/VISC/PRONY
Block Format Keyword This is an isotropic viscoelastic Maxwell model that can be used to add viscoelasticity to certain shell and solid element material models. The viscoelasticity is input using a Prony series.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/VISC/PRONY/mat_ID/unit_ID  
M  ${K}_{v}$ 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

${G}_{i}$  ${\beta}_{i}$  ${K}_{i}$  ${\beta}_{ki}$ 
Definitions
Field  Contents  SI Unit Example 

mat_ID  Material identifier which
refers to the viscosity card (Integer, maximum 10 digits) 

unit_ID  Unit Identifier (Integer, maximum 10 digits) 

M  Maxwell model order
(number of Prony coefficients). Default = 0 (Integer) 

${K}_{v}$  Viscous bulk modulus.
3 Only used if
${K}_{i}=0$
. Default = 0. (Real) 
$\left[\text{Pa}\cdot \text{s}\right]$ 
${G}_{i}$  Shear relaxation modulus
for i^{th} term
(i=1,
M). (Real) 
$\left[\text{Pa}\right]$ 
${\beta}_{i}$  Decay shear constant for
i^{th} term
(i=1,
M). (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
${K}_{i}$  Bulk relaxation modulus
for i^{th} term
(i=1, M). 3 (Real) 
$\left[\text{Pa}\right]$ 
${\beta}_{ki}$  Decay bulk constant for
i^{th} term
(i=1,
M). (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Comments
 For shell elements this model is
available with /MAT/LAW66 and
/MAT/LAW25 (COMPSH).
For solid elements it is available with material laws /MAT/LAW38 (VISC_TAB), /MAT/LAW42 (OGDEN), /MAT/LAW69, /MAT/LAW70 (FOAM_TAB), /MAT/LAW82, /MAT/LAW88, /MAT/LAW90, /MAT/LAW92, /MAT/LAW103 (HENSELSPITTEL), and /MAT/LAW106 (JCOOK_ALM).
 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) $${S}_{\mathit{ij}}={\displaystyle \underset{0}{\overset{t}{\int}}2G(ts)\frac{\partial \mathit{dev}\left[{\epsilon}_{\mathit{ij}}\right]}{\partial s}}\mathit{ds}$$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, ${\tau}_{i}$ is the relaxation time.
 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) $$P={K}_{v}{\dot{\epsilon}}_{vol}$$
 If the bulk relaxation modulus is
${K}_{i}>0$
, the viscous pressure is computed
as:
 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.