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Tendon

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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 “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 Force Displacement ( x) Toe region Linear region

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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 “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 Force Displacement ( x) Toe region Linear region

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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 From Wainwright et al. (1976). “Mechanical design in organisms”. Displacement Force Fast stretch Slow stretch At faster stretching velocities: 1. More force needed to rupture tendon

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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 Time 2-10 min ( 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 From Wainwright et al. (1976). “Mechanical design in organisms”. Displacement Force Fast stretch Slow stretch At faster stretching velocities: 1. More force needed to rupture tendon 2. More energy is stored

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

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For normal stretches, 90-95% of the elastic energy stored in tendons & ligaments is returned Force Displacement Energy lost 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 Force Displacement Elastic energy stored during stretch Area = A) ½ Fx B) ½ kx C) ½ kx 2 D) A & B E)A & C (x,F)

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Achilles elastic energy storage during stance phase of run FgFg Example of important equations: U elastic = 0.5 k ( L) 2 F = k L Known: k Achilles = 260 kN/m F = 4700 N U elastic = ? A)2.34 B)42120 C) 42 D)0.042 E) None of the above F Achilles

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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 ( ) = L-L o /L o – L: current length – L o :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 ( )=Force/Area

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Stress/Strain ( ) 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 Injury Failure (rupture) Stress (MN/m 2 ) Strain 100 8% Elastic region Plastic region Toe region E yield failure

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Stress vs. Strain for tendon/ligament Similar for all mammalian tendons & ligaments Elastic modulus: slope E=stress/strain, = – units of Pascals (N/m 2 ), same as stress – kPa, Mpa, GPa Strain (%) Stress (MN/m 2 ) Stretch Recoil 70 5 35 2.5 0 0

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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= 1000 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 / m 2 = Pa Strain (%) = (displacement / rest length) 100 = ( L / 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 causes k

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Extending the stress-strain relationship to injurious loads for tendon/ligament Injury Failure (rupture) Stress (MN/m 2 ) Strain 100 8% Elastic region Plastic region

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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 (P 0 ): ~ 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. 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 Wrist extensor – k = 15 kN/m – F (muscle) = 60 N – L = F / k = 0.004 m Achilles tendon – k = 260 kN/m – F (muscle) = 4.7 kN – L = F / k = 0.018 m Force Displacement Achilles Wrist ext.

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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 / m 2

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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 10 9 N/m 2 for tendon stress = F/A strain = L/L 10,000 cm 2 = 1 m 2 Useful tendon equations

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E = 1.5 x 10 9 N/m 2 for both tendons wristAchilles L = 0.17 mL = 0.29m A = 1.67 x 10 -6 m 2 A = 0.00005 m 2 k = EA/L = 15 kN/m k = EA/L = 260 kN/m elongation for 60N load? elongation for 4,700N load? L = F/k = 0.004mF/k = 0.018m Strain? = L/L = 0.004 / 0.17 = 2.4% = 0.018 / 0.29 = 6.2% Human Tendons Compared

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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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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

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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: A achilles = 0.00005 m 2 ; L wrist = 0.17m Practice Problem

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