viscoelastic stress strain curves

[jurs]

Hi,

Does anyone know methods to transform a relaxation shear modulus function to a standard viscoelastic stress strain curve?

I have a relaxation function G(t) and a bulk modulus.

Thanks in advance,
CJ

 

[brad]

Viscoelasticity theory assumes that there is an instantaneous modulus Eo and a longterm modulus Einfinity ("Ei" for this discussion). They are related to each other through a relaxation function:

Ei = Eo - E(t) = Eo (1- R)

where R is the relaxation function, often described in the form:
R = Ai e^(-t/TAUi).

Ai and TAUi are fit to the physical test data. One can see that as t goes to infinity, R goes to 0 and we arrive at Eo. As t goes to zero, Ai is the normalized difference between Ei and Eo. For better fits, there may be multiple A/TAU combinations (A1/TAU1, A2/TAU2, etc). The time constant TAU is then varied by decade or decades.

For your purpose, you need to determine first whether your given modulus is Eo or Ei.

What code are you using to do this analysis?
Brad

 

[jurs]

Brad,

Thanks for your response, you sound very knowledgable on the topic. Here is the deal, I am currently creating a piliminary impact model of a rubber/cork composite material in LSDyna [material#6]. In the literature I happened to come across a relaxation function for a material that is very close to mine (but not exact). The relaxation function is in the form.

G(t) = G_inf +(G_o - G_inf)e^(-B*t)

and all the variables are given, including an assumed bulk modulus, K. It would be nice to convert this to a stress strain curve (or series of stress strain curves), which is what I am fimilar with. I realize this may be a naive endeavor.
I really want to understand how this function is dervived experimentally, and how I can devise an experiment of my own to extract a relationship for my material. I dont have access to all sorts of fancy machines so the test I make will have to be improvised. I do have acess to an MTS static and fatique test machine (uniaxial compression and tension), but thats about it.

Thanks in advance for your help,
CJ

 

[brad]

It's the same description.
Your
G(t) = E(t)
G_o = Eo
G_inf = Ei
B = 1/TAU
(G_o = G_inf) = R

You are using a more standard convention; I couldn't remember the exact convention so I made up my own terms.

You cannot really do a "stress-strain" curve. The G(t) essentially can be normalized, and thought of as a multiplier to your elastic modulus. This multiplier changes with time.

I have done this in the past with ABAQUS. With only a MTS machine, you can slowly apply load to arrive at G_inf. You can then do a load- or displacement-controlled analysis with a quick application followed by holding at the same load/displacement until the response decays to G_inf. From such a displacement-vs-time or load-vs-time curve, one can back-calculate the constants. This could be done by hand; ABAQUS does it fairly seamlessly for you. I can't say how DYNA handles this.
Brad

 

[jurs]

Thanks Brad, you have been a lot of help.

Do you have any suggestion on how slow to apply the load to extract E_inf. I assume thew closer the vel is to zero the better value you will get, but how slow is slow enough? I dont want to sit around until the next decade :/)

For clarification, when you say "holding at the same load/displacement until the response decays " do you mean applying the load suddenly and then holding it at some value of desplacement until your stress decays down to E_inf?

Craig