Let us consider two bodies in contact so that the resultant force between them is P. The contact is only over points, which belong to a small region of the surface of each body. One of the bodies can be considered as absolutely rigid. If the body is rigid, then it is called a punch. Hertz (1882a) considered 3D frictionless contact of two isotropic, linear elastic solids. It is assumed in the classic nonadhesive Hertz problem formulation that two smooth and convex elastic solids initially contact at one point.

Let us use both the Cartesian and the cylindrical coordinate frames, namely, x₁ = x, x₂ = y, x₃ = z and r, φ, z, where $r=\sqrt{x^2+y^2}$ and x = r cos φ, y = r sin φ. Let us place the origin (O) of the Cartesian x₁, x₂, x⁺₃ and x₁, x₂, x⁻₃ coordinates at the point of initial contact between two bodies, and the axes of x⁺₃ and x⁻₃ are directed along normals to the two bodies drawn toward the inside of each (Love, 1893). Let us assume that axis x₁ and axis x₂ are the same for both bodies.

The quantities referring to the body x⁺₃ ≥ 0 are denoted by a superscript plus sign, and those referring to the second body are denoted by a superscript minus sign. Then the surfaces are described by the shape functions f⁺ and f⁻, respectively, i.e.,

$x_3^{+}=f^{+}\left(x_1,x_2\right),x_3^{-}=f^{-}\left(x_1,x_2\right).$

where the common tangent plane at O is taken as the x₁, x₂ plane.

If the bodies are in contact with each other, then displacements u⁺_i, u⁻_i and stresses σ⁺_ij, σ⁻_ij are generated. In the general case, if a body is blunt, this means that the contact problem can be considered in a geometrically linear formulation and the boundary value problem is formulated for a positive half-space, i.e., each of the solids is considered as a half-space.

Let us consider two points m⁺ and m⁻, one on each surface, that initially were not in contact and having the same coordinates (x₁, x₂). A very important assumption of the Hertz contact problem is that only the vertical displacements u⁻₃ and u⁺₃ of the boundaries are taken into account, while displacements u⁻₁, u⁻₂, u⁺₁, and u⁺₂ are disregarded. This means that if the points m⁺ and m⁺ contact with each other after application of the load, then the following geometrical condition has to be satisfied:

$u_3^{+}\left(x_1,x_2\right)={\delta }^{-}-f^{-}\left(x_1,x_2\right),u_3^{-}\left(x_1,x_2\right)={\delta }^{+}-f^{+}\left(x_1,x_2\right),$

  (3.1)

where δ± are the approaches of the distant points of each body toward the contact plane x^±₃ = 0.

Expression (3.1) can be written as

$u_3^{+}+u_3^{-}=\delta -\left(f^{+}+f^{-}\right),\delta ={\delta }^{+}+{\delta }^{-}.$

  (3.2)

Hertz (1882a) described the general shape of a 3D body as an elliptic paraboloid. It follows from the formulation of the problem that f(0, 0) = 0. It is also known that the first derivatives of a smooth function at an extremum (in our case at a maximum) of the function are zero. If a decomposition of f⁺ and f⁻ into a Taylor series is not degenerative for second terms, then the truncated decomposition can be written as

$z^{+}=\left(A^{+}{x}_1^2+B^{+}{x}_2^2+2C^{+}{x}_1{x}_2\right),z^{-}=\left(A^{-}{x}_1^2+B^{-}{x}_2^2+2C^{-}{x}_1{x}_2\right),$

  (3.3)

where A^±, B^±, and C^± are constants of the decompositions. Substituting Eq. (3.3) into Eq. (3.1), one gets

$u_3^{+}+u_3^{-}=\delta -\left[\left(A^{+}+A^{-}\right)x_1^2+\left(B^{+}+B^{-}\right)x_2^2+2\left(C^{+}+C^{-}\right)x_1{x}_2\right].$

  (3.4)

After rotating axis x₁ and axis x₂ into the properly chosen axes x and y, one can rewrite Eq. (3.4) as

$u_3^{+}+u_3^{-}=\delta -A{x}^2-B{y}^2.$

  (3.5)

Thus, it was assumed in the classic Hertz formulation that the shape function f is described as an elliptic paraboloid. The equation z⁺ + z⁻ = const describes a curve within the xy plane such that all points of the curve correspond to the same height. Hertz (1882a) showed that the contact region for contacting solids whose shape is described by Eq. (3.3) can be described by z⁺ + z⁻ = Ax² + By² = const and the region is an ellipse.

2.1.2 The Hertz-Type Boundary Value Problem

It is supposed that the shape of the punch f and the external parameter of the problem $\mathcal{P}$ are given and one has to find the bounded region G on the boundary plane x₃ = 0 of the half-space at the points where the punch and the medium are in mutual contact, displacements u_i, and stresses σ_ij. If the pressing force P is taken as the external parameter $\mathcal{P}$, then one has to find the depth of indentation δ and the characteristic size of the contact region l. If δ is taken as $\mathcal{P}$, then one has to find P and l.

In the general case of a 3D Hertz-type contact problem, it is not assumed that the punch shape is described by an elliptic paraboloid and the contact region is an ellipse as Hertz (1882a) did. The Hertz-type contact problems mean that the formulation of the problem is geometrically linear, the contact region is unknown, only vertical displacements of the boundary are taken into account, and we have the same boundary conditions within and outside the contact region as in classic Hertz contact problems (Hertz, 1882a). Hence, one must find the finite region G of the points at which the bodies are in mutual contact, the relative approach of the bodies δ > 0, the displacements u⁺_i and u⁻_i, and the stresses σ⁺_ij and σ⁻_ij.

One can see from Eq. (3.1) that the problem formulation is mathematically equivalent to the problem of contact between a positive half-space and an indenter whose shape function f is equal to the initial distance (the gap) between the surfaces, i.e., f = f⁺ + f⁻. As we will see, this problem, in turn, can be reduced to the problem of contact between a rigid indenter (a punch) and an elastic half-space with effective elastic constants. For the sake of simplicity, let us consider the case when a rigid punch is pressed by the force P into the positive half-space x₃ ≥ 0. The quantities sought must satisfy the following:

1. equations of equilibrium

${\sigma }_{\mathit{ij},j}=0,i,j=1,2,3;$

  (3.6)

2. constitutive relations

${\sigma }_{\mathit{ij}}=F\left({\epsilon }_{\mathit{ij}}\right),{\epsilon }_{\mathit{ij}}=\left(u_{i,j}+u_{j,i}\right)/2;$

  (3.7)

3. conditions at infinity

$\mathbf{u}\left(\mathbf{x},\mathcal{P}\right)\to 0,\left|\mathbf{x}\right|\to \infty ;$

  (3.8)

4. integral condition

${\displaystyle {\int }_{R^2}{\sigma }_{33}\left(x_1,x_2,0,\mathcal{P}\right)\text{d}{x}_1\text{d}{x}_2}=-P;$

  (3.9)

5. boundary conditions outside the contact region $G\left(\mathcal{P}\right)$

${\sigma }_{3j}\left(x_1,x_2,0,\mathcal{P}\right)=0,\left(x_1,x_2\right)\in R^{2}G\left(\mathcal{P}\right);$

  (3.10)

6. three boundary conditions within the contact region G(P).

Here and henceforth, a comma before the subscript denotes the derivative with respect to the corresponding coordinate; and summation is assumed over repeated Latin subscripts that take values from 1 to 3, while there is no summation over repeated Greek subscripts that take values from 1 to 2.

The contact region G is defined as an open region such that if x ∈ G, then the gap (u₃ − g) between the punch and the half-space is equal to zero (see Eq. 3.2) and surface stresses are nonpositive, while outside the contact region, i.e., for $\mathbf{x}\in {\mathbb{R}}^{2}G$, the gap is positive and the stresses are equal to zero. This can be written as

$\begin{array}{lll}{u}_3\left(\mathbf{x};\mathcal{P}\right)=g\left(\mathbf{x};\mathcal{P}\right),\hfill & {\sigma }_{33}\left(\mathbf{x};\mathcal{P}\right)\le 0,\hfill & \mathbf{x}\in G\left(\mathcal{P}\right),\hfill \\ u_3\left(\mathbf{x};\mathcal{P}\right)>g\left(\mathbf{x};\mathcal{P}\right),\hfill & {\sigma }_{3i}\left(\mathbf{x};\mathcal{P}\right)=0,\hfill & \mathbf{x}\in {\mathbb{R}}^{2}G\left(\mathcal{P}\right).\hfill \end{array}$

  (3.11)

For the general case of the problem of vertical pressing, we have

$g\left(\mathbf{x};\mathcal{P}\right)=\delta \left(\mathcal{P}\right)-f\left(x_1,x_2\right).$

  (3.12)

The choice of two other conditions (3.13), (3.14), or (3.15) depends on the friction within the contact region.

Remarks.

1. The above formulations are also valid for time-dependent materials. For such materials, one should add the variable t to the arguments of all functions, e.g., one should write write $\mathbf{u}\left(\mathbf{x},t,\mathcal{P}\right)$ instead of $\mathbf{u}\left(\mathbf{x},\mathcal{P}\right)$.

2. Depending on the form of the operator of constitutive relations F in Eq. (3.7), the material behavior of the medium may be anisotropic or isotropic, and linear or nonlinear with different rheological properties.

2.1.3 The Types of Contact Boundary Conditions

It is supposed that ${\sigma }_{33}\left(\mathbf{x},\mathcal{P}\right)\le 0$ when $\mathbf{x}\in G\left(\mathcal{P}\right)$. Then depending on the frictional conditions within the contact region, the problem can be frictionless, nonslipping (it is also called “adhesive”), or frictional (Johnson, 1985).

In the frictionless problem, these conditions can be written as

${\sigma }_{31}^{\pm }\left(x_1,x_2,0,\mathcal{P}\right)={\sigma }_{32}^{\pm }\left(x_1,x_2,0,\mathcal{P}\right)=0,\left(x_1,x_2\right)\in G\left(\mathcal{P}\right).$

  (3.13)

In the nonslipping contact problem, there is no relative slip between the bodies within the contact region. If the values

$v_1\left(x_1,x_2,\mathcal{P}\right)\equiv u_1^{+}\left(x_1,x_2,0,\mathcal{P}\right)-u_1^{-}\left(x_1,x_2,0,\mathcal{P}\right)$

and

$v_2\left(x_1,x_2,\mathcal{P}\right)\equiv u_2^{+}\left(x_1,x_2,0,\mathcal{P}\right)-u_2^{-}\left(x_1,x_2,0,\mathcal{P}\right)$

are introduced, then the condition within this region is that these values do not change with augmentation of the external parameter $\mathcal{P}$. These conditions of no relative slip can be expressed by

$\frac{\partial }{\partial \mathcal{P}}{v}_{\alpha }\left(x_1,x_2,\mathcal{P}\right)=0,\text{d}\mathcal{P}>0.$

  (3.14)

In the frictional contact problem, it is usually assumed (Bryant & Keer, 1982; Galin, 1945) that the contact region consists of the following parts: the inner part G₁, in which the interfacial friction must be sufficient to prevent any slip taking place between bodies, i.e., Eq. (3.14) holds; the outer part G G₁, in which the friction must satisfy the Amontons–Kotelnikov (or Coulomb) frictional law (in fact, Amontons (1699) gave a verbal formulation of the law as three relations observed for an optical lens polishing process, while Kotelnikov (1774) introduced the notion of the coefficient of friction and presented the law as a formula).

Let us define the vector of tangential stresses:

${\tau }^{\pm }\left(x_1,x_2,0,\mathcal{P}\right)\equiv \left({\sigma }_{31}^{\pm }\left(x_1,x_2,0,\mathcal{P}\right),{\sigma }_{32}^{\pm }\left(x_1,x_2,0,\mathcal{P}\right)\right).$

If μ_f is the coefficient of friction, then the frictional contact conditions can be written as (Spektor, 1981; Vermeulen & Johnson, 1964)

$\begin{array}{c}\frac{\partial }{\partial \mathcal{P}}{v}_{\alpha }\left(x_1,x_2,\mathcal{P}\right)=0,\text{d}\mathcal{P}>0,\left(x_1,x_2\right)\in G_1,\\ {\tau }^{\pm }\left(x_1,x_2,0,\mathcal{P}\right)=-{\mu }_{\text{f}}{\sigma }_{33}^{\pm }\left(x_1,x_2,0,\mathcal{P}\right)\left[\frac{\mathbf{v}\left(x_1,x_2,\mathcal{P}\right)}{\left|\mathbf{v}\left(x_1,x_2,\mathcal{P}\right)\right|}\right],\left(x_1,x_2\right)\in G{G}_1,\end{array}$

  (3.15)

where $\mathbf{v}\left(x_1,x_2,\mathcal{P}\right)=\left(v_1\left(x_1,x_2,\mathcal{P}\right),v_2\left(x_1,x_2,\mathcal{P}\right)\right)$.

In addition to the above formulations of the Hertz-type contact problems, sometimes a more general formulation of the axisymmetric mixed boundary value contact problems is considered. In the general formulation (see, e.g., G. I. Popov, 1973), it is assumed that in the system subjected to a normal contact force P, the displacements $u_r\left(r,0,\mathcal{P}\right)$ and $u_z\left(r,0,\mathcal{P}\right)$ are known within the contact region, and the solids are not loaded outside the contact region, i.e.,

$u_r\left(r,0,\mathcal{P}\right)=s\left(r\right),u_z\left(r,0,\mathcal{P}\right)=g\left(r\right),\text{for}r\le a;$

  (3.16)

${\sigma }_{\mathit{rz}}\left(r,0,P\right)={\sigma }_{\mathit{zz}}\left(r,0,P\right)=0,\text{for}r>a;$

  (3.17)

where s(r) and g(r) are known functions of the radial and normal displacements, respectively. The condition for the given radial displacements $u_r\left(r,0,\mathcal{P}\right)$ can be reformulated as the condition for mismatch strain distributions ε(r) between the contact surfaces (Guo, Jin, & Gao, 2011).

2.1.4 The Harmonic Function Formulation of the Frictionless Problem

It is well known (see, e.g., Galin, 1961, 2008; Rabotnov, 1977) that the components of the displacement vector u in a linear elastic solid can be expressed through four harmonic functions. This is the so-called Papkovich–Neuber representation. Further, if all tangential stresses σ_3α on the boundary plane of a linear elastic half-space are equal to zero, then all displacements and stresses can be represented by a single harmonic function Φ (see, e.g., Galin, 1961; Rabotnov, 1977), i.e., δΦ = 0, where δ is the Laplace operator:

$\text{Δ}=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}.$

  (3.18)

In this case one has

$u_3=2\left(1-v\right)\text{Φ}\left(x,y,0\right),{\sigma }_{33}=\frac{E}{1+v}\frac{\partial \text{Φ}\left(x,y,0\right)}{\partial z}.$

Thus, the boundary conditions of the contact problem can be expressed in terms of the harmonic function Φ: one knows the function within the contact region G, while its normal derivative is given on the rest of the half-space boundary:

$\begin{array}{c}\text{Φ}\left(x,y,0\right)=\frac{w\left(x,y\right)}{2\left(1-\nu \right)},\left(x,y\right)\in G;\\ \frac{\partial \text{Φ}\left(x,y,0\right)}{\partial z}=-\frac{\left(1+\nu \right)p\left(x,y\right)}{E},\left(x,y\right)\in {\mathbb{R}}^{2}G;\end{array}$

where w and p are normal displacements and pressure given within and outside the contact region G, respectively. Hence, the contact problem is a mixed boundary value problem for a harmonic function Φ.

If a normal pressure p(x₁, x₂) is applied only to the finite region G, then the harmonic function Φ can be written as the so-called single layer potential (see, e.g., Galin, 2008)

$\text{Φ}\left(x,y,z\right)=\frac{1}{2\pi }{\displaystyle \int {\displaystyle \underset{G}{\int }\frac{p\left(\xi ,\eta \right)\text{d}\xi \text{d}\eta }{\sqrt{{\left(x-\xi \right)}^2+{\left(y-\eta \right)}^2+z^2}}}},$

and one has

$u_3\left(x,y,0\right)=\frac{1}{\pi E^{*}}{\displaystyle \int {\displaystyle \underset{G}{\int }\frac{p\left(\xi ,\eta \right)\text{d}\xi \text{d}\eta }{\sqrt{{\left(x-\xi \right)}^2+{\left(y-\eta \right)}^2}}}},$

  (3.19)

where E* = E/(1 − ν²) is the reduced elastic modulus.

If one considers contact of two elastic solids, then

$u_3^{+}\left(x,y,0\right)=\frac{1-{\left({\nu }^{+}\right)}^2}{\pi E^{+}}{\displaystyle \int {\displaystyle \underset{G}{\int }\frac{p\left(\xi ,\eta \right)\text{d}\xi \text{d}\eta }{\sqrt{{\left(x-\xi \right)}^2+{\left(y-\eta \right)}^2}}}}$

and

$u_3^{-}\left(x,y,0\right)=\frac{1-{\left({\nu }^{-}\right)}^2}{\pi E^{-}}{\displaystyle \int {\displaystyle \underset{G}{\int }\frac{p\left(\xi ,\eta \right)\text{d}\xi \text{d}\eta }{\sqrt{{\left(x-\xi \right)}^2+{\left(y-\eta \right)}^2}}}}.$

Substituting the above expressions into Eq. (3.2), one obtains

$\delta -\left(f^{+}+f^{-}\right)=\frac{1}{\pi E^{*}}{\displaystyle \int {\displaystyle \underset{G}{\int }\frac{p\left(\xi ,\eta \right)\text{d}\xi \text{d}\eta }{\sqrt{{\left(x-\xi \right)}^2+{\left(y-\eta \right)}^2}}}},$

  (3.20)

where the reduced elastic modulus is

$\frac{1}{E^{*}}=\frac{1-{\left({\nu }^{+}\right)}^2}{E^{+}}+\frac{1-{\left({\nu }^{-}\right)}^2}{E^{-}}.$

  (3.21)

If the indenter is rigid, i.e., E⁻ = ∞, then E* = E/(1 − ν²), where E = E⁺ and ν = ν⁺ are the elastic modulus and the Poisson ratio of the half-space, respectively.

It follows from the above consideration that the frictionless Hertz-type contact problem reduces to the problem of determination of the pressure distribution satisfying Eq. (3.20).

2.1.5 Incompatibility of the Hertz-Type Problem Formulations

One needs to be aware that the Hertz formulation of the contact problems leads to incompatibility of displacement fields. There are two types of incompatibility of the contact problems: (1) penetration of the upper material layer into the lower one owing to the geometrically linear formulation of the problem; (2) penetration of the material into the punch owing to the disregarding of the tangential displacements in the formulation of the Hertz-type contact problems (see, e.g., Borodich, Galanov, & Suarez-Alvarez, 2014; Rvachev & Protsenko, 1977).

Considering the classic Boussinesq problem for a concentrated load and the Abramov (1937) problem for a two-dimensional (2D) nonslipping contact between a flat punch and an isotropic linear elastic half-space, Rvachev and Protsenko (1977) discussed both types of incompatibility. To avoid or at least to reduce the incompatibility, one needs to employ the geometrically nonlinear formulation of the contact problem that includes the Signorini–Fichera conditions of impenetrability of the material points (Fichera, 1972; Signorini, 1933) along with accounting for the boundary tangential displacements (Kindrachuk & Galanov, 2014; Kindrachuk, Galanov, Kartuzov, & Dub, 2009).

Contact problems with the conditions of impenetrability linearized with respect to boundary tangential displacements were studied by Galanov (1983) and Galanov, Krivonos (1984a). If one takes into account the conditions of impenetrability

$u_z\left(r,0,P\right)-\delta +f\left[r+u_r\left(r,0,\mathcal{P}\right)\right]\ge 0,$

then instead of Eq. (3.12) in the first condition Eq. (3.11), the following one has to be written:

$u_z\left(r,0,P\right)-\delta +f\left[(r+u_r\left(r,0,\mathcal{P}\right)\right]=0.$

  (3.22)

The above equation is normally nonlinear; hence, the condition within the contact region can be linearized with respect to u_r and can be written as

$u_z\left(r,0,P\right)-\delta +f\left(r\right)+L\left(r\right)u_r\left(r,0,P\right)=0,$

  (3.23)

where L(r) is obtained by linearization of f with respect to u_r.

As an attempt to reduce the degree of this incompatibility, Galanov (1983) considered a refined formulation of the Hertz-type contact problem when lateral displacements are taken into account. It was shown that the use of this more rigorous formulation than the Hertzian one substantially reduces the degree of the displacement incompatibility observed at the contact region and under the region. The contact problem with the nonlinear boundary condition (3.22) was studied by Galanov and Krivonos (1984b). A formulation of unilateral contact between two deformable continuous solids undergoing large displacements with friction at the interface was presented by Curnier et al. (1995).

Using a known expression for the potential of an ellipsoid, Hertz (1882a) solved the isotropic linear elastic contact problem (3.20) for an elliptic paraboloid, and showed that the contact region is an ellipse with semiaxes a and b. He wrote condition (3.5) as

$\delta -A{x}^2-B{y}^2=\frac{3P}{4\pi E^{*}}{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{1-\frac{x^2}{a^2+k}-\frac{y^2}{b^2+k}}{\sqrt{k\left(a^2+k\right)\left(b^2+k\right)}}\mathit{dk}},$

  (3.24)

which can be split into three equation for δ, a, and b. Hertz solved the equations explicitly.

Boussinesq (1885) noted that to get a unique solution for the unknown contact region, the normal component of stress in the frictionless Hertz-type contact problems must be equal to zero at the border line of the contact region. This condition gives the depth of indentation δ of the convex punch; another value δ₀ of the depth of indentation would correspond to the superposition of the rigid displacement δ − δ₀ of the punch and to the corresponding singular stress field at the edge of the contact region. As Maugis and Barquins (1983) commented for this condition, such a superposition was used later by Johnson (1985) and Johnson et al. (1971) to study adhesive contact of spheres.

After careful examination of the Hertz contact problem, it was shown by Dinnik (1952) in 1908 for a circular contact region and in 1917 by Belyaev (1924) for an elliptic contact region that the point of maximum shearing stresses and consequently the point of first yield in the Hertz contact theory are beneath the contact surface. A very useful analysis of the Hertz problem was presented by Sackfield and Hills (1983).

We will not present the Hertz solution here. It is important for us that the following power law expressions for the a–P and δ–P relations can be derived:

$a=C_1{P}^{1/3},\delta =C_2{P}^{2/3},$

  (3.25)

where C₁ and C₂ are constants that may be derived explicitly using the Hertz solution.

Shtaerman (1941) solved the contact problem for an indenter that was closer to the surface than the Hertzian punch, i.e., he considered not f = Ax² + By² as Hertz did but f = (Ax² + By²)². It follows from the Shtaerman solution that

$a=C_3{P}^{1/5},\delta =C_{4}P/a=C_5{P}^{4/5},$

  (3.26)

where C₃, C₄, and C₅ are constants that were derived explicitly by Shtaerman (1941).

2.2.2 Dimensional Analysis of Anisotropic Problems

Note that the Hertz equation (3.24) and the corresponding solutions are also valid not only for isotropic linear elastic materials but also for some anisotropic and nonlinear prestressed materials described by linearized models.

The mechanics of anisotropic elastic materials is a quite developed research field (see, e.g., Lekhnitskii, 1981; Ting, 1996). The contact problems for anisotropic solids were studied in many papers. However, as Willis (1966) noted, it is difficult to expect to obtain an analytical solution to the contact problems for generally anisotropic solids except for transversely isotropic materials. For arbitrary anisotropic, linear elastic media, the constitutive relations have the form of Hooke’s law:

${\sigma }_{\mathit{ij}}=c_{\mathit{ijkl}}{\epsilon }_{\mathit{kl}}\text{or}{\sigma }_{\mathit{ij}}=c_{\mathit{ijkl}}{u}_{k,l},c_{\mathit{ijkl}}=c_{\mathit{ijkl}}=c_{\mathit{klij}},$

  (3.27)

where c_ijkl are components of the tensor of elastic constants.

With use of dimensional analysis, it can easily be shown (Borodich, 1990e) that the displacement of any surface point of an anisotropic linear elastic half-space under the influence of a point load P is proportional to the ratio, P/r, where r is the distance to the point of the application of the load. Indeed, the central theorem of dimensional analysis or the Π theorem says (see, e.g., Barenblatt, 1996):

If a physical law is written as a relationship between some dimensional quantity and several dimensional governing parameters, then

– it is possible to rewrite this relationship as a relationship between some dimensionless parameter and several dimensionless products of the governing parameters;

– the number of dimensionless products is equal to the total number of the governing parameters minus the number of governing parameters with independent dimensions.

Let a concentrated force P be applied normally to the surface of the half-space at the point 0, i.e., it is directed along the z axis. If the material of the half-space is linear elastic anisotropic with Hooke’s law (3.27), then this is the Boussinesq problem for a concentrated load acting on an anisotropic elastic half-space.

Let us apply the Π theorem. The normal displacements (u₃) at points of the boundary are given by

$u_3={\text{Φ}}_u\left(P,r,ɸ,c_{\mathit{ijlm}}\right),$

where Φ_u is some functional relation. For the physical dimensions of the governed and the governing parameters, one can write

$\left[u_3\right]=L,\left[r\right]=L,\left[P\right]=F,\left[ɸ\right]=1,\left[c_{\mathit{ijlm}}\right]=F{L}^{-2},$

where L and F denote the physical dimensions of length and force, respectively. In the above case, the number of governing parameters (n) is 24 because there are 21 independent components of the tensor c_ijlm. The first two governing parameters have independent dimensions, and [u₃/r] = 1. Hence, the number of governing parameters with independent dimensions (k) is two, and n − k = 22, and we have the following dimensioniess parameters

$\text{Π}=\frac{u_3}{r},{\text{Π}}_1=\frac{P}{c_{1111}{r}^2},{\text{Π}}_2=ɸ,{\text{Π}}_p=\frac{c_{\mathit{ijlm}}}{c_{1111}},p=3,\dots ,22,$

and we have

$\text{Π}={\text{Φ}}_1\left({\text{Π}}_1,{\text{Π}}_2,{\text{Π}}_p\right).$

The problem is not axially symmetric because of anisotropy of the material. Taking into account that the problem is linear, one obtains

$\text{Π}={\text{Π}}_1{\text{Φ}}_2\left({\text{Π}}_2,{\text{Π}}_p\right)\text{or}{u}_3=\frac{P}{r{c}_{1111}}{\text{Φ}}_2\left(\phi ,\frac{c_{\mathit{ijlm}}}{c_{1111}}\right).$

  (3.28)

In particular, if the half-space is isotropic, then there are only two elastic constants [E] = FL⁻² and [ν] = 1. In addition, the problem is axially symmetric because of isotropy of the material. Hence, Φ₂ is independent of Π₂ and one has the known result

$u_3=\frac{P}{\mathit{Er}}{\text{Φ}}_2\left(\nu \right).$

If we solved the problem exactly, then we could see that Φ₂(ν) = (1 − ν²)/(π). The detailed expression for the Green’s function for an anisotropic elastic half-space can be found in Vlassak and Nix (1994).

Using dimensional arguments similar to the ones above, let us consider the Boussinesq relation for a flat-ended cylindrical indenter of radius a contacting an arbitrary anisotropic elastic half-space. Assuming that the base is horizontal and the total load is P, one immediately obtains the following relation for the depth of indentation (see, e.g., Borodich, Galanov, Keer, & Suarez-Alvarez, 2014):

$\delta =\frac{P}{a{c}_{1111}}{\text{Φ}}_{\text{c}}\left(\frac{c_{\mathit{ijkl}}}{c_{1111}}\right),$

  (3.29)

where Φ_c is a function of the dimensionless elastic material constants.

Willis (1966, 1967) obtained much more detailed results. He studied both Hertzian and Boussinesq contact problems for anisotropic bodies and showed that the functional form of the pressure distribution between the contacting bodies can be found explicitly, and therefore the problem is reduced to finding the displacements due to a pressure distribution of this form. With use of these results, the JKR theory was extended to transversely isotropic elastic materials (Borodich, Galanov, Keer, & Suarez-Alvarez, 2014; Espinasse, Keer, Borodich, Yu, & Wang, 2010).

2.2.3 Solutions for Transversely Isotropic Materials

Transverse isotropy is a very important case of anisotropy because many natural and artificial materials behave effectively as transversely isotropic elastic solids. For example, modern tribological coatings or layered composite materials can often be described as having transversely isotropic properties. Various problems and approaches to the mechanics of transversely isotropic materials were discussed in Ding, Chen, and Zhang (2006). Numerous papers have been devoted to the study of various problems of mechanics for transversely isotropic materials (see, e.g., Kaplunov, Kossovich, & Rogerson, 2000; Kuo & Keer, 1992; Lekhnitskii, 1940), in particular to contact and indentation problems (see, e.g., Argatov, 2011; Borodich, 1989; Delafargue & Ulm, 2004). Referring to Sveklo (1964), Willis (1966) argued that in the case of transversely isotropic materials, the Hertz-type contact problem can be reduced to one of potential theory. Green and Zerna (1968) also showed the similarities between isotropic and transversely isotropic frictionless 2D indentation problems. A similar result was presented earlier by Lekhnitskii (1940, 1981), who computed the stresses in the interior of a transversely isotropic half-space loaded by a normal concentrated load. Independently of Willis, Conway, Farnham, and Ku (1967) presented an analytical expression for the solution for the frictionless Hertz-type contact between transversely isotropic solids using the results obtained by Lekhnitskii (1940, 1981).

To describe the general anisotropy of elastic solids, the tensor of elastic constants c_ijkl has been used; however, the matrix form is more convenient for the description of transversely isotropic solids. This is because the tensor of elastic constants is reduced to five elastic constants of the material: a₁₁, a₁₂, a₁₃, a₃₃, and a₄₄. Let the z axis be taken normal to a plane of isotropy, i.e., the z axis is the axis of rotational symmetry, then Hooke’s law (3.27) becomes

$\left(\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ {\epsilon }_{12}\\ {\epsilon }_{13}\\ {\epsilon }_{23}\end{array}\right)=\left(\begin{array}{cccccc}{a}_{11}& a_{12}& a_{13}& 0& 0& 0\\ a_{12}& a_{11}& a_{13}& 0& 0& 0\\ a_{13}& a_{13}& a_{33}& 0& 0& 0\\ 0& 0& 0& a_{11}-a_{12}& 0& \\ 0& 0& 0& 0& a_{44}/2& 0\\ 0& 0& 0& 0& 0& a_{44}/2\end{array}\right)\left(\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{12}\\ {\sigma }_{13}\\ {\sigma }_{23}\end{array}\right).$

  (3.30)

In cylindrical coordinates Hooke’s law for a transversely isotropic elastic material becomes

$\left(\begin{array}{c}{\epsilon }_{\mathit{rr}}\\ {\epsilon }_{ɸɸ}\\ {\epsilon }_{\mathit{zz}}\\ {\epsilon }_{rɸ}\\ {\epsilon }_{\mathit{rz}}\\ {\epsilon }_{ɸz}\end{array}\right)=\left(\begin{array}{cccccc}{a}_{11}& a_{12}& a_{13}& 0& 0& 0\\ a_{12}& a_{11}& a_{13}& 0& 0& 0\\ a_{13}& a_{13}& a_{33}& 0& 0& 0\\ 0& 0& 0& a_{11}-a_{12}& 0& \\ 0& 0& 0& 0& a_{44}/2& 0\\ 0& 0& 0& 0& 0& a_{44}/2\end{array}\right)\left(\begin{array}{c}{\sigma }_{\mathit{rr}}\\ {\sigma }_{ɸɸ}\\ {\sigma }_{\mathit{zz}}\\ {\sigma }_{rɸ}\\ {\sigma }_{\mathit{rz}}\\ {\sigma }_{ɸz}\end{array}\right)$

  (3.31)

or

$\left(\begin{array}{c}{\sigma }_{\mathit{rr}}\\ {\sigma }_{ɸɸ}\\ {\sigma }_{\mathit{zz}}\\ {\sigma }_{rɸ}\\ {\sigma }_{\mathit{rz}}\\ {\sigma }_{ɸz}\end{array}\right)=\left(\begin{array}{cccccc}{A}_{11}& A_{12}& A_{13}& 0& 0& 0\\ A_{12}& A_{11}& A_{13}& 0& 0& 0\\ A_{13}& A_{13}& A_{33}& 0& 0& 0\\ 0& 0& 0& A_{11}-A_{12}& 0& \\ 0& 0& 0& 0& 2A_{44}& 0\\ 0& 0& 0& 0& 0& 2A_{44}\end{array}\right)\left(\begin{array}{c}{\epsilon }_{\mathit{rr}}\\ {\epsilon }_{ɸɸ}\\ {\epsilon }_{\mathit{zz}}\\ {\epsilon }_{rɸ}\\ {\epsilon }_{\mathit{rz}}\\ {\epsilon }_{ɸz}\end{array}\right),$

  (3.32)

where A₁₁, A₁₂, A₁₃, A₃₃, and A₄₄ are five elastic constants of the material,

$A_{11}=\frac{a_{11}{a}_{33}-a_{13}^2}{\left(a_{11}-a_{12}\right)m},A_{12}=\frac{a_{13}^2-a_{12}{a}_{33}}{\left(a_{11}-a_{12}\right)m},A_{13}=\frac{-a_{13}}{m},$

  (3.33)

$A_{33}=\frac{a_{11}+a_{12}}{m},A_{44}=\frac{1}{a_{44}}{A}_{11}-A_{12}=\frac{1}{a_{11}-a_{12}},$

  (3.34)

and

$m=\left(a_{11}+a_{12}\right)a_{33}+2a_{13}^2.$

Using the above expressions and the results obtained by Lekhnitskii (1940, 1981), one can show that the solution of the Boussinesq problem for a concentrated load P is

$u_3\left(r,0\right)=\frac{P}{\pi E_{\text{TI}}r},$

  (3.35)

where the coefficient (E_TI)⁻¹ is

${\left(E_{\text{TI}}\right)}^{-1}=-\frac{S_1+S_2}{2D^{1/2}\left(\mathit{AC}-D\right)}\left[\left(D-2\mathit{BD}+\mathit{AC}\right)a_{11}-\left(2D-\mathit{BD}-\mathit{AC}\right)a_{12}\right]$

  (3.36)

and

$\begin{array}{c}A=\frac{a_{13}\left(a_{11}-a_{12}\right)}{a_{11}{a}_{33}-a_{13}^2},B=\frac{a_{13}\left(a_{13}+a_{44}\right)=a_{12}{a}_{33}}{a_{11}{a}_{33}-a_{13}^2},\\ C=\frac{a_{13}\left(a_{11}-a_{12}\right)=a_{11}{a}_{44}}{a_{11}{a}_{33}-a_{13}^2},D=\frac{a_{11}^2-a_{12}^2}{a_{11}{a}_{33}-a_{13}^2},\\ 2D{S}_{1,2}^2=A+C\pm {\left[{\left(A+C\right)}^2-4D\right]}^{1/2},\\ S_{1,2}=\sqrt{\frac{A+C}{2D}\pm {\left[{\left(\frac{A+C}{2D}\right)}^2-\frac{1}{D}\right]}^{1/2}}.\end{array}$

For isotropic materials E_TI is equal to E*.

Thus, one can write a general equation for the pressure distribution under an indenter acting on a linear elastic half-space having rotational symmetry of its elastic properties,

$u_3\left(x,y,0\right)=\frac{1}{\pi K^{*}}{\displaystyle \int {\displaystyle \underset{G}{\int }\frac{p\left(\xi ,\eta \right)\text{d}\xi \text{d}\eta }{\sqrt{{\left(x-\xi \right)}^2+{\left(y-\eta \right)}^2}}}},$

  (3.37)

that is the same up to a constant as the pressure distribution within an isotropic linear elastic half-space. It follows from the above consideration that one needs just to replace the reduced elastic modulus E* in Eq. (3.19) by K* = E_TI (do not confuse the modulus K* with the modulus K = (4/3)E* of an isotropic elastic material).

2.2.4 Solutions for Prestressed Materials

Nowadays indentation techniques are often applied to biological and/or prestressed nonlinear materials (see, e.g., Kendall, Kendall, & Rehfeldt, 2011; Scherge & Gorb, 2001; Sen, Subramanian, & Discher, 2005; Yang, 2004). Information concerning mechanical problems for homogeneously deformed nonlinear elastic solids can be found elsewhere. For example, Biot (1965), Eringen and Suhubi (1975), and Guz (1986a, 1986b) gave detailed descriptions of static and dynamic problems for prestressed elastic media. Usually various potential functions are used to describe the stress–strain relations in nonlinear materials. These include harmonic, Mooney-Rivlin, Treloar (neo-Hookean solid), Bartenev-Khazanovich, and other potentials (see, e.g., Babich, Guz, & Rudnitskii, 2004; Guz, 1986a). The equations of the theory of nonlinear elasticity may be written using a variety of stress and strain tensors, and linearized equations for bodies with initial stresses may be also written in a variety of forms. A comprehensive review of current state-of-the-art research in the area of contact problems for prestressed solids was presented by Babich et al. (2004). They noted that the overwhelming number of authors writing about contact problems for prestressed solids assume (1) first the elastic solid is prestressed and then comes into contact with a punch, (2) the stress field due to the contact is just a small perturbation of the large initial stresses, (3) the initial stress field can be considered as homogeneous, and (4) the linearized elastic contact problem has a unique solution.

Let σ_ij(x) be components in the Cartesian system of coordinates of the nonsymmetric stress tensor σ, linearly related to the components of the Kirchhoff stress tensor. The components of σ are related to the area units in the initial static stress state. The component σ_nl is the lth component of the stress vector σ_n acting on the positive side of the coordinate surface with the nth component of the normal, i.e., σ_n = σ_nli_l. The vector u is the vector of the small perturbations of the displacements and i_l is the unit basis vectors.

For homogeneously prestressed elastic solids, the linearized constitutive relations and the equations of equilibrium can be written as (Guz, 1986a, 1986b)

${\sigma }_{\mathit{ij}}={\omega }_{\mathit{ijkl}}^{*}\frac{\partial u_k}{\partial x_l},{\omega }_{\mathit{ijkl}}^{*}\frac{{\partial }^2{u}_k}{\partial x_i{x}_l}=0.$

Here ω*_ijkl are the components of the proportionality between the small perturbations of the stresses and deformations, which, in the general case, have lesser symmetry properties than the tensor C_ijkl of anisotropic linear elasticity: ω*_ijkl = ω*_lkji, ω*_ijkl ≠ ω*_ijlk, ω*_ijkl ≠ ω*_klij.

A 2D contact problem for a prestressed plane whose elastic properties are described by the Mooney potential was solved in a pioneering paper by Filippova (1973). Later she extended her results to 3D solids (Filippova, 1978) and considered contact problems for incompressible materials of the neo-Hookean type:

$W=\frac{1}{2}\mu \left({\lambda }_1^2-1+{\lambda }_2^2-1+{\lambda }_3^2-1\right),$

where W is the Treloar potential, μ is the Lamé constant, which is equal to the shear modulus when the deformations are small (the shear modulus μ = E/2(1 + ν)), and λ_i is the extension ratio in the x_i direction. Because the material is incompressible, one has

${\lambda }_1{\lambda }_2{\lambda }_3=1.$

  (3.38)

Filippova (1978) considered not only the case of homogeneously prestressed solids (λ₁ = λ₂), but also the case λ₁ ≠ λ₂. She noted also that in the case when the initial extensions are the same in both directions (λ₁ = λ₂ = λ_r = λ), the solution of the Boussinesq problem for a concentrated load P is

$u_3\left(r,0\right)=\frac{P}{4\pi \mu r}{N}_{\text{F}}\left(\lambda \right),$

where

$N_{\text{F}}\left(\lambda \right)=\frac{2{\lambda }^4\left(1+{\lambda }^3\right)}{{\lambda }^9+{\lambda }^6+3{\lambda }^3-1}.$

  (3.39)

Because for incompressible solids ν = 0.5, one has

$u_3\left(r,0\right)=\frac{P}{\pi E^{*}r}{N}_{\text{F}}\left(\lambda \right).$

Hence, the integral equation of an arbitrary contact problem for equally and uniformly prestressed solids differs from the integral equation of the corresponding classic contact problem only by a constant coefficient N(λ) = N_F(λ).

Practically simultaneously with the publication of the above-mentioned paper, Dhaliwal and Singh (1978) considered axisymmetric contact problems for uniformly prestressed neo-Hookean solids. Their solution was based on the paper by Sneddon (1965), who considered classic axisymmetric contact problems (see above). Dhaliwal and Singh (1978) noted that the contact problem solution has to be multiplied by a coefficient that depends on the initial tension (compression). They presented the following coefficient:

$N_{\text{DS}}=-2\frac{1-k^2}{{\lambda }_z^2\left[{\left(1+k^2\right)}^2-4k\right]},$

  (3.40)

where k = λ_r/λ_z.

Let us show that the coefficient N_DS introduced by Dhaliwal and Singh (1978) and the coefficient N_F(λ) introduced by Filippova (1978) are the same. If the initial extensions are the same in the x₁ and x₂ directions, then it follows from the condition of incompressibility (Eq. 3.38) that k = λ³ and λ_z = λ⁻². Substituting these values into Eq. (3.40), one obtains

$N_{\text{DS}}\left(\lambda \right)=-2\frac{\left(1-{\lambda }^6\right){\lambda }^4}{{\left(1+{\lambda }^6\right)}^2-4{\lambda }^3}=-2\frac{\left(1-{\lambda }^6\right){\lambda }^4}{{\lambda }^{12}+2{\lambda }^6-4{\lambda }^3+1}$

or

$N_{\text{DS}}\left(\lambda \right)=-\frac{2{\lambda }^4\left(1-{\lambda }^3\right)\left(1+{\lambda }^3\right)}{-\left(1-{\lambda }^3\right)\left({\lambda }^9+{\lambda }^6+3{\lambda }^3-1\right)}=N_{\text{F}}\left(\lambda \right).$

Babich and Guz (1984) showed that all Hertz-type contact problems for contact between a punch and a nonlinear elastic homogeneously prestressed half-space coincide with the mixed problem for the harmonic potential of the contact problem for an isotropic linear elastic half-space up to a constant multiplier. Babich et al. (2004) gave examples of the multipliers for other potentials of nonlinear materials, e.g., (1) for a nonlinear material with the harmonic potential

$N\left(\lambda \right)=N_{\text{H}}=\frac{{\lambda }^2}{2+\nu }{\left(\lambda -\frac{1+\nu }{2+\nu }\right)}^{-1},$

and (2) for a nonlinear material with the Bartenev-Khazanovich potential

$N\left(\lambda \right)=N_{\text{BK}}=2{\lambda }^{7/2}{\left(3{\lambda }^3-1\right)}^{-1}.$

The effective contact modulus E_PS for a nonlinear elastic homogeneously prestressed half-space is E_PS = E*/N(λ), where N(λ) depends on the initial deformations λ within the xy plane and the nonlinear strain potential of the material.

Thus, Eq. (3.37) is still valid for homogeneously prestressed solids if one takes K* = E_PS. As noted by Borodich (1990a), if one considers contact between a transversely isotropic indenter and a prestressed half-space, then Eq. (3.37) is still valid; however (K*)⁻¹ = (E_TI)⁻¹ + (E_PS)⁻¹.

I would like to repeat that the materials of all the above-considered frictionless Hertz-type contact problems that are reduced to the mixed problem for the harmonic potential with an appropriate effective modulus K* have rotational symmetry of their elastic properties. I will use the effective modulus K* in all frictionless problems of this type without further explanations that the results obtained are applicable to all these materials having rotational symmetry of the properties.

2.2.5 Axisymmetric Frictionless Contact

In the problem of vertical indentation of a medium with rotational symmetry of the elastic properties by an axisymmetric punch, the contact region is always a circle. This fact simplifies analysis of the problem.

Boussinesq (1885) solved the problem of contact between a flat-ended circular punch and an elastic half-space. The Boussinesq relation for a flat-ended cylindrical indenter of radius a is

$P=\frac{2E}{1-v^2}a\delta \equiv 2K^{*}a\delta .$

  (3.41)

In 1939 two very important results were presented. Love (1939) considered the Boussinesq problem for a rigid cone f(r) = B₁r of the included semi-vertical angle α. The formulation of the Love problem was similar to the formulation of the frictionless Hertz-type contact problem. It was assumed that the cone over the part between the vertex and a certain circular section of radius a is in contact with the elastic half-space. Then the pressed region is given by r ≤ a at z = 0, and the vertical displacement on the pressed region is given by u₃ = δ − r cot α. Love (1939) used the same linearized formulation of the boundary value problem as Hertz and all boundary conditions were formulated for the z = 0 plane. Using this formulation, he obtained

$P=\frac{\pi }{2}{E}^{*}{B}_1{a}^2,\delta =\frac{\pi }{2}{B}_{1}a.$

  (3.42)

Hence, one has

$a=A{P}^{1/2},\delta =B{P}^{1/2},$

  (3.43)

where A and B are constants that may be derived explicitly using the Love solution (Eq. 3.42).

Although Sneddon was accurate in his citation of early papers of his predecessors, researchers in the materials science community attribute to him many results obtained by other researchers. In particular, it is quite often stated that Sneddon (1948) solved the contact problem for a cone and derived formula (Eq. 3.42).

Shtaerman (1939) presented a solution to the problem of an elastic contact for an axisymmetric solid whose shape is given by a power-law function (a monomial) of even degree

$f\left(r\right)=B_{2n}{r}^{2n},B_{2n}=\text{const},$

  (3.44)

where B_2n is a constant of the shape and the integer degree is given by n ≥ 1.

Thus, many explicit solutions to the frictionless problem for isotropic, linear elastic solids were obtained in the case when the distance between contacting bodies is a homogeneous function. These are solutions obtained by Hertz (1882a) for d = 2, Love (1939) for d = 1, Shtaerman (1939) for d = 2n, Galin (1946) for d = m/n (where m and n are natural numbers), and others. Willis (1966) studied in detail problems for anisotropic, linear elastic solids in the case d = 2.

Let us consider next the axisymmetric Hertz-type contact problems with nonslipping boundary conditions. If the external parameter of the problem $\mathcal{P}$ is gradually increased, then the surface displacements $u_r\left(r,0,\mathcal{P}\right)$ and $u_z\left(r,0,\mathcal{P}\right)$ will be functions of both r and the parameter of the problem $\mathcal{P}$. Once the point of the surface contacts with the indenter, its radial displacement does not change further with $\mathcal{P}$. Hence, instead of the conditions (Eq. 3.13), one can write the nonslipping condition (Eq. 3.14) within the contact region, which in the axisymmetric case can be written as

$\frac{\partial u_r}{\partial \mathcal{P}}\left(r,0,\mathcal{P}\right)=0.$

  (3.62)

Hence, the radial displacements within the contact region do not change with augmentation of the external parameter of the problem. In this formulation, the normal and radial displacements are consistent with the punch shape, and therefore it follows from Eq. (3.12) that g(r) = δ − f(r) and the radial displacements $u_r\left(r,0,\mathcal{P}\right)$ cannot be arbitrary.

The analysis of the nonslipping contact problems was performed first incrementally (Goodman, 1962; Mossakovskii, 1954, 1963) for a growth in the contact radius a. Mossakovskii noted self-similarity of the problem for punches described by monomial functions (Eq. 3.53). However, only Spence (1968) pointed out that the solution can be obtained directly without application of the incremental techniques (Gladwell, 1980; Johnson, 1985). Self-similarity of a general 3D frictional Hertz-type contact problem was shown later by Borodich (1993a).

Mossakovskii (1954, 1963) considered only two particular examples of nonslipping contact problems; namely, the problems for a flat-ended cylinder and a parabolic punch. Spence (1968) introduced an alternative method for solution of the problems, corrected some misprints in the Mossakovskii examples, and presented also the solution to the problem for a conical punch.

Following Mossakovskii and Spence, let us take the contact radius a as the external parameter of the problem $\mathcal{P}$.

2.4.2 The Mossakovskii Solution for Nonslipping Contact

In 1954 Mossakovskii presented the solution to a mixed boundary value problem for an elastic half-space when the line separating the boundary conditions is a circle. As an example, he gave a solution for a flat-ended circular punch of radius a under the condition of nonslipping contact. In this case, the compressing normal stresses σ_zz under a flat-ended punch of radius a is

${\sigma }_{\mathit{zz}}^0\left(r,0,a\right)=C_{\text{M}}{\delta }_0\frac{1}{r}\frac{\text{d}}{\text{d}r}{\displaystyle \underset{0}{\overset{r}{\int }}\text{sin}\left(\beta \text{ln}\frac{a-x}{a+x}\right)\frac{x}{\sqrt{r^2-x^2}}\text{d}x}.$

  (3.63)

Here δ₀ is the depth of indentation of the punch and

$\beta =\frac{1}{2\pi }\text{ln}\left(3-4\nu \right),C_{\text{M}}=\frac{8\mu \left(1-\nu \right)}{\pi \left(1-2\nu \right)\sqrt{3-4\nu }}.$

The correctness of formula (Eq. 3.63) was later checked by Keer (1967) and Spence (1968). Speaking about the further calculations of the compressing stress by Mossakovskii, Spence (1968) remarked that a factor of 2 was omitted throughout his paper of 1963, beginning with his equation (2.16). Indeed, Mossakovskii’s papers have various misprints; for example, Mossakovskii’s expression for the contact force for the Mossakovskii–Boussinesq problem obtained by integration of the pressure (Eq. 3.63) over the contact region was not presented in the correct form. The corrected Mossakovskii formula is (Khadem & Keer, 1974; Spence, 1968)

$P=4\mu {\delta }_{0}a\frac{\ln \left(3-4\nu \right)}{1-2\nu }.$