Why is the stress on a body not a vector?

Question

In my textbook, Physics, Part II—Textbook for Class XI, there's a line which talks about why stress is not a vector:

Stress is not a vector quantity since, unlike a force, stress cannot be assigned a specific direction. Force acting on the portion of a body on a specified side of a section has a definite direction.

It does not elaborate why stress cannot be assigned a specific direction. I know that the stress on a body is the restoring force (applicable when the body is deformed) per unit area. Has it got something to do with the fact that area itself is a vector? Moreover, we often say that tensile (or compressive) stress is applied perpendicularly to the surface. That's specifying direction, isn't it?

An intuitive explanation (instead of a rigorous mathematical one) is highly appreciated.

Answer 1

Draw a square on an elastomer strip and stretch it:

"OK, I get this:"

The lengthwise load (comprising two force vectors, to the left and to the right) applies a stress state on the shape. What kind of stress?

"I'll assume the stress state can be expressed as a vector. I guess the vector corresponds to normal stresses to the left and to the right (i.e., forces acting perpendicular to the left and right sides). This is consistent with normal stresses changing side lengths of infinitesimal elements. I guess I'll call the vector [1 0 0], where I've normalized by the load magnitude."

Now consider drawing not a square but a diamond, for the same load.

"OK, now I get this:"

What is the stress state?

"It now includes some shear stress, since interior angles are now changing. Effectively, some forces on the sides are now parallel instead of solely perpendicular. The vector [1 0 0] doesn't capture this change, nor can I transform it rotationally to scale with [1 1 0] or [1 -1 0], say, because the diamond doesn't deform that way either; it stretches more to the left and right than it shrinks up and down. Hmm.

"Nature doesn't care which way we draw our coordinate systems, so we need a mathematical representation that transforms correctly. I have to conclude that a vector is incapable of representing the stress. However, a tensor would work:

$$\left[\begin{array}{ccc} 1 &0 &0\\0& 0& 0\\0& 0 &0\end{array}\right]$$

would transform upon a 45° rotation into

$$\left[\begin{array}{ccc} 1/2 &1/2 &0\\1/2& 1/2& 0\\0& 0 &0\end{array}\right],$$

which is consistent with the observed deformation of the diamond. Specifically, the side lengths stretch equally from an equibiaxial stress—from the diagonal elements—of 1/2, and this is superimposed on a shape change from a shear stress—from the off-diagonal elements—of 1/2.

"Furthermore, the tensor satisfies the standard requirements, such as invariance of the trace (here, 1) and two other invariants. These invariants capture the true essence of the stress state, which must be coordinate independent."

Why not just list those indices as, say, [½ ½ 0 ½ ½ 0 0 0 0] to make a vector?

"That not a true Cartesian vector, which has three elements and a well-defined direction. It's just a list."

One more question. When we apply a load on a surface, the resulting stress state has a well-defined direction that corresponds to the load. Why isn't a tensor needed here?

"A tensor is still needed to describe the stress state because of the above reasoning, but neither the surface nor the load are free to rotate, so any infinitesimal element aligned with the surface is constrained. Although this appears to suggest that stress has a single direction, its a particular result of the constraint and doesn't hold in general."

(Images from my site, adapted from a photograph by Nelson Fitness.)

Answer 2

We can put a wire under tensile stress by pulling each end with a force of equal magnitude. If the wire has an East-West alignment we need to pull its eastern end to the East and its western end to the West (even though one of these forces may be inconspicuously applied by a fixed anchorage to which one end of the wire is attached). You need both these forces to put the wire under uniform stress. So the stress doesn't point to the East or to the West. It has an alignment rather than a direction. That doesn't stop us from calculating the tensile stress by dividing the magnitude of either force by the area over which it acts.

Stress, then, is not a vector quantity. [Instead it can be represented by a mathematical object called a second rank tensor. Tensors enable us to represent not just tensile stress but other sorts of stress (such as shear stress), or combinations of different sorts of stress.]

Answer 3

Suppose a ball so full of air that is almost exploding. Due to the symmetry of the situation, it is clear that each small area of its surface is being stretched equally to all tangential directions. It is not possible to represent such a stress state to an entity with a modulus and a given direction as it is the case of a vector.

However it is possible to define a $x$ and $y$ axis for a small squared region along $2$ orthogonal tangential directions. There is an equal force $F$ for each of this directions. As forces are vectors, when we project and add them to a direction $45^{\circ}$, the result is $F\sqrt{2}$.

But if we divide $F$ by the side of the square, and divide $F\sqrt{2}$ by its diagonal (because it is the length associated with an angle of $45^{\circ}$), the result is the same, as should be due to the symmetry.

It is possible to see in this example that the state of stress can not be adequately represented by one vector, but by a stress tensor formed with $2$ tensile stresses in mutually orthogonal directions.

Answer 4

Stress is not a vector because it needs more information than what a vector can provide.

Stress can be represented as a $2^{nd}$ order tensor quantity $\mathbb{T}$ that relates two vectors, the unit normal vector, $\mathbf{\hat{n}}$, of a surface passing through a chosen point and the force per unit surface acting on it, $\mathbf{t_n}$,

$\mathbf{t_n} = \mathbf{\hat{n}} \cdot \mathbb{T}$.

If you change the orientation of the surface, the force per unit surface changes as well. This can be interpreted with the equilibrium of the Cauchy tetrahedron, see wiki page https://en.m.wikipedia.org/wiki/Cauchy_stress_tensor