Mechanics of deformable solids Definition

Stress deviator tensor

Builds on Cauchy stress tensor Trace of a matrix

Definition

Let $\sigma$ be a Cauchy stress tensor at a point, a real symmetric $3 \times 3$ matrix, and let $I$ be the identity matrix. The mean stress is $\sigma_m = \tfrac13 \operatorname{tr} \sigma$ , where $\operatorname{tr}$ is the trace. The stress deviator tensor is $s = \sigma - \sigma_m I,$ and $\sigma_m I$ is the hydrostatic or volumetric part of $\sigma$ .

Remarks

The deviator is symmetric with $\operatorname{tr} s = 0$ , and it has the same principal directions as $\sigma$ , with principal values $\sigma_k - \sigma_m$ . Its invariant $J_2 = \tfrac12 \sum_{i,j} s_{ij}^2$ enters the von Mises yield criterion. For uniaxial tension $\sigma = \operatorname{diag}(s_0, 0, 0)$ , one gets $\sigma_m = s_0/3$ , $s = \operatorname{diag}(2s_0/3, -s_0/3, -s_0/3)$ and $J_2 = s_0^2/3$ . For a hydrostatic stress $\sigma = -p I$ , the deviator is $0$ . In fluid mechanics the pressure is often taken as $p = -\sigma_m$ , so that $\sigma = -p I + s$ .

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Tag	`HJ19X19N0X`
URI	`m2://HJ19X19N0X`
Kind	definition
Writing	W0 stub
MSC	74A10, 74C05
Defines	Stress deviator tensor
Uses	Cauchy stress tensor, Trace of a matrix
Changed	3 November 2026

Stress deviator tensor

Definition

Remarks

Reading path

Further reading

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