/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 (I_smstr=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)
$γ_{i}$		$τ_{i}$

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)
$γ_{i}$	Shear relaxation modulus for $i$ ^th term ( $i$ =1, `M`). (Real)
$τ_{i}$	Relaxation time for $i$ ^th term ( $i$ =1, `M`). (Real)	$[s]$

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 non-compatible material or strain formulation.
Coefficients ( $G_{i}$ , $η_{i}$ ) are used to describe rate effects through the Maxwell model.

Figure 1.
The initial shear modulus given by the formula below, it corresponds to the shear modulus of material law.
$G_{0} {=G}_{\infty} + \sum_{i} G_{i}$
and
$η_{i} = \frac{1}{τ_{i}}$
The stiffness ratio is defined using:
$γ_{\infty} = \frac{G_{\infty}}{G_{0}} =1- \sum_{i} γ_{i}$
Where, $γ_{i} = \frac{G_{i}}{G_{0}}$ .
The viscosity effect is taken into account by using a Prony series. The Kirchhoff viscous stress is computed using:
$τ (t) = τ_{0} (t) - \int_{0}^{t} \dot{γ} (s) \cdot τ_{0} (t - s) d s$
with $γ (t) = \sum_{i = 1}^{M} γ_{i} e^{(\frac{- τ_{i}}{t})}$ .