I have seen some theories saying stress is a scalar. I don't understand why. Anyone can explain.

  • 2
    $\begingroup$ Neither. It's a second-order tensor. $\endgroup$ – alephzero May 3 '19 at 22:37
  • $\begingroup$ can we do vector sum on stress? $\endgroup$ – Christian777 May 3 '19 at 23:08
  • $\begingroup$ Whatever you mean by "a vector sum", the answer is probably no. $\endgroup$ – alephzero May 4 '19 at 0:48

As a mechanical engineer, I've come across the stress being represented mainly as a second order tensor, the Cauchy stress tensor, however vector representations are also common. As for scalar representations? This might be common in scenarios of uniaxial stress, such as a bar under uniform tensile stress, where there is only one component of stress.

Cauchy stress tensor

In general, for a given coordinate system $O(x,y,z)$, there are 6 components of stress:

$$\sigma_{xx}\quad \sigma_{yy}\quad \sigma_{zz}\quad \sigma_{xy}\quad \sigma_{xz}\quad \sigma_{yz}$$

Let's consider a solid object of arbitrary shape. It can be a beam, a brick, a gear, etc. In general, the stress will vary throughout the solid.

To consider the stresses at a point in the solid body, let's surgicaly remove an infinitesimally small cube from within the solid at the point of interest, to observe the stress acting on the faces of the cube. The faces of the cube is aligned with the $x$-, $y$- and $z$-axes of our coordinate system. On each face of the six faces of the cube, three components of stress act: two components lie tangential to the face (these components are called shear stresses), and one component is perpendicular to the face (this component is called the normal stress). The diagram below shows the stresses acting on three of the faces.

enter image description here

Six faces, each with three stress components, might suggest a total of 18 components. However, balance of forces (stress $\times$ infinitesimal face area) onto the cube means pairs of stress components on opposite faces are equal and opposite ($\rightarrow$ 9 unique components).

The convention for defining stress components is to use the form $\sigma_{ij}$, where subscript $i$ indicates the face on the cube which the stress component acts, and $j$ indicates the direction of the stress. ($\sigma_{xy}$ acts in the $y$ direction on the $x$-face of the cube).

By balance of moments, it can be shown that $$\sigma_{xy} = \sigma_{yx}, \qquad \sigma_{xz} = \sigma_{zx}, \qquad \sigma_{yz} = \sigma_{zy}$$ reducing the number of unique stress components to six.

Therefore, at any point in a 3D solid, the stress state is fully described by six numbers:

$\sigma_{xx}\quad \sigma_{yy}\quad \sigma_{zz}$ are the normal stress components

$\sigma_{xy}\quad \sigma_{xz}\quad \sigma_{yz}$ are the shear stress components.

(Frequently, shear stresses (but not normal stresses) are alternatively expressed as $\tau_{ij}$ instead of $\sigma_{ij}$)

A natural means of expressing this stress state is as a Cauchy stress tensor:

$$[\sigma] = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \\ \end{bmatrix}$$

or, equivalently, to highlight the symmetric nature of the stress tensor as well as using only the 6 unique components:

$$[\sigma] = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{xy} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{xz} & \sigma_{yz} & \sigma_{zz} \\ \end{bmatrix}$$

In 2D stress problems where, say, stresses in the $z$-direction are not of concern, this tensor is often reduced to

$$[\sigma] = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{xy} & \sigma_{yy} \\ \end{bmatrix}$$

The advantage of the Cauchy tensor is that it is possible to re-express the stress components in a different frame of reference $O(x',y',z')$ by performing a change of basis on the tensor. This is done by performing a matrix transformation on the array of elements $[\sigma]$ to obtain the array of stress components in the new coordinate system $[\sigma']$. That is, if a vector with components $[a]$ expressed in $O(x,y,z)$ is related to the components $[a']$ re-expressed in $O(x',y',z')$ via the equation

$$[a'] = [Q][a]$$

where $[Q]$ is an orthogonal matrix, then the Cauncy stress tensor elements in both systems are interrelated by

$$[\sigma'] = [Q][\sigma][Q]^T$$

There is nothing stopping you from re-arranging the 6 stress components into a 6-dimension column vector, such as

$[\sigma] = \begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz} \\ \sigma_{xy} \\ \sigma_{xz} \\ \sigma_{yz} \\ \end{bmatrix}$

This is convenient for expressing material laws in linear elasticity problems, as in this form, the stress vector is related to a similarly defined strain vector via multiplication with a $6\times 6$ constitution matrix (as opposed to the stress tensor being related with the strain tensor via appropriate multiplication of a fourth order tensor!). However, re-expressing this vector in different coordinate systems is not a readily available action. A transformation matrix $[Q]$ cannot be used directly here, least to say that $[Q]$ is a $3\times 3$ matrix, and the stress vector here is $6 \times 1$!

The Cauchy stress vector

In the first case, we used the unit cube to describe the stress state at a point by observing the faces of an infinitesimal cube. An alternative way to examine stress is to analyse how the stress act on an imaginary surface at that point. Imagine now that we cut the solid body about a plane that passes through the point, and the plane has a normal vector

$$[n] = \begin{bmatrix}n_x \\ n_y \\ n_z\end{bmatrix}$$

enter image description here

Now let's analyse the stress that act on this plane at the point of interest.

Like for the case of the cube, three perpendicular components of stress act on the cutting plane, and these are

$$t_x \quad t_y \quad t_z$$

In fact, the total stress acting on this surface can be represented as a vector, in the form

$$[t] = \begin{bmatrix}t_x \\ t_y \\ t_z\end{bmatrix}$$

This vector is a true vector is the sense that it can be transformed to re-expressed its elements in a new coordinate system. Considering the transformation matrix $[Q]$ from earlier, the stress vector representations in the two coordinate systems are interrelated by

$$[t'] = [Q][t]$$

This stress vector is referred to as the Cauchy stress vector. Note that expression of stress as a vector requires a reference surface of a given normal $[n]$ (The Cauchy stress tensor can be defined without providing a reference surface).

The Cauchy stress vector is related to the Cauchy stress tensor by the Cauchy stress equation:

$$[\sigma][n] = [t]$$

In full:

$$\begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{xy} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{xz} & \sigma_{yz} & \sigma_{zz} \\ \end{bmatrix} \begin{bmatrix}n_x \\ n_y \\ n_z\end{bmatrix} = \begin{bmatrix}t_x \\ t_y \\ t_z\end{bmatrix} $$

Stress vs traction

To define our stress vector, we had to cut at a plane to obtain a surface on which to describe stress. However, what happens if we consider a point on the outer surface of our solid? A similar vector can be computed by considering the force per unit acting on the outer surface of the solid, but, as a mechanical engineer, I would argue that this vector is called a traction vector instead of a stress vector at this point, since now this vector arises due to external forces rather that internal. Stresses are internal forces per unit area, whereas tractions are external forces per unit area. However, this is just a matter of terminology, and other disciplines may use different terminologies.

To summarise

Stress can be expressed as a vector or a tensor. Stress can only really be described by a scalar for simple uniaxial stress problems.

  • $\begingroup$ In the interest of accuracy, the phrase ''In general, for a given coordinate system..." doesn't seem to be quite correct, because in general there are nine components, as far as we never leave the linear elastic region, then by making some assumptions and approximations the constitutive relations are true, and we have only six components. One should know where the approximation lies. $\endgroup$ – Sam Farjamirad May 7 '19 at 16:46
  • $\begingroup$ I will stress that there are 6 and only 6 unique components of stress, which arise by balance of linear momentum (or balance of forces for static cases) and balance of angular momentum (or balance of moments for static), which I mention in my answer. This is independent of any constitutive relationships or compatibility relationships. This holds in general for linear elasticity. $\endgroup$ – Involute May 7 '19 at 17:30
  • $\begingroup$ The way you described is correct "balance of linear momentum (or balance of forces for static cases) and balance of angular momentum (or balance of moments for static)" is actually the right start point, but if you write down the moment equilibrium, you'll see the stresses are not symmetric, however if you ignore the second order terms, then yes the tensor is symmetric then, as i mentioned this is true in general linear elastic deformation. I don't recall i said anything about compatibility. $\endgroup$ – Sam Farjamirad May 7 '19 at 18:05
  • $\begingroup$ About my tone: this is very standard question, and you made some solid points, the answer is rich in details, i'm trying to add more details, i don't have to prove anything to you or anybody else, but i'm the only one so far went through your answer and upvoted it, loot at here imgur.com/QZKUQ2p $\endgroup$ – Sam Farjamirad May 7 '19 at 18:15
  • $\begingroup$ I appreciate your intentions! :) Still, I’m still a little intrigued where the can possibly be a case where the stress tensor is not symmetric. Surely if the stresses are observed on an infinitesimal cube, then any second order terms can be made arbitrarily small as the cube size tends toward zero. I’d be eager if you could point out contrary cases, though $\endgroup$ – Involute May 7 '19 at 19:36

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