1. Topic 5: 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
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. Linear Elasticity Constitutive Law
In terms of the Strain Energy, W:

4. Elasticity Tensor, Cijkl
fourth-order tensor of elastic constants
34 = 81 components
symmetry conditions:
Tij = Tji ekl = elk  6x6 = 36 independent constants
2W = Cijkleijekl = Cklijekleij
 Cijkl = Cklij  leaves 21 independent constants
simplest special case – Isotropy:
l and m are the Lamé constants

5. Isotropic Hookean Solids: Technical Constants
Measured from standard tests
Uniaxial test:
Young’s modulus, E = slope of the stress-strain curve
Poisson ratio, n = (-) ratio transverse:axial strain
T11 = Ee11= l(e11 + e22 + e33) + 2me11
= l(e11 – ne11 – ne11) + 2me11
 E = l(1 – 2n) + 2m
T22 = 0 = l(e11+e22+e33) + 2me22
= l(–e22/n + e22 + e22) + 2me22

6. Isotropic Hookean Solids: Technical Constants
Shear modulus, G = half slope of the shear stress vs. shear strain curve
For i  j, Tij = 2meij
 G = m
Bulk modulus, K = mean stress s0 divided by volume change, D (dilatation)

7. Stiffness Matrix [cij]
Represent the stress and strain tensors as column matrices
[si] = [T11, T22, T33, T23, T13, T12]T
[ei] = [e11, e22, e33, 2e23, 2e13, 2e12]T
[si] = [cij][ej]
[cij] is the (6x6) stiffness matrix
e.g. for isotropic Hookean materials

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

9. 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

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

11. Orthotropy: Compliance Matrix

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

13. Transverse Isotropy E1 and E2 are similar compared with E3
Similarly, n31 and n32 are close compared with n21
 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:
c11=c22
c13=c23
c44=c55
c66=0.5(c11-c12)

14. Transverse Isotropy: Compliance Matrix[ ] ú û ù ê ë é - = f t ij 1 G E s n

15. Equations of Linear Elastostatics
Equilibrium Equations
Constitutive Law
(isotropic)
Kinematic Relations

16. Navier’s Equations

17. Example Problem in Linear Elasticity: Simple Torsion of a Circular Shaft
Displacements in cylindrical polar coordinates:
a = twist per unit length
Displacements in rectangular Cartesian coordinates:
See section 12.4 (pages in YC Fung. A First Course in Continuum Mechanics, 3rd Ed., 1994

18. Simple Torsion of a Circular Shaft: Stress and Strain Components
Stress components (from the Constitutive Law):

19. Simple Torsion of a Circular Shaft: Equilibrium and Boundary Conditions
Equilibrium Equations (no body forces):
Boundary Conditions:

20. Resultant Forces and Moments
Integrating stresses over cross-section to get force resultants:
Resultant Torsional Moment, M:
G = shear modulus
GJ = torsional rigidity
a = twist per unit length

21. 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.
The equilibrium equations, together with the constitutive equation for linear elasticity and the strain-displacement relation give us Navier’s equations of linear elastostatics.
They are used to solve boundary value problems for bone.