Algebraic Inversion of Stress-Strain Relations?

[Hypatio]

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?

 

[Studiot]

Which particular relations are you thinking of?

 

[Hypatio]

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.

 

[AlephZero]

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.

 

[Hypatio]

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.