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Experimental and numerical study of ball size effect on restitution coefficient in low velocity impacts
International Journal of Impact Engineering 37 (2010) 1037e1044
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International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Experimental and numerical study of ball size effect on restitution coefficient in low velocity impacts A. Aryaei a, K. Hashemnia b, K. Jafarpur b, * a b
K. N. Toosi University of Technology, Tehran 193951999, Iran Shiraz University, Shiraz, Fars 7134851154, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 July 2009 Received in revised form 19 April 2010 Accepted 25 April 2010 Available online 26 May 2010
In this article, the Coefficient of Restitution (COR) and Energy Loss Percentage (ELP) of one-dimensional impacts are determined experimentally for different ball sizes using a drop test apparatus. Ball diameters range from 6 to 12 mm, made of steel and aluminum dropped on steel and aluminum sheets. Furthermore, effects of ball sizes on COR and contact time duration are studied numerically. In addition, time variation of displacement of the midpoint of sheet’s top surface and vibration of the ball’s center are investigated. This is done using a finite element model for elasto-plastic collisions. In this work, dynamic and explicit analyses have been carried out by using the LS-DYNA module of ANSYS. The numerical results are validated by the present experimental data. Moreover, the present results are compared with previous works. Results show that COR decreases as balls’ diameters increase. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Drop test Finite element method Low velocity impact Restitution coefficient
1. Introduction Impact of two spherical bodies or a spherical body with a plane surface is of fundamental importance in numerous engineering applications ranging from macroscopic mechanical engineering to microscopic particle technology. However, impact between two deformable bodies is a very complicated phenomenon. The major characteristics of the problem are the short duration and large localized stresses generated in the vicinity of the contact area. Therefore, plastic deformation is often involved for most practical impact problems [1]. When plastic deformation encountered, the collision/contact problems become so complicated that an accurate theoretical solution is difficult to obtain. In addition, in most collisions, plastic deformation occurs, causing energy to be dissipated, and resulting in a Coefficient of Restitutions (COR) less than unity. COR is a coefficient that indicates collision type. This coefficient is between zero and one. The more this coefficient is closer to one the more the impact is closer to an elastic impact and vice versa. COR for several materials have been measured experimentally [2e8]. Numerical simulations using the Finite Element * Corresponding author. Tel.: þ98 711 6133020; fax: þ98 711 6473511. E-mail addresses: [email protected] (A. Aryaei), k_hashemnia@ yahoo.com (K. Hashemnia), [email protected], [email protected] (K. Jafarpur). 0734-743X/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2010.04.005
Method (FEM) have been performed for determination of impact properties of colliding bodies [9e21]. In these works various models have been used to simulate the impact phenomenon such as elasto-perfectly plastic impact, elasto-plastic impacts, elastic impacts and elasto-viscoplastic impacts. Furthermore, many aspects of impact phenomenon were under investigation. For example, effect of material strain rate sensitivity and impact velocity on COR, contact pressure, contact duration and contact force for low velocity impact between two identical steel spheres were studied by Minamoto et al. [9] experimentally and numerically. Impact velocity dependence of COR has been investigated by Zhang and Vu-Quoc [10] using elastic and elasto-plastic collision models. In Zhang’s work, nonlinear FEM code of ABAQUS was used to model impact between a deformable sphere and a rigid, frictionless planer surface. Coaplen et al. [11] have studied the impact between bodies with dissimilar materials to determine composite COR theoretically. It’s worth noting that composite COR represents the energy dissipated during a collision between two bodies with dissimilar materials. Collision and recoil of elasto-plastic particles in low velocity impact have been studied theoretically by Weir and Tallon [12]. The modeling of impact mechanism has been presented in some recent papers [13e21]. Wong et al. [13] have modeled the contact between particles using Discrete Element Method (DEM). In their work, springs, dampers and sliding friction interfaces are used to predict impact behavior. Additionally, they have performed some
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Nomenclature a d1d2 d1d2 E1E2 DE %DE F g h1 h2
radius of contact area Ball’s diameters mass of spheres young modulii of spheres energy loss value energy loss percent contact force gravitational acceleration height of dropping height of restitution
m* R R* t
effective mass sphere radius effective radius time duration of collision r Ball’s density sy yield stress n1 ; n2 Poisson ratios of colliding balls Ball normal speed before collision vin Ball normal speed after collision vout vn1 ; vn2 colliding Balls’ normal velocities before collision v’n1 ; v’n2 colliding Balls’ normal velocities after collision
experiments to determine COR for selected materials in impacts with relatively low velocities [13]. For elasto-plastic collisions, Walton and Braun [20] proposed a simplified linear model based on a finite element analysis (FEA). Another model was suggested by Thornton [14]. Recently, a new elasto-plastic Normal Force-Displacement (NFD) model based on an additive decomposition of the contact radius and a generalization of Hertzian contact mechanics to the nonlinear materials is proposed by Vu-Quoc and Zhang [15]. Moreover, Vu-Quoc et al. [21] developed an accurate new NFD model for contacting spheres that is based on an additive decomposition of the contact e area radius, a correction of the local curvature of the particles at the contact point, and a formulation inspired from continuum theory of elastoplasticity. Their results were compared with Hertz theory and a FEA model. The COR obtained by their model have good agreement with FEA results. Additionally, they presented an accurate elasto-plastic frictional and another tangential force-displacement (TFD) model for granular-flow simulations, respectively [22,23]. The tangential COR obtained by their model have good agreement with the FEA results. Besides, experimental results were presented by Goldsmith [2] and Kangur and Kleis [16]. For oblique impacts of deformable bodies, friction plays an important role in the COR; for more details one can refer to Stronge [17] and Vu-Quoc et al. [18]. On the other hand, in the case of normal impacts, friction has no considerable effect on COR [18,21e23]. Impact theories are widely used in shot peening process and multi-body dynamics. Due to wide use of the shot peening process in different industries such as aircraft industry, many projects have been modeled and simulated in shot peening processes [19]. Because of some difficulties in analytical solutions, use of FEM is very common [24e28]. Two typical commercial finite element codes for modeling shot peening are LS-DYNA and ABAQUS-explicit [29e32]. Therefore, precise determination of COR is very important in analyzing impact properties of two colliding bodies. For this reason many researchers use theoretical, numerical, and experimental methods to determine COR [2,18]. In almost all previous researches, the ball size effect on COR was not under consideration. However, in
the present work, this effect on COR and contact time duration are studied numerically with two different element shapes. Also, time variation displacement of midpoint of sheet’s top surface and vibration of the ball’s center are investigated, as well. Moreover, through experiments, steel and Aluminum balls with different diameters have been tested to determine CORs in their impact with steel and aluminum sheets in relatively low velocities by use of a drop test apparatus.
Fig. 1. A ball colliding with a planar surface at rest.
Fig. 2. Two elastic spheres subjected to a concentric force.
2. Theory Velocity of particles after impact can’t be determined solely by solving the linear momentum equation during impact between two particles because of their plastic deformation. Therefore a coefficient called COR _e_ is introduced. It is defined as [33]:
e ¼
v’n2 v’n1 vn1 vn2
(1)
where vn1 and vn2 are particles’ velocities before impact and v’n1 and v’n2 are particles’ velocities after impact. So COR is defined as the ratio between relative velocities after and before impact. This coefficient is between zero and one. The impact is fully plastic if and only if COR is zero; it is fully elastic if and only if COR is one. In the
A. Aryaei et al. / International Journal of Impact Engineering 37 (2010) 1037e1044
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Fig. 4. Eight-node element for meshing plate and ball.
sffiffiffiffiffiffi h2 e ¼ h1
Fig. 3. Schematic of the test apparatus.
latter case no energy loss is occurred. In actual cases this coefficient is neither zero nor one. According to Fig. 1, if one of the particles is at rest, relation (1) is reduced to its new form:
vout e ¼ vin
(2)
where vin ; vout are the initial and final particle speeds, respectively. Thus, if a particle moving with speed vin collides to a resting surface, it rebounds with speed vout .
vout ¼ evin
(3)
Now, consider that a ball drops over a surface from a height of h1 . Through comparing the magnitude of the forces involved, it can be shown that air resistance and buoyancy forces have no significant effect on the results of the present experimental data. Moreover, in papers in which collisions between two bodies have been investigated, air resistance is neglected during two bodies’ travel [9,12,13]. Therefore:
vin
pffiffiffiffiffiffiffiffiffiffiffi ¼ 2gh1
(4)
It is clear that its speed after impact would be:
vout ¼ e
pffiffiffiffiffiffiffiffiffiffiffi 2gh1
(5)
Then it rebounds to height h2 over the surface. Therefore COR can be determined from:
h2 ¼ e2 h1
(6)
Or:
(7)
The energy loss during this impact is:
DE ¼ mgðh1 h2 Þ ¼ mgh1 1 e2 %DE ¼
1 e2 100
Density (kg/m3)
Hardness (BHN)
Young modules (GPa)
Poisson’s ratio
Yield strength (MPa)
Steel Aluminum
7566.5 2639.7
111 29
207 71.7
0.294 .333
225 100
(9)
Friction in direct (straight) impacts doesn’t have significant role; this is due to the fact that there is no tangential force interacting between two bodies. This issue has been pointed out in several references [18,21e23]. However, in oblique impacts friction should be considered because of the existence of tangential forces. Thus, in the present work, since we have direct impact and consequently only normal forces are exerted to two bodies, friction forces are not that important and can be neglected.
3. Extension of Hertz theory to impact mechanics Hertz theory is a contact mechanics theory that predicts deformations of two elastic bodies subjected to a concentric force. Also it determines the radius of a circular or width of rectangular contact area for the cases related to two spheres or two cylinders in contact, respectively. Fig. 2 indicates the contact area with radius a when two spheres compressed into each other by a force F [34]. If E1 ; n1 and d1 represent young modulus, Poisson’s ratio and diameter of sphere #1, respectively and the same parameters for sphere #2 are denoted by E2 ; n2 and d2 , then the radius of contact area a between two spheres would be calculated as follows:
d1
Material
(8)
And the Energy Loss Percentage (ELP) is calculated from:
2 3 ð1n21 Þ ð1n22 Þ 1=3 þ 3F E1 E2 5 a ¼ 4 1 þ 1 8
Table 1 Mechanical properties of steel sheet’s material.
(10)
d2
Effective radius R* and effective young modulus E* can be defined as [3]:
1 1 1 ¼ 2 þ d1 d2 R*
(11)
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Fig. 5. (a) Mesh picture of sheet quarter with brick element (isometric view). (b) Mesh picture of ball quarter with tetragonal element. (c) Mesh picture of ball quarter with brick element. (d) Mesh picture of sheet quarter with brick element and ball quarter with tetragonal element (front view).
1 n21 1 n22 1 ¼ þ E1 E2 E*
(12)
So Eq. (9) can be rewritten as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3F R* a ¼ * 4 E
(13)
Eqs. (10)e(13) can be used to solve a contact problem between a sphere (ball) and a plane surface (sheet). For a plane surfaced ¼ Nis assumed. So if the contacting sphere and plane surface are made of the same material then Eq. (10) would reduce to:
Fig. 6. Mesh independency check for 7 mm in diameter steel ball dropped on steel sheet.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3F 1 n2 d 1 a ¼ E 4
(14)
where E; y are young modulus and Poisson’s ratio of the material, respectively and is diameter of the sphere. A theoretical expression for the COR of elasto-plastic impact was developed by Stronge [3]:
2
e ¼
Vy Vin
2
!0:75 8 Vin 2 3 5 Vy 5
(15)
In the above equation, Vy is the impact velocity which is just sufficient to initiate yielding. Now, if we consider two bodies with the same materials:
Fig. 7. Time independency check for 11 mm in diameter steel ball dropped on steel sheet.
A. Aryaei et al. / International Journal of Impact Engineering 37 (2010) 1037e1044
Fig. 8. Present numerical and experimental data of COR variation vs. balls’ diameters for steel balls dropped on steel sheet.
Vy2
4p 3p 4 1:1sy 4 1:1sy R*3 ¼ 5 4 E* m*
(16)
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Fig. 10. COR variation with respect to ball diameter for Aluminum ball dropped on steel sheet.
pffiffiffi !0:4 1:25 2pr 1 y2 R t ¼ 2:94 E ð2Vin Þ0:2
(20)
sy is the yield strength of bodies’ materials. Effective mass m* is defined as:
1 1 1 ¼ þ m1 m2 m*
4. Test procedure description
(17)
where m1 and m2 are the masses of colliding spheres. Now, for the problem of collision of a sphere with a plane surface with the same materials we have:
E* ¼
E 4 ; R* ¼ R; m* ¼ m ¼ pR3 r 3 2 1 y2
(18)
In Eq. (18), R and r are sphere radius and sphere’s material density, respectively. Therefore:
Vy2 ¼
!4 sy 3 3p 4 2:2 1 y2 sy 1:1 r 5 4 E
(19)
From Eqs. (15) to (19), it can be concluded that from a theoretical point of view, spheres’ sizes do not have any effect on the COR related to these impact problems. In an elastic normal collision between a sphere and a rigid surface the duration of collision t is given by Ref. [10].
Fig. 9. Present numerical and experimental data of ELP variation vs. balls’ diameters for steel balls dropped on steel sheet.
Impact experiments were done using a test apparatus consists of a wooden frame. As shown in Fig. 3, a ball is sitting on the top of the 1.5 m height frame from which has the chance to be dropped on a sheet rested on the bottom of the frame. Steel balls and aluminum balls are held with the use of a magnet and a gripper, respectively. After a ball collides with a sheet it rebounds to a height less than the initial height. During ball dropping and rebounding, a film is taken using a DVD recorder. Then films are seen in slow motion mode and heights of restitution were read from the scale. It should be noticed that a ball shouldn’t collide with the former balls impact location because of the plastic deformation (permanent trace) on the sheet. Steel balls were obtained from ball bearings. There are three almost identical balls in each size for the experimental part. Each ball was dropped several times on sheets. Some of the mechanical properties of steel and aluminum sheet’s material are presented in Table 1. Sheet’s hardness is measured through Brinell hardness test. Also, sheet’s density is measured by a simple experiment and presented in the same table. Other properties are taken from available data [34].
Fig. 11. COR variation with impact velocity for steel balls on steel sheets. Fitted lines are based on experimental data in Ref. [13].
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Table 2 Comparison between CORs measured in Ref. [13] and in the present work. COR¼.56
COR¼.54 COR¼.52
Based on correlation derived from Ref. [13]’s diagram for a 1.5 mm ball diameter v ¼ 5.4 m/s Based on correlation derived from Ref. [13]’s diagram for a 3 mm ball diameter v ¼ 5.4 m/s Our experimental results for a 6 mm ball diameter v ¼ 5.4 m/s
5. Finite elements model and analysis The numerical simulation has been carried out by using a threedimensional finite element elasto-plastic model. The analysis was performed by the commercial finite element code ANSYS. In this work, dynamic and explicit analysis has been done using LS-DYNA module of ANSYS. In order to take into account the effect of balls’ sizes, the modeling and analysis of different steel balls dropped on steel plates are presented here. Plates and balls were considered homogeneous and isotropic. SOLID164 shown in Fig. 4 were used for meshing the ball and sheet. This linear element is defined by eight nodes having translations, velocities and accelerations in the nodal X, Y and Z directions as the degrees of freedom [35]. Two different shapes of SOLID164 were used for ball meshing which are in brick and tetragonal shapes while for sheet meshing, only brick shape elements were used. Mesh configuration related to tetragonal and brick elements are indicated from different views for sheet and ball in Fig. 5(bed). Tetragonal and brick element shapes are referred as the “element shape No.1” and “element shape No. 2”, respectively. Moreover, the analysis is symmetric in mesh and geometry with respect to the two perpendicular planes crossing the sheet’s thickness. To set boundary conditions, all degrees of freedom for nodes attached to the bottom of the plate have been constrained. Additionally, elements near impact zone have been fined enough for better convergence and more precision as shown in Fig. 5(a). The steel sheet dimensions are 210 mm 210 mm 17 mm. Ball diameters range from 6 mm to 12 mm as stated above. Fig. 6 shows that after about 16 000 elements of sheet and ball, COR converges to about 0.5. Therefore in all of the following diagrams related to numerical results (Figs. 8, 9, 13e16), the same number of elements has been considered. Fig. 6 belongs to collision of a 7 mm diameter steel ball to a steel sheet. In dynamic investigations, time increments independency must also be checked. As Fig. 7 reveals, COR related to a 11 mm in diameter steel ball dropped on a steel sheet converges to 0.5 approximately
Fig. 12. COR variation vs. ball diameter in aluminum ball dropped on aluminum sheet.
Fig. 13. Contact duration time variation with ball diameter for steel balls dropped on steel sheet obtained numerically and theoretically.
after 250 time increments; in all of the following diagrams related to numerical results, 1000 time increments have been used.
6. Results and discussion Results including diagrams indicating COR and ELP with respect to ball diameters are presented. COR and ELP variation vs. dropping balls’ diameters for steel balls dropped on steel sheets obtained experimentally and numerically can be seen in Figs. 8 and 9. As these figures demonstrate, the present numerical and experimental results are in good agreements. Average values of COR determined numerically is lower than experimental values. The trend of these three data sets is the same. The average value of COR obtained experimentally is 0.5 while this value is 0.48 and 0.45 calculated based on the present numerical model with “Element Shape No.1” and “Element Shape No. 2”, respectively. Experimental average value of ELP is 75% while numerical values are 77% and 79% related to “Element Shape No.1” and “Element Shape No. 2”, respectively. Data obtained from finite element analyses don’t indicate a special and uniform trend in spite of data obtained from experiments, although numerical results are in the same range as experimental results. Therefore it can be concluded that numerical analyses don’t give us results coinciding with our expectation in specific problems all the time. Fig. 10 shows the COR variation with respect to Aluminum balls’ diameters in their collision with steel sheets.
Fig. 14. Variation of COR with ball diameters for steel balls dropped on steel sheet with various number of time steps.
A. Aryaei et al. / International Journal of Impact Engineering 37 (2010) 1037e1044
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Fig. 15. Time variation of velocity of ball’s center for 10 mm in diameter ball dropped on steel sheet obtained numerically.
Fig. 16. Variation of deformation of midpoint of steel sheet top surface with time under collision of 10 mm in diameter steel ball (numerical study).
Clearly, the slope of the fitted line, indicates the overall trend of data is showing even greater value of slope related to the impact of steel balls on steel sheets. Therefore, according to the data presented, we can conclude that the ball size has more significant effect on COR in impacts between a ball and a sheet made from different materials than in those occurred between a ball and a sheet with the same materials. However, we think that this claim needs more investigations to be supported and accepted. As Figs. 8, 9 and 10 show, the present experimental and numerical results indicate the effect of balls’ sizes on COR in spite of those determined theoretically using Eq. (15) and (19). The existing theories can’t predict any size effect on COR for this kind of impact. In addition, we compared the present results with those of Ref. [13] as illustrated in Fig. 11. Emphasizing again that the obtained curve is based on the extrapolation of Ref. [13]’s results, the agreement of the present results and those of Ref. [13] are quite good. Furthermore, a closer look at Fig. 11 given in Ref. [13], clearly shows the effect of ball size in low velocity impacts. Here, in Fig. 11, we have generated COR vs. impact velocity based on Ref. [13]. The equation that is obtained by logarithmic curve fitting for 3 mm size diameter is COR ¼ 0.09Ln(v) þ 0.691 with vto be the impact velocity. A comparison between CORs measured in Ref. [13]’s experimentally and ours is presented in Table 2. Also some similar experiments have been done on aluminum balls dropping on aluminum sheets as shown in Fig. 12. In addition, the present results for impact of aluminum balls on aluminum sheets can be compared with the results given in Refs. [10,12]. Current experimental results are in good agreement with those presented in Refs. [10,12].
To compare the present numerical with theoretical results obtained from Eq. (19), time duration of impact with respect to ball diameters for impacts between steel balls and steel sheets are presented in Fig. 13. It can be observed that the overall trends of the two data sets are the same but some fluctuations can be seen among the numerical data points. However, the average time of the two sets are different. Fig. 14 shows COR variation vs. ball sizes obtained numerically with brick elements for steel balls dropped on steel sheets for different time increment. As illustrated in this figure, data points related to these time increments are very close to each other. The time variation of velocity of center of a 10 mm in diameter steel ball during impact on a steel sheet through 0.001 s time interval with 1000 time increments is shown in Fig. 15. Fig. 16 indicates time variation deformation of midpoint of steel sheet top surface under collision of a 10 mm in diameter steel ball through 0.001 s time interval for 1000 time increments. The jump of the curve approximately at 400 micro seconds shows the elastic rebound after elasto-plastic deformation and the final displacement-the permanent set- is about 0.0334 mm. The small amplitude vibration of ball’s center and midpoint of sheet’s top surface after rebound of ball are shown in Figs. 15 and 16, respectively; the jump could be due to the assumption of ball deformability.
7. Conclusions The effect of ball size on COR in low velocity impacts is investigated through experiments. In addition, the dependence of COR and contact time duration on ball sizes are studied numerically, and
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the midpoint’s displacement time variation of sheet’s top surface and vibration of the ball’s center are examined. Based on the present experimental results, COR decrease as balls’ diameters increase in spite of what existing theories indicate. However, this conclusion needs more investigations. Increase in balls’ diameters may leads to more effective plastic deformation which is one of the main causes of energy loss. The present numerical results using FEM were validated by the experimental data of this work. However, there are some differences between these two studies. Moreover, the effects of deformability and damping of ball’s and sheet’s materials on vibration generation and suppression are well recognized according to the current numerical results. Acknowledgment Authors are very thankful to Mr. Keramat Hashemnia for correcting the English writing. Also, he and Mr. Soroush Aryaei are acknowledged for their help and preparation of instruments and materials required for the test apparatus.
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Contents lists available at ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Experimental and numerical study of ball size effect on restitution coefficient in low velocity impacts A. Aryaei a, K. Hashemnia b, K. Jafarpur b, * a b
K. N. Toosi University of Technology, Tehran 193951999, Iran Shiraz University, Shiraz, Fars 7134851154, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 July 2009 Received in revised form 19 April 2010 Accepted 25 April 2010 Available online 26 May 2010
In this article, the Coefficient of Restitution (COR) and Energy Loss Percentage (ELP) of one-dimensional impacts are determined experimentally for different ball sizes using a drop test apparatus. Ball diameters range from 6 to 12 mm, made of steel and aluminum dropped on steel and aluminum sheets. Furthermore, effects of ball sizes on COR and contact time duration are studied numerically. In addition, time variation of displacement of the midpoint of sheet’s top surface and vibration of the ball’s center are investigated. This is done using a finite element model for elasto-plastic collisions. In this work, dynamic and explicit analyses have been carried out by using the LS-DYNA module of ANSYS. The numerical results are validated by the present experimental data. Moreover, the present results are compared with previous works. Results show that COR decreases as balls’ diameters increase. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Drop test Finite element method Low velocity impact Restitution coefficient
1. Introduction Impact of two spherical bodies or a spherical body with a plane surface is of fundamental importance in numerous engineering applications ranging from macroscopic mechanical engineering to microscopic particle technology. However, impact between two deformable bodies is a very complicated phenomenon. The major characteristics of the problem are the short duration and large localized stresses generated in the vicinity of the contact area. Therefore, plastic deformation is often involved for most practical impact problems [1]. When plastic deformation encountered, the collision/contact problems become so complicated that an accurate theoretical solution is difficult to obtain. In addition, in most collisions, plastic deformation occurs, causing energy to be dissipated, and resulting in a Coefficient of Restitutions (COR) less than unity. COR is a coefficient that indicates collision type. This coefficient is between zero and one. The more this coefficient is closer to one the more the impact is closer to an elastic impact and vice versa. COR for several materials have been measured experimentally [2e8]. Numerical simulations using the Finite Element * Corresponding author. Tel.: þ98 711 6133020; fax: þ98 711 6473511. E-mail addresses: [email protected] (A. Aryaei), k_hashemnia@ yahoo.com (K. Hashemnia), [email protected], [email protected] (K. Jafarpur). 0734-743X/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2010.04.005
Method (FEM) have been performed for determination of impact properties of colliding bodies [9e21]. In these works various models have been used to simulate the impact phenomenon such as elasto-perfectly plastic impact, elasto-plastic impacts, elastic impacts and elasto-viscoplastic impacts. Furthermore, many aspects of impact phenomenon were under investigation. For example, effect of material strain rate sensitivity and impact velocity on COR, contact pressure, contact duration and contact force for low velocity impact between two identical steel spheres were studied by Minamoto et al. [9] experimentally and numerically. Impact velocity dependence of COR has been investigated by Zhang and Vu-Quoc [10] using elastic and elasto-plastic collision models. In Zhang’s work, nonlinear FEM code of ABAQUS was used to model impact between a deformable sphere and a rigid, frictionless planer surface. Coaplen et al. [11] have studied the impact between bodies with dissimilar materials to determine composite COR theoretically. It’s worth noting that composite COR represents the energy dissipated during a collision between two bodies with dissimilar materials. Collision and recoil of elasto-plastic particles in low velocity impact have been studied theoretically by Weir and Tallon [12]. The modeling of impact mechanism has been presented in some recent papers [13e21]. Wong et al. [13] have modeled the contact between particles using Discrete Element Method (DEM). In their work, springs, dampers and sliding friction interfaces are used to predict impact behavior. Additionally, they have performed some
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Nomenclature a d1d2 d1d2 E1E2 DE %DE F g h1 h2
radius of contact area Ball’s diameters mass of spheres young modulii of spheres energy loss value energy loss percent contact force gravitational acceleration height of dropping height of restitution
m* R R* t
effective mass sphere radius effective radius time duration of collision r Ball’s density sy yield stress n1 ; n2 Poisson ratios of colliding balls Ball normal speed before collision vin Ball normal speed after collision vout vn1 ; vn2 colliding Balls’ normal velocities before collision v’n1 ; v’n2 colliding Balls’ normal velocities after collision
experiments to determine COR for selected materials in impacts with relatively low velocities [13]. For elasto-plastic collisions, Walton and Braun [20] proposed a simplified linear model based on a finite element analysis (FEA). Another model was suggested by Thornton [14]. Recently, a new elasto-plastic Normal Force-Displacement (NFD) model based on an additive decomposition of the contact radius and a generalization of Hertzian contact mechanics to the nonlinear materials is proposed by Vu-Quoc and Zhang [15]. Moreover, Vu-Quoc et al. [21] developed an accurate new NFD model for contacting spheres that is based on an additive decomposition of the contact e area radius, a correction of the local curvature of the particles at the contact point, and a formulation inspired from continuum theory of elastoplasticity. Their results were compared with Hertz theory and a FEA model. The COR obtained by their model have good agreement with FEA results. Additionally, they presented an accurate elasto-plastic frictional and another tangential force-displacement (TFD) model for granular-flow simulations, respectively [22,23]. The tangential COR obtained by their model have good agreement with the FEA results. Besides, experimental results were presented by Goldsmith [2] and Kangur and Kleis [16]. For oblique impacts of deformable bodies, friction plays an important role in the COR; for more details one can refer to Stronge [17] and Vu-Quoc et al. [18]. On the other hand, in the case of normal impacts, friction has no considerable effect on COR [18,21e23]. Impact theories are widely used in shot peening process and multi-body dynamics. Due to wide use of the shot peening process in different industries such as aircraft industry, many projects have been modeled and simulated in shot peening processes [19]. Because of some difficulties in analytical solutions, use of FEM is very common [24e28]. Two typical commercial finite element codes for modeling shot peening are LS-DYNA and ABAQUS-explicit [29e32]. Therefore, precise determination of COR is very important in analyzing impact properties of two colliding bodies. For this reason many researchers use theoretical, numerical, and experimental methods to determine COR [2,18]. In almost all previous researches, the ball size effect on COR was not under consideration. However, in
the present work, this effect on COR and contact time duration are studied numerically with two different element shapes. Also, time variation displacement of midpoint of sheet’s top surface and vibration of the ball’s center are investigated, as well. Moreover, through experiments, steel and Aluminum balls with different diameters have been tested to determine CORs in their impact with steel and aluminum sheets in relatively low velocities by use of a drop test apparatus.
Fig. 1. A ball colliding with a planar surface at rest.
Fig. 2. Two elastic spheres subjected to a concentric force.
2. Theory Velocity of particles after impact can’t be determined solely by solving the linear momentum equation during impact between two particles because of their plastic deformation. Therefore a coefficient called COR _e_ is introduced. It is defined as [33]:
e ¼
v’n2 v’n1 vn1 vn2
(1)
where vn1 and vn2 are particles’ velocities before impact and v’n1 and v’n2 are particles’ velocities after impact. So COR is defined as the ratio between relative velocities after and before impact. This coefficient is between zero and one. The impact is fully plastic if and only if COR is zero; it is fully elastic if and only if COR is one. In the
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Fig. 4. Eight-node element for meshing plate and ball.
sffiffiffiffiffiffi h2 e ¼ h1
Fig. 3. Schematic of the test apparatus.
latter case no energy loss is occurred. In actual cases this coefficient is neither zero nor one. According to Fig. 1, if one of the particles is at rest, relation (1) is reduced to its new form:
vout e ¼ vin
(2)
where vin ; vout are the initial and final particle speeds, respectively. Thus, if a particle moving with speed vin collides to a resting surface, it rebounds with speed vout .
vout ¼ evin
(3)
Now, consider that a ball drops over a surface from a height of h1 . Through comparing the magnitude of the forces involved, it can be shown that air resistance and buoyancy forces have no significant effect on the results of the present experimental data. Moreover, in papers in which collisions between two bodies have been investigated, air resistance is neglected during two bodies’ travel [9,12,13]. Therefore:
vin
pffiffiffiffiffiffiffiffiffiffiffi ¼ 2gh1
(4)
It is clear that its speed after impact would be:
vout ¼ e
pffiffiffiffiffiffiffiffiffiffiffi 2gh1
(5)
Then it rebounds to height h2 over the surface. Therefore COR can be determined from:
h2 ¼ e2 h1
(6)
Or:
(7)
The energy loss during this impact is:
DE ¼ mgðh1 h2 Þ ¼ mgh1 1 e2 %DE ¼
1 e2 100
Density (kg/m3)
Hardness (BHN)
Young modules (GPa)
Poisson’s ratio
Yield strength (MPa)
Steel Aluminum
7566.5 2639.7
111 29
207 71.7
0.294 .333
225 100
(9)
Friction in direct (straight) impacts doesn’t have significant role; this is due to the fact that there is no tangential force interacting between two bodies. This issue has been pointed out in several references [18,21e23]. However, in oblique impacts friction should be considered because of the existence of tangential forces. Thus, in the present work, since we have direct impact and consequently only normal forces are exerted to two bodies, friction forces are not that important and can be neglected.
3. Extension of Hertz theory to impact mechanics Hertz theory is a contact mechanics theory that predicts deformations of two elastic bodies subjected to a concentric force. Also it determines the radius of a circular or width of rectangular contact area for the cases related to two spheres or two cylinders in contact, respectively. Fig. 2 indicates the contact area with radius a when two spheres compressed into each other by a force F [34]. If E1 ; n1 and d1 represent young modulus, Poisson’s ratio and diameter of sphere #1, respectively and the same parameters for sphere #2 are denoted by E2 ; n2 and d2 , then the radius of contact area a between two spheres would be calculated as follows:
d1
Material
(8)
And the Energy Loss Percentage (ELP) is calculated from:
2 3 ð1n21 Þ ð1n22 Þ 1=3 þ 3F E1 E2 5 a ¼ 4 1 þ 1 8
Table 1 Mechanical properties of steel sheet’s material.
(10)
d2
Effective radius R* and effective young modulus E* can be defined as [3]:
1 1 1 ¼ 2 þ d1 d2 R*
(11)
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Fig. 5. (a) Mesh picture of sheet quarter with brick element (isometric view). (b) Mesh picture of ball quarter with tetragonal element. (c) Mesh picture of ball quarter with brick element. (d) Mesh picture of sheet quarter with brick element and ball quarter with tetragonal element (front view).
1 n21 1 n22 1 ¼ þ E1 E2 E*
(12)
So Eq. (9) can be rewritten as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3F R* a ¼ * 4 E
(13)
Eqs. (10)e(13) can be used to solve a contact problem between a sphere (ball) and a plane surface (sheet). For a plane surfaced ¼ Nis assumed. So if the contacting sphere and plane surface are made of the same material then Eq. (10) would reduce to:
Fig. 6. Mesh independency check for 7 mm in diameter steel ball dropped on steel sheet.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3F 1 n2 d 1 a ¼ E 4
(14)
where E; y are young modulus and Poisson’s ratio of the material, respectively and is diameter of the sphere. A theoretical expression for the COR of elasto-plastic impact was developed by Stronge [3]:
2
e ¼
Vy Vin
2
!0:75 8 Vin 2 3 5 Vy 5
(15)
In the above equation, Vy is the impact velocity which is just sufficient to initiate yielding. Now, if we consider two bodies with the same materials:
Fig. 7. Time independency check for 11 mm in diameter steel ball dropped on steel sheet.
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Fig. 8. Present numerical and experimental data of COR variation vs. balls’ diameters for steel balls dropped on steel sheet.
Vy2
4p 3p 4 1:1sy 4 1:1sy R*3 ¼ 5 4 E* m*
(16)
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Fig. 10. COR variation with respect to ball diameter for Aluminum ball dropped on steel sheet.
pffiffiffi !0:4 1:25 2pr 1 y2 R t ¼ 2:94 E ð2Vin Þ0:2
(20)
sy is the yield strength of bodies’ materials. Effective mass m* is defined as:
1 1 1 ¼ þ m1 m2 m*
4. Test procedure description
(17)
where m1 and m2 are the masses of colliding spheres. Now, for the problem of collision of a sphere with a plane surface with the same materials we have:
E* ¼
E 4 ; R* ¼ R; m* ¼ m ¼ pR3 r 3 2 1 y2
(18)
In Eq. (18), R and r are sphere radius and sphere’s material density, respectively. Therefore:
Vy2 ¼
!4 sy 3 3p 4 2:2 1 y2 sy 1:1 r 5 4 E
(19)
From Eqs. (15) to (19), it can be concluded that from a theoretical point of view, spheres’ sizes do not have any effect on the COR related to these impact problems. In an elastic normal collision between a sphere and a rigid surface the duration of collision t is given by Ref. [10].
Fig. 9. Present numerical and experimental data of ELP variation vs. balls’ diameters for steel balls dropped on steel sheet.
Impact experiments were done using a test apparatus consists of a wooden frame. As shown in Fig. 3, a ball is sitting on the top of the 1.5 m height frame from which has the chance to be dropped on a sheet rested on the bottom of the frame. Steel balls and aluminum balls are held with the use of a magnet and a gripper, respectively. After a ball collides with a sheet it rebounds to a height less than the initial height. During ball dropping and rebounding, a film is taken using a DVD recorder. Then films are seen in slow motion mode and heights of restitution were read from the scale. It should be noticed that a ball shouldn’t collide with the former balls impact location because of the plastic deformation (permanent trace) on the sheet. Steel balls were obtained from ball bearings. There are three almost identical balls in each size for the experimental part. Each ball was dropped several times on sheets. Some of the mechanical properties of steel and aluminum sheet’s material are presented in Table 1. Sheet’s hardness is measured through Brinell hardness test. Also, sheet’s density is measured by a simple experiment and presented in the same table. Other properties are taken from available data [34].
Fig. 11. COR variation with impact velocity for steel balls on steel sheets. Fitted lines are based on experimental data in Ref. [13].
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Table 2 Comparison between CORs measured in Ref. [13] and in the present work. COR¼.56
COR¼.54 COR¼.52
Based on correlation derived from Ref. [13]’s diagram for a 1.5 mm ball diameter v ¼ 5.4 m/s Based on correlation derived from Ref. [13]’s diagram for a 3 mm ball diameter v ¼ 5.4 m/s Our experimental results for a 6 mm ball diameter v ¼ 5.4 m/s
5. Finite elements model and analysis The numerical simulation has been carried out by using a threedimensional finite element elasto-plastic model. The analysis was performed by the commercial finite element code ANSYS. In this work, dynamic and explicit analysis has been done using LS-DYNA module of ANSYS. In order to take into account the effect of balls’ sizes, the modeling and analysis of different steel balls dropped on steel plates are presented here. Plates and balls were considered homogeneous and isotropic. SOLID164 shown in Fig. 4 were used for meshing the ball and sheet. This linear element is defined by eight nodes having translations, velocities and accelerations in the nodal X, Y and Z directions as the degrees of freedom [35]. Two different shapes of SOLID164 were used for ball meshing which are in brick and tetragonal shapes while for sheet meshing, only brick shape elements were used. Mesh configuration related to tetragonal and brick elements are indicated from different views for sheet and ball in Fig. 5(bed). Tetragonal and brick element shapes are referred as the “element shape No.1” and “element shape No. 2”, respectively. Moreover, the analysis is symmetric in mesh and geometry with respect to the two perpendicular planes crossing the sheet’s thickness. To set boundary conditions, all degrees of freedom for nodes attached to the bottom of the plate have been constrained. Additionally, elements near impact zone have been fined enough for better convergence and more precision as shown in Fig. 5(a). The steel sheet dimensions are 210 mm 210 mm 17 mm. Ball diameters range from 6 mm to 12 mm as stated above. Fig. 6 shows that after about 16 000 elements of sheet and ball, COR converges to about 0.5. Therefore in all of the following diagrams related to numerical results (Figs. 8, 9, 13e16), the same number of elements has been considered. Fig. 6 belongs to collision of a 7 mm diameter steel ball to a steel sheet. In dynamic investigations, time increments independency must also be checked. As Fig. 7 reveals, COR related to a 11 mm in diameter steel ball dropped on a steel sheet converges to 0.5 approximately
Fig. 12. COR variation vs. ball diameter in aluminum ball dropped on aluminum sheet.
Fig. 13. Contact duration time variation with ball diameter for steel balls dropped on steel sheet obtained numerically and theoretically.
after 250 time increments; in all of the following diagrams related to numerical results, 1000 time increments have been used.
6. Results and discussion Results including diagrams indicating COR and ELP with respect to ball diameters are presented. COR and ELP variation vs. dropping balls’ diameters for steel balls dropped on steel sheets obtained experimentally and numerically can be seen in Figs. 8 and 9. As these figures demonstrate, the present numerical and experimental results are in good agreements. Average values of COR determined numerically is lower than experimental values. The trend of these three data sets is the same. The average value of COR obtained experimentally is 0.5 while this value is 0.48 and 0.45 calculated based on the present numerical model with “Element Shape No.1” and “Element Shape No. 2”, respectively. Experimental average value of ELP is 75% while numerical values are 77% and 79% related to “Element Shape No.1” and “Element Shape No. 2”, respectively. Data obtained from finite element analyses don’t indicate a special and uniform trend in spite of data obtained from experiments, although numerical results are in the same range as experimental results. Therefore it can be concluded that numerical analyses don’t give us results coinciding with our expectation in specific problems all the time. Fig. 10 shows the COR variation with respect to Aluminum balls’ diameters in their collision with steel sheets.
Fig. 14. Variation of COR with ball diameters for steel balls dropped on steel sheet with various number of time steps.
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Fig. 15. Time variation of velocity of ball’s center for 10 mm in diameter ball dropped on steel sheet obtained numerically.
Fig. 16. Variation of deformation of midpoint of steel sheet top surface with time under collision of 10 mm in diameter steel ball (numerical study).
Clearly, the slope of the fitted line, indicates the overall trend of data is showing even greater value of slope related to the impact of steel balls on steel sheets. Therefore, according to the data presented, we can conclude that the ball size has more significant effect on COR in impacts between a ball and a sheet made from different materials than in those occurred between a ball and a sheet with the same materials. However, we think that this claim needs more investigations to be supported and accepted. As Figs. 8, 9 and 10 show, the present experimental and numerical results indicate the effect of balls’ sizes on COR in spite of those determined theoretically using Eq. (15) and (19). The existing theories can’t predict any size effect on COR for this kind of impact. In addition, we compared the present results with those of Ref. [13] as illustrated in Fig. 11. Emphasizing again that the obtained curve is based on the extrapolation of Ref. [13]’s results, the agreement of the present results and those of Ref. [13] are quite good. Furthermore, a closer look at Fig. 11 given in Ref. [13], clearly shows the effect of ball size in low velocity impacts. Here, in Fig. 11, we have generated COR vs. impact velocity based on Ref. [13]. The equation that is obtained by logarithmic curve fitting for 3 mm size diameter is COR ¼ 0.09Ln(v) þ 0.691 with vto be the impact velocity. A comparison between CORs measured in Ref. [13]’s experimentally and ours is presented in Table 2. Also some similar experiments have been done on aluminum balls dropping on aluminum sheets as shown in Fig. 12. In addition, the present results for impact of aluminum balls on aluminum sheets can be compared with the results given in Refs. [10,12]. Current experimental results are in good agreement with those presented in Refs. [10,12].
To compare the present numerical with theoretical results obtained from Eq. (19), time duration of impact with respect to ball diameters for impacts between steel balls and steel sheets are presented in Fig. 13. It can be observed that the overall trends of the two data sets are the same but some fluctuations can be seen among the numerical data points. However, the average time of the two sets are different. Fig. 14 shows COR variation vs. ball sizes obtained numerically with brick elements for steel balls dropped on steel sheets for different time increment. As illustrated in this figure, data points related to these time increments are very close to each other. The time variation of velocity of center of a 10 mm in diameter steel ball during impact on a steel sheet through 0.001 s time interval with 1000 time increments is shown in Fig. 15. Fig. 16 indicates time variation deformation of midpoint of steel sheet top surface under collision of a 10 mm in diameter steel ball through 0.001 s time interval for 1000 time increments. The jump of the curve approximately at 400 micro seconds shows the elastic rebound after elasto-plastic deformation and the final displacement-the permanent set- is about 0.0334 mm. The small amplitude vibration of ball’s center and midpoint of sheet’s top surface after rebound of ball are shown in Figs. 15 and 16, respectively; the jump could be due to the assumption of ball deformability.
7. Conclusions The effect of ball size on COR in low velocity impacts is investigated through experiments. In addition, the dependence of COR and contact time duration on ball sizes are studied numerically, and
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the midpoint’s displacement time variation of sheet’s top surface and vibration of the ball’s center are examined. Based on the present experimental results, COR decrease as balls’ diameters increase in spite of what existing theories indicate. However, this conclusion needs more investigations. Increase in balls’ diameters may leads to more effective plastic deformation which is one of the main causes of energy loss. The present numerical results using FEM were validated by the experimental data of this work. However, there are some differences between these two studies. Moreover, the effects of deformability and damping of ball’s and sheet’s materials on vibration generation and suppression are well recognized according to the current numerical results. Acknowledgment Authors are very thankful to Mr. Keramat Hashemnia for correcting the English writing. Also, he and Mr. Soroush Aryaei are acknowledged for their help and preparation of instruments and materials required for the test apparatus.
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