Appendix

Abstract

A significant number of contact problems concerning rotationally symmetric bodies can be solved via reduction to the non-adhesive frictionless normal contact problem. The problem of an adhesive normal contact is thereby reduced entirely to the solution of a non-adhesive normal contact. Likewise, the tangential contact problem in the Cattaneo–Mindlin approximation can be reduced to the frictionless normal contact problem using the principle of superposition by Cattaneo–Mindlin–Jäger–Ciavarella. And finally, contacts of viscoelastic bodies can be reduced to the corresponding elastic problem, i.e. the non-adhesive elastic normal contact, using the method by Lee und Radok.

This problem of the elastic, frictionless, non-adhesive normal contact of rotationally symmetric bodies can in turn be reduced to the problem of the indentation by a rigid cylindrical indenter. If the solution for the contact between a rigid cylindrical punch of an arbitrary radius and a given elastic medium is known, then the solution for an arbitrarily shaped punch consists of a simple superposition of indentations by punches of varying radii. The problem of an indentation by a flat cylindrical punch cannot be further reduced and thus requires an actual formal solution. In this appendix, we perform this basic step and describe the following “construction of the building” of the MDR.

In the closing section, the historical (first) solution of the contact problem for an arbitrary axisymmetric profile with a compact contact area according to Föppl and Schubert is given.

A significant number of contact problems concerning rotationally symmetric bodies can be solved via reduction to the non-adhesive frictionless normal contact problem. The problem of an adhesive normal contact is thereby reduced entirely to the solution of a non-adhesive normal contact. Likewise, the tangential contact problem in the Cattaneo–Mindlin approximation can be reduced to the frictionless normal contact problem using the principle of superposition by Cattaneo–Mindlin–Jäger–Ciavarella. And finally, contacts of viscoelastic bodies can be reduced to the corresponding elastic problem, i.e., the non-adhesive elastic normal contact, using the method by Lee und Radok.

This problem of the elastic, frictionless, non-adhesive normal contact of rotationally symmetric bodies can in turn be reduced via the MDR to the problem of the indentation by a rigid cylindrical indenter. If the solution for the contact between a rigid cylindrical punch of an arbitrary radius and a given elastic medium is known, then the solution for an arbitrarily shaped punch consists of a simple superposition of indentations by punches of varying radii.

The problem of an indentation by a flat cylindrical punch cannot be further reduced and thus requires an actual formal solution. In this appendix, we perform this basic step and describe the “construction of the building” of the MDR. In doing this, we follow the appendix of the book by Popov (2015). In Sect. 11.8 of this chapter an overview of the special functions used in this book is provided. In (the closing) Sect. 11.9, the historical solution of the contact problem for an arbitrary axisymmetric profile with a compact contact area according to Föppl (1941) and Schubert (1942) is given.

11.1 The Flat Punch Solution for Homogeneous Materials

This section demonstrates that the pressure distribution at the surface of an elastic half-space

$$p(r;a)=p_{0}\left(1-\frac{r^{2}}{a^{2}}\right)^{-1/2},\quad r^{2}=x^{2}+y^{2},\ r<a$$

(11.1)

leads to a constant normal displacement of the circle $r<a$ and, therefore, corresponds to the indentation by a rigid circular punch. The basis for the calculation is the general relationship between an arbitrary pressure distribution $p(x^{\prime},y^{\prime})$ and the resulting surface deformation $w$ given by the fundamental solution of Boussinesq :

$$w(x,y)=\frac{1}{\pi E^{*}}\iint p(x^{\prime},y^{\prime})\frac{\mathrm{d}x^{\prime}\mathrm{d}y^{\prime}}{r},\quad r=\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}},$$

(11.2)

with the effective elasticity modulus $E^{*}$. The integral (11.2) is calculated using the system of coordinates and notations shown in Fig. 11.1. Due to rotational symmetry of the stress distribution, the vertical displacement of a surface point is dependent solely on the distance r of this point from the origin O. Therefore, it is sufficient to determine the displacements of the points on the $x$-axis. In the following, we will calculate the vertical deformation at point A. For this, we must determine the displacement at point A caused by the stress in the “varying” point B, and subsequently integrate over all possible positions of point B within the load zone. Due to rotational symmetry, the stress at point B also depends solely on the distance $t$ of the point to the origin O. For this distance we obtain $t^{2}=r^{2}+s^{2}+2rs\cos\varphi$. Therefore, the pressure distribution is:

$$\begin{aligned}\displaystyle p(s,\varphi)&\displaystyle=p_{0}\left(1-\frac{r^{2}+s^{2}+2rs\cos\varphi}{a^{2}}\right)^{-1/2}\\ \displaystyle&\displaystyle=p_{0}a(a^{2}-r^{2}-s^{2}-2rs\cos\varphi)^{-1/2}\\ \displaystyle&\displaystyle=p_{0}a(\alpha^{2}-2\beta s-s^{2})^{-1/2},\end{aligned}$$

(11.3)

where we have introduced $\alpha^{2}=a^{2}-r^{2}$ and $\beta=r\cos\varphi$.

Fig. 11.1

Calculation of the vertical deformation caused by normal stresses acting on a circular area: (a) at a location within the load zone, (b) at a location outside of the load zone

For the $z$-component of the displacement of a point within the load zone we obtain:

$$w=\frac{1}{\pi E^{*}}p_{0}a\int_{0}^{2\pi}\left(\int_{0}^{s_{1}}(\alpha^{2}-2\beta s-s^{2})^{-1/2}\mathrm{d}s\right)\mathrm{d}\varphi.$$

(11.4)

Here, $s_{1}$ is the positive root of the equation $\alpha^{2}-2\beta s-s^{2}=0$. The integral over $\mathrm{d}s$ is calculated to:

$$\int_{0}^{s_{1}}(\alpha^{2}-2\beta s-s^{2})^{-1/2}\mathrm{d}s=\frac{\pi}{2}-\arctan(\beta/\alpha).$$

(11.5)

It is apparent that $\arctan(\beta(\varphi)/\alpha)=-\arctan(\beta(\varphi+\pi)/\alpha)$. Thus, by integrating over $\varphi$, the term with $\arctan$ is eliminated. Therefore,

$$w(r;a)=\frac{1}{\pi E^{*}}p_{0}a\int_{0}^{2\pi}\frac{\pi}{2}d\varphi=\frac{\pi p_{0}a}{E^{*}}=:d=\text{const},\quad r<a,$$

(11.6)

where the indentation depth d is introduced.

We now consider a point $A$ outside the load zone (Fig. 11.1b). In this case:

$$p(s,\varphi)=p_{0}a(\alpha^{2}+2\beta s-s^{2})^{-1/2}.$$

(11.7)

The displacement is then given by the equation:

$$w=\frac{1}{\pi E^{*}}p_{0}a\int_{-\varphi_{1}}^{\varphi_{1}}\left(\int_{s_{1}}^{s_{2}}\left(\alpha^{2}+2\beta s-s^{2}\right)^{-1/2}\mathrm{d}s\right)\mathrm{d}\varphi$$

(11.8)

where $s_{1}$ and $s_{2}$ are the roots of the equation:

$$\alpha^{2}+2\beta s-s^{2}=0.$$

(11.9)

Accordingly, it follows that:

$$\int_{s_{1}}^{s_{2}}\left(\alpha^{2}+2\beta s-s^{2}\right)^{-1/2}\mathrm{d}s=\pi.$$

(11.10)

The remaining integration in (11.8) now results trivially in $w=\frac{2}{E^{*}}p_{0}a\varphi_{1}$ or, taking into account the obvious geometric relation from Fig. 11.1b, $\varphi_{1}=\arcsin(a/r)$,

$$w(r;a)=\frac{2}{E^{*}}p_{0}a\cdot\arcsin(a/r),\quad r\geq a,$$

(11.11)

which, accounting for (11.6), can also be written as:

$$w(r;a)=\frac{2}{\pi}d\cdot\arcsin(a/r),\quad r\geq a.$$

(11.12)

From the result (11.6) it follows directly that the assumed pressure distribution is generated by an indentation by a rigid cylindrical punch. The total force acting on the load zone equals:

$$F_{N}=\int_{0}^{a}p_{0}\left(1-\frac{r^{2}}{a^{2}}\right)^{-1/2}2\pi r\mathrm{d}r=2\pi p_{0}a^{2}.$$

(11.13)

The contact stiffness is defined as the relation of force to displacement:

$$k_{z}=2aE^{*}.$$

(11.14)

The pressure distribution (11.1) can also be represented in the following form with the consideration of (11.6):

$$p(r)=\frac{1}{\pi}\frac{E^{*}d}{\sqrt{a^{2}-r^{2}}}.$$

(11.15)

11.2 Normal Contact of Axisymmetric Profiles with a Compact Contact Area

In this section we will present the solution for the normal contact problem for axisymmetric profiles with a compact contact area in its general form. For this, we will use the solution of the contact problem of a rigid flat cylindrical punch obtained in the preceding section. Simultaneously, a derivation of the fundamental equations of the MDR will be performed.

We will now consider the contact between a rigid indenter of the shape $\tilde{z}=f(r)$ and an elastic half-space characterized by the effective elasticity modulus $E^{*}$. Let the indentation depth under the effect of the normal force $F_{N}$ be d and the contact radius a. For a given profile shape, any of these three quantities uniquely determines the other two. The indentation depth is a unique function of the contact radius, which is denoted by:

$$d=g(a).$$

(11.16)

Let us examine the process of the indentation from the first touch to the final indentation depth d, denoting the values of the force, the indentation depth, and the contact radius during the indentation by $\tilde{F}_{N}$, $\tilde{d}$ and $\tilde{a}$, respectively. The process can then be viewed as a change in the indentation depth from $\tilde{d}=0$ to $\tilde{d}=d$, with the contact radius changing from $\tilde{a}=0$ to $\tilde{a}=a$, and the contact force from $\tilde{F}_{N}=0$ to $\tilde{F}_{N}=F_{N}$. The normal force at the end of the process can be written in the following form:

$$F_{N}=\int_{0}^{F_{N}}\mathrm{d}\tilde{F}_{N}=\int_{0}^{a}\frac{\mathrm{d}\tilde{F}_{N}}{\mathrm{d}\tilde{d}}\frac{\mathrm{d}\tilde{d}}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}.$$

(11.17)

Taking into account that the differential stiffness of a zone of radius $\tilde{a}$ is given by:

$$\frac{\mathrm{d}\tilde{F}_{N}}{\mathrm{d}\tilde{d}}=2E^{*}\tilde{a}$$

(11.18)

(see (11.14)), and using the notation (11.16), we obtain:

$$F_{N}=2E^{*}\int_{0}^{a}\tilde{a}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}.$$

(11.19)

Integration by parts now results in:

$$\begin{aligned}\displaystyle F_{N}&\displaystyle=2E^{*}\left[a\cdot g(a)-\int_{0}^{a}g(\tilde{a})\mathrm{d}\tilde{a}\right]=2E^{*}\left[\int_{0}^{a}[g(a)-g(\tilde{a})]\mathrm{d}\tilde{a}\right]\\ \displaystyle&\displaystyle=2E^{*}\left[\int_{0}^{a}[d-g(\tilde{a})]\mathrm{d}\tilde{a}\right].\end{aligned}$$

(11.20)

We now turn our attention to calculating the pressure distribution in the contact area. An infinitesimal indentation of an area of radius $\tilde{a}$ generates the following contribution to the pressure distribution (see (11.15)):

$$\mathrm{d}p(r)=\frac{1}{\pi}\frac{E^{*}}{\sqrt{\tilde{a}^{2}-r^{2}}}\mathrm{d}\tilde{d},\quad\text{for}\ r<\tilde{a}.$$

(11.21)

The pressure distribution at the end of the indentation process equals the sum of the incremental pressure distributions

$$p(r)=\int_{d(r)}^{d}\frac{1}{\pi}\frac{E^{*}}{\sqrt{\tilde{a}^{2}-r^{2}}}\mathrm{d}\tilde{d}=\int_{r}^{a}\frac{1}{\pi}\frac{E^{*}}{\sqrt{\tilde{a}^{2}-r^{2}}}\frac{\mathrm{d}\tilde{d}}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a},$$

(11.22)

or under consideration of the notation (11.16)

$$p(r)=\frac{E^{*}}{\pi}\int_{r}^{a}\frac{1}{\sqrt{\tilde{a}^{2}-r^{2}}}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}.$$

(11.23)

The function $g(a)$ from (11.16), therefore, uniquely defines both the normal force (11.20) and the pressure distribution (11.23). The normal contact problem is reduced to the determination of the function $g(a)$ (11.16).

The function $d=g(a)$ can be determined as follows. The infinitesimal surface displacement at the point $r=a$ for infinitesimal indentation by $\mathrm{d}\tilde{d}$ of a contact area of radius $\tilde{a}<a$ is, according to (11.12), equal to:

$$\mathrm{d}w(a)=\frac{2}{\pi}\arcsin\left(\frac{\tilde{a}}{a}\right)\mathrm{d}\tilde{d}.$$

(11.24)

The total vertical displacement at the end of the indentation process is, therefore, equal to:

$$w(a)=\frac{2}{\pi}\int_{0}^{d}\arcsin\left(\frac{\tilde{a}}{a}\right)\mathrm{d}\tilde{d}=\frac{2}{\pi}\int_{0}^{a}\arcsin\left(\frac{\tilde{a}}{a}\right)\frac{\mathrm{d}\tilde{d}}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a},$$

(11.25)

or with the notation (11.16), it is:

$$w(a)=\frac{2}{\pi}\int_{0}^{a}\arcsin\left(\frac{\tilde{a}}{a}\right)\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}.$$

(11.26)

This vertical displacement, however, is obviously equal to $w(a)=d-f(a)$:

$$d-f(a)=\frac{2}{\pi}\int_{0}^{a}\arcsin\left(\frac{\tilde{a}}{a}\right)\frac{\mathrm{d}g\left(\tilde{a}\right)}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}.$$

(11.27)

Integration by parts and consideration of (11.16) leads to the equation:

$$f(a)=\frac{2}{\pi}\int_{0}^{a}\frac{g(\tilde{a})}{\sqrt{a^{2}-\tilde{a}^{2}}}\mathrm{d}\tilde{a}.$$

(11.28)

This is Abel’s integral equation, which is solved with respect to $g(a)$ in the following way (Bracewell 1965):

$$g(a)=a\int_{0}^{a}\frac{f^{\prime}(\tilde{a})}{\sqrt{a^{2}-\tilde{a}^{2}}}\mathrm{d}\tilde{a}.$$

(11.29)

With the determination of the function $g(a)$, the contact problem is completely solved.

It is easy to see that (11.16), (11.20), (11.23), (11.28), and (11.29) coincide exactly with the equations for the MDR (Chap. 2, (2.17)) , which proves the validity of the MDR.

11.3 Adhesive Contact of Axisymmetric Profiles with a Compact Contact Area

The MDR described in Sect. 11.2 for the solution of non-adhesive contact of rotationally symmetric bodies can easily be generalized to adhesive contacts. The generalization is based on the fundamental idea of Johnson, Kendall, and Roberts that the contact with adhesion can be determined from the contact without adhesion plus a rigid-body translation. In other words, the configuration of the adhesive contact can be obtained by initially pressing the body without consideration of the adhesion to a certain contact radius a (Fig. 11.2a) and, subsequently, retracting to a certain critical height $\Updelta l$ while maintaining the constant contact area (Fig. 11.2b). Since both the indentation of any rotationally symmetric profile and the subsequent rigid body translation are mapped correctly by the MDR, this is valid also for the superposition of these two movements.

Fig. 11.2

Qualitative representation of the pressing and retraction process of a one-dimensional indenter with an elastic foundation, which exactly represents the characteristics of the adhesive contact between a rigid spherical punch and an elastic half-space. a Indentation without accounting for adhesion; b retraction with constant contact radius

The still unknown critical length $\Updelta l=\Updelta l(a)$ can be determined using the principle of virtual work. According to this principle, the system is in equilibrium when the energy does not change for small variations of its generalized coordinates. Applied to the adhesive contact, it means that the change in the elastic energy for a small reduction of the contact radius from a to $a-\Updelta x$ is equal to the change in the surface energy $2\pi a\Updelta x\Updelta\gamma$, where $\Updelta\gamma$ is the separation work of the contacting surfaces per unit area. Since the MDR maps the relation of force to displacement exactly, the elastic energy is also reproduced exactly. The change in the elastic energy can, therefore, be calculated directly in the MDR model. Due to the detachment of each spring at the edge of the contact, the elastic energy is reduced by $E^{*}\Updelta x\Updelta l^{2}$. Balance of the changes in the elastic and the adhesive energy results in:

$$2\pi a\Updelta x\Updelta\gamma=E^{*}\Updelta x\Updelta l^{2}.$$

(11.30)

It follows that:

$$\Updelta l=\sqrt{\frac{2\pi a\Updelta\gamma}{E^{*}}}.$$

(11.31)

This criterion for the detachment of the edge springs in the adhesive MDR model was found by Heß (2011) and is known as the rule of Heß.

11.4 The Flat Punch Solution for FGMs

In Sect. 11.1, the solution of the contact problem between a rigid, cylindrical flat punch and an homogeneous elastic half-space was derived via the Boussinesq fundamental solution. Analogously, the corresponding solution can be found for an elastically inhomogeneous half-space with the variable elastic modulus

$$E(z)=E_{0}\left(\frac{z}{c_{0}}\right)^{k}\quad\text{with}\ -1<k<1.$$

(11.32)

The fundamental solution in this case is given by:

$$w(r)=\frac{(1-\nu^{2})\cos\left(\frac{k\pi}{2}\right)c_{0}^{k}}{h_{N}(k,\nu)\pi E_{0}}\frac{F_{N}}{r^{1+k}},$$

(11.33)

with $h_{N}(k,\nu)$, which was introduced in (9.5) in Chap. 9. Using the superposition principle and (11.33) the normal displacement of the half-space surface can be determined for any pressure distribution $p(x^{\prime},y^{\prime})$:

$$\begin{gathered}\displaystyle w(x,y)=\frac{(1-\nu^{2})\cos\left(\frac{k\pi}{2}\right)c_{0}^{k}}{h_{N}(k,\nu)\pi E_{0}}\iint p(x^{\prime},y^{\prime})\frac{\mathrm{d}x^{\prime}\mathrm{d}y^{\prime}}{r^{1+k}}\\ \displaystyle\text{with}\ r=\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}}.\end{gathered}$$

(11.34)

In the following we will show that a pressure distribution

$$p(r;a)=p_{0}\left(1-\frac{r^{2}}{a^{2}}\right)^{\frac{k-1}{2}},\quad r^{2}=x^{2}+y^{2},\ r<a$$

(11.35)

generates a constant normal displacement inside the circle with radius a. Utilizing the transformed variables depicted in Fig. 11.1a and by introducing the short-cuts $\alpha^{2}=a^{2}-r^{2}$ and $\beta=r\cos\varphi$, (11.35) can be written in the form:

$$p(s,\varphi)=\frac{p_{0}a^{1-k}}{\left(\sqrt{\alpha^{2}-2\beta s-s^{2}}\right)^{1-k}}.$$

(11.36)

The surface normal displacement of a point within the circular area stressed by the pressure distribution (11.36) is determined by (11.34):

$$w(r;a)=\frac{(1-\nu^{2})\cos\left(\frac{k\pi}{2}\right)c_{0}^{k}p_{0}a^{1-k}}{h_{N}(k,\nu)\pi E_{0}}\int_{0}^{2\pi}\int_{0}^{s_{1}}\frac{1}{\left(\sqrt{\alpha^{2}-2\beta s-s^{2}}\right)^{1-k}}\frac{\mathrm{d}s}{s^{k}}\mathrm{d}\varphi,$$

(11.37)

where $s_{1}=-\beta+\sqrt{\beta^{2}+\alpha^{2}}$ is the positive root of the denominator under the integral in (11.37). The inner integral over s results in

$$\begin{aligned}\displaystyle&\displaystyle\int_{0}^{s_{1}}\frac{1}{\left(\sqrt{\alpha^{2}-2\beta s-s^{2}}\right)^{1-k}}\frac{\mathrm{d}s}{s^{k}}=\\ \displaystyle&\displaystyle\frac{\pi}{2\cos\left(\frac{k\pi}{2}\right)}\\ \displaystyle&\displaystyle{}\quad-\frac{\Upgamma\left(1-\frac{k}{2}\right)\Upgamma\left(\frac{1+k}{2}\right)}{\sqrt{\pi}}\frac{\beta}{\sqrt{\alpha^{2}+\beta^{2}}}\,{}_{2}\mathrm{F}_{1}\left(\frac{1}{2},\frac{1+k}{2};\frac{3}{2};\frac{\beta^{2}}{\alpha^{2}+\beta^{2}}\right).\end{aligned}$$

(11.38)

With the symmetry relation $\beta(\varphi+\pi)=-\beta(\varphi)$ it is clear that the second, hypergeometric term will not contribute after integrating over ${\varphi}$. Hence, the normal displacement inside the pressured ring is given by:

$$w(r;a)=\frac{(1-\nu^{2})c_{0}^{k}\pi p_{0}a^{1-k}}{h_{N}(k,\nu)E_{0}}=:d=\text{const},\quad r<a.$$

(11.39)

As such, the pressure distribution (11.35) results from the indentation by a rigid flat cylindrical punch.

We now proceed to the calculation of the normal displacement of a point outside the loading zone. Taking into account the variables introduced in Fig. 11.1b, the pressure distribution (11.35) then takes the form:

$$p(s,\varphi)=\frac{p_{0}a^{1-k}}{\left(\sqrt{\alpha^{2}+2\beta(\varphi)s-s^{2}}\right)^{1-k}},$$

(11.40)

and the normal displacements can be determined from the integral:

$$w(r;a)=\frac{(1-\nu^{2})\cos\left(\frac{k\pi}{2}\right)c_{0}^{k}p_{0}a^{1-k}}{h_{N}(k,\nu)\pi E_{0}}\int_{-\varphi_{1}}^{\varphi_{1}}\int_{s_{1}}^{s_{2}}\frac{1}{\left(\sqrt{\alpha^{2}+2\beta s-s^{2}}\right)^{1-k}}\frac{\mathrm{d}s}{s^{k}}\mathrm{d}\varphi,$$

(11.41)

whereas $s_{1/2}=\beta\mp\sqrt{\beta^{2}+\alpha^{2}}$. The inner integral resolves to:

$$\begin{aligned}\displaystyle&\displaystyle\int_{s_{1}}^{s_{2}}\frac{1}{\left(\sqrt{\alpha^{2}+2\beta s-s^{2}}\right)^{1-k}}\frac{\mathrm{d}s}{s^{k}}=\\ \displaystyle&\displaystyle\frac{\sqrt{\pi}\Upgamma\left(\frac{1+k}{2}\right)}{\Upgamma\left(1+\frac{k}{2}\right)}\left(\frac{\alpha^{2}+\beta^{2}}{\beta^{2}}\right)^{k/2}\,{}_{2}\mathrm{F}_{1}\left(\frac{k}{2},\frac{1+k}{2};1+\frac{k}{2};\frac{\alpha^{2}+\beta^{2}}{\beta^{2}}\right).\end{aligned}$$

(11.42)

The subsequent integration over ${\varphi}$ after some transformations leads to the sought normal displacements outside the loaded area:

$$\begin{aligned}\displaystyle&\displaystyle w(r;a)=\\ \displaystyle&\displaystyle\frac{2(1-\nu^{2})\cos\left(\frac{k\pi}{2}\right)c_{0}^{k}p_{0}a^{1-k}}{h_{N}(k,\nu)E_{0}(1+k)}\left(\frac{a}{r}\right)^{k+1}\,{}_{2}\mathrm{F}_{1}\left(\frac{1+k}{2},\frac{1+k}{2};\frac{3+k}{2};\frac{a^{2}}{r^{2}}\right),\end{aligned}$$

(11.43)

which (taking into account (11.39)) can be written in the form:

$$w(r;a)=\frac{2d\cos\left(\frac{k\pi}{2}\right)}{\pi(1+k)}\left(\frac{a}{r}\right)^{k+1}\,{}_{2}\mathrm{F}_{1}\left(\frac{1+k}{2},\frac{1+k}{2};\frac{3+k}{2};\frac{a^{2}}{r^{2}}\right).$$

(11.44)

Integration of the pressure distribution over the contact area will give the total normal force:

$$F_{N}(a)=\int_{0}^{a}p_{0}\left(1-\frac{r^{2}}{a^{2}}\right)^{\frac{k-1}{2}}2\pi r\mathrm{d}r=\frac{2\pi p_{0}a^{2}}{k+1}.$$

(11.45)

The normal contact stiffness results from (11.39) and (11.45):

$$k_{z}:=\frac{\mathrm{d}F_{N}}{\mathrm{d}d}=\frac{2h_{N}(k,\nu)E_{0}}{(1-\nu^{2})c_{0}^{k}(1+k)}a^{1+k}.$$

(11.46)

Application of (11.39) also allows for the reformulation of the pressure distribution in terms of the indentation depth:

$$p(r;a)=\frac{h_{N}(k,\nu)E_{0}d}{\pi(1-\nu^{2})c_{0}^{k}(a^{2}-r^{2})^{\frac{1-k}{2}}}.$$

(11.47)

11.5 Normal Contact of Axisymmetric Profiles with a Compact Contact Area for FGMs

Starting from the contact solution for the indentation of a power-law graded elastic half-space by a rigid, flat-ended cylindrical punch, we derive the solution of an arbitrarily shaped, axisymmetric contact problem with a compact contact area. Analogous to the derivation for elastically homogeneous materials from Sect. 11.2, we first assume that the indentation depth can be written as an unambiguous function of the contact radius:

$$d=g(a).$$

(11.48)

In the following, formulas for the normal force, pressure distribution, and the normal displacements of the surface are derived, all of which depend only on the function g. For this purpose we will make use of an idea by Mossakovskii (1963), according to which the solution of the general axisymmetric contact problem results from a superposition of (differential) flat-ended punch solutions. As a final step the determination of the function g will be shown.

For the calculation of the normal force, we assume the incremental contact stiffness according to (11.46) which, taking (11.48) into account, will lead to:

$$\mathrm{d}\tilde{F}_{N}=\frac{2h_{N}(k,\nu)E_{0}}{(1-\nu^{2})c_{0}^{k}(1+k)}\tilde{a}^{1+k}\mathrm{d}\tilde{d}=\frac{2h_{N}(k,\nu)E_{0}}{(1-\nu^{2})c_{0}^{k}(1+k)}\tilde{a}^{1+k}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}.$$

(11.49)

The indentation process can be viewed as a change in the indentation depth from $\tilde{d}=0$ to $\tilde{d}=d$, with the contact radius changing from $\tilde{a}=0$ to $\tilde{a}=a$ and the contact force from $\tilde{F}_{N}=0$ to $\tilde{F}_{N}=F_{N}$. The normal force at the end of the process can be calculated from (11.49) by integration:

$$F_{N}(a)=\frac{2h_{N}(k,\nu)E_{0}}{(1-\nu^{2})c_{0}^{k}(1+k)}\int_{0}^{a}\tilde{a}^{1+k}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}.$$

(11.50)

Integration by parts results in:

$$\begin{aligned}\displaystyle F_{N}(a)&\displaystyle=\frac{2h_{N}(k,\nu)E_{0}}{(1-\nu^{2})c_{0}^{k}}\int_{0}^{a}\tilde{a}^{k}\left[g(a)-g(\tilde{a})\right]\mathrm{d}\tilde{a}\\ \displaystyle&\displaystyle=\int_{-a}^{a}c_{N}(\tilde{a})\left[d-g(\tilde{a})\right]\mathrm{d}\tilde{a},\end{aligned}$$

(11.51)

wherein $c_{N}(\tilde{a})$ denotes the foundation modulus defined by (9.4).

The calculation of the pressure distribution is done in the same way. An infinitesimal indentation of an area of radius $\tilde{a}$ generates the following contribution to the pressure distribution (see (11.47)):

$$\begin{aligned}\displaystyle\mathrm{d}p(r;\tilde{a})&\displaystyle=\frac{h_{N}(k,\nu)E_{0}}{\pi(1-\nu^{2})c_{0}^{k}(\tilde{a}^{2}-r^{2})^{\frac{1-k}{2}}}\mathrm{d}\tilde{d}\\ \displaystyle&\displaystyle=\frac{h_{N}(k,\nu)E_{0}}{\pi(1-\nu^{2})c_{0}^{k}(\tilde{a}^{2}-r^{2})^{\frac{1-k}{2}}}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}\quad\text{for}\ r<\tilde{a}.\end{aligned}$$

(11.52)

The pressure distribution at the end of the indentation process equals the sum of the incremental pressure distributions:

$$p(r)=\frac{h_{N}(k,\nu)E_{0}}{\pi(1-\nu^{2})c_{0}^{k}}\int_{r}^{a}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\frac{\mathrm{d}\tilde{a}}{(\tilde{a}^{2}-r^{2})^{\frac{1-k}{2}}}=-\frac{c_{N}(c_{0})}{\pi c_{0}^{k}}\int_{r}^{a}\frac{w^{\prime}_{1D}(\tilde{a})}{(\tilde{a}^{2}-r^{2})^{\frac{1-k}{2}}}\mathrm{d}\tilde{a},$$

(11.53)

wherein $w_{1D}$ denotes the displacement of the Winkler foundation introduced by (9.9).

For the calculation of the normal displacements outside the loaded area, we first need the displacement proportion caused by an incremental indentation with the radius $\tilde{a}$. Thereby, (11.44) will provide the relation

$$\mathrm{d}w(r;\tilde{a})=\begin{cases}\mathrm{d}\tilde{d}&\text{for}\ 0\leq r<\tilde{a}\\ \displaystyle\frac{2\cos\left(\frac{k\pi}{2}\right)}{\pi(1+k)}\left(\frac{\tilde{a}}{r}\right)^{1+k}&\\ \displaystyle\quad{}\cdot{}_{2}\mathrm{F}_{1}\left(\frac{1+k}{2},\frac{1+k}{2};\frac{3+k}{2};\frac{\tilde{a}^{2}}{r^{2}}\right)\mathrm{d}\tilde{d}&\text{for}\ r> \tilde{a}.\end{cases}$$

(11.54)

Summation over all incremental contributions $0\leq\tilde{a}\leq a$, accounting for the condition $r> a$, gives:

$$\begin{aligned}\displaystyle&\displaystyle w(r;a)=\\ \displaystyle&\displaystyle\frac{2\cos\left(\frac{k\pi}{2}\right)}{\pi(1+k)}\int_{0}^{a}\left(\frac{\tilde{a}}{r}\right)^{1+k}\,{}_{2}\mathrm{F}_{1}\left(\frac{1+k}{2},\frac{1+k}{2};\frac{3+k}{2};\frac{\tilde{a}^{2}}{r^{2}}\right)\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a},\end{aligned}$$

(11.55)

which (after integrating by parts) results in the determining relation for the displacements outside the contact area:

$$\begin{aligned}\displaystyle w(r;a)&\displaystyle=\frac{2\cos\left(\frac{k\pi}{2}\right)}{\pi}\int_{0}^{a}\frac{\tilde{a}^{k}[g(a)-g(\tilde{a})]}{(r^{2}-\tilde{a}^{2})^{\frac{1+k}{2}}}\mathrm{d}\tilde{a}\\ \displaystyle&\displaystyle=\frac{2\cos\left(\frac{k\pi}{2}\right)}{\pi}\int_{0}^{a}\frac{\tilde{a}^{k}w_{1D}(\tilde{a})}{(r^{2}-\tilde{a}^{2})^{\frac{1+k}{2}}}\mathrm{d}\tilde{a}\quad\text{for}\ r> a.\end{aligned}$$

(11.56)

Equations (11.51), (11.53), and (11.56) reproduce the MDR relations (9.11) und (9.12) for the normal contact of power-law graded elastic bodies. They only depend on the yet not determined function $g(\tilde{a})$. To give an expression for this function we first calculate the normal displacements inside the contact area based on an incremental formulation and (11.54). We obtain:

$$\begin{aligned}\displaystyle w(r;a)=&\displaystyle\int_{r}^{a}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}\\ \displaystyle&\displaystyle{}+\frac{2\cos\left(\frac{k\pi}{2}\right)}{\pi(1+k)}\int_{0}^{r}\left(\frac{\tilde{a}}{r}\right)^{1+k}\,{}_{2}\mathrm{F}_{1}\left(\frac{1+k}{2},\frac{1+k}{2};\frac{3+k}{2};\frac{\tilde{a}^{2}}{r^{2}}\right)\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}\\ \displaystyle&\displaystyle\quad\text{for}\ r<a.\end{aligned}$$

(11.57)

Integration by parts results in:

$$w(r;a)=g(a)-\frac{2\cos\left(\frac{k\pi}{2}\right)}{\pi}\int_{0}^{r}\frac{\tilde{a}^{k}g(\tilde{a})}{(r^{2}-\tilde{a}^{2})^{\frac{1+k}{2}}}\mathrm{d}\tilde{a}\quad\text{for}\ 0<r<a,$$

(11.58)

which of course must equal the normal displacements due to the indenting profile:

$$w(r;a)=d-f(r)\quad\text{for}\ 0<r<a.$$

(11.59)

Hence, the functions f and g are connected via the Abel transform

$$f(r)=\frac{2\cos\left(\frac{k\pi}{2}\right)}{\pi}\int_{0}^{r}\frac{\tilde{a}^{k}g(\tilde{a})}{(r^{2}-\tilde{a}^{2})^{\frac{1+k}{2}}}\mathrm{d}\tilde{a}.$$

(11.60)

Inverse transform of this equation yields a determining relation for the function $g(\tilde{a})$:

$$g(\tilde{a})=|\tilde{a}|^{1-k}\int_{0}^{|\tilde{a}|}\frac{f^{\prime}(r)}{(\tilde{a}^{2}-r^{2})^{\frac{1-k}{2}}}\mathrm{d}r,$$

(11.61)

which naturally corresponds to (9.2) in Chap. 9.

11.6 Adhesive Contact of Axisymmetric Profiles with a Compact Contact Area for FGMs

The calculation of adhesive contacts with FGMs is performed completely analogously to the case of adhesive contact of homogeneous materials. Only the Winkler foundation is redefined by (9.3):

$$\Updelta k_{z}=c_{N}(x)\cdot\Updelta x$$

(11.62)

with $c_{N}(x)$ defined by (9.4). The energy balance now takes on the form:

$$2\pi a\Updelta x\Updelta\gamma=c_{N}(a)\Updelta x\Updelta l^{2}.$$

(11.63)

It follows that:

$$\Updelta l=\sqrt{\frac{2\pi a\Updelta\gamma}{c_{N}(a)}}.$$

(11.64)

For a rigid indenter,

$$c_{N}(x)=h_{N}(k,\nu)E^{*}\left(\frac{|x|}{c_{0}}\right)^{k}$$

(11.65)

and (11.64) can be written in the following form:

$$\Updelta l=\sqrt{\frac{2\pi a\Updelta\gamma c_{0}^{k}a^{1-k}}{E^{*}h_{N}(k,\nu)}}$$

(11.66)

which justifies (9.45).

11.7 Tangential Contact of Axisymmetric Profiles with a Compact Contact Area

There exists a very close analogy between normal and tangential contact problems. The normal force and pressure distribution for an indentation $d$ of a flat cylindrical indenter of radius $a$ are given by the equations:

$$\begin{aligned}\displaystyle F_{N}&\displaystyle=2E^{*}ad,\\ \displaystyle p(r)&\displaystyle=\frac{1}{\pi}\frac{E^{*}d}{\sqrt{a^{2}-r^{2}}},\end{aligned}$$

(11.67)

(see Sect. 11.1). A tangential displacement $u^{(0)}$ leads to the tangential force and stress distribution given by Johnson (1985):

$$\begin{aligned}\displaystyle F_{x}&\displaystyle=2G^{*}au^{(0)},\\ \displaystyle\tau(r)&\displaystyle=\frac{1}{\pi}\frac{G^{*}u^{(0)}}{\sqrt{a^{2}-r^{2}}},\end{aligned}$$

(11.68)

which differs from those for the normal contact only in the notation. Now we consider the simultaneous impression of an axisymmetric profile $z=f(r)$ in the normal and tangential directions and characterize these movements through the normal and tangential displacements as functions of the contact radius:

$$d=g(a),\quad u^{(0)}=h(a).$$

(11.69)

The force and stress during the indentation starting at initial contact are then:

$$F_{N}=2E^{*}\int_{0}^{a}\tilde{a}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a},\quad p(r)=\frac{E^{*}}{\pi}\int_{r}^{a}\frac{1}{\sqrt{\tilde{a}^{2}-r^{2}}}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}$$

(11.70)

and

$$F_{x}=2G^{*}\int_{0}^{a}\tilde{a}\frac{\mathrm{d}h(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a},\quad\tau(r)=\frac{G^{*}}{\pi}\int_{r}^{a}\frac{1}{\sqrt{\tilde{a}^{2}-r^{2}}}\frac{\mathrm{d}h(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}.$$

(11.71)

We now consider the following two-step process. The punch is initially pressed in a purely normal motion until the contact radius c is reached, and then further pressed in such a way that it moves simultaneously normally and tangentially until contact radius a is reached, whereby

$$\mathrm{d}h=\lambda\cdot\mathrm{d}g.$$

(11.72)

The normal force and the pressure distribution at the end of this process is still given by (11.70), while the tangential force and stress distribution obviously result in:

$$F_{x}=2G^{*}\int_{c}^{a}\tilde{a}\frac{\mathrm{d}h(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}=2G^{*}\lambda\int_{c}^{a}\tilde{a}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}$$

(11.73)

and

$$\tau(r)=\begin{cases}\displaystyle\frac{G^{*}}{\pi}\lambda\int\nolimits_{c}^{a}\frac{1}{\sqrt{\tilde{a}^{2}-r^{2}}}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a},&\text{for}\ r<c,\\ \displaystyle\frac{G^{*}}{\pi}\lambda\int\nolimits_{r}^{a}\frac{1}{\sqrt{\tilde{a}^{2}-r^{2}}}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a},&\text{for}\ c<r<a.\end{cases}$$

(11.74)

In the area $c<r<a$, the normal and tangential stress distribution have the same form:

$$\tau(r)=\lambda\frac{G^{*}}{E^{*}}p(r).$$

(11.75)

If we set

$$\lambda\frac{G^{*}}{E^{*}}=\mu$$

(11.76)

then the examined contact will have the following properties:

$$\begin{aligned}\displaystyle u(r)&\displaystyle=u^{(0)}=\text{const},&\displaystyle&\displaystyle\text{for}\ r<c,\\ \displaystyle\tau(r)&\displaystyle=\mu p(r),&\displaystyle&\displaystyle\text{for}\ c<r<a.\end{aligned}$$

(11.77)

These conditions correspond exactly to the stick and slip conditions in a tangential contact with the coefficient of friction ${\mu}$. Therefore, the force (11.73) and the stress distribution (11.74) under consideration of (11.75) solve the tangential contact problem:

$$\begin{aligned}\displaystyle F_{x}&\displaystyle=2\mu E^{*}\int_{c}^{a}\tilde{a}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a},\\ \displaystyle\tau(r)&\displaystyle=\begin{cases}\displaystyle\mu\frac{E^{*}}{\pi}\Bigg(\int\nolimits_{r}^{a}\frac{1}{\sqrt{\tilde{a}^{2}-r^{2}}}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}&\\ \displaystyle\qquad\quad{}-\int\nolimits_{r}^{c}\frac{1}{\sqrt{\tilde{a}^{2}-r^{2}}}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a}\Bigg),&\text{for}\ r<c,\\ \displaystyle\mu\frac{E^{*}}{\pi}\int\nolimits_{r}^{a}\frac{1}{\sqrt{\tilde{a}^{2}-r^{2}}}\frac{\mathrm{d}g(\tilde{a})}{\mathrm{d}\tilde{a}}\mathrm{d}\tilde{a},&\text{for}\ c<r<a,\end{cases}\end{aligned}$$

(11.78)

or

$$\tau(r)=\mu\begin{cases}p(r;a)-p(r;c),&\text{for}\ r<c,\\ p(r;a),&\text{for}\ c<r<a,\end{cases}$$

(11.79)

where we denote the normal pressure distribution at the contact radii a and c by $p(r;a)$ and $p(r;c)$, respectively. The tangential displacement in the contact is obtained by integration of (11.72):

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

(11.80)

It is easy to see that (11.78) and (11.80) coincide with (4.39), thereby proving the validity of the MDR for tangential contacts.

11.8 Definitions of Special Functions Used in this Book

At certain points in this book, non-elementary functions have been utilized. For ease of reference, we have provided a short summary of their definitions and significant properties.

11.8.1 Elliptical Integrals

Several definite integrals that cannot be solved in the form of elementary functions are themselves defined as (non-elementary) functions. The so-called elliptical integrals belong to this class of functions. Let $k$ be an elliptical modulus with $0\leq k^{2}\leq 1$ and $0\leq\theta\leq\pi/2$. The incomplete elliptical integrals of the first and second kind are defined as:

$$\begin{aligned}\displaystyle\mathrm{F}(\theta,k)&\displaystyle:=\int_{0}^{\theta}\frac{\mathrm{d}\varphi}{\sqrt{1-k^{2}\sin^{2}\varphi}},\\ \displaystyle\mathrm{E}(\theta,k)&\displaystyle:=\int_{0}^{\theta}\sqrt{1-k^{2}\sin^{2}\varphi}\mathrm{d}\varphi.\end{aligned}$$

(11.81)

For the case of $\theta=\pi/2$, this results in the so-called complete elliptical integrals:

$$\begin{aligned}\displaystyle\mathrm{K}(k)&\displaystyle:=\int_{0}^{\pi/2}\frac{\mathrm{d}\varphi}{\sqrt{1-k^{2}\sin^{2}\varphi}}=\mathrm{F}\left(\frac{\pi}{2},k\right),\\ \displaystyle\mathrm{E}(k)&\displaystyle:=\int_{0}^{\pi/2}\sqrt{1-k^{2}\sin^{2}\varphi}\mathrm{d}\varphi=\mathrm{E}\left(\frac{\pi}{2},k\right).\end{aligned}$$

(11.82)

It should be noted that several mathematical databases provide the elliptical integrals as a function of the modulus $m=k^{2}$.

11.8.2 The Gamma Function

Euler’s Gamma function can be defined in an integral form by the expression:

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

(11.83)

It is easy to prove the recursion property

$$\Upgamma(z)=(z-1)\Upgamma(z-1).$$

(11.84)

The Gamma function is, therefore, a generalization of the faculty function for arbitrary complex arguments, since (11.84) immediately returns the following relationship for positive integer arguments of the function:

$$\Upgamma(n)=(n-1)!,\quad n\in\mathbb{N}^{*}.$$

(11.85)

The complete Gamma function defined in (11.83) can be generalized to the “lower” or “upper” incomplete Gamma function by restricting the integration limits,

$$\begin{aligned}\displaystyle\Upgamma_{o}(z,a)&\displaystyle:=\int_{a}^{\infty}t^{z-1}\exp(-t)\mathrm{d}t,\quad\mathop{\mathrm{Re}}\{z\}> 0,\ a\in\mathbb{R}^{+}\\ \displaystyle\Upgamma_{u}(z,a)&\displaystyle:=\int_{0}^{a}t^{z-1}\exp(-t)\mathrm{d}t,\quad\mathop{\mathrm{Re}}\{z\}> 0,\ a\in\mathbb{R}^{+}\end{aligned}$$

(11.86)

with the obvious property

$$\Upgamma_{o}(z,a)+\Upgamma_{u}(z,a)=\Upgamma(z).$$

(11.87)

11.8.3 The Beta Function

The complete Beta function relates to the Gamma function according to the definition:

$$\mathrm{B}(x,y):=\frac{\Upgamma(x)\Upgamma(y)}{\Upgamma(x,y)},\quad\mathop{\mathrm{Re}}\{x,y\}> 0.$$

(11.88)

The integral definition can also be shown in this representation:

$$\mathrm{B}(x,y)=\int_{0}^{1}u^{x-1}(1-u)^{y-1}\mathrm{d}u,\quad\mathop{\mathrm{Re}}\{x,y\}> 0.$$

(11.89)

The Beta function is clearly symmetric, with $\mathrm{B}(x,y)=\mathrm{B}(y,x)$. From (11.85) and (11.88) it is apparent that the binomial coefficient

$$\begin{aligned}\displaystyle\binom{n}{k}&\displaystyle:=\frac{n!}{k!(n-k)!}=\frac{\Upgamma(n+1)}{\Upgamma(k+1)\Upgamma(n-k+1)}\\ \displaystyle&\displaystyle\phantom{:}=\frac{1}{(n+1)\mathrm{B}(k+1,n-k+1)},\quad n,k\in\mathbb{N},\ n\geq k\end{aligned}$$

(11.90)

can be expressed by the Beta function. From the definition of the Beta function also follows the recursion property

$$\mathrm{B}(x,y)=\mathrm{B}(x+1,y)+\mathrm{B}(x,y+1).$$

(11.91)

By restricting the integration limits in (11.89), the complete Beta function can be generalized to the “lower” or “upper” incomplete Beta function. Usually the upper limit is set, resulting in the definition

$$\mathrm{B}(z;x,y)=\int_{0}^{z}u^{x-1}(1-u)^{y-1}\mathrm{d}u,\quad\mathop{\mathrm{Re}}\{x,y\}> 0,\ z\in[0;1].$$

(11.92)

11.8.4 The Hypergeometric Function

Many different hypergeometric functions exist. In this book, only the Gaussian hypergeometric function

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

(11.93)

is used. It is the solution of the hypergeometric differential equation

$$z(1-z)y^{\prime\prime}(z)+[c-(a+b+1)z]y^{\prime}(z)-aby(z)=0.$$

(11.94)

The derivative of this function is given by:

$$\frac{\mathrm{d}}{\mathrm{d}z}\left[{}_{2}\mathrm{F}_{1}(a,b;c;z)\right]=\frac{ab}{c}\,{}_{2}\mathrm{F}_{1}(a+1,b+1;c+1;z).$$

(11.95)

Certain special cases of the hypergeometric function can be expressed by elementary functions, such as:

$${}_{2}\mathrm{F}_{1}(1,1;1;z)=\frac{1}{1-z}.$$

(11.96)

11.8.5 The Struve H-Function

The Struve H-function is a Bessel-type function. It can be expanded to the power series

$$\mathrm{H}_{n}(z):=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{\Upgamma\left(k+\frac{3}{2}\right)\Upgamma\left(k+n+\frac{3}{2}\right)}\left(\frac{z}{2}\right)^{2k+n+1},\quad z\in\mathbb{C}$$

(11.97)

and is the solution of the inhomogeneous Bessel differential equation

$$y^{\prime\prime}(z)z^{2}+y^{\prime}(z)z+(z^{2}-n^{2})y(z)=\frac{4}{\sqrt{\pi}\Upgamma(n+1/2)}\left(\frac{z}{2}\right)^{n+1}.$$

(11.98)

Additionally, the following differentiation property of the Struve H-function can be demonstrated:

$$\frac{\mathrm{d}}{\mathrm{dz}}\left[\mathrm{H}_{n}(z)\right]=\mathrm{H}_{n-1}(z)-\frac{n}{z}\mathrm{H}_{n}(z).$$

(11.99)

11.9 Solutions to Axisymmetric Contact Problems According to Föppl and Schubert

In this section, we will present the derivation of the solution for the contact problem of an arbitrary axially symmetric indenter, which was published by Föppl (1941 ) and Schubert (1942 ). This is the first known publication to perform the derivation of the equations that lay the foundation for this book. Over the course of the history of contact mechanics, they were derived multiple times using different approaches. For instance, they were later found by Galin (1946 ) as well (likely independently). They reached a high degree of international recognition through the paper of Sneddon (1965) —one of the most cited publications in the history of contact mechanics. Yet Sneddon merely offered a different derivation approach to obtain already known solutions, including those of Galin (whom he cites). And much later, new interpretations and derivations of the same equations were published multiple times. Some of these were quite useful since they provided a new perspective on the issues and facilitated different generalizations and developments. This includes the interpretations by Jäger (1995 ), who treated the indentation of a curved body as the superposition of infinitesimal indentations of flat cylindrical punches (although even this idea was not original and was previously used by Mossakovskii). The MDR is also based on the equations of Föppl–Schubert–Galin–Sneddon–Jäger, however, it offers an intuitive physical–mnemonic interpretation, which can be directly generalized to many other contact problems.

Although 77 years have passed since the publications of Föppl and Schubert, this first historic derivation still surprisingly remains the most direct and simple of all. It is valuable to understand this derivation both for historic and didactic reasons. The presentation of the derivation of Föppl and Schubert very closely follows the original publication, however, modified notations are used to highlight the direct connection to the equations of the MDR.

We consider an axially symmetric pressure distribution $p(r)$ in a circle of radius a (see Fig. 11.3). We calculate the displacement at the “point of observation” A caused by an infinitesimally small force in the “source point” B and then integrate over all source points. The location of the source point is parametrized by the linear coordinate $s$ and the angle ${\varphi}$. The vertical displacement of point A by the force $\mathrm{d}F_{N}=p(\rho)s\mathrm{d}s\mathrm{d}\varphi$ in point B is given by the fundamental solution (2.2):

$$\mathrm{d}w(r)=\frac{p(\rho)s\mathrm{d}s\mathrm{d}\varphi}{\pi E^{*}s}=\frac{1}{\pi E^{*}}p(\rho)\mathrm{d}s\mathrm{d}\varphi.$$

(11.100)

Fig. 11.3

Schematic diagram to display the notation proposed by Föppl (1941)

The total vertical displacement caused by the entire pressure distribution results from the integration of

$$w(r)=\frac{1}{\pi}\int_{0}^{\pi}\left(\frac{1}{E^{*}}\int_{s_{1}}^{s_{2}}p(\rho)\mathrm{d}s\right)\mathrm{d}\varphi.$$

(11.101)

In his paper, Föppl (1941) proposed replacing the parametrization s and ${\varphi}$ with new variables ${\rho}$ and ${\xi}$, which uniquely define the location of point B too, and relate to s and ${\varphi}$ according to the following equations:

$$\begin{aligned}\displaystyle s&\displaystyle=\sqrt{\rho^{2}-\xi^{2}}+\sqrt{r^{2}-\xi^{2}},&\displaystyle&\displaystyle\xi\leq\rho\leq a,\\ \displaystyle\varphi&\displaystyle=\arcsin\left(\frac{\xi}{r}\right),&\displaystyle&\displaystyle 0\leq\xi\leq r.\end{aligned}$$

(11.102)

Accordingly, the derivations are:

$$\begin{aligned}\displaystyle\frac{\partial s}{\partial\rho}&\displaystyle=\frac{\rho}{\sqrt{\rho^{2}-\xi^{2}}},\\ \displaystyle\frac{\partial\varphi}{\partial\xi}&\displaystyle=\frac{1}{\sqrt{r^{2}-\xi^{2}}}.\end{aligned}$$

(11.103)

Denoting the expression in parenthesis in (11.101) by $w_{1D}(\xi)$:

$$w_{1D}(\xi)=\frac{1}{E^{*}}\int_{s_{1}}^{s_{2}}p(\rho)\mathrm{d}s=\frac{2}{E^{*}}\int_{\xi}^{a}\frac{p(\rho)\rho\mathrm{d}\rho}{\sqrt{\rho^{2}-\xi^{2}}}.$$

(11.104)

For the vertical displacement (11.101), we then obtain:

$$w(r)=\frac{2}{\pi}\int_{0}^{\pi/2}w_{1D}(\xi)\mathrm{d}\varphi=\frac{2}{\pi}\int_{0}^{r}\frac{w_{1D}(\xi)\mathrm{d}\xi}{\sqrt{r^{2}-\xi^{2}}}.$$

(11.105)

Both (11.104) and (11.105) are identical to (2.16) and (2.14) of the MDR. These were already derived in the paper by Föppl (1941), as previously described in this chapter, and enable the calculation of the displacement field resulting from a known pressure distribution. In his paper, Föppl examined the pressure distributions $(1-r^{2}/a^{2})^{-1/2}$, $(1-r^{2}/a^{2})^{1/2}$ and a constant pressure distribution, demonstrating that the former corresponds to a constant displacement and the latter corresponds to a parabolic indenter.

The contribution of his doctoral candidate, Schubert (1942), was the inversion of the integral equations (11.104) and (11.105). Since these are Abel transforms, Schubert found the solutions

$$w_{1D}(\xi)=\xi\int_{0}^{\xi}\frac{w^{\prime}(\rho)}{\sqrt{\xi^{2}-\rho^{2}}}\mathrm{d}\rho$$

(11.106)

and

$$p(\rho)=-\frac{E^{*}}{\pi}\int_{\rho}^{a}\frac{w^{\prime}_{1D}(\xi)}{\sqrt{\xi^{2}-\rho^{2}}}\mathrm{d}\xi,$$

(11.107)

which are identical to (2.6) and (2.13) of the MDR.

Equations (11.106) and (11.107) completely solve the contact problem: with three-dimensional form $w(\rho)$ and using (11.106), one can calculate the auxiliary function $w_{1D}(\xi)$, which then determines the pressure distribution by (11.107). Schubert used this approach to solve the contact problems of the flat punch, the cone, and the power-law profiles of second, fourth, and sixth-order, the concave power-law profiles of the second and fourth-order, and the cylindrical indenter with rounded edges.

The publications of Föppl and Schubert of course did not contain the MDR interpretations of their equations, which requires a couple of additional steps. In the interpretation of the MDR, $w_{1D}(\xi)$ is the vertical displacement in the equivalent MDR model. The necessary property for the transition to the MDR interpretation is the relation $w_{1D}(\xi=0)=w(r=0)$ (which guarantees that indentation depth of the three-dimensional profile is also the indentation depth of the equivalent MDR profile). It follows from (11.105), when the limit $w_{1D}(\xi=0)$ is substituted for $w_{1D}(\xi)$ in the limit case $r\rightarrow 0$ and the identity

$$\frac{2}{\pi}\int_{0}^{r}\frac{\mathrm{d}\xi}{\sqrt{r^{2}-\xi^{2}}}\equiv 1.$$

(11.108)

is taken into account. Then

$$\begin{aligned}\displaystyle w(r=0)&\displaystyle=\lim_{r\rightarrow 0}[w(r)]=w_{1D}(\xi=0)\cdot\frac{2}{\pi}\int_{0}^{r}\frac{\mathrm{d}\xi}{\sqrt{r^{2}-\xi^{2}}}\\ \displaystyle&\displaystyle=w_{1D}(\xi=0).\end{aligned}$$

(11.109)

It follows trivially from (11.104) that the contact radius is determined by the equation $w_{1D}(a)=0$. The equation determining the force (2.11) follows from (11.107):

$$\begin{aligned}\displaystyle F_{N}&\displaystyle=2\pi\int_{0}^{a}p(r)r\mathrm{d}r=-2E^{*}\int_{0}^{a}\left(\int_{\rho}^{a}\frac{w^{\prime}_{1D}(\xi)}{\sqrt{\xi^{2}-r^{2}}}\mathrm{d}\xi\right)r\mathrm{d}r\\ \displaystyle&\displaystyle=-2E^{*}\int_{0}^{a}w^{\prime}_{1D}(\xi)\left(\int_{0}^{\xi}\frac{r\mathrm{d}r}{\sqrt{\xi^{2}-r^{2}}}\right)\mathrm{d}\xi\\ \displaystyle&\displaystyle=-2E^{*}\int_{0}^{a}\xi w^{\prime}_{1D}(\xi)\mathrm{d}\xi=2E^{*}\int_{0}^{a}w_{1D}(\xi)\mathrm{d}\xi.\end{aligned}$$

(11.110)

Thus, all fundamental equations of the MDR are determined and the only task remaining is to “put them into words”.

In closing, it should be noted that the publication by Schubert also contained the complete solution of the plane contact problem, which he applied to the following profiles: flat punch with symmetric load, flat punch with axisymmetric load, wedge profiles, parabolically rounded wedge profile, power-law profiles of the second, fourth, and sixth-order, concave power-law profiles of the second and fourth-order, and the flat punch with rounded edges.

It is most regrettable that this excellent work, which in itself nearly represents a small “handbook of contact mechanics”, remained nearly unknown for a long time and has only just been “rediscovered” in recent years.