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 importantNot 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 movementStore elastic energyTendon properties affect force transmitted from muscle to bone
5 Structure Primarily collagen : a structural protein Collagen fibril -> fascicle->tendonBad blood supply -> slow to heal
6 Parallel bundles of collagen fibers Resist stretching along long axis of tendonSufficiently flexible
7 Tendon Outline: Function Structure Mechanical Properties Significance to movement
8 Mechanical Properties Many experiments on isolated tendonsShow same mechanical property across different tendons
10 Tendon or ligament Linear Toe region Force Displacement (Dx) “J-shaped”Stiffness (k) = slopeunits = N/mStiffness: force required to stretch tendon/ligament by a unit distanceForce per change in lengthHooke’s LawF=kxF=elastic forcex=amount of stretchk=stiffness
11 Tendons/ligaments are viscoelastic Purely elastic materialsforce-displacement relationship does NOT depend on velocity of stretch or time held at a length or loadViscoelastic materialsforce-displacement relationship DOES depend on:Velocity of stretchingTime held at a given length or loadThink of other materials that are viscoelastic?
12 Tendons are viscoelastic Nonlinear responseHysteresisVelocity dependent loadingCreepLoad relaxation
13 Viscoelasticity trait #1: Nonlinear Response ForceDisplacement (Dx)ToeregionLinear“J-shaped”Stiffness (k) = slopeunits = N/mStiffness: force required to stretch tendon/ligament by a unit distanceForce per change in lengthHooke’s LawF=kxF=elastic forcex=amount of stretchk=stiffness
14 Viscoelasticity trait #2: Hysteresis (Stretch & recoil: ) Displacement (x)ForceStretchRecoilHysteresis:Force vs. displacement different for stretch & recoil
15 Viscoelasticity trait #3: velocity dependent stiffness Fast stretchSlow stretchForceAt faster stretching velocities:1. More force needed to rupture tendonDisplacementFrom Wainwright et al. (1976). “Mechanical design in organisms”.
16 Viscoelasticity trait #4: Creep DisplacementTimeStretched with a constant force & displacement measuredLength increases with time
17 Viscoelasticity trait #5: Load relaxation Specimen held at a constant length & force measuredForce2-10 minTime( N & F, Fig 3-10)
18 Elastic energyStretch: mechanical work done on tendon/ligament equals elastic energy storageArea under force - displacement curveForceDisplacementElastic energy stored during stretch
19 Viscoelasticity trait #3: velocity dependent stiffness Fast stretchSlow stretchForceAt faster stretching velocities:1. More force needed to rupture tendon2. More energy is storedDisplacementFrom Wainwright et al. (1976). “Mechanical design in organisms”.
20 Elastic energyStretch: mechanical work done on tendon/ligament equals elastic energy storageArea under force - displacement curveRecoil: material returns some (most) of energy stored elastically during stretch
21 Mechanical energy stored & returned by tendon/ligament ForceForceDisplacementDisplacementElastic energystored duringstretchElastic energyreturned duringrecoil
22 For normal stretches, 90-95% of the elastic energy stored in tendons & ligaments is returned Energy lostForceDisplacementLarger hysteresis loop - greater energy loss • Hysteresis: indicates “viscoelasticity”
23 Elastic energyStretch: mechanical work done on tendon/ligament equals elastic energy storageArea under force - displacement curveArea =½ Fx½ kx½ kx2A & BA & C(x,F)ForceDisplacementElastic energy stored during stretch
24 Achilles elastic energy storage during stance phase of run Example of important equations:Uelastic = 0.5 k (DL)2F = kDLKnown:kAchilles = 260 kN/mF = 4700 NUelastic = ?A)2.34B)42120C) 42D)0.042E) None of the aboveFAchillesFg
25 Strain Can measure length change in terms of mm But more useful as % of original length, so can compare tendons of different lengthsStrain (e) = L-Lo/LoL: current lengthLo:original length‘stretchiness’
26 StressBecause tendons have different thickness, want to normalize force as wellThicker tendons need more force and vice versaSo normalize by areaStress (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 shapeMeasure intrinsic tendon properties
28 Stress/Strain Relation for Tendon/Ligament PlasticregionStress s (MN/m2)syield100sfailureElastic regionFailure(rupture)Toe regionInjuryEse8%Strain e
29 Stress vs. Strain for tendon/ligament (MN/m2)StretchRecoil705352.5Similar for all mammalian tendons & ligamentsElastic modulus: slopeE=stress/strain, =s/eunits of Pascals (N/m2), same as stresskPa, 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 GPaSoft wood (pine) E = 0.6 GPaPassive muscle E = 10kPaRubber E = 20kPaBone E= 20 GPaWalnut E= 15 GpaDiamond E= GpaJello E = 1Pa
32 Stress vs. strain: material not geometry Two important definitions:Stress = F / AF = force; A = cross-sect. areaUnits = N / m2 = PaStrain (%) = (displacement / rest length) • 100= (DL / L) • 100
33 Stiffness vs. Elastic Modulus Elastic Modulus (a.k.a. “Young’s Modulus”)Slope of stress-strain relationshipa material propertyStiffnessSlope of force-displacement relationshipdepends on :material (modulus) & geometryStructural property
34 Stress/Strain vs Force/Length Material property vs. structural propertyStress/Strain ind of geometryForce/Length (stiffness) depends on geometry.
35 Geometry effects Stress = Elastic modulus • Strain F / A = E • ∆L / L Force = Stiffness • displacementF = k∆LCombine (1) & (2) to find: k = EA/LE: similar in all tendons/ligamentsA or L causesk
36 Extending the stress-strain relationship to injurious loads for tendon/ligament PlasticregionStress (MN/m2)100Elastic regionFailure(rupture)Injury8%Strain
37 Stress/Strain vs Force/Length Material property vs. structural propertyStress/Strain ind of geometryForce/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 propertiesdifferent geometry (architecture)
41 High force vs. versus fine control Muscles in arm/hand demand fine controlprecision more important than energySlinky vs. rope
43 Ankle extensor tendon vs. wrist extensor tendon k = 15 kN/mF (muscle) = 60 NDL = F / k = mAchilles tendonk = 260 kN/mF (muscle) = 4.7 kNDL = F / k = mAchillesForceWrist 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. tendonk: 17 times greaterGeometric differences?A: 30 times greaterL: 1.75 times longerk = EA/LE ~ 1.5 GN / m2
46 Useful tendon equations F = k LElastic Energy = 0.5 k (L)2Elastic Energy = 1/2 F Lk = A/L elastic modulus = stress/strain~ 1.5 x 109 N/m2 for tendonstress = F/Astrain = L/L10,000 cm2 = 1 m2
47 Human Tendons Compared E = 1.5 x 109 N/m2 for both tendonswrist AchillesL = 0.17 m L = 0.29mA = 1.67 x 10-6 m A = m2k = EA/L = 15 kN/m k = EA/L = 260 kN/melongation for 60N load? elongation for 4,700N load?L = F/k = 0.004m F/k = 0.018mStrain?= L/L = / 0.17 = 2.4% = / 0.29 = 6.2%
48 Problem Solving Approach Write down what is givenWrite down what you need to findWrite down the equations you will useShow work!Step by step
49 Practice ProblemDesign 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 ProblemIf 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