7.1 Introduction
7.1.1 The “material properties” of the musculoskeletal tissues
- The “material properties” of the musculoskeletal tissues depend on the underlying micro-structures
- They are typically inhomogeneous
- The depend on age, disease, and other factors
- Thus, we must consider
- Tissue level (inhomogeneous, anisotropic)
- Organ level (homogenized, isotropic or anisotropic)
- Knowledge of the scale of interest is critical to our material modeling approach
7.1.2 Challenges in biological material testing
- Challenges in material testing include:
- What is our scale of interest?
- Isolation of homogeneous tissue
- Measurement of deformation (porous tissue)
- Control of environment (in-vivo vs ex-vivo)
- Inter-specimen variability
- We attempt generalization based on:
- Density
- Organic composition
- Mineralization
- …
- Biological tissues adapt to their environment at the cellular,
structural, and molecular level
- ie bone density and geometry adapt to load
- Feedback loop (in time)
- Mechanics respond to biology
- Biology responds to mechanics
- An important aspect of modern implant design is prediction of
in-vivo mechanobiologic response
- How will the bone respond to the implant?
- ie: will stress shielding be a problem?
- Currently, we have little control and seek to understand
- In future, we may also try to influence this response
7.1.3 Composition of bone
- Bone composed of organics and inorganics
- Loosely defined, organic compounds contain carbon covalently bonded to hydrogen, oxygen or nitrogen, etc
- By mass, bone is approximately:
- 60% inorganic material (calcium phosphate, etc)
- 30% organic
- 10% water
7.1.4 Collagen
@Bartel2006
- Organic content of bone is mostly collagen
- Collagen is the strongest and most abundant protein in the body
- Rod-shaped molecules are about 300 nm long and 1.5 nm in diameter
- Arranged in a quarter stagger pattern into fibrils which have 20-40nm diameter
@Bartel2006
“Starting with a single helical protein chain consisting of a variety of amino acids connected by peptide bonds (top), three of these polypeptide chains are combined to form the triple helix tropocollagen molecule (second down), referred to most often as simply the collagen molecule.”15
@Bartel2006
The different types of collagen have different types of polypeptide chains. The collagen molecules are arranged in parallel in a regular quarter-stagger arrangement to comprise the collagen fibril (bottom).16
7.1.5 Bone is a hierarchical composite material
@Bartel2006
- Bone is a hierarchical composite material
The four levels of bone microstructure, from the level of mineralized collagen fibrils to cortical and trabecular bone. It is generally assumed that at the former level, all bone is equal, although there can be subtle differences in the nature of the lamellar architecture and degree of mineralization between cortical and Trabecular bone.17
7.1.6 Lowest hierarchical level
@Bartel2006
- Collagen fibril (\(\approx 0.1\) micron)
- Sheet lamellae of uni-directional fibrils (most common) or blocks of pseudo random “woven” fibrils (\(\approx 10\) micron)
7.1.7 Highest hierarchical level
- Cortical bone
- Trabecular bone
7.1.8 Highest hierarchical level, diaphysis
@Tortora1983
Diagram of a sector of the shaft of a long bone, showing the different types of cortical bone, trabecular bone, and the various channels. The osteons are located between the outer and inner circumferential lamellae.18
7.1.9 Cortical bone
- Laminar bone (sandwich sheets of lamellar bone layers)
- Haversian
bone (10-15
lamellae in
a cylinder surrounding a
Haversian
canal)
- Contains blood vessels, nerves, and lymphatics
- Osteon (substructure which includes the
Haversian
canal) 1-3 mm long by 200 microns in diameter
- This is a “unit cell” (non-medical term) discrete structure for mechanical study
- Osteons are continually being torn down and replaced
- Process takes months for individual osteon, thus time is required (stress fracture if overused without adaptation time)
- Osteons are bound to each other by a cement line
- It is “weak”
- It is analogous to matrix in a composite material
- Passes shear loads, dissipates energy
7.1.10 Damage detection
- Pores exist in the bone cells
- Fluid resides around bone cells in holes (called lacunae)
- These lacunae are interconnected by tiny channels (canaliculi) and meet at gap junctions
- Small molecules including ions pass between the cells and are thought to help sense damage
@Bartel2006
7.1.11 Trabecular architecture
@Bartel2006
7.1.12 Differences between cortical and trabecular bone
- The biggest difference is the cellular-like (spongy
structure of trabecular bone
- Holes filled with marrow
- The rods and plates tend to remodel more often, thus, trabecular bone is less mineralized than the “older” cortical bone
- The material properties are thus slightly worse on the tissue level (even worse on the structural level due to cellular structures)
7.1.13 Elastic anisotropy
7.1.14 Thought experiment: isotropic material
7.1.15 Uniform rectangular block pulled on both ends
Public Domain Kerina yin 2011
Block subject to normal stress
What strains do you expect for \({\sigma_{xx}}\ne 0\) (all others stresses = 0)?
- We are all familiar with Hooke’s Law: \[\begin{equation*} {\sigma_{xx}}= E {\varepsilon_{xx}} \end{equation*}\]
- Rearranging: \[\begin{equation*} {\varepsilon_{xx}}= \frac{{\sigma_{xx}}}{E} \end{equation*}\]
- But what are the other strains?
- But what are the other strains? \[\begin{align} {\varepsilon_{yy}}=& \, -\nu {\varepsilon_{xx}}\cr {\varepsilon_{zz}}=& \, -\nu {\varepsilon_{xx}}\cr =& \, -\nu \frac{{\sigma_{xx}}}{E} \cr {\varepsilon_{ij}}=& \, 0 \hspace{5mm} \mbox{ for } i\neq j \cr \end{align}\]
- Similarly, we can obtain similar equations in the other directions:
For \({\sigma_{yy}}\ne 0\), all others 0? \[\begin{align} {\varepsilon_{yy}}=& \frac{{\sigma_{yy}}}{E} \cr {\varepsilon_{xx}}=& -\nu \frac{{\sigma_{yy}}}{E} \cr {\varepsilon_{zz}}=& -\nu \frac{{\sigma_{yy}}}{E} \cr {\varepsilon_{ij}}=& \; 0 \hspace{5mm} \mbox{ for } i\neq j \cr \end{align}\]
For \({\sigma_{zz}}\ne 0\), all others 0? \[\begin{align} {\varepsilon_{zz}}=& \, \frac{{\sigma_{zz}}}{E} \cr {\varepsilon_{xx}}=& \, -\nu \frac{{\sigma_{zz}}}{E} \cr {\varepsilon_{yy}}=& \, -\nu \frac{{\sigma_{zz}}}{E} \cr {\varepsilon_{ij}}=& \, 0 \hspace{5mm} \mbox{ for } i\neq j \cr \end{align}\]
- We’ve found a pattern for the normal stress-normal strain response
- What about for shear?
Block subject to shear stress
For \({\tau_{xy}}\ne 0\), all others 0? \[\begin{align} {\gamma_{xy}}=& \, \frac{{\tau_{xy}}}{G} \cr {\gamma_{xz}}=& \, 0 \cr {\gamma_{yz}}=& \, 0 \cr {\varepsilon_{ii}}=& \, 0 \hspace{5mm} \mathrm{No \; sum} \cr \end{align}\]
For \({\tau_{xz}}\ne 0\), all others 0? \[\begin{align} {\gamma_{xz}}=& \, \frac{{\tau_{xz}}}{G} \cr {\gamma_{xy}}=& \, 0 \cr {\gamma_{yz}}=& \, 0 \cr {\varepsilon_{ii}}=& \, 0 \hspace{5mm} \mathrm{No \; sum} \cr \end{align}\]
Block subject to shear stress
For \({\tau_{yz}}\ne 0\), all others 0? \[\begin{align} {\gamma_{yz}}=& \, \frac{{\tau_{yz}}}{G} \cr {\gamma_{xy}}=& \, 0 \cr {\gamma_{xz}}=& \, 0 \cr {\varepsilon_{ii}}=& \, 0 \hspace{5mm} \mathrm{No \; sum} \cr \end{align}\]
For multiple simultaneous stresses: use superposition
7.1.16 Isotropic constitutive behavior
\[\left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} \frac{1}{E} & \frac{-\nu}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \\ \frac{-\nu}{E} & \frac{1}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \\ \frac{-\nu}{E} & \frac{-\nu}{E} & \frac{1}{E} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2 G} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2 G} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2 G} \\ \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\}\] Note also: the shear modulus is: \[\begin{split} %% G =& \mu \\ G =& \frac{E}{2 (1+\nu)}\\ \end{split}\] Thus there are only two constants that describe the behavior of an isotropic material
7.1.17 Is this how all materials behave?
7.1.18 Origin of anisotropic behavior in bone
- In cortical bone, osteons align parallel to loads
- In trabecular bone, rods and plates also align with loads
- Bone properties have directionality (called anisotropy)
7.1.19 Principal material coordinate system
@Bartel2006
Principal material coordinate system for an orthotropic material. This coordinate system (right) is aligned with the mutually orthogonal “grain” axes of the material’s microstructure (left. As a class of anisotropic materials, orthotropic materials have their grain along three mutually perpendicular axes.
@Bartel2006
Spatial variations in the orientation of the principal material coordinate system can occur. In many cases, a local coordinate system in cylindrical coordinates can be used to describe such spatial variations.
7.1.20 Anisotropic behavior
Woven fibrous composite
7.1.21 Anisotropic behavior
- Fibrous composites exhibit a more complex constitutive response
- Consider the following material description: \[\left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16} \\ S_{21} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26} \\ S_{31} & S_{32} & S_{33} & S_{34} & S_{35} & S_{36} \\ S_{41} & S_{42} & S_{43} & S_{44} & S_{45} & S_{46} \\ S_{51} & S_{52} & S_{53} & S_{54} & S_{55} & S_{56} \\ S_{61} & S_{62} & S_{63} & S_{64} & S_{65} & S_{66} \\ \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\}\]
- This is generalized Hooke’s law (applicable to any linear elastic material–called anisotropic).
- For simplicity, we can write generalized Hooke’s law as: \[\{{\varepsilon}\} = [S] \{\sigma\}\]
- the values of \(S\) are called “elastic compliances”
7.1.22 Elastic constants
- It is important to be able to consider the inversion of this system: \[\begin{split} \{{\varepsilon}\} =& [S] \{\sigma\} \\ \{\sigma\} =& [S]^-1 \{{\varepsilon}\} \\ \{\sigma\} =& [C] \{{\varepsilon}\} \\ \end{split}\]
- The values of \(C\) are called Elastic constants
- \([C]\) and \([S]\) are fully populated for an anisotropic material
7.1.23 For isotropic material:
\[\left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\} = \frac{E}{(1+\nu)(1-2\nu)} \left[ \begin{array}{cccccc} 1-\nu & \nu & \nu & 0 & 0 & 0 \\ \nu & 1-\nu & \nu & 0 & 0 & 0 \\ \nu & \nu & 1-\nu & 0 & 0 & 0 \\ 0 & 0 & 0 & 1-2\nu & 0 & 0 \\ 0 & 0 & 0 & 0 & 1-2\nu & 0 \\ 0 & 0 & 0 & 0 & 0 & 1-2\nu \\ \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\}\]
7.1.24 Lamé constants
\[\left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} 2 \mu + \lambda & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & 2 \mu + \lambda & \lambda & 0 & 0 & 0 \\ \lambda & \lambda &2 \mu + \lambda & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 \mu \\ \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\}\]
Where: \[\begin{split} \mu =& \frac{E}{2 (1-\nu)} \\ \lambda =& \frac{\nu E}{(1+\nu)(1-2 \nu)} \\ \end{split}\]
- Finally, we can also use indicial notation to quickly write our constitutive relationship:
\[{\varepsilon_{ij}}= \frac{1}{E} \left[(1+\nu) {\sigma_{ij}}- \nu \delta_{ij} {\sigma_{kk}}\right]\]
\[{\sigma_{ij}}= 2 \mu {\varepsilon_{ij}}+ \lambda \delta_{ij} {\varepsilon_{kk}}\] \(\mu\) and \(\lambda\) are called the Lamé constants. They can be found in standard texts but will not be discussed further.
- This method of expressing the equations is powerful and worthy of study, however, we will not discuss it further
7.1.25 Other material descriptions
- There are materials that fit between total anisotropy (21 constants) and isotropic (2 constants).
- In the aerospace world, a critical one is “orthotropic” (9 constants) \[\left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} \frac{1}{E_{xx}} & -\frac{\nu_{yx}}{E_{yy}} & -\frac{\nu_{zx}}{E_{zz}} & 0 & 0 & 0 \\ -\frac{\nu_{xy}}{E_{xx}} & \frac{1}{E_{yy}} & -\frac{\nu_{zy}}{E_{zz}} & 0 & 0 & 0 \\ -\frac{\nu_{xz}}{E_{xx}} & -\frac{\nu_{yz}}{E_{yy}} & \frac{1}{E_{zz}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2 G_{yz}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2 G_{zx}} & 0 \\ 0 & 0 & 0 & & 0 & \frac{1}{2 G_{xy}} \\ \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\}\]
- Due to symmetry: \[\begin{split} \frac{\nu_{yx}}{E_{yy}} =& \frac{\nu_{xy}}{E_{xx}} \\ \frac{\nu_{zx}}{E_{zz}} =& \frac{\nu_{xz}}{E_{xx}} \\ \frac{\nu_{zy}}{E_{zz}} =& \frac{\nu_{yz}}{E_{yy}} \\ \end{split}\]
\[\left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ {} & C_{22} & C_{23} & 0 & 0 & 0 \\ {} & {}& C_{33} & 0 & 0 & 0 \\ {} & {}& {} & C_{44} & 0 & 0 \\ {} & {}& {} & {} & C_{55} & 0 \\ {} & {}& {} & {} & {} & C_{66} \\ \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\}\]
- This is often the best description of a composite ply.
- It works for bone in some cases too
- Also:
- Monoclinic (13 constants)
- Transversely isotropic (5 constants-ex: unidirectional composites and bone)
- Cubic (3 constants-ex: silicon)
- Finally, this entire description is referred to as “generalized Hooke’s law” (Robert Hooke, Late 17th century)
\[{\sigma_{ij}}= E_{ijkl} \, \varepsilon_{kl}\]
7.1.26 Cortical bone is well described as transversely isotropic
ch03_08
7.1.27 Transversely isotropic
7.1.27.1 5 constants
\[\left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} \frac{1}{E_T} & -\frac{\nu_T}{E_T} & -\frac{\nu_L}{E_L} & 0 & 0 & 0 \\ -\frac{\nu_T}{E_T} & \frac{1}{E_T} & -\frac{\nu_L}{E_L} & 0 & 0 & 0 \\ -\frac{\nu_L}{E_L} & -\frac{\nu_L}{E_L} & \frac{1}{E_L} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2 G} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2 G} & 0 \\ 0 & 0 & 0 & & 0 & \frac{1+\nu_T}{E_T} \\ \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\}\]
- \(E_T\) and \(E_L\) are transverse (in plane) and longitudinal (out of plane) modulus
- \(\nu_T\) and \(\nu_L\) are transverse (in plane) and longitudinal (out of plane) Poisson’s ratios
- \(G\) is the out of plane shear modulus
Table 3.1 @Bartel2006
- Note: stresses and strains must be transformed into the local coordinate system to apply the constitutive law
- Alternatively, the stiffness or compliance matrix must be transformed into the global coordinate system
- See Bartel’s book or me for additional info
7.1.28 Have we covered it all?
- Bone is anisotropic and inhomogeneous
- Bone is rate dependent
- Bone constitutive response depends on fatigue, age, damage, and plasticity
- In models of bone, it is often convenient to assume isotropic
properties
- Difficult to obtain a “better” description
- Transversely isotropic (and orthotropic) may be more appropriate