# The Applicability of the Hertzian Formulas to Point Contacts of Spheres and Spherical Caps

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hertzian Formulas for Circular Contacts

- the bodies are homogeneous, isotropic, and non-conforming;
- the bodies are elastic and the strains are small so that the theory of elasticity can be applied;
- the elastic bodies in contact are assumed to be semi-infinite elastic half-spaces; this means that contact dimensions and extent of deformation are far smaller than the geometrical dimensions of the contacting bodies;
- the profiles may be represented by a second order law;
- the surfaces are frictionless;
- adhesive forces are ignored;
- the surfaces have negligible roughness.
- Two additional timely assumptions may be included:
- there are no thin coatings on the surfaces;
- the surfaces are relatively hard.

_{1}, and Poisson’s ratio ν

_{1}pressed with a load F against a plane (E

_{2}, ν

_{2}), as reported in [26], will be now considered. In the ideally rigid case (not deformed bodies) the sphere can be well approximated with a paraboloid in the zone close to the contact point. Details of the development of the Hertzian formulas for the above case are given in Appendix A, while only the commonly used resulting ones are reported also in this section.

_{1}= R and R

_{2}= R + c.

## 3. Experimental Results for a Conformal Real Spherical Pair

_{a}< 1 μm). The cap was 20.3 mm high. The seat was 88.8 mm wide and 23.5 mm high, with a spherical 10.28 mm deep cup on top. The diametral clearance range between cap and seat was from 0.046 to 0.104 mm corresponding to relative clearances of 0.036% and 0.082%, respectively.

## 4. Numerical Simulation of Non-Conformal Contacts

^{−4}of the sphere radius. The total number of elements employed varied between simulations and was approximately 11,000 (3000 for conformal contacts). The plane was modelled by a fixed rigid cylinder, represented by a rectangle in the axisymmetric model. The spheres were linear elastic and the plane was infinitely stiff. All contacts were frictionless unless otherwise specified. Moreover, the contact penetration tolerance was set to 0.1 nm.

_{Hz}/E’) of 2.5 × 10

^{−3}, calculated theoretically using Equation (1) and Equation (4). This pressure corresponded to 1.1 GPa in steel, 0.6 GPa in titanium, and 0.4 GPa in aluminium. The displacement steps were 100, equally spaced.

## 5. Extension to Conformal Contacts

## 6. Conclusions

- The results regarding the relative displacement of non-conformal spherical bodies were in good agreement with the theoretical ones.
- Regarding the conformal spherical bodies, the radial clearance and the dimensions of both the cap and the seat showed a great influence on their non-Hertzian behavior. The differences in relative displacement increased with decreasing geometrical quantities. Moreover, friction, in this case, showed a non-negligible effect.
- For all the considered cases, Hertz’s formula always provided the greatest rigid body approach which means the lowest contact stiffness.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{Hz}is the maximum Hertzian pressure, and x is the distance from the center of the contact.

**Figure A1.**Deformations of a plane and a sphere pressed together. The displacements w of the two bodies are indicated inside and outside the contact area of radius a. From [26].

_{1}(x) and w

_{2}(x) are the displacements of the surface of each body in the direction normal to the plane tangent to the spheres at their theoretical contact point (considered positive in the direction of the inward-surface normal) and δ

_{1}e δ

_{2}are the macroscopic displacements of the bodies corresponding to w

_{1}(0) and w

_{2}(0), respectively. Using relation (A3) one obtains:

_{s}) and the plane (z

_{p}) inside and outside the contact area are, respectively:

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**Figure 1.**Deformed elastic sphere and elastic plane (colored figures), undeformed sphere (green line), and distance between the two deformed surfaces or equivalent deformed elastic sphere on rigid plane (black line). δ is the rigid body approach, a the radius of the circular contact area.

**Figure 2.**Tilting pad journal bearing with ball-and-socket pivots. From [18].

**Figure 3.**Experimental displacement vs. applied load compared with minimum and maximum clearance Hertzian displacement. From [18].

**Figure 4.**Dimensionless experimental and Hertzian displacements vs. dimensionless load for the minimum and maximum clearances.

**Figure 6.**Examples of contacts with imposed displacement or load. δ is the rigid body approach, F the applied load.

**Figure 7.**Deformation maps in the vertical direction for a non-conformal case (steel spheres, R

_{1}= 25 mm, R

_{2}= 125 mm, F = 195 N); (

**a**) whole cap, (

**b**) zoom of the contact area.

**Figure 8.**Comparison of simulation and theoretical values: (

**a**) radius of the contact area, (

**b**) maximum contact pressure, and (

**c**) applied force vs. displacement for a 25 mm radius steel sphere on rigid plane.

**Figure 9.**Schematic of cap and plane non-conformal contact. δ is the rigid body approach, h the height of the cap.

**Figure 10.**Schematic of conformal contact with imposed displacement or load. δ is the rigid body approach, F the applied load, h the height of the cap, R

_{1}the radius of the cap, b the minimum thickness of the seat.

**Figure 11.**Deformation maps in the vertical direction for a conformal case (steel sphere and cavity; R = 25 mm, C = 0.25%, F = 195 N).

**Figure 12.**Displacement vs. force for a spherical cap–seat contact for different relative clearances. Continuous lines are obtained by imposing a uniform downward displacement to the upper surface. Dashed lines are obtained by imposing a uniform compressing force to the upper surface.

**Figure 15.**Deformation maps in the vertical direction for a conformal case (steel sphere and cavity; R = 25 mm, C = 0.25%). (

**a**) h/R = 1, (

**b**) h/R = 1/8.

**Figure 16.**Force vs. imposed displacement for different seat boundary conditions, C = 0.25% and b/R = 0.125.

R_{1} [mm] | R_{2} [mm] | Material 1 | Material 2 | Mean Error % on F |
---|---|---|---|---|

5 | ∞ | Steel | Rigid | 0.22 |

25 | ∞ | Steel | Rigid | 0.48 |

25 | ∞ | Titanium | Rigid | 0.40 |

25 | ∞ | Aluminium | Rigid | 0.46 |

125 | ∞ | Steel | Rigid | 0.37 |

5 | 125 | Steel | Steel | 0.24 |

25 | 125 | Steel | Steel | 0.24 |

25 | 125 | Titanium | Steel | 0.29 |

25 | 125 | Aluminium | Steel | 0.26 |

125 | 125 | Steel | Steel | 0.25 |

Material | E [GPa] | ν |
---|---|---|

Steel | 200 | 0.3 |

Titanium | 96 | 0.36 |

Aluminium | 71 | 0.33 |

C [%] | R’ [mm] | h/R | b/R | Notes |
---|---|---|---|---|

0.05 | 50,025 | 1 | 1 | |

0.25 | 10,025 | 1 | 1 | |

1.25 | 2025 | 1 | 1 | |

0.25 | 10,025 | 0.5 | 1 | |

0.25 | 10,025 | 0.25 | 1 | |

0.25 | 10,025 | 0.125 | 1 | |

0.25 | 10,025 | 1 | 0.5 | |

0.25 | 10,025 | 1 | 0.25 | |

0.25 | 10,025 | 1 | 0.125 | |

0.25 | 10,025 | 0.125 | 1 | Fixed support |

0.05 | 50,025 | 1 | 1 | Friction = 0.8 |

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**MDPI and ACS Style**

Ciulli, E.; Betti, A.; Forte, P.
The Applicability of the Hertzian Formulas to Point Contacts of Spheres and Spherical Caps. *Lubricants* **2022**, *10*, 233.
https://doi.org/10.3390/lubricants10100233

**AMA Style**

Ciulli E, Betti A, Forte P.
The Applicability of the Hertzian Formulas to Point Contacts of Spheres and Spherical Caps. *Lubricants*. 2022; 10(10):233.
https://doi.org/10.3390/lubricants10100233

**Chicago/Turabian Style**

Ciulli, Enrico, Alberto Betti, and Paola Forte.
2022. "The Applicability of the Hertzian Formulas to Point Contacts of Spheres and Spherical Caps" *Lubricants* 10, no. 10: 233.
https://doi.org/10.3390/lubricants10100233