# A Mixed Elasto-Hydrodynamic Lubrication Model for Wear Calculation in Artificial Hip Joints

^{*}

## Abstract

**:**

## 1. Introduction

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- The progressive wear phenomenon in the lubricating gap calculation through a modified Archard law;
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- The non-Newtonian synovial fluid behavior through the cross-viscosity model;
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- The cup surface deflection using a constrained column model.

## 2. Materials and Methods

#### 2.1. The Hip Joint during the Gait

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- The Flexion–Extension rotation (FE) around the Medio–Lateral axis (ML) perpendicular to the sagittal plane;
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- The Adduction–Abduction rotation (AA) around the Anterior–Posterior axis (AP) perpendicular to the frontal plane;
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- The Internal–External Rotation (IER) around the Proximo–Distal axis (PD) perpendicular to the horizontal plane.

^{®}software was chosen for the simulations allowing the calculation of the three hip angles evolution during the gait cycle and the angular velocities, which are directly involved in the lubrication analysis.

#### 2.2. The Modified Reynolds Lubrication Equation

#### 2.3. The Spherical Joint

#### 2.4. Discrete Reynolds Equation

^{®}environment.

## 3. Results and Discussion

- -
- Only the time evolution of the $y$ component of the dimensionless eccentricity $\mathit{n}$ from 0 to 1.1 over 0.1 s;
- -
- Only the time evolution of the $z$ component of the angular velocity $\mathbf{\omega}$ as a constant value equal to 500 rad/s, so that a faster relative sliding motion was considered.$$\begin{array}{cc}\mathit{n}=\left[\begin{array}{c}0\\ 1.1\frac{t}{0.1}\\ 0\end{array}\right]& \mathbf{\omega}=\left[\begin{array}{c}0\\ 0\\ 500\end{array}\right]\end{array}\frac{rad}{s}$$

## 4. Conclusions

^{®}during a normal gait cycle.

- -
- -
- -
- The simulation conducted in the framework of the radial clearance sensitivity analysis showed the expected tribological behavior in terms of classical EHL shapes.

- -
- Build a finite element model to calculate the deformation field of both acetabular cup and femoral head, in order to analyze the tribological behavior of other types of THRs in which both contact bodies must be considered deformable;
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- Consider the surfaces’ topography in the gap calculation in order to analyze a more realistic surface in the mixed lubrication mode and also to consider material transfer phenomena;
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- Improve the wear modeling in order to consider more specific wear modes, such as, for example, delamination effects.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

FE, AA, IER | Flexion–Extension, Adduction–Abduction, Internal–External Rotation |

ML, AP, PD | Medio–Lateral, Anterior–Posterior, Proximo–Distal hip axes |

$x$, $y$, $z$ | Cartesian axes |

$\mathit{N}$ | Load vector |

$\mathbf{\omega}$ | Angular velocity vector |

${\alpha}_{in}$, ${\beta}_{av}$ | Acetabular cup inclination angle and anteversion angle |

${\mathit{R}}_{{x}_{i}}\left(\theta \right)$ | Rotation matrix for a rotation $\theta $ around the ${x}_{i}$ axis |

${\mathit{R}}_{c}$ | Rotation matrix from the hip reference frame to the cup one |

$p$ | Fluid pressure |

$h$ | Fluid film thickness |

$\rho $ | Fluid density |

${\rho}_{0}$ | Nominal fluid density |

${a}_{\rho}$, ${b}_{\rho}$ | Dowson–Higginson coefficients |

$\mu $ | Fluid viscosity |

${\mu}_{eff}$ | Effective nominal viscosity |

${\alpha}_{\mu}$ | Barus exponential coefficient |

${\mu}_{0}$, ${\mu}_{\infty}$ | Cross viscosities in correspondence of $0$ and theoretically $\infty $ shear rate |

$k$, $n$ | Cross viscosity model parameters |

$\overline{\gamma}$ | Average shear rate |

${h}_{g}$ | Geometrical gap |

$\delta $ | Total surfaces’ deformation |

${\delta}_{f}$, ${\delta}_{c}$ | Deformation due to fluid pressure and contact deformation |

${u}_{w}$ | Linear wear |

${\mathsf{\Delta}}_{b}$ | Boundary layer thickness |

${p}_{c}$ | Contact pressure |

${p}_{t}$ | Total pressure |

${\tau}_{w}$ | Wear rate |

${k}_{w}$, ${k}_{{w}_{0}}$ | Wear factor function and nominal wear factor |

${\alpha}_{w}$ | Modified Archard model exponential coefficient |

${R}_{a}$ | Average roughness |

${V}_{w}$ | Wear volume |

$\theta $, $\phi $ | Spherical angles |

$t$ | Time |

$\widehat{\mathit{r}}$ | Radial unit vector |

$r$, $R$ | Femoral head and acetabular cup radii |

$c$ | Radial clearance |

$H$ | Acetabular cup thickness |

$E$, $\nu $ | Acetabular cup Young modulus and Poisson coefficient |

${k}_{d}$ | Constraint column model constant |

${p}_{0}$ | Boundary pressure |

$\mathit{e}$, $\mathit{n}$ | Eccentricity vector and its dimensionless form |

$\mathit{v}$, ${\mathit{v}}_{sl}$ | Entraining and sliding velocity vectors |

${\mathit{R}}_{s}$ | Spherical rotation matrix |

$i$, $j$, $n$ | Finite difference subscripts |

$\mathit{p}$ | Discretized pressure vector |

$\mathit{R}$, ${\mathit{J}}_{\mathit{R}}$ | Discretized Reynolds equation vector and its analytical Jacobian matrix |

${\lambda}_{r}$, ${u}_{r}$, ${\tau}_{r}$ | Relaxation factor and its parameters |

$res$, $tol$ | Iterative cycles residual and tolerance |

$\mathit{F}$, ${\mathit{J}}_{\mathit{F}}$ | Load difference function vector and its numerical Jacobian matrix |

${V}_{w,r}$ | Volumetric wear rate |

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**Figure 1.**Hip joint reference frames. FE: Flexion–Extension rotation; ML: Medio–Lateral axis; AA: Adduction–Abduction rotation; AP: Anterior–Posterior axis; IER: Internal–External Rotation; PD: Proximo–Distal axis.

**Figure 3.**(

**a**) Hip loads in the cup reference frame; (

**b**) hip angular velocities in the cup reference frame.

**Figure 11.**(

**a**) Pressure field in the full film lubrication phase; (

**b**) gap field in the full film lubrication phase.

**Figure 12.**(

**a**) Pressure and gap in correspondence of the pressure peak point in the full film lubrication phase along the θ direction; (

**b**) pressure and gap in correspondence of the pressure peak point in the full film lubrication phase along the φ direction.

**Figure 13.**(

**a**) Pressure field in the mixed lubrication phase;

**(b)**gap field in the mixed lubrication phase.

**Figure 14.**(

**a**) Pressure and gap in correspondence with the pressure peak point in the mixed lubrication phase along the θ direction;

**(b)**pressure and gap in correspondence with the pressure peak point in the mixed lubrication phase along the φ direction.

**Figure 16.**(

**a–d**) Fluid pressure transition from the full film lubrication phase to the mixed one; (

**e–h**) contact pressure transition from the full film lubrication phase to the mixed one.

**Figure 17.**(

**a–d**) Total pressure field evolution from 12% of the cycle to 98%; (

**e–h**) linear wear field evolution from 12% of the cycle to 98%.

**Figure 21.**(

**a**) Pressure, (

**b**) gap and (

**c**) linear wear profiles in the rotation plane in correspondence with different radial clearance values.

Parameter | Value |
---|---|

Acetabular cup inclination angle ${\alpha}_{in}$ | ${45}^{\xb0}$ [10] |

Acetabular cup anteversion angle ${\beta}_{av}$ | ${90}^{\xb0}$ |

Femoral head radius $r$ | $14\text{}mm$ [10] |

Radial clearance $c$ | $100\text{}\mu m$ [10] |

Acetabular cup thickness $H$ | $9.5\text{}mm$ [15] |

Acetabular cup Young modulus $E$ | $1\text{}GPa$ [10] |

Acetabular cup Poisson ratio $\nu $ | $0.4$ [10] |

Barus model exponential factor ${\alpha}_{\mu}$ | $19.8\xb7{10}^{-9}\text{}P{a}^{-1}$ [31] |

Cross model upper limit viscosity ${\mu}_{0}$ | $40\text{}Pa\text{}s$ [32] |

Cross model lower limit viscosity ${\mu}_{\infty}$ | $9\text{}mPa\text{}s$ [32] |

Cross model parameter $k$ | $9.54$ [32] |

Cross model parameter $n$ | $0.73$ [32] |

Synovial fluid nominal density ${\rho}_{0}$ | $850\text{}kg/{m}^{3}$ |

Density model parameter ${a}_{\rho}$ | $5.9\xb7{10}^{8}\text{}Pa$ [31] |

Density model parameter ${b}_{\rho}$ | $1.34$ [31] |

Boundary pressure ${p}_{0}$ | $0\text{}Pa$ |

Boundary layer thickness ${\mathsf{\Delta}}_{b}$ | $30\text{}nm$ [19,35] |

Roughness ${R}_{a}$ | $0.1\text{}\mu m$ [10] |

Nominal wear factor ${k}_{w0}$ | ${10}^{-7}\text{}m{m}^{3}/N/m$ [10] |

Archard model exponential factor ${\alpha}_{w}$ | $2.24$ [34] |

Spherical domain mesh density | $100\times 100$ |

Time domain steps number | $70$ |

Pressure tolerance $to{l}_{p}$ | ${10}^{-5}$ |

Load tolerance $to{l}_{N}$ | ${10}^{-2}$ |

Relaxation parameter ${u}_{r}$ | $0.2$ |

Relaxation parameter ${\tau}_{r}$ | $0.2$ |

Increment for the numerical Jacobian $\epsilon $ | ${10}^{-4}$ |

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**MDPI and ACS Style**

Ruggiero, A.; Sicilia, A.
A Mixed Elasto-Hydrodynamic Lubrication Model for Wear Calculation in Artificial Hip Joints. *Lubricants* **2020**, *8*, 72.
https://doi.org/10.3390/lubricants8070072

**AMA Style**

Ruggiero A, Sicilia A.
A Mixed Elasto-Hydrodynamic Lubrication Model for Wear Calculation in Artificial Hip Joints. *Lubricants*. 2020; 8(7):72.
https://doi.org/10.3390/lubricants8070072

**Chicago/Turabian Style**

Ruggiero, Alessandro, and Alessandro Sicilia.
2020. "A Mixed Elasto-Hydrodynamic Lubrication Model for Wear Calculation in Artificial Hip Joints" *Lubricants* 8, no. 7: 72.
https://doi.org/10.3390/lubricants8070072