IMPACT MODELS AND COEFFICIENT OF RESTITUTION: A REVIEW

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1 IMPACT MODELS AND COEFFICIENT OF RESTITUTION: A REVIEW M. Ahmad 1, K.A. Ismail 2 and F. Mat 1 1 School of Mechatronic Engineering, Universiti Malaysia Perlis (UniMAP), Kamus Alam Pauh Putra, Arau, Perlis, Malaysia 2 School of Manufacturing Engineering, Universiti Malaysia Perlis (UniMAP), Kamus Alam Pauh Putra, Arau, Perlis, Malaysia masniezam@yahoo.com ABSTRACT Imact is a comlicated henomenon that occurs when two or more bodies collide at a short time eriod. Imact model is used to determine the resonses of the contact bodies due to the imact event. Furthermore, energy loss due to imact can be determined by coefficient of restitution (COR). Although numerous imact models have been reorted in the literatures, the works to imrove the model are continuously exlored to achieve erfection of the imact model. The aims of this aer are to resent the methodologies that have been used to obtain the imact models and COR, evaluate on the ros and cons of the revious imact models and determine the otential area to imrove the current imact models and COR. The methods to obtain COR from exeriment and finite element method (FEM) are briefly discussed. Besides that, several significant imact models develoed by the researchers are also comared. It is found that more works to determine COR in oblique and reeated imacts should be erformed. Until now, the viscolastic imact model is considered to be the most reliable in imact alication. This review is intended to assist the derivation of imact models in the future that can imrove the accuracy and solve the roblem in the revious imact models to be alied in various imact alications. Keywords: imact model, coefficient of restitution, contact law. INTRODUCTION Imact is defined as the collision between two bodies at an instant of time (Stronge, 2004). During imact, the bodies exerience a high force level at very brief duration, while energy is raidly dissiated and high acceleration and deceleration occured (Gilardi and Sharf, 2002). Two hases haen during imact; the comression hase occur when the bodies initially start to contact and comressed against each other and the restitution hase takes lace when the bodies start to searate but still in contact (Gharib and Hurmuzlu, 2012). The latter hase ends when the bodies are comletely searated. Imact model is theoretically derived based on the comression and restitution hases. In general, the imact model can be divided into several tyes; erfectly elastic (no energy loss), artially elastic (energy loss with no ermanent deformation), erfectly lastic (all energy loss with ermanent deformation) and artially lastic (some energy loss with ermanent deformation). A linear imact model is develoed by a comliance of linear stiffness or daming element. On the other hand, the nonlinear imact model is develoed by a comliance of nonlinear stiffness or daming element. The addition of daming element (also known as viscous element) in the comliance model causes the imacted bodies to be deformed in both elastic and viscous characteristics and it is known as viscoelastic or viscolastic models resectively. The objective of develoing the imact model is to obtain the resonse of the bodies during the ostimact, based on the known arameter during the reimact (Gilardi and Sharf, 2002). However, since the imact model deends on many arameters, the work to obtain the final solution is very comlex. To overcome this roblem, coefficient of restitution is used to get the solution in the imact model. Coefficient of restitution, COR, e, is a arameter used to determine the energy loss during imact. The value of e is in the range 0 e 1, where (e = 0) indicates a erfectly lastic collision (total energy loss) while (e = 1) indicates a erfectly elastic collision (no energy loss). However, in ractices, it is quite imossible to create a erfectly elastic collision because some of energy is dissiated to sound, heat, strain energy and etc. COR value is deends on the imact seed (linear and angular movements), material roerties, geometry, sliding friction and contact eriod. The first COR model was develoed by Newton (kinematic), followed by Poisson (kinetic) and finally by Stronge (energetic). v Kinematic COR: e = - f (1) v 0 f - Kinetic COR: e = - c c Wr Energetic COR: e = - (3) W c where v f and v 0 are the final relative velocity and initial relative velocity resectively, f and c are the (2) 6549

2 normal imulse for restitution and comression resectively and Wr and Wc are the work done during restitution and comression hases, resectively. These definitions of COR are equivalent, unless if the configuration of imact involve with friction and the direction of the sli changes during collision, only the energetic COR can rovides the correct solution (Ismail and Stronge, 2008). The imact models and COR are widely used to solve the roblems in sorts engineering (Cross, 2010), (Goodwill and Haake, 2001), (Cross, 2014), geology (Ashayer, 2007), (Imre et al., 2008), coal gasification industry (Gibson et al., 2013), automotive (Batista, 2006), robotic (Vasilooulos et al., 2014) and many more. Previously, extensive studies have been reorted on develoment and modification of the imact models. However, these imact models have their own limitation and the solution in imact models are not usually straightforward. To address this concern, many researchers are still working to create and modify the imact models in order to imrove the current imact models (Jacobs and Waldron, 2015), (Rathboneet al., 2015), (Alves et al., 2015). The aims of this aer are to resent the methodologies that have been used to obtain the imact models and COR, discuss on the ros and cons of the revious imact models and determine the otential area to imrove the current imact models and COR. PREVIOUS STUDIES OF COR Previous researchers use exeriment and finite element analysis (FEA) to obtain the value of COR in order to solve the analytical model of imact models. Three tyes of imact are always erformed; direct, oblique and reeated imacts. Several tyes of balls had been vertically dro on a brass rod by Cross (Cross, 2010) to obtain the change of seed during imact and dynamic hysteresis curve at low seed of imact was resented. Even at low imact seed, the local damage and lastic deformation were still exerienced by the ball, where the similar result was obtained by Johnson (Johnson, 1985). Furthermore, the energy loss on the ball was greater in a dynamic analysis comared to the static analysis. Besides that, Aryaei et al. (Aryaei et al., 2010) studied the effect of ball size to the COR value by dro test exeriment and FEA. They found that the ball size gave a significant effect to the COR result esecially when the ball and the imacted surface were made of different materials. However, this result is contradict with the theoretical exression by Stronge (Stronge, 2004) where the ball size did not effect on the COR result in direct imact roblems. Most of researchers used a shere shae as the object for imact but only a few studies analysed the COR by using the other shaes. Buzzi et al. (Buzzi et al., 2012) exerimentally comared the result of kinematic COR for circular, ellitic, square and entagonal shaes of blocks. These blocks were droed with initial sin through the ram and imacting a rigid landing block in an oblique direction. Besides that, the measurement of COR for the irregular shaed article had been exerimentally erformed by several studies (Hastie, 2013), (Li et al., 2004). A high seed camera and a mirror were used to obtain the article s velocity in three dimension and the article was assumed as shere and cylinder shae for the analytical solution (Hastie, 2013). For the analysis in oblique imact, normal and tangential COR were considered due to change of imact direction in two different axes. Cross (Cross, 2002) used tennis ball and suerball to imact the wood, emery and rebound ace surfaces in oblique direction. As a result, the angle of incidence affect greatly to the COR. Furthermore, negative value of tangential COR could be obtained due to low angle of incidence and coefficient of sliding friction that encourage the ball to slide throughout the imact. Besides that, Dong and Moys (Dong and Moys, 2006) erformed the oblique imact with initial sin (clockwise and counter-clockwise direction) was introduced on the steel ball. They found that at low incidence angle, the value of horizontal COR could be negative or more than one due to the angular direction of the sinning balls. Minamoto and Kawamura (Minamoto and Kawamura, 2011) erformed a high seed imact (u to 20 m/s) between two steel sheres by using air gun in exeriment and it was validated by FEA. According to their results, COR reduces when the imact seed is increased, however, the rate of reducing COR is larger in low imact velocity (below 2.5 m/s). During imact, the kinetic energy is dissiated by lastic deformation and stress wave roagation. For the artially elastic imact, the kinetic energy is dissiated by the roagation of elastic wave (Mangwandi et al., 2007). Hunter (Hunter, 1957) found that the energy loss due to elastic wave roagation is less than 1% of the initial kinetic energy for a steel ball imacting a large steel or glass. Besides that, Wu et al. (Wu et al., 2005) erformed the imact of elastic shere with the elastic and elasticerfectly lastic substrates by using FEA. They found that the energy dissiation due to wave roagation is less than 3% of the kinetic energy and it deends on the size of the imacted surface. Moreover, the kinetic energy loss due to wave roagation is higher when the imact involve with slender bodies such as rods, beams, lates and shells (Seifried et al., 2005). The study of COR for reeated imacts between two bodies is still given less attention by the revious researchers. Seifried et al. (Seifried et al., 2005) erformed reeated imact between steel shere and aluminium rod by using exeriment and FEA. They found that when the number of imact is increased, the imact force and COR are also increase due to residual stress that reduce the energy loss. Furthermore, Minamoto et al. (Minamoto et al., 2014) also obtain the similar results for the imact develoed by viscolastic model in FEA. 6550

3 (a) (b) (c) (d) Figure-1: Force-indentation curve for (a) Nonlinear elastic model, (b) Nonlinear viscoelastic model, (c) Nonlinear elastolastic model and (d) Linear viscolastic model. Adated from Gharib and Hurmuzlu, EVOLUTION OF IMPACT MODEL Imact between two rigid bodies can be treated by using discrete or continuous model. In discrete model, the velocity is instantaneously changed during the imact and it always measured by the imulse-momentum method. Meanwhile, the continuous model reresent the collision in a finite duration. The force-indentation relationshi during imact could be measured by using the continuous model, so the imact behaviour could be analysed. The evolution of the discrete and continuous models will be discussed in the next sections. Discrete model Imulse-momentum rincile is a discrete model that was conventionally used to solve the roblem in imact. This rincile assumed that the imact is haen at very brief of eriod and there is no change in the configuration of the imacting bodies during the imact. The analysis is divided into two intervals, which are during re and ost imact. COR or the imulse ratio that was obtained from the exeriment is emloyed into this imulse-momentum rincile in order to determine the ost imact velocity and energy loss during imact. In general, the conservation of momentum is alied in the imulse-momentum rincile, which is given by: m 1u 1 + m 2u 2 = m 1v 1 + m 2v 2 (4) where m 1 and m 2 are the mass of the first and second objects resectively, while u 1 and u 2 are the initial velocity and v 1 and v 2 are the final velocity of the first and second objects resectively. However, the results obtain by this rincile is inconsistent when there is a friction on the bodies during imact and COR value is difficult to be determined for the imact between the flexible bodies (Wang and Mason, 1992), (Escalona et al., 1998). Continuous model Continuous model, also known as comliance based method is roosed to overcome the roblems in the discrete model. During collision between two bodies, the comliance of a small contact region around the initial contact oint is reresented by the lumed arameter, which are stiffness and daming elements. As the contact force between the bodies is continuously act during imact, so the force-indentation rofile is the rimary outut in this continuous model. It could be derived from the equation of motions that was develoed from the comliance at the contact oint. The simlest elastic imact model is exressed by Hooke s Law. This linear erfectly elastic model is reresented by a linear sring element in which the sring embodies the elasticity of the contacting surfaces and the force is exressed by: F (5) where is the stiffness of the sring and reresents the relative indentation between the imacting bodies. Besides that, Hertz (Hertz et al., 1896) roosed a nonlinear erfectly elastic contact between two isotroic sheres and the force-indentation curve is shown in Figure-1(a). A nonlinear sring element is reresented as a comliance at the contact oint and the imact force, F is given by: F n (6) where n is the nonlinear ower exonent. This is a nonlinear erfectly elastic model where the alication is limited to low imact velocity and only alicable in elastic deformation and hard bodies. For most imact cases, the initial kinetic energy is dissiated due to wave roagation, lastic deformation or other factors. However, Hooke s law and Hertz model are not account for energy dissiation, thus the COR is unity. Because of that, daming element (dashot) is added into the Hertz model to account for energy dissiation during imact. The combination of the stiffness and daming elements is called viscoelastic constitutive model and it is limited to low range of relative imact velocity. Furthermore, this model is only valid for elastic deformation as similar with Hertz model, thus, the indention value return to zero at the end of imact as shown in Figure-1(b). Kelvin-Voigt model is the first linear viscoelastic imact model that consists of a linear sring and a linear dashot element connected in arallel configuration as shown in Figure-2(a). 6551

4 (a) (b) into the Hertz model (Yigit et al., 2011). Furthermore, the contact arameters (material roerties and geometric condition) are difficult to determine when using a nonlinear daming model, so these arameters are usually adjusted based on the exerimental results (Gilardi and Sharf, 2002). In contrast with Kelvin-Voigt model, Maxwell model is a linear viscoelastic constitutive model that combining a nonlinear sring and daming elements in series comliance as shown in Figure-2(c). Hence, the imact force is defined by: F c (9) (c) Figure-2: (a) Kelvin-Voigt model, (b) Force-indentation curve for Kelvin-Voigt model and (c) Maxwell model. Adated from Gilardi and Sharf, 2002 and Vladimír, This model is a linear viscoelastic and ratedeendent imact model, where the imact force is defined as: F c (7) where c and are the daming coefficient and rate of indentation between the imacting bodies resectively. However, according to the force-indentation relationshi in Figure-2(b), the contact force is nonzero at initial and final imact due to resence of daming element. To encounter this roblem, Hunt and Crossley (Hunt and Crossley, 1975) roosed a nonlinear viscoelastic imact model based on the Hertz and Kelvin- Voigt models. They relace the linear sring and daming elements in the Kelvin-Voigt model with nonlinear sring and daming elements, thus the imact force is exressed by: F n n c (8) Throughout this nonlinear viscoelastic model, the force at the beginning and final are zero, which successfully solve the roblem in Kelvin-Voigt model. In addition, the daming coefficient can be exressed as a function of the coefficient of friction since both are related to energy dissiation arameter during imact (Gilardi and Sharf, 2002). Thus, many researchers had roosed the exression of the daming coefficient and the nonlinear ower exonent, n through various exerimental and theoretical aroaches, which had been comared by Alves et al. (Alves et al., 2015). However, Yigit et al. (Yigit et al., 2011) claims that this model is only considered in a articular roblem and it does not deend on any hysical exlanation. In addition, the solution is more comlex due to additional of nonlinear arameter During imact, the normal force is smoothly increases with normal comression on the deforming region and the same force is act in both sring and daming element (Stronge, 2004). Maxwell model is differ from the Kelvin-Voigt model, where the force at the beginning and final imact is equal to zero. Nonetheless, this model results in COR that is indeendent of imact velocity, which deny the exerimental and analytical evidences that COR is deendent to imact velocity (Stronge, 2004), (Yigit et al., 2011). The revious imact models (elastic and viscoelastic) are only alicable for very low imact velocity where the material deformation is in elastic region. However, most of the imact ractice are resulting the bodies to deform in both elastic and lastic region. For examle, Johnson (Johnson, 1985) erformed an imact exeriment between two metallic bodies and found that the contact stress is high enough to cause material yielding and lastic deformation even at imact velocity of 0.14 m/s. Hence, numerous researchers had develoed elastolastic imact models that include lastic or ermanent deformation of the materials (Yigit and Christoforou, 1994), (Thornton, 2013), (Lim and Stronge, 1999), (Thornton et al., 2013), (Christoforou, 1993). Goldsmith (Goldsmith, 1960) roosed a nonlinear elastolastic model based on the Hertz model that can reresent the lastic deformation during the restitution hase as shown in Figure-1(c) which is exressed by: F Fmax max n (10) where Fmax and max are the maximum normal force and indentation during comression hase and is the ermanent indentation after searation. Then, Lankarani and Nikravesh (Lankarani and Nikravesh, 1994) rovide an easier solution to determine max and. On the other hand, Yigit and Christoforou (Yigit and Christoforou, 1994) roosed a nonlinear elastolastic contact law based on the Hertz and Johnson imact models. They divided the imact event into three hases. 6552

5 At first, the elastic loading is considered as Hertzian contact, followed by elastic-lastic loading until a yield oint and lastly is the Hertzian elastic unloading where the ermanent deformation is considered. In their elastolastic model, the arameters are easily obtained from material roerties and contact geometry and this model also account with the ermanent deformation. However, the energy loss due to wave roagation is not considered in this elastolastic model (Yigit et al., 2011). To encounter this roblem, Ismail and Stronge (Ismail and Stronge, 2008) roosed a linear viscolastic imact model, based on the Maxwell model. The forceindentation curve for this viscolastic model is shown in Figure-1(d). They relaced the elastic element (linear sring) with an elastolastic element (bilinear sring) and connects them with a linear daming element in series. Thus, the energy loss due to lastic deformation, wave roagation and other factors are accounted in this viscolastic imact model. Furthermore, this model can be solved by analytical and numerical analyses while the elastolastic imact model that was roosed by Yigit and Christoforou (Yigit and Christoforou, 1994) can only be solved by numerical analysis. Nonetheless, as similar with the Maxwell model, this model results in COR that is indeendent of imact velocity, which deny the exerimental and mathematical evidence that COR is deendent to imact velocity (Stronge, 2004), (Yigit et al., 2011). So, in 2011, Yigit et al. (Yigit et al., 2011) imrove the lastic loss factor coefficient in the Ismail viscolastic model that imosed the COR to be deendence on the imact velocity. Furthermore, they also develoed a nonlinear viscolastic model according to their revious elastolastic imact model and obtained an almost similar results with Ismail. The disadvantage of this model is observed in the force-indentation curve for elastic imact (at imact velocity of 0.1 m/s), where there is nonzero indentation value at the end of imact. This is haen due to ermanent deformation of daming element and there is no elastic element arallel to the daming element (Yigit et al., 2011). Thus, this viscolastic model can be imroved by adding a sring element arallel to the daming element as shown in Figure-3. Through the addition of this sring element, the deformation in the elastic imact could be restored at the end of imact. This model is a combination of Maxwell and Kelvin-Voigt model and it is similar to the standard solid model. However, the used of this model for the viscolastic imact has not been exlored by any researchers until now. CONCLUSIONS AND RECOMMENDATIONS Review on the coefficient of restitution (COR) and imact models have been resented in this aer. Until now, energetic COR that was introduced by Stronge is the most consistent and alicable in a wider alication. The exerimental works to determine COR in oblique and reeated imacts are still given less attention by the revious researchers. Although the analysis of imact is easier by using numerous simulation software that available today, however, the accuracy of this numerical analysis actually deends on the theory of the imact model itself. This is the reason of researchers are still develoing and modifying the imact models until today. Throughout the revious discussions, every imact model has their own advantage and disadvantages. Until now, the viscolastic imact model is claimed to be the most reliable in imact alication. For the future work, it is worth to imrove the current viscolastic imact model as this model still has a certain limitation. It could be done by develoing a viscolastic imact model based on the standard solid model. This model is hyothetically valid in both elastic and lastic imact that has not been accurately redicted by the revious imact models. However, the validity of this roosed model has to be exerimentally and numerically verified in the forthcoming works. REFERENCES Alves, J., Peixinho, N., da Silva, M. T., Flores, P., & Lankarani, H. M. (2015). A comarative study of the viscoelastic constitutive models for frictionless contact interfaces in solids. Mechanism and Machine Theory, 85, Aryaei, A., Hashemnia, K., & Jafarur, K. (2010). Exerimental and numerical study of ball size effect on restitution coefficient in low velocity imacts. International Journal of Imact Engineering, 37(10), Ashayer, P. (2007). Alication of rigid body imact mechanics and discrete element modeling to rockfall simulation. University of Toronto. Figure-3: Addition of sring element arallel to the daming element in viscolastic model. Batista, M. (2006). On the Mutual Coefficient of Restitution in Two Car Collinear Collisions, 16. Retrieved from htt://arxiv.org/abs/hysics/ Buzzi, O., Giacomini, A., & Sadari, M. (2012). Laboratory investigation on high values of restitution 6553

6 coefficients. Rock Mechanics and Rock Engineering, 45(1), Christoforou, A. P. (1993). On the contact of a sherical indenter and a thin comosite laminate. Comosite Structures, 26(1-2), Cross, R. (2002). Measurements of the horizontal coefficient of restitution for a suerball and a tennis ball. American Journal of Physics, 70(5), 482. Cross, R. (2010). Enhancing the Bounce of a Ball. The Physics Teacher, 48(7), 450. Cross, R. (2014). Imact of sorts balls with striking imlements. Sorts Engineering, 17(1), Dong, H., & Moys, M. H. (2006). Exerimental study of oblique imacts with initial sin. Powder Technology, 161(1), Escalona, J., Mayo, J., & Dominguez, J. (1998). A Critical Study of the Use of the Generalized Imulse Momentum Balance Equations in Flexible Multibody Systems. Journal of Sound and Vibration, 217(3), Gharib, M., & Hurmuzlu, Y. (2012). A new contact force model for low coefficient of restitution imact. Journal of Alied Mechanics, 79(6), Gibson, L. M., Goalan, B., Pisuati, S. V., & Shadle, L. J. (2013). Image analysis measurements of article coefficient of restitution for coal gasification alications. Powder Technology, 247, Gilardi, G., & Sharf, I. (2002). Literature survey of contact dynamics modelling. Mechanism and Machine Theory, 37(10), Goldsmith, W., 1960, Imact, Edward Arnold, London. Goodwill, S. R., & Haake, S. J. (2001). Sring damer model of an imact between a tennis ball and racket. Hastie, D. B. (2013). Exerimental measurement of the coefficient of restitution of irregular shaed articles imacting on horizontal surfaces. Chemical Engineering Science, 101, Hertz, H. R., Jones, D.., & Schott, G.. (1896). Miscellaneous aers. Macmillan and Co. Ltd (Vol. 92). Hunt, K. H., & Crossley, F. R. E. (1975). Coefficient of Restitution Interreted as Daming in Vibroimact. Journal of Alied Mechanics, 42(2), 440. Hunter, S. C. (1957). Energy absorbed by elastic waves during imact. Journal of the Mechanics and Physics of Solids, 5(3), Imre, B., Räbsamen, S., & Sringman, S. M. (2008). A coefficient of restitution of rock materials. Comuters & Geosciences, 34(4), Ismail, K. A., & Stronge, W. J. (2008). Imact of Viscolastic Bodies: Dissiation and Restitution. Journal of Alied Mechanics, 75(6), Jacobs, D. A., & Waldron, K. J. (2015). Modeling Inelastic Collisions With the Hunt Crossley Model Using the Energetic Coefficient of Restitution. Journal of Comutational and Nonlinear Dynamics, 10(2), Johnson, K. L. (1985). Contact mechanics. Cambridge University Press. Lankarani, H. M., & Nikravesh, P. E. (1994). Continuous Contact Force Models for Imact Analysis in Multibody Systems. Nonlinear Dynamics, 5(2), Li, T., Zhang, J., & Ge, W. (2004). Simle measurement of restitution coefficient of irregular articles, 2(6), Lim, C. T., & Stronge, W. J. (1999). Oblique elastic lastic imact between rough cylinders in lane strain. International Journal of Engineering Science, 37(1), Mangwandi, C., Cheong, Y. S., Adams, M. J., Hounslow, M. J., & Salman, A. D. (2007). The coefficient of restitution of different reresentative tyes of granules. Chemical Engineering Science, 62(1-2), Minamoto, H., & Kawamura, S. (2011). Moderately high seed imact of two identical sheres. International Journal of Imact Engineering, 38(2-3), Minamoto, H., Seifried, R., Eberhard, P., & Kawamura, S. (2014). Exerimental and Numerical Analysis of Reeated Imacts between Two Sheres. Alied Mechanics and Materials (Vol. 566, ). Rathbone, D., Marigo, M., Dini, D., & van Wachem, B. (2015). An accurate force dislacement law for the modelling of elastic lastic contacts in discrete element simulations. Powder Technology. Seifried, R., Schiehlen, W., & Eberhard, P. (2005). Numerical and exerimental evaluation of the coefficient of restitution for reeated imacts. International Journal of Imact Engineering, 32(1-4), Stronge, W. J. (2004). Imact Mechanics. Cambridge University Press. Thornton, C. (2013). Coefficient of restitution for collinear collisions of elastic-erfectly lastic sheres. Journal of Alied Mechanics, 64(June 1997),

7 Thornton, C., Cummins, S. J., & Cleary, P. W. (2013). An investigation of the comarative behaviour of alternative contact force models during inelastic collisions. Powder Technology, 233, Vasilooulos, V., Paraskevas, I. S., & Paadooulos, E. G. (2014). Comliant Terrain Legged Locomotion Using a Viscolastic Aroach, (Iros), Vladimír, G. (2010). Testing and alication of new henomenological material model for foam materials. Elektrotechnika, Retrieved from htt://scholar.google.com/scholar?hl=en&btng=search&q =intitle:testing+and+alication+of+new+phenomenolo gical+material+model+for+foam+materials#0 Wang, Y., & Mason, M. T. (1992). Two-Dimensional Rigid-Body Collisions With Friction. Journal of Alied Mechanics, 59(3), 635. Wu, C.-Y., Li, L.-Y., & Thornton, C. (2005). Energy dissiation during normal imact of elastic and elastic lastic sheres. International Journal of Imact Engineering, 32(1-4), Yigit, A. S., & Christoforou, A. P. (1994). On the imact of a sherical indenter and an elastic-lastic transversely isotroic half sace. Comosites Engineering, 4(11), Yigit, A. S., Christoforou, A. P., & Majeed, M. A. (2011). A nonlinear visco-elastolastic imact model and the coefficient of restitution. Nonlinear Dynamics, 66(4),