3D Stress-Strain Tensor Relations

​
initial state
tensors
σ
11
=
0
σ
12​
=
0
σ
13
=
0
σ
22​
=
0
σ
23
=
0
σ
33
=
0
σ =
0.
0.
0.
0.
0.
0.
0.
0.
0.
Pa
Young's modulus (Pa)
1
Poisson's ratio
0
ε =
0.
0.
0.
0.
0.
0.
0.
0.
0.
rotation method
rotation about
current axes
original axes
1st rotation axis
x
y
z
2nd rotation axis
x
y
z
3rd rotation axis
x
y
z
calculate rotated stress and strain
1st rotation angle
0
2nd rotation angle
0
3rd rotation angle
0
This Demonstration considers stress-strain relationships in the mechanical behavior of materials. Stress is represented by a second-rank tensor
σ
ij
with nine components. However, only six components are independent, since the tensor is symmetrical. We calculate the relationship between uniaxial stress/strain in 3D space. You can select values of Young's modulus
Y
and Poisson's ratio
v
, which determine the strain state, represented by the strain tensor
ε
ij
.
As well, consider rotation about the three Cartesian axes through selected angles. The stress and strain tensors are correspondingly transformed to
σ'
and
ε'
(not shown), while the physical situation is not changed.

References

[1] J. P. Steimel. "Materials Science and Engineering." Pacific Open Texts. (May 11, 2022) scholarlycommons.pacific.edu/open-textbooks/8.
[2] Mathematica Stack Exchange. "Resize a Manipulate by Grabbing a Corner." (May 11, 2022) mathematica.stackexchange.com/questions/199234/resize-a-manipulate-by-grabbing-a-corner.

External Links

Composing Rotations via Hamilton Turns
Combining Two 3D Rotations
Mohr's Circle and Stress Transformations

Permanent Citation

Richard Jerue, Leon Tran, Daniel Balerite, Mark Michael, Tanraj Shergill, Joshua Steimel
​
​"3D Stress-Strain Tensor Relations" from the Wolfram Demonstrations Project http://demonstrations.wolfram.com/3DStressStrainTensorRelations/​
​Published: May 16, 2022
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