/VISC/PRONY

Format

Definitions

Comments

1. For shell elements this model is available with /MAT/LAW66 and /MAT/LAW25 (COMPSH).

   For solid elements it is available with hyperelastic material laws /MAT/LAW42 (OGDEN), /MAT/LAW69, /MAT/LAW82, /MAT/LAW88 and /MAT/LAW92.

2. The viscosity effect is taken into account by using a Prony series. The deviatoric viscous stress is given by the convolution integral of the form:(1)
   $$S_{\mathit{ij}}={\displaystyle \underset{0}{\overset{t}{\int }}2G(t-s)\frac{\partial \mathit{dev}\left[{\epsilon }_{\mathit{ij}}\right]}{\partial s}}\mathit{ds}$$
   with(2)

   $$G(t)={\displaystyle \sum _{i=1}^M{G}_i{e}^{-{\beta }_{i}t}}$$

   and $\mathit{dev}\left[{\epsilon }_{\mathit{ij}}\right]$ denotes the deviatoric part of strain tensor.

   Shear decay:(3)

   $${\beta }_i=\left(\frac{1}{{\tau }_i}\right)$$
   Where,

   ${\tau }_i$

   Relaxation time

3. For the viscous pressure, two formulations are available:
   • If the bulk relaxation modulus is $K_i>0$ , the viscous pressure is computed as:(4)
     $$P=-{\displaystyle {\int }_0^{t}K\left(s\right){\dot{\epsilon }}_{vol}}ds$$

     with ${\dot{\epsilon }}_{vol}=trace\left(\dot{\epsilon }\right)={\dot{\epsilon }}_{xx}+{\dot{\epsilon }}_{yy}+{\dot{\epsilon }}_{zz}$ and $K\left(t\right)={\displaystyle {\sum }_1^M{K}_i{e}^{-{\beta }_{ki}t}}$

   • If the bulk relaxation modulus is $K_i=0$ and the viscous bulk modulus $K_{\nu }>0$ , the viscous pressure is computed as:(5)
     $$P=-K_v{\dot{\epsilon }}_{vol}$$
4. Starting with Radioss version 2017, identical results are obtained using the same Prony coefficents G_i in /VISC/PRONY and viscoelastic materials /MAT/LAW34 (BOLTZMAN), /MAT/LAW40 (KELVINMAX), and /MAT/LAW42 (OGDEN). In previous Radioss versions, 2 G_i had to be input into /VISC/PRONY to get equivalent results.