/PROP/TYPE4 (SPRING)

Block Format Keyword Defines spring property with one translational DOF. This spring accounts for nonlinear stiffness, damping and different unloading. Deformation based failure criteria is available.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/PROP/TYPE4/prop_ID/unit_ID or /PROP/SPRING/prop_ID/unit_ID
prop_title
Mass       sens_ID Isflag Ileng    
K1 C1 A1 B1 D1
fct_ID11 H1 fct_ID21 fct_ID31 fct_ID41   δ min 1 δ max 1
F1 E1 Ascale1 Hscale1    

Definition

Field Contents SI Unit Example
prop_ID Property identifier.

(Integer, maximum 10 digits)

 
unit_ID Unit Identifier.

(Integer, maximum 10 digits)

 
prop_title Property title.

(Character, maximum 100 characters)

 
Mass Mass.

(Real)

[ kg m ]
sens_ID Sensor identifier used for spring activation or deactivation.
= 0
Spring is active.

(Integer)

 
Isflag Sensor flag. 2
=0
Spring element activated when sens_ID activates and cannot be deactivated.
=1
Spring element deactivated when sens_ID activates and cannot be activated.
=2
Spring elements are activated, or deactivated state matches the sensor state and can switch back and forth. The spring initial length ( l 0 ) is based the spring length at the activation time.

(Integer)

 
Ileng Input per unit length flag.
=0
Spring properties are input as explained in the definition table.
=1
Spring mass and inertia input are per unit length. Spring stiffness is a function of engineering strain. 3 4

(Integer)

 
K1 If f c t _ I D 11 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaam 4yaiaadshacaGGFbGaamysaiaadseadaWgaaWcbaGaaGymaiaaigda aeqaaOGaeyypa0JaaGimaaaa@401A@ : Linear loading and unloading stiffness.

If f c t _ I D 11 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaam 4yaiaadshacaGGFbGaamysaiaadseadaWgaaWcbaGaaGymaiaaigda aeqaaOGaeyiyIKRaaGimaaaa@40DB@ : Only used as unloading stiffness for elasto-plastic springs.

(Real)

[ N m ]
C1 Damping.

(Real)

[ Ns m ]
A1 Nonlinear stiffness function scale factor.

Default = 1.0 (Real)

[ N ]
B1 Logarithmic rate effects scale factor.

(Real)

[ N ]
D1 Logarithmic rate effects scale factor.

Default = 1.0 (Real)

[ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@
fct_ID11 Function identifier defining f ( δ ) . 4
= 0
Linear spring with stiffness K1.

If H1 =4: Function defines upper yield curve.

If H1 =8: Function is mandatory and defines the force versus spring length.

(Integer)

 
H1 Spring Hardening flag for nonlinear spring.
=0
Elastic spring.
=1
Nonlinear elastic plastic spring with isotropic hardening.
=2
Nonlinear elastic plastic spring with uncoupled hardening.
=4
Nonlinear elastic plastic spring with kinematic hardening.
=5
Nonlinear elastic plastic spring with nonlinear unloading.
=6
Nonlinear elastic plastic spring with isotropic hardening and nonlinear unloading.
=7
Nonlinear elastic spring with elastic hystersis.
=8
Nonlinear elastic spring with total length function.

(Integer)

 
fct_ID21 Function identifier defining force as a function of spring velocity, g ( δ ˙ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbWaae WaaeaacuaH0oazgaGaaaGaayjkaiaawMcaaaaa@3A83@ .

(Integer)

 
fct_ID31 Function identifier.

If H1 =4: Defines lower yield curve

If H1 =5: Defines residual displacement versus maximum displacement

If H1 =6: Defines nonlinear unloading curve

If H1 =7: Defines nonlinear unloading curve

(Integer)

 
fct_ID41 Function identifier for nonlinear damping h ( δ ˙ ) .

(Integer)

 
δ min 1 Negative failure displacement.

Default = -1030 (Real)

[ m ]
δ max 1 Positive failure displacement.

Default = 1030 (Real)

[ m ]
F1 Abscissa scale factor for damping functions for g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374C@ and h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374C@ .

Default = 1.0 (Real)

[ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@
E1 Ordinate scale factor for the damping function g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374C@ .

(Real)

[ N ]
Ascale1 Abscissa scale factor for the stiffness function f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374C@ .

Default = 1.0 (Real)

[ m ]
Hscale1 Ordinate scale factor for the damping function h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374C@ .

Default = 1.0 (Real)

 

Example (Seatbelt)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/2
unit for prop
                  kg                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/PROP/SPRING/2/2
Seatbelt 
#                  M                               sensor_ID    Isflag     Ileng
                5E-5                                       0         0         1
#                 K1                  C1                  A1                  B1                  D1
               0.001                   0                   0                   0                   0
# fct_ID11        H1  fct_ID21  fct_ID31  fct_ID41                     delta_min           delta_max
         1         2         0                   0                             0                   0
#                 F1                  E1             Ascale1             Hscale1
                   0                   0                   0                   0
/MOVE_FUNCT/1
Seatbelt 
#           Ascale_x            Fscale_y            Ashift_x            Fshift_y
                                   0.001
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/1
Seatbelt loading force vs engineering strain 
#                  X                   Y
                  0.                  0.
               0.005                700.
                0.02               3100.
                0.03               5500.
                0.15              17000.
               1000.              17000.
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA

Comments

  1. The spring has one translational degree of freedom in the local x direction which is defined between node N1 and N2 of the spring.
  2. Spring is activated and/or deactivated by sensor defined in sens_ID and depends on Isflag:
    • If Isflag = 0, the spring element is activated by the sens_ID and cannot be deactivated. The initial length of the spring is based on the spring length at time=0.
    • If Isflag = 1, the spring element is deactivated by the sens_ID and cannot be activated. The initial length of the spring is based on the spring length at time=0.
    • If Isflag = 2, the spring is activated and/or deactivated by sens_ID and can switch activation state multiple times. If sensor is activated, the spring is active; if sensor is deactivated, spring is deactivated. The spring initial length, l 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGSbWaaSbaaSqaaiaaicdaaeqaaaaa@3AAE@ , is the distance between spring nodes at the time of sensor activation.
  3. If Ileng = 1, the spring properties are based on the initial spring length. The input should be entered as:
    M = m l 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaey ypa0ZaaSaaaeaacaWGTbaabaGaamiBamaaBaaaleaacaaIWaaabeaa aaaaaa@3B0F@ K = k * l 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbGaey ypa0Jaam4AaiaacQcacaWGSbWaaSbaaSqaaiaaicdaaeqaaaaa@3BA9@
    C = c * l 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbGaey ypa0Jaam4yaiaacQcacaWGSbWaaSbaaSqaaiaaicdaaeqaaaaa@3B99@  
    Each spring will then have the following properties in the model:
    m = M l 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGTbGaey ypa0JaamytaiabgwSixlaadYgadaWgaaWcbaGaaGimaaqabaaaaa@3D49@ k = K l 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbGaey ypa0ZaaSaaaeaacaWGlbaabaGaamiBamaaBaaaleaacaaIWaaabeaa aaaaaa@3B0B@
    c = C l 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbGaey ypa0ZaaSaaaeaacaWGdbaabaGaamiBamaaBaaaleaacaaIWaaabeaa aaaaaa@3AFB@  
    Where,
    M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@ , K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@ and C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@
    Spring values entered in the spring property fields
    m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@ , k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@ and c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@
    Spring’s actual physical mass, stiffness and damping
    l 0
    Initial spring length which is the distance between node N1 and N2 of the spring
    δ min 1  and  δ max 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH0oazda qhaaWcbaGaciyBaiaacMgacaGGUbaabaGaaGymaaaakiaabccacaqG HbGaaeOBaiaabsgacaqGGaGaeqiTdq2aa0baaSqaaiGac2gacaGGHb GaaiiEaaqaaiaaigdaaaaaaa@452A@
    Failure values entered as engineering strain
  4. Force computation. For additional information, refer to Stiffness Formulation in the User Guide.
    • If Ileng =0, the value of force F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbaaaa@3729@ in the spring is computed as:
      For H1 = 1,2,4,5,6,7:(1)
      F = f ( δ A s c a l e 1 ) [ A 1 + B 1 ln ( max ( 1 , | δ ˙ D 1 | ) ) + E 1 g ( δ ˙ F 1 ) ] + C 1 δ ˙ + H s c a l e 1 h ( δ ˙ F 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaey ypa0JaciOzamaabmaabaWaaSaaaeaacqaH0oazaeaacaWGbbGaam4C aiaadogacaWGHbGaamiBaiaadwgadaWgaaWcbaGaaGymaaqabaaaaa GccaGLOaGaayzkaaWaamWaaeaacaWGbbWaaSbaaSqaaiaaigdaaeqa aOGaey4kaSIaamOqamaaBaaaleaacaaIXaaabeaakiGacYgacaGGUb WaaeWaaeaaciGGTbGaaiyyaiaacIhadaqadaqaaiaaigdacaGGSaWa aqWaaeaadaWcaaqaaiqbes7aKzaacaaabaGaamiramaaBaaaleaaca aIXaaabeaaaaaakiaawEa7caGLiWoaaiaawIcacaGLPaaaaiaawIca caGLPaaacqGHRaWkcaWGfbWaaSbaaSqaaiaaigdaaeqaaOGaci4zam aabmaabaWaaSaaaeaacuaH0oazgaGaaaqaaiaadAeadaWgaaWcbaGa aGymaaqabaaaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaGaey4kaS Iaam4qamaaBaaaleaacaaIXaaabeaakiqbes7aKzaacaGaey4kaSIa amisaiaadohacaWGJbGaamyyaiaadYgacaWGLbWaaSbaaSqaaiaaig daaeqaaOGaciiAamaabmaabaWaaSaaaeaacuaH0oazgaGaaaqaaiaa dAeadaWgaaWcbaGaaGymaaqabaaaaaGccaGLOaGaayzkaaaaaa@72CB@

    Where, δ = l l 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH0oazcq GH9aqpcaWGSbGaeyOeI0IaamiBamaaBaaaleaacaaIWaaabeaaaaa@3CBF@ is the difference between the current length and the initial length of the spring element.

    with l 0 < δ < + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislca WGSbWaaSbaaSqaaiaaicdaaeqaaOGaeyipaWJaeqiTdqMaeyipaWJa ey4kaSIaeyOhIukaaa@3F2D@

    If H1 = 8:(2)
    F = f ( l A s c a l e 1 ) [ A 1 + B 1 ln ( max ( 1 , | δ ˙ D 1 | ) ) + E 1 g ( δ ˙ F 1 ) ] + C 1 δ ˙ + H s c a l e 1 h ( δ ˙ F 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaey ypa0JaciOzamaabmaabaWaaSaaaeaacaWGSbaabaGaamyqaiaadoha caWGJbGaamyyaiaadYgacaWGLbWaaSbaaSqaaiaaigdaaeqaaaaaaO GaayjkaiaawMcaamaadmaabaGaamyqamaaBaaaleaacaaIXaaabeaa kiabgUcaRiaadkeadaWgaaWcbaGaaGymaaqabaGcciGGSbGaaiOBam aabmaabaGaciyBaiaacggacaGG4bWaaeWaaeaacaaIXaGaaiilamaa emaabaWaaSaaaeaacuaH0oazgaGaaaqaaiaadseadaWgaaWcbaGaaG ymaaqabaaaaaGccaGLhWUaayjcSdaacaGLOaGaayzkaaaacaGLOaGa ayzkaaGaey4kaSIaamyramaaBaaaleaacaaIXaaabeaakiGacEgada qadaqaamaalaaabaGafqiTdqMbaiaaaeaacaWGgbWaaSbaaSqaaiaa igdaaeqaaaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaiabgUcaRi aadoeadaWgaaWcbaGaaGymaaqabaGccuaH0oazgaGaaiabgUcaRiaa dIeacaWGZbGaam4yaiaadggacaWGSbGaamyzamaaBaaaleaacaaIXa aabeaakiGacIgadaqadaqaamaalaaabaGafqiTdqMbaiaaaeaacaWG gbWaaSbaaSqaaiaaigdaaeqaaaaaaOGaayjkaiaawMcaaaaa@7217@

    Where, 0 < l < MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqGWaGaey ipaWJaamiBaiabgYda8iabg6HiLcaa@3B7B@ and l 0 < δ < + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislca WGSbWaaSbaaSqaaiaaicdaaeqaaOGaeyipaWJaeqiTdqMaeyipaWJa ey4kaSIaeyOhIukaaa@3F2D@

    If Ileng = 1, the value of force F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbaaaa@3729@ in the spring is computed as:(3)
    F = f ( ε A s c a l e 1 ) [ A 1 + B 1 ln ( max ( 1 , | ε ˙ D 1 | ) ) + E 1 g ( ε ˙ F 1 ) ] + C 1 ε ˙ + H s c a l e 1 h ( ε ˙ F 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaey ypa0JaciOzamaabmaabaWaaSaaaeaacqaH1oqzaeaacaWGbbGaam4C aiaadogacaWGHbGaamiBaiaadwgadaWgaaWcbaGaaGymaaqabaaaaa GccaGLOaGaayzkaaWaamWaaeaacaWGbbWaaSbaaSqaaiaaigdaaeqa aOGaey4kaSIaamOqamaaBaaaleaacaaIXaaabeaakiGacYgacaGGUb WaaeWaaeaaciGGTbGaaiyyaiaacIhadaqadaqaaiaaigdacaGGSaWa aqWaaeaadaWcaaqaaiqbew7aLzaacaaabaGaamiramaaBaaaleaaca aIXaaabeaaaaaakiaawEa7caGLiWoaaiaawIcacaGLPaaaaiaawIca caGLPaaacqGHRaWkcaWGfbWaaSbaaSqaaiaaigdaaeqaaOGaci4zam aabmaabaWaaSaaaeaacuaH1oqzgaGaaaqaaiaadAeadaWgaaWcbaGa aGymaaqabaaaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaGaey4kaS Iaam4qamaaBaaaleaacaaIXaaabeaakiqbew7aLzaacaGaey4kaSIa amisaiaadohacaWGJbGaamyyaiaadYgacaWGLbWaaSbaaSqaaiaaig daaeqaaOGaciiAamaabmaabaWaaSaaaeaacuaH1oqzgaGaaaqaaiaa dAeadaWgaaWcbaGaaGymaaqabaaaaaGccaGLOaGaayzkaaaaaa@72D5@
    Where,
    ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzaa a@3805@
    Engineering strain
    ε = δ l 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzcq GH9aqpdaWcaaqaaiabes7aKbqaaiaadYgadaWgaaWcbaGaaGimaaqa baaaaaaa@3C97@
    f ( ε ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaae WaaeaacqaH1oqzaiaawIcacaGLPaaaaaa@3A7A@
    Nonlinear stiffness is a function of engineering strain
    g ( ε ˙ )  and  h ( ε ˙ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbWaae WaaeaacuaH1oqzgaGaaaGaayjkaiaawMcaaiaabccacaqGHbGaaeOB aiaabsgacaqGGaGaamiAamaabmaabaGafqyTduMbaiaaaiaawIcaca GLPaaaaaa@42AC@
    Damping is a function of engineering strain rate
  5. If H1 > 0 and fct_ID11 = 0, f ( δ ) = 1  or  f ( ε ) = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaae WaaeaacqaH0oazaiaawIcacaGLPaaacqGH9aqpcaaIXaGaaeiiaiaa b+gacaqGYbGaaeiiaiGacAgadaqadaqaaiabew7aLbGaayjkaiaawM caaiabg2da9iaaigdaaaa@4543@ .