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TECHNICAL PAPERS: Soft Tissue

# A Penetration-Based Finite Element Method for Hyperelastic 3D Biphasic Tissues in Contact: Part 1-Derivation of Contact Boundary Conditions

[+] Author and Article Information
Kerem Ün, Robert L. Spilker

Department of Biomedical Engineering and Scientific Computation Research Center,  Rensselaer Polytechnic Institute, Troy, NY 12180-3590

In the undeformed configuration of penetrating tissues, a vector that is normal to tissue $A$ is, in general, not exactly normal to tissue $B$. However, for physiological geometries the error made through this assumption is negligibly small.

J Biomech Eng 128(1), 124-130 (Sep 22, 2005) (7 pages) doi:10.1115/1.2133769 History: Received November 16, 2004; Revised September 22, 2005

## Abstract

In this study, we extend the penetration method, previously introduced to simulate contact of linear hydrated tissues in an efficient manner with the finite element method, to problems of nonlinear biphasic tissues in contact. This paper presents the derivation of contact boundary conditions for a biphasic tissue with hyperelastic solid phase using experimental kinematics data. Validation of the method for calculating these boundary conditions is demonstrated using a canonical biphasic contact problem. The method is then demonstrated on a shoulder joint model with contacting humerus and glenoid tissues. In both the canonical and shoulder examples, the resulting boundary conditions are found to satisfy the kinetic continuity requirements of biphasic contact. These boundary conditions represent input to a three-dimensional nonlinear biphasic finite element analysis; details of that finite element analysis will be presented in a manuscript to follow.

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## Figures

Figure 1

Graphical depiction of the idea behind penetration method

Figure 2

Picture of overlapping cartilage models as obtained from the solid modeler (top). Definitions of geometric parameters in overlapped models (bottom—illustrated in 2D, although analysis is done in 3D).

Figure 3

The geometry of the shoulder joint with glenoid and humeral cartilage layers

Figure 4

Canonical contact geometry and the geometric parameters for configurations CT (constant thickness) and VT (varying thickness) used in the validation studies

Figure 5

Configuration CT: pressure distribution on the contacting cartilage pairs as calculated through nonlinear penetration method displayed as function of radial distance from the axis

Figure 6

Configuration VT: pressure distribution on the contacting cartilage pairs, as calculated through nonlinear penetration method, displayed as a function of radial distance from the axis

Figure 7

Comparison of the magnitude of the total traction (upper) and pressure (lower) on the contact faces of the humeral head (left) and glenoid (right) cartilages at an arm elevation angle of 40deg as calculated by the nonlinear penetration procedure

Figure 8

Comparison of the magnitude of the total traction (upper) and pressure (lower) on the contact faces of the humeral head (left) and glenoid (right) cartilages at an arm elevation angle of 60deg as calculated by the nonlinear penetration procedure

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