Wear

Abstract

Along with the closely related phenomenon of fatigue, wear is a central aspect in estimating the service life of any technical system. The microscopic and mesoscopic mechanisms causing the macroscopically observable phenomenon of wear are extremely varied and range from abrasive or adhesive debris formation, to the reintegration of previously removed material, to oxidation, and to chemical or mechanical intermixing of the involved surfaces. Accordingly, the formulation of a general wear law is quite difficult. Analogous to the Amontons–Coulomb law of dry friction, a common approach involves an elementary, linear relationship between wear intensity and normal load introduced by Reye, Archard and Hirst, as well as Khrushchov and Babichev. However, many of the results in this chapter do not depend on the exact form of the law of wear. We only assume that the wear is sufficiently continuous so that no wear particles are released whose linear dimensions are comparable to the characteristic lengths of the contact problem. In this chapter, we will first examine the wear by complete sliding of the surfaces involved and then the wear in contacts subjected to small amplitude oscillations (fretting).

Wear is the mechanical or chemical degradation of surfaces. Along with the closely related phenomenon of fatigue, wear is a central aspect in estimating the service life of any technical system. As is the case with friction, the microscopic and mesoscopic mechanisms causing the macroscopically observable phenomenon of wear are extremely varied and range from abrasive or adhesive debris formation, to the reintegration of previously removed material, to oxidation, and to chemical or mechanical intermixing of the involved surfaces. Accordingly, the formulation of a general wear law is quite difficult. Analogous to the Amontons–Coulomb law of dry friction, a common approach involves an elementary, linear relationship introduced by Reye (1860 ), Archard and Hirst (1956 ), as well as Khrushchov and Babichev (1960 ). When using this law, one must bear in mind that it is a very rough approximation.

In this chapter, we consider rotationally symmetric profiles that wear while retaining rotational symmetry. Trivially, wear is axially-symmetric for an axially symmetric load (such as a torsional load around the center line of the profile, which was studied by Galin and Goryacheva (1977) ). However, the retention of axial symmetry of the profile does not necessarily require axial symmetry of the load condition.

Wear can even be rotationally-symmetric during movements in directions which violate the symmetry of the system. This is the case, for example, when a rotationally symmetric profile is in a state of gross slip at a constant speed relative to an elastic base, where only the profile yet not the elastic base is the subject of wear. Since (in this case) the pressure is axially-symmetric and the relative slip speed of both surfaces is constant in all points of contact, the wear intensity is also axially-symmetric. At least, this is valid for all local wear laws where the wear intensity is only dependent on the pressure and relative slip rate. If the counterpart exhibits wear, the system loses its axial symmetry.

Approximate axial symmetry is also retained in the practically relevant case of wear due to low-amplitude oscillations (fretting). In this case, the stress tensor is not isotropic. For a unidirectional oscillation in the Cattaneo–Mindlin approximation , the stress tensor only has a tangential component which, however, is solely dependent on the polar radius. In the same Cattaneo–Mindlin approximation, the relative displacements of the surfaces also only depend on the radius. Wear determined by the local pressure and the relative displacements is thus axially-symmetric, regardless of the particular form of the law of wear. This leads to the common phenomenon of ring-shaped wear patterns.

Many of the results in this chapter do not depend on the exact form of the law of wear. The only assumption held is that the wear occurs continually to prevent the generation of wear particles of the same characteristic length as the contact problem. We further assume that the law of wear is local, i.e., the wear intensity at a location depends only on the pressure and the velocity occurring at this location. In the rare cases demanding a concrete law of wear, we will assume the simplest law by Reye–Archard–Khrushchov , which says that the local wear intensity is proportional to the local pressure and the local relative velocity. With the profile shape of the wearing body at the point in time t denoted by $f(r,t)$, the law can be written as:

$$\dot{f}(r,t)\sim p(r,t)\left|v_{\text{rel}}(r,t)\right|.$$

(6.1)

Here, $v_{\text{rel}}$ is the relative velocity between the contacting surfaces and $p(r,t)$ is the local pressure. The dot above the quantity signifies the differentiation with respect to time.

In this chapter, we initially examine wear for gross slip of the contacting surfaces and then wear due to fretting.

6.1 Wear Caused by Gross Slip

We consider an axially symmetric punch with the profile shape $f(r,t)$ and characterize its normal load by the (generally time-dependent) indentation depth d(t). The profile shape and the indentation depth uniquely define the entire solution of the normal contact problem, including the contact radius a, the pressure distribution $p(r,t)$, and the normal force:

$$F_{N}(t)=2\pi\int_{0}^{a}p(r,t)r\mathrm{d}r.$$

(6.2)

The two common load types we consider in the following sections are the force-controlled loading (i.e., $F_{N}$ is given) and displacement-controlled loading (i.e., the “height” d is given).

In the context of wear problems, the “indentation depth” in a contact mechanical sense is measured from the lowest point of the current (i.e., worn) profile. Therefore, its definition changes with progressing wear.

6.1.1 Wear at Constant Height

The simplest case of wear is realized when an indenter with the original profile $f_{0}(r)$ is initially pressed to the indentation depth $d_{0}$, and subsequently displaced tangentially at this height. While the calculation of the process of wear, in this case, remains a relatively complicated contact mechanical task (see the work of Dimaki et al. (2016)) with the process also depending on the explicit type of the law of wear, the final state is simply defined by the condition that the profile just barely remains in contact at all points with the base. This state is reached when the original profile is “cut off” at the height $d_{0}$. This conclusion is not bound to the axial symmetry, nor to the type of movement (as long as a constant height is maintained): any profile displaced arbitrarily at a constant height (ensuring a relative displacement in all contact points) tends to assume the shape of the original profile cut off at the height $d_{0}$.

6.1.2 Wear at Constant Normal Force

During force-controlled processes, the normal force remains constant and the wear process never ceases. The case of cylindrical indenting bodies, where the contact area remains constant despite wear, is particularly simple. Pressing a cylindrical punch into the base with subsequent tangential displacement at a constant speed leads into a state of steady wear after a transient period, in which all points of the punch wear at the same linear rate. Under the assumption of any local law of wear in which the wear intensity is proportional to the product of the pressure and the sliding velocity, it follows that in the state of steady wear, the pressure is constant in the entire contact area and equal to $p_{0}=F_{N}/A$, where $A$ is the (constant) contact area. The profile shape that generates a constant pressure was previously determined in Sect. 2.5.6:

$$w(r;a,p_{0})=\frac{4p_{0}a}{\pi E^{*}}\mathrm{E}\left(\frac{r}{a}\right),\quad r\leq a.$$

(6.3)

Here, $\mathrm{E}(\cdot)$ refers to the complete elliptical integral of the second kind:

$$\mathrm{E}(k):=\int_{0}^{\pi/2}\sqrt{1-k^{2}\sin^{2}\varphi}\mathrm{d}\varphi.$$

(6.4)

The expression (6.3) is shown in Fig. 6.1 in normalized variables.

Fig. 6.1

Shape of the punch in the phase of steady wear during unidirectional slip at a constant rate

For general cylindrical punches (with any arbitrary, not necessarily circular face), this analysis can be generalized to heterogeneous punches with wear coefficients $k=k(x,y)$ that depend on the position in the contact area (characterized by the coordinates x and y), but not on the z-coordinate. One such example is a fiber-composite pin, whose fibers run parallel to the longitudinal axis. Once the system settles into a steady state, the wear intensity is constant at all points of the contact area:

$$\dot{f}=k(x,y)\cdot v_{0}\cdot p(x,y)=C.$$

(6.5)

It follows that

$$p(x,y)=\frac{C}{k(x,y)\cdot v_{0}}.$$

(6.6)

The constant C is obtained from the condition:

$$F_{N}=\int p(x,y)\mathrm{d}x\mathrm{d}y.$$

(6.7)

The pressure distribution is then determined from (6.5) to be:

$$p(x,y)=\frac{F_{N}}{k(x,y)\int k(x,y)^{-1}\mathrm{d}x\mathrm{d}y}.$$

(6.8)

Thus, the pressure distribution for a given normal force is explicitly determined by the heterogeneity of the wear coefficient. The shape of the wear surface in the steady state is given by the shape of the elastic continuum under the effect of the pressure distribution given in (6.8).

The steady state does not exist for non-cylindrical indenters. Yet qualitatively, the situation is unchanged. In this case, the stress distribution in the worn contact area approaches a constant value as well, determined by the normal force and the current contact area (Dimaki et al. 2016 ). However, the state of constant pressure is only reached in approximation.

A special case of gross slip wear is the wear of a rotationally symmetric profile under the effect of axial twisting (torsion). A cylindrical indenter of radius a twisting with the angular velocity ${\Upomega}$ has a local slip speed given by $v_{\text{rel}}(r,t)=\Upomega r$. Accordingly, the linear wear rate is:

$$\dot{f}=k\Upomega r\cdot p(r)=C,$$

(6.9)

and it is constant in all points of the contact area in the steady state. The constant C is obtained from the condition:

$$F_{N}=2\pi\int_{0}^{a}p(r)r\mathrm{d}r.$$

(6.10)

The pressure distribution is determined from (6.9) and (6.10):

$$p(r)=\frac{F_{N}}{2\pi ra}.$$

(6.11)

The profile corresponding to this stress distribution can be determined using the MDR. In the first step, the distributed load in the MDR space is calculated with the following transformation:

$$\begin{aligned}\displaystyle q_{z}(x)&\displaystyle=2\int_{x}^{\infty}\frac{rp(r)}{\sqrt{r^{2}-x^{2}}}\mathrm{d}r=\frac{F_{N}}{\pi a}\int_{x}^{a}\frac{1}{\sqrt{r^{2}-x^{2}}}\mathrm{d}r\\ \displaystyle&\displaystyle=\frac{F_{N}}{\pi a}\ln\left(\frac{a+\sqrt{a^{2}-x^{2}}}{x}\right).\end{aligned}$$

(6.12)

This gives the result for the vertical displacement in the one-dimensional MDR model:

$$w_{1D}(x)=\frac{q_{z}(x)}{E^{*}}=\frac{F_{N}}{\pi aE^{*}}\ln\left(\frac{a+\sqrt{a^{2}-x^{2}}}{x}\right),$$

(6.13)

and subsequently the displacements in the original three-dimensional space (Fig. 6.2):

$$\begin{aligned}\displaystyle w(r)&\displaystyle=\frac{2}{\pi}\int_{0}^{r}\frac{u_{z}(x)}{\sqrt{r^{2}-x^{2}}}\mathrm{d}x\\ \displaystyle&\displaystyle=\frac{F_{N}}{\pi aE^{*}}\frac{2}{\pi}\int_{0}^{r}\ln\left(\frac{a+\sqrt{a^{2}-x^{2}}}{x}\right)\frac{\mathrm{d}x}{\sqrt{r^{2}-x^{2}}}\\ \displaystyle&\displaystyle=\frac{F_{N}}{\pi aE^{*}}\Uppsi(r/a),\end{aligned}$$

(6.14)

where

$$\Uppsi(\zeta)=\frac{2}{\pi}\int_{0}^{\zeta}\ln\left(\frac{1+\sqrt{1^{2}-\xi^{2}}}{\xi}\right)\frac{\mathrm{d}\xi}{\sqrt{\zeta^{2}-\xi^{2}}}.$$

(6.15)

Fig. 6.2

Shape of the profile generated by wear due to rotation around the vertical axis at a constant angular velocity (see (6.14))

For small radii, this function exhibits a logarithmic singularity in the form

$$\Uppsi(\zeta)\approx 3.420544234-\ln(\zeta),\quad\text{for small}\ \zeta.$$

(6.16)

Galin and Goryacheva (1977 ) also investigated non-cylindrical, rotationally symmetric profiles in their study of torsional wear. Soldatenkov (2010 ) examined various rotationally symmetric wear problems of technical relevance (including stochastic ones), such as ball bearings or the wheel-rail contact.

6.2 Fretting Wear

Many technical or biological systems are part of a periodically working mechanism or are subjected to vibrations. The tribological contacts in such systems are loaded in an oscillating manner due to the periodicity. In general, the displacement amplitudes of these oscillations are sufficiently small to avoid gross slip of the contact. However, the edge of the contact area unavoidably sees the formation of a slip zone, where the contacting surfaces wear and fatigue as a result of the relative displacement. This effect is called fretting and is of marked importance for the operational lifespan of tribological systems.

Just like all wear phenomena, fretting wear has very diverse mechanical and chemical mechanisms. We also distinguish between different fretting modes and fretting regimes. The fretting modes are distinguished by the type of the underlying contact problem: an oscillation normal to the contact plane is called radial fretting. The classical case of oscillations in the contact plane is called tangential fretting and the case of an oscillating rotation around the normal axis of the contact plane is called torsional fretting. An oscillating rolling contact leads to rotational fretting. However, the different modes barely differ in their qualitative behavior (Zhou and Zhu 2011 ).

Different fretting regimes are distinguished based on the behavior of the contact during an oscillation, e.g., near complete stick, partial slip, near complete gross slip, or significant gross slip. These were first systematically examined by Vingsbo and Søderberg (1988 ) using fretting maps . The authors determined that depending on the regime, either wear or fatigue was the dominant form of material degradation caused by fretting.

Even under the assumption of the simplest laws of friction and wear, the analytical calculation of the particular wear dynamics is always extremely complicated and usually impossible. As shown by Ciavarella and Hills (1999 ), certain cases have a final “shakedown” state where no further wear occurs. Contrary to the wear process, the worn profile in the shakedown state does not depend on the particular law of wear nor on the fretting mode and can often be determined analytically. A qualitative explanation for the existence of such a limiting profile is easily given: if within one oscillation period a part of the contact area fulfills the no-slip condition while other areas experience at least transient slip, the stick zone will not wear while the slip zone will experience progressing wear. Intuitively, this leads to a lower pressure in the slip zone, with the stick zone taking on the additional load. Over the course of further oscillations, the stick zone continues sticking while the pressure in the slip zone decreases continually until vanishing completely, at which point the surfaces are in incipient contact with no load. Based on this fact, Popov (2014 ) was able to determine the general solution for this limiting profile in the case of a rotationally symmetric indenter. These results were experimentally confirmed by Dmitriev et al. (2016 ). The existence of such a limiting profile is a universal conclusion and is not bound to the assumption of axial symmetry.

The following section is dedicated to the calculation of the limiting profiles for various initial indenter shapes, as shown by Popov (2014). Let $f(r)$ be the rotationally symmetric profile of a rigid indenter which is pressed by d into an elastic half-space with an effective elastic modulus of $E^{*}$. Moreover, let the contact area have the radius a and the pressure distribution in the contact be $p(r)$. The (oscillating) tangential or torsional load causes the formation of a periodically changing slip zone in the contact. The radius of the area of permanent stick is referred to as c. The aforementioned limiting state is determined by the conditions:

$$\begin{aligned}\displaystyle f_{\infty}(r)&\displaystyle=f_{0}(r),\quad r\leq c\\ \displaystyle p_{\infty}(r)&\displaystyle=0,\quad r> c.\end{aligned}$$

(6.17)

The index “${\infty}$” designates the shakedown state and the index “0” the unworn initial state. As previously explained in this chapter, the radius c results from the no-slip condition for the oscillation of the unworn profile and remains unchanged during the progression of wear. The calculation of the limiting profile consists of three steps.

• 1. Determination of the radius c for the area of permanent stick for the original profiles $f_{0}(r)$.

• 2. Determination of the limiting profile using the equations derived by Popov (2014), as follows:

  $$f_{\infty}(r)=\begin{cases}f_{0}(r),&r\leq c,\ r> a,\\ \displaystyle\frac{2}{\pi}\left[\int\nolimits_{0}^{c}\frac{g_{0}(x)\mathrm{d}x}{\sqrt{r^{2}-x^{2}}}+d\arccos\left(\frac{c}{r}\right)\right],&c<r\leq a,\end{cases}$$

  (6.18)

  where

  $$g_{0}(x)=|x|\int_{0}^{|x|}\frac{f_{0}^{\prime}(r)\mathrm{d}r}{\sqrt{x^{2}-r^{2}}}.$$

  (6.19)

• 3. Determination of the contact radius $a_{\infty}$ in the limiting state from the condition:

  $$f_{\infty}(a_{\infty})=f_{0}(a_{\infty}).$$

  (6.20)

The solution structure clearly shows that the radius c of the permanent stick zone is the only load-dependent and material-dependent parameter to contribute to the solution of the limiting profile. The stick radius is calculated for the unworn profile and is valid for the entire wear process. Solution (6.18) does not yet define the outer radius of the worn area $a_{\infty}$. In the final step, $a_{\infty}$ is determined using (6.20).

In the following sections, we first explain how to determine the radius of the permanently sticking contact area. Subsequently, we give the form of the punch in its final shakedown state for an assumed known radius c of the permanent stick zone. We will focus on a selection of indenter profiles from Chap. 2, which is justified by their technical relevance.

6.2.1 Determining the Radius of the Permanent Stick Zone

6.2.1.1 Horizontal Oscillations at Constant Indentation Depth

The easiest way to calculate the radius c of the permanent stick zone is by using the MDR. Following the steps of the MDR (4.21) (see Sect. 4.4 of this book), we define an elastic foundation and an MDR modified profile $g_{0}(x)$ according to (6.19) on which further contact mechanical calculations are performed instead of the original three-dimensional system.

When the profile is displaced in the tangential direction by $u^{(0)}$, the springs are loaded normally and tangentially. The radius of the stick zone is given by the following equation, which sets the maximum tangential force equal to ${\mu}$ multiplied with the normal force:

$$G^{*}u^{(0)}=\mu E^{*}[d-g(c)].$$

(6.21)

For oscillations in the tangential direction in accordance with the law $u^{(0)}(t)=\Updelta u^{(0)}\cos(\omega t)$, the smallest stick radius (and thus the radius of the permanent stick zone) is reached at the maximum horizontal displacement:

$$G^{*}\Updelta u^{(0)}=\mu E^{*}[d-g(c)].$$

(6.22)

The form of the function $g(c)$ was determined for a great number of profiles in Chap. 2. We will forego repeating them here.

6.2.1.2 Bimodal Oscillations

Let us now consider cases in which the punch experiences simultaneous oscillations in horizontal and vertical directions:

$$\begin{aligned}\displaystyle u^{(0)}(t)&\displaystyle=\Updelta u^{(0)}\cos(\omega_{1}t),\\ \displaystyle d(t)&\displaystyle=d_{0}+\Updelta d\cos(\omega_{2}t-\varphi).\end{aligned}$$

(6.23)

The first thing to note is that, in this case, the limiting profile is still given by (6.18). However, for d we must now insert the maximum indentation depth over the course of one oscillation period:

$$f_{\infty}(r)=\begin{cases}f_{0}(r),&r\leq c,\ r> a,\\ \displaystyle\frac{2}{\pi}\left[\int\nolimits_{0}^{c}\frac{g_{0}(x)\mathrm{d}x}{\sqrt{r^{2}-x^{2}}}+d_{\text{max}}\arccos\left(\frac{c}{r}\right)\right],&c<r\leq a,\end{cases}$$

(6.24)

with

$$d_{\text{max}}=d_{0}+\Updelta d.$$

(6.25)

Let us now calculate c. In the MDR representation, the normal and tangential forces at the coordinate c are given by:

$$\begin{aligned}\displaystyle\Updelta F_{x}&\displaystyle=G^{*}\Updelta x\cdot u^{(0)}(t)=G^{*}\Updelta x\cdot\Updelta u^{(0)}\cos(\omega_{1}t),\\ \displaystyle\Updelta F_{z}&\displaystyle=E^{*}\Updelta x\cdot d(t)=E^{*}\Updelta x\cdot\left[d_{0}+\Updelta d\cos(\omega_{2}t-\varphi)\right].\end{aligned}$$

(6.26)

The radius of the permanent stick zone is determined from the condition that the absolute value of the tangential force at the location c must never exceed the product of the normal force and the coefficient of friction:

$$\left|G^{*}\Updelta u^{(0)}\cos(\omega_{1}t)\right|\leq\mu E^{*}\left[d_{0}+\Updelta d\cos(\omega_{2}t-\varphi)-g(c)\right].$$

(6.27)

If the frequencies $\omega_{1}$ and $\omega_{2}$ are incommensurable, or alternatively, if the phase $\varphi$ is not locked (i.e., it is slowly changing), this condition is satisfied only when the maximum value of the left-hand side of the inequality is smaller than the minimal value of the right-hand side. The critical value is reached when these two values are equal:

$$G^{*}\Updelta u^{(0)}=\mu E^{*}[d_{0}-\Updelta d-g(c)].$$

(6.28)

The only difference in this equation compared to (6.22) is that it features the minimal indentation depth $d_{\text{min}}=d_{0}-\Updelta d$ instead of simply d.

For commensurable or equal frequencies and locked phase shift, the solution is generally very complicated and it can be referenced in Mao et al. (2016).

6.2.2 The Cone

The unworn profile of a conical indenter can be written as:

$$f_{0}(r)=r\tan\theta.$$

(6.29)

Here ${\theta}$ is the slope of the cone. The shakedown profile is then given by:

$$f_{\infty}(r)=\begin{cases}r\tan\theta,&r\leq c,\ r> a_{\infty},\\ \displaystyle r\tan\theta\left(1-\sqrt{1-\frac{c^{2}}{r^{2}}}\right)+\frac{2d}{\pi}\arccos\left(\frac{c}{r}\right),&c<r\leq a_{\infty},\end{cases}$$

(6.30)

with an indentation depth d and the radius of the permanent stick zone c.

The post-shakedown contact radius $a_{\infty}$ is obtained as the solution of the equation:

$$\frac{2d}{\pi}\arccos\left(\frac{c}{a_{\infty}}\right)=\tan\theta\sqrt{{a_{\infty}}^{2}-c^{2}}.$$

(6.31)

The profile resulting from (6.30) is shown in Fig. 6.3.

Fig. 6.3

Shakedown profile normalized to $a_{\infty}\tan\theta$, for different values of the permanent stick radius $c$ for fretting wear of a conical indenter

6.2.3 The Paraboloid

For the paraboloid with the radius of curvature R and corresponding unworn profile

$$f_{0}(r)=\frac{r^{2}}{2R},$$

(6.32)

the limiting profile is described as a function of the indentation depth d and the radius c of the permanent stick area is given by:

$$f_{\infty}(r)=\begin{cases}\dfrac{r^{2}}{2R},&r\leq c,\ r> a_{\infty},\\ \displaystyle\frac{2}{\pi}\Bigg[\frac{r^{2}}{2R}\arcsin\left(\frac{c}{r}\right)-\frac{c}{2R}\sqrt{r^{2}-c^{2}}&\\ \quad\displaystyle{}+d\arccos\left(\frac{c}{r}\right)\Bigg],&c<r\leq a_{\infty}.\end{cases}$$

(6.33)

This dependency is shown in Fig. 6.4. The contact radius in the worn state results from the solution of the equation:

$$\frac{a_{\infty}^{2}}{2R}=\frac{2}{\pi}\left[\frac{a_{\infty}^{2}}{2R}\arcsin\left(\frac{c}{a_{\infty}}\right)-\frac{c}{2R}\sqrt{a_{\infty}^{2}-c^{2}}+d\arccos\left(\frac{c}{a_{\infty}}\right)\right].$$

(6.34)

Fig. 6.4

Shakedown-profile normalized to $a_{\infty}^{2}/(2R)$, for various values of permanent stick radius c for the fretting wear of a parabolic indenter

6.2.4 The Profile in the Form of a Lower Law

We now consider indenter profiles of the general power-law form:

$$f_{0}(r)=br^{n},\quad n\in\mathbb{R}^{+},$$

(6.35)

with a (dimensional) constant b and a positive real number n. In Chap. 2 (see Sect. 2.5.8) it has already been shown, that the equivalent profile in the MDR is given by a stretched power function with the same exponent n:

$$g_{0}(x)=\kappa(n)b|x|^{n}.$$

(6.36)

The stretch factor was given as:

$$\kappa(n):=\sqrt{\pi}\frac{\Upgamma(n/2+1)}{\Upgamma\left[(n+1)/2\right]},$$

(6.37)

with the gamma function

$$\Upgamma(z):=\int_{0}^{\infty}t^{z-1}\exp(-t)\mathrm{d}t.$$

(6.38)

Equation (5.18) then gives the following shakedown profile at the end of the wear process:

$$\begin{aligned}\displaystyle f_{\infty}(r)&\displaystyle=\begin{cases}br^{n},&r\leq c,\ r> a,\\ \displaystyle\frac{2}{\pi}\Bigg[\kappa(n)b\int\nolimits_{0}^{c}\frac{x^{n}\mathrm{d}x}{\sqrt{r^{2}-x^{2}}}+d\arccos\left(\frac{c}{r}\right)\Bigg],&c<r\leq a,\end{cases}\\ \displaystyle&\displaystyle=\begin{cases}br^{n},&r\leq c,\ r> a,\\ \displaystyle\frac{2}{\pi}\Bigg[\kappa(n)b\frac{c^{n+1}}{(n+1)r}{}_{2}{\mathrm{F}}_{1}\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\frac{c^{2}}{r^{2}}\right)&\\ \displaystyle\quad{}+d\arccos\left(\frac{c}{r}\right)\Bigg],&c<r\leq a,\end{cases}\end{aligned}$$

(6.39)

with the indentation depth d and the permanent stick radius c. The notation ${}_{2}\mathrm{F}_{1}(\cdot,\cdot;\cdot;\cdot)$ is used for the hypergeometric function

$${}_{2}\mathrm{F}_{1}(a,b;c;z):=\sum_{n=0}^{\infty}\frac{\Upgamma(a+n)\Upgamma(b+n)\Upgamma(c)}{\Upgamma(a)\Upgamma(b)\Upgamma(c+n)}\frac{z^{n}}{n!}.$$

(6.40)

For $n=1$ the case of the cone is from Sect. 6.2.2 and is reproduced, and for $n=2$ it is the solution for the paraboloid (Sect. 6.2.3).

6.2.5 The Truncated Cone

In the previous chapters, the truncated cone and paraboloid have already been discussed several times. For the truncated cone with the profile function

$$f_{0}(r)=\begin{cases}0,&r\leq b,\\ (r-b)\tan\theta,&r> b,\end{cases}$$

(6.41)

with the radius b at the blunt end and the slope angle ${\theta}$, the following equivalent plane profile was determined in Chap. 2 (Sect. 2.5.9):

$$g_{0}(x)=\begin{cases}0,&|x|\leq b,\\ |x|\tan\theta\arccos\left(\dfrac{b}{|x|}\right),&|x|> b.\end{cases}$$

(6.42)

It has been shown in Chap. 4 (Sect. 4.6.5) that the radius of the stick zone c cannot fall below the value of b. Therefore, the worn boundary profile, with (6.18) and the indentation depth d, is described by

$$f_{\infty}(r)=\begin{cases}f_{0}(r),&r\leq c,\ r> a_{\infty},\\ \displaystyle\frac{2}{\pi}\Bigg[\tan\theta\int\nolimits_{b}^{c}x\arccos\left(\frac{b}{x}\right)\frac{\mathrm{d}x}{\sqrt{r^{2}-x^{2}}}&\\ \displaystyle\quad{}+d\arccos\left(\frac{c}{r}\right)\Bigg],&c<r\leq a_{\infty}.\end{cases}$$

(6.43)

The contact radius at the end of the wear process is, as always, defined by the relationship (6.20). The profile of (6.43) is shown in normalized form in Fig. 6.5. For $b=0$, of course, the solution of the complete cone from Sect. 6.2.2 is recovered.

Fig. 6.5

Shakedown profile normalized to $\left(a_{\infty}-b\right)\tan\theta$, for $b=0.1a_{\infty}$ and various values of the permanent stick radius c for fretting wear of a truncated cone

6.2.6 The Truncated Paraboloid

Let us now consider the truncated parabolic, whose unworn profile is described by the rule:

$$f(r)=\begin{cases}0,&r\leq b,\\ \dfrac{r^{2}-b^{2}}{2R},&r> b,\end{cases}$$

(6.44)

Here R denotes the radius of curvature of the parabolic base body and b the radius at the flattened tip. In Chap. 2 (Sect. 2.5.10), the equivalent profile has already been found:

$$g(x)=\begin{cases}0,&|x|\leq b,\\ \displaystyle\frac{|x|}{R}\sqrt{x^{2}-b^{2}},&|x|> b.\end{cases}$$

(6.45)

As in the case of the truncated cone of the radius of the stick zone, c cannot fall below the value of b Equation (6.18). (d denotes the indentation depth, as always) allows the shakedown profile to be described as:

$$\begin{aligned}\displaystyle f_{\infty}(r)&\displaystyle=\begin{cases}f_{0}(r),&r\leq c,\ r> a_{\infty},\\ \displaystyle\frac{2}{\pi}\Bigg[\frac{1}{R}\int\nolimits_{b}^{c}x\sqrt{x^{2}-b^{2}}\frac{\mathrm{d}x}{\sqrt{r^{2}-x^{2}}}+d\arccos\left(\frac{c}{r}\right)\Bigg],&c<r\leq a_{\infty},\end{cases}\\ \displaystyle&\displaystyle=\begin{cases}f_{0}(r),&r\leq c,\ r> a_{\infty},\\ \displaystyle\frac{1}{\pi R}\Bigg[(r^{2}-b^{2})\arcsin\left(\frac{\sqrt{c^{2}-b^{2}}}{\sqrt{r^{2}-b^{2}}}\right)&\\ \displaystyle\qquad{}-\sqrt{c^{2}-b^{2}}\sqrt{r^{2}-c^{2}}+2dR\arccos\left(\frac{c}{r}\right)\Bigg],&c<r\leq a_{\infty}.\end{cases}\end{aligned}$$

(6.46)

This is shown in a normalized manner in Fig. 6.6. For $b=0$ we obtain, of course, the solution of the complete paraboloid from Sect. 6.2.3.

Fig. 6.6

Shakedown profile normalized to $(a_{\infty}^{2}-b^{2})/(2R)$, for $b=0.1a_{\infty}$ and various values of the permanent stick radius c for the fretting wear of a truncated paraboloid

6.2.7 Further Profiles

In Chap. 2, it was shown that the equivalent profile functions g(x) of various technically relevant indenters considered in the literature can be considered as the superposition of the elementary bodies paraboloid, truncated cone, and truncated paraboloid. Since the integral expression in (6.18) is linear in $g_{0}(x)$, for the superposition the solutions of the integrals for the elementary bodies mentioned, given in the previous sections, can simply be summed up. We will, therefore, only specify the superposition rules for the functions $g_{0}$ and graph the profiles at the end of the fretting process. We will refrain from providing the complete solution in order to avoid redundancies.

6.2.7.1 The Cone with Parabolic Cap

For a cone with the slope angle $\theta$, which at point $r=b$ differentiably passes into a parabolic cap with the radius of curvature $R:=b/\tan\theta$, the rotationally symmetric profile can be written as follows:

$$f(r)=\begin{cases}\dfrac{r^{2}\tan\theta}{2b},&r\leq b,\\ r\tan\theta-\dfrac{b}{2}\tan\theta,&r> b,\end{cases}$$

(6.47)

For the equivalent profile in MDR, the following superposition has been shown in Chap. 2 (see Sect. 2.5.12):

$$\begin{aligned}\displaystyle g_{0}(x;b,\theta)=&\displaystyle g_{0,P}\left(x;R=\frac{b}{\tan\theta}\right)+g_{0,\mathit{KS}}(x;b,\theta)\\ \displaystyle&\displaystyle{}-g_{0,\mathit{PS}}\left(x;b,R=\frac{b}{\tan\theta}\right).\end{aligned}$$

(6.48)

Here, $g_{0,P}$ denotes the unworn, equivalent profile of a paraboloid (see Sect. 6.2.3), $g_{0,\mathit{KS}}$ that of a truncated cone (see Sect. 6.2.5), and $g_{0,\mathit{PS}}$ that of a truncated paraboloid (see Sect. 6.2.6). The no-slip radius can fall below the value of b, but then the limiting profile is the same as in the case of the simple paraboloid. For this reason, some variants of the shakedown profile with $c\geq b$ are shown in Fig. 6.7.

Fig. 6.7

Normalized shakedown profile for $b=0.5a_{\infty}$, and various values of the permanent stick radius c for the fretting wear of a cone with a rounded tip

6.2.7.2 The Paraboloid with Paraboloid Cap

The rotationally symmetric profile of this body is described by the function

$$f(r)=\begin{cases}\dfrac{r^{2}}{2R_{1}},&r\leq b,\\ \dfrac{r^{2}-h^{2}}{2R_{2}},&r> b.\end{cases}$$

(6.49)

The radius of the cap is $R_{1}$, and that of the main body is $R_{2}$. The continuity of $f$ at the position of $r=b$ requires

$$h^{2}=b^{2}\left(1-\frac{R_{2}}{R_{1}}\right),$$

(6.50)

and one can introduce an effective radius of curvature:

$$R^{*}=\frac{R_{1}R_{2}}{R_{1}-R_{2}}.$$

(6.51)

From a contact mechanical point of view, this body can be described as a superposition:

$$g_{0}(x;b,R_{1},R_{2})=g_{0,P}(x;R=R_{1})+g_{0,\mathit{PS}}(x;b,R=R^{*}),$$

(6.52)

as can be looked up in Chap. 2 (Sect. 2.5.13). The permanent stick radius can, again, drop below the value of b, but the limiting profile is the same as in the case of the simple paraboloid. In Fig. 6.8 some variants of the shakedown profile are shown by way of example.

Fig. 6.8

Normalized shakedown profile for $b=0.3a_{\infty}$, $R_{1}=R^{*}$, and various values of the permanent stick radius c for the fretting wear of a paraboloid with parabolic cap

6.2.7.3 The Cylindrical Flat Punch with a Rounded Edge

The indenter has the axisymmetric profile

$$f(r)=\begin{cases}0,&r\leq b,\\ \dfrac{\left(r-b\right)^{2}}{2R},&r> b,\end{cases}$$

(6.53)

with the radius $b$, for which the flat base of the punch passes into the rounded edge with the radius of curvature R. The transformed profile g can be thought of as the sum (see Sect. 2.5.14) $g_{0}(x;b,R)=g_{0,\mathit{PS}}(x;b,R)-g_{0,\mathit{KS}}(x;b,\tan\theta=b/R)$. The indices “KS” and “PS” refer to the respective results of the truncated cone and paraboloid. Figure 6.9 has been used to illustrate some profiles after the fretting process.

Fig. 6.9

Normalized shakedown profile for $b=0.5a_{\infty}$, and various values of the permanent stick radius c for the fretting wear of a flat cylindrical punch with rounded corners