Deviatoric stress and invariants

The stress tensor can be expressed as the sum of two stress tensors, namely: the hydrostatic stress tensor and the deviatoric stress tensor. In this article we will define the hydrostatic and the deviatoric part of the stress tensor and we will calculate the invariants of the stress deviator tensor. The invariants of the deviatoric stress are used frequently in failure criteria.

Consider a stress tensor $\sigma_{ij}$ acting on a body. The stressed body tends to change both its volume and its shape. The part of the stress tensor that tends to change the volume of the body is called mean hydrostatic stress tensor or volumetric stress tensor. The part that tends to distort the body is called stress deviator tensor. Hence, the stress tensor may expressed as:

 $\sigma_{ij}=s_{ij}+p\delta_{ij}$ (1)

where $\delta_{ij}$ is the Kronecker delta (with $\delta_{ij}=1$ if $i=j$ and $\delta_{ij}=0$ if $i\neq j$), $p$ is the mean stress given by:

 $p=\frac{1}{3}\sigma_{kk}=\frac{1}{3}\left(\sigma_{11}+\sigma_{22}+\sigma_{33}\right)=\frac{1}{3}I_{1}$ (2)

where $I_{1}$ is the first invariant of the stress tensor (see also: Principal stresses and stress invariants). The product $p\delta_{ij}$ is the hydrostatic stress tensor and contains only normal stresses. The deviatoric stress tensor can be obtained by subtracting the hydrostatic stress tensor from the stress tensor:

 $\begin{array}{rl}s_{ij}=&\sigma_{ij}-p\delta_{ij}\\=&\left[\begin{array}{ccc}\sigma_{11}-p & \sigma_{12} & \sigma_{13}\\ \sigma_{21} & \sigma_{22}-p & \sigma_{23}\\ \sigma_{31} & \sigma_{32} & \sigma_{33}-p\end{array}\right]\end{array}$ (3)

In order to calculate the invariants of the stress deviator tensor we will follow the same procedure used in the article Principal stresses and stress invariants. It must be mentioned that the principal directions of the stress deviator tensor coincide with the principal directions of the stress tensor. The characteristic equation for $s_{ij}$ is:

 $\left| s_{ij}-s\delta_{ij}\right|=s^{3}-J_{1}s^{2}-J_{2}s-J_{3}=0$ (4)

where $J_{1}$, $J_{2}$ and $J_{3}$ are the first, second and third deviatoric stress invariants, respectively. The roots of the polynomial are the three principal deviatoric stresses $s_{1}$, $s_{2}$ and $s_{3}$. $J_{1}$, $J_{2}$ and $J_{3}$ may be calculated by the following expressions:

 $\begin{array}{rl}J_{1}=&s_{kk}=0\\J_{2}=&\frac{1}{2}s_{ij}s_{ji}\\=&\frac{1}{6}\left[\left(\sigma_{11}-\sigma_{22}\right)^2+\left(\sigma_{22}-\sigma_{33}\right)^2+\left(\sigma_{33}-\sigma_{11}\right)^2\right]\\&+\sigma_{12}^2+\sigma_{23}^2+\sigma_{31}^2\\=&\frac{1}{3}I_{1}^{2}-I_{2}\\J_{3}=&det(s_{ij})\\=&\frac{1}{3}s_{ij}s_{jk}s_{ki}\\=&\frac{2}{27}I_{1}^{3}-\frac{1}{3}I_{1}I_{2}+I_{3}\end{array}$ (5)

where $I_{1}$, $I_{2}$ and $I_{3}$ are the three invariants of the stress tensor and $det(s_{ij})$ is the determinant of $s_{ij}$. It should be mentioned that since $J_{1}=s_{kk}=0$, the stress deviator tensor describes a state of pure shear.

Example

Calculate the stress deviator tensor and its invariants for the following stress tensor:

 $\sigma_{ij}=\left[\begin{array}{ccc}2&-3&4\\-3&-5&1\\4&1&6\end{array}\right]$ (6)

Solution

Firstly we calculate the mean pressure $p$:

 $p=\frac{1}{3}\left(2-5+6\right)=1$ (7)

From equation (3) we calculate the stress deviator tensor:

 $s_{ij}=\left[\begin{array}{ccc}1&-3&4\\-3&-6&1\\4&1&5\end{array}\right]$ (8)

For the stress deviator tensor invariants we will use equations (5) and we get:

 $\begin{array}{rl}J_{1}=&0\\J_{2}=&\frac{1}{2}s_{ij}s_{ji}\\=&\frac{1}{2}\left(s_{11}^2+s_{22}^2+s_{33}^2+2s_{12}^2+2s_{23}^2+2s_{13}^2\right)\\=&\frac{1}{2}\left(1^2+(-6)^2+5^2+2*(-3)^2+2*1^2+2*4^2\right)\\=&57\\J_{3}=&\frac{1}{3}s_{ij}s_{jk}s_{ki}\\=&s_{11}s_{22}s_{33}+2s_{12}s_{23}s_{13}-s_{11}s_{23}^2-s_{22}s_{13}^2-s_{33}s_{12}^2\\=&1*(-6)*5+2*(-3)*1*4\\&-1*1^2-(-6)*4^2-5*(-3)^2\\=&-4\end{array}$ (9)

Finally the characteristic equation is:

 $s^3-57s+4=0$ (10)

Suggested Bibliography

Y.C. Fung and P. Tong. Classical and Computational Solid Mechanics. World Scientific, Singapore, 2001.

L.E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice Hall, Englewood Cliffs, New Jersey, 1969.

W.F. Chen and D.J. Han. Plasticity for Structural Engineers. Springer-Verlag,  New York, 1988.

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5 Responses to Deviatoric stress and invariants

1. pramodh says:

the subject is very clearly explained and the example helped in avoiding confusion……. thanks for the clarification

2. Leo says:

I think there’s a mistake at step 8, s12 should be unchanged so it’s -1, and J2 I got 49

• rockmechs says:

You are right. There was a mistake on equation (6), not on equation (8). The stress tensor must be symmetric, so s12 is -3, not -1. I fixed it. Thank you for your remark.

3. Kava says:

I’d like to know what this measure of stress is used for. It seems to be used primarily by geotechs (based on a Google search). I see the derivation over and over but no practical application – can you please describe the usage?

• rockmechs says:

Deviatoric stress is used primarly in failure criteria (and especially the second invariant J2 of the deviatoric stress tensor). It represents the shear part of the stress tensor, the part that tends to distort the shape of the body. You will find more info if you read more on failure criteria. I am planning to write soon a few articles on well known failure criteria.