A constitutive law for trabecular bone
Motivated by applications in orthopaedic surgery, new constitutive laws for trabecular (or spongious) bone are developed in the framework of continuum mechanics, implemented in a mechanical analysis computer program, validated by a number of in vitro experiments and illustrated by the simulation of a femoral total hip component. Current knowledge about the morphological and mechanical properties of trabecular bone is reviewed for setting the background and clarifying the contributions of the thesis. Comprehensive 1D and 3D theoretical models based on the approach of standard generalized materials are developed with a specific attention towards irreversible phenomena. The 1D model includes linear elasticity and rate-independent as well as rate-dependent plastic strain flow with damage. Based on a second order fabric tensor, the 3D model includes inhomogeneous, orthotropic linear elasticity and rate-independent plasticity with damage. In order to solve boundary value problems involving complex bone or bone-implant structures, implicit projection algorithms are developed for integrating the plastic flow rules with damage and implemented in the computer program TACT combining the finite element method, the linear iteration method and the finite difference method. The resulting numerical models are illustrated by the means of traction, bending and torsion benchmark tests. A number of pilot in vitro experiments are undertaken on human and bovine trabecular bone specimens in order to validate the theoretical models and identify the material constants. Quasistatic uniaxial and torsion experiments are performed with a method avoiding artefacts due to the inhomogeneous boundary conditions associated with porosity. Anisotropic elasticity, plasticity and damage of trabecular bone prove to be successfully described by the models in terms of structural density and morphology. Finally, the 3D constitutive law is applied to the biomechanical problem of primary stability of a cementless femoral total hip component in order to illustrate its potential.