Biomedical Engineering

1. Mechanical Properties of Materials
Alan Cottenden, January 2014
MSc in Physics and Engineering in Medicine
Module MPHY3B21 / MPHYMB21 / MPHYGB21

2. Extension (x)
Load(P)
P
P
x
Fig 1: Tensile Load / Extension Curve for a Typical Ductile
Material

3. Fig 2: Load / Extension Curves for Different Size Rods
Extension (x)
Load(P)
Cross section
area, A
Length, L
Decreasing L, constant A
DecreasingA,constantL

4. L
Fig 3: Definitions of Stress and Strain for Tension
A
PStress == σ
L
xStrain == ε
P
P
A
x
Tension

5. Fig 4: Young’s Modulus and Yield Point for a Typical Ductile
Material
Strain (x/L)
Stress(P/A)
Y
Slope, E

6. Fig 5: Tensile Loading / Unloading Curve for a Typical Ductile
Material
Strain
Stress
Loading Unloading
Permanent
strain
Y

7. Fig 6: Typical Tensile Stress / Strain Curve for a Ductile
Material
Strain
Stress
Y
U

8. Fig 7: Typical Tensile Stress /Strain Curve for a Brittle Material
Strain
Stress
Y
U

9. Extension (x)
Load(P)
P
P
x
Fig 8: Work Done from a Load / Extension Curve
Area under
curve = total
work done

10. Fig 9: Work Done per Unit Original Volume of Material
Strain
Stress
Area under
curve = work
done per unit
volume

11. Fig 10: Energy Consumed by Plastic Deformation
Strain
Stress
Area in loop = energy
cost per unit volume
Loading Unloading

12. Fig 11: Hysteresis from Loading /Unloading Cycle on Rubber
Area in loop
= energy cost
per unit
volume per
cycle
Strain
Stress Loading
Unloading

13. L
A
P
P
x
Fig 12: Definitions of Stress and Strain for Tension and
Compression
A
PStress == σ
L
xStrain == ε
P
P
A
x
Tension Compression

14. Fig 13: Form of Typical Compression Stress / Strain Curve
Strain
Stress

15. Fig 14: Definitions of Stress and Strain for Shear
A
PStress == σ
L
xStrain == ε
x
P
P
A
L
Shear

16. Fig 15: Bend is a Combination of Tension (Outside of the
Curve) and Compression (Inside of Curve)
Inertia
Rugby tackle
Inertia
Tension
Compression

17. Torsion (shear)
Compression
Fig 16: Combination of Compression and Torsion in a Leg
Bone from Landing Heavily and Twisting Simultaneously

18. Fig 17: Leg Orthosis Containing Carbon Fibre / Polyester
Strips to Impart Toughness
Blunting by PET (yellow)
of a crack starting in
carbon fibre bundle (black)

19. Fig 18: Isotropic (left) Versus Anisotropic (right)
Mechanical Properties
Εr
Εr
Εz
Εz
Εy
ΕxΕx
Εy
Εy
Εr
Εr
Εz
Εz
Εy
ΕxΕx
Εy
rzyx EEEE === rzyx EEEE ≠≠≠

20. Fig 19: Micrograph of Osteons in Cortical Bone
300 µm

21. Fig 20: Osteon Structure (Expanded Telescopically for
Clarity)
Concentric lamellae (yellow)
showing preferred direction of
collagen fibres (black arrows)
varying between lamellae.

22. Fig 21: Typical Tensile Stress / Strain Curve for Cortical Bone
0
10
20
30
40
50
60
70
80
90
100
0 0.005 0.01 0.015
Strain
Stress(Mpa)

23. Fig 22: Equal Strain Model for Composites
σc σh
σb
σb
σc
σh
εb = εc = εh

24. Fig 23: Equal Stress Model for Composites
εc εh
εb
εb
εc
εh
σb = σc = σh

25. Fig 24: Cancellous Bone
1 mm

26. Fig 25: Cortical and Cancellous Bone in a long bone

27. Fig 26: X-ray Image of Bone Plates on the ulna and radius, in
the arm.

28. l
l
Radius of
curvature at
crack tip, r
Fig 27: Crack Geometry Parameters

29. Fig 28: Stress Distribution Round a Crack
σo
σo
σ








+=
r
l
oc .21σσ
l
r = radius of
curvature at
crack tip
σo

30. Fig 29: Model for Crack Blunting in Bone
Hydroxyapatite
Collagen-rich
tissues

31. Fig 30: Typical Stress / Strain Curves for Soft Tissues
)( βσα
ε
σ
+=
d
d )1)(exp( −= αεβσ
σ
ε
σ
d
d
σ
ε

32. Fig 31: Creep and Relaxation
Time
Displacement
Force
Time
Creep Relaxation
Constant
force
Constant
displacement

33. Fig 32: Spring characteristic
Force,F Displacement, uF
L
u
ukF .=

34. Fig 33: Dashpot Characteristic
Force,F
Displacement rate, u
Fu
uF .η=
Viscous fluid

35. Fig 34: Maxwell Model
Displacement
Time
Load
k
F
k
F
u u F
k
η

36. Fig 35: Voigt Model
u u F Displacement
Time
Load
k
F
k η
η
F
Initial
gradient

37. Fig 36: Kelvin Model
Displacement
Time
Load
k1
u u
k2
F
( )2k
F
( )21 kk
F
+

38. Weight off leg Reaction from ground
Weight of body
Fig 37: Bone plate prone to fatigue failure

39. Fig 38: Prosthetic hip joint