1. Mechanics of Bone
BME 615

2. Why study bone mechanics?
Why Test Bone? – Interesting Mechanical Properties
Why study bone mechanics?
Interesting Mechanical Properties
In vivo load leads to complex stress distributions

3. Why study bone mechanics?
Why Test Bone? – Interesting Mechanical Properties
Why study bone mechanics?
Important structural function
Lots of pathologies cause mechanical compromise and clincial issues.
Obvious mechano-transduction
Important for fixation systems for trauma or joint prosthetics
Good demo of “apparent” properties, orthotropic behaviors
Why not?

4. Bone will remodel to adapt to loads
Why Test Bone? – Interesting Mechanical Properties
Wolff’s Law
Bone will remodel to adapt to loads

5. Bone will remodel to adapt to loads
Why Test Bone? – Interesting Mechanical Properties
Wolff’s Law
Bone will remodel to adapt to loads
J. Wolff

6. Ankylosed Knee – J. Wolff

7. Two types of bone to accommodate complex loading
Why Test Bone? – Interesting Mechanical Properties
Bone Types
Two types of bone to accommodate complex loading
1. Compact (Cortical) Bone
Relatively high Young’s modulus
Higher resistance to torsion and bending
Outer shell of bones
Slow turnover

8. Two types of bone to accommodate complex loading
Why Test Bone? – Interesting Mechanical Properties
Bone Types
Two types of bone to accommodate complex loading
1. Cancellous (trabecular) Bone
Lower apparent modulus
Higher resistance compression
More elastic than cortical
Inner portion of bone
Higher turnover

9. Apparent Modulus “Apparent Modulus”
Why Test Bone? – Interesting Mechanical Properties
Apparent Modulus
“Apparent Modulus”
“Trabecular bone” does not fill entire space
Spaces filled with bone marrow
Mechanical properties reflect what’s going on for the entire sample
“apparent” modulus  what the overall mechanical properties are doing (in this case, bone + what’s filling the gaps)
How will the bone marrow affect the mechanical test results?

10. Bone Elastic Modulus, E Cortical Bone E ≈ 17.9 GPa
4/28/2017
4/28/2017
Bone Elastic Modulus, E
Cortical Bone
E ≈ 17.9 GPa
σult ≈ 170 MPa (compression)
σult ≈ 120 MPa (tension)
Trabecular Bone
E ≈ GPa
σult ≈ 2.2 MPa (compression)
Note: Bone is non-homogeneous and assumed orthotropic
(spatial & direction dependence)

11. Mechanics
9 independent parameters require 9 independent tests

12. How to test bone? Sample Selection
Cancellous bone with different orientation
Different locations may lead to different mechanical properties
Try not to test off of principal axes
Cortical bone
In vivo load leads to complex stress distributions

13. Structural Testing Three-point Testing Advantages:
How to test bone? – Three-point testing
Structural Testing
Three-point Testing
Advantages:
easy specimen preparation
(relatively) easy testing
Disadvantage:
Sensitive to geometry (specimen-specific)
Calculated parameters: flexural stress and strain, flexural modulus, fracture toughness

14. Structural Testing Three-point Testing For hollow cylinder:
How to test bone? – Three-point testing
Structural Testing
Three-point Testing
For hollow cylinder:
parameters
Stress intensity factor at crack tip:

15. Tensile Testing Apparent properties Apparent properties
Bone cement to embed ends in metal cap
Apparent properties
Apparent properties

16. Compression Testing Advantages: Disadvantages: Calculated parameters:
How to test bone? – Compression testing
Compression Testing
Advantages:
Easy calculations
Uniaxial
Disadvantages:
Careful specimen preparation
Principal material axes?
End conditions?
Spatially changing properties
Specimen size requirements
Calculated parameters:
Apparent compressive strength,
Apparent modulus of elasticity

17. Behaviors from Testing
How to test bone? – Compression testing
Behaviors from Testing
Cortical Bone
Trabecular Bone
Stronger in compression than tension
Different properties with different bone types

18. Compression Testing Analysis
How to test bone? – Compression testing
Compression Testing Analysis
Modulus Estimation
Elastic modulus can be estimated from apparent density using a power law relationship
α=2.5 for ρ<1.2 g/cm3
α=3.2 for ρ>1.2 g/cm3
ρ = Tissue Mass/Bulk Volume
Reference: DR Carter, GS Beaupre, Skeletal Function and Form: Mechanobiology of Skeletal Development, Aging and Regeneration. Cambridge University Press 2007

19. Simulated Compression Test
Virtual Lab Demo
Simulated Compression Test
2cm x 2cm x 2cm cube of bone taken from inside of pig femoral head
Cancellous/trabecular bone
Bone marrow left in place inside the cube X Y Z
How will this likely affect testing?

20. Simulated Compression Test
Virtual Lab Demo
Simulated Compression Test
Steady Compression Applied
Actuator
Bone Sample
Testing Stage
Strain
Time

21. Simulated Compression Test
Virtual Lab Demo
Simulated Compression Test
Steady Compression Applied
Actuator
Bone Sample
Testing Stage
Specimen rotated until all three sides tested

22. Simulated Compression Test
Virtual Lab Demo
Simulated Compression Test
Gather “apparent” mechanical properties
Approximating properties as if homogeneous, continuous substance
centimeters
millimeters
At what scale does the continuum assumption break down?

23. Indentation Nano-indentation Density measurements Ultrasound
Other testing methods
Equations given previously in BVP section
Indentation
Nano-indentation
Density measurements
Ultrasound
Longitudinal wave propagation velocity in an elastic, homogeneous material

24. Structural Components
Why Test Bone? – Interesting Mechanical Properties
Structural Components
Two major components to accommodate complex loading
Collagen: 90% of the organic matrix
(~36% of total dry weight)
Provides tensile strength to bone
Primarily type I collagen
Calcium hydroxyapatite: inorganic matrix
(~60% of total dry weight)
Provides compressive strength to bone
Responsible for bone mineralization

25. Mixture Theory Voigt Upper Bound:
Why Test Bone? – Interesting Mechanical Properties
Mixture Theory
Mi = Modulus of ith constituent fi = volume fraction of ith constituent
Voigt Upper Bound:
Model as parallel components in mixture
Component 1
Component 2 HA C
f1M1
f2M2
Not for apparent behaviors

26. Mixture Theory Reuss Lower Bound:
Why Test Bone? – Interesting Mechanical Properties
Mixture Theory
Mi = Modulus of ith constituent fi = volume fraction of ith constituent
Reuss Lower Bound:
Model as serial components in mixture
Component 1
Component 2 HA C
f1M1
f2M2

27. Mixture Theory Apparent Modulus Volume Fraction of HA
Why Test Bone? – Interesting Mechanical Properties
Mixture Theory
Mi = Modulus of ith constituent fi = volume fraction of ith constituent
MHA
Apparent Modulus
Voigt
Reuss Mc 1
Volume Fraction of HA

28. Expectations after this section
Why study bone mechanics?
Methods of testing to obtain orthotropic mechanical properties
“Apparent” properties
Mixture theory for homogeneous elastic composites
What can you do with properties?

29. Further References Class Website: “Bone Introduction”
BME 315 Notes: Bone, Beams, and Torsion
Ref F (Bone)
Ref F (more Bone Mechanics)
“Bounds on bone stiffness”
“Bone Issues and Pathology”
“von Mises Stress”
“Failure and Fatigue in Bone”
R. Lakes’ Website: silver.neep.wisc.edu/~lakes/

30. Parameters σf = Stress in outer fibers at midpoint, (MPa)
εf = Strain in the outer surface, (mm/mm)
P = load at a given point on the load deflection curve, (N)
L = Support span, (mm)
y = distance from neutral axis
ρ = radius of curvature
back
P is the applied load,
B is the thickness of the specimen,
a is the crack length,
W is the width of the specimen