Contributed by:
Normal stress, Poisson's ratio, Shear stress, Uniaxial test, Stress-Strain curves, True stress and strain
2.
Definition of normal stress
(axial stress)
F
A
3.
Definition of normal strain
L
L0
5.
Definition of shear stress
F
A0
6.
Definition of shear strain
x
tan
l
9.
Stress-Strain Curves
http://www.uoregon.edu/~struct/courseware/461/461_lectures/
461_lecture24/461_lecture24.html
10.
Stress-Strain Curve
(ductile material)
http://www.shodor.org/~jingersoll/weave/tutorial/node4.html
11.
Stress-Strain Curve
(brittle material)
12.
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
13.
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
14.
True stress and true strain
True stress and true strain are based upon
instantaneous values of cross sectional
area and gage length
15.
The Region of Stress-Strain Curve
Stress Strain Curve
Volume
Volume
Pressure
• Similar to Pressure-Volume Curve
• Area = Work
16.
Uni-axial Stress State
Elastic analysis
17.
Stress-Strain Relationship
Hooke’s Law:
E
E -- Young’s modulus
G
G -- shear modulus
18.
Stresses on Inclined Planes
19.
Thermal Strain
Straincaused by temperature changes. α is a
material characteristic called the coefficient of
thermal expansion.
20.
Strains caused by temperature changes and strains
caused by applied loads are essentially independent.
Therefore, the total amount of strain may be expressed as
21.
Bi-axial state elastic analysis
22.
(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
23.
Transformation of Plane Stress
24.
Mohr’s Circle (Plane Stress)
http://www.tecgraf.puc-rio.br/etools/mohr/mohreng.html
25.
Mohr’s Circle (Plane Stress)
26.
Instruction to draw Mohr’s Circle
1. Determine the point on the body in which the principal stresses are to be
2. Treating the load cases independently and calculated the stresses for the point
3. Choose a set of x-y reference axes and draw a square element centered on the
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.
27.
Mohr’s Circle (Plane Stress)
http://www.egr.msu.edu/classes/me423/aloos/
lecture_notes/lecture_4.pdf
30.
Stress-Strain Relationship
(Plane stress)
x 1 0 x
E
y 2
1 0 y
1 0 0 1
xy
xy 2
1
z ( )( x y )
E
http://www4.eas.asu.edu/concrete/elasticity2_95/sld001.htm
32.
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
33.
Mohr's Circle (Plane Strain)
(εxx' - εavg)2 + ( γx'y' / 2 )2 = R2
εxx + εyy
εavg =
2
http://www.shodor.org/~jingersoll/weave4/tutorial/tutorial.html
34.
Principal Strain
http://www.efunda.com/formulae/
solid_mechanics/mat_mechanics/
calc_principal_strain.cfm
36.
Stress-Strain Relationship
(Plane strain)
1 0
x 1 x
E (1 )
y 1 0
(1 )(1 2 ) 1 y
z 1 2 z
0 0
2(1 )
E
z 1 2 ( x y )
1
37.
Tri-axial stress state
elastic analysis
38.
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
39.
Stress and strain components
40.
The stress on a inclined plane
(l, m, n)
z
n
3
2 p
y
x 1 n
2 3 2 3 2 2
( n ) n2 ( 2 ) l ( 1 2 )( 1 3 )
2 2
1 2 1 2
( n 3 ) n2 ( 3 ) m 2 ( 2 3 )( 2 1 )
2 2
2 2 2 2
( n 1 ) n2 ( 1 ) n 2 ( 3 1 )( 3 2 )
2 2
41.
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
42.
Stress-Strain Relationship
Generalized Hooke’s Law:
1 0 0 0
x 1 0 0 0 x 1
1
y 1 0 0 0
y
1 2
E 0 z ET 1
z
0 0 0
2
0
xy (1 )(1 2 ) 1 2 xy 1 2 0
yz 0 0 0 0 0 yz 0
2
zx 1 2 zx 0
0 0 0 0 0
2
For isotropic materials
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