Stress (mechanics) - Stress Deviator Tensor

Stress Deviator Tensor

The stress tensor can be expressed as the sum of two other stress tensors:

  1. a mean hydrostatic stress tensor or volumetric stress tensor or mean normal stress tensor, which tends to change the volume of the stressed body; and
  2. a deviatoric component called the stress deviator tensor, which tends to distort it.

So:

where is the mean stress given by

Note that convention in solid mechanics differs slightly from what is listed above. In solid mechanics, pressure is generally defined as negative one-third the trace of the stress tensor.

The deviatoric stress tensor can be obtained by subtracting the hydrostatic stress tensor from the stress tensor:

\begin{align} \ s_{ij} &= \sigma_{ij} - \frac{\sigma_{kk}}{3}\delta_{ij},\,\\ \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} p & 0 & 0 \\ 0 & p & 0 \\ 0 & 0 & p \\ \end{matrix}}\right] \\ &=\left[{\begin{matrix} \sigma_{11}-p & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22}-p & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33}-p \\ \end{matrix}}\right]. \\ \end{align}

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