Materials

1. Stress FYI: All three types of stress are measured in newtons / meter2 but all have different effects on solids. Materials Solids are often placed under stress - for example, a weight is hung from a steel cable, or a steel column supports the weight of a building or a floor. Structural engineers, who make sure that a structure such as a plane, a bridge, or a sky scraper, is safe and functional, need to study the properties of solids. We first define stress, which is a measure of the magnitude of a load that is placed on a material. F A stress = There are three types of stress: tension is a lengthening stress, compression is a shortening stress and shear is a cutting or bending stress. shear tension compression

2. Strain L L0 FYI: Typical building materials return to their original dimensions when the stress is removed, as long as that stress does not exceed a particular limit, characteristic of the material. Materials Structural engineers can measure the amount of strain, which is the amount of deformation a solid is exhibiting under the action of a stress. change in length original length L L0 = strain = Engineers study stress-strain curves, which are characteristic of each material. To create a stress-strain curve, engineers subject a material to ever-greater stress forces, and measure the strain (deformation): F A

3. Hooke’s Law Hooke’s Law states that... “within the elastic limit extension of a body is directly proportional to the applied load.” Extension  Force Applied L F F=k L k-elastic const or spring const F (N) Theelastic const or spring const is the force per unit extension and it gives how stiff a spring is. L (m)

4. How can we find K experimentally? Take the gradient of the graph : F2 – F1 L2 – L1 F (N) F L L (m) Note: Make the “gradient triangle” as large as possible

5. When we undertake an experiment we should only change one variable at a time to make it a fair test. We call this the “independent variable” L (m) Quantities we measure, (and subsequently calculate) are called “dependent variables”. All other variables which are kept the same are called the “control variables” Often graphs have the independent variable along the bottom and the dependent up the side Hooke’s law is a notable exception

6. material A is stiffer than B & C F (N) L (m) C stops obeying Hooke’s law... After the limit of proportionality the material behaves in a ductile fashion. The material stretches more with a small extra force.

7. Strain Energy Strain Energy is the energy stored in a body due to change of shape. The stretched spring has elastic potential energyWork has been done because the force moves through a distance. Workdone=Average Force x extension Workdone=Average Force x extension ==½ (k L)L =½ k (L)2=½ k (x)2 The force involved ranges from 0 to F and so the average is F/2.but F=k L Average Force=½ (k L)LThe distance moved by the force is L Workdone=Strain EnergyStrain Energy =½ k (x)2 This means that the strain energy is represented by the area under the line on a graph of load( y axis) against extension (x-axia)

8. Materials :Definitions 1 Elasticity: The ability of a solid to regain its shape after it has been deformed or distorted Tensile : Deformation due to stretching Compressive Deformation: deformation due to compression Ductile: The ability to be drawn into a wire (Copper is a good example) Brittle: Material breaks without any “give”. Cannot be permanently stretched

9. Stress -Strain Graphs Graphs of Stress against Strain are useful. They provide a method of comparing materials of different thicknesses and original lengths

10. Stress -Strain Graphs • Linear region where Hooke’s law is obeyed • The limit of proportionality • Elastic Limit – point where the material stops returning to its original length • Yield point(s) where the material ‘necks’ • Ultimate Tensile Stress (U.T.S.) • B. breaking point

11. Materials :Definitions 2 Elastic Limit: The maximum amount a material can be stretched by a force and still return to its original shape and size. The material has no permanent change in shape or size Yield Point: Beyond the elastic limit, a point is reached at which there is a noticeably larger permanent change in length. This results in plastic behaviour Ultimate Tensile Strength: The maximum stress that can be applied without breaking

12. Plasticity: A plastic material does not return to its original size and shape when the force is removed. There is a permanent stretching and change of shape • Stiffness: A measure of how difficult it is to change the size or shape of a material. • Thick steel wire is stiffer than thin steel wire of the same length. • Short steel wire is stiffer than longer steel wire of the same diameter. • Steel is stiffer than copper of the same diameter and length, because copper extends more per unit force

13. The linear region where Hooke’s law is obeyed is of interest and allows us to compare materials. This is known as Young’s modulus

14. Stress -Strain curves Comparing stress/strain graphs of brittle and ductile materials for example glass and copper

15. Stress -Strain curves

16. Young's Modulus (for tension and compression) Deformation Under Compression or Tension Young’s Modulus YOUNG'S MODULUS - CHANGE IN LENGTH Both tension and compression act to change the length of a material - tension stretches, and compression shortens. The elastic modulus for tension and compression (not shear) is called Young's modulus (Y). Thus stress strain Y = F/A L/L0 Y = FL0 AL Y = We can solve the above equation for L to get a useful relationship: FL0 AY L=

17. yield point elastic limit breaking point permanent set FYI: Since the region of the graph up to the elastic limit is linear, we can characterize a material by the slope: stress / strain. We call this ratio the ELASTIC MODULUS. FYI: Thus tables showing properties of materials can simply show a single number for that material, rather than a graph. 40000 n/m2 5010-8 stress strain FYI: For this particular material the elastic modulus is = 8 1010n/m2 = elastic modulus = Stress strain curves FYI: Structural engineers NEVER design a structural component to exceed its ELASTIC LIMIT. failure region yield region elastic region plastic region 70000 60000 50000 40000 30000 20000 10000 0 0 50 100 150 200 250 300 350 400 strain (10-8) FYI: If a material is stressed beyond its YIELD POINT, it will become permanently deformed. As the stress is removed, it will follow the PERMANENT SET line, as illustrated: stress (N/m2) stress strain elastic modulus =

18. A A L L0 L FYI: Since the cable is under TENSION it STRETCHES so that its length under the load is 15.0 + 2. 4 = 17. 4 m. Solids and Fluids9-1 Solids and Elastic Moduli FYI: The typical stretches for cables (and compressions for columns) is extremely small. YOUNG'S MODULUS - CHANGE IN LENGTH A 15.0-meter long steel cable with a diameter of 1 cm has a 500 metric ton mass hanging from it. How much does it stretch under this tension load? Assume Young's modulus for steel is Y = 401010 n/m2. A = r2 F = mg 2 d 2 F = (500)(1000)(10) A =  F = 5106 n 2 .01 2 A =  A = 7.85410-5m2 FL0 AY (5106)(15) (7.85410-5)(401010) L= = L= 2.4 m

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