3Tendon Outline: Function Structure Mechanical Properties Significance to movement
4Function 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
5Structure Primarily collagen : a structural protein Collagen fibril -> fascicle->tendonBad blood supply -> slow to heal
6Parallel bundles of collagen fibers Resist stretching along long axis of tendonSufficiently flexible
7Tendon Outline: Function Structure Mechanical Properties Significance to movement
8Mechanical Properties Many experiments on isolated tendonsShow same mechanical property across different tendons
10Tendon 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
11Tendons/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?
12Tendons are viscoelastic Nonlinear responseHysteresisVelocity dependent loadingCreepLoad relaxation
13Viscoelasticity 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
14Viscoelasticity trait #2: Hysteresis (Stretch & recoil: ) Displacement (x)ForceStretchRecoilHysteresis:Force vs. displacement different for stretch & recoil
15Viscoelasticity 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”.
16Viscoelasticity trait #4: Creep DisplacementTimeStretched with a constant force & displacement measuredLength increases with time
17Viscoelasticity trait #5: Load relaxation Specimen held at a constant length & force measuredForce2-10 minTime( N & F, Fig 3-10)
18Elastic energyStretch: mechanical work done on tendon/ligament equals elastic energy storageArea under force - displacement curveForceDisplacementElastic energy stored during stretch
19Viscoelasticity 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”.
20Elastic 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
21Mechanical energy stored & returned by tendon/ligament ForceForceDisplacementDisplacementElastic energystored duringstretchElastic energyreturned duringrecoil
22For normal stretches, 90-95% of the elastic energy stored in tendons & ligaments is returned Energy lostForceDisplacementLarger hysteresis loop - greater energy loss • Hysteresis: indicates “viscoelasticity”
23Elastic 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
24Achilles 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
25Strain 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’
26StressBecause tendons have different thickness, want to normalize force as wellThicker tendons need more force and vice versaSo normalize by areaStress (s)=Force/Area
27Stress/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
28Stress/Strain Relation for Tendon/Ligament PlasticregionStress s (MN/m2)syield100sfailureElastic regionFailure(rupture)Toe regionInjuryEse8%Strain e
29Stress 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
30Compare 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
31Can 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
32Stress 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
33Stiffness 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
34Stress/Strain vs Force/Length Material property vs. structural propertyStress/Strain ind of geometryForce/Length (stiffness) depends on geometry.
35Geometry 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
36Extending the stress-strain relationship to injurious loads for tendon/ligament PlasticregionStress (MN/m2)100Elastic regionFailure(rupture)Injury8%Strain
37Stress/Strain vs Force/Length Material property vs. structural propertyStress/Strain ind of geometryForce/Length (stiffness) depends on geometry.
38Tendon strain Achilles tendon during running: ~ 6% close to strain where injury occurs (~ 8%)Wrist extensor due to muscle force (P0): ~ 2%
39Tendon Outline: Function Structure Mechanical Properties Significance to movement
40We 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)
41High force vs. versus fine control Muscles in arm/hand demand fine controlprecision more important than energySlinky vs. rope
43Ankle 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
44Basis for tendon stiffness variation? different material properties?different geometry (architecture)?
45Achilles 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
46Useful 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
47Human 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%
48Problem Solving Approach Write down what is givenWrite down what you need to findWrite down the equations you will useShow work!Step by step
49Practice 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
50Practice 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