Thermal expansion
In this tutorial, we will see how we can model thermal expansion using Elasticipy.
Isotropic case
At first, let’s consider steel as an example, assuming an isotropic behaviour:
>>> from Elasticipy.ThermalExpansion import ThermalExpansionTensor as ThEx
>>> alpha = ThEx.isotropic(11e-6)
>>> print(alpha)
Thermal expansion tensor
[[1.1e-05 0.0e+00 0.0e+00]
[0.0e+00 1.1e-05 0.0e+00]
[0.0e+00 0.0e+00 1.1e-05]]
To compute the strain due to an increase of temperature, just multiply alpha by this amount:
>>> dT = 10
>>> alpha * dT
Strain tensor
[[0.00011 0. 0. ]
[0. 0.00011 0. ]
[0. 0. 0.00011]]
We can also compute the strain values from a series (i.e. an array) of temperature increases, and get an array of strain with the corresponding shape:
>>> eps = alpha * [0,1,2]
>>> print(eps)
Strain tensor
Shape=(3,)
>>> eps[0]
Strain tensor
[[0. 0. 0.]
[0. 0. 0.]
[0. 0. 0.]]
>>> eps[-1]
Strain tensor
[[2.2e-05 0.0e+00 0.0e+00]
[0.0e+00 2.2e-05 0.0e+00]
[0.0e+00 0.0e+00 2.2e-05]]
See here for details about multidimensional arrays of tensors.
Anisotropic case
The anisotropic case just works as before. For instance, if we consider carbon fibers:
>>> alpha = ThEx.transverse_isotropic(alpha_11=5.6e-6, alpha_33=-0.4e-6)
>>> print(alpha)
Thermal expansion tensor
[[ 5.6e-06 0.0e+00 0.0e+00]
[ 0.0e+00 5.6e-06 0.0e+00]
[ 0.0e+00 0.0e+00 -4.0e-07]]
As is, we consider that the fiber is parallel to the Z axis. We can obviously rotate it. For instance, if we want to align the fiber with the X axis:
>>> from scipy.spatial.transform import Rotation
>>> rotation = Rotation.from_euler('Y', 90, degrees=True) # Rotate of 90° around Y axis
>>> alpha * rotation
Thermal expansion tensor
[[-4.00000000e-07 0.00000000e+00 1.33226763e-21]
[ 0.00000000e+00 5.60000000e-06 0.00000000e+00]
[ 1.33226763e-21 0.00000000e+00 5.60000000e-06]]
If we want to consider multiple orientations at once, we can create an array of thermal expansion tensors, just as we do with strain/stress tensors:
>>> rotations = Rotation.random(10000, random_state=123) # 10k random orientations. random_state ensure reproducibility
>>> alpha_rotated = alpha * rotations
>>> print(alpha_rotated)
Thermal expansion tensor
Shape=(10000,)
If we want to compute the strain for each combination of orientations and temperatures in [0,1,2] (as done above),
we can use the matmul operator:
>>> eps = alpha_rotated.matmul([0,1,2])
>>> print(eps)
Strain tensor
Shape=(10000, 3)
For instance we can check out the maximum value for initial (0°) and final (2°) temperatures:
>>> eps[:,0].max() # 0 because it corresponds to 0°
Strain tensor
[[ 0. -0. -0.]
[-0. 0. -0.]
[-0. -0. 0.]]
>>> eps[:,-1].max()
Strain tensor
[[1.12000000e-05 5.99947076e-06 5.99905095e-06]
[5.99947076e-06 1.11999999e-05 5.99621436e-06]
[5.99905095e-06 5.99621436e-06 1.11999992e-05]]
We see that the maximum value for the shear strain is consistent with the Mohr circle, as we have: