Mechanics of Biomaterials

1
Lecture 3 Mechanics of Biomaterials
Course Web
http//www.aeromech.usyd.edu.au/people/academic/qi
ngli/MECH4981.htm
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 Model
F
M
T
Recall Lecture 1 statics/dynamics methods to
determine force/moment/torque
T
M
F
4
Introductory Mechanics Model Stress Analysis

• Sport injury?
• Bone damage?

Normal stress
Motion Measurement
Pure bending analysis
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
• Hookes law, Youngs modulus, Poissons ratio
  etc
• Hookes Law (uniaxial)
• ? the strain is directly proportional to the
  stress
• Hookes Law (General)
• ? Stress tensor ?
• ? Strain tensor ?
• ? Stiffness tensor S (Stiffness tensor)

?L
L
? Compliance tensor CS-1
7
Elastic Constants Youngs Modulus

• Youngs Modulus E
• Relationship between tensile or compressive
  stress and strain
• Applies for small strains (within the elastic
  range)

http//www.lib.umich.edu/dentlib/Dental_tables/t
oc.html
8
Uniaxial Test Finite Large Deformation

• Undeformed Configuration
• ? length L
• ? Undeformed area A
• Deformed Configuration
• ? length l
• ? Deformed area a

Density ? 0

• Cauchy Stress (True stress)

• Nominal Stress (Engineering Stress)

• Second Piola-Kirchhoff Stress

9
Elastic Constants (other 4 constants)

• Poissons 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)
• Lames constant ? from tensor production

10
Relationship Between the Elastic Constants

• Youngs modulus (E)
• Poissons ratio (?)
• Bulk modulus (K)
• Shear modulus (G)
• Lames constant (?)
• For an isotropic material, elastic constants are
  CONSTANT

11
Hookes 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
Hookes 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)
16
Orthotropic Properties Cortical Bone

• E1 6.91 - 18.1 GPa
• E2 8.51 - 19.4 GPa
• E3 17.0 - 26.5 GPa
• G12 2.41 - 7.22 GPa
• G13 3.28 - 8.65 GPa
• G23 3.28 - 8.67 GPa
• ?ij 0.12 - 0.62

Youngs Moduli
Shear Moduli
Poissons 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
18
Material Constitutive Models Isotropy
2 independent components
19
Hookes Law for an Isotropic Elastic Material
Stress-Strain Relationship
Strain-Stress Relationship
20
Hookes Law (Isotropic) Contd
where ?ij Kronecker delta, ?ij 1 if ij,
otherwise (i?j), ?ij 0. That is
e.g.
21
Mechanics Model of Introductory Example
y (2)
x (1)
en
ez
z (3)
et
22
Mechanics of Introductory Example Contd
x (1)
en
ez
F3
F3
z (3)
et
23
Mechanics of Introductory Example Contd
Pure Bending
y (2)
x (1)
Myy
ez
z (3)
et
Mxx
Total stress in zz
Eccentric Axial Loading
24
Equilibrium Equations (General)
Where
div - Divergence
Dynamic equilibrium
25
Biomechanical Test Method
Site-specific test
Femoral neck test
26
Finite Element Method


Femur
Knee
Hip
27
CT-Based Finite Element Modelling Procedure
Molar
PDL
FE model
d) FE model
a) CT Image Segmentation
c) CAD model
b) Sectional curves
Whole Jaw model Computationally more accurate
Part of model Computationally more efficient
28
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

Section S-S
y
y
F
T
B
S
Cancellous
Fixed
R
yh
z
x
A
r
S
M
x
Cortical
l
l

• Submission of tutorial question of callus
  formation mechanics 10