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Research Papers

The Phan-Thien and Tanner Model Applied to the Lubrication of Knee Prostheses

[+] Author and Article Information
Brenda A. Weiss

Facultad de Ingeniería,
Universidad Nacional de Entre Ríos,
CONICET, GBC-FI,
Ruta Prov. 11, km 10,
Oro Verde C.P. 3100, Argentina
e-mail: bweiss@ingenieria.uner.edu.ar

Benyebka Bou-Saïd

Fellow ASME
Université de Lyon,
CNRS INSA-Lyon,
LaMCoS, UMR5259,
Lyon F-69621, France
e-mail: benyebka.bou-said@insa-lyon.fr

Sebastián Ubal

Grupo Biomecánica Computacional,
Facultad de Ingeniería,
Universidad Nacional de Entre Ríos,
Oro Verde 3100, Argentina and
IBB,
Universidad Nacional de Entre Ríos,
CONICET,
Ruta Prov. 11, km 10,
Oro Verde C.P. 3100, Argentina
e-mail: subal@ingenieria.uner.edu.ar

José Di Paolo

Grupo Biomecánica Computacional,
Facultad de Ingeniería,
Universidad Nacional de Entre Ríos,
Ruta Prov. 11 km 10,
Oro Verde C.P. 3100, Argentina
e-mail: jdipaolo@ingenieria.uner.edu.ar

Manuscript received October 29, 2018; final manuscript received February 14, 2019; published online May 6, 2019. Assoc. Editor: Guy M. Genin.

J Biomech Eng 141(8), 081008 (May 06, 2019) (10 pages) Paper No: BIO-18-1469; doi: 10.1115/1.4043032 History: Received October 29, 2018; Revised February 14, 2019

This work aims to provide a contribution to determine a proper model for the study of fluid film lubrication for the reduction of knee prostheses failure due to polyethylene wear. The Phan-Thien and Tanner (PTT) rheological law and the elastic deformation of the articular surfaces were considered in this modeling. The governing equations were solved numerically for different geometries and different Weissenberg numbers. The lubrication approximation applied to the PTT rheological law leads to an expression for the apparent viscosity similar to the Cross model. The results attest the importance of considering the non-Newtonian behavior of the synovial fluid, the elastic deformation, and the geometrical features of the prostheses to obtain quantitative information.

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References

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Figures

Grahic Jump Location
Fig. 1

(a) Ellipsoid on plane model, coordinate system, components of the velocity vector u,v,w and considered domain (area within dotted lines), see Tables 2 and 3. (b) x-z plane view. (c) x-y view, considered domain (gray) and boundary conditions where y=0 is a symmetry line.

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Fig. 2

Dimensionless apparent viscosity η̃ versus Weγ˙x̃ (γ˙x̃ is a dimensionless rate of deformation and We is the Weissenberg number), at the symmetry line (y=0, see Fig. 1(c))

Grahic Jump Location
Fig. 3

Dimensionless load versus dimensionless time for plane on plane case (PoP, actually Rx,Ry≫L). The results of Yousfi et al. are shown in black symbols. The arrow indicates increasing Weissenberg number (We).

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Fig. 4

Dimensionless load versus dimensionless time. Results for rigid plane on plane case (PoP, Rx,Ry≫L) and EoPR. The arrow indicates increasing Weissenberg number (We).

Grahic Jump Location
Fig. 5

Relative load (W/WN) versus dimensionless time. Results for plane on plane (PoP, Rx,Ry≫L) and EoPR. The arrow indicates increasing Weissenberg number (We).

Grahic Jump Location
Fig. 6

Dimensionless total stress distribution (πzz̃=πzzh02/η0W2L) along the symmetry line (y=0, see Fig. 1(c)) for both plane on plane (PoP, Rx,Ry≫L) and EoPR, at different dimensionless instants of time (t̃, legend) and Weissenberg number We=10−4

Grahic Jump Location
Fig. 7

Dimensionless load versus dimensionless time. Results for EoPR and EoPE cases. The arrow indicates increasing Weissenberg number (We).

Grahic Jump Location
Fig. 8

Minimum dimensionless film thickness (hm̃=hm/h0) versus dimensionless load-carrying capacity (W̃). Results for EoPE case. The arrow indicates increasing Weissenberg number (We).

Grahic Jump Location
Fig. 9

Relative load (W/WN) versus dimensionless time. Results for EoPE case. The arrow indicates increasing Weissenberg number (We).

Grahic Jump Location
Fig. 10

Relative friction coefficient (fy/fyN) versus dimensionless time. Results for EoPR and EoPE cases. The arrows indicate increasing Weissenberg number (We).

Tables

Errata

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