/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)
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 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E1@ th term ( i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E1@ =1, M).

(Real)
τ i Decay shear constant for i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E1@ th term ( i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E1@ =1, M).

(Real)

 [ s ]

Comments

  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 ( G i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqaaaaaaaaaWdbe aacaqGhbWaaSbaaSqaaiaabMgaaeqaaaaa@37F6@ , η i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaH3oaAdaWgaaWcbaGaamyAaaqabaaaaa@38D9@ ) 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 ∞ + ∑ i G i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGhbWaaSbaaSqaaiaabcdaaeqaaOGaaeypaiaabEeadaWgaaWc baGaeyOhIukabeaakiaabUcadaaeqbqaaiaabEeadaWgaaWcbaGaae yAaaqabaaabaGaamyAaaqab0GaeyyeIuoaaaa@408E@

    and

    η i = 1 τ i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaH3oaAdaWgaaWcbaGaamyAaaqabaGccaqG9aWaaSaaaeaacaaI XaaabaGaeqiXdq3aaSbaaSqaaiaadMgaaeqaaaaaaaa@3D4D@

    The stiffness ratio is defined using:

    γ ∞ = G ∞ G 0 =1- ∑ i γ i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHZoWzdaWgaaWcbaGaeyOhIukabeaakiaab2dadaWcaaqaaiaa bEeadaWgaaWcbaGaeyOhIukabeaaaOqaaiaabEeadaWgaaWcbaGaaG imaaqabaaaaOGaaeypaiaabgdacaqGTaWaaabuaeaacqaHZoWzdaWg aaWcbaGaaeyAaaqabaaabaGaamyAaaqab0GaeyyeIuoaaaa@4646@

    Where, γ i = G i G 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHZoWzdaWgaaWcbaGaamyAaaqabaGccaqG9aWaaSaaaeaacaqG hbWaaSbaaSqaaiaadMgaaeqaaaGcbaGaae4ramaaBaaaleaacaaIWa aabeaaaaaaaa@3D4D@ .

  5. The viscosity effect is taken into account by using a Prony series. The Kirchhoff viscous stress is computed using:
    τ t = τ 0 t − ∫ 0 t γ ˙ s ⋅ τ 0 t − s d s MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHepaDdaqadaqaaiaadshaaiaawIcacaGLPaaacaqG9aGaeqiX dq3aaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaay zkaaGaeyOeI0Yaa8qmaeaacuaHZoWzgaGaamaabmaabaGaam4CaaGa ayjkaiaawMcaaaWcbaGaaGimaaqaaiaadshaa0Gaey4kIipakiabgw Sixlabes8a0naaBaaaleaacaaIWaaabeaakmaabmaabaGaamiDaiab gkHiTiaadohaaiaawIcacaGLPaaacaWGKbGaam4Caaaa@549D@

    with γ t = ∑ i = 1 M γ i e − τ i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHZoWzdaqadaqaaiaadshaaiaawIcacaGLPaaacaqG9aWaaabC aeaacqaHZoWzdaWgaaWcbaGaamyAaaqabaGccaWGLbWaaWbaaSqabe aadaqadaqaamaalaaabaGaeyOeI0IaeqiXdq3aaSbaaWqaaiaadMga aeqaaaWcbaGaamiDaaaaaiaawIcacaGLPaaaaaaabaGaamyAaiabg2 da9iaaigdaaeaacaWGnbaaniabggHiLdaaaa@4B00@ .