Deformation of Solids

Deformation of Solids • So far, we have assumed that solids always retain their original shape and cannot be deformed (“rigid bodies”) • However, the stretching, squeezing, and twisting of real objects when forces are applied are often too important to ignore (causing deformations) • Stress characterizes the strength of the forces causing the deformation (force per unit area) • Strain describes the resulting deformation • When the stress and strain are small enough, the two are directly proportional to each other: Stress / Strain = Elastic modulus (Hooke’s Law) • There are three general types of stresses and strains: tensile, bulk, and shear

Tensile Stress and Strain A F F A • A solid under tensile stress (or tension) produces a distortion due to stretching (i.e. rope under tension): • Tensile stress = F / A • Scalar quantity sinceFis the magnitude of the force • SI unit of stress is the pascal: 1 pascal = 1 Pa = 1 N/m2 • In the British system, stress is measured by the pound per square inch (psi): 1 psi = 6895 Pa • Note that units of stress are same as pressure • Tensile strain (stretch / unit length) = (l – l0) / l0 = Dl / l0 • Pure (dimensionless) number with no units l0 l

Tensile/Compressive Stress and Strain • For sufficiently small tensile stress, stress and strain are proportional through Young’s modulusY: • The stress can compress the object: • Compressive strain defined same way as tensile strain, butthen Dlcorresponds to decrease in length • For many materials, Young’s modulus has same value for both tensile and compressive stresses • Materials with largeYare relatively unstretchable F A A F l0 l

Bulk Stress and Strain • When an object experiences uniform pressure on all sides, its shape remains the same but its volume changes • We characterize a volume compression with bulk modulusB: • P = pressure = F / A • Minus sign included because increase in pressure always causes decrease in volume(ifDPis positive,DVis negative) • Reciprocal of bulk modulus is compressibility (a material having a large bulk modulus does not compress easily) (The object is being squeezed! Such is the case with an object under water.)

Shear Stress and Strain (Examples of shear stress are a ribbon being deformed by scissors and a book being deformed while pushing across the top of one cover) • Another kind of deformation results when forces are applied tangent to opposite surfaces of an object • The ratio of shear stress to shear strain is called the shear modulusS: • Material having large shear modulus is difficult to bend • Shear stress, shear strain, and shear modulus apply to solid materials only • Liquids and gasses, not having definite shape or elastic behavior, would not respond the same way as solids

CQ1: A single steel column is to support a mass of 1.5 ×108 kg. If the yield strength for steel is 2.5 × 108 N/m2 and safety regulations require the column to withstand five times the weight it presently holds, what should be the approximate cross-sectional area of the base of the column? • 0.6 m2 • 3 m2 • 6 m2 • 30 m2

CQ2: The sole of a certain tennis shoe has a shear modulus of 4 × 107. If the height of the sole is doubled, the strain will: • decrease by a factor of two. • remain the same. • increase by a factor of two. • increase by a factor of four.

F F F F Example Problem #9.6 Solution (details given in class): 22 N (directed downward in the diagram) Wire: A A l0 l A stainless-steel orthodontic wire is applied to a tooth as shown above. The wire has an unstretched length of 3.1 cm and a diameter of 0.22 mm. If the wire is stretched 0.10 mm, find the magnitude and direction of the force on the tooth. Disregard the width of the tooth, and assume that Young’s modulus for stainless steel is 18  1010 Pa.

Density and Pressure • Density of uniform substance is defined as: • Under normal conditions, densities of solids and liquids are about 1000 times greater than the densities of gases • Some densities vary within the material  above formula then gives average density • In general, density depends on environmental factors such as temperature and pressure • Specific gravity is ratio of material’s density to density of water at 4°C (1000 kg/m3) • Pressure = the amount of force per unit area exerted on an object • Force exerted by fluid (at rest) on object is always perpendicular to the surfaces of the object

Fluids at Rest • The force associated with fluid pressure is due to molecules of fluid colliding with their surroundings • SI units of pressure are the pascal (1 Pa = 1 N/m2) • Although pressure and force are used interchangeably in everyday life, they are quite different in physics • Pressure is a scalar quantity • Fluid pressure acts perpendicular to any surface in the fluid, no matter how the surface is oriented • Pressure depends on force and area over which force is applied • In order for a fluid to be in equilibrium, all points at the same depth must be at the same pressure (otherwise a given portion of fluid would accelerate left or right) F1 F2

Fluids at Rest • The pressurePat a depthhbelow the surface of a fluid is greater than the external pressureP0at the surface (independent of the shape of the container): • Normal atmospheric pressure at sea level isP0 = 1.013  105 Pa • Liquids have nearly uniform densities, since their density is nearly independent of pressure • Gases only have uniform densities over short vertical distances • Gauge pressure measuresP – P0 • Tire pressure gauges typically measure gauge pressure • Negative for partial vacuums • Total pressure is called absolute pressure (assumes uniform density)

Pressure Measurements • Simplest pressure gauge: open-tube manometer • Unknown pressure given byP = P0 + rgh • Mercury barometer is another common pressure gauge • Measures atmospheric pressureP0 = rgh, whereP≈ 0 (top of tube filled only with some mercury vapor),r = density of mercury, h = height of mercury column • Sphygmomanometer used as blood- pressure gauge • Specialized mercury-filled manometer • Measurements taken just below maximum (systolic pressure) and minimum (diastolic pressure) value produced by heart • Depends on blood flow through brachial artery

CQ3: Mercury has a specific gravity of 13.6. The column of mercury in the barometer below has a height h = 76 cm. If a similar barometer were made with water, what would be the approximate height h of the column of water? • 5.6 cm • 76 cm • 154 cm • 1034 cm

CQ4: Two identical discs sit at the bottom of a 3 m pool of water whose surface is exposed to atmospheric pressure. The first disc acts as a plug to seal the drain as shown. The second disc covers a container containing nearly a perfect vacuum. If each disc has an area of 1 m2, what is the approximate difference in force necessary to open the containers? (Note: 1 atm = 101,300 Pa) • There is no difference. • 3000 N • 101,300 N • 104,300 N

Pascal’s Law • If we increaseP0at top surface (say through use of piston pushing down on fluid),Pat any depth increases by exactly the same amount • Pascal’s Law: Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel. • This concept is used in hydraulic lifts to multiply force (dentist’s chairs, car lifts and jacks, elevators, hydraulic brakes) (DPis applied pressure increase)

Example Problem #9.28 Piston 1 in the figure at right has a diameter of 0.25 in; piston 2 has a diameter of 1.5 in. In the absence of friction, determine the force F necessary to support the 500-lb weight. Solution (details given in class): 2.3 lb

Buoyancy • Archimedes’s principle: When a body is completely or partially immersed in a fluid, the fluid exerts an upward force (“buoyant force”) on the body equal to the weight of the fluid displaced by the body (the fluid that makes way for the body) • Buoyant force arises because fluid pressure below an object is larger than pressure above it • Buoyant force is the same for objects with different densities but same volume (Force B is the same whether cube is steel, aluminum, lead, or water!)

Buoyancy • Case I: Totally submerged object • Unsupported object will sink or float depending on whetherrobjis greater than or less thanrfluid • Explains how hot air balloons work • Case II: Floating Object • Object is partially submerged and in equilibrium (B = Wobj) • Explains why icebergs are mostly (≈ 90%) submerged, and why brain is mostly supported by cerebrospinal fluid Net force  Buoyant Force Demo

CQ5: A helium balloon will rise into the atmosphere until: • The temperature of the helium inside the balloon is equal to the temperature of the air outside the balloon. • The mass of the helium inside the balloon is equal to the mass of the air outside the balloon. • The weight of the balloon is equal to the force of the upward air current. • The density of the helium in the balloon is equal to the density of the air surrounding the balloon.

Example Problem #9.41 A sample of an unknown material appears to weigh 300 N in air and 200 N when immersed in alcohol of specific gravity 0.700. What are (a) the volume and (b) the density of the material? Solution (details given in class): • 1.46  10–2 m3 • 2.10  103 kg/m3

CQ6: A brick with a density of 1.4 × 103 kg/m3 is placed on top of a piece of Styrofoam floating on water. If one half the volume of the Styrofoam sinks below the water, what is the ratio of the volume of the Styrofoam compared to the volume of the brick? (Assume the Styrofoam is massless.) • 0.7 • 1.4 • 2.8 • 5.6

Fluid Flow • We will use the simple idealized model of an ideal fluid to describe fluid motion • Fluid is incompressible (its density doesn’t change) • Fluid has no internal friction (viscosity) • Fluid motion is steady (no change in velocity, density, and pressure at each point in fluid with time) • Fluid flow is streamline (or laminar), i.e. not turbulent (a small wheel placed in fluid would translate but not rotate) • Since the mass of a moving fluid doesn’t change as it flows, and since the flow is steady: • Amount of fluid entering pipe in given time interval = amount of fluid leaving pipe in same interval • Explains flow from garden hoses, faucets, even winds downtown with tall buildings (equation of continuity)

Fluid Flow • Energy conservation applied to an ideal fluid gives Bernoulli’s equation: • P1 , P2 : Pressure at points 1 and 2 • ½rv2 : Kinetic energy per unit volume • rgy : Potential energy per unit volume • Bernoulli’s equation can be used to understand: • Lift on airplane wings • Lift of golf balls/drop of curveballs • Household plumbing • Blood flow in arteries • Consistent with Newton’s 3rd law/cons. of momentum

CQ7: If the container pictured below is filled with an ideal fluid, which point in the fluid most likely has the greatest pressure? • Point A • Point B • Point C • Point D

Example Problem #9.54 A large storage tank, open to the atmosphere at the top and filled with water, develops a small hole in its side at a point 16.0 m below the water level. If the rate of flow from the leak is 2.50  10–3 m3/min, determine (a) the speed at which the water leaves the hole and (b) the diameter of the hole. Solution (details given in class): • 17.7 m/s • 1.73 mm

CQ8: A spigot is to be placed on a water tank below the surface of the water. Which of the following gives the distance of the spigot below the surface h compared to the velocity with which the water will run through the spigot? • Figure A • Figure B • Figure C • Figure D

Deformation of Solids