/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  ${K}_{v}$  Itab  Ishape 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

${G}_{i}$  ${\beta}_{i}$  ${K}_{i}$  ${\beta}_{ki}$ 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

Ifunc_G  XGscale  YGscale  
Ifunc_K  XKscale  YKscale 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

Ifunc_Gs  XGs_scale  YGs_scale  
Ifunc_Gl  XGl_scale  YGl_scale  
Ifunc_Ks  XKs_scale  YKs_scale  
Ifunc_Kl  XKl_scale  YKl_scale 
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) 
$\left[\text{Pa}\cdot \text{s}\right]$ 
Itab  Tabulated formulation flag
(Integer) 

Ishape  Tabulated Prony series
shape flag (Only if Itab ≠0)
(Integer) 

${G}_{i}$  Shear relaxation modulus
for i^{th} term
(i=1,
M). (Real) 
$\left[\text{Pa}\right]$
$\left[\text{s}\right]$ 
${\beta}_{i}$  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) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Ifunc_G  Relaxation test data curve
for shear modulus. (Integer) 

XGscale  Time scale factor for
shear modulus relaxation test data curve. Default = 1.0 (Real) 
$\left[\text{s}\right]$ 
YGscale  Scale factor for shear
modulus relaxation test data curve. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Ifunc_K  Relaxation test data curve
for bulk modulus. (Integer) 

XKscale  Time scale factor for bulk
modulus relaxation test data curve. Default = 1.0 (Real) 
$\left[\text{s}\right]$ 
YKscale  Scale factor for bulk
modulus relaxation test data curve. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Ifunc_Gs  Shear storage modulus data
curve. (Integer) 

XGs_scale  Frequency scale factor for
shear storage modulus test data curve. Default = 1.0 (Real) 
$\text{[Hz]}$ 
YGs_Scale  Scale factor for shear
storage modulus test data curve. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Ifunc_Gl  Shear loss modulus data
curve. (Integer) 

XGl_scale  Frequency scale factor for
shear loss modulus test data curve. Default = 1.0 (Real) 
$\text{[Hz]}$ 
YGl_Scale  Scale factor for shear
loss modulus test data curve. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Ifunc_Ks  Bulk storage modulus data
curve. (Integer) 

XKs_scale  Frequency scale factor for
bulk storage modulus test data curve. Default = 1.0 (Real) 
$\text{[Hz]}$ 
YKs_scale  Scale factor for bulk
storage modulus test data curve. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Ifunc_Kl  Bulk loss modulus data
curve. (Integer) 

XKl_scale  Frequency scale factor for
bulk loss modulus test data curve. Default = 1.0 (Real) 
$\text{[Hz]}$ 
YKl_scale  Scale factor for bulk loss
modulus test data curve. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
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.
 Prony series parameters can be
automatically fit from test data using the flag Itab:
 If Itab = 1, prony series parameters are fitted from relaxation tests data, i.e moduli versus time curves.
 If Itab = 2, Prony series parameters are fitted from Dynamic Mechanical Analysis (DMA) tests data, i.e storage and loss moduli versus frequency curves.
Note: The convergence of the least square fit may be hard to achieve for very high orders.  The shape of the fitted Prony
series (only in case where Itab ≠ 0) can be chosen by you:
 If Ishape =
0, the shape of the fitted Prony series are the
same as the one given above, so as:
(6) $$G(t)={\displaystyle \sum _{i=1}^{M}{G}_{i}{e}^{{\beta}_{i}t}}$$and(7) $$K(t)={\displaystyle \sum _{i=1}^{M}{K}_{i}{e}^{{\beta}_{ki}t}}$$  If Ishape =
1, the shape of the fitted Prony series is modified
to consider the infinite values of the moduli, so as:
(8) $$G(t)={G}_{\infty}+{\displaystyle \sum _{i=1}^{M}{G}_{i}{e}^{{\beta}_{i}t}}$$and(9) $$K(t)={K}_{\infty}+{\displaystyle \sum _{i=1}^{M}{K}_{i}{e}^{{\beta}_{ki}t}}$$
Note: In this case, the infinite value of the moduli is taken as the last value of the relaxation test data curve if Itab = 1, or the first storage modulus value if Itab = 2.  If Ishape =
0, the shape of the fitted Prony series are the
same as the one given above, so as: