1. Mechanics of Biomaterials
Course Web

2. Objectives Establish biomaterial constitutive models
Determine the biomechanical response to load
Analyse the prosthetic design
Estimate the health status of living tissues under stress

3. Introductory Mechanics ModelF M T
Recall “Lecture 1”: statics/dynamics methods to determine force/moment/torque T M F

4. Dynamics analysis to determine load
Introductory Mechanics Model – Stress Analysis
Sport injury?
Bone damage?
Normal stress
Motion Measurement
Pure bending analysis M T F M y x z
Dynamics analysis to determine load

5. Methods of Biomechanics
Analytical Method – Solid Mechanics I and II
Biomechanical Experiment – Test
Numerical Techniques – FEM

6. Elastic Behavior Basic element representing an elastic material
Hooke’s law, Young’s modulus, Poisson’s ratio etc
Hooke’s Law (uniaxial):
 the strain is directly proportional to the stress
Hooke’s Law (General):
 Stress tensor []
 Strain tensor []
 Stiffness tensor [S] (Stiffness tensor) L L
 Compliance tensor [C]=[S]-1

7. Elastic Constants – Young’s Modulus
Young’s Modulus E:
Relationship between tensile or compressive stress and strain
Applies for small strains (within the elastic range)
Biomaterials (Isotropic)
E (GPa)*
Cancellous bone
0.49
Cortical bone
14.7
Long bone - Femur
17.2
Long bone - Humerus
Long bone - Radius
18.6
Long bone - Tibia
18.1
Vertebrae - Cervical
0.23
Vertebrae - Lumbar
0.16 *

8. Uniaxial Test – Finite Large Deformation
Undeformed Configuration
 length = L
 Undeformed area = A
Deformed Configuration
 length = l
 Deformed area = a L A
Density  0 a F l
Density  L L
Cauchy Stress (True stress)
Nominal Stress (Engineering Stress)
Second Piola-Kirchhoff Stress

9. Elastic Constants – (other 4 constants)
Poisson’s ratio
Describe lateral deformation in response to an axial load
Shear Modulus
Describes relationship between applied torque and angle of deformation
Bulk Modulus
Describes the change in volume in response to hydrostatic pressure
(equal stresses in all directions)
Lame’s constant  – from tensor production

10. Relationship Between the Elastic Constants
Young’s modulus (E)
Poisson’s ratio ()
Bulk modulus (K)
Shear modulus (G)
Lame’s constant ()
For an isotropic material, elastic constants are CONSTANT

11. Hooke’s Law – Tensor Representation
(1  x, 2  y, 3  z) or
Remarks:
Stress tensor and strain tensor are the 2nd order tensors
[S] and [C] are the fourth order tensor

12. Hooke’s Law – Matrix Representation
Compliance Matrix

13. Material Constitutive Models
Anisotropy
21 independent components elasticity matrix
Orthotropy
9 independent components to elasticity matrix
Transverse isotropy
5 independent components
Isotropy
2 independent components

14. Material Constitutive Models – Anisotropy
(Most likely) 21 independent components in elasticity matrix
Symmetric matrix

15. Material Constitutive Models – Orthotropy
9 independent components to elasticity matrix (along 3 directions) 1 2 3

16. Orthotropic Properties – Cortical Bone
Young’s Moduli
E1: GPa
E2 : GPa
E3 : GPa
G12: GPa
G13: GPa
G23: GPa
ij:
Shear Moduli
Poisson’s Ratios
Remarks: the high standard deviations in property values seen in one are not necessarily (although may possibly be) due to experimental error
 E: 15%
 G: 10%
  : 30%

17. Material Constitutive Models – Transversely Isotropy
5 independent components 1 2 3

18. Material Constitutive Models – Isotropy
2 independent components 1 2 3

19. Hooke’s Law for an Isotropic Elastic Material
Stress-Strain Relationship
Strain-Stress Relationship

20. Hooke’s Law (Isotropic) – Cont’d
where ij – Kronecker delta, ij =1 if i=j, otherwise (i≠j), ij =0. That is
e.g.

21. Mechanics Model of Introductory Exampleen
x (1) ez
z (3) et

22. Mechanics of Introductory Example – Cont’den
x (1) ez F3 F3
z (3) et

23. Mechanics of Introductory Example – Cont’d
Pure Bending
y (2)
x (1)
Myy ez
z (3) et
Mxx
Total stress in zz: x y
Eccentric Axial Loading

24. Equilibrium Equations (General)
Where:
div - Divergence
Dynamic equilibrium:

25. Biomechanical Test Method
Femoral neck test
Site-specific test

26. Finite Element Method
Femur
Knee
Hip

27. CT-Based Finite Element Modelling Procedure
Molar
PDL
FE model
a) CT Image Segmentation
b) Sectional curves
c) CAD model
d) FE model
Whole Jaw model
Computationally more accurate
Part of model
Computationally more efficient

28. 3 unit all-ceramic dental bridge analysis
Finite Element Modelling Example
3 unit all-ceramic dental bridge analysis
Solid model
VM stress Contour

29. Assignment
Approximately use engineering beam theory to calculate principal stresses – 60%
 Mohr circles
 Nature of stress (tension or compression)
Apply 3D finite element method to calculate the principal stress – 30%
 Selection of elements and mesh density
 Contours of principal stress
 Comparison against analytical solution from Beam Theory y
Section S-S y T F
Fixed S B
Cancellous yh R z x A r S M x
Cortical l l
Submission of tutorial question of callus formation mechanics – 10%