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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
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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)
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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
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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.
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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%.
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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.
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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 %.
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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
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1. The Long and Short of Bows
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Battle of AgincourtOct. 25,1415
Henry V
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length
forc
e
biomechanics of ‘long bows’
human reach(0.6 meters)
humanStrength(350 N)
energy stored In bow =105 Joules
Europeanyew
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tendon
horn or bonewood
resist tension
resist compression
biomechanics of composite bows
Odysseus stringing his bow
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length
forc
e
biomechanics of composite bows
human reach(0.6 meters)
humanstrength(350 N)
energy stored In bow =170 Joules
Initialtension
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biomechanics of true catapults
Roman BallistaProjectile: 2-150 kg ballsRange: 400 meters
600040 kg stone ballsfound at Carthage
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Build your own!
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Lecture #5: Material Properties II(breaking stuff )
Outline:Part 1: AneurismsPart 2: CracksPart 3: Collagen
Benefits of the ‘J-shaped’ curve
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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