Finite Element Modeling of Tissues
Porous-permeable tissues have often been modeled using porous media theories such as the biphasic theory. Finite element contact analyses of porous deformable media under large deformations are not generally available in existing commercial codes. Therefore we performed a study to examine the equivalence of the short-time biphasic and incompressible elastic responses for arbitrary deformations and constitutive relations from first principles. This equivalence was illustrated in problems of unconfined compression of a disk, and of articular contact under finite deformation, using two different constitutive relations for the solid matrix of cartilage, one of which accounts for the large disparity observed between the tensile and compressive moduli in this tissue. Demonstrating this equivalence under general conditions provided a rationale for using available finite element codes for incompressible elastic materials as a practical substitute for biphasic analyses, so long as only the short-time biphasic response is sought.
Since existing commercial codes do not provide a consistent implementation of finite element contact for porous deformable media, we subsequently formulated and implemented a finite element contact algorithm for solid-fluid (biphasic) mixtures, accommodating both finite deformation and sliding. The finite element source code is currently available to the public. The algorithm uses a penalty method regularized with an augmented Lagrangian method to enforce the continuity of contact traction and normal component of fluid flux across the contact interface. The formulation addressed the need to automatically enforce free-draining conditions outside of the contact interface. The accuracy of the implementation was verified using contact problems for which exact solutions were obtained by alternative analyses. Illustrations were also provided that demonstrate large deformations and sliding under configurations relevant to biomechanical applications such as articular contact. This study addressed an important computational need in the biomechanics of porous-permeable soft tissues. Placing the source code in the public domain will provide a useful resource to the biomechanics community.