Definition of normal stress
(axial stress)
F

A
Definition of normal strain
L

L0
Poisson’s ratio
Definition of shear stress
F

A0
Definition of shear strain
x
  tan  
l
Tensile Testing
Stress-Strain Curves
Stress-Strain Curves
http://www.uoregon.edu/~struct/courseware/461/461_lectures/4
61_lecture24/461_lecture24.html
Stress-Strain Curve
(ductile material)
http://www.shodor.org/~jingersoll/weave/tutorial/node4.html
Stress-Strain Curve
(brittle material)
Example: stress-strain curve for low-carbon steel
•1 - Ultimate Strength
•2 - Yield Strength
•3 - Rupture
•4 - Strain hardening region
•5 - Necking region
Hooke's law is only valid for the
portion of the curve between the
origin and the yield point.
http://en.wikipedia.org/wiki/Hooke's_law
σPL ⇒ Proportional Limit - Stress above which stress is not longer proportional to strain.
σEL ⇒ Elastic Limit - The maximum stress that can be applied without resulting in permanent
deformation when unloaded.
σYP ⇒ Yield Point - Stress at which there are large increases in strain with little or no increase in
stress. Among common structural materials, only steel exhibits this type of response.
σYS ⇒ Yield Strength - The maximum stress that can be applied without exceeding a specified
value of permanent strain (typically .2% = .002 in/in).
OPTI 222 Mechanical Design in Optical Engineering 21
σU ⇒ Ultimate Strength - The maximum stress the material can withstand (based on the original
area)
True stress and true strain
True stress and true strain are based upon
instantaneous values of cross sectional
area and gage length
The Region of Stress-Strain Curve
Stress Strain Curve
Volume
Volume
Pressure
• Similar to Pressure-Volume Curve
• Area = Work
Uni-axial Stress State
Elastic analysis
Stress-Strain Relationship
Hooke’s Law:
  E
E -- Young’s modulus
  G
G -- shear modulus
Stresses on Inclined Planes
Thermal Strain
Straincaused by temperature changes. α is a
material characteristic called the coefficient of
thermal expansion.
Strains caused by temperature changes and strains
caused by applied loads are essentially independent.
Therefore, the total amount of strain may be expressed as
follows:
Bi-axial state elastic analysis
(1) Plane stress
• State of plane stress occurs in a thin plate subjected to forces acting in the mid-plane of the
plate
• State of plane stress also occurs on the free surface of a structural element or machine
component, i.e., at any point of the surface not subjected to an external force.
Transformation of Plane Stress
Mohr’s Circle (Plane Stress)
http://www.tecgraf.puc-rio.br/etools/mohr/mohreng.html
Mohr’s Circle (Plane Stress)
Instruction to draw Mohr’s Circle
1. Determine the point on the body in which the principal stresses are to be
determined.
2. Treating the load cases independently and calculated the stresses for the point
chosen.
3. Choose a set of x-y reference axes and draw a square element centered on the
axes.
4. Identify the stresses σx, σy, and τxy = τyx and list them with the proper sign.
5. Draw a set of σ - τ coordinate axes with σ being positive to the right and τ being
positive in the
upward direction. Choose an appropriate scale for the each axis.
6. Using the rules on the previous page, plot the stresses on the x face of the element
in this coordinate system (point V). Repeat the process for the y face (point H).
7. Draw a line between the two point V and H. The point where this line crosses the
σ axis establishes the center of the circle.
8. Draw the complete circle.
9. The line from the center of the circle to point V identifies the x axis or reference
axis for angle measurements (i.e. θ = 0).
Note: The angle between the reference axis and the σ axis is equal to 2θp.
Mohr’s Circle (Plane Stress)
http://www.egr.msu.edu/classes/me423/aloos/lecture
_notes/lecture_4.pdf
Principal Stresses
Maximum shear stress
Stress-Strain Relationship
(Plane stress)

 x 
0
1 
E 
 
 1
0
 y  
2 
  1   0 0 1 
 xy 
2

 
 x
  

y 




 xy 

z 
1
( )( x   y )
E
http://www4.eas.asu.edu/concrete/elasticity2_95/sld001.htm
(2) Plane strain
Coordinate Transformation
The transformation of strains with respect to the {x,y,z} coordinates to
the strains with respect to {x',y',z'} is performed via the equations
Mohr's Circle (Plane Strain)
(εxx' - εavg)2 + ( γx'y' / 2 )2 = R2
εavg =
εxx + εyy
2
http://www.shodor.org/~jingersoll/weave4/tutorial/tutorial.html
Principal Strain
http://www.efunda.com/formulae/solid_mechani
cs/mat_mechanics/calc_principal_strain.cfm
Maximum shear strain
Stress-Strain Relationship
(Plane strain)

 1
 x 

E (1  )  
 
 y  
 1 
(
1


)(
1

2

)
 

 z
 0


1 
1
0

0 
 x 
 

0

 y 
 
1  2  z 

2(1  ) 
z 
E
1 
 

(



)
y 
1  2 x

Tri-axial stress state
elastic analysis
3D stress at a point
three (3) normal stresses may act on faces of the cube, as well
as, six (6) components of shear stress
Stress and strain components
The stress on a inclined plane
(l, m, n)
z
3
n
2
p
y
x
( n 
1
 2 3
n
) 2   n2  (
 2  3
) 2  l 2 ( 1   2 )( 1   3 )
2
2
 3  1 2 2  3 1 2
( n 
)  n  (
)  m 2 ( 2   3 )( 2   1 )
2
2
   2 2 2 1  2 2
( n  1
)  n  (
)  n 2 ( 3   1 )( 3   2 )
2
2
3-D Mohr’s Circle
D
* The 3 circles expressed by the 3 equations intersect in point D,
and the value of coordinates of D is the stresses of the inclined
plane
Stress-Strain Relationship
Generalized Hooke’s Law:


1 
 
 x 
1 


 
 

1 
 y

 z 
E
0
0
 0
 
 xy  (1   )(1  2 ) 
 0
0
0
 yz 

 
 zx 

0
0
 0
0
0
0
0
0
1  2
2
0
0
0
1  2
2
0
0
0 
0    x 
 

0  y

0    z   ET
  xy  1  2
0   yz 
 
1  2   zx 
2 
For isotropic materials
1
1
 
1
 
0
0
 
0