/VISC/LPRONY
Block Format Keyword This model describes an isotropic viscoelastic Maxwell model that can be used to add viscoelasticity to solid element with total strain formulation (I_{smstr}=10 or 12).
The viscoelasticity is input using a Prony series.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/VISC/LPRONY/mat_ID/unit_ID  
M  Form  flag_visc 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

${\gamma}_{i}$  ${\tau}_{i}$ 
Definition
Field  Contents  SI Unit Example 

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

unit_ID  (Optional) Unit
identifier. (Integer, maximum 10 digits) 

M  Maxwell model order (number of
Prony coefficients). Maximum order is 100. Default = 1 (Integer) 

Form  Initial viscoelastic modulus
formulation used.
(Integer) 

flag_visc  Viscous formulation flag.
(Integer) 

${\gamma}_{i}$  Shear relaxation modulus for
$i$
^{th} term (
$i$
=1,
M). (Real) 

${\tau}_{i}$  Decay shear constant for
$i$
^{th} term (
$i$
=1,
M). (Real) 
$\left[\text{s}\right]$ 
Comments
 This viscous model is available only for total strain formulation with I_{smstr}=10 or 12 in the solid property).
 Form=1 is available only for material law /MAT/LAW42 (OGDEN), /MAT/LAW62 (VISC_HYP) and /MAT/LAW69.
 The viscosity model is ignored in case it is applied on a noncompatible material or strain formulation.
 Coefficients (
${\text{G}}_{\text{i}}$
,
${\eta}_{i}$
) are used to describe rate effects through
the Maxwell model.
The initial shear modulus given by the formula below, it corresponds to the shear modulus of material law.
$${\text{G}}_{\text{0}}{\text{=G}}_{\infty}\text{+}{\displaystyle \sum _{i}{\text{G}}_{\text{i}}}$$and
$${\eta}_{i}\text{=}\frac{1}{{\tau}_{i}}$$The stiffness ratio is defined using:
$${\gamma}_{\infty}\text{=}\frac{{\text{G}}_{\infty}}{{\text{G}}_{0}}\text{=1}{\displaystyle \sum _{i}{\gamma}_{\text{i}}}$$Where, ${\gamma}_{i}\text{=}\frac{{\text{G}}_{i}}{{\text{G}}_{0}}$ .
 The viscosity effect is taken into account by using a Prony series.
The Kirchhoff viscous stress is computed using:$$\tau \left(t\right)\text{=}{\tau}_{0}\left(t\right){\displaystyle {\int}_{0}^{t}\dot{\gamma}\left(s\right)}\cdot {\tau}_{0}\left(ts\right)ds$$
with $\gamma \left(t\right)\text{=}{\displaystyle \sum _{i=1}^{M}{\gamma}_{i}{e}^{\left(\frac{{\tau}_{i}}{t}\right)}}$ .