# Algebraic Inversion of Stress-Strain Relations?

## Main Question or Discussion Point

How is this accomplished? How can one derive equations for stress in terms of strain from equations of strain in terms of stress or vice versa?

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Which particular relations are you thinking of?

Which particular relations are you thinking of?

I think that the stress-strain relations for linear elasticity are

$$\sigma_{ij}=\lambda \epsilon_{kk} +2\mu\epsilon_{ij}$$
and
$$\epsilon_{ij}=\frac{1+v}{E}\sigma_{ij}-\frac{v}{E}\sigma_{kk}$$

Inversion is briefly discussed in 'section' 3.2.8 here:

http://solidmechanics.org/text/Chapter3_2/Chapter3_2.htm

but I do not comprehend how exactly the inversion is performed

My actual goal is to invert the viscoelastic stress-strain relation here:

$$\dot{\sigma_{ij}}+\frac{\mu}{\eta}\sigma_{ij}=2\mu\dot{\epsilon_{ij}}+\delta_{ij}\left [ \lambda\dot{\epsilon_{kk}}-k\alpha_V \dot{T}+k\frac{\mu}{\eta}(\epsilon_{kk}-\alpha_V T) \right ]$$

which obviously has some additional terms. But I'm not sure how trivial this is.

Last edited:
AlephZero
Homework Helper
The basic idea is to write the stress-strain equations as a 6x6 matrix equation

$$\varepsilon = D \sigma$$

and then invert the matrix. The matrix inversion is easy for an isotropic solid.

For the thermal example in your link, you then write the complete equation in matrix form

$$\varepsilon = D \sigma + A \Delta t$$

where A is a vector. Then

$$D^{-1} \varepsilon = \sigma + D^{-1} A \Delta t$$

which is usually written as

$$C \varepsilon = \sigma + C A \Delta t$$

so

$$\sigma = C \varepsilon - C A \Delta t$$

And you can multiply out CA to get the equations in your link.

Thanks, although I'm not sure how to treat some of the other terms which appear here:

$$\sigma_{ij}=2\mu\epsilon_{ij}+\epsilon\lambda+\delta_{ij}\left [\epsilonk\int_0^t\frac{\mu}{\eta}dt-k\alpha_V T-k\int_0^t\alpha_V\dot{T}\frac{\mu}{\eta}dt-\int_0^t\sigma_{ij}\frac{\mu}{\eta}dt \right ]$$

Can all of the terms be introduced into the same matrix $$C=D^{-1}$$ as you have done with $$k\alpha_V T$$ or do they have to be described with a unique matrix? I'm specifically unsure about how to deal with the last term in which $$\sigma_{ij}$$ appears. How is this introduced into the matrix formula?

Thanks again.