# Why is the deviatoric stress the stress that causes distortions?

I'm reading this text on stress.

It is mentioned that the Cauchy stress tensor can be split into a sum of two other tensors: hydrostatic pressure $$\pi$$ and deviatoric stress. Hydrostatic pressure is defined as the mean of the normal stresses. Deviatoric stress tensor is what we get when we subtract a tensor with the pressure on diagonal from the original Cauchy stress tensor.

\begin{align} \ \,\\ \left[{\begin{matrix} s_{11} & s_{12} & s_{13} \\ s_{21} & s_{22} & s_{23} \\ s_{31} & s_{32} & s_{33} \end{matrix}}\right] &=\left[{\begin{matrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{matrix}}\right]-\left[{\begin{matrix} \pi & 0 & 0 \\ 0 & \pi & 0 \\ 0 & 0 & \pi \end{matrix}}\right] \\ &=\left[{\begin{matrix} \sigma_{11}-\pi & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22}-\pi & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33}-\pi \end{matrix}}\right]. \end{align}

where

$$\pi=\frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3}$$

It is then said:

The second component is the Deviatoric stress and is what actually causes distortion of the body.

My question is: Why do the diagonal stresses in the deviatoric stress tensor cause distortions, considering they are still normal stresses?

I've been taught that normal stresses cause only volumetric changes, and don't cause distortions, that is, don't change the shape of the element. It is the shear stresses, which lay off-diagonal, that cause shape changes to the element.

The shear stresses are present unchanged on the deviator tensor, but the normal stresses are now differences between the normal stresses on the element and their average. But they are still normal stresses, so why are they present in the deviator tensor, if the purpose of the deviator tensor is to include only stresses that cause distortions?

• They cause purely volumetric strains only if they are all equal (i.e., an isotropic state of stress). Deviations from them all being equal is what causes distortions. Jun 22, 2019 at 0:14