Stress/strain Relationship for Solids

Strength Of Materilas

2. Definition of normal stress
(axial stress)
A
F
=s

3. Definition of normal strain
0L
LD
=e

4. Poisson’s ratio

5. Definition of shear stress
0A
F
=t

6. Definition of shear strain
l
xD
==qgtan

7. Tensile Testing

8. Stress-Strain Curves

9. Stress-Strain Curves
http://www.uoregon.edu/~struct/courseware/461/461_lectures/4
61_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
area)

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
•Similar to Pressure-Volume Curve
•Area = Work
Volume
Pressure
Volume

16. Uni-axial Stress State
Elastic analysis

17. Stress-Strain Relationship
esE=
E -- Young’s modulus
gtG=
G -- shear modulus
Hooke’s Law:

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
follows:

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 force.

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
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.

27. Mohr’s Circle (Plane Stress)
http://www.egr.msu.edu/classes/me423/aloos/lectur
e_notes/lecture_4.pdf

28. Principal Stresses

29. Maximum shear stress

30. http://www4.eas.asu.edu/concrete/elasticity2_95/sld001.htm
Stress-Strain Relationship
(Plane stress)
))((
1
yxz
E
ssne +-=
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31. (2) Plane strain

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)
(ε
xx'
- ε
avg
)
2
+ ( γ
x'y'
/ 2 )
2
= R
2
ε
avg
=
ε
xx
+ ε
yy

2
http://www.shodor.org/~jingersoll/weave4/tutorial/tutorial.html

34. http://www.efunda.com/formulae/solid_mechan
ics/mat_mechanics/calc_principal_strain.cfm
Principal Strain

35. Maximum shear strain

36. Stress-Strain Relationship
(Plane strain)
ú
û
ù
ê
ë
é
+
-+
= )(
211
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E
ee
n
n
n
s
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nn
n
s
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)1(2
21
00
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1
0
1
1
)21)(1(
)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
))(()
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(
3121
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ss
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ss
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-
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+
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(
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22132213
ssss
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-
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(
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ssss
ss
t
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s --+
-
=+
+
+ n
nn
y
x
z
(l, m, n)
p
2
s
3
s
1
s
n
t
n
s

41. 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

42. Stress-Strain Relationship
For isotropic materials
Generalized Hooke’s Law:
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