structuralelement(bone)

tensileelement(ligament/tendon)

elasticelement(tendon)

actuator(muscle)

Lecture #4: Material Properties I

Outline:Part 1: Performance of MaterialsPart 2: The Long and Short of Bows

force

length

‘stuff’ tester

1. How do we measure and assess performance of mechanical elements?

AreaL

L

normalize force and length:

stress () = force / cross sectional areastrain () = 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 (1789-1857)

length

force

canonical stress-strain plot

strain ()

stre

ss ()

failure

many important material properties are derived from stress-strain plot:

1. stiffness or modulus2. strength3. extensibility4. toughness5. resiliance

force

length

‘stuff’ tester

strain ()

stre

ss ()

stiffness (or modulus)

failurestiffness, E:

E =

A.k.a. ‘Young’s Modulus’

note, has unitsof stress (Pa)

slope at a givenpoint = stiffness

materialcan have differentstiffness at different% strain

e.g. substance X has a modulus of 750 Pa.

strain ()

stre

ss ()

strength and extensibility

failure

strength= stress at failure(‘breaking stress’) units of stress (Pa)

extensibility= strain at failure(‘breaking strain’)

units of strainextensibility

strength

e.g. substance X has a strength of 25 kPa and an extensibility of 7%.

strain ()

stre

ss ()

toughness

failure

extensibility

strength

Work = force dx

Thus, area understress-strain curveis a form of normalizedwork.

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 ()

stre

ss ()

resilience

Work ofextension

work ofcontraction

net work

e.g. substance X exhibits 85% resilance.

Reslience =

work of contractionwork of extension

A measurement ofenergy recovered fromelastic storage.

Dimensionless valueexpressed as %.

Different types of deformation

test section

F

L

TENSION

‘tensile modulus’, E

COMPRESSION

‘compressive modulus’, E

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 AgincourtOct. 25,1415

Henry V

length

forc

e

biomechanics of ‘long bows’

human reach(0.6 meters)

humanStrength(350 N)

energy stored In bow =105 Joules

Europeanyew

tendon

horn or bonewood

resist tension

resist compression

biomechanics of composite bows

Odysseus stringing his bow

length

forc

e

biomechanics of composite bows

human reach(0.6 meters)

humanstrength(350 N)

energy stored In bow =170 Joules

Initialtension

biomechanics of true catapults

Roman BallistaProjectile: 2-150 kg ballsRange: 400 meters

600040 kg stone ballsfound at Carthage

Build your own!

Lecture #5: Material Properties II(breaking stuff )

Outline:Part 1: AneurismsPart 2: CracksPart 3: Collagen

Benefits of the ‘J-shaped’ curve

force

length

AreaL

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)

1L

‘Engineering’ vs. ‘True’ stress and strain