Tendon
Tendon
• Outline:
– Function
– Structure
– Mechanical Properties
– Significance to movement
Function
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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
Structure
• Primarily collagen : a structural protein
• Collagen fibril -> fascicle->tendon
• Bad blood supply -> slow to heal
• Parallel bundles of collagen fibers
• Resist stretching along long axis of tendon
• Sufficiently flexible
Tendon
• Outline:
– Function
– Structure
– Mechanical Properties
– Significance to movement
Mechanical Properties
• Many experiments on isolated tendons
• Show same mechanical property across
different tendons
Tendon or ligament
Force
Linear
region
• “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
Toe
region
Displacement (Dx)
– F=kx
• F=elastic force
• x=amount of stretch
• k=stiffness
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?
Tendons are viscoelastic
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Nonlinear response
Hysteresis
Velocity dependent loading
Creep
Load relaxation
Viscoelasticity trait #1:
Nonlinear Response
Force
Linear
region
• “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
Toe
region
Displacement (Dx)
– F=kx
• F=elastic force
• x=amount of stretch
• k=stiffness
Viscoelasticity trait #2:
Hysteresis (Stretch & recoil: )
Force
Stretch
Recoil
• Hysteresis:
Force vs. displacement
different for
stretch & recoil
Displacement (x)
Viscoelasticity trait #3:
velocity dependent stiffness
Fast stretch
Force
Slow stretch
At faster stretching velocities:
1. More force needed to rupture tendon
Displacement
From Wainwright et al. (1976). “Mechanical design in organisms”.
Viscoelasticity trait #4:
Creep
Displacement
• Stretched with a
constant force &
displacement
measured
– Length increases
with time
Time
Viscoelasticity trait #5:
Load relaxation
• Specimen held at a
constant length &
force measured
Force
2-10 min
Time
( N & F, Fig 3-10)
Elastic energy
• Stretch: mechanical work done on
tendon/ligament equals elastic energy storage
Force
– Area under force - displacement curve
Displacement
Elastic energy stored during stretch
Viscoelasticity trait #3:
velocity dependent stiffness
Fast stretch
Force
Slow stretch
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”.
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
Mechanical energy stored & returned by
tendon/ligament
Force
Force
Displacement
Elastic energy
stored during
stretch
Displacement
Elastic energy
returned during
recoil
For normal stretches, 90-95% of the elastic energy stored in
tendons & ligaments is returned
Energy lost
Force
• Larger hysteresis loop greater energy loss
• Hysteresis: indicates
“viscoelasticity”
Displacement
Elastic energy
• Stretch: mechanical work done on
tendon/ligament equals elastic energy storage
– Area under force - displacement curve
(x,F)
A) ½ Fx
B) ½ kx
C) ½ kx2
D) A & B
E) A & C
Force
Area =
Displacement
Elastic energy stored during stretch
Achilles elastic energy storage during stance
phase of run
Example of important equations:
Uelastic = 0.5 k (DL)2
F = kDL
FAchilles
Known:
kAchilles = 260 kN/m
F = 4700 N
Uelastic = ?
Fg
A)2.34
B)42120
C) 42
D)0.042
E) None of the above
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’
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
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
Stress/Strain Relation for
Tendon/Ligament
Plastic
region
Stress s (MN/m2)
100
syield
Toe region
Elastic region
sfailure
Failure
(rupture)
E
s
Injury
e
8%
Strain e
Stress vs. Strain for tendon/ligament
• Similar for all
mammalian tendons &
ligaments
• Elastic modulus: slope
• E=stress/strain, =s/e
(MN/m2)
70
Stretch
35
– units of Pascals (N/m2),
same as stress
– kPa, Mpa, GPa
Recoil
0
0
2.5
Strain (%)
5
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
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= 1000 Gpa
Jello E = 1Pa
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
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
Stress/Strain vs Force/Length
• Material property vs. structural property
• Stress/Strain ind of geometry
• Force/Length (stiffness) depends on geometry.
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
Extending the stress-strain relationship to
injurious loads for tendon/ligament
Stress (MN/m2)
100
Plastic
region
Elastic region
Failure
(rupture)
Injury
8%
Strain
Stress/Strain vs Force/Length
• Material property vs. structural property
• Stress/Strain ind of geometry
• Force/Length (stiffness) depends on geometry.
Tendon strain
• Achilles tendon during running: ~ 6%
– close to strain where injury occurs (~ 8%)
• Wrist extensor due to muscle force (P0): ~ 2%
Tendon
• Outline:
– Function
– Structure
– Mechanical Properties
– Significance to movement
We need tendons with different
stiffnesses for different functions.
How is this accomplished?
Possibilities:
– different material properties
– different geometry (architecture)
High force vs. versus fine control
• Muscles in arm/hand demand fine control
– precision more important than energy
Slinky vs. rope
Ankle extensor tendon vs. wrist
extensor tendon
• Wrist extensor
– k = 15 kN/m
– F (muscle) = 60 N
– DL = F / k = 0.004 m
• Achilles tendon
– k = 260 kN/m
– F (muscle) = 4.7 kN
– DL = F / k = 0.018 m
Force
Achilles
Wrist ext.
Displacement
Basis for tendon stiffness variation?
– different material properties?
– different geometry (architecture)?
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
Useful tendon equations
F = k DL
Elastic Energy = 0.5 k (DL)2
Elastic Energy = 1/2 F DL
k = A/L
 elastic modulus = stress/strain
~ 1.5 x 109 N/m2 for tendon
stress = F/A
strain = DL/L
10,000 cm2 = 1 m2
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 m2
A = 0.00005 m2
k = EA/L = 15 kN/m
k = EA/L = 260 kN/m
elongation for 60N load?
DL = F/k = 0.004m
Strain?
= DL/L = 0.004 / 0.17 = 2.4%
elongation for 4,700N load?
F/k = 0.018m
= 0.018 / 0.29 = 6.2%
Problem Solving Approach
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Write down what is given
Write down what you need to find
Write down the equations you will use
Show work!
– Step by step
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
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 = 0.00005 m2; Lwrist = 0.17m