be independent of the impact conditions. Fig. 8 shows the erosion rate as a function of impact velocity and impingement angle for the 150 μm semi-rounded sand impacting carbon steel 1018 plate. The impact velocities are set as 9 m/s, 18 m/s and 28 m/s. It is important to note that the model is developed with experimental data for the impact
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velocity of 28 m/s, and three estimated coefficients n, c~C and c~D are applied to the erosion rate for
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the impact velocity of 9 m/s and 18 m/s. In three cases, n= 0.55, c~C = 11.8 and c~D = 4.2·106. Very
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good agreement is observed between the measured results and the present model for all velocities.
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Since the model predictions for 18 m/s have an error than measured, the error can be controlled in
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the allowable range. This represents that the factors are independent of the impact velocity, and are
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only concerned with material properties. Besides, the black dashed lines represent the maximum erosion angle. It is found that the trend of the maximum erosion angle can be predicted for different
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impact velocities. The larger the impact velocity is, the larger the maximum erosion angle will be. The model shows good agreement with the trend. In second set of experiments, the model predictions are compared to experimental data of aluminum alloy 6061 eroded with 150 μm semi-rounded sand. In Fig. 9, the impact velocities are set as the same velocities in the last experiments. Similarly, the model is developed with experimental data for the impact velocity of 28 m/s. In three cases, n= 0.81, c~C = 0.29 and c~D = 1.74·106. The predictions of the present model show agreement with the experimental data. It indicates that the estimated coefficients n, c~C and c~D are only concerned with the material properties as well.
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4.2 Hardness ratio effect
Since the coefficients n, c~C and c~D have been demonstrated to be relevant to the material properties, we hope the coefficients is related with very few properties, including material properties and impact conditions. In this paper, impact conditions are verified to have no effect on the coefficients in Fig. 8 and Fig. 9, and the material properties are supposed to only be the hardness
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ratio. Although other material properties affect the coefficients, their influences can be neglected. ~
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Fig. 10 shows the relationship between the hardness ratio and the empirical exponent n. For H
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<0.15, the linear trend line agrees with the exponent n of the present models. The R- square of the
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fitting in Fig. 10 is 0.963. It can be seen that the larger the hardness ratio is, the smaller the
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exponent n is. The maximum erosion angle, combined with the analysis in Section 3.2, becomes ~
larger as the hardness ratio increases. However, for H >0.15, the value of exponent n stabilizes 0.5.
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It may be due to that the current status is not in pure erosion, rather than mixed erosion and
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breakage. The hardness ratio demonstrates the gap between the target material and the particle: the larger the hardness ratio is, the smaller the gap is. Once the gap is relatively small, the particle will be broken [32]. Then a part of the absorbed energy used for erosion is applied to the particle breakage. The range of exponent n is 0.5~1.0, which has been identified in Ref. [9]. Fig. 11 presents the relationship between the hardness ratio and the cutting erosion factor c~C in the double logarithm coordinate. The solid line is a trend line. The red dashed line is the upper boundary and the blue dashed line is the lower boundary. The boundaries may be due to the measuring error or the target roughness. However, these do not affect the objective law. The cutting erosion factor c~C has a linear relation with the hardness ratio in the double logarithm coordinate. In Fig. 11, the R-square of the fitting is 0.927. It is believed that the cutting erosion factor c~C is 16
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dominated by the hardness ratio, and the influences of other material properties can be neglected. It can be seen that the cutting erosion factor c~C rises exponentially with the hardness ratio increased in the Cartesian coordinate. Fig. 12 presents the relationship between the hardness ratio and the deformation erosion factor c~D .
The trend line and the boundaries are indicated by the solid line and the dashed lines. The
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deformation erosion factor c~D rises with the hardness ratio because it is proportional to the material brittleness. It is important to note that it is more vulnerable to crack propagation with larger
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hardness ratio. Therefore, the deformation mechanism is significant for the impact of the larger
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hardness ratio. This phenomenon can be found in Fig. 6 and Fig. 7. In Fig. 12, the R-square of the
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fitting is 0.958.
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4.3 Velocity exponents
It is generally believed that the erosion rate is proportional to the power-law relation with U 0fC
, and QD
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particle velocity, i.e QC
U 0f D
. Previously, it has been found that fC and fD have
no concern with the impingement angle, and fC and fD are constant for the same experiment. Fig. 13 shows the relationship between the hardness ratio and the velocity exponents fC and fD. It can be ~
seen that fC is in the range of 2.0~2.7. Ref.[9] is almost unanimous support for the results. For H ~
<0.15, the trend shows a linear growth, while for H >0.15, fC remains the same. fD is nearly ~
~
constant at 2.8 in the range of H <0.15, while decreasing in the range of H >0.15. It may be due to the particle breakage. Unless it never occurs the particle breakage, fC and fD maintain the original trend of growth, and exceed the present values. It means a part of absorbed energy is applied to the particle breakage, rather than pure erosion.
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4.4 Empirical model and adaptability
The relationships between empirical coefficients and hardness ratio have been found in the previous content. Thus, the mechanistic model can be developed through the relationships. Eqs.(21)~(23) give the expressions of empirical coefficients. In these expressions, c~C and c~D give ~
an estimated range in which the upper and the lower boundaries refer to the fitting range. For H
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<0.15, the model adaptability is discussed in the next part.
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~ n -3.24 H 0.98
-p
~ c~C 10 3.19 lg(H )b1
(21) (22)
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~ c~D 2.68 107 H b2
(20)
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where, b1 is in the range of 3.9 ~ 5.0, b2 is in the range of -4105~ 106. Fig. 14 shows the model adaptability for different materials. It can be seen that the present model can estimate the erosion
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rate. The visible error is related with the estimated range of empirical coefficients. It is definitely
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worth that the new model integrated the empirical correlations shows the same order of erosion rate, which makes progress and assists the industrial design in erosion.
5. Conclusions
In the current study, a new phenomenological erosion model was developed based on some studies in the literature to calculate the erosion of various ductile materials. The model is composed of two parts, cutting and deformation with two indentation sizes and three empirical coefficients. An accurate impact model is introduced to predict indentation sizes by DEM simulations and empirical coefficients are obtained by fitting experimental data for different materials. The introduction of DEM guarantees that the empirical coefficients only depend on hardness ratio, while other 18
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influential factors are incorporated into the indentation sizes. Moreover, this paper validates the relationship and summarizes the empirical equations of empirical coefficients. The following ~
conclusions on empirical coefficients may be drawn: For H <0.15, the larger the hardness ratio is, the smaller the exponent n is. However, empirical coefficients c~C and c~D rise with the hardness ~
ratio increased. For H >0.15, because of the mixed erosion and breakage, n keeps at the value of
of
0.5. In addition, cutting removal has an approximate 2.0~2.7 power-law relation with particle velocity which is consistent of experimental findings. Deformation damage removal has a 2.8 ~
-p
~ H >0.15.
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power-law relation with particle velocity for H <0.15, while the velocity exponent decreases for
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Furthermore, since empirical coefficients n, c~C and c~D are defined as the functions of the
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hardness ratio, this paper has given the empirical correlations. If the new model integrated the
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empirical correlations, a visible error exists between the new model and experiments. It may be due to that limited experimental data can be provided, but does not affect the conclusion of this paper. It
Acknowledgements
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is definitely worth that the new model makes progress and helps the industrial design in erosion.
The authors appreciate the financial supports provided by the National Natural Science Foundation of China (Grant No. 51976031) and Qinglan Project of Jiangsu Province of China.
Nomenclature
Symbol Description
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empirical exponent
c
empirical coefficient
d
particle size (m)
f
velocity exponent
m
mass (kg)
n
empirical exponent
nM
Meyer hardness exponent
p0
uniform pressure at the start of fully plastic loading stage (Pa)
py
yield strain (Pa)
vi
particle velocity at the ith moment
w
indentation width (m)
we
indentation width under elastic deformation (m)
wp,
indentation width under plastic deformation (m)
wp,full
indentation width at the start of fully plastic deformation (m)
wy
indentation width at yield point (m)
y
indentation depth (m)
Ay
projected contact area in the normal direction (m2)
E
Young's modulus (GPa)
E*
equivalent Young's modulus (GPa)
Er
erosion rate (m3/kg)
Fn
normal force (N)
H
hardness (kgf/mm2)
Jo ur
na
lP
re
-p
ro
of
b
20
Journal Pre-proof
indentation length (m)
Nf
mean number of impacts needed to cause the material removal
R*
equivalent radius (m)
U0
initial velocity of the particle (m/s)
Ux0
initial tangential velocity of the particle (m/s)
α
impingement angle (deg)
β
particle sharpness factor
Γ
interface energy (J/m2)
δi
displacement at the ith moment
δm
maximum displacement (m)
δn
normal displacement (m)
δp
limited displacement at the start of fully plastic loading (m)
δy
limited displacement at the end of the elastic loading (m)
εc
fatigue strength (MPa)
ξ
empirical coefficient
ψ
empirical coefficient
△Q
volume loss (m3)
△QC
cutting material removal (m3)
△QD
deformation damage material removal (m3)
△QE
elementary volume of the material removal after Nf normal impacts (m3)
△εp
strain magnitude (MPa)
c~
erosion factor
Jo ur
na
lP
re
-p
ro
of
L
21
Journal Pre-proof ~ H
hardness ratio
Subscript Description effective
max
maximum
p
particle
tar
target surface
C
cutting
D
deformation
-p
ro
of
eff
lP
[1]
re
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na
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impact integrated adhesive, elastoplastic and microslip behaviors, Adv. Powder Technol. 31 (2020) In Press. doi:10.1016/j.apt.2020.08.003.
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damage caused by solid particle impact, Wear. 203–204 (1997) 573–579.
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doi:10.1016/S0043-1648(96)07430-3. M.R.W. Brake, An analytical elastic plastic contact model with strain hardening and frictional effects for normal and oblique impacts, Int. J. Solids Struct. 62 (2015) 104–123. doi:10.1016/j.ijsolstr.2015.02.018. [29]
V. Heuer, G. Walter, I.M. Hutchings, A study of the erosive wear of fibrous ceramic components by solid particle impact, Wear. 225–229 (1999) 493–501. doi:10.1016/S0043-1648(98)00373-1.
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V. Hadavi, C.E. Moreno, M. Papini, Numerical and experimental analysis of particle fracture during solid particle erosion, Part II: Effect of incident angle, velocity and abrasive size, Wear. 356–357 (2016) 146–157. doi:10.1016/j.wear.2016.03.009. S. Yerramareddy, S. Bahadur, Effect of operational variables, microstructure and mechanical
of
[33]
properties on the erosion of Ti-6Al-4V, Wear. 142 (1991) 253–263.
-p
J. Malik, I.H. Toor, W.H. Ahmed, Z.M. Gasem, M.A. Habib, R. Ben-Mansour, H.M. Badr,
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Evaluating the effect of hardness on erosion characteristics of aluminum and steels, J. Mater.
na
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Eng. Perform. 23 (2014) 2274–2282. doi:10.1007/s11665-014-1004-x.
Jo ur
[34]
ro
doi:10.1016/0043-1648(91)90168-T.
26
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Credit author statement Yunshan Dong: Investigation, Methodology, Writing - original draft. Fengqi Si: Supervision, Conceptualization, Funding acquisition. Wei Jin: Investigation, Validation. Yue Cao: Validation, Writing - original draft.
Jo ur
na
lP
re
-p
ro
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Shaojun Ren: Validation.
27
Journal Pre-proof Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Jo ur
na
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-p
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
28
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Figures
ax
α
ym
ax
wm
Le
Lm
ro
geometry for a single impact
-p
Fig. 1 Indentation
ax
of
ff
lP
re
y
na
crack
Jo ur
Fig. 2 Sketch of crack propagation
Compressor
Feeder Cylinderical lens
Mirror
Pulsed laser
Feeder
Synchronizer
Nozzle Target
Computer
CCD Camera
Fig. 3 Experimental set of abrasive erosion
29
Journal Pre-proof
P
of
Fig. 4 Hard flat indenting a deformable sphere contradicts
ro -p re
1.0
0.5
0
Experiment [26] Chang et al. Du et al. Jamari et al. Prediction
100 200 Contact force,Fn [N]
300
na
0.0
lP
Indentation width,w [m]
1.5
5
68
0
0 30 60 90 Impingement Angle, [deg]
(a)
Experiment [7] Model Component_x Component_y
10
5 64
0
15
3
3
10
15
Volumn Loss, Q [mm /53gSiC]
Experiment [7] Model Component_x Component_y
Volumn Loss, Q [mm /53gSiC]
3
Volumn Loss, Q [mm /23gSiC]
15
Jo ur
Fig. 5 Indentation width as a function of the normal load of aluminum sphere.
0 30 60 90 Impingement Angle, [deg]
(b)
Experiment [7] Model Component_x Component_y
10
5 62
0
0 30 60 90 Impingement Angle, [deg]
(c)
Fig. 6 Erosion rate as a function of impingement angle. Target materials are: (a) aluminum, n=0.93 (b) copper, n=0.895
30
Journal Pre-proof (c) SAE-1055 stainless steel, n=0.675. Impingement velocity is 107m/s. Abrasive particle is 250 μm SiC.
-6
10
Erosion Rate,Er [m /kg]
1.5
lead-Experiment [27] aluminum-Experiment [27] Iron-Experiment [27] Grey cast iron-Experiment [27] lead-Model aluminum-Model Iron-Model Grey cast iron-Model
without small angle effect
3
1.0
0
30 60 Impingement Angle, [deg]
90
ro
0.0
of
0.5
-p
Fig. 7 Erosion rate as a function of impingement angle.
re lP
4
U=28m/s (Experiment [10]) U=18m/s (Experiment [10]) U=9m/s (Experiment [10]) U=28m/s (Model) U=18m/s (Model) U=9m/s (Model)
na
3
Erosion Rate,Er [mm /kg]
-9 6 10
Jo ur
2
0
0
30
60
Impingement Angle, [deg]
90
Fig. 8 Erosion rate of carbon steel 1018 at different impact velocities: n= 0.55, c~C = 11.8 and c~D = 4.2·106. -9
3
Erosion Rate,Er [mm /kg]
12
10
U=28m/s (Experiment [10]) U=18m/s (Experiment [10]) U=9m/s (Experiment [10]) U=28m/s (Model) U=18m/s (Model) U=9m/s (Model)
9 6 3 0
0
30 60 Impingement Angle, [deg]
31
90
Journal Pre-proof Fig. 9 Erosion rate of aluminum alloy 6061 at different impact velocities: n= 0.81, c~C = 0.29 and c~D = 1.74·106.
1.0 2
R =0.963
0.8
n
0.6 0.4 Model Trendline
0.0 0.0
0.3
ro
0.1 0.2 ~ Hardness ratio, H
of
0.2
-p
Fig. 10 Empirical exponent n as a function of hardness ratio.
re
1000 100
lP
10 1
~
cC
2
R =0.927
na
0.1
Model Trendline
Jo ur
0.01
1E-3 1E-3
0.01
0.1
1
10
~ Hardness ratio, H
Fig.11 Cutting erosion factor c~C as a function of hardness ratio. 6
10.0
10
D
2
R =0.958
5.0
c
~
7.5
2.5 Model Trendline
0.0 0.0
0.1
0.2 ~
Hardness ratio, H
32
0.3
Journal Pre-proof Fig.12 Deformation erosion factor c~D as a function of hardness ratio.
3.0
3.5
3.0
fD
fC
2.5
2.5
2.0 2.0
Model Trendline
0.05
0.10 0.15 0.20 ~ Hardness ratio, H
0.25
1.5 0.0
0.30
0.1 0.2 ~ Hardness ratio, H
of
1.5 0.00
Model Trendline
(b)
ro
(a)
0.3
-p
Fig. 13 Velocity exponents as a function of hardness ratio: (a) cutting velocity exponent fC. (b) deformation velocity
na
2
0
0
30 60 Impingement Angle, [deg]
3
4
Erosion Rate,Er [m /kg]
lP
Experiment [33] Prediction
Jo ur
3
Erosion Rate,Er [m /kg]
6
re
exponent fD
15 Experiment [34] Prediction
10
5
0
90
0
(a)
30 60 Impingement Angle, [deg]
90
(b)
Fig. 14 Predictions of erosion rate using estimated empirical coefficients: (a) Ti-6Al-4V alloy eroded with 125 μm angular SiC particle at velocity of 55 m/s [33]. (b) carbon steel 1020 eroded with 50 μm angular alumina particle at velocity of 60 m/s [34].
33
Journal Pre-proof
Tables Table 1 Particle mechanical properties. Young's Density
Particle
Poisson’s modulus
3
[kg/m ]
SiO2 [10,27]
3,200
a2
2,300
a2
ratio
a1
Adopted from Ref. [28].
a2
Adopted from Ref. [13].
0.17
Shape
[kgf/mm ]
0.345 0.192
strength [mm]
exponent
75.2
a2
410
a2
80
a1
26
a2
a2
2.15
2,500
a2
1,122
a2
2.2 2.2
[MPa] 40 a1
3
rounded
10,000
a2
0.25
angular
11,000
a2
0.15,0.326
sub-rounded
of
SiC [7]
2,750
hardness
Table 2 Target material properties.
ro
Al [26]
Yield Particle size
2
[GPa] a1
Meyer Hardness
Young's
Particle
Poisson’s
[kg/m3]
ratio
-p
Target
Density
modulus
Interface Hardness
Yield strength
[kgf/mm2]
[MPa]
2,485
10,000 b1
energy [J/m2]
Al
3,200
b1
Al [7]
SiC
2,700
Copper [7]
SiC
8,890 b1
SAE-1055 steel [7]
SiC
7,850 b2
Iron [27] Grey cast iron [27] Carbon steel 1018 [10] Al 6061 [10]
SiO2 SiO2 SiO2
11,400
na
Al 1050-H16 [27]
SiO2
0.35
2,700 7,870 7,500
Jo ur
Lead [27]
0.17
SiO2 SiO2
b1
7,870 2,700
430
0.35
31
372.4
0.4
117 b1
63
646.8
0.28
0.28 b2
200 b2
200
587.6
0.25
20
b4
5.5
b5
0.4
35
b6
124
b6
0.4
0.42
b3
0.33
b6
0.291
b7
0.29
b8
0.29
b1
0.33
b1
70
b1
0.33 b1
lP
SiC [26]
re
[GPa]
b1
14
b3
69
b6
200
b7
130
b4
205
b1
70
b1
153 306 131 31
b1
Adopted from Ref. [13].
b2
Adopted from www.iron-foundry.com/AISI-1055-SAE-UNS-G10550-Carbon-Steel-Foundry.html.
b3
Adopted from www.matweb.com/search/DataSheet.aspx?MatGUID=ebd6d2cdfdca4fc285885cc4749c36b1.
b4
Adopted from Ref. [11].
b5
Adopted from http://www.goodfellow.com/E/Lead.html.
b6
Adopted from www.matweb.com/search/DataSheet.aspx?MatGUID=f54812ceded24176b44c4e0af379ea39.
b7
Adopted from http://www.matweb.com/search/DataSheet.aspx?MatGUID=654ca9c358264b5392d43315d8535b7d.
b8
Adopted from www.vonroll-casting.ch/en/grey-cast-iron.html.
34
50
b7
260
3,700 55
0.3
b8
b1
b1
0.3 0.35 0.35
Journal Pre-proof Table 3
Model parameters. Model parameters calibration Reference erosion experiments
Hardness ratio
Reference
Particle
wall
~ H
[26]
Al
SiC
-
[7]
SiC
Al
0.012
Cutting
Deformation
Velocity
mechanism
mechanism
exponent
n
cC
-
-
cD -
fC
fD
-
-
0.93
6.9·10
-2
1.54·10
5
2.096
2.777
3.57·10
5
2.146
2.794
1.81·10
6
2.443
2.818
2.104
2.792
[7]
SiC
Copper
0.025
0.895
7.9·10
-2
[7]
SiC
SAE-1055 steel
0.08
0.675
7.9·10
-1
[27]
SiO2
Lead
0.0178
0.925
1.22·10
[27]
SiO2
Al 1050-H16
0.031
0.87
1.48·10-2
6.36·105
2.182
2.803
[27]
SiO2
Iron
0.136
0.575
2.5
3.15·106
2.595
2.806
28.9
7.08·10
11.8
4.2·10
0.81
35
5.9·10
of ro
0.0624
0.55
2.9·10
-p
Al 6061
0.117
0.5
re
SiO2
Carbon steel 1018
0.273
lP
[10]
SiO2
Grey cast iron
na
[10]
SiO2
Jo ur
[27]
-2
-1
5
6
2.663
2.545
6
2.632
2.812
1.74·10
2.274
2.792
6
Journal Pre-proof
Graphical Abstract -6
3
Erosion Rate,Er [m /kg]
1.5
10
lead-Experiment [27] aluminum-Experiment [27] Iron-Experiment [27] Grey cast iron-Experiment [27] lead-Model aluminum-Model Iron-Model Grey cast iron-Model
without small angle effect
1.0
0.5
0
30 60 Impingement Angle, [deg]
90
of
0.0
Jo ur
na
lP
re
-p
ro
Erosion rate predictions as a function of impingement angle
36
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Highlights
of ro -p
re
lP
na
A new mechanistic model of abrasive erosion using DEM was developed Erosion rate with coefficients was taken as a new function of indentation sizes Empirical coefficients were validated to be mainly related with the hardness ratio Relationships between hardness ratio, erosion and breakage were studied Velocity exponents for two erosion mechanisms were discussed
Jo ur
37
Yunshan Dong, Fengqi Si, Wei Jin, Yue Cao, Shaojun Ren PII:
S0032-5910(20)31070-6
DOI:
https://doi.org/10.1016/j.powtec.2020.11.017
Reference:
PTEC 15994
To appear in:
Powder Technology
Received date:
17 April 2020
Revised date:
29 September 2020
Accepted date:
11 November 2020
Please cite this article as: Y. Dong, F. Si, W. Jin, et al., A new mechanistic model for abrasive erosion using discrete element method, Powder Technology (2020), https://doi.org/10.1016/j.powtec.2020.11.017
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© 2020 Published by Elsevier.
Journal Pre-proof
A new mechanistic model for abrasive erosion using discrete element method Yunshan Dong, Fengqi Si*, Wei Jin, Yue Cao, Shaojun Ren Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of
Corresponding author
School of Energy & Environment, Southeast University, No.2 Sipailou
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Address
Prof. Fengqi Si
re
*
-p
ro
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Energy and Environment, Southeast University, Nanjing 210096, China.
na
Road, Nanjing, 210096, China +86 25 83794521
E-mail address
[email protected]
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Tel number
Declarations of interest
None
1
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Abstract
In this investigation an approach to abrasion erosion prediction is developed with consideration of cutting and deformation mechanisms using the discrete element method (DEM). Erosion rate is divided into two parts, cutting and deformation, which are related to the indentation sizes that are
of
predicted by DEM. Empirical coefficients, of course, are introduced to establish the relevance between erosion rate and indentation volume. The proposed model is validated against experiments
ro
and results show the model accurately predicts abrasive erosion, especially for the maximum
-p
erosion angle. Furthermore, empirical coefficients are deemed to have more associations with ~
re
hardness ratio H , and their relevance and influence on erosion are discussed. Cutting removal has
~
lP
a 2.0~2.7 power-law relation with particle velocity and deformation damage removal has a 2.8 ~
na
power-law relation for H <0.15, decreasing for H >0.15. Finally, the universal correlation with
application.
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empirical coefficients is integrated to the proposed model, which makes advances in the industrial
Keyword: Abrasive erosion; Discrete element method; Mechanistic model; Particle impact
1. Introduction
Abrasive erosion due to repetitive impacts of solid particles has been recognized as one form of material degradation in many industries, such as power plant, petroleum and aerospace [1–4]. The erosion damage inevitably reduces the service life of industrial equipments, increases the risk of equipment failure, and may even result in great loss to the industries. So, it is paramount to quantify the erosion and distinguish the location which is most at risk. Prediction of erosion with
2
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high accuracy is vital to prevent production problems and maintain the operational security, even help in designing the equipment with minimum erosion damage. In order to predict the volume of material removal, efforts of developing analytical erosion models go half a century back and many erosion equations were derived under different assumptions and methodologies. The earliest work of this research direction is from Finnie [5]. In
of
his pioneering work, the treatment of erosion is valid only at low incidence angles and fails in predicting the erosion at normal incidence by treating as deformation erosion, which is the primary
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erosion mechanism at this angle, as a completely different physical phenomenon from cutting
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erosion. Bitter [6,7] took deformation erosion to be significant at high angles. However,
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experimental validations for two distinct types of erosion were poor and Bitter model presented
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little justification for his assumptions. Later, Hutchings [8] introduced the mechanism of low-cycle
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fatigue to determine the effect on the erosion mechanism during oblique impact. Many later erosion models, such as Huang et al. [9], Arabnejad et al. [10], Ben-Ami et al. [11] and Uzi et al. [12,13]
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adopted these two forms of erosion mechanisms. Due to the immature mechanism, almost all of the erosion models incorporate empirical coefficients which need to be calibrated in direct measurement. Huang et al. [9] derived the cutting and deformation erosion with empirical coefficients, which are related to hardness and ductility, into the tangential and normal velocity effect. However, its incapability is to predict a change in the maximum erosion angle [11]. Arabnejad et al. [10] combined Finnie's cutting model and Bitter's deformation model together with empirical coefficients and find the relation of these coefficients to material hardness. Ben-Ami et al. [11] found the empirical coefficients are the function of target material hardness from the impact experiment of same particle. The aforementioned models neglect the particle hardness, not giving
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the more general regularity of coefficients. In fact, the coefficients are related to many other parameters except the target hardness and cause the inapplicability for some cases. For instance, many pre-existing analytical erosion models are inappropriate for the erosion of wide screening particles, which may cause the difference between the prediction and the experiment in Ref, [14]. Uzi et al. [12,13] investigated the energy dissipation model in particle impact and found the
of
material removal is related with the tangential kinetic energy loss. In their study, the empirical coefficients changed with the particle sizes, which was appropriate for simulation of multi-scale
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particles. Besides, Zhao et al. [15] compared the shear impact energy to the erosion energy, found
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that the ratio of the shear impact energy to the erosion energy changed with the incident angle.
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Ismail et al. [16] applied the finite element to calculate the deformation erosion and found the
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removal.
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empirical coefficient varies with particle when studying the effect of particle size on the material
As mentioned above, the empirical coefficients are associated with many parameters, such as
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the particle size, the target hardness and even more unknown parameters in the pre-existing analytical erosion models. It leads to the inapplicability of the analytical models when changing some operation boundaries. A fully-empirical model [17–19] can be applied to some erosion cases, but the number of empirical coefficients is much more than the analytical models and it is hard to find the patterns of empirical coefficients. Thus, a comprehensive erosion model should guarantee that the empirical coefficients are related with less material properties and operation condition. It can ensure that the proposed model is applicative for more situations, such as the particle size changing, and even different impact case. DEM is a way to solve the problem. In the work of Zhao et al. [15] DEM is applied to simulate the erosion energy, but the varied coefficients show that the
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energy dissipation cannot ensure the commonality of the erosion model for the different angle. Then, the volume method may be another way to eliminate this difficulty like Ref. [9]. In using DEM, the volume method integrates more influential factors into the indentation sizes, rather than the empirical coefficients, ensuring the adaptability of erosion model. In this paper, we focus on developing a new mechanistic model for abrasive erosion using
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DEM. The newly developed model investigates the mechanisms of cutting removal and deformation damage removal. Both two removals are concerned with unknown indentation sizes
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that are calculated by solving the simultaneous equations of micro-particle impact. Furthermore,
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empirical coefficients are still introduced to establish the relevance between the erosion rate and
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indentation volume. The relevance is validated through sufficient experimental data. Finally, these
adaptability.
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2. Mechanistic model
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empirical coefficients are discussed, and developed to be the universal correlations and verify their
As mentioned before, it is common to attribute the abrasive erosion to two mechanisms. Fig.1 shows the indentation geometry for a single impact. Generally, cutting removal is the main mechanism of erosion at low impingement angles, as the material can be squeezed ahead and to the sides to form ridges at low particle velocities or directly cut at high particle velocities [20]. The ridges can be further chipped away from the surface after subsequent impacts. Deformation damage removal is the main mechanism of erosion at high impingement angles. This mechanism attributes to a brittle behavior, whereas more vulnerability at low impact angles is conventionally referred as ductile behavior. Numerous experiments [21,22] show that brittle materials can neglect the cutting
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removal, whereas ductile materials need consider these two types of removals. Therefore, the total removal of the ductile material is as follows: Q QC QD
(1)
2.1 Cutting removal
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Generally, the cutting material removal is proportional to the volume swept by the particle, as follows:
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QC wmax y max Leff
(2)
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The effective length Leff, rather than the maximum length, is introduced to the effective volume
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of indentation. It is noted that because of the elastic recovery, the indentation swept by the particle
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is equivalent to the crater volume. Owing to these complexities, only a part of the volume is taken
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away from the target surface. For the harder target eroded by the same particle, the effective length of the indentation is larger. The formalization of effective length in Ref. [9] is applied to Eq.(2), as
Leff
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follows: m pU x20 n n wmax y max
(3)
In Eq.(2), QC represents the effective volume of cutting. The mechanism of effective length is so complicated that the empirical correlation of Leff is presented. In fact, the exponent should be 0.5~1.0, which will be examined below. At low impingement angles, it is believed that the particle rotation cuts down the cutting removal [11]. Hence, a modified function is integrated to Eq.(2), as follows: f 1 - e -200
2
(4)
Integrating Eqs.(2) ~ (4), the final cutting removal yields: 6
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(5)
In Eq.(5), the coefficient cC is associated with the material properties.
2.2 Deformation damage removal
At low (<~10 m/s) and high impact velocities (~100 m/s), the erosion is due to the plastic
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deformation. While at even higher impact velocity (~1000 m/s), the erosion comes from melt-driven erosion [23]. The field of present study lies in the low and higher impact velocities, and the low
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cycle fatigue model is taken as a comprehensive method to solve the deformation damage removal.
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The low cycle fatigue model regards that the crack propagation is from the multiple impacts, and is
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the primary cause for the deformation damage removal, as shown in Fig. 2. Since the erosion
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process involves an accumulation of plastic deformation in the surface layers of the target, it is
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proposed that a suitable criterion may be a function of the fatigue strength. The fatigue strength is a property of the material and may be thought of as a measure of its ductility under erosion conditions.
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Coffin-Manson failure criterion is adopted to determine the removal of surface material. Coffin-Manson criterion is expressed as [8]: c 2 p N bf
(6)
where b is generally in range of 0.5~0.7, and assumed to be 0.5 in this paper. Besides, the strain magnitude is a function of the maximum depth of the indentation and the effective particle size, as follows: p y max d eff
(7)
where deff is concerned with the particle shape, e.g. deff=βdp. Substituting Eq.(7) into Eq.(6), Nf is defined as follows:
7
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b2
(8)
According to the failure criterion, an elementary removed volume for Nf impacts equals the indentation volume for a single normal impact, which is equal to: QE
ymax
0
(9)
Ay dy
where Ay is defined as: Ay d eff y
of
(10)
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Substituting Eq.(10) into Eq.(9), the elementary volume is as follows: (11)
-p
2 QE d eff ymax
Therefore, the amount of deformation damage material removal at a single impact is: QE
re
QD
Nf
(12)
1b 2 2b 2 d eff y max c1 b
(13)
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Q D c D
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Substituting Eq.(8) and Eq.(11) into Eq.(12), QD is expressed as follows:
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The deformation erosion factor cD is parallel to the cutting erosion factor cC. It is associated with the material properties. By neglecting mutual effect of two mechanisms of erosion, the total erosion rate consists of two parts: - n 1- n E r c~C f w1max y maxU x20 c~D
where
E r Q m p
1b 2 2b 2 d eff y max c1 b
(14)
, c~C c C , and c~D c D m p c1 b . The erosion rate is expressed in terms of volume
loss to the mass of erodent particle. It means that the volume loss is identical under the same condition of particle impact.
2.3 micro-particle impact
Section 2.1 and Section 2.2 have shown the mechanisms of the cutting removal and the 8
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deformation damage removal in the abrasive erosion. However, the indentation sizes wmax and ymax in Eq.(14) are unknown. These two sizes are concerned with the model of particle impact. Therefore, a model of micro-particle impact is introduced to discretely solve wmax and ymax by DEM simulations. The adhesive and elastoplastic behaviors have been integrated into the impact process. The loading phase has been divided into perfectly-elastic region, elastoplastic region and fully plastic region. In
of
this paper, the fully plastic region is incorporated into elastoplastic region. The unloading phase is regarded as the release of elasticity. The model has demonstrated a high degree of accuracy than
-p
This gives force-displacement relationship as follows:
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other model in Ref. [24] . The process of DEM simulation has been referred in other papers[11,25].
lP
re
3 4E w 3 12 if n y in loading or unloading under el astic loading 3R 8E w n y 4E 3 w sech e p , full y 3R n y 12 w nM p 0 Fn 1 - sech 8E w 3 if n y in loading nM 2 p , full y w p , full nM m y 4E 3 1 - sech m y 2 p 0 wm w sech e p , full y w nM - 2 p , full y 3R p , full 2 w w w 12 8E w 3 sin 1 1 if unloading under el astoplastic loading w m wm wm
(15)
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na
where E* = EpEtar/(Ep+Etar), R* = 0.5deff for particle-wall impact. So as it can be imagine, the angular particles bring about more material removal, whereas the smooth particles cause the less removal. In addition, Ref. [24] gives the value of the empirical coefficient of Eq. (15). In Eq.(15), the corresponding contact radius is defined as: we if y in loading or unloading under el astic loading n y y w 1 - sech n w w sech if y in loading p y e p y p m y y we 1 - sech m w if unloading under el astoplastic loading sech p y p y p
9
(16)
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where we and wp satisfy the following relationships: we2 2we R E
n
2 n
12
(17)
2we R E
12
w 2p
(18)
In Eq. (16), both critical parameters δy and δp are unknown, which have to find the quantity by the corresponding radius. The corresponding radius can be obtained as follows:
R
w p, full
2E w y
12
(19)
of
2E wy
3 R p0 21 2 E
(20)
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py
-p
where both py and p0 are the physical property parameter. Substituting wy into Eq. (17), δy is
re
obtained. Substituting wp,full into Eq. (19), δp is obtained.
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Equivalent yield strength Py is an important parameter to measure the change of contact process from elastic contact to elastic-plastic contact, which satisfies the relation as follows:
(21)
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p y min p y，p , p y，tar
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The contact pressure P0 for fully plastic deformation can be obtained from the hardness of the two contact objects, as follows: -1
2 2 p0 g 10 6 H p H tar
(22)
The advantage to adopt DEM simulations is adding more influential factors to the indentation sizes. On the other hand, the empirical coefficients n, c~C and c~D are independent of other influential factors, but correlated with individual material properties. For example, wmax and ymax integrate the information of adhesion in Eq.(14), elastoplasticity, and even particle shape. Therefore, the empirical coefficients may be independent of these three influential factors. The present study hopes these empirical coefficients are concerned with only one influential factor. That is, for the
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given particle and target, the empirical coefficients remain const for any impacts. To extend simulations to the particle dynamics, we use the discrete-element method, implemented here in software MATLAB, which have the high efficiency. The particle impact process satisfies Newton's second law of motion, so the corresponding solutions are as follows: vi 1 vi
Fn t mp
Fn t 2 mp
(24)
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of
i 1 i vi t
(23)
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3. Relevance validations
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The complication associated with measurements compels the validations of the present model
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to be done, where sufficient experimental data is available for the erosion rate. The common
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experimental set refers to Fig.3. Compressed air carries the solid particle to strike the target surface. The particle velocity is measured through CCD camera. Various experimental results with different
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materials are collected from the literature and the relevant material properties are provided for the particles in Table 1 and for the target materials in Table 2. Note that these properties are also the ones that are used in sections of validations and results. Besides, the relevance validation focus on the applicability of Eq.(14), giving the relevance between the erosion rate and indentation volume rather than a complete expression including the empirical coefficients. In this section, the empirical coefficients n, c~C and c~D are obtained by fitting the experimental data. The fitting results are demonstrated in Table 3.
3.1 Indentation geometry
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Before verifying the relevance between the erosion rate and indentation volume, it need to validate the accuracy of indentation geometry from Eqs.(15)~(18). Since there are some tests measuring the indentation profile for the abrasive mass flow rate [29], no experiments measure the indentation geometry for a single impact. Whether for the static experiment or for the dynamic process, the indentation geometry depends on Eq.(16). Therefore, a static experiment could be used
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to substitute the dynamic impacting process. It doesn't affect the conclusion. The experiment studies the flattening of an aluminum sphere by an approximately rigid flat silicon carbide surface in Fig. 4.
ro
An optical interference microscope is used to measure the width of the indentation. Due to the
-p
limitations of the experiment, only indentation width is measurable as a comparison. In Fig. 5, the
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width is regarded as a function of the normal load. In the case, the present model of micro-particle
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impact is shown to agree strongly with the experiment data. As the data is compared to the other
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models, a different trend is observed: a model that shows poor agreement with the measurements is Du et al. [30] showing the most discrepancy between the predictions and the measurements. The
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remaining models by Chang et al. [31] and Jamari et al. [26] show agreement with the experiment, but show significant discrepancies for the relationship between the width and the normal contact force. This step ensures the indentation sizes are accurate and further applied to the solution of material removal reasonably.
3.2 Erosion rate
As mentioned before, the existing models predict a discrepancy in the maximum erosion angle. Fig. 6 shows the volume loss with Finnie’s experiment data [7] and corresponding values from the present model. The experiments were conducted using 250 μm angular silicon carbide particles
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impacting aluminum, copper and SAE-1055 stainless steel target materials. The impact velocity of abrasive particles is set as 107 m/s. The average error is below 5%, and the fair agreement is observed between the model predictions and experiments. In these three cases, the advantage is presented by introducing the exponent n, which governs the maximum erosion angle. It can be seen that the smaller the exponent n is, the larger the maximum erosion angle will be. It is noted that the
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hardest material is SAE-1055 stainless steel, and the softest material is aluminum. It seems that the exponent n is related to the target material hardness.
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At small impingement angles, abrasive erosion is dominated by the cutting removal. However
-p
at the large impingement angle, abrasive erosion is dominated by the deformation damage removal.
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In Fig. 5, the open circle represents that the cutting removal equals the deformation damage
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removal. It is believed that the smaller the corresponding angle in the open circle is, the earlier the
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impact goes through the dominated region of the deformation damage removal. The corresponding angles in the open circle are 68 [deg], 64 [deg] and 62 [deg] for aluminum, copper and SAE-1055
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stainless steel respectively. It seems that the trend is also related to the target material hardness. For the larger hardness, the erosion is more likely to be dominated by the deformation damage removal. Besides, it is generally believed that the brittle materials have a larger hardness. This explains why the brittle materials can neglect the cutting removal. In another comparison, model predictions are compared to experimental data of lead, aluminum, iron and grey cast iron eroded with 326 μm semi-rounded quartz particles at the impact velocity of 100 m/s [27]. Fig. 7 presents the erosion rate as a function of impingement angle. The average error is below 6%. Both the experimental data and the model predictions show very good agreement. It is important to note that lead has a smallest hardness, and its maximum erosion angle
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is the minimum. While grey cast iron has a largest hardness, and its maximum erosion angle is maximum. Similarly, this proved the relationship between the maximum erosion angle and the target material hardness. In Fig. 7, the dashed line is the model prediction of lead without small angle effect. The results have an average 40% error than measured at low impingement angle. When considering the effect of the low impingement angle, these discrepancies will be eliminated. These
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two cases show the present model has good relevance between the erosion rate and indentation
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volume.
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4. Results and discussion
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Various experimental results with different materials are collected from the literature and the
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corresponding coefficients of the erosion model are provided in Table 3. As stated before, the
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exponent n is related with the target material hardness and each of previous experiments has the same abrasive particles individually in Fig. 6 and Fig. 7. However, almost experiments have
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different particles impacting different target materials. Therefore, the hardness ratio is introduced to discuss its effect on the exponent n below, and the parameter contains the information of particle hardness and target material hardness. The hardness ratio is defined as: H ~ H tar Hp
(25)
Generally, when the hardness ratio stays below a certain critical value, pure abrasive erosion takes place. The critical value of hardness ratio will be discussed below. In addition, the relationships between other empirical coefficients c~C and c~D and the hardness ratio are also discussed below.
4.1 Impact velocity effect 14
Journal Pre-proof In order to validate that the coefficients n, c~C and c~D are only concerned with the material properties, a set of experiments are conducted by varying the impact velocity and the impingement angle. It means that c~C and c~D should be verified to be independent of the impact conditions. Fig. 8 shows the erosion rate as a function of impact velocity and impingement angle for the 150 μm semi-rounded sand impacting carbon steel 1018 plate. The impact velocities are set as 9 m/s, 18 m/s and 28 m/s. It is important to note that the model is developed with experimental data for the impact
of
velocity of 28 m/s, and three estimated coefficients n, c~C and c~D are applied to the erosion rate for
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the impact velocity of 9 m/s and 18 m/s. In three cases, n= 0.55, c~C = 11.8 and c~D = 4.2·106. Very
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good agreement is observed between the measured results and the present model for all velocities.
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Since the model predictions for 18 m/s have an error than measured, the error can be controlled in
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the allowable range. This represents that the factors are independent of the impact velocity, and are
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only concerned with material properties. Besides, the black dashed lines represent the maximum erosion angle. It is found that the trend of the maximum erosion angle can be predicted for different
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impact velocities. The larger the impact velocity is, the larger the maximum erosion angle will be. The model shows good agreement with the trend. In second set of experiments, the model predictions are compared to experimental data of aluminum alloy 6061 eroded with 150 μm semi-rounded sand. In Fig. 9, the impact velocities are set as the same velocities in the last experiments. Similarly, the model is developed with experimental data for the impact velocity of 28 m/s. In three cases, n= 0.81, c~C = 0.29 and c~D = 1.74·106. The predictions of the present model show agreement with the experimental data. It indicates that the estimated coefficients n, c~C and c~D are only concerned with the material properties as well.
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4.2 Hardness ratio effect
Since the coefficients n, c~C and c~D have been demonstrated to be relevant to the material properties, we hope the coefficients is related with very few properties, including material properties and impact conditions. In this paper, impact conditions are verified to have no effect on the coefficients in Fig. 8 and Fig. 9, and the material properties are supposed to only be the hardness
of
ratio. Although other material properties affect the coefficients, their influences can be neglected. ~
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Fig. 10 shows the relationship between the hardness ratio and the empirical exponent n. For H
-p
<0.15, the linear trend line agrees with the exponent n of the present models. The R- square of the
re
fitting in Fig. 10 is 0.963. It can be seen that the larger the hardness ratio is, the smaller the
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exponent n is. The maximum erosion angle, combined with the analysis in Section 3.2, becomes ~
larger as the hardness ratio increases. However, for H >0.15, the value of exponent n stabilizes 0.5.
na
It may be due to that the current status is not in pure erosion, rather than mixed erosion and
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breakage. The hardness ratio demonstrates the gap between the target material and the particle: the larger the hardness ratio is, the smaller the gap is. Once the gap is relatively small, the particle will be broken [32]. Then a part of the absorbed energy used for erosion is applied to the particle breakage. The range of exponent n is 0.5~1.0, which has been identified in Ref. [9]. Fig. 11 presents the relationship between the hardness ratio and the cutting erosion factor c~C in the double logarithm coordinate. The solid line is a trend line. The red dashed line is the upper boundary and the blue dashed line is the lower boundary. The boundaries may be due to the measuring error or the target roughness. However, these do not affect the objective law. The cutting erosion factor c~C has a linear relation with the hardness ratio in the double logarithm coordinate. In Fig. 11, the R-square of the fitting is 0.927. It is believed that the cutting erosion factor c~C is 16
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dominated by the hardness ratio, and the influences of other material properties can be neglected. It can be seen that the cutting erosion factor c~C rises exponentially with the hardness ratio increased in the Cartesian coordinate. Fig. 12 presents the relationship between the hardness ratio and the deformation erosion factor c~D .
The trend line and the boundaries are indicated by the solid line and the dashed lines. The
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deformation erosion factor c~D rises with the hardness ratio because it is proportional to the material brittleness. It is important to note that it is more vulnerable to crack propagation with larger
ro
hardness ratio. Therefore, the deformation mechanism is significant for the impact of the larger
-p
hardness ratio. This phenomenon can be found in Fig. 6 and Fig. 7. In Fig. 12, the R-square of the
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re
fitting is 0.958.
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4.3 Velocity exponents
It is generally believed that the erosion rate is proportional to the power-law relation with U 0fC
, and QD
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particle velocity, i.e QC
U 0f D
. Previously, it has been found that fC and fD have
no concern with the impingement angle, and fC and fD are constant for the same experiment. Fig. 13 shows the relationship between the hardness ratio and the velocity exponents fC and fD. It can be ~
seen that fC is in the range of 2.0~2.7. Ref.[9] is almost unanimous support for the results. For H ~
<0.15, the trend shows a linear growth, while for H >0.15, fC remains the same. fD is nearly ~
~
constant at 2.8 in the range of H <0.15, while decreasing in the range of H >0.15. It may be due to the particle breakage. Unless it never occurs the particle breakage, fC and fD maintain the original trend of growth, and exceed the present values. It means a part of absorbed energy is applied to the particle breakage, rather than pure erosion.
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4.4 Empirical model and adaptability
The relationships between empirical coefficients and hardness ratio have been found in the previous content. Thus, the mechanistic model can be developed through the relationships. Eqs.(21)~(23) give the expressions of empirical coefficients. In these expressions, c~C and c~D give ~
an estimated range in which the upper and the lower boundaries refer to the fitting range. For H
of
<0.15, the model adaptability is discussed in the next part.
ro
~ n -3.24 H 0.98
-p
~ c~C 10 3.19 lg(H )b1
(21) (22)
re
~ c~D 2.68 107 H b2
(20)
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where, b1 is in the range of 3.9 ~ 5.0, b2 is in the range of -4105~ 106. Fig. 14 shows the model adaptability for different materials. It can be seen that the present model can estimate the erosion
na
rate. The visible error is related with the estimated range of empirical coefficients. It is definitely
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worth that the new model integrated the empirical correlations shows the same order of erosion rate, which makes progress and assists the industrial design in erosion.
5. Conclusions
In the current study, a new phenomenological erosion model was developed based on some studies in the literature to calculate the erosion of various ductile materials. The model is composed of two parts, cutting and deformation with two indentation sizes and three empirical coefficients. An accurate impact model is introduced to predict indentation sizes by DEM simulations and empirical coefficients are obtained by fitting experimental data for different materials. The introduction of DEM guarantees that the empirical coefficients only depend on hardness ratio, while other 18
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influential factors are incorporated into the indentation sizes. Moreover, this paper validates the relationship and summarizes the empirical equations of empirical coefficients. The following ~
conclusions on empirical coefficients may be drawn: For H <0.15, the larger the hardness ratio is, the smaller the exponent n is. However, empirical coefficients c~C and c~D rise with the hardness ~
ratio increased. For H >0.15, because of the mixed erosion and breakage, n keeps at the value of
of
0.5. In addition, cutting removal has an approximate 2.0~2.7 power-law relation with particle velocity which is consistent of experimental findings. Deformation damage removal has a 2.8 ~
-p
~ H >0.15.
ro
power-law relation with particle velocity for H <0.15, while the velocity exponent decreases for
re
Furthermore, since empirical coefficients n, c~C and c~D are defined as the functions of the
lP
hardness ratio, this paper has given the empirical correlations. If the new model integrated the
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empirical correlations, a visible error exists between the new model and experiments. It may be due to that limited experimental data can be provided, but does not affect the conclusion of this paper. It
Acknowledgements
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is definitely worth that the new model makes progress and helps the industrial design in erosion.
The authors appreciate the financial supports provided by the National Natural Science Foundation of China (Grant No. 51976031) and Qinglan Project of Jiangsu Province of China.
Nomenclature
Symbol Description
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empirical exponent
c
empirical coefficient
d
particle size (m)
f
velocity exponent
m
mass (kg)
n
empirical exponent
nM
Meyer hardness exponent
p0
uniform pressure at the start of fully plastic loading stage (Pa)
py
yield strain (Pa)
vi
particle velocity at the ith moment
w
indentation width (m)
we
indentation width under elastic deformation (m)
wp,
indentation width under plastic deformation (m)
wp,full
indentation width at the start of fully plastic deformation (m)
wy
indentation width at yield point (m)
y
indentation depth (m)
Ay
projected contact area in the normal direction (m2)
E
Young's modulus (GPa)
E*
equivalent Young's modulus (GPa)
Er
erosion rate (m3/kg)
Fn
normal force (N)
H
hardness (kgf/mm2)
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na
lP
re
-p
ro
of
b
20
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indentation length (m)
Nf
mean number of impacts needed to cause the material removal
R*
equivalent radius (m)
U0
initial velocity of the particle (m/s)
Ux0
initial tangential velocity of the particle (m/s)
α
impingement angle (deg)
β
particle sharpness factor
Γ
interface energy (J/m2)
δi
displacement at the ith moment
δm
maximum displacement (m)
δn
normal displacement (m)
δp
limited displacement at the start of fully plastic loading (m)
δy
limited displacement at the end of the elastic loading (m)
εc
fatigue strength (MPa)
ξ
empirical coefficient
ψ
empirical coefficient
△Q
volume loss (m3)
△QC
cutting material removal (m3)
△QD
deformation damage material removal (m3)
△QE
elementary volume of the material removal after Nf normal impacts (m3)
△εp
strain magnitude (MPa)
c~
erosion factor
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na
lP
re
-p
ro
of
L
21
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hardness ratio
Subscript Description effective
max
maximum
p
particle
tar
target surface
C
cutting
D
deformation
-p
ro
of
eff
lP
[1]
re
References
E. Bousser, L. Martinu, J.E. Klemberg-Sapieha, Solid particle erosion mechanisms of
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protective coatings for aerospace applications, Surf. Coatings Technol. 257 (2014) 165–181.
[2]
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doi:10.1016/j.surfcoat.2014.08.037.
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[34]
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doi:10.1016/0043-1648(91)90168-T.
26
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Credit author statement Yunshan Dong: Investigation, Methodology, Writing - original draft. Fengqi Si: Supervision, Conceptualization, Funding acquisition. Wei Jin: Investigation, Validation. Yue Cao: Validation, Writing - original draft.
Jo ur
na
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re
-p
ro
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Shaojun Ren: Validation.
27
Journal Pre-proof Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Jo ur
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
28
Journal Pre-proof
Figures
ax
α
ym
ax
wm
Le
Lm
ro
geometry for a single impact
-p
Fig. 1 Indentation
ax
of
ff
lP
re
y
na
crack
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Fig. 2 Sketch of crack propagation
Compressor
Feeder Cylinderical lens
Mirror
Pulsed laser
Feeder
Synchronizer
Nozzle Target
Computer
CCD Camera
Fig. 3 Experimental set of abrasive erosion
29
Journal Pre-proof
P
of
Fig. 4 Hard flat indenting a deformable sphere contradicts
ro -p re
1.0
0.5
0
Experiment [26] Chang et al. Du et al. Jamari et al. Prediction
100 200 Contact force,Fn [N]
300
na
0.0
lP
Indentation width,w [m]
1.5
5
68
0
0 30 60 90 Impingement Angle, [deg]
(a)
Experiment [7] Model Component_x Component_y
10
5 64
0
15
3
3
10
15
Volumn Loss, Q [mm /53gSiC]
Experiment [7] Model Component_x Component_y
Volumn Loss, Q [mm /53gSiC]
3
Volumn Loss, Q [mm /23gSiC]
15
Jo ur
Fig. 5 Indentation width as a function of the normal load of aluminum sphere.
0 30 60 90 Impingement Angle, [deg]
(b)
Experiment [7] Model Component_x Component_y
10
5 62
0
0 30 60 90 Impingement Angle, [deg]
(c)
Fig. 6 Erosion rate as a function of impingement angle. Target materials are: (a) aluminum, n=0.93 (b) copper, n=0.895
30
Journal Pre-proof (c) SAE-1055 stainless steel, n=0.675. Impingement velocity is 107m/s. Abrasive particle is 250 μm SiC.
-6
10
Erosion Rate,Er [m /kg]
1.5
lead-Experiment [27] aluminum-Experiment [27] Iron-Experiment [27] Grey cast iron-Experiment [27] lead-Model aluminum-Model Iron-Model Grey cast iron-Model
without small angle effect
3
1.0
0
30 60 Impingement Angle, [deg]
90
ro
0.0
of
0.5
-p
Fig. 7 Erosion rate as a function of impingement angle.
re lP
4
U=28m/s (Experiment [10]) U=18m/s (Experiment [10]) U=9m/s (Experiment [10]) U=28m/s (Model) U=18m/s (Model) U=9m/s (Model)
na
3
Erosion Rate,Er [mm /kg]
-9 6 10
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2
0
0
30
60
Impingement Angle, [deg]
90
Fig. 8 Erosion rate of carbon steel 1018 at different impact velocities: n= 0.55, c~C = 11.8 and c~D = 4.2·106. -9
3
Erosion Rate,Er [mm /kg]
12
10
U=28m/s (Experiment [10]) U=18m/s (Experiment [10]) U=9m/s (Experiment [10]) U=28m/s (Model) U=18m/s (Model) U=9m/s (Model)
9 6 3 0
0
30 60 Impingement Angle, [deg]
31
90
Journal Pre-proof Fig. 9 Erosion rate of aluminum alloy 6061 at different impact velocities: n= 0.81, c~C = 0.29 and c~D = 1.74·106.
1.0 2
R =0.963
0.8
n
0.6 0.4 Model Trendline
0.0 0.0
0.3
ro
0.1 0.2 ~ Hardness ratio, H
of
0.2
-p
Fig. 10 Empirical exponent n as a function of hardness ratio.
re
1000 100
lP
10 1
~
cC
2
R =0.927
na
0.1
Model Trendline
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0.01
1E-3 1E-3
0.01
0.1
1
10
~ Hardness ratio, H
Fig.11 Cutting erosion factor c~C as a function of hardness ratio. 6
10.0
10
D
2
R =0.958
5.0
c
~
7.5
2.5 Model Trendline
0.0 0.0
0.1
0.2 ~
Hardness ratio, H
32
0.3
Journal Pre-proof Fig.12 Deformation erosion factor c~D as a function of hardness ratio.
3.0
3.5
3.0
fD
fC
2.5
2.5
2.0 2.0
Model Trendline
0.05
0.10 0.15 0.20 ~ Hardness ratio, H
0.25
1.5 0.0
0.30
0.1 0.2 ~ Hardness ratio, H
of
1.5 0.00
Model Trendline
(b)
ro
(a)
0.3
-p
Fig. 13 Velocity exponents as a function of hardness ratio: (a) cutting velocity exponent fC. (b) deformation velocity
na
2
0
0
30 60 Impingement Angle, [deg]
3
4
Erosion Rate,Er [m /kg]
lP
Experiment [33] Prediction
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3
Erosion Rate,Er [m /kg]
6
re
exponent fD
15 Experiment [34] Prediction
10
5
0
90
0
(a)
30 60 Impingement Angle, [deg]
90
(b)
Fig. 14 Predictions of erosion rate using estimated empirical coefficients: (a) Ti-6Al-4V alloy eroded with 125 μm angular SiC particle at velocity of 55 m/s [33]. (b) carbon steel 1020 eroded with 50 μm angular alumina particle at velocity of 60 m/s [34].
33
Journal Pre-proof
Tables Table 1 Particle mechanical properties. Young's Density
Particle
Poisson’s modulus
3
[kg/m ]
SiO2 [10,27]
3,200
a2
2,300
a2
ratio
a1
Adopted from Ref. [28].
a2
Adopted from Ref. [13].
0.17
Shape
[kgf/mm ]
0.345 0.192
strength [mm]
exponent
75.2
a2
410
a2
80
a1
26
a2
a2
2.15
2,500
a2
1,122
a2
2.2 2.2
[MPa] 40 a1
3
rounded
10,000
a2
0.25
angular
11,000
a2
0.15,0.326
sub-rounded
of
SiC [7]
2,750
hardness
Table 2 Target material properties.
ro
Al [26]
Yield Particle size
2
[GPa] a1
Meyer Hardness
Young's
Particle
Poisson’s
[kg/m3]
ratio
-p
Target
Density
modulus
Interface Hardness
Yield strength
[kgf/mm2]
[MPa]
2,485
10,000 b1
energy [J/m2]
Al
3,200
b1
Al [7]
SiC
2,700
Copper [7]
SiC
8,890 b1
SAE-1055 steel [7]
SiC
7,850 b2
Iron [27] Grey cast iron [27] Carbon steel 1018 [10] Al 6061 [10]
SiO2 SiO2 SiO2
11,400
na
Al 1050-H16 [27]
SiO2
0.35
2,700 7,870 7,500
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Lead [27]
0.17
SiO2 SiO2
b1
7,870 2,700
430
0.35
31
372.4
0.4
117 b1
63
646.8
0.28
0.28 b2
200 b2
200
587.6
0.25
20
b4
5.5
b5
0.4
35
b6
124
b6
0.4
0.42
b3
0.33
b6
0.291
b7
0.29
b8
0.29
b1
0.33
b1
70
b1
0.33 b1
lP
SiC [26]
re
[GPa]
b1
14
b3
69
b6
200
b7
130
b4
205
b1
70
b1
153 306 131 31
b1
Adopted from Ref. [13].
b2
Adopted from www.iron-foundry.com/AISI-1055-SAE-UNS-G10550-Carbon-Steel-Foundry.html.
b3
Adopted from www.matweb.com/search/DataSheet.aspx?MatGUID=ebd6d2cdfdca4fc285885cc4749c36b1.
b4
Adopted from Ref. [11].
b5
Adopted from http://www.goodfellow.com/E/Lead.html.
b6
Adopted from www.matweb.com/search/DataSheet.aspx?MatGUID=f54812ceded24176b44c4e0af379ea39.
b7
Adopted from http://www.matweb.com/search/DataSheet.aspx?MatGUID=654ca9c358264b5392d43315d8535b7d.
b8
Adopted from www.vonroll-casting.ch/en/grey-cast-iron.html.
34
50
b7
260
3,700 55
0.3
b8
b1
b1
0.3 0.35 0.35
Journal Pre-proof Table 3
Model parameters. Model parameters calibration Reference erosion experiments
Hardness ratio
Reference
Particle
wall
~ H
[26]
Al
SiC
-
[7]
SiC
Al
0.012
Cutting
Deformation
Velocity
mechanism
mechanism
exponent
n
cC
-
-
cD -
fC
fD
-
-
0.93
6.9·10
-2
1.54·10
5
2.096
2.777
3.57·10
5
2.146
2.794
1.81·10
6
2.443
2.818
2.104
2.792
[7]
SiC
Copper
0.025
0.895
7.9·10
-2
[7]
SiC
SAE-1055 steel
0.08
0.675
7.9·10
-1
[27]
SiO2
Lead
0.0178
0.925
1.22·10
[27]
SiO2
Al 1050-H16
0.031
0.87
1.48·10-2
6.36·105
2.182
2.803
[27]
SiO2
Iron
0.136
0.575
2.5
3.15·106
2.595
2.806
28.9
7.08·10
11.8
4.2·10
0.81
35
5.9·10
of ro
0.0624
0.55
2.9·10
-p
Al 6061
0.117
0.5
re
SiO2
Carbon steel 1018
0.273
lP
[10]
SiO2
Grey cast iron
na
[10]
SiO2
Jo ur
[27]
-2
-1
5
6
2.663
2.545
6
2.632
2.812
1.74·10
2.274
2.792
6
Journal Pre-proof
Graphical Abstract -6
3
Erosion Rate,Er [m /kg]
1.5
10
lead-Experiment [27] aluminum-Experiment [27] Iron-Experiment [27] Grey cast iron-Experiment [27] lead-Model aluminum-Model Iron-Model Grey cast iron-Model
without small angle effect
1.0
0.5
0
30 60 Impingement Angle, [deg]
90
of
0.0
Jo ur
na
lP
re
-p
ro
Erosion rate predictions as a function of impingement angle
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Highlights
of ro -p
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lP
na
A new mechanistic model of abrasive erosion using DEM was developed Erosion rate with coefficients was taken as a new function of indentation sizes Empirical coefficients were validated to be mainly related with the hardness ratio Relationships between hardness ratio, erosion and breakage were studied Velocity exponents for two erosion mechanisms were discussed
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37