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Technical Briefs

An Augmented Lagrangian Method for Sliding Contact of Soft Tissue

[+] Author and Article Information
Hongqiang Guo1

 Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180muerghq@gmail.com

Jeffrey C. Nickel, Laura R. Iwasaki, Robert L. Spilker

Departments of Orthodontics and Dentofacial Orthopedics and Oral Biology, School of Dentistry,  University of Missouri-Kansas City, Kansas City, MO 64108Department of Biomedical Engineering,  Rensselaer Polytechnic Institute, Troy, NY 12180

1

Corresponding author.

J Biomech Eng 134(8), 084503 (Aug 06, 2012) (6 pages) doi:10.1115/1.4007177 History: Received January 09, 2012; Revised June 11, 2012; Posted July 18, 2012; Published August 06, 2012; Online August 06, 2012

Despite the importance of sliding contact in diarthrodial joints, only a limited number of studies have addressed this type of problem, with the result that the mechanical behavior of articular cartilage in daily life remains poorly understood. In this paper, a finite element formulation is developed for the sliding contact of biphasic soft tissues. The augmented Lagrangian method is used to enforce the continuity of contact traction and fluid pressure across the contact interface. The resulting method is implemented in the commercial software COMSOL Multiphysics. The accuracy of the new implementation is verified using an example problem of sliding contact between a rigid, impermeable indenter and a cartilage layer for which analytical solutions have been obtained. The new implementation’s capability to handle a complex loading regime is verified by modeling plowing tests of the temporomandibular joint (TMJ) disc.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figure 2

Experimentally measured indenter motion in the horizontal direction versus time. The loading regime consisted of stage I: quasi-static preload ramped over 0≤t≤0.887s to 7.6 N, and stage II: dynamic horizontal motion. In stage II, the preload was held constant and a displacement control protocol of sinusoidal motion was specified in two sub-phases. In stage IIa, the indenter moved 2.9 mm to the right, and back to the starting position over a time period of 0.093 s. The indenter continued moving in stage IIb with a sinusoidal magnitude of 4 mm and a period of 0.35 s.

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Figure 3

Deformation of the cartilage layer construct and associated fluid pressure distribution at different values of the Peclet number (Pe) during sliding of a rigid impermeable indenter on the cartilage. The direction of movement by the indenter is to the right. For (a), (b), and (c): The color key for pressure (kPa) is shown above the cartilage layer construct. The black outline of the construct illustrates the original geometry. Arrows indicate vectors of fluid flow.

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Figure 4

Steady-state response of the total normal stress σyT and the fluid pressure p, along the top surface of the cartilage layer construct at (a) Pe = 0.01, (b) Pe = 1, and (c) Pe = 100 for the sliding of a rigid impermeable indenter on the cartilage. Here, x = 0 corresponds to the geometric midpoint of the contact area. The contact area moves toward the positive direction of the x axis. The lines are the results of the finite element solution using the augmented Lagrangian method and the symbols are the results of the semianalytical solution [6].

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Figure 5

Model predictions using the true sliding contact solution and the moving mesh solution [1], respectively, and the experimental results for the total normal stress at pressure gauges (a) PG4, (b) PG5, and (c) PG6 through one full cycle of stage IIb. The sign of the total normal stress was reversed for display purposes. PG5 is located in the middle of the TMJ disc, PG4 is 3 mm to the left of PG5, and PG6 is 3 mm to the right of PG5.

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Figure 6

Distributions of the total normal stress (a) σyT, and fluid pressure (b) p, respectively, at t = 0.887 s (end of stage I) on the geometry of the TMJ disc construct deformed by a loaded indenter (not shown) centered on the disc’s top surface, similar to the setup in Fig. 1. Arrows indicate the extent of the normal stress in the upper (solid line) and lower (dotted line) portions of the construct. The black outline of the construct illustrates the original geometry.

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Figure 7

Distributions of the total normal stress (in kPa) σyT on the deformed geometry of the TMJ disc at times (a) t = 0.9335 s (indenter full right in stage IIa), (b) t = 1.0675 s (indenter full left in stage IIb), (c) t = 1.2425 s (indenter full right in stage IIb), and (d) t = 1.33 s (indenter back to starting position). The black outline of the construct illustrates the original geometry.

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Figure 8

Distributions of the normal strain εy on the deformed geometry of the TMJ disc at times (a) t = 0.887 s (end of stage I), (b) t = 0.9335 s (indenter full right in stage IIa), (c) t = 1.0675 s (indenter full left in stage IIb), (d) t = 1.2425 s (indenter full right in stage IIb), and (e) t = 1.33 s (indenter back to starting position). The black outline of the construct illustrates the original geometry.

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Figure 1

A schematic diagram of the sliding contact with rigid impermeable indenter. Only the tip of the indenter with radius Rind is illustrated. This tip contacts the cartilage layer construct with height h and width w, which are not shown to modeled scale. Below, the cartilage layer is attached to (subchondral) bone.

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