Modelo de Coef de Rest

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International Journal of Impact Engineering 27 (2002) 317341

Modeling the dependence of the coefficient of restitution on the

impact velocity in elasto-plastic collisions

Xiang Zhang1, Loc Vu-Quoc*

Aerospace Engineering, Mechanics & Engineering Science, University of Florida, Gainesville, FL 32611, USA

Received 17 January 1999; received in revised form 18 February 2001; accepted 16 August 2001

Abstract

We discuss the modeling of the coefficient of restitution as a function of the incoming velocity in elasto-

plastic collisions with normal frictionless impact, and compare the results from nonlinear finite-element

analysis to those of two recent normal force displacement models: One by Thornton (ASME J. Appl. Mech.

64 (1997) 383) and one by Vu-Quoc and Zhang (Proc. R. Soc. London, Ser. A 455 (1999) 4013) which is the

displacement-driven counterpart of the force-driven model proposed by Vu-Quoc, Zhang, and Lesburg

(ASME. J. Appl. Mech. 67 (2000) 363). The resulting values of the coefficient of restitution are also

compared to those from the model proposed in Stronge (in: R.C. Batra, A.K. Mal, G.P. MacSithigh (Eds.),

Impact Waves and Fractures, ASME AMD 205 (1995) 351). The relationships among the coefficient of

restitution, the incoming velocity, the collision time, the contact force/displacement, the normal pressure

distribution are presented and discussed. These results establish the better accuracy provided by the model

proposed by Vu-Quoc, Zhang, and Lesburg, when compared to previously proposed models. r 2002

Published by Elsevier Science Ltd.

Keywords: Elasto-plastic collision; Finite element analysis; Contact mechanics; Coefficient of restitution

1. Introduction

The collision between deformable objects has been the subject of intensive investigation by

many researchers using theoretical, numerical, and experimental methods (e.g., [15]). Our work is

motivated primarily by the need to develop more accurate and reliable contact forcedisplacement

(FD) models for granular flow simulations using the discrete-element method (DEM) (see [6,9]).

*Corresponding author. Tel.: +1-352-392-6227; fax: +1-352-392-7303. URL: www.aero.ufl.edu/~vql/.

E-mail address: [email protected] (L. Vu-Quoc).1Graduate research assistant; now with Siemens Corporate Research, Princeton, New Jersey.

0734-743X/02/$ - see front matterr 2002 Published by Elsevier Science Ltd.

PII: S 0 7 3 4 - 7 4 3 X ( 0 1 )0 0 0 5 2 - 5


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For elastic contact, Hertzian contact mechanics (see [7,8]) provides an accurate nonlinear elastic

model. In granular flow simulations, often much simpler (linear) models are used.

When plastic deformation is involved, the collision/contact problems become so complicated

that an accurate theoretical solution is difficult to obtain. In most collisions, plastic deformationoccurs, causing energy to be dissipated, and resulting in a coefficient of restitution less than unity.

For elasto-plastic collisions, Walton and Braun proposed a simplified linear model based on finite

element analysis (FEA) results. A more refined model was proposed in Thornton [10]. More

recently, a new elasto-plastic normal FD (NFD) model based on an additive decomposition of the

contact radius and a generalization of Hertzian contact mechanics to the nonlinear materials is

proposed in Vu-Quoc and Zhang [11]. Even though experimental results were presented in

Goldsmith [1] and in Kangur and Kleis [4], the material and geometry properties were not given in

detail for use in a model (which could be either a finite element model or a forcedisplacement

model for granular flow simulations).

In the present work, we use the nonlinear FEA code ABAQUS [12] to model the dynamicprocess of the collision between a deformable sphere and a rigid, frictionless planar surface. Both

elastic material and elasto-plastic material are considered. The results from the elastic material are

compared to Hertz contact to calibrate the FEA model. After we switch to an elasto-plastic

material in the FEA model, the results are then used to compare to those obtained from the

elasto-plastic FD models by Thornton [10] and by Vu-Quoc and Zhang [11]. Such a comparison

can be viewed as a validation of these FE models for granular flow simulations. Note that we are

considering here the case of normal frictionless impact of spheres. For oblique impacts of

deformable bodies, friction plays an important role in the coefficients of restitution; we refer to

Stronge [13] and Vu-Quoc et al. [5] for more details.

2. Finite element model

Fig. 1 shows a sphere colliding against a frictionless rigid planar surface, a situation equivalent

to two identical spheres with the same velocity amplitude colliding against each other. In our

nonlinear dynamic FEA, the size and the material properties of the sphere are chosen to be, radius

Fig. 1. A sphere colliding with a frictionless rigid planar surface.

X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) 317341318


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R 0:1 m; Youngs modulus E 7:0 1010 N=m2; Poissons ratio n 0:3; and density r 2:699 103 kg=m3: For elasto-plastic collisions, elasto-perfectly plastic model with von Misesyield criterion is employed. The yield stress of the material is chosen to be sY 1:0 10

8 N=m2:

Since there is no rotation of the particle about itself in the collision that we study, axisymmetricFE models are employed to carry out the analyses. All axisymmetric elements used are CAX6

elements of the nonlinear FE code ABAQUS [12]. Fig. 2 shows one of the meshes employed in our

FEA. In this FE model, the half sphere is discretized into 1640 axisymmetric six-node triangular

elements (Fig. 2(a)) with a total of 3288 nodes, and with three levels of mesh refinement around

the contact area (Fig. 2(b)). The nodes 13288 were numbered from top to bottom of the half

circle shown in Fig. 2(a), with nodes 12503288 concentrated in the more refined area around the

contact point (Fig. 2(b)). We designate this FE model as model B. The other two FE models,

models A and C, employed in our analyses are similar to the one shown in Fig. 2, but with

different number of levels of mesh refinement and different number of elements. Model A has 928

axisymmetric six-node triangular elements and 1886 nodes, with two levels of mesh refinementaround the contact area. Model C has 2951 axisymmetric six-node triangular elements and 5892

1

2

3 1

2

(a)

(b)

Fig. 2. Axisymmetric finite element model of the sphere. (a) Sphere discretization. (b) Zoomed-in view around the

contact area.

X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) 317341 319


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nodes, with four levels of mesh refinement around the contact area. When not specified, the FEA

results presented later are obtained using model B (Fig. 2).

For low velocity impacts, the deformation of the sphere during a collision is concentrated in a

small region around the contact area (see later results, such as the contours of Mises stress

presented in Sections 3.2 and 4.2). In order to accurately represent the overall response, we refine

the FE mesh in the small region close to the contact point (Fig. 2(b)). The mesh refinement is

achieved by the use of incompatible interelement matching at the boundary of the different zones

of refined mesh, as illustrated in Fig. 3. In our FE models, these incompatible elements are

connected to each other using multi-point constraints (MPCs). In Fig. 3, the second ordertriangular elements ; ; and are connected using quadratic MPCs. Node m of element andnode n of element do not have independent degrees of freedom (DOF); their displacements are

determined by the quadratic functions of the displacements of nodes i, j, and k, which are

common to elements ; ; and :The contact detection and contact analysis between the sphere and the rigid surface are carried

out using the 1D IRS21A contact elements of ABAQUS [12]. For the FE model shown in Fig. 2,

the size of the contact elements around the contact area is about 1 :02 104 m (half of the size ofthe triangular element), which is much less than the radius of the contact area. For example, the

maximum radius of contact area is about 2:32 103 m for the elasto-plastic collision with an

incoming velocity vin 0:10 m=s (see Section 4.2). Therefore, it can be concluded that thediscretization of the sphere is fine enough to describe the collision behavior accurately.

3. Elastic collisions

Using the nonlinear FE code ABAQUS [12], with the FE models described in Section 2, we

carry out a series of dynamic FEA for elastic collisions between an elastic sphere and a frictionless

rigid surface with different incoming velocities. As mentioned in Section 1, the behavior of such

collisions can be solved theoretically using Hertz theory through a quasi-static procedure. In this

i

m

j

n

k

Fig. 3. Incompatible elements connected by multi-point constraints (MPCs).



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section, we compare our dynamic FEA results of elastic collisions with the corresponding results

obtained by applying the Hertz theory through a quasi-static procedure. In addition, the

errorFwhich may be caused by the energy dissipation due to wave propagation and possible

numerical stability problem in the case of extremely soft materialFis discussed by comparing the

results of collision.

3.1. The Hertz theory for elastic contact

Fig. 4 depicts the contact between two spheres subjected to normal load P: The equivalent

elastic modulus En

and the equivalent contact curvature 1Rn are given as follows:2

En :1 i n

2

iE

1 j n2

jE

13:1

and

1

Rn:

1

iR

1

jR

; 3:2

Fig. 4. Two spheres in contact, subjected to normal load P:

2The symbol : designates equal by definition.



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where iR is the radius of sphere i; in and iE the Poisson ratio and Youngs modulus of the

material of sphere i; respectively. Similarly, jR; jn , and jE are the same properties for sphere j:The contact area is a circle radius a (Fig. 5(a)). On the contact surface, the distribution of the

Hertz normal pressure p is axisymmetric and shaped as half of an ellipse. At a point A

of a distance r from the center of the contact area (Fig. 5(a)), the normal pressure pr can be

expressed as

pr pm 1 r

a

2 1=2: 3:3

The normal pressure is related to the normal force P by

pm 3P

2pa2: 3:4

Fig. 5(b) depicts the elliptic profile of the Hertz normal pressure across the diameter of the contact

area.

With the radius a of the contact area given by [8, Eq. 4:22]

a 3PRn

4En

1=3; 3:5

the approach of two distant points on the two spheres can be expressed as [8, Eq. 4:23]

ia j a a2

Rn

9P2

16RnEn2 1=3

: 3:6

Introducing Eq. (3.5) into Eq. (3.4), we obtain

pm 3P

2pa2

6PEn2

p3Rn2

1=3: 3:7

Hertzs theory assumes that the contact area is much smaller than the size of the spheres, i.e.,

a5iR and a5jR: Vu-Quoc et al. [5] present a number of FEA results dealing with elastic andelasto-plastic contact of two identical spheres, and a comparison to Hertzian contact results. Their

goal is to use these numerical experiments to construct forcedisplacement models for elasto-

plastic contact based on a generalization of Hertz contact mechanics. Consider the case when an

Fig. 5. Contact area and Hertz normal pressure. (a) Circular contact area. (b) Hertz normal pressure p at Section B-B:



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elasto-perfectly plastic sphere is in contact with a frictionless rigid surface. According to the Hertz

theory with the von Mises yield criterion, the relationship between the yield stress sY and the yield

normal load PYFi.e., the normal load at which an incipient yield occurs inside the sphereFcan

be expressed as follows (see [14])

PY p

3R21 n22

6E2AYnsY

3; 3:8

where AYn is a function of the Poisson ratio n: In order to give an idea of the magnitude ofAYn; for a material with n 0:3; we have AY0:3 1:613; and for n 0:4; we haveAY0:4 1:738:

In an elastic normal collision between a sphere and a rigid surface, the forcedisplacement

relation during the collision can be described using the Hertz theory as if there is a nonlinear

spring acting between two objects; the duration of the collision is given by [3,15].

t 2:94

1:25 ffiffiffi2ppr1 n2E

!2=5 R2vin

1=5; 3:9

where t is the contact time during the collision, r the density of the sphere material, and vin the

incoming velocity. Relation (3.9) between the contact duration time t and the incoming velocity

vin for elastic collisions is validated by Walton [3] using dynamic FEA.

3.2. FEA results

An important result from our dynamic FEA of elastic collisions is the coefficient of restitution.

By definition, the coefficient of restitution in a collision in the normal direction (Fig. 1) can be

calculated using3

e :vout

vin: 3:10

In our FEA, the incoming velocity vin of the sphere is an input parameter, whereas the outgoing

velocity vout of the sphere is obtained by averaging the velocities of all nodes of the FE mesh at a

time right after the sphere is separated from the rigid surface. The coefficient of restitution

obtained from FEA is presented in Table 1. The results show that by using either model A, B, or

C, the coefficient of restitution obtained from FEA of elastic collision is close to one, i.e., more

Table 1

Coefficient of restitution for elastic collisions

vin m=s Model A Model B Model C

0.02 0.9976 0.9997 1.000

0.06 0.9983 0.9970 1.000

0.10 0.9923 0.9984 0.9975

0.20 0.9993 0.9999 1.000

3 In this paper, vin and vout are the magnitudes of the velocity (i.e., positive value), and not the algebraic quantity.



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than 99% of the kinetic energy is recovered from the collision, and the sphere rebounds with an

outgoing velocity of about the same magnitude as that of the incoming velocity, but in the

opposite direction. The following conclusion can be drawn: For low-velocity elastic collision

between a sphere (with properties described in Section 2) and a frictionless rigid surface, theenergy dissipation caused by the elastic wave inside the sphere is very small, thus can be ignored.

More results and discussion on this issue will be presented in Section 3.3 below.

Fig. 6 depicts the plot of the normal contact force P versus the normal displacement4 a for the

elastic collision with incoming velocity vin 0:20 m=s: The unloading path of the forcedisplacement (FD) curve from FEA results is almost on top of the loading path of the FD curve,

meaning that there is almost no energy dissipation. The loading curve produced using the Hertz

theory by Eq. (3.7) is also presented in Fig. 6, which shows that the FEA results agree well with

the Hertz theory for elastic contact. At the points with the highest normal contact force Pmax

1:144 104 N; the maximum normal displacement obtained from FEA results is amax 4:95 105 m; while the corresponding normal displacement produced using the Hertz theory by

Eq. (3.7) is amaxHz 4:99 105 m: The difference between the result from dynamic FEA andthe result from the Hertz theory is only 0.8%.

We also extracted the collision duration time t from our dynamic FEA by subtracting the time

tc; when the sphere comes into contact with the rigid surface, from the time ts; when the spherecompletely separates from the rigid surface, i.e., t ts tc: The maximum possible errorcommitted on t is the integration time-step size, which is very small; for example, the time-step

Fig. 6. Forcedisplacement curve for elastic collision with incoming velocity vin 0:20 m=s:

4 In a granular-flow simulation using the soft-particle technique, the normal displacement a is the normal penetration

into the particle in question, and is either ia or ja in Eq. (3.6). For identical spheres, we have a i a j a: See [6] formore details on the discrete element method (DEM) employed.



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size around the separation of the sphere for the collision with incoming velocity vin 0:10 m=susing FE model B is 3:207 106 s: The results from FEA using different models are presented inTable 2 and Fig. 7, and are compared with the theoretical prediction using the Hertz theory, i.e.,

Eq. (3.9). Compared to the time-step size, the error of the computed collision duration time forthe collision with incoming velocity vin 0:10 m=s is about 0.38%, or less than 1%. Again, ourFEA results agree closely with the theoretical prediction using the Hertz theory with a quasi-static

procedure.

Fig. 8(a) and (b) show the distributions of the normal pressure on the contact surface obtained

from our dynamic FEA results. In these figures, the Hertz normal pressure by Eq. (3.3) is

represented by the solid line; the original FEA results are represented by the small circles 3. In

order to remove the spurious oscillations in the original FEA results, we perform an averaging

process to produce a much smoother curve, shown by the symbols x. The oscillations in the

original FEA results are probably due to the type of contact elements employed. The contact

Table 2

Collision duration (in s) versus incoming velocity for elastic collisions

vin m=s Hertz theory Model A Model B Model C

0.02 1:200 103 1:16 103 1:16 103 1:16 103

0.06 9:298 104 9:33 104 9:27 104 9:30 104

0.10 8:395 104 8:38 104 8:36 104 8:38 104

0.20 7:308 104 7:28 104 7:28 104 7:30 104

Fig. 7. Collision duration versus incoming velocity for elastic collisions.



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radius a from FEA results is obtained by using cubic spline interpolation at zero normal pressure

p 0: This cubic spline is based on the two averaged data points that are closest to the r-axis, andtheir two mirror symmetric points about the r-axis. The distribution of the normal pressure p for

the elastic collision with the incoming velocity vin 0:02 m=s at the time when the normal contactforce P reaches its maximum value Pmax 716:6 N is shown in Fig. 8(a). Even though the originalFEA results oscillate around the Hertz normal pressure, the averaged normal pressure from FEA

results agrees closely with the Hertz theory. In addition, the contact area radius a 0:000877 mfrom FEA results agrees with that from the Hertz theory, ahz 0:000887 m; with a smalldifference of 1.1%. In this case, from Fig. 8(a), there are nine contact elements involved in the

contact. Similarly, the distribution of the normal pressure p of the collision with incoming velocity

vin 0:10 m=s at the time when the normal contact force P is at its maximum value Pmax 4938 N is presented in Fig. 8(b). Again, we observe good agreements between FEA results and the

Hertz theory. In Fig. 8(b), there are 17 contact elements involved in the contact.

Figs. 9a, b shows the contour of the Mises equivalent stress inside the sphere during the

collision at some selected time stations. The Mises equivalent stress is defined as (see [12])

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2S : S

r; 3:11

where S is the stress deviatoric tensor. This Mises equivalent stress q is actually another form of

the second invariant J2 in terms of the stress deviator. In elasto-plastic problem, the Mises

equivalent stress q by Eq. (3.11) can be used to check the plastic deformation. When the von Mises

yield criterion (see [15,16]) is applied, the region where qosY is elastic, and the region where

q sY is plastic.

Fig. 9(a) shows the contour of the Mises equivalent stress inside the sphere for the elastic

collision with incoming velocity vin 0:10 m=s at time t 9:90 105 s when the normal contact

Fig. 8. Normal pressure profile over contact area. Original dynamic FEA data (3); averaged data (). (a) Collision with

vin 0:02 m=s; at maximum normal force Pmax 716:6 N: (b) Collision with vin 0:10 m=s; at maximum normal forcePmax 4938 N:



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force increases to P 999:1 N: It shows that the highest Mises stress is not on the contact surface,but inside the sphere, at about half of the radius a of contact area above the contact

surface. During a collision, high stress levels are concentrated in a small region close to the contact

surface.

Fig. 9(b) shows the contour of the Mises equivalent stress inside the sphere for the elastic

collision with incoming velocity vin 0:20 m=s at time t 5:06 104 s when the normal contact

force decreases to P 8279 N after it reaches its maximum value. We observe that the stress state

inside the sphere changes gradually, i.e., without any dramatic changes, going from loading tounloading. This feature in elastic collisions is very different from that of elasto-plastic collisions

(see Section 4.2 below).

3.3. FEA results for the elastic collision of soft spheres

In order to show the effect of the elastic modulus on the collision behavior, we also carry out

dynamic FEA using the FE model B described in Section 2, with exactly the same impact velocity

and material properties except the Young moduli.

Fig. 9. Contour of Mises stress inside the sphere during elastic collisions. (a) Collision with vin 0:10 m=s; loading att 9:90 105 s and P 999:1 N: (b) Collision with vin 0:20 m=s; unloading at t 5:06 10

4 s and P 8279 N:



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Fig. 10 shows the distributions of the rebounding velocity of all the nodes in the FE model right

after the sphere separates from the rigid surface. When the Youngs modulus of the sphere

material is E 7:0 1010 N=m2; the one that we use for most dynamic FEA in this paper, therebounding velocities of most nodes are almost equal to the impact velocity, except at a very small

region close to the contact area, where there is some fluctuation in the magnitude of the

rebounding velocity. Similar results can be observed from Fig. 10(b) for a softer material with

E 7:0 106 N=m2: Even with a Youngs modulus as 104 times soft as that of the material weemployed in most of the simulations that we present in this paper, we observe that the energy

dissipation caused by the internal elastic wave propagation is still very small. The coefficient ofrestitution obtained from FEA by averaging the rebounding velocity of all nodes in the model is

shown in Table 3.

Clearly, from Fig. 10, the computation of the coefficient of restitution as shown in Eq. (3.10)

depends on the material properties such as Youngs modulus E; mass density r; etc. In otherwords, when the ratio E=rR is smaller, the time for the elastic wave propagating across thesphere is longer; thus, the effect of elastic wave propagation on average rebounding velocity vout is

larger. For most of the elastic and elasto-plastic collisions studied in this paper, the ratio E=rR islarge, resulting in a much higher elastic wave propagation speed. In the case where Youngs

modulus E 7:0 1010 N=m2; mass density r 2:699 103 kg=m3; and sphere radius R

Fig. 10. Rebounding velocity in the sphere versus node numbers for elastic collision with incoming velocity vin

0:10 m=s: (a) Sphere with E 7:0 1010 N=m2; at time t 9:24 104 s: (b) Sphere with E 7:0 106 N=m2; at timet 3:27 102 s:

Table 3

Coefficient of restitution for the sphere with different Youngs moduli in elastic collisions, vin 0:10 m=s

E N=m2 7:0 1010 7:0 106

e 0.9984 0.9890



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0:1 m; the collision duration time t is hundreds of times longer than the time for the elastic waveto propagate across the size of the sphere, thus validating the use of a quasi-static force

displacement (FD) model at the contact point. Such an FD model will be presented shortly in

Section 4.

4. Elasto-plastic collisions

In this section, we present the dynamic FEA results for elasto-plastic collisions between a sphere

of elasto-perfectly plastic material and a frictionless rigid planar surface. In addition, we compare

our FEA results with the results of DEM simulation using the Vu-Quoc and Zhang [11] elasto-

plastic NFD model to show the correctness of the NFD model. At first, a brief introduction of the

Vu-Quoc and Zhang [11] elasto-plastic NFD model is given below.

4.1. Elasto-plastic NFD models for DEM simulation

The elasto-plastic NFD model proposed in Vu-Quoc and Zhang [11] (displacement-driven

version) is developed for simulating elasto-plastic contact between two spheres; in the present

version, the spheres have the same material properties. For two spheres in contact as shown in

Fig. 4, when the normal contact force P is less than, or equal to, the yield normal force PY given

by Eq. (3.8), the behavior of the contact can be determined by the Hertz theory as described in

Section 3.1.

In the case when P > PY; i.e., plastic deformation occurs, to obtain correct simulation results,the effect of the plastic deformation on the forcedisplacement relation should be accounted for.

Let us first consider the case in which the normal force P increases (i.e., the loading case). When

the normal force P is greater than the yield normal force PY; plastic deformation occurs, andcauses the contact area radius to be larger than that in elastic contact. Let aep be the contact area

radius for elasto-plastic contact. We split the elasto-plastic contact radius aep into

aep ae ap; 4:1

where the radius ae corresponds to the elastically recoverable part, and the radius ap is the plastic

correction part, which can be modeled according to

ap 0 for PpPY;

CaP PY for P > PY;( 4:2

based on our FEA results (see [5,11]), and where Ca is a constant depending on the properties of

the spheres. For the elasto-plastic contact between the sphere (with the same properties as

described in Section 2), in contact with a frictionless rigid surface, we obtain Ca 2:33 107 N=m: Further, we assume that the relationship between aep and a still follows Eq. (3.6), butwith a modified radius of local contact curvature Rnep; i.e.,

a aep2

2Rnep; 4:3



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where

Rnep CRRn; 4:4

CR 1:0 for PpPY;

1:0 KcP PY for P > PY;

(4:5

with Kc being a constant determined by the properties of the contacting spheres. For example, for

the elasto-plastic contact between two identical spheres with the same properties as those

described in Section 2, we obtain Kc 2:69 104 1=N (see [11]).

In the displacement-driven version of this new NFD model, with the known input parameters

PY; Ca; Kc; and with a given normal displacement a; we can construct a nonlinear equation interms of the unknown normal contact force P; by combining Eq. (4.1) to Eq. (4.5). This nonlinearequation is then solved by using the NewtonRaphson method for the normal contact force P:

Now let us consider the case where the normal contact force P (and also the normaldisplacement a due to contact) is decreasing (unloading). If the maximum force Pmax is greater

than the yield force PY; there will be plastic deformation, and the residual normal displacementares should be computed by

ares amax aepmax a

pmax2

2CRmaxRn

; 4:6

where CRmax is determined by Eq. (4.5) with P Pmax: Similar to the plastic strain during astress unloading in the continuum plasticity theory, the plastic correction contact radius ap of an

elasto-plastic contact remains constant during unloading, i.e.,

ap apmax 0 for PmaxpPY;

CaPmax PY for Pmax > PY:

(4:7

Unloading is performed elastically following the Hertz theory, but accounting for the plastic

deformation that have occurred.5 Therefore, the elastic contact radius ae during unloading can be

expressed as a function of the normal contact displacement a and the normal residual

displacement ares given in Eq. (4.6) as follows:

ae 2CRmaxRna ares

1=2: 4:8

Note that in both Eqs. (4.6) and (4.8), the elastic radius of curvature Rn is used, instead of the

elasto-plastic radius of curvature Rnep defined by Eq. (4.4). The normal contact force during

unloading can be computed using the Hertz theory as follows:

P 2E

3Rn1 n2

ae3: 4:9

Thornton [10] proposed an NFD model that also accounts for the effect of plastic deformation

on the NFD relationship. In this NFD model, Thornton [10] assumed that quasi-static contact

mechanics theory is valid during a collision between two spheres. In this model, during elastic

5Elastic unloading following the Hertz theory is a good approximation of the numerical-experiment results. For more

details, see [11,14].



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loading, the normal traction (i.e., the distribution of normal pressure on the contact area) and theNFD relationship follow the Hertz theory; when plastic deformation occurs, the normal traction

is equal to a contact yield stress denoted by sYTh everywhere inside the contact area, as shown in

Fig. 11.6 Based on the above assumption, Thornton [10] derived a linear relationship between the

normal displacement a and the normal contact force P after the incipient plastic deformation. For

unloading after the plastic deformation had occurred, Thornton [10] followed the NFD

relationship in elastic Hertzian contact mechanics, but with a larger radius of relative contact

curvature Rnp that resulted from irreversible plastic deformation.

The coefficient of restitution eTh from the Thornton [10] NFD model can thus be derived to be a

function of the incoming velocity vin; and is expressed as follows:

eTh 6ffiffiffi

3p5

!1

1

6

vY

vin

2" #( )1=2vY

vin

vY

vin

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:2 0:2

vY

vin

2s24351

8

Modelo de Coef de Rest

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