Lecture #5: Material Properties II(breaking stuff )

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

Benefits of the ‘J-shaped’ curve

Work of fracture (a.k.a ‘toughness’)is inversely proportional to stiffness

1. Stiff materials tend to be brittle.2. Compliant materials often

display ‘J-shaped’ curve

What are advantages of ‘J-shaped curve’ ?

1) Stability in biological plumbing

2) Resistance to crack propagation

Consider vessel with small bulge:If pressure increases,what will happen?

p x r

p = pressurer = radius T = tension(units: Force/length)

Tension

r

pp0

‘hoop’ tension governed by LaPlace’s Law:

(1749 -1827)

Part I: Aneurisms

This is why we don’t make square plumbing!!!

stableresponse

stiffness

stiffnessstiffness

anuerism

p x r

p = pressurer = radius T = tension(units: Force/length)

Tension

r

pp0

Part II: Cracksmaterial properties do not predict failure

vs.Cracked materialmuch more vulnerableto failure

force

‘stresslines’ indirection of strain stress

concentration:force/area ishigher

r

L

backgroundstress,

concentrated stress = + 2(L/r)

How much is stressincreased?

C.E. Inglis1875-1952

One solutions to crack propagation

stress lines

increase r

bone

echinodermossicles

This is why ships & Airplanes have portholes,not rectangular windows.

energy balance of crack propagation

tensileload

stress

Energy required to open crack of length, L = Work of Fracture x Length(energy/area x distance)

Energy liberated by opening crack of = Strain Energy x Area

Thus energy cost rises linearly with L,Energy gain rises with L2 (A proportional to L2)

stressrelievedIn area A

length of new crack = L

L

0

energycost(-)

energygain(+)

crack length (L)

ener

gy

net energy

critical crack length

d(Energy)/dL= 0Lc = Work of fracture Strain energy

A.A. Griffith1893-1963

Strain energy = ½

Lc = 2 Wf

2 Wf E

or

energy balance of crack propagation

compositematerial

Lc = 2 Wf

2 Wf E

=

How to avoid crack propagation?

1. Increase work of fracture:

increasedcrack length

Lamellar frameworks create tough materials.

tough materials

Tough materialsgive rough breaks.shatter vs. cleave

Wood will do bothdepending on direction.

e.g. wood

3 microns

e.g. nacre(mother of pearl)

tough materials

2. Decrease strain energy at fracture extension:

‘J-shapedcurve’

Less energy stored in materialto ‘drive crack’

consider balloon:

Elevated elastic energy drives a catastrophic failure from a pin prick.

Lc = 2 Wf

2 Wf E

=

How to avoid crack propagation?

How do you make a material with a ‘J-shaped’ curve?

1. In series (e.g. helix-loop-helix)

Mix rods and springs in clever way:

e.g. titin

2. In disordered linked array

e.g. nuchal ligaments

elastin ‘springs’linking

collagen ‘rods’

Part III: Collagen

Part III: Collagen

Most common protein in vertebrate body BY FAR!20% of a mouse by weight.

33% glycine, 20% hydroxyproline

Each tropo-collagen fiber held together by hydrogen bonds involving central glycines:

1 2 3 1

glycine

fiberwithin fiberconstruction:

Julian Voss-Andreae's sculpture Unraveling Collagen (2005)

safety factor = material strength

maximum stress experienced