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Can someone explain to me why the stress tensor is symmetrical. I understand that Sij=Sji , but can someone give me the assumption or the physical reason why this is true. Thanks.
PerennialII said:[tex]
e_{ijk}\sigma_{jk}=0
[/tex]
However, in the presence of couple-stresses, i.e. moments per unit volume, the stress tensor is non-symmetric. This also is the case when the Knudsen number is close to one, K_{n} -> 1, e.g. Non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as polymers.
quasi426 said:Can someone explain to me why the stress tensor is symmetrical. I understand that Sij=Sji , but can someone give me the assumption or the physical reason why this is true.
Symmetry under spacetime translations implies (by Noether theorem) that the canonical energy-momentum (or stress) tensor
[tex]
T_{ab} = \frac{\partial L}{\partial \partial_{a} \phi} \partial_{b} \phi - \eta_{ab}L
[/tex]
is conserved;
[tex] \partial^{a} T_{ab} = 0[/tex]
But it is not, in general, symmetric! Well, it is not unique either, for you could define a new tensor
[tex]
\Theta_{ab} = T_{ab} + \partial^{c} X_{cab}
[/tex]
which is also conserved, [itex]\partial^{a}\Theta_{ab} = 0[/itex], provided that
[tex]X_{cab} = - X_{acb}[/tex]
In a Lorentz invariant theories, we may choose [itex]X_{cab}[/itex] to make [ the new stress tensor] [itex]\Theta_{ab}[/itex] symmetric.
So, your question should have been: Why do we want the stress tensor to be symmetric?
There are two reasons for this:
1) In general relativity, the matter fields couple to gravity via the stress tensor and this is given by the Einstein equations
[tex]R_{ab} - \frac{1}{2} g_{ab} R = - k \Theta_{ab}[/tex]
Since the (geometrical) Ricci tensor [itex]R_{ab}[/itex] and the metric tensor [itex]g_{ab}[/itex] are both symmetric, so [itex]\Theta_{ab}[/itex] must be also.
2) The second reason for requiring a symmetric stress tensor comes from Lorentz symmetry:
Lorentz invariance implies that the ungular momentum tensor;
[tex]\mathcal{M}_{cab} = \Theta_{ca} x_{b} - \Theta_{cb} x_{a}[/tex]
is conserved! But
[tex]\partial^{c} \mathcal{M}_{cab} = \Theta_{ab} - \Theta_{ba}[/tex]
Thus, conservation of ungular momentum requires the stress tensor to be symmetric;
[tex]\Theta_{ab} = \Theta_{ba}[/tex]
regards
sam
The stress tensor is a mathematical concept used in physics and engineering to describe the distribution of internal forces within a material. It is important because it helps us understand the behavior of materials under different types of stress, such as tension, compression, and shear.
The stress tensor is symmetrical because it follows the laws of conservation of energy and momentum. In other words, the total force acting on a material in one direction must be balanced by an equal and opposite force in the opposite direction, resulting in a symmetrical distribution of stress.
Symmetry is related to the stress tensor because it ensures that the forces acting on a material are balanced and there is no net force that could cause the material to deform or break. The symmetrical distribution of stress also helps in simplifying the mathematical equations used to describe the behavior of materials under different types of stress.
Yes, in certain situations, the stress tensor can be non-symmetrical. This usually occurs when there are external forces acting on the material or when the material is undergoing dynamic changes. However, in most practical scenarios, the stress tensor is assumed to be symmetrical for simplicity and ease of calculation.
The symmetry of the stress tensor is measured using mathematical techniques such as matrix algebra and tensor analysis. These methods involve calculating the components of the stress tensor and checking if they satisfy the conditions for symmetry. If the components are equal, the stress tensor is considered to be symmetrical.