structural element (bone) tensile element (ligament /tendon) elastic element (tendon) actuator (muscle) Lecture #4: Material Properties I Outline: Part 1: Performance of Materials Part 2: The Long and Short of Bows

force length ‘stuff’ tester 1. How do we measure and assess performance of mechanical elements? Area LL L normalize force and length: stress (  ) = force / cross sectional area strain (  ) = change in length / total length Force Units: Stress: force per unit area (M T -2 L -1 ) SI unit = Pascal (newton per square meter) Strain: length per unit length (dimensionless!) typically given in percent: e.g. 50 mm increase in length of 1 meter structure is a strain of + 5%. Augustin Cauchy ( )

length force canonical stress-strain plot strain (  ) stress (  ) failure many important material properties are derived from stress-strain plot: 1.stiffness or modulus 2.strength 3.extensibility 4.toughness 5.resiliance force length ‘stuff’ tester

strain (  ) stress (  ) stiffness (or modulus) failure stiffness, E: E =  A.k.a. ‘Young’s Modulus’ note, has units of stress (Pa) slope at a given point = stiffness material can have different stiffness at different % strain e.g. substance X has a modulus of 750 Pa.

strain (  ) stress (  ) strength and extensibility failure strength = stress at failure (‘breaking stress’) units of stress (Pa) extensibility = strain at failure (‘breaking strain’) units of strain extensibility strength e.g. substance X has a strength of 25 kPa and an extensibility of 7%.

strain (  ) stress (  ) toughness failure extensibility strength Work =  force dx Thus, area under stress-strain curve is a form of normalized work. units of stress (Pa) Toughness fits best our intuition of ‘strength’. area = work required break substance =TOUGHNESS e.g. substance X has a toughness of 1000 Pa.

strain (  ) stress (  ) resilience Work of extension work of contraction net work e.g. substance X exhibits 85% resilance. Reslience = work of contraction work of extension A measurement of energy recovered from elastic storage. Dimensionless value expressed as %.

Different types of deformation test section FF LL TENSION ‘tensile modulus’ , E COMPRESSION ‘compressive modulus’ , E FF A SHEAR ‘shear strain,  ’ (angular deflection) ‘shear stress,  ’ ‘shear modulus, G’  shear stress,  = force/area

1. The Long and Short of Bows

Battle of Agincourt Oct. 25, 1415 Henry V

length force biomechanics of ‘long bows’ human reach (0.6 meters) human Strength (350 N) energy stored In bow = 105 Joules European yew

tendon horn or bone wood resist tension resist compression biomechanics of composite bows Odysseus stringing his bow

length force biomechanics of composite bows human reach (0.6 meters) human strength (350 N) energy stored In bow = 170 Joules Initial tension

biomechanics of true catapults Roman Ballista Projectile: kg balls Range: 400 meters kg stone balls found at Carthage

Build your own!

Lecture #5: Material Properties II (breaking stuff ) Outline: Part 1: Aneurisms Part 2: Cracks Part 3: Collagen Benefits of the ‘J-shaped’  curve

force length Area LL L stress (  ) = F / A 0 strain (  ) =  L / L 0 Force Engineering units But…what if strain is large? Area will decrease and we will underestimate stress. True units: stress (  ) = F / A (  ) strain (  ) = ln ( L / L 0 ) strain (  ) = dL = ln ( L / L 0 ) 1L1L ‘Engineering’ vs. ‘True’ stress and strain