StressStrain Relationship the constant G is called the shear modulus and relates the shear stress an

1
Stress/strain Relationship for Solids
2
Definition of normal stress (axial stress)
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Definition of normal strain
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Poissons ratio
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Definition of shear stress
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Definition of shear strain
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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/n
ode4.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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sPL ? Proportional Limit - Stress above which
stress is not longer proportional to strain. sEL
? Elastic Limit - The maximum stress that can be
applied without resulting in permanent
deformation when unloaded. sYP ? 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. sYS ? 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 sU ?
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
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Uni-axial Stress State Elastic analysis
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Stress-Strain Relationship
Hookes Law
E -- Youngs modulus
G -- shear modulus
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Stresses on Inclined Planes
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Thermal Strain
Straincaused by temperature changes. a 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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Mohrs Circle (Plane Stress)
http//www.tecgraf.puc-rio.br/etools/mohr/mohreng.
html
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Mohrs Circle (Plane Stress)
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Instruction to draw Mohrs 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 sx, sy, and txy tyx and list them
with the proper sign. 5. Draw a set of s - t
coordinate axes with s being positive to the
right and t 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 s 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. ?
0). Note The angle between the reference axis
and the s axis is equal to 2?p.
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Mohrs 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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Stress-Strain Relationship(Plane stress)
http//www4.eas.asu.edu/concrete/elasticity2_95/sl
d001.htm
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(2) Plane strain
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Coordinate Transformation
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Mohr's Circle (Plane Strain)
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
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Maximum shear strain
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Stress-Strain Relationship (Plane strain)
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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
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3-D Mohrs 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
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Stress-Strain Relationship
Generalized Hookes Law
For isotropic materials