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Viscoelastic material: Prony series in Abaqus


I’m using
Prony series to model viscoelasticity of an asphalt material in a pavement. I
have defined the material properties through time-domain Prony series in
Abaqus. The parameters are:

g_i (shear
relaxation modulus ratio): 0.8

k_i (bulk
relaxation modulus ratio): 0

(relaxation time): 1

I am using
*VISCO procedure for the analysis and the trapezoidal loading is applied by
giving amplitude-time definition in tabular form. The loading is given below:

                    Time                  Amplitude

                        0                                0

                     0.002                        100

                     0.0088                     100

                     0.0108                       0

                     1.0                               0

Likewise I
have simulated 5 cycles and analyzed pavement’s response in real time scale. I
changed g_i and tau_i values and observed the results.

For g_i=
0.8, the tensile elastic strain is smaller and tensile inelastic strain is
larger than g_i= 0.9 (keeping k_i and tau_i unchanged).

For tau_i=1,
the tensile elastic strain is smaller and tensile inelastic strain is larger
than tau_i=1.2 (keeping g_i and k_i unchanged).

I am not
able to understand what exactly we do when we increase g_i or tau_i.  When we increase material relaxation time
parameter (which is actually the ratio of coefficient of viscosity to modulus
of elasticity of the material) are we making the material more viscous? Also,
can you please tell me what is the difference between loading relaxation time
(1.0 – 0.0108 = 0.9892sec) and material relaxation time (tau_i= 1sec)? If we do
not give any loading relaxation time then material will not relax? Can anybody
please tell me what the significance of each term is? And how the system
response affects by variation of these parameters? 

please give me a push in the right direction here.

Thanks in



I suggest that you read up a bit on viscoelasticity.  My favorite introductory book on the subject is

"Viscoelastic properties of polymers " by John. D. Ferry .

Another useful book is

"Viscoelastic solids " by R. S. Lakes.

For a nice continuum mechanics discussion see

"Continuum mechanics: elasticity, plasticity,
" by Ellis Harold Dill

For an explanation of thr Prony series approach you can see

"Polymer engineering science and
viscoelasticity: an introduction
" By Hal F.
Brinson, L. Catherine Brinson

-- Biswajit


please iam new in abaqus and i would like to model a viscoelastic contact problem i don't know how to model a material(maxwell generalised model) using prony series.(if i have 3 branches for example)

please i need your help

thanks a lot

tarkes's picture


Kindly go through the book the initial few chapters of the book

Mechanical Response of Polymers by Alan S. Wineman, K. R. Rajagopal. This tells all the derivations behind modelling of viscoelastic materials. I hope it helps

pghasemi's picture

Dear all,

I want to simulate Dynamic Modulus test in ABAQUS. I have some lab data of dynamic modulus (E*) I calculated the storage modulus (E') , relaxation modulus (E(t)) and complex shear modulus (G(t)). I know that I should calculate Prony coefficients, but I don't know how shoudl I do that. would you please help me? 



tarkes's picture

This is a very common problem in viscoelasticity. Infact there are many articles as well few softwares are available to convert this data. In ABAQUS you will find provision of frequency viscoelastic data. I think this converts suitably into Prony series. You have to look into manuals how to feed the data in suitable form to ABAQUS. Another such similar software I am aware of is Viscodata ( These softwares automatically converts into suitable Prony series.

Also if you search, you will find papers on conversion of the dynamic viscoelastic data into Prony series. If you are good at coding, you can try implementing those ideas in the code. Basics of conversion of data using laplace transforms and other details you will find in chapter 6 of Mechanical Response of Polymers by Alan S. Wineman, K. R. Rajagopal

Hope it helps.


pghasemi's picture

Thanks for the help.

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