Algebraic Inversion of Stress-Strain Relations?

  • Thread starter Hypatio
  • Start date
  • #1
151
1

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?
 

Answers and Replies

  • #2
5,439
7
Which particular relations are you thinking of?
 
  • #3
151
1
Which particular relations are you thinking of?

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

[tex]\sigma_{ij}=\lambda \epsilon_{kk} +2\mu\epsilon_{ij}[/tex]
and
[tex]\epsilon_{ij}=\frac{1+v}{E}\sigma_{ij}-\frac{v}{E}\sigma_{kk}[/tex]

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:

[tex]\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 ][/tex]

which obviously has some additional terms. But I'm not sure how trivial this is.
 
Last edited:
  • #4
AlephZero
Science Advisor
Homework Helper
6,994
291
The basic idea is to write the stress-strain equations as a 6x6 matrix equation

[tex]\varepsilon = D \sigma[/tex]

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

[tex]\varepsilon = D \sigma + A \Delta t[/tex]

where A is a vector. Then

[tex]D^{-1} \varepsilon = \sigma + D^{-1} A \Delta t[/tex]

which is usually written as

[tex]C \varepsilon = \sigma + C A \Delta t[/tex]

so

[tex]\sigma = C \varepsilon - C A \Delta t[/tex]

And you can multiply out CA to get the equations in your link.
 
  • #5
151
1
Thanks, although I'm not sure how to treat some of the other terms which appear here:

[tex]\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 ][/tex]

Can all of the terms be introduced into the same matrix [tex]C=D^{-1}[/tex] as you have done with [tex]k\alpha_V T[/tex] 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 [tex]\sigma_{ij}[/tex] appears. How is this introduced into the matrix formula?

Thanks again.
 

Related Threads on Algebraic Inversion of Stress-Strain Relations?

  • Last Post
Replies
4
Views
12K
  • Last Post
Replies
1
Views
1K
Replies
2
Views
4K
  • Last Post
Replies
2
Views
6K
Replies
1
Views
622
  • Last Post
Replies
6
Views
20K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
5
Views
10K
  • Last Post
Replies
10
Views
10K
Top