Elastic impact of sphere on large plate

https://doi.org/10.1016/j.jmps.2021.104604Get rights and content

Highlights

  • The Zener impact was solved by using homotopy analysis method.

  • Structural compliance breaks contact force symmetry.

  • Zener energy loss is proportional to the impact velocity to the power of 0.2.

Abstract

Different from the Hertz impact, a sphere impacting on a plate has an extra dissipation by the flexural wave propagating on the plate. This wave dissipation has been noticed by Zener (1941), who introduced this dissipation term into the governing equation. Because of this extra dissipation term by the flexural wave, the Zener equation cannot be treated as the same way as in the Hertz impact problem. As a result, in the past 80 years, except few numerical tries for Zener equation, no analytical solution has been provided for the evolution of compressive displacement or contact force during the Zener impact. In this paper, by using homotopy analysis method, we analytically solved the Zener impact problem. After constructing an auxiliary linear operator, we derived an analytical solution for the evolution of contact force up to the first order of the embedding homotopy parameter. Present model can take full account of the elongation of contact time by plate compliance, which is impossible by using Hertz impact model. Comparisons with finite element simulation show that the prediction of the contact force history is of high accuracy. With the solution of force history, we derived an explicit expression for the energy loss of Zener type, this Zener loss has a different dependence on the impact velocity as the Hunter loss does. Also, both the motions of the sphere and the plate are able to be predicted with high accuracy. In addition, with the zeroth-order solution, we derived explicit expressions for both the coefficient of restitution and the contact duration.

Introduction

The impact of an elastic sphere to a half-space was first studied by Hertz (1882). In his approach, the system was conservative so that the general form of controlling equation was written asd2δdt2+kcδ32=0where δ is the displacement of sphere, t the time, and kc a coefficient depending on the material properties and geometry. By integrating the controlling equation, Hertz obtained an explicit expression for the contact duration given byTHertz=2.94(1516m1mhm1+mh1E*)25R115V015where E* is the effective modules given by 1/E*=(1ν12)/E1+(1νh2)/Eh, m1 the mass of the sphere, mh the mass of the half-space, R the radius of the sphere, E1 the Young's modulus of the sphere, Eh the Young's modulus of the half-space, ν1 the Poisson's ratio of the sphere, νh the Poisson's ratio of the half-space, and V0 the initial velocity.

Although Eq. (1) neglected the energy dissipation by plasticity or viscosity, Hertz's impact model was widely used to study the energy loss of impact. However, Hertz was not able to give an explicit solution of displacement to Eq. (1). What is more, to authors’ knowledge, no explicit solution has been derived so far. Based on Hertz's numerical solution of impact, Hunter studied the energy loss by stress wave in the half-space (often cited as Hunter loss) during the impact. In his approach, he assumed that the history of contact force pulse obeyed the Hertzian impact, and the energy loss was determined by the force pulse over the free surface (Hunter, 1957). Thus, the history of contact force must be known a priori.

In lack of explicit expression for the contact force history, Hunter first assumed a history profile for the force pulse by numerically integrating Eq. (1), and then formulated the energy loss by the dynamic response of the half-space due to the assumed force profile on the surface. To complete this formulation, he adopted from Hertz impact solution both the amplitude and the contact period, corresponding to the maximum compressive relative displacement and the contact duration, respectively. The maximum compressive displacement, δm, was expressed as:δm=(1516m1mhm1+mh1E*)25R15V045

By comparing with numerical results of Eq. (1), Hunter found out that sinusoidal profile could provide enough accuracy for the displacement history, expressed as:δ=δmsin(1.068V0δmt)

As a result, the profile of impact force history, F(t), was determined by Newton's law:F(t)=m1δ¨=m1δm(1.068V0δm)2sin(1.068V0δmt)

Using this assumption of force history, the Hunter loss, due to stress wave effect as a result of the force pulse on the impact surface of a half-space, was presented explicitly. The predicted coefficient of restitution was compared with experiments by Tillett (1954); agreement for the cases with energy loss less than 5% is well, however, large deviations was found for cases with energy loss more than 5%.

Noticing these deviations, Reed improved Hunter's assumption of the force history. Instead of Newton's law of motion, he used the Hertzian contact law to produce the force history, which in consequence has a sinusoidal profile with a fractional power (Reed, 1985):F(t)=43R12E*δm32sin32(1.068V0δmt)

With Eq. (6), Reed could provide a more accurate prediction for the impact experiment (Tillett, 1954) than Hunter's model did.

The sinusoidal profile or sinusoidal profile with fractional powers were also summarized in the study of the impact of two spheres (Johnson, 1985), and in the study of the impact between a sphere and a slender beam (Stronge, 2019). Their work showed this approximation was acceptable if only a small fraction of kinetic energy was converted into elastic wave.

These sinusoidal profiles share a common characteristic: symmetry with respect to the time line t=1.471δm/V0, which means the history of contact force are identical for both compressing stage and recovering stage. In general, the compliance of the structure prolongs the contact duration and reduces the maximum force in any impact (Stronge, 2019). As a result, when the compliance of the structure plays an important part during the impact, the symmetry feature of previous assumption for force profiles (Hunter, 1957; Reed, 1985) cannot hold any more. This viewpoint has been noticed by Lee in the study the impact of a sphere on a very slender beam (Lee, 1940), the compliance of the beam elongates the contact duration, and thus the sinusoidal profile is obviously no longer correct. Such symmetry break by compliance was also noticed in the classic book on impact dynamics (Stronge, 2019). However, till now, there is still no analytical solution to consider the structure compliance.

The Hunter's treatment is more like a one-way coupling approach: the contact pulse determines the energy loss by stress wave propagating in the half-space, but the contact pulse is not affected by the stiffness of the half-space. This assumption can be accepted for a half-space in some cases, but for the structure with high compliance such as a beam or a plate, large deviation may appear and thus a correct model considering the compliance of structure is required.

For the elastic impact of a sphere on a flexible structure, for example, a plate, the energy loss comprises of not only the Hunter loss but also a secondary loss due to the flexural oscillations on the plate, illustrated by Fig. 1. When the diameter-to-thickness ratio exceeds 0.2, Koller showed that the prediction by Hunter's model deviates from experiments (Koller and Kolsky, 1987). In such case, the flexural wave must be considered and the compliance of the plate, similar to the compliance of a beam, can significantly elongate the duration of recovery and break the symmetrical sinusoidal assumption (Zener, 1941). As a result, such process can hardly be explained with the theories by Hunter (1957), Aboudi (1978), Hutchings (1979), or Reed (1985).

To model the impact on a flexible plate, Zener derived a coupled controlling equation depending on an inelasticity parameter λ. In his model, the plate was sufficiently large such that the impacting process was not affected by the reflected flexural wave from the boundary. With numerical solution, Zener obtained a relationship of the COR e with λ; the result showed good agreement with Raman's experiments (Raman, 1920) and later confirmed by another experiments (Tillett, 1954). However, until now, we have not found any analytical solution for Zener's equation except for some linearization approaches in the literature. For example, Mueller et al. (2015) replaced the nonlinear contact law with a linear one and hence obtained an approximated solution, which can agree well for λ<0.85 and e>0.2. Such linearization method was used to study the contact time (Muller et al., 2016) and the energy dissipation (Boettcher et al., 2017b).

For the Hertz impact of a sphere on a half space, Hunter has proposed a sinusoidal profile to fit Eq. (1). Recently, for the Zener impact of a sphere on a thick plate, Boettcher et al. recalculated the model of Hunter using Reed's more accurate force-time approximation, obtaining more accurate results (Boettcher et al., 2017a). For general cases, which includes the cases with large λ, similar to Hunter's approach, we tried to seek a force profile which fits Zener's numerical solution. Unlike the treatment by Hunter (1957) and Reed (1985), as λ varies, the breaking of symmetry leads to that no universal representative function could represent the history of Zener's impact force so far.

The symmetry of contact force history will also break due to contact plasticity (Lifshitz and Kolsky, 1964; Patil and Higgs, 2017; Stronge, 2000; Thornton, 1997; Wu et al., 2003, 2005), contact adhesion (Ciavarella et al., 2019; Feng et al., 2009; Peng et al., 2020; Thornton et al., 2017), and contact viscosity (Flores et al., 2011; Hunter, 1960; Lankarani and Nikravesh, 1990; Maier and Tsai, 1974). For example, on the study of high-speed elastoplastic impact of a sphere on a half-space, Hutchings showed that asymmetrical force history was evident; Hutchings combined sinusoidal and cosinusoidal profiles with different periods for the loading and unloading of contact force, respectively (Hutchings, 1979). In addition, plasticity-induced asymmetry was also noticed in our previous study (Peng et al., 2021). These plastic or viscous situations often imply high-speed impacts or viscous materials, and thus the Hertzian contact law can hardly be used to establish the controlling equation.

In sum, the breaking of symmetry by compliance has been noticed for almost 80 years, but either the profile of contact force history has not been provided universally, or the assumed profiles lack of theoretical foundation. In this paper, we analytically solved Zener's equation using homotopy analysis method. After constructing an auxiliary linear operator, we derived an analytical solution for the contact force history up to the first order of the embedding homotopy parameter. With the force solution, we studied the energy loss of Zener type impact theoretically and proposed a semi-analytical model. Further, the motion of sphere, the deflection of plate, and the coefficient of restitution were also investigated; finite element simulations were also carried out to verify the proposed solution.

Section snippets

Zener's equation

Considering the flexural vibration of plate due to a point load, Zener derived the controlling equation to model the elastic impact between a sphere and a plate. Assuming the plate is sufficiently large, he thus ignored the effect of reflected flexural wave from the boundary. The controlling equation was:d2δdt2+F(δ)m*+αddtF(δ)=0with initial conditions:{δ|t=0=0δ˙|t=0=V0where V0 is the initial velocity of the sphere, δ the compressive displacement during contact, δ˙ the time derivative of δ, m*

Results and discussion

Using the explicit expressions, Eqs. (35) and (42), we can obtain the contact force history, with which the feature of asymmetry will be checked in the section. Further, the contact force history can be used to calculate the motions of sphere and plate, and in turn the energy loss of Zener type. This section is organized as follows: first, we formulate the contact forces based on the displacement variation during the impact; second, we compare the present contact force history with the one in

Conclusion

Using homotopy analysis method, we proposed an analytical solution to Zener's equation to predict the elastic impact between a sphere and a large plate. Using the obtained contact force history, we studied the energy loss due to the flexural wave, as well as the motion of the sphere and plate during the impact. The conclusions are as follows:

  • 1

    By solving Zener's equation, we obtained the contact force history up to the first order of the homotopy embedding parameter (Eqs. (35) and (42)). These

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (No. 11772334, 11890681, 12022210, 12032001), by the Youth Innovation Promotion Association CAS (2018022), by the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB22040501). We would like to acknowledge anonymous reviewers and Prof. X.H. Shi for their helpful comments and discussions.

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