What is the correct formula to calculate the effective strain under plane stress conditions?

In plane stress it is assumed that the out of plane stresses are zero. The out of plane strains are not zero but these tend to be ignored. Wikipedia gives the equation for equivalent strain as:

$\epsilon_{eq} = \sqrt{\frac{2}{3}{\epsilon}^{dev}_{ij}:{\epsilon}^{dev}_{ij}}$

But from the 2/3 in the equation above it is clear that this was derived from the full strain tensor. How is the effective strain calculated if the out of plain strains are not calculated in the plane stress approximation? Do these have to be calculated to then calculate the effective stress or is it common practice to just neglect these in the formula above?

(I am using the terms effective and equivalent here interchangeably, please correct me if this is wrong as I can't seem to clear up what the difference is if plasticity is not involved).


This is my current thinking. If stresses are calculated from strains using a plane stress simplification then in vector notation (from wiki):

$ \begin{bmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{12} & \sigma_{22} \end{bmatrix} =\frac{E}{1-v^2}\left((1-v) \begin{bmatrix} \epsilon_{11} & \epsilon_{12} \\ \epsilon_{12} & \epsilon_{22} \end{bmatrix} +v\mathbf{I}(\epsilon_{11}+\epsilon_{22})\right) $

And $\epsilon_{33}$ can be calculated from $\frac{-v}{E}(\sigma_{11}+\sigma_{22})$. Considering that all of the strain components have now been calculated it is possible to then calculate the deviatoric strain using the conventional 3D formulas for the hydrostatic and deviatoric components. Does this seem correct?

  • $\begingroup$ out-of-plane strains are not "unknown" in a plane stress analysis, they can be computed from the constitutive relation. $\endgroup$
    – agentp
    Mar 7, 2017 at 19:07
  • $\begingroup$ Agreed. Maybe I should rephrase the question. So they must be calculated then? $\endgroup$ Mar 7, 2017 at 19:17
  • $\begingroup$ Maybe this [related question] contains the answer you are looking for? [related question]: engineering.stackexchange.com/questions/13528/… $\endgroup$
    – JLo
    Mar 7, 2017 at 22:29
  • $\begingroup$ It's not obvious to me how the question you've linked contains the answer. Could you please elaborate? $\endgroup$ Mar 7, 2017 at 22:41


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