Lecture #5: Material Properties II (breaking stuff ) Outline: Part 1: Aneurisms Part 2: Cracks Part 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 = pressure r = radius T = tension (units: Force/length) Tension r p p0p0 ‘hoop’ tension governed by LaPlace’s Law: ( ) Part I: Aneurisms This is why we don’t make square plumbing!!!

      stable response stiffness anuerism  p x r p = pressure r = radius T = tension (units: Force/length) Tension r p p0p0

Part II: Cracks material properties do not predict failure vs. Cracked material much more vulnerable to failure force ‘stress lines’ in direction of strain stress concentration: force/area is higher r L background stress,  concentrated stress =  + 2   (L/r) How much is stress increased? C.E. Inglis

One solutions to crack propagation stress lines increase r bone echinoderm ossicles This is why ships & Airplanes have portholes, not rectangular windows.

energy balance of crack propagation tensile load 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 L 2 (A proportional to L 2 ) stress relieved In area A length of new crack = L

L 0 energy cost (-) energy gain (+) crack length (L) energy net energy critical crack length d(Energy)/dL= 0 L c = Work of fracture  Strain energy A.A. Griffith Strain energy = ½  L c = 2 W f   2 W f E    or energy balance of crack propagation

composite material L c = 2 W f   2 W f E    = How to avoid crack propagation? 1. Increase work of fracture: increased crack length Lamellar frameworks create tough materials.

tough materials Tough materials give rough breaks. shatter vs. cleave Wood will do both depending on direction. e.g. wood

3 microns e.g. nacre (mother of pearl) tough materials

2. Decrease strain energy at fracture extension:   ‘J-shaped curve’ Less energy stored in material to ‘drive crack’ consider balloon: Elevated elastic energy drives a catastrophic failure from a pin prick. L c = 2 W f   2 W f 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

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: 1231 glycine

fiber within fiber construction:

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

safety factor = material strength maximum stress experienced