Visco-elasto Materials for Foams (LAW33)

This material law can be used to model low density closed cell polyurethane foams, impactors, impact limiters. It can only be used with solid elements.

The main assumptions in this law are:
  • The components of the stress tensor are uncoupled until full volumetric compaction is achieved (Poisson's ratio = 0.0).
  • The material is isotropic.
  • The effect of the enclosed air is considered via a separate Pressure versus Volumetric Strain relation:
    図 1.
    P a i r = − P 0 ⋅ γ 1 + γ − Φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGHbGaamyAaiaadkhaaeqaaOGaeyypa0JaeyOeI0YaaSaa aeaacaWGqbWaaSbaaSqaaiaaicdaaeqaaOGaeyyXICTaeq4SdCgaba GaaGymaiabgUcaRiabeo7aNjabgkHiTiabfA6agbaaaaa@4730@
    with:
    図 2.
    γ = V V 0 − 1 + γ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey ypa0ZaaSaaaeaacaWGwbaabaGaamOvamaaBaaaleaacaaIWaaabeaa aaGccqGHsislcaaIXaGaey4kaSIaeq4SdC2aaSbaaSqaaiaaicdaae qaaaaa@4071@
    Where,
    γ
    Volumetric strain
    Φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyeaaa@3771@
    Porosity
    P 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaaIWaaabeaaaaa@37B2@
    Initial air pressure
    γ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS baaSqaaiaaicdaaeqaaaaa@3884@
    Initial volumetric strain
  • The structural stresses σ follow the Maxwell-Kelvin-Voight viscoelastic model (Generalized Kelvin-Voigt Model (LAW35), 式 12 before the limiting yield curve is reached):
    図 3. Maxwell-Kelvin-Voight Model


    図 4.
    σ i j ( t + Δ t ) = σ i j ( t ) + [ E ε ˙ i j − ( E + E t η σ i j s ( t ) ) + E ⋅ E t η ε i j ] ⋅ Δ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadMgacaWGQbaabaaaaOWaaeWaaeaacaWG0bGaey4kaSIa euiLdqKaamiDaaGaayjkaiaawMcaaiabg2da9iabeo8aZnaaDaaale aacaWGPbGaamOAaaqaaaaakmaabmaabaGaamiDaaGaayjkaiaawMca aiabgUcaRmaadmaabaGaamyraiqbew7aLzaacaWaaSbaaSqaaiaadM gacaWGQbaabeaakiabgkHiTmaabmaabaWaaSaaaeaacaWGfbGaey4k aSIaamyramaaBaaaleaacaWG0baabeaaaOqaaiabeE7aObaacqaHdp WCdaqhaaWcbaGaamyAaiaadQgaaeaacaWGZbaaaOWaaeWaaeaacaWG 0baacaGLOaGaayzkaaaacaGLOaGaayzkaaGaey4kaSYaaSaaaeaaca WGfbGaeyyXICTaamyramaaBaaaleaacaWG0baabeaaaOqaaiabeE7a ObaacqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLBbGaay zxaaGaeyyXICTaeuiLdqKaamiDaaaa@6D87@
  • The Young's modulus used in the calculation is: E = max ( E , E 1 ε ˙ + E 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2 da9iGac2gacaGGHbGaaiiEaiaacIcacaWGfbGaaiilaiaadweadaWg aaWcbaGaaGymaaqabaGccuaH1oqzgaGaaiabgUcaRiaadweadaWgaa WcbaGaaGOmaaqabaGccaGGPaaaaa@4377@
  • Yield is defined by a user-defined curve versus volumetric strain, γ , or σ = A + B ( 1 + C γ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0JaamyqaiabgUcaRiaadkeadaqadaqaaiaaigdacqGHRaWkcaWG dbGaeq4SdCgacaGLOaGaayzkaaaaaa@40C4@
  • Yield is applied to the principal structural stresses.
  • Unloading follows Young's modulus, which results in viscous unloading.
  • The full stress tensor is obtained by adding air pressure to the structual stresses:
    図 5.
    σ total ij ( t )= σ ij ( t )− P air δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaW baaSqabeaacaWG0bGaam4BaiaadshacaWGHbGaamiBaaaakmaaBaaa leaacaWGPbGaamOAaaqabaGcdaqadaqaaiaadshaaiaawIcacaGLPa aacqGH9aqpcqaHdpWCdaqhaaWcbaGaamyAaiaadQgaaeaaaaGcdaqa daqaaiaadshaaiaawIcacaGLPaaacqGHsislcaWGqbWaaSbaaSqaai aadggacaWGPbGaamOCaaqabaGccqaH0oazdaWgaaWcbaGaamyAaiaa dQgaaeqaaaaa@5113@