Recap There are 4 aggregates states of matter: - Solid: Strong interatomic bonds, particles cannot move freely. - Liquid: Weaker bonds, particles move more freely - Gas: No interatomic bonds, particles move freely. - Plasma: Ionized gas, electrons and ions are separated. Transitions from one state into another are initiated by heating/cooling the material. Density is mass per volume: Pressure is force per area:
Today s lecture Solids and Fluids Variation of Pressure with Depth Deformation of Solids
Pressure as a function of depth Goal: Calculate the pressure at a given depth. Let s consider a stationary fluid contained within the volume indicated in the plot. Three forces act on it and must balance each other. Otherwise the fluid would flow, i.e. not be stationary: (i) Weight of the fluid contained in the volume: (ii) Downward force exerted by the fluid above it: (iii) Upwards force exerted by the fluid below it: Force balance:
Pressure as a function of depth We can apply this equation to a particular scenario: y1 = 0 m (water surface). Then, P1 is the atmospheric air pressure above the water level typically called: Then (P1 = P0, P2 = P): h is the depth below the water surface level. P0 is the result of a similar calculation considering a column of air from the surface of the earth to the edge of space.
Pressure vs. depth
Example problem: Pressure vs. Depth Calculate the absolute pressure at the bottom of a freshwater lake at a depth of 27.5 m. Assume the density of the water is 1000 kg/m 3 and the air above is at a pressure of 101.3 kpa. What force is exerted by the water on the window of an underwater vehicle at this depth if the window is circular and has a diameter of 35 cm?
Pascal s principle The pressure in a fluid depends on depth and on the value of the pressure at its surface, P0. If P0 is increased, for instance by pressing on a piston in a hydraulic press, this pressure will be transmitted through the entire fluid, i.e. the pressure on the second piston will increase accordingly. If both pistons are at different heights, the second term will cause a pressure difference between both fluid columns. We neglect this difference. Then: If A2 > A1: F2 > F1! This allows to lift heavy objects using small forces.
Elastic Properties of Materials All objects are deformable (whether shape or size or both) through the application of external forces. When the forces are removed, the object will tend to its original shape, exhibiting elastic behavior, if the applied force was not too strong. A sufficiently strong force will permanently deform or break a given material. Stress is the force per unit area causing the deformation: Stress = F/A Strain is a measure of the amount of deformation: For example, deformation in Length: Strain = ΔL/L0 where L0 is the material length and ΔL the change in length.
How can we relate stress to strain? A solid material consists of atom/molecules that are bond to each other. We can model these bonds as springs. We know Hooke s law for springs: Solid bodies: Increasing the surface area of the solid body results in a shorter extension for a given force. This is similar to two parallel springs (each feels only F/2). The change in length, ΔL, will be proportional to the applied force, if the force is not too high. For a given force increasing the initial length, L0, results in a bigger extension, ΔL. This is analogous to a series of 2 springs (each feels F). In conclusion we find:
Elasticity in Length When an external force is applied to a bar of length L o perpendicular to its cross section, it will be stretched to a greater length L o + ΔL. The external force is balanced by internal forces (springs). Stress is proportional to strain until a maximum stress (elastic limit) is reached. Beyond this point the deformation remains permanent. The material breaks at an even higher stress (breaking point). Here, F/A is the tensile stress and ΔL/L0 is the tensile strain. Y is the elastic (Young s) modulus.
Tensile stress test Y is different for different materials. A large Y means that a material is difficult to stretch.
Example problem: Tensile stress Let s consider two cylindrical rods of length L0 = 1 m and radius r = 0.5 cm. One rod is made of steel, the other is made of nylon. We now hang a mass of 500 kg to the bottom of the rod. What will be the change of length of (a) the steel and (b) the nylon rod? 500 kg YSteel = 20 10 10 N m -3 YNylon = 0.36 10 10 N m -3
Elasticity in shape Now, a force, F, is exerted on an object parallel to its top surface, A, while the bottom surface is held fixed. The top surface will move a distance, Δx, relative to the bottom surface the object will be sheared. Here: F/A is the shear stress and Δx/h is the shear strain. h is the height of the object. S is a constant of proportionality called the shear modulus.
Elasticity of Volume Now, equal forces are applied perpendicularly to all surfaces (e.g. underwater pressure). The object will undergo a change in volume, ΔV, without a change in shape. Here, ΔP = ΔF/A is the ratio of the change in the magnitude of the applied force to the surface area. V is the original volume of the object. B is the Bulk Modulus (material dependent constant). ΔP is the volume stress and ΔV/V the volume strain.
Elasticity of Solids vs. Liquids Length: Shape: Volume: Solids have all three moduli; Bulk, Shear, and Young s. Liquids have only bulk moduli, they will not undergo a shearing or tensile stress, but will flow instead: - If a fluid is at rest in a container, all portions of the fluid are in static equilibrium. - All points at the same depth must be at the same pressure, otherwise the fluid will flow.
Summary All objects are deformable. Until a maximum stress (F/A) the object will tend to its original shape, when the force is removed (elasticity). There are 3 types of elasticities: (i) Elasticity of length: (ii) Elasticity of shape: (iii) Elasticity of volume: Pressure depends on depth: Pascal s principle: (hydraulic press)
Pressure measurements Manometer: One end of the U-shaped tube is open to the atmosphere. The other end is connected to the pressure to be measured. If P in the system is greater than atmospheric pressure, h is positive (if less, then h is negative). h is a measure for the gauge pressure, P - P0. This also what you measure, when you measure the pressure in tires.
Blood pressure measurements The pressure in the cuff is increased until the flow of blood through the brachial artery is stopped. The pressure is measured simultaneously by a manometer. Then a valve is opened to decrease the pressure. The measurer listens to the flow of blood through this artery using a stethoscope. When the pressure in the cuff is the same as the pressure produced by a heartbeat (systolic pressure), the artery opens momentarily. This can be heard and the pressure is written down. The normal value is 120 mm of mercury. The pressure is further lowered until it is lower than the minimum heart pressure and the blood flows continuously. This can be heard again and the corresponding pressure is written down (normal: 80 mm of mercury).
Example problem: Shear stress A defensive lineman of mass m = 125 kg makes a flying tackle at vi = 4 m/s on a stationary quarterback of mass m = 85 kg. The lineman s helmet makes solid contact with the quarterback s femur. (a) What is the speed vf of the two athletes immediately after the contact? - Assume a perfectly inelastic linear collision. (b) If the collision lasts for 0.1 s, estimate the average force exerted on the quarterback s femur. (c) If the cross-sectional area of the femur is 5 x 10-4 m2, calculate the shear stress exerted on the bone in the collision.
Elastic moduli and ultimate strengths