Constitutive Modeling of the Mechanical Behavior of Trabecular Bone – Continuum Mechanical Approaches

Abstract

Typical uniaxial stress-strain curves for different types of bones are shown in Figure 6.1 for the compressive regime. For all the curves, an initial linear elastic behavior can be observed. The common approach is to describe this elastic part on the basis of Hooke's law, cf. Sections 6.3.1-6.3.4. This elastic range is followed by a strong nonlinear behavior of almost constant stress (so-called stress plateau). At higher strains, some curves show a strong increase in the stress where densification begins.These macroscopic stress-strain curves are similar to the behavior known from completely different types of materials such as cellular polymers and metals or even concrete. Although the deformation mechanism on the microlevel can be completely different, a common approach is to use the constitutive equations of metal plasticity to describe the nonelastic behavior, cf. [1-3]. As we see in this chapter, bones have some kind of cellular or porous structure, and the classical equations of full dense metals (e.g., von Mises or Tresca) must be extended by at least the hydrostatic pressure to account for the fact that such a material is even in the plastic range compressible. This general theory of a yield or failure surface based on stress invariants is introduced in Section 6.3.5. Many extensions of this theory are known for bones. However, the main focus is to thoroughly introduce the concept of a yield and limit surface so that possible extensions (e.g., by damage variables or the consideration of anisotropy) are easier to incorporate.Many different approaches to derive new constitutive equations are known. Nowadays, the finite element method is the standard tool in computational engineering and advanced analysis tools (e.g., µCT) allow an extremely detailed imaging of bone structure. More and more powerful computer hardware (RAM and CPU) enables and supports this trend. However, there are approaches based on simplified model structures that reveal some advantages compared to these highly computerized approaches. Thus, some classical model structures are presented in the second part of the chapter. These simpler models are, in many cases, able to consider the major physical effect and may finally yield a mathematical equation to

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