Agricultural Science 1B Principles & Processes in Agriculture Mike Wheatland (m.wheatland@physics.usyd.edu.au)
Outline - Lectures weeks 9-12 Chapter 6: Balance in nature - description of energy balance in the atmosphere Chapter 7: Properties of water - surface tension, pressure, specific heat capacity Chapter 8: Materials: Elasticity & Viscosity - stress and strain, rheology Chapter 9: Farm machinery: Friction & Lubrication - friction Chapter 10: Farm machinery: Stability - Newton s laws, torque Chapter 11: Farm machinery: Vibrations - oscillations, resonance
Amended timetable
Chapter 8: Materials: Elasticity & Viscosity
Stress and strain Elasticity: deformation of bodies under pairs of (small) applied forces Strain: relative change in dimensions of body (no units) Stress: applied force per unit area (same units as pressure, e.g. Pa) stress/strain = a property of the material (a `modulus ) Three simple deformations are linear extension, uniform compression, and shear
A = h w Linear tension or compression L " L +!L Stress S = F / A w " w -!L F h " h -!h L w L +!L Strain e =!L / L Stress/strain = Young s modulus Y so Y = S/e = FL/(AΔL) F h Transverse contraction described by Poisson ratio σ:!h/h =!w/w = - # e = -# (!L/L)
Uniform compression Stress = p Strain = -!V/ V (the significance of the minus sign is that the volume decreases as the pressure increases) p Example: hydrostatic pressure stress/strain = bulk modulus B
"L Shear F L! Area, A stress/strain = shear modulus G F Stress = F / A strain =! ΔL/L = "L / L! is small = tan -1 θ θ (small θ)
Linear extension: - Young s modulus Y = stress/strain = (F/A)/(ΔL/L) - Force proportional to extension (Hooke s law) Uniform compression: - Bulk modulus B = stress/strain = -p/(δv/v) Shear: - Shear modulus (modulus of rigidity) G = stress/strain = (F/A)/θ These moduli are fundamental properties of materials Substance Young s Modulus Y / 10 10 Pa Bulk Modulus B/ 10 10 Pa Shear Modulus G / 10 10 Pa aluminum 7.05 6 to 8 steel 19 to 21 16 to 18 8 to 9 glass (crown) 6.5 to 7.8 4 to 6 2 to 3 water 0.2 0 mercury 2.1 0 air (atmospheric pressure) 1.4!10-5 0
Elastic and non-elastic behaviour Mild steel subject to linear extension OP: elastic regime P: elastic limit Y: yield point JK: fluid behaviour KL: elastic LM: plastic flow B: fracture
Elastic regime: Hooke s law applies (F = kδl) - if stress is removed, strain returns to zero - body returns to original length Beyond the elastic regime, the body is permanently deformed - if stress is removed, strain does not return to zero Plastic flow: material behaves like a fluid - strain increases at a fixed stress - stress/strain = 0 This is ductile behaviour - other materials are brittle, e.g. concrete stress P Y J A brittle material strain
Strength and fatigue Strength is the stress which causes a material to break - linear extension: compressive and tensile strength - shear: shear strength Materials can fail at stresses << strength if subject to repeated stresses - e.g. bend a paper click back and forth - the material becomes `fatigued - very important to monitor for fatigue e.g. in aircraft bodies
More complicated deformations Example: horizontal bar under a force - the top of the bar is in compression, the bottom in tension - bending strength depends on shape of bar as well as elastic properties of material
- beams like a. maximise area A at top, bottom and so minimize stress F/A there a. I shape beam b. Tube shape beam c. Solid beam d. Solid beam - a. commonly used in building - hollow beams are more resistant to bending than solid beams, for a given weight per unit length
Failure by buckling single walled roll of paper roll of paper with several walls Hollow tubes are prone to buckling under (linear) compression
Agricultural applications 1. Trees as structural elements Trees are weight-bearing columns Observed that diameter D, height h are related: D α h 1.5 (i.e. D = Ch 1.5, C is constant) This is NOT due to the requirement that the stress due to weight at base < strength - in that case height would be independent of D Column fails by bending: turns out that this implies h < CD 2/3, C is constant
2. Mammalian bones and scaling Compressional strength of bones is 200 MPa - peak stresses in mammalian bones measured to be 50 MPa From: http://fathom.lib.uchicago.edu/2/21701757/ Stress "(mass) 1/ 3
Safety factor of 3-5 No relationship between peak stress and body mass If large mammals were `scaled up versions of small mammals, we would expect peak stress α (mass) 1/3 Large mammals have different proportions From: http://fathom.lib.uchicago.edu/2/21701757/ (From Galileo, Two New Sciences 1638)
Solids, liquids, & rheological materials Solids subject to shear forces deform quickly and then cease deforming (in the elastic regime) - when the forces are removed they return to their original state (they recover) F F Liquids subject to shear forces deform and continue to deform - when the forces are removed they do not recover
Newton s law of viscosity "L F Area, A L! F Stress = F / A strain =! = "L / L! is small For Newtonian fluids: shear stress = constant x rate of shear deformation F / A = " d# dt viscosity
Viscosity η is a measure of the `thickness of liquids - units are Pa s - originates in intermolecular forces Viscosity controls flow in a pipe, leading to Poiseuille s law: Q = " R4 #P 8$L where Q is volume flow rate, R is pipe radius, L is pipe length, and ΔP is pressure drop along pipe - strong dependence on radius - double radius: flow rate is 16x larger! - explains why narrowing of arteries increases blood pressure
Rheological materials Rheological materials cover plastics and non-newtonian fluids Plasticity: there is a threshold stress above which flow occurs Non-Newtonian fluids: do not obey Newton s law - viscosity may depend on applied stress rate - viscosity may depend on duration of applied stress Visco-elastic materials: exhibit viscous and elastic behaviours - e.g. recover for short applied stresses, not for longer
10 Viscosity of blood Blood is non-newtonian - red blood cells aggregate at low shear, break up at higher Viscosity / Pa.s 1 NORMAL MEN INFARCTION AND THROMBOSIS 0.1 0.01 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Rate of shear / s -1
Summary Elasticity - linear extension - uniform compression - shear - stress-strain curves Agricultural applications - support of trees - bones and scaling Solids, liquids, rheological materials - property of viscosity - Poiseuille s law - non-newtonian fluids, plasticity