9 Bartel  Chapter 3

9.0.1 Introduction

9.0.1.0.1 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

9.0.1.0.2 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

9.0.2 Composition of bone

9.0.2.0.1 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

9.0.2.0.2 Collagen

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  • 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

9.0.2.0.3 Bone is a hierarchical composite material

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9.0.2.0.4 Lowest hierarchical level

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  • Bone is a hierarchical composite material

    • Collagen fibril ($$0.1 micron)

    • Sheet lamellae of uni-directional fibrils (most common) or blocks of pseudo random “woven” fibrils ($$10 micron)

9.0.2.0.5 Highest hierarchical level

  • Cortical bone

    • Tightly packed lamellar, Haversian, laminar, or woven bone

    • Lots of osteons

  • Trabecular bone

    • Highly porous rods and plates interspersed with marrow spaces

    • Less well organized packets of lamellae

    • Very few osteons if any

9.0.2.0.6 Highest hierarchical level, diaphysis

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9.0.2.0.7 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

9.0.2.0.8 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

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9.0.2.0.9 Trabecular architecture

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9.0.2.0.10 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)

9.0.3 Elastic anisotropy

9.0.3.1 Thought experiment: isotropic material

9.0.3.1.1 Uniform rectangular block pulled on both ends

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: \[{\sigma_{xx}}= E {\varepsilon_{xx}}\]

  • Rearranging: \[{\varepsilon_{xx}}= \frac{{\sigma_{xx}}}{E}\]

  • But what are the other strains? \[\begin{split} {\varepsilon_{yy}}=& \, -\nu {\varepsilon_{xx}}\\ {\varepsilon_{zz}}=& \, -\nu {\varepsilon_{xx}}\\ =& \, -\nu \frac{{\sigma_{xx}}}{E} \\ {\varepsilon_{ij}}=& \, 0 \hspace{5mm} i\neq j\\ \end{split}\]

  • Similarly, we can obtain similar equations in the other directions:

For \({\sigma_{yy}}\ne 0\), all others 0? \[\begin{split} {\varepsilon_{yy}}=& \frac{{\sigma_{yy}}}{E} \\ {\varepsilon_{xx}}=& -\nu \frac{{\sigma_{yy}}}{E} \\ {\varepsilon_{zz}}=& -\nu \frac{{\sigma_{yy}}}{E} \\ {\varepsilon_{ij}}=& \; 0 \hspace{5mm} i\neq j\\ \end{split}\]

For \({\sigma_{zz}}\ne 0\), all others 0? \[\begin{split} {\varepsilon_{zz}}=& \, \frac{{\sigma_{zz}}}{E} \\ {\varepsilon_{xx}}=& \, -\nu \frac{{\sigma_{zz}}}{E} \\ {\varepsilon_{yy}}=& \, -\nu \frac{{\sigma_{zz}}}{E} \\ {\varepsilon_{ij}}=& \, 0 \hspace{5mm} i\neq j\\ \end{split}\]

  • We’ve found a pattern for the normal stress-normal strain response

  • What about for shear?

Block subject to shear stress

0.5 For \({\tau_{xy}}\ne 0\), all others 0? \[\begin{split} {\gamma_{xy}}=& \, \frac{{\tau_{xy}}}{G} \\ {\gamma_{xz}}=& \, 0 \\ {\gamma_{yz}}=& \, 0 \\ {\varepsilon_{ii}}=& \, 0 \hspace{5mm} \mathrm{No sum}\\ \end{split}\]

For \({\tau_{xz}}\ne 0\), all others 0? \[\begin{split} {\gamma_{xz}}=& \, \frac{{\tau_{xz}}}{G} \\ {\gamma_{xy}}=& \, 0 \\ {\gamma_{yz}}=& \, 0 \\ {\varepsilon_{ii}}=& \, 0 \hspace{5mm} \mathrm{No sum}\\ \end{split}\]

0.5 For \({\tau_{yz}}\ne 0\), all others 0? \[\begin{split} {\gamma_{yz}}=& \, \frac{{\tau_{yz}}}{G} \\ {\gamma_{xy}}=& \, 0 \\ {\gamma_{xz}}=& \, 0 \\ {\varepsilon_{ii}}=& \, 0 \hspace{5mm} \mathrm{No sum}\\ \end{split}\]

  • For multiple simultaneous stresses: use superposition
9.0.3.1.2 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

9.0.3.1.3 Is this how all materials behave?
9.0.3.1.4 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)

9.0.3.2 Principal material coordinate system

9.0.3.2.1 Principal material coordinate system

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9.0.3.3 Anisotropic behavior

Woven fibrous composite

9.0.3.3.1 Anisotropic behavior

Fibrous composites exhibit a more complex constitutive response

Consider the following material description: \[\left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \\ a_{61} & a_{62} & a_{63} & a_{64} & a_{65} & a_{66} \\ \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\}\] This is generalized Hooke’s law (applicable to any linear elastic material–called anisotropic).

  • For simplicity, we can write generalized Hooke’s law as: \[\{{\varepsilon}\} = [a] \{\sigma\}\]

  • the values of a are called “elastic compliances”

9.0.3.3.2 Elastic constants

  • It is important to be able to consider the inversion of this system: \[\begin{split} \{{\varepsilon}\} =& [a] \{\sigma\} \\ \{\sigma\} =& [a]^-1 \{{\varepsilon}\} \\ \{\sigma\} =& [c] \{{\varepsilon}\} \\ \end{split}\]

  • The values of \(c\) are called Elastic constants

  • \([c]\) and \([a]\) are fully populated for an anisotropic material

9.0.3.3.3 For isotropic material:

\[\left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\} = \frac{E}{(1+\nu)(1-2\nu)} \left[ \begin{array}{cccccc} 1-\nu & \nu & \nu & 0 & 0 & 0 \\ \nu & 1-\nu & \nu & 0 & 0 & 0 \\ \nu & \nu & 1-\nu & 0 & 0 & 0 \\ 0 & 0 & 0 & 1-2\nu & 0 & 0 \\ 0 & 0 & 0 & 0 & 1-2\nu & 0 \\ 0 & 0 & 0 & 0 & 0 & 1-2\nu \\ \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\}\]

9.0.3.3.4 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

9.0.3.3.5 Other material descriptions
9.0.3.3.6 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)

\[{\sigma_{ij}}= E_{ijkl} \, \varepsilon_{kl}\]

9.0.3.3.7 Cortical bone is well described as transversely isotropic

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9.0.3.3.8 Transversely isotropic
9.0.3.3.8.1 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

imageTable 3.1

  • 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

9.0.3.3.9 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

9.0.4 Material properties of cortical bone

9.0.4.0.1 Asymmetric stiffness and strength

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9.0.4.0.2 Asymmetric stiffness and strength

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9.0.4.0.3 Stiffness and strength with age

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9.0.4.0.4 Volume fraction

  • Porosity is present in both cortical and trabecular bone

  • Define volume fraction (\(V_f\)) as the volume of actual bone tissue to the bulk volume

  • Cortical 70-95%

  • Trabecular 5-60%

  • Extremes are the young adult and elderly

9.0.4.0.5 Bone density

  • Density is strongly dependent on porosity and volume fraction

  • It is also a primary indicator of bone strength and stiffness

  • Apparent density is mass per bulk volume

  • Common measure of apparent density include:

    • Hydrated

    • De-hydrated

    • De-organified

  • Tissue density is mass per volume of actual bone tissue (2.0g/cc)

    • Importantly, this volume excludes vascular pore spaces
9.0.4.0.6 Relationship between bone density and volume fraction

  • The volume fraction, tissue density, and apparent density are related by \[\rho_{\mathrm{app}} = \rho_{\mathrm{tiss}} V_f\]

  • Apparent densities

    • Cortical – \(\approx\) 1.85 g/cc

    • Trabecular – 0.10-0.50 g/cc

  • Trabecular density decreases about 2% per decade after skeletal maturity

  • Note also the cortical bone wall thickness decreases as you age

9.0.4.0.7 Heterogeneity and variability
9.0.4.0.7.1 Strength

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9.0.4.0.8 Mineral content

  • Mineral content is also important for mechanical properties

  • It is measured after heating bone to 700C for 24 hours (de-organification and drying)

  • Content increases during skeletal growth and remains fairly constant thereafter

9.0.4.0.9 Heterogeneity and variability
9.0.4.0.9.1 Stiffness

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9.0.4.0.10 Density and strength

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9.0.4.0.11 Fatigue

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9.0.4.0.12 Minor’s rule for fatigue

  • Hypothesis is that the fractional fatigue lives sum together and predict failure

  • \(\displaystyle\sum_{i=1}^n \frac{N_i}{N_{Fi}} = 1\)

    • Where there are \(n\) different load levels

  • Works well in most brittle metals (implants!)

  • Not validated for bone!!!

  • In bone, common assumption to replace stress with strain

    • Account for various levels of porosity, etc (also reduces variability in the test data)

  • What about bone remodeling?

9.0.4.0.13 Creep

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  • De-vitalized bone exhibits creep

    • Resistance better in compression then tension

    • Difficult to test in-vivo response

  • Metals creep – may/may not be significant for ortho

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9.0.4.0.14 Creep

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9.0.4.0.15 Plasticity and micro-structural damage

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  • Note the change in slope after yield – microdamage!
9.0.4.0.16 Strain rate sensitivity

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  • Only a factor of two – mildly rate dependent

  • Probably not critical for most physiologic loads

9.0.5 Material properties of trabecular bone

9.0.5.0.1 Trabecular bone behavior and large variability

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  • Large range in strength (note the axis!)
9.0.5.0.2 Trabecular bone apparent density

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9.0.5.0.3 Trabecular bone crush strength and age

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9.0.5.0.4 Trabecular bone yield asymmetry

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9.0.5.0.5 Trabecular bone yield anisotropy
9.0.5.0.5.1 Yield stress anisotropic – yield strain isotropic

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9.0.5.0.6 Fatigue of trabecular bone

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9.0.5.0.7 Post-yield damage of trabecular bone

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9.0.5.0.8 Failure prediction

  • Von Mises stress criterion is not good for bone (particularly when shear stresses are high)

  • Tsai-Wu is a much better metric of strength, however, requires significantly more experiments to establish the criterion \[\begin{split} F_1 \sigma_1 + F_2 \sigma_2 + F_3 \sigma_3 &\\ + F_{11} \sigma_1^2 + F_{22} \sigma_2^2 + F_{33} \sigma_3^2 &\\ + 2 F_{12} \sigma_1 \sigma_2 + 2 F_{13} \sigma_1 \sigma_3 + 2 F_{23} \sigma_2 \sigma_3 &\\ + F_{44} \sigma_4^2 + F_{55} \sigma_5^2 + F_{66} \sigma_6^2 &=1 \end{split}\] (Note vector representation of stress 1-6)

  • Requires tension, compression, and torsion tests in longitudinal and transverse specimens

  • This is a big challenge in practical prediction of bone failure

    • Most studies only compare stresses without predicting failure
9.0.5.0.9

9.1 image