Viscous Materials

General case of viscous materials represents a time-dependent inelastic behavior.

However, special attention is paid to the viscoelastic materials such as polymers exhibiting a rate- and time-dependent behavior. The viscoelasticity can be represented by a recoverable instantaneous elastic deformation and a non-recoverable viscous part occurring over the time. The characteristic feature of viscoelastic material is its fading memory. In a perfectly elastic material, the deformation is proportional to the applied load. In a perfectly viscous material, the rate of change of the deformation over time is proportional to the load. When an instantaneous constant tensile stress σ 0 is applied to a viscoelastic material, a slow continuous deformation of the material is observed. When the resulting time dependent strain ε ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aae WaaeaacaWG0baacaGLOaGaayzkaaaaaa@3A1F@ , is measured, the tensile creep compliance is defined as:(1)
D ( t ) = ε ( t ) σ 0
The creep behavior is mainly composed of three phases:
  • Primary creep with fast decrease in creep strain rate
  • Secondary creep with slow decrease in creep strain rate
  • Tertiary creep with fast increase in creep strain rate.
The creep strain rate is the slope of creep strain to time curve.
Another kind of loading concerns viscoelastic materials subjected to a constant tensile strain, ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaaicdaaeqaaaaa@3883@ . In this case, the stress, σ ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aae WaaeaacaWG0baacaGLOaGaayzkaaaaaa@3A3B@ which is called stress relaxation, gradually decreases. The tensile relaxation modulus is then defined as:(2)
E ( t ) = σ ( t ) ε 0
Because viscoelastic response is a combination of elastic and viscous responses, the creep compliance and the relaxation modulus are often modeled by combinations of springs and dashpots. A simple schematic model of viscoelastic material is given by the Maxwell model shown in 図 1. The model is composed of an elastic spring with the stiffness E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@ and a dashpot assigned a viscosity μ . It is assumed that the total strain is the sum of the elastic and viscous strains:(3)
ε = ε e + ε v


図 1. Maxwell Model
The time derivation of the last expression gives the expression of the total strain rate:(4)
ε ˙ = ε ˙ e + ε ˙ v
As the dashpot and the spring are in series, the stress is the same in the two parts:(5)
σ e = σ v = σ
The constitutive relations for linear spring and dashpot are written as:(6)

σ = E ε e then σ ˙ = E ε ˙ e

(7)
σ = μ ε ˙ v
Combining 式 4, 式 6 and 式 7, an ordinary differential equation for stress is obtained:(8)

σ ˙ = E ( ε ˙ σ μ ) or σ ˙ = E ε ˙ σ τ

Where, τ = μ E is the relaxation time. A solution to the differential equation is given by the convolution integral:(9)
σ( t )= t E e ( tt' ) τ dε( t' ) dt' dt' = t R( tt' ) dε( t' ) dt' dt' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qadaqaaiaadshaaiaawIcacaGLPaaacqGH9aqpdaWdXaqaaiaadwea caWGLbWaaWbaaSqabeaadaWcaaqaaiabgkHiTmaabmaabaGaamiDai abgkHiTiaadshacaGGNaaacaGLOaGaayzkaaaabaGaeqiXdqhaaaaa kmaalaaabaGaamizaiabew7aLnaabmaabaGaamiDaiaacEcaaiaawI cacaGLPaaaaeaacaWGKbGaamiDaiaacEcaaaGaamizaiaadshacaGG NaaaleaacqGHsislcqGHEisPaeaacaWG0baaniabgUIiYdGccqGH9a qpdaWdXaqaaiaadkfadaqadaqaaiaadshacqGHsislcaWG0bGaai4j aaGaayjkaiaawMcaamaalaaabaGaamizaiabew7aLnaabmaabaGaam iDaiaacEcaaiaawIcacaGLPaaaaeaacaWGKbGaamiDaiaacEcaaaGa amizaiaadshacaGGNaaaleaacqGHsislcqGHEisPaeaacaWG0baani abgUIiYdaaaa@6E3B@
Where, R ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaabm aabaGaamiDaaGaayjkaiaawMcaaaaa@394F@ is the relaxation modulus. The last equation is valid for the special case of Maxwell one-dimensional model. It can be extended to the multi-axial case by:(10)
σ( t )= t C ijkl ( tt' ) dε( t' ) dt' dt' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qadaqaaiaadshaaiaawIcacaGLPaaacqGH9aqpdaWdXaqaaiaadoea daWgaaWcbaGaamyAaiaadQgacaWGRbGaamiBaaqabaGcdaqadaqaai aadshacqGHsislcaWG0bGaai4jaaGaayjkaiaawMcaamaalaaabaGa amizaiabew7aLnaabmaabaGaamiDaiaacEcaaiaawIcacaGLPaaaae aacaWGKbGaamiDaiaacEcaaaGaamizaiaadshacaGGNaaaleaacqGH sislcqGHEisPaeaacaWG0baaniabgUIiYdaaaa@56E9@

Where, C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaadMgacaWGQbGaam4AaiaadYgaaeqaaaaa@3C1A@ are the relaxation moduli. The Maxwell model represents reasonably the material relaxation. But it is only accurate for secondary creep as the viscous strains after unloading are not taken into account.

Another simple schematic model for viscoelastic materials is given by Kelvin-Voigt solid. The model is represented by a simple spring-dashpot system working in parallel as shown in 図 2.


図 2. Kelvin-Voigt Model
The mathematical relation of Kelvin-Voigt solid is written as:(11)
σ = E ε + η ε ˙

When η = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH3oaAcq GH9aqpcaaIWaaaaa@3AD4@ (no dashpot), the system is a linearly elastic system. When E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@ =0 (no spring), the material behavior is expressed by Newton's equation for viscous fluids. In the above relation, a one-dimensional model is considered. For multiaxial situations, the equations can be generalized and rewritten in tensor form.

The Maxwell and Kelvin-Voigt models are appropriate for ideal stress relaxation and creep behaviors. They are not adequate for most of physical materials. A generalization of these laws can be obtained by adding other springs to the initial models as shown in 図 3 and 図 4. The equations related to the generalized Maxwell model are given as:(12)
σ = σ i = σ 1
(13)
ε = σ i E i
(14)
ε ˙ = σ ˙ 1 E 1 + σ 1 η 1
The mathematical relations which hold the generalized Kelvin-Voigt model are: (15)
ε = ε e + ε k

σ = σ e + σ k

ε e = σ E ; ε k = σ V E t ; ε ˙ k = σ V η

The combination of these equations enables to obtain the expression of stress and strain rates: (16)
ε ˙ = ε ˙ e + ε ˙ k = σ ˙ E + ε ˙ k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH1oqzga Gaaiabg2da9iqbew7aLzaacaWaaWbaaSqabeaacaWGLbaaaOGaey4k aSIafqyTduMbaiaadaahaaWcbeqaaiaadUgaaaGccqGH9aqpdaWcaa qaaiqbeo8aZzaacaaabaGaamyraaaacqGHRaWkcuaH1oqzgaGaamaa CaaaleqabaGaam4Aaaaaaaa@4803@
(17)
σ = η ε ˙ k + E t ε k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCcq GH9aqpcqaH3oaAcuaH1oqzgaGaamaaCaaaleqabaGaam4Aaaaakiab gUcaRiaadweadaWgaaWcbaGaamiDaaqabaGccqaH1oqzdaahaaWcbe qaaiaadUgaaaaaaa@4453@
(18)
σ ˙ =E ε ˙ ( E E t )σ η + E E t ε η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga Gaaiabg2da9iaadweacuaH1oqzgaGaaiabgkHiTmaalaaabaWaaeWa aeaacaWGfbGaeyOeI0IaamyramaaBaaaleaacaWG0baabeaaaOGaay jkaiaawMcaaiabeo8aZbqaaiabeE7aObaacqGHRaWkdaWcaaqaaiaa dweacaWGfbWaaSbaaSqaaiaadshaaeqaaOGaeqyTdugabaGaeq4TdG gaaaaa@4D61@


図 3. Generalized Maxwell Model


図 4. Generalized Kelvin-Voigt Model

The models described above concern the viscoelastic materials. The plasticity can be introduced in the models by using a plastic spring. The plastic element is inactive when the stress is less than the yield value. The modified model is able to reproduce creep and plasticity behaviors. The viscoplasticity law (LAW33) in Radioss will enable to implement very general constitutive laws useful for a large range of applications as low density closed cells polyurethane foam, honeycomb, impactors and impact limiters.

The behavior of viscoelastic materials can be generalized to three dimensions by separating the stress and strain tensors into deviatoric and pressure components:(19)
s i j = 0 t 2 Ψ ( t τ ) e i j τ d τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9maapedabaGaaGOmaiab fI6aznaabmaabaGaamiDaiabgkHiTiabes8a0bGaayjkaiaawMcaam aalaaabaGaeyOaIyRaamyzamaaBaaaleaacaWGPbGaamOAaaqabaaa keaacqGHciITcqaHepaDaaGaamizaiabes8a0bWcbaGaaGimaaqaai aadshaa0Gaey4kIipaaaa@5120@
(20)
σ kk ( t )= 0 t 3K( tτ ) ε kk τ dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaam4AaiaadUgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGa ayzkaaGaeyypa0Zaa8qmaeaacaaIZaGaam4samaabmaabaGaamiDai abgkHiTiabes8a0bGaayjkaiaawMcaamaalaaabaGaeyOaIyRaeqyT du2aaSbaaSqaaiaadUgacaWGRbaabeaaaOqaaiabgkGi2kabes8a0b aacaWGKbGaeqiXdqhaleaacaaIWaaabaGaamiDaaqdcqGHRiI8aaaa @5472@

Where, s i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS baaSqaaiaadMgacaWGQbaabeaaaaa@3A69@ and e i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGLbWaaS baaSqaaiaadMgacaWGQbaabeaaaaa@3A5B@ are the stress and strain deviators. ε k k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaam4AaiaadUgaaeqaaaaa@3B1B@ , Ψ ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHOoqwda qadaqaaiaadshaaiaawIcacaGLPaaaaaa@3B79@ and K( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaae WaaeaacaWG0baacaGLOaGaayzkaaaaaa@3ABA@ are respectively the dilatation and the shear and bulk relaxation moduli.