Non Linear Behavior of Articular Ligaments: Formulation with the Microplane Theory

E. Blangino, S. Valente, and M. Barba (Argentina)

Keywords

Constitutive equations; microplane theory; finite strain; hyperelasticity.

Abstract

To model the behavior of biological soft tissues, as ligaments and tendons, in the frame of the continuum mechanics, several hypothesis must be made in order to simplify the tensorial expressions and to facilitate the numerical treatment. Ligaments are composite. The mathematical model should include the characteristics of the aqueous matrix, the stretch of the reinforcing fibers and the interactions between them. In normal conditions, ligaments work in the elastic domain and exhibit different non-linear behavior in uniaxial tension in longitudinal and transverse directions. The hyperelastic formulation is preferred, assuming non-linearity and large strains. The matrix is considered as isotropic, homogeneous and non linear. The stretch of the reinforcing fibers can be partially represented by an exponential function. The microplane theory allows an (at most) vectorial formulation in “units” called microplanes for mecano continuum constitutive relations. This fact simplifies the analytical and numerical treatment. Macroscopical relations are obtained performing a suitable integration. This formulation favors, in a natural way, the incorporation of complex features like anisotropy. The aim of this work is to present constitutive relations for ligaments formulated within the microplane theory and to recover some known tensorial expressions. It will allow to establish more complete models for ligaments.

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