Solutions of the Problems of a Viscoelastic Dynamic Contact between Smooth Curvilinear Surfaces of two Solid Bodies by the Application of the “ Method of the Specific Forces ”

Solutions of the problems of a viscoelastic dynamic contact between smooth curvilinear surfaces of two solid bodies by the application of the “Method of the specific forces” have been given in the article, and the new conception for the definition of the elastic and the viscous forces in the common case of dynamics of a viscoelastic contact is proposed here by the further development of this method. Essence of this method is that, the forces of viscosity and the forces of elasticity can be found by integration of the specific forces acting inside an elementary volume of the contact zone. It is shown here, that this method allows finding the viscoelastic forces for any theoretical or experimental dependencies between the distance of mutual approach of two solid bodies and the diameter of the contact area. Also, the derivation of the integral equations of the viscoelastic forces, the equations for pressure in the contact is presented. Viscoelastic dynamic contacts between two spherical bodies, and between a spherical solid body and a semi-space at impact have been examined. Work and Energy in the phases of compression and restitution, and at the rolling shear have been derived. Approximate solutions for the differential equations of movement (displacement) by using the method of equivalent work have been derived. Equations for the normal contact stresses have been obtained. Also, equations for kinematic and dynamic parameters of the viscoelastic collision have been derived in this article. Examples of the comparison of theoretical results and conclusions have been given in the paper.


Introduction
The objective of this paper is the further development of the application of the "Method of the specific forces (MSF)" for solutions to the problems of a viscoelastic dynamic contact between smooth curvilinear surfaces of two solid bodies.The new conception is proposed here, how to find the elastic and the viscous forces by an integration of the specific forces in the boundaries of the contact area, which can be found by considering of the geometry of the contact.This method has been already used by author for case of the collision between a spherical solid body and a semi-space (Goloshchapov, 2003(Goloshchapov, , 2015)).
It is assumed here that the surfaces of contact are smooth and in this case we are not considering the influence of roughness on the contact forces, and the initial velocities of contact V x and Vτ is less than the effective sound speed in the volume of deformation (Figure 1).Also the effect of the adhesive forces and a plastic deformations have not been considered in this paper.
It is obviously that during the time of a dynamic contact, in the initial point A, the curvilinear surfaces are moving relative to each other with the normal relative velocity V x and the tangential relative velocity V τ between them under action of the normal force F n and the tangential shear force F τ .Also let: F τy and F τz are the tangential forces along axes Y and Z; x is the size of the mutual approach of the bodies along the axis X; r is a current radius of the contact area; h xa is the depth of the contact surface in the plane XAY , or other words it is the depth of indentation of the more hard body into the surface of more soft body; h xb is the depth of the contact surface in the plane XAZ (This size is not shown in the Figure 1); 1 x and 2 x are normal compression deformations of the contacting surfaces relative to the initial point of the contact A; R 1a and R 2a are the radiuses of curvature of the contacting surfaces on the border of contact area in the plane XAY; V τy and V τz are the projections of the tangential relative velocity V τ by axes Y and Z, or they are the tangential relative velocities of the displacement of surfaces of bodies along axes Y and Z; a is the big axis of an elliptic (oval) contact area; b is the small axis of an elliptic (oval) contact area; O 1 and O 2 are the centres of curvature of the contacting surfaces in the initial point of contact A; R 1b and R 2b are the radiuses of curvature of the contacting surfaces on the border of contact area in the plane XAZ.
It is obvious, that in a general case the area of contact is taking the shape of an oval or an ellipse, and in the same time, the contact surface takes the identical shape with the harder surface, and the surface, which has less hardness slips by a surface of the harder body.It is a micro-slip and we do not take in account the energy of this dissipative process in this paper.You can say that in process of sliding deformation under action of the tangential force, the area of contact does not take the elliptical shape, but it is the oval or other asymmetric figure.
But indeed, if the tangential force equal zero the contact area takes the right elliptical shape, but in the case of sliding deformation under the action of the tangential force, the elliptic shape of the contact area is transformed in the oval.Therefore, we can suppose, since the normal force stays same independently of the action of the tangential force, the normal contact pressure (or the normal contact stress) is not changed and stays same, and therefore the size of square of the contact is not changed too.According to this statement, we approximately can consider the area of contact like having the elliptical shape during all the contact time.Also in the case of an impact between the contact surfaces, when the rolling shear between them has place, we can state that the area of contact approximately keeps the elliptical shape too.Moreover, as we will see further, it does not matter, the area of contact is an ellipse or it is an oval, in the both cases, the derivation of the sizes of the contact area can be done in the same manner.
The viscoelastic forces can be found as the sums of the elastic forces and the viscous forces: Where: is the normal elastic force; is the normal viscous force; is the tangential viscous force by axis Y; is the tangential elastic force by axis Y; is the tangential viscous force by axis Z; is the tangential elastic force by axis Z.

Where
, , , y b , are variable parameters depending of the displacement x, y and z , and where: is the effective parameter of viscosity (damping parameter), is the effective parameter of elasticity (stiffness); , are the effective parameters viscosity at a shift; , are the effective parameters of elasticity (stiffness) at a shift.
are the relative displacements (deformations) between contacting bodies or between contacting surfaces in the initial point A; are relative velocities of displacements (deformations) between contacting bodies or between contacting surfaces in the initial point A.
In the past many old papers and others published recently (Mindlin, 1949;Simon, 1967;Johnson, 1985 ;Goloshchapov, 2003Goloshchapov, , 2015;;Cundall & Strack,1979;Hertzsch, Spahn , Schafer, Dippel, & Wolf, 1996;Ramírez, Poeschel, Brilliantov, & Schwager, 1999;Stronge, 2000;Roylance, 2001;Brilliantov & Poeschel, 2004;Makse, Gland, Jnohnson, & Schwartz, 2004;Schwager & Poschel, Van Zeebroeck, 2005;Dintwa, 2006;Schwager & Poschel, 2007, 2008;Cheng, Subic, & Monir Takla, 2008;Becker, Schwagerand, & Pöschel, 2008;Schwager & Poschel, 2008;Thornton, 2009;Cummins, Thornton, & Cleary, 2012) have been used traditional theoretical rheological models, such as the "Linear Spring Dashpot Model"-(LS+D), the "Hertz Mindlin Spring Dashpot Model"-(HM+D), and the "Discrete Elements Method" -(DEM) and others .In all of these methods and models, for the definition of the effective parameter of elasticity (Some authors name it like a stiffness, or spring parameter), the Hertz's theory of elastic contact between two surfaces (Landau andLifshitz, 1944, 1965) has been used.Also, for the purpose of finding the tangential forces, the coefficient of friction was taken as a constant value.The more comprehensive analysis and review of these already known models and methods can be found, for example, in the monographs of the authors, such as: Stronge (2000); Van Zeebroeck (2005); Dintwa (2006); Li (2006).But, the most basic problem in the finding of the mechanical dynamic parameters of viscoelasticity , , , in the equations ( 3) is that, they are not the constant values.They all are variable magnitudes, because all dynamic mechanical and physical properties of the materials depend on dynamic conditions of loading (displacements, a velocity and a frequency) and temperature.But, the Hertz theory allows only the finding the normal elastic force.The existing methods still cannot give the complete answer, how these nonlinear parameters of viscoelasticity can be found for the practical application by using the dynamic modules of elasticity and viscosity, which usually can be found by using the known methods (Ferry, 1948(Ferry, , 1963;;Moore, 1975;Van Krevelen 1972;Nilsen, 1978Nilsen, , 1994) ) For example as we know according the Hertz theory (Landau & Lifshtz, 1944, 1965) for the contact of two spherical surfaces . Where E is effective elasticity modulus, R is the effective radius of contact curvature.As we can see, the stiffness cx is the nonlinear function of displacement x, but we still have a problem in definition of others parameters , , , , , which are the variable nonlinear functions too.
And also in already existing researches, the coefficient of friction usually is taken like a constant value for the purpose of finding the tangential forces, but according to Equations ( 1), ( 2) and (3) it can be defined as follows: As we can see from this equation, the coefficient of friction is not a constant value, but it changes during the time of contact, because the dynamic contact between two bodies is a non-equilibrium process, and all dynamic mechanical and physical properties of the materials depend on dynamic conditions of loading and temperature.For example, the dynamic elasticity modules are very yieldable to a changing of a velocity and temperature of the matter in the area of deformations (Ferry, 1948(Ferry, , 1963;;Lee, 1962;Van Krevelen, 1972;Moor 1978;Nilsen, 1978Nilsen, , 1994;;Lakes, 1998;Meyers, 1994;Menard, 1999;Roylance, 2001;Goloshchapov, 2001Goloshchapov, , 2003Goloshchapov, , 2015;;Hosford, 2005;Popov, 2010, Popov & Hess, 2015).Also, it is necessary to mention that, the researches in the field of the collision of viscoelastic particles (granules) with identical mechanical properties have been made by Brilliantov, Spahn, Hertzsch and Pöschel (1996).They have obtained the equation for the normal viscous force with variable viscosity parameter is the damping viscous parameter, and where 1 η and 2 η are the viscous constants.But this theoretical result can only be used for the contact of the bodies with the same physical-mechanical properties, and in this case we have the problem of finding the viscous constants " 1 η " and " 2 η ".If the contacting surfaces have different physical-mechanical properties this conception does not give the answer, because this is a yet more difficult problem.
Also the interesting method -"Method of Dimensionality Reduction (MDR)" has been presented by Popov (2015), but it can be used only in the case of contact between three-dimensional, axial-symmetric bodies and a foundation (a semi-space).In all these papers, to find the equations for tangential forces, the coefficient of friction again was taken as a constant value.
Thus, as we can see, the many problems still exist now in these research areas.Therefore, especially for the solving of these problems, such as the definition of the normal viscous force and the all tangential viscoelastic forces, and for the finding of the kinematic and the dynamic mechanical parameters between two contacting surfaces, such as the elasticity modulus and the viscosity modulus, the theoretical and experimental ways have been developed and represented in this article below.

Derivation of the Equations for the Viscoelastic Forces by the "Method of the Specific Forces (MSF)"
Let us assume that in the infinitesimal period of the time dt, when the mutual approach between a body and a semi-space is the infinitesimal magnitude dx (Figure 2), inside the elementary infinitesimal volume dV, which is arising around the initial point of the contact A (Figure1 and Figure2), the infinitesimal viscoelastic forces dF n , dF τy and dF τz are beginning to act.
Where: da and db the big and the small axes of the infinitesimal contact area; is the normal effective specific viscoelastic force; is the tangential effective specific viscoelastic force by axis Z; is the tangential effective specific viscoelastic force by axis Y.
Also it is very important to understand indeed that, these specific forces Fxi, Fyi and Fzi are the elementary axial loads distributed on the infinitesimal sizes, da, db and dx parallel to axes X, Y and Z (Figure2).The name "specific force" already was applied by author of article (Goloshchapov, 2015), and this name is the short name, which usually is used to denote the elementary distributed axial loads.

The Effective Dynamic Modules and the Effective Dynamic Viscosities
According to the "Newton's Third Law" the effective specific forces and the specific forces between bodies have to be equal: Also let us suppose that the volume of deformation is the system of an infinitely large number of elementary discrete elements (Figure 3.) connected among themselves definitely.And also, in this case let us assume, that for the infinitesimal period of the contact time dt all deformations inside of each elementary discrete element are changing linearly and therefore all specific forces are changing linearly too.Based on this, the equations for all specific forces can be written as the linear functions: (7) Figure 3. Illustration of the "Elementary discrete elements model (EDEM)": a. the elementary discrete element of the normal contact between two bodies; b. the effective elementary discrete element of the normal contact Where: E′ is the effective dynamic elasticity modulus; E η ′ is the effective dynamic viscosity; is effective dynamic elasticity modulus at the shear; is the effective dynamic viscosity at the shear; and are the dynamic elasticity modules.and is the dynamic elasticity modules at the shear.and is the dynamic viscosities at the shear.
In the proposed model, each elementary deformation between two bodies develops analogically like the deformation of the elementary discrete element, which is depicted in Figure 3.a.It is a simple case of the linear model of deformations of elementary discrete elements, and instead this model with four elements we can use its analogy -the model with two effective elements depicted in Figure 3.b.Also the "Elementary discrete elements model" for the normal forces can be used for the tangential forces in the same manner.
It is obvious that the according to the initial conditions, when t = 0, х = 0, the specific elastic forces acting by X are equal at the initial instant of the contact (in this point they are equal zero); and the according to the boundary conditions, when х = х m , they are equal at the instant of the maximum compression between two surfaces (in this point they reach the maximum value).But at the same time, the specific viscous forces are equal at the initial instant of the contact, when t = 0, х = 0 (in this point they are equal to the maximum value, because the velocity has the maximum value), and they are equal at the instant of the maximum compression between two surfaces, when х = х m (in this point they are equal zero, because the velocity ).All of these forces in the Equation 7 And also we can write that and (10) Where: and are the coefficients of deformation.

Integral Equations for the Viscoelastic Forces
Now, since according to ( 5) and ( 6) we can write that and according to (7), since , , the twelve expressions for the infinitesimal viscoelastic forces can be written separately as: And since the limits of integration h xa and h xb are known, the integral equations for all viscoelastic forces can be written respectively as

The Geometry of the Area of the Contact and the Pressure Distribution
And now, the important moment, it can be seen that for a finding of the solutions for all Equations (12) we have to know only the equations or the formulas for a = f(x), b = f(x), and for h xa = f(x) and h xb = f(x).For example, we can use that 2 / 1 ) (Rx r = according to the Hertz theory, but according to this theory, the area of contact is a flat surface and the depth of indentation (the depth of the contact surface) h x = 0.But in reality the area of contact usually is not a flat, it is a curvilinear surface.In Hertz's theoretical models has been taken that the contacting surfaces deform together without of the micro-sliding, but in reality each surface deforms independently.Therefore, to find the radiuses of the contact area r a and r b in reality, first of all, let us consider the geometry of contact between two curvilinear surfaces in the normal section in the plane XAY, like it is depicted in the illustration in Figure1.It is obviously that, in the time of indentation of more hard surface into a soft surface, the contact surface takes a curvilinear shape, where the point B (see Figure1) is a special point where the deformations always equal zero, and the border of the area of contact always pass through this point B. According to this statement, the distance O 2 B between this point and the centre of curvature O 2 of the surface of more hard body will not be changed in the period of time of contact.This distance always equals to the radius of curvature R 2 .Also the distance O 1 B between this point and the centre of curvature O 1 of the surface of less hard body will not be changed in the period of time of contact too.This distance always equals to the radius of curvature R 1 .Hence, obviously that , and since as , after a simple calculation, if to neglect by members of smallest order, we get the next equation for the radius of contact area r a = f(x) equals a/2, and then analogically in same way we can find the radius of contact area is the effective initial radius of contact curvature in the plane XAZ.The Equation ( 13) is not convenient for using and therefore, let us rewrite them as are the correlation coefficients.Practically for the solution of the contact problems, the correlation coefficient can be found by the method of iterations and a consecutive approximation.
Obvious from ( 13) that the surface of the contact takes the elliptical shapes in the planes XAY and XAZ, but for a simplification, its shape can be approximated by the parabolic functions ( 14) respectively as Since the surface of the contact has approximately the parabolic shape, let us to take that the radial distribution of the pressure inside of this area changes analogically according to the parabolic function as Where: r y is a current radius of the contact area by axis Y; r z is a current radius of the contact area by axis Z; P c is the maximum magnitude of the pressure in the centre of the contact area.
Further since the square under these functions in the Equation17 and the square under the linear function of the mean pressure P m in the contact area are equal, we can write that Then after the integration follows and finally the ratio between maximum and the mean pressure in the contact zone can be found as Now let us to define the depths of the contact surface (see Figure 1) in the planes XAY and XAZ.The expressions for the radiuses of contact area can be found also as follows After a simple geometric calculation, if to neglect by members the smallest order, we obtain the next equations for the radiuses of contact area: Then after the comparison equations ( 22) and ( 13) we can write that , the formulas for xa h and xb h can be written as follows are the coefficients of depth of the contact surface.
Also let us remark that, in the case a very small deformations, when But, according to the Hertz theory (Landau andLifshitz 1944, 1965) , but it is possible only in the one case when

The Equations for all Viscoelastic Forces
Since as , according to the Equation ( 14) we can write that The taking of the derivatives da/dx and db/dx gives us and then after an integration of the integral equations for the normal forces from ( 12), we get where and if the contact area is a circle R a = R b = R we have the same solution that have been obtained for the contact between spherical surfaces by using the Hertz theory (Landau & Lifshitz, 1944, 1965) Thus, it is obvious that the proposed method of the finding of the normal elastic forces definitely is valid and correct.It can be seen that, if we know a functional dependency between r and x, we can always find the elastic force.But, if this method is correct for the definition this force, hence it should be valid for the definition of all viscoelastic forces in the equations ( 12).The equation for the normal viscous forces can be found in the same way by integration: are the function linear independent from x, and it cannot be integrated by x, and it stays outside of integrals.And since as bbn abn bn In the case when the contact area is a circle Since as x and y are linearly independent and , after an integration of the equations for the tangential forces from Equation ( 12) their solutions can be written as follows: Here z z y y , , ,   are the functions linear independent from x, and they cannot be integrated by x, and they stay outside of integrals.
The equations for the tangential elastic and viscous forces can be written now, as the sum of the elastic and the viscous tangential forces from Equation (34): Where: Thus finaly, according to Equations ( 1), ( 31), ( 34) and (35) the next system of equations for general viscoelastic forces can be written respectively: And thus, we can write the expressions for the variable viscoelasticity parameters as follows: It is not possible the finding of the viscoelastic forces separately for each contacting body if to use the Hertz theory and others already existing theories, but it is possible by using the "Method of the specific forces (MSF)".The integral Equation ( 12) separately for each contacting body can be written as follows: Where usually the index i = 1 use for a soft body and i = 2 use for a hard body.

=
, then after differentiation we get: After a substituting the expressions (40) into (39) and then after their integration we can get the equations for the viscoelastic forces separately for each body.If we can find the viscoelastic forces separately for each body hence we can find separately the viscoelastic stresses for each body too.It is not possible to do by using the Hertz theory or by using others already existing theories and methods.
The normal viscoelastic stresses, which equal to the mean pressure P m in the contact area, can be found as  20), ( 37) and ( 41) the expression for maximum value of the normal contact pressure (stresses) can be written as follows:

Dynamic Contact between two Spherical Bodies at Impact
In the case of the contact between two spheres, when the contact area is a circle R a =R b = R, r a = r b =r, a = b=2r, F τz = 0, F τ = F τy , F bτ = F b τy , F cτ = F cτy (see Figure1) follows that , and , 2 / 1 2 R k p = ψ , and then according to Equations ( 1) and (37) we get the next system of equations for the main viscoelastic forces: Where , and where the expressions for the variable viscoelasticity parameters in the Equation (3) can be written as follows Also, for this case, the expression for the depth of the contact surface can be written as follows

Contact between a Spherical Solid Body and a Semi-Space at Impact
This example is given here, because the results, which can be obtained in this case, can be used also for others cases of a dynamic contacts, such as for the dynamic contact at impact between two spherical bodies or between two cylindrical bodies.
Let a spherical body, having the average statistical mass , the radius R and the initial velocity V 0 , comes into viscoelastic contact under an arbitrary angle of attack α to the surface of semi-space at the initial instant of the time t = 0, at the initial point of contact 0 (Figure 4).And let the vectors of velocities are applied to the centre of mass of the body (the point C 0 ).Also in Figure4 are designated: V 0x = V 0 sinα, V 0y = V 0 cosα are the initial normal and tangential velocities of a body; M = F τ l x is the reactive moment; l x is the shoulder of tangential force; ω is the angular velocity and ε is the angular acceleration around of the centre of mass of a body; V d is the volume of deformations, which is forming in the course of contact.Also the geometry of surface of the contact zone is characterised by the geometrical parameters, such as ( Figure 4): is the normal deformation of the surface of semi-space in the middle of the contact area, which as well, is the approach of a semi-space relative to a body; is the normal deformation of the surface of a body in the middle of the contact area, which as well, is the approach of a semi-space relative to a semi-space; r -the radius of the contact area; h x is the depth of the contact surface, or other words, h x is the depth of indentation of a body into the surface of semi-space.Also, it is seen here (Figure4) that at the initial instant of the time, the body with the centre of mass in the point C 0 comes into contact with the surface of semi-space at the initial point of the contact 0 with coordinates х = 0 and y = 0, but at the instant of the time t, the centre of mass of a body (the point C t ) takes the position with coordinates x and y.Where: m is the effective or reduced mass of the contacting bodies; -the accelerations of the centre of mass of the body; J z is the moment of inertia of a body; the angle of rotation of the body around the centre of mass; is the angular acceleration of a body around the centre of mass.
We can see here that is the distance of the mutual approach (the total deformation) between a body and a semi-space, and as well, in the same instant of the time, it is the displacement of the centre of mass of a body relative to the initial point of contact 0 by axis X.At impact of two bodies, the effective mass m enters usually like the mass of the third body, and the movement (the displacement) x of the centre of mass of this third body takes equal to the distance of the mutual approach (or a compression, an overlapping) of the colliding bodies.At impact of two bodies, according to the second law of Newton, we can write that , where V 0x = V 1x +V 2x .These equations are valid only for the movement of the centres of mass of the bodies.All authors, who use these equations, for example, Stronge (2000), Dintwa, (2006), Bordbar, Hyppänen (2007), Antypov, Elliott, Hancock (2011), Landau (1944Landau ( , 1965)), Brilliantov (1996) take that x is the displacement of centre mass of this third body and x is the mutual approach (a compression, an overlapping) between the bodies too.From these two expressions follows that and since as in the case of the collision of a body and a semi-space , and hence we can take that .As well, if the semi-space is immovable, when the velocity of a semi-space V 1x = 0, follows that .Hence it is proved that in the case of collision between a body and a semi-space, the mass of a body 2 m is equal to the effective mass m.Further in this paper, the mass of body is designated by the symbol m.Consequently, the distance of the mutual approach between a body and a semi-space is equal to the displacement x of the centre of mass of a body.Analogically as for x we can write that is the displacement of the centre of mass of a body relative to the initial point of contact 0 by axis Y, where -is the tangential deformation of the surface of a semi-space is the tangential deformation of the surface of a body.
It is obvious that, in the case of the contact between a spherical body and a semi-space, the effective radius R and the radius of a body are equal, and if take in account ( 25) and that M = F τ l x , then if the dynamic viscosities replace by the dynamic viscosity modulus according to the known expressions (Ferry, 1948(Ferry, , 1963;;Lee, 1962;Van Krevelen, 1972;Moor 1978;Nilsen, 1978Nilsen, , 1994;;Lakes 1998;Meyers 1994;Menard 1999;Goloshchapov, 2001Goloshchapov, , 2003Goloshchapov, , 2015;;Hosford 2005;Popov 2010, Popov andHess 2015) and , the equations for the normal and the tangential viscoelastic forces and for the reactive moment can be written as follows: (47) Where is the effective viscosity modulus under, is the effective viscosity modulus at shear, is the normal angular frequency of damped oscillations of the volume of deformation Vd by the axis X, is the tangential angular frequency of damped oscillations of the volume of deformation Vd by the axis Y.
We have to mark that in the case of contact between a spherical body and a semi-space, when R 2 = R (see Equation 44*) follows that Viscosity modules can be found by using the known (Ferry,1948;Moore,1975;Van Krevelen, 1972;Nilsen, 1978;Landel, 1994) where is the angle of mechanical losses.
Let us notice that often the dynamic modulus of elasticity is named yet as the accumulation (storage) modulus, and the dynamic modulus of viscosity is named yet as the loss modulus.
And now, according to ( 45) and ( 47), the differential equations of the movement (displacement) of the centre of mass of a body can be written as follows: Or it can be also written in the canonical form as ( 5 0 ) Where, the formulas for the variable viscoelasticity parameters in the system of equation ( 50) can be written as

Work and Energy
As we know the period of time at impact includes two principally different phases such as, the phase of the compression and the phase of the restitution.Also in the duration of a collision, the full initial kinetic energy of a (where V ty is the tangential velocity of the centre of mass of a body in the instant of rebound).Therefore, the description of the processes of the compression and the restitution along the axis X, and the shear along the axis Y are given independently in this part of the paper.The basic problems here are the finding the equations for the work of viscoelastic forces, the dissipative energy in the phases of the compression, the restitution and at the shear.Also it is necessary to determine the coefficients of restitution, the maximum size of the compression between a body and a semi-space, the dynamic modules of elasticity and viscosity.

Work and Energy in the Phases of Compression and Restitution
The graphical illustration of the functional dependences between the normal viscoelastic forces and the displacement of the centre of mass of a body is depicted in Figure 5: (a).Also the "Rheological model of Kelvin-Vogt", which usually is used for the viscoelastic contact, is represented in Figure 5: (b).As we can see, the viscosity force has the extremum in some point of the time equal τb.
( ) and Analogically the works A xct and A xbt in the restitution phase can be found as follows: Where: x is the period time of the contact; 1 τ is the period time of the compression; 2 τ is the period time of the restitution; x m is the maximum magnitude of the compression between a body and a semi-space (also it is the maximum displacement of the centre of mass of a body, which is equal to the maximum of mutual approach between a body and a semi-space).
According ( 52), ( 53), ( 54), ( 55) and ( 56) the equations for the work of the compression and the restitution can be written as follows: and according ( 48) and ( 57) we can write , and by using the first of the equations (58), we get the formula for x m respectively Also, we can define the energetic coefficient of restitution e x , which equals to the square of the kinematic coefficient of restitution k x (further it will be named simply the coefficient of restitution), like the ratio between W tx and W 0x : Since as we can take that we get that and using ( 60) and ( 62) we get that Thus, we have got the equation, which binds the coefficient of restitution and the tangent of the angle of mechanical losses.So, if k x = 1, tgβ → 0 we get the totally elastic impact, but if k x = 0, tgβ → ∞ then we get the totally viscous impact.Using (63) we can write the formula for the restitution coefficient as If to compare the Equations ( 59) and ( 64) we can finally get the expression for the maximum magnitude of the compression between a body and a semi-space respectively as In the case of a totally elastic impact, when k x = 1 and k p = 1 we get the same result, as it has been obtained by L. Landau (1944Landau ( , 1965) ) according to the Hertz theory for the absolutely elastic contact.

Work and Energy at the Rolling Shear
It is obvious that, in the during time of the displacement and the rolling shear along axis Y, the tangential initial kinetic energy of a body W0y is spent for the work A y of the tangential viscoelastic force τ F .The work A y can be found as the sum of the works A yb and A yc , where A yb is the work of the tangential viscous force τ b F and A yc is the work of the tangential elastic force.But on other hand, it is obvious as well, that the work A yb is transformed into the dissipative energy Q ω and the work A yc is transformed into the work A ω of the rotation of the body around the centre of mass of a body.Thus, according to the "Law of preservation of energy for a non-conservative (dissipative) mechanical systems", we can write the equations for the displacement of the centre of mass of a body and for the rotation of a body relative to the centre of mass of a body, as follows below: Where: , and where Also at the point of the rebound, when , all these works in (74) can be found by integration, as follows: The full changing of the energy of the dissipative system at the rolling shear can be found as the difference between A y and A yc from (74): According to the Equation ( 74) the conclusion can be drawn that the work A yc is transformed into the kinetic energy of the rotation of the body relative its centre of mass, but on the other hand the work A yb is transformed into dissipative energy Q ω in the process of the internal friction.Accordingly, using ( 74) and ( 75

Approximate solution to the Differential Equations of the Displacement by using the Method of the Equivalent Works
For practical application of the differential Equation ( 50) with the variable viscoelasticity parameters, we can find their approximate solutions in the same manner as for the equations with the equivalent constant viscoelasticity parameters, if we choose the equivalent constant parameters B x , C x and B y , C y so that the work A xcm and A xbm , A ycm and A ybm with the variable viscoelasticity parameters c x , b x and c y , b y will be equal to the works with the constant viscoelasticity parameters.Thus, according to this statement, and since as the work A xcm and A xbm , A ycm and A ybm are known from Equations ( 53), ( 54) and (75), we can write next equations in the phase of the compression and also in the phase of the rolling shear for the period of the compression time Hence, according to the results obtained in (82) and in (83), we can write the expressions for the equivalent constant viscoelasticity parameters, respectively as: Thus, the Equation ( 50) with variable parameters can be rewritten as the equations with constant parameters as follows: ( ) The Equation ( 85) are the equations of the damped oscillations and the solutions to these equations are known: It is obviously that the period of time of the contact τ x is equal to the semi-period of damped oscillations T x /2 by axis X. x and also by using Equations ( 60), ( 62),( 63) and ( 87) we get: The equation for the restitution coefficient we can write now as follows: If tgβ = 0 hence k x = 1 , it is a totally elastic impact, but if tgβ = π/2 hence k x = 0 and x = 0 , it is absolutely plastic impact.Both of these two cases are not possible in nature.
Finally, from ( 65) and (89) follows that  9), ( 46) and ( 48) tgβ can be calculated by formula ) ( The equations for the velocities of the centre of mass of a body can be received by differentiation of ( 86): where and since tgβ is known from (88), by using ( 65),( 93) and (94) we get

Determination of the Dynamics Modules by the Method of the "Temperature -Time Superposition"
The dynamic elasticity and viscosity modules for high velocities of the collision can be found, if to follow the principles of the "Time-temperature superposition" according to the equation of the "WLF" Williams -Landel -Ferry or Arrhenius (Ferry, 1963;Van Krevelen, 1972;Moore, 1975;Nilsen and Landel, 1994).First of all we have to define experimentally the effect of temperature for the period of the contact time τ x , and for the coefficient of restitution k x at the fixed initial velocity of impact.For example, if we define these parameters for velocity at 2 m/s, then using the principles of the "Time-temperature superposition" we can determine their values for any velocities interesting for us, for example for velocity 100 m/c and for temperature 100 0C.After this, when τ x and k x will be known, we can find the value of tgβ and the dynamic modules E′′ and E′.If to use the equation ( 95), the expression for the calculation of the effective dynamic elasticity module can be written as follows And, if to use (48), ( 88) and ( 96) we get the formula for the calculation of the effective dynamic viscosity module Obviously, if k x = 0, then E′′ = 0 too.We can find G′ and G′′ in the analogical way.

Analyse and Conclusion
It is a very important now to confirm of the correctness of the offered theories and methods, obtained in this article, if to compare them with others already available.For example, the equation for the elastic force have been obtained by the using of the "Method of the specific forces".In the case, when k p = 1 we have the same solution like in the Hertz theory.Hence the Hertz theory gives the partial results in comparisons with proposed "MSF" method, because MSF" let to find the viscoelastic forces for any curvilinear contact between two surfaces, but Hertz case can be using only for the flat contact.The obtained result proves us that, the "Method of the specific forces" is definitely valid for finding of the normal elastic force, because if we know a functional dependency between r and x, we can always find the elastic force.It is obviously that, if it gives the correct way for the definition elastic force, and also as it was represented, it is valid for the definition of the viscous force and the tangential viscoelastic forces.We cannot find viscous force and the tangential viscoelastic forces by using the Hertz's theoretical model, but we can do this by using the "Method of the Specific Forces".It is obviously that, for the finding the normal viscous and the tangential viscoelastic forces, we can take k p = 1, like according the Hertz theory, but we should be aware that, in this case, the contact area is a flat surface according to 2D tensor of deformations and h x -the depth of indentation has to be equal to zero.But in reality, the contact surface takes the curvilinear shape, therefore, alternatively in this paper, the way of the finding of the radius of the contact area, by considering the geometry of contact between two curvilinear surfaces, have been proposed.It was received that the radius of contact area r = f(x)can be find by the equation .Since this equation is not convenient in using, and therefore it was proposed the finding r as , where is the correlation coefficient, which can be found by the method of iterations and consecutive approximations.If a deformation is small, when , hence we can take .And, if contact area is a flat, when , follows from Equation 47* that .Practically the area of contact can be considered as a flat surface only in the case, when the surface of a semi-space in many times harder than the surface of a body.For example, it is possible in the case of impact between a rubber ball and a steel plane.Hence, it is obviously, that the "MSF" is the universal method, which can be used for any functional dependencies between the radius or the diameter of the contact area and the distance of the mutual approach (the total deformation) between two curvilinear surfaces.But nevertheless, we still have the question: What kind of the equation is better to take for finding of the radius of contact area, by the Hertz theory or directly by the way of consideration the geometry of the contact, like it is proposed in this article?Objectively to answer this question, we have to analyse simply logically the way as these equations were received.It was taken according to the Hertz theory that the contact surface is a flat, and the deformations are very small, the contact pressure is distributed analogically as an electrical potential ( Remark: an electrical potential is the scalar function, but a pressure is the vector function), and then, on the basis of this main statements, the equation between the radius of the contact area and the normal elastic force, and the equation between the distance of the mutual approach and the normal elastic force as the effect have been obtained.Then only after that, in result of the comparison of these two equations by excluding the normal force (Landau, 1944(Landau, , 1965)), the expression have been received.But in this article the analogical functional dependence have been proposed as the cause, in the result of the direct consideration the geometry of the contact.It was shown that, in the time of indentation of more hard surface into a soft surface, the contact surface takes a curvilinear elliptical shape (the function is ), where the point B (see Figure1) is a special point where the deformations always equal zero, and the border of the area of contact always pass through this point B. Therefore, the proposed geometrical method is more exact, than according to the Hertz's theoretical model.
On the other hand, the equation for the normal viscous force gives the similar result as it has place in the contact between two bodies with identical mechanical properties in the equation (4*), which have been obtained by Brilliantov, N. V., Spahn, F., Hertzsch, J.-M., and Poeschel, T. (1996).After the comparison of two these equations, since as and we obtain the following ( 9 8 ) But since as and in the quasi-static conditions we get ( 9 9 ) If hence too, it is a totally elastic impact.Thus we can find the parameter A by a very simple way using the "Method of the Specific Forces".Also the equation ( 65) to determine the maximum displacement x m have been derived.It is obvious, that in the case of k x = 1 and k p =1 we have the same result, as was obtained by Landau (1944Landau ( ,1965) ) for a totally elastic impact by using the Hertz Theory.It proves the correctness of the way of finding the Equation 65.But we have to understand that, this equation has the borders of application which can be found if to solve the next equation In the case when the plastic deformations will be have place in the zone of the contact.
Also it is necessary to proof the correctness of the definition of the work for the normal viscous force in Equation ( 54) and in Equation ( 82), because two ways in the order of integrations are possible to apply here such as, that have been taken for finding the solution in Equation ( 54) and in Equation ( 82) and like it is shown below (101) It is simple to proof that the 1-st variant of the order of integration in Equation ( 54) and in Equation ( 82) is correct and the second variant in Equation ( 101) is not correct.It obviously that the attitude between the normal viscous force and the normal elastic force can be find as follows As we can see this attitude is a constant value and it should not change at dependence by the order of integration.If we take the second variant of integration according to Equation ( 101), the values of the constants of viscoelasticity C x and B x will be changed.Thus therefore, in the first variant of integration as it is taken in Equation ( 54) and in Equation ( 82) we have a valid solution.
In conclusion, first of all, let us to mark, that the method of specific viscoelastic forces allows to find the equations for all viscoelastic forces.The proposed method is a principally different with others in which are using the Hertz's theory, the classical theory of elasticity and the tensor algebra.In this method the new conception is proposed, how to find the elastic and viscous forces by an integration of the specific forces in the infinitesimal boundaries of the contact area.The radius of contact area can be taken according the Hertz theory or can be found by the considering the geometry of the contact.This method can be used in researches of the contact dynamics of any shape of contacting surfaces.Also in the article the method of the solution of the differential equations of a movement has been proposed and they have been solved.This method also can be used for determination of the dynamic mechanical properties of materials, and it can be used in the design of wear-resistant elements and coverings for components of machines and equipment, which are working in harsh conditions where they are subjected to the action of flow or jet abrasive particles.Also the theoretical and experimental statements which are presented here can be useful in the design of elements and details machines and mechanism which are being in the conditions of the dynamic contact.The results of the experimental and theoretical research and the method of the specific forces presented in this article can be used for the determination of the viscoelastic forces, contact stresses, durability and fatigue life for a wide spectrum of the tasks relevant to collisions between solid bodies under different loading conditions.Opportunities exist to use the obtained results practically in the design and development of new advanced materials, wear-resistant elastic coatings and elements for pneumatic and hydraulic systems, stop valves, fans, centrifugal pumps, injectors, valves, gate valves and in other installations.Also the using of this theory gives an opportunity for the development of analytical and experimental methods allowing optimising the basic dynamic and mechanical visco-elastic qualities already existing materials and in the development new advanced materials and elements of machines.Also this theory can be used not only for visco-elastic contact and also for any other kind of contacts, such as the elasto-plastic contact and for the elasto-visco-plastic contact too.

Figure 1 .
Figure1.Illustration of the mutual approach between two curvilinear surfaces of two solid bodies along the axis X relative to the initial point of contact A = , Y is the Young modulus or the elasticity modul, ν is the Poisson ratio,

Figure 2 .
Figure 2. Illustration of the action of the specific elementary viscoelastic forces inside the infinitesimal volume dV in the vicinity of point A are linear continuous functions and if they are equal for these two values of the argument x, they have to be equal for any other values as well, or by other words, they are equal in any instant of the time of the contact.Analogically this conclusion is valid for other specific forces acting by axes Y and Z.Thus consequently we can we can write the equations for the effective dynamic modules and the effective dynamic viscosities as , , , area, and according to Equation (26) follows

Figure 4 .
Figure 4. Schematic illustration of the indentation of a spherical body into a semi-space the two independent parts such as, the normal initial kinetic energy of a where V t is the velocity of the centre of mass of a body in the instant of rebound) is included two independent parts such as, the normal kinetic energy of a tx is the normal velocity of the centre of mass of a body in the instant of the rebound ) and the tangential kinetic energy of a body at the instant of rebound Figure 5: (a) -The graphical illustration of the functional dependences between the normal viscoelastic forces and the displacement x of the centre of mass of a body; (b) -The "Nonlinear Rheological Model of Kelvin-Vogt", where c x and b x are not the constant magnitudes It is obvious that A xcm = A xct and hence the potential energy which has been accumulated inside of the elastic element (spring) fully returns back to the body in the instant of rebound.The works A xcm and A xbm at the compression can be found by integration:

y
my is the velocity at the instant of the time 1 τ = t; ω m is the angular velocity relative to the centre of mass of a body at the instant of the time 1 is displacement of the centre of mass of a body along axis Y at the instant of the time 1

F
for the angular velocity at the instant time of rebound can be written as follows the work A yb of the viscous tangential force τ b is equal to the dissipative energy Q ω , using Equation (75) we get frequency of the harmonic oscillations by axis Y.
a very simple way to calculate x m , if we know the value of tgβ.According to the Equation ( for the velocity, the duration of the time of the impact equals to the period of the time of the contact can be found now from the conditions , which can be approximated by the parabolic function