ABSTRACT of the Doctoral Thesis

A Constitutive Law for Softening and Densification of Trabecular Bone in Compression

by Mathieu Charlebois (2008)


Within the body, bone is a material with high stiffness and low resilience, which provides load bearing, support to the body and protection of major organs. Moreover, bone is a dynamic material that constantly renews itself through a remodelling process. This remodelling demands, for a constant bone mass, the rate of formation to be equal to the rate of resorption. The unbalance of these rates is the source of a major disease called osteoporosis. This disease drives bone research.

There are two types of bone: compact and trabecular bone. While compact bone is dense and composes the surface of the bones, trabecular bone is an open-cell porous structure found mainly in the epiphysis of long bones, within the short bone, e.g. vertebrae, the flat bones, e.g. skull, and tendon attachment sites. The mechanical properties of trabecular bone depend largely on its structure, which present considerable variation between anatomical site, age, and individuals. Therefore, architectural parameters describing the structure of trabecular bone have been developed and incorporated into relationships between morphology and mechanical properties. These relationships, namely density and fabric-based compliance tensor and yield criterion, are integrated into constitutive law developed specifically to trabecular bone.

Three-dimensional images provided by computed tomography can be converted into finite element models, which in turn provide the possibility of evaluating bone properties, e.g. stiffness and strength. These models require a mathematical relationship between stress and strain, i.e. a constitutive law, that captures the main features of the mechanical response of bone.

The current work consisted of developing a constitutive law that captures the main features of the stress-strain curve of trabecular bone in large compression. In this loading condition, trabecular bone presents softening, consisting of a reduction of stress passed the ultimate point, and rehardening, consisting of an increase of stress at higher strains. These softening-rehardening characteristics involves a non-zero minimum point. Following the minimum point, rehardening consists of a quasi-linear stress increase with strain followed by a fast increase at very large strain. These two phases are associated with a progressive pore collapse and with pore closure, respectively. In addition, passed its yield point, trabecular bone accumulates irreversible strains and loses stiffness. Accounting for these features, an elastoplastic model coupled with damage was developed in the framework of standard generalized material, to ensure its thermodynamic consistency. The model was implemented into a finite element software. To prevent mesh-dependency of the solution, the model was enhanced with an integral-type nonlocal averaging technique. Finally, the parameters of the model depend on the bone volume fraction and fabric tensor, describing respectively bone density and anisotropy.

In order to provide data for the parameter identification of the proposed constitutive law, mechanical experiments reaching large strain compression were performed on human trabecular bone originating from several anatomical sites using confined and unconfined setups. Curve parameters were defined, computed and analyzed using regression models developed to include morphological parameters. The complete results and analysis are presented.

Finally, the parameters of the constitutive law were identified from the mechanical test results using a novel procedure composed of two steps. First, the parameters were identified using an assumption of homogeneity. Second, heterogeneous voxel-based continuum FE models, created from five chosen specimens were used to adjust the parameters using the nonlocal formulation. Finally, the constitutive law was applied to voxel-based models of vertebral bodies in order to show an example of potential applications. Comparing the results with a previous study provided strong insight about the effect of softening in the constitutive law. Briefly, the accumulation of the inelastic process is concentrated in a single layer within the vertebral body, as opposed to models without softening that can predict several of these layers.

In conclusion, the current study proposes a novel constitutive law to describe trabecular bones mechanical behaviour in large compressive strain and mechanical testing was conducted to identify the parameters of the law. Finally, using the law into a continuum FE model showed the potential of the law. Future work is oriented towards providing an automated procedure to apply this numerical method to studies requiring large numbers of simulations.


revised 081124 (hjb)