Download - Stress/strain Relationship for Solids
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Definition of normal stress (axial stress)
AF
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Definition of normal strain
0LL
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Poisson’s ratio
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Definition of shear stress
0AF
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Definition of shear strain
lx
tan
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Tensile Testing
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Stress-Strain Curves
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Stress-Strain Curves
http://www.uoregon.edu/~struct/courseware/461/461_lectures/461_lecture24/461_lecture24.html
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Stress-Strain Curve (ductile material)
http://www.shodor.org/~jingersoll/weave/tutorial/node4.html
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Stress-Strain Curve (brittle material)
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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
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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)
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True stress and true strain
True stress and true strain are based upon instantaneous values of cross sectional
area and gage length
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The Region of Stress-Strain Curve
Stress Strain Curve
• Similar to Pressure-Volume Curve• Area = Work
Volume
Pressure
Volume
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Uni-axial Stress State Elastic analysis
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Stress-Strain Relationship
EE -- Young’s modulus
GG -- shear modulus
Hooke’s Law:
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Stresses on Inclined Planes
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Thermal Strain
Straincaused by temperature changes. α is a material characteristic called the coefficient of thermal expansion.
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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:
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Bi-axial state elastic analysis
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(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.
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Transformation of Plane Stress
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Mohr’s Circle (Plane Stress)
http://www.tecgraf.puc-rio.br/etools/mohr/mohreng.html
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Mohr’s Circle (Plane Stress)
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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 theupward 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.
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Mohr’s Circle (Plane Stress)
http://www.egr.msu.edu/classes/me423/aloos/lecture_notes/lecture_4.pdf
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Principal Stresses
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Maximum shear stress
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http://www4.eas.asu.edu/concrete/elasticity2_95/sld001.htm
Stress-Strain Relationship(Plane stress)
))((1yxz E
xy
y
x
xy
y
x E
2100
0101
1 2
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(2) Plane strain
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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
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Mohr's Circle (Plane Strain)
(εxx' - εavg)2 + ( γx'y' / 2 )2 = R2
εavg = εxx + εyy
2http://www.shodor.org/~jingersoll/weave4/tutorial/tutorial.html
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http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/calc_principal_strain.cfm
Principal Strain
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Maximum shear strain
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Stress-Strain Relationship (Plane strain)
)(
211 yxzE
z
y
x
z
y
x E
)1(22100
011
01
1
)21)(1()1(
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Tri-axial stress state elastic analysis
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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
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Stress and strain components
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The stress on a inclined plane
))(()2
()2
( 312122322232
lnn
))(()2
()2
( 123222132213
mnn
))(()2
()2
( 231322212221
nnn
y
x
z
(l, m, n)
p2
3
1n
n
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3-D Mohr’s Circle
* 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
D
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Stress-Strain Relationship
For isotropic materials
Generalized Hooke’s Law:
000111
21
22100000
02210000
00221000
000100010001
)21)(1(
TEE
zx
yz
xy
z
y
x
zx
yz
xy
z
y
x