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Tendon

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**Tendon Outline: Function Structure Mechanical Properties**

Significance to movement

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**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

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**Structure Primarily collagen : a structural protein**

Collagen fibril -> fascicle->tendon Bad blood supply -> slow to heal

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**Parallel bundles of collagen fibers**

Resist stretching along long axis of tendon Sufficiently flexible

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**Tendon Outline: Function Structure Mechanical Properties**

Significance to movement

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**Mechanical Properties**

Many experiments on isolated tendons Show same mechanical property across different tendons

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**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

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**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?

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**Tendons are viscoelastic**

Nonlinear response Hysteresis Velocity dependent loading Creep Load relaxation

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**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

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**Viscoelasticity trait #2: Hysteresis (Stretch & recoil: )**

Displacement (x) Force Stretch Recoil Hysteresis: Force vs. displacement different for stretch & recoil

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**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”.

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**Viscoelasticity trait #4: Creep**

Displacement Time Stretched with a constant force & displacement measured Length increases with time

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**Viscoelasticity trait #5: Load relaxation**

Specimen held at a constant length & force measured Force 2-10 min Time ( N & F, Fig 3-10)

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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

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**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”.

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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

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**Mechanical energy stored & returned by tendon/ligament**

Force Force Displacement Displacement Elastic energy stored during stretch Elastic energy returned during recoil

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**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”

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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

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**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

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**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’

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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

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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

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**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

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**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

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**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

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**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

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**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

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**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

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**Stress/Strain vs Force/Length**

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

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**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 causesk

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**Extending the stress-strain relationship to injurious loads for tendon/ligament**

Plastic region Stress (MN/m2) 100 Elastic region Failure (rupture) Injury 8% Strain

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**Stress/Strain vs Force/Length**

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

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**Tendon strain Achilles tendon during running: ~ 6%**

close to strain where injury occurs (~ 8%) Wrist extensor due to muscle force (P0): ~ 2%

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**Tendon Outline: Function Structure Mechanical Properties**

Significance to movement

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**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)

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**High force vs. versus fine control**

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

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**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

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**Basis for tendon stiffness variation?**

different material properties? different geometry (architecture)?

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**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

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**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

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**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%

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**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

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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

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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

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