/VISC/PRONY

Block Format Keyword This is an isotropic visco-elastic Maxwell model that can be used to add visco-elasticity to certain shell and solid element material models. The visco-elasticity 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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaaS baaSqaaiaadAhaaeqaaaaa@3855@ Itab Ishape        
If Itab = 0, ready only if M > 0, each pair of shear relaxation and shear decay per line
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
G i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGPbaabeaaaaa@37DE@ β i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda WgaaWcbaGaamyAaaqabaaaaa@3A23@ K i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGPbaabeaaaaa@37DE@ β k i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda WgaaWcbaGaam4AaiaadMgaaeqaaaaa@3B13@    
If Itab = 1,
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Ifunc_G XGscale YGscale          
Ifunc_K XKscale YKscale          
If Itab = 2,
(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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaaS baaSqaaiaadAhaaeqaaaaa@3855@ Viscous bulk modulus. 3 Only used if K i = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGPbaabeaakiabg2da9iaaicdaaaa@39A8@ .

Default = 0. (Real)

[ Pas ]
Itab Tabulated formulation flag
= 0 (Default)
No tabulated functions
= 1
Relaxation test tabulated functions
= 2
DMA tests tabulated functions

(Integer)

 
Ishape Tabulated Prony series shape flag (Only if Itab0)
= 0
Classic shape of Prony series
= 1
Infinite value shape

(Integer)

 
G i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGPbaabeaaaaa@37DE@ Shear relaxation modulus for ith term (i=1, M).

(Real)

[ Pa ]

[ s ]

β i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda WgaaWcbaGaamyAaaqabaaaaa@3A23@ Decay shear constant for ith term (i=1, M).

(Real)

[ 1 s ]
K i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGPbaabeaaaaa@37DE@ Bulk relaxation modulus for ith term (i=1, M). 3

(Real)

[ Pa ]
β k i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda WgaaWcbaGaam4AaiaadMgaaeqaaaaa@3B13@ Decay bulk constant for ith term (i=1, M).

(Real)

[ 1 s ]
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)

[ s ]
YGscale Scale factor for shear modulus relaxation test data curve.

Default = 1.0 (Real)

[ Pa ]
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)

[ s ]
YKscale Scale factor for bulk modulus relaxation test data curve.

Default = 1.0 (Real)

[ Pa ]
Ifunc_Gs Shear storage modulus data curve.

(Integer)

 
XGs_scale Frequency scale factor for shear storage modulus test data curve.

Default = 1.0 (Real)

[Hz]
YGs_Scale Scale factor for shear storage modulus test data curve.

Default = 1.0 (Real)

[ Pa ]
Ifunc_Gl Shear loss modulus data curve.

(Integer)

 
XGl_scale Frequency scale factor for shear loss modulus test data curve.

Default = 1.0 (Real)

[Hz]
YGl_Scale Scale factor for shear loss modulus test data curve.

Default = 1.0 (Real)

[ Pa ]
Ifunc_Ks Bulk storage modulus data curve.

(Integer)

 
XKs_scale Frequency scale factor for bulk storage modulus test data curve.

Default = 1.0 (Real)

[Hz]
YKs_scale Scale factor for bulk storage modulus test data curve.

Default = 1.0 (Real)

[ Pa ]
Ifunc_Kl Bulk loss modulus data curve.

(Integer)

 
XKl_scale Frequency scale factor for bulk loss modulus test data curve.

Default = 1.0 (Real)

[Hz]
YKl_scale Scale factor for bulk loss modulus test data curve.

Default = 1.0 (Real)

[ Pa ]

Comments

  1. 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 (HENSEL-SPITTEL), and /MAT/LAW106 (JCOOK_ALM).

  2. 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 ij = 0 t 2 G ( t s ) dev [ ε ij ] s ds
    with(2)
    G ( t ) = i = 1 M G i e β i t

    and dev [ ε ij ] denotes the deviatoric part of strain tensor.

    Shear decay:(3)
    β i = ( 1 τ i )

    Where, τ i is the relaxation time.

  3. For the viscous pressure, two formulations are available:
    • If the bulk relaxation modulus is K i > 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaaS baaSqaaiaadMgaaeqaaOGaeyOpa4JaaGimaaaa@3A14@ , the viscous pressure is computed as:(4)
      P = 0 t K ( s ) ε ˙ v o l d s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0JaeyOeI0Yaa8qmaeaacaWGlbWaaeWaaeaacaWGZbaacaGLOaGa ayzkaaGafqyTduMbaiaadaWgaaWcbaGaamODaiaad+gacaWGSbaabe aaaeaacaaIWaaabaGaamiDaaqdcqGHRiI8aOGaamizaiaadohaaaa@46EF@

      with ε ˙ v o l = t r a c e ( ε ˙ ) = ε ˙ x x + ε ˙ y y + ε ˙ z z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH1oqzga GaamaaBaaaleaacaWG2bGaam4BaiaadYgaaeqaaOGaeyypa0JaamiD aiaadkhacaWGHbGaam4yaiaadwgadaqadaqaaiqbew7aLzaacaaaca GLOaGaayzkaaGaeyypa0JafqyTduMbaiaadaWgaaWcbaGaamiEaiaa dIhaaeqaaOGaey4kaSIafqyTduMbaiaadaWgaaWcbaGaamyEaiaadM haaeqaaOGaey4kaSIafqyTduMbaiaadaWgaaWcbaGaamOEaiaadQha aeqaaaaa@5271@ and K ( t ) = 1 M K i e β k i t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaae WaaeaacaWG0baacaGLOaGaayzkaaGaeyypa0ZaaabmaeaacaWGlbWa aSbaaSqaaiaadMgaaeqaaOGaamyzamaaCaaaleqabaGaeyOeI0Iaeq OSdi2aaSbaaWqaaiaadUgacaWGPbaabeaaliaadshaaaaabaGaaGym aaqaaiaad2eaa0GaeyyeIuoaaaa@46E2@

    • If the bulk relaxation modulus is K i = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaaS baaSqaaiaadMgaaeqaaOGaeyypa0JaaGimaaaa@3A12@ and the viscous bulk modulus K ν > 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacqaH9oGBaeqaaOGaeyOpa4JaaGimaaaa@3A76@ , the viscous pressure is computed as:(5)
      P = K v ε ˙ v o l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0JaeyOeI0Iaam4samaaBaaaleaacaWG2baabeaakiqbew7aLzaa caWaaSbaaSqaaiaadAhacaWGVbGaamiBaaqabaaaaa@3FE3@
  4. Starting with Radioss version 2017, identical results are obtained using the same Prony coefficents Gi in /VISC/PRONY and viscoelastic materials /MAT/LAW34 (BOLTZMAN), /MAT/LAW40 (KELVINMAX), and /MAT/LAW42 (OGDEN). In previous Radioss versions, 2 Gi had to be input into /VISC/PRONY to get equivalent results.
  5. 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.


      Figure 1. Example of Prony series fitting on shear modulus relaxation test data
    • If Itab = 2, Prony series parameters are fitted from Dynamic Mechanical Analysis (DMA) tests data, i.e storage and loss moduli versus frequency curves.


      Figure 2. Example of Prony series fitting on shear storage and loss modulus DMA test data. data taken from [Tapia-Romero et al.,2020]
    In both cases, an automatic least-square fit is realized to find the parameters ( G i , β i , K i , β k i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbWaaS baaSqaaiaadMgaaeqaaOGaaiilaiabek7aInaaBaaaleaacaWGPbaa beaakiaacYcacaWGlbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiabek 7aInaaBaaaleaacaWGRbGaamyAaaqabaaaaa@42C2@ ) in agreement with the order M of the Prony series defined by you. It is highly recommended to start with a small order and then increase it if the precision of the fitted curve is not sufficient.
    Note: The convergence of the least square fit may be hard to achieve for very high orders.
  6. 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 ) = i = 1 M G i e β i t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbGaai ikaiaadshacaGGPaGaeyypa0ZaaabCaeaacaWGhbWaaSbaaSqaaiaa dMgaaeqaaOGaamyzamaaCaaaleqabaGaeyOeI0IaeqOSdi2aaSbaaW qaaiaadMgaaeqaaSGaamiDaaaaaeaacaWGPbGaeyypa0JaaGymaaqa aiaad2eaa0GaeyyeIuoaaaa@47EE@
      and (7)
      K ( t ) = i = 1 M K i e β k i t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbGaai ikaiaadshacaGGPaGaeyypa0ZaaabCaeaacaWGlbWaaSbaaSqaaiaa dMgaaeqaaOGaamyzamaaCaaaleqabaGaeyOeI0IaeqOSdi2aaSbaaW qaaiaadUgacaWGPbaabeaaliaadshaaaaabaGaamyAaiabg2da9iaa igdaaeaacaWGnbaaniabggHiLdaaaa@48E6@
    • 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 + i = 1 M G i e β i t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbGaai ikaiaadshacaGGPaGaeyypa0Jaam4ramaaBaaaleaacqGHEisPaeqa aOGaey4kaSYaaabCaeaacaWGhbWaaSbaaSqaaiaadMgaaeqaaOGaam yzamaaCaaaleqabaGaeyOeI0IaeqOSdi2aaSbaaWqaaiaadMgaaeqa aSGaamiDaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad2eaa0Gaey yeIuoaaaa@4B43@
      and(9)
      K ( t ) = K + i = 1 M K i e β k i t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbGaai ikaiaadshacaGGPaGaeyypa0Jaam4samaaBaaaleaacqGHEisPaeqa aOGaey4kaSYaaabCaeaacaWGlbWaaSbaaSqaaiaadMgaaeqaaOGaam yzamaaCaaaleqabaGaeyOeI0IaeqOSdi2aaSbaaWqaaiaadUgacaWG PbaabeaaliaadshaaaaabaGaamyAaiabg2da9iaaigdaaeaacaWGnb aaniabggHiLdaaaa@4C3F@
    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.