# /VISC/LPRONY

Block Format Keyword This model describes an isotropic visco-elastic Maxwell model that can be used to add visco-elasticity to solid element with total strain formulation (Ismstr=10 or 12).

The visco-elasticity 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
Read each pair of shear relaxation and shear decay per line
(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 visco-elastic modulus formulation used.
= 1 (Default)
Initial elastic modulus is the long-term rigidity.
= 2
Initial elastic modulus is instantaneous rigidity.

(Integer)

flag_visc Viscous formulation flag.
= 1 (Default)
Viscous stress is accounted for in both the deviatoric and volumetric stress which enables the lateral expansion effect for the entered Poisson’s ratio.
= 2
Viscous stress is accounted for in the deviatoric stress only and thus should only be used for incompressible materials with Poisson’s ratio close to 0.5.

(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)

$\left[\text{s}\right]$

1. This viscous model is available only for total strain formulation with Ismstr=10 or 12 in the solid property).
2. Form=1 is available only for material law /MAT/LAW42 (OGDEN), /MAT/LAW62 (VISC_HYP) and /MAT/LAW69.
3. The viscosity model is ignored in case it is applied on a non-compatible material or strain formulation.
4. 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.

${\text{G}}_{\text{0}}{\text{=G}}_{\infty }\text{+}\sum _{i}{\text{G}}_{\text{i}}$

and

${\eta }_{i}\text{=}\frac{1}{{\tau }_{i}}$

The stiffness ratio is defined using:

${\gamma }_{\infty }\text{=}\frac{{\text{G}}_{\infty }}{{\text{G}}_{0}}\text{=1-}\sum _{i}{\gamma }_{\text{i}}$

Where, ${\gamma }_{i}\text{=}\frac{{\text{G}}_{i}}{{\text{G}}_{0}}$ .

5. The viscosity effect is taken into account by using a Prony series. The Kirchhoff viscous stress is computed using:
$\tau \left(t\right)\text{=}{\tau }_{0}\left(t\right)-{\int }_{0}^{t}\stackrel{˙}{\gamma }\left(s\right)\cdot {\tau }_{0}\left(t-s\right)ds$

with $\gamma \left(t\right)\text{=}\sum _{i=1}^{M}{\gamma }_{i}{e}^{\left(\frac{-{\tau }_{i}}{t}\right)}$ .