Stress-strain relation and constitutive equation

[sara_87]

Homework Statement

A voigt solid is a model viscoelastic material which in uniaxial tension has the stress-strain relation $$\sigma$$=E₀($$\epsilon$$+t₀$$\dot{\epsilon}$$),
where E₀ and t₀ are constants.
1) sketch the creep and stress relaxation curves for this material.
2) show that the relaxation function is E₀{1+t₀$$\delta$$(t-$$\tau)$$}
3) give a 3-dimensional generalisation of the above constitutive equation for an incompressible isotropic material.

Homework Equations

$$\delta$$T_ij(t)=G_ijkl(t-$$\tau$$)$$\delta$$E_kl($$\tau$$)
where G_ijkl(t-$$\tau$$) is the relaxation function.

The Attempt at a Solution

Can someone please help me on how to start this question because i have no idea how to do it.
For (2), do i have to rearrange the equation above to find the relaxation function??.
Thank you

 

[Mapes]

I'm not sure what you've covered in class, but I'd tend to approach the problem either by (1) figuring out what viscoelastic components make up the system from the constitutive equation or (2) using Laplace transforms. Do either of these sound familiar?

 

[sara_87]

yes, the first approach is what i am supposed to use.

 

[Mapes]

So what do you think the different components are, and how are they arranged? There's a certain arrangement that results in the stress being the sum of two or more terms.

 

[sara_87]

sorry, i don't know what you mean. which constitutive equation are where supposed to get the components from.
I haven't done this in class, but teaching myself from a book so I'm a bit confused.
Thank you

 

[Mapes]

What viscoelastic components exist, and what is the relation between stress and strain for each of them? How does the stress combine when two elements are attached in parallel? (I.e., what is the stress-strain relationship for the connected elements?) How about when they're connected in series? How about the strain? This is essential stuff to know when working with viscoelastic models.

which constitutive equation are where supposed to get the components from.

Huh?

 

[sara_87]

the stress-strain relationship for the connected element is $$\sigma$$ = E₀($$\epsilon$$ + t₀$$\dot{\epsilon}$$) from the question.

is there a certain form of vicoelastic components or do different materials have different components? and how am i supposed to find the viscoelastic components from the question given? is there a method like a formula or something?
thanks

 

[Mapes]

the stress-strain relationship for the connected element is $$\sigma$$ = E₀($$\epsilon$$ + t₀$$\dot{\epsilon}$$) from the question.

But you would also like to know the strain in terms of stress so you can sketch the creep behavior. That's why it's useful to understand what the individual components are and how they work together.

and how am i supposed to find the viscoelastic components from the question given? is there a method like a formula or something?
thanks

My best recommendation is to try to answer the questions in my post #6. It's not meant to pile more work on you; your assignment assumes that you know the answers to all of these questions already. It's material that you'd typically learn in a first lecture on viscoelasticity.

 

[sara_87]

Yes, i think you're right :)
In the book, it says:
''The superposition principle is assumed, according to which the total stress at time t is obtained by superimposing the effect at time t of all the strain increments at times $$\tau$$< t. thus:
T_ij(t)=$$\int$$ {G_ijkl(t-\tau)}\frac{dE_kl(\tau)}{d\tau} d$$\tau$$
This is the constitutive equation for linear viscoelasticity. The functions G_ijkl are called relaxation functions.''
I think this is supposed to help me but I am quite confused.
There arent any examples, that's why I am stuck.

 

[Mapes]

Does your book say nothing about springs and dashpots? These are the fundamental components of any viscoelastic system. The Voigt solid comprises a system of these components.

 

[sara_87]

no, unfortunately not, nothing about springs or dashpots.
I had a feeling that the book lacked more theory after i read this question.
so am i supposed to know the components?

 

[Mapes]

You may want to get a hold of a good book on mechanics of materials such as Hosford or Courtney. Alternatively, Wikipedia and many other online sites describe simple viscoelastic models.

 

[sara_87]

ok, I'm going to read up more about viscoelasticity and if i still don't know how to do this question, i will come back to this thread :)
thank you

 

[sara_87]

On wikipedia, it explains what a voigt material is and it gives the stress-strain relation as given in the question but it doesn't say anything about the components. is there a way of determining the relaxation function from the stress-strain relation?

 

[Mapes]

On wikipedia, it explains what a voigt material is and it gives the stress-strain relation as given in the question but it doesn't say anything about the components.

? "The...Voigt model...can be represented by a purely viscous damper and purely elastic spring connected in parallel..." It's right there in the article.

is there a way of determining the relaxation function from the stress-strain relation?

The relaxation function is the stress profile over time following a unit step in strain.

 

[sara_87]

oh, so the relaxation function is the stress?
'following a unit step in strain'?? what do you mean by that?

 

[Mapes]

A unit step is a sudden increase from zero to a positive value. Like anchoring one end of a material sample and yanking and holding the other end at some specific position.