Physics 6A. Stress, Strain and Elastic Deformations. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

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Thomasina Hodges
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1 Physics 6 Stress, Strain and Elastic Deforations

2 When a force is applied to an object, it will defor. If it snaps back to its original shape when the force is reoved, then the deforation was ELSTIC. We already know about springs - reeber Hooke s Law : F spring = -k Δx Hooke s Law is a special case of a ore general rule involving stress and strain. Stress Strain (const.) The constant will depend on the aterial that the object is ade fro, and it is called an ELSTIC MODULUS. In the case of tension (stretching) or copression we will call it Young s Modulus*. So our basic forula will be: Y Stress Strain *Bonus Question who is this forula naed for? Click here for the answer

3 To use our forula we need to define what we ean by Stress and Strain. STRESS is the sae idea as PRESSURE. In fact it is the sae forula: Stress Force rea

4 To use our forula we need to define what we ean by Stress and Strain. STRESS is the sae idea as PRESSURE. In fact it is the sae forula: Stress Force rea STRIN is a easure of how uch the object defors. We divide the change in the length by the original length to get strain: Strain L L 0

5 To use our forula we need to define what we ean by Stress and Strain. STRESS is the sae idea as PRESSURE. In fact it is the sae forula: Stress Force rea STRIN is a easure of how uch the object defors. We divide the change in the length by the original length to get strain: Strain L L 0 Now we can put these together to get our forula for the Young s Modulus: Y F L L 0

6 EXMPLE: nylon rope used by ountaineers elongates 1.10 under the weight of a 65.0kg cliber. If the rope is initially 45.0 in length and 7.0 in diaeter, what is Young s odulus for this nylon?

7 EXMPLE: nylon rope used by ountaineers elongates 1.10 under the weight of a 65.0kg cliber. If the rope is initially 45.0 in length and 7.0 in diaeter, what is Young s odulus for this nylon? L 0 =45 ΔL=1.1

8 EXMPLE: nylon rope used by ountaineers elongates 1.10 under the weight of a 65.0kg cliber. If the rope is initially 45.0 in length and 7.0 in diaeter, what is Young s odulus for this nylon? couple of quick calculations and we can just plug in to our forula: Y F L L 0 L 0 =45 ΔL=1.1

9 EXMPLE: nylon rope used by ountaineers elongates 1.10 under the weight of a 65.0kg cliber. If the rope is initially 45.0 in length and 7.0 in diaeter, what is Young s odulus for this nylon? couple of quick calculations and we can just plug in to our forula: Y F L L 0 L 0 =45 ΔL=1.1 F g 65kg N s 7 r ( ) Don t forget to cut the diaeter in half.

10 EXMPLE: nylon rope used by ountaineers elongates 1.10 under the weight of a 65.0kg cliber. If the rope is initially 45.0 in length and 7.0 in diaeter, what is Young s odulus for this nylon? couple of quick calculations and we can just plug in to our forula: Y F L L 0 L 0 =45 Y 637N N N ΔL=1.1 F g 65kg N s 7 r ( ) Don t forget to cut the diaeter in half.

11 EXMPLE: steel wire.00 long with circular cross-section ust stretch no ore than 0.5c when a 400.0N weight is hung fro one of its ends. What iniu diaeter ust this wire have?

12 EXMPLE: steel wire.00 long with circular cross-section ust stretch no ore than 0.5c when a 400.0N weight is hung fro one of its ends. What iniu diaeter ust this wire have? dia=? L 0 = ΔL=0.5c 400N

13 EXMPLE: steel wire.00 long with circular cross-section ust stretch no ore than 0.5c when a 400.0N weight is hung fro one of its ends. What iniu diaeter ust this wire have? We have ost of the inforation for our forula. We can look up Young s 11 odulus for steel in a table: Y N steel 10 dia=? L 0 = ΔL=0.5c 400N

14 EXMPLE: steel wire.00 long with circular cross-section ust stretch no ore than 0.5c when a 400.0N weight is hung fro one of its ends. What iniu diaeter ust this wire have? We have ost of the inforation for our forula. We can look up Young s 11 odulus for steel in a table: Y N steel 10 Y F L L 0 The only piece issing is the area we can rearrange the forula dia=? L 0 = ΔL=0.5c 400N

15 EXMPLE: steel wire.00 long with circular cross-section ust stretch no ore than 0.5c when a 400.0N weight is hung fro one of its ends. What iniu diaeter ust this wire have? We have ost of the inforation for our forula. We can look up Young s 11 odulus for steel in a table: Y N steel 10 Y F L L 0 The only piece issing is the area we can rearrange the forula dia=? L 0 = F L Y L 0 ΔL=0.5c 400N

16 EXMPLE: steel wire.00 long with circular cross-section ust stretch no ore than 0.5c when a 400.0N weight is hung fro one of its ends. What iniu diaeter ust this wire have? We have ost of the inforation for our forula. We can look up Young s 11 odulus for steel in a table: Y N steel 10 Y F L L 0 The only piece issing is the area we can rearrange the forula dia=? L 0 = F L0 Y L 400N N N ΔL=0.5c

17 EXMPLE: steel wire.00 long with circular cross-section ust stretch no ore than 0.5c when a 400.0N weight is hung fro one of its ends. What iniu diaeter ust this wire have? We have ost of the inforation for our forula. We can look up Young s 11 odulus for steel in a table: Y N steel 10 Y F L F L0 Y L L 0 400N N The only piece issing is the area we can rearrange the forula One last step we need the diaeter, and we have the area: 6 400N dia=? L 0 = ΔL=0.5c r circle

18 EXMPLE: steel wire.00 long with circular cross-section ust stretch no ore than 0.5c when a 400.0N weight is hung fro one of its ends. What iniu diaeter ust this wire have? We have ost of the inforation for our forula. We can look up Young s 11 odulus for steel in a table: Y N steel 10 Y F L F L0 Y L L 0 400N N The only piece issing is the area we can rearrange the forula One last step we need the diaeter, and we have the area: 6 400N dia=? L 0 = ΔL=0.5c r 6 4 circle r

19 EXMPLE: steel wire.00 long with circular cross-section ust stretch no ore than 0.5c when a 400.0N weight is hung fro one of its ends. What iniu diaeter ust this wire have? We have ost of the inforation for our forula. We can look up Young s 11 odulus for steel in a table: Y N steel 10 Y F L F L0 Y L L 0 400N N The only piece issing is the area we can rearrange the forula One last step we need the diaeter, and we have the area: 6 400N dia=? L 0 = ΔL=0.5c r 6 4 circle r double the radius to get the diaeter: d

20 EXMPLE: When a weight is hung fro a cylindrical wire of diaeter D, it produces a tensile stress X in the wire. If the sae weight is hung fro a wire having twice the diaeter as the first one, the tensile stress in this wire will be x a) x b) x c) 4 d)x e)4x

21 EXMPLE: When a weight is hung fro a cylindrical wire of diaeter D, it produces a tensile stress X in the wire. If the sae weight is hung fro a wire having twice the diaeter as the first one, the tensile stress in this wire will be x a) x b) x c) 4 d)x e)4x We can do this one just by staring at the forula for stress: Stress Force rea

22 EXMPLE: When a weight is hung fro a cylindrical wire of diaeter D, it produces a tensile stress X in the wire. If the sae weight is hung fro a wire having twice the diaeter as the first one, the tensile stress in this wire will be x a) x b) x c) 4 d)x e)4x We can do this one just by staring at the forula for stress: Stress Force rea The force is the sae in both cases because it says they use the sae weight. The area is related to the square of the radius (or diaeter), so when the diaeter doubles the area goes up by a factor of 4.

23 EXMPLE: When a weight is hung fro a cylindrical wire of diaeter D, it produces a tensile stress X in the wire. If the sae weight is hung fro a wire having twice the diaeter as the first one, the tensile stress in this wire will be x a) x b) x c) 4 d)x e)4x We can do this one just by staring at the forula for stress: Stress Force rea The force is the sae in both cases because it says they use the sae weight. The area is related to the square of the radius (or diaeter), so when the diaeter doubles the area goes up by a factor of 4. Thus the stress should go down by a factor of 4 (area is in the denoinator) nswer c)

24 Bulk Modulus and Volue Changes When pressure is applied to an object fro all directions, its volue will change accordingly. Think of squishing a foa ball, or inflating a balloon. In this case we use a 3-diensional version of Young s odulus. We call it BULK MODULUS, and it is defined in a siilar way: Bulk Modulus Pressure change p B V Volue change V 0

25 Bulk Modulus and Volue Changes Exaple: When water freezes into ice it expands in volue by 9.05 percent. Suppose a volue of water is in a household water pipe or a cavity in a rock. If the water freezes, what pressure ust be exerted on it to keep its volue fro expanding? (If the pipe or rock cannot supply this pressure, the pipe will burst or the rock will split.) The bulk odulus for ice is 8x10 9 N/. nswser: 6.6x10 8 N/

1 (a) On the axes of Fig. 7.1, sketch a stress against strain graph for a typical ductile material. stress. strain. Fig. 7.1 [2]

1 (a) On the axes of Fig. 7.1, sketch a stress against strain graph for a typical ductile material. stress strain Fig. 7.1 [2] (b) Circle from the list below a material that is ductile. jelly c amic gl