1. Bone Mechanics Bone is a hard connective tissue Forms rigid skeleton
Yield strain is small < 0.01
Elastic modulus is high (18 GPa) compared with normal working stresses
Stress-strain relation is linear in elastic range
Strain-rate dependence of stress is minor in normal conditions
Bone is frequently approximated as a linear (Hookean) elastic material
2011: This is too long to cover the boundary value solution
19 January 1999
This fits comfortably in one lecture (80 mins). Not time for Bone remodeling which should be a separate lecture. Rather than cram in more stuff, add some explanatory detail to existing material such as:
1. index and/or direct notation form for orthotropic or transversely isotropic strain energy or stress-strain relation (Decided against it jan 2000)
2. Stiffness or compliance matrix for transverse isotropy (added for TI jan 2000)
3. more explanation of boundary conditions in torsion example (looks ok, jan 2000)
It would be worth adding simple beam problem as an alternative example that could either be (a) shown if time or (b) shown instead of torsion problem in years when torsion is given as homework or design assignment.

2. Bone Anisotropy Bone is a composite Bone has organized microstructure
mineral matrix
collagen fibers
Bone has organized microstructure
lamellar (layered)
Haversian (tubular)
trabecular (spongy, fabric-like)
Elastic moduli vary with type of loading:
tension – compression
bending – shear
Elastic moduli vary with orientation
transverse vs. axial
Bone is anisotropic
requires more than two elastic constants
It is hard to allow enough time for the example but topic 6 is shorter so there is plenty of time in the two lectures combined. Plan to revisit the example the next time before getting into bone remodeling. That worked well this time. However additions are needed:
Generic introduction to boundary conditions after Navier’s equations
More explanation of torsion problem including a separate slide showing boundary conditions.
A more direct derivation? Maybe mainly in polar coords
A more detailed discussion of GJ and the moment twist relation plus an analogy with the moment curvature relation and I. Equations for J (and I) for other cross-sections.
A more stepwise approach to the derivation with each step on a separate slide.

3. Constitutive Law for Linear (Hookean) Elasticity
Elasticity tensor, Cijkl, a 4th-order tensor of material constants
34 = 81 components
symmetry conditions:
Tij = Tji εkl = εlk → 6x6 = 36 independent constants
Increment of work (strain energy) dW =Tijdεij =Cijklεkldεij
⇒ W = ∫Cijklεkldεij = ½Cijklεklεij = ½Cklijεijεkl
⇒ Cijkl = Cklij → leaves 21 independent constants
simplest special case – Isotropy:
λ and μ are the Lamé constants

4. Isotropic Hookean Solids: Technical Constants
Measured from standard tests
Uniaxial test:
Young’s modulus, E = slope of the stress-strain curve
Poisson ratio, ν = (-) ratio transverse:axial strain
T11 = Eε11=λ(ε11 + ε22 + ε33) + 2με11
= λ(ε11 – νε11 – νε11) + 2με11
→ E = λ(1 – 2ν) + 2μ
T22 = 0 = λ(ε11+ε22+ε33) + 2με22
= λ(–ε22/ν + ε22 + ε22) + 2με22

5. Isotropic Hookean Solids: Technical Constants
Shear modulus, G = half slope of the shear stress vs. shear strain curve
For i ≠ j, Tij = 2μεij
⇒ G = μ
Bulk modulus, K = mean stress σ0 divided by volume change, Δ (dilatation)

6. Stiffness Matrix [cij]
Represent the stress and strain tensors as column matrices
[σi] = [T11, T22, T33, T23, T13, T12]T
[ei] = [ε11, ε22, ε33, 2ε23, 2ε13, 2ε12]T
[σi] = [cij][ej]
[cij] is the (6x6) stiffness matrix
e.g. for isotropic Hookean materials

7. Compliance Matrix [sij]
The inverse of the stiffness matrix
[ei] = [sij] [σj]
[sij] is the (6x6) compliance matrix
e.g. for isotropic Hookean solids, in terms of the technical constants:

8. Orthotropy bone often assumed to be orthotropic
different properties in the three mutually perpendicular directions: 3 Young's moduli; 3 shear moduli; 3 independent Poisson ratios
→3 uniaxial tests and 3 plane shear tests
structural axes of orthotropic symmetry are defined by bone microstructure
Long bone structural axes
(1) radial
(2) circumferential
(3) longitudinal
As for isotropy, stiffness matrix has 12 non-zero components, but 9 independent values instead of 2

9. Orthotropy: Stiffness Matrix
Technical constants
3 Young's moduli for uniaxial strain along each axis, Ei
6 Poisson ratios, νij for strain in the j-direction when loaded in the i-direction (i ≠ j)
νijEj = νjiEi (no sum) leaving 3 independent Poisson ratios
3 shear moduli, Gij = Gji for shear in the i-j plane

10. Orthotropy: Compliance Matrix

11. Transverse Isotropy E1 and E2 are similar compared with E3
Similarly, ν31 and ν32 are close compared with ν21
⇒ greater differences between axial and transverse directions than between radial and circumferential
Transversely isotropic materials
one preferred (“fiber”) axis, i.e. long axis of the bone
in long bones, the "fibers" are the osteons
isotropic properties in plane transverse to fibers
stiffness matrix simplifies from 9 to 5 independent constants Et, Ef, 𝜈f, 𝜈t, Gf:
c11=c22
c13=c23
c44=c55
c66=0.5(c11-c12)

12. Technical Constants for Human Bone
From SC Cowin, Chapter 2 in Handbook of Bioengineering, 1987

13. Bone Growth and Remodeling
Bone continually remodels
growth, reinforcement, resorption
depends on stress and strain
There is an optimal range of stress for maximum strength
understressed or overstressed bone can weaken
stresses on fractured bone affect healing
stress-dependent remodeling affects surgical implant and prosthesis design, e.g. fracture fixation plates, surgical screws, artificial joints
1978: radiographic evidence of bone resorption seen in 70% of total hip replacement patients
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

14. Stress-Dependent Remodeling
Osteoclasts - cells responsible for resorption
Osteoblasts - cells responsible for growth
compressive stress stimulates formation of new bone and is important for fracture healing
loss of normal stress → loss of calcium and reduced bone density
Time scales:
fastest remodeling is due to change in mineral content
healing - weeks
remodeling - months/years
growth/maturation - years
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

15. Types of Bone Remodeling
Two types of remodeling in bone:
1. surface (external) remodeling
change in bone shape and dimensions
deposition on to or resorption of bone material from inner or outer surfaces
2. internal remodeling
change in:
bulk density
trabecular size
orientation
osteon size, etc.
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

16. Functional Adaptation and Optimal Design
Principal of Functional Adaptation, Roux (1895):
“the ability of organs (and cells, tissues and organisms) to adapt their capacity to function in response to altered demands by practice”
Functional adaptation in bone is remodeling of structure, shape & mechanical properties in response to altered loading
Related to the engineering concept of optimal design, e.g.
Theory of Uniform Strength — attempts to produce the same maximum normal stress (brittle material) or shear stress (ductile material) throughout the body for a specific loading
Theory of Trajectorial Architecture — concentrates material in the paths of force transmission, such as principal stress lines, e.g. fiber reinforcing of composite (kevlar-mylar) sails
Principle of Maximum-Minimum Design — maximize strength for minimum weight or cost
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

17. Stress Adaptation of Trabecular Bone
G.H. von Meyer’s trabecular bone architecture in human femur (1867)
Principal stress trajectories of Culmann’s crane
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a few more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures as they become available.

18. Remodeling of Trabecular Bone: Wolff's Law
Wolff (1872): when loads are changed by trauma or change in activity, functional remodeling reorients bone trabeculae so they align with the new principal stress axes
Wolff never actually proved this
Wolff's “law of bone transformation” (1884): “there is a perfect mathematical correspondence between the structure of cancellous bone in proximal femur and Culmann’s trajectories”
Culmann’s trajectories and other of Wolff’s assertions were suspect, but photoelastic studies (Pauwels,1954) confirmed Wolff's law
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

19. Bone Mechanics: Key Points
Under physiological loads, bone can be assumed Hookean elastic with a high elastic modulus (10-20 GPa)
The microstructure of the bone composite makes the material response anisotropic.
Compared with an isotropic Hookean elastic solid which has two independent technical constants, transversely isotropic linearly elastic solids have five independent elastic constants and orthotropic Hookean solids have nine.
For human cortical bone orthotropy is a somewhat better assumption than transverse isotropy, but transverse isotropy is a much better approximation than isotropy.
Internal remodeling results in altered bone properties
External remodeling results in altered bone size or shape
Bone growth and remodeling is stress-adaptive
Wolff’s law described how trabecular bone reorients when principal stress axes change