Prony series to model viscoelastic behaviour 1

[Wyb]

I'd like to model a viscoelastic material in Prony series terms in Abaqus 6.9-2.

I have DMA measurements (shear loss and shear storage moduli vs frequency) for various viscoelastic materials in terms of frequency Prony series, so g_i's and tau_i's.

In Abaqus, I create a material in terms of 'general' (density), 'mechanical\elasticity\elastic' (Young's modulus and Poisson's ratio) and 'mechical\elasticity\viscoelastic' (g_i and tau_i).

What do I need to fill in for the Young's modulus, the G_0 (capital G, loss modulus is defined as G_s(omega) = G_0*[1-sum etc)? Or is the Young's modulus G_0/3 or...? I assume Poisson's ratio is almost 0.5, but can I derive that from the DMA measurements?

Finally, I'm not sure what I need to fill in for the k_i's ('mechical\elasticity\viscoelastic')? Zeros?

What's the relationship between the Maxwell model (spring and damper in series (or multiples) and the Prony series?

Thank you in advance for your response!

PS: I posted this topic already on the ABAQUS Engtips forum, but I think it fits better here.

 

[GrahamBennett]

You don't say what sort of materials you are modelling but you should consider whether you wish to use small strain (as you describe) or large strain (hyperelastic) elasticity to describe the elastic behaviour. If you are using small strain elasticity there are well-known relationships between the different elastic constants; for an incompressible material, Poisson's ratio =0.5 and hence G=E/3. You cannot deduce Poisson's ratio from your shear data but it may well be reasonable to assume incompressibility ie Poisson's ratio = 0.5. Similarly for an incompressible material it would be reasonable to put k_is as zero because bulk relaxation will be negligible. This is all explained in the Abaqus User Manual sections 18.7.1 and 18.7.2. If you use a hyperelastic model the elastic behaviour is described in terms of the shear and bulk modulus so you do not need to define E or Poisson's ratio.

G0 is the instantaneous shear modulus, the value for an infinitely fast test (infinitely high frequency). Alternatively you can use G infinity, the value for an infinitely slow test, when all relaxation has taken place. Both of these are physically abstract but you should be able to make a reasonable guess by extrapolating your DMA data.

The Prony series is an array of Maxwell elements in parallel with an extra spring (without a dashpot) also in parallel. Each term of the Prony series can be represented by one Maxwell element.

You questions suggest you need to do some background reading about viscoelastic behaviour in order to understand properly the behaviour you are trying to model. I would recommend Ferry, J.D. (1980) "Viscoelastic properties of polymers", 3rd Edition, Wiley.

If you are interested in our work on modelling rubbers you could read "DYNAMIC PROPERTIES OF FILLED RUBBER - PART I: SIMPLE MODEL, EXPERIMENTAL DATA AND SIMULATED RESULTS" H.R. AHMADI, J.G.R. KINGSTON, A.H. MUHR, Rubber Chemistry and Technology, vol 81, pp1-18. But you should be aware that the viscoelastic model is of very limited use for modelling filled rubbers.

 

[Mottahedi]

Hello,
I tried one time modeling a rubber by using prony series and using an eigenfrequency analysis in ANSYS12. However, I did not get any real part for the frequencies, meaning no damping effect was detected. On the other hand, by defining simply damping coefficient, in addition to E,v, for a material I can get the damping and hence hysteresis. I can not understand if we really could not model hysteresis or unloading effects in a prony rubber?

 

[Wyb]

Hello again. Thank you for your response.

I have the Prony series now in term of G0, g1 to g5 and tau1 to tau5. Now I'm confused what to do with the G0 term? I cannot put in the viscoelastic data table, since it starts with g1. I used the definition of the Prony series from the Abaqus manual:

Gs = G0 (1- sum(gi)) + G0 * sum( (gi*taui^2*w^2)/(1+taui^2*w^2)

and

Gl = G0 * sum( (gi*taui*w)/(1+taui^2*w^2)

So should I put G0 as Young's modulus in the elastic definition? (i.e. E=G0*3)

Or should I use the shear modulus at omega is zero times three here?

I'd like to note that G0 is not the same as the shear modulus at omega is zero.

Thank you in advance!

 

[Wyb]

To be clear:

I don't know where to put the G0 from the Prony series.

I'd like to create a viscoelastic material using Prony series. In the Abaqus 6.9-2 CAE manual I found I have to define a viscoelastic material by the next material behaviours:
1. Density: Density
2. Elastic: Young's modulus and Poisson's ratio (Note: Moduli time scale I put to Instantanious since I want to do a direct steady state dynamics analysis)
3. Viscoelastic: Prony (frequency): gi's and taui's (ki=0)

From this I have only one place where to put G0 and that is as Young's modulus, since E=3*G I put 3*G0 there. Is that correct?

Thank you for your response.