Elastic-Plastic Orthotropic Composite Solids

The material LAW14 (COMPSO) in Radioss allows to simulate orthotropic elasticity, Tsai-Wu plasticity with damage, brittle rupture and strain rate effects. The constitutive law applies to only one layer of lamina. Therefore, each layer needs to be modeled by a solid mesh. A layer is characterized by one direction of the fiber or material. The overall behavior is assumed to be elasto-plastic orthotropic.

Direction 1 is the fiber direction, defined with respect to the local reference frame r , s , t as shown in 図 1.


図 1. Local Reference Frame

For the case of unidirectional orthotropy (that is, E 33 = E 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIZaGaaG4maaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaa ikdacaaIYaaabeaaaaa@3BE4@ and G 31 = G 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIZaGaaG4maaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaa ikdacaaIYaaabeaaaaa@3BE4@ ) the material LAW53 in Radioss allows to simulate an orthotropic elastic-plastic behavior by using a modified Tsai-Wu criteria.

Linear Elasticity

When the lamina has a purely linear elastic behavior, the stress calculation algorithm:
  • Transform the lamina stress, σ i j ( t ) , and strain rate, d ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38E8@ , from global reference frame to fiber reference frame.
  • Compute lamina stress at time t + Δ t by explicit time integration:
    (1) σ ij ( t+Δt )= σ ij ( t )+ D ijkl d kl Δt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOWaaeWaaeaacaWG 0bGaey4kaSIaaeiLdiaadshaaiaawIcacaGLPaaacqGH9aqpcqaHdp WCdaWgaaWcbaGaamyAaiaadQgaaeqaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaGaey4kaSIaamiramaaBaaaleaacaWGPbGaamOAaiaadU gacaWGSbaabeaakiaayIW7caWGKbWaaSbaaSqaaiaadUgacaWGSbaa beaakiaayIW7caqGuoGaamiDaaaa@5756@
  • Transform the lamina stress, σ i j ( t + Δ t ) , back to global reference frame.
The elastic constitutive matrix C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaaaa@36BE@ of the lamina relates the non-null components of the stress tensor to those of strain tensor:(2) { σ } = [ D ] { ε }
The inverse relation is generally developed in term of the local material axes and nine independent elastic constants:(3) { ε 11 ε 22 ε 33 γ 12 γ 23 γ 31 } = [ 1 E 11 ν 21 E 22 ν 31 E 33 0 0 0 1 E 22 ν 32 E 33 0 0 0 1 E 33 0 0 0 1 2 G 12 0 0 S y m m . 1 2 G 23 0 1 2 G 31 ] { σ 11 σ 22 σ 33 σ 12 σ 23 σ 31 }
Where,
E i j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38E8@
Young's modulus
G i j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38E8@
Shear modulus
ν i j
Poisson's ratios
γ i j
Strain components due to the distortion


図 2. Strain Components and Distortion

Orthotropic Plasticity

Lamina yield surface defined by Tsai-Wu yield criteria is used for each layer:(4) F = f ( W p ) = F 1 σ 1 + F 2 σ 2 + F 3 σ 3 + F 11 σ 1 2 + F 22 σ 2 2 + F 33 σ 3 2 + F 44 σ 12 2 + F 55 σ 23 2 + F 66 σ 31 2 + 2 F 12 σ 1 σ 2 + 2 F 23 σ 2 σ 3 + 2 F 13 σ 1 σ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGgb Gaeyypa0JaamOzaiaacIcacaWGxbWaaSbaaSqaaiaadchaaeqaaOGa aiykaiabg2da9iaadAeadaWgaaWcbaGaaGymaaqabaGccqaHdpWCda WgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGgbWaaSbaaSqaaiaaikda aeqaaOGaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaamOram aaBaaaleaacaaIZaaabeaakiabeo8aZnaaBaaaleaacaaIZaaabeaa kiabgUcaRiaadAeadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeq4Wdm 3aa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaSIaamOramaaBaaa leaacaaIYaGaaGOmaaqabaGccqaHdpWCdaqhaaWcbaGaaGOmaaqaai aaikdaaaGccqGHRaWkcaWGgbWaaSbaaSqaaiaaiodacaaIZaaabeaa kiabeo8aZnaaDaaaleaacaaIZaaabaGaaGOmaaaakiabgUcaRiaadA eadaWgaaWcbaGaaGinaiaaisdaaeqaaOGaeq4Wdm3aa0baaSqaaiaa igdacaaIYaaabaGaaGOmaaaaaOqaaiabgUcaRiaadAeadaWgaaWcba GaaGynaiaaiwdaaeqaaOGaeq4Wdm3aa0baaSqaaiaaikdacaaIZaaa baGaaGOmaaaakiabgUcaRiaadAeadaWgaaWcbaGaaGOnaiaaiAdaae qaaOGaeq4Wdm3aa0baaSqaaiaaiodacaaIXaaabaGaaGOmaaaakiab gUcaRiaaikdacaWGgbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabeo 8aZnaaBaaaleaacaaIXaaabeaakiabeo8aZnaaBaaaleaacaaIYaaa beaakiabgUcaRiaaikdacaWGgbWaaSbaaSqaaiaaikdacaaIZaaabe aakiabeo8aZnaaBaaaleaacaaIYaaabeaakiabeo8aZnaaBaaaleaa caaIZaaabeaakiabgUcaRiaaikdacaWGgbWaaSbaaSqaaiaaigdaca aIZaaabeaakiabeo8aZnaaBaaaleaacaaIXaaabeaakiabeo8aZnaa BaaaleaacaaIZaaabeaaaaaa@93C6@

with:

F i = 1 σ i y c + 1 σ i y t ( i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@ =1,2,3);

F 11 = 1 σ 1 y c σ 1 y t ; F 22 = 1 σ 2 y c σ 2 y t ; F 33 = 1 σ 3 y c σ 3 y t ;

F 44 = 1 σ 12 y c σ 12 y t ; F 55 = 1 σ 23 y c σ 23 y t ; F 66 = 1 σ 31 y c σ 31 y t ;

F 12 = 1 2 ( F 11 F 22 ) ; F 23 = 1 2 F 22

Where, σ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCdaWgaaWcbaGaamyAaaqabaaaaa@3BB4@ is the yield stress in direction i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@ , c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@ and t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@ denote respectively for compression and tension. f ( W p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaam4vamaaBaaaleaacaWGWbaabeaaaOGaayjkaiaawMcaaaaa @3A71@ represents the yield envelope evolution during work hardening with respect to strain rate effects:(5) f ( W p ) = ( 1 + B W p n ) ( 1 + c .1 n ( ε ˙ ε ˙ 0 ) )
Where,
W p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa aaleaacaWGWbaabeaaaaa@37F3@
Plastic work
B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@
Hardening parameter
n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@
Hardening exponent
c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@
Strain rate coefficient
f ( W p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaam4vamaaBaaaleaacaWGWbaabeaaaOGaayjkaiaawMcaaaaa @3A71@ is limited by a maximum value f max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaaciGGTbGaaiyyaiaacIhaaeqaaaaa@39E1@ :(6) f ( W p ) f max = ( σ max σ y ) 2

If the maximum value is reached the material is failed.

In 式 5, the strain rate effects on the evolution of yield envelope. However, it is also possible to take into account the strain rate ε ˙ effects on the maximum stress σ max as shown in 図 3.
(a) Strain rate effect on σ max (b) No strain rate effect on σ max




σ = σ y ( 1 + c .1 n ( ε ˙ ε ˙ 0 ) )

σ max = σ max 0 ( 1 + c .1 n ( ε ˙ ε ˙ 0 ) )

f max = ( σ max σ y ) 2

σ = σ y ( 1 + c .1 n ( ε ˙ ε ˙ 0 ) )

σ max = σ max 0

図 3. Strain Rate Dependency

Unidirectional Orthotropy

LAW 53 in Radioss provides a simple model for unidirectional orthotropic solids with plasticity. The unidirectional orthotropy condition implies:(7) E 33 = E 22 G 31 = G 12
The orthotropic plasticity behavior is modeled by a modified Tsai-Wu criterion (Orthotropic Plasticity, 式 4) in which:(8) F 12 = 2 ( σ y 45 c ) 2 1 2 ( F 11 + F 22 + F 44 ) + F 1 + F 2 σ y 45 c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake GabaaVqiaadAeadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0Za aSaaaeaacaaIYaaabaWaaeWaaeaacqaHdpWCdaqhaaWcbaGaamyEaa qaaiaaisdacaaI1aGaam4yaaaaaOGaayjkaiaawMcaamaaCaaaleqa baGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaam aabmaabaGaamOramaaBaaaleaacaaIXaGaaGymaaqabaGccqGHRaWk caWGgbWaaSbaaSqaaiaaikdacaaIYaaabeaakiabgUcaRiaadAeada WgaaWcbaGaaGinaiaaisdaaeqaaaGccaGLOaGaayzkaaGaey4kaSYa aSaaaeaacaWGgbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamOram aaBaaaleaacaaIYaaabeaaaOqaaiabeo8aZnaaDaaaleaacaWG5baa baGaaGinaiaaiwdacaWGJbaaaaaaaaa@5D97@
Where, σ 45 y c is yield stress in 45° unidirectional test. The yield stresses in direction 11, 22, 12, 13 and 45° are defined by independent curves obtained by unidirectional tests (図 4). The curves give the stress variation in function of a so-called strain ε v :(9) ε ν = 1 ( T r a c e [ ε ] ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiabe27aUbqabaGccqGH9aqpcaaIXaGaeyOeI0YaaeWaaeaa caWGubGaamOCaiaadggacaWGJbGaamyzamaadmaabaGaeqyTdugaca GLBbGaayzxaaaacaGLOaGaayzkaaaaaa@45E3@


図 4. Yield Stress Curve for a Unidirectional Orthotropic Material