1. Tendon

3. Tendon Outline: Function Structure Mechanical Properties
Significance to movement

4. Function Connect muscle to bone, but are not rigid Are quite stretchy
Passive but important
Not just rigid, passive structural links b/n muscle and bone, but also affect movement through the overall function of the muscle-tendon-unit.
Function:
transmit muscle force and slide during movement
Store elastic energy
Tendon properties affect force transmitted from muscle to bone

5. Structure Primarily collagen : a structural protein
Collagen fibril -> fascicle->tendon
Bad blood supply -> slow to heal

6. Parallel bundles of collagen fibers
Resist stretching along long axis of tendon
Sufficiently flexible

7. Tendon Outline: Function Structure Mechanical Properties
Significance to movement

8. Mechanical Properties
Many experiments on isolated tendons
Show same mechanical property across different tendons

10. Tendon or ligament Linear Toe region Force Displacement (Dx)
“J-shaped”
Stiffness (k) = slope
units = N/m
Stiffness: force required to stretch tendon/ligament by a unit distance
Force per change in length
Hooke’s Law
F=kx
F=elastic force
x=amount of stretch
k=stiffness

11. Tendons/ligaments are viscoelastic
Purely elastic materials
force-displacement relationship does NOT depend on velocity of stretch or time held at a length or load
Viscoelastic materials
force-displacement relationship DOES depend on:
Velocity of stretching
Time held at a given length or load
Think of other materials that are viscoelastic?

12. Tendons are viscoelastic
Nonlinear response
Hysteresis
Velocity dependent loading
Creep
Load relaxation

13. Viscoelasticity trait #1: Nonlinear Response
Force
Displacement (Dx)
Toe
region
Linear
“J-shaped”
Stiffness (k) = slope
units = N/m
Stiffness: force required to stretch tendon/ligament by a unit distance
Force per change in length
Hooke’s Law
F=kx
F=elastic force
x=amount of stretch
k=stiffness

14. Viscoelasticity trait #2: Hysteresis (Stretch & recoil: )
Displacement (x)
Force
Stretch
Recoil
Hysteresis:
Force vs. displacement different for stretch & recoil

15. Viscoelasticity trait #3: velocity dependent stiffness
Fast stretch
Slow stretch
Force
At faster stretching velocities:
1. More force needed to rupture tendon
Displacement
From Wainwright et al. (1976). “Mechanical design in organisms”.

16. Viscoelasticity trait #4: Creep
Displacement
Time
Stretched with a constant force & displacement measured
Length increases with time

17. Viscoelasticity trait #5: Load relaxation
Specimen held at a constant length & force measured
Force
2-10 min
Time
( N & F, Fig 3-10)

18. Elastic energy
Stretch: mechanical work done on tendon/ligament equals elastic energy storage
Area under force - displacement curve
Force
Displacement
Elastic energy stored during stretch

19. Viscoelasticity trait #3: velocity dependent stiffness
Fast stretch
Slow stretch
Force
At faster stretching velocities:
1. More force needed to rupture tendon
2. More energy is stored
Displacement
From Wainwright et al. (1976). “Mechanical design in organisms”.

20. Elastic energy
Stretch: mechanical work done on tendon/ligament equals elastic energy storage
Area under force - displacement curve
Recoil: material returns some (most) of energy stored elastically during stretch

21. Mechanical energy stored & returned by tendon/ligament
Force
Force
Displacement
Displacement
Elastic energy
stored during
stretch
Elastic energy
returned during
recoil

22. For normal stretches, 90-95% of the elastic energy stored in tendons & ligaments is returned
Energy lost
Force
Displacement
Larger hysteresis loop - greater energy loss • Hysteresis: indicates “viscoelasticity”

23. Elastic energy
Stretch: mechanical work done on tendon/ligament equals elastic energy storage
Area under force - displacement curve
Area =
½ Fx
½ kx
½ kx2
A & B
A & C
(x,F)
Force
Displacement
Elastic energy stored during stretch

24. Achilles elastic energy storage during stance phase of run
Example of important equations:
Uelastic = 0.5 k (DL)2
F = kDL
Known:
kAchilles = 260 kN/m
F = 4700 N
Uelastic = ?
A)2.34
B)42120
C) 42
D)0.042
E) None of the above
FAchilles Fg

25. Strain Can measure length change in terms of mm
But more useful as % of original length, so can compare tendons of different lengths
Strain (e) = L-Lo/Lo
L: current length
Lo:original length
‘stretchiness’

26. Stress
Because tendons have different thickness, want to normalize force as well
Thicker tendons need more force and vice versa
So normalize by area
Stress (s)=Force/Area

27. Stress/Strain (s/e)
By normalizing stress and strain, can now compare properties of materials of different sizes and shapes, regardless of absolute shape
Measure intrinsic tendon properties

28. Stress/Strain Relation for Tendon/Ligament
Plastic
region
Stress s (MN/m2)
syield
100
sfailure
Elastic region
Failure
(rupture)
Toe region
Injury E s e 8%
Strain e

29. Stress vs. Strain for tendon/ligament
(MN/m2)
Stretch
Recoil 70 5 35
2.5
Similar for all mammalian tendons & ligaments
Elastic modulus: slope
E=stress/strain, =s/e
units of Pascals (N/m2), same as stress
kPa, Mpa, GPa

30. Compare the stiffness of a rubber band and a block of soft wood
A) rubber band is more stiff B) rubber band is less stiff C) stiffness is similar D) Not enough information

31. Can compare different materials easily
Tendon E = 1 GPa
Soft wood (pine) E = 0.6 GPa
Passive muscle E = 10kPa
Rubber E = 20kPa
Bone E= 20 GPa
Walnut E= 15 Gpa
Diamond E= Gpa
Jello E = 1Pa

32. Stress vs. strain: material not geometry
Two important definitions:
Stress = F / A
F = force; A = cross-sect. area
Units = N / m2 = Pa
Strain (%) = (displacement / rest length) • 100
= (DL / L) • 100

33. Stiffness vs. Elastic Modulus
Elastic Modulus (a.k.a. “Young’s Modulus”)
Slope of stress-strain relationship
a material property
Stiffness
Slope of force-displacement relationship
depends on :
material (modulus) & geometry
Structural property

34. Stress/Strain vs Force/Length
Material property vs. structural property
Stress/Strain ind of geometry
Force/Length (stiffness) depends on geometry.

35. Geometry effects Stress = Elastic modulus • Strain F / A = E • ∆L / L
Force = Stiffness • displacement
F = k∆L
Combine (1) & (2) to find: k = EA/L
E: similar in all tendons/ligaments
A or L causesk

36. Extending the stress-strain relationship to injurious loads for tendon/ligament
Plastic
region
Stress (MN/m2)
100
Elastic region
Failure
(rupture)
Injury 8%
Strain

37. Stress/Strain vs Force/Length
Material property vs. structural property
Stress/Strain ind of geometry
Force/Length (stiffness) depends on geometry.

38. Tendon strain Achilles tendon during running: ~ 6%
close to strain where injury occurs (~ 8%)
Wrist extensor due to muscle force (P0): ~ 2%

39. Tendon Outline: Function Structure Mechanical Properties
Significance to movement

40. We need tendons with different stiffnesses for different functions
We need tendons with different stiffnesses for different functions. How is this accomplished?
Possibilities:
different material properties
different geometry (architecture)

41. High force vs. versus fine control
Muscles in arm/hand demand fine control
precision more important than energy
Slinky vs. rope

43. Ankle extensor tendon vs. wrist extensor tendon
k = 15 kN/m
F (muscle) = 60 N
DL = F / k = m
Achilles tendon
k = 260 kN/m
F (muscle) = 4.7 kN
DL = F / k = m
Achilles
Force
Wrist ext.
Displacement

44. Basis for tendon stiffness variation?
different material properties?
different geometry (architecture)?

45. Achilles tendon vs. wrist extensor tendon
Achilles tendon vs. wrist ext. tendon
k: 17 times greater
Geometric differences?
A: 30 times greater
L: 1.75 times longer
k = EA/L
E ~ 1.5 GN / m2

46. Useful tendon equations
F = k L
Elastic Energy = 0.5 k (L)2
Elastic Energy = 1/2 F L
k = A/L
 elastic modulus = stress/strain
~ 1.5 x 109 N/m2 for tendon
stress = F/A
strain = L/L
10,000 cm2 = 1 m2

47. Human Tendons Compared
E = 1.5 x 109 N/m2 for both tendons
wrist Achilles
L = 0.17 m L = 0.29m
A = 1.67 x 10-6 m A = m2
k = EA/L = 15 kN/m k = EA/L = 260 kN/m
elongation for 60N load? elongation for 4,700N load?
L = F/k = 0.004m F/k = 0.018m
Strain?
= L/L = / 0.17 = 2.4% = / 0.29 = 6.2%

48. Problem Solving Approach
Write down what is given
Write down what you need to find
Write down the equations you will use
Show work!
Step by step

49. Practice Problem
Design a wrist extensor tendon that when loaded with 60N of force will undergo the same %strain (6.2%) as the Achilles tendon.
(Given L, determine A)
L=0.17 m

50. Practice Problem
If the wrist extensor tendon in the example had a cross sectional area = to the Achilles tendon example, what would be the absolute length change with a load of 60 N?
Given: Aachilles = m2; Lwrist = 0.17m