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Lecture 01 stress, strain and elasticity

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- 1. Lecture 1 Stress, strain and elasticity
- 2. Linear deformation of a solid L0 + ∆L F F ∆L = Cross-sectional area A 1 F L0 Y A Proportionality factor Young’s modulus F Y = A ∆L L0 stress strain (deformation)
- 3. Elastic moduli Hooke’s law: stress strain = constant (elastic modulus) This works for small strains. Remember springs? F =k ∆x (We absorbed A and L0 into k) Unit of stress: SI: Pascal 1 Pa = 1 N/m2 US: psi (pounds per square inch)
- 4. Young’s modulus Y = F A ∆L L0 Measure of stiffness Material Young’s modulus (GPa) Rubber (small strain) 0.01-0.1 wood 1-10 bone 9-16 concrete 20 steel 200
- 5. In-class example: Young’s modulus When a tensile stress of 5 × 106 Pa is applied to the ends of a bar, it is desired that the strain be about 5 × 10−4. The most appropriate material to use for this bar would be: A. Rubber ( Y ~ 3 × 107 Pa) B. Wood ( Y ~ 1 × 1010 Pa) C. Brass ( Y ~ 1 × 1011 Pa) D. New material ( Y ~ 1012 Pa) E. None of these come close. stress 5 × 106 Pa Y = = = 1010 Pa strain 5 × 10 −4
- 6. Pressure in a fluid DEMO: Plastic glass with cover Fluid = liquid or gas Particles are always moving i.e., hitting surfaces i.e., exerting (perpendicular) forces on surfaces Pressure F⊥ p= A Surface of area A F⊥ (units: Pa) • does not depend on orientation of surface • increases with depth
- 7. Bulk stress and strain F⊥ Put it at the bottom of Michigan lake pressure p0 pressure p0 + ∆p volume V0 ∆p Bulk B =− modulus ∆V V0 volume V0 + ∆V stress strain (∆V < 0) Compressibility 1 k = B
- 8. Bulk modulus GAS: compressible, small B (density depends a lot on pressure) LIQUID and SOLIDS: (nearly) incompressible, large B (density almost “constant”) H2O Gas (STP) 104 105 Pb 106 107 108 109 Bulk modulus (Pa) 1010 Steel 1011
- 9. Shear modulus Top (or bottom) area A F// h F// Shear modulus FP S = x A h stress strain x Shear angle θ strain = tan θ = x h
- 10. Example: Jello cube N A Jello cube of side d = 4 cm is placed on a 10° incline. It tilts to an angle of 12°. What is the shear modulus of Jello? θ = 12° fS fS mg φ = 10° FP = mg sin ϕ = fS m = d 3 ρJello ≈ d 3 ρwater A =d 2 strain = tan θ d 3 ρwater g sin ϕ d 2 = d ρwater g sin ϕ S = tan θ tan θ 4 × 10 −2 m 103 kg/m3 10 m/s2 sin10° = ~ 300 Pa tan12° ( )( )( ) mg sinφ
- 11. Beyond Hooke’s law (Permanent deformations) (Reversible deformations)
- 12. A little more realistic
- 13. Compressive vs. tensile strength Material Tensile strength (MPa) Compressive strength (MPa) steel 500 500 cast iron 170 550 concrete 2 20 marble - 80 wood (parallel to grain) 40 35 bone 130 170
- 14. Arches Material Tensile strength (MPa) Compressive strength (MPa) concrete 2 20 marble - 80 wood (parallel to grain) 40 35 compression Mostly compression tension Ok with wood, not stone Good design for stone

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