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Elasticity - Stress and Strain

Table of Contents
Introduction
Problems
Answers

      
Introduction

When a force is exerted on the suspended metal to the left, the length of the object changes.  As long as the amount of elongation, /\L, is small compared to its length, the elongation is directly proportional to the force.  This was first noted by Robert Hooke.  

F = k/\L    

This is not the most useful form because the value of k is different not just for each material but also for different cross-sectional areas.

         1    F
/\L = ---  --- Lo
         E    A

E is the elastic modulus or Young's modulus and is only dependent on the material.  Units are N/m2. 
Physics Charts - Elastic Moduli

A is the cross-sectional area.

Lo is the original length.

       

As the above graph shows, as force is added the a material, the relationship is linear in the Elastic Region.  

In the Plastic Region, the material does not change in a linear fashion.

If stretched to the Elastic Limit or beyond, it does not return to its original length.

If stretched to the Breaking Point, the material will break into two pieces.

Stress Force        F
--------  =  ----         Directly proportional to the /\L.       Units   N/m2   
 Area        A

    
Strain      Change in Length        /\L
------------------------  =  -------     Measure of how much it has been deformed.    Unit less ratio.
 Original Length           Lo   

As shown in the diagram above, Tension is where the object is being pulled in opposite directions and Compression is where the object is being pushed from opposite directions.  The equations are the same for Tension and Compression.  The values of the elastic modulus are generally the same for Tension and Compression.

In Shear Stress, the object has equal and opposite forces applied across its opposite faces.  

            1  F
/\L  =  --- --- Lo                      G is the Shear Modulus and has units of N/m2.    Physics Charts - Shear Moduli
           G  A

Shear Moduli are generally 1/2 to 1/3 the value of Elastic Moduli.

    

If an object is subjected to forces from all sides, the volume will change.

/\V         1                                         /\P
-----  = - --- /\P         or      B  =  -  ------------                      Physics Charts - Bulk Moduli
 Vo         B                                      /\V / Vo    

   
B is the Bulk Modulus.  Since liquids and gases do not have a definite shape, only the Bulk Modulus applies to liquids and solids.   Elastic, Shear, and Bulk all apply to solids.

Table of Contents

     

Problems

1.      A 15 cm long animal tendon was found to stretch 3.7 mm by a force of 13.4 N.  The tendon was approximately round with an average diameter of 8.5 mm.  Calculate the elastic modulus of this tendon.
   
2. How much pressure is needed to compress the volume of an iron block by 0.10 percent?  Express answer in N/m2, and compare it to atmospheric pressure (1.0 x 105 N/m2).   Physics Charts - Bulk Moduli
    
3. A depths of 2.00 x 10 3 m in the sea, the pressure is about 200 times atmospheric pressure.  By what percentage does an iron bathysphere's volume change at this depth?   Physics Charts - Bulk Moduli
   
4. A nylon tennis string on a racquet is under a tension of 250. N.  If its diameter is 1.00 mm, by how much is it lengthened from its untensioned length of 30.0 cm?
  
5. A vertical steel girder with a cross-sectional area of 0.15 m2 has a 1550 kg sign hanging from its end.  (a) What is the stress within the girder?  (b) What is the strain on the girder?  (c) If the girder is 9.50 m long, how much is it lengthened?  (Ignore the mass of the girder itself.)
   
6. A scallop forces open its shell with an elastic material called abductin, whose elastic modulus is 2.0 x 10 6 N/m2.  If this piece of abductin is 3.0 mm thick, and has a cross-sectional area of 0.50 cm2, how much potential energy does it store when compressed 1.0 mm?

Table of Contents

    

Answers

1.      A 15 cm long animal tendon was found to stretch 3.7 mm by a force of 13.4 N.  The tendon was approximately round with an average diameter of 8.5 mm.  Calculate the elastic modulus of this tendon.
   
   
         1   F
/\L = --- --- Lo                      Convert Units and solve for E.  F shoud be in N and area in m2.  L and /\L same unit.
         E   A

   
Lo  =  15 cm     /\L = 0.37 cm       r   =  0.00425 m    
   

A = r2  =  ( 0.00425 m )2  =  5.7 x 10 -5 m2   

           1    F                    1           13.4 N
E  =  ----- --- Lo   =  ------------  --------------------- 15 cm  =  9.5 x 10 6 N/m2  
         /\L   A             0.37 cm      5.7 x 10 -5 m2 
   
2. How much pressure is needed to compress the volume of an iron block by 0.10 percent?  Express answer in N/m2, and compare it to atmospheric pressure (1.0 x 105 N/m2).   Physics Charts - Bulk Moduli
    
   
/\V         1                          
-----  = - --- /\P           Solve for /\P .    
 Vo         B     
   

             B /\V                                        -  0.10
/\P =  -  --------  =   -  90 x 10 9 N/m2  x ---------  =  9.0 x 107 N/m2  
              Vo                                              100
   

 9.0 x 107 N/m2  
----------------------  =  900 times greater than atmospheric pressure
 1.0 x 105 N/m2  
   
3. A depths of 2.00 x 10 3 m in the sea, the pressure is about 200 times atmospheric pressure.  By what percentage does an iron bathysphere's volume change at this depth?    Physics Charts - Bulk Moduli
   
   
/\V         1                            1        
-----  = - --- /\P  =  -  ----------------------  x  2.0 x 10 7 N/m2  =   - 2.2 x 10 -4  
 Vo         B                 90 x 10 9 N/m2   

   
Percentage = - 2.2 x 10 -4 x 100  =  - 2.2 x 10 -2  
   
4. A nylon tennis string on a racquet is under a tension of 250. N.  If its diameter is 1.00 mm, by how much is it lengthened from its untensioned length of 30.0 cm?      Physics Charts - Elastic Moduli
  
   
A =  r2  =  ( .000500 m )2  =  7.9 x 10 -7 m2  

     
         1    F                     1                             250. N
/\L = ---  --- Lo   =  -----------------------  --------------------- 30.0 cm  =  1.9 cm
         E    A             5 x 10 9 N m -2       7.9 x 10 -7 m2  
   
5. A vertical steel girder with a cross-sectional area of 0.15 m2 has a 1550 kg sign hanging from its end.  (a) What is the stress within the girder?  (b) What is the strain on the girder?  (c) If the girder is 9.50 m long, how much is it lengthened?  (Ignore the mass of the girder itself.)     Physics Charts - Elastic Moduli
   
   
F  =  mg  =  1550 kg x 9.80 m s -2  =  15200 N

   
                  F         15200 N
Stress  =  ----  =  -------------  =  1.0 x 10 5 N/m 2  
                  A          0.15 m2 
   

                   /\L          1                                 1
Strain  =  -------  =  ----- x Stress  =  ----------------------- x 1.0 x 10 5 N/m 2  =  5.0 x 10 -7  
                   Lo          E                        200 x 10 9 N/m2   

   

/\L  =  Strain x Lo  =   5.0 x 10 -7  x  9.50 m  =  4.8 x 10 -6 m
   
6. A scallop forces open its shell with an elastic material called abductin, whose elastic modulus is 2.0 x 10 6 N/m2.  If this piece of abductin is 3.0 mm thick, and has a cross-sectional area of 0.50 cm2, how much potential energy does it store when compressed 1.0 mm?
   
   
         1    F                                                        F
/\L = ---  --- Lo      and   F = k/\L  =>  /\L  =  -------    and  PE  =  k/\L2       set  /\L  =  /\L
         E    A                                                       k 

   

  1    F                F
----  ---- Lo  =  ------                    Solve for k.    1 m = 100 cm    1 m2  =  1000 cm2  
 E    A                k

   

           E   A           2.0 x 10 6 N/m2  x  5.0 x 10 -5 m2 
k  =  ----------   =  ---------------------------------------------  =  3.3 x 10 4 N/m
             Lo                               3.0 x 10 -3 m

   

 PE  =  k/\L2   =  x 3.3 x 10 4 N/m x ( 1.0 x 10 -3 m )2  =  0.017 Joules
         

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