On Mechanical and Motion Behavior of the Normal Impact Interface between a Rigid Sphere and Elastic Half-Space

Abstract

This paper presents exact solutions for the mechanical behavior of the interface during the normal collision between a rigid sphere and an elastic half-space based on kinematics and particle dynamics theory. The interfacial contact stress is significantly different from the static solution obtained from the Hertz contact theory. Firstly, according to the kinematics theory, the elastic half-space interface deformation of the sphere and the half-space interface in the collision process is divided into the deformation of the contact area and the non-contact area. The curve of the non-contract area is strictly antisymmetric to the deformation curve of the contact area, and the lateral deformation for each particle at the interface can be neglected when the collision depth is relatively small. Then, the vertical deformation equation of the contact area of the half-space interface, the dynamic equation of the rigid sphere, and the dynamic equation of the interface particle in the contact region are established. The proportional relationship between the stress or strain of any particle in the contact area, the stress or strain of the collision center point, and the method for determining the maximum collision depth are obtained. The equal deformation depth in the contact and non-contact regions, and the proportional relationship between the stress or strain in the contact area and the center point of the collision at any moment are consistent with the Hertz contact theory, which verifies the reliability of the current study. Taking the sphere of no initial velocity collides with the half-space under pure gravity as an example, when the elastic half-space Poisson’s ratio is taken as 0.2–0.4, the ratio of the maximum contact stress determined by the Hertz theory and the current solution when the sphere reaches the maximum collision depth is 0.58–0.54. Based on this study, the contact stress and its distribution in the interfacial contact region can be obtained when the motion state of the sphere and the interface are determined.

1. Introduction

The interaction of objects is divided into contact and non-contact according to the action distance. The work forms of non-contact action mainly refer to electromagnetic action and gravitational action, while the work forms of contact action refer to static contact and dynamic collision action. Hertz et al. gave solutions for the problems of stress and strain at the static contact interface. The obtained main theoretical achievements include the relationship between the maximum contact depth of the interface and the acting force, the relative relationship between any point of the interface and the center point stress (stress and strain distribution law), and the relationship between the maximum contact width of the interface and the maximum contact depth [1]. The most remarkable feature of the traditional contact theory represented by Hertz is that the contact between objects is static, and the determination of the physical quantities of the contact interface is independent of time, which is characterized by a static problem based on elastic mechanics.

When people study the collision problem, the commonly used method is still based on the Hertz theory. In this theory, the reaction force of the deformation state at the interface is determined and the mechanical (motion) physical parameters of the contact interface are obtained in the collision process by taking the deformation of the impact center as a displacement condition and the distribution law of interface stress as the basis [2,3]. Based on the Hertz theory, the contact stress in plenty of wheel–rail dynamic interaction is employed [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. Although the contact stress value is much smaller than the yield strength of the material, a thin layer of plastic damage layer appears on the surface of the rail in practical engineering (that is, the actual contact stress is higher than the calculated value and higher than the yield strength); therefore, the traditional contact theory fails to explain this phenomenon.

From the point of view of kinematics (dynamics), any static state has undergone a dynamic process; that is, a static state is a particular dynamic state. If the dynamic behavior of the contact interface is considered, the determination of its physical parameters may be different from that of the static theory. Particle dynamics (wave dynamics) [29] reveals that forces propagate as stress waves in a medium after generation. Since the continuous change of the contact interface collision caused, the short time collision lasting and the stress wave collision generating in the interface taking time to propagate in the medium (collision body and collided body), energy dissipation would occur during the propagation process [2,3]. The mechanical effect of the interface is likely significantly stronger than the static contact during the collision process. The Hertzian state can be formed only when the force generated at the interface propagates sufficiently in the medium (collision body and collided body) in the form of a stress wave. The affected medium is fully involved, and the dynamic properties disappear (the dynamic energy is completely dissipated) during the collision.

Hence, it is very necessary to study the non-static (collision) contact problem of objects. The collision problem studied in this paper includes the dynamic interaction between the rigid sphere and the elastic half-space under gravity (without initial collision velocity) and under the action of both gravity and initial collision velocity. They are all in motion during the activation process. Then, a set of governing equations is established, including the half-space interface (contact zone) particle dynamics equation, the dynamic sphere equation, and the half-space interface (contact zone) deformation distribution equation. Moreover, the stress–strain distribution law of each transient interface during the collision process, the motion state of the collision body, and its corresponding interface collision depth and stress–strain values are obtained.

Although the theoretical basis of this study is completely different from traditional contact mechanics, some key conclusions describing the mechanical (deformation) state of the half-space interface deformation zone (including the contact zone and the non-contact zone) are consistent. For example, the conclusions that the deformation area of the half-space interface (including the contact area and the non-contact area) that is composed of the maximum contact boundary equally divides the collision depth of the collision center at any time, the relative relationship between the stress (strain) of any particle in the contact area, and the stress (strain) of the collision center point at the moment are consistent with the conclusion of Hertz theory under static conditions, which verifies the reliability of the research theory in this paper. However, it should be noted that during the collision process between a rigid sphere and elastic half-space under the pure action of gravity without the initial velocity, the normal stress at the collision center is much greater than that obtained by the static-equilibrium-based traditional contact theory when the collision depth reaches the maximum value, which is very useful for understanding the possible extreme values of the normal stress at each particle of the collision interface during the motion.

2. Formation of Mathematical Physics Equation Sets

2.1. Basic Conditions and Related Assumptions

Basic conditions: For the sphere, its radius is R; its mass is m; its gravitational acceleration is g, and its initial velocity is v₀. For the elastic half-space, its elastic modulus is E; its Poisson’s ratio is μ and its mass density is ρ. The plane formed by the displacement of the sphere’s centroid, the center point of the collision interface, relative to the contact boundary point is u₀, and the displacement from the plane formed by the boundary point to the half-space interface is δ.

Basic assumptions: The sphere is rigid, the elastic half-space is in an elastic working state, and the sphere has a normal collision with the elastic half-space. During the collision process, the sphere and the elastic half-space are kept close, and the lateral deformation of the interface of the contact area between them is ignored. The normal interface stress outside the collision contact area is zero. That is, the part of the interface beyond the contact region does not provide a reaction force. The outline of the contact region is tangent to that of the non-contact area at the contact boundary.

Coordinate system establishment: One coordinate system (x, y, z) is established with the center of the just-colliding rigid sphere as the coordinate origin and t as the time coordinate. The other coordinate system (x′, y′, z′) is established with the collision center of the elastic half-space interface as the coordinate origin O′. The half-space interface equation is z = R(z′ = 0), t′ is the time when each point on the sphere starts to collide with the half-space (that is, the time of the centroid of the sphere moving), as shown in Figure 1a.

2.2. Determination of Deformation Function of Collision Interface

From Figure 1a, it can be known that the relationship between the O-xyz coordinate system and the O′-x′y′z′ coordinate system is:

$$\{\begin{array}{l}{x}^{\prime }=x\\ y^{\prime }=y\\ z^{\prime }=z-R\end{array}$$

(1)

Let the vertical displacement of the sphere at any point A on the collision contact surface relative to the plane the contact boundary points forming after the collision be u and the vertical displacement of the boundary point be δ, then according to the geometric relationship between u and u₀ shown in Figure 1b, and equations can be obtained:

$$x^2+y^2+{(R-u_0+u)}^2=R^2$$

(2)

$${x^{\prime }}^2+{y^{\prime }}^2+{(R-u_0+u)}^2=R^2$$

(3)

From Equation (2) that when u = 0, it is the maximum boundary of the contact between the small sphere and the elastic half-space, and the value of its radius r₀ (see Figure 1b) is:

$$r_0=\sqrt{x^2+y^2}=\sqrt{2R{u}_0-u_0^2}$$

(4)

From Equations (1) and (2), in the O-xyz coordinate system, the expression of the vertical displacement of any point on the contact surface (z = R) of the collision interface relative to the collision center is:

$$u\left(x,y\right)=\left[u_0-(R-\sqrt{(R^2-x^2-y^2)})\right]$$

(5)

2.3. Determination of Deformation Function of Collision Interface

2.3.1. Time Dependency of u₀, δ, ω₀

Let $r=\sqrt{x^2+y^2}$, it can be proven that:

$$\omega =\omega (r)=R-\sqrt{R^2-r^2}$$

(6)

Taking the sphere as the research object, let ${\omega }_0=R-\sqrt{R^2-r_0{}^2}$, r₀ is the maximum half width of contact, and u₀ = ω₀, then:

$${\left(R-{\omega }_0\right)}^2+r_0{}^2=R^2$$

(7)

After a time Δt, motion characteristics in the center of mass of the sphere and collision center increase Δu₀, Δδ, and Δω₀ respectively, and equations can be obtained:

$$\Delta {\omega }_0=\Delta u_0=\frac{r_0\Delta r_0}{\sqrt{R^2-r_0{}^2}}$$

(8)

Moreover, the deformation equation in the zOr coordinate system can be written as:

$${\left(Z-u_0-\delta \right)}^2+r^2=R^2$$

(9)

Then:

$$\left(Z-u_0-\delta \right)\Delta u_0=r\Delta r$$

(10)

When $Z=R+\delta$ and r = r₀, the point (z, r₀) is the contact point between the sphere and interface, substitute it into Equations (10)–(12), and we can derive:

$$\left(R-u_0\right)\Delta u_0=r_0\Delta r_0$$

(11)

$$\Delta u_0=\frac{r_0}{R-u_0}\Delta r_0=\frac{r_0}{\sqrt{R^2-r_0{}^2}}\Delta r_0$$

(12)

From Equation (9), it can be derived that:

$$\left(Z-u_0-\delta \right)\Delta \delta =r\Delta r$$

(13)

When $Z=R+\delta$, it reaches the maximum contact boundary r = r₀, it can be rewritten as:

$$\left(R-u_0\right)\Delta \delta =r_0\Delta r_0$$

(14)

Deduced further and we can obtain:

$$\Delta \delta =\frac{r_0}{R-u_0}\Delta r_0=\frac{r_0}{\sqrt{R^2-r_0{}^2}}\Delta r_0$$

(15)

From the derivation above, after any micro time Δt, Δu₀ = Δδ = Δω₀ can always be derived, also be written as:

$$\frac{\Delta {\omega }_0}{\Delta t}=\frac{\Delta \delta }{\Delta t}\Rightarrow \underset{\Delta t\to 0}{\lim }\frac{\Delta {\omega }_0}{\Delta t}=\underset{\Delta t\to 0}{\lim }\frac{\Delta \delta }{\Delta t}\Rightarrow \frac{\partial {\omega }_0}{\partial t}-\frac{\partial \delta }{\partial t}=0\Rightarrow \frac{\partial \left({\omega }_0-\delta \right)}{\partial t}=0\Rightarrow {\omega }_0-\delta =a$$

(16)

where a is a constant.

When t = 0, ${\omega }_0=u_0=\delta =0$, and then a = 0; it can be proven that:

$${\omega }_0=u_0=\delta$$

(17)

It also means that at any time point δ = u₀ = ω₀, as shown in Figure 2.

2.3.2. Determination of the Deformation Curve in Non-Contact Area

(i)

Boundary point curve equation of contact area of collision process interface

The sphere moves from O₁ to O₂ in micro time Δt, the boundary point in the contact area of a sphere and half-space are B, C′, in the coordinate system ω₀B₀r₀; the sphere’s motion(coordinate) state parameters are u₀, δ, ω₀, r₀; the delta state parameters are Δu₀, Δδ, Δω₀, Δr₀—see Figure 3.

Let the curve equation of sphere curve be ${\omega }_0={\omega }_0\left(r_0\right)$, the boundary points B and C′ lies on curve $\stackrel{⌢}{B{C}^{\prime }}$, in the coordinate system $\omega B{r}^{\prime }$, let $\stackrel{⌢}{B{C}^{\prime }}$ equation be ${\omega }_1={\omega }_1\left(r\right)$.

In order to obtain the curve at any moment in the non-contact area, we first assume that it remains unchanged. After obtaining its equation, the rationality and uniqueness of the assumed curve will be proven according to the physical and mathematical logic. Based on the assumption, we have $\stackrel{⌢}{B_{-1}{C}_{-1}}\cong \stackrel{⌢}{C^{\prime }{C}_{-1}}\cong \stackrel{⌢}{B_{-1}B}$, and let the equation of the curve be ${\omega }_2={\omega }_2\left(r\right)$, then:

$$\{\begin{array}{l}{\omega }_0={\omega }_0\left(r_0\right)\\ {\omega }_1={\omega }_1\left(r\right)\\ {\omega }_2={\omega }_2\left(r\right)\end{array}$$

(18)

According to the known condition: the sphere’s equation of the sphere determines the contact boundary point should satisfy Equation (7); we find that:

$$\Delta {\omega }_0=\frac{r_0\Delta r_0}{\sqrt{R^2-r_0{}^2}}$$

(19)

Because the point C′ is also the contact boundary point, it satisfies $\Delta \delta =\Delta {\omega }_0=\Delta u_0$ (shown in the previous context), then:

$$\Delta {\omega }_1+\Delta {\omega }_0=0$$

(20)

This can also be written as:

$$\Delta {\omega }_1=-\Delta {\omega }_0=\frac{-r_0\Delta r_0}{\sqrt{R^2-r_0{}^2}}=\frac{d{\omega }_1\left(r\right)}{dr}\Delta r$$

(21)

In the coordinate system $\omega B{r}^{\prime }$, $\Delta r_0=\Delta r$, then:

$$\frac{d{\omega }_1\left(r\right)}{dr}=\frac{-r_0}{\sqrt{R^2-r_0{}^2}}\Rightarrow \Delta {\omega }_1=-\frac{r_0\Delta r_0}{\sqrt{R^2-r_0{}^2}}=\frac{-r_0\Delta r_0}{\sqrt{R^2-r_0{}^2}}$$

(22)

According to the initial condition, when r₀ = 0, it can be derived that: ${\omega }_0={\omega }_1=0$. The equation of curve $\stackrel{⌢}{B{C}^{\prime }}$ in the coordinate system ${\omega }_0{B}_0{r}_0$ can be expressed as:

$${\left(R+{\omega }_1\right)}^2+r_0{}^2=R^2$$

(23)

For any boundary point, Equation (24) can be satisfied:

$$\{\begin{array}{l}{\left(R+{\omega }_1\right)}^2+r_0{}^2=R^2\\ {\left(R-{\omega }_0\right)}^2+r_0{}^2=R^2\end{array}\Rightarrow {\omega }_1=-{\omega }_0$$

(24)

The result shows that the curve ${\omega }_1$ is decided by ${\left(R+{\omega }_1\right)}^2+r_0{}^2=R^2$, and is symmetrical with ${\left(R-{\omega }_0\right)}^2+r_0{}^2=R^2$ with ${\omega }_0\left(\delta \right)$ as the symmetry axis.

(ii)

Deformation Curve Equation of Interface’s Non-contact Area

For any moment, the non-contact area curve ω₂, to satisfy the continuity of deformation assumption, must be tangent to the small sphere’s curve ω₀ at point B, i.e., $d{\omega }_0/d{r}_0=d{\omega }_2/dr$, which can be written as:

$${\omega }_0^{\prime }\left(r_0\right)={\omega }_2^{\prime }\left(r\right)|{}_{r=0}$$

(25)

The maximum contact boundary point from B to C′, on the curve ω₂ from point B₋₁ to point B, it can be proven that:

$$\{\begin{array}{l}\Delta {\omega }_2=-{\Delta {\omega }_2^{\prime }\left(r\right)|}_{r=0}\Delta r=-\Delta \delta \hfill \\ \Delta \delta ={{\omega }_0^{\prime }\left(r\right)|}_{r=0}\Delta r_0=\Delta \delta \hfill \end{array}$$

(26)

Then:

$$-{\omega }_2^{\prime }\left(r\right)|{}_{r=0}\Delta r={\omega }_0^{\prime }\left(r_0\right)\Delta r_0\Rightarrow \{\begin{array}{l}\Delta {\omega }_2=\Delta {\omega }_0\\ \Delta r=-\Delta r_0\end{array}$$

(27)

Further:

$${\omega }_2^{\prime }\left(r\right)|{}_{r=0}={\omega }_0^{\prime }\left(r_0\right)=\frac{r_0}{\sqrt{R^2-r_0{}^2}}$$

(28)

Then:

$$\Delta {\omega }_2\left(r\right)=\frac{r_0}{\sqrt{R^2-r_0{}^2}}\Delta r=\frac{r_0}{\sqrt{R^2-r_0{}^2}}\left(-\Delta r_0\right)$$

(29)

According to r₀ = 0, ω₀ = 0, r = 0 and ω₂ = 0, it can be proven that ω₀ in range $r\in \left[0,r_0\right]$, is the unsymmetrical curve of ω₂ in range $r\in \left[-r_0,0\right]$.

From the above derivation, we can deduce three curve equations in zOr coordinate system; the equations correspond to the spherical equation of the sphere, the equation of the contact boundary point position curve, and the equation of the deformation outside the contact area at any moment, respectively:

$$\{\begin{array}{l}{\left(z-u_0-\delta \right)}^2+r^2=R^2\\ {\left(z-2R\right)}^2+r^2=R^2\\ {\left(z-2R\right)}^2+{\left(r-2r_0\right)}^2=R^2\end{array}$$

(30)

The above conclusion is based on the non-contact area deformation curve line shape, which remains unchanged; if the line shape changes, it can be concluded that when $\Delta r=\Delta r_0$, it can be deduced that $\Delta {\omega }_2\ne \Delta {\omega }_0$. This leads to the conclusion that the contact point, contact boundary point, and deformation boundary point of the sphere and the half-space interface do not coincide at the moment of the initial collision, i.e., the initial values of the above curves are not consistent, and do not satisfy the known conditions, so the non-contact zone line shape must remain stable during the collision.

2.3.3. Demonstration for the Existence of Lateral Deformation in the Deformation Zone of the Half-Space Interface

The collision interface between the rigid sphere and the elastic half-space produces spatial stresses. In order to evaluate the influence of the deformation, stress, and strain of the interfacial particles in two directions other than the normal direction on the normal stress, the magnitude should be compared with the normal value to determine that it can be reasonably neglected when the normal deformation is small.

In this paper, we adopt the assumption that the incremental lateral displacement and the incremental normal displacement of the particle in the interface contact area during the collision are determined by the centroid of the sphere and the direction of the radius of the particle, so as to obtain the representation of the lateral deformation, and then prove the reasonableness and uniqueness of the assumption.

It can be deduced that $\Delta u_r\ll \Delta u_0+\Delta \delta$ based on $\Delta u_r=\left(\Delta u_0+\Delta \delta \right)\frac{\sqrt{R^2-{\left(r+u_r\right)}^2}}{R^2}\left(r+u_r\right)$. Moreover, due to the axisymmetric feature, the deformation in the circumferential direction formed by the particle and the collision center can be expressed as $\Delta u_{\phi }=2\pi \Delta u_r$. It is assumed that u_r can be neglected when $u_0+\delta$ is extremely small, i.e., the influence of lateral deformation on normal stress can be reasonably ignored.

If we consider that each particle point of the interface produces only vertical displacement and take any point A as the object of study, as shown in Figure 4, its relative displacement to the elastic half-space interface (Z = R) is u_z:

$$u_z=Z-R=u_0+\delta -\left(R-\sqrt{R^2-r^2}\right)$$

(31)

If any interface particle point A at u₀ and δ state has lateral displacement u_r, the interface mass satisfies the spherical equation; let it move to C after Δu₀, Δδ, Δt, and C is located on the radius O₁A’s extension of the sphere, then:

$${\left(Z^{\prime }-u_0-\delta \right)}^2+{\left(r+u_r\right)}^2=R^2\Rightarrow Z^{\prime }=u_0+\delta +\sqrt{R^2-{\left(r+u_r\right)}^2}$$

(32)

Let $\omega =R-\sqrt{R^2-{\left(r+u_r\right)}^2}$, we have

$$\Delta \omega =\frac{r+u_r}{\sqrt{R^2-{\left(r+u_r\right)}^2}}\Delta u_r$$

(33)

Then:

$$\Delta u_{z^{\prime }}=\Delta u_0+\Delta \delta -\Delta \omega$$

(34)

C is located on the radius O₁As extension of the sphere, then:

$$\frac{\Delta u_r}{\Delta u_{z^{\prime }}}=\frac{r+u_r}{\sqrt{R^2-{\left(r+u_r\right)}^2}}$$

(35)

It can be derived:

$$\Delta \omega =\frac{r+u_r}{\sqrt{R^2-{\left(r+u_r\right)}^2}}·\frac{r+u_r}{\sqrt{R^2-{\left(r+u_r\right)}^2}}\Delta u_{z^{\prime }}$$

(36)

Then:

$$\Delta u_{z^{\prime }}=\Delta u_0+\Delta \delta -\frac{r+u_r}{\sqrt{R^2-{\left(r+u_r\right)}^2}}·\frac{r+u_r}{\sqrt{R^2-{\left(r+u_r\right)}^2}}\Delta u_{z^{\prime }}$$

(37)

It can be concluded from the above derivations:

$$\begin{array}{c}\Delta u_{z^{\prime }}\frac{R^2}{R^2 - {\left(r + u_r\right)}^2}=\Delta u_0+\Delta \delta \\ \frac{\sqrt{R^2 - {\left(r + u_r\right)}^2}}{r + u_r}\Delta u_r=\Delta u_0+\Delta \delta -\frac{r + u_r}{\sqrt{R^2 - {\left(r + u_r\right)}^2}}\Delta u_r\end{array}$$

(38)

Then:

$$\Delta u_r=\left(\Delta u_0+\Delta \delta \right)\frac{\sqrt{R^2-{\left(r+u_r\right)}^2}}{R^2}\left(r+u_r\right)$$

(39)

Use the known condition of no lateral shift of the center point to determine the reasonableness of the assumption of the direction of point C.

When r + u_r = 0, i.e., the point is the center point, it can be proven that:

$$\{\begin{array}{l}\Delta u_r=0\\ \Delta u_{z^{\prime }}=\Delta u_0+\Delta \delta \end{array}$$

(40)

Therefore, there is no lateral shift of the centroid, and its displacement increment is consistent with the center of the sphere, in accordance with the assumption.

If point C is not on the extension of the radius of the sphere, then:

$$\frac{\Delta u_r}{\Delta u_{z^{\prime }}}>\frac{r+u_r}{\sqrt{R^2-{\left(r+u_r\right)}^2}} \mathrm{or} \frac{\Delta u_r}{\Delta u_{z^{\prime }}}<\frac{r+u_r}{\sqrt{R^2-{\left(r+u_r\right)}^2}}$$

(41)

That is

$$\{\begin{array}{l}\Delta u_r>\frac{r + u_r}{\sqrt{R^2 - {\left(r + u_r\right)}^2}}\Delta u_{z^{\prime }} \mathrm{or} \Delta u_r<\frac{r + u_r}{\sqrt{R^2 - {\left(r + u_r\right)}^2}}\Delta u_{z^{\prime }}\\ \Delta u_{z^{\prime }}>\frac{\sqrt{R^2 - {\left(r + u_r\right)}^2}}{r + u_r}\Delta u_r \mathrm{or} \Delta u_{z^{\prime }}<\frac{\sqrt{R^2 - {\left(r + u_r\right)}^2}}{r + u_r}\Delta u_r\end{array}$$

(42)

Further,

$$\Delta u_{z^{\prime }}=\Delta u_0+\Delta \delta -\frac{r+u_r}{\sqrt{R^2-{\left(r+u_r\right)}^2}}\Delta u_r$$

(43)

That is,

$$\{\begin{array}{l}\Delta u_0+\Delta \delta -\frac{r + u_r}{\sqrt{R^2 - {\left(r + u_r\right)}^2}}\Delta u_r>\frac{\sqrt{R^2 - {\left(r + u_r\right)}^2}}{r + u_r}\Delta u_r \mathrm{or}\\ \Delta u_0+\Delta \delta -\frac{r + u_r}{\sqrt{R^2 - {\left(r + u_r\right)}^2}}\Delta u_r<\frac{\sqrt{R^2 - {\left(r + u_r\right)}^2}}{r + u_r}\Delta u_r\end{array}$$

(44)

It can be concluded that:

$$\{\begin{array}{l}\Delta u_r>\frac{\sqrt{R^2 - {\left(r + u_r\right)}^2}}{R^2}\left(r + u_r\right)\left(\Delta u_0+\Delta \delta \right) \mathrm{or}\\ \Delta u_r<\frac{\sqrt{R^2 - {\left(r + u_r\right)}^2}}{R^2}\left(r + u_r\right)\left(\Delta u_0+\Delta \delta \right)\end{array}$$

(45)

$$\{\begin{array}{l}\Delta u_{z^{\prime }}>\frac{R^2 - {\left(r + u_r\right)}^2}{R^2}\left(\Delta u_0+\Delta \delta \right) \mathrm{or}\\ \Delta u_{z^{\prime }}<\frac{R^2 - {\left(r + u_r\right)}^2}{R^2}\left(\Delta u_0+\Delta \delta \right)\end{array}$$

(46)

When r + Δu_r = 0, i.e., the center point, it is true that: Δu_r < 0 or Δu_r > 0, Δu_z′ < Δu₀ + Δδ or Δu_z′ > Δu₀ + Δδ. That is, the lateral shift increment at the center point is not consistent with the center motion of the sphere, which is not compatible with the assumption. Therefore, if there is a lateral shift increment at each particle point of the half-space contact interface, the lateral shift increment is determined by the circular angle γ corresponding to the arc formed by this particle point and the collision center when the particle point is at u₀, δ state, and its direction points to the non-contact area. Since the deformation curve in the non-contact area at any moment must be antisymmetric with the deformation curve in the contact area, it can be proved that if there is a lateral deformation increment in the non-contact area and the lateral shift is determined by the angle of the arc corresponding to the farthest deformation point when the mass point is at u₀, δ state, and its direction points to the collision center.

According to the known conditions, at any moment, the interface collision center O″ has no lateral shift, that is,$u_r|{}_{o^{″}}=0$, interface deformation zone boundary point O-2 has neither lateral nor vertical displacement, that is, $u_r|{}_{o-2}=0$, the cumulative value of lateral shift in the contact area is $\sum u_r|{}_{0\le r\le r_0}$, the cumulative value of the lateral shift in the non-contact area is $\sum u_r|{}_{r_0\le r\le 2r_0}$, and then:

$$\sum u_r|{}_{0\le r\le r_0}+\sum u_r|{}_{r_0\le r\le 2r_0}=0$$

(47)

That is

$$\sum u_r|{}_{0\le r\le r_0}=-\sum u_r|{}_{r_0\le r\le 2r_0}$$

(48)

Due to the deformation curve in the contact zone $\stackrel{⌢}{o^{″}{c}^{\prime }}$ is unsymmetrical with $\stackrel{⌢}{c^{\prime }{o}_{-2}}$, it is not difficult to conclude that there is no lateral displacement of each mass point at the interface of the whole deformation zone (contact and non-contact zones).

So far, it can be concluded that the half-space interface (deformation zone) has only vertical displacement during the collision with the sphere.

2.4. Formation of Dynamic Equations of Particles at Collision Interface

Any particle on the elastic half-space and interface can satisfy the following equation (particle dynamics equation):

$$\rho \frac{{\partial }^2\left(u+\delta \right)}{\partial t^2}|{}_{t\ge t^{\prime }}=\frac{E}{2\left(1+\mu \right)}\left[\frac{1}{1-2\mu }\frac{\partial \theta }{\partial z^{\prime }}+\frac{{\partial }^2\left(u+\delta \right)}{\partial {x^{\prime }}^2}+\frac{{\partial }^2\left(u+\delta \right)}{\partial {y^{\prime }}^2}+\frac{{\partial }^2\left(u+\delta \right)}{\partial {z^{\prime }}^2}\right]$$

(49)

where $\theta =\frac{\partial \left(u+\delta \right)}{\partial z^{\prime }}+\frac{\partial v}{\partial x^{\prime }}+\frac{\partial w}{\partial y^{\prime }}$, u, v and w are the displacements of the particle in the x′, y′, and z′ directions, respectively. Since the sphere only displaces in the z-direction, v = 0, w = 0, $\theta =\frac{\partial \left(u+\delta \right)}{\partial z^{\prime }}$, then Equation (49) can be written as:

$$\rho \frac{{\partial }^2\left(u+\delta \right)}{\partial t^2}|{}_{t\ge t^{\prime }}= \frac{E}{2(1+\mu )}\left[\frac{{\partial }^2\left(u+\delta \right)}{\partial {x^{\prime }}^2}+\frac{{\partial }^2\left(u+\delta \right)}{\partial {y^{\prime }}^2}+\frac{2-2\mu }{1-2\mu }\frac{{\partial }^2\left(u+\delta \right)}{\partial {z^{\prime }}^2}\right]$$

(50)

$$\frac{{\partial }^2\left(u+\delta \right)}{\partial t^2}|{}_{t\ge t^{\prime }}=c^2\left[\frac{{\partial }^2\left(u+\delta \right)}{\partial {x^{\prime }}^2}+\frac{{\partial }^2\left(u+\delta \right)}{\partial {y^{\prime }}^2}+\eta \frac{{\partial }^2\left(u+\delta \right)}{\partial {z^{\prime }}^2}\right]$$

(51)

in which

$$c^2=\frac{E}{2\rho \left(1+\mu \right)}$$

(52)

$$\eta =\frac{2-2\mu }{1-2\mu }$$

(53)

where E is the elastic modulus of the elastic half-space; μ is the Poisson’s ratio of the elastic half-space, ρ is the half-space mass density, and for every point, the time t′ to start to collision is constant. According to the relationship of the coordinate system, the equations can be obtained:

$$\{\begin{array}{l}\frac{\partial \left(u + \delta \right)}{\partial x^{\prime }}=\frac{\partial \left(u + \delta \right)}{\partial x}\\ \frac{\partial \left(u + \delta \right)}{\partial y^{\prime }}=\frac{\partial \left(u + \delta \right)}{\partial y}\\ \frac{\partial \left(u + \delta \right)}{\partial z^{\prime }}=\frac{\partial \left(u + \delta \right)}{\partial \left(z - R\right)}=\frac{\partial \left(u + \delta \right)}{\partial z}\end{array}$$

(54)

$$\{\begin{array}{l}\frac{{\partial }^2\left(u + \delta \right)}{\partial {x^{\prime }}^2}=\frac{{\partial }^2\left(u + \delta \right)}{\partial x^2}\\ \frac{{\partial }^2\left(u + \delta \right)}{\partial {y^{\prime }}^2}=\frac{{\partial }^2\left(u + \delta \right)}{\partial y^2}\\ \frac{{\partial }^2\left(u + \delta \right)}{\partial {z^{\prime }}^2}=\frac{{\partial }^2\left(u + \delta \right)}{\partial {\left(z - R\right)}^2}=\frac{{\partial }^2\left(u + \delta \right)}{\partial z^2}\end{array}$$

(55)

It can be derived from Equation (55) that:

$$\frac{{\partial }^2\left(u+\delta \right)}{\partial t^2}|{}_{t\ge t^{\prime }}=c^2\left[\frac{{\partial }^2\left(u+\delta \right)}{\partial x^2}+\frac{{\partial }^2\left(u+\delta \right)}{\partial y^2}+\eta \frac{{\partial }^2\left(u+\delta \right)}{\partial z^2}\right]$$

(56)

Equation (56) is equivalent to Equation (51) and can also be obtained by using the invariance theory of coordinate translation function (equation).

2.5. Formation of the Equation of Motion of the Sphere

Taking the sphere as the research object, the reaction force of the collision interface to the sphere is f:

$$f={\displaystyle \iint \left[{\lambda }_1\frac{\partial \left(u+\delta \right)}{\partial x}dydz+{\lambda }_2\frac{\partial \left(u+\delta \right)}{\partial y}dzdx+{\lambda }_3\frac{\partial \left(u+\delta \right)}{\partial z}dxdy\right]}$$

(57)

From the rigid assumption of the sphere, it can be known that the motion state at the centroid of the sphere is consistent with the center point of the collision interface, and the motion state of each particle of the collision interface is constant with the center point. It can be known from Newton’s law:

$$m\frac{{\partial }^2\left(u_0+\delta \right)}{\partial t^2}=mg-{\displaystyle \iint \left[{\lambda }_1\frac{\partial \left(u+\delta \right)}{\partial x}dydz+{\lambda }_2\frac{\partial \left(u+\delta \right)}{\partial y}dzdx+{\lambda }_3\frac{\partial \left(u+\delta \right)}{\partial z}dxdy\right]}$$

(58)

where λ₁, λ₂, λ₃ are Lame constants.

Since the collision is on the interface and the contact surface is axisymmetric, the equation of motion of the sphere can be rewritten as:

$$m\frac{{\partial }^2\left(u_0+\delta \right)}{\partial t^2}=mg-{\displaystyle \iint {\lambda }_3\frac{\partial \left(u+\delta \right)}{\partial z}dxdy}$$

(59)

$${\lambda }_3=\frac{1-\mu }{\left(1+\mu \right)\left(1-2\mu \right)}E$$

(60)

2.6. A Set of Governing Mathematical and Physical Equations Describing the Collision between a Sphere and Elastic Half-Space

Equation (61) is a set of mathematical physics equations to study this problem. The three equations in Equation (61) are the interface particle dynamics equation, the sphere’s motion equation, and the distribution function of vertical deformation of the collision interface, respectively.

$$\{\begin{array}{l}\frac{{\partial }^2\left(u + \delta \right)}{\partial t^2}|{}_{t\ge t^{\prime }}=c^2\left[\frac{{\partial }^2\left(u + \delta \right)}{\partial x^2}+\frac{{\partial }^2\left(u + \delta \right)}{\partial y^2}+\eta \frac{{\partial }^2\left(u + \delta \right)}{\partial z^2}\right]\hfill \\ m\frac{{\partial }^2\left(u_0 + \delta \right)}{\partial t^2}=mg-{\displaystyle \iint {\lambda }_3\frac{\partial \left(u+\delta \right)}{\partial z}dxdy}\hfill \\ u=u_0-(R-\sqrt{R^2-x^2-y^2})\hfill \end{array}$$

(61)

The initial conditions of Equation (61) can be given by:

$$\{\begin{array}{c}{\left(u_0+\delta \right)|}_{t=0}=0\\ {\frac{\partial \left(u_0 + \delta \right)}{\partial t}|}_{t=0}=v_0\\ {\frac{{\partial }^2\left(u_0 + \delta \right)}{\partial t^2}|}_{t=0}=g\end{array}$$

(62)

3. Solution Procedure of Mathematical Physics Equations

3.1. Strain Distribution Law of Collision Interface

3.1.1. Dynamic Equation of Particles of Collision Interface

(i)

Dynamic equation of arbitrary particles in collision interface

For any moment, in the collision interface, u₀ is independent of x, y, but only related to z and t, that is:

$$\{\begin{array}{l}\frac{\partial u_0}{\partial x}=\frac{\partial \delta }{\partial x}=0\\ \frac{{\partial }^2{u}_0}{\partial x^2}=\frac{{\partial }^2\delta }{\partial x^2}=0\\ \frac{\partial u_0}{\partial y}=\frac{\partial \delta }{\partial y}=0\\ \frac{{\partial }^2{u}_0}{\partial y^2}=\frac{{\partial }^2\delta }{\partial y^2}=0\end{array}$$

(63)

Therefore, substituting Equations (5) and (6) into Equation (49):

$$\frac{{\partial }^2\left(u+\delta \right)}{\partial t^2}|{}_{t\ge t^{\prime }}=\frac{{\partial }^2{u}_0}{\partial t^2}|{}_{t\ge t^{\prime }}=c^2\left[\eta \frac{{\partial }^2\left(u_0+\delta -\omega \right)}{\partial z^2}-\frac{{\partial }^2\omega }{\partial x^2}-\frac{{\partial }^2\omega }{\partial y^2}\right]$$

(64)

Considering

$$\begin{array}{l}\frac{\partial \omega }{\partial x}=\frac{x}{\sqrt{R^2 - x^2 - y^2}}\\ \frac{{\partial }^2\omega }{\partial x^2}=\frac{1}{\sqrt{R^2 - x^2 - y^2}}+\frac{x^2}{{\left(R^2 - x^2 - y^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\\ \frac{{\partial }^2\omega }{\partial y^2}=\frac{1}{\sqrt{R^2 - x^2 - y^2}}+\frac{y^2}{{\left(R^2 - x^2 - y^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\end{array}$$

(65)

Substituting Equation (65) into Equation (64) and setting $u+\delta =u_{\mathrm{T}}$, the dynamic equation of any particle at the collision interface can be further expressed as:

$$\frac{{\partial }^2{u}_0}{\partial t^2}=c^2\left\{\eta \frac{{\partial }^2{u}_{\mathrm{T}}}{\partial z^2}-\frac{1}{R-\left(u_0+\delta \right)+u_{\mathrm{T}}}-\frac{R^2}{{\left[R-\left(u_0+\delta \right)+u_{\mathrm{T}}\right]}^3}\right\}$$

(66)

(ii)

Dynamic equation of center point of collision interface

When x = y = 0, the dynamic equation at the center of the collision interface is:

$$\frac{{\partial }^2{u}_0}{\partial t^2}=c^2\left[{\eta \frac{{\partial }^2(u_0+\delta -\omega )}{\partial z^2}|}_{\omega =0}-\frac{2}{R}\right]$$

(67)

At the center point, ω ≡ 0, so:

$$\eta \frac{{\partial }^2(u_0+\delta -\omega )}{\partial z^2}=\eta \frac{{\partial }^2\left(u_0+\delta \right)}{\partial z^2}$$

(68)

Therefore, the dynamic equation of the center point can be expressed as:

$$\frac{{\partial }^2{u}_0}{\partial t^2}=c^2\left[\eta \frac{{\partial }^2\left(u_0+\delta \right)}{\partial z^2}-\frac{2}{R}\right]$$

(69)

According to the initial value conditions, it can be known:

$$g=c^2\left[{\frac{{\partial }^2\left(u_0+\delta \right)}{\partial z^2}|}_{u_0=0}-\frac{2}{R}\right]$$

(70)

(iii)

Dynamic equation of boundary point of collision interface

When $x^2+y^2=2R{u}_0-u_0{}^2$, it is the dynamic equation of the collision boundary point particle, and ω = u₀:

$$\frac{{\partial }^2{u}_0}{\partial t^2}=c^2\left[{\eta \frac{{\partial }^2\left(u_0+\delta -\omega \right)}{\partial z^2}|}_{\omega =u_0}-\frac{1}{R-u_0}-\frac{R^2}{{\left(R-u_0\right)}^3}\right]$$

(71)

At any time t, the collision depth of the center point u₀ and the velocity and acceleration of each particle point on the interface is consistent, and the dynamic equation of the center point, boundary point, and particle point at any point can be formed:

$$\{\begin{array}{l}\frac{{\partial }^2{u}_0}{\partial t^2}=c^2\left[\eta \frac{{\partial }^2\left(u_0 + \delta \right)}{\partial z^2}-\frac{2}{R}\right]\hfill \\ \begin{array}{l}\frac{{\partial }^2{u}_0}{\partial t^2}=c^2\left[{\eta \frac{{\partial }^2(u_0 + \delta - \omega )}{\partial z^2}|}_{\omega =u_0}-\frac{1}{R - u_0}-\frac{R^2}{{\left(R-u_0\right)}^3}\right]\\ \frac{{\partial }^2{u}_0}{\partial t^2}=c^2\left[\eta \frac{{\partial }^2{u}_{\mathrm{T}}}{\partial z^2}-\frac{1}{R - \left(u_0 + \delta \right) + u_{\mathrm{T}}}-\frac{R^2}{{\left[R - \left(u_0 + \delta \right) + u_{\mathrm{T}}\right]}^3}\right]\end{array}\hfill \end{array}$$

(72)

3.1.2. Strain Relationship between Any Point in the Contact Region and the Center Point

At any time t, the deformation of the center point u₀, in terms of the relative relationship of the physical quantities between the remaining mass points and the center point in the collision interface, ${\partial }^2{u}_0/\partial t^2$ can be regarded as a constant, and by defining $\partial u_{\mathrm{T}}/\partial z=P^{\prime }$, Equation (33) can be rewritten as:

$$\frac{{\partial }^2{u}_0}{\partial t^2}=c^2\left\{\eta P^{\prime }\frac{d{P}^{\prime }}{d{u}_{\mathrm{T}}}-\frac{1}{R-\left(u_0+\delta \right)+u_{\mathrm{T}}}-\frac{R^2}{{\left[R-\left(u_0+\delta \right)+u_{\mathrm{T}}\right]}^3}\right\}$$

(73)

$${\displaystyle {\int }_{\delta }^{u_{\mathrm{T}}}\frac{{\partial }^2{u}_0}{\partial t^2}}d{u}_{\mathrm{T}}=c^2\left\{\eta {\displaystyle {\int }_{{P^{\prime }}_0}^{P^{\prime }}{P}^{\prime }d{P}^{\prime }-{\displaystyle {\int }_{\delta }^{u_{\mathrm{T}}}\frac{1}{R-\left(u_0+\delta \right)+u_{\mathrm{T}}}d{u}_{\mathrm{T}}}-}{\displaystyle {\int }_{\delta }^{u_{\mathrm{T}}}\frac{R^2}{{\left[R-\left(u_0+\delta \right)+u_{\mathrm{T}}\right]}^3}d{u}_{\mathrm{T}}}\right\}$$

(74)

Then,

$$\frac{{\partial }^2{u}_0}{\partial t^2}u=c^2\eta \frac{{\left(\frac{\partial u_{\mathrm{T}}}{\partial z}\right)}^2-{\left({\frac{\partial u_{\mathrm{T}}}{\partial z}|}_{\begin{array}{l}\omega =u_0\\ u_{\mathrm{T}}=0\end{array}}\right)}^2}{2}-c^2\ln \frac{R-u_0+u}{R-u_0}+\frac{R^2{c}^2}{2}\left[\frac{1}{{\left(R-u_0+u\right)}^2}-\frac{1}{{\left(R-u_0\right)}^2}\right]$$

(75)

At the center point, u = u₀,

$$\frac{{\partial }^2{u}_0}{\partial t^2}{u}_0=c^2\eta \frac{{\left[\frac{\partial \left(u_0+\delta \right)}{\partial z}\right]}^2-{\left({\frac{\partial u_{\mathrm{T}}}{\partial z}|}_{\begin{array}{l}\omega =u_0\\ u_{\mathrm{T}}=\delta \end{array}}\right)}^2}{2}-c^2\ln \frac{R}{R-u_0}+\frac{R^2{c}^2}{2}\left[\frac{1}{R^2}-\frac{1}{{\left(R-u_0\right)}^2}\right]$$

(76)

Since $u,u_0\ll R$ in the collision process, the following approximate calculations are taken into account:

$$\ln \frac{R-u_0+u}{R-u_0}\approx \frac{u}{R}$$

(77)

$$\ln \frac{R}{R-u_0}\approx \frac{u_0}{R}$$

(78)

$$\frac{R^2}{2}\left[\frac{1}{{\left(R-u_0+u\right)}^2}-\frac{1}{{\left(R-u_0\right)}^2}\right]\approx -\frac{u}{R}$$

(79)

$$\frac{R^2}{2}\left[\frac{1}{R^2}-\frac{1}{{(R-u_0)}^2}\right]\approx -\frac{u_0}{R}$$

(80)

Substitute the above results into Equations (75) and (76):

$$\frac{{\partial }^2{u}_0}{\partial t^2}=c^2\eta \frac{{\left(\frac{\partial u_{\mathrm{T}}}{\partial z}\right)}^2-{\left({\frac{\partial u_{\mathrm{T}}}{\partial z}|}_{\begin{array}{l}\omega =u_0\\ u_{\mathrm{T}}=\delta \end{array}}\right)}^2}{2}-\frac{2u{c}^2}{R}$$

(81)

$$\frac{{\partial }^2{u}_0}{\partial t^2}=c^2\eta \frac{{\left[\frac{\partial \left(u_0 + \delta \right)}{\partial z}\right]}^2-{\left({\frac{\partial u_{\mathrm{T}}}{\partial z}|}_{\begin{array}{l}\omega =u_0\\ u_{\mathrm{T}}=\delta \end{array}}\right)}^2}{2}-\frac{2u_0{c}^2}{R}$$

(82)

When $u_{\mathrm{T}}=\delta$ it is the boundary point. According to the assumption, there are ${\frac{\partial u_{\mathrm{T}}}{\partial z}|}_{\begin{array}{l}\omega =u_0\\ u_{\mathrm{T}}=\delta \end{array}}={\frac{\partial \delta }{\partial z}|}_{\begin{array}{l}\omega =u_0\\ u_{\mathrm{T}}=\delta \end{array}}=\frac{\partial \delta }{\partial z}=0$ and ${\frac{\partial \left(u_0-\omega \right)}{\partial z}|}_{\omega =u_0}=0$,

$$\{\begin{array}{l}\frac{{\partial }^2{u}_0}{\partial t^2}u=c^2\left[\frac{\eta }{2}{\left(\frac{\partial u}{\partial z}\right)}^2-\frac{2u}{R}\right]\hfill \\ \frac{{\partial }^2{u}_0}{\partial t^2}{u}_0=c^2\left[\frac{\eta }{2}{\left(\frac{\partial u_0}{\partial z}\right)}^2-\frac{2u_0}{R}\right]\hfill \end{array}$$

(83)

According to Equation (83), the relative relationship between the strain (stress) of any point on the collision interface and the center point at any time can be determined. Each case (${\partial }^2{u}_0/\partial t^2=0$ or ${\partial }^2{u}_0/\partial t^2\ne 0$) can be considered separately.

When $\frac{{\partial }^2{u}_0}{\partial t^2}\ne 0$, it can be obtained:

$$\frac{u}{u_0}=\frac{\frac{\eta }{2}{\left(\frac{\partial u}{\partial z}\right)}^2-\frac{2u}{R}}{\frac{\eta }{2}{\left(\frac{\partial u_0}{\partial z}\right)}^2-\frac{2u_0}{R}}$$

(84)

Therefore, the following equation can be obtained:

$$\frac{{\left(\frac{\partial u}{\partial z}\right)}^2}{{\left(\frac{\partial u_0}{\partial z}\right)}^2}=\frac{u}{u_0}=1-\frac{\omega }{u_0}\approx \frac{\frac{x^2+y^2}{2R}}{\frac{r_0^2}{2R}}=1-\frac{r^2}{r_0^2}, r^2=x^2+y^2$$

(85)

Therefore

$$\frac{\frac{\partial u}{\partial z}}{\frac{\partial u_0}{\partial z}}=\sqrt{1-\frac{r^2}{r_0^2}}$$

(86)

When $\frac{{\partial }^2{u}_0}{\partial t^2}=0$, we have $\frac{\eta }{2}{\left(\frac{\partial u}{\partial z}\right)}^2=\frac{2u}{R}$ and $\frac{\eta }{2}{\left(\frac{\partial u_0}{\partial z}\right)}^2=\frac{2u_0}{R}$

$$\frac{\frac{\partial u}{\partial z}}{\frac{\partial u_0}{\partial z}}=\sqrt{\frac{u}{u_0}}=\sqrt{1-\frac{\omega }{u_0}}\approx \sqrt{1-\frac{r^2}{r_0^2}}$$

(87)

From the above analysis, it is known that:

$$\{\begin{array}{l}\frac{{\partial }^2{u}_0}{\partial t^2}\ne 0:\frac{\frac{\partial u}{\partial z}}{\frac{\partial u_0}{\partial z}}=\sqrt{1-\frac{r^2}{r_0^2}}\\ \frac{{\partial }^2{u}_0}{\partial t^2}=0:\frac{\frac{\partial u}{\partial z}}{\frac{\partial u_0}{\partial z}}=\sqrt{1-\frac{r^2}{r_0^2}}\end{array}$$

(88)

3.2. Analysis of Sphere (Interface) Movement Shape Parameters

The parameters of the sphere movement form include acceleration, displacement (collision velocity), movement time, and the normal stress of the interface particle. To obtain the relationship between the parameters, the following solution paths are set:

Let the reaction force of the collision interface be f at any time,

$$f={\displaystyle \underset{x^2+y^2\le 2R{u}_0-u_0^2}{\iint }{\lambda }_3\frac{\partial u}{\partial z}dxdy}={\displaystyle \underset{r^2\le 2R{u}_0-u_0^2=r_0^2}{\iint }{\lambda }_3\frac{\partial u}{\partial z}rdrd\phi }=\pi {\lambda }_3\frac{\partial u_0}{\partial z}\cdot {\displaystyle {\int }_0^{2R{u}_0-u_0^2}\sqrt{1-\frac{r^2}{r_0^2}}d{r}^2}=\frac{2\pi {\lambda }_3}{3}\frac{\partial u_0}{\partial z}(2R{u}_0-u_0^2)\approx \frac{4\pi {\lambda }_3}{3}\frac{\partial u_0}{\partial z}R{u}_0$$

(89)

The equation of motion of the sphere can be described as:

$$m\frac{{\partial }^2{u}_0}{\partial t^2}=mg-\frac{4\pi R{\lambda }_3}{3}{u}_0\frac{\partial u_0}{\partial z}$$

(90)

Together with Equation (83), the equation set can be formed as:

$$\{\begin{array}{l}\frac{{\partial }^2{u}_0}{\partial t^2}=g-\frac{4\pi R{\lambda }_3}{3m}{u}_0\frac{\partial u_0}{\partial z}\\ \frac{{\partial }^2{u}_0}{\partial t^2}{u}_0=c^2\left[\frac{\eta }{2}{\left(\frac{\partial u_0}{\partial z}\right)}^2-\frac{2u_0}{R}\right]\end{array}$$

(91)

The following equation can be obtained:

$$\eta {\left(\frac{\partial u_0}{\partial z}\right)}^2+\frac{8\pi R{\lambda }_3}{3m{c}^2}{u}_0^2\frac{\partial u_0}{\partial z}-\frac{4u_0}{R}-\frac{2g{u}_0}{c^2}=0$$

(92)

The relationship between the interface collision center point strain ($\partial u/\partial z$) and the corresponding collision depth u₀ can be described as:

$$\begin{array}{ll}\frac{\partial u_0}{\partial z}& =\frac{\sqrt{{(\frac{8\pi R{\lambda }_3}{3m{c}^2}{u}_0^2)}^2+\frac{16u_0\eta }{R}+\frac{8g{u}_0\eta }{c^2}}-\frac{8\pi R{\lambda }_3}{3m{c}^2}{u}_0^2}{2\eta }\\ & =\sqrt{\frac{4\eta c^2{u}_0 + 2gR{u}_0\eta }{R{c}^2{\eta }^2}}\sqrt{\frac{16{\pi }^2{R}^2{\lambda }_3^2{u}_0^4}{9m^2{c}^4{\eta }^2}\cdot \frac{R{\eta }^2{c}^2}{4\eta c^2{u}_0 + 2gR{u}_0\eta }+1}-\frac{4\pi R{\lambda }_3}{3m{c}^2\eta }{u}_0^2\\ & =\sqrt{\frac{4c^2{u}_0 + 2gR{u}_0}{R{c}^2\eta }}\sqrt{\frac{8{\pi }^2{R}^3{\lambda }_3^2{u}_0^3}{9m^2{c}^2(2\eta c^2 + gR\eta )}+1}-\frac{4\pi R{\lambda }_3}{3m{c}^2\eta }{u}_0^2\end{array}$$

(93)

Considering $\frac{8{\pi }^2{R}^3{\lambda }_3^2{u}_0^3}{9m^2{c}^2(2\eta c^2+gR\eta )}\ll 1$, and therefore we have

$$\frac{\partial u_0}{\partial z}\approx \sqrt{\frac{4c^2+2gR}{R{c}^2\eta }}\cdot \sqrt{u_0}-\frac{4\pi R{\lambda }_3}{3m{c}^2\eta }{u}_0^2$$

(94)

In order to obtain the relationship between the velocity of the sphere and the depth of collision, substitute Equation (60) with Equation (56):

$$m\frac{{\partial }^2{u}_0}{\partial t^2}=mg-\frac{4\pi {\lambda }_3}{3}R{u}_0\frac{\partial u_0}{\partial z}=mg-\frac{4\pi R{\lambda }_3}{3}\sqrt{\frac{4c^2+2gR}{R{c}^2\eta }}\cdot u_0^{\frac{3}{2}}+\frac{16{\pi }^2{R}^2{\lambda }_3^2{u}_0^3}{9m{c}^2\eta }$$

(95)

$$m\frac{\partial u_0}{\partial t}\frac{d\left(\frac{\partial u_0}{\partial t}\right)}{d{u}_0}=mg-\frac{4\pi R{\lambda }_3}{3}\sqrt{\frac{4c^2+2gR}{R{c}^2\eta }}\cdot u_0^{\frac{3}{2}}+\frac{16{\pi }^2{R}^2{\lambda }_3^2{u}_0^3}{9m{c}^2\eta }$$

(96)

By defining $v=\partial u_0/\partial t$, we have

$$mvdv=mgd{u}_0-\frac{4\pi R{\lambda }_3}{3}\sqrt{\frac{4c^2+2gR}{R{c}^2\eta }}\cdot u_0^{\frac{3}{2}}d{u}_0+\frac{16{\pi }^2{R}^2{\lambda }_3^2{u}_0^3}{9m{c}^2\eta }d{u}_0$$

(97)

$$v^2=2g{u}_0+v_0^2-\frac{16\pi R{\lambda }_3}{15m}\sqrt{\frac{4c^2+2gR}{R{c}^2\eta }}\cdot u_0^{\frac{5}{2}}+\frac{8{\pi }^2{R}^2{\lambda }_3^2{u}_0^4}{9m^2{c}^2\eta }$$

(98)

Considering

$$\frac{\frac{8{\pi }^2{R}^2{\lambda }_3^2{u}_0^4}{9m^2{c}^2\eta }}{\frac{16\pi R{\lambda }_3}{15m}{u}_0^{\frac{5}{2}}}\cdot \sqrt{\frac{R\eta }{4}}=\frac{5\pi {\lambda }_3{u}_0^{\frac{3}{2}}{R}^{\frac{3}{2}}}{6m{c}^2\sqrt{\eta }}\cdot \frac{1}{2}=\frac{5\pi {\lambda }_3}{12m{c}^2\sqrt{\eta }}{(R{u}_0)}^{\frac{2}{3}}\ll 1$$

(99)

$$2gR\ll 4c^2$$

(100)

we have

$$v\approx \sqrt{2g{u}_0+v_0^2-\frac{16\pi R{\lambda }_3}{15m}\sqrt{\frac{4}{R\eta }}\cdot u_0^{\frac{5}{2}}}$$

(101)

$$t={\displaystyle {\int }_0^{u_0}\frac{d{u}_0}{\sqrt{2g{u}_0+v_0^2-\frac{16\pi R{\lambda }_3}{15m}\sqrt{\frac{4}{R\eta }}\cdot u_0^{\frac{5}{2}}}}}$$

(102)

Based on Equations (101) and (102), the array diagram of the excited waves at the collision interface is schematically shown in Figure 5.

3.3. Determination of Eigenvalues of Sphere (Interface) Motion Shape Parameters

The characteristic values of the motion shape parameters of the center point of the collision interface include the maximum collision depth and the maximum velocity.

When $\frac{{\partial }^2{u}_0}{\partial t^2}=0$, that is $\{\begin{array}{l}g-\frac{4\pi R{\lambda }_3}{3m}{u}_0\frac{\partial u_0}{\partial z}=0\\ \frac{\eta }{2}{\left(\frac{\partial u_0}{\partial z}\right)}^2-\frac{2u_0}{R}=0\end{array}$, and one is led to

$$u_0={\left(\frac{9m^2{g}^2\eta }{64{\pi }^{2}R{\lambda }_3^2}\right)}^{\frac{1}{3}}$$

(103)

At this time, v = v_max. Substitute u₀ into the following equation:

$$v_{\max }=\sqrt{2g{u}_0+v_0^2-\frac{16\pi R{\lambda }_3}{15m}\sqrt{\frac{4}{R\eta }}\cdot u_0^{\frac{5}{2}}}$$

(104)

When v = 0, u₀ = u_0max. u_0max can be obtained by solving the following equation:

$$2g{u}_0+v_0^2-\frac{16\pi R{\lambda }_3}{15m}\sqrt{\frac{4}{R\eta }}\cdot u_0^{\frac{5}{2}}=0$$

(105)

At this time, $\frac{{\partial }^2{u}_0}{\partial t^2}={\frac{{\partial }^2{u}_0}{\partial t^2}|}_{\max }=g-\frac{4\pi {\lambda }_3}{3m}R{u}_{0\max }{\frac{\partial u_0}{\partial z}|}_{\max }$ and ${\frac{\partial u_0}{\partial z}|}_{\max }$ can also be obtained.

4. Discussions and Conclusions

4.1. Main Conclusions Drawn from This Paper

4.1.1. The Relationship between the Collision Interface Deformation Curve at Any Time

The maximum deformation depth of the contact area u₀ is equal to the maximum deformation depth of the non-contact area δ, that is δ = u₀; the deformation curve of the non-contact region is called against the deformation curve of the contact region; the interfacial particles of the deformation region have no lateral shift.

4.1.2. The Distribution of Interfacial Stress and Strain at Any Time

The distribution of interfacial stress and strain at any time can be expressed as

$$\frac{\sigma }{{\sigma }_0}=\frac{{\lambda }_3\frac{\partial u}{\partial z}}{{\lambda }_3\frac{\partial u_0}{\partial z}}=\frac{\frac{\partial u}{\partial z}}{\frac{\partial u_0}{\partial z}}=\sqrt{1-\frac{r^2}{r_0^2}}$$

(106)

$$\frac{\partial u_0}{\partial z}=\sqrt{\frac{4c^2+2gR}{R{c}^2\eta }}\cdot \sqrt{u_0}-\frac{4\pi R{\lambda }_3}{3m{c}^2}{u}_0^2$$

(107)

where $\sigma$ and ${\sigma }_0$ are the stress at any point of the collision interface and the stress at the collision center point, respectively.

4.1.3. Stress at the Center Point of the Interface When the Acceleration of the Sphere Is Zero, and the Motion Velocity Is Maximum

$\frac{{\partial }^2{u}_0}{\partial t^2}=0$ refers to the equilibrium state, and at this time $v=v_{\max }$,

$$\{\begin{array}{l}{u}_0={\left(\frac{9m^2{g}^2\eta }{64{\pi }^{2}R{\lambda }_3^2}\right)}^{\frac{1}{3}}\\ \frac{\partial u_0}{\partial z}=\sqrt{\frac{4u_0}{R\eta }}\end{array}$$

(108)

The stress of the center point:

$${\sigma }_0={\lambda }_3\frac{\partial u_0}{\partial z}={\left(\frac{3mg}{\pi }\right)}^{\frac{1}{3}}{\eta }^{-\frac{1}{3}}{R}^{-\frac{2}{3}}{\left[\frac{(1-\mu )E}{(1+\mu )(1-2\mu )}\right]}^{\frac{2}{3}}$$

(109)

When v₀ = 0 (the initial velocity of the collision is 0), v = 0, u₀ = u_0max

$$2g{u}_0-\frac{16\pi R{\lambda }_3}{15m}\sqrt{\frac{4}{R\eta }}{u}_0^{\frac{5}{2}}=0$$

(110)

then

$$u_0=u_{0\max }={\left(\frac{225m^2{g}^2\eta }{256{\pi }^{2}R{\lambda }_3^2}\right)}^{\frac{1}{3}}$$

(111)

Therefore,

$$\frac{\partial u_0}{\partial z}=\sqrt{\frac{4u_0}{R\eta }}=2{\left(\frac{15mg}{16\pi {\lambda }_3}\right)}^{\frac{1}{3}}{\eta }^{-\frac{1}{3}}{R}^{-\frac{2}{3}}$$

(112)

$${\sigma }_0={\lambda }_3\frac{\partial u_0}{\partial z}={\left(\frac{15mg}{2\pi }\right)}^{\frac{1}{3}}{\eta }^{-\frac{1}{3}}{R}^{-\frac{2}{3}}{\left[\frac{(1-\mu )E}{(1+\mu )(1-2\mu )}\right]}^{\frac{2}{3}}$$

(113)

4.2. Comparison with Traditional Hertz Theory

4.2.1. Conclusions about Hertz Contact Theory

Some basic conclusions in Hertz contact theory are listed as follows.

$$\{\begin{array}{l}\frac{\frac{\partial u^{\prime }}{\partial z}}{\frac{\partial u_0^{\prime }}{\partial z}}=\sqrt{1-\frac{r^2}{r_0^2}}\hfill \\ u_0^{\prime }={\left(\frac{9P^2}{16R{E}^{\ast 2}}\right)}^{\frac{1}{3}}={\left(\frac{6mg}{{\pi }^3}\right)}^{\frac{1}{3}}{R}^{-\frac{2}{3}}{\left(\frac{E}{1-{\mu }^2}\right)}^{\frac{2}{3}}\left(E^{\ast 2}=\frac{E}{1-{\mu }^2}\right)\hfill \\ {\sigma }_0^{\prime }={\left(\frac{6P{E}^{\ast 2}}{{\pi }^3{R}^2}\right)}^{\frac{1}{3}}={\left(\frac{6mg{E}^{\ast 2}}{{\pi }^3{R}^2}\right)}^{\frac{1}{3}}\hfill \\ u_0^{\prime }={\delta }^{\prime }\hfill \end{array}$$

(114)

4.2.2. Analysis and Comparison

When $\frac{{\partial }^2{u}_0}{\partial t^2}=0$ and $v=v_{\max }$(equilibrium state), we have

$$\frac{{\sigma }_0^{\prime }}{{\sigma }_0}=\frac{{\left(\frac{6mg}{{\pi }^3}\right)}^{\frac{1}{3}}{R}^{-\frac{2}{3}}{\left(\frac{E}{1 - {\mu }^2}\right)}^{\frac{2}{3}}}{{\left(\frac{3mg}{\pi }\right)}^{\frac{1}{3}}{\eta }^{-\frac{1}{3}}{R}^{-\frac{2}{3}}{\left[\frac{(1 - \mu )E}{(1 + \mu )(1-2\mu )}\right]}^{\frac{2}{3}}}={\left(\frac{2}{{\pi }^2}\right)}^{\frac{1}{3}}{\left(\frac{2}{1-\mu }\right)}^{\frac{1}{3}}{\left[\frac{1-2\mu }{{(1 - \mu )}^2}\right]}^{\frac{1}{3}}$$

(115)

It can be derived that ${\sigma }_0^{\prime }/{\sigma }_0$ varies from 0.72 to 0.78 with μ varying from 0.2 to 0.4.

When v = 0, v₀ = 0, and u₀ = u_max,

$$\frac{{\sigma }_0^{\prime }}{{\sigma }_0}={\left(\frac{4}{5{\pi }^2}\right)}^{\frac{1}{3}}{\left(\frac{2}{1-\mu }\right)}^{\frac{1}{3}}{\left[\frac{1-2\mu }{{(1-\mu )}^2}\right]}^{\frac{1}{3}}$$

(116)

It can be derived that ${\sigma }_0^{\prime }/{\sigma }_0$ varies from 0.54 to 0.58 with μ varying from 0.2 to 0.4.

The strain ratio between any point in the contact area and the collision center point is the same between the current study and Hertz contact theory:

$$\frac{\frac{\partial u}{\partial z}}{\frac{\partial u_0}{\partial z}}=\frac{\frac{\partial u^{\prime }}{\partial z}}{\frac{\partial u_0^{\prime }}{\partial z}}$$

(117)

Moreover, $u_0=\delta$ derived in this paper is consistent with $u_0^{\prime }={\delta }^{\prime }$ those obtained from the Hertz contact theory.

4.3. Analysis and Evaluation

This paper presents exact solutions for the interface deformation (contact and non-contact regions) and mechanical behavior (contact region) between a rigid sphere and elastic half-space during the normal collision process according to the kinematics and dynamics theory. The results are compared with the traditional contact theory (i.e., Hertz theory), and part of the conclusions are consistent, which are summarized as follows: (1) the maximum deformation depth of the contact region is consistent with the maximum deformation depth of the non-contact region and (2) the stress (strain) ratio between any point in the contact area and the collision center point is determined by the ratio of the contact radius to the maximum contact radius.

Notably, for the contact in the state of motion, the non-contact deformation curve of the interface deformation area at any time is strictly antisymmetric to the deformation curve of the contact area. Moreover, it is worth mentioning that the mechanical effect of the interface is significantly stronger than that of the static contact during the collision process. As a result, there is a certain risk of analyzing the behavior of the mechanical effect of the contact interface under the moving state by applying the traditional contact theory analysis.

The mathematical and physical methods used in this paper to describe and analyze the collision process are taken for granted under the dictates of theoretical logic, but the series of results obtained in this paper are based on the basic assumptions that the rigid sphere is closely attached to the elastic half-space, and the non-contact area is tangent to the contact area at the intersection point (at the contact boundary at that moment), and the lateral deformation of the masses at the collision interface on the interface is neglected in very low-velocity collisions. The research results need to be verified and corrected in practice. We are now working on the accurate representation of the interface mechanics and kinematic behavior in the case of medium velocity collisions, where the lateral deformation of the interface particles is not neglected, in an effort to break through the traditional static contact theory and obtain more realistic results of the interface mechanics and deformation behavior during the collision.

The theories and methods adopted in this paper could help further reveal other phenomena existing in the collision process.