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Stress-Energy Tensor
--------------------
Tensors are geometric objects that describe linear
relations between vectors, scalars, and other tensors.
They are the counterparts of 4-vectors in Special
Relativity.
Consider the stress, σ, inside a material. How can
we describe the forces, F, inside of it. Imagine a
cubic region with faces of area, A, within the material.
The corresponding stress tensor is:
- -
| σ_{xx} σ_{xy} σ_{xz} |
σ = | σ_{yx} σ_{yy} σ_{yz} |
| σ_{zx} σ_{zy} σ_{zz} |
- -
Stress is a vector defined as F/A. In geometry we
can represent an area as a vector. The vector is
perpendicular to the area and the length of the vector
is proportional to the area. The forces can be written
as the dot product of the stress and the area vector.
Thus:
F_{x} = x component from x face + x comp. from y face
+ x comp. from z face
= σ_{xx}.A_{x} + σ_{yx}.A_{y} + σ_{zx}.A_{z}
F_{y} = ....
F_{z} = ....
Total force: F = F_{x} + F_{y} + F_{z}
The off diagonal components represent shear stresses.
Tensors are defined in terms of ranks as follows.
Rank-0 tensor = scalar - no indeces
Rank-1 tensor = vector = 1d matrix - 1 index
Rank-2 tensor = 2d matrix - 2 indeces
Rank-3 tensor = 3d matrix (cube) - 3 indeces
Rank 2 Tensor
-------------
Simplest way to visualize is in terms of the OUTER
(Dyadic) product of 2 vectors A,B.
T^{mn} = A^{m}B^{n}
- -
_{ } | A_{1}B_{1} A_{1}B_{2} A_{1}B_{3} |
_{ } = | A_{2}B_{1} A_{2}B_{2} A_{2}B_{3} |
_{ } | A_{3}B_{1} A_{3}B_{2} A_{3}B_{3} |
- -