Biomedical Engineering Reference
In-Depth Information
6
Constitutive Modeling of the Mechanical Behavior of Trabecular
Bone - Continuum Mechanical Approaches
Andreas Ochsner and Seyed Mohammad Hossein Hosseini
6.1
Introduction
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 engi-
neering and advanced analysis tools (e.g.,
CT) 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 sim-
plified 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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