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Numerical study of impact erosion of multiple solid particle
Applied Surface Science 423 (2017) 176–184
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Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Full Length Article
Numerical study of impact erosion of multiple solid particle Chao Zheng, Yonghong Liu ∗ , Cheng Chen, Jie Qin, Renjie Ji, Baoping Cai College of Mechanical and Electronic Engineering, China University of Petroleum, Qingdao 266580, China
a r t i c l e
i n f o
Article history: Received 29 March 2017 Received in revised form 17 May 2017 Accepted 12 June 2017 Available online 15 June 2017 Keywords: Numerical study Material erosion Erosion characteristics Geometry evolution
a b s t r a c t Material erosion caused by continuous particle impingement during hydraulic fracturing results in significant economic loss and increased production risks. The erosion process is complex and has not been clearly explained through physical experiments. To address this problem, a multiple particle model in a 3D configuration was proposed to investigate the dynamic erosion process. This approach can significantly reduce experiment costs. The numerical model considered material damping and elastic-plastic material behavior of target material. The effects of impact parameters on erosion characteristics, such as plastic deformation, contact time, and energy loss rate, were investigated. Based on comprehensive studies, the dynamic erosion mechanism and geometry evolution of eroded crater was obtained. These findings can provide a detailed erosion process of target material and insights into the material erosion caused by multiple particle impingement. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Erosion wear caused by the impacts of particles is one of the most common harmful behaviors of equipment in the multiphase flow [1–3]. The development of unconventional oil and gas resources with low permeability increased in recent years with the consumption of conventional resources. Hydraulic fracturing technology with high flow rate and high sand ratio is crucial to the development of low-permeability or extra low-permeability reservoirs [4–6]. Erosion wear due to proppant impingement is one of the most important issues in the fracturing process. Pipeline pressure, which reaches up to tens to hundreds of MPa, create cracks for downhole formation. Unpredictable consequences would occur if the conveying pipelines suffer from erosion failure. Proppant in the fracturing fluid affects tool surface and causes erosion scars and cracks, which severely affect production safety [7–9]. Thus, the erosion mechanism of substrate material or deposited coating under fracturing working conditions should be urgently studied [10–12]. Considerable research investigated erosion wear [13–15]. Erosion is a complex problem caused by many factors, such as particle velocity, impact angle, particle shape, and particle size [16,17]. Various experiments were conducted to explain the erosion process. Experiments usually provide limited information, such as erosion rate and eroded morphologies. These limited data cannot provide sufficient information to further analyze the dynamic
∗ Corresponding author. E-mail address: [email protected] (Y. Liu). http://dx.doi.org/10.1016/j.apsusc.2017.06.132 0169-4332/© 2017 Elsevier B.V. All rights reserved.
erosion process. Numerous experiments that aim to study the influence of each factor on the erosion mechanisms experimentally consume substantial time and money. In addition, the dynamic impact process occurs briefly; thus, capturing the erosion behavior of an erodent and a target material is a difficult process. Some researchers used high-speed cameras to capture images during particle impingement and studied its dynamic characteristics. However, expensive and sophisticated instruments are needed, which substantially increases testing cost. However, numerical technique allows researchers to model the erosion process with different parameters. The numerical model can provide detailed information of the particle and target material, such as stresses, interaction forces, and deformation [18–21]. The numerical values need to adjusted and recalculated to investigate the effect of different parameters on the dynamic process. Hence, the effects of these parameters on the erosion characteristics can be investigated. Some researchers studied erosion wear caused by abrasive particle collisions [22–24]. Aponte et al. used computational fluid dynamics to predict the erosion behavior of geometries in a fluid environment that contains abrasive particles [25]. Liu et al. studied the erosion of pipeline caused by sand particles in the oil and gas transportation system [26]. In their studies, the computational fluid dynamics helped provide insight into the movement of particle and fracturing fluid and predict erosion rate. However, the interactions between the particle and target material cannot be considered, which is the core issue of erosion wear mechanism. The continuous impingement between the particles and target material causes severe erosion. To address this problem, the present study proposes a multiple particle impact model to solve the dynamic erosion pro-
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Table 1 Mechanical material properties. Material
Density (kg/m3 )
Young’s modulus (GPa)
Yield strength (MPa)
Tensile strength (MPa)
Poisson’s ratio
Q345
7850
210
345
490
0.28
Fig 1. Mesh generation of the multiple particle impact model.
cess. The effect of impact parameters on erosion characteristics was investigated. Detailed change information of target material under multiple particles impingement was obtained for further analyses. The numerical results of the dynamic erosion process would help explain the erosion mechanism and provide scientific basis for material erosion due to multiple particle impingements during fracturing. The rest of this paper is organized as follows. The numerical details are introduced in Section 2. Subsequently, results and discussions are presented in Section 3. Finally, main conclusions are briefly presented in Section 4.
where εy ≤ ε ≤ εu , εy is the yield strains, εu is the ultimate strains, Dy and Du are the coefficients associated with εy and εu , respectively. Du is a coefficient evaluated from the strain rate sensitivity properties at the ultimate tensile strength of a material. In the modified equation, the values of Dy , Du , and q were 40, 6340, and 5, respectively. In Eq. (2), the strain hardening of the target material was not considered. Eq. (2) was modified by assuming the presence of linear strain hardening between yield strain εy and ultimate strain εu . To consider strain hardening, the strain hardening relationship between yield strain εy and ultimate strain εu was added in the equation and expressed as
⎡ ⎢
2. Numerical details
= 0 · (F + Gε) · ⎣1 +
2.1. Constitutive models The strain of target material changes dramatically during the dynamic erosion process. Plastic strain and strain rate are important parameters that affect material behavior during particle collision. To study the dynamic erosion process, the plastic flow behavior of a target material was modeled using the Cowper–Symonds equation [27,28]. The Cowper–Symonds equation considers strain-hardening and strain-rate sensitivity. In the present study, the constitutive model is given as:
⁄ = 1 + ε˙ ⁄Dy 0
1⁄q (1)
where is the dynamic flow stresses, 0 is the static flow stresses, respectively, ε˙ is the strain rate, Dy and q are constants of target material with values of 40 and 5, respectively. Under the multi-axial stress condition, and ε˙ correspond to the equivalent dynamic flow stress and the associated equivalent strain rate, respectively. To solve stress analysis, a modified CS model was developed for the uniaxial stress condition, which is described as:
⎡
⎢ = 0 · ⎣1 +
εu − εy ε˙
ε − εy Du + (εu − ε) Dy
1⁄q
⎤ ⎥ ⎦
(2)
εu − εy ε˙
ε − εy Du + (εu − ε) Dy
1⁄q
⎤ ⎥ ⎦
(3)
F = 1 − Gεy
(4)
G=
(5)
o
⁄
where 0 and u are the uniaxial static flow stresses at εy and εu , respectively. 2.2. Multiple particle impact Material erosions that occur in most industrial applications are caused by continuous particle collision. During the fracturing process, proppants were carried in the fracturing fluid and injected through the pipeline under high pressure pump. The particles impacted the tool surface continuously and caused severe erosion wear. A multiple particle model was established to explore the dynamic process and mechanism. The continuous effect of an erodent can be considered in this model, which can reflect actual working conditions. 2.3. Material properties Multiple particle erosion was simulated on the Q345 material. The mechanical properties of the target material, such as Poisson’s ratio, yield strength, and Young’s modulus at a temperature of 20 ◦ C, are given in Table 1. The materials are assumed to be isotropic,
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Fig. 2. Eroded craters with plastic strain contours under different impact velocity and angle.
linear elastic, and plastic, following the von Mises yield criterion. To simulate the fracturing working conditions, the erodent was a spherical particle with a density of 2100 kg/m3 and diameter of 800 m. 2.4. Mesh generation Fig. 1 shows the mesh generation of the 3D geometry. Hexahedral structured grids were used to mesh the target material. To
improve calculation accuracy and save computational resources, the contact region was meshed with a fine grid. The particle was produced using meshed O-type mesh generation method. Grid independency was conducted to guarantee that the mesh size was neither time-consuming nor low in accuracy. Table 2 shows wear depth profiles with different grid numbers under an impact velocity of 15 m/s and 90◦ impact angle. The eroded depth values indicated that the solution was essentially grid-independent when the number of grids was more than 250,000. The multiple particle
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Table 2 Grid independency study. Grid index
Grid number
Maximum eroded depth (m)
1 2 3 4 5
62,500 125,316 250,000 499,849 1,000,000
5.43 6.27 6.52 6.53 6.53
Fig. 4. Depth of eroded crater with plastic strain contours during multiple collisions.
Fig. 3. Von Mises stress distribued in Z axis direction after multiple collisions.
impact process was simulated in the commercially available software ANSYS/LS-DYNA, which was one of the most popular solvers for explicit dynamics problems. 2.5. Boundary conditions The particle was set as a rigid body, which meant that the strain of particle was zero during the contact process. To save on calculation time and computer resource, only a half-model was established. Symmetry conditions were applied to the symmetry plane. The symmetry plane was constrained to move in the X direction, the bottom plane was fixed, and multiple particles were constrained from displacement in the X direction and rotation around the Y and Z axes. 3. Results and discussion 3.1. Characteristics of eroded craters Fig. 2 shows the plastic strain contours under impact velocities of 15 m/s, 20 m/s, and 25 m/s and an impact angle of 45◦ and 90◦ . Under the impact angle of 45◦ , the eroded crater demonstrated obvious shearing lip with asymmetric characteristics. The eroded morphologies became increasingly evident with the increase of impact velocity because of shear and extrusion effects. Under the impact angle of 90◦ , eroded crater was symmetrical and the depth of the crater and the height of shearing lips around the crater increased with impact velocity. The size and depth of crater were related to the impact parameters and target material properties. In general, when the impact velocity is high, the eroded crater was large. As mentioned above, an eroded crater appeared on the surface of target material after multiple particle impingements. The stress distributed in the target material exceeds the yield limit of the material. Fig. 3 shows that the stress distribution in the Z axis after three particle collisions. When the impact velocity increased from 15 m/s to 25 m/s, the maximum stress region appeared to have a downward trend, which can be confirmed in Fig. 3. Similarly, the
Fig. 5. Effect of impact velocity on the contact time.
thickness of the residual stress layer was high under high impact velocity. This finding was similar to the shot peening process, which indicated that the existence of residual stress layer can significantly improve fatigue performance [29,30]. The position of maximum residual stress is closely related with the impact parameters, such as impact velocity and particle size. Fig. 4 presents the depth of eroded craters with the plastic strain contours under different impact velocities. The depth of the crater was defined as a vertical distance between the bottom of the crater and the top surface. The figure shows that the total plastic deformation increased with impact number and velocity and the depth value of each impingement decreased with the increase in impact number. Maximum depth value was obtained after the first impingement. Compared with that in the first impingement, the depth value of the second and third was smaller. This result can be attributed to surface hardening effects. The surface of the target material presented a lower plastic performance and higher surface hardness after multiple impingements. 3.2. Contact time and energy loss Contact time between the erodent and the target surface was difficult to achieve through physical experiments. Contact time can be recorded clearly in the numerical model. Fig. 5 shows the contact time under different impact velocities. Contact time decreased with increased impact velocity. Fig. 6 shows the contact time under different impact angles. Contact time was large under low-impact angle because of the
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Fig. 6. Effect of impact angle on the contact time.
Fig. 8. Effect of impact angle on the energy loss rate.
with the increase of impact velocity, which indicated that additional energy transferred to the target material. Maximum energy loss rate reached 76% under the impact velocity of 25 m/s. Energy loss for subsequent collisions decreased because of high surface hardness and small contact time, which can be confirmed from previous analyses. Fig. 8 plots the energy loss curve after multiple particle collisions under different impact angles. The figure shows that the energy loss increased with impact angle. In general, energy loss was small under an impact angle of 0◦ to 15◦ because the tangential component of velocity was large. From 15◦ to 75◦ , energy loss increased linearly impact angle. From 75◦ to 90◦ , the energy loss after first collision increased slowly. Energy loss after second and third collision changed slightly. This result can be attributed to the surface hardening effect. 3.3. Erosion process analyses
Fig. 7. Effect of impact velocity on the energy loss rate.
large tangential component of velocity, which helped form the ploughing characteristics on the target surface. Collision occurred in less than 10−5 s. The target material deformed quickly. Thus, heat generated by plastic deformation did not have sufficient time to transfer around. Therefore, the dynamic erosion process occurred in an adiabatic environment and resulted in temperature increase. Contact time decreased with increased impact number. This phenomenon occurred because surface hardening after multiple particle impingements increased surface harness and stiffness coefficient, thereby decreasing contact time. Therefore, the erosion process was affected by the thermal effect and strain hardening phenomenon. Energy exchange occurred between the particles and the target material during the solid particle collision. Hutchings et al. used high-speed photography to examine energy distribution during collision [31]. Part of kinetic energy was transferred to the material during collision [32]. This part of transferred energy consisted of plastic deformation and elastic wave energy. The energy used for plastic deformation would finally become heat energy and part of stored energy. The kinetic energy loss rate was calculated using the following equation. =
Ek 1 − Ek 2Ek1
(6)
where is the kinetic energy loss rate, Ek1 is the initial kinetic energy before collision, and Ek2 is the final kinetic energy after collision. Fig. 7 shows the rate of energy loss of particles after multiple collisions under different impact velocities. Energy loss rate increased
3.3.1. Dynamic erosion characteristics Fig. 9 shows the geometry evolution of eroded crater with the plastic strain contours after multiple particle collisions. The deformation region expanded with the increase in impact number, which emerged from continuous collision. The maximum plastic deformation was located in the contact region. During the collision, the contact force increased rapidly and resulted in the removal of the target material. After the fourth collision, the outer region of the eroded crater exhibited high plastic deformation, which indicated that part of material was extruded and formed lips. After the continuous collisions, the target material would finally detach the surface because of material failure. To further study the collision evolution, specified grids were chosen for quantitative analyses. Fig. 10 shows the comparison of plastic strains for specified grids. Fig. 10a–e plots the plastic strain curve with the impact number. The analysis of the plastic deformation curve indicates that the plastic deformation in the central region increased dramatically at the first particle collision, as shown in Fig. 10a-b. Grid No. 30465 exhibited the maximum plastic deformation strain at the second collision, as shown in Fig. 10c. Grid Nos. 30466 and 30467 exhibited the maximum plastic deformation strain at the third collision, as shown in Fig. 10d-e. Plastic deformation expanded from the central grid to the outer grid. Similarly, the accumulation of plastic deformation was evident at the outer region. The geometric parameters were studied to examine the geometry evolution of the eroded crater. Fig. 11 presents the evolution of crater parameters with the increase of collision numbers. The width and depth with the plastic strain contours were also defined,
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Fig. 9. Geometry evolution of eroded crater during multiple collisions.
as shown in Fig. 11. Width and depth can be used to characterize the eroded crater because of the symmetry of model. The plastic deformation value of the first collision was the largest. Growth rates then became slower with the further increase of impact number. This trend was similar to the change of maximum plastic deformation value, as shown in Fig. 9. This result can be attributed to the effect of strain-hardening and residual stress inner the target material.
3.3.2. Dynamic erosion mechanism Interaction forces were the main factors of material failure during the erosion process. According to the momentum theorem, the
contact forces between the erodent and the target material can be defined as follows: Ft =
d(m · v · cos ˛) dt
(7)
Fn =
d(m · v · sin ˛) dt
(8)
where Ft is the tangential contact force, Fn is the normal contact force, m is the mass of erodent, v is the impact velocity, and ␣ is the impact angle. Fig. 12 shows the dynamic erosion process under 45◦ impact angle and 50 m/s impact velocity. At 1.2 s, the particle began to contact the target material, and the contact region exhibited severe plastic deformation. The contact forces consisted of normal force and tangential force due to normal velocity component and
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Fig. 11. Evolution of geometric parameters of eroded crater after multiple collisions. Fig. 10. Plastic strains of specified grids during multiple particles collisions.
tangential velocity component. At 1.5 s, the target surface was squeezed under the effect of normal force. Meanwhile, tangential force applied on the material then caused flow of target material [33]. At 1.8 s, the eroded region presented obvious shear characteristics. At 2.1 s, the normal velocity component slowed down to
zero, and the tangential velocity component was the dominant erosion factor. Fig. 12d–f shows that tangential cutting was the main characteristic under low impact angle. Fig. 13 shows the dynamic erosion process under 90◦ impact angle and 50 m/s impact velocity. The first particle began to impact
Fig. 12. Dynamic erosion process under 45◦ impact angle and 50 m/s impact velocity.
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Fig. 13. Dynamic erosion process under 90◦ impact angle and 50 m/s impact velocity.
the target surface at 0.9 s. According to Eqs. (7) and (8), contact force was large because of the large normal velocity and short contact time. From 1.2 to 1.8 s, the dynamic erosion process completed and the particle bounced off the surface. The target material was extruded and formed lips around the crater under normal contact force. Under repetitive particle collisions, the target material would cause surface cracking and fatigue. The lips and debris formed on the surface were removed by the subsequent collisions. The continuous collisions contributed to the detaching of target material, which can be validated by the experimental study reported [34]. Therefore, the erosion mechanisms under different eroded parameters can be identified. Fig. 14 shows the dynamic evolution of contact force imposed on the particle under 45◦ impact angle and 50 m/s impact velocity. The tangential and normal contact forces gained several maximum and minimum values during the dynamic erosion process. In addition, the normal force was larger than the tangential force, which meant that the vertical impact was more severe. Fig. 15 shows the dynamic evolution of contact force imposed on the particle under 90◦ impact angle and 50 m/s impact velocity. The tangential force was nearly zero. Normal force increased quickly with the time, then decreased to zero during the dynamic erosion process. The time to reach the peak value of the normal contact force under 90◦ impact angle was shorter than that under 45◦
Fig. 14. Dynamic evolution of contact force under 45◦ impact angle and 50 m/s impact velocity.
impact angle because vertical velocity was large under high impact angle, thereby resulting in fast deformation.
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Fig. 15. Dynamic evolution of contact force under 90◦ impact angle and 50 m/s impact velocity.
4. Conclusions This study investigated the dynamic erosion process using multiple particle model in 3D. The effect of impact parameters on the dynamic erosion process was investigated. The salient features of the study are summarized as follows. (1) Maximum plastic deformation occurred at the first particle collision and decreased with the increase of collision number. During multiple particle collisions, strain hardening phenomenon was observed, which significantly influenced the impact process. (2) The impact parameters significantly affected the contact time between erodent and target surface. Energy loss increased with impact velocity and angle, which indicates severe collision during erosion process. The width and depth of eroded crater first increased sharply with the collision number, then increased gradually with the further increase of collision number. (3) The dynamic erosion process can be identified by analyzing the geometry evolution at different contact times. Under low impact angle, both the tangential and normal contact force had several maximum and minimum values during the dynamic erosion process. The main erosion mechanisms were plastic deformation and cutting due to tangential shear. Under high impact angle, the normal force increased quickly with the time, then decreased to zero during the dynamic erosion process. Vertical cracking and fatigue of surface material were the primary erosion mechanisms because of repetitive solid particle collisions. Acknowledgments The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 51675535), the Fundamental Research Funds for the Central Universities (Grant No. 15CX08007A, 17CX02058, 17CX05022). References [1] N. Ojala, K. Valtonen, A. Antikainen, A. Kemppainen, J. Minkkinen, O. Oja, V.T. Kuokkala, Wear performance of quenched wear resistant steels in abrasive slurry erosion, Wear 354-355 (2016) 21–31. [2] R. Tarodiya, B.K. Gandhi, Hydraulic performance and erosive wear of centrifugal slurry pumps-a review, Powder Technol. 305 (2017) 27–38. [3] F. Darihaki, E. Hajidavalloo, A. Ghasemzadeh, G.A. Safian, Erosion prediction for slurry flow in choke geometry, Wear 372-373 (2017) 42–53.
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Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Full Length Article
Numerical study of impact erosion of multiple solid particle Chao Zheng, Yonghong Liu ∗ , Cheng Chen, Jie Qin, Renjie Ji, Baoping Cai College of Mechanical and Electronic Engineering, China University of Petroleum, Qingdao 266580, China
a r t i c l e
i n f o
Article history: Received 29 March 2017 Received in revised form 17 May 2017 Accepted 12 June 2017 Available online 15 June 2017 Keywords: Numerical study Material erosion Erosion characteristics Geometry evolution
a b s t r a c t Material erosion caused by continuous particle impingement during hydraulic fracturing results in significant economic loss and increased production risks. The erosion process is complex and has not been clearly explained through physical experiments. To address this problem, a multiple particle model in a 3D configuration was proposed to investigate the dynamic erosion process. This approach can significantly reduce experiment costs. The numerical model considered material damping and elastic-plastic material behavior of target material. The effects of impact parameters on erosion characteristics, such as plastic deformation, contact time, and energy loss rate, were investigated. Based on comprehensive studies, the dynamic erosion mechanism and geometry evolution of eroded crater was obtained. These findings can provide a detailed erosion process of target material and insights into the material erosion caused by multiple particle impingement. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Erosion wear caused by the impacts of particles is one of the most common harmful behaviors of equipment in the multiphase flow [1–3]. The development of unconventional oil and gas resources with low permeability increased in recent years with the consumption of conventional resources. Hydraulic fracturing technology with high flow rate and high sand ratio is crucial to the development of low-permeability or extra low-permeability reservoirs [4–6]. Erosion wear due to proppant impingement is one of the most important issues in the fracturing process. Pipeline pressure, which reaches up to tens to hundreds of MPa, create cracks for downhole formation. Unpredictable consequences would occur if the conveying pipelines suffer from erosion failure. Proppant in the fracturing fluid affects tool surface and causes erosion scars and cracks, which severely affect production safety [7–9]. Thus, the erosion mechanism of substrate material or deposited coating under fracturing working conditions should be urgently studied [10–12]. Considerable research investigated erosion wear [13–15]. Erosion is a complex problem caused by many factors, such as particle velocity, impact angle, particle shape, and particle size [16,17]. Various experiments were conducted to explain the erosion process. Experiments usually provide limited information, such as erosion rate and eroded morphologies. These limited data cannot provide sufficient information to further analyze the dynamic
∗ Corresponding author. E-mail address: [email protected] (Y. Liu). http://dx.doi.org/10.1016/j.apsusc.2017.06.132 0169-4332/© 2017 Elsevier B.V. All rights reserved.
erosion process. Numerous experiments that aim to study the influence of each factor on the erosion mechanisms experimentally consume substantial time and money. In addition, the dynamic impact process occurs briefly; thus, capturing the erosion behavior of an erodent and a target material is a difficult process. Some researchers used high-speed cameras to capture images during particle impingement and studied its dynamic characteristics. However, expensive and sophisticated instruments are needed, which substantially increases testing cost. However, numerical technique allows researchers to model the erosion process with different parameters. The numerical model can provide detailed information of the particle and target material, such as stresses, interaction forces, and deformation [18–21]. The numerical values need to adjusted and recalculated to investigate the effect of different parameters on the dynamic process. Hence, the effects of these parameters on the erosion characteristics can be investigated. Some researchers studied erosion wear caused by abrasive particle collisions [22–24]. Aponte et al. used computational fluid dynamics to predict the erosion behavior of geometries in a fluid environment that contains abrasive particles [25]. Liu et al. studied the erosion of pipeline caused by sand particles in the oil and gas transportation system [26]. In their studies, the computational fluid dynamics helped provide insight into the movement of particle and fracturing fluid and predict erosion rate. However, the interactions between the particle and target material cannot be considered, which is the core issue of erosion wear mechanism. The continuous impingement between the particles and target material causes severe erosion. To address this problem, the present study proposes a multiple particle impact model to solve the dynamic erosion pro-
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Table 1 Mechanical material properties. Material
Density (kg/m3 )
Young’s modulus (GPa)
Yield strength (MPa)
Tensile strength (MPa)
Poisson’s ratio
Q345
7850
210
345
490
0.28
Fig 1. Mesh generation of the multiple particle impact model.
cess. The effect of impact parameters on erosion characteristics was investigated. Detailed change information of target material under multiple particles impingement was obtained for further analyses. The numerical results of the dynamic erosion process would help explain the erosion mechanism and provide scientific basis for material erosion due to multiple particle impingements during fracturing. The rest of this paper is organized as follows. The numerical details are introduced in Section 2. Subsequently, results and discussions are presented in Section 3. Finally, main conclusions are briefly presented in Section 4.
where εy ≤ ε ≤ εu , εy is the yield strains, εu is the ultimate strains, Dy and Du are the coefficients associated with εy and εu , respectively. Du is a coefficient evaluated from the strain rate sensitivity properties at the ultimate tensile strength of a material. In the modified equation, the values of Dy , Du , and q were 40, 6340, and 5, respectively. In Eq. (2), the strain hardening of the target material was not considered. Eq. (2) was modified by assuming the presence of linear strain hardening between yield strain εy and ultimate strain εu . To consider strain hardening, the strain hardening relationship between yield strain εy and ultimate strain εu was added in the equation and expressed as
⎡ ⎢
2. Numerical details
= 0 · (F + Gε) · ⎣1 +
2.1. Constitutive models The strain of target material changes dramatically during the dynamic erosion process. Plastic strain and strain rate are important parameters that affect material behavior during particle collision. To study the dynamic erosion process, the plastic flow behavior of a target material was modeled using the Cowper–Symonds equation [27,28]. The Cowper–Symonds equation considers strain-hardening and strain-rate sensitivity. In the present study, the constitutive model is given as:
⁄ = 1 + ε˙ ⁄Dy 0
1⁄q (1)
where is the dynamic flow stresses, 0 is the static flow stresses, respectively, ε˙ is the strain rate, Dy and q are constants of target material with values of 40 and 5, respectively. Under the multi-axial stress condition, and ε˙ correspond to the equivalent dynamic flow stress and the associated equivalent strain rate, respectively. To solve stress analysis, a modified CS model was developed for the uniaxial stress condition, which is described as:
⎡
⎢ = 0 · ⎣1 +
εu − εy ε˙
ε − εy Du + (εu − ε) Dy
1⁄q
⎤ ⎥ ⎦
(2)
εu − εy ε˙
ε − εy Du + (εu − ε) Dy
1⁄q
⎤ ⎥ ⎦
(3)
F = 1 − Gεy
(4)
G=
(5)
o
⁄
where 0 and u are the uniaxial static flow stresses at εy and εu , respectively. 2.2. Multiple particle impact Material erosions that occur in most industrial applications are caused by continuous particle collision. During the fracturing process, proppants were carried in the fracturing fluid and injected through the pipeline under high pressure pump. The particles impacted the tool surface continuously and caused severe erosion wear. A multiple particle model was established to explore the dynamic process and mechanism. The continuous effect of an erodent can be considered in this model, which can reflect actual working conditions. 2.3. Material properties Multiple particle erosion was simulated on the Q345 material. The mechanical properties of the target material, such as Poisson’s ratio, yield strength, and Young’s modulus at a temperature of 20 ◦ C, are given in Table 1. The materials are assumed to be isotropic,
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Fig. 2. Eroded craters with plastic strain contours under different impact velocity and angle.
linear elastic, and plastic, following the von Mises yield criterion. To simulate the fracturing working conditions, the erodent was a spherical particle with a density of 2100 kg/m3 and diameter of 800 m. 2.4. Mesh generation Fig. 1 shows the mesh generation of the 3D geometry. Hexahedral structured grids were used to mesh the target material. To
improve calculation accuracy and save computational resources, the contact region was meshed with a fine grid. The particle was produced using meshed O-type mesh generation method. Grid independency was conducted to guarantee that the mesh size was neither time-consuming nor low in accuracy. Table 2 shows wear depth profiles with different grid numbers under an impact velocity of 15 m/s and 90◦ impact angle. The eroded depth values indicated that the solution was essentially grid-independent when the number of grids was more than 250,000. The multiple particle
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Table 2 Grid independency study. Grid index
Grid number
Maximum eroded depth (m)
1 2 3 4 5
62,500 125,316 250,000 499,849 1,000,000
5.43 6.27 6.52 6.53 6.53
Fig. 4. Depth of eroded crater with plastic strain contours during multiple collisions.
Fig. 3. Von Mises stress distribued in Z axis direction after multiple collisions.
impact process was simulated in the commercially available software ANSYS/LS-DYNA, which was one of the most popular solvers for explicit dynamics problems. 2.5. Boundary conditions The particle was set as a rigid body, which meant that the strain of particle was zero during the contact process. To save on calculation time and computer resource, only a half-model was established. Symmetry conditions were applied to the symmetry plane. The symmetry plane was constrained to move in the X direction, the bottom plane was fixed, and multiple particles were constrained from displacement in the X direction and rotation around the Y and Z axes. 3. Results and discussion 3.1. Characteristics of eroded craters Fig. 2 shows the plastic strain contours under impact velocities of 15 m/s, 20 m/s, and 25 m/s and an impact angle of 45◦ and 90◦ . Under the impact angle of 45◦ , the eroded crater demonstrated obvious shearing lip with asymmetric characteristics. The eroded morphologies became increasingly evident with the increase of impact velocity because of shear and extrusion effects. Under the impact angle of 90◦ , eroded crater was symmetrical and the depth of the crater and the height of shearing lips around the crater increased with impact velocity. The size and depth of crater were related to the impact parameters and target material properties. In general, when the impact velocity is high, the eroded crater was large. As mentioned above, an eroded crater appeared on the surface of target material after multiple particle impingements. The stress distributed in the target material exceeds the yield limit of the material. Fig. 3 shows that the stress distribution in the Z axis after three particle collisions. When the impact velocity increased from 15 m/s to 25 m/s, the maximum stress region appeared to have a downward trend, which can be confirmed in Fig. 3. Similarly, the
Fig. 5. Effect of impact velocity on the contact time.
thickness of the residual stress layer was high under high impact velocity. This finding was similar to the shot peening process, which indicated that the existence of residual stress layer can significantly improve fatigue performance [29,30]. The position of maximum residual stress is closely related with the impact parameters, such as impact velocity and particle size. Fig. 4 presents the depth of eroded craters with the plastic strain contours under different impact velocities. The depth of the crater was defined as a vertical distance between the bottom of the crater and the top surface. The figure shows that the total plastic deformation increased with impact number and velocity and the depth value of each impingement decreased with the increase in impact number. Maximum depth value was obtained after the first impingement. Compared with that in the first impingement, the depth value of the second and third was smaller. This result can be attributed to surface hardening effects. The surface of the target material presented a lower plastic performance and higher surface hardness after multiple impingements. 3.2. Contact time and energy loss Contact time between the erodent and the target surface was difficult to achieve through physical experiments. Contact time can be recorded clearly in the numerical model. Fig. 5 shows the contact time under different impact velocities. Contact time decreased with increased impact velocity. Fig. 6 shows the contact time under different impact angles. Contact time was large under low-impact angle because of the
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Fig. 6. Effect of impact angle on the contact time.
Fig. 8. Effect of impact angle on the energy loss rate.
with the increase of impact velocity, which indicated that additional energy transferred to the target material. Maximum energy loss rate reached 76% under the impact velocity of 25 m/s. Energy loss for subsequent collisions decreased because of high surface hardness and small contact time, which can be confirmed from previous analyses. Fig. 8 plots the energy loss curve after multiple particle collisions under different impact angles. The figure shows that the energy loss increased with impact angle. In general, energy loss was small under an impact angle of 0◦ to 15◦ because the tangential component of velocity was large. From 15◦ to 75◦ , energy loss increased linearly impact angle. From 75◦ to 90◦ , the energy loss after first collision increased slowly. Energy loss after second and third collision changed slightly. This result can be attributed to the surface hardening effect. 3.3. Erosion process analyses
Fig. 7. Effect of impact velocity on the energy loss rate.
large tangential component of velocity, which helped form the ploughing characteristics on the target surface. Collision occurred in less than 10−5 s. The target material deformed quickly. Thus, heat generated by plastic deformation did not have sufficient time to transfer around. Therefore, the dynamic erosion process occurred in an adiabatic environment and resulted in temperature increase. Contact time decreased with increased impact number. This phenomenon occurred because surface hardening after multiple particle impingements increased surface harness and stiffness coefficient, thereby decreasing contact time. Therefore, the erosion process was affected by the thermal effect and strain hardening phenomenon. Energy exchange occurred between the particles and the target material during the solid particle collision. Hutchings et al. used high-speed photography to examine energy distribution during collision [31]. Part of kinetic energy was transferred to the material during collision [32]. This part of transferred energy consisted of plastic deformation and elastic wave energy. The energy used for plastic deformation would finally become heat energy and part of stored energy. The kinetic energy loss rate was calculated using the following equation. =
Ek 1 − Ek 2Ek1
(6)
where is the kinetic energy loss rate, Ek1 is the initial kinetic energy before collision, and Ek2 is the final kinetic energy after collision. Fig. 7 shows the rate of energy loss of particles after multiple collisions under different impact velocities. Energy loss rate increased
3.3.1. Dynamic erosion characteristics Fig. 9 shows the geometry evolution of eroded crater with the plastic strain contours after multiple particle collisions. The deformation region expanded with the increase in impact number, which emerged from continuous collision. The maximum plastic deformation was located in the contact region. During the collision, the contact force increased rapidly and resulted in the removal of the target material. After the fourth collision, the outer region of the eroded crater exhibited high plastic deformation, which indicated that part of material was extruded and formed lips. After the continuous collisions, the target material would finally detach the surface because of material failure. To further study the collision evolution, specified grids were chosen for quantitative analyses. Fig. 10 shows the comparison of plastic strains for specified grids. Fig. 10a–e plots the plastic strain curve with the impact number. The analysis of the plastic deformation curve indicates that the plastic deformation in the central region increased dramatically at the first particle collision, as shown in Fig. 10a-b. Grid No. 30465 exhibited the maximum plastic deformation strain at the second collision, as shown in Fig. 10c. Grid Nos. 30466 and 30467 exhibited the maximum plastic deformation strain at the third collision, as shown in Fig. 10d-e. Plastic deformation expanded from the central grid to the outer grid. Similarly, the accumulation of plastic deformation was evident at the outer region. The geometric parameters were studied to examine the geometry evolution of the eroded crater. Fig. 11 presents the evolution of crater parameters with the increase of collision numbers. The width and depth with the plastic strain contours were also defined,
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Fig. 9. Geometry evolution of eroded crater during multiple collisions.
as shown in Fig. 11. Width and depth can be used to characterize the eroded crater because of the symmetry of model. The plastic deformation value of the first collision was the largest. Growth rates then became slower with the further increase of impact number. This trend was similar to the change of maximum plastic deformation value, as shown in Fig. 9. This result can be attributed to the effect of strain-hardening and residual stress inner the target material.
3.3.2. Dynamic erosion mechanism Interaction forces were the main factors of material failure during the erosion process. According to the momentum theorem, the
contact forces between the erodent and the target material can be defined as follows: Ft =
d(m · v · cos ˛) dt
(7)
Fn =
d(m · v · sin ˛) dt
(8)
where Ft is the tangential contact force, Fn is the normal contact force, m is the mass of erodent, v is the impact velocity, and ␣ is the impact angle. Fig. 12 shows the dynamic erosion process under 45◦ impact angle and 50 m/s impact velocity. At 1.2 s, the particle began to contact the target material, and the contact region exhibited severe plastic deformation. The contact forces consisted of normal force and tangential force due to normal velocity component and
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Fig. 11. Evolution of geometric parameters of eroded crater after multiple collisions. Fig. 10. Plastic strains of specified grids during multiple particles collisions.
tangential velocity component. At 1.5 s, the target surface was squeezed under the effect of normal force. Meanwhile, tangential force applied on the material then caused flow of target material [33]. At 1.8 s, the eroded region presented obvious shear characteristics. At 2.1 s, the normal velocity component slowed down to
zero, and the tangential velocity component was the dominant erosion factor. Fig. 12d–f shows that tangential cutting was the main characteristic under low impact angle. Fig. 13 shows the dynamic erosion process under 90◦ impact angle and 50 m/s impact velocity. The first particle began to impact
Fig. 12. Dynamic erosion process under 45◦ impact angle and 50 m/s impact velocity.
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Fig. 13. Dynamic erosion process under 90◦ impact angle and 50 m/s impact velocity.
the target surface at 0.9 s. According to Eqs. (7) and (8), contact force was large because of the large normal velocity and short contact time. From 1.2 to 1.8 s, the dynamic erosion process completed and the particle bounced off the surface. The target material was extruded and formed lips around the crater under normal contact force. Under repetitive particle collisions, the target material would cause surface cracking and fatigue. The lips and debris formed on the surface were removed by the subsequent collisions. The continuous collisions contributed to the detaching of target material, which can be validated by the experimental study reported [34]. Therefore, the erosion mechanisms under different eroded parameters can be identified. Fig. 14 shows the dynamic evolution of contact force imposed on the particle under 45◦ impact angle and 50 m/s impact velocity. The tangential and normal contact forces gained several maximum and minimum values during the dynamic erosion process. In addition, the normal force was larger than the tangential force, which meant that the vertical impact was more severe. Fig. 15 shows the dynamic evolution of contact force imposed on the particle under 90◦ impact angle and 50 m/s impact velocity. The tangential force was nearly zero. Normal force increased quickly with the time, then decreased to zero during the dynamic erosion process. The time to reach the peak value of the normal contact force under 90◦ impact angle was shorter than that under 45◦
Fig. 14. Dynamic evolution of contact force under 45◦ impact angle and 50 m/s impact velocity.
impact angle because vertical velocity was large under high impact angle, thereby resulting in fast deformation.
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Fig. 15. Dynamic evolution of contact force under 90◦ impact angle and 50 m/s impact velocity.
4. Conclusions This study investigated the dynamic erosion process using multiple particle model in 3D. The effect of impact parameters on the dynamic erosion process was investigated. The salient features of the study are summarized as follows. (1) Maximum plastic deformation occurred at the first particle collision and decreased with the increase of collision number. During multiple particle collisions, strain hardening phenomenon was observed, which significantly influenced the impact process. (2) The impact parameters significantly affected the contact time between erodent and target surface. Energy loss increased with impact velocity and angle, which indicates severe collision during erosion process. The width and depth of eroded crater first increased sharply with the collision number, then increased gradually with the further increase of collision number. (3) The dynamic erosion process can be identified by analyzing the geometry evolution at different contact times. Under low impact angle, both the tangential and normal contact force had several maximum and minimum values during the dynamic erosion process. The main erosion mechanisms were plastic deformation and cutting due to tangential shear. Under high impact angle, the normal force increased quickly with the time, then decreased to zero during the dynamic erosion process. Vertical cracking and fatigue of surface material were the primary erosion mechanisms because of repetitive solid particle collisions. Acknowledgments The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 51675535), the Fundamental Research Funds for the Central Universities (Grant No. 15CX08007A, 17CX02058, 17CX05022). References [1] N. Ojala, K. Valtonen, A. Antikainen, A. Kemppainen, J. Minkkinen, O. Oja, V.T. Kuokkala, Wear performance of quenched wear resistant steels in abrasive slurry erosion, Wear 354-355 (2016) 21–31. [2] R. Tarodiya, B.K. Gandhi, Hydraulic performance and erosive wear of centrifugal slurry pumps-a review, Powder Technol. 305 (2017) 27–38. [3] F. Darihaki, E. Hajidavalloo, A. Ghasemzadeh, G.A. Safian, Erosion prediction for slurry flow in choke geometry, Wear 372-373 (2017) 42–53.
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