A question about the Ogden model in viscoelasticity case

[ws18]

Dear FEAP team,

I am trying to understand the constitutive equations for the Ogden-Viscoelastic model as used in FEAP (specifically version 85, and I have access to version 86). In the Ogden model, the sum of the product of parameters is related to the shear modulus for elastic materials, denoted as Sum(C_j * a_j) = 2G. However, I am uncertain about how this relation changes in the viscoelastic case. In FEAP, Prony's series is employed as the relaxation function, which provides an instantaneous modulus, G (Equation 7.62), as well as the modulus in the equilibrium state, u_0 * G.

Based on my preliminary understanding, I believe the Ogden model should be associated with the material response in the equilibrium state. Consequently, I assume that the sum of the product of parameters in the Ogden model is related to the equilibrium shear modulus, resulting in Sum(C_j * a_j) = 2u_0 * G. I have read an introduction to the Ogden-viscoelastic model in COMSOL (refer to the attached figure), which seems to align with this understanding. However, it appears that this interpretation is not applicable in FEAP, as FEAP relates the sum of the product of parameters to the instantaneous modulus, Sum(C_j * a_j) = 2G.

Can someone confirm whether my understanding is correct or provide a reference that can help me gain a better understanding of the implementation of the Ogden-Viscoelastic model in FEAP? Specifically, I would like to know the procedure for converting the C_j and a_j parameters used in FEAP to the properties of the equilibrium state. I ask this question because I need to include a table of identified constitutive parameters in my paper. I would like to clearly explain the meaning of these parameters in FEAP so that researchers using our model can understand how to convert our parameters in other finite element analysis (FEA) packages in order to reproduce our results.

Thank you,
Wenya

[FEAP_Admin]

You should have a look at the FEAP User manual and the FEAP Theory manual; they go over how the model works.   They work just like what you have attached, just using different symbols.  Notwithstanding, in FEAP you will do two things. 

1. You will define the modulus of the material using ELAStic OGDEn (see user manual section 7.3.5).  These numbers will give you G0 and K0, the short-time moduli of the material.

2. Then you will define relative relaxation strengths mu_i and relaxation times tau_i with VISCoelastic DEVIatoric (and optionally K_i and relaxation times tauk_i with VISCoelastic VOLUmetric).  These values are used to build a mu_0 = 1 - sum_i  mu_i  which you need to ensure is greater than 0.  If you want bulk viscoelastic behavior, then  when defining the K_i make sure that K_0 = 1 - sum_i K_i is greater than zero.

3. The model that will be used is G(t) = G0 * (mu_0 + sum_i mu_i exp[-t/tau_i]) and for the bulk K(t) = K0 * (K_0 + sum_i K_i exp[-t/tauk_i].

Most people leave the bulk response elastic.  Note, you can use up to 3 VISC DEVI lines; similarly one can use up to 3 VISC VOLU lines.