Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions

Transcription

1 International Journal of Impact Engineering 27 (2002) Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions Xiang Zhang 1, 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 elastoplastic 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., [1 5]). Our work is motivated primarily by the need to develop more accurate and reliable contact force displacement (FD) models for granular flow simulations using the discrete-element method (DEM) (see [6,9]). *Corresponding author. Tel.: ; fax: URL: address: vu-quoc@ufl.edu (L. Vu-Quoc). 1 Graduate research assistant; now with Siemens Corporate Research, Princeton, New Jersey X/02/$ - see front matter r 2002 Published by Elsevier Science Ltd. PII: S X ( 0 1 )

2 318 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) 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 deformation occurs, 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 force displacement model for granular flow simulations). In the present work, we use the nonlinear FEA code ABAQUS [12] to model the dynamic process 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.

3 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) R ¼ 0:1 m; Young s modulus E ¼ 7: N=m 2 ; Poisson s ratio n ¼ 0:3; and density r ¼ 2: kg=m 3 : For elasto-plastic collisions, elasto-perfectly plastic model with von Mises yield criterion is employed. The yield stress of the material is chosen to be s Y ¼ 1: N=m 2 : Since there is no rotation of the particle about itself in the collision that we study, axisymmetric FE 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 were numbered from top to bottom of the half circle shown in Fig. 2(a), with nodes 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 refinement around the contact area. Model C has 2951 axisymmetric six-node triangular elements and 5892 (a) Fig. 2. Axisymmetric finite element model of the sphere. (a) Sphere discretization. (b) Zoomed-in view around the contact area. (b)

4 320 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) i m j n k Fig. 3. Incompatible elements connected by multi-point constraints (MPCs). 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 order triangular elements ; ; and are connected using quadratic MPCs. Node m of element and node 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: m (half of the size of the triangular element), which is much less than the radius of the contact area. For example, the maximum radius of contact area is about 2: m for the elasto-plastic collision with an incoming velocity v in ¼ 0:10 m=s (see Section 4.2). Therefore, it can be concluded that the discretization 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

5 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) 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 The Hertz theory for elastic contact Fig. 4. Two spheres in contact, subjected to normal load P: Fig. 4 depicts the contact between two spheres subjected to normal load P: The equivalent elastic modulus E n and the equivalent contact curvature 1 R are given as follows: 2 n E n :¼ 1 ðiþ n 2 ðiþe þ 1 ð jþ n 2 1 ð3:1þ ð jþe and 1 R :¼ 1 n ðiþr þ 1 ; ð3:2þ ð jþr 2 The symbol :¼ designates equal by definition.

6 322 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Fig. 5. Contact area and Hertz normal pressure. (a) Circular contact area. (b) Hertz normal pressure p at Section B-B: where ðiþ R is the radius of sphere i; ðiþn and ðiþ E the Poisson ratio and Young s modulus of the material of sphere i; respectively. Similarly, ðjþ R; ðjþ n, and ðjþ E 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 pðrþ can be expressed as pðrþ ¼p m 1 r 2 1=2 : ð3:3þ a The normal pressure is related to the normal force P by p m ¼ 3P 2pa 2: ð3:4þ 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. ð4:22þ] 1=3 a ¼ 3PRn ; ð3:5þ 4E n the approach of two distant points on the two spheres can be expressed as [8, Eq. ð4:23þ] 1=3 : ð3:6þ ðiþa þ ðjþ a ¼ a2 R ¼ 9P 2 n 16R n ðe n Þ 2 Introducing Eq. (3.5) into Eq. (3.4), we obtain p m ¼ 3P 2pa 2 ¼ 6PðEn Þ 2 1=3 p 3 ðr n Þ 2 : ð3:7þ Hertz s theory assumes that the contact area is much smaller than the size of the spheres, i.e., a5 ðiþ R and a5 ð jþ R: Vu-Quoc et al. [5] present a number of FEA results dealing with elastic and elasto-plastic contact of two identical spheres, and a comparison to Hertzian contact results. Their goal is to use these numerical experiments to construct force displacement models for elastoplastic contact based on a generalization of Hertz contact mechanics. Consider the case when an

7 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Table 1 Coefficient of restitution for elastic collisions v in ðm=sþ Model A Model B Model C 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 s Y and the yield normal load P Y Fi.e., the normal load at which an incipient yield occurs inside the spherefcan be expressed as follows (see [14]) P Y ¼ p3 R 2 ð1 n 2 Þ 2 6E 2 ½A Y ðnþs Y Š 3 ; ð3:8þ where A Y ðnþ is a function of the Poisson ratio n: In order to give an idea of the magnitude of A Y ðnþ; for a material with n ¼ 0:3; we have A Y ð0:3þ ¼1:613; and for n ¼ 0:4; we have A Y ð0:4þ ¼1:738: In an elastic normal collision between a sphere and a rigid surface, the force displacement 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]. pffiffiffi! 2=5 1:25 2 prð1 n 2 Þ R t ¼ 2:94 E ð2v in Þ 1=5; ð3:9þ where t is the contact time during the collision, r the density of the sphere material, and v in the incoming velocity. Relation (3.9) between the contact duration time t and the incoming velocity v in for elastic collisions is validated by Walton [3] using dynamic FEA 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 using 3 e :¼ v out : ð3:10þ v in In our FEA, the incoming velocity v in of the sphere is an input parameter, whereas the outgoing velocity v out 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 3 In this paper, v in and v out are the magnitudes of the velocity (i.e., positive value), and not the algebraic quantity.

8 324 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Fig. 6. Force displacement curve for elastic collision with incoming velocity v in ¼ 0:20 m=s: 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, the energy 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 displacement 4 a for the elastic collision with incoming velocity v in ¼ 0:20 m=s: The unloading path of the force displacement (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 P max ¼ 1: N; the maximum normal displacement obtained from FEA results is a max ¼ 4: m; while the corresponding normal displacement produced using the Hertz theory by Eq. (3.7) is ða max Þ Hz ¼ 4: m: The difference between the result from dynamic FEA and the 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 t c ; when the sphere comes into contact with the rigid surface, from the time t s ; when the sphere completely separates from the rigid surface, i.e., t ¼ t s t c : The maximum possible error committed on t is the integration time-step size, which is very small; for example, the time-step 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 ðiþ a or ðjþ a in Eq. (3.6). For identical spheres, we have a ¼ ðiþ a ¼ ðjþ a: See [6] for more details on the discrete element method (DEM) employed.

9 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Table 2 Collision duration (in s) versus incoming velocity for elastic collisions v in ðm=sþ Hertz theory Model A Model B Model C : : : : : : : : : : : : : : : : Fig. 7. Collision duration versus incoming velocity for elastic collisions. size around the separation of the sphere for the collision with incoming velocity v in ¼ 0:10 m=s using FE model B is 3: s: The results from FEA using different models are presented in Table 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 for the collision with incoming velocity v in ¼ 0:10 m=s is about 0.38%, or less than 1%. Again, our FEA 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

10 326 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Fig. 8. Normal pressure profile over contact area. Original dynamic FEA data (3); averaged data (). (a) Collision with v in ¼ 0:02 m=s; at maximum normal force P max ¼ 716:6 N: (b) Collision with v in ¼ 0:10 m=s; at maximum normal force P max ¼ 4938 N: radius a from FEA results is obtained by using cubic spline interpolation at zero normal pressure p ¼ 0: This cubic spline is based on the two averaged data points that are closest to the r-axis, and their two mirror symmetric points about the r-axis. The distribution of the normal pressure p for the elastic collision with the incoming velocity v in ¼ 0:02 m=s at the time when the normal contact force P reaches its maximum value P max ¼ 716:6 N is shown in Fig. 8(a). Even though the original FEA 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 ¼ 0: m from FEA results agrees with that from the Hertz theory, a hz ¼ 0: m; with a small difference 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 v in ¼ 0:10 m=s at the time when the normal contact force P is at its maximum value P max ¼ 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]) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 q ¼ ðs : SÞ; ð3:11þ 2 where S is the stress deviatoric tensor. This Mises equivalent stress q is actually another form of the second invariant J 2 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 qos Y is elastic, and the region where q ¼ s Y is plastic. Fig. 9(a) shows the contour of the Mises equivalent stress inside the sphere for the elastic collision with incoming velocity v in ¼ 0:10 m=s at time t ¼ 9: s when the normal contact

11 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Fig. 9. Contour of Mises stress inside the sphere during elastic collisions. (a) Collision with v in ¼ 0:10 m=s; loading at t ¼ 9: s and P ¼ 999:1 N: (b) Collision with v in ¼ 0:20 m=s; unloading at t ¼ 5: s and P ¼ 8279 N: force increases to P ¼ 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 v in ¼ 0:20 m=s at time t ¼ 5: s when the normal contact force decreases to P ¼ 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 to unloading. This feature in elastic collisions is very different from that of elasto-plastic collisions (see Section 4.2 below) 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.

12 328 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Fig. 10. Rebounding velocity in the sphere versus node numbers for elastic collision with incoming velocity v in ¼ 0:10 m=s: (a) Sphere with E ¼ 7: N=m 2 ; at time t ¼ 9: s: (b) Sphere with E ¼ 7: N=m 2 ; at time t ¼ 3: s: Table 3 Coefficient of restitution for the sphere with different Young s moduli in elastic collisions, v in ¼ 0:10 m=s E ðn=m 2 Þ 7: : e 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 Young s modulus of the sphere material is E ¼ 7: N=m 2 ; the one that we use for most dynamic FEA in this paper, the rebounding 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 ¼ 7: N=m 2 : Even with a Young s modulus as 10 4 times soft as that of the material we employed 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 of restitution 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 Young s modulus E; mass density r; etc. In other words, when the ratio E=ðrRÞ is smaller, the time for the elastic wave propagating across the sphere is longer; thus, the effect of elastic wave propagation on average rebounding velocity v out is larger. For most of the elastic and elasto-plastic collisions studied in this paper, the ratio E=ðrRÞ is large, resulting in a much higher elastic wave propagation speed. In the case where Young s modulus E ¼ 7: N=m 2 ; mass density r ¼ 2: kg=m 3 ; and sphere radius R ¼

13 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) :1 m; the collision duration time t is hundreds of times longer than the time for the elastic wave to 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 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] elastoplastic 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 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 P Y 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 > P Y ; i.e., plastic deformation occurs, to obtain correct simulation results, the effect of the plastic deformation on the force displacement 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 P Y ; plastic deformation occurs, and causes the contact area radius to be larger than that in elastic contact. Let a ep be the contact area radius for elasto-plastic contact. We split the elasto-plastic contact radius a ep into a ep ¼ a e þ a p ; where the radius a e corresponds to the elastically recoverable part, and the radius a p is the plastic correction part, which can be modeled according to ( a p ¼ 0 for PpP Y; ð4:2þ C a ðp P Y Þ for P > P Y ; based on our FEA results (see [5,11]), and where C a 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 C a ¼ 2: N=m: Further, we assume that the relationship between a ep and a still follows Eq. (3.6), but with a modified radius of local contact curvature ðr n Þ ep ; i.e., a ¼ ðaep Þ 2 2ðR n Þ ep ; ð4:3þ ð4:1þ

14 330 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) where ðr n Þ ep ¼ C R R n ; ( C R ¼ 1:0 for PpP Y; 1:0 þ K c ðp P Y Þ for P > P Y ; ð4:4þ ð4:5þ with K c 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 K c ¼ 2: =N (see [11]). In the displacement-driven version of this new NFD model, with the known input parameters P Y ; C a ; K c ; and with a given normal displacement a; we can construct a nonlinear equation in terms of the unknown normal contact force P; by combining Eq. (4.1) to Eq. (4.5). This nonlinear equation is then solved by using the Newton Raphson method for the normal contact force P: Now let us consider the case where the normal contact force P (and also the normal displacement a due to contact) is decreasing (unloading). If the maximum force P max is greater than the yield force P Y ; there will be plastic deformation, and the residual normal displacement a res should be computed by a res ¼ a max ½ðaep Þ max ða p Þ max Š 2 ; ð4:6þ 2ðC R Þ max R n where ðc R Þ max is determined by Eq. (4.5) with P ¼ P max : Similar to the plastic strain during a stress unloading in the continuum plasticity theory, the plastic correction contact radius a p of an elasto-plastic contact remains constant during unloading, i.e., ( a p ¼ða p Þ max ¼ 0 for P maxpp Y ; ð4:7þ C a ðp max P Y Þ for P max > P Y : Unloading is performed elastically following the Hertz theory, but accounting for the plastic deformation that have occurred. 5 Therefore, the elastic contact radius a e during unloading can be expressed as a function of the normal contact displacement a and the normal residual displacement a res given in Eq. (4.6) as follows: a e ¼½2ðC R Þ max R n ða a res ÞŠ 1=2 : ð4:8þ Note that in both Eqs. (4.6) and (4.8), the elastic radius of curvature R n is used, instead of the elasto-plastic radius of curvature ðr n Þ ep defined by Eq. (4.4). The normal contact force during unloading can be computed using the Hertz theory as follows: P ¼ 2E 3R n ð1 n 2 Þ ða e Þ 3 : 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 5 Elastic unloading following the Hertz theory is a good approximation of the numerical-experiment results. For more details, see [11,14]. ð4:9þ

15 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) loading, the normal traction (i.e., the distribution of normal pressure on the contact area) and the NFD relationship follow the Hertz theory; when plastic deformation occurs, the normal traction is equal to a contact yield stress denoted by ðs Y Þ Th everywhere inside the contact area, as shown in Fig 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 R n p that resulted from irreversible plastic deformation. The coefficient of restitution e Th from the Thornton [10] NFD model can thus be derived to be a function of the incoming velocity v in ; and is expressed as follows: ( pffiffi! " 6 3 e Th ¼ 1 1 # ) 8 2 v 2 1=2 Y v sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 < Y vy þ 2 1:2 0:2 v 2 4 Y v in : v in v in v in 1 9 = ; 1=4 ; ð4:10þ where v Y is defined as the yield velocity, i.e., the relative incoming velocity when incipient plastic deformation develops (below this velocity, no plastic deformation occurs), and is given by 1=2 v Y ¼ 3:194 ðs YÞ 5 Th ðrn Þ 3 Fig. 11. Normal traction in the Thornton [10] NFD model. ðe n Þ 4 m n ; ð4:11þ where m n is the equivalent mass defined as m n ¼ðð1= ðiþ mþþð1= ð jþ mþþ 1 : The contact yield stress ðs Y Þ Th is the maximum normal pressure on the contact area ðp 0 Þ when yield begins. We employ the Hertz theory together with the von Mises criterion to obtain ðs Y Þ Th ¼ 1:61 s Y (see [11,14]), where s Y is the yield stress of the sphere material. The radius R n of the relative contact curvature and the equivalent Young s modulus E n are given by Eqs. (3.2) and (3.1), respectively, according to the Hertz theory. 6 The subscript Th in ðs Y Þ Th is mnemonic for Thornton.

16 332 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Table 4 Coefficient of restitution for elasto-plastic collisions v in ðm=sþ Model A Model B Model C FEA results Similar to the FEA of elastic collisions presented in Section 3.2, we carry out dynamic FEA of elasto-plastic collisions between the sphere described in Section 2 and a frictionless rigid surface with various incoming velocities using the nonlinear FE code ABAQUS [12]. Elasto-perfectly plastic material properties as described in Section 2 are used. The coefficient of restitution for collisions with different incoming velocities obtained from FEA results are listed in Table 4. The coefficient of restitution in Table 4 are all less than one, showing that there are energy dissipations caused by plastic deformation. The coefficient of restitution decreases with increasing incoming velocity, because the larger the incoming velocity, the higher the level of plastic deformation, and thus the higher the level of energy dissipation. Fig. 12 shows coefficient of restitution versus the incoming velocity with results coming from (i) FEA, (ii) discrete element method (DEM) simulations using the Vu-Quoc and Zhang [11] elastoplastic NFD model (denoted by VZ NFD model ), (iii) the Thornton [10] NFD model, and (vi) the Stronge model [17]. 7 The results from FEA using FE models A, B, and C (see Section 2) are presented using the symbols þ, *, and 3, respectively. For collisions with incoming velocities v in ¼ 0:02; 0:06 and 0:10 m=s; the results from DEM simulation using the Vu-Quoc and Zhang [11] NFD model agree well with the FEA results. For the collision with incoming velocity v in ¼ 0:20 m=s; which is about 120 times of the yield velocity 8 v Y and for which the maximum normal contact force reaches P max ¼ 8373 N; the coefficient of restitution from FEA is e ¼ 0:5853; while that from the Vu-Quoc and Zhang [11] NFD model is e vz ¼ 0:4773; the difference between the two results is 18.5%. We will discuss the cause of this difference later. The coefficient of restitution from the Thornton [10] NFD model given by Eq. (4.10) is also shown in Fig. 12 for comparison. Note that in Eqs. (4.10) and (4.11), the incoming velocity v in and yield velocity v Y are the relative incoming velocities of two spheres in collision, we need to double the incoming velocity when considering the collision between one sphere and a frictionless rigid surface which is equivalent to the collision of two identical spheres with doubled relative incoming velocity. In comparison with our dynamic FEA results, the coefficient of restitution from DEM simulation using the Vu-Quoc and Zhang [11] NFD model is clearly superior to that obtained from the Thornton [10] NFD model, for the incoming velocity in the range shown in Fig. 12, 7 The results of Stronge [17] are obtained using Eqs. ð24þ ð26þ; from Stronge [17] with k ¼ 1:1: 8 Plastic deformation begin to develop inside the sphere for the incoming velocity v in ¼ v Y ¼ 1: m=s computed by Eq. (4.11).

17 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Fig. 12. Coefficient of restitution versus incoming velocity, as obtained from various models. Fig. 13. Normal contact force P versus normal displacement a: (a) Collision with v in ¼ 0:02 m=s: (b) Collision with v in ¼ 0:06 m=s: namely, when v in =v Y p120: We refer the readers to Vu-Quoc and Zhang [11] for a more detailed comparison of the Vu-Quoc and Zhang [11] NFD model and the Thornton [10] NFD model. Fig. 13 shows the normal contact force P versus the normal displacement a for the collisions with incoming velocity v in ¼ 0:02 m=s (v in =v Y ¼ 12) and v in ¼ 0:06 m=s (v in =v Y ¼ 36). The results

18 334 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Table 5 Maximum force P max ; maximum displacement a max ; and residual displacement a res versus incoming velocity v in v in P max ðnþ a max ðmþ a res ðmþ ðm=sþ FEA VZ NFD FEA VZ NFD FEA VZ NFD : : : : : : : : : : : : : : : : Fig. 14. Normal contact force P versus normal displacement a: (a) Collision with v in ¼ 0:10 m=s: (b) Collision with v in ¼ 0:20 m=s: produced by the DEM simulations using the Vu-Quoc and Zhang [11] elasto-plastic NFD model agree closely with the corresponding FEA results. The maximum normal contact force P max ; the maximum normal displacement a max ; and the residual normal displacement a res caused by plastic deformation are listed in Table 5. Similar to Fig. 13, Fig. 14 shows the normal contact force P versus the normal displacement a for the collisions with incoming velocities v in ¼ 0:10 m=s (v in =v Y ¼ 60) and v in ¼ 0:20 m=s (v in =v Y ¼ 120). In Fig. 14(a), with v in =v Y ¼ 60; the NFD relation produced by the DEM simulations using the Vu-Quoc and Zhang [11] elasto-plastic NFD model agrees reasonably well with the corresponding FEA results. By Eq. (3.8), the incipient-yield contact force is P Y ¼ 36:45 N for the sphere with properties given in Section 2. In this case, the maximum normal contact force is P max ¼ 3922 N from FEA results (Table 5), and thus the ratio P max =P Y ¼ 107:6: In Fig. 14(b), with v in =v Y ¼ 120; there is a clear departure of the results using the Vu-Quoc and Zhang [11] NFD model from the FEA results. In this case, the maximum normal contact force P max ¼ 8373 N is much larger than the yield normal force P Y by a ratio of P max =P Y ¼ 229:7: Such a large maximum normal force is the reason for the departure of the results using the Vu-Quoc

19 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Fig. 15. Collision duration time for elasto-plastic collision versus incoming velocity. Table 6 Comparison of collision time t versus incoming velocity v in v in ðm=sþ Hertz (elastic) FEA (elasto-plastic) VZ NFD (elasto-plastic) : : : : : : : : : : : : and Zhang [11] NFD model from FEA results, because the parameters C a and K c employed in the Vu-Quoc and Zhang [11] NFD model were extracted for maximum normal force less than 1500 N (i.e., P max =P Y ¼ 41:1). Clearly, the maximum normal force P max ¼ 8373 N is way outside the range of validity of the parameters employed. In other words, in order to make the results using the Vu-Quoc and Zhang [11] NFD model work up to P max ¼ 8373 N; we need to extend the range of P max to this force level in the extraction of the model parameters C a and K c : The collision duration time t for the elasto-plastic collision obtained from our FEA and from the DEM simulations using the Vu-Quoc and Zhang [11] NFD model is shown in Fig. 15 and in Table 6. The computation method is the same as that explained in Section 3.2. Since the integration time-step size for the nonlinear dynamic FEA is automatically chosen by ABAQUS, and the time-step size for elasto-plastic analysis is even smaller compared to that for elastic analysis, the error for the data presented in Fig. 15 and in Table 6 is much smaller than 1%. For example, the time-step size around the separation time of the FE model B from the rigid planar

20 336 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) surface, for v in ¼ 0:10 m=sis7: s; making the error on t of about 0.09%. For the results using DEM simulation, the time-step size is fixed at 1: s; thus making the error on t about 0.13%. The collision duration time for elastic collision obtained using the Hertz theory as given by Eq. (3.9) is also presented for comparison. Again, when v in =v Y p60; the results from DEM simulation using the Vu-Quoc and Zhang [11] NFD model agree accurately with those from FEA. It is also observed that the collision duration time from FEA for elasto-plastic collision is very close to the collision time for elastic collision from the Hertz theory, meaning that the effect of plastic deformation on the collision duration time is very small, at least for the cases in which 12pv in =v Y p120: Such an observation deserves further study. Fig. 16 shows the distribution of the normal pressure on the contact surface obtained from our dynamic FEA for elasto-plastic collisions. The Hertz normal pressure as given by Eq. (3.3) (for elastic contact) is also presented; the original FEA data are presented by the symbols 3. Similar to that presented in Fig. 8, we average the original oscillating FEA data to obtain a much smoother curve shown in Fig. 8 using the symbols x. The contact area radius a from FEA results is obtained by using cubic spline interpolation in the same manner as for results shown in Fig. 8. The distribution of the normal pressure p for the elasto-plastic collision with incoming velocity v in ¼ 0:02 m=s at the time when the normal contact force P reaches its maximum value P max ¼ 647:8 N is shown in Fig. 16(a). The contact area radius a ¼ 1: m from the FEA results for the elasto-plastic collision is larger than the contact area radius a hz ¼ 8: m from the Hertz theory for the elastic collision under the same normal force level. Similarly, the distribution of the normal pressure p for the elasto-plastic collision with incoming velocity v in ¼ 0:10 m=s at the time when the normal contact force P reaches its maximum value P max ¼ 3922 N is presented in Fig. 8(b). Again, we observe that the contact area radius from the FEA results for the elastoplastic collision is larger than that from the Hertz theory for the elastic collision under the same normal force level. In both Fig. 16(a) and (b), the magnitude of the normal pressure on the Fig. 16. Normal pressure over the contact area for elasto-plastic collisions. (a) Collision with v in ¼ 0:02 m=s; at maximum normal force P max ¼ 647:8 N: (b) Collision with v in ¼ 0:10 m=s; at maximum normal force P max ¼ 3922 N:

21 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Fig. 17. Normal pressure distribution on contact surface from a static FEA at P ¼ 1500 N: contact surface inside the contact area is roughly constant, and is more than twice that of the material yield stress s Y : The normal pressure distribution on the contact surface from the dynamic FEA results agrees with that obtained from the static FEA results for elasto-plastic contact problems presented in Vu-Quoc et al. [5] (Fig. 17). We refer the readers to Vu-Quoc et al. [5] and Vu-Quoc et al. [14] for more detailed discussions, on the static FEA results of elasto-plastic contact problems. Remark 4.1. The static results shown in Fig. 17 were obtained with an FE mesh that was much denser than the FE meshes in models A, B, and C employed in the present paper. Even in this static analysis, one can still see some oscillations, which are due to the properties of the finiteelement formulation employed, and clearly not due to numerical instability in dynamic analyses; see also Section 3.3. Fig. 18(a) shows the contour of the Mises equivalent stress inside the sphere for the elastoplastic collision with incoming velocity v in ¼ 0:10 m=s at time t ¼ 1: s when the normal contact force increases to P ¼ 999:7 N: The sphere material is modeled by the elasto-perfectly plastic model with the von Mises yield criterion, according to which the material will yield when the Mises equivalent stress q (given by Eq. (3.11)) is equal to the yield stress s Y : With s Y ¼ 1: N=m 2 ; the area encircled by the contour line marked a in Fig. 18(a) is the plastic zone. The same applies to Figs. 18(b), 19(b), and (a). Fig. 18(a) shows that the plastic deformation is first developed not on the contact surface but at a point inside the sphere, and close to the contact area. When the normal contact force increases, the plastic zone expands to reach the contact surface, beginning from the edge of the circular contact area, while the material on the contact surface and around the center of the contact area remains elastic. Comparing Fig. 18(a) with Fig. 9(a), we see that the maximum Mises stress inside a sphere with elastic-perfectly plastic material is limited to the material yield stress s Y ; which is much lower than the stress reached in an elastic collision.

22 338 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Fig. 18. Contour of Mises stress inside the sphere during elasto-plastic collisions. (a) Collision with v in ¼ 0:10 m=s; loading at t ¼ 1: s and P ¼ 999:7 N: (b) Collision with v in ¼ 0:20 m=s; loading at t ¼ 4: s and P ¼ 8373 N: Fig. 18(b) shows the contour of Mises equivalent stress inside the sphere for the elasto-plastic collision with incoming velocity v in ¼ 0:20 m=s at time t ¼ 4: s when the normal contact force increases to P ¼ 8373 N: When the normal loading is high, the plastic zone develops to the contact surface on most parts inside the contact area. Fig. 19(b) shows the contour of Mises equivalent stress inside the sphere for the elasto-plastic collision with incoming velocity v in ¼ 0:20 m=s at time t ¼ 4: s when the normal contact force decreases to P ¼ 8347 N; right after the normal contact force reaches its highest value, and when the sphere starts to separate from the rigid surface. At this time, even though the normal contact force P ¼ 8347 N is much larger than the incipient yield force P Y ¼ 36:45 N; once the normal force starts unloading, the Mises equivalent stress inside the sphere immediately decreases to a level qos Y everywhere. With the plastic deformation frozen, the material behaves elastically during the normal contact force unloading session. Fig. 19(a) shows the contour of Mises equivalent stress inside the sphere for the elasto-plastic collision with incoming velocity v in ¼ 0:20 m=s at time t ¼ 6: s when the normal contact force decreases to P ¼ 1017 N after this force reaches its highest value and after the sphere is separated from the rigid surface. Comparing Fig. 19(a) (P ¼ 1017 N; unloading) and Fig. 18(a)

23 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) Fig. 19. Contour of Mises stress inside the sphere during elasto-plastic collisions. (a) Collision with v in ¼ 0:20 m=s; unloading at t ¼ 6: s and P ¼ 1017 N: (b) Collision with v in ¼ 0:20 m=s; beginning unloading at t ¼ 4: s and P ¼ 8347 N: (P ¼ 999:7 N; loading), we observe that the stress distribution during unloading with plastic deformation can be very different from the stress distribution during loading, even at the same normal force level. Unlike elastic collision, the distribution of stress inside the sphere in an elastoplastic collision is dependent on the loading history. 5. Conclusion We presented the dynamic simulations of the collisions between a sphere and a frictionless rigid planar surface using the nonlinear finite-element code ABAQUS. Such collisions are equivalent to the collisions between two identical spheres. The results of the collisions using an elastic sphere obtained from our FEA agreed closely with the results produced by applying the Hertz theory in various aspects. Such an agreement validated the reliability of the FEA models used in our FEA.

24 340 X. Zhang, L. Vu-Quoc / International Journal of Impact Engineering 27 (2002) The distribution of the Mises stress (3.11) produced by FEA confirmed the prediction based on the Hertz theory and von Mises yield criterion that the first yield during a contact occurs not on the contact surface, but inside the sphere (see [14] for more details). In addition, we carried out a series of FEA of the elastic collision using spheres with different elastic moduli. Observations based on FEA results supported the conclusion that the energy dissipation caused by the elastic wave propagation inside the sphere body could be ignored for most of the collisions discussed in this paper, except for one case in which the sphere material was extremely soft (see Section 3.3). 9 We also presented dynamic FEA results of elasto-plastic collisions. We compared both the results from DEM simulationsfusing the elasto-plastic normal force displacement (NFD) model presented in Vu-Quoc and Zhang [11], the Thornton [10] NFD model, and the Stronge [13] modelfto the FEA results. The comparison among the coefficients of restitution obtained using different methods showed a close agreement between the FEA results and the results from the DEM simulation using the Vu-Quoc and Zhang [11] NFD model, even with the model parameters obtained from an analysis with only low-level contact-force for elasto-plastic collisions, especially for the collisions with incoming velocity v in =v y p60: Such an agreement is also observed when comparing the force displacement (FD) curves from FEA and those from the Vu-Quoc and Zhang [11] NFD model. The dynamic FEA validated the Vu-Quoc and Zhang [11] NFD model, and also revealed that the application range of Vu-Quoc and Zhang [11] NFD model may be affected by the choice of model parameters. It was also observed that plastic deformation inside the sphere did not affect the collision time much, under low-velocity collision v in =v y p120: We also provided information on the development of the plastic zone inside the sphere. Other interesting phenomena described in this paper deserve further investigation. For a comparison of our NFD model and the resulting coefficient of restitution with experiments, we refer the readers to Zhang and Vu-Quoc [19]. The readers are also referred to references [20 23], which are related to the same topic discussed in the present paper. Acknowledgements We thank our colleagues, Dr. Otis Walton and Mr. Lee Lesburg, for their discussions, and Mr. Tam Trinh for his help in post-processing of the FEA results. We also thank Dr. W.J. Stronge for his papers and discussion. The support of the National Science Foundation is gratefully acknowledged. References [1] Goldsmith W. Impact: the theory and physical behaviour of colliding solids. London: Edward Arnold, [2] Ning Z, Thornton C. Elastic-plastic impact of fine particles with a surface. In: Thornton C, editor. Powders and Grains. Netherlands: Balkema, Rotterdam, p [3] Walton OR. Numerical simulation of inelastic, frictional particle particle interactions. In: Roco MC, editor. Particulate Two-Phase Flow, Stoneham, MA: Butterworth-Heinemann, p [chapter 25]. 9 A reviewer informed us that the issue of whether quasi-static analysis was representative of dynamic deformation was also addressed in Tsai [18].