Tabulated Strain Rate Dependent Law for Viscoelastic Materials (LAW38)

The law incorporated in Radioss can only be used with solid elements.

It can be used to model:
  • polymers
  • elastomers
  • foam seat cushions
  • dummy paddings
  • hyperfoams
  • hypoelastic materials
In compression, the nominal stress-strain curves for different strain rates are defined by you (図 1). Up to 5 curves may be input. The curves represent nominal stresses versus engineering strains.


図 1. Nominal Stress-strain Curves Defined by User Input Functions
The first curve is considered to represent the static loading. All values of the strain rate lower than the assumed static curve are replaced by the strain rate of the static curve. It is reasonable to set the strain rate corresponding to the first curve equal to zero. For strain rates higher than the last curve, values of the last curve are used. For a given value of ε ε ˙ , two values of function at for the two immediately lower ε ˙ _ 1 and higher ε ˙ _ 2 strain rates are read. The related stress is then computed as:(1)
σ = σ 2 + ( σ 1 σ 2 ) [ 1 ( ε ˙ ε ˙ 1 ε ˙ 2 ε ˙ 1 ) a ] b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0Jaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaOGaey4kaSYaaeWaaeaa cqaHdpWCdaWgaaWcbaGaaGymaaqabaGccqGHsislcqaHdpWCdaWgaa WcbaGaaGOmaaqabaaakiaawIcacaGLPaaadaWadaqaaiaaigdacqGH sisldaqadaqaamaalaaabaGafqyTduMbaiaacqGHsislcuaH1oqzga GaamaaBaaaleaacaaIXaaabeaaaOqaaiqbew7aLzaacaWaaSbaaSqa aiaaikdaaeqaaOGaeyOeI0IafqyTduMbaiaadaWgaaWcbaGaaGymaa qabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaWGHbaaaaGccaGL BbGaayzxaaWaaWbaaSqabeaacaWGIbaaaaaa@5708@

Parameters a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@ and b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@ define the shape of the interpolation functions. If a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@ = b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@ = 1, then the interpolation is linear.

図 2 shows the influence of a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@ and b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@ parameters.


図 2. Influence of a and b Parameters
The coupling between the principal nominal stresses in tension is computed using anisotropic Poisson's ratio:(2)
ν i j = ν c + ( ν t ν c ) ( 1 exp ( R v | ε i j | ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd42aaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9iabe27aUnaaBaaaleaa caWGJbaabeaakiabgUcaRmaabmaabaGaeqyVd42aaSbaaSqaaiaads haaeqaaOGaeyOeI0IaeqyVd42aaSbaaSqaaiaadogaaeqaaaGccaGL OaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0IaciyzaiaacIhacaGGWb WaaeWaaeaacqGHsislcaWGsbWaaSbaaSqaaiaadAhaaeqaaOWaaqWa aeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLhWUaay jcSdaacaGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@5819@

Where, ν t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd42aaS baaSqaaiaadshaaeqaaaaa@38D4@ is the maximum Poisson's ratio in tension, ν c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd42aaS baaSqaaiaadshaaeqaaaaa@38D4@ being the maximum Poisson's ratio in compression, and R v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWG2baabeaaaaa@37F5@ , the exponent for the Poisson's ratio computation (in compression, Poisson's ratio is always equal to ν c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd42aaS baaSqaaiaadshaaeqaaaaa@38D4@ ).

In compression, material behavior is given by nominal stress versus nominal strain curves as defined by you for different strain rates. Up to 5 curves may be input.

The algorithm of the formulation follows several steps:
  1. Compute principal nominal strains and strain rates.
  2. Find corresponding stress value from the curve network for each principal direction.
  3. Compute principal Cauchy stress.
  4. Compute global Cauchy stress.
  5. Compute instantaneous modulus, viscosity and stable time step.
Stress, strain and strain rates must be positive in compression. Unloading may be either defined with an unloading curve, or else computed using the "static" curve, corresponding to the lowest strain rate (図 3 and 図 4).


図 3. Unloading Behavior (No Unloading Curve Defined)


図 4. Unloading Behavior (Unloading Curve Defined)
It should be noted that for stability reasons, damping is applied to strain rates with a damping factor:(3)
ε ˙ n 1 + R D ( ε ˙ n ε ˙ n 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbai aadaahaaWcbeqaaiaad6gacqGHsislcaaIXaaaaOGaey4kaSIaamOu amaaBaaaleaacaWGebaabeaakmaabmaabaGafqyTduMbaiaadaahaa Wcbeqaaiaad6gaaaGccqGHsislcuaH1oqzgaGaamaaCaaaleqabaGa amOBaiabgkHiTiaaigdaaaaakiaawIcacaGLPaaaaaa@4703@

The stress recovery may be applied to the model in order to ensure that the stress tensor is equal to zero, in an undeformed state.

An hysteresis decay may be applied when loading, unloading or in both phases by:(4)
σ = σ H min ( 1 , ( 1 e β ε ( t ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0Jaeq4WdmNaeyyXICTaamisaiabgwSixlGac2gacaGGPbGaaiOB amaabmaabaGaaGymaiaacYcadaqadaqaaiaaigdacqGHsislcaWGLb WaaWbaaSqabeaacqGHsislcqaHYoGycqaH1oqzdaqadaqaaiaadsha aiaawIcacaGLPaaaaaaakiaawIcacaGLPaaaaiaawIcacaGLPaaaaa a@50B3@
Where,
H MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaaaa@36C4@
Hysteresis coefficient
β
Relaxation parameter
Confined air content may be taken into account, either by using a user-defined function, or using the following relation:(5)
P a i r = P 0 ( 1 V V 0 ) ( V V 0 Φ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGHbGaamyAaiaadkhaaeqaaOGaeyypa0JaamiuamaaBaaa leaacaaIWaaabeaakmaalaaabaWaaeWaaeaacaaIXaGaeyOeI0YaaS aaaeaacaWGwbaabaGaamOvamaaBaaaleaacaaIWaaabeaaaaaakiaa wIcacaGLPaaaaeaadaqadaqaamaalaaabaGaamOvaaqaaiaadAfada WgaaWcbaGaaGimaaqabaaaaOGaeyOeI0IaeuOPdyeacaGLOaGaayzk aaaaaaaa@4935@
The relaxation may be applied to air pressure: (6)
P a i r = min ( P a i r , P max ) exp ( R p t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGHbGaamyAaiaadkhaaeqaaOGaeyypa0JaciyBaiaacMga caGGUbWaaeWaaeaacaWGqbWaaSbaaSqaaiaadggacaWGPbGaamOCaa qabaGccaGGSaGaamiuamaaBaaaleaaciGGTbGaaiyyaiaacIhaaeqa aaGccaGLOaGaayzkaaGaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsi slcaWGsbWaaSbaaSqaaiaadchaaeqaaOGaamiDaaGaayjkaiaawMca aaaa@4FDF@