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TECHNICAL PAPERS: Soft Tissue

# A Penetration-Based Finite Element Method for Hyperelastic 3D Biphasic Tissues in Contact. Part II: Finite Element Simulations

[+] Author and Article Information
Kerem Ün

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

Robert L. Spilker1

Department of Biomedical Engineering and Scientific Computation Research Center,  Rensselaer Polytechnic Institute, Troy, NY 12180-3590spilker@rpi.edu

1

Corresponding author.

J Biomech Eng 128(6), 934-942 (May 10, 2006) (9 pages) doi:10.1115/1.2354203 History: Received November 17, 2005; Revised May 10, 2006

## Abstract

The penetration method allows for the efficient finite element simulation of contact between soft hydrated biphasic tissues in diarthrodial joints. Efficiency of the method is achieved by separating the intrinsically nonlinear contact problem into a pair of linked biphasic finite element analyses, in which an approximate, spatially and temporally varying contact traction is applied to each of the contacting tissues. In Part I of this study, we extended the penetration method to contact involving nonlinear biphasic tissue layers, and demonstrated how to derive the approximate contact traction boundary conditions. The traction derivation involves time and space dependent natural boundary conditions, and requires special numerical treatment. This paper (Part II) describes how we obtain an efficient nonlinear finite element procedure to solve for the biphasic response of the individual contacting layers. In particular, alternate linearization of the nonlinear weak form, as well as both velocity-pressure, $v‐p$, and displacement-pressure, $u‐p$, mixed formulations are considered. We conclude that the $u‐p$ approach, with linearization of both the material law and the deformation gradients, performs best for the problem at hand. The nonlinear biphasic contact solution will be demonstrated for the motion of the glenohumeral joint of the human shoulder joint.

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## Figures

Figure 1

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

Figure 2

Maximum tensile stress distribution on the contacting face of the humeral tissue layer at arm elevations of 40deg (top left), 50deg (top right), and 60deg (bottom). The maximum tensile stress distribution shifts superiorly, and the maximum value increases, with increasing arm elevation angle. Joint orientation with respect to anterior (A), posterior (P), superior (S), and inferior (I) directions is indicated.

Figure 3

Maximum tensile stress distribution on the bone interface of the humerus at arm elevations of 40deg (top left), 50deg (top right), and 60deg (bottom). The maximum tensile stress distribution shifts superiorly, and the maximum value increases, with increasing arm elevation angle. The presence of the glenoid is visible from the circular stress pattern. Joint orientation with respect to anterior (A), posterior (P), superior (S), and inferior (I) directions is indicated.

Figure 4

Maximum compressive stress distribution on the bone interface of the humerus at arm elevations of 40deg (top left), 50deg (top right), and 60deg (bottom). The maximum compressive stress distribution shifts superiorly, and the maximum value increases, with increasing arm elevation angle. The presence of the glenoid is visible from the circular stress pattern. Joint orientation with respect to anterior (A), posterior (P), superior (S), and inferior (I) directions is indicated.

Figure 5

Maximum shear stress distribution on the bone interface of the humerus at arm elevations of 40deg (top left), 50deg (top right), and 60deg (bottom). The maximum shear stress distribution shifts superiorly, and the maximum value increases, with increasing arm elevation angle. The presence of the glenoid is visible from the circular stress pattern. Joint orientation with respect to anterior (A), posterior (P), superior (S), and inferior (I) directions is indicated.

Figure 6

Maximum tensile stress distribution on the contacting face of the glenoid at arm elevations of 40deg (left) and 50deg (right). The change in the angle increases the maximum tensile stress, but there is no visible shift in the distribution. Joint orientation with respect to anterior (A), posterior (P), superior (S), and inferior (I) directions is indicated.

Figure 7

Maximum tensile stress distribution on the bone interface of the glenoid at arm elevations of 40deg (left) and 50deg (right). The change in the angle increases the maximum tensile stress, but there is no visible shift in the distribution. Joint orientation with respect to anterior (A), posterior (P), superior (S), and inferior (I) directions is indicated.

Figure 8

Maximum compressive stress distribution on the bone interface of the glenoid at arm elevations of 40deg (left) and 50deg (right). The change in the angle does not significantly affect either the peak values of the maximum compressive stress or its distribution. Joint orientation with respect to anterior (A), posterior (P), superior (S), and inferior (I) directions is indicated.

Figure 9

Maximum shear stress distribution on the bone interface of the glenoid at arm elevations of 40deg (left) and 50deg (right). The change in the angle slightly increases the maximum shear stress, but there is no visible shift in the distribution. Joint orientation with respect to anterior (A), posterior (P), superior (S), and inferior (I) directions is indicated.

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