Elastic plastic damage laws for cortical bone

Motivated by applications in orthopaedic and maxillo-facial surgery, the mechanical behaviour of cortical bone tissue in cyclic overloads at physiological strain rates is investigated. The emphasis is on the development of appropriate constitutive laws that faithfully reproduce the loading, unloading, and reloading sequence observed during experimental in vitro uniaxial testing. To this end, the models include three distinct modes of evolution, namely a linear elastic mode due to bone cohesion, a damage mode where microcracks are generated and a plastic mode corresponding to sliding at the microcracks. The proposed models use the internal state variable approach common in continuum damage mechanics and allow a straightforward interpretation of the constitutive behaviour of cortical bone. They are derived within the generalized standard materials formalism and are thus thermodynamically consistent. The mathematical formulation of the models is based on the definition of two internal state variables: a damage variable that represents the microcrack density reducing the tissue stiffness, and a plastic strain variable representing the deformation associated with these microcracks. Firstly, two one-dimensional models describing the uniaxial quasistatic behaviour of cortical bone are developed. The first one includes a single scalar damage variable, whereas the second one is based on tensile and compressive damage variables, which improves the simulation results. Both models are then extended into rate-dependent alternatives by relating the rate of damage accumulation to some high power of the damage threshold stress. All four models consider different tensile and compressive damage threshold stresses as it is the case for cortical bone. Secondly, the material constants characterizing the one-dimensional models are identified on experimental grounds. To this end, a series of in vitro uniaxial overloading tests were carried out on bovine cortical bone. Reliable measurements were obtained in tension using dumbbell specimens, avoiding thus undesirable boundary effects. Thirdly, a three-dimensional rate-independent constitutive law inspired by the one-dimensional models is formulated and implemented in a finite element code. It includes porous fabric-based orthotropic elasticity and rate-independent plasticity with damage. The onset of damage is characterized by an orthotropic stress-based damage criterion described by porosity and fabric, which takes into account distinct tensile and compressive damage threshold stresses. Finally, the potential of the new three-dimensional elastic plastic damage constitutive law for cortical bone is demonstrated by means of a finite element analysis of the compression of a vertebra.