# 12.3 Stress, strain, and elastic modulus  (Page 2/26)

 Page 2 / 26
$\begin{array}{c}1\phantom{\rule{0.2em}{0ex}}\text{psi}=6895\phantom{\rule{0.2em}{0ex}}\text{Pa}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}1\phantom{\rule{0.2em}{0ex}}\text{Pa}=1.450\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\text{psi}\\ 1\phantom{\rule{0.2em}{0ex}}\text{atm}=1.013\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\text{Pa}=14.7\phantom{\rule{0.2em}{0ex}}\text{psi.}\end{array}$

An object or medium under stress becomes deformed. The quantity that describes this deformation is called strain    . Strain is given as a fractional change in either length (under tensile stress) or volume (under bulk stress) or geometry (under shear stress). Therefore, strain is a dimensionless number. Strain under a tensile stress is called tensile strain    , strain under bulk stress is called bulk strain    (or volume strain ), and that caused by shear stress is called shear strain    .

The greater the stress, the greater the strain; however, the relation between strain and stress does not need to be linear. Only when stress is sufficiently low is the deformation it causes in direct proportion to the stress value. The proportionality constant in this relation is called the elastic     modulus . In the linear limit of low stress values, the general relation between stress and strain is

$\text{stress}=\text{(elastic modulus)}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{strain.}$

As we can see from dimensional analysis of this relation, the elastic modulus has the same physical unit as stress because strain is dimensionless.

We can also see from [link] that when an object is characterized by a large value of elastic modulus, the effect of stress is small. On the other hand, a small elastic modulus means that stress produces large strain and noticeable deformation. For example, a stress on a rubber band produces larger strain (deformation) than the same stress on a steel band of the same dimensions because the elastic modulus for rubber is two orders of magnitude smaller than the elastic modulus for steel.

The elastic modulus for tensile stress is called Young’s modulus    ; that for the bulk stress is called the bulk modulus    ; and that for shear stress is called the shear modulus    . Note that the relation between stress and strain is an observed relation, measured in the laboratory. Elastic moduli for various materials are measured under various physical conditions, such as varying temperature, and collected in engineering data tables for reference ( [link] ). These tables are valuable references for industry and for anyone involved in engineering or construction. In the next section, we discuss strain-stress relations beyond the linear limit represented by [link] , in the full range of stress values up to a fracture point. In the remainder of this section, we study the linear limit expressed by [link] .

Approximate elastic moduli for selected materials
Material Young’s modulus
$×\phantom{\rule{0.2em}{0ex}}{10}^{10}\text{Pa}$
Bulk modulus
$×\phantom{\rule{0.2em}{0ex}}{10}^{10}\text{Pa}$
Shear modulus
$×\phantom{\rule{0.2em}{0ex}}{10}^{10}\text{Pa}$
Aluminum 7.0 7.5 2.5
Bone (tension) 1.6 0.8 8.0
Bone (compression) 0.9
Brass 9.0 6.0 3.5
Brick 1.5
Concrete 2.0
Copper 11.0 14.0 4.4
Crown glass 6.0 5.0 2.5
Granite 4.5 4.5 2.0
Hair (human) 1.0
Hardwood 1.5 1.0
Iron 21.0 16.0 7.7
Lead 1.6 4.1 0.6
Marble 6.0 7.0 2.0
Nickel 21.0 17.0 7.8
Polystyrene 3.0
Silk 6.0
Steel 20.0 16.0 7.5
Acetone 0.07
Ethanol 0.09
Glycerin 0.45
Mercury 2.5
Water 0.22

## Tensile or compressive stress, strain, and young’s modulus

Tension or compression occurs when two antiparallel forces of equal magnitude act on an object along only one of its dimensions, in such a way that the object does not move. One way to envision such a situation is illustrated in [link] . A rod segment is either stretched or squeezed by a pair of forces acting along its length and perpendicular to its cross-section. The net effect of such forces is that the rod changes its length from the original length ${L}_{0}$ that it had before the forces appeared, to a new length L that it has under the action of the forces. This change in length $\text{Δ}L=L-{L}_{0}$ may be either elongation (when L is larger than the original length ${L}_{0}\right)$ or contraction (when L is smaller than the original length ${L}_{0}\right).$ Tensile stress and strain occur when the forces are stretching an object, causing its elongation, and the length change $\text{Δ}L$ is positive. Compressive stress and strain occur when the forces are contracting an object, causing its shortening, and the length change $\text{Δ}L$ is negative.

#### Questions & Answers

At time to = 0 the current to the DC motor is reverse, resulting in angular displacement of the motor shafts given by angle = (198rad/s)t - (24rad/s^2)t^2 - (2rad/s^3)t^3 At what time is the angular velocity of the motor shaft zero
3s
Basit
what is angular velocity
In three experiments, three different horizontal forces are ap- plied to the same block lying on the same countertop. The force magnitudes are F1 " 12 N, F2 " 8 N, and F3 " 4 N. In each experi- ment, the block remains stationary in spite of the applied force. Rank the forces according to (a) the
state Hooke's law of elasticity
Hooke's law states that the extension produced is directly proportional to the applied force provided that the elastic limit is not exceeded. F=ke;
Shaibu
thanks
Aarti
You are welcome
Shaibu
thnx
Junaid
what is drag force
Junaid
A backward acting force that tends to resist thrust
Ian
solve:A person who weighs 720N in air is lowered in to tank of water to about chin level .He sits in a harness of negligible mass suspended from a scale that reads his apparent weight .He then dumps himself under water submerging his body .If his weight while submerged is 34.3N. find his density
Ian
The weight inside the tank is lesser due to the buoyancy force by the water displaced. Weight of water displaced = His weight outside - his weight inside tank = 720 - 34.3 = 685.7N Now, the density of water = 997kg/m³ (this is a known value) Volume of water displaced = Mass/Density (next com)
Sharath
density or relative density
Shaibu
density
Ian
Upthrust =720-34.3=685.7N mass of water displayed = 685.7/g vol of water displayed = 685.7/g/997 hence, density of man = 720/g / (685.7/g/997) =1046.6 kg/m3
1046.8
R.d=weight in air/upthrust in water =720/34.3=20.99 R.d=density of substance/density of water 20.99=x/1 x=20.99g/cm^3
Shaibu
Kg /cubic meters
Shaibu
Upthrust = 720-34.3=685.7N vol of water = 685.7/g/density of water = 685.7/g/997 so density of man = 720/g /(685.7/g/997) =1046.8 kg/m3
is there anyway i can see your calculations
Ian
Upthrust =720-34.3=685.7
Upthrust 720-34.3
=685.7N
Vol of water = 685.7/g/997
Hence density of man = 720/g / (685.7/g/997)
=1046.8 kg/m3
so the density of water is 997
Shaibu
Yes
Okay, thanks
Shaibu
try finding the volume then
Ian
Vol of man = vol of water displayed
I've done that; I got 0.0687m^3
Shaibu
okay i got it thanks
Ian
u welcome
Shaibu
HELLO kindly assist me on this...(MATHS) show that the function f(x)=[0 for xor=0]is continuous from the right of x->0 but not from the left of x->0
I do not get the question can you make it clearer
Ark
Same here, the function looks very ambiguous. please restate the question properly.
Sharath
please help me solve this problem.a hiker begins a trip by first walking 25kmSE from her car.she stops and sets her tent for the night . on the second day, she walks 40km in a direction 60°NorthofEast,at which she discovers a forest ranger's tower.find components of hiker's displacement for each day
Take a paper. put a point (name is A), now draw a line in the South east direction from A. Assume the line is 25 km long. that is the first stop (name the second point B) From B, turn 60 degrees to the north of East and draw another line, name that C. that line is 40 km long. (contd.)
Sharath
Now, you know how to calculate displacements, I hope? the displacement between two points is the shortest distance between the two points. go ahead and do the calculations necessary. Good luck!
Sharath
thank you so much Sharath Kumar
Liteboho
thank you, have also learned alot
Duncan
No issues at all. I love the subject and teaching it is fun. Cheers!
Sharath
cheers!
Liteboho
cheers too
Duncan
hii
Lakshya
hii too
Siciid
haye
Siciid
yes
Siciid
yes
Lakshya
shggggg
Lakshya
you mean
Siciid
solution problems
Siciid
what is the definition of model
please is there any way that i can understand physics very well i know am not support to ask this kind of question....
matthew
yes
Duncan
prove using vector algebra that the diagonals of a rhombus perpendicular to each other.
A projectile is thrown with a speed of v at an angle of theta has a range of R on the surface of the earth. For same v and theta,it's range on the surface of moon will be
0
Keshav
what is soln..
Keshav
o
Duncan
Using some kinematics, time taken for the projectile to reach ground is (2*v*g*Sin (∆)) (here, g is gravity on Earth and ∆ is theta) therefore, on Earth, R = 2*v²*g*Sin(∆)*Cos(∆) on moon, the only difference is the gravity. Gravity on moon = 0.166*g substituting that value in R, we get the new R
Sharath
Some corrections to my old post. Time taken to reach ground = 2*v*Sin (∆)/g R = (2*v²*Sin(∆)*Cos(∆))/g I put the g in the numerator by mistake in my old post. apologies for that. R on moon = (R on Earth)/(0.166)
Sharath
state Newton's first law of motion
Every body will continue in it's state of rest or of uniform motion in a straight line, unless it is compelled to change that state by an external force.
Kumaga
if you want this to become intuitive to you then you should state it
Shii
changing the state of rest or uniform motion of a body
koffi
if a body is in rest or motion it is always rest or motion, upto external force appied on it. it explains inertia
Omsai
what is a vector
smith
a ship move due north at 100kmhr----1 on a River flowing be due east on at 25kmperhr. cal the magnitude of the resultant velocity of the ship.
The result is a simple vector addition. The angle between the vectors is 90 degrees, so we can use Pythagoras theorem to get the result. V magnitude = sqrt(100*100 + 25*25) = 103.077 km/hr. the direction of the resultant vector can be found using trigonometry. Tan (theta) = 25/100.
Kumar
103.077640640442km/h
Peter
state Newton's first law of motion
An object continues to be in its state of rest or motion unless compelled by some external force
Alem
First law (law of inertia)- If a body is at rest, it would remain at rest and if the body is in the motion, it would be moving with the same velocity until or unless no external force is applied on it. If force F^=0 acceleration a^=0 or v^=0 or constant.
Govindsingh
how would you measure displacement in your car?
what is constellation
The product of a. (vector b× vector a)
I want to join the conversation
ok
Kamal
Ok
Bishal
ok
Rohit
Two charges 1uc and 3uc are separated 4m apart. find the point on the line connecting them at which their electric field intensity balances each other
Chukwurah
hmmm
JMPSCL