In the research paper "I. Krakovský et al., A few remarks on the electrostriction of elastomers, 1999" the authors stated the equations of the principle stresses that are produced in an isotropic dielectric, as follows:

$$\sigma_{xx} = \sigma_{yy} = \frac{1}{2} \varepsilon_0 \varepsilon E^2 \left(1 + \frac{a_2}{\varepsilon}\right)$$

$$\sigma_{zz} = - \frac{1}{2} \varepsilon_0 \varepsilon E^2 \left(1 - \frac{a_1+a_2}{\varepsilon}\right)$$

where $\sigma_{xx}$, $\sigma_{yy}$, and $\sigma_{zz}$ are the principal stresses. $\varepsilon_0$ is vacuum permittivity and $\varepsilon$ is the relative permittivity. $E$ is the electric field, and $a_1$ and $a_2$ are two parameters describing the change of dielectric properties of material in shear and bulk deformation.

I would like to compare them with other equations of stress from another paper, but in the other paper the stress tensor is stated in its tensor form containing all the elements including shear stress, so without the principle stresses presentation.

Thus, I would like to know how to convert the principles stresses back and get the full stress tensor, so that I could see if both papers have same stress equations.


1 Answer 1


Given a set of principal stresses, it is not possible to get a unique state of stress. This is because the stress tensor has 6 independent components, whereas there are only 3 principal stresses. However, given a state of stress, you can calculate the unique principal stresses. Your best bet would be to calculate the principal stresses from the other paper and match it with these. To do that, you would just have to get the eigen-values of the stress tensor.


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