Tendon
Tendon Outline: Function Structure Mechanical Properties Significance to movement
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
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 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
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 Nonlinear response Hysteresis Velocity dependent loading Creep Load relaxation
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
Viscoelasticity trait #2: Hysteresis (Stretch & recoil: ) Displacement (x) Force Stretch Recoil Hysteresis: Force vs. displacement different for stretch & recoil
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”.
Viscoelasticity trait #4: Creep Displacement Time Stretched with a constant force & displacement measured Length increases with 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 Area under force - displacement curve Force Displacement Elastic energy stored during stretch
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”.
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 Displacement Elastic energy stored during stretch Elastic energy returned during recoil
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”
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
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
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) syield 100 sfailure Elastic region Failure (rupture) Toe region Injury E s e 8% Strain e
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
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 causesk
Extending the stress-strain relationship to injurious loads for tendon/ligament Plastic region Stress (MN/m2) 100 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 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 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 Achilles Force 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 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
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? elongation for 4,700N load? L = F/k = 0.004m F/k = 0.018m Strain? = L/L = 0.004 / 0.17 = 2.4% = 0.018 / 0.29 = 6.2%
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
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